TSTP Solution File: ITP131^2 by Leo-III---1.7.12

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%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : ITP131^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n016.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 22:23:13 EDT 2024

% Result   : Theorem 59.37s 21.24s
% Output   : Refutation 60.01s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    7
%            Number of leaves      :  309
% Syntax   : Number of formulae    :  607 ( 194 unt;  35 typ;   0 def)
%            Number of atoms       : 1513 ( 580 equ;   2 cnn)
%            Maximal formula atoms :    4 (   2 avg)
%            Number of connectives : 6002 ( 108   ~;  33   |;  44   &;5261   @)
%                                         (   0 <=>; 556  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   15 (   6 avg)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :   48 (  48   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   37 (  34 usr;   5 con; 0-3 aty)
%            Number of variables   : 1594 (  58   ^1489   !;  18   ?;1594   :)
%                                         (  29  !>;   0  ?*;   0  @-;   0  @+)

% Comments : 
%------------------------------------------------------------------------------
thf(nat_type,type,
    nat: $tType ).

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

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

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

thf(ab_group_add_type,type,
    ab_group_add: 
      !>[TA: $tType] : $o ).

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

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

thf(semigroup_mult_type,type,
    semigroup_mult: 
      !>[TA: $tType] : $o ).

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

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

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

thf(ab_semigroup_mult_type,type,
    ab_semigroup_mult: 
      !>[TA: $tType] : $o ).

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

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

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

thf(cancel146912293up_add_type,type,
    cancel146912293up_add: 
      !>[TA: $tType] : $o ).

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

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

thf(canoni770627133id_add_type,type,
    canoni770627133id_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(ordere623563068d_diff_type,type,
    ordere623563068d_diff: 
      !>[TA: $tType] : $o ).

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

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

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

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

thf(times_times_type,type,
    times_times: 
      !>[TA: $tType] : ( TA > TA > TA ) ).

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

thf(number2016821345itions_type,type,
    number2016821345itions: ( nat > nat ) > nat > $o ).

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(i_type,type,
    i: nat ).

thf(k_type,type,
    k: nat ).

thf(n_type,type,
    n: nat ).

thf(p_type,type,
    p: nat > nat ).

thf(130,axiom,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( minus_minus @ TA @ ( plus_plus @ TA @ B @ C ) @ A )
            = ( plus_plus @ TA @ ( minus_minus @ TA @ B @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_127_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2) ).

thf(645,plain,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( minus_minus @ TA @ ( plus_plus @ TA @ B @ C ) @ A )
            = ( plus_plus @ TA @ ( minus_minus @ TA @ B @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(125,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] : ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_122_le__iff__diff__le__0) ).

thf(629,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] : ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(37,axiom,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = ( plus_plus @ TA @ A @ A ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_34_double__zero__sym) ).

thf(365,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)],[37]) ).

thf(164,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/sandbox2/benchmark/theBenchmark.p',fact_161_measure__induct__rule) ).

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

thf(31,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/sandbox2/benchmark/theBenchmark.p',fact_28_zero__eq__add__iff__both__eq__0) ).

thf(347,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)],[31]) ).

thf(206,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_203_add__left__mono) ).

thf(893,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(169,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_add @ TA )
     => ( ( plus_plus @ TA )
        = ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_166_add_Ocommute) ).

thf(775,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)],[169]) ).

thf(776,plain,
    ! [TA: $tType] :
      ( ~ ( ab_semigroup_add @ TA )
      | ( ( plus_plus @ TA )
        = ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
    inference(cnf,[status(esa)],[775]) ).

thf(777,plain,
    ! [TA: $tType] :
      ( ( ( plus_plus @ TA )
        = ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) )
      | ~ ( ab_semigroup_add @ TA ) ),
    inference(lifteq,[status(thm)],[776]) ).

thf(225,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B
       != ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_222_gr__implies__not0) ).

thf(962,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B
       != ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(72,axiom,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ A ) @ ( zero_zero @ TA ) )
          = ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_69_double__add__le__zero__iff__single__add__le__zero) ).

thf(468,plain,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ A ) @ ( zero_zero @ TA ) )
          = ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(143,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ A )
       => ( ord_less @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_140_diff__less__mono) ).

thf(683,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ A )
       => ( ord_less @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(1,conjecture,
    $false,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).

thf(2,negated_conjecture,
    ~ $false,
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(276,plain,
    ~ $false,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(268,axiom,
    comm_monoid_diff @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__diff) ).

thf(1068,plain,
    comm_monoid_diff @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(18,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
        & ( ord_less @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_15_nat__mult__less__cancel__disj) ).

thf(308,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
        & ( ord_less @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(200,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_197_gr__zeroI) ).

thf(878,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)],[200]) ).

thf(159,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ D )
          = ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ E ) )
        = ( D
          = ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_156_nat__eq__add__iff2) ).

thf(741,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ D )
          = ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ E ) )
        = ( D
          = ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(273,axiom,
    wellorder @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Owellorder) ).

thf(1073,plain,
    wellorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(120,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus @ nat @ ( plus_plus @ nat @ A @ B ) @ A )
      = B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_117_diff__add__inverse) ).

thf(614,plain,
    ! [A: nat,B: nat] :
      ( ( minus_minus @ nat @ ( plus_plus @ nat @ A @ B ) @ A )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(141,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
       => ( ord_less @ nat @ ( minus_minus @ nat @ B @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_138_diff__less) ).

thf(677,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
       => ( ord_less @ nat @ ( minus_minus @ nat @ B @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(100,axiom,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( minus_minus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_97_diff__diff__add) ).

thf(550,plain,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( minus_minus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(152,axiom,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ ( minus_minus @ nat @ B @ C ) )
      = ( ~ ( ( ( ord_less @ nat @ B @ C )
              & ~ ( A @ ( zero_zero @ nat ) ) )
            | ? [D: nat] :
                ( ( B
                  = ( plus_plus @ nat @ C @ D ) )
                & ~ ( A @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_149_nat__diff__split__asm) ).

thf(715,plain,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ ( minus_minus @ nat @ B @ C ) )
      = ( ~ ( ( ( ord_less @ nat @ B @ C )
              & ~ ( A @ ( zero_zero @ nat ) ) )
            | ? [D: nat] :
                ( ( B
                  = ( plus_plus @ nat @ C @ D ) )
                & ~ ( A @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(181,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/sandbox2/benchmark/theBenchmark.p',fact_178_infinite__descent__measure) ).

thf(814,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)],[181]) ).

thf(213,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
          = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_210_comm__monoid__add__class_Oadd__0) ).

thf(913,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)],[213]) ).

thf(263,axiom,
    ordere779506340up_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add) ).

thf(1063,plain,
    ordere779506340up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(220,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/sandbox2/benchmark/theBenchmark.p',fact_217_add__mono__thms__linordered__field_I2_J) ).

thf(930,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)],[220]) ).

thf(50,axiom,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( minus_minus @ TA @ ( plus_plus @ TA @ A @ B ) @ A )
          = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_47_add__diff__cancel__left_H) ).

thf(404,plain,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( minus_minus @ TA @ ( plus_plus @ TA @ A @ B ) @ A )
          = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(188,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_185_less__not__refl) ).

thf(841,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(203,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] :
            ? [C: TA] :
              ( B
              = ( plus_plus @ TA @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_200_le__iff__add) ).

thf(885,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] :
            ? [C: TA] :
              ( B
              = ( plus_plus @ TA @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(10,axiom,
    i != k,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_7__092_060open_062i_A_092_060noteq_062_Ak_092_060close_062) ).

thf(286,plain,
    i != k,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(287,plain,
    i != k,
    inference(polarity_switch,[status(thm)],[286]) ).

thf(288,plain,
    k != i,
    inference(lifteq,[status(thm)],[287]) ).

thf(132,axiom,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( minus_minus @ TA @ ( plus_plus @ TA @ C @ B ) @ A )
            = ( plus_plus @ TA @ C @ ( minus_minus @ TA @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_129_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc) ).

thf(651,plain,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( minus_minus @ TA @ ( plus_plus @ TA @ C @ B ) @ A )
            = ( plus_plus @ TA @ C @ ( minus_minus @ TA @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(22,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/sandbox2/benchmark/theBenchmark.p',fact_19_less__add__same__cancel1) ).

thf(320,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)],[22]) ).

thf(105,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
            = ( minus_minus @ TA @ B @ C ) )
          = ( ( plus_plus @ TA @ A @ C )
            = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_102_eq__diff__eq) ).

thf(565,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
            = ( minus_minus @ TA @ B @ C ) )
          = ( ( plus_plus @ TA @ A @ C )
            = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(45,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_42_diff__self) ).

thf(389,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(84,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ C @ ( minus_minus @ nat @ B @ A ) )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ C @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_81_Nat_Odiff__diff__right) ).

thf(504,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ C @ ( minus_minus @ nat @ B @ A ) )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ C @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(112,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ B ) @ ( minus_minus @ nat @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_109_diff__le__mono2) ).

thf(587,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ B ) @ ( minus_minus @ nat @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(221,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/sandbox2/benchmark/theBenchmark.p',fact_218_add__mono__thms__linordered__field_I5_J) ).

thf(934,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)],[221]) ).

thf(259,axiom,
    ordere623563068d_diff @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff) ).

thf(1059,plain,
    ordere623563068d_diff @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(177,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_mult @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) )
          = ( times_times @ TA @ B @ ( times_times @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_174_mult_Oleft__commute) ).

thf(802,plain,
    ! [TA: $tType] :
      ( ( ab_semigroup_mult @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) )
          = ( times_times @ TA @ B @ ( times_times @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(30,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
          = ( ord_less_eq @ TA @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_27_add__le__cancel__left) ).

thf(344,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
          = ( ord_less_eq @ TA @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(182,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ~ ( ord_less @ nat @ A @ B )
       => ( ord_less @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_179_linorder__neqE__nat) ).

thf(817,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)],[182]) ).

thf(103,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
          = ( minus_minus @ TA @ ( plus_plus @ TA @ A @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_100_diff__diff__eq2) ).

thf(559,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
          = ( minus_minus @ TA @ ( plus_plus @ TA @ A @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(191,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
      | ( ord_less_eq @ nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_188_nat__le__linear) ).

thf(853,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
      | ( ord_less_eq @ nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(55,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_52_less__nat__zero__code) ).

thf(419,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(420,plain,
    ~ ? [A: nat] : ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(miniscope,[status(thm)],[419]) ).

thf(421,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(cnf,[status(esa)],[420]) ).

thf(227,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_224_not__less0) ).

thf(969,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(11,axiom,
    ord_less @ nat @ ( minus_minus @ nat @ n @ k ) @ i,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_8__092_060open_062n_A_N_Ak_A_060_Ai_092_060close_062) ).

thf(289,plain,
    ord_less @ nat @ ( minus_minus @ nat @ n @ k ) @ i,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(242,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/sandbox2/benchmark/theBenchmark.p',fact_239_less__add__eq__less) ).

thf(1013,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)],[242]) ).

thf(8,axiom,
    ( ( p @ i )
   != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_5__092_060open_062p_Ai_A_092_060noteq_062_A0_092_060close_062) ).

thf(282,plain,
    ( ( p @ i )
   != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(283,plain,
    ( ( p @ i )
   != ( zero_zero @ nat ) ),
    inference(polarity_switch,[status(thm)],[282]) ).

thf(284,plain,
    ( ( p @ i )
   != ( zero_zero @ nat ) ),
    inference(lifteq,[status(thm)],[283]) ).

thf(201,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
         => ( ord_less_eq @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_198_add__le__imp__le__right) ).

thf(881,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
         => ( ord_less_eq @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(54,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_51_bot__nat__0_Onot__eq__extremum) ).

thf(416,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(218,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/sandbox2/benchmark/theBenchmark.p',fact_215_add__strict__mono) ).

thf(924,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)],[218]) ).

thf(80,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_diff @ TA )
     => ! [A: TA,B: TA] :
          ( ( minus_minus @ TA @ A @ ( plus_plus @ TA @ A @ B ) )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_77_diff__add__zero) ).

thf(492,plain,
    ! [TA: $tType] :
      ( ( comm_monoid_diff @ TA )
     => ! [A: TA,B: TA] :
          ( ( minus_minus @ TA @ A @ ( plus_plus @ TA @ A @ B ) )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(251,axiom,
    ! [A: nat] :
      ( ( times_times @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_248_mult__0) ).

thf(1040,plain,
    ! [A: nat] :
      ( ( times_times @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(123,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times @ nat @ A @ ( minus_minus @ nat @ B @ C ) )
      = ( minus_minus @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_120_diff__mult__distrib2) ).

thf(623,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times @ nat @ A @ ( minus_minus @ nat @ B @ C ) )
      = ( minus_minus @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(244,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_241_trans__less__add1) ).

thf(1020,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)],[244]) ).

thf(255,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_252_add__le__mono1) ).

thf(1051,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(76,axiom,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ A )
          = ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_73_add__le__same__cancel1) ).

thf(480,plain,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ A )
          = ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(33,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
            = ( plus_plus @ TA @ A @ B ) )
          = ( B
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_30_add__cancel__right__right) ).

thf(353,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)],[33]) ).

thf(226,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_223_less__zeroE) ).

thf(966,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(6,axiom,
    ord_less @ nat @ n @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_3_lower__bound) ).

thf(280,plain,
    ord_less @ nat @ n @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(13,axiom,
    ! [A: nat > nat,B: nat,C: nat] :
      ( ( number2016821345itions @ A @ B )
     => ( ord_less_eq @ nat @ ( A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_10_partitions__bounds) ).

thf(293,plain,
    ! [A: nat > nat,B: nat,C: nat] :
      ( ( number2016821345itions @ A @ B )
     => ( ord_less_eq @ nat @ ( A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(135,axiom,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ C @ ( minus_minus @ TA @ ( plus_plus @ TA @ B @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_132_le__add__diff) ).

thf(660,plain,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ C @ ( minus_minus @ TA @ ( plus_plus @ TA @ B @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(231,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_228_bot__nat__0_Oextremum__unique) ).

thf(981,plain,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(166,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/sandbox2/benchmark/theBenchmark.p',fact_163_add__right__imp__eq) ).

thf(766,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)],[166]) ).

thf(272,axiom,
    semigroup_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Osemigroup__add) ).

thf(1072,plain,
    semigroup_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(67,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ( B
          = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_64_mult__is__0) ).

thf(453,plain,
    ! [A: nat,B: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ( B
          = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(274,axiom,
    monoid_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Omonoid__add) ).

thf(1074,plain,
    monoid_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(216,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/sandbox2/benchmark/theBenchmark.p',fact_213_add__strict__right__mono) ).

thf(920,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)],[216]) ).

thf(42,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_39_diff__zero) ).

thf(380,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(86,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( plus_plus @ nat @ C @ ( minus_minus @ nat @ B @ A ) )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ C @ B ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_83_Nat_Oadd__diff__assoc) ).

thf(512,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( plus_plus @ nat @ C @ ( minus_minus @ nat @ B @ A ) )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ C @ B ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(99,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = C )
         => ( A
            = ( minus_minus @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_96_add__implies__diff) ).

thf(546,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = C )
         => ( A
            = ( minus_minus @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(94,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( minus_minus @ TA @ A @ B )
              = ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_91_eq__iff__diff__eq__0) ).

thf(534,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( minus_minus @ TA @ A @ B )
              = ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(267,axiom,
    ab_semigroup_mult @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oab__semigroup__mult) ).

thf(1067,plain,
    ab_semigroup_mult @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(257,axiom,
    ordere516151231imp_le @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le) ).

thf(1057,plain,
    ordere516151231imp_le @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(212,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_209_add_Ocomm__neutral) ).

thf(910,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)],[212]) ).

thf(184,axiom,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ! [D: nat] :
              ( ( ord_less @ nat @ D @ C )
             => ( A @ D ) )
         => ( A @ C ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_181_nat__less__induct) ).

thf(825,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)],[184]) ).

thf(170,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/sandbox2/benchmark/theBenchmark.p',fact_167_add_Oright__cancel) ).

thf(778,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)],[170]) ).

thf(16,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ C @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
        & ( ord_less @ nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_13_mult__less__cancel2) ).

thf(302,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ C @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
        & ( ord_less @ nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(194,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_191_le__trans) ).

thf(862,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(183,axiom,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ~ ( A @ C )
         => ? [D: nat] :
              ( ( ord_less @ nat @ D @ C )
              & ~ ( A @ D ) ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_180_infinite__descent) ).

thf(820,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)],[183]) ).

thf(46,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/sandbox2/benchmark/theBenchmark.p',fact_43_add__less__cancel__right) ).

thf(392,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)],[46]) ).

thf(59,axiom,
    ! [A: nat] :
      ( ( plus_plus @ nat @ A @ ( zero_zero @ nat ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_56_Nat_Oadd__0__right) ).

thf(429,plain,
    ! [A: nat] :
      ( ( plus_plus @ nat @ A @ ( zero_zero @ nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(156,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ D ) @ ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ E ) )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_153_nat__diff__add__eq1) ).

thf(729,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ D ) @ ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ E ) )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(51,axiom,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
          = ( minus_minus @ TA @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_48_add__diff__cancel__left) ).

thf(407,plain,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
          = ( minus_minus @ TA @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(146,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ ( minus_minus @ nat @ B @ C ) )
      = ( ord_less @ nat @ ( plus_plus @ nat @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_143_less__diff__conv) ).

thf(693,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ ( minus_minus @ nat @ B @ C ) )
      = ( ord_less @ nat @ ( plus_plus @ nat @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(214,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/sandbox2/benchmark/theBenchmark.p',fact_211_add__less__imp__less__right) ).

thf(916,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)],[214]) ).

thf(190,axiom,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ D )
           => ( ord_less_eq @ nat @ D @ C ) )
       => ? [D: nat] :
            ( ( A @ D )
            & ! [E: nat] :
                ( ( A @ E )
               => ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_187_Nat_Oex__has__greatest__nat) ).

thf(847,plain,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ D )
           => ( ord_less_eq @ nat @ D @ C ) )
       => ? [D: nat] :
            ( ( A @ D )
            & ! [E: nat] :
                ( ( A @ E )
               => ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(248,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/sandbox2/benchmark/theBenchmark.p',fact_245_add__less__mono) ).

thf(1032,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)],[248]) ).

thf(61,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/sandbox2/benchmark/theBenchmark.p',fact_58_nat__add__left__cancel__less) ).

thf(435,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)],[61]) ).

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/sandbox2/benchmark/theBenchmark.p',fact_22_add__right__cancel) ).

thf(329,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(137,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
          = ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_134_le__diff__eq) ).

thf(665,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
          = ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(163,axiom,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = A )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_160_zero__reorient) ).

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

thf(68,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
      = ( ord_less_eq @ nat @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_65_nat__add__left__cancel__le) ).

thf(456,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
      = ( ord_less_eq @ nat @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(9,axiom,
    ord_less @ nat @ ( zero_zero @ nat ) @ ( p @ k ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_6_assms_I2_J) ).

thf(285,plain,
    ord_less @ nat @ ( zero_zero @ nat ) @ ( p @ k ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(89,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_86_diff__commute) ).

thf(522,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(185,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_182_less__irrefl__nat) ).

thf(830,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(21,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/sandbox2/benchmark/theBenchmark.p',fact_18_add__less__same__cancel2) ).

thf(317,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)],[21]) ).

thf(243,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_240_trans__less__add2) ).

thf(1017,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)],[243]) ).

thf(264,axiom,
    cancel146912293up_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add) ).

thf(1064,plain,
    cancel146912293up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(178,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_mult @ TA )
     => ( ( times_times @ TA )
        = ( ^ [A: TA,B: TA] : ( times_times @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_175_mult_Ocommute) ).

thf(805,plain,
    ! [TA: $tType] :
      ( ( ab_semigroup_mult @ TA )
     => ( ( times_times @ TA )
        = ( ^ [A: TA,B: TA] : ( times_times @ TA @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(806,plain,
    ! [TA: $tType] :
      ( ~ ( ab_semigroup_mult @ TA )
      | ( ( times_times @ TA )
        = ( ^ [A: TA,B: TA] : ( times_times @ TA @ B @ A ) ) ) ),
    inference(cnf,[status(esa)],[805]) ).

thf(807,plain,
    ! [TA: $tType] :
      ( ( ( times_times @ TA )
        = ( ^ [A: TA,B: TA] : ( times_times @ TA @ B @ A ) ) )
      | ~ ( ab_semigroup_mult @ TA ) ),
    inference(lifteq,[status(thm)],[806]) ).

thf(1110,plain,
    ! [TA: $tType] :
      ( ( ( times_times @ TA )
        = ( ^ [A: TA,B: TA] : ( times_times @ TA @ B @ A ) ) )
      | ( ( ab_semigroup_mult @ nat )
       != ( ab_semigroup_mult @ TA ) ) ),
    inference(paramod_ordered,[status(thm)],[1067,807]) ).

thf(1111,plain,
    ( ( times_times @ nat )
    = ( ^ [A: nat,B: nat] : ( times_times @ nat @ B @ A ) ) ),
    inference(pattern_uni,[status(thm)],[1110:[bind_type(TA,$thf( nat ))]]) ).

thf(155,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ D ) @ ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ E ) )
        = ( minus_minus @ nat @ D @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_152_nat__diff__add__eq2) ).

thf(725,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ D ) @ ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ E ) )
        = ( minus_minus @ nat @ D @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(34,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ A ) )
          = ( B
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_31_add__cancel__right__left) ).

thf(356,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)],[34]) ).

thf(197,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
          = ( A
           != ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_194_zero__less__iff__neq__zero) ).

thf(869,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)],[197]) ).

thf(234,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus @ nat @ A @ B )
        = A )
     => ( B
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_231_add__eq__self__zero) ).

thf(989,plain,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus @ nat @ A @ B )
        = A )
     => ( B
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(88,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = ( minus_minus @ TA @ C @ D ) )
         => ( ( A = B )
            = ( C = D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_85_diff__eq__diff__eq) ).

thf(519,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = ( minus_minus @ TA @ C @ D ) )
         => ( ( A = B )
            = ( C = D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(109,axiom,
    ! [A: nat,B: nat] :
      ( ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
     => ( ( ( minus_minus @ nat @ B @ A )
          = ( zero_zero @ nat ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_106_diffs0__imp__equal) ).

thf(578,plain,
    ! [A: nat,B: nat] :
      ( ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
     => ( ( ( minus_minus @ nat @ B @ A )
          = ( zero_zero @ nat ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(73,axiom,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( plus_plus @ TA @ B @ A ) )
          = ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_70_le__add__same__cancel2) ).

thf(471,plain,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( plus_plus @ TA @ B @ A ) )
          = ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(269,axiom,
    ab_semigroup_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oab__semigroup__add) ).

thf(1069,plain,
    ab_semigroup_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(1081,plain,
    ! [TA: $tType] :
      ( ( ( plus_plus @ TA )
        = ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) )
      | ( ( ab_semigroup_add @ nat )
       != ( ab_semigroup_add @ TA ) ) ),
    inference(paramod_ordered,[status(thm)],[1069,777]) ).

thf(1082,plain,
    ( ( plus_plus @ nat )
    = ( ^ [A: nat,B: nat] : ( plus_plus @ nat @ B @ A ) ) ),
    inference(pattern_uni,[status(thm)],[1081:[bind_type(TA,$thf( nat ))]]) ).

thf(162,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ D ) @ ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ E ) )
        = ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_159_nat__less__add__iff1) ).

thf(753,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ D ) @ ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ E ) )
        = ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(95,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( minus_minus @ TA @ A @ C ) @ ( minus_minus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_92_diff__strict__right__mono) ).

thf(537,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( minus_minus @ TA @ A @ C ) @ ( minus_minus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(129,axiom,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( minus_minus @ TA @ C @ ( minus_minus @ TA @ B @ A ) )
            = ( minus_minus @ TA @ ( plus_plus @ TA @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_126_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right) ).

thf(642,plain,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( minus_minus @ TA @ C @ ( minus_minus @ TA @ B @ A ) )
            = ( minus_minus @ TA @ ( plus_plus @ TA @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(79,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
          = ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_76_diff__gt__0__iff__gt) ).

thf(489,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
          = ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(113,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ B )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ B @ C ) )
          = ( ord_less_eq @ nat @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_110_le__diff__iff_H) ).

thf(590,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ B )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ B @ C ) )
          = ( ord_less_eq @ nat @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(144,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ord_less @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( ord_less @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_141_less__diff__iff) ).

thf(686,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ord_less @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( ord_less @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(275,axiom,
    zero @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ozero) ).

thf(1075,plain,
    zero @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(187,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_184_less__not__refl2) ).

thf(837,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(1097,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( p @ k ) )
     != ( ord_less @ nat @ A @ ( zero_zero @ nat ) ) ),
    inference(paramod_ordered,[status(thm)],[285,421]) ).

thf(1098,plain,
    ! [A: nat] :
      ( ( ( zero_zero @ nat )
       != A )
      | ( ( p @ k )
       != ( zero_zero @ nat ) ) ),
    inference(simp,[status(thm)],[1097]) ).

thf(1099,plain,
    ( ( p @ k )
   != ( zero_zero @ nat ) ),
    inference(simp,[status(thm)],[1098]) ).

thf(133,axiom,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( plus_plus @ TA @ C @ ( minus_minus @ TA @ B @ A ) )
            = ( minus_minus @ TA @ ( plus_plus @ TA @ C @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_130_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc) ).

thf(654,plain,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( plus_plus @ TA @ C @ ( minus_minus @ TA @ B @ A ) )
            = ( minus_minus @ TA @ ( plus_plus @ TA @ C @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(260,axiom,
    ordere223160158up_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) ).

thf(1060,plain,
    ordere223160158up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(232,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_229_le__0__eq) ).

thf(984,plain,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(154,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less @ nat @ ( minus_minus @ nat @ B @ A ) @ C )
        = ( ord_less @ nat @ B @ ( plus_plus @ nat @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_151_less__diff__conv2) ).

thf(721,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less @ nat @ ( minus_minus @ nat @ B @ A ) @ C )
        = ( ord_less @ nat @ B @ ( plus_plus @ nat @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(238,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less @ nat @ A @ B )
        | ( A = B ) )
     => ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_235_less__or__eq__imp__le) ).

thf(1002,plain,
    ! [A: nat,B: nat] :
      ( ( ( ord_less @ nat @ A @ B )
        | ( A = B ) )
     => ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(39,axiom,
    ! [TA: $tType] :
      ( ( monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_36_add_Oright__neutral) ).

thf(371,plain,
    ! [TA: $tType] :
      ( ( monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(85,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( plus_plus @ nat @ ( minus_minus @ nat @ B @ A ) @ C )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ B @ C ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_82_Nat_Oadd__diff__assoc2) ).

thf(508,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( plus_plus @ nat @ ( minus_minus @ nat @ B @ A ) @ C )
        = ( minus_minus @ nat @ ( plus_plus @ nat @ B @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(19,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/sandbox2/benchmark/theBenchmark.p',fact_16_add__gr__0) ).

thf(311,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)],[19]) ).

thf(97,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = ( minus_minus @ TA @ C @ D ) )
         => ( ( ord_less @ TA @ A @ B )
            = ( ord_less @ TA @ C @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_94_diff__eq__diff__less) ).

thf(541,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = ( minus_minus @ TA @ C @ D ) )
         => ( ( ord_less @ TA @ A @ B )
            = ( ord_less @ TA @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(134,axiom,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ ( minus_minus @ TA @ B @ A ) )
            = ( ord_less_eq @ TA @ ( plus_plus @ TA @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_131_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2) ).

thf(657,plain,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ ( minus_minus @ TA @ B @ A ) )
            = ( ord_less_eq @ TA @ ( plus_plus @ TA @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(256,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ D )
       => ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_253_add__le__mono) ).

thf(1054,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ D )
       => ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(271,axiom,
    semigroup_mult @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Osemigroup__mult) ).

thf(1071,plain,
    semigroup_mult @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(140,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( ord_less @ TA @ A @ ( plus_plus @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_137_diff__less__eq) ).

thf(674,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( ord_less @ TA @ A @ ( plus_plus @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(246,axiom,
    ! [A: nat,B: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ B ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_243_not__add__less2) ).

thf(1026,plain,
    ! [A: nat,B: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(92,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( minus_minus @ TA @ C @ B ) @ ( minus_minus @ TA @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_89_diff__left__mono) ).

thf(530,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( minus_minus @ TA @ C @ B ) @ ( minus_minus @ TA @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(219,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/sandbox2/benchmark/theBenchmark.p',fact_216_add__mono__thms__linordered__field_I1_J) ).

thf(926,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)],[219]) ).

thf(48,axiom,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( minus_minus @ TA @ ( plus_plus @ TA @ A @ B ) @ B )
          = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_45_add__diff__cancel__right_H) ).

thf(398,plain,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( minus_minus @ TA @ ( plus_plus @ TA @ A @ B ) @ B )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(249,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
     => ( ord_less @ nat @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_246_add__lessD1) ).

thf(1035,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)],[249]) ).

thf(136,axiom,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( plus_plus @ TA @ ( minus_minus @ TA @ B @ A ) @ A )
            = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_133_ordered__cancel__comm__monoid__diff__class_Odiff__add) ).

thf(662,plain,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( plus_plus @ TA @ ( minus_minus @ TA @ B @ A ) @ A )
            = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(153,axiom,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ ( minus_minus @ nat @ B @ C ) )
      = ( ( ( ord_less @ nat @ B @ C )
         => ( A @ ( zero_zero @ nat ) ) )
        & ! [D: nat] :
            ( ( B
              = ( plus_plus @ nat @ C @ D ) )
           => ( A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_150_nat__diff__split) ).

thf(718,plain,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ ( minus_minus @ nat @ B @ C ) )
      = ( ( ( ord_less @ nat @ B @ C )
         => ( A @ ( zero_zero @ nat ) ) )
        & ! [D: nat] :
            ( ( B
              = ( plus_plus @ nat @ C @ D ) )
           => ( A @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(204,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_201_add__right__mono) ).

thf(888,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(57,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_54_bot__nat__0_Oextremum) ).

thf(425,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(160,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ D )
          = ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ E ) )
        = ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ D )
          = E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_157_nat__eq__add__iff1) ).

thf(745,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ D )
          = ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ E ) )
        = ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ D )
          = E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(5,axiom,
    ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) @ n,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_2_upper__bound) ).

thf(279,plain,
    ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( p @ i ) @ i ) @ ( times_times @ nat @ ( p @ k ) @ k ) ) @ n,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(14,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ C @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
       => ( ord_less_eq @ nat @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_11_mult__le__cancel2) ).

thf(296,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ C @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
       => ( ord_less_eq @ nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(64,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( times_times @ nat @ C @ B ) )
      = ( ( A = C )
        | ( B
          = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_61_mult__cancel2) ).

thf(444,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( times_times @ nat @ C @ B ) )
      = ( ( A = C )
        | ( B
          = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(157,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ D ) @ ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ E ) )
        = ( ord_less_eq @ nat @ D @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_154_nat__le__add__iff2) ).

thf(733,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ D ) @ ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ E ) )
        = ( ord_less_eq @ nat @ D @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(111,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ C )
       => ( ord_less @ nat @ ( minus_minus @ nat @ C @ B ) @ ( minus_minus @ nat @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_108_diff__less__mono2) ).

thf(584,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ C )
       => ( ord_less @ nat @ ( minus_minus @ nat @ C @ B ) @ ( minus_minus @ nat @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(239,axiom,
    ( ( ord_less_eq @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less @ nat @ A @ B )
          | ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_236_le__eq__less__or__eq) ).

thf(1007,plain,
    ( ( ord_less_eq @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less @ nat @ A @ B )
          | ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(168,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/sandbox2/benchmark/theBenchmark.p',fact_165_add_Oleft__commute) ).

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

thf(198,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B
           != ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_195_gr__implies__not__zero) ).

thf(872,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)],[198]) ).

thf(873,plain,
    ! [TA: $tType,B: TA,A: TA] :
      ( ~ ( canoni770627133id_add @ TA )
      | ~ ( ord_less @ TA @ A @ B )
      | ( B
       != ( zero_zero @ TA ) ) ),
    inference(cnf,[status(esa)],[872]) ).

thf(874,plain,
    ! [TA: $tType,B: TA,A: TA] :
      ( ( B
       != ( zero_zero @ TA ) )
      | ~ ( canoni770627133id_add @ TA )
      | ~ ( ord_less @ TA @ A @ B ) ),
    inference(lifteq,[status(thm)],[873]) ).

thf(875,plain,
    ! [TA: $tType,A: TA] :
      ( ~ ( canoni770627133id_add @ TA )
      | ~ ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ),
    inference(simp,[status(thm)],[874]) ).

thf(7,axiom,
    ord_less_eq @ nat @ k @ n,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_4__092_060open_062k_A_092_060le_062_An_092_060close_062) ).

thf(281,plain,
    ord_less_eq @ nat @ k @ n,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(53,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( minus_minus @ TA @ ( plus_plus @ TA @ A @ B ) @ B )
          = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_50_add__diff__cancel) ).

thf(413,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( minus_minus @ TA @ ( plus_plus @ TA @ A @ B ) @ B )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(104,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
          = ( minus_minus @ TA @ ( plus_plus @ TA @ A @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_101_add__diff__eq) ).

thf(562,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
          = ( minus_minus @ TA @ ( plus_plus @ TA @ A @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(229,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_226_gr0I) ).

thf(975,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(222,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/sandbox2/benchmark/theBenchmark.p',fact_219_infinite__descent0__measure) ).

thf(936,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)],[222]) ).

thf(82,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_79_diff__is__0__eq_H) ).

thf(498,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(258,axiom,
    strict2144017051up_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add) ).

thf(1058,plain,
    strict2144017051up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(27,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_24_le__zero__eq) ).

thf(335,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(149,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ ( plus_plus @ nat @ C @ B ) @ A )
        = ( plus_plus @ nat @ C @ ( minus_minus @ nat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_146_Nat_Odiff__add__assoc) ).

thf(704,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ ( plus_plus @ nat @ C @ B ) @ A )
        = ( plus_plus @ nat @ C @ ( minus_minus @ nat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(32,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/sandbox2/benchmark/theBenchmark.p',fact_29_add__eq__0__iff__both__eq__0) ).

thf(350,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)],[32]) ).

thf(119,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus @ nat @ ( plus_plus @ nat @ A @ B ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_116_diff__add__inverse2) ).

thf(611,plain,
    ! [A: nat,B: nat] :
      ( ( minus_minus @ nat @ ( plus_plus @ nat @ A @ B ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(173,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/sandbox2/benchmark/theBenchmark.p',fact_170_group__cancel_Oadd2) ).

thf(787,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)],[173]) ).

thf(172,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/sandbox2/benchmark/theBenchmark.p',fact_169_add_Oassoc) ).

thf(784,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)],[172]) ).

thf(254,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_251_trans__le__add1) ).

thf(1048,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(41,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_38_cancel__comm__monoid__add__class_Odiff__cancel) ).

thf(377,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(36,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = B )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_33_add__cancel__left__left) ).

thf(362,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)],[36]) ).

thf(158,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ D ) @ ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ E ) )
        = ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_155_nat__le__add__iff1) ).

thf(737,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ D ) @ ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ E ) )
        = ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ D ) @ E ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(252,axiom,
    ( ( ord_less_eq @ nat )
    = ( ^ [A: nat,B: nat] :
        ? [C: nat] :
          ( B
          = ( plus_plus @ nat @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_249_nat__le__iff__add) ).

thf(1043,plain,
    ( ( ord_less_eq @ nat )
    = ( ^ [A: nat,B: nat] :
        ? [C: nat] :
          ( B
          = ( plus_plus @ nat @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(116,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( minus_minus @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( minus_minus @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_113_Nat_Odiff__diff__eq) ).

thf(599,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( minus_minus @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( minus_minus @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(241,axiom,
    ( ( ord_less @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq @ nat @ A @ B )
          & ( A != B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_238_nat__less__le) ).

thf(1011,plain,
    ( ( ord_less @ nat )
    = ( ^ [A: nat,B: nat] :
          ( ( ord_less_eq @ nat @ A @ B )
          & ( A != B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(205,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ~ ! [C: TA] :
                ( B
               != ( plus_plus @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_202_less__eqE) ).

thf(890,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ~ ! [C: TA] :
                ( B
               != ( plus_plus @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(75,axiom,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ B )
          = ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_72_add__le__same__cancel2) ).

thf(477,plain,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ B )
          = ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(180,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_mult @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( times_times @ TA @ A @ B ) @ C )
          = ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_177_ab__semigroup__mult__class_Omult__ac_I1_J) ).

thf(811,plain,
    ! [TA: $tType] :
      ( ( ab_semigroup_mult @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( times_times @ TA @ A @ B ) @ C )
          = ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(193,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_190_eq__imp__le) ).

thf(858,plain,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(23,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/sandbox2/benchmark/theBenchmark.p',fact_20_less__add__same__cancel2) ).

thf(323,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)],[23]) ).

thf(114,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_111_diff__le__self) ).

thf(594,plain,
    ! [A: nat,B: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(176,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/sandbox2/benchmark/theBenchmark.p',fact_173_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(799,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)],[176]) ).

thf(77,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
          = ( ord_less_eq @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_74_diff__ge__0__iff__ge) ).

thf(483,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
          = ( ord_less_eq @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(148,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ ( plus_plus @ nat @ B @ C ) @ A )
        = ( plus_plus @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_145_Nat_Odiff__add__assoc2) ).

thf(700,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ ( plus_plus @ nat @ B @ C ) @ A )
        = ( plus_plus @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(124,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( minus_minus @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_121_diff__mult__distrib) ).

thf(626,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( minus_minus @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(167,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/sandbox2/benchmark/theBenchmark.p',fact_164_add__left__imp__eq) ).

thf(769,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)],[167]) ).

thf(223,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_220_bot__nat__0_Oextremum__strict) ).

thf(943,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(138,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( ord_less_eq @ TA @ A @ ( plus_plus @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_135_diff__le__eq) ).

thf(668,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( ord_less_eq @ TA @ A @ ( plus_plus @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(247,axiom,
    ! [A: nat,B: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_244_not__add__less1) ).

thf(1029,plain,
    ! [A: nat,B: nat] :
      ~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(207,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_204_add__mono) ).

thf(895,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(70,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( minus_minus @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_67_diff__diff__left) ).

thf(462,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( minus_minus @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(261,axiom,
    ordere236663937imp_le @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) ).

thf(1061,plain,
    ordere236663937imp_le @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(1044,plain,
    ( ( ord_less_eq @ nat )
    = ( ^ [A: nat,B: nat] :
        ? [C: nat] :
          ( B
          = ( plus_plus @ nat @ A @ C ) ) ) ),
    inference(lifteq,[status(thm)],[1043]) ).

thf(202,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
         => ( ord_less_eq @ TA @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_199_add__le__imp__le__left) ).

thf(883,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
         => ( ord_less_eq @ TA @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(15,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
       => ( ord_less_eq @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_12_nat__mult__le__cancel__disj) ).

thf(299,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
       => ( ord_less_eq @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(62,axiom,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ A )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_59_diff__self__eq__0) ).

thf(438,plain,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ A )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(217,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/sandbox2/benchmark/theBenchmark.p',fact_214_add__strict__left__mono) ).

thf(922,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)],[217]) ).

thf(266,axiom,
    cancel_semigroup_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__semigroup__add) ).

thf(1066,plain,
    cancel_semigroup_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(4,axiom,
    number2016821345itions @ p @ n,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1_assms_I1_J) ).

thf(278,plain,
    number2016821345itions @ p @ n,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(40,axiom,
    ! [TA: $tType] :
      ( ( monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
          = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_37_add_Oleft__neutral) ).

thf(374,plain,
    ! [TA: $tType] :
      ( ( monoid_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(131,axiom,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( plus_plus @ TA @ ( minus_minus @ TA @ B @ A ) @ C )
            = ( minus_minus @ TA @ ( plus_plus @ TA @ B @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_128_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2) ).

thf(648,plain,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( plus_plus @ TA @ ( minus_minus @ TA @ B @ A ) @ C )
            = ( minus_minus @ TA @ ( plus_plus @ TA @ B @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(161,axiom,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ D ) @ ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ E ) )
        = ( ord_less @ nat @ D @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ E ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_158_nat__less__add__iff2) ).

thf(749,plain,
    ! [A: nat,B: nat,C: nat,D: nat,E: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A @ C ) @ D ) @ ( plus_plus @ nat @ ( times_times @ nat @ B @ C ) @ E ) )
        = ( ord_less @ nat @ D @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ B @ A ) @ C ) @ E ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(101,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ A @ ( plus_plus @ TA @ B @ C ) )
          = ( minus_minus @ TA @ ( minus_minus @ TA @ A @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_98_diff__add__eq__diff__diff__swap) ).

thf(553,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ A @ ( plus_plus @ TA @ B @ C ) )
          = ( minus_minus @ TA @ ( minus_minus @ TA @ A @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(175,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/sandbox2/benchmark/theBenchmark.p',fact_172_add__mono__thms__linordered__semiring_I4_J) ).

thf(795,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)],[175]) ).

thf(142,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus @ nat @ A @ ( plus_plus @ nat @ A @ B ) )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_139_diff__add__0) ).

thf(680,plain,
    ! [A: nat,B: nat] :
      ( ( minus_minus @ nat @ A @ ( plus_plus @ nat @ A @ B ) )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(121,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ C @ B ) )
      = ( minus_minus @ nat @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_118_diff__cancel2) ).

thf(617,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ C @ B ) )
      = ( minus_minus @ nat @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(87,axiom,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( minus_minus @ TA @ ( minus_minus @ TA @ A @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_84_cancel__ab__semigroup__add__class_Odiff__right__commute) ).

thf(516,plain,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( minus_minus @ TA @ ( minus_minus @ TA @ A @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(90,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = ( minus_minus @ TA @ C @ D ) )
         => ( ( ord_less_eq @ TA @ A @ B )
            = ( ord_less_eq @ TA @ C @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_87_diff__eq__diff__less__eq) ).

thf(525,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = ( minus_minus @ TA @ C @ D ) )
         => ( ( ord_less_eq @ TA @ A @ B )
            = ( ord_less_eq @ TA @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(151,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_148_le__diff__conv) ).

thf(712,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(192,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ A )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_189_le__antisym) ).

thf(855,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ A )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(145,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less @ nat @ A @ B )
     => ( ( plus_plus @ nat @ B @ ( minus_minus @ nat @ A @ B ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_142_add__diff__inverse__nat) ).

thf(690,plain,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less @ nat @ A @ B )
     => ( ( plus_plus @ nat @ B @ ( minus_minus @ nat @ A @ B ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(270,axiom,
    comm_monoid_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__add) ).

thf(1070,plain,
    comm_monoid_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(174,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/sandbox2/benchmark/theBenchmark.p',fact_171_group__cancel_Oadd1) ).

thf(791,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)],[174]) ).

thf(63,axiom,
    ! [A: nat] :
      ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_60_diff__0__eq__0) ).

thf(441,plain,
    ! [A: nat] :
      ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(28,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_25_not__gr__zero) ).

thf(338,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)],[28]) ).

thf(179,axiom,
    ! [TA: $tType] :
      ( ( semigroup_mult @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( times_times @ TA @ A @ B ) @ C )
          = ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_176_mult_Oassoc) ).

thf(808,plain,
    ! [TA: $tType] :
      ( ( semigroup_mult @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( times_times @ TA @ A @ B ) @ C )
          = ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(35,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( plus_plus @ TA @ A @ B )
            = A )
          = ( B
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_32_add__cancel__left__right) ).

thf(359,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)],[35]) ).

thf(98,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ D )
           => ( ord_less @ TA @ ( minus_minus @ TA @ A @ D ) @ ( minus_minus @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_95_diff__strict__mono) ).

thf(544,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ D )
           => ( ord_less @ TA @ ( minus_minus @ TA @ A @ D ) @ ( minus_minus @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(211,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
          = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_208_add_Ogroup__left__neutral) ).

thf(907,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( zero_zero @ TA ) @ A )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(245,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/sandbox2/benchmark/theBenchmark.p',fact_242_add__less__mono1) ).

thf(1023,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)],[245]) ).

thf(1008,plain,
    ( ( ^ [A: nat,B: nat] :
          ( ( ord_less @ nat @ A @ B )
          | ( A = B ) ) )
    = ( ord_less_eq @ nat ) ),
    inference(lifteq,[status(thm)],[1007]) ).

thf(20,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/sandbox2/benchmark/theBenchmark.p',fact_17_add__less__same__cancel1) ).

thf(314,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)],[20]) ).

thf(107,axiom,
    ! [TA: $tType] :
      ( ( ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ C ) )
         => ( ( minus_minus @ TA @ A @ D )
            = ( plus_plus @ TA @ B @ ( minus_minus @ TA @ C @ D ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_104_group__cancel_Osub1) ).

thf(571,plain,
    ! [TA: $tType] :
      ( ( ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( A
            = ( plus_plus @ TA @ B @ C ) )
         => ( ( minus_minus @ TA @ A @ D )
            = ( plus_plus @ TA @ B @ ( minus_minus @ TA @ C @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(199,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_196_not__less__zero) ).

thf(876,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(230,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
     => ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_227_bot__nat__0_Oextremum__uniqueI) ).

thf(978,plain,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
     => ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(44,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_41_diff__0__right) ).

thf(386,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(71,axiom,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ A ) )
          = ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_68_zero__le__double__add__iff__zero__le__single__add) ).

thf(465,plain,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( plus_plus @ TA @ A @ A ) )
          = ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(66,axiom,
    ! [A: nat] :
      ( ( times_times @ nat @ A @ ( zero_zero @ nat ) )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_63_mult__0__right) ).

thf(450,plain,
    ! [A: nat] :
      ( ( times_times @ nat @ A @ ( zero_zero @ nat ) )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(102,axiom,
    ! [TA: $tType] :
      ( ( ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( minus_minus @ TA @ ( plus_plus @ TA @ A @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_99_diff__add__eq) ).

thf(556,plain,
    ! [TA: $tType] :
      ( ( ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( plus_plus @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( minus_minus @ TA @ ( plus_plus @ TA @ A @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(96,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( minus_minus @ TA @ C @ B ) @ ( minus_minus @ TA @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_93_diff__strict__left__mono) ).

thf(539,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( minus_minus @ TA @ C @ B ) @ ( minus_minus @ TA @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(139,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
          = ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_136_less__diff__eq) ).

thf(671,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
          = ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(210,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
            & ( C = D ) )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_207_add__mono__thms__linordered__semiring_I3_J) ).

thf(903,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
            & ( C = D ) )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(110,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_107_less__imp__diff__less) ).

thf(581,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(81,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_78_zero__less__diff) ).

thf(495,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(38,axiom,
    ! [TA: $tType] :
      ( ( linord219039673up_add @ TA )
     => ! [A: TA] :
          ( ( ( plus_plus @ TA @ A @ A )
            = ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_35_double__zero) ).

thf(368,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)],[38]) ).

thf(189,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
      = ( ( ord_less @ nat @ A @ B )
        | ( ord_less @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_186_nat__neq__iff) ).

thf(844,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)],[189]) ).

thf(262,axiom,
    canoni770627133id_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add) ).

thf(1062,plain,
    canoni770627133id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(224,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/sandbox2/benchmark/theBenchmark.p',fact_221_infinite__descent0) ).

thf(946,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)],[224]) ).

thf(126,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ( ( ord_less @ TA )
        = ( ^ [A: TA,B: TA] : ( ord_less @ TA @ ( minus_minus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_123_less__iff__diff__less__0) ).

thf(632,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ( ( ord_less @ TA )
        = ( ^ [A: TA,B: TA] : ( ord_less @ TA @ ( minus_minus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(49,axiom,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
          = ( minus_minus @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_46_add__diff__cancel__right) ).

thf(401,plain,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
          = ( minus_minus @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(171,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/sandbox2/benchmark/theBenchmark.p',fact_168_add_Oleft__cancel) ).

thf(781,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)],[171]) ).

thf(236,axiom,
    ! [A: nat > nat,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less @ nat @ D @ E )
         => ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ ( A @ B ) @ ( A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_233_less__mono__imp__le__mono) ).

thf(995,plain,
    ! [A: nat > nat,B: nat,C: nat] :
      ( ! [D: nat,E: nat] :
          ( ( ord_less @ nat @ D @ E )
         => ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ ( A @ B ) @ ( A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(106,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = C )
          = ( A
            = ( plus_plus @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_103_diff__eq__eq) ).

thf(568,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = C )
          = ( A
            = ( plus_plus @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(12,axiom,
    ! [A: nat > nat] :
      ( ( number2016821345itions @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( ^ [B: nat] : ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_9_partitions__zero) ).

thf(290,plain,
    ! [A: nat > nat] :
      ( ( number2016821345itions @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( ^ [B: nat] : ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(165,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/sandbox2/benchmark/theBenchmark.p',fact_162_measure__induct) ).

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

thf(43,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_diff @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_40_zero__diff) ).

thf(383,plain,
    ! [TA: $tType] :
      ( ( comm_monoid_diff @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(209,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( A = B )
            & ( ord_less_eq @ TA @ C @ D ) )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_206_add__mono__thms__linordered__semiring_I2_J) ).

thf(899,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( A = B )
            & ( ord_less_eq @ TA @ C @ D ) )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(253,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_250_trans__le__add2) ).

thf(1045,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(65,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( times_times @ nat @ A @ C ) )
      = ( ( B = C )
        | ( A
          = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_62_mult__cancel1) ).

thf(447,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( times_times @ nat @ A @ C ) )
      = ( ( B = C )
        | ( A
          = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(60,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        & ( B
          = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_57_add__is__0) ).

thf(432,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)],[60]) ).

thf(56,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_53_neq0__conv) ).

thf(422,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(196,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_193_zero__le) ).

thf(867,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(195,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_192_le__refl) ).

thf(865,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(117,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( ord_less_eq @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_114_le__diff__iff) ).

thf(603,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( ord_less_eq @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(237,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( A != B )
       => ( ord_less @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_234_le__neq__implies__less) ).

thf(999,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( A != B )
       => ( ord_less @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(122,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
      = ( minus_minus @ nat @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_119_Nat_Odiff__cancel) ).

thf(620,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
      = ( minus_minus @ nat @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(186,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_183_less__not__refl3) ).

thf(833,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(52,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( plus_plus @ TA @ ( minus_minus @ TA @ A @ B ) @ B )
          = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_49_diff__add__cancel) ).

thf(410,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( plus_plus @ TA @ ( minus_minus @ TA @ A @ B ) @ B )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(17,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ A @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
        & ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_14_nat__0__less__mult__iff) ).

thf(305,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ 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)],[17]) ).

thf(24,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/sandbox2/benchmark/theBenchmark.p',fact_21_double__add__less__zero__iff__single__add__less__zero) ).

thf(326,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)],[24]) ).

thf(215,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/sandbox2/benchmark/theBenchmark.p',fact_212_add__less__imp__less__left) ).

thf(918,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)],[215]) ).

thf(29,axiom,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
          = ( ord_less_eq @ TA @ A @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_26_add__le__cancel__right) ).

thf(341,plain,
    ! [TA: $tType] :
      ( ( ordere236663937imp_le @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
          = ( ord_less_eq @ TA @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(127,axiom,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ A @ B )
           => ( ( ( minus_minus @ TA @ B @ A )
                = C )
              = ( B
                = ( plus_plus @ TA @ C @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_124_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add) ).

thf(635,plain,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ A @ B )
           => ( ( ( minus_minus @ TA @ B @ A )
                = C )
              = ( B
                = ( plus_plus @ TA @ C @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(147,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ( minus_minus @ nat @ B @ A )
          = C )
        = ( B
          = ( plus_plus @ nat @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_144_Nat_Ole__imp__diff__is__add) ).

thf(696,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ( minus_minus @ nat @ B @ A )
          = C )
        = ( B
          = ( plus_plus @ nat @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(208,axiom,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
            & ( ord_less_eq @ TA @ C @ D ) )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_205_add__mono__thms__linordered__semiring_I1_J) ).

thf(897,plain,
    ! [TA: $tType] :
      ( ( ordere779506340up_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
            & ( ord_less_eq @ TA @ C @ D ) )
         => ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(74,axiom,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( plus_plus @ TA @ A @ B ) )
          = ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_71_le__add__same__cancel1) ).

thf(474,plain,
    ! [TA: $tType] :
      ( ( ordere516151231imp_le @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( plus_plus @ TA @ A @ B ) )
          = ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(93,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ D ) @ ( minus_minus @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_90_diff__mono) ).

thf(532,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ D ) @ ( minus_minus @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(265,axiom,
    cancel1352612707id_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add) ).

thf(1065,plain,
    cancel1352612707id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(47,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/sandbox2/benchmark/theBenchmark.p',fact_44_add__less__cancel__left) ).

thf(395,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)],[47]) ).

thf(233,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_230_less__eq__nat_Osimps_I1_J) ).

thf(987,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(26,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/sandbox2/benchmark/theBenchmark.p',fact_23_add__left__cancel) ).

thf(332,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)],[26]) ).

thf(108,axiom,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ ( zero_zero @ nat ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_105_minus__nat_Odiff__0) ).

thf(575,plain,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ ( zero_zero @ nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(58,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_55_le0) ).

thf(427,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(115,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_112_diff__le__mono) ).

thf(596,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(91,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ C ) @ ( minus_minus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_88_diff__right__mono) ).

thf(528,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ C ) @ ( minus_minus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(69,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ B @ ( minus_minus @ nat @ B @ A ) )
        = A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_66_diff__diff__cancel) ).

thf(459,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ B @ ( minus_minus @ nat @ B @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(235,axiom,
    ! [A: nat] :
      ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_232_plus__nat_Oadd__0) ).

thf(992,plain,
    ! [A: nat] :
      ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(3,axiom,
    ord_less_eq @ nat @ i @ n,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_0__092_060open_062i_A_092_060le_062_An_092_060close_062) ).

thf(277,plain,
    ord_less_eq @ nat @ i @ n,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(83,axiom,
    ! [A: nat,B: nat] :
      ( ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_80_diff__is__0__eq) ).

thf(501,plain,
    ! [A: nat,B: nat] :
      ( ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(118,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ( minus_minus @ nat @ B @ A )
            = ( minus_minus @ nat @ C @ A ) )
          = ( B = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_115_eq__diff__iff) ).

thf(607,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ( minus_minus @ nat @ B @ A )
            = ( minus_minus @ nat @ C @ A ) )
          = ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

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/sandbox2/benchmark/theBenchmark.p',fact_75_zero__less__double__add__iff__zero__less__single__add) ).

thf(486,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(240,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_237_less__imp__le__nat) ).

thf(1009,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(128,axiom,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( plus_plus @ TA @ A @ ( minus_minus @ TA @ B @ A ) )
            = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_125_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse) ).

thf(639,plain,
    ! [TA: $tType] :
      ( ( ordere623563068d_diff @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( plus_plus @ TA @ A @ ( minus_minus @ TA @ B @ A ) )
            = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(150,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ ( minus_minus @ nat @ B @ A ) )
        = ( ord_less_eq @ nat @ ( plus_plus @ nat @ C @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_147_Nat_Ole__diff__conv2) ).

thf(708,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ ( minus_minus @ nat @ B @ A ) )
        = ( ord_less_eq @ nat @ ( plus_plus @ nat @ C @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(228,axiom,
    ! [A: nat] :
      ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_225_not__gr0) ).

thf(972,plain,
    ! [A: nat] :
      ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(250,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( times_times @ nat @ A @ C ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ( B = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_247_nat__mult__eq__cancel__disj) ).

thf(1037,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( times_times @ nat @ A @ C ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ( B = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(60209,plain,
    $false,
    inference(cvc4,[status(thm)],[645,629,365,760,347,893,777,962,468,683,276,1068,308,878,741,1073,614,677,550,715,814,913,1063,930,404,841,885,288,651,320,565,389,504,587,934,1059,802,344,817,559,853,421,969,289,1013,284,881,416,924,492,1040,623,1020,1051,480,353,966,280,293,660,981,766,1072,453,1074,920,380,512,546,534,1067,1057,910,825,778,302,862,820,392,429,729,407,693,916,847,1032,435,329,665,757,456,285,522,830,317,1017,1064,1111,725,356,869,989,519,578,471,1082,753,537,642,489,590,686,1075,837,1099,654,1060,984,721,1002,371,805,508,311,541,657,1054,1071,674,1026,530,926,398,1035,662,718,888,425,745,279,296,444,733,584,1007,772,286,875,281,413,562,975,936,498,1058,335,704,350,611,787,784,1048,377,362,737,1043,599,1011,419,890,477,811,858,323,594,799,483,700,626,769,943,668,1029,895,462,1061,1044,883,299,438,922,1066,278,374,648,749,553,795,680,617,516,525,712,855,690,282,1070,872,791,441,338,808,359,544,907,1023,1008,314,571,876,978,386,465,450,556,539,671,903,581,495,368,844,1062,946,632,401,781,995,568,290,763,383,899,1045,447,432,422,867,865,603,999,620,833,410,305,326,918,341,807,635,696,897,474,532,1065,395,987,332,575,427,596,528,459,992,277,501,607,486,1009,639,1069,708,972,1037,775]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12  % Problem  : ITP131^2 : TPTP v8.2.0. Released v7.5.0.
% 0.11/0.16  % Command  : run_Leo-III %s %d
% 0.16/0.37  % Computer : n016.cluster.edu
% 0.16/0.37  % Model    : x86_64 x86_64
% 0.16/0.37  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.37  % Memory   : 8042.1875MB
% 0.16/0.37  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.16/0.37  % CPULimit : 300
% 0.16/0.37  % WCLimit  : 300
% 0.16/0.37  % DateTime : Sat May 18 17:04:39 EDT 2024
% 0.16/0.37  % CPUTime  : 
% 0.98/0.88  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 1.73/1.11  % [INFO] 	 Parsing done (223ms). 
% 1.73/1.11  % [INFO] 	 Running in sequential loop mode. 
% 2.33/1.33  % [INFO] 	 eprover registered as external prover. 
% 2.61/1.33  % [INFO] 	 cvc4 registered as external prover. 
% 2.61/1.34  % [INFO] 	 Scanning for conjecture ... 
% 2.78/1.42  % [INFO] 	 Found a conjecture (or negated_conjecture) and 273 axioms. Running axiom selection ... 
% 2.78/1.44  % [INFO] 	 Axiom selection finished. Selected 273 axioms (removed 0 axioms). 
% 3.21/1.61  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 3.21/1.63  % [INFO] 	 Type checking passed. 
% 3.21/1.63  % [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 ... 
% 59.37/21.24  % External prover 'cvc4' found a proof!
% 59.37/21.24  % [INFO] 	 Killing All external provers ... 
% 59.37/21.24  % Time passed: 20700ms (effective reasoning time: 20122ms)
% 59.37/21.24  % 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)>
% 59.37/21.24  % Axioms used in derivation (273): fact_33_add__cancel__left__left, fact_152_nat__diff__add__eq2, fact_252_add__le__mono1, fact_222_gr__implies__not0, fact_168_add_Oleft__cancel, fact_207_add__mono__thms__linordered__semiring_I3_J, fact_239_less__add__eq__less, fact_50_add__diff__cancel, fact_169_add_Oassoc, fact_122_le__iff__diff__le__0, fact_85_diff__eq__diff__eq, fact_21_double__add__less__zero__iff__single__add__less__zero, fact_212_add__less__imp__less__left, fact_107_less__imp__diff__less, fact_27_add__le__cancel__left, fact_200_le__iff__add, fact_155_nat__le__add__iff1, fact_82_Nat_Oadd__diff__assoc2, fact_143_less__diff__conv, fact_86_diff__commute, fact_162_measure__induct, fact_10_partitions__bounds, fact_225_not__gr0, fact_3_lower__bound, fact_156_nat__eq__add__iff2, fact_202_less__eqE, fact_83_Nat_Oadd__diff__assoc, fact_62_mult__cancel1, fact_79_diff__is__0__eq_H, fact_11_mult__le__cancel2, fact_213_add__strict__right__mono, fact_56_Nat_Oadd__0__right, tcon_Nat_Onat___Groups_Ocomm__monoid__diff, fact_91_eq__iff__diff__eq__0, fact_175_mult_Ocommute, fact_206_add__mono__thms__linordered__semiring_I2_J, fact_34_double__zero__sym, fact_159_nat__less__add__iff1, fact_35_double__zero, fact_54_bot__nat__0_Oextremum, fact_211_add__less__imp__less__right, fact_31_add__cancel__right__left, fact_192_le__refl, fact_181_nat__less__induct, fact_48_add__diff__cancel__left, fact_15_nat__mult__less__cancel__disj, fact_218_add__mono__thms__linordered__field_I5_J, tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, fact_106_diffs0__imp__equal, fact_215_add__strict__mono, fact_145_Nat_Odiff__add__assoc2, fact_65_nat__add__left__cancel__le, fact_4__092_060open_062k_A_092_060le_062_An_092_060close_062, fact_14_nat__0__less__mult__iff, fact_160_zero__reorient, fact_140_diff__less__mono, fact_59_diff__self__eq__0, fact_165_add_Oleft__commute, fact_128_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le, fact_74_diff__ge__0__iff__ge, fact_63_mult__0__right, fact_53_neq0__conv, fact_238_nat__less__le, fact_229_le__0__eq, fact_147_Nat_Ole__diff__conv2, fact_78_zero__less__diff, fact_98_diff__add__eq__diff__diff__swap, fact_227_bot__nat__0_Oextremum__uniqueI, fact_157_nat__eq__add__iff1, fact_109_diff__le__mono2, fact_66_diff__diff__cancel, fact_49_diff__add__cancel, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, fact_28_zero__eq__add__iff__both__eq__0, fact_131_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2, fact_1_assms_I1_J, fact_199_add__le__imp__le__left, fact_64_mult__is__0, fact_176_mult_Oassoc, fact_73_add__le__same__cancel1, fact_76_diff__gt__0__iff__gt, fact_81_Nat_Odiff__diff__right, tcon_Nat_Onat___Groups_Ocomm__monoid__add, fact_43_add__less__cancel__right, fact_205_add__mono__thms__linordered__semiring_I1_J, fact_12_nat__mult__le__cancel__disj, fact_36_add_Oright__neutral, fact_135_diff__le__eq, fact_241_trans__less__add1, fact_226_gr0I, fact_25_not__gr__zero, fact_246_add__lessD1, fact_51_bot__nat__0_Onot__eq__extremum, fact_69_double__add__le__zero__iff__single__add__le__zero, fact_89_diff__left__mono, fact_46_add__diff__cancel__right, tcon_Nat_Onat___Groups_Ocancel__semigroup__add, fact_60_diff__0__eq__0, fact_171_group__cancel_Oadd1, tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff, fact_234_le__neq__implies__less, fact_243_not__add__less2, fact_240_trans__less__add2, fact_88_diff__right__mono, tcon_Nat_Onat___Groups_Oab__semigroup__mult, fact_80_diff__is__0__eq, fact_55_le0, fact_37_add_Oleft__neutral, tcon_Nat_Onat___Groups_Omonoid__add, fact_146_Nat_Odiff__add__assoc, fact_101_add__diff__eq, fact_8__092_060open_062n_A_N_Ak_A_060_Ai_092_060close_062, fact_142_add__diff__inverse__nat, fact_245_add__less__mono, fact_163_add__right__imp__eq, fact_26_add__le__cancel__right, fact_136_less__diff__eq, fact_190_eq__imp__le, fact_114_le__diff__iff, fact_221_infinite__descent0, fact_203_add__left__mono, fact_105_minus__nat_Odiff__0, fact_95_diff__strict__mono, fact_41_diff__0__right, fact_129_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc, fact_19_less__add__same__cancel1, fact_195_gr__implies__not__zero, fact_223_less__zeroE, fact_166_add_Ocommute, fact_97_diff__diff__add, fact_39_diff__zero, fact_208_add_Ogroup__left__neutral, fact_149_nat__diff__split__asm, fact_138_diff__less, fact_42_diff__self, tcon_Nat_Onat___Orderings_Owellorder, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, fact_57_add__is__0, tcon_Nat_Onat___Groups_Ozero, fact_154_nat__le__add__iff2, fact_133_ordered__cancel__comm__monoid__diff__class_Odiff__add, fact_113_Nat_Odiff__diff__eq, fact_197_gr__zeroI, fact_178_infinite__descent__measure, fact_52_less__nat__zero__code, fact_61_mult__cancel2, fact_184_less__not__refl2, fact_177_ab__semigroup__mult__class_Omult__ac_I1_J, fact_100_diff__diff__eq2, fact_18_add__less__same__cancel2, fact_244_not__add__less1, fact_231_add__eq__self__zero, fact_121_diff__mult__distrib, fact_191_le__trans, fact_182_less__irrefl__nat, fact_45_add__diff__cancel__right_H, fact_209_add_Ocomm__neutral, fact_174_mult_Oleft__commute, fact_126_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right, fact_148_le__diff__conv, fact_249_nat__le__iff__add, fact_67_diff__diff__left, fact_16_add__gr__0, fact_132_le__add__diff, fact_139_diff__add__0, fact_228_bot__nat__0_Oextremum__unique, fact_32_add__cancel__left__right, fact_68_zero__le__double__add__iff__zero__le__single__add, fact_112_diff__le__mono, fact_118_diff__cancel2, fact_125_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse, fact_72_add__le__same__cancel2, tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add, fact_188_nat__le__linear, fact_235_less__or__eq__imp__le, fact_115_eq__diff__iff, fact_214_add__strict__left__mono, fact_217_add__mono__thms__linordered__field_I2_J, fact_0__092_060open_062i_A_092_060le_062_An_092_060close_062, fact_194_zero__less__iff__neq__zero, fact_161_measure__induct__rule, fact_44_add__less__cancel__left, fact_144_Nat_Ole__imp__diff__is__add, fact_116_diff__add__inverse2, fact_164_add__left__imp__eq, fact_248_mult__0, fact_153_nat__diff__add__eq1, fact_96_add__implies__diff, fact_93_diff__strict__left__mono, fact_22_add__right__cancel, fact_196_not__less__zero, fact_224_not__less0, fact_20_less__add__same__cancel2, fact_119_Nat_Odiff__cancel, fact_185_less__not__refl, fact_6_assms_I2_J, fact_2_upper__bound, fact_167_add_Oright__cancel, fact_70_le__add__same__cancel2, fact_150_nat__diff__split, fact_71_le__add__same__cancel1, fact_180_infinite__descent, fact_237_less__imp__le__nat, tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, fact_111_diff__le__self, fact_210_comm__monoid__add__class_Oadd__0, fact_134_le__diff__eq, fact_251_trans__le__add1, fact_247_nat__mult__eq__cancel__disj, fact_99_diff__add__eq, fact_17_add__less__same__cancel1, fact_9_partitions__zero, fact_232_plus__nat_Oadd__0, fact_102_eq__diff__eq, fact_130_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc, fact_90_diff__mono, fact_183_less__not__refl3, fact_193_zero__le, fact_29_add__eq__0__iff__both__eq__0, fact_189_le__antisym, fact_198_add__le__imp__le__right, fact_242_add__less__mono1, fact_127_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2, fact_24_le__zero__eq, fact_38_cancel__comm__monoid__add__class_Odiff__cancel, fact_173_ab__semigroup__add__class_Oadd__ac_I1_J, fact_87_diff__eq__diff__less__eq, fact_187_Nat_Oex__has__greatest__nat, fact_236_le__eq__less__or__eq, fact_104_group__cancel_Osub1, fact_47_add__diff__cancel__left_H, fact_186_nat__neq__iff, tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add, tcon_Nat_Onat___Groups_Osemigroup__add, fact_94_diff__eq__diff__less, fact_170_group__cancel_Oadd2, tcon_Nat_Onat___Groups_Osemigroup__mult, fact_13_mult__less__cancel2, fact_219_infinite__descent0__measure, tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, fact_216_add__mono__thms__linordered__field_I1_J, fact_141_less__diff__iff, fact_123_less__iff__diff__less__0, fact_58_nat__add__left__cancel__less, fact_137_diff__less__eq, fact_110_le__diff__iff_H, fact_117_diff__add__inverse, fact_103_diff__eq__eq, fact_92_diff__strict__right__mono, fact_77_diff__add__zero, fact_40_zero__diff, fact_158_nat__less__add__iff2, tcon_Nat_Onat___Groups_Oab__semigroup__add, fact_204_add__mono, fact_230_less__eq__nat_Osimps_I1_J, fact_233_less__mono__imp__le__mono, fact_151_less__diff__conv2, fact_120_diff__mult__distrib2, fact_5__092_060open_062p_Ai_A_092_060noteq_062_A0_092_060close_062, fact_23_add__left__cancel, fact_30_add__cancel__right__right, fact_201_add__right__mono, fact_108_diff__less__mono2, fact_7__092_060open_062i_A_092_060noteq_062_Ak_092_060close_062, fact_172_add__mono__thms__linordered__semiring_I4_J, fact_179_linorder__neqE__nat, fact_253_add__le__mono, fact_84_cancel__ab__semigroup__add__class_Odiff__right__commute, fact_250_trans__le__add2, fact_75_zero__less__double__add__iff__zero__less__single__add, fact_124_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add, fact_220_bot__nat__0_Oextremum__strict
% 59.37/21.24  % No. of inferences in proof: 572
% 59.37/21.24  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 20700 ms resp. 20122 ms w/o parsing
% 60.01/21.41  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 60.01/21.41  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------