TSTP Solution File: ITP026^2 by Leo-III---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : ITP026^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n002.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 : Mon May 20 22:22:11 EDT 2024

% Result   : Theorem 62.60s 45.49s
% Output   : Refutation 63.66s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  338
% Syntax   : Number of formulae    :  624 ( 204 unt;  54 typ;   0 def)
%            Number of atoms       : 2118 ( 378 equ;   2 cnn)
%            Maximal formula atoms :   17 (   3 avg)
%            Number of connectives : 11285 ( 134   ~;  22   |;  80   &;10289   @)
%                                         (   0 <=>; 760  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   28 (   9 avg)
%            Number of types       :    9 (   8 usr)
%            Number of type conns  :  341 ( 341   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   54 (  51 usr;  16 con; 0-9 aty)
%            Number of variables   : 1869 ( 206   ^1524   !;  58   ?;1869   :)
%                                         (  81  !>;   0  ?*;   0  @-;   0  @+)

% Comments : 
%------------------------------------------------------------------------------
thf(product_unit_type,type,
    product_unit: $tType ).

thf(nat_type,type,
    nat: $tType ).

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

thf(e_type,type,
    e: $tType ).

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

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

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

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

thf(zero_type,type,
    zero: 
      !>[TA: $tType] : $o ).

thf(order_type,type,
    order: 
      !>[TA: $tType] : $o ).

thf(group_add_type,type,
    group_add: 
      !>[TA: $tType] : $o ).

thf(wellorder_type,type,
    wellorder: 
      !>[TA: $tType] : $o ).

thf(comm_monoid_diff_type,type,
    comm_monoid_diff: 
      !>[TA: $tType] : $o ).

thf(ordered_ab_group_add_type,type,
    ordered_ab_group_add: 
      !>[TA: $tType] : $o ).

thf(cancel1352612707id_add_type,type,
    cancel1352612707id_add: 
      !>[TA: $tType] : $o ).

thf(canoni770627133id_add_type,type,
    canoni770627133id_add: 
      !>[TA: $tType] : $o ).

thf(carrier_type,type,
    carrier: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ TA ) > ( set @ TB ) ) ).

thf(ring_type,type,
    ring: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > $o ) ).

thf(zero2_type,type,
    zero2: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ TA ) ) > TB ) ).

thf(ideal_type,type,
    ideal: 
      !>[TA: $tType,TB: $tType] : ( ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TA ) ) ) > ( set @ TB ) > $o ) ).

thf(annihilator_type,type,
    annihilator: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TD ) ) ).

thf(hom_type,type,
    hom: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType] : ( ( carrier_ext @ TF @ ( aGroup_ext @ TF @ ( ring_ext @ TF @ TE ) ) ) > ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TF @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TF @ TA ) ) ) > ( carrier_ext @ ( TD > TB ) @ ( aGroup_ext @ ( TD > TB ) @ ( module_ext @ ( TD > TB ) @ TF @ product_unit ) ) ) ) ).

thf(module_type,type,
    module: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TC @ TB ) ) ) > ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( ring_ext @ TC @ TA ) ) ) > $o ) ).

thf(free_generator_type,type,
    free_generator: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TB ) > $o ) ).

thf(generator_type,type,
    generator: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TB ) > $o ) ).

thf(l_comb_type,type,
    l_comb: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > nat > ( nat > TD ) > ( nat > TB ) > TB ) ).

thf(linear_span_type,type,
    linear_span: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TD ) > ( set @ TB ) > ( set @ TB ) ) ).

thf(mdl_type,type,
    mdl: 
      !>[TA: $tType,TB: $tType,TC: $tType] : ( ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TB @ TA ) ) ) > ( set @ TC ) > ( carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TB @ product_unit ) ) ) ) ).

thf(misomorphic_type,type,
    misomorphic: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType] : ( ( carrier_ext @ TF @ ( aGroup_ext @ TF @ ( ring_ext @ TF @ TE ) ) ) > ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TF @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TF @ TA ) ) ) > $o ) ).

thf(quotie135653060odules_type,type,
    quotie135653060odules: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TB ) > ( set @ TB ) > ( set @ TD ) ) ).

thf(smodule_ideal_coeff_type,type,
    smodule_ideal_coeff: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TD ) > ( set @ TB ) ) ).

thf(submodule_type,type,
    submodule: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TB ) > $o ) ).

thf(algebr833503410le_fgs_type,type,
    algebr833503410le_fgs: 
      !>[TA: $tType,TB: $tType,TC: $tType,TD: $tType] : ( ( carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TC ) ) ) > ( carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ TD @ TA ) ) ) > ( set @ TB ) > ( set @ TB ) ) ).

thf(pi_type,type,
    pi: 
      !>[TA: $tType,TB: $tType] : ( ( set @ TB ) > ( TB > ( set @ TA ) ) > ( set @ ( TB > TA ) ) ) ).

thf(minus_minus_type,type,
    minus_minus: 
      !>[TA: $tType] : ( TA > TA > TA ) ).

thf(zero_zero_type,type,
    zero_zero: 
      !>[TA: $tType] : TA ).

thf(suc_type,type,
    suc: nat > nat ).

thf(bot_bot_type,type,
    bot_bot: 
      !>[TA: $tType] : TA ).

thf(ord_less_type,type,
    ord_less: 
      !>[TA: $tType] : ( TA > TA > $o ) ).

thf(ord_less_eq_type,type,
    ord_less_eq: 
      !>[TA: $tType] : ( TA > TA > $o ) ).

thf(collect_type,type,
    collect: 
      !>[TA: $tType] : ( ( TA > $o ) > ( set @ TA ) ) ).

thf(insert_type,type,
    insert: 
      !>[TA: $tType] : ( TA > ( set @ TA ) > ( set @ TA ) ) ).

thf(member_type,type,
    member: 
      !>[TA: $tType] : ( TA > ( set @ TA ) > $o ) ).

thf(h_type,type,
    h: set @ a ).

thf(h1_type,type,
    h1: set @ a ).

thf(m_type,type,
    m: carrier_ext @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) ).

thf(n_type,type,
    n: carrier_ext @ e @ ( aGroup_ext @ e @ ( module_ext @ e @ b @ f ) ) ).

thf(r_type,type,
    r: carrier_ext @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) ).

thf(g_type,type,
    g: nat > a ).

thf(h2_type,type,
    h2: a ).

thf(m2_type,type,
    m2: nat ).

thf(t_type,type,
    t: nat > b ).

thf(x_type,type,
    x: a ).

thf(xa_type,type,
    xa: nat ).

thf(111,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_Suc__inject) ).

thf(645,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(218,axiom,
    ! [A: set @ b,B: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( linear_span @ b @ d @ a @ c @ r @ m @ A @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) )
          = ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_l__spanA__l__span) ).

thf(1024,plain,
    ! [A: set @ b,B: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( linear_span @ b @ d @ a @ c @ r @ m @ A @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) )
          = ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(17,axiom,
    order @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oorder_1) ).

thf(333,plain,
    order @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(276,axiom,
    ! [A: set @ b,B: nat > b,C: nat,D: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( member @ ( nat > b ) @ B
          @ ( pi @ nat @ b
            @ ( collect @ nat
              @ ^ [E: nat] : ( ord_less_eq @ nat @ E @ C ) )
            @ ^ [E: nat] : A ) )
       => ( ( member @ ( nat > a ) @ D
            @ ( pi @ nat @ a
              @ ( collect @ nat
                @ ^ [E: nat] : ( ord_less_eq @ nat @ E @ C ) )
              @ ^ [E: nat] : ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) )
         => ( ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ D )
            = ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_linear__comb0__2) ).

thf(1179,plain,
    ! [A: set @ b,B: nat > b,C: nat,D: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( member @ ( nat > b ) @ B
          @ ( pi @ nat @ b
            @ ( collect @ nat
              @ ^ [E: nat] : ( ord_less_eq @ nat @ E @ C ) )
            @ ^ [E: nat] : A ) )
       => ( ( member @ ( nat > a ) @ D
            @ ( pi @ nat @ a
              @ ( collect @ nat
                @ ^ [E: nat] : ( ord_less_eq @ nat @ E @ C ) )
              @ ^ [E: nat] : ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) )
         => ( ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ D )
            = ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(155,axiom,
    comm_monoid_diff @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__diff) ).

thf(777,plain,
    comm_monoid_diff @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(248,axiom,
    ! [A: set @ a] :
      ( ( free_generator @ b @ d @ a @ c @ r @ m @ A )
     => ( generator @ b @ d @ a @ c @ r @ m @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_free__generator__generator) ).

thf(1110,plain,
    ! [A: set @ a] :
      ( ( free_generator @ b @ d @ a @ c @ r @ m @ A )
     => ( generator @ b @ d @ a @ c @ r @ m @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(209,axiom,
    ideal @ b @ d @ r @ ( annihilator @ b @ d @ a @ c @ r @ m ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_Ann__is__ideal) ).

thf(1005,plain,
    ideal @ b @ d @ r @ ( annihilator @ b @ d @ a @ c @ r @ m ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(121,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
           => ( B @ C ) ) )
      = ( ( B @ ( zero_zero @ nat ) )
        & ! [C: nat] :
            ( ( ord_less @ nat @ C @ A )
           => ( B @ ( suc @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_All__less__Suc2) ).

thf(683,plain,
    ! [A: nat,B: nat > $o] :
      ( ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
           => ( B @ C ) ) )
      = ( ( B @ ( zero_zero @ nat ) )
        & ! [C: nat] :
            ( ( ord_less @ nat @ C @ A )
           => ( B @ ( suc @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(119,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
      = ( A
        = ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_subset__empty) ).

thf(677,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
      = ( A
        = ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(96,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ ( suc @ B ) )
        = ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_not__less__less__Suc__eq) ).

thf(597,plain,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ ( suc @ B ) )
        = ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(175,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ B )
      = ( ord_less @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_Suc__le__eq) ).

thf(861,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ B )
      = ( ord_less @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(273,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ~ ( member @ TA @ A @ B )
     => ( ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ C ) )
        = ( minus_minus @ ( set @ TA ) @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_Diff__insert0) ).

thf(1173,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ~ ( member @ TA @ A @ B )
     => ( ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ C ) )
        = ( minus_minus @ ( set @ TA ) @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(263,axiom,
    ! [A: set @ b,B: set @ a,C: nat > b,D: nat,E: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( member @ ( nat > b ) @ C
            @ ( pi @ nat @ b
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
              @ ^ [F: nat] : A ) )
         => ( ( member @ ( nat > a ) @ E
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : B ) )
           => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ D @ C @ E ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_l__comb__mem__linear__span) ).

thf(1142,plain,
    ! [A: set @ b,B: set @ a,C: nat > b,D: nat,E: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( member @ ( nat > b ) @ C
            @ ( pi @ nat @ b
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
              @ ^ [F: nat] : A ) )
         => ( ( member @ ( nat > a ) @ E
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : B ) )
           => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ D @ C @ E ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(35,axiom,
    order @ product_unit,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Ounit___Orderings_Oorder_5) ).

thf(379,plain,
    order @ product_unit,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(237,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( C
         != ( bot_bot @ ( set @ TA ) ) )
       => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
         => ( ( member @ TA @ D @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ C ) )
           => ? [E: nat,F: nat > TA] :
                ( ( member @ ( nat > TA ) @ F
                  @ ( pi @ nat @ TA
                    @ ( collect @ nat
                      @ ^ [G: nat] : ( ord_less_eq @ nat @ G @ E ) )
                    @ ^ [G: nat] : C ) )
                & ? [G: nat > TD] :
                    ( ( member @ ( nat > TD ) @ G
                      @ ( pi @ nat @ TD
                        @ ( collect @ nat
                          @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                        @ ^ [H: nat] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ) @ B ) ) )
                    & ( D
                      = ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ G @ F ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_Module_Omem__fgs__l__comb) ).

thf(1078,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( C
         != ( bot_bot @ ( set @ TA ) ) )
       => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
         => ( ( member @ TA @ D @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ C ) )
           => ? [E: nat,F: nat > TA] :
                ( ( member @ ( nat > TA ) @ F
                  @ ( pi @ nat @ TA
                    @ ( collect @ nat
                      @ ^ [G: nat] : ( ord_less_eq @ nat @ G @ E ) )
                    @ ^ [G: nat] : C ) )
                & ? [G: nat > TD] :
                    ( ( member @ ( nat > TD ) @ G
                      @ ( pi @ nat @ TD
                        @ ( collect @ nat
                          @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                        @ ^ [H: nat] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ) @ B ) ) )
                    & ( D
                      = ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ G @ F ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(38,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ B @ C )
       => ( ord_less @ nat @ ( suc @ A ) @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_less__trans__Suc) ).

thf(385,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ B @ C )
       => ( ord_less @ nat @ ( suc @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(77,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( ! [C: nat] :
            ( ( A @ ( suc @ C ) )
           => ( A @ C ) )
       => ( A @ ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_zero__induct) ).

thf(500,plain,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( ! [C: nat] :
            ( ( A @ ( suc @ C ) )
           => ( A @ C ) )
       => ( A @ ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(132,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_48_less__nat__zero__code) ).

thf(715,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(167,axiom,
    ! [A: nat] :
      ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_diff__0__eq__0) ).

thf(814,plain,
    ! [A: nat] :
      ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(144,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
          = ( ord_less_eq @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_diff__ge__0__iff__ge) ).

thf(747,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
          = ( ord_less_eq @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(83,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_less__imp__le__nat) ).

thf(538,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(271,axiom,
    ! [A: set @ a,B: a] :
      ( ( A
       != ( bot_bot @ ( set @ a ) ) )
     => ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( member @ a @ B @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) )
         => ? [C: nat,D: nat > a] :
              ( ( member @ ( nat > a ) @ D
                @ ( pi @ nat @ a
                  @ ( collect @ nat
                    @ ^ [E: nat] : ( ord_less_eq @ nat @ E @ C ) )
                  @ ^ [E: nat] : A ) )
              & ? [E: nat > b] :
                  ( ( member @ ( nat > b ) @ E
                    @ ( pi @ nat @ b
                      @ ( collect @ nat
                        @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
                      @ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
                  & ( B
                    = ( l_comb @ b @ d @ a @ c @ r @ m @ C @ E @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_mem__fgs__l__comb) ).

thf(1164,plain,
    ! [A: set @ a,B: a] :
      ( ( A
       != ( bot_bot @ ( set @ a ) ) )
     => ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( member @ a @ B @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) )
         => ? [C: nat,D: nat > a] :
              ( ( member @ ( nat > a ) @ D
                @ ( pi @ nat @ a
                  @ ( collect @ nat
                    @ ^ [E: nat] : ( ord_less_eq @ nat @ E @ C ) )
                  @ ^ [E: nat] : A ) )
              & ? [E: nat > b] :
                  ( ( member @ ( nat > b ) @ E
                    @ ( pi @ nat @ b
                      @ ( collect @ nat
                        @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
                      @ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
                  & ( B
                    = ( l_comb @ b @ d @ a @ c @ r @ m @ C @ E @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(162,axiom,
    ! [A: set @ a,B: set @ a,C: set @ b] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ B )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( ideal @ b @ d @ r @ C )
         => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ C @ A ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_l__span__gen__mono) ).

thf(797,plain,
    ! [A: set @ a,B: set @ a,C: set @ b] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ B )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( ideal @ b @ d @ r @ C )
         => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ C @ A ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(130,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_not0__implies__Suc) ).

thf(709,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(7,axiom,
    ! [TA: $tType] : ( order @ ( set @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oorder_2) ).

thf(301,plain,
    ! [TA: $tType] : ( order @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(54,axiom,
    ! [A: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_zero__less__Suc) ).

thf(436,plain,
    ! [A: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(113,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( ord_less @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_Suc__less__eq) ).

thf(651,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( ord_less @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(27,axiom,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ! [D: nat] :
              ( ( ord_less_eq @ nat @ ( suc @ D ) @ C )
             => ( A @ D ) )
         => ( A @ C ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_full__nat__induct) ).

thf(357,plain,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ! [D: nat] :
              ( ( ord_less_eq @ nat @ ( suc @ D ) @ C )
             => ( A @ D ) )
         => ( A @ C ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(230,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: nat > TD,E: nat,F: nat > TA,G: nat > TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
       => ( ( member @ ( nat > TD ) @ D
            @ ( pi @ nat @ TD
              @ ( collect @ nat
                @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
              @ ^ [H: nat] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ) @ B ) ) )
         => ( ( member @ ( nat > TA ) @ F
              @ ( pi @ nat @ TA
                @ ( collect @ nat
                  @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                @ ^ [H: nat] : C ) )
           => ( ( member @ ( nat > TA ) @ G
                @ ( pi @ nat @ TA
                  @ ( collect @ nat
                    @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                  @ ^ [H: nat] : C ) )
             => ( ! [H: nat] :
                    ( ( member @ nat @ H
                      @ ( collect @ nat
                        @ ^ [I: nat] : ( ord_less_eq @ nat @ I @ E ) ) )
                   => ( ( F @ H )
                      = ( G @ H ) ) )
               => ( ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ F )
                  = ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ G ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_Module_Olinear__comb__eq) ).

thf(1059,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: nat > TD,E: nat,F: nat > TA,G: nat > TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
       => ( ( member @ ( nat > TD ) @ D
            @ ( pi @ nat @ TD
              @ ( collect @ nat
                @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
              @ ^ [H: nat] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ) @ B ) ) )
         => ( ( member @ ( nat > TA ) @ F
              @ ( pi @ nat @ TA
                @ ( collect @ nat
                  @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                @ ^ [H: nat] : C ) )
           => ( ( member @ ( nat > TA ) @ G
                @ ( pi @ nat @ TA
                  @ ( collect @ nat
                    @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                  @ ^ [H: nat] : C ) )
             => ( ! [H: nat] :
                    ( ( member @ nat @ H
                      @ ( collect @ nat
                        @ ^ [I: nat] : ( ord_less_eq @ nat @ I @ E ) ) )
                   => ( ( F @ H )
                      = ( G @ H ) ) )
               => ( ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ F )
                  = ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ G ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(85,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
     => ( ord_less @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_Suc__less__SucD) ).

thf(542,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
     => ( ord_less @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(105,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_Zero__not__Suc) ).

thf(619,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(173,axiom,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ ( suc @ D ) )
           => ( A @ D ) )
       => ( A @ ( minus_minus @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_zero__induct__lemma) ).

thf(838,plain,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ ( suc @ D ) )
           => ( A @ D ) )
       => ( A @ ( minus_minus @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(168,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( A @ C )
           => ( A @ ( suc @ C ) ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_old_Onat_Oinducts) ).

thf(817,plain,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( A @ C )
           => ( A @ ( suc @ C ) ) )
       => ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(198,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ? [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
            & ( B @ C ) ) )
      = ( ( B @ ( zero_zero @ nat ) )
        | ? [C: nat] :
            ( ( ord_less @ nat @ C @ A )
            & ( B @ ( suc @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_Ex__less__Suc2) ).

thf(969,plain,
    ! [A: nat,B: nat > $o] :
      ( ( ? [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
            & ( B @ C ) ) )
      = ( ( B @ ( zero_zero @ nat ) )
        | ? [C: nat] :
            ( ( ord_less @ nat @ C @ A )
            & ( B @ ( suc @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(213,axiom,
    ( ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( bot_bot @ ( set @ a ) ) )
    = ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_empty__fgs) ).

thf(1013,plain,
    ( ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( bot_bot @ ( set @ a ) ) )
    = ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(252,axiom,
    ord_less_eq @ nat @ xa @ ( minus_minus @ nat @ m2 @ ( suc @ ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_19) ).

thf(1115,plain,
    ord_less_eq @ nat @ xa @ ( minus_minus @ nat @ m2 @ ( suc @ ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(247,axiom,
    member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_module__inc__zero) ).

thf(1109,plain,
    member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(34,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_le__0__eq) ).

thf(376,plain,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(106,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_old_Onat_Oexhaust) ).

thf(623,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(260,axiom,
    ! [A: set @ a,B: a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( member @ a @ B @ A )
       => ( member @ a @ B @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_h__in__linear__span) ).

thf(1136,plain,
    ! [A: set @ a,B: a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( member @ a @ B @ A )
       => ( member @ a @ B @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(266,axiom,
    ! [A: set @ b,B: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_lin__span__coeff__mono) ).

thf(1151,plain,
    ! [A: set @ b,B: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(25,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_Suc__mono) ).

thf(353,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(277,axiom,
    ! [A: set @ a,B: nat > b,C: nat,D: nat > a,E: nat > a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( ( member @ ( nat > b ) @ B
            @ ( pi @ nat @ b
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
              @ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
          & ( member @ ( nat > a ) @ D
            @ ( pi @ nat @ a
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
              @ ^ [F: nat] : A ) )
          & ( member @ ( nat > a ) @ E
            @ ( pi @ nat @ a
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
              @ ^ [F: nat] : A ) )
          & ! [F: nat] :
              ( ( member @ nat @ F
                @ ( collect @ nat
                  @ ^ [G: nat] : ( ord_less_eq @ nat @ G @ C ) ) )
             => ( ( D @ F )
                = ( E @ F ) ) ) )
       => ( ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ D )
          = ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ E ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_linear__comb__eqTr) ).

thf(1183,plain,
    ! [A: set @ a,B: nat > b,C: nat,D: nat > a,E: nat > a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( ( member @ ( nat > b ) @ B
            @ ( pi @ nat @ b
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
              @ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
          & ( member @ ( nat > a ) @ D
            @ ( pi @ nat @ a
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
              @ ^ [F: nat] : A ) )
          & ( member @ ( nat > a ) @ E
            @ ( pi @ nat @ a
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
              @ ^ [F: nat] : A ) )
          & ! [F: nat] :
              ( ( member @ nat @ F
                @ ( collect @ nat
                  @ ^ [G: nat] : ( ord_less_eq @ nat @ G @ C ) ) )
             => ( ( D @ F )
                = ( E @ F ) ) ) )
       => ( ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ D )
          = ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(226,axiom,
    ! [A: set @ a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_submodule__inc__0) ).

thf(1051,plain,
    ! [A: set @ a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(70,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_nat_Oinject) ).

thf(480,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(151,axiom,
    ! [A: set @ b,B: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_linear__span__sub) ).

thf(766,plain,
    ! [A: set @ b,B: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(139,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TA @ TD ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),C: carrier_ext @ TF @ ( aGroup_ext @ TF @ ( module_ext @ TF @ TA @ TE ) )] :
      ( ( module @ TC @ TA @ TD @ TB @ A @ B )
     => ( ( module @ TF @ TA @ TE @ TB @ C @ B )
       => ( module @ ( TC > TF ) @ TA @ product_unit @ TB @ ( hom @ TA @ TB @ TC @ TD @ TF @ TE @ B @ A @ C ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_Module_OHOM__is__module) ).

thf(734,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TA @ TD ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),C: carrier_ext @ TF @ ( aGroup_ext @ TF @ ( module_ext @ TF @ TA @ TE ) )] :
      ( ( module @ TC @ TA @ TD @ TB @ A @ B )
     => ( ( module @ TF @ TA @ TE @ TB @ C @ B )
       => ( module @ ( TC > TF ) @ TA @ product_unit @ TB @ ( hom @ TA @ TB @ TC @ TD @ TF @ TE @ B @ A @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(5,axiom,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ! [D: nat] :
              ( ( ord_less @ nat @ D @ C )
             => ( A @ D ) )
         => ( A @ C ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_nat__less__induct) ).

thf(293,plain,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ! [D: nat] :
              ( ( ord_less @ nat @ D @ C )
             => ( A @ D ) )
         => ( A @ C ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(177,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
      = ( ? [C: nat] :
            ( ( B
              = ( suc @ C ) )
            & ( ord_less @ nat @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_Suc__less__eq2) ).

thf(866,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
      = ( ? [C: nat] :
            ( ( B
              = ( suc @ C ) )
            & ( ord_less @ nat @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(235,axiom,
    ! [A: set @ a,B: set @ a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( ( submodule @ b @ d @ a @ c @ r @ m @ B )
       => ( ( ord_less_eq @ ( set @ a ) @ A @ B )
         => ( submodule @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_submodule__of__mdl) ).

thf(1072,plain,
    ! [A: set @ a,B: set @ a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( ( submodule @ b @ d @ a @ c @ r @ m @ B )
       => ( ( ord_less_eq @ ( set @ a ) @ A @ B )
         => ( submodule @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(154,axiom,
    ! [A: nat] :
      ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_not__gr0) ).

thf(774,plain,
    ! [A: nat] :
      ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(115,axiom,
    ! [A: nat,B: nat,C: nat > nat > $o] :
      ( ( ord_less @ nat @ A @ B )
     => ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
       => ( ! [D: nat,E: nat,F: nat] :
              ( ( ord_less @ nat @ D @ E )
             => ( ( ord_less @ nat @ E @ F )
               => ( ( C @ D @ E )
                 => ( ( C @ E @ F )
                   => ( C @ D @ F ) ) ) ) )
         => ( C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_less__Suc__induct) ).

thf(660,plain,
    ! [A: nat,B: nat,C: nat > nat > $o] :
      ( ( ord_less @ nat @ A @ B )
     => ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
       => ( ! [D: nat,E: nat,F: nat] :
              ( ( ord_less @ nat @ D @ E )
             => ( ( ord_less @ nat @ E @ F )
               => ( ( C @ D @ E )
                 => ( ( C @ E @ F )
                   => ( C @ D @ F ) ) ) ) )
         => ( C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(207,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TD,D: TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( ideal @ TD @ TB @ B @ C )
       => ( ( member @ TA @ D @ ( smodule_ideal_coeff @ TD @ TB @ TA @ TC @ B @ A @ C ) )
         => ? [E: nat,F: nat > TD] :
              ( ( member @ ( nat > TD ) @ F
                @ ( pi @ nat @ TD
                  @ ( collect @ nat
                    @ ^ [G: nat] : ( ord_less_eq @ nat @ G @ E ) )
                  @ ^ [G: nat] : C ) )
              & ? [G: nat > TA] :
                  ( ( member @ ( nat > TA ) @ G
                    @ ( pi @ nat @ TA
                      @ ( collect @ nat
                        @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                      @ ^ [H: nat] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) ) )
                  & ( D
                    = ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ F @ G ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_Module_Omem__smodule__ideal__coeff) ).

thf(998,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TD,D: TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( ideal @ TD @ TB @ B @ C )
       => ( ( member @ TA @ D @ ( smodule_ideal_coeff @ TD @ TB @ TA @ TC @ B @ A @ C ) )
         => ? [E: nat,F: nat > TD] :
              ( ( member @ ( nat > TD ) @ F
                @ ( pi @ nat @ TD
                  @ ( collect @ nat
                    @ ^ [G: nat] : ( ord_less_eq @ nat @ G @ E ) )
                  @ ^ [G: nat] : C ) )
              & ? [G: nat > TA] :
                  ( ( member @ ( nat > TA ) @ G
                    @ ( pi @ nat @ TA
                      @ ( collect @ nat
                        @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                      @ ^ [H: nat] : ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) ) )
                  & ( D
                    = ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ F @ G ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(204,axiom,
    ! [A: set @ a,B: set @ a,C: nat > a,D: nat,E: nat > b] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ A )
       => ( ( member @ ( nat > a ) @ C
            @ ( pi @ nat @ a
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
              @ ^ [F: nat] : B ) )
         => ( ( member @ ( nat > b ) @ E
              @ ( pi @ nat @ b
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
           => ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ B ) ) @ D @ E @ C )
              = ( l_comb @ b @ d @ a @ c @ r @ m @ D @ E @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_eSum__in__Submodule) ).

thf(988,plain,
    ! [A: set @ a,B: set @ a,C: nat > a,D: nat,E: nat > b] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ A )
       => ( ( member @ ( nat > a ) @ C
            @ ( pi @ nat @ a
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
              @ ^ [F: nat] : B ) )
         => ( ( member @ ( nat > b ) @ E
              @ ( pi @ nat @ b
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
           => ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ B ) ) @ D @ E @ C )
              = ( l_comb @ b @ d @ a @ c @ r @ m @ D @ E @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(284,axiom,
    ! [A: set @ a,B: set @ a,C: set @ a] :
      ( ( free_generator @ b @ d @ a @ c @ r @ m @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ C )
       => ( ( ord_less_eq @ ( set @ a ) @ C @ A )
         => ( ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ B ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_fgs__mono) ).

thf(1201,plain,
    ! [A: set @ a,B: set @ a,C: set @ a] :
      ( ( free_generator @ b @ d @ a @ c @ r @ m @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ C )
       => ( ( ord_less_eq @ ( set @ a ) @ C @ A )
         => ( ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ B ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(245,axiom,
    ! [A: set @ a,B: set @ a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( ( submodule @ b @ d @ a @ c @ r @ m @ B )
       => ( ideal @ b @ d @ r @ ( quotie135653060odules @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_quotient__of__submodules__is__ideal) ).

thf(1104,plain,
    ! [A: set @ a,B: set @ a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( ( submodule @ b @ d @ a @ c @ r @ m @ B )
       => ( ideal @ b @ d @ r @ ( quotie135653060odules @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(181,axiom,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( C @ A )
       => ( ! [D: nat] :
              ( ( ord_less_eq @ nat @ A @ D )
             => ( ( C @ D )
               => ( C @ ( suc @ D ) ) ) )
         => ( C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_nat__induct__at__least) ).

thf(877,plain,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( C @ A )
       => ( ! [D: nat] :
              ( ( ord_less_eq @ nat @ A @ D )
             => ( ( C @ D )
               => ( C @ ( suc @ D ) ) ) )
         => ( C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(124,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( A != B )
       => ( ord_less @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_le__neq__implies__less) ).

thf(692,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( A != B )
       => ( ord_less @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(36,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ C )
       => ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_Module_Osubmodule__subset) ).

thf(380,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ C )
       => ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(87,axiom,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ A )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_diff__self__eq__0) ).

thf(546,plain,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ A )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(283,axiom,
    ! [A: set @ a] :
      ( ( free_generator @ b @ d @ a @ c @ r @ m @ A )
     => ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_free__generator__sub) ).

thf(1199,plain,
    ! [A: set @ a] :
      ( ( free_generator @ b @ d @ a @ c @ r @ m @ A )
     => ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(233,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA,D: TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ C )
       => ( ( member @ TA @ D @ C )
         => ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_Module_Osubmodule__subset1) ).

thf(1067,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA,D: TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ C )
       => ( ( member @ TA @ D @ C )
         => ( member @ TA @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(229,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ b @ TA ) )] :
      ( ( module @ TB @ b @ TA @ d @ A @ r )
     => ( ( misomorphic @ b @ d @ a @ c @ TB @ TA @ r @ m @ A )
       => ( misomorphic @ b @ d @ TB @ TA @ a @ c @ r @ A @ m ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_misom__sym) ).

thf(1057,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ b @ TA ) )] :
      ( ( module @ TB @ b @ TA @ d @ A @ r )
     => ( ( misomorphic @ b @ d @ a @ c @ TB @ TA @ r @ m @ A )
       => ( misomorphic @ b @ d @ TB @ TA @ a @ c @ r @ A @ m ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(18,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less @ nat @ B @ C )
           => ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_lift__Suc__mono__less) ).

thf(334,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less @ nat @ B @ C )
           => ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(185,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less_eq @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less_eq @ nat @ B @ C )
           => ( ord_less_eq @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_lift__Suc__mono__le) ).

thf(910,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less_eq @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less_eq @ nat @ B @ C )
           => ( ord_less_eq @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(240,axiom,
    ! [A: set @ b,B: set @ a,C: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( submodule @ b @ d @ a @ c @ r @ m @ B )
       => ( ( ord_less_eq @ ( set @ a ) @ C @ B )
         => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ C ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_l__span__sub__submodule) ).

thf(1091,plain,
    ! [A: set @ b,B: set @ a,C: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( submodule @ b @ d @ a @ c @ r @ m @ B )
       => ( ( ord_less_eq @ ( set @ a ) @ C @ B )
         => ( ord_less_eq @ ( set @ a ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ C ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(156,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_cancel__comm__monoid__add__class_Odiff__cancel) ).

thf(778,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(223,axiom,
    ! [A: set @ a,B: a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( member @ a @ B @ A )
       => ( member @ a @ B @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_elem__fgs) ).

thf(1042,plain,
    ! [A: set @ a,B: a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( member @ a @ B @ A )
       => ( member @ a @ B @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(40,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_diff__is__0__eq_H) ).

thf(392,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(159,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ ( suc @ B ) )
      = ( ( ord_less_eq @ nat @ A @ B )
        | ( A
          = ( suc @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_le__Suc__eq) ).

thf(788,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ ( suc @ B ) )
      = ( ( ord_less_eq @ nat @ A @ B )
        | ( A
          = ( suc @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(137,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_less__imp__diff__less) ).

thf(729,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(16,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( A != B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_less__not__refl3) ).

thf(329,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(170,axiom,
    ! [A: set @ a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_submodule__subset) ).

thf(830,plain,
    ! [A: set @ a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(269,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
       => ( ( member @ TA @ D @ C )
         => ( member @ TA @ D @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_Module_Oelem__fgs) ).

thf(1159,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
       => ( ( member @ TA @ D @ C )
         => ( member @ TA @ D @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(280,axiom,
    ! [A: set @ a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ord_less_eq @ ( set @ a ) @ A @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_l__span__cont__H) ).

thf(1195,plain,
    ! [A: set @ a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ord_less_eq @ ( set @ a ) @ A @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(257,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: TA] :
      ( ( ~ ( member @ TA @ A @ B )
       => ( A = C ) )
     => ( member @ TA @ A @ ( insert @ TA @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_insertCI) ).

thf(1128,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: TA] :
      ( ( ~ ( member @ TA @ A @ B )
       => ( A = C ) )
     => ( member @ TA @ A @ ( insert @ TA @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(62,axiom,
    ! [A: set @ a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_fgs__sub__carrier) ).

thf(456,plain,
    ! [A: set @ a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(200,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less_eq @ TA @ ( A @ ( suc @ D ) ) @ ( A @ D ) )
         => ( ( ord_less_eq @ nat @ B @ C )
           => ( ord_less_eq @ TA @ ( A @ C ) @ ( A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_lift__Suc__antimono__le) ).

thf(974,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less_eq @ TA @ ( A @ ( suc @ D ) ) @ ( A @ D ) )
         => ( ( ord_less_eq @ nat @ B @ C )
           => ( ord_less_eq @ TA @ ( A @ C ) @ ( A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(142,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less @ nat @ B @ ( suc @ A ) )
        = ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_le__less__Suc__eq) ).

thf(742,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less @ nat @ B @ ( suc @ A ) )
        = ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(254,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ( member @ TA @ A @ B )
     => ( ( minus_minus @ ( set @ TA ) @ ( insert @ TA @ A @ C ) @ B )
        = ( minus_minus @ ( set @ TA ) @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_insert__Diff1) ).

thf(1121,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ( member @ TA @ A @ B )
     => ( ( minus_minus @ ( set @ TA ) @ ( insert @ TA @ A @ C ) @ B )
        = ( minus_minus @ ( set @ TA ) @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(231,axiom,
    ! [TA: $tType,A: TA] : ( member @ TA @ A @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_singletonI) ).

thf(1064,plain,
    ! [TA: $tType,A: TA] : ( member @ TA @ A @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(210,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ m2 )
     => ! [B: nat] :
          ( ( ord_less_eq @ nat @ B @ m2 )
         => ( ( ( g @ A )
              = ( g @ B ) )
           => ( A = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_1) ).

thf(1006,plain,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ m2 )
     => ! [B: nat] :
          ( ( ord_less_eq @ nat @ B @ m2 )
         => ( ( ( g @ A )
              = ( g @ B ) )
           => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(39,axiom,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_Suc__neq__Zero) ).

thf(388,plain,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(178,axiom,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( insert @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) )
      = ( insert @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_insert__Diff__single) ).

thf(869,plain,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( insert @ TA @ A @ ( minus_minus @ ( set @ TA ) @ B @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) )
      = ( insert @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(126,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ord_less @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( ord_less @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_less__diff__iff) ).

thf(697,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ord_less @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( ord_less @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(239,axiom,
    ! [A: set @ a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_fgs__submodule) ).

thf(1089,plain,
    ! [A: set @ a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(219,axiom,
    ! [A: set @ a,B: nat > b,C: nat,D: nat > a,E: nat > a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( member @ ( nat > b ) @ B
          @ ( pi @ nat @ b
            @ ( collect @ nat
              @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
            @ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
       => ( ( member @ ( nat > a ) @ D
            @ ( pi @ nat @ a
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
              @ ^ [F: nat] : A ) )
         => ( ( member @ ( nat > a ) @ E
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
                @ ^ [F: nat] : A ) )
           => ( ! [F: nat] :
                  ( ( member @ nat @ F
                    @ ( collect @ nat
                      @ ^ [G: nat] : ( ord_less_eq @ nat @ G @ C ) ) )
                 => ( ( D @ F )
                    = ( E @ F ) ) )
             => ( ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ D )
                = ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_linear__comb__eq) ).

thf(1028,plain,
    ! [A: set @ a,B: nat > b,C: nat,D: nat > a,E: nat > a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( member @ ( nat > b ) @ B
          @ ( pi @ nat @ b
            @ ( collect @ nat
              @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
            @ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) )
       => ( ( member @ ( nat > a ) @ D
            @ ( pi @ nat @ a
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
              @ ^ [F: nat] : A ) )
         => ( ( member @ ( nat > a ) @ E
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
                @ ^ [F: nat] : A ) )
           => ( ! [F: nat] :
                  ( ( member @ nat @ F
                    @ ( collect @ nat
                      @ ^ [G: nat] : ( ord_less_eq @ nat @ G @ C ) ) )
                 => ( ( D @ F )
                    = ( E @ F ) ) )
             => ( ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ D )
                = ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(73,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ? [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
            & ( B @ C ) ) )
      = ( ( B @ A )
        | ? [C: nat] :
            ( ( ord_less @ nat @ C @ A )
            & ( B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_Ex__less__Suc) ).

thf(488,plain,
    ! [A: nat,B: nat > $o] :
      ( ( ? [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
            & ( B @ C ) ) )
      = ( ( B @ A )
        | ? [C: nat] :
            ( ( ord_less @ nat @ C @ A )
            & ( B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(20,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
      | ( ord_less_eq @ nat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_nat__le__linear) ).

thf(339,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
      | ( ord_less_eq @ nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(108,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( minus_minus @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A )
      = ( bot_bot @ ( set @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_empty__Diff) ).

thf(637,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( minus_minus @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A )
      = ( bot_bot @ ( set @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(215,axiom,
    ! [TA: $tType,A: TA] :
      ~ ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_empty__iff) ).

thf(1016,plain,
    ! [TA: $tType,A: TA] :
      ~ ( member @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(122,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_diff @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_zero__diff) ).

thf(686,plain,
    ! [TA: $tType] :
      ( ( comm_monoid_diff @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(236,axiom,
    ! [A: set @ b] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( linear_span @ b @ d @ a @ c @ r @ m @ A @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) )
        = ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_l__span__zero) ).

thf(1075,plain,
    ! [A: set @ b] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( linear_span @ b @ d @ a @ c @ r @ m @ A @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) )
        = ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(251,axiom,
    ord_less_eq @ ( set @ a ) @ h1 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_10) ).

thf(1114,plain,
    ord_less_eq @ ( set @ a ) @ h1 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(243,axiom,
    ( ( ( g @ xa )
      = h2 )
    | ( member @ a @ ( g @ xa ) @ h1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_20) ).

thf(1099,plain,
    ( ( ( g @ xa )
      = h2 )
    | ( member @ a @ ( g @ xa ) @ h1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(127,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_less__Suc__eq__le) ).

thf(701,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(80,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: set @ TA,E: set @ TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( free_generator @ TD @ TB @ TA @ TC @ B @ A @ C )
       => ( ( ord_less_eq @ ( set @ TA ) @ D @ E )
         => ( ( ord_less_eq @ ( set @ TA ) @ E @ C )
           => ( ord_less_eq @ ( set @ TA ) @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ D ) @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_Module_Ofgs__mono) ).

thf(515,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: set @ TA,E: set @ TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( free_generator @ TD @ TB @ TA @ TC @ B @ A @ C )
       => ( ( ord_less_eq @ ( set @ TA ) @ D @ E )
         => ( ( ord_less_eq @ ( set @ TA ) @ E @ C )
           => ( ord_less_eq @ ( set @ TA ) @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ D ) @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(114,axiom,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ D )
           => ( ord_less_eq @ nat @ D @ C ) )
       => ? [D: nat] :
            ( ( A @ D )
            & ! [E: nat] :
                ( ( A @ E )
               => ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_Nat_Oex__has__greatest__nat) ).

thf(654,plain,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ D )
           => ( ord_less_eq @ nat @ D @ C ) )
       => ? [D: nat] :
            ( ( A @ D )
            & ! [E: nat] :
                ( ( A @ E )
               => ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(279,axiom,
    ! [A: set @ a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ product_unit ) ) @ ( mdl @ a @ b @ c @ m @ A ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_mdl__carrier) ).

thf(1192,plain,
    ! [A: set @ a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ product_unit ) ) @ ( mdl @ a @ b @ c @ m @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(22,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
           => ( B @ C ) ) )
      = ( ( B @ A )
        & ! [C: nat] :
            ( ( ord_less @ nat @ C @ A )
           => ( B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_All__less__Suc) ).

thf(343,plain,
    ! [A: nat,B: nat > $o] :
      ( ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
           => ( B @ C ) ) )
      = ( ( B @ A )
        & ! [C: nat] :
            ( ( ord_less @ nat @ C @ A )
           => ( B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(165,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ideal @ TB @ TD @ B @ ( annihilator @ TB @ TD @ TA @ TC @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_Module_OAnn__is__ideal) ).

thf(805,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ideal @ TB @ TD @ B @ ( annihilator @ TB @ TD @ TA @ TC @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(42,axiom,
    ! [A: nat,B: nat] :
      ( ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
     => ( ( ( minus_minus @ nat @ B @ A )
          = ( zero_zero @ nat ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_diffs0__imp__equal) ).

thf(398,plain,
    ! [A: nat,B: nat] :
      ( ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
     => ( ( ( minus_minus @ nat @ B @ A )
          = ( zero_zero @ nat ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(118,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ( minus_minus @ ( set @ TA ) @ A @ B )
        = ( bot_bot @ ( set @ TA ) ) )
      = ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_Diff__eq__empty__iff) ).

thf(674,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ( minus_minus @ ( set @ TA ) @ A @ B )
        = ( bot_bot @ ( set @ TA ) ) )
      = ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(161,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ( suc @ A )
         != B )
       => ( ord_less @ nat @ ( suc @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_Suc__lessI) ).

thf(794,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ( suc @ A )
         != B )
       => ( ord_less @ nat @ ( suc @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(133,axiom,
    ! [A: nat,B: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ A @ B ) @ ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_diff__less__Suc) ).

thf(718,plain,
    ! [A: nat,B: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ A @ B ) @ ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(6,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_less__not__refl) ).

thf(298,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(50,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA,D: set @ TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ C )
       => ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ D )
         => ( ideal @ TB @ TD @ B @ ( quotie135653060odules @ TB @ TD @ TA @ TC @ B @ A @ C @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_Module_Oquotient__of__submodules__is__ideal) ).

thf(425,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA,D: set @ TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ C )
       => ( ( submodule @ TB @ TD @ TA @ TC @ B @ A @ D )
         => ( ideal @ TB @ TD @ B @ ( quotie135653060odules @ TB @ TD @ TA @ TC @ B @ A @ C @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(143,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( submodule @ TB @ TD @ TA @ TC @ B @ A @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_Module_Osubmodule__whole) ).

thf(745,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( submodule @ TB @ TD @ TA @ TC @ B @ A @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(123,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ C )
       => ( ord_less @ nat @ ( minus_minus @ nat @ C @ B ) @ ( minus_minus @ nat @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_diff__less__mono2) ).

thf(689,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ C )
       => ( ord_less @ nat @ ( minus_minus @ nat @ C @ B ) @ ( minus_minus @ nat @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(57,axiom,
    zero @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ozero) ).

thf(444,plain,
    zero @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(98,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( generator @ TB @ TD @ TA @ TC @ B @ A @ C )
       => ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_Module_Ogenerator__sub__carrier) ).

thf(601,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( generator @ TB @ TD @ TA @ TC @ B @ A @ C )
       => ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(196,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ ( suc @ B ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_less__antisym) ).

thf(965,plain,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ ( suc @ B ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(145,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_le__SucI) ).

thf(750,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(79,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
       => ( ord_less_eq @ ( set @ TA ) @ ( algebr833503410le_fgs @ TB @ TD @ TA @ TC @ B @ A @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_Module_Ofgs__sub__carrier) ).

thf(513,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
       => ( ord_less_eq @ ( set @ TA ) @ ( algebr833503410le_fgs @ TB @ TD @ TA @ TC @ B @ A @ C ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(52,axiom,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) )
        = ( insert @ TA @ B @ C ) )
      = ( ( B = A )
        & ( ord_less_eq @ ( set @ TA ) @ C @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_singleton__insert__inj__eq) ).

thf(430,plain,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) )
        = ( insert @ TA @ B @ C ) )
      = ( ( B = A )
        & ( ord_less_eq @ ( set @ TA ) @ C @ ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(267,axiom,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( member @ TA @ A @ ( insert @ TA @ B @ C ) )
      = ( ( A = B )
        | ( member @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_insert__iff) ).

thf(1154,plain,
    ! [TA: $tType,A: TA,B: TA,C: set @ TA] :
      ( ( member @ TA @ A @ ( insert @ TA @ B @ C ) )
      = ( ( A = B )
        | ( member @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(183,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ A )
       => ( ord_less @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_diff__less__mono) ).

thf(904,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ A )
       => ( ord_less @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(138,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ( ord_less @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_Suc__lessD) ).

thf(732,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ( ord_less @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(256,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( ! [B: TA] :
            ~ ( member @ TA @ B @ A ) )
      = ( A
        = ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_all__not__in__conv) ).

thf(1125,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( ! [B: TA] :
            ~ ( member @ TA @ B @ A ) )
      = ( A
        = ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(153,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ B ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_Diff__subset) ).

thf(772,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( minus_minus @ ( set @ TA ) @ A @ B ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(1,conjecture,
    member @ a @ ( g @ xa ) @ h1,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_21) ).

thf(2,negated_conjecture,
    ~ ( member @ a @ ( g @ xa ) @ h1 ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(286,plain,
    ~ ( member @ a @ ( g @ xa ) @ h1 ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(194,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq @ nat @ A @ B ) )
      = ( ord_less_eq @ nat @ ( suc @ B ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_not__less__eq__eq) ).

thf(958,plain,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq @ nat @ A @ B ) )
      = ( ord_less_eq @ nat @ ( suc @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(255,axiom,
    ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ h1 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_11) ).

thf(1124,plain,
    ord_less_eq @ ( set @ a ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ h1 ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(46,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less @ nat @ A @ B )
        | ( A = B ) )
     => ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_less__or__eq__imp__le) ).

thf(413,plain,
    ! [A: nat,B: nat] :
      ( ( ( ord_less @ nat @ A @ B )
        | ( A = B ) )
     => ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(180,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_le__imp__less__Suc) ).

thf(875,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(93,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_neq0__conv) ).

thf(577,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(261,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TB,D: set @ TA,E: nat] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( ideal @ TB @ TD @ B @ C )
       => ( ( ord_less_eq @ ( set @ TA ) @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
         => ! [F: nat > TB,G: nat > TA] :
              ( ( ( member @ ( nat > TB ) @ F
                  @ ( pi @ nat @ TB
                    @ ( collect @ nat
                      @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                    @ ^ [H: nat] : C ) )
                & ( member @ ( nat > TA ) @ G
                  @ ( pi @ nat @ TA
                    @ ( collect @ nat
                      @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                    @ ^ [H: nat] : D ) ) )
             => ( member @ TA @ ( l_comb @ TB @ TD @ TA @ TC @ B @ A @ E @ F @ G ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_Module_Oliear__comb__memTr) ).

thf(1139,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TB,D: set @ TA,E: nat] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( ideal @ TB @ TD @ B @ C )
       => ( ( ord_less_eq @ ( set @ TA ) @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
         => ! [F: nat > TB,G: nat > TA] :
              ( ( ( member @ ( nat > TB ) @ F
                  @ ( pi @ nat @ TB
                    @ ( collect @ nat
                      @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                    @ ^ [H: nat] : C ) )
                & ( member @ ( nat > TA ) @ G
                  @ ( pi @ nat @ TA
                    @ ( collect @ nat
                      @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                    @ ^ [H: nat] : D ) ) )
             => ( member @ TA @ ( l_comb @ TB @ TD @ TA @ TC @ B @ A @ E @ F @ G ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(37,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_subset__antisym) ).

thf(382,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(91,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
     => ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_bot__nat__0_Oextremum__uniqueI) ).

thf(572,plain,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
     => ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(11,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( order @ TA )
     => ( order @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder) ).

thf(313,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( order @ TA )
     => ( order @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(102,axiom,
    ( ( ord_less_eq @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less @ nat @ A @ B )
          | ( A = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_le__eq__less__or__eq) ).

thf(611,plain,
    ( ( ord_less_eq @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less @ nat @ A @ B )
          | ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(246,axiom,
    ! [A: set @ b] :
      ( ( ideal @ b @ d @ r @ A )
     => ( submodule @ b @ d @ a @ c @ r @ m @ ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_smodule__ideal__coeff__is__Submodule) ).

thf(1107,plain,
    ! [A: set @ b] :
      ( ( ideal @ b @ d @ r @ A )
     => ( submodule @ b @ d @ a @ c @ r @ m @ ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(66,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_gr0__implies__Suc) ).

thf(466,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(128,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_zero__less__diff) ).

thf(704,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(190,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_le__zero__eq) ).

thf(947,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(24,axiom,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ ( zero_zero @ nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_minus__nat_Odiff__0) ).

thf(350,plain,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ ( zero_zero @ nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(84,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_less__eq__nat_Osimps_I1_J) ).

thf(540,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(117,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,TH: $tType,A: carrier_ext @ TG @ ( aGroup_ext @ TG @ ( module_ext @ TG @ TB @ TH ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: carrier_ext @ TF @ ( aGroup_ext @ TF @ ( module_ext @ TF @ TB @ TE ) ),D: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) )] :
      ( ( module @ TG @ TB @ TH @ TD @ A @ B )
     => ( ( module @ TF @ TB @ TE @ TD @ C @ B )
       => ( ( module @ TA @ TB @ TC @ TD @ D @ B )
         => ( ( misomorphic @ TB @ TD @ TF @ TE @ TG @ TH @ B @ C @ A )
           => ( ( misomorphic @ TB @ TD @ TG @ TH @ TA @ TC @ B @ A @ D )
             => ( misomorphic @ TB @ TD @ TF @ TE @ TA @ TC @ B @ C @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_Module_Omisom__trans) ).

thf(672,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,TH: $tType,A: carrier_ext @ TG @ ( aGroup_ext @ TG @ ( module_ext @ TG @ TB @ TH ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: carrier_ext @ TF @ ( aGroup_ext @ TF @ ( module_ext @ TF @ TB @ TE ) ),D: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) )] :
      ( ( module @ TG @ TB @ TH @ TD @ A @ B )
     => ( ( module @ TF @ TB @ TE @ TD @ C @ B )
       => ( ( module @ TA @ TB @ TC @ TD @ D @ B )
         => ( ( misomorphic @ TB @ TD @ TF @ TE @ TG @ TH @ B @ C @ A )
           => ( ( misomorphic @ TB @ TD @ TG @ TH @ TA @ TC @ B @ A @ D )
             => ( misomorphic @ TB @ TD @ TF @ TE @ TA @ TC @ B @ C @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(58,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TA @ TC ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),C: set @ TA] :
      ( ( module @ TD @ TA @ TC @ TB @ A @ B )
     => ( ( ideal @ TA @ TB @ B @ C )
       => ( submodule @ TA @ TB @ TD @ TC @ B @ A @ ( smodule_ideal_coeff @ TA @ TB @ TD @ TC @ B @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_Module_Osmodule__ideal__coeff__is__Submodule) ).

thf(445,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ TA @ TC ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) ),C: set @ TA] :
      ( ( module @ TD @ TA @ TC @ TB @ A @ B )
     => ( ( ideal @ TA @ TB @ B @ C )
       => ( submodule @ TA @ TB @ TD @ TC @ B @ A @ ( smodule_ideal_coeff @ TA @ TB @ TD @ TC @ B @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(189,axiom,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( ord_less @ nat @ A @ B )
     => ( ! [D: nat] :
            ( ( B
              = ( suc @ D ) )
           => ( C @ D ) )
       => ( ! [D: nat] :
              ( ( ord_less @ nat @ D @ B )
             => ( ( C @ ( suc @ D ) )
               => ( C @ D ) ) )
         => ( C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_strict__inc__induct) ).

thf(936,plain,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( ord_less @ nat @ A @ B )
     => ( ! [D: nat] :
            ( ( B
              = ( suc @ D ) )
           => ( C @ D ) )
       => ( ! [D: nat] :
              ( ( ord_less @ nat @ D @ B )
             => ( ( C @ ( suc @ D ) )
               => ( C @ D ) ) )
         => ( C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(202,axiom,
    ! [A: set @ b,B: set @ a,C: nat > b,D: nat,E: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( ( member @ ( nat > b ) @ C
              @ ( pi @ nat @ b
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : A ) )
            & ( member @ ( nat > a ) @ E
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) ) ) )
         => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ D @ C @ E ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_l__span__closed2Tr) ).

thf(979,plain,
    ! [A: set @ b,B: set @ a,C: nat > b,D: nat,E: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( ( member @ ( nat > b ) @ C
              @ ( pi @ nat @ b
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : A ) )
            & ( member @ ( nat > a ) @ E
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) ) ) )
         => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ D @ C @ E ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(158,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( minus_minus @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( minus_minus @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_Nat_Odiff__diff__eq) ).

thf(784,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( minus_minus @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( minus_minus @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(28,axiom,
    ord_less @ nat @ ( zero_zero @ nat ) @ m2,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_16) ).

thf(362,plain,
    ord_less @ nat @ ( zero_zero @ nat ) @ m2,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(13,axiom,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ~ ( A @ C )
         => ? [D: nat] :
              ( ( ord_less @ nat @ D @ C )
              & ~ ( A @ D ) ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_infinite__descent) ).

thf(318,plain,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ~ ( A @ C )
         => ? [D: nat] :
              ( ( ord_less @ nat @ D @ C )
              & ~ ( A @ D ) ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(112,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_bot__nat__0_Oextremum) ).

thf(649,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(212,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ b @ TC ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ b @ TB ) )] :
      ( ( module @ TD @ b @ TC @ d @ A @ r )
     => ( ( module @ TA @ b @ TB @ d @ B @ r )
       => ( ( misomorphic @ b @ d @ TD @ TC @ a @ c @ r @ A @ m )
         => ( ( misomorphic @ b @ d @ a @ c @ TA @ TB @ r @ m @ B )
           => ( misomorphic @ b @ d @ TD @ TC @ TA @ TB @ r @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_misom__trans) ).

thf(1011,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( module_ext @ TD @ b @ TC ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ b @ TB ) )] :
      ( ( module @ TD @ b @ TC @ d @ A @ r )
     => ( ( module @ TA @ b @ TB @ d @ B @ r )
       => ( ( misomorphic @ b @ d @ TD @ TC @ a @ c @ r @ A @ m )
         => ( ( misomorphic @ b @ d @ a @ c @ TA @ TB @ r @ m @ B )
           => ( misomorphic @ b @ d @ TD @ TC @ TA @ TB @ r @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(48,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( suc @ ( minus_minus @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_Suc__pred) ).

thf(419,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( suc @ ( minus_minus @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(225,axiom,
    ! [A: set @ a,B: set @ a] :
      ( ( generator @ b @ d @ a @ c @ r @ m @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( ord_less_eq @ ( set @ a ) @ A @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) )
         => ( generator @ b @ d @ a @ c @ r @ m @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_generator__generator) ).

thf(1048,plain,
    ! [A: set @ a,B: set @ a] :
      ( ( generator @ b @ d @ a @ c @ r @ m @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( ord_less_eq @ ( set @ a ) @ A @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) )
         => ( generator @ b @ d @ a @ c @ r @ m @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(140,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ord_less @ nat @ ( minus_minus @ nat @ A @ ( suc @ B ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_diff__Suc__less) ).

thf(736,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ord_less @ nat @ ( minus_minus @ nat @ A @ ( suc @ B ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(104,axiom,
    ! [A: nat] :
      ~ ( ord_less_eq @ nat @ ( suc @ A ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_Suc__n__not__le__n) ).

thf(616,plain,
    ! [A: nat] :
      ~ ( ord_less_eq @ nat @ ( suc @ A ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(8,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( wellorder @ TB )
     => ! [A: TA > TB,B: TA > $o,C: TA] :
          ( ! [D: TA] :
              ( ! [E: TA] :
                  ( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
                 => ( B @ E ) )
             => ( B @ D ) )
         => ( B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_measure__induct) ).

thf(303,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( wellorder @ TB )
     => ! [A: TA > TB,B: TA > $o,C: TA] :
          ( ! [D: TA] :
              ( ! [E: TA] :
                  ( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
                 => ( B @ E ) )
             => ( B @ D ) )
         => ( B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(69,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_old_Onat_Oinject) ).

thf(477,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(75,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ B @ ( minus_minus @ nat @ B @ A ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_diff__diff__cancel) ).

thf(494,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ B @ ( minus_minus @ nat @ B @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(148,axiom,
    wellorder @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Owellorder) ).

thf(758,plain,
    wellorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(101,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( free_generator @ TD @ TB @ TA @ TC @ B @ A @ C )
       => ( generator @ TD @ TB @ TA @ TC @ B @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_Module_Ofree__generator__generator) ).

thf(609,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( free_generator @ TD @ TB @ TA @ TC @ B @ A @ C )
       => ( generator @ TD @ TB @ TA @ TC @ B @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(14,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ~ ( ord_less @ nat @ A @ B )
       => ( ord_less @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_linorder__neqE__nat) ).

thf(323,plain,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ~ ( ord_less @ nat @ A @ B )
       => ( ord_less @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(95,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
          = ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_diff__gt__0__iff__gt) ).

thf(594,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
          = ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(136,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_le__trans) ).

thf(726,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(71,axiom,
    ! [TA: $tType,A: TA > $o] :
      ( ( ( bot_bot @ ( set @ TA ) )
        = ( collect @ TA @ A ) )
      = ( ! [B: TA] :
            ~ ( A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_empty__Collect__eq) ).

thf(483,plain,
    ! [TA: $tType,A: TA > $o] :
      ( ( ( bot_bot @ ( set @ TA ) )
        = ( collect @ TA @ A ) )
      = ( ! [B: TA] :
            ~ ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(187,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ ( suc @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_not__less__eq) ).

thf(915,plain,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ ( suc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(152,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ B ) @ ( minus_minus @ nat @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_diff__le__mono2) ).

thf(769,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ B ) @ ( minus_minus @ nat @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(217,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TB,D: set @ TA,E: nat > TB,F: nat,G: nat > TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( ideal @ TB @ TD @ B @ C )
       => ( ( ord_less_eq @ ( set @ TA ) @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
         => ( ( member @ ( nat > TB ) @ E
              @ ( pi @ nat @ TB
                @ ( collect @ nat
                  @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ F ) )
                @ ^ [H: nat] : C ) )
           => ( ( member @ ( nat > TA ) @ G
                @ ( pi @ nat @ TA
                  @ ( collect @ nat
                    @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ F ) )
                  @ ^ [H: nat] : D ) )
             => ( member @ TA @ ( l_comb @ TB @ TD @ TA @ TC @ B @ A @ F @ E @ G ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_Module_Ol__comb__mem) ).

thf(1022,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TB,D: set @ TA,E: nat > TB,F: nat,G: nat > TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( ideal @ TB @ TD @ B @ C )
       => ( ( ord_less_eq @ ( set @ TA ) @ D @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) )
         => ( ( member @ ( nat > TB ) @ E
              @ ( pi @ nat @ TB
                @ ( collect @ nat
                  @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ F ) )
                @ ^ [H: nat] : C ) )
           => ( ( member @ ( nat > TA ) @ G
                @ ( pi @ nat @ TA
                  @ ( collect @ nat
                    @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ F ) )
                  @ ^ [H: nat] : D ) )
             => ( member @ TA @ ( l_comb @ TB @ TD @ TA @ TC @ B @ A @ F @ E @ G ) @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(275,axiom,
    ! [A: set @ a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( generator @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_fgs__generator) ).

thf(1177,plain,
    ! [A: set @ a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( generator @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(249,axiom,
    module @ a @ b @ c @ d @ m @ r,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_Module__axioms) ).

thf(1112,plain,
    module @ a @ b @ c @ d @ m @ r,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(30,axiom,
    module @ e @ b @ f @ d @ n @ r,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_4) ).

thf(367,plain,
    module @ e @ b @ f @ d @ n @ r,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(89,axiom,
    ! [A: nat] :
      ( A
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_n__not__Suc__n) ).

thf(552,plain,
    ! [A: nat] :
      ( A
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(174,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( ~ ( A @ ( zero_zero @ nat ) )
       => ? [C: nat] :
            ( ( ord_less_eq @ nat @ C @ B )
            & ! [D: nat] :
                ( ( ord_less @ nat @ D @ C )
               => ~ ( A @ D ) )
            & ( A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_ex__least__nat__le) ).

thf(848,plain,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( ~ ( A @ ( zero_zero @ nat ) )
       => ? [C: nat] :
            ( ( ord_less_eq @ nat @ C @ B )
            & ! [D: nat] :
                ( ( ord_less @ nat @ D @ C )
               => ~ ( A @ D ) )
            & ( A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(82,axiom,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( C @ A )
       => ( ! [D: nat] :
              ( ( ord_less_eq @ nat @ A @ D )
             => ( ( ord_less @ nat @ D @ B )
               => ( ( C @ D )
                 => ( C @ ( suc @ D ) ) ) ) )
         => ( C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_dec__induct) ).

thf(520,plain,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( C @ A )
       => ( ! [D: nat] :
              ( ( ord_less_eq @ nat @ A @ D )
             => ( ( ord_less @ nat @ D @ B )
               => ( ( C @ D )
                 => ( C @ ( suc @ D ) ) ) ) )
         => ( C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(186,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
     => ( ~ ( ord_less @ nat @ A @ B )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_less__SucE) ).

thf(912,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
     => ( ~ ( ord_less @ nat @ A @ B )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(282,axiom,
    member @ a @ h2 @ h,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_14) ).

thf(1198,plain,
    member @ a @ h2 @ h,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(169,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ A )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_le__antisym) ).

thf(827,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ A )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(125,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: carrier_ext @ TF @ ( aGroup_ext @ TF @ ( module_ext @ TF @ TB @ TE ) )] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( module @ TF @ TB @ TE @ TD @ C @ B )
       => ( ( misomorphic @ TB @ TD @ TA @ TC @ TF @ TE @ B @ A @ C )
         => ( misomorphic @ TB @ TD @ TF @ TE @ TA @ TC @ B @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_Module_Omisom__sym) ).

thf(695,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: carrier_ext @ TF @ ( aGroup_ext @ TF @ ( module_ext @ TF @ TB @ TE ) )] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( module @ TF @ TB @ TE @ TD @ C @ B )
       => ( ( misomorphic @ TB @ TD @ TA @ TC @ TF @ TE @ B @ A @ C )
         => ( misomorphic @ TB @ TD @ TF @ TE @ TA @ TC @ B @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(232,axiom,
    member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ h1 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_12) ).

thf(1066,plain,
    member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ h1 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(214,axiom,
    ord_less_eq @ ( set @ a ) @ h @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_9) ).

thf(1015,plain,
    ord_less_eq @ ( set @ a ) @ h @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(274,axiom,
    ( member @ ( nat > a ) @ g
    @ ( pi @ nat @ a
      @ ( collect @ nat
        @ ^ [A: nat] : ( ord_less_eq @ nat @ A @ m2 ) )
      @ ^ [A: nat] : ( insert @ a @ h2 @ h1 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).

thf(1176,plain,
    ( member @ ( nat > a ) @ g
    @ ( pi @ nat @ a
      @ ( collect @ nat
        @ ^ [A: nat] : ( ord_less_eq @ nat @ A @ m2 ) )
      @ ^ [A: nat] : ( insert @ a @ h2 @ h1 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(9,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B != A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_less__not__refl2) ).

thf(306,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(250,axiom,
    submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ ( bot_bot @ ( set @ a ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_18) ).

thf(1113,plain,
    submodule @ b @ d @ a @ c @ r @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ ( insert @ a @ h2 @ ( bot_bot @ ( set @ a ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(244,axiom,
    ( ( g @ m2 )
    = h2 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_3) ).

thf(1102,plain,
    ( ( g @ m2 )
    = h2 ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(55,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_not__less0) ).

thf(438,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(68,axiom,
    ! [A: nat > nat,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less @ nat @ D @ E )
         => ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ ( A @ B ) @ ( A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_less__mono__imp__le__mono) ).

thf(473,plain,
    ! [A: nat > nat,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less @ nat @ D @ E )
         => ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ ( A @ B ) @ ( A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(220,axiom,
    ! [A: set @ a,B: a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( ( member @ a @ B @ A )
       => ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_submodule__subset1) ).

thf(1034,plain,
    ! [A: set @ a,B: a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( ( member @ a @ B @ A )
       => ( member @ a @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(120,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_bot__nat__0_Oextremum__unique) ).

thf(680,plain,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(259,axiom,
    ( x
    = ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_13) ).

thf(1134,plain,
    ( x
    = ( l_comb @ b @ d @ a @ c @ r @ m @ m2 @ t @ g ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(26,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( free_generator @ TB @ TD @ TA @ TC @ B @ A @ C )
       => ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_Module_Ofree__generator__sub) ).

thf(355,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) ),C: set @ TA] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( ( free_generator @ TB @ TD @ TA @ TC @ B @ A @ C )
       => ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(264,axiom,
    ! [A: set @ b,B: set @ a,C: nat > b,D: nat,E: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( member @ ( nat > b ) @ C
            @ ( pi @ nat @ b
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
              @ ^ [F: nat] : A ) )
         => ( ( member @ ( nat > a ) @ E
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) ) )
           => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ D @ C @ E ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_l__span__closed2) ).

thf(1145,plain,
    ! [A: set @ b,B: set @ a,C: nat > b,D: nat,E: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( member @ ( nat > b ) @ C
            @ ( pi @ nat @ b
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
              @ ^ [F: nat] : A ) )
         => ( ( member @ ( nat > a ) @ E
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ B ) ) )
           => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ D @ C @ E ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(216,axiom,
    ! [A: set @ b,B: nat > b,C: nat,D: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ( member @ ( nat > b ) @ B
            @ ( pi @ nat @ b
              @ ( collect @ nat
                @ ^ [E: nat] : ( ord_less_eq @ nat @ E @ C ) )
              @ ^ [E: nat] : A ) )
          & ( member @ ( nat > a ) @ D
            @ ( pi @ nat @ a
              @ ( collect @ nat
                @ ^ [E: nat] : ( ord_less_eq @ nat @ E @ C ) )
              @ ^ [E: nat] : ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ) )
       => ( ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ D )
          = ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_linear__comb0__2Tr) ).

thf(1018,plain,
    ! [A: set @ b,B: nat > b,C: nat,D: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ( member @ ( nat > b ) @ B
            @ ( pi @ nat @ b
              @ ( collect @ nat
                @ ^ [E: nat] : ( ord_less_eq @ nat @ E @ C ) )
              @ ^ [E: nat] : A ) )
          & ( member @ ( nat > a ) @ D
            @ ( pi @ nat @ a
              @ ( collect @ nat
                @ ^ [E: nat] : ( ord_less_eq @ nat @ E @ C ) )
              @ ^ [E: nat] : ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ) ) ) )
       => ( ( l_comb @ b @ d @ a @ c @ r @ m @ C @ B @ D )
          = ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(197,axiom,
    wellorder @ product_unit,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Ounit___Orderings_Owellorder_4) ).

thf(968,plain,
    wellorder @ product_unit,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(63,axiom,
    ring @ b @ d @ r,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_sc__Ring) ).

thf(458,plain,
    ring @ b @ d @ r,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(94,axiom,
    ! [A: nat,B: nat,C: nat > nat > $o] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ! [D: nat] : ( C @ D @ D )
       => ( ! [D: nat,E: nat,F: nat] :
              ( ( C @ D @ E )
             => ( ( C @ E @ F )
               => ( C @ D @ F ) ) )
         => ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
           => ( C @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_transitive__stepwise__le) ).

thf(580,plain,
    ! [A: nat,B: nat,C: nat > nat > $o] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ! [D: nat] : ( C @ D @ D )
       => ( ! [D: nat,E: nat,F: nat] :
              ( ( C @ D @ E )
             => ( ( C @ E @ F )
               => ( C @ D @ F ) ) )
         => ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
           => ( C @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(109,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_diff__0__right) ).

thf(640,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(179,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
      = ( ( ord_less @ nat @ A @ B )
        | ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_less__Suc__eq) ).

thf(872,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
      = ( ( ord_less @ nat @ A @ B )
        | ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(129,axiom,
    ! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_empty__subsetI) ).

thf(707,plain,
    ! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ ( bot_bot @ ( set @ TA ) ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(193,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ B )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ B @ C ) )
          = ( ord_less_eq @ nat @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_le__diff__iff_H) ).

thf(954,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ B )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ B @ C ) )
          = ( ord_less_eq @ nat @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(141,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_less__zeroE) ).

thf(739,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(10,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) )
            = ( ord_less @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_lift__Suc__mono__less__iff) ).

thf(310,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) )
            = ( ord_less @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(131,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_diff__zero) ).

thf(712,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(208,axiom,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ b @ TA ) )] :
      ( ( module @ TB @ b @ TA @ d @ A @ r )
     => ( module @ ( a > TB ) @ b @ product_unit @ d @ ( hom @ b @ d @ a @ c @ TB @ TA @ r @ m @ A ) @ r ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_HOM__is__module) ).

thf(1003,plain,
    ! [TA: $tType,TB: $tType,A: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( module_ext @ TB @ b @ TA ) )] :
      ( ( module @ TB @ b @ TA @ d @ A @ r )
     => ( module @ ( a > TB ) @ b @ product_unit @ d @ ( hom @ b @ d @ a @ c @ TB @ TA @ r @ m @ A ) @ r ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(227,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( member @ TA @ C @ B ) )
     => ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_subsetI) ).

thf(1053,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( member @ TA @ C @ B ) )
     => ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(160,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_less__Suc0) ).

thf(791,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(56,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_gr0I) ).

thf(441,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(65,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ B )
     => ? [C: nat] :
          ( B
          = ( suc @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_Suc__le__D) ).

thf(463,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ B )
     => ? [C: nat] :
          ( B
          = ( suc @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(238,axiom,
    ! [A: set @ b,B: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( generator @ b @ d @ a @ c @ r @ m @ B )
       => ( ( ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A )
            = ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
         => ( ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B )
            = ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_l__span__closed3) ).

thf(1085,plain,
    ! [A: set @ b,B: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( generator @ b @ d @ a @ c @ r @ m @ B )
       => ( ( ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A )
            = ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
         => ( ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B )
            = ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(192,axiom,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( insert @ TA @ A @ ( insert @ TA @ A @ B ) )
      = ( insert @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_insert__absorb2) ).

thf(951,plain,
    ! [TA: $tType,A: TA,B: set @ TA] :
      ( ( insert @ TA @ A @ ( insert @ TA @ A @ B ) )
      = ( insert @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(110,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_diff__le__self) ).

thf(643,plain,
    ! [A: nat,B: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(222,axiom,
    ! [A: set @ a,B: set @ a,C: nat > a,D: nat,E: nat > b] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ A )
       => ( ( ( member @ ( nat > a ) @ C
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : B ) )
            & ( member @ ( nat > b ) @ E
              @ ( pi @ nat @ b
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ) )
         => ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ B ) ) @ D @ E @ C )
            = ( l_comb @ b @ d @ a @ c @ r @ m @ D @ E @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_eSum__in__SubmoduleTr) ).

thf(1038,plain,
    ! [A: set @ a,B: set @ a,C: nat > a,D: nat,E: nat > b] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ A )
       => ( ( ( member @ ( nat > a ) @ C
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : B ) )
            & ( member @ ( nat > b ) @ E
              @ ( pi @ nat @ b
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ) )
         => ( ( l_comb @ b @ d @ a @ product_unit @ r @ ( mdl @ a @ b @ c @ m @ ( algebr833503410le_fgs @ b @ d @ a @ c @ r @ m @ B ) ) @ D @ E @ C )
            = ( l_comb @ b @ d @ a @ c @ r @ m @ D @ E @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(32,axiom,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ! [C: TA] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect @ TA @ A )
        = ( collect @ TA @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_Collect__cong) ).

thf(370,plain,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ! [C: TA] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect @ TA @ A )
        = ( collect @ TA @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(86,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
       => ( submodule @ TD @ TB @ TA @ TC @ B @ A @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_Module_Ofgs__submodule) ).

thf(544,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
       => ( submodule @ TD @ TB @ TA @ TC @ B @ A @ ( algebr833503410le_fgs @ TD @ TB @ TA @ TC @ B @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(184,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ ( suc @ B ) )
     => ( ~ ( ord_less_eq @ nat @ A @ B )
       => ( A
          = ( suc @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_le__SucE) ).

thf(907,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ ( suc @ B ) )
     => ( ~ ( ord_less_eq @ nat @ A @ B )
       => ( A
          = ( suc @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(19,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
      = ( ( ord_less @ nat @ A @ B )
        | ( ord_less @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_nat__neq__iff) ).

thf(336,plain,
    ! [A: nat,B: nat] :
      ( ( A != B )
      = ( ( ord_less @ nat @ A @ B )
        | ( ord_less @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(262,axiom,
    misomorphic @ b @ d @ a @ c @ a @ c @ r @ m @ m,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_misom__self) ).

thf(1141,plain,
    misomorphic @ b @ d @ a @ c @ a @ c @ r @ m @ m,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(103,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ? [C: nat] :
            ( ( A
              = ( suc @ C ) )
            & ( ord_less @ nat @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_less__Suc__eq__0__disj) ).

thf(613,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ? [C: nat] :
            ( ( A
              = ( suc @ C ) )
            & ( ord_less @ nat @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(97,axiom,
    cancel1352612707id_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add) ).

thf(600,plain,
    cancel1352612707id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(176,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_Suc__leD) ).

thf(864,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(241,axiom,
    ! [TA: $tType,A: TA,B: TA > $o] :
      ( ( member @ TA @ A @ ( collect @ TA @ B ) )
      = ( B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_mem__Collect__eq) ).

thf(1094,plain,
    ! [TA: $tType,A: TA,B: TA > $o] :
      ( ( member @ TA @ A @ ( collect @ TA @ B ) )
      = ( B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(76,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( minus_minus @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_diff__Suc__Suc) ).

thf(497,plain,
    ! [A: nat,B: nat] :
      ( ( minus_minus @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( minus_minus @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(107,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( A @ C )
           => ( A @ ( suc @ C ) ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_nat__induct) ).

thf(627,plain,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( A @ C )
           => ( A @ ( suc @ C ) ) )
       => ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(60,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( ord_less_eq @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_le__diff__iff) ).

thf(450,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( ord_less_eq @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(268,axiom,
    ! [A: set @ a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( module @ a @ b @ product_unit @ d @ ( mdl @ a @ b @ c @ m @ A ) @ r ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_mdl__is__module) ).

thf(1157,plain,
    ! [A: set @ a] :
      ( ( submodule @ b @ d @ a @ c @ r @ m @ A )
     => ( module @ a @ b @ product_unit @ d @ ( mdl @ a @ b @ c @ m @ A ) @ r ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(201,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_diff__le__mono) ).

thf(976,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(149,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_old_Onat_Odistinct_I1_J) ).

thf(759,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(182,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( ~ ( A @ ( zero_zero @ nat ) )
       => ? [C: nat] :
            ( ( ord_less @ nat @ C @ B )
            & ! [D: nat] :
                ( ( ord_less_eq @ nat @ D @ C )
               => ~ ( A @ D ) )
            & ( A @ ( suc @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_ex__least__nat__less) ).

thf(891,plain,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( ~ ( A @ ( zero_zero @ nat ) )
       => ? [C: nat] :
            ( ( ord_less @ nat @ C @ B )
            & ! [D: nat] :
                ( ( ord_less_eq @ nat @ D @ C )
               => ~ ( A @ D ) )
            & ( A @ ( suc @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(278,axiom,
    ! [A: set @ a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) )
        = ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_l__span__l__span) ).

thf(1189,plain,
    ! [A: set @ a] :
      ( ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
     => ( ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) )
        = ( linear_span @ b @ d @ a @ c @ r @ m @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(31,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( misomorphic @ TB @ TD @ TA @ TC @ TA @ TC @ B @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_Module_Omisom__self) ).

thf(368,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TB @ TC ) ),B: carrier_ext @ TB @ ( aGroup_ext @ TB @ ( ring_ext @ TB @ TD ) )] :
      ( ( module @ TA @ TB @ TC @ TD @ A @ B )
     => ( misomorphic @ TB @ TD @ TA @ TC @ TA @ TC @ B @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(29,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ( minus_minus @ nat @ B @ A )
            = ( minus_minus @ nat @ C @ A ) )
          = ( B = C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_eq__diff__iff) ).

thf(363,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ( minus_minus @ nat @ B @ A )
            = ( minus_minus @ nat @ C @ A ) )
          = ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(90,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
           => ( ~ ( A @ C )
             => ? [D: nat] :
                  ( ( ord_less @ nat @ D @ C )
                  & ~ ( A @ D ) ) ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_infinite__descent0) ).

thf(556,plain,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
           => ( ~ ( A @ C )
             => ? [D: nat] :
                  ( ( ord_less @ nat @ D @ C )
                  & ~ ( A @ D ) ) ) )
       => ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(195,axiom,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_old_Onat_Odistinct_I2_J) ).

thf(961,plain,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(81,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
       => ( ord_less @ nat @ ( minus_minus @ nat @ B @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_diff__less) ).

thf(517,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
       => ( ord_less @ nat @ ( minus_minus @ nat @ B @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(43,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_Zero__neq__Suc) ).

thf(401,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(47,axiom,
    canoni770627133id_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add) ).

thf(418,plain,
    canoni770627133id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(157,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_bot__nat__0_Onot__eq__extremum) ).

thf(781,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(206,axiom,
    ! [A: set @ b,B: set @ a,C: nat > b,D: nat,E: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( member @ ( nat > b ) @ C
            @ ( pi @ nat @ b
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
              @ ^ [F: nat] : A ) )
         => ( ( member @ ( nat > a ) @ E
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) )
           => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ D @ C @ E ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_l__span__closed) ).

thf(995,plain,
    ! [A: set @ b,B: set @ a,C: nat > b,D: nat,E: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( member @ ( nat > b ) @ C
            @ ( pi @ nat @ b
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
              @ ^ [F: nat] : A ) )
         => ( ( member @ ( nat > a ) @ E
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) )
           => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ D @ C @ E ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(4,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_not__gr__zero) ).

thf(290,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(116,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( B
         != ( suc @ A ) )
       => ~ ! [C: nat] :
              ( ( ord_less @ nat @ A @ C )
             => ( B
               != ( suc @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_Nat_OlessE) ).

thf(667,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( B
         != ( suc @ A ) )
       => ~ ! [C: nat] :
              ( ( ord_less @ nat @ A @ C )
             => ( B
               != ( suc @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(150,axiom,
    ! [A: nat,B: nat] :
      ( ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_diff__is__0__eq) ).

thf(763,plain,
    ! [A: nat,B: nat] :
      ( ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(171,axiom,
    ! [A: set @ b,B: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( submodule @ b @ d @ a @ c @ r @ m @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_linear__span__subModule) ).

thf(832,plain,
    ! [A: set @ b,B: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( submodule @ b @ d @ a @ c @ r @ m @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(23,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ~ ! [C: nat] :
            ( ( ord_less @ nat @ A @ C )
           => ( B
             != ( suc @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_Suc__lessE) ).

thf(346,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ~ ! [C: nat] :
            ( ( ord_less @ nat @ A @ C )
           => ( B
             != ( suc @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(88,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( suc @ ( minus_minus @ nat @ B @ ( suc @ A ) ) )
        = ( minus_minus @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_Suc__diff__Suc) ).

thf(549,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( suc @ ( minus_minus @ nat @ B @ ( suc @ A ) ) )
        = ( minus_minus @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(224,axiom,
    ! [A: set @ b,B: set @ a,C: nat > b,D: nat,E: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( member @ ( nat > b ) @ C
            @ ( pi @ nat @ b
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
              @ ^ [F: nat] : A ) )
         => ( ( member @ ( nat > a ) @ E
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : B ) )
           => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ D @ C @ E ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_l__comb__mem) ).

thf(1045,plain,
    ! [A: set @ b,B: set @ a,C: nat > b,D: nat,E: nat > a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( ( member @ ( nat > b ) @ C
            @ ( pi @ nat @ b
              @ ( collect @ nat
                @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
              @ ^ [F: nat] : A ) )
         => ( ( member @ ( nat > a ) @ E
              @ ( pi @ nat @ a
                @ ( collect @ nat
                  @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ D ) )
                @ ^ [F: nat] : B ) )
           => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ D @ C @ E ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(53,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: TA] :
      ( ( ( insert @ TA @ A @ B )
        = ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) )
      = ( ( A = C )
        & ( ord_less_eq @ ( set @ TA ) @ B @ ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_singleton__insert__inj__eq_H) ).

thf(433,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: TA] :
      ( ( ( insert @ TA @ A @ B )
        = ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) )
      = ( ( A = C )
        & ( ord_less_eq @ ( set @ TA ) @ B @ ( insert @ TA @ C @ ( bot_bot @ ( set @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(78,axiom,
    ! [TA: $tType,A: TA] :
      ( ( collect @ TA @ ( (=) @ TA @ A ) )
      = ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_singleton__conv2) ).

thf(510,plain,
    ! [TA: $tType,A: TA] :
      ( ( collect @ TA @ ( (=) @ TA @ A ) )
      = ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(270,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( member @ TA @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_Collect__mem__eq) ).

thf(1161,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( member @ TA @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(163,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ A ) @ B ) @ ( suc @ C ) )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_Suc__diff__diff) ).

thf(800,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ A ) @ B ) @ ( suc @ C ) )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(3,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( wellorder @ TB )
     => ! [A: TA > TB,B: TA > $o,C: TA] :
          ( ! [D: TA] :
              ( ! [E: TA] :
                  ( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
                 => ( B @ E ) )
             => ( B @ D ) )
         => ( B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_measure__induct__rule) ).

thf(287,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( wellorder @ TB )
     => ! [A: TA > TB,B: TA > $o,C: TA] :
          ( ! [D: TA] :
              ( ! [E: TA] :
                  ( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
                 => ( B @ E ) )
             => ( B @ D ) )
         => ( B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(253,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: nat > TD,E: nat,F: nat > TA,G: nat > TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
       => ( ( ( member @ ( nat > TD ) @ D
              @ ( pi @ nat @ TD
                @ ( collect @ nat
                  @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                @ ^ [H: nat] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ) @ B ) ) )
            & ( member @ ( nat > TA ) @ F
              @ ( pi @ nat @ TA
                @ ( collect @ nat
                  @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                @ ^ [H: nat] : C ) )
            & ( member @ ( nat > TA ) @ G
              @ ( pi @ nat @ TA
                @ ( collect @ nat
                  @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                @ ^ [H: nat] : C ) )
            & ! [H: nat] :
                ( ( member @ nat @ H
                  @ ( collect @ nat
                    @ ^ [I: nat] : ( ord_less_eq @ nat @ I @ E ) ) )
               => ( ( F @ H )
                  = ( G @ H ) ) ) )
         => ( ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ F )
            = ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ G ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_Module_Olinear__comb__eqTr) ).

thf(1116,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ),B: carrier_ext @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ),C: set @ TA,D: nat > TD,E: nat,F: nat > TA,G: nat > TA] :
      ( ( module @ TA @ TD @ TC @ TB @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ C @ ( carrier @ TA @ ( aGroup_ext @ TA @ ( module_ext @ TA @ TD @ TC ) ) @ A ) )
       => ( ( ( member @ ( nat > TD ) @ D
              @ ( pi @ nat @ TD
                @ ( collect @ nat
                  @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                @ ^ [H: nat] : ( carrier @ TD @ ( aGroup_ext @ TD @ ( ring_ext @ TD @ TB ) ) @ B ) ) )
            & ( member @ ( nat > TA ) @ F
              @ ( pi @ nat @ TA
                @ ( collect @ nat
                  @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                @ ^ [H: nat] : C ) )
            & ( member @ ( nat > TA ) @ G
              @ ( pi @ nat @ TA
                @ ( collect @ nat
                  @ ^ [H: nat] : ( ord_less_eq @ nat @ H @ E ) )
                @ ^ [H: nat] : C ) )
            & ! [H: nat] :
                ( ( member @ nat @ H
                  @ ( collect @ nat
                    @ ^ [I: nat] : ( ord_less_eq @ nat @ I @ E ) ) )
               => ( ( F @ H )
                  = ( G @ H ) ) ) )
         => ( ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ F )
            = ( l_comb @ TD @ TB @ TA @ TC @ B @ A @ E @ D @ G ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(258,axiom,
    submodule @ b @ d @ a @ c @ r @ m @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_submodule__0) ).

thf(1133,plain,
    submodule @ b @ d @ a @ c @ r @ m @ ( insert @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( bot_bot @ ( set @ a ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(146,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_bot__nat__0_Oextremum__strict) ).

thf(752,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(59,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_diff__self) ).

thf(447,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(12,axiom,
    ! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
      ( ! [D: TA] :
          ( ~ ( A @ D )
         => ? [E: TA] :
              ( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
              & ~ ( A @ E ) ) )
     => ( A @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_infinite__descent__measure) ).

thf(315,plain,
    ! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
      ( ! [D: TA] :
          ( ~ ( A @ D )
         => ? [E: TA] :
              ( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
              & ~ ( A @ E ) ) )
     => ( A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(45,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_diff__commute) ).

thf(410,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(44,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( suc @ B ) )
     => ( A
       != ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_nat_OdiscI) ).

thf(405,plain,
    ! [A: nat,B: nat] :
      ( ( A
        = ( suc @ B ) )
     => ( A
       != ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(265,axiom,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ A @ B ) @ C )
      = ( ( member @ TA @ A @ C )
        & ( ord_less_eq @ ( set @ TA ) @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_insert__subset) ).

thf(1148,plain,
    ! [TA: $tType,A: TA,B: set @ TA,C: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ ( insert @ TA @ A @ B ) @ C )
      = ( ( member @ TA @ A @ C )
        & ( ord_less_eq @ ( set @ TA ) @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(49,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ ( suc @ B ) @ A )
        = ( suc @ ( minus_minus @ nat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_Suc__diff__le) ).

thf(422,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ ( suc @ B ) @ A )
        = ( suc @ ( minus_minus @ nat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(191,axiom,
    ideal @ b @ d @ r @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_8) ).

thf(950,plain,
    ideal @ b @ d @ r @ ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(242,axiom,
    ~ ( member @ a @ h2 @ h1 ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_15) ).

thf(1097,plain,
    ~ ( member @ a @ h2 @ h1 ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(172,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( minus_minus @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_Diff__empty) ).

thf(835,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( minus_minus @ ( set @ TA ) @ A @ ( bot_bot @ ( set @ TA ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(33,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
      = ( ? [B: nat] :
            ( A
            = ( suc @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_gr0__conv__Suc) ).

thf(373,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
      = ( ? [B: nat] :
            ( A
            = ( suc @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(99,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B
       != ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_gr__implies__not0) ).

thf(603,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B
       != ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(15,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_less__irrefl__nat) ).

thf(326,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(188,axiom,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( C @ B )
       => ( ! [D: nat] :
              ( ( ord_less_eq @ nat @ A @ D )
             => ( ( ord_less @ nat @ D @ B )
               => ( ( C @ ( suc @ D ) )
                 => ( C @ D ) ) ) )
         => ( C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_inc__induct) ).

thf(918,plain,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( C @ B )
       => ( ! [D: nat] :
              ( ( ord_less_eq @ nat @ A @ D )
             => ( ( ord_less @ nat @ D @ B )
               => ( ( C @ ( suc @ D ) )
                 => ( C @ D ) ) ) )
         => ( C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(21,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TA @ TD ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( module @ TC @ TA @ TD @ TB @ A @ B )
     => ( ring @ TA @ TB @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_Module_Osc__Ring) ).

thf(341,plain,
    ! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: carrier_ext @ TC @ ( aGroup_ext @ TC @ ( module_ext @ TC @ TA @ TD ) ),B: carrier_ext @ TA @ ( aGroup_ext @ TA @ ( ring_ext @ TA @ TB ) )] :
      ( ( module @ TC @ TA @ TD @ TB @ A @ B )
     => ( ring @ TA @ TB @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(166,axiom,
    ! [TA: $tType,A: TA > nat,B: TA > $o,C: TA] :
      ( ! [D: TA] :
          ( ( ( A @ D )
            = ( zero_zero @ nat ) )
         => ( B @ D ) )
     => ( ! [D: TA] :
            ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ D ) )
           => ( ~ ( B @ D )
             => ? [E: TA] :
                  ( ( ord_less @ nat @ ( A @ E ) @ ( A @ D ) )
                  & ~ ( B @ E ) ) ) )
       => ( B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_infinite__descent0__measure) ).

thf(807,plain,
    ! [TA: $tType,A: TA > nat,B: TA > $o,C: TA] :
      ( ! [D: TA] :
          ( ( ( A @ D )
            = ( zero_zero @ nat ) )
         => ( B @ D ) )
     => ( ! [D: TA] :
            ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ D ) )
           => ( ~ ( B @ D )
             => ? [E: TA] :
                  ( ( ord_less @ nat @ ( A @ E ) @ ( A @ D ) )
                  & ~ ( B @ E ) ) ) )
       => ( B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(272,axiom,
    free_generator @ b @ d @ a @ c @ r @ m @ h,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_5) ).

thf(1172,plain,
    free_generator @ b @ d @ a @ c @ r @ m @ h,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(228,axiom,
    ord_less_eq @ ( set @ a ) @ h1 @ h,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_6) ).

thf(1056,plain,
    ord_less_eq @ ( set @ a ) @ h1 @ h,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(281,axiom,
    submodule @ b @ d @ a @ c @ r @ m @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_submodule__whole) ).

thf(1197,plain,
    submodule @ b @ d @ a @ c @ r @ m @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(134,axiom,
    ! [TA: $tType,A: TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( B = A ) )
      = ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_singleton__conv) ).

thf(720,plain,
    ! [TA: $tType,A: TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( B = A ) )
      = ( insert @ TA @ A @ ( bot_bot @ ( set @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(41,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_Suc__le__mono) ).

thf(395,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(92,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( suc @ A ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_Suc__leI) ).

thf(575,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( suc @ A ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(51,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
       => ( ( minus_minus @ ( set @ TA ) @ B @ ( minus_minus @ ( set @ TA ) @ C @ A ) )
          = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_double__diff) ).

thf(427,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
       => ( ( minus_minus @ ( set @ TA ) @ B @ ( minus_minus @ ( set @ TA ) @ C @ A ) )
          = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(64,axiom,
    ! [A: nat > nat > $o,B: nat,C: nat] :
      ( ! [D: nat] : ( A @ D @ ( zero_zero @ nat ) )
     => ( ! [D: nat] : ( A @ ( zero_zero @ nat ) @ ( suc @ D ) )
       => ( ! [D: nat,E: nat] :
              ( ( A @ D @ E )
             => ( A @ ( suc @ D ) @ ( suc @ E ) ) )
         => ( A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_diff__induct) ).

thf(459,plain,
    ! [A: nat > nat > $o,B: nat,C: nat] :
      ( ! [D: nat] : ( A @ D @ ( zero_zero @ nat ) )
     => ( ! [D: nat] : ( A @ ( zero_zero @ nat ) @ ( suc @ D ) )
       => ( ! [D: nat,E: nat] :
              ( ( A @ D @ E )
             => ( A @ ( suc @ D ) @ ( suc @ E ) ) )
         => ( A @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(205,axiom,
    ! [A: set @ b,B: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_linear__span__inc__0) ).

thf(992,plain,
    ! [A: set @ b,B: set @ a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ( member @ a @ ( zero2 @ a @ ( module_ext @ a @ b @ c ) @ m ) @ ( linear_span @ b @ d @ a @ c @ r @ m @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(135,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( A != B )
       => ( ord_less @ ( set @ TA ) @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_psubsetI) ).

thf(723,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( A != B )
       => ( ord_less @ ( set @ TA ) @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(203,axiom,
    ! [A: set @ b,B: a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( member @ a @ B @ ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A ) )
       => ? [C: nat,D: nat > b] :
            ( ( member @ ( nat > b ) @ D
              @ ( pi @ nat @ b
                @ ( collect @ nat
                  @ ^ [E: nat] : ( ord_less_eq @ nat @ E @ C ) )
                @ ^ [E: nat] : A ) )
            & ? [E: nat > a] :
                ( ( member @ ( nat > a ) @ E
                  @ ( pi @ nat @ a
                    @ ( collect @ nat
                      @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
                    @ ^ [F: nat] : ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) )
                & ( B
                  = ( l_comb @ b @ d @ a @ c @ r @ m @ C @ D @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_mem__smodule__ideal__coeff) ).

thf(982,plain,
    ! [A: set @ b,B: a] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( member @ a @ B @ ( smodule_ideal_coeff @ b @ d @ a @ c @ r @ m @ A ) )
       => ? [C: nat,D: nat > b] :
            ( ( member @ ( nat > b ) @ D
              @ ( pi @ nat @ b
                @ ( collect @ nat
                  @ ^ [E: nat] : ( ord_less_eq @ nat @ E @ C ) )
                @ ^ [E: nat] : A ) )
            & ? [E: nat > a] :
                ( ( member @ ( nat > a ) @ E
                  @ ( pi @ nat @ a
                    @ ( collect @ nat
                      @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
                    @ ^ [F: nat] : ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) )
                & ( B
                  = ( l_comb @ b @ d @ a @ c @ r @ m @ C @ D @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(100,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_less__SucI) ).

thf(607,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(72,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_le0) ).

thf(486,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(164,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ B )
     => ( ord_less @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_Suc__le__lessD) ).

thf(803,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ B )
     => ( ord_less @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(67,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_nat_Odistinct_I1_J) ).

thf(469,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(211,axiom,
    ! [A: set @ a] :
      ( ( generator @ b @ d @ a @ c @ r @ m @ A )
     => ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_generator__sub__carrier) ).

thf(1009,plain,
    ! [A: set @ a] :
      ( ( generator @ b @ d @ a @ c @ r @ m @ A )
     => ( ord_less_eq @ ( set @ a ) @ A @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(221,axiom,
    ( member @ ( nat > b ) @ t
    @ ( pi @ nat @ b
      @ ( collect @ nat
        @ ^ [A: nat] : ( ord_less_eq @ nat @ A @ m2 ) )
      @ ^ [A: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_2) ).

thf(1037,plain,
    ( member @ ( nat > b ) @ t
    @ ( pi @ nat @ b
      @ ( collect @ nat
        @ ^ [A: nat] : ( ord_less_eq @ nat @ A @ m2 ) )
      @ ^ [A: nat] : ( carrier @ b @ ( aGroup_ext @ b @ ( ring_ext @ b @ d ) ) @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(147,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( minus_minus @ ( set @ TA ) @ A @ A )
      = ( bot_bot @ ( set @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_Diff__cancel) ).

thf(755,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( minus_minus @ ( set @ TA ) @ A @ A )
      = ( bot_bot @ ( set @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(74,axiom,
    ! [TA: $tType,A: TA > $o] :
      ( ( ( collect @ TA @ A )
        = ( bot_bot @ ( set @ TA ) ) )
      = ( ! [B: TA] :
            ~ ( A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_Collect__empty__eq) ).

thf(491,plain,
    ! [TA: $tType,A: TA > $o] :
      ( ( ( collect @ TA @ A )
        = ( bot_bot @ ( set @ TA ) ) )
      = ( ! [B: TA] :
            ~ ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(285,axiom,
    member @ a @ h2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_17) ).

thf(1204,plain,
    member @ a @ h2 @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(199,axiom,
    ( ( ord_less @ nat )
    = ( ^ [A: nat] : ( ord_less_eq @ nat @ ( suc @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_less__eq__Suc__le) ).

thf(972,plain,
    ( ( ord_less @ nat )
    = ( ^ [A: nat] : ( ord_less_eq @ nat @ ( suc @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(61,axiom,
    ( ( ord_less @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq @ nat @ A @ B )
          & ( A != B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_nat__less__le) ).

thf(454,plain,
    ( ( ord_less @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq @ nat @ A @ B )
          & ( A != B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(234,axiom,
    ! [A: set @ b,B: set @ a,C: nat] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ! [D: nat > b,E: nat > a] :
            ( ( ( member @ ( nat > b ) @ D
                @ ( pi @ nat @ b
                  @ ( collect @ nat
                    @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
                  @ ^ [F: nat] : A ) )
              & ( member @ ( nat > a ) @ E
                @ ( pi @ nat @ a
                  @ ( collect @ nat
                    @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
                  @ ^ [F: nat] : B ) ) )
           => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ C @ D @ E ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_liear__comb__memTr) ).

thf(1069,plain,
    ! [A: set @ b,B: set @ a,C: nat] :
      ( ( ideal @ b @ d @ r @ A )
     => ( ( ord_less_eq @ ( set @ a ) @ B @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) )
       => ! [D: nat > b,E: nat > a] :
            ( ( ( member @ ( nat > b ) @ D
                @ ( pi @ nat @ b
                  @ ( collect @ nat
                    @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
                  @ ^ [F: nat] : A ) )
              & ( member @ ( nat > a ) @ E
                @ ( pi @ nat @ a
                  @ ( collect @ nat
                    @ ^ [F: nat] : ( ord_less_eq @ nat @ F @ C ) )
                  @ ^ [F: nat] : B ) ) )
           => ( member @ a @ ( l_comb @ b @ d @ a @ c @ r @ m @ C @ D @ E ) @ ( carrier @ a @ ( aGroup_ext @ a @ ( module_ext @ a @ b @ c ) ) @ m ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(1205,plain,
    $false,
    inference(e,[status(thm)],[645,1024,333,1179,777,1110,1005,683,677,597,861,1173,1142,379,1078,385,500,715,814,747,538,1164,797,709,301,436,651,357,1059,542,619,838,817,969,1013,1115,1109,376,623,1136,1151,353,1183,1051,480,766,734,293,866,1072,774,660,998,988,1201,1104,877,692,380,546,1199,1067,1057,334,910,1091,778,1042,392,788,729,329,830,1159,1195,1128,456,974,742,1121,1064,1006,388,869,697,1089,1028,488,339,637,1016,686,1075,1114,1099,701,515,654,1192,343,805,398,674,794,718,298,425,745,689,444,601,965,750,513,430,1154,904,732,1125,772,286,958,1124,413,875,577,1139,382,572,313,611,1107,466,704,947,350,540,672,445,936,979,784,362,318,649,1011,419,1048,736,616,303,477,494,758,609,323,594,726,483,915,769,1022,1177,1112,367,552,848,520,912,1198,827,695,1066,1015,1176,306,1113,1102,438,473,1034,680,1134,355,1145,1018,968,458,580,640,872,707,954,739,310,712,1003,1053,791,441,463,1085,951,643,1038,370,544,907,336,1141,613,600,864,1094,497,627,450,1157,976,759,891,1189,368,363,556,961,517,401,418,781,995,290,667,763,832,346,549,1045,433,510,1161,800,287,1116,1133,752,447,315,410,405,1148,422,950,1097,835,373,603,326,918,341,807,1172,1056,1197,720,395,575,427,459,992,723,982,607,486,803,469,1009,1037,755,491,1204,972,454,1069]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12  % Problem  : ITP026^2 : TPTP v8.2.0. Released v7.5.0.
% 0.14/0.15  % Command  : run_Leo-III %s %d
% 0.16/0.36  % Computer : n002.cluster.edu
% 0.16/0.36  % Model    : x86_64 x86_64
% 0.16/0.36  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.36  % Memory   : 8042.1875MB
% 0.16/0.36  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.16/0.36  % CPULimit : 300
% 0.16/0.36  % WCLimit  : 300
% 0.16/0.36  % DateTime : Sat May 18 18:34:54 EDT 2024
% 0.16/0.36  % CPUTime  : 
% 0.86/0.83  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 1.85/1.10  % [INFO] 	 Parsing done (265ms). 
% 1.85/1.11  % [INFO] 	 Running in sequential loop mode. 
% 2.72/1.32  % [INFO] 	 eprover registered as external prover. 
% 2.72/1.32  % [INFO] 	 cvc4 registered as external prover. 
% 2.72/1.33  % [INFO] 	 Scanning for conjecture ... 
% 2.87/1.43  % [INFO] 	 Found a conjecture (or negated_conjecture) and 285 axioms. Running axiom selection ... 
% 3.45/1.53  % [INFO] 	 Axiom selection finished. Selected 283 axioms (removed 2 axioms). 
% 4.10/1.76  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 4.48/1.78  % [INFO] 	 Type checking passed. 
% 4.48/1.79  % [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 ... 
% 62.60/45.48  % External prover 'e' found a proof!
% 62.60/45.48  % [INFO] 	 Killing All external provers ... 
% 62.60/45.48  % Time passed: 45026ms (effective reasoning time: 44370ms)
% 62.60/45.48  % 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)>
% 62.60/45.49  % Axioms used in derivation (283): fact_159_lift__Suc__mono__less__iff, fact_51_Suc__diff__diff, fact_103_Zero__neq__Suc, fact_129_less__SucI, fact_99_not0__implies__Suc, fact_4_free__generator__sub, fact_37_nat_Oinject, fact_249_l__span__closed, fact_111_nat_Odistinct_I1_J, fact_67_Nat_Oex__has__greatest__nat, fact_204_empty__iff, fact_241_lin__span__coeff__mono, fact_9_linear__comb__eq, fact_41_le0, fact_144_le__diff__iff_H, fact_168_Suc__le__eq, fact_165_Suc__le__lessD, fact_57_less__Suc0, fact_238_singletonI, fact_185_diff__Suc__less, fact_22_Module_Omem__fgs__l__comb, fact_85_transitive__stepwise__le, fact_25_HOM__is__module, fact_187_Module_Ofree__generator__sub, tcon_Nat_Onat___Groups_Ocomm__monoid__diff, fact_11_l__comb__mem, conj_9, conj_11, fact_245_l__span__closed3, fact_147_Nat_Odiff__diff__eq, fact_184_ex__least__nat__less, fact_69_le__antisym, fact_142_gr0I, fact_218_l__span__cont__H, conj_3, tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, fact_220_l__span__gen__mono, fact_60_Module_Omisom__self, fact_149_eq__diff__iff, fact_93_le__SucE, conj_14, fact_29_Module_Ol__comb__mem, fact_171_less__Suc__eq__0__disj, fact_216_submodule__of__mdl, fact_23_Module_Ofgs__sub__carrier, fact_235_empty__subsetI, fact_2_submodule__whole, conj_10, fact_181_diff__less, fact_12_liear__comb__memTr, fact_7_fgs__mono, fact_152_diffs0__imp__equal, fact_160_lift__Suc__mono__less, fact_101_old_Onat_Oexhaust, fact_153_less__imp__diff__less, fact_72_Suc__inject, fact_182_Module_Osubmodule__subset1, fact_169_Suc__leI, fact_163_less__Suc__eq__le, fact_200_subset__antisym, fact_202_empty__Diff, tcon_Product__Type_Ounit___Orderings_Owellorder_4, fact_180_Suc__diff__Suc, fact_91_Suc__le__D, fact_92_le__SucI, fact_94_Suc__leD, fact_248_l__comb__mem__linear__span, fact_18_mem__smodule__ideal__coeff, fact_199_subsetI, fact_164_le__less__Suc__eq, fact_161_le__imp__less__Suc, fact_219_l__span__l__span, fact_122_Suc__less__SucD, fact_224_submodule__0, fact_114_less__or__eq__imp__le, fact_98_less__eq__nat_Osimps_I1_J, fact_190_singleton__insert__inj__eq_H, fact_188_eSum__in__SubmoduleTr, fact_112_less__mono__imp__le__mono, fact_73_infinite__descent__measure, fact_186_Module_Osubmodule__subset, fact_206_Collect__empty__eq, fact_77_less__irrefl__nat, fact_16_quotient__of__submodules__is__ideal, fact_80_less__not__refl, tcon_Product__Type_Ounit___Orderings_Oorder_5, fact_26_Suc__pred, fact_14_mem__fgs__l__comb, conj_18, fact_232_cancel__comm__monoid__add__class_Odiff__cancel, fact_88_not__less__eq__eq, fact_3_submodule__subset, fact_113_le__neq__implies__less, fact_139_less__zeroE, fact_177_diff__less__mono, fact_31_zero__less__diff, conj_2, fact_36_old_Onat_Oinject, fact_28_free__generator__generator, fact_138_gr__implies__not0, fact_21_Module_Ofgs__submodule, conj_15, fact_146_diff__le__mono, fact_134_Nat_OlessE, fact_240_l__spanA__l__span, fact_84_Module_Oquotient__of__submodules__is__ideal, fact_203_Diff__cancel, fact_140_not__less0, fact_189_eSum__in__Submodule, fact_68_nat__le__linear, fact_10_linear__comb__eqTr, fact_59_Module_Omisom__trans, fact_194_l__span__closed2Tr, fact_8_fgs__submodule, fact_137_infinite__descent0, fact_81_nat__neq__iff, tcon_Nat_Onat___Groups_Ozero, fact_52_diff__Suc__Suc, fact_131_Suc__lessI, fact_74_linorder__neqE__nat, fact_205_all__not__in__conv, tcon_Nat_Onat___Orderings_Owellorder, fact_24_misom__self, fact_225_empty__fgs, fact_53_diff__self__eq__0, fact_243_generator__generator, fact_167_dec__induct, fact_237_insert__Diff__single, fact_65_measure__induct__rule, fact_106_diff__induct, fact_35_diff__is__0__eq_H, fact_229_diff__0__right, fact_208_Diff__insert0, fact_110_old_Onat_Odistinct_I2_J, fact_47_bot__nat__0_Onot__eq__extremum, fact_252_Diff__subset, fact_135_infinite__descent0__measure, fact_148_le__diff__iff, fact_63_Module_Ogenerator__sub__carrier, fact_192_diff__gt__0__iff__gt, fact_230_zero__diff, fact_87_full__nat__induct, fact_158_lift__Suc__mono__le, fact_1_submodule__subset1, fact_150_zero__induct__lemma, fact_251_double__diff, fact_166_inc__induct, fact_78_less__not__refl3, fact_127_less__Suc__eq, fact_62_Module_Ofree__generator__generator, fact_212_insert__absorb2, fact_217_h__in__linear__span, conj_16, fact_54_diff__0__eq__0, fact_75_infinite__descent, fact_236_insert__subset, fact_191_singleton__insert__inj__eq, fact_64_Module_OAnn__is__ideal, fact_71_n__not__Suc__n, fact_132_Suc__lessE, fact_197_module__inc__zero, fact_125_All__less__Suc, fact_231_diff__zero, fact_221_linear__span__sub, fact_228_diff__self, fact_100_old_Onat_Oinducts, fact_82_diff__commute, fact_210_insertCI, fact_242_linear__span__inc__0, fact_30_Module_Oliear__comb__memTr, fact_117_nat__less__le, fact_176_Suc__diff__le, fact_130_less__SucE, fact_234_subset__empty, conj_6, fact_214_mdl__carrier, fact_179_diff__less__Suc, fact_124_Suc__less__eq2, fact_223_fgs__generator, fact_120_less__Suc__induct, fact_173_All__less__Suc2, conj_20, fact_157_lift__Suc__antimono__le, conj_1, fact_34_diff__is__0__eq, fact_95_bot__nat__0_Oextremum__uniqueI, fact_193_diff__ge__0__iff__ge, fact_155_Module_Osc__Ring, tcon_Set_Oset___Orderings_Oorder_2, fact_247_singleton__conv2, fact_97_le__0__eq, fact_45_Collect__mem__eq, conj_19, fact_49_neq0__conv, fact_70_le__trans, fact_40_bot__nat__0_Oextremum, fact_213_submodule__inc__0, fact_76_nat__less__induct, fact_56_zero__less__Suc, fact_61_Module_Omisom__sym, fact_244_l__span__zero, fact_201_Diff__empty, fact_108_nat_OdiscI, fact_162_less__eq__Suc__le, fact_105_zero__induct, fact_17_smodule__ideal__coeff__is__Submodule, fact_50_diff__diff__cancel, fact_13_Ann__is__ideal, fact_15_Module__axioms, fact_33_Module_Olinear__comb__eqTr, fact_96_bot__nat__0_Oextremum__unique, fact_6_fgs__sub__carrier, fact_246_singleton__conv, fact_115_le__eq__less__or__eq, fact_83_Module_Osmodule__ideal__coeff__is__Submodule, conj_8, fact_90_le__Suc__eq, fact_172_gr0__implies__Suc, fact_175_Ex__less__Suc2, fact_239_linear__span__subModule, conj_17, fact_107_nat__induct, fact_178_less__diff__iff, fact_143_diff__le__mono2, fact_170_ex__least__nat__le, fact_174_gr0__conv__Suc, fact_102_Zero__not__Suc, fact_116_less__imp__le__nat, fact_133_Suc__lessD, conj_12, fact_145_diff__le__self, tcon_fun___Orderings_Oorder, fact_38_misom__sym, tcon_Nat_Onat___Orderings_Oorder_1, fact_104_Suc__neq__Zero, fact_79_less__not__refl2, fact_250_linear__comb0__2, fact_156_Module_Omem__smodule__ideal__coeff, fact_233_Diff__eq__empty__iff, fact_55_misom__trans, conj_4, fact_128_Ex__less__Suc, fact_226_le__zero__eq, fact_109_old_Onat_Odistinct_I1_J, fact_196_linear__comb0__2Tr, fact_126_not__less__eq, fact_151_minus__nat_Odiff__0, fact_5_elem__fgs, conj_13, fact_207_empty__Collect__eq, fact_42_Suc__less__eq, tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, fact_209_insert__Diff1, conj_5, fact_195_l__span__closed2, fact_227_not__gr__zero, fact_19_Module_Oelem__fgs, conj_0, fact_183_Module_Osubmodule__whole, fact_119_strict__inc__induct, fact_141_not__gr0, fact_32_Module_Olinear__comb__eq, fact_43_Suc__mono, fact_89_Suc__n__not__le__n, fact_20_Module_Ofgs__mono, fact_0_sc__Ring, fact_211_insert__iff, fact_123_less__antisym, fact_86_nat__induct__at__least, fact_39_Suc__le__mono, fact_27_generator__sub__carrier, fact_121_less__trans__Suc, fact_222_l__span__sub__submodule, fact_118_not__less__less__Suc__eq, fact_48_less__nat__zero__code, fact_215_mdl__is__module, fact_66_measure__induct, fact_198_psubsetI, fact_154_diff__less__mono2, fact_46_Collect__cong, fact_136_bot__nat__0_Oextremum__strict, fact_58_Module_OHOM__is__module, fact_44_mem__Collect__eq
% 62.60/45.49  % No. of inferences in proof: 570
% 62.60/45.49  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 45026 ms resp. 44370 ms w/o parsing
% 63.66/45.76  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 63.66/45.76  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------