TSTP Solution File: ITP126^2 by Leo-III---1.7.12

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%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : ITP126^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n004.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 22:23:10 EDT 2024

% Result   : Theorem 78.78s 38.08s
% Output   : Refutation 79.01s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  376
% Syntax   : Number of formulae    :  689 ( 332 unt;  65 typ;   0 def)
%            Number of atoms       : 1512 ( 675 equ;   4 cnn)
%            Maximal formula atoms :    6 (   2 avg)
%            Number of connectives : 5588 ( 194   ~;  26   |; 103   &;4757   @)
%                                         (   0 <=>; 508  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   6 avg)
%            Number of types       :    6 (   5 usr)
%            Number of type conns  :  166 ( 166   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   72 (  68 usr;  16 con; 0-6 aty)
%            Number of variables   : 1624 (  46   ^1452   !;  72   ?;1624   :)
%                                         (  54  !>;   0  ?*;   0  @-;   0  @+)

% Comments : 
%------------------------------------------------------------------------------
thf(i_type,type,
    i: $tType ).

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

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

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

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

thf(size_type,type,
    size: 
      !>[TA: $tType] : $o ).

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

thf(neg_numeral_type,type,
    neg_numeral: 
      !>[TA: $tType] : $o ).

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

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

thf(semiring_1_type,type,
    semiring_1: 
      !>[TA: $tType] : $o ).

thf(monoid_add_type,type,
    monoid_add: 
      !>[TA: $tType] : $o ).

thf(semiring_char_0_type,type,
    semiring_char_0: 
      !>[TA: $tType] : $o ).

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

thf(semigroup_add_type,type,
    semigroup_add: 
      !>[TA: $tType] : $o ).

thf(linordered_idom_type,type,
    linordered_idom: 
      !>[TA: $tType] : $o ).

thf(comm_monoid_add_type,type,
    comm_monoid_add: 
      !>[TA: $tType] : $o ).

thf(linordered_field_type,type,
    linordered_field: 
      !>[TA: $tType] : $o ).

thf(ab_semigroup_add_type,type,
    ab_semigroup_add: 
      !>[TA: $tType] : $o ).

thf(cancel_semigroup_add_type,type,
    cancel_semigroup_add: 
      !>[TA: $tType] : $o ).

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

thf(linord219039673up_add_type,type,
    linord219039673up_add: 
      !>[TA: $tType] : $o ).

thf(ordere216010020id_add_type,type,
    ordere216010020id_add: 
      !>[TA: $tType] : $o ).

thf(ordere779506340up_add_type,type,
    ordere779506340up_add: 
      !>[TA: $tType] : $o ).

thf(linord1659791738miring_type,type,
    linord1659791738miring: 
      !>[TA: $tType] : $o ).

thf(archim1804426504_field_type,type,
    archim1804426504_field: 
      !>[TA: $tType] : $o ).

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

thf(strict797366125id_add_type,type,
    strict797366125id_add: 
      !>[TA: $tType] : $o ).

thf(ordere236663937imp_le_type,type,
    ordere236663937imp_le: 
      !>[TA: $tType] : $o ).

thf(ordere223160158up_add_type,type,
    ordere223160158up_add: 
      !>[TA: $tType] : $o ).

thf(strict2144017051up_add_type,type,
    strict2144017051up_add: 
      !>[TA: $tType] : $o ).

thf(ordere516151231imp_le_type,type,
    ordere516151231imp_le: 
      !>[TA: $tType] : $o ).

thf(semiri456707255roduct_type,type,
    semiri456707255roduct: 
      !>[TA: $tType] : $o ).

thf(bNF_Greatest_shift_type,type,
    bNF_Greatest_shift: 
      !>[TA: $tType,TB: $tType] : ( ( ( list @ TB ) > TA ) > TB > ( list @ TB ) > TA ) ).

thf(plus_plus_type,type,
    plus_plus: 
      !>[TA: $tType] : ( TA > TA > TA ) ).

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

thf(if_type,type,
    if: 
      !>[TA: $tType] : ( $o > TA > TA > TA ) ).

thf(drop_type,type,
    drop: 
      !>[TA: $tType] : ( nat > ( list @ TA ) > ( list @ TA ) ) ).

thf(find_type,type,
    find: 
      !>[TA: $tType] : ( ( TA > $o ) > ( list @ TA ) > ( option @ TA ) ) ).

thf(gen_length_type,type,
    gen_length: 
      !>[TA: $tType] : ( nat > ( list @ TA ) > nat ) ).

thf(cons_type,type,
    cons: 
      !>[TA: $tType] : ( TA > ( list @ TA ) > ( list @ TA ) ) ).

thf(list_ex_type,type,
    list_ex: 
      !>[TA: $tType] : ( ( TA > $o ) > ( list @ TA ) > $o ) ).

thf(nth_type,type,
    nth: 
      !>[TA: $tType] : ( ( list @ TA ) > nat > TA ) ).

thf(c_MFOTL_Oformula_OEq_type,type,
    c_MFOTL_Oformula_OEq: 
      !>[TA: $tType] : ( ( trm @ TA ) > ( trm @ TA ) > ( formula @ TA ) ) ).

thf(fvi_type,type,
    fvi: 
      !>[TA: $tType] : ( nat > ( formula @ TA ) > ( set @ nat ) ) ).

thf(monito518325957Exists_type,type,
    monito518325957Exists: 
      !>[TA: $tType] : ( ( monito748960549ormula @ TA ) > ( monito748960549ormula @ TA ) ) ).

thf(monito320348316_MNext_type,type,
    monito320348316_MNext: 
      !>[TA: $tType] : ( i > ( monito748960549ormula @ TA ) > $o > ( list @ nat ) > ( monito748960549ormula @ TA ) ) ).

thf(monito1987675900_MPrev_type,type,
    monito1987675900_MPrev: 
      !>[TA: $tType] : ( i > ( monito748960549ormula @ TA ) > $o > ( list @ ( set @ ( list @ ( option @ TA ) ) ) ) > ( list @ nat ) > ( monito748960549ormula @ TA ) ) ).

thf(monito1197352414ormula_type,type,
    monito1197352414ormula: 
      !>[TA: $tType] : ( ( TA > nat ) > ( monito748960549ormula @ TA ) > nat ) ).

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

thf(semiring_1_of_nat_type,type,
    semiring_1_of_nat: 
      !>[TA: $tType] : ( nat > TA ) ).

thf(size_size_type,type,
    size_size: 
      !>[TA: $tType] : ( TA > nat ) ).

thf(nat_triangle_type,type,
    nat_triangle: nat > nat ).

thf(none_type,type,
    none: 
      !>[TA: $tType] : ( option @ TA ) ).

thf(some_type,type,
    some: 
      !>[TA: $tType] : ( TA > ( option @ TA ) ) ).

thf(size_option_type,type,
    size_option: 
      !>[TA: $tType] : ( ( TA > nat ) > ( option @ TA ) > nat ) ).

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

thf(phi_type,type,
    phi: formula @ a ).

thf(b2_type,type,
    b2: nat ).

thf(n_type,type,
    n: nat ).

thf(v_type,type,
    v: list @ ( option @ b ) ).

thf(x_type,type,
    x: b ).

thf(182,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_68_Ex__less__Suc) ).

thf(809,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)],[182]) ).

thf(251,axiom,
    strict2144017051up_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_4) ).

thf(1024,plain,
    strict2144017051up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(164,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_87_All__less__Suc2) ).

thf(760,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)],[164]) ).

thf(312,axiom,
    ! [TA: $tType,A: TA,B: list @ TA,C: nat] :
      ( ( nth @ TA @ ( cons @ TA @ A @ B ) @ ( suc @ C ) )
      = ( nth @ TA @ B @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_nth__Cons__Suc) ).

thf(1237,plain,
    ! [TA: $tType,A: TA,B: list @ TA,C: nat] :
      ( ( nth @ TA @ ( cons @ TA @ A @ B ) @ ( suc @ C ) )
      = ( nth @ TA @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).

thf(231,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_85_less__Suc__eq__0__disj) ).

thf(962,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)],[231]) ).

thf(99,axiom,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat] :
          ( ( ( zero_zero @ TA )
            = ( semiring_1_of_nat @ TA @ A ) )
          = ( ( zero_zero @ nat )
            = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_of__nat__0__eq__iff) ).

thf(555,plain,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat] :
          ( ( ( zero_zero @ TA )
            = ( semiring_1_of_nat @ TA @ A ) )
          = ( ( zero_zero @ nat )
            = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(75,axiom,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = A )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_zero__reorient) ).

thf(479,plain,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = A )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(278,axiom,
    ! [TA: $tType,A: nat,B: nat > TA > $o] :
      ( ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ A )
           => ? [D: TA] : ( B @ C @ D ) ) )
      = ( ? [C: list @ TA] :
            ( ( ( size_size @ ( list @ TA ) @ C )
              = A )
            & ! [D: nat] :
                ( ( ord_less @ nat @ D @ A )
               => ( B @ D @ ( nth @ TA @ C @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_Skolem__list__nth) ).

thf(1105,plain,
    ! [TA: $tType,A: nat,B: nat > TA > $o] :
      ( ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ A )
           => ? [D: TA] : ( B @ C @ D ) ) )
      = ( ? [C: list @ TA] :
            ( ( ( size_size @ ( list @ TA ) @ C )
              = A )
            & ! [D: nat] :
                ( ( ord_less @ nat @ D @ A )
               => ( B @ D @ ( nth @ TA @ C @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(307,axiom,
    ! [TA: $tType] :
      ( ( strict797366125id_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_pos__add__strict) ).

thf(1211,plain,
    ! [TA: $tType] :
      ( ( strict797366125id_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).

thf(89,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less @ TA @ A @ B )
            & ( C = D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_add__mono__thms__linordered__field_I1_J) ).

thf(523,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less @ TA @ A @ B )
            & ( C = D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(157,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
         => ( ord_less @ TA @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_add__less__imp__less__left) ).

thf(741,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
         => ( ord_less @ TA @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(304,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_17_measure__induct) ).

thf(1205,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)],[304]) ).

thf(29,axiom,
    ! [TA: $tType,A: monito748960549ormula @ TA,B: monito748960549ormula @ TA] :
      ( ( ( monito518325957Exists @ TA @ A )
        = ( monito518325957Exists @ TA @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_mformula_Oinject_I5_J) ).

thf(379,plain,
    ! [TA: $tType,A: monito748960549ormula @ TA,B: monito748960549ormula @ TA] :
      ( ( ( monito518325957Exists @ TA @ A )
        = ( monito518325957Exists @ TA @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(137,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ C @ D )
       => ( ord_less @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_add__less__mono) ).

thf(677,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ C @ D )
       => ( ord_less @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(257,axiom,
    zero @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ozero_20) ).

thf(1041,plain,
    zero @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(203,axiom,
    ! [TA: $tType,A: option @ TA,B: TA] :
      ( ( A
        = ( some @ TA @ B ) )
     => ( A
       != ( none @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_option_OdiscI) ).

thf(878,plain,
    ! [TA: $tType,A: option @ TA,B: TA] :
      ( ( A
        = ( some @ TA @ B ) )
     => ( A
       != ( none @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(129,axiom,
    ! [A: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_zero__less__Suc) ).

thf(655,plain,
    ! [A: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(32,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_add_Ocomm__neutral) ).

thf(384,plain,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(151,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_not0__implies__Suc) ).

thf(715,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(72,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
            = ( plus_plus @ TA @ A @ B ) )
          = ( B
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_add__cancel__right__right) ).

thf(472,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
            = ( plus_plus @ TA @ A @ B ) )
          = ( B
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(145,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        & ( B
          = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_add__is__0) ).

thf(698,plain,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        & ( B
          = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(135,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_Zero__not__Suc) ).

thf(670,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(294,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_less__zeroE) ).

thf(1164,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(178,axiom,
    ! [TA: $tType,A: nat,B: nat,C: formula @ TA] :
      ( ( member @ nat @ A @ ( fvi @ TA @ ( suc @ B ) @ C ) )
      = ( member @ nat @ ( suc @ A ) @ ( fvi @ TA @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_fvi__Suc) ).

thf(797,plain,
    ! [TA: $tType,A: nat,B: nat,C: formula @ TA] :
      ( ( member @ nat @ A @ ( fvi @ TA @ ( suc @ B ) @ C ) )
      = ( member @ nat @ ( suc @ A ) @ ( fvi @ TA @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(171,axiom,
    ! [TA: $tType,A: TA] :
      ( ( none @ TA )
     != ( some @ TA @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_option_Odistinct_I1_J) ).

thf(782,plain,
    ! [TA: $tType,A: TA] :
      ( ( none @ TA )
     != ( some @ TA @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(149,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_71_Suc__lessI) ).

thf(709,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)],[149]) ).

thf(5,axiom,
    semiri456707255roduct @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) ).

thf(320,plain,
    semiri456707255roduct @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(54,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ A @ C ) )
          = ( B = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_add_Oleft__cancel) ).

thf(436,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ A @ C ) )
          = ( B = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(189,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_add__less__mono1) ).

thf(829,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(275,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ~ ! [C: nat] :
            ( B
           != ( suc @ ( plus_plus @ nat @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_less__natE) ).

thf(1095,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ~ ! [C: nat] :
            ( B
           != ( suc @ ( plus_plus @ nat @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(40,axiom,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ A @ C ) )
         => ( B = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_add__left__imp__eq) ).

thf(404,plain,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ A @ C ) )
         => ( B = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(206,axiom,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( ( some @ TA @ A )
        = ( some @ TA @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_option_Oinject) ).

thf(885,plain,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( ( some @ TA @ A )
        = ( some @ TA @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(260,axiom,
    ! [TA: $tType,A: trm @ TA,B: trm @ TA] :
      ( ( size_size @ ( formula @ TA ) @ ( c_MFOTL_Oformula_OEq @ TA @ A @ B ) )
      = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( trm @ TA ) @ A ) @ ( size_size @ ( trm @ TA ) @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_formula_Osize_I11_J) ).

thf(1046,plain,
    ! [TA: $tType,A: trm @ TA,B: trm @ TA] :
      ( ( size_size @ ( formula @ TA ) @ ( c_MFOTL_Oformula_OEq @ TA @ A @ B ) )
      = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( trm @ TA ) @ A ) @ ( size_size @ ( trm @ TA ) @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(172,axiom,
    ! [A: nat,B: nat] : ( ord_less @ nat @ A @ ( suc @ ( plus_plus @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_less__add__Suc1) ).

thf(785,plain,
    ! [A: nat,B: nat] : ( ord_less @ nat @ A @ ( suc @ ( plus_plus @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(83,axiom,
    ! [TA: $tType,A: TA > nat] :
      ( ( size_option @ TA @ A @ ( none @ TA ) )
      = ( suc @ ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_option_Osize__gen_I1_J) ).

thf(504,plain,
    ! [TA: $tType,A: TA > nat] :
      ( ( size_option @ TA @ A @ ( none @ TA ) )
      = ( suc @ ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(117,axiom,
    ! [A: nat] :
      ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ A )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_plus__nat_Oadd__0) ).

thf(619,plain,
    ! [A: nat] :
      ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(160,axiom,
    order @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oorder_19) ).

thf(751,plain,
    order @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(14,axiom,
    linordered_idom @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Rings_Olinordered__idom) ).

thf(344,plain,
    linordered_idom @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(222,axiom,
    ! [TA: $tType] :
      ( ( strict2144017051up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ D )
           => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_add__strict__mono) ).

thf(934,plain,
    ! [TA: $tType] :
      ( ( strict2144017051up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ D )
           => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(226,axiom,
    ! [TA: $tType,A: list @ TA,B: list @ TA] :
      ( ( ( size_size @ ( list @ TA ) @ A )
       != ( size_size @ ( list @ TA ) @ B ) )
     => ( A != B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_neq__if__length__neq) ).

thf(949,plain,
    ! [TA: $tType,A: list @ TA,B: list @ TA] :
      ( ( ( size_size @ ( list @ TA ) @ A )
       != ( size_size @ ( list @ TA ) @ B ) )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(264,axiom,
    ! [TA: $tType] :
      ( ( list_ex @ TA )
      = ( ^ [A: TA > $o,B: list @ TA] :
          ? [C: nat] :
            ( ( ord_less @ nat @ C @ ( size_size @ ( list @ TA ) @ B ) )
            & ( A @ ( nth @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_list__ex__length) ).

thf(1059,plain,
    ! [TA: $tType] :
      ( ( list_ex @ TA )
      = ( ^ [A: TA > $o,B: list @ TA] :
          ? [C: nat] :
            ( ( ord_less @ nat @ C @ ( size_size @ ( list @ TA ) @ B ) )
            & ( A @ ( nth @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(286,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
         => ( ord_less @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_add__less__imp__less__right) ).

thf(1132,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
         => ( ord_less @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(212,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( bNF_Greatest_shift @ TA @ TB )
      = ( ^ [A: ( list @ TA ) > TB,B: TA,C: list @ TA] : ( A @ ( cons @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_BNF__Greatest__Fixpoint_Oshift__def) ).

thf(902,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( bNF_Greatest_shift @ TA @ TB )
      = ( ^ [A: ( list @ TA ) > TB,B: TA,C: list @ TA] : ( A @ ( cons @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(47,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_comm__monoid__add__class_Oadd__0) ).

thf(421,plain,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(234,axiom,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ A )
          = ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_add__less__same__cancel1) ).

thf(969,plain,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ A )
          = ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(281,axiom,
    member @ nat @ ( zero_zero @ nat ) @ ( fvi @ a @ b2 @ phi ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).

thf(1115,plain,
    member @ nat @ ( zero_zero @ nat ) @ ( fvi @ a @ b2 @ phi ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(123,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( semiring_1_of_nat @ TA @ A ) )
          = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_of__nat__0__less__iff) ).

thf(638,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( semiring_1_of_nat @ TA @ A ) )
          = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(195,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_trans__less__add2) ).

thf(853,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(73,axiom,
    ordere779506340up_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Oordered__ab__semigroup__add) ).

thf(475,plain,
    ordere779506340up_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(174,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_verit__comp__simplify1_I1_J) ).

thf(789,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(57,axiom,
    semiring_1 @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Rings_Osemiring__1) ).

thf(443,plain,
    semiring_1 @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(6,axiom,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ C @ B ) )
         => ( A = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_add__right__imp__eq) ).

thf(321,plain,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ C @ B ) )
         => ( A = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(27,axiom,
    monoid_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Omonoid__add) ).

thf(376,plain,
    monoid_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(288,axiom,
    ! [A: nat,B: nat,C: int] :
      ( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ B ) @ C ) )
      = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A @ B ) ) @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_zadd__int__left) ).

thf(1136,plain,
    ! [A: nat,B: nat,C: int] :
      ( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ B ) @ C ) )
      = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A @ B ) ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(16,axiom,
    ! [TA: $tType,A: monito748960549ormula @ TA,B: i,C: monito748960549ormula @ TA,D: $o,E: list @ ( set @ ( list @ ( option @ TA ) ) ),F: list @ nat] :
      ( ( monito518325957Exists @ TA @ A )
     != ( monito1987675900_MPrev @ TA @ B @ C @ D @ E @ F ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_mformula_Odistinct_I53_J) ).

thf(348,plain,
    ! [TA: $tType,A: monito748960549ormula @ TA,B: i,C: monito748960549ormula @ TA,D: $o,E: list @ ( set @ ( list @ ( option @ TA ) ) ),F: list @ nat] :
      ( ( monito518325957Exists @ TA @ A )
     != ( monito1987675900_MPrev @ TA @ B @ C @ D @ E @ F ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(181,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_nat_Oinject) ).

thf(806,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(213,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_less__SucI) ).

thf(905,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(121,axiom,
    semigroup_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Osemigroup__add_15) ).

thf(634,plain,
    semigroup_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(155,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_add__strict__left__mono) ).

thf(734,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(268,axiom,
    ! [TA: $tType,A: list @ TA,B: nat] :
      ( ( ( size_size @ ( list @ TA ) @ A )
        = ( suc @ B ) )
      = ( ? [C: TA,D: list @ TA] :
            ( ( A
              = ( cons @ TA @ C @ D ) )
            & ( ( size_size @ ( list @ TA ) @ D )
              = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_length__Suc__conv) ).

thf(1072,plain,
    ! [TA: $tType,A: list @ TA,B: nat] :
      ( ( ( size_size @ ( list @ TA ) @ A )
        = ( suc @ B ) )
      = ( ? [C: TA,D: list @ TA] :
            ( ( A
              = ( cons @ TA @ C @ D ) )
            & ( ( size_size @ ( list @ TA ) @ D )
              = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(283,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_34_old_Onat_Oinducts) ).

thf(1119,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)],[283]) ).

thf(131,axiom,
    semiring_1 @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__1_18) ).

thf(660,plain,
    semiring_1 @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(43,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_verit__sum__simplify) ).

thf(411,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(84,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_16_measure__induct__rule) ).

thf(507,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)],[84]) ).

thf(90,axiom,
    size @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Nat_Osize) ).

thf(527,plain,
    size @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(238,axiom,
    ! [TA: $tType,A: trm @ TA] :
      ( ( size_size @ ( trm @ TA ) @ A )
     != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_trm_Osize__neq) ).

thf(981,plain,
    ! [TA: $tType,A: trm @ TA] :
      ( ( size_size @ ( trm @ TA ) @ A )
     != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(166,axiom,
    ! [A: nat] :
      ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_not__gr0) ).

thf(766,plain,
    ! [A: nat] :
      ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(62,axiom,
    ! [TA: $tType] : ( size @ ( formula @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_MFOTL_Oformula___Nat_Osize_26) ).

thf(452,plain,
    ! [TA: $tType] : ( size @ ( formula @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(55,axiom,
    ! [A: int] :
      ( ( plus_plus @ int @ A @ ( zero_zero @ int ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_plus__int__code_I1_J) ).

thf(439,plain,
    ! [A: int] :
      ( ( plus_plus @ int @ A @ ( zero_zero @ int ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(132,axiom,
    ! [A: nat,B: nat] :
      ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A @ B ) )
      = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A ) @ ( semiring_1_of_nat @ int @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_int__ops_I5_J) ).

thf(661,plain,
    ! [A: nat,B: nat] :
      ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A @ B ) )
      = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A ) @ ( semiring_1_of_nat @ int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(291,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_67_less__Suc__eq) ).

thf(1153,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)],[291]) ).

thf(23,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) )
          = ( plus_plus @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_add_Oleft__commute) ).

thf(366,plain,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) )
          = ( plus_plus @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(93,axiom,
    ! [TA: $tType,A: option @ TA] :
      ( ( ! [B: TA] :
            ( A
           != ( some @ TA @ B ) ) )
      = ( A
        = ( none @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_not__Some__eq) ).

thf(534,plain,
    ! [TA: $tType,A: option @ TA] :
      ( ( ! [B: TA] :
            ( A
           != ( some @ TA @ B ) ) )
      = ( A
        = ( none @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(159,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_24_nat__less__induct) ).

thf(746,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)],[159]) ).

thf(250,axiom,
    ! [TA: $tType,A: TA > nat,B: monito748960549ormula @ TA] :
      ( ( monito1197352414ormula @ TA @ A @ ( monito518325957Exists @ TA @ B ) )
      = ( plus_plus @ nat @ ( monito1197352414ormula @ TA @ A @ B ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_mformula_Osize__gen_I5_J) ).

thf(1021,plain,
    ! [TA: $tType,A: TA > nat,B: monito748960549ormula @ TA] :
      ( ( monito1197352414ormula @ TA @ A @ ( monito518325957Exists @ TA @ B ) )
      = ( plus_plus @ nat @ ( monito1197352414ormula @ TA @ A @ B ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(258,axiom,
    semiring_char_0 @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Nat_Osemiring__char__0_16) ).

thf(1042,plain,
    semiring_char_0 @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(198,axiom,
    ! [TA: $tType] :
      ( ( '?' @ ( option @ TA ) )
      = ( ^ [A: ( option @ TA ) > $o] :
            ( ( A @ ( none @ TA ) )
            | ? [B: TA] : ( A @ ( some @ TA @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_split__option__ex) ).

thf(862,plain,
    ! [TA: $tType] :
      ( ( '?' @ ( option @ TA ) )
      = ( ^ [A: ( option @ TA ) > $o] :
            ( ( A @ ( none @ TA ) )
            | ? [B: TA] : ( A @ ( some @ TA @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(186,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_Suc__inject) ).

thf(820,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(35,axiom,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ A @ C ) )
          = ( B = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_add__left__cancel) ).

thf(392,plain,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ A @ C ) )
          = ( B = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(48,axiom,
    cancel_semigroup_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Ocancel__semigroup__add) ).

thf(424,plain,
    cancel_semigroup_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(41,axiom,
    ! [TA: $tType] :
      ( ( semiri456707255roduct @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
            = ( plus_plus @ TA @ A @ B ) )
          = ( B
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_add__0__iff) ).

thf(407,plain,
    ! [TA: $tType] :
      ( ( semiri456707255roduct @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
            = ( plus_plus @ TA @ A @ B ) )
          = ( B
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(51,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_add_Ogroup__left__neutral) ).

thf(429,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(184,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_Zero__neq__Suc) ).

thf(815,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(229,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) )
         => ( ord_less @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_of__nat__less__imp__less) ).

thf(957,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) )
         => ( ord_less @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(209,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_not__less0) ).

thf(894,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(309,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_61_less__trans__Suc) ).

thf(1217,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)],[309]) ).

thf(110,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_bot__nat__0_Onot__eq__extremum) ).

thf(598,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(215,axiom,
    ! [TA: $tType,A: TA > nat,B: i,C: monito748960549ormula @ TA,D: $o,E: list @ nat] :
      ( ( monito1197352414ormula @ TA @ A @ ( monito320348316_MNext @ TA @ B @ C @ D @ E ) )
      = ( plus_plus @ nat @ ( monito1197352414ormula @ TA @ A @ C ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_mformula_Osize__gen_I7_J) ).

thf(916,plain,
    ! [TA: $tType,A: TA > nat,B: i,C: monito748960549ormula @ TA,D: $o,E: list @ nat] :
      ( ( monito1197352414ormula @ TA @ A @ ( monito320348316_MNext @ TA @ B @ C @ D @ E ) )
      = ( plus_plus @ nat @ ( monito1197352414ormula @ TA @ A @ C ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(77,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ C ) )
         => ( ( plus_plus @ TA @ D @ A )
            = ( plus_plus @ TA @ B @ ( plus_plus @ TA @ D @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_group__cancel_Oadd2) ).

thf(484,plain,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ C ) )
         => ( ( plus_plus @ TA @ D @ A )
            = ( plus_plus @ TA @ B @ ( plus_plus @ TA @ D @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(254,axiom,
    ! [TA: $tType,A: i,B: monito748960549ormula @ TA,C: $o,D: list @ nat] :
      ( ( size_size @ ( monito748960549ormula @ TA ) @ ( monito320348316_MNext @ TA @ A @ B @ C @ D ) )
      = ( plus_plus @ nat @ ( size_size @ ( monito748960549ormula @ TA ) @ B ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_mformula_Osize_I16_J) ).

thf(1032,plain,
    ! [TA: $tType,A: i,B: monito748960549ormula @ TA,C: $o,D: list @ nat] :
      ( ( size_size @ ( monito748960549ormula @ TA ) @ ( monito320348316_MNext @ TA @ A @ B @ C @ D ) )
      = ( plus_plus @ nat @ ( size_size @ ( monito748960549ormula @ TA ) @ B ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(119,axiom,
    ! [A: nat > nat > $o,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( A @ D @ E )
          = ( A @ E @ D ) )
     => ( ! [D: nat] : ( A @ D @ ( zero_zero @ nat ) )
       => ( ! [D: nat,E: nat] :
              ( ( A @ D @ E )
             => ( A @ D @ ( plus_plus @ nat @ D @ E ) ) )
         => ( A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_Euclid__induct) ).

thf(625,plain,
    ! [A: nat > nat > $o,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( A @ D @ E )
          = ( A @ E @ D ) )
     => ( ! [D: nat] : ( A @ D @ ( zero_zero @ nat ) )
       => ( ! [D: nat,E: nat] :
              ( ( A @ D @ E )
             => ( A @ D @ ( plus_plus @ nat @ D @ E ) ) )
         => ( A @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(297,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ ( zero_zero @ nat ) )
        = ( plus_plus @ nat @ A @ B ) )
      = ( ( ( A
            = ( suc @ ( zero_zero @ nat ) ) )
          & ( B
            = ( zero_zero @ nat ) ) )
        | ( ( A
            = ( zero_zero @ nat ) )
          & ( B
            = ( suc @ ( zero_zero @ nat ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_one__is__add) ).

thf(1178,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ ( zero_zero @ nat ) )
        = ( plus_plus @ nat @ A @ B ) )
      = ( ( ( A
            = ( suc @ ( zero_zero @ nat ) ) )
          & ( B
            = ( zero_zero @ nat ) ) )
        | ( ( A
            = ( zero_zero @ nat ) )
          & ( B
            = ( suc @ ( zero_zero @ nat ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(293,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ! [C: TA] :
                ( ( B
                  = ( plus_plus @ TA @ A @ C ) )
               => ( C
                  = ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_canonically__ordered__monoid__add__class_OlessE) ).

thf(1159,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ! [C: TA] :
                ( ( B
                  = ( plus_plus @ TA @ A @ C ) )
               => ( C
                  = ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).

thf(306,axiom,
    ordere779506340up_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_9) ).

thf(1210,plain,
    ordere779506340up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).

thf(163,axiom,
    ! [TA: $tType,A: monito748960549ormula @ TA] :
      ( ( size_size @ ( monito748960549ormula @ TA ) @ A )
     != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_mformula_Osize__neq) ).

thf(757,plain,
    ! [TA: $tType,A: monito748960549ormula @ TA] :
      ( ( size_size @ ( monito748960549ormula @ TA ) @ A )
     != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(139,axiom,
    ( ( nat_triangle @ ( zero_zero @ nat ) )
    = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_triangle__0) ).

thf(682,plain,
    ( ( nat_triangle @ ( zero_zero @ nat ) )
    = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(65,axiom,
    ordere223160158up_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add) ).

thf(456,plain,
    ordere223160158up_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(7,axiom,
    ! [TA: $tType,A: i,B: monito748960549ormula @ TA,C: $o,D: list @ ( set @ ( list @ ( option @ TA ) ) ),E: list @ nat,F: i,G: monito748960549ormula @ TA,H: $o,I: list @ nat] :
      ( ( monito1987675900_MPrev @ TA @ A @ B @ C @ D @ E )
     != ( monito320348316_MNext @ TA @ F @ G @ H @ I ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_mformula_Odistinct_I61_J) ).

thf(324,plain,
    ! [TA: $tType,A: i,B: monito748960549ormula @ TA,C: $o,D: list @ ( set @ ( list @ ( option @ TA ) ) ),E: list @ nat,F: i,G: monito748960549ormula @ TA,H: $o,I: list @ nat] :
      ( ( monito1987675900_MPrev @ TA @ A @ B @ C @ D @ E )
     != ( monito320348316_MNext @ TA @ F @ G @ H @ I ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(290,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_41_nat__induct) ).

thf(1143,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)],[290]) ).

thf(249,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_72_Suc__lessE) ).

thf(1017,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)],[249]) ).

thf(154,axiom,
    ! [A: $o,B: nat,C: nat] :
      ( ( A
       => ( ( semiring_1_of_nat @ int @ ( if @ nat @ A @ B @ C ) )
          = ( semiring_1_of_nat @ int @ B ) ) )
      & ( ~ A
       => ( ( semiring_1_of_nat @ int @ ( if @ nat @ A @ B @ C ) )
          = ( semiring_1_of_nat @ int @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_int__if) ).

thf(725,plain,
    ! [A: $o,B: nat,C: nat] :
      ( ( A
       => ( ( semiring_1_of_nat @ int @ ( if @ nat @ A @ B @ C ) )
          = ( semiring_1_of_nat @ int @ B ) ) )
      & ( ~ A
       => ( ( semiring_1_of_nat @ int @ ( if @ nat @ A @ B @ C ) )
          = ( semiring_1_of_nat @ int @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(39,axiom,
    order @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oorder_22) ).

thf(403,plain,
    order @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(244,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_62_Suc__less__SucD) ).

thf(1006,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)],[244]) ).

thf(108,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ A @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
        | ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_add__gr__0) ).

thf(593,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ A @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
        | ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(200,axiom,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ( ( semiring_1_of_nat @ TA @ ( zero_zero @ nat ) )
        = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_of__nat__0) ).

thf(869,plain,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ( ( semiring_1_of_nat @ TA @ ( zero_zero @ nat ) )
        = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(114,axiom,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ A ) @ ( zero_zero @ TA ) )
          = ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_double__add__less__zero__iff__single__add__less__zero) ).

thf(610,plain,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ A ) @ ( zero_zero @ TA ) )
          = ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(205,axiom,
    ordere223160158up_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_5) ).

thf(884,plain,
    ordere223160158up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(253,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B != A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_less__not__refl2) ).

thf(1028,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(78,axiom,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ A ) )
          = ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_zero__less__double__add__iff__zero__less__single__add) ).

thf(488,plain,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ A ) )
          = ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(94,axiom,
    ! [TA: $tType,A: i,B: monito748960549ormula @ TA,C: $o,D: list @ ( set @ ( list @ ( option @ TA ) ) ),E: list @ nat] :
      ( ( size_size @ ( monito748960549ormula @ TA ) @ ( monito1987675900_MPrev @ TA @ A @ B @ C @ D @ E ) )
      = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( list @ ( set @ ( list @ ( option @ TA ) ) ) ) @ D ) @ ( size_size @ ( monito748960549ormula @ TA ) @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_mformula_Osize_I15_J) ).

thf(537,plain,
    ! [TA: $tType,A: i,B: monito748960549ormula @ TA,C: $o,D: list @ ( set @ ( list @ ( option @ TA ) ) ),E: list @ nat] :
      ( ( size_size @ ( monito748960549ormula @ TA ) @ ( monito1987675900_MPrev @ TA @ A @ B @ C @ D @ E ) )
      = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( list @ ( set @ ( list @ ( option @ TA ) ) ) ) @ D ) @ ( size_size @ ( monito748960549ormula @ TA ) @ B ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(96,axiom,
    ! [TA: $tType] : ( size @ ( list @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_List_Olist___Nat_Osize_23) ).

thf(547,plain,
    ! [TA: $tType] : ( size @ ( list @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(148,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_114_zero__less__iff__neq__zero) ).

thf(706,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)],[148]) ).

thf(269,axiom,
    strict797366125id_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_7) ).

thf(1075,plain,
    strict797366125id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(74,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_44_Collect__cong) ).

thf(476,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)],[74]) ).

thf(248,axiom,
    wellorder @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Owellorder) ).

thf(1016,plain,
    wellorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(107,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_gr0I) ).

thf(590,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(118,axiom,
    ! [TA: $tType,A: TA > nat,B: TA] :
      ( ( size_option @ TA @ A @ ( some @ TA @ B ) )
      = ( plus_plus @ nat @ ( A @ B ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_option_Osize__gen_I2_J) ).

thf(622,plain,
    ! [TA: $tType,A: TA > nat,B: TA] :
      ( ( size_option @ TA @ A @ ( some @ TA @ B ) )
      = ( plus_plus @ nat @ ( A @ B ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(126,axiom,
    ! [TA: $tType] :
      ( ( ordere216010020id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_add__pos__pos) ).

thf(647,plain,
    ! [TA: $tType] :
      ( ( ordere216010020id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(276,axiom,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( plus_plus @ TA @ B @ A ) )
          = ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_less__add__same__cancel2) ).

thf(1099,plain,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( plus_plus @ TA @ B @ A ) )
          = ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(146,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat] :
          ~ ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_of__nat__less__0__iff) ).

thf(701,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat] :
          ~ ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(66,axiom,
    ! [TA: $tType] : ( size @ ( monito748960549ormula @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Monitor__Mirabelle__pzlrlsievl_Omformula___Nat_Osize_28) ).

thf(457,plain,
    ! [TA: $tType] : ( size @ ( monito748960549ormula @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(239,axiom,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ B )
          = ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_add__less__same__cancel2) ).

thf(984,plain,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ B )
          = ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(153,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_nat_Odistinct_I1_J) ).

thf(721,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(104,axiom,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_48_old_Onat_Odistinct_I2_J) ).

thf(569,plain,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(170,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus @ nat @ A @ B )
        = ( suc @ ( zero_zero @ nat ) ) )
      = ( ( ( A
            = ( suc @ ( zero_zero @ nat ) ) )
          & ( B
            = ( zero_zero @ nat ) ) )
        | ( ( A
            = ( zero_zero @ nat ) )
          & ( B
            = ( suc @ ( zero_zero @ nat ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_add__is__1) ).

thf(779,plain,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus @ nat @ A @ B )
        = ( suc @ ( zero_zero @ nat ) ) )
      = ( ( ( A
            = ( suc @ ( zero_zero @ nat ) ) )
          & ( B
            = ( zero_zero @ nat ) ) )
        | ( ( A
            = ( zero_zero @ nat ) )
          & ( B
            = ( suc @ ( zero_zero @ nat ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(303,axiom,
    ! [A: nat,B: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_not__add__less1) ).

thf(1202,plain,
    ! [A: nat,B: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(21,axiom,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = ( plus_plus @ TA @ A @ A ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_double__zero__sym) ).

thf(360,plain,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = ( plus_plus @ TA @ A @ A ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(130,axiom,
    ! [TA: $tType,A: TA] :
      ( ( size_size @ ( option @ TA ) @ ( some @ TA @ A ) )
      = ( suc @ ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_option_Osize_I4_J) ).

thf(657,plain,
    ! [TA: $tType,A: TA] :
      ( ( size_size @ ( option @ TA ) @ ( some @ TA @ A ) )
      = ( suc @ ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(255,axiom,
    ! [TA: $tType,A: option @ TA] :
      ( ( A
       != ( none @ TA ) )
     => ~ ! [B: TA] :
            ( A
           != ( some @ TA @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_option_Oexhaust) ).

thf(1035,plain,
    ! [TA: $tType,A: option @ TA] :
      ( ( A
       != ( none @ TA ) )
     => ~ ! [B: TA] :
            ( A
           != ( some @ TA @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(100,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_84_lift__Suc__mono__less) ).

thf(558,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)],[100]) ).

thf(136,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_63_less__antisym) ).

thf(674,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)],[136]) ).

thf(295,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_old_Onat_Odistinct_I1_J) ).

thf(1167,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(227,axiom,
    monoid_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Omonoid__add_17) ).

thf(953,plain,
    monoid_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(176,axiom,
    cancel_semigroup_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__semigroup__add_12) ).

thf(794,plain,
    cancel_semigroup_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(218,axiom,
    ! [TA: $tType] :
      ( ( archim1804426504_field @ TA )
     => ! [A: TA] :
        ? [B: nat] : ( ord_less @ TA @ A @ ( semiring_1_of_nat @ TA @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_reals__Archimedean2) ).

thf(926,plain,
    ! [TA: $tType] :
      ( ( archim1804426504_field @ TA )
     => ! [A: TA] :
        ? [B: nat] : ( ord_less @ TA @ A @ ( semiring_1_of_nat @ TA @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(152,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_less__nat__zero__code) ).

thf(718,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(207,axiom,
    ! [A: nat,B: nat] : ( ord_less @ nat @ A @ ( suc @ ( plus_plus @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_less__add__Suc2) ).

thf(888,plain,
    ! [A: nat,B: nat] : ( ord_less @ nat @ A @ ( suc @ ( plus_plus @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(232,axiom,
    ! [TA: $tType,A: ( list @ TA ) > $o,B: list @ TA] :
      ( ! [C: list @ TA] :
          ( ! [D: list @ TA] :
              ( ( ord_less @ nat @ ( size_size @ ( list @ TA ) @ D ) @ ( size_size @ ( list @ TA ) @ C ) )
             => ( A @ D ) )
         => ( A @ C ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_length__induct) ).

thf(965,plain,
    ! [TA: $tType,A: ( list @ TA ) > $o,B: list @ TA] :
      ( ! [C: list @ TA] :
          ( ! [D: list @ TA] :
              ( ( ord_less @ nat @ ( size_size @ ( list @ TA ) @ D ) @ ( size_size @ ( list @ TA ) @ C ) )
             => ( A @ D ) )
         => ( A @ C ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(49,axiom,
    semiring_char_0 @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Nat_Osemiring__char__0) ).

thf(425,plain,
    semiring_char_0 @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(86,axiom,
    ! [A: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ A )
     => ~ ! [B: nat] :
            ( ( A
              = ( semiring_1_of_nat @ int @ B ) )
           => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_pos__int__cases) ).

thf(513,plain,
    ! [A: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ A )
     => ~ ! [B: nat] :
            ( ( A
              = ( semiring_1_of_nat @ int @ B ) )
           => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(187,axiom,
    ! [TA: $tType,A: TA > $o,B: list @ TA,C: TA] :
      ( ( ( find @ TA @ A @ B )
        = ( some @ TA @ C ) )
      = ( ? [D: nat] :
            ( ( ord_less @ nat @ D @ ( size_size @ ( list @ TA ) @ B ) )
            & ( A @ ( nth @ TA @ B @ D ) )
            & ( C
              = ( nth @ TA @ B @ D ) )
            & ! [E: nat] :
                ( ( ord_less @ nat @ E @ D )
               => ~ ( A @ ( nth @ TA @ B @ E ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_find__Some__iff) ).

thf(824,plain,
    ! [TA: $tType,A: TA > $o,B: list @ TA,C: TA] :
      ( ( ( find @ TA @ A @ B )
        = ( some @ TA @ C ) )
      = ( ? [D: nat] :
            ( ( ord_less @ nat @ D @ ( size_size @ ( list @ TA ) @ B ) )
            & ( A @ ( nth @ TA @ B @ D ) )
            & ( C
              = ( nth @ TA @ B @ D ) )
            & ! [E: nat] :
                ( ( ord_less @ nat @ E @ D )
               => ~ ( A @ ( nth @ TA @ B @ E ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(196,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus @ nat @ ( suc @ A ) @ B )
      = ( plus_plus @ nat @ A @ ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_add__Suc__shift) ).

thf(856,plain,
    ! [A: nat,B: nat] :
      ( ( plus_plus @ nat @ ( suc @ A ) @ B )
      = ( plus_plus @ nat @ A @ ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(105,axiom,
    ! [TA: $tType,TB: $tType,A: option @ TB,B: ( option @ TB ) > ( option @ TA ) > $o,C: option @ TA] :
      ( ( ( A
          = ( none @ TB ) )
       => ( B @ A @ C ) )
     => ( ( ( C
            = ( none @ TA ) )
         => ( B @ A @ C ) )
       => ( ! [D: TB,E: TA] :
              ( ( A
                = ( some @ TB @ D ) )
             => ( ( C
                  = ( some @ TA @ E ) )
               => ( B @ A @ C ) ) )
         => ( B @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_combine__options__cases) ).

thf(573,plain,
    ! [TA: $tType,TB: $tType,A: option @ TB,B: ( option @ TB ) > ( option @ TA ) > $o,C: option @ TA] :
      ( ( ( A
          = ( none @ TB ) )
       => ( B @ A @ C ) )
     => ( ( ( C
            = ( none @ TA ) )
         => ( B @ A @ C ) )
       => ( ! [D: TB,E: TA] :
              ( ( A
                = ( some @ TB @ D ) )
             => ( ( C
                  = ( some @ TA @ E ) )
               => ( B @ A @ C ) ) )
         => ( B @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(141,axiom,
    ( ( ord_less @ nat )
    = ( ^ [A: nat,B: nat] :
        ? [C: nat] :
          ( B
          = ( suc @ ( plus_plus @ nat @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_less__iff__Suc__add) ).

thf(689,plain,
    ( ( ord_less @ nat )
    = ( ^ [A: nat,B: nat] :
        ? [C: nat] :
          ( B
          = ( suc @ ( plus_plus @ nat @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(58,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ( ( plus_plus @ TA )
        = ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_add_Ocommute) ).

thf(444,plain,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ( ( plus_plus @ TA )
        = ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(111,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
         => ( ~ ( ord_less @ TA @ A @ B )
           => ( ord_less @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_linorder__neqE__linordered__idom) ).

thf(601,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
         => ( ~ ( ord_less @ TA @ A @ B )
           => ( ord_less @ TA @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(185,axiom,
    linord1659791738miring @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_8) ).

thf(819,plain,
    linord1659791738miring @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(236,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
      = ( ord_less @ nat @ B @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_nat__add__left__cancel__less) ).

thf(975,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
      = ( ord_less @ nat @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(143,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ nat @ A @ B )
         => ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_less__imp__of__nat__less) ).

thf(694,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ nat @ A @ B )
         => ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(193,axiom,
    ! [TA: $tType,A: TA > $o,B: TA,C: list @ TA] :
      ( ( ( A @ B )
       => ( ( find @ TA @ A @ ( cons @ TA @ B @ C ) )
          = ( some @ TA @ B ) ) )
      & ( ~ ( A @ B )
       => ( ( find @ TA @ A @ ( cons @ TA @ B @ C ) )
          = ( find @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_find_Osimps_I2_J) ).

thf(843,plain,
    ! [TA: $tType,A: TA > $o,B: TA,C: list @ TA] :
      ( ( ( A @ B )
       => ( ( find @ TA @ A @ ( cons @ TA @ B @ C ) )
          = ( some @ TA @ B ) ) )
      & ( ~ ( A @ B )
       => ( ( find @ TA @ A @ ( cons @ TA @ B @ C ) )
          = ( find @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(168,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_64_Suc__less__eq2) ).

thf(772,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)],[168]) ).

thf(158,axiom,
    ! [TA: $tType] :
      ( ( '!' @ ( option @ TA ) )
      = ( ^ [A: ( option @ TA ) > $o] :
            ( ( A @ ( none @ TA ) )
            & ! [B: TA] : ( A @ ( some @ TA @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_split__option__all) ).

thf(743,plain,
    ! [TA: $tType] :
      ( ( '!' @ ( option @ TA ) )
      = ( ^ [A: ( option @ TA ) > $o] :
            ( ( A @ ( none @ TA ) )
            & ! [B: TA] : ( A @ ( some @ TA @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(202,axiom,
    ! [TA: $tType,A: TA,B: TA > $o] :
      ( ( member @ TA @ A @ ( collect @ TA @ B ) )
      = ( B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_mem__Collect__eq) ).

thf(875,plain,
    ! [TA: $tType,A: TA,B: TA > $o] :
      ( ( member @ TA @ A @ ( collect @ TA @ B ) )
      = ( B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(124,axiom,
    ! [TA: $tType,A: nat,B: TA,C: list @ TA] :
      ( ( gen_length @ TA @ A @ ( cons @ TA @ B @ C ) )
      = ( gen_length @ TA @ ( suc @ A ) @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_gen__length__code_I2_J) ).

thf(641,plain,
    ! [TA: $tType,A: nat,B: TA,C: list @ TA] :
      ( ( gen_length @ TA @ A @ ( cons @ TA @ B @ C ) )
      = ( gen_length @ TA @ ( suc @ A ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(289,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( A = B )
            & ( ord_less @ TA @ C @ D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_add__mono__thms__linordered__field_I2_J) ).

thf(1139,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( A = B )
            & ( ord_less @ TA @ C @ D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(223,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_60_less__Suc__induct) ).

thf(936,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)],[223]) ).

thf(30,axiom,
    linord219039673up_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Olinordered__ab__group__add) ).

thf(382,plain,
    linord219039673up_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(1,conjecture,
    ( ( ( size_size @ ( list @ ( option @ b ) ) @ ( cons @ ( option @ b ) @ ( some @ b @ x ) @ v ) )
      = ( suc @ n ) )
    & ! [A: nat] :
        ( ~ ( ord_less @ nat @ A @ ( suc @ n ) )
        | ( ( ( nth @ ( option @ b ) @ ( cons @ ( option @ b ) @ ( some @ b @ x ) @ v ) @ A )
            = ( none @ b ) )
          = ( ~ ( member @ nat @ A @ ( fvi @ a @ b2 @ phi ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_2) ).

thf(2,negated_conjecture,
    ~ ( ( ( size_size @ ( list @ ( option @ b ) ) @ ( cons @ ( option @ b ) @ ( some @ b @ x ) @ v ) )
        = ( suc @ n ) )
      & ! [A: nat] :
          ( ~ ( ord_less @ nat @ A @ ( suc @ n ) )
          | ( ( ( nth @ ( option @ b ) @ ( cons @ ( option @ b ) @ ( some @ b @ x ) @ v ) @ A )
              = ( none @ b ) )
            = ( ~ ( member @ nat @ A @ ( fvi @ a @ b2 @ phi ) ) ) ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(313,plain,
    ~ ( ( ( size_size @ ( list @ ( option @ b ) ) @ ( cons @ ( option @ b ) @ ( some @ b @ x ) @ v ) )
        = ( suc @ n ) )
      & ! [A: nat] :
          ( ~ ( ord_less @ nat @ A @ ( suc @ n ) )
          | ( ( ( nth @ ( option @ b ) @ ( cons @ ( option @ b ) @ ( some @ b @ x ) @ v ) @ A )
              = ( none @ b ) )
            = ( ~ ( member @ nat @ A @ ( fvi @ a @ b2 @ phi ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(173,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_not__less__zero) ).

thf(787,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(70,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = A )
          = ( B
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_add__cancel__left__right) ).

thf(466,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = A )
          = ( B
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(11,axiom,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( if @ TA @ $false @ A @ B )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_If_2_1_T) ).

thf(335,plain,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( if @ TA @ $false @ A @ B )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(216,axiom,
    ! [TA: $tType] :
      ( ( gen_length @ TA )
      = ( ^ [A: nat,B: list @ TA] : ( plus_plus @ nat @ A @ ( size_size @ ( list @ TA ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_gen__length__def) ).

thf(919,plain,
    ! [TA: $tType] :
      ( ( gen_length @ TA )
      = ( ^ [A: nat,B: list @ TA] : ( plus_plus @ nat @ A @ ( size_size @ ( list @ TA ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(95,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_50_infinite__descent0__measure) ).

thf(540,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)],[95]) ).

thf(112,axiom,
    ! [TA: $tType,A: TA,B: list @ TA] :
      ( ( size_size @ ( list @ TA ) @ ( cons @ TA @ A @ B ) )
      = ( suc @ ( size_size @ ( list @ TA ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_length__Cons) ).

thf(604,plain,
    ! [TA: $tType,A: TA,B: list @ TA] :
      ( ( size_size @ ( list @ TA ) @ ( cons @ TA @ A @ B ) )
      = ( suc @ ( size_size @ ( list @ TA ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(246,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( member @ TA @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_Collect__mem__eq) ).

thf(1011,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( member @ TA @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(156,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( suc @ B ) )
     => ( A
       != ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_nat_OdiscI) ).

thf(736,plain,
    ! [A: nat,B: nat] :
      ( ( A
        = ( suc @ B ) )
     => ( A
       != ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(61,axiom,
    cancel1352612707id_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Ocancel__comm__monoid__add) ).

thf(451,plain,
    cancel1352612707id_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(3,axiom,
    ! [A: $o] :
      ( ( A = $true )
      | ( A = $false ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_If_3_1_T) ).

thf(318,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(116,axiom,
    ! [TA: $tType,A: formula @ TA] :
      ( ( size_size @ ( formula @ TA ) @ A )
     != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_formula_Osize__neq) ).

thf(616,plain,
    ! [TA: $tType,A: formula @ TA] :
      ( ( size_size @ ( formula @ TA ) @ A )
     != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(28,axiom,
    ( ( (=) @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( semiring_1_of_nat @ int @ A )
          = ( semiring_1_of_nat @ int @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_nat__int__comparison_I1_J) ).

thf(377,plain,
    ( ( (=) @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( semiring_1_of_nat @ int @ A )
          = ( semiring_1_of_nat @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(241,axiom,
    ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_less__int__code_I1_J) ).

thf(990,plain,
    ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(127,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_gr0__implies__Suc) ).

thf(649,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(259,axiom,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ! [A: nat,B: nat] :
          ( ( semiring_1_of_nat @ TA @ ( plus_plus @ nat @ A @ B ) )
          = ( plus_plus @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_of__nat__add) ).

thf(1043,plain,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ! [A: nat,B: nat] :
          ( ( semiring_1_of_nat @ TA @ ( plus_plus @ nat @ A @ B ) )
          = ( plus_plus @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(208,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_old_Onat_Oexhaust) ).

thf(890,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(33,axiom,
    ! [TA: $tType,A: i,B: monito748960549ormula @ TA,C: $o,D: list @ nat,E: i,F: monito748960549ormula @ TA,G: $o,H: list @ nat] :
      ( ( ( monito320348316_MNext @ TA @ A @ B @ C @ D )
        = ( monito320348316_MNext @ TA @ E @ F @ G @ H ) )
      = ( ( A = E )
        & ( B = F )
        & ( C = G )
        & ( D = H ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_mformula_Oinject_I7_J) ).

thf(387,plain,
    ! [TA: $tType,A: i,B: monito748960549ormula @ TA,C: $o,D: list @ nat,E: i,F: monito748960549ormula @ TA,G: $o,H: list @ nat] :
      ( ( ( monito320348316_MNext @ TA @ A @ B @ C @ D )
        = ( monito320348316_MNext @ TA @ E @ F @ G @ H ) )
      = ( ( A = E )
        & ( B = F )
        & ( C = G )
        & ( D = H ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(120,axiom,
    ! [TA: $tType,A: nat,B: list @ TA,C: nat] :
      ( ( ord_less_eq @ nat @ A @ ( size_size @ ( list @ TA ) @ B ) )
     => ( ( nth @ TA @ ( drop @ TA @ A @ B ) @ C )
        = ( nth @ TA @ B @ ( plus_plus @ nat @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_nth__drop) ).

thf(631,plain,
    ! [TA: $tType,A: nat,B: list @ TA,C: nat] :
      ( ( ord_less_eq @ nat @ A @ ( size_size @ ( list @ TA ) @ B ) )
     => ( ( nth @ TA @ ( drop @ TA @ A @ B ) @ C )
        = ( nth @ TA @ B @ ( plus_plus @ nat @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(167,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_less__Suc0) ).

thf(769,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(224,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_29_nat__neq__iff) ).

thf(943,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)],[224]) ).

thf(9,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( order @ TA )
     => ( order @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder) ).

thf(330,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( order @ TA )
     => ( order @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(280,axiom,
    ! [A: nat,B: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_not__add__less2) ).

thf(1112,plain,
    ! [A: nat,B: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(80,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_bot__nat__0_Oextremum__strict) ).

thf(494,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(272,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( A != B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_less__not__refl3) ).

thf(1080,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(15,axiom,
    ! [TA: $tType] :
      ( ( monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_add_Oright__neutral) ).

thf(345,plain,
    ! [TA: $tType] :
      ( ( monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(98,axiom,
    ! [TA: $tType] :
      ( ( size_size @ ( list @ TA ) )
      = ( gen_length @ TA @ ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_length__code) ).

thf(552,plain,
    ! [TA: $tType] :
      ( ( size_size @ ( list @ TA ) )
      = ( gen_length @ TA @ ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(194,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_23_infinite__descent) ).

thf(848,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)],[194]) ).

thf(88,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
          = ( ord_less @ TA @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_add__less__cancel__left) ).

thf(520,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
          = ( ord_less @ TA @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(68,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ C @ B ) )
          = ( A = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_add_Oright__cancel) ).

thf(462,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ C @ B ) )
          = ( A = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(221,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_less__numeral__extra_I3_J) ).

thf(932,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(296,axiom,
    ! [TA: $tType,A: nat > TA > $o,B: nat > TA > TA > $o] :
      ( ? [C: TA] : ( A @ ( zero_zero @ nat ) @ C )
     => ( ! [C: TA,D: nat] :
            ( ( A @ D @ C )
           => ? [E: TA] :
                ( ( A @ ( suc @ D ) @ E )
                & ( B @ D @ C @ E ) ) )
       => ? [C: nat > TA] :
          ! [D: nat] :
            ( ( A @ D @ ( C @ D ) )
            & ( B @ D @ ( C @ D ) @ ( C @ ( suc @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_dependent__nat__choice) ).

thf(1171,plain,
    ! [TA: $tType,A: nat > TA > $o,B: nat > TA > TA > $o] :
      ( ? [C: TA] : ( A @ ( zero_zero @ nat ) @ C )
     => ( ! [C: TA,D: nat] :
            ( ( A @ D @ C )
           => ? [E: TA] :
                ( ( A @ ( suc @ D ) @ E )
                & ( B @ D @ C @ E ) ) )
       => ? [C: nat > TA] :
          ! [D: nat] :
            ( ( A @ D @ ( C @ D ) )
            & ( B @ D @ ( C @ D ) @ ( C @ ( suc @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(37,axiom,
    strict2144017051up_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add) ).

thf(399,plain,
    strict2144017051up_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(230,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus @ nat @ A @ B )
        = A )
     => ( B
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_add__eq__self__zero) ).

thf(959,plain,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus @ nat @ A @ B )
        = A )
     => ( B
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(64,axiom,
    ordere216010020id_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Oordered__comm__monoid__add) ).

thf(455,plain,
    ordere216010020id_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(277,axiom,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( plus_plus @ TA @ A @ B ) )
          = ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_less__add__same__cancel1) ).

thf(1102,plain,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( plus_plus @ TA @ A @ B ) )
          = ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(217,axiom,
    ! [A: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ A )
     => ? [B: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
          & ( A
            = ( semiring_1_of_nat @ int @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_zero__less__imp__eq__int) ).

thf(922,plain,
    ! [A: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ A )
     => ? [B: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
          & ( A
            = ( semiring_1_of_nat @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(308,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ? [C: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
          & ( ( plus_plus @ nat @ A @ C )
            = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_less__imp__add__positive) ).

thf(1213,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ? [C: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
          & ( ( plus_plus @ nat @ A @ C )
            = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).

thf(188,axiom,
    ! [A: nat] : ( ord_less @ nat @ A @ ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_lessI) ).

thf(827,plain,
    ! [A: nat] : ( ord_less @ nat @ A @ ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(270,axiom,
    cancel1352612707id_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_11) ).

thf(1076,plain,
    cancel1352612707id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(50,axiom,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( if @ TA @ $true @ A @ B )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_If_1_1_T) ).

thf(426,plain,
    ! [TA: $tType,A: TA,B: TA] :
      ( ( if @ TA @ $true @ A @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(92,axiom,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat] :
          ( ( semiring_1_of_nat @ TA @ ( suc @ A ) )
         != ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_of__nat__neq__0) ).

thf(531,plain,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat] :
          ( ( semiring_1_of_nat @ TA @ ( suc @ A ) )
         != ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(287,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_add__strict__right__mono) ).

thf(1134,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(183,axiom,
    ! [TA: $tType,A: TA,B: list @ TA] :
      ( ( cons @ TA @ A @ B )
     != B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_not__Cons__self2) ).

thf(812,plain,
    ! [TA: $tType,A: TA,B: list @ TA] :
      ( ( cons @ TA @ A @ B )
     != B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(162,axiom,
    ! [A: nat] :
      ( ( plus_plus @ nat @ A @ ( zero_zero @ nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_Nat_Oadd__0__right) ).

thf(754,plain,
    ! [A: nat] :
      ( ( plus_plus @ nat @ A @ ( zero_zero @ nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(261,axiom,
    ! [TA: $tType,A: nat,B: nat,C: list @ TA] :
      ( ( drop @ TA @ A @ ( drop @ TA @ B @ C ) )
      = ( drop @ TA @ ( plus_plus @ nat @ A @ B ) @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_drop__drop) ).

thf(1049,plain,
    ! [TA: $tType,A: nat,B: nat,C: list @ TA] :
      ( ( drop @ TA @ A @ ( drop @ TA @ B @ C ) )
      = ( drop @ TA @ ( plus_plus @ nat @ A @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(305,axiom,
    ( ( ord_less @ nat )
    = ( ^ [A: nat,B: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A ) @ ( semiring_1_of_nat @ int @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_nat__int__comparison_I2_J) ).

thf(1208,plain,
    ( ( ord_less @ nat )
    = ( ^ [A: nat,B: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A ) @ ( semiring_1_of_nat @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(106,axiom,
    canoni770627133id_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add) ).

thf(589,plain,
    canoni770627133id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(177,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ( ord_less @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_Suc__lessD) ).

thf(795,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ( ord_less @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(138,axiom,
    ! [TA: $tType] : ( size @ ( option @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Option_Ooption___Nat_Osize_27) ).

thf(680,plain,
    ! [TA: $tType] : ( size @ ( option @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(233,axiom,
    ordere236663937imp_le @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_6) ).

thf(968,plain,
    ordere236663937imp_le @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(298,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_21_infinite__descent__measure) ).

thf(1181,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)],[298]) ).

thf(150,axiom,
    ! [A: nat] :
      ( ( nat_triangle @ ( suc @ A ) )
      = ( plus_plus @ nat @ ( nat_triangle @ A ) @ ( suc @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_triangle__Suc) ).

thf(712,plain,
    ! [A: nat] :
      ( ( nat_triangle @ ( suc @ A ) )
      = ( plus_plus @ nat @ ( nat_triangle @ A ) @ ( suc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(252,axiom,
    ! [TA: $tType,A: monito748960549ormula @ TA] :
      ( ( size_size @ ( monito748960549ormula @ TA ) @ ( monito518325957Exists @ TA @ A ) )
      = ( plus_plus @ nat @ ( size_size @ ( monito748960549ormula @ TA ) @ A ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_mformula_Osize_I14_J) ).

thf(1025,plain,
    ! [TA: $tType,A: monito748960549ormula @ TA] :
      ( ( size_size @ ( monito748960549ormula @ TA ) @ ( monito518325957Exists @ TA @ A ) )
      = ( plus_plus @ nat @ ( size_size @ ( monito748960549ormula @ TA ) @ A ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(201,axiom,
    ! [TA: $tType,A: nat,B: list @ TA] :
      ( ( ord_less @ nat @ A @ ( size_size @ ( list @ TA ) @ B ) )
     => ( ( cons @ TA @ ( nth @ TA @ B @ A ) @ ( drop @ TA @ ( suc @ A ) @ B ) )
        = ( drop @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_Cons__nth__drop__Suc) ).

thf(872,plain,
    ! [TA: $tType,A: nat,B: list @ TA] :
      ( ( ord_less @ nat @ A @ ( size_size @ ( list @ TA ) @ B ) )
     => ( ( cons @ TA @ ( nth @ TA @ B @ A ) @ ( drop @ TA @ ( suc @ A ) @ B ) )
        = ( drop @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(292,axiom,
    ! [TA: $tType,A: TA > $o,B: TA,C: list @ TA] :
      ( ( list_ex @ TA @ A @ ( cons @ TA @ B @ C ) )
      = ( ( A @ B )
        | ( list_ex @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_list__ex__simps_I1_J) ).

thf(1156,plain,
    ! [TA: $tType,A: TA > $o,B: TA,C: list @ TA] :
      ( ( list_ex @ TA @ A @ ( cons @ TA @ B @ C ) )
      = ( ( A @ B )
        | ( list_ex @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(44,axiom,
    group_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Ogroup__add) ).

thf(414,plain,
    group_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(8,axiom,
    ! [TA: $tType,A: monito748960549ormula @ TA,B: i,C: monito748960549ormula @ TA,D: $o,E: list @ nat] :
      ( ( monito518325957Exists @ TA @ A )
     != ( monito320348316_MNext @ TA @ B @ C @ D @ E ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_mformula_Odistinct_I55_J) ).

thf(327,plain,
    ! [TA: $tType,A: monito748960549ormula @ TA,B: i,C: monito748960549ormula @ TA,D: $o,E: list @ nat] :
      ( ( monito518325957Exists @ TA @ A )
     != ( monito320348316_MNext @ TA @ B @ C @ D @ E ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(228,axiom,
    ! [TA: $tType,A: option @ TA] :
      ( ( A
       != ( none @ TA ) )
      = ( ? [B: TA] :
            ( A
            = ( some @ TA @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_not__None__eq) ).

thf(954,plain,
    ! [TA: $tType,A: option @ TA] :
      ( ( A
       != ( none @ TA ) )
      = ( ? [B: TA] :
            ( A
            = ( some @ TA @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(175,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) )
          = ( ord_less @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_of__nat__less__iff) ).

thf(791,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) )
          = ( ord_less @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(300,axiom,
    ! [TA: $tType,A: list @ TA,B: list @ TA] :
      ( ( ( size_size @ ( list @ TA ) @ A )
        = ( size_size @ ( list @ TA ) @ B ) )
     => ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ ( size_size @ ( list @ TA ) @ A ) )
           => ( ( nth @ TA @ A @ C )
              = ( nth @ TA @ B @ C ) ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_nth__equalityI) ).

thf(1188,plain,
    ! [TA: $tType,A: list @ TA,B: list @ TA] :
      ( ( ( size_size @ ( list @ TA ) @ A )
        = ( size_size @ ( list @ TA ) @ B ) )
     => ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ ( size_size @ ( list @ TA ) @ A ) )
           => ( ( nth @ TA @ A @ C )
              = ( nth @ TA @ B @ C ) ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(192,axiom,
    ! [TA: $tType,A: nat] :
    ? [B: list @ TA] :
      ( ( size_size @ ( list @ TA ) @ B )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_Ex__list__of__length) ).

thf(840,plain,
    ! [TA: $tType,A: nat] :
    ? [B: list @ TA] :
      ( ( size_size @ ( list @ TA ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(12,axiom,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ( plus_plus @ TA @ A @ A )
            = ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_double__zero) ).

thf(338,plain,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ( plus_plus @ TA @ A @ A )
            = ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(20,axiom,
    comm_monoid_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Ocomm__monoid__add) ).

thf(359,plain,
    comm_monoid_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(285,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
            | ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_add__less__zeroD) ).

thf(1130,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
            | ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(310,axiom,
    comm_monoid_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__add_14) ).

thf(1220,plain,
    comm_monoid_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).

thf(214,axiom,
    ! [A: nat > $o] :
      ( ~ ( A @ ( zero_zero @ nat ) )
     => ( ? [B: nat] : ( A @ B )
       => ? [B: nat] :
            ( ~ ( A @ B )
            & ( A @ ( suc @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_exists__least__lemma) ).

thf(907,plain,
    ! [A: nat > $o] :
      ( ~ ( A @ ( zero_zero @ nat ) )
     => ( ? [B: nat] : ( A @ B )
       => ? [B: nat] :
            ( ~ ( A @ B )
            & ( A @ ( suc @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(256,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_95_not__gr__zero) ).

thf(1038,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)],[256]) ).

thf(25,axiom,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ C @ B ) )
          = ( A = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_add__right__cancel) ).

thf(370,plain,
    ! [TA: $tType] :
      ( ( cancel_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( plus_plus @ TA @ C @ B ) )
          = ( A = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(147,axiom,
    ! [TA: $tType,A: nat,B: list @ TA] :
      ( ( ( suc @ A )
        = ( size_size @ ( list @ TA ) @ B ) )
      = ( ? [C: TA,D: list @ TA] :
            ( ( B
              = ( cons @ TA @ C @ D ) )
            & ( ( size_size @ ( list @ TA ) @ D )
              = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_Suc__length__conv) ).

thf(703,plain,
    ! [TA: $tType,A: nat,B: list @ TA] :
      ( ( ( suc @ A )
        = ( size_size @ ( list @ TA ) @ B ) )
      = ( ? [C: TA,D: list @ TA] :
            ( ( B
              = ( cons @ TA @ C @ D ) )
            & ( ( size_size @ ( list @ TA ) @ D )
              = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(115,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_83_lift__Suc__mono__less__iff) ).

thf(613,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)],[115]) ).

thf(311,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_52_infinite__descent0) ).

thf(1221,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)],[311]) ).

thf(262,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_40_diff__induct) ).

thf(1052,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)],[262]) ).

thf(245,axiom,
    ! [TA: $tType,A: TA,B: list @ TA] :
      ( ( size_size @ ( list @ TA ) @ ( cons @ TA @ A @ B ) )
      = ( plus_plus @ nat @ ( size_size @ ( list @ TA ) @ B ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_list_Osize_I4_J) ).

thf(1008,plain,
    ! [TA: $tType,A: TA,B: list @ TA] :
      ( ( size_size @ ( list @ TA ) @ ( cons @ TA @ A @ B ) )
      = ( plus_plus @ nat @ ( size_size @ ( list @ TA ) @ B ) @ ( suc @ ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(81,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ( plus_plus @ nat @ C @ B )
          = ( plus_plus @ nat @ A @ D ) )
       => ( ord_less @ nat @ C @ D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_less__add__eq__less) ).

thf(497,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ( plus_plus @ nat @ C @ B )
          = ( plus_plus @ nat @ A @ D ) )
       => ( ord_less @ nat @ C @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(197,axiom,
    ! [TA: $tType,A: TA,B: list @ TA,C: TA,D: list @ TA] :
      ( ( ( cons @ TA @ A @ B )
        = ( cons @ TA @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_list_Oinject) ).

thf(859,plain,
    ! [TA: $tType,A: TA,B: list @ TA,C: TA,D: list @ TA] :
      ( ( ( cons @ TA @ A @ B )
        = ( cons @ TA @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(274,axiom,
    semiri456707255roduct @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_2) ).

thf(1094,plain,
    semiri456707255roduct @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(237,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_58_not__less__less__Suc__eq) ).

thf(978,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)],[237]) ).

thf(69,axiom,
    linord1659791738miring @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Rings_Olinordered__nonzero__semiring) ).

thf(465,plain,
    linord1659791738miring @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(22,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ A ) )
          = ( B
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_add__cancel__right__left) ).

thf(363,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ A ) )
          = ( B
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(220,axiom,
    ordere216010020id_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_10) ).

thf(931,plain,
    ordere216010020id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(76,axiom,
    ! [TA: $tType] : ( order @ ( set @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oorder_21) ).

thf(482,plain,
    ! [TA: $tType] : ( order @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(279,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A
        = ( plus_plus @ nat @ B @ C ) )
     => ( ( suc @ A )
        = ( plus_plus @ nat @ B @ ( suc @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_nat__arith_Osuc1) ).

thf(1108,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A
        = ( plus_plus @ nat @ B @ C ) )
     => ( ( suc @ A )
        = ( plus_plus @ nat @ B @ ( suc @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(60,axiom,
    semigroup_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Osemigroup__add) ).

thf(450,plain,
    semigroup_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(273,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_39_zero__induct) ).

thf(1084,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)],[273]) ).

thf(87,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
          = ( ord_less @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_add__less__cancel__right) ).

thf(517,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
          = ( ord_less @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(4,axiom,
    order @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Orderings_Oorder_1) ).

thf(319,plain,
    order @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(265,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ? [C: nat] :
          ( B
          = ( suc @ ( plus_plus @ nat @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_less__imp__Suc__add) ).

thf(1062,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ? [C: nat] :
          ( B
          = ( suc @ ( plus_plus @ nat @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(225,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_22_linorder__neqE__nat) ).

thf(946,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)],[225]) ).

thf(34,axiom,
    ! [TA: $tType] : ( size @ ( trm @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_MFOTL_Otrm___Nat_Osize_24) ).

thf(390,plain,
    ! [TA: $tType] : ( size @ ( trm @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(46,axiom,
    ! [A: nat,B: nat] :
      ( ( ( semiring_1_of_nat @ int @ A )
        = ( semiring_1_of_nat @ int @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_int__int__eq) ).

thf(418,plain,
    ! [A: nat,B: nat] :
      ( ( ( semiring_1_of_nat @ int @ A )
        = ( semiring_1_of_nat @ int @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(243,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_59_strict__inc__induct) ).

thf(995,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)],[243]) ).

thf(128,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_70_less__SucE) ).

thf(652,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)],[128]) ).

thf(17,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
          = ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(351,plain,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
          = ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(103,axiom,
    ab_semigroup_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oab__semigroup__add_13) ).

thf(568,plain,
    ab_semigroup_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(134,axiom,
    ! [TA: $tType,A: option @ TA] :
      ( ( size_size @ ( option @ TA ) @ A )
     != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_option_Osize__neq) ).

thf(667,plain,
    ! [TA: $tType,A: option @ TA] :
      ( ( size_size @ ( option @ TA ) @ A )
     != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(301,axiom,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_Suc__neq__Zero) ).

thf(1193,plain,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(271,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ ( suc @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_not__less__eq) ).

thf(1077,plain,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ ( suc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(165,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_8_Suc__less__eq) ).

thf(763,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)],[165]) ).

thf(31,axiom,
    order @ i,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Interval_O_092_060I_062___Orderings_Oorder_25) ).

thf(383,plain,
    order @ i,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(211,axiom,
    ! [TA: $tType] :
      ( ( drop @ TA @ ( zero_zero @ nat ) )
      = ( ^ [A: list @ TA] : A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_drop0) ).

thf(899,plain,
    ! [TA: $tType] :
      ( ( drop @ TA @ ( zero_zero @ nat ) )
      = ( ^ [A: list @ TA] : A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(97,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus @ nat @ ( suc @ A ) @ B )
      = ( suc @ ( plus_plus @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_add__Suc) ).

thf(549,plain,
    ! [A: nat,B: nat] :
      ( ( plus_plus @ nat @ ( suc @ A ) @ B )
      = ( suc @ ( plus_plus @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(179,axiom,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat] :
          ( ( ( semiring_1_of_nat @ TA @ A )
            = ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_of__nat__eq__0__iff) ).

thf(800,plain,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat] :
          ( ( ( semiring_1_of_nat @ TA @ A )
            = ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(53,axiom,
    ! [A: int] :
      ( ( plus_plus @ int @ ( zero_zero @ int ) @ A )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_plus__int__code_I2_J) ).

thf(433,plain,
    ! [A: int] :
      ( ( plus_plus @ int @ ( zero_zero @ int ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(85,axiom,
    ! [TA: $tType,A: TA,B: list @ TA] :
      ( ( nth @ TA @ ( cons @ TA @ A @ B ) @ ( zero_zero @ nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_nth__Cons__0) ).

thf(510,plain,
    ! [TA: $tType,A: TA,B: list @ TA] :
      ( ( nth @ TA @ ( cons @ TA @ A @ B ) @ ( zero_zero @ nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(190,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_neq0__conv) ).

thf(832,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(140,axiom,
    ! [TA: $tType,A: ( option @ TA ) > $o,B: option @ TA] :
      ( ( A @ ( none @ TA ) )
     => ( ! [C: TA] : ( A @ ( some @ TA @ C ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_option_Oinducts) ).

thf(684,plain,
    ! [TA: $tType,A: ( option @ TA ) > $o,B: option @ TA] :
      ( ( A @ ( none @ TA ) )
     => ( ! [C: TA] : ( A @ ( some @ TA @ C ) )
       => ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(282,axiom,
    ! [A: nat,B: nat] :
      ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A @ B ) )
      = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A ) @ ( semiring_1_of_nat @ int @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_int__plus) ).

thf(1116,plain,
    ! [A: nat,B: nat] :
      ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A @ B ) )
      = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A ) @ ( semiring_1_of_nat @ int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(161,axiom,
    ! [TA: $tType] :
      ( ( ordere216010020id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_add__neg__neg) ).

thf(752,plain,
    ! [TA: $tType] :
      ( ( ordere216010020id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(59,axiom,
    ! [TA: $tType,A: i,B: monito748960549ormula @ TA,C: $o,D: list @ ( set @ ( list @ ( option @ TA ) ) ),E: list @ nat,F: i,G: monito748960549ormula @ TA,H: $o,I: list @ ( set @ ( list @ ( option @ TA ) ) ),J: list @ nat] :
      ( ( ( monito1987675900_MPrev @ TA @ A @ B @ C @ D @ E )
        = ( monito1987675900_MPrev @ TA @ F @ G @ H @ I @ J ) )
      = ( ( A = F )
        & ( B = G )
        & ( C = H )
        & ( D = I )
        & ( E = J ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_mformula_Oinject_I6_J) ).

thf(447,plain,
    ! [TA: $tType,A: i,B: monito748960549ormula @ TA,C: $o,D: list @ ( set @ ( list @ ( option @ TA ) ) ),E: list @ nat,F: i,G: monito748960549ormula @ TA,H: $o,I: list @ ( set @ ( list @ ( option @ TA ) ) ),J: list @ nat] :
      ( ( ( monito1987675900_MPrev @ TA @ A @ B @ C @ D @ E )
        = ( monito1987675900_MPrev @ TA @ F @ G @ H @ I @ J ) )
      = ( ( A = F )
        & ( B = G )
        & ( C = H )
        & ( D = I )
        & ( E = J ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(52,axiom,
    zero @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Ozero) ).

thf(432,plain,
    zero @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(42,axiom,
    ab_semigroup_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Oab__semigroup__add) ).

thf(410,plain,
    ab_semigroup_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(191,axiom,
    ( ( ( size_size @ ( list @ ( option @ b ) ) @ v )
      = n )
    & ! [A: nat] :
        ( ( ord_less @ nat @ A @ n )
       => ( ( ( nth @ ( option @ b ) @ v @ A )
            = ( none @ b ) )
          = ( ~ ( member @ nat @ A @ ( fvi @ a @ ( suc @ b2 ) @ phi ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_1) ).

thf(835,plain,
    ( ( ( size_size @ ( list @ ( option @ b ) ) @ v )
      = n )
    & ! [A: nat] :
        ( ( ord_less @ nat @ A @ n )
       => ( ( ( nth @ ( option @ b ) @ v @ A )
            = ( none @ b ) )
          = ( ~ ( member @ nat @ A @ ( fvi @ a @ ( suc @ b2 ) @ phi ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(199,axiom,
    ! [A: nat] :
      ( A
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_n__not__Suc__n) ).

thf(865,plain,
    ! [A: nat] :
      ( A
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(26,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( zero_zero @ TA ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            & ( B
              = ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_add__eq__0__iff__both__eq__0) ).

thf(373,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = ( zero_zero @ TA ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            & ( B
              = ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(102,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B
           != ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_gr__implies__not__zero) ).

thf(564,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B
           != ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(13,axiom,
    ! [TA: $tType] :
      ( ( monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_add_Oleft__neutral) ).

thf(341,plain,
    ! [TA: $tType] :
      ( ( monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(19,axiom,
    strict797366125id_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add) ).

thf(358,plain,
    strict797366125id_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(210,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
     => ( ord_less @ nat @ A @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_add__lessD1) ).

thf(897,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
     => ( ord_less @ nat @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(122,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus @ nat @ A @ ( suc @ B ) )
      = ( suc @ ( plus_plus @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_add__Suc__right) ).

thf(635,plain,
    ! [A: nat,B: nat] :
      ( ( plus_plus @ nat @ A @ ( suc @ B ) )
      = ( suc @ ( plus_plus @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(266,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_list__decode_Ocases) ).

thf(1065,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(144,axiom,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less @ TA @ A @ B )
            & ( ord_less @ TA @ C @ D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_add__mono__thms__linordered__field_I5_J) ).

thf(696,plain,
    ! [TA: $tType] :
      ( ( ordere223160158up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less @ TA @ A @ B )
            & ( ord_less @ TA @ C @ D ) )
         => ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(263,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_89_Ex__less__Suc2) ).

thf(1056,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)],[263]) ).

thf(133,axiom,
    ! [TA: $tType,A: nat,B: list @ TA,C: TA,D: list @ TA] :
      ( ( ( drop @ TA @ A @ B )
        = ( cons @ TA @ C @ D ) )
     => ( ( nth @ TA @ B @ A )
        = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_nth__via__drop) ).

thf(664,plain,
    ! [TA: $tType,A: nat,B: list @ TA,C: TA,D: list @ TA] :
      ( ( ( drop @ TA @ A @ B )
        = ( cons @ TA @ C @ D ) )
     => ( ( nth @ TA @ B @ A )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(38,axiom,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat,B: nat] :
          ( ( ( semiring_1_of_nat @ TA @ A )
            = ( semiring_1_of_nat @ TA @ B ) )
          = ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_of__nat__eq__iff) ).

thf(400,plain,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat,B: nat] :
          ( ( ( semiring_1_of_nat @ TA @ A )
            = ( semiring_1_of_nat @ TA @ B ) )
          = ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(302,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_74_Nat_OlessE) ).

thf(1197,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)],[302]) ).

thf(219,axiom,
    ! [TA: $tType,A: nat,B: TA,C: list @ TA] :
      ( ( drop @ TA @ ( suc @ A ) @ ( cons @ TA @ B @ C ) )
      = ( drop @ TA @ A @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_drop__Suc__Cons) ).

thf(928,plain,
    ! [TA: $tType,A: nat,B: TA,C: list @ TA] :
      ( ( drop @ TA @ ( suc @ A ) @ ( cons @ TA @ B @ C ) )
      = ( drop @ TA @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(45,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = B )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_add__cancel__left__left) ).

thf(415,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = B )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(36,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( A = B )
            & ( C = D ) )
         => ( ( plus_plus @ TA @ A @ C )
            = ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_add__mono__thms__linordered__semiring_I4_J) ).

thf(395,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( A = B )
            & ( C = D ) )
         => ( ( plus_plus @ TA @ A @ C )
            = ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(240,axiom,
    ! [TA: $tType,A: nat,B: formula @ TA,C: nat] :
      ( ! [D: nat] :
          ( ( member @ nat @ D @ ( fvi @ TA @ ( suc @ A ) @ B ) )
         => ( ord_less @ nat @ D @ C ) )
     => ! [D: nat] :
          ( ( member @ nat @ D @ ( fvi @ TA @ A @ B ) )
         => ( ord_less @ nat @ D @ ( suc @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_fvi__Suc__bound) ).

thf(987,plain,
    ! [TA: $tType,A: nat,B: formula @ TA,C: nat] :
      ( ! [D: nat] :
          ( ( member @ nat @ D @ ( fvi @ TA @ ( suc @ A ) @ B ) )
         => ( ord_less @ nat @ D @ C ) )
     => ! [D: nat] :
          ( ( member @ nat @ D @ ( fvi @ TA @ A @ B ) )
         => ( ord_less @ nat @ D @ ( suc @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(109,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_7_Suc__mono) ).

thf(596,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)],[109]) ).

thf(91,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_trans__less__add1) ).

thf(528,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(125,axiom,
    ! [TA: $tType,A: list @ TA] :
      ( ( drop @ TA @ ( zero_zero @ nat ) @ A )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_drop__0) ).

thf(644,plain,
    ! [TA: $tType,A: list @ TA] :
      ( ( drop @ TA @ ( zero_zero @ nat ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(67,axiom,
    ! [TA: $tType] :
      ( ( neg_numeral @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
          = ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_is__num__normalize_I1_J) ).

thf(459,plain,
    ! [TA: $tType] :
      ( ( neg_numeral @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
          = ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(242,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
      = ( ? [B: nat] :
            ( A
            = ( suc @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_gr0__conv__Suc) ).

thf(992,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
      = ( ? [B: nat] :
            ( A
            = ( suc @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(142,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_less__irrefl__nat) ).

thf(691,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(56,axiom,
    ordere516151231imp_le @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le) ).

thf(442,plain,
    ordere516151231imp_le @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(82,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_65_All__less__Suc) ).

thf(501,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)],[82]) ).

thf(10,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( zero_zero @ TA )
            = ( plus_plus @ TA @ A @ B ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            & ( B
              = ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_zero__eq__add__iff__both__eq__0) ).

thf(332,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( zero_zero @ TA )
            = ( plus_plus @ TA @ A @ B ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            & ( B
              = ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(284,axiom,
    ordere516151231imp_le @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_3) ).

thf(1129,plain,
    ordere516151231imp_le @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(24,axiom,
    neg_numeral @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Num_Oneg__numeral) ).

thf(369,plain,
    neg_numeral @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(113,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_gr__zeroI) ).

thf(607,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(180,axiom,
    ! [TA: $tType] :
      ( ( (=) @ ( list @ TA ) )
      = ( ^ [A: list @ TA,B: list @ TA] :
            ( ( ( size_size @ ( list @ TA ) @ A )
              = ( size_size @ ( list @ TA ) @ B ) )
            & ! [C: nat] :
                ( ( ord_less @ nat @ C @ ( size_size @ ( list @ TA ) @ A ) )
               => ( ( nth @ TA @ A @ C )
                  = ( nth @ TA @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_list__eq__iff__nth__eq) ).

thf(803,plain,
    ! [TA: $tType] :
      ( ( (=) @ ( list @ TA ) )
      = ( ^ [A: list @ TA,B: list @ TA] :
            ( ( ( size_size @ ( list @ TA ) @ A )
              = ( size_size @ ( list @ TA ) @ B ) )
            & ! [C: nat] :
                ( ( ord_less @ nat @ C @ ( size_size @ ( list @ TA ) @ A ) )
               => ( ( nth @ TA @ A @ C )
                  = ( nth @ TA @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(71,axiom,
    ! [TA: $tType] :
      ( ( semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
          = ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_add_Oassoc) ).

thf(469,plain,
    ! [TA: $tType] :
      ( ( semigroup_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
          = ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(18,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ C ) )
         => ( ( plus_plus @ TA @ A @ D )
            = ( plus_plus @ TA @ B @ ( plus_plus @ TA @ C @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_group__cancel_Oadd1) ).

thf(354,plain,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ C ) )
         => ( ( plus_plus @ TA @ A @ D )
            = ( plus_plus @ TA @ B @ ( plus_plus @ TA @ C @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(299,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ? [C: TA] :
                ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
                & ( ord_less @ TA @ C @ A )
                & ( ord_less @ TA @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_field__lbound__gt__zero) ).

thf(1184,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ? [C: TA] :
                ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
                & ( ord_less @ TA @ C @ A )
                & ( ord_less @ TA @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(79,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_less__not__refl) ).

thf(491,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(204,axiom,
    ( ( ord_less @ int )
    = ( ^ [A: int,B: int] :
        ? [C: nat] :
          ( B
          = ( plus_plus @ int @ A @ ( semiring_1_of_nat @ int @ ( suc @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_zless__iff__Suc__zadd) ).

thf(882,plain,
    ( ( ord_less @ int )
    = ( ^ [A: int,B: int] :
        ? [C: nat] :
          ( B
          = ( plus_plus @ int @ A @ ( semiring_1_of_nat @ int @ ( suc @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(169,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B
       != ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_gr__implies__not0) ).

thf(775,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B
       != ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(101,axiom,
    ! [TA: $tType] :
      ( ( size @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( size_size @ TA @ A )
           != ( size_size @ TA @ B ) )
         => ( A != B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_size__neq__size__imp__neq) ).

thf(560,plain,
    ! [TA: $tType] :
      ( ( size @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( size_size @ TA @ A )
           != ( size_size @ TA @ B ) )
         => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(63,axiom,
    ordere236663937imp_le @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(454,plain,
    ordere236663937imp_le @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(267,axiom,
    ! [TA: $tType] :
      ( ( size_size @ ( option @ TA ) @ ( none @ TA ) )
      = ( suc @ ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_option_Osize_I3_J) ).

thf(1069,plain,
    ! [TA: $tType] :
      ( ( size_size @ ( option @ TA ) @ ( none @ TA ) )
      = ( suc @ ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(247,axiom,
    ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
    = ( zero_zero @ int ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_int__ops_I1_J) ).

thf(1014,plain,
    ( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
    = ( zero_zero @ int ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(235,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_old_Onat_Oinject) ).

thf(972,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(1240,plain,
    $false,
    inference(cvc4,[status(thm)],[809,1024,760,1237,962,555,479,1105,1211,523,741,1205,379,677,1041,878,655,384,715,472,698,670,1164,797,782,709,320,436,829,1095,404,885,1046,785,504,619,751,344,934,949,1059,1132,902,421,969,1115,638,853,475,789,443,321,376,1136,348,806,905,634,734,1072,1119,660,411,507,527,981,766,452,439,661,1153,366,534,746,1021,1042,862,820,392,424,407,429,815,957,894,1217,598,916,484,1032,625,1178,1159,1210,757,682,456,324,1143,1017,725,403,1006,593,869,610,884,1028,488,537,547,706,1075,476,1016,590,622,647,1099,701,457,984,721,569,779,1202,360,657,1035,558,674,1167,953,794,926,718,888,965,425,513,824,856,573,689,444,601,819,975,694,843,772,743,875,641,1139,936,382,313,787,466,335,919,540,604,1011,736,451,318,616,377,990,649,1043,890,387,631,769,943,330,1112,494,1080,345,552,848,520,462,932,1171,399,959,455,1102,922,1213,827,1076,426,531,1134,812,754,1049,1208,589,795,680,968,1181,712,1025,872,1156,414,327,954,791,1188,840,338,359,1130,1220,907,1038,370,703,613,1221,1052,1008,497,859,1094,978,465,363,931,482,1108,450,1084,517,319,1062,946,390,418,995,652,351,568,667,1193,1077,763,383,899,549,800,433,510,832,684,1116,752,447,432,410,835,865,373,564,341,358,897,635,1065,696,1056,664,400,1197,928,415,395,987,596,528,644,459,992,691,442,501,332,1129,369,607,803,469,354,1184,491,882,775,560,454,1069,1014,972]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11  % Problem  : ITP126^2 : TPTP v8.2.0. Released v7.5.0.
% 0.06/0.13  % Command  : run_Leo-III %s %d
% 0.13/0.34  % Computer : n004.cluster.edu
% 0.13/0.34  % Model    : x86_64 x86_64
% 0.13/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34  % Memory   : 8042.1875MB
% 0.13/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34  % CPULimit : 300
% 0.13/0.34  % WCLimit  : 300
% 0.13/0.34  % DateTime : Sat May 18 18:31:24 EDT 2024
% 0.13/0.34  % CPUTime  : 
% 1.03/0.98  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 1.97/1.35  % [INFO] 	 Parsing done (364ms). 
% 2.18/1.36  % [INFO] 	 Running in sequential loop mode. 
% 3.16/1.73  % [INFO] 	 eprover registered as external prover. 
% 3.16/1.73  % [INFO] 	 cvc4 registered as external prover. 
% 3.16/1.74  % [INFO] 	 Scanning for conjecture ... 
% 3.51/1.89  % [INFO] 	 Found a conjecture (or negated_conjecture) and 311 axioms. Running axiom selection ... 
% 3.87/2.03  % [INFO] 	 Axiom selection finished. Selected 310 axioms (removed 1 axioms). 
% 4.85/2.24  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 4.85/2.26  % [INFO] 	 Type checking passed. 
% 4.85/2.26  % [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 ... 
% 78.78/38.07  % External prover 'cvc4' found a proof!
% 78.78/38.07  % [INFO] 	 Killing All external provers ... 
% 78.78/38.08  % Time passed: 37577ms (effective reasoning time: 36710ms)
% 78.78/38.08  % 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)>
% 78.78/38.08  % Axioms used in derivation (310): fact_207_add__is__1, fact_46_nat_OdiscI, fact_216_list_Osize_I4_J, tcon_Monitor__Mirabelle__pzlrlsievl_Omformula___Nat_Osize_28, fact_68_Ex__less__Suc, fact_231_nat__int__comparison_I1_J, fact_219_pos__int__cases, fact_96_fvi__Suc__bound, fact_142_add__Suc__right, fact_71_Suc__lessI, fact_93_Skolem__list__nth, fact_174_drop__0, tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, fact_171_add__Suc__shift, fact_144_drop0, fact_159_drop__Suc__Cons, fact_85_less__Suc__eq__0__disj, tcon_Nat_Onat___Groups_Oab__semigroup__add_13, fact_238_triangle__0, tcon_Nat_Onat___Nat_Osize, fact_128_zero__eq__add__iff__both__eq__0, fact_206_nth__via__drop, fact_37_Zero__neq__Suc, fact_94_nth__equalityI, fact_209_less__imp__add__positive, tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add, fact_244_mformula_Osize__gen_I7_J, fact_24_nat__less__induct, fact_76_combine__options__cases, fact_81_option_OdiscI, fact_80_option_Oexhaust, fact_129_add__eq__0__iff__both__eq__0, fact_246_mformula_Oinject_I5_J, tcon_Int_Oint___Groups_Ozero, fact_169_add__Suc, fact_0_nth__Cons__Suc, fact_5_not__Some__eq, fact_224_plus__int__code_I1_J, tcon_MFOTL_Oformula___Nat_Osize_26, fact_54_less__zeroE, fact_53_gr__implies__not0, tcon_Interval_O_092_060I_062___Orderings_Oorder_25, fact_16_measure__induct__rule, fact_184_add_Ocomm__neutral, fact_195_add_Oleft__commute, fact_90_length__Suc__conv, tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_5, fact_9_neq0__conv, fact_39_zero__induct, fact_84_lift__Suc__mono__less, tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_10, fact_208_one__is__add, fact_201_of__nat__less__imp__less, fact_10_less__nat__zero__code, tcon_Nat_Onat___Groups_Ozero_20, fact_177_add__strict__right__mono, fact_167_add__less__mono, fact_139_add__less__cancel__left, fact_127_of__nat__eq__iff, fact_23_infinite__descent, fact_27_less__not__refl2, fact_214_less__imp__Suc__add, fact_19_n__not__Suc__n, fact_88_gr0__conv__Suc, fact_162_trans__less__add2, fact_136_add_Oright__neutral, fact_189_group__cancel_Oadd1, fact_187_ab__semigroup__add__class_Oadd__ac_I1_J, fact_197_add__right__imp__eq, fact_14_nat_Oinject, fact_186_reals__Archimedean2, fact_69_less__SucI, fact_92_list__eq__iff__nth__eq, fact_192_add_Oleft__cancel, fact_217_add__less__zeroD, fact_239_mformula_Osize_I15_J, fact_149_less__add__same__cancel1, fact_212_less__add__Suc2, fact_155_of__nat__eq__0__iff, fact_104_dependent__nat__choice, fact_191_add_Oassoc, tcon_Int_Oint___Groups_Oordered__comm__monoid__add, fact_196_add__left__imp__eq, fact_202_add__neg__neg, fact_204_canonically__ordered__monoid__add__class_OlessE, fact_30_size__neq__size__imp__neq, fact_55_not__less0, fact_225_zless__iff__Suc__zadd, fact_156_of__nat__less__iff, tcon_Int_Oint___Groups_Ocancel__semigroup__add, fact_3_zero__less__Suc, fact_138_add__less__cancel__right, fact_241_mformula_Oinject_I6_J, fact_109_zero__reorient, fact_15_option_Oinject, fact_99_fvi__Suc, tcon_Int_Oint___Groups_Omonoid__add, tcon_Int_Oint___Orderings_Oorder_1, fact_41_nat__induct, fact_73_Suc__lessD, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_6, tcon_Int_Oint___Groups_Osemigroup__add, fact_215_gen__length__def, fact_117_less__numeral__extra_I3_J, tcon_Int_Oint___Groups_Oab__semigroup__add, tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_7, tcon_HOL_Obool___Orderings_Oorder_22, tcon_Nat_Onat___Groups_Osemigroup__add_15, fact_131_add__cancel__right__left, fact_12_list_Oinject, fact_130_add__cancel__right__right, fact_163_trans__less__add1, fact_106_trm_Osize__neq, fact_52_infinite__descent0, fact_235_int__plus, fact_33_not0__implies__Suc, fact_44_Collect__cong, fact_133_add__cancel__left__left, fact_89_Ex__less__Suc2, fact_111_gr__zeroI, fact_32_Ex__list__of__length, fact_181_add__mono__thms__linordered__field_I2_J, fact_74_Nat_OlessE, fact_26_less__not__refl3, fact_124_of__nat__0__less__iff, help_If_1_1_T, fact_122_Cons__nth__drop__Suc, fact_123_list__ex__length, fact_79_option_Oinducts, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_9, fact_56_not__gr0, fact_105_formula_Osize__neq, fact_226_less__int__code_I1_J, tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, fact_194_add_Ocommute, fact_112_not__less__zero, tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le, fact_83_lift__Suc__mono__less__iff, fact_121_option_Osize__gen_I2_J, fact_151_double__add__less__zero__iff__single__add__less__zero, fact_114_zero__less__iff__neq__zero, fact_211_less__add__Suc1, fact_66_not__less__eq, fact_120_gen__length__code_I2_J, tcon_MFOTL_Otrm___Nat_Osize_24, fact_102_find__Some__iff, fact_145_drop__drop, tcon_Int_Oint___Rings_Olinordered__idom, fact_8_Suc__less__eq, fact_86_gr0__implies__Suc, fact_4_not__None__eq, fact_59_strict__inc__induct, tcon_Nat_Onat___Orderings_Owellorder, fact_7_Suc__mono, fact_60_less__Suc__induct, fact_178_add__strict__left__mono, fact_98_option_Osize__gen_I1_J, fact_249_mformula_Osize__gen_I5_J, fact_61_less__trans__Suc, fact_140_add__is__0, fact_242_mformula_Oinject_I7_J, fact_232_int__if, fact_166_not__add__less1, fact_17_measure__induct, fact_175_add__less__imp__less__right, fact_218_Euclid__induct, tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_4, tcon_Nat_Onat___Nat_Osemiring__char__0_16, tcon_List_Olist___Nat_Osize_23, fact_240_mformula_Osize_I16_J, fact_150_less__add__same__cancel2, fact_126_add__right__cancel, fact_64_Suc__less__eq2, fact_223_plus__int__code_I2_J, fact_25_less__irrefl__nat, fact_75_length__induct, fact_62_Suc__less__SucD, fact_152_zero__less__double__add__iff__zero__less__single__add, fact_108_option_Osize__neq, fact_43_Collect__mem__eq, tcon_Int_Oint___Groups_Oordered__ab__semigroup__add, fact_172_add__eq__self__zero, tcon_Option_Ooption___Nat_Osize_27, fact_193_add_Oright__cancel, tcon_Int_Oint___Rings_Osemiring__1, fact_157_of__nat__add, fact_248_mformula_Odistinct_I55_J, fact_18_not__Cons__self2, tcon_Int_Oint___Groups_Ocomm__monoid__add, fact_134_double__zero__sym, fact_35_old_Onat_Oexhaust, fact_29_nat__neq__iff, fact_143_nat__add__left__cancel__less, fact_91_Suc__length__conv, fact_78_split__option__ex, fact_77_split__option__all, fact_135_double__zero, fact_168_add__lessD1, fact_119_length__code, tcon_Int_Oint___Groups_Ogroup__add, tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_11, fact_243_mformula_Odistinct_I61_J, fact_22_linorder__neqE__nat, fact_179_add__strict__mono, fact_36_Zero__not__Suc, fact_227_linorder__neqE__linordered__idom, fact_229_int__ops_I1_J, help_If_3_1_T, fact_13_old_Onat_Oinject, fact_228_nat__int__comparison_I2_J, help_If_2_1_T, fact_158_add__gr__0, fact_147_add__less__same__cancel1, tcon_Int_Oint___Num_Oneg__numeral, fact_82_option_Odistinct_I1_J, fact_110_find_Osimps_I2_J, fact_118_field__lbound__gt__zero, tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_8, fact_72_Suc__lessE, fact_222_int__int__eq, fact_176_add__less__imp__less__left, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_3, fact_230_verit__comp__simplify1_I1_J, fact_1_nth__Cons__0, fact_173_plus__nat_Oadd__0, fact_234_int__ops_I5_J, tcon_Int_Oint___Groups_Olinordered__ab__group__add, fact_164_add__less__mono1, tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le, fact_49_nat_Odistinct_I1_J, tcon_Set_Oset___Orderings_Oorder_21, fact_70_less__SucE, fact_21_infinite__descent__measure, fact_97_length__Cons, fact_100_option_Osize_I4_J, fact_51_bot__nat__0_Oextremum__strict, fact_20_Suc__inject, fact_58_not__less__less__Suc__eq, fact_190_group__cancel_Oadd2, conj_1, fact_11_bot__nat__0_Onot__eq__extremum, tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_2, fact_95_not__gr__zero, fact_57_gr0I, fact_65_All__less__Suc, fact_185_comm__monoid__add__class_Oadd__0, fact_87_All__less__Suc2, fact_141_Nat_Oadd__0__right, fact_237_add__0__iff, fact_47_old_Onat_Odistinct_I1_J, tcon_Nat_Onat___Groups_Ocomm__monoid__add_14, fact_251_nth__drop, tcon_Nat_Onat___Rings_Osemiring__1_18, fact_233_verit__sum__simplify, tcon_Int_Oint___Nat_Osemiring__char__0, fact_63_less__antisym, fact_247_mformula_Odistinct_I53_J, fact_146_list__ex__simps_I1_J, fact_113_gr__implies__not__zero, tcon_fun___Orderings_Oorder, fact_40_diff__induct, fact_2_less__Suc0, fact_182_add__mono__thms__linordered__field_I5_J, fact_183_add_Ogroup__left__neutral, fact_137_add_Oleft__neutral, fact_50_infinite__descent0__measure, fact_116_list__decode_Ocases, fact_148_add__less__same__cancel2, fact_213_less__iff__Suc__add, fact_154_of__nat__0__eq__iff, fact_125_add__left__cancel, fact_31_neq__if__length__neq, fact_107_mformula_Osize__neq, fact_180_add__mono__thms__linordered__field_I1_J, fact_221_zadd__int__left, fact_153_of__nat__0, tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, fact_28_less__not__refl, fact_210_less__natE, fact_67_less__Suc__eq, fact_38_Suc__neq__Zero, fact_250_formula_Osize_I11_J, fact_6_lessI, tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add, fact_160_is__num__normalize_I1_J, fact_199_of__nat__neq__0, fact_42_mem__Collect__eq, fact_236_triangle__Suc, fact_220_zero__less__imp__eq__int, tcon_Nat_Onat___Groups_Ocancel__semigroup__add_12, fact_198_of__nat__less__0__iff, fact_132_add__cancel__left__right, fact_101_option_Osize_I3_J, tcon_Nat_Onat___Orderings_Oorder_19, fact_48_old_Onat_Odistinct_I2_J, conj_0, fact_203_add__pos__pos, fact_205_pos__add__strict, tcon_Nat_Onat___Groups_Omonoid__add_17, fact_103_exists__least__lemma, fact_165_not__add__less2, tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, fact_188_add__mono__thms__linordered__semiring_I4_J, tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add, fact_245_mformula_Osize_I14_J, fact_200_less__imp__of__nat__less, fact_115_BNF__Greatest__Fixpoint_Oshift__def, fact_34_old_Onat_Oinducts, fact_161_less__add__eq__less, fact_170_nat__arith_Osuc1
% 78.78/38.08  % No. of inferences in proof: 624
% 78.78/38.08  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 37577 ms resp. 36710 ms w/o parsing
% 79.01/38.24  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 79.01/38.24  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------