TSTP Solution File: ITP103^2 by Leo-III---1.7.12

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%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : ITP103^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n017.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Mon May 20 22:22:56 EDT 2024

% Result   : Theorem 18.39s 4.72s
% Output   : Refutation 18.84s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  321
% Syntax   : Number of formulae    :  604 ( 179 unt;  40 typ;   0 def)
%            Number of atoms       : 1793 ( 614 equ;   0 cnn)
%            Maximal formula atoms :    6 (   3 avg)
%            Number of connectives : 7131 ( 212   ~;  66   |; 130   &;5915   @)
%                                         (   0 <=>; 808  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   7 avg)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   82 (  82   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   42 (  40 usr;   6 con; 0-4 aty)
%            Number of variables   : 1561 (  20   ^1460   !;  46   ?;1561   :)
%                                         (  35  !>;   0  ?*;   0  @-;   0  @+)

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

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

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

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

thf(field_type,type,
    field: 
      !>[TA: $tType] : $o ).

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

thf(mult_zero_type,type,
    mult_zero: 
      !>[TA: $tType] : $o ).

thf(linorder_type,type,
    linorder: 
      !>[TA: $tType] : $o ).

thf(ordered_ring_type,type,
    ordered_ring: 
      !>[TA: $tType] : $o ).

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

thf(division_ring_type,type,
    division_ring: 
      !>[TA: $tType] : $o ).

thf(semiring_bits_type,type,
    semiring_bits: 
      !>[TA: $tType] : $o ).

thf(semidom_divide_type,type,
    semidom_divide: 
      !>[TA: $tType] : $o ).

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

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

thf(linordered_ring_type,type,
    linordered_ring: 
      !>[TA: $tType] : $o ).

thf(ordered_semiring_type,type,
    ordered_semiring: 
      !>[TA: $tType] : $o ).

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

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

thf(ordered_semiring_0_type,type,
    ordered_semiring_0: 
      !>[TA: $tType] : $o ).

thf(ordere1490568538miring_type,type,
    ordere1490568538miring: 
      !>[TA: $tType] : $o ).

thf(linord581940658strict_type,type,
    linord581940658strict: 
      !>[TA: $tType] : $o ).

thf(semiri1193490041visors_type,type,
    semiri1193490041visors: 
      !>[TA: $tType] : $o ).

thf(linord20386208strict_type,type,
    linord20386208strict: 
      !>[TA: $tType] : $o ).

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

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

thf(linord893533164strict_type,type,
    linord893533164strict: 
      !>[TA: $tType] : $o ).

thf(semiri1923998003cancel_type,type,
    semiri1923998003cancel: 
      !>[TA: $tType] : $o ).

thf(euclid191655569cancel_type,type,
    euclid191655569cancel: 
      !>[TA: $tType] : $o ).

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

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

thf(product_type,type,
    product: 
      !>[TA: $tType,TB: $tType] : ( ( list @ TB ) > ( list @ TA ) > ( list @ ( product_prod @ TB @ TA ) ) ) ).

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

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

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

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

thf(divide_divide_type,type,
    divide_divide: 
      !>[TA: $tType] : ( TA > TA > TA ) ).

thf(k_type,type,
    k: nat ).

thf(n_type,type,
    n: nat ).

thf(xs_type,type,
    xs: list @ a ).

thf(132,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_135_mult__neg__neg) ).

thf(645,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(126,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
      = ( ? [C: nat] :
            ( ( B
              = ( suc @ C ) )
            & ( ord_less @ nat @ A @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_68_Suc__less__eq2) ).

thf(629,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
      = ( ? [C: nat] :
            ( ( B
              = ( suc @ C ) )
            & ( ord_less @ nat @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(69,axiom,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( divide_divide @ TA @ ( divide_divide @ TA @ A @ B ) @ C )
          = ( divide_divide @ TA @ A @ ( times_times @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_134_divide__divide__eq__left_H) ).

thf(479,plain,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( divide_divide @ TA @ ( divide_divide @ TA @ A @ B ) @ C )
          = ( divide_divide @ TA @ A @ ( times_times @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(196,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
            = ( ord_less @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_146_mult__less__cancel__left__pos) ).

thf(846,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
            = ( ord_less @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(249,axiom,
    ! [TA: $tType] :
      ( ( ordere1490568538miring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_228_ordered__comm__semiring__class_Ocomm__mult__left__mono) ).

thf(1031,plain,
    ! [TA: $tType] :
      ( ( ordere1490568538miring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(207,axiom,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_46_Suc__neq__Zero) ).

thf(893,plain,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(269,axiom,
    ! [A: nat,B: nat,C: nat > nat > $o] :
      ( ( ord_less @ nat @ A @ B )
     => ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
       => ( ! [D: nat,E: nat,F: nat] :
              ( ( ord_less @ nat @ D @ E )
             => ( ( ord_less @ nat @ E @ F )
               => ( ( C @ D @ E )
                 => ( ( C @ E @ F )
                   => ( C @ D @ F ) ) ) ) )
         => ( C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_64_less__Suc__induct) ).

thf(1083,plain,
    ! [A: nat,B: nat,C: nat > nat > $o] :
      ( ( ord_less @ nat @ A @ B )
     => ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
       => ( ! [D: nat,E: nat,F: nat] :
              ( ( ord_less @ nat @ D @ E )
             => ( ( ord_less @ nat @ E @ F )
               => ( ( C @ D @ E )
                 => ( ( C @ E @ F )
                   => ( C @ D @ F ) ) ) ) )
         => ( C @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(277,axiom,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( ( times_times @ TA @ B @ A )
              = C )
           => ( B
              = ( divide_divide @ TA @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_167_eq__divide__imp) ).

thf(1110,plain,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( ( times_times @ TA @ B @ A )
              = C )
           => ( B
              = ( divide_divide @ TA @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(30,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_30_Suc__inject) ).

thf(352,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(144,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ( suc @ A )
         != B )
       => ( ord_less @ nat @ ( suc @ A ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_75_Suc__lessI) ).

thf(683,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ( suc @ A )
         != B )
       => ( ord_less @ nat @ ( suc @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(264,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_34_nat__less__induct) ).

thf(1068,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)],[264]) ).

thf(236,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ? [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
            & ( B @ C ) ) )
      = ( ( B @ ( zero_zero @ nat ) )
        | ? [C: nat] :
            ( ( ord_less @ nat @ C @ A )
            & ( B @ ( suc @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_89_Ex__less__Suc2) ).

thf(994,plain,
    ! [A: nat,B: nat > $o] :
      ( ( ? [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
            & ( B @ C ) ) )
      = ( ( B @ ( zero_zero @ nat ) )
        | ? [C: nat] :
            ( ( ord_less @ nat @ C @ A )
            & ( B @ ( suc @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(265,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( divide_divide @ nat @ B @ A ) @ C )
       => ( ord_less @ nat @ B @ ( times_times @ nat @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_184_div__less__imp__less__mult) ).

thf(1073,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( divide_divide @ nat @ B @ A ) @ C )
       => ( ord_less @ nat @ B @ ( times_times @ nat @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(36,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_236_mult__right__mono) ).

thf(379,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(55,axiom,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( times_times @ TA @ B @ A ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_141_mult__pos__neg2) ).

thf(440,plain,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( times_times @ TA @ B @ A ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(142,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(677,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)],[142]) ).

thf(253,axiom,
    semigroup_mult @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Osemigroup__mult) ).

thf(1041,plain,
    semigroup_mult @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(38,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( divide_divide @ TA @ A @ B ) @ C )
          = ( divide_divide @ TA @ ( times_times @ TA @ A @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_123_times__divide__eq__left) ).

thf(384,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( divide_divide @ TA @ A @ B ) @ C )
          = ( divide_divide @ TA @ ( times_times @ TA @ A @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(96,axiom,
    ab_semigroup_mult @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oab__semigroup__mult) ).

thf(550,plain,
    ab_semigroup_mult @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(77,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ ( zero_zero @ nat ) )
        = ( times_times @ nat @ A @ B ) )
      = ( ( A
          = ( suc @ ( zero_zero @ nat ) ) )
        & ( B
          = ( suc @ ( zero_zero @ nat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_9_one__eq__mult__iff) ).

thf(500,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ ( zero_zero @ nat ) )
        = ( times_times @ nat @ A @ B ) )
      = ( ( A
          = ( suc @ ( zero_zero @ nat ) ) )
        & ( B
          = ( suc @ ( zero_zero @ nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(67,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ~ ! [C: nat] :
            ( ( ord_less @ nat @ A @ C )
           => ( B
             != ( suc @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_76_Suc__lessE) ).

thf(472,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ~ ! [C: nat] :
            ( ( ord_less @ nat @ A @ C )
           => ( B
             != ( suc @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(262,axiom,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ B @ C ) @ ( times_times @ TA @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_237_mult__right__mono__neg) ).

thf(1063,plain,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ B @ C ) @ ( times_times @ TA @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(140,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B
       != ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_57_gr__implies__not0) ).

thf(670,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B
       != ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(152,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_205_list__decode_Ocases) ).

thf(709,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(229,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ C @ B ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
              & ( ord_less @ TA @ A @ C ) )
            | ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
              & ( ord_less @ TA @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_152_mult__less__cancel__right__disj) ).

thf(973,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ C @ B ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
              & ( ord_less @ TA @ A @ C ) )
            | ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
              & ( ord_less @ TA @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(104,axiom,
    ordere1490568538miring @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Oordered__comm__semiring) ).

thf(565,plain,
    ordere1490568538miring @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(53,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less @ nat @ B @ C )
           => ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_84_lift__Suc__mono__less) ).

thf(436,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less @ nat @ B @ C )
           => ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(259,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( divide_divide @ TA @ A @ B ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_160_zero__less__divide__iff) ).

thf(1055,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( divide_divide @ TA @ A @ B ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(113,axiom,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_151_mult__strict__right__mono) ).

thf(587,plain,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(150,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ( divide_divide @ nat @ B @ A )
          = ( zero_zero @ nat ) )
        = ( ord_less @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_171_div__eq__0__conv_H) ).

thf(702,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ( divide_divide @ nat @ B @ A )
          = ( zero_zero @ nat ) )
        = ( ord_less @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(26,axiom,
    linorder @ $o,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Olinorder_2) ).

thf(344,plain,
    linorder @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(93,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(542,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)],[93]) ).

thf(100,axiom,
    semidom_divide @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemidom__divide) ).

thf(559,plain,
    semidom_divide @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(46,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ? [C: TA] :
                ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
                & ( ord_less @ TA @ C @ A )
                & ( ord_less @ TA @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_208_field__lbound__gt__zero) ).

thf(416,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ? [C: TA] :
                ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
                & ( ord_less @ TA @ C @ A )
                & ( ord_less @ TA @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(5,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( divide_divide @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_247_divide__nonpos__nonneg) ).

thf(289,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( divide_divide @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(58,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
       => ( ord_less @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_93_mult__less__mono1) ).

thf(448,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
       => ( ord_less @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(242,axiom,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_148_mult__strict__left__mono) ).

thf(1013,plain,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(3,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_213_le__zero__eq) ).

thf(284,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)],[3]) ).

thf(169,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
     => ( ~ ( ord_less @ nat @ A @ B )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_74_less__SucE) ).

thf(770,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
     => ( ~ ( ord_less @ nat @ A @ B )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(176,axiom,
    ! [TA: $tType] :
      ( ( euclid191655569cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( A
              = ( zero_zero @ TA ) )
           => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
              = ( zero_zero @ TA ) ) )
          & ( ( A
             != ( zero_zero @ TA ) )
           => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
              = ( divide_divide @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_24_div__mult__mult1__if) ).

thf(789,plain,
    ! [TA: $tType] :
      ( ( euclid191655569cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( A
              = ( zero_zero @ TA ) )
           => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
              = ( zero_zero @ TA ) ) )
          & ( ( A
             != ( zero_zero @ TA ) )
           => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
              = ( divide_divide @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(17,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( divide_divide @ TA @ A @ B ) @ ( zero_zero @ TA ) )
          = ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_252_divide__le__0__iff) ).

thf(321,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( divide_divide @ TA @ A @ B ) @ ( zero_zero @ TA ) )
          = ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(35,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_142_zero__less__mult__iff) ).

thf(376,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(83,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_79_nat__mult__eq__cancel__disj) ).

thf(514,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)],[83]) ).

thf(220,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( divide_divide @ nat @ A @ B )
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_3_div__less) ).

thf(937,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( divide_divide @ nat @ A @ B )
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(28,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_12_old_Onat_Oinject) ).

thf(348,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(182,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_230_mult__nonneg__nonpos2) ).

thf(806,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(71,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_45_Zero__neq__Suc) ).

thf(485,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(192,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ B @ C )
       => ( ord_less @ nat @ ( suc @ A ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_65_less__trans__Suc) ).

thf(834,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ B @ C )
       => ( ord_less @ nat @ ( suc @ A ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(128,axiom,
    ! [TA: $tType] :
      ( ( semidom_divide @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ A )
            = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_105_nonzero__mult__div__cancel__left) ).

thf(634,plain,
    ! [TA: $tType] :
      ( ( semidom_divide @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ A )
            = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(42,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_38_less__not__refl) ).

thf(396,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(137,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_36_less__not__refl3) ).

thf(660,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(80,axiom,
    ! [TA: $tType] :
      ( ( semiri1193490041visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( zero_zero @ TA ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_114_mult__eq__0__iff) ).

thf(507,plain,
    ! [TA: $tType] :
      ( ( semiri1193490041visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( zero_zero @ TA ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(87,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] :
            ! [C: TA] :
              ( ( ord_less @ TA @ B @ C )
             => ( A != C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_225_le__greater__neq__conv) ).

thf(527,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] :
            ! [C: TA] :
              ( ( ord_less @ TA @ B @ C )
             => ( A != C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(119,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_33_infinite__descent) ).

thf(602,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)],[119]) ).

thf(168,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ( times_times @ nat @ B @ A )
          = ( times_times @ nat @ C @ A ) )
        = ( B = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_188_mult__cancel2__gr0) ).

thf(766,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ( times_times @ nat @ B @ A )
          = ( times_times @ nat @ C @ A ) )
        = ( B = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(275,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_32_linorder__neqE__nat) ).

thf(1104,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)],[275]) ).

thf(197,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ? [C: nat] :
            ( ( A
              = ( suc @ C ) )
            & ( ord_less @ nat @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_85_less__Suc__eq__0__disj) ).

thf(849,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ? [C: nat] :
            ( ( A
              = ( suc @ C ) )
            & ( ord_less @ nat @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(12,axiom,
    ! [TA: $tType] :
      ( ( semiring_bits @ TA )
     => ! [A: TA] :
          ( ( divide_divide @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_185_bits__div__0) ).

thf(307,plain,
    ! [TA: $tType] :
      ( ( semiring_bits @ TA )
     => ! [A: TA] :
          ( ( divide_divide @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(222,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( divide_divide @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_156_divide__pos__neg) ).

thf(942,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( divide_divide @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(94,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less @ TA @ ( divide_divide @ TA @ A @ C ) @ ( divide_divide @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_161_divide__strict__right__mono) ).

thf(546,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less @ TA @ ( divide_divide @ TA @ A @ C ) @ ( divide_divide @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(184,axiom,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ C @ B ) @ ( times_times @ TA @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_240_mult__left__mono__neg) ).

thf(810,plain,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ C @ B ) @ ( times_times @ TA @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(37,axiom,
    ! [TA: $tType,TB: $tType,A: list @ TB,B: list @ TA] :
      ( ( size_size @ ( list @ ( product_prod @ TB @ TA ) ) @ ( product @ TB @ TA @ A @ B ) )
      = ( times_times @ nat @ ( size_size @ ( list @ TB ) @ A ) @ ( size_size @ ( list @ TA ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_210_length__product) ).

thf(381,plain,
    ! [TA: $tType,TB: $tType,A: list @ TB,B: list @ TA] :
      ( ( size_size @ ( list @ ( product_prod @ TB @ TA ) ) @ ( product @ TB @ TA @ A @ B ) )
      = ( times_times @ nat @ ( size_size @ ( list @ TB ) @ A ) @ ( size_size @ ( list @ TA ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(90,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( times_times @ TA @ B @ C ) @ ( times_times @ TA @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_150_mult__strict__right__mono__neg) ).

thf(534,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( times_times @ TA @ B @ C ) @ ( times_times @ TA @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(178,axiom,
    order @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oorder_1) ).

thf(798,plain,
    order @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(22,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( divide_divide @ TA @ A @ B )
            = ( divide_divide @ TA @ A @ C ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_120_divide__cancel__left) ).

thf(334,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( divide_divide @ TA @ A @ B )
            = ( divide_divide @ TA @ A @ C ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(212,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring @ 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 @ ( zero_zero @ TA ) @ B )
             => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
               => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_244_mult__mono) ).

thf(910,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring @ 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 @ ( zero_zero @ TA ) @ B )
             => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
               => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(161,axiom,
    ! [A: nat] : ( ord_less @ nat @ A @ ( suc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_19_lessI) ).

thf(746,plain,
    ! [A: nat] : ( ord_less @ nat @ A @ ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(172,axiom,
    ! [TA: $tType,A: nat] :
    ? [B: list @ TA] :
      ( ( size_size @ ( list @ TA ) @ B )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_203_Ex__list__of__length) ).

thf(778,plain,
    ! [TA: $tType,A: nat] :
    ? [B: list @ TA] :
      ( ( size_size @ ( list @ TA ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(254,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ B )
       => ( ord_less @ nat @ A @ ( times_times @ nat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_98_n__less__m__mult__n) ).

thf(1042,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ B )
       => ( ord_less @ nat @ A @ ( times_times @ nat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(10,axiom,
    ! [TA: $tType] :
      ( ( semiring_bits @ TA )
     => ! [A: TA] :
          ( ( divide_divide @ TA @ A @ ( zero_zero @ TA ) )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_186_bits__div__by__0) ).

thf(302,plain,
    ! [TA: $tType] :
      ( ( semiring_bits @ TA )
     => ! [A: TA] :
          ( ( divide_divide @ TA @ A @ ( zero_zero @ TA ) )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(117,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
         => ( ~ ( ord_less @ TA @ A @ B )
           => ( ord_less @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_126_linorder__neqE__linordered__idom) ).

thf(598,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
         => ( ~ ( ord_less @ TA @ A @ B )
           => ( ord_less @ TA @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(188,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ B @ A ) @ ( times_times @ TA @ C @ A ) )
            = ( divide_divide @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_103_nonzero__mult__divide__mult__cancel__right) ).

thf(820,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ B @ A ) @ ( times_times @ TA @ C @ A ) )
            = ( divide_divide @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(8,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_13_nat_Oinject) ).

thf(297,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(49,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( divide_divide @ TA @ A @ B ) @ ( zero_zero @ TA ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_158_divide__less__0__iff) ).

thf(424,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( divide_divide @ TA @ A @ B ) @ ( zero_zero @ TA ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(147,axiom,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ A @ ( zero_zero @ TA ) )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_113_mult__zero__right) ).

thf(693,plain,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ A @ ( zero_zero @ TA ) )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(186,axiom,
    ! [TA: $tType] :
      ( ( euclid191655569cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
            = ( divide_divide @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_26_div__mult__mult1) ).

thf(815,plain,
    ! [TA: $tType] :
      ( ( euclid191655569cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
            = ( divide_divide @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(105,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_214_bot__nat__0_Oextremum) ).

thf(566,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(33,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( A
          = ( zero_zero @ nat ) )
       => ( ( divide_divide @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
          = ( zero_zero @ nat ) ) )
      & ( ( A
         != ( zero_zero @ nat ) )
       => ( ( divide_divide @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
          = ( divide_divide @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_2_nat__mult__div__cancel__disj) ).

thf(361,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( A
          = ( zero_zero @ nat ) )
       => ( ( divide_divide @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
          = ( zero_zero @ nat ) ) )
      & ( ( A
         != ( zero_zero @ nat ) )
       => ( ( divide_divide @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
          = ( divide_divide @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(205,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B @ A ) @ C )
          & ( ord_less @ nat @ C @ ( times_times @ nat @ ( suc @ B ) @ A ) ) )
        = ( ( divide_divide @ nat @ C @ A )
          = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_212_le__less__div__conv) ).

thf(879,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B @ A ) @ C )
          & ( ord_less @ nat @ C @ ( times_times @ nat @ ( suc @ B ) @ A ) ) )
        = ( ( divide_divide @ nat @ C @ A )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(198,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ ( divide_divide @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_154_divide__neg__neg) ).

thf(852,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ ( divide_divide @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(112,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
        = ( ord_less @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_90_nat__mult__less__cancel1) ).

thf(583,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
        = ( ord_less @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(63,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_18_Suc__mono) ).

thf(461,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(74,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( divide_divide @ TA @ B @ C ) @ ( divide_divide @ TA @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_162_divide__strict__right__mono__neg) ).

thf(493,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( divide_divide @ TA @ B @ C ) @ ( divide_divide @ TA @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(166,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( A
              = ( zero_zero @ TA ) )
           => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
              = ( zero_zero @ TA ) ) )
          & ( ( A
             != ( zero_zero @ TA ) )
           => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
              = ( divide_divide @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_107_mult__divide__mult__cancel__left__if) ).

thf(757,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( A
              = ( zero_zero @ TA ) )
           => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
              = ( zero_zero @ TA ) ) )
          & ( ( A
             != ( zero_zero @ TA ) )
           => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
              = ( divide_divide @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(97,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_86_gr0__implies__Suc) ).

thf(551,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(18,axiom,
    ! [TA: $tType,TB: $tType] : ( size @ ( product_prod @ TB @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Nat_Osize_5) ).

thf(324,plain,
    ! [TA: $tType,TB: $tType] : ( size @ ( product_prod @ TB @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(267,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ B @ ( times_times @ TA @ C @ A ) )
           => ( ord_less @ TA @ ( divide_divide @ TA @ B @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_179_mult__imp__div__pos__less) ).

thf(1079,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ B @ ( times_times @ TA @ C @ A ) )
           => ( ord_less @ TA @ ( divide_divide @ TA @ B @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(247,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_209_less__numeral__extra_I3_J) ).

thf(1027,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(159,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ B @ ( divide_divide @ TA @ C @ A ) )
            = ( ord_less @ TA @ C @ ( times_times @ TA @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_176_neg__less__divide__eq) ).

thf(742,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ B @ ( divide_divide @ TA @ C @ A ) )
            = ( ord_less @ TA @ C @ ( times_times @ TA @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(134,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_39_nat__neq__iff) ).

thf(650,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)],[134]) ).

thf(272,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ ( divide_divide @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_157_divide__pos__pos) ).

thf(1096,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ ( divide_divide @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(31,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_232_mult__nonneg__nonpos) ).

thf(356,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(174,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( divide_divide @ TA @ A @ ( divide_divide @ TA @ B @ C ) )
          = ( divide_divide @ TA @ ( times_times @ TA @ A @ C ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_110_divide__divide__eq__right) ).

thf(783,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( divide_divide @ TA @ A @ ( divide_divide @ TA @ B @ C ) )
          = ( divide_divide @ TA @ ( times_times @ TA @ A @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(47,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ A @ B ) )
      = ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ A )
        & ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_217_one__le__mult__iff) ).

thf(420,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ A @ B ) )
      = ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ A )
        & ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(20,axiom,
    ! [A: nat] :
      ( A
     != ( suc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_29_n__not__Suc__n) ).

thf(328,plain,
    ! [A: nat] :
      ( A
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(24,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_220_le__numeral__extra_I3_J) ).

thf(339,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(219,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ( times_times @ nat @ A @ B )
          = ( times_times @ nat @ A @ C ) )
        = ( B = C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_91_nat__mult__eq__cancel1) ).

thf(933,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ( times_times @ nat @ A @ B )
          = ( times_times @ nat @ A @ C ) )
        = ( B = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(98,axiom,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_112_mult__zero__left) ).

thf(554,plain,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(131,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
          = ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_229_zero__le__mult__iff) ).

thf(642,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
          = ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(129,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
           => ( B @ C ) ) )
      = ( ( B @ ( zero_zero @ nat ) )
        & ! [C: nat] :
            ( ( ord_less @ nat @ C @ A )
           => ( B @ ( suc @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_87_All__less__Suc2) ).

thf(637,plain,
    ! [A: nat,B: nat > $o] :
      ( ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
           => ( B @ C ) ) )
      = ( ( B @ ( zero_zero @ nat ) )
        & ! [C: nat] :
            ( ( ord_less @ nat @ C @ A )
           => ( B @ ( suc @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(72,axiom,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_239_mult__nonpos__nonpos) ).

thf(489,plain,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(244,axiom,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( divide_divide @ TA @ A @ B )
            = C )
          = ( ( ( B
               != ( zero_zero @ TA ) )
             => ( A
                = ( times_times @ TA @ C @ B ) ) )
            & ( ( B
                = ( zero_zero @ TA ) )
             => ( C
                = ( zero_zero @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_164_divide__eq__eq) ).

thf(1016,plain,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( divide_divide @ TA @ A @ B )
            = C )
          = ( ( ( B
               != ( zero_zero @ TA ) )
             => ( A
                = ( times_times @ TA @ C @ B ) ) )
            & ( ( B
                = ( zero_zero @ TA ) )
             => ( C
                = ( zero_zero @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(151,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(706,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)],[151]) ).

thf(145,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_44_Zero__not__Suc) ).

thf(686,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(68,axiom,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
            = ( divide_divide @ TA @ B @ C ) )
          = ( ( ( C
               != ( zero_zero @ TA ) )
             => ( ( times_times @ TA @ A @ C )
                = B ) )
            & ( ( C
                = ( zero_zero @ TA ) )
             => ( A
                = ( zero_zero @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_165_eq__divide__eq) ).

thf(476,plain,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
            = ( divide_divide @ TA @ B @ C ) )
          = ( ( ( C
               != ( zero_zero @ TA ) )
             => ( ( times_times @ TA @ A @ C )
                = B ) )
            & ( ( C
                = ( zero_zero @ TA ) )
             => ( A
                = ( zero_zero @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(193,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ ( suc @ B ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_67_less__antisym) ).

thf(837,plain,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ ( suc @ B ) )
       => ( B = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(233,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ( A
          = ( suc @ ( zero_zero @ nat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_227_nat__leSuc0__conv) ).

thf(984,plain,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ( A
          = ( suc @ ( zero_zero @ nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(107,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_28_measure__induct) ).

thf(569,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)],[107]) ).

thf(78,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( B
             != ( zero_zero @ TA ) )
           => ( ( ( divide_divide @ TA @ C @ A )
                = ( divide_divide @ TA @ D @ B ) )
              = ( ( times_times @ TA @ C @ B )
                = ( times_times @ TA @ D @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_163_frac__eq__eq) ).

thf(503,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( B
             != ( zero_zero @ TA ) )
           => ( ( ( divide_divide @ TA @ C @ A )
                = ( divide_divide @ TA @ D @ B ) )
              = ( ( times_times @ TA @ C @ B )
                = ( times_times @ TA @ D @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(41,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
      = ( ( ord_less @ nat @ A @ B )
        | ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_71_less__Suc__eq) ).

thf(393,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
      = ( ( ord_less @ nat @ A @ B )
        | ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(278,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_55_bot__nat__0_Oextremum__strict) ).

thf(1114,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(133,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_6_nat__0__less__mult__iff) ).

thf(647,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)],[133]) ).

thf(237,axiom,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ A @ ( divide_divide @ TA @ B @ C ) )
          = ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_109_times__divide__eq__right) ).

thf(997,plain,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ A @ ( divide_divide @ TA @ B @ C ) )
          = ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(257,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
           => ( B @ C ) ) )
      = ( ( B @ A )
        & ! [C: nat] :
            ( ( ord_less @ nat @ C @ A )
           => ( B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_69_All__less__Suc) ).

thf(1050,plain,
    ! [A: nat,B: nat > $o] :
      ( ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
           => ( B @ C ) ) )
      = ( ( B @ A )
        & ! [C: nat] :
            ( ( ord_less @ nat @ C @ A )
           => ( B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(61,axiom,
    canoni770627133id_add @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add) ).

thf(457,plain,
    canoni770627133id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(216,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_54_infinite__descent0__measure) ).

thf(921,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)],[216]) ).

thf(136,axiom,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( ( times_times @ TA @ A @ B )
              = ( times_times @ TA @ A @ C ) )
            = ( B = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_130_mult__left__cancel) ).

thf(657,plain,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( ( times_times @ TA @ A @ B )
              = ( times_times @ TA @ A @ C ) )
            = ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(251,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_219_nat__mult__le__cancel__disj) ).

thf(1035,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)],[251]) ).

thf(141,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_15_less__nat__zero__code) ).

thf(674,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(88,axiom,
    ordered_semiring @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Oordered__semiring) ).

thf(530,plain,
    ordered_semiring @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(170,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( ord_less @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_17_Suc__less__eq) ).

thf(773,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( ord_less @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(163,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( times_times @ TA @ C @ B ) @ ( times_times @ TA @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_147_mult__strict__left__mono__neg) ).

thf(750,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( times_times @ TA @ C @ B ) @ ( times_times @ TA @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(200,axiom,
    ! [TA: $tType] : ( size @ ( list @ TA ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_List_Olist___Nat_Osize_4) ).

thf(856,plain,
    ! [TA: $tType] : ( size @ ( list @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(160,axiom,
    linord1659791738miring @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring) ).

thf(745,plain,
    linord1659791738miring @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(156,axiom,
    ! [TA: $tType] :
      ( ( euclid191655569cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ B @ A ) @ ( times_times @ TA @ C @ A ) )
            = ( divide_divide @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_25_div__mult__mult2) ).

thf(733,plain,
    ! [TA: $tType] :
      ( ( euclid191655569cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ B @ A ) @ ( times_times @ TA @ C @ A ) )
            = ( divide_divide @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(118,axiom,
    ordered_semiring_0 @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Oordered__semiring__0) ).

thf(601,plain,
    ordered_semiring_0 @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(82,axiom,
    linord893533164strict @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict) ).

thf(513,plain,
    linord893533164strict @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(51,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ B @ A ) @ ( times_times @ TA @ A @ C ) )
            = ( divide_divide @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_101_nonzero__mult__divide__mult__cancel__right2) ).

thf(430,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ B @ A ) @ ( times_times @ TA @ A @ C ) )
            = ( divide_divide @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(210,axiom,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( ( divide_divide @ TA @ B @ A )
              = C )
            = ( B
              = ( times_times @ TA @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_168_nonzero__divide__eq__eq) ).

thf(904,plain,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( ( divide_divide @ TA @ B @ A )
              = C )
            = ( B
              = ( times_times @ TA @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(48,axiom,
    size @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Nat_Osize) ).

thf(423,plain,
    size @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(44,axiom,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( ord_less @ nat @ A @ B )
     => ( ! [D: nat] :
            ( ( B
              = ( suc @ D ) )
           => ( C @ D ) )
       => ( ! [D: nat] :
              ( ( ord_less @ nat @ D @ B )
             => ( ( C @ ( suc @ D ) )
               => ( C @ D ) ) )
         => ( C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_63_strict__inc__induct) ).

thf(402,plain,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( ord_less @ nat @ A @ B )
     => ( ! [D: nat] :
            ( ( B
              = ( suc @ D ) )
           => ( C @ D ) )
       => ( ! [D: nat] :
              ( ( ord_less @ nat @ D @ B )
             => ( ( C @ ( suc @ D ) )
               => ( C @ D ) ) )
         => ( C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(6,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( divide_divide @ TA @ A @ B )
            = ( divide_divide @ TA @ C @ B ) )
          = ( ( B
              = ( zero_zero @ TA ) )
            | ( A = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_121_divide__cancel__right) ).

thf(291,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( divide_divide @ TA @ A @ B )
            = ( divide_divide @ TA @ C @ B ) )
          = ( ( B
              = ( zero_zero @ TA ) )
            | ( A = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(45,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_190_mult_Oleft__commute) ).

thf(413,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)],[45]) ).

thf(195,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_16_neq0__conv) ).

thf(843,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(282,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( divide_divide @ nat @ ( times_times @ nat @ A @ B ) @ A )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1_div__mult__self1__is__m) ).

thf(1125,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( divide_divide @ nat @ ( times_times @ nat @ A @ B ) @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(204,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( divide_divide @ nat @ B @ A ) @ C )
        = ( ord_less @ nat @ B @ ( times_times @ nat @ C @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_183_div__less__conv) ).

thf(875,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( divide_divide @ nat @ B @ A ) @ C )
        = ( ord_less @ nat @ B @ ( times_times @ nat @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(190,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( suc @ B ) )
     => ( A
       != ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_50_nat_OdiscI) ).

thf(826,plain,
    ! [A: nat,B: nat] :
      ( ( A
        = ( suc @ B ) )
     => ( A
       != ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(110,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_23_mult__is__0) ).

thf(577,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)],[110]) ).

thf(57,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ ( suc @ B ) )
        = ( A = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_62_not__less__less__Suc__eq) ).

thf(445,plain,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ ( suc @ B ) )
        = ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(108,axiom,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( B
              = ( times_times @ TA @ C @ A ) )
           => ( ( divide_divide @ TA @ B @ A )
              = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_166_divide__eq__imp) ).

thf(572,plain,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( B
              = ( times_times @ TA @ C @ A ) )
           => ( ( divide_divide @ TA @ B @ A )
              = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(14,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_221_zero__le) ).

thf(313,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(122,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_124_linordered__field__no__lb) ).

thf(611,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(65,axiom,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( ( times_times @ TA @ B @ A )
              = ( times_times @ TA @ C @ A ) )
            = ( B = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_131_mult__right__cancel) ).

thf(466,plain,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( ( times_times @ TA @ B @ A )
              = ( times_times @ TA @ C @ A ) )
            = ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(270,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_31_infinite__descent__measure) ).

thf(1090,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)],[270]) ).

thf(92,axiom,
    ! [TA: $tType] :
      ( ( linord893533164strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_153_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono) ).

thf(540,plain,
    ! [TA: $tType] :
      ( ( linord893533164strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(153,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( A @ C )
           => ( A @ ( suc @ C ) ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_49_nat__induct) ).

thf(713,plain,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( A @ C )
           => ( A @ ( suc @ C ) ) )
       => ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(76,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_198_gr__implies__gr0) ).

thf(498,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(260,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( suc @ ( zero_zero @ nat ) ) )
      = ( ( A
          = ( suc @ ( zero_zero @ nat ) ) )
        & ( B
          = ( suc @ ( zero_zero @ nat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_8_mult__eq__1__iff) ).

thf(1058,plain,
    ! [A: nat,B: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( suc @ ( zero_zero @ nat ) ) )
      = ( ( A
          = ( suc @ ( zero_zero @ nat ) ) )
        & ( B
          = ( suc @ ( zero_zero @ nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(276,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( divide_divide @ TA @ ( divide_divide @ TA @ A @ B ) @ C )
          = ( divide_divide @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_111_divide__divide__eq__left) ).

thf(1107,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( divide_divide @ TA @ ( divide_divide @ TA @ A @ B ) @ C )
          = ( divide_divide @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(231,axiom,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( times_times @ TA @ A @ C ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_115_mult__cancel__left) ).

thf(979,plain,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( times_times @ TA @ A @ C ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(209,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_43_old_Onat_Oexhaust) ).

thf(900,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(191,axiom,
    ! [A: nat] :
      ( ( times_times @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_80_mult__0) ).

thf(831,plain,
    ! [A: nat] :
      ( ( times_times @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(279,axiom,
    ! [A: nat] :
      ( ( times_times @ nat @ A @ ( zero_zero @ nat ) )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_22_mult__0__right) ).

thf(1117,plain,
    ! [A: nat] :
      ( ( times_times @ nat @ A @ ( zero_zero @ nat ) )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(59,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_35_less__irrefl__nat) ).

thf(451,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(16,axiom,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = A )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_189_zero__reorient) ).

thf(318,plain,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = A )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(281,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide @ nat @ A @ ( times_times @ nat @ B @ C ) )
      = ( divide_divide @ nat @ ( divide_divide @ nat @ A @ B ) @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_82_div__mult2__eq) ).

thf(1122,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide @ nat @ A @ ( times_times @ nat @ B @ C ) )
      = ( divide_divide @ nat @ ( divide_divide @ nat @ A @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(256,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( times_times @ TA @ B @ A ) @ C )
           => ( ord_less @ TA @ B @ ( divide_divide @ TA @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_180_mult__imp__less__div__pos) ).

thf(1048,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( times_times @ TA @ B @ A ) @ C )
           => ( ord_less @ TA @ B @ ( divide_divide @ TA @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(157,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_218_mult__le__cancel2) ).

thf(736,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)],[157]) ).

thf(39,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ( ( ord_less @ TA )
        = ( ^ [A: TA,B: TA] :
            ! [C: TA] :
              ( ( ord_less_eq @ TA @ B @ C )
             => ( A != C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_223_less__ge__neq__conv) ).

thf(387,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ( ( ord_less @ TA )
        = ( ^ [A: TA,B: TA] :
            ! [C: TA] :
              ( ( ord_less_eq @ TA @ B @ C )
             => ( A != C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(248,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_215_le0) ).

thf(1029,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(226,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ ( divide_divide @ TA @ B @ C ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
             => ( ord_less @ TA @ ( times_times @ TA @ A @ C ) @ B ) )
            & ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
             => ( ( ( ord_less @ TA @ C @ ( zero_zero @ TA ) )
                 => ( ord_less @ TA @ B @ ( times_times @ TA @ A @ C ) ) )
                & ( ~ ( ord_less @ TA @ C @ ( zero_zero @ TA ) )
                 => ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_174_less__divide__eq) ).

thf(964,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ ( divide_divide @ TA @ B @ C ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
             => ( ord_less @ TA @ ( times_times @ TA @ A @ C ) @ B ) )
            & ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
             => ( ( ( ord_less @ TA @ C @ ( zero_zero @ TA ) )
                 => ( ord_less @ TA @ B @ ( times_times @ TA @ A @ C ) ) )
                & ( ~ ( ord_less @ TA @ C @ ( zero_zero @ TA ) )
                 => ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(261,axiom,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_144_zero__less__mult__pos2) ).

thf(1061,plain,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(27,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( divide_divide @ TA @ A @ B )
            = ( zero_zero @ TA ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_118_divide__eq__0__iff) ).

thf(345,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( divide_divide @ TA @ A @ B )
            = ( zero_zero @ TA ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(201,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( A @ C )
           => ( A @ ( suc @ C ) ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_42_old_Onat_Oinducts) ).

thf(858,plain,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( A @ C )
           => ( A @ ( suc @ C ) ) )
       => ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(179,axiom,
    semiring_bits @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Parity_Osemiring__bits) ).

thf(799,plain,
    semiring_bits @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(155,axiom,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
           != ( zero_zero @ TA ) )
         => ( ( A
             != ( zero_zero @ TA ) )
            & ( B
             != ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_127_mult__not__zero) ).

thf(726,plain,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
           != ( zero_zero @ TA ) )
         => ( ( A
             != ( zero_zero @ TA ) )
            & ( B
             != ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(125,axiom,
    ! [TA: $tType] :
      ( ( semidom_divide @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ B @ A ) @ A )
            = B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_102_nonzero__mult__div__cancel__right) ).

thf(626,plain,
    ! [TA: $tType] :
      ( ( semidom_divide @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ B @ A ) @ A )
            = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(214,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
            = ( ord_less @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_145_mult__less__cancel__left__neg) ).

thf(915,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
            = ( ord_less @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(43,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ ( suc @ A ) @ B )
        = ( times_times @ nat @ ( suc @ A ) @ C ) )
      = ( B = C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_81_Suc__mult__cancel1) ).

thf(399,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ ( suc @ A ) @ B )
        = ( times_times @ nat @ ( suc @ A ) @ C ) )
      = ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(202,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( divide_divide @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
        = ( divide_divide @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_100_nat__mult__div__cancel1) ).

thf(868,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( divide_divide @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
        = ( divide_divide @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(206,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( ! [C: nat] :
            ( ( A @ ( suc @ C ) )
           => ( A @ C ) )
       => ( A @ ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_47_zero__induct) ).

thf(883,plain,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( ! [C: nat] :
            ( ( A @ ( suc @ C ) )
           => ( A @ C ) )
       => ( A @ ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(243,axiom,
    semiri1193490041visors @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors) ).

thf(1015,plain,
    semiri1193490041visors @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(121,axiom,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( times_times @ TA @ C @ B ) )
          = ( ( B
              = ( zero_zero @ TA ) )
            | ( A = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_116_mult__cancel__right) ).

thf(608,plain,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( times_times @ TA @ C @ B ) )
          = ( ( B
              = ( zero_zero @ TA ) )
            | ( A = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(102,axiom,
    semiri1923998003cancel @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel) ).

thf(563,plain,
    semiri1923998003cancel @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(238,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ B @ ( divide_divide @ TA @ C @ A ) )
            = ( ord_less @ TA @ ( times_times @ TA @ B @ A ) @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_178_pos__less__divide__eq) ).

thf(1000,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ B @ ( divide_divide @ TA @ C @ A ) )
            = ( ord_less @ TA @ ( times_times @ TA @ B @ A ) @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(54,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( suc @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_73_less__SucI) ).

thf(438,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(135,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_51_old_Onat_Odistinct_I1_J) ).

thf(653,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(213,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
      = ( ? [B: nat] :
            ( A
            = ( suc @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_88_gr0__conv__Suc) ).

thf(912,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
      = ( ? [B: nat] :
            ( A
            = ( suc @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(85,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( divide_divide @ TA @ B @ A ) @ C )
            = ( ord_less @ TA @ B @ ( times_times @ TA @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_177_pos__divide__less__eq) ).

thf(521,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( divide_divide @ TA @ B @ A ) @ C )
            = ( ord_less @ TA @ B @ ( times_times @ TA @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(266,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ B )
       => ( ord_less @ nat @ A @ ( times_times @ nat @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_97_n__less__n__mult__m) ).

thf(1076,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ B )
       => ( ord_less @ nat @ A @ ( times_times @ nat @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(114,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ? [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
            & ( B @ C ) ) )
      = ( ( B @ A )
        | ? [C: nat] :
            ( ( ord_less @ nat @ C @ A )
            & ( B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_72_Ex__less__Suc) ).

thf(589,plain,
    ! [A: nat,B: nat > $o] :
      ( ( ? [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
            & ( B @ C ) ) )
      = ( ( B @ A )
        | ? [C: nat] :
            ( ( ord_less @ nat @ C @ A )
            & ( B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(95,axiom,
    ( ( divide_divide @ nat @ ( size_size @ ( list @ a ) @ xs ) @ k )
    = ( suc @ n ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).

thf(548,plain,
    ( ( divide_divide @ nat @ ( size_size @ ( list @ a ) @ xs ) @ k )
    = ( suc @ n ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(185,axiom,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) )
         => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_241_split__mult__pos__le) ).

thf(812,plain,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) )
         => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(268,axiom,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_143_zero__less__mult__pos) ).

thf(1081,plain,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(124,axiom,
    ! [A: nat > $o] :
      ( ~ ( A @ ( zero_zero @ nat ) )
     => ( ? [B: nat] : ( A @ B )
       => ? [B: nat] :
            ( ~ ( A @ B )
            & ( A @ ( suc @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_207_exists__least__lemma) ).

thf(617,plain,
    ! [A: nat > $o] :
      ( ~ ( A @ ( zero_zero @ nat ) )
     => ( ? [B: nat] : ( A @ B )
       => ? [B: nat] :
            ( ~ ( A @ B )
            & ( A @ ( suc @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(165,axiom,
    ( k
   != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_1) ).

thf(754,plain,
    ( k
   != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(89,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_59_not__less0) ).

thf(531,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(223,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_56_infinite__descent0) ).

thf(944,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)],[223]) ).

thf(177,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ ( divide_divide @ TA @ B @ A ) @ C )
            = ( ord_less @ TA @ ( times_times @ TA @ C @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_175_neg__divide__less__eq) ).

thf(795,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ ( divide_divide @ TA @ B @ A ) @ C )
            = ( ord_less @ TA @ ( times_times @ TA @ C @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(143,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( divide_divide @ TA @ A @ B ) @ ( divide_divide @ TA @ C @ B ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
             => ( ord_less @ TA @ A @ C ) )
            & ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
             => ( ord_less @ TA @ C @ A ) )
            & ( B
             != ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_159_divide__less__cancel) ).

thf(680,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( divide_divide @ TA @ A @ B ) @ ( divide_divide @ TA @ C @ B ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
             => ( ord_less @ TA @ A @ C ) )
            & ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
             => ( ord_less @ TA @ C @ A ) )
            & ( B
             != ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(62,axiom,
    ! [A: nat] :
      ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_60_not__gr0) ).

thf(458,plain,
    ! [A: nat] :
      ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(189,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ A )
      = ( ( A
         != ( zero_zero @ nat ) )
        & ( A
         != ( suc @ ( zero_zero @ nat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_226_nat__geSucSuc0__conv) ).

thf(823,plain,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ A )
      = ( ( A
         != ( zero_zero @ nat ) )
        & ( A
         != ( suc @ ( zero_zero @ nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(171,axiom,
    ! [A: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_11_zero__less__Suc) ).

thf(776,plain,
    ! [A: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(111,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_137_mult__less__0__iff) ).

thf(580,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less @ TA @ ( zero_zero @ TA ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(146,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) )
          = ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_235_mult__le__0__iff) ).

thf(690,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) )
          = ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(130,axiom,
    ! [TA: $tType] :
      ( ( linordered_ring @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ ( times_times @ TA @ A @ A ) @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_136_not__square__less__zero) ).

thf(640,plain,
    ! [TA: $tType] :
      ( ( linordered_ring @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ ( times_times @ TA @ A @ A ) @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(203,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ C @ A ) )
            = ( divide_divide @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_104_nonzero__mult__divide__mult__cancel__left2) ).

thf(872,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ C @ A ) )
            = ( divide_divide @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(228,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_233_mult__nonneg__nonneg) ).

thf(971,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(158,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_61_gr0I) ).

thf(739,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(99,axiom,
    ! [TA: $tType] :
      ( ( linordered_ring @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_242_zero__le__square) ).

thf(557,plain,
    ! [TA: $tType] :
      ( ( linordered_ring @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(239,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( B
         != ( suc @ A ) )
       => ~ ! [C: nat] :
              ( ( ord_less @ nat @ A @ C )
             => ( B
               != ( suc @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_78_Nat_OlessE) ).

thf(1003,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( B
         != ( suc @ A ) )
       => ~ ! [C: nat] :
              ( ( ord_less @ nat @ A @ C )
             => ( B
               != ( suc @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(273,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_7_nat__mult__less__cancel__disj) ).

thf(1098,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)],[273]) ).

thf(13,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( divide_divide @ TA @ A @ B ) )
          = ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_250_zero__le__divide__iff) ).

thf(310,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( divide_divide @ TA @ A @ B ) )
          = ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(258,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(1053,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(252,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ( ( ord_less @ TA )
        = ( ^ [A: TA,B: TA] :
            ! [C: TA] :
              ( ( ord_less_eq @ TA @ C @ A )
             => ( B != C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_224_greater__le__neq__conv) ).

thf(1038,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ( ( ord_less @ TA )
        = ( ^ [A: TA,B: TA] :
            ! [C: TA] :
              ( ( ord_less_eq @ TA @ C @ A )
             => ( B != C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(194,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_20_mult__cancel2) ).

thf(840,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)],[194]) ).

thf(64,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( times_times @ TA @ ( divide_divide @ TA @ A @ B ) @ ( divide_divide @ TA @ C @ D ) )
          = ( divide_divide @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_132_times__divide__times__eq) ).

thf(463,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( times_times @ TA @ ( divide_divide @ TA @ A @ B ) @ ( divide_divide @ TA @ C @ D ) )
          = ( divide_divide @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(109,axiom,
    mult_zero @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Omult__zero) ).

thf(576,plain,
    mult_zero @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(183,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring @ 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 @ ( zero_zero @ TA ) @ A )
             => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
               => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_243_mult__mono_H) ).

thf(808,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring @ 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 @ ( zero_zero @ TA ) @ A )
             => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
               => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(211,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_41_not0__implies__Suc) ).

thf(907,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(246,axiom,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_52_old_Onat_Odistinct_I2_J) ).

thf(1023,plain,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(123,axiom,
    ! [A: nat > nat > $o,B: nat,C: nat] :
      ( ! [D: nat] : ( A @ D @ ( zero_zero @ nat ) )
     => ( ! [D: nat] : ( A @ ( zero_zero @ nat ) @ ( suc @ D ) )
       => ( ! [D: nat,E: nat] :
              ( ( A @ D @ E )
             => ( A @ ( suc @ D ) @ ( suc @ E ) ) )
         => ( A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_48_diff__induct) ).

thf(613,plain,
    ! [A: nat > nat > $o,B: nat,C: nat] :
      ( ! [D: nat] : ( A @ D @ ( zero_zero @ nat ) )
     => ( ! [D: nat] : ( A @ ( zero_zero @ nat ) @ ( suc @ D ) )
       => ( ! [D: nat,E: nat] :
              ( ( A @ D @ E )
             => ( A @ ( suc @ D ) @ ( suc @ E ) ) )
         => ( A @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(240,axiom,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_138_mult__neg__pos) ).

thf(1008,plain,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(116,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( divide_divide @ TA @ A @ B ) @ C )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
             => ( ord_less @ TA @ A @ ( times_times @ TA @ C @ B ) ) )
            & ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
             => ( ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
                 => ( ord_less @ TA @ ( times_times @ TA @ C @ B ) @ A ) )
                & ( ~ ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
                 => ( ord_less @ TA @ ( zero_zero @ TA ) @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_173_divide__less__eq) ).

thf(595,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( divide_divide @ TA @ A @ B ) @ C )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
             => ( ord_less @ TA @ A @ ( times_times @ TA @ C @ B ) ) )
            & ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
             => ( ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
                 => ( ord_less @ TA @ ( times_times @ TA @ C @ B ) @ A ) )
                & ( ~ ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
                 => ( ord_less @ TA @ ( zero_zero @ TA ) @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(235,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less @ TA @ B @ C ) )
            | ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less @ TA @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_149_mult__less__cancel__left__disj) ).

thf(991,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
          = ( ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less @ TA @ B @ C ) )
            | ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less @ TA @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(218,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ( ord_less @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_77_Suc__lessD) ).

thf(931,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ( ord_less @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(70,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) )
            = ( ord_less @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_83_lift__Suc__mono__less__iff) ).

thf(482,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) )
            = ( ord_less @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(230,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ A )
      = ( ( A
         != ( zero_zero @ nat ) )
        & ( A
         != ( suc @ ( zero_zero @ nat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_200_nat__grSuc0__conv) ).

thf(976,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ A )
      = ( ( A
         != ( zero_zero @ nat ) )
        & ( A
         != ( suc @ ( zero_zero @ nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(86,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_58_less__zeroE) ).

thf(524,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(75,axiom,
    ! [TA: $tType] :
      ( ( semiri1193490041visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( zero_zero @ TA ) )
         => ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_128_divisors__zero) ).

thf(495,plain,
    ! [TA: $tType] :
      ( ( semiri1193490041visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( zero_zero @ TA ) )
         => ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(241,axiom,
    ! [TA: $tType,A: ( list @ TA ) > $o,B: list @ TA] :
      ( ! [C: list @ TA] :
          ( ! [D: list @ TA] :
              ( ( ord_less @ nat @ ( size_size @ ( list @ TA ) @ D ) @ ( size_size @ ( list @ TA ) @ C ) )
             => ( A @ D ) )
         => ( A @ C ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_202_length__induct) ).

thf(1010,plain,
    ! [TA: $tType,A: ( list @ TA ) > $o,B: list @ TA] :
      ( ! [C: list @ TA] :
          ( ! [D: list @ TA] :
              ( ( ord_less @ nat @ ( size_size @ ( list @ TA ) @ D ) @ ( size_size @ ( list @ TA ) @ C ) )
             => ( A @ D ) )
         => ( A @ C ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(149,axiom,
    ! [A: nat,B: nat] :
      ( ( ( divide_divide @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ( B
          = ( zero_zero @ nat ) )
        | ( ord_less @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_170_div__eq__0__conv) ).

thf(699,plain,
    ! [A: nat,B: nat] :
      ( ( ( divide_divide @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ( B
          = ( zero_zero @ nat ) )
        | ( ord_less @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(162,axiom,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_140_mult__pos__pos) ).

thf(748,plain,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(274,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ ( suc @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_70_not__less__eq) ).

thf(1101,plain,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ ( suc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(29,axiom,
    order @ $o,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oorder_3) ).

thf(351,plain,
    order @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(40,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ord_less @ nat @ B @ ( times_times @ nat @ ( suc @ ( divide_divide @ nat @ B @ A ) ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_108_less__div__Suc__mult) ).

thf(390,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ord_less @ nat @ B @ ( times_times @ nat @ ( suc @ ( divide_divide @ nat @ B @ A ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(127,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less @ TA @ ( divide_divide @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_155_divide__neg__pos) ).

thf(632,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less @ TA @ ( divide_divide @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(175,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( divide_divide @ TA @ ( divide_divide @ TA @ A @ B ) @ ( divide_divide @ TA @ C @ D ) )
          = ( divide_divide @ TA @ ( times_times @ TA @ A @ D ) @ ( times_times @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_133_divide__divide__times__eq) ).

thf(786,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( divide_divide @ TA @ ( divide_divide @ TA @ A @ B ) @ ( divide_divide @ TA @ C @ D ) )
          = ( divide_divide @ TA @ ( times_times @ TA @ A @ D ) @ ( times_times @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(173,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ ( times_times @ nat @ B @ C ) )
     => ( ord_less @ nat @ ( divide_divide @ nat @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_96_Euclidean__Division_Oless__mult__imp__div__less) ).

thf(781,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ ( times_times @ nat @ B @ C ) )
     => ( ord_less @ nat @ ( divide_divide @ nat @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(225,axiom,
    wellorder @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Owellorder) ).

thf(963,plain,
    wellorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(84,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B != A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_37_less__not__refl2) ).

thf(517,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(106,axiom,
    euclid191655569cancel @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel) ).

thf(568,plain,
    euclid191655569cancel @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(139,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_192_mult_Oassoc) ).

thf(667,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)],[139]) ).

thf(167,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
       => ( ord_less @ nat @ ( times_times @ nat @ C @ A ) @ ( times_times @ nat @ C @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_92_mult__less__mono2) ).

thf(763,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
       => ( ord_less @ nat @ ( times_times @ nat @ C @ A ) @ ( times_times @ nat @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(91,axiom,
    ! [TA: $tType,A: list @ TA,B: list @ TA] :
      ( ( ( size_size @ ( list @ TA ) @ A )
       != ( size_size @ ( list @ TA ) @ B ) )
     => ( A != B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_204_neq__if__length__neq) ).

thf(536,plain,
    ! [TA: $tType,A: list @ TA,B: list @ TA] :
      ( ( ( size_size @ ( list @ TA ) @ A )
       != ( size_size @ ( list @ TA ) @ B ) )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(255,axiom,
    ! [A: nat] :
      ( ( divide_divide @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_4_div__by__Suc__0) ).

thf(1045,plain,
    ! [A: nat] :
      ( ( divide_divide @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(52,axiom,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ ( divide_divide @ nat @ B @ C ) )
      = ( ( ( C
            = ( zero_zero @ nat ) )
          & ( A @ ( zero_zero @ nat ) ) )
        | ? [D: nat] :
            ( ( ord_less_eq @ nat @ ( times_times @ nat @ C @ D ) @ B )
            & ( ord_less @ nat @ B @ ( times_times @ nat @ C @ ( suc @ D ) ) )
            & ( A @ D ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_211_split__div_H) ).

thf(433,plain,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ ( divide_divide @ nat @ B @ C ) )
      = ( ( ( C
            = ( zero_zero @ nat ) )
          & ( A @ ( zero_zero @ nat ) ) )
        | ? [D: nat] :
            ( ( ord_less_eq @ nat @ ( times_times @ nat @ C @ D ) @ B )
            & ( ord_less @ nat @ B @ ( times_times @ nat @ C @ ( suc @ D ) ) )
            & ( A @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(81,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_14_bot__nat__0_Onot__eq__extremum) ).

thf(510,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(180,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ( A
          = ( suc @ ( zero_zero @ nat ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_199_nat__lessSucSuc0__conv) ).

thf(800,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ( A
          = ( suc @ ( zero_zero @ nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(4,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( divide_divide @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_248_divide__nonneg__nonpos) ).

thf(287,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( divide_divide @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(9,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( divide_divide @ TA @ A @ C ) @ ( divide_divide @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_251_divide__right__mono) ).

thf(300,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( divide_divide @ TA @ A @ C ) @ ( divide_divide @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(164,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_238_mult__left__mono) ).

thf(752,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(15,axiom,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA] :
          ( ( divide_divide @ TA @ A @ ( zero_zero @ TA ) )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_122_division__ring__divide__zero) ).

thf(315,plain,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA] :
          ( ( divide_divide @ TA @ A @ ( zero_zero @ TA ) )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(227,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_53_nat_Odistinct_I1_J) ).

thf(967,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(187,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
           => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ B @ A ) )
             => ( ord_less @ TA @ ( divide_divide @ TA @ C @ B ) @ ( divide_divide @ TA @ C @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_181_divide__strict__left__mono) ).

thf(818,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
           => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ B @ A ) )
             => ( ord_less @ TA @ ( divide_divide @ TA @ C @ B ) @ ( divide_divide @ TA @ C @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(199,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
     => ( ord_less @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_66_Suc__less__SucD) ).

thf(854,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
     => ( ord_less @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(103,axiom,
    linord20386208strict @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Olinordered__semiring__strict) ).

thf(564,plain,
    linord20386208strict @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(11,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( divide_divide @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_246_divide__nonpos__nonpos) ).

thf(305,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( divide_divide @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(19,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( divide_divide @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_249_divide__nonneg__nonneg) ).

thf(326,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( divide_divide @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(25,axiom,
    ! [TA: $tType] :
      ( ( semidom_divide @ TA )
     => ! [A: TA] :
          ( ( divide_divide @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_117_div__0) ).

thf(341,plain,
    ! [TA: $tType] :
      ( ( semidom_divide @ TA )
     => ! [A: TA] :
          ( ( divide_divide @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(79,axiom,
    zero @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ozero) ).

thf(506,plain,
    zero @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(32,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_27_measure__induct__rule) ).

thf(358,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)],[32]) ).

thf(208,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( times_times @ TA @ B @ A ) @ ( times_times @ TA @ C @ A ) )
            = ( ord_less @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_201_mult__less__iff1) ).

thf(897,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( times_times @ TA @ B @ A ) @ ( times_times @ TA @ C @ A ) )
            = ( ord_less @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(138,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ A ) @ B ) @ ( times_times @ nat @ ( suc @ A ) @ C ) )
      = ( ord_less @ nat @ B @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_94_Suc__mult__less__cancel1) ).

thf(664,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ A ) @ B ) @ ( times_times @ nat @ ( suc @ A ) @ C ) )
      = ( ord_less @ nat @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(250,axiom,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_139_mult__pos__neg) ).

thf(1033,plain,
    ! [TA: $tType] :
      ( ( linord20386208strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(224,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_21_mult__cancel1) ).

thf(960,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)],[224]) ).

thf(263,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) )
         => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_234_split__mult__neg__le) ).

thf(1065,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) )
         => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(148,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_10_less__Suc0) ).

thf(696,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(1,conjecture,
    ord_less @ nat @ ( times_times @ nat @ n @ k ) @ ( size_size @ ( list @ a ) @ xs ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_2) ).

thf(2,negated_conjecture,
    ~ ( ord_less @ nat @ ( times_times @ nat @ n @ k ) @ ( size_size @ ( list @ a ) @ xs ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(283,plain,
    ~ ( ord_less @ nat @ ( times_times @ nat @ n @ k ) @ ( size_size @ ( list @ a ) @ xs ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(217,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_191_mult_Ocommute) ).

thf(928,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)],[217]) ).

thf(7,axiom,
    ! [TA: $tType] :
      ( ( semidom_divide @ TA )
     => ! [A: TA] :
          ( ( divide_divide @ TA @ A @ ( zero_zero @ TA ) )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_119_div__by__0) ).

thf(294,plain,
    ! [TA: $tType] :
      ( ( semidom_divide @ TA )
     => ! [A: TA] :
          ( ( divide_divide @ TA @ A @ ( zero_zero @ TA ) )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(215,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_193_ab__semigroup__mult__class_Omult__ac_I1_J) ).

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

thf(271,axiom,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( B
              = ( divide_divide @ TA @ C @ A ) )
            = ( ( times_times @ TA @ B @ A )
              = C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_169_nonzero__eq__divide__eq) ).

thf(1093,plain,
    ! [TA: $tType] :
      ( ( division_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( B
              = ( divide_divide @ TA @ C @ A ) )
            = ( ( times_times @ TA @ B @ A )
              = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(234,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( divide_divide @ nat @ ( times_times @ nat @ B @ A ) @ A )
        = B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_0_div__mult__self__is__m) ).

thf(987,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( divide_divide @ nat @ ( times_times @ nat @ B @ A ) @ A )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(21,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( order @ TA )
     => ( order @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder) ).

thf(332,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( order @ TA )
     => ( order @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(50,axiom,
    ! [A: nat,B: nat] :
      ( ( ( divide_divide @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ( ord_less @ nat @ A @ B )
        | ( B
          = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_95_Euclidean__Division_Odiv__eq__0__iff) ).

thf(427,plain,
    ! [A: nat,B: nat] :
      ( ( ( divide_divide @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ( ord_less @ nat @ A @ B )
        | ( B
          = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(245,axiom,
    ! [TA: $tType] :
      ( ( size @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( size_size @ TA @ A )
           != ( size_size @ TA @ B ) )
         => ( A != B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_40_size__neq__size__imp__neq) ).

thf(1019,plain,
    ! [TA: $tType] :
      ( ( size @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( size_size @ TA @ A )
           != ( size_size @ TA @ B ) )
         => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(154,axiom,
    ! [TA: $tType] :
      ( ( semiri1193490041visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( B
             != ( zero_zero @ TA ) )
           => ( ( times_times @ TA @ A @ B )
             != ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_129_no__zero__divisors) ).

thf(723,plain,
    ! [TA: $tType] :
      ( ( semiri1193490041visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( B
             != ( zero_zero @ TA ) )
           => ( ( times_times @ TA @ A @ B )
             != ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(232,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_125_linordered__field__no__ub) ).

thf(982,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(56,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_187_not__gr__zero) ).

thf(442,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)],[56]) ).

thf(280,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ ( zero_zero @ TA ) )
           => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
             => ( ord_less @ TA @ ( divide_divide @ TA @ C @ A ) @ ( divide_divide @ TA @ C @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_182_divide__strict__left__mono__neg) ).

thf(1120,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ ( zero_zero @ TA ) )
           => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
             => ( ord_less @ TA @ ( divide_divide @ TA @ C @ A ) @ ( divide_divide @ TA @ C @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(34,axiom,
    ! [TA: $tType,A: nat > TA > $o,B: nat > TA > TA > $o] :
      ( ? [C: TA] : ( A @ ( zero_zero @ nat ) @ C )
     => ( ! [C: TA,D: nat] :
            ( ( A @ D @ C )
           => ? [E: TA] :
                ( ( A @ ( suc @ D ) @ E )
                & ( B @ D @ C @ E ) ) )
       => ? [C: nat > TA] :
          ! [D: nat] :
            ( ( A @ D @ ( C @ D ) )
            & ( B @ D @ ( C @ D ) @ ( C @ ( suc @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_206_dependent__nat__choice) ).

thf(369,plain,
    ! [TA: $tType,A: nat > TA > $o,B: nat > TA > TA > $o] :
      ( ? [C: TA] : ( A @ ( zero_zero @ nat ) @ C )
     => ( ! [C: TA,D: nat] :
            ( ( A @ D @ C )
           => ? [E: TA] :
                ( ( A @ ( suc @ D ) @ E )
                & ( B @ D @ C @ E ) ) )
       => ? [C: nat > TA] :
          ! [D: nat] :
            ( ( A @ D @ ( C @ D ) )
            & ( B @ D @ ( C @ D ) @ ( C @ ( suc @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(120,axiom,
    linorder @ nat,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Olinorder) ).

thf(607,plain,
    linorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(181,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_216_Suc__le__mono) ).

thf(803,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(23,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( divide_divide @ TA @ B @ C ) @ ( divide_divide @ TA @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_245_divide__right__mono__neg) ).

thf(337,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( divide_divide @ TA @ B @ C ) @ ( divide_divide @ TA @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(101,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_5_mult__less__cancel2) ).

thf(560,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)],[101]) ).

thf(73,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_231_mult__nonpos__nonneg) ).

thf(491,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(115,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ A )
     => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ B )
       => ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_99_one__less__mult) ).

thf(592,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ A )
     => ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ B )
       => ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(60,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] :
            ! [C: TA] :
              ( ( ord_less @ TA @ C @ A )
             => ( B != C ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_222_ge__less__neq__conv) ).

thf(454,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] :
            ! [C: TA] :
              ( ( ord_less @ TA @ C @ A )
             => ( B != C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(66,axiom,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
            = ( divide_divide @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_106_nonzero__mult__divide__mult__cancel__left) ).

thf(469,plain,
    ! [TA: $tType] :
      ( ( field @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( divide_divide @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) )
            = ( divide_divide @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(221,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ ( times_times @ nat @ B @ C ) )
     => ( ord_less @ nat @ ( divide_divide @ nat @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_172_Util__Div_Oless__mult__imp__div__less) ).

thf(940,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ ( times_times @ nat @ B @ C ) )
     => ( ord_less @ nat @ ( divide_divide @ nat @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(1211,plain,
    $false,
    inference(e,[status(thm)],[645,629,479,846,1031,893,1083,1110,352,683,1068,994,1073,379,440,677,1041,384,550,500,472,1063,670,709,973,565,436,1055,587,702,344,542,559,416,289,448,1013,284,770,789,321,376,514,937,348,806,485,834,634,396,660,507,527,602,766,1104,849,307,942,546,810,381,534,798,334,910,746,778,1042,302,598,820,297,424,693,815,566,361,879,852,583,461,493,757,551,324,1079,1027,742,650,1096,356,783,420,328,339,933,554,642,637,489,1016,706,686,476,837,984,569,503,393,1114,647,997,1050,457,921,657,1035,674,530,773,750,856,745,733,601,513,430,904,423,402,291,413,843,1125,875,826,577,445,572,313,611,466,1090,540,713,498,1058,1107,979,900,831,1117,451,318,1122,1048,736,387,1029,964,1061,345,858,799,726,626,915,399,868,883,1015,608,563,1000,438,653,912,521,1076,589,548,812,1081,617,754,531,944,795,680,458,823,776,580,690,640,872,971,739,557,1003,1098,310,1053,1038,840,463,576,808,907,1023,613,1008,595,991,931,482,976,524,495,1010,699,748,1101,351,390,632,786,781,963,517,568,667,763,536,1045,433,510,800,287,300,752,315,967,818,854,564,305,326,341,506,358,897,664,1033,960,1065,696,283,928,294,918,1093,987,332,427,1019,723,982,442,1120,369,607,803,337,560,491,592,454,469,940]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12  % Problem  : ITP103^2 : TPTP v8.2.0. Released v7.5.0.
% 0.11/0.15  % Command  : run_Leo-III %s %d
% 0.14/0.36  % Computer : n017.cluster.edu
% 0.14/0.36  % Model    : x86_64 x86_64
% 0.14/0.36  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36  % Memory   : 8042.1875MB
% 0.14/0.36  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36  % CPULimit : 300
% 0.14/0.36  % WCLimit  : 300
% 0.14/0.36  % DateTime : Sat May 18 15:40:24 EDT 2024
% 0.14/0.36  % CPUTime  : 
% 0.97/0.85  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 1.78/1.10  % [INFO] 	 Parsing done (242ms). 
% 1.78/1.11  % [INFO] 	 Running in sequential loop mode. 
% 2.60/1.32  % [INFO] 	 eprover registered as external prover. 
% 2.60/1.32  % [INFO] 	 cvc4 registered as external prover. 
% 2.60/1.33  % [INFO] 	 Scanning for conjecture ... 
% 2.75/1.42  % [INFO] 	 Found a conjecture (or negated_conjecture) and 280 axioms. Running axiom selection ... 
% 3.19/1.51  % [INFO] 	 Axiom selection finished. Selected 280 axioms (removed 0 axioms). 
% 3.82/1.67  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 3.82/1.69  % [INFO] 	 Type checking passed. 
% 3.82/1.69  % [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 ... 
% 18.39/4.70  % External prover 'e' found a proof!
% 18.39/4.71  % [INFO] 	 Killing All external provers ... 
% 18.39/4.71  % Time passed: 4191ms (effective reasoning time: 3598ms)
% 18.39/4.71  % 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)>
% 18.39/4.72  % Axioms used in derivation (280): fact_26_div__mult__mult1, fact_166_divide__eq__imp, fact_113_mult__zero__right, fact_214_bot__nat__0_Oextremum, fact_20_mult__cancel2, fact_6_nat__0__less__mult__iff, fact_165_eq__divide__eq, fact_108_less__div__Suc__mult, fact_41_not0__implies__Suc, fact_19_lessI, fact_69_All__less__Suc, fact_67_less__antisym, fact_85_less__Suc__eq__0__disj, fact_188_mult__cancel2__gr0, tcon_Product__Type_Oprod___Nat_Osize_5, fact_181_divide__strict__left__mono, fact_134_divide__divide__eq__left_H, fact_160_zero__less__divide__iff, fact_136_not__square__less__zero, fact_98_n__less__m__mult__n, fact_138_mult__neg__pos, fact_40_size__neq__size__imp__neq, fact_14_bot__nat__0_Onot__eq__extremum, tcon_Nat_Onat___Orderings_Olinorder, fact_236_mult__right__mono, fact_70_not__less__eq, tcon_Nat_Onat___Nat_Osize, fact_217_one__le__mult__iff, fact_140_mult__pos__pos, fact_77_Suc__lessD, fact_215_le0, fact_43_old_Onat_Oexhaust, fact_129_no__zero__divisors, fact_95_Euclidean__Division_Odiv__eq__0__iff, fact_51_old_Onat_Odistinct_I1_J, fact_63_strict__inc__induct, fact_240_mult__left__mono__neg, fact_114_mult__eq__0__iff, fact_153_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono, fact_163_frac__eq__eq, fact_110_divide__divide__eq__right, fact_60_not__gr0, fact_94_Suc__mult__less__cancel1, fact_159_divide__less__cancel, fact_84_lift__Suc__mono__less, fact_82_div__mult2__eq, fact_224_greater__le__neq__conv, fact_167_eq__divide__imp, fact_1_div__mult__self1__is__m, fact_101_nonzero__mult__divide__mult__cancel__right2, fact_190_mult_Oleft__commute, fact_142_zero__less__mult__iff, fact_28_measure__induct, fact_189_zero__reorient, tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors, fact_178_pos__less__divide__eq, fact_97_n__less__n__mult__m, fact_23_mult__is__0, fact_52_old_Onat_Odistinct_I2_J, fact_88_gr0__conv__Suc, fact_99_one__less__mult, fact_135_mult__neg__neg, fact_122_division__ring__divide__zero, fact_115_mult__cancel__left, fact_73_less__SucI, fact_17_Suc__less__eq, fact_231_mult__nonpos__nonneg, fact_172_Util__Div_Oless__mult__imp__div__less, tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, fact_176_neg__less__divide__eq, fact_10_less__Suc0, tcon_Nat_Onat___Rings_Oordered__semiring, fact_216_Suc__le__mono, fact_76_Suc__lessE, fact_21_mult__cancel1, fact_184_div__less__imp__less__mult, fact_208_field__lbound__gt__zero, fact_9_one__eq__mult__iff, fact_198_gr__implies__gr0, fact_221_zero__le, fact_105_nonzero__mult__div__cancel__left, fact_248_divide__nonneg__nonpos, fact_55_bot__nat__0_Oextremum__strict, tcon_List_Olist___Nat_Osize_4, fact_36_less__not__refl3, fact_132_times__divide__times__eq, fact_124_linordered__field__no__lb, fact_187_not__gr__zero, fact_205_list__decode_Ocases, tcon_Nat_Onat___Groups_Oab__semigroup__mult, fact_202_length__induct, fact_168_nonzero__divide__eq__eq, fact_120_divide__cancel__left, fact_92_mult__less__mono2, fact_173_divide__less__eq, fact_141_mult__pos__neg2, fact_90_nat__mult__less__cancel1, fact_243_mult__mono_H, fact_74_less__SucE, fact_107_mult__divide__mult__cancel__left__if, fact_200_nat__grSuc0__conv, fact_118_divide__eq__0__iff, fact_180_mult__imp__less__div__pos, fact_53_nat_Odistinct_I1_J, fact_8_mult__eq__1__iff, fact_89_Ex__less__Suc2, fact_46_Suc__neq__Zero, fact_123_times__divide__eq__left, fact_25_div__mult__mult2, fact_148_mult__strict__left__mono, fact_239_mult__nonpos__nonpos, fact_218_mult__le__cancel2, fact_242_zero__le__square, fact_226_nat__geSucSuc0__conv, tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel, fact_30_Suc__inject, fact_93_mult__less__mono1, fact_195_gr__implies__not__zero, fact_183_div__less__conv, fact_126_linorder__neqE__linordered__idom, fact_83_lift__Suc__mono__less__iff, fact_27_measure__induct__rule, fact_152_mult__less__cancel__right__disj, fact_62_not__less__less__Suc__eq, fact_86_gr0__implies__Suc, fact_220_le__numeral__extra_I3_J, fact_116_mult__cancel__right, fact_186_bits__div__by__0, tcon_Nat_Onat___Orderings_Owellorder, fact_229_zero__le__mult__iff, fact_162_divide__strict__right__mono__neg, tcon_Nat_Onat___Groups_Ozero, fact_197_gr__zeroI, fact_35_less__irrefl__nat, fact_75_Suc__lessI, fact_119_div__by__0, fact_16_neq0__conv, fact_223_less__ge__neq__conv, fact_102_nonzero__mult__div__cancel__right, fact_4_div__by__Suc__0, fact_64_less__Suc__induct, fact_39_nat__neq__iff, fact_175_neg__divide__less__eq, fact_161_divide__strict__right__mono, fact_112_mult__zero__left, fact_185_bits__div__0, tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict, fact_203_Ex__list__of__length, fact_244_mult__mono, fact_147_mult__strict__left__mono__neg, fact_228_ordered__comm__semiring__class_Ocomm__mult__left__mono, fact_170_div__eq__0__conv, fact_171_div__eq__0__conv_H, fact_174_less__divide__eq, fact_49_nat__induct, fact_192_mult_Oassoc, fact_206_dependent__nat__choice, fact_37_less__not__refl2, fact_232_mult__nonneg__nonpos, fact_79_nat__mult__eq__cancel__disj, fact_131_mult__right__cancel, fact_32_linorder__neqE__nat, fact_80_mult__0, tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel, fact_149_mult__less__cancel__left__disj, fact_104_nonzero__mult__divide__mult__cancel__left2, fact_117_div__0, fact_96_Euclidean__Division_Oless__mult__imp__div__less, tcon_Nat_Onat___Parity_Osemiring__bits, fact_29_n__not__Suc__n, tcon_Nat_Onat___Rings_Omult__zero, tcon_Nat_Onat___Rings_Oordered__semiring__0, fact_158_divide__less__0__iff, fact_251_divide__right__mono, fact_103_nonzero__mult__divide__mult__cancel__right, fact_209_less__numeral__extra_I3_J, fact_72_Ex__less__Suc, fact_201_mult__less__iff1, fact_234_split__mult__neg__le, fact_204_neq__if__length__neq, fact_100_nat__mult__div__cancel1, fact_194_zero__less__iff__neq__zero, fact_109_times__divide__eq__right, fact_42_old_Onat_Oinducts, fact_18_Suc__mono, fact_213_le__zero__eq, fact_13_nat_Oinject, fact_5_mult__less__cancel2, fact_196_not__less__zero, fact_34_nat__less__induct, fact_250_zero__le__divide__iff, fact_143_zero__less__mult__pos, fact_252_divide__le__0__iff, fact_139_mult__pos__neg, fact_210_length__product, tcon_HOL_Obool___Orderings_Oorder_3, fact_155_divide__neg__pos, fact_44_Zero__not__Suc, fact_179_mult__imp__div__pos__less, fact_235_mult__le__0__iff, fact_121_divide__cancel__right, fact_81_Suc__mult__cancel1, conj_1, fact_61_gr0I, fact_199_nat__lessSucSuc0__conv, fact_207_exists__least__lemma, fact_219_nat__mult__le__cancel__disj, fact_3_div__less, tcon_HOL_Obool___Orderings_Olinorder_2, fact_50_nat_OdiscI, fact_225_le__greater__neq__conv, fact_233_mult__nonneg__nonneg, fact_45_Zero__neq__Suc, fact_144_zero__less__mult__pos2, fact_65_less__trans__Suc, fact_87_All__less__Suc2, fact_157_divide__pos__pos, fact_7_nat__mult__less__cancel__disj, fact_133_divide__divide__times__eq, fact_130_mult__left__cancel, fact_177_pos__divide__less__eq, fact_212_le__less__div__conv, fact_48_diff__induct, fact_56_infinite__descent0, fact_24_div__mult__mult1__if, fact_227_nat__leSuc0__conv, fact_78_Nat_OlessE, fact_156_divide__pos__neg, fact_59_not__less0, fact_241_split__mult__pos__le, fact_151_mult__strict__right__mono, fact_247_divide__nonpos__nonneg, fact_191_mult_Ocommute, fact_145_mult__less__cancel__left__neg, fact_111_divide__divide__eq__left, tcon_Nat_Onat___Rings_Oordered__comm__semiring, tcon_fun___Orderings_Oorder, fact_0_div__mult__self__is__m, fact_57_gr__implies__not0, tcon_Nat_Onat___Orderings_Oorder_1, fact_66_Suc__less__SucD, fact_54_infinite__descent0__measure, fact_91_nat__mult__eq__cancel1, fact_58_less__zeroE, tcon_Nat_Onat___Groups_Osemigroup__mult, fact_12_old_Onat_Oinject, tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, fact_238_mult__left__mono, fact_2_nat__mult__div__cancel__disj, fact_164_divide__eq__eq, tcon_Nat_Onat___Rings_Osemidom__divide, fact_249_divide__nonneg__nonneg, fact_237_mult__right__mono__neg, fact_38_less__not__refl, tcon_Nat_Onat___Rings_Olinordered__semiring__strict, fact_128_divisors__zero, fact_154_divide__neg__neg, fact_22_mult__0__right, fact_71_less__Suc__eq, fact_31_infinite__descent__measure, conj_0, fact_33_infinite__descent, fact_150_mult__strict__right__mono__neg, fact_169_nonzero__eq__divide__eq, fact_246_divide__nonpos__nonpos, fact_68_Suc__less__eq2, fact_15_less__nat__zero__code, fact_222_ge__less__neq__conv, fact_146_mult__less__cancel__left__pos, fact_182_divide__strict__left__mono__neg, fact_127_mult__not__zero, fact_211_split__div_H, fact_47_zero__induct, fact_245_divide__right__mono__neg, fact_125_linordered__field__no__ub, fact_106_nonzero__mult__divide__mult__cancel__left, fact_193_ab__semigroup__mult__class_Omult__ac_I1_J, fact_11_zero__less__Suc, fact_230_mult__nonneg__nonpos2, fact_137_mult__less__0__iff
% 18.39/4.72  % No. of inferences in proof: 564
% 18.39/4.72  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 4191 ms resp. 3598 ms w/o parsing
% 18.84/4.87  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 18.84/4.87  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------