TSTP Solution File: ITP151^2 by Leo-III---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : ITP151^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n004.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Mon May 20 22:23:25 EDT 2024

% Result   : Theorem 84.03s 24.31s
% Output   : Refutation 84.30s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  349
% Syntax   : Number of formulae    :  655 ( 206 unt;  45 typ;   0 def)
%            Number of atoms       : 1698 ( 447 equ;   8 cnn)
%            Maximal formula atoms :   13 (   2 avg)
%            Number of connectives : 5810 ( 204   ~;  30   |;  76   &;4618   @)
%                                         (   0 <=>; 882  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   7 avg)
%            Number of types       :    4 (   3 usr)
%            Number of type conns  :  144 ( 144   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   48 (  44 usr;   7 con; 0-3 aty)
%            Number of variables   : 1617 (  36   ^1480   !;  62   ?;1617   :)
%                                         (  39  !>;   0  ?*;   0  @-;   0  @+)

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

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

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

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

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

thf(ord_type,type,
    ord: 
      !>[TA: $tType] : $o ).

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

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

thf(no_bot_type,type,
    no_bot: 
      !>[TA: $tType] : $o ).

thf(no_top_type,type,
    no_top: 
      !>[TA: $tType] : $o ).

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

thf(monoid_mult_type,type,
    monoid_mult: 
      !>[TA: $tType] : $o ).

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

thf(preorder_type,type,
    preorder: 
      !>[TA: $tType] : $o ).

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

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

thf(dense_order_type,type,
    dense_order: 
      !>[TA: $tType] : $o ).

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

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

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

thf(semiring_1_cancel_type,type,
    semiring_1_cancel: 
      !>[TA: $tType] : $o ).

thf(dense_linorder_type,type,
    dense_linorder: 
      !>[TA: $tType] : $o ).

thf(linordered_semidom_type,type,
    linordered_semidom: 
      !>[TA: $tType] : $o ).

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

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

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

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

thf(semiri134348788visors_type,type,
    semiri134348788visors: 
      !>[TA: $tType] : $o ).

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

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

thf(condit1656338222tinuum_type,type,
    condit1656338222tinuum: 
      !>[TA: $tType] : $o ).

thf(condit1037483654norder_type,type,
    condit1037483654norder: 
      !>[TA: $tType] : $o ).

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

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

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

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

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

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

thf(polyno498386536degree_type,type,
    polyno498386536degree: 
      !>[TA: $tType] : ( ( polyno1783536151e_poly @ TA ) > nat ) ).

thf(polyno367318022egreen_type,type,
    polyno367318022egreen: 
      !>[TA: $tType] : ( ( polyno1783536151e_poly @ TA ) > nat > nat ) ).

thf(polyno2122670676snpoly_type,type,
    polyno2122670676snpoly: 
      !>[TA: $tType] : ( ( polyno1783536151e_poly @ TA ) > $o ) ).

thf(polyno86455060npolyh_type,type,
    polyno86455060npolyh: 
      !>[TA: $tType] : ( ( polyno1783536151e_poly @ TA ) > nat > $o ) ).

thf(power_power_type,type,
    power_power: 
      !>[TA: $tType] : ( TA > nat > TA ) ).

thf(n_type,type,
    n: nat ).

thf(p_type,type,
    p: polyno1783536151e_poly @ a ).

thf(166,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: nat,C: TA] :
          ( ( ord_less_eq @ TA @ ( power_power @ TA @ A @ ( suc @ B ) ) @ ( power_power @ TA @ C @ ( suc @ B ) ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_power__le__imp__le__base) ).

thf(809,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: nat,C: TA] :
          ( ( ord_less_eq @ TA @ ( power_power @ TA @ A @ ( suc @ B ) ) @ ( power_power @ TA @ C @ ( suc @ B ) ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(73,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TA )
     => ( ord @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oord) ).

thf(518,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TA )
     => ( ord @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(87,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_less__asym_H) ).

thf(555,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(124,axiom,
    no_top @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Orderings_Ono__top) ).

thf(666,plain,
    no_top @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(13,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
           => ( A = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_antisym) ).

thf(352,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
           => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(114,axiom,
    linord1659791738miring @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Rings_Olinordered__nonzero__semiring) ).

thf(645,plain,
    linord1659791738miring @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(193,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ B ) @ ( minus_minus @ nat @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_diff__le__mono2) ).

thf(892,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ B ) @ ( minus_minus @ nat @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(61,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
           => ~ ( ord_less_eq @ TA @ B @ C ) )
         => ( ( ( ord_less_eq @ TA @ B @ A )
             => ~ ( ord_less_eq @ TA @ A @ C ) )
           => ( ( ( ord_less_eq @ TA @ A @ C )
               => ~ ( ord_less_eq @ TA @ C @ B ) )
             => ( ( ( ord_less_eq @ TA @ C @ B )
                 => ~ ( ord_less_eq @ TA @ B @ A ) )
               => ( ( ( ord_less_eq @ TA @ B @ C )
                   => ~ ( ord_less_eq @ TA @ C @ A ) )
                 => ~ ( ( ord_less_eq @ TA @ C @ A )
                     => ~ ( ord_less_eq @ TA @ A @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_le__cases3) ).

thf(479,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( ord_less_eq @ TA @ A @ B )
           => ~ ( ord_less_eq @ TA @ B @ C ) )
         => ( ( ( ord_less_eq @ TA @ B @ A )
             => ~ ( ord_less_eq @ TA @ A @ C ) )
           => ( ( ( ord_less_eq @ TA @ A @ C )
               => ~ ( ord_less_eq @ TA @ C @ B ) )
             => ( ( ( ord_less_eq @ TA @ C @ B )
                 => ~ ( ord_less_eq @ TA @ B @ A ) )
               => ( ( ( ord_less_eq @ TA @ B @ C )
                   => ~ ( ord_less_eq @ TA @ C @ A ) )
                 => ~ ( ( ord_less_eq @ TA @ C @ A )
                     => ~ ( ord_less_eq @ TA @ A @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(264,axiom,
    zero @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ozero_19) ).

thf(1105,plain,
    zero @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(212,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_nat_Odistinct_I1_J) ).

thf(962,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(58,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_order_Ostrict__trans) ).

thf(468,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(301,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( semiring_1_of_nat @ TA @ A ) )
          = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_of__nat__0__less__iff) ).

thf(1211,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( semiring_1_of_nat @ TA @ A ) )
          = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).

thf(130,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ B )
     => ( ! [C: nat] :
            ( ( A @ ( suc @ C ) )
           => ( A @ C ) )
       => ( A @ ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_zero__induct) ).

thf(683,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)],[130]) ).

thf(142,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_less__irrefl__nat) ).

thf(730,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(225,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_diff__le__mono) ).

thf(994,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(3,axiom,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ( type @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',subcl_Groups_Ozero___HOL_Otype) ).

thf(308,plain,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ( type @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(188,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_bot__nat__0_Onot__eq__extremum) ).

thf(878,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(241,axiom,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ( ( semiring_1_of_nat @ TA @ ( zero_zero @ nat ) )
        = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_of__nat__0) ).

thf(1041,plain,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ( ( semiring_1_of_nat @ TA @ ( zero_zero @ nat ) )
        = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(287,axiom,
    linord1659791738miring @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_6) ).

thf(1173,plain,
    linord1659791738miring @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(282,axiom,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat,B: nat,C: nat] :
          ( ( ( semiring_1_of_nat @ TA @ A )
            = ( power_power @ TA @ ( semiring_1_of_nat @ TA @ B ) @ C ) )
          = ( A
            = ( power_power @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_of__nat__power__eq__of__nat__cancel__iff) ).

thf(1158,plain,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat,B: nat,C: nat] :
          ( ( ( semiring_1_of_nat @ TA @ A )
            = ( power_power @ TA @ ( semiring_1_of_nat @ TA @ B ) @ C ) )
          = ( A
            = ( power_power @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(152,axiom,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ~ ( A @ C )
         => ? [D: nat] :
              ( ( ord_less @ nat @ D @ C )
              & ~ ( A @ D ) ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_infinite__descent) ).

thf(762,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)],[152]) ).

thf(25,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_le__cases) ).

thf(384,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(274,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_bot__nat__0_Oextremum__strict) ).

thf(1137,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(85,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
            = ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_antisym__conv) ).

thf(550,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
            = ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(133,axiom,
    ! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
      ( ! [D: TA] :
          ( ~ ( A @ D )
         => ? [E: TA] :
              ( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
              & ~ ( A @ E ) ) )
     => ( A @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_infinite__descent__measure) ).

thf(698,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)],[133]) ).

thf(80,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_less__imp__not__less) ).

thf(538,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(195,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_not__less0) ).

thf(898,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(227,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less_eq @ TA @ ( A @ ( suc @ D ) ) @ ( A @ D ) )
         => ( ( ord_less_eq @ nat @ B @ C )
           => ( ord_less_eq @ TA @ ( A @ C ) @ ( A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_lift__Suc__antimono__le) ).

thf(1001,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less_eq @ TA @ ( A @ ( suc @ D ) ) @ ( A @ D ) )
         => ( ( ord_less_eq @ nat @ B @ C )
           => ( ord_less_eq @ TA @ ( A @ C ) @ ( A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(271,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
          = ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_diff__gt__0__iff__gt) ).

thf(1127,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
          = ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(137,axiom,
    no_top @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Ono__top_16) ).

thf(709,plain,
    no_top @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(6,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_order_Ostrict__implies__order) ).

thf(320,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(93,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_Suc__inject) ).

thf(565,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(27,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_dual__order_Oasym) ).

thf(389,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(95,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
         => ( ~ ( ord_less @ TA @ A @ B )
           => ( ord_less @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_neqE) ).

thf(570,plain,
    ! [TA: $tType] :
      ( ( linorder @ 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)],[95]) ).

thf(202,axiom,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat] :
          ( ( ( semiring_1_of_nat @ TA @ A )
            = ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_of__nat__eq__0__iff) ).

thf(917,plain,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat] :
          ( ( ( semiring_1_of_nat @ TA @ A )
            = ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(247,axiom,
    wellorder @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Owellorder) ).

thf(1059,plain,
    wellorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(246,axiom,
    ! [A: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ A )
     => ~ ! [B: nat] :
            ( ( A
              = ( semiring_1_of_nat @ int @ B ) )
           => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_pos__int__cases) ).

thf(1055,plain,
    ! [A: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ A )
     => ~ ! [B: nat] :
            ( ( A
              = ( semiring_1_of_nat @ int @ B ) )
           => ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(140,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_diff__is__0__eq_H) ).

thf(724,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(207,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
           => ( ~ ( A @ C )
             => ? [D: nat] :
                  ( ( ord_less @ nat @ D @ C )
                  & ~ ( A @ D ) ) ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_infinite__descent0) ).

thf(934,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)],[207]) ).

thf(16,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ A @ B )
              & ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_order__class_Oorder_Oeq__iff) ).

thf(357,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ A @ B )
              & ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(294,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ B @ C )
       => ( ord_less @ nat @ ( suc @ A ) @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_less__trans__Suc) ).

thf(1191,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)],[294]) ).

thf(82,axiom,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ( ( '?' @ TA )
        = ( ^ [A: TA > $o] :
            ? [B: TA] :
              ( ( A @ B )
              & ! [C: TA] :
                  ( ( ord_less @ TA @ C @ B )
                 => ~ ( A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_exists__least__iff) ).

thf(542,plain,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ( ( '?' @ TA )
        = ( ^ [A: TA > $o] :
            ? [B: TA] :
              ( ( A @ B )
              & ! [C: TA] :
                  ( ( ord_less @ TA @ C @ B )
                 => ~ ( A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(54,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( minus_minus @ TA @ A @ C ) @ ( minus_minus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_diff__strict__right__mono) ).

thf(460,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( minus_minus @ TA @ A @ C ) @ ( minus_minus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(90,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_dual__order_Oirrefl) ).

thf(559,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(180,axiom,
    monoid_mult @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Omonoid__mult_14) ).

thf(853,plain,
    monoid_mult @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(215,axiom,
    ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_less__int__code_I1_J) ).

thf(969,plain,
    ~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(154,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: nat,C: TA] :
          ( ( ord_less @ TA @ ( power_power @ TA @ A @ B ) @ ( power_power @ TA @ C @ B ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_power__less__imp__less__base) ).

thf(770,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: nat,C: TA] :
          ( ( ord_less @ TA @ ( power_power @ TA @ A @ B ) @ ( power_power @ TA @ C @ B ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(189,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ( suc @ A )
         != B )
       => ( ord_less @ nat @ ( suc @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_Suc__lessI) ).

thf(881,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)],[189]) ).

thf(269,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less_eq @ nat @ A @ B )
         => ( ord_less_eq @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_of__nat__mono) ).

thf(1115,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less_eq @ nat @ A @ B )
         => ( ord_less_eq @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(262,axiom,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat] :
          ( ( semiring_1_of_nat @ TA @ ( suc @ A ) )
         != ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_of__nat__neq__0) ).

thf(1100,plain,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat] :
          ( ( semiring_1_of_nat @ TA @ ( suc @ A ) )
         != ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(159,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_less__numeral__extra_I3_J) ).

thf(789,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(266,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: nat] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( power_power @ TA @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_zero__le__power) ).

thf(1109,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: nat] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( power_power @ TA @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(22,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_less__irrefl) ).

thf(376,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(115,axiom,
    ! [A: nat] :
      ( A
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_n__not__Suc__n) ).

thf(646,plain,
    ! [A: nat] :
      ( A
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(175,axiom,
    ! [A: nat > $o] :
      ( ~ ( A @ ( zero_zero @ nat ) )
     => ( ? [B: nat] : ( A @ B )
       => ? [B: nat] :
            ( ~ ( A @ B )
            & ( A @ ( suc @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_exists__least__lemma) ).

thf(834,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)],[175]) ).

thf(12,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_ord__eq__less__trans) ).

thf(348,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(165,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
      = ( ? [C: nat] :
            ( ( B
              = ( suc @ C ) )
            & ( ord_less @ nat @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_Suc__less__eq2) ).

thf(806,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)],[165]) ).

thf(213,axiom,
    linordered_semidom @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Olinordered__semidom_9) ).

thf(966,plain,
    linordered_semidom @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(291,axiom,
    ( ( zero_zero @ int )
    = ( zero_zero @ int ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_zero__integer_Orsp) ).

thf(1183,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(220,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: TA,C: nat] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
             => ( ( ord_less_eq @ TA @ ( power_power @ TA @ A @ C ) @ ( power_power @ TA @ B @ C ) )
                = ( ord_less_eq @ TA @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_power__mono__iff) ).

thf(981,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: TA,C: nat] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
             => ( ( ord_less_eq @ TA @ ( power_power @ TA @ A @ C ) @ ( power_power @ TA @ B @ C ) )
                = ( ord_less_eq @ TA @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(252,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( ord_less_eq @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_le__diff__iff) ).

thf(1072,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( ord_less_eq @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(36,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ A @ B )
              & ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_eq__iff) ).

thf(411,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ A @ B )
              & ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(67,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = ( minus_minus @ TA @ C @ D ) )
         => ( ( ord_less_eq @ TA @ A @ B )
            = ( ord_less_eq @ TA @ C @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_diff__eq__diff__less__eq) ).

thf(507,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = ( minus_minus @ TA @ C @ D ) )
         => ( ( ord_less_eq @ TA @ A @ B )
            = ( ord_less_eq @ TA @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(285,axiom,
    semiri134348788visors @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5) ).

thf(1169,plain,
    semiri134348788visors @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(30,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ B @ A )
              & ( ord_less_eq @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_dual__order_Oeq__iff) ).

thf(397,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ B @ A )
              & ( ord_less_eq @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(42,axiom,
    ! [TA: $tType] :
      ( ( dense_order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ? [C: TA] :
              ( ( ord_less @ TA @ A @ C )
              & ( ord_less @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_dense) ).

thf(428,plain,
    ! [TA: $tType] :
      ( ( dense_order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ? [C: TA] :
              ( ( ord_less @ TA @ A @ C )
              & ( ord_less @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(297,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_zero__less__diff) ).

thf(1201,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(203,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( power_power @ TA @ ( semiring_1_of_nat @ TA @ A ) @ B ) )
          = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
            | ( B
              = ( zero_zero @ nat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_of__nat__zero__less__power__iff) ).

thf(920,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( power_power @ TA @ ( semiring_1_of_nat @ TA @ A ) @ B ) )
          = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
            | ( B
              = ( zero_zero @ nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(128,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: TA,C: TA] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
             => ( ( ( power_power @ TA @ B @ A )
                  = ( power_power @ TA @ C @ A ) )
                = ( B = C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_power__eq__iff__eq__base) ).

thf(678,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: TA,C: TA] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
             => ( ( ( power_power @ TA @ B @ A )
                  = ( power_power @ TA @ C @ A ) )
                = ( B = C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(76,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
         => ( ( ord_less_eq @ TA @ A @ B )
           => ( ord_less @ TA @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_order_Onot__eq__order__implies__strict) ).

thf(529,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
         => ( ( ord_less_eq @ TA @ A @ B )
           => ( ord_less @ TA @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(120,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( minus_minus @ TA @ C @ B ) @ ( minus_minus @ TA @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_diff__strict__left__mono) ).

thf(661,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less @ TA @ ( minus_minus @ TA @ C @ B ) @ ( minus_minus @ TA @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(280,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B != A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_less__not__refl2) ).

thf(1153,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B != A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(78,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( minus_minus @ TA @ C @ B ) @ ( minus_minus @ TA @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_diff__left__mono) ).

thf(534,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( minus_minus @ TA @ C @ B ) @ ( minus_minus @ TA @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(243,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ ( suc @ B ) )
        = ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_not__less__less__Suc__eq) ).

thf(1047,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)],[243]) ).

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

thf(512,plain,
    group_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(47,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_linordered__field__no__ub) ).

thf(439,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(200,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less @ nat @ B @ C )
           => ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_lift__Suc__mono__less) ).

thf(910,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)],[200]) ).

thf(305,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_diff__zero) ).

thf(1223,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).

thf(145,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( minus_minus @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( minus_minus @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_Nat_Odiff__diff__eq) ).

thf(746,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( minus_minus @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( minus_minus @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(172,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( minus_minus @ TA @ A @ B )
              = ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_eq__iff__diff__eq__0) ).

thf(825,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ( ( (=) @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( minus_minus @ TA @ A @ B )
              = ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(209,axiom,
    ! [A: nat,B: nat] :
      ( ( ( power_power @ nat @ A @ B )
        = ( suc @ ( zero_zero @ nat ) ) )
      = ( ( B
          = ( zero_zero @ nat ) )
        | ( A
          = ( suc @ ( zero_zero @ nat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_nat__power__eq__Suc__0__iff) ).

thf(952,plain,
    ! [A: nat,B: nat] :
      ( ( ( power_power @ nat @ A @ B )
        = ( suc @ ( zero_zero @ nat ) ) )
      = ( ( B
          = ( zero_zero @ nat ) )
        | ( A
          = ( suc @ ( zero_zero @ nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(288,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
      = ( ? [B: nat] :
            ( A
            = ( suc @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_gr0__conv__Suc) ).

thf(1174,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
      = ( ? [B: nat] :
            ( A
            = ( suc @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).

thf(303,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_cancel__comm__monoid__add__class_Odiff__cancel) ).

thf(1217,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).

thf(91,axiom,
    zero @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Ozero) ).

thf(561,plain,
    zero @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(131,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat] :
          ~ ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_of__nat__less__0__iff) ).

thf(693,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat] :
          ~ ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(178,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_less__not__refl) ).

thf(847,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(238,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
          = ( A
           != ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_zero__less__iff__neq__zero) ).

thf(1032,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)],[238]) ).

thf(45,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_nat_Oinject) ).

thf(435,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(123,axiom,
    ordered_ab_group_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Oordered__ab__group__add) ).

thf(665,plain,
    ordered_ab_group_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(192,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ~ ( ord_less @ nat @ A @ B )
       => ( ord_less @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_linorder__neqE__nat) ).

thf(889,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)],[192]) ).

thf(254,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ A )
       => ( ord_less @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_diff__less__mono) ).

thf(1079,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ A )
       => ( ord_less @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(138,axiom,
    ! [A: nat] :
      ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ A )
      = ( suc @ ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_power__Suc__0) ).

thf(710,plain,
    ! [A: nat] :
      ( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ A )
      = ( suc @ ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(265,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_not__gr__zero) ).

thf(1106,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)],[265]) ).

thf(100,axiom,
    cancel1352612707id_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Ocancel__comm__monoid__add) ).

thf(583,plain,
    cancel1352612707id_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(75,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( A
            = ( B @ C ) )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less_eq @ TA @ E @ F )
                 => ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_ord__eq__le__subst) ).

thf(522,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( A
            = ( B @ C ) )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less_eq @ TA @ E @ F )
                 => ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(62,axiom,
    order @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oorder_22) ).

thf(493,plain,
    order @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(300,axiom,
    ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_less__eq__int__code_I1_J) ).

thf(1210,plain,
    ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).

thf(98,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = ( minus_minus @ TA @ C @ D ) )
         => ( ( A = B )
            = ( C = D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_diff__eq__diff__eq) ).

thf(578,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = ( minus_minus @ TA @ C @ D ) )
         => ( ( A = B )
            = ( C = D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(258,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) )
          = ( ord_less @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_48_of__nat__less__iff) ).

thf(1089,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) )
          = ( ord_less @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(5,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ~ ( ord_less @ TA @ A @ B ) )
          = ( ( ord_less @ TA @ B @ A )
            | ( A = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_not__less__iff__gr__or__eq) ).

thf(317,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ~ ( ord_less @ TA @ A @ B ) )
          = ( ( ord_less @ TA @ B @ A )
            | ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(116,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_less__imp__neq) ).

thf(650,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(234,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ( B
         != ( suc @ A ) )
       => ~ ! [C: nat] :
              ( ( ord_less @ nat @ A @ C )
             => ( B
               != ( suc @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_Nat_OlessE) ).

thf(1017,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)],[234]) ).

thf(249,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_le__numeral__extra_I3_J) ).

thf(1064,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(267,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ nat @ A @ B )
         => ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_less__imp__of__nat__less) ).

thf(1111,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ nat @ A @ B )
         => ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(33,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ ( B @ C ) )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less_eq @ TA @ E @ F )
                 => ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_order__subst1) ).

thf(403,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ ( B @ C ) )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less_eq @ TA @ E @ F )
                 => ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(260,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_diff__le__self) ).

thf(1096,plain,
    ! [A: nat,B: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(15,axiom,
    semiri134348788visors @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors) ).

thf(356,plain,
    semiri134348788visors @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(104,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: $o] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ B @ A )
           => C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_less__imp__triv) ).

thf(593,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: $o] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ B @ A )
           => C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(110,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less @ TA @ A @ B )
              | ( B = A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_dual__order_Oorder__iff__strict) ).

thf(610,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less @ TA @ A @ B )
              | ( B = A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(65,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_order_Ostrict__implies__not__eq) ).

thf(499,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(182,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
          = ( ord_less_eq @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_diff__ge__0__iff__ge) ).

thf(857,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
          = ( ord_less_eq @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(157,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_gr0I) ).

thf(783,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(39,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_order__trans) ).

thf(420,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(190,axiom,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_old_Onat_Odistinct_I2_J) ).

thf(884,plain,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(8,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( wellorder @ TB )
     => ! [A: TA > TB,B: TA > $o,C: TA] :
          ( ! [D: TA] :
              ( ! [E: TA] :
                  ( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
                 => ( B @ E ) )
             => ( B @ D ) )
         => ( B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_measure__induct__rule) ).

thf(339,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( wellorder @ TB )
     => ! [A: TA > TB,B: TA > $o,C: TA] :
          ( ! [D: TA] :
              ( ! [E: TA] :
                  ( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
                 => ( B @ E ) )
             => ( B @ D ) )
         => ( B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(196,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_not__less__zero) ).

thf(901,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(113,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
         => ( ~ ( ord_less @ TA @ A @ B )
           => ( ord_less @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_linorder__neqE__linordered__idom) ).

thf(642,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)],[113]) ).

thf(302,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
      = ( ( ord_less @ nat @ A @ B )
        | ( ord_less @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_nat__neq__iff) ).

thf(1214,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)],[302]) ).

thf(136,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ A ) @ B ) @ ( suc @ C ) )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_Suc__diff__diff) ).

thf(706,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ A ) @ B ) @ ( suc @ C ) )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(237,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ( minus_minus @ nat @ B @ A )
            = ( minus_minus @ nat @ C @ A ) )
          = ( B = C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_eq__diff__iff) ).

thf(1028,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ( minus_minus @ nat @ B @ A )
            = ( minus_minus @ nat @ C @ A ) )
          = ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(255,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] : ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_le__iff__diff__le__0) ).

thf(1082,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ( ( ord_less_eq @ TA )
        = ( ^ [A: TA,B: TA] : ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(299,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: nat,C: TA] :
          ( ( ( power_power @ TA @ A @ ( suc @ B ) )
            = ( power_power @ TA @ C @ ( suc @ B ) ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
           => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
             => ( A = C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_power__inject__base) ).

thf(1207,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: nat,C: TA] :
          ( ( ( power_power @ TA @ A @ ( suc @ B ) )
            = ( power_power @ TA @ C @ ( suc @ B ) ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
           => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
             => ( A = C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).

thf(103,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( ord_less @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ A @ B )
              & ( B != A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_dual__order_Ostrict__iff__order) ).

thf(590,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ( ( ord_less @ TA )
        = ( ^ [A: TA,B: TA] :
              ( ( ord_less_eq @ TA @ A @ B )
              & ( B != A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(226,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( A != B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_less__not__refl3) ).

thf(997,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( A != B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(134,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ B @ ( minus_minus @ nat @ B @ A ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_diff__diff__cancel) ).

thf(701,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ B @ ( minus_minus @ nat @ B @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(117,axiom,
    ! [TA: $tType] :
      ( ( dense_linorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ! [D: TA] :
                ( ( ord_less @ TA @ A @ D )
               => ( ( ord_less @ TA @ D @ B )
                 => ( ord_less_eq @ TA @ D @ C ) ) )
           => ( ord_less_eq @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_dense__le__bounded) ).

thf(654,plain,
    ! [TA: $tType] :
      ( ( dense_linorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ! [D: TA] :
                ( ( ord_less @ TA @ A @ D )
               => ( ( ord_less @ TA @ D @ B )
                 => ( ord_less_eq @ TA @ D @ C ) ) )
           => ( ord_less_eq @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(221,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat,C: nat] :
          ( ( ord_less_eq @ TA @ ( power_power @ TA @ ( semiring_1_of_nat @ TA @ A ) @ B ) @ ( semiring_1_of_nat @ TA @ C ) )
          = ( ord_less_eq @ nat @ ( power_power @ nat @ A @ B ) @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_of__nat__le__of__nat__power__cancel__iff) ).

thf(984,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat,C: nat] :
          ( ( ord_less_eq @ TA @ ( power_power @ TA @ ( semiring_1_of_nat @ TA @ A ) @ B ) @ ( semiring_1_of_nat @ TA @ C ) )
          = ( ord_less_eq @ nat @ ( power_power @ nat @ A @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(94,axiom,
    semiring_1 @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Rings_Osemiring__1) ).

thf(569,plain,
    semiring_1 @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(71,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_less__not__sym) ).

thf(515,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(279,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_neq0__conv) ).

thf(1150,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(248,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
           => ? [C: TA] :
                ( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
                & ( ord_less @ TA @ C @ A )
                & ( ord_less @ TA @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_field__lbound__gt__zero) ).

thf(1060,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)],[248]) ).

thf(244,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B
           != ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_gr__implies__not__zero) ).

thf(1050,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)],[244]) ).

thf(53,axiom,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o,B: TA] :
          ( ! [C: TA] :
              ( ! [D: TA] :
                  ( ( ord_less @ TA @ D @ C )
                 => ( A @ D ) )
             => ( A @ C ) )
         => ( A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_less__induct) ).

thf(457,plain,
    ! [TA: $tType] :
      ( ( wellorder @ TA )
     => ! [A: TA > $o,B: TA] :
          ( ! [C: TA] :
              ( ! [D: TA] :
                  ( ( ord_less @ TA @ D @ C )
                 => ( A @ D ) )
             => ( A @ C ) )
         => ( A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(66,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( B = C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_ord__less__eq__trans) ).

thf(503,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( B = C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(108,axiom,
    ! [TA: $tType] :
      ( ( archim1804426504_field @ TA )
     => ! [A: TA] :
        ? [B: nat] : ( ord_less @ TA @ A @ ( semiring_1_of_nat @ TA @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_reals__Archimedean2) ).

thf(605,plain,
    ! [TA: $tType] :
      ( ( archim1804426504_field @ TA )
     => ! [A: TA] :
        ? [B: nat] : ( ord_less @ TA @ A @ ( semiring_1_of_nat @ TA @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(161,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_le__zero__eq) ).

thf(792,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)],[161]) ).

thf(257,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_less__Suc0) ).

thf(1086,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(17,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( A
            = ( B @ C ) )
         => ( ( ord_less @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less @ TA @ E @ F )
                 => ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_ord__eq__less__subst) ).

thf(360,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( A
            = ( B @ C ) )
         => ( ( ord_less @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less @ TA @ E @ F )
                 => ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(245,axiom,
    order @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oorder_17) ).

thf(1054,plain,
    order @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(286,axiom,
    ! [A: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A )
     => ? [B: nat] :
          ( A
          = ( semiring_1_of_nat @ int @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_zero__le__imp__eq__int) ).

thf(1170,plain,
    ! [A: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A )
     => ? [B: nat] :
          ( A
          = ( semiring_1_of_nat @ int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(239,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_diff__0__right) ).

thf(1035,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(89,axiom,
    preorder @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Orderings_Opreorder_1) ).

thf(558,plain,
    preorder @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(127,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B
       != ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_gr__implies__not0) ).

thf(674,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( B
       != ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(102,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( B = C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_ord__le__eq__trans) ).

thf(586,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( B = C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(191,axiom,
    semiring_1_cancel @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__1__cancel_10) ).

thf(888,plain,
    semiring_1_cancel @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(205,axiom,
    ! [A: nat > $o,B: nat] :
      ( ! [C: nat] :
          ( ! [D: nat] :
              ( ( ord_less @ nat @ D @ C )
             => ( A @ D ) )
         => ( A @ C ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_nat__less__induct) ).

thf(926,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)],[205]) ).

thf(146,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_bot__nat__0_Oextremum) ).

thf(750,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(41,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
          | ( A = B )
          | ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_less__linear) ).

thf(425,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
          | ( A = B )
          | ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(70,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_dual__order_Ostrict__implies__order) ).

thf(513,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(263,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less_eq @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less_eq @ nat @ B @ C )
           => ( ord_less_eq @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_lift__Suc__mono__le) ).

thf(1103,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less_eq @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less_eq @ nat @ B @ C )
           => ( ord_less_eq @ TA @ ( A @ B ) @ ( A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(96,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = ( minus_minus @ TA @ C @ D ) )
         => ( ( ord_less @ TA @ A @ B )
            = ( ord_less @ TA @ C @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_diff__eq__diff__less) ).

thf(573,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ( minus_minus @ TA @ A @ B )
            = ( minus_minus @ TA @ C @ D ) )
         => ( ( ord_less @ TA @ A @ B )
            = ( ord_less @ TA @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(143,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( A @ C )
           => ( A @ ( suc @ C ) ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_nat__induct) ).

thf(733,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)],[143]) ).

thf(101,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB,C: TA] :
          ( ( ord_less_eq @ ( TA > TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_le__funE) ).

thf(584,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB,C: TA] :
          ( ( ord_less_eq @ ( TA > TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(170,axiom,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: polyno1783536151e_poly @ TA,B: nat,C: nat] :
          ( ( polyno86455060npolyh @ TA @ A @ B )
         => ( ( ord_less @ nat @ C @ B )
           => ( ( polyno367318022egreen @ TA @ A @ C )
              = ( zero_zero @ nat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_degreen__0) ).

thf(819,plain,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: polyno1783536151e_poly @ TA,B: nat,C: nat] :
          ( ( polyno86455060npolyh @ TA @ A @ B )
         => ( ( ord_less @ nat @ C @ B )
           => ( ( polyno367318022egreen @ TA @ A @ C )
              = ( zero_zero @ nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(230,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ? [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
            & ( B @ C ) ) )
      = ( ( B @ A )
        | ? [C: nat] :
            ( ( ord_less @ nat @ C @ A )
            & ( B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_Ex__less__Suc) ).

thf(1007,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)],[230]) ).

thf(176,axiom,
    zero @ a,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tfree_0) ).

thf(843,plain,
    zero @ a,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(155,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ~ ! [C: nat] :
            ( ( ord_less @ nat @ A @ C )
           => ( B
             != ( suc @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_Suc__lessE) ).

thf(772,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)],[155]) ).

thf(144,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat,C: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( power_power @ TA @ ( semiring_1_of_nat @ TA @ B ) @ C ) )
          = ( ord_less @ nat @ A @ ( power_power @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_of__nat__power__less__of__nat__cancel__iff) ).

thf(743,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat,C: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( power_power @ TA @ ( semiring_1_of_nat @ TA @ B ) @ C ) )
          = ( ord_less @ nat @ A @ ( power_power @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(211,axiom,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_Suc__neq__Zero) ).

thf(958,plain,
    ! [A: nat] :
      ( ( suc @ A )
     != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(83,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( A = B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_eq__refl) ).

thf(545,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( A = B )
         => ( ord_less_eq @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(92,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
           => ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_dual__order_Oantisym) ).

thf(562,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
           => ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(198,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: nat > TA,B: nat,C: nat] :
          ( ! [D: nat] : ( ord_less @ TA @ ( A @ D ) @ ( A @ ( suc @ D ) ) )
         => ( ( ord_less @ TA @ ( A @ B ) @ ( A @ C ) )
            = ( ord_less @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_lift__Suc__mono__less__iff) ).

thf(904,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)],[198]) ).

thf(218,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
           => ( B @ C ) ) )
      = ( ( B @ ( zero_zero @ nat ) )
        & ! [C: nat] :
            ( ( ord_less @ nat @ C @ A )
           => ( B @ ( suc @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_All__less__Suc2) ).

thf(975,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)],[218]) ).

thf(112,axiom,
    preorder @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Opreorder_20) ).

thf(641,plain,
    preorder @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(28,axiom,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat,B: nat] :
          ( ( ( semiring_1_of_nat @ TA @ A )
            = ( semiring_1_of_nat @ TA @ B ) )
          = ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_of__nat__eq__iff) ).

thf(391,plain,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat,B: nat] :
          ( ( ( semiring_1_of_nat @ TA @ A )
            = ( semiring_1_of_nat @ TA @ B ) )
          = ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(49,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_less__trans) ).

thf(445,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ B @ C )
           => ( ord_less @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(24,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
          | ( ord_less_eq @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_linear) ).

thf(382,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
          | ( ord_less_eq @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(139,axiom,
    ! [A: nat,B: nat,C: nat > $o] :
      ( ( ord_less @ nat @ A @ B )
     => ( ! [D: nat] :
            ( ( B
              = ( suc @ D ) )
           => ( C @ D ) )
       => ( ! [D: nat] :
              ( ( ord_less @ nat @ D @ B )
             => ( ( C @ ( suc @ D ) )
               => ( C @ D ) ) )
         => ( C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_strict__inc__induct) ).

thf(713,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)],[139]) ).

thf(259,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_polypow_Ocases) ).

thf(1092,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(135,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_Suc__mono) ).

thf(704,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)],[135]) ).

thf(81,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_less__asym) ).

thf(540,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(295,axiom,
    ! [A: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ A )
     => ? [B: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
          & ( A
            = ( semiring_1_of_nat @ int @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_zero__less__imp__eq__int) ).

thf(1194,plain,
    ! [A: int] :
      ( ( ord_less @ int @ ( zero_zero @ int ) @ A )
     => ? [B: nat] :
          ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
          & ( A
            = ( semiring_1_of_nat @ int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(34,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less @ TB @ A @ B )
         => ( ( ord_less @ TA @ ( C @ B ) @ D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less @ TB @ E @ F )
                 => ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_order__less__subst2) ).

thf(406,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less @ TB @ A @ B )
         => ( ( ord_less @ TA @ ( C @ B ) @ D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less @ TB @ E @ F )
                 => ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(174,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( minus_minus @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_diff__Suc__Suc) ).

thf(831,plain,
    ! [A: nat,B: nat] :
      ( ( minus_minus @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( minus_minus @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(129,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ( ord_less @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_Suc__lessD) ).

thf(681,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ B )
     => ( ord_less @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(270,axiom,
    ! [A: nat > $o,B: nat] :
      ( ( A @ ( zero_zero @ nat ) )
     => ( ! [C: nat] :
            ( ( A @ C )
           => ( A @ ( suc @ C ) ) )
       => ( A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_old_Onat_Oinducts) ).

thf(1117,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)],[270]) ).

thf(232,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_not0__implies__Suc) ).

thf(1011,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(223,axiom,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = A )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_zero__reorient) ).

thf(990,plain,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = A )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(51,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ B )
         => ( ( ord_less_eq @ TA @ ( C @ B ) @ D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less_eq @ TB @ E @ F )
                 => ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_order__subst2) ).

thf(451,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ B )
         => ( ( ord_less_eq @ TA @ ( C @ B ) @ D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less_eq @ TB @ E @ F )
                 => ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(276,axiom,
    linorder @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Olinorder_13) ).

thf(1144,plain,
    linorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(106,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_ord__eq__le__trans) ).

thf(599,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A = B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(184,axiom,
    ! [A: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_zero__less__Suc) ).

thf(863,plain,
    ! [A: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(60,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_order_Oasym) ).

thf(477,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ~ ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(167,axiom,
    polyno86455060npolyh @ a @ p @ n,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_1) ).

thf(811,plain,
    polyno86455060npolyh @ a @ p @ n,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(63,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ C ) @ ( minus_minus @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_diff__right__mono) ).

thf(494,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ C ) @ ( minus_minus @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(150,axiom,
    preorder @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Opreorder_12) ).

thf(758,plain,
    preorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(169,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A ) @ ( semiring_1_of_nat @ int @ B ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_zle__int) ).

thf(816,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A ) @ ( semiring_1_of_nat @ int @ B ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(74,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB,C: TA] :
          ( ( ord_less_eq @ ( TA > TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_le__funD) ).

thf(520,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB,C: TA] :
          ( ( ord_less_eq @ ( TA > TB ) @ A @ B )
         => ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(194,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_less__nat__zero__code) ).

thf(895,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(235,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less @ nat @ A @ B )
     => ( ( ord_less @ nat @ A @ ( suc @ B ) )
       => ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_less__antisym) ).

thf(1022,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)],[235]) ).

thf(289,axiom,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ( ( polyno2122670676snpoly @ TA )
        = ( ^ [A: polyno1783536151e_poly @ TA] : ( polyno86455060npolyh @ TA @ A @ ( zero_zero @ nat ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_isnpoly__def) ).

thf(1177,plain,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ( ( polyno2122670676snpoly @ TA )
        = ( ^ [A: polyno1783536151e_poly @ TA] : ( polyno86455060npolyh @ TA @ A @ ( zero_zero @ nat ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(55,axiom,
    cancel146912293up_add @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add) ).

thf(462,plain,
    cancel146912293up_add @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(10,axiom,
    linorder @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Orderings_Olinorder) ).

thf(345,plain,
    linorder @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(18,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( ord_less @ TB @ A @ ( B @ C ) )
         => ( ( ord_less @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less @ TA @ E @ F )
                 => ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_order__less__subst1) ).

thf(367,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( order @ TA )
        & ( order @ TB ) )
     => ! [A: TB,B: TA > TB,C: TA,D: TA] :
          ( ( ord_less @ TB @ A @ ( B @ C ) )
         => ( ( ord_less @ TA @ C @ D )
           => ( ! [E: TA,F: TA] :
                  ( ( ord_less @ TA @ E @ F )
                 => ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
             => ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(201,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( suc @ B ) )
     => ( A
       != ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_nat_OdiscI) ).

thf(912,plain,
    ! [A: nat,B: nat] :
      ( ( A
        = ( suc @ B ) )
     => ( A
       != ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(296,axiom,
    ! [TA: $tType] :
      ( ( semiri134348788visors @ TA )
     => ! [A: TA,B: nat] :
          ( ( ( power_power @ TA @ A @ B )
            = ( zero_zero @ TA ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            & ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_power__eq__0__iff) ).

thf(1198,plain,
    ! [TA: $tType] :
      ( ( semiri134348788visors @ TA )
     => ! [A: TA,B: nat] :
          ( ( ( power_power @ TA @ A @ B )
            = ( zero_zero @ TA ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            & ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(250,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( ord_less @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_Suc__less__eq) ).

thf(1066,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)],[250]) ).

thf(268,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: TA,C: nat] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
           => ( ord_less_eq @ TA @ ( power_power @ TA @ A @ C ) @ ( power_power @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_power__mono) ).

thf(1113,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: TA,C: nat] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
           => ( ord_less_eq @ TA @ ( power_power @ TA @ A @ C ) @ ( power_power @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(46,axiom,
    semiring_1_cancel @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Rings_Osemiring__1__cancel) ).

thf(438,plain,
    semiring_1_cancel @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(43,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB] :
          ( ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) )
         => ( ord_less_eq @ ( TA > TB ) @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_le__funI) ).

thf(431,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ! [A: TA > TB,B: TA > TB] :
          ( ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) )
         => ( ord_less_eq @ ( TA > TB ) @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(118,axiom,
    linordered_semidom @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Rings_Olinordered__semidom) ).

thf(658,plain,
    linordered_semidom @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(242,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
     => ( ~ ( ord_less @ nat @ A @ B )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_less__SucE) ).

thf(1044,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)],[242]) ).

thf(132,axiom,
    ! [A: nat] :
      ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_not__gr0) ).

thf(695,plain,
    ! [A: nat] :
      ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(29,axiom,
    ! [A: nat,B: nat] :
      ( ( ( semiring_1_of_nat @ int @ A )
        = ( semiring_1_of_nat @ int @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_int__int__eq) ).

thf(394,plain,
    ! [A: nat,B: nat] :
      ( ( ( semiring_1_of_nat @ int @ A )
        = ( semiring_1_of_nat @ int @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(1,conjecture,
    ( ( polyno498386536degree @ a @ p )
    = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_2) ).

thf(2,negated_conjecture,
    ( ( polyno498386536degree @ a @ p )
   != ( zero_zero @ nat ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(306,plain,
    ( ( polyno498386536degree @ a @ p )
   != ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(253,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( suc @ ( minus_minus @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_Suc__pred) ).

thf(1076,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ( ( suc @ ( minus_minus @ nat @ A @ ( suc @ ( zero_zero @ nat ) ) ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(162,axiom,
    ord @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oord_18) ).

thf(795,plain,
    ord @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(121,axiom,
    order @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Orderings_Oorder_2) ).

thf(663,plain,
    order @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(21,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_order__refl) ).

thf(374,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(273,axiom,
    ! [TA: $tType] :
      ( ( semiri134348788visors @ TA )
     => ! [A: TA,B: nat] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( power_power @ TA @ A @ B )
           != ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_power__not__zero) ).

thf(1134,plain,
    ! [TA: $tType] :
      ( ( semiri134348788visors @ TA )
     => ! [A: TA,B: nat] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( power_power @ TA @ A @ B )
           != ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(168,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ B )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ B @ C ) )
          = ( ord_less_eq @ nat @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_le__diff__iff_H) ).

thf(812,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ B )
       => ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ B @ C ) )
          = ( ord_less_eq @ nat @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(9,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less @ TA @ A @ B )
         => ( ( A != B )
           => ( ord_less @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_linorder__cases) ).

thf(342,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less @ TA @ A @ B )
         => ( ( A != B )
           => ( ord_less @ TA @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(277,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ A )
     => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_nat__one__le__power) ).

thf(1145,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ A )
     => ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

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

thf(355,plain,
    monoid_mult @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(148,axiom,
    canoni770627133id_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add) ).

thf(754,plain,
    canoni770627133id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(86,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ D ) @ ( minus_minus @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_diff__mono) ).

thf(553,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ D ) @ ( minus_minus @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(156,axiom,
    ! [A: nat,B: nat,C: nat > nat > $o] :
      ( ( ord_less @ nat @ A @ B )
     => ( ! [D: nat] : ( C @ D @ ( suc @ D ) )
       => ( ! [D: nat,E: nat,F: nat] :
              ( ( ord_less @ nat @ D @ E )
             => ( ( ord_less @ nat @ E @ F )
               => ( ( C @ D @ E )
                 => ( ( C @ E @ F )
                   => ( C @ D @ F ) ) ) ) )
         => ( C @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_less__Suc__induct) ).

thf(776,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)],[156]) ).

thf(236,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat,C: nat] :
          ( ( ord_less @ TA @ ( power_power @ TA @ ( semiring_1_of_nat @ TA @ A ) @ B ) @ ( semiring_1_of_nat @ TA @ C ) )
          = ( ord_less @ nat @ ( power_power @ nat @ A @ B ) @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_of__nat__less__of__nat__power__cancel__iff) ).

thf(1025,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat,C: nat] :
          ( ( ord_less @ TA @ ( power_power @ TA @ ( semiring_1_of_nat @ TA @ A ) @ B ) @ ( semiring_1_of_nat @ TA @ C ) )
          = ( ord_less @ nat @ ( power_power @ nat @ A @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(35,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_dual__order_Orefl) ).

thf(409,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(224,axiom,
    ord_less @ nat @ ( zero_zero @ nat ) @ n,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).

thf(993,plain,
    ord_less @ nat @ ( zero_zero @ nat ) @ n,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(88,axiom,
    semiring_char_0 @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Nat_Osemiring__char__0) ).

thf(557,plain,
    semiring_char_0 @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(228,axiom,
    semiring_char_0 @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Nat_Osemiring__char__0_11) ).

thf(1003,plain,
    semiring_char_0 @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(37,axiom,
    linorder @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Olinorder_21) ).

thf(414,plain,
    linorder @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(216,axiom,
    cancel146912293up_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_7) ).

thf(971,plain,
    cancel146912293up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(261,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( suc @ A ) @ ( suc @ B ) )
     => ( ord_less @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_Suc__less__SucD) ).

thf(1098,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)],[261]) ).

thf(4,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less @ TB @ A @ B )
         => ( ( ( C @ B )
              = D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less @ TB @ E @ F )
                 => ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_ord__less__eq__subst) ).

thf(310,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less @ TB @ A @ B )
         => ( ( ( C @ B )
              = D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less @ TB @ E @ F )
                 => ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(160,axiom,
    semiring_1 @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__1_15) ).

thf(791,plain,
    semiring_1 @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(293,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ? [C: nat] :
            ( ( A
              = ( suc @ C ) )
            & ( ord_less @ nat @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_less__Suc__eq__0__disj) ).

thf(1188,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)],[293]) ).

thf(48,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_less__imp__not__eq) ).

thf(441,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(56,axiom,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_linordered__field__no__lb) ).

thf(463,plain,
    ! [TA: $tType] :
      ( ( linordered_field @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(272,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ord_less @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( ord_less @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_less__diff__iff) ).

thf(1130,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ A @ C )
       => ( ( ord_less @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
          = ( ord_less @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(304,axiom,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ! [A: nat] :
          ( ( power_power @ TA @ ( zero_zero @ TA ) @ ( suc @ A ) )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_power__0__Suc) ).

thf(1220,plain,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ! [A: nat] :
          ( ( power_power @ TA @ ( zero_zero @ TA ) @ ( suc @ A ) )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).

thf(199,axiom,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ! [A: nat,B: nat] :
          ( ( semiring_1_of_nat @ TA @ ( power_power @ nat @ A @ B ) )
          = ( power_power @ TA @ ( semiring_1_of_nat @ TA @ A ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_of__nat__power) ).

thf(907,plain,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ! [A: nat,B: nat] :
          ( ( semiring_1_of_nat @ TA @ ( power_power @ nat @ A @ B ) )
          = ( power_power @ TA @ ( semiring_1_of_nat @ TA @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(240,axiom,
    ! [A: nat,B: nat] :
      ( ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_diff__is__0__eq) ).

thf(1038,plain,
    ! [A: nat,B: nat] :
      ( ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(19,axiom,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( minus_minus @ TA @ ( minus_minus @ TA @ A @ C ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_diff__right__commute) ).

thf(370,plain,
    ! [TA: $tType] :
      ( ( cancel146912293up_add @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( minus_minus @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( minus_minus @ TA @ ( minus_minus @ TA @ A @ C ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(256,axiom,
    comm_monoid_diff @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__diff) ).

thf(1085,plain,
    comm_monoid_diff @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(59,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ B )
         => ( ( ( C @ B )
              = D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less_eq @ TB @ E @ F )
                 => ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_ord__le__eq__subst) ).

thf(470,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ( ord @ TA )
        & ( ord @ TB ) )
     => ! [A: TB,B: TB,C: TB > TA,D: TA] :
          ( ( ord_less_eq @ TB @ A @ B )
         => ( ( ( C @ B )
              = D )
           => ( ! [E: TB,F: TB] :
                  ( ( ord_less_eq @ TB @ E @ F )
                 => ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
             => ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(97,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( preorder @ TA )
     => ( preorder @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Opreorder) ).

thf(576,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( preorder @ TA )
     => ( preorder @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(111,axiom,
    ! [TA: $tType] :
      ( ( condit1037483654norder @ TA )
     => ! [A: TA,B: TA,C: TA > $o] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( C @ A )
           => ( ~ ( C @ B )
             => ? [D: TA] :
                  ( ( ord_less_eq @ TA @ A @ D )
                  & ( ord_less_eq @ TA @ D @ B )
                  & ! [E: TA] :
                      ( ( ( ord_less_eq @ TA @ A @ E )
                        & ( ord_less @ TA @ E @ D ) )
                     => ( C @ E ) )
                  & ! [E: TA] :
                      ( ! [F: TA] :
                          ( ( ( ord_less_eq @ TA @ A @ F )
                            & ( ord_less @ TA @ F @ E ) )
                         => ( C @ F ) )
                     => ( ord_less_eq @ TA @ E @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_complete__interval) ).

thf(613,plain,
    ! [TA: $tType] :
      ( ( condit1037483654norder @ TA )
     => ! [A: TA,B: TA,C: TA > $o] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( C @ A )
           => ( ~ ( C @ B )
             => ? [D: TA] :
                  ( ( ord_less_eq @ TA @ A @ D )
                  & ( ord_less_eq @ TA @ D @ B )
                  & ! [E: TA] :
                      ( ( ( ord_less_eq @ TA @ A @ E )
                        & ( ord_less @ TA @ E @ D ) )
                     => ( C @ E ) )
                  & ! [E: TA] :
                      ( ! [F: TA] :
                          ( ( ( ord_less_eq @ TA @ A @ F )
                            & ( ord_less @ TA @ F @ E ) )
                         => ( C @ F ) )
                     => ( ord_less_eq @ TA @ E @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(141,axiom,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat] :
          ( ( ( zero_zero @ TA )
            = ( semiring_1_of_nat @ TA @ A ) )
          = ( ( zero_zero @ nat )
            = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_of__nat__0__eq__iff) ).

thf(727,plain,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat] :
          ( ( ( zero_zero @ TA )
            = ( semiring_1_of_nat @ TA @ A ) )
          = ( ( zero_zero @ nat )
            = A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(105,axiom,
    ! [TA: $tType] :
      ( ( dense_linorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ! [D: TA] :
                ( ( ord_less @ TA @ A @ D )
               => ( ( ord_less @ TA @ D @ B )
                 => ( ord_less_eq @ TA @ C @ D ) ) )
           => ( ord_less_eq @ TA @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_dense__ge__bounded) ).

thf(595,plain,
    ! [TA: $tType] :
      ( ( dense_linorder @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ! [D: TA] :
                ( ( ord_less @ TA @ A @ D )
               => ( ( ord_less @ TA @ D @ B )
                 => ( ord_less_eq @ TA @ C @ D ) ) )
           => ( ord_less_eq @ TA @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(26,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less @ TA @ A @ B )
         => ( ( ~ ( ord_less @ TA @ B @ A ) )
            = ( B = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_antisym__conv3) ).

thf(386,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ~ ( ord_less @ TA @ A @ B )
         => ( ( ~ ( ord_less @ TA @ B @ A ) )
            = ( B = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(219,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_diff @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_zero__diff) ).

thf(978,plain,
    ! [TA: $tType] :
      ( ( comm_monoid_diff @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(275,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_old_Onat_Oexhaust) ).

thf(1140,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
     => ~ ! [B: nat] :
            ( A
           != ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(57,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
           => ( A = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_order__class_Oorder_Oantisym) ).

thf(465,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ A )
           => ( A = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(290,axiom,
    ! [TA: $tType] :
      ( ( semiring_1_cancel @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less_eq @ nat @ A @ B )
         => ( ( semiring_1_of_nat @ TA @ ( minus_minus @ nat @ B @ A ) )
            = ( minus_minus @ TA @ ( semiring_1_of_nat @ TA @ B ) @ ( semiring_1_of_nat @ TA @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_of__nat__diff) ).

thf(1180,plain,
    ! [TA: $tType] :
      ( ( semiring_1_cancel @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less_eq @ nat @ A @ B )
         => ( ( semiring_1_of_nat @ TA @ ( minus_minus @ nat @ B @ A ) )
            = ( minus_minus @ TA @ ( semiring_1_of_nat @ TA @ B ) @ ( semiring_1_of_nat @ TA @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(206,axiom,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: polyno1783536151e_poly @ TA,B: nat] :
          ( ( polyno86455060npolyh @ TA @ A @ ( suc @ B ) )
         => ( ( polyno498386536degree @ TA @ A )
            = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_degree__isnpolyh__Suc) ).

thf(931,plain,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: polyno1783536151e_poly @ TA,B: nat] :
          ( ( polyno86455060npolyh @ TA @ A @ ( suc @ B ) )
         => ( ( polyno498386536degree @ TA @ A )
            = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(126,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_diff__commute) ).

thf(671,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( minus_minus @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(281,axiom,
    ( ( zero_zero @ nat )
    = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_zero__natural_Orsp) ).

thf(1157,plain,
    $true,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(197,axiom,
    cancel1352612707id_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_8) ).

thf(903,plain,
    cancel1352612707id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(99,axiom,
    ! [TA: $tType] :
      ( ( condit1656338222tinuum @ TA )
     => ! [A: TA] :
        ? [B: TA] :
          ( ( ord_less @ TA @ A @ B )
          | ( ord_less @ TA @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_ex__gt__or__lt) ).

thf(581,plain,
    ! [TA: $tType] :
      ( ( condit1656338222tinuum @ TA )
     => ! [A: TA] :
        ? [B: TA] :
          ( ( ord_less @ TA @ A @ B )
          | ( ord_less @ TA @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(151,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_Suc__le__mono) ).

thf(759,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)],[151]) ).

thf(231,axiom,
    condit1037483654norder @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_4) ).

thf(1010,plain,
    condit1037483654norder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(177,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ! [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
           => ( B @ C ) ) )
      = ( ( B @ A )
        & ! [C: nat] :
            ( ( ord_less @ nat @ C @ A )
           => ( B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_All__less__Suc) ).

thf(844,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)],[177]) ).

thf(72,axiom,
    ord @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oord_23) ).

thf(517,plain,
    ord @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(32,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ A )
           => ( ord_less_eq @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_dual__order_Otrans) ).

thf(401,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ A )
           => ( ord_less_eq @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(158,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: nat,C: TA] :
          ( ( ( power_power @ TA @ A @ B )
            = ( power_power @ TA @ C @ B ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
           => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
             => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
               => ( A = C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_power__eq__imp__eq__base) ).

thf(786,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: TA,B: nat,C: TA] :
          ( ( ( power_power @ TA @ A @ B )
            = ( power_power @ TA @ C @ B ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
           => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
             => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
               => ( A = C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(23,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B != A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_dual__order_Ostrict__implies__not__eq) ).

thf(378,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(125,axiom,
    ! [A: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A )
     => ~ ! [B: nat] :
            ( A
           != ( semiring_1_of_nat @ int @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_nonneg__int__cases) ).

thf(667,plain,
    ! [A: int] :
      ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A )
     => ~ ! [B: nat] :
            ( A
           != ( semiring_1_of_nat @ int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(11,axiom,
    ! [TA: $tType] :
      ( ( archim1804426504_field @ TA )
     => ! [A: TA] :
        ? [B: nat] : ( ord_less_eq @ TA @ A @ ( semiring_1_of_nat @ TA @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_real__arch__simple) ).

thf(346,plain,
    ! [TA: $tType] :
      ( ( archim1804426504_field @ TA )
     => ! [A: TA] :
        ? [B: nat] : ( ord_less_eq @ TA @ A @ ( semiring_1_of_nat @ TA @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(84,axiom,
    no_bot @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Orderings_Ono__bot) ).

thf(549,plain,
    no_bot @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(79,axiom,
    ! [TA: $tType] :
      ( ( no_top @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_gt__ex) ).

thf(536,plain,
    ! [TA: $tType] :
      ( ( no_top @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(283,axiom,
    ! [A: nat > nat > $o,B: nat,C: nat] :
      ( ! [D: nat] : ( A @ D @ ( zero_zero @ nat ) )
     => ( ! [D: nat] : ( A @ ( zero_zero @ nat ) @ ( suc @ D ) )
       => ( ! [D: nat,E: nat] :
              ( ( A @ D @ E )
             => ( A @ ( suc @ D ) @ ( suc @ E ) ) )
         => ( A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_diff__induct) ).

thf(1161,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)],[283]) ).

thf(44,axiom,
    ! [TA: $tType] :
      ( ( no_bot @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_lt__ex) ).

thf(433,plain,
    ! [TA: $tType] :
      ( ( no_bot @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(68,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( order @ TA )
     => ( order @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder) ).

thf(510,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( order @ TA )
     => ( order @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(147,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_less__SucI) ).

thf(752,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ B )
     => ( ord_less @ nat @ A @ ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(50,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B != A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_less__imp__not__eq2) ).

thf(447,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(214,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_le0) ).

thf(967,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(208,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_zero__le) ).

thf(950,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(181,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ ( suc @ B ) @ A )
        = ( suc @ ( minus_minus @ nat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_Suc__diff__le) ).

thf(854,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( minus_minus @ nat @ ( suc @ B ) @ A )
        = ( suc @ ( minus_minus @ nat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(278,axiom,
    ! [A: nat] : ( ord_less @ nat @ A @ ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_lessI) ).

thf(1148,plain,
    ! [A: nat] : ( ord_less @ nat @ A @ ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(40,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( wellorder @ TB )
     => ! [A: TA > TB,B: TA > $o,C: TA] :
          ( ! [D: TA] :
              ( ! [E: TA] :
                  ( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
                 => ( B @ E ) )
             => ( B @ D ) )
         => ( B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_measure__induct) ).

thf(422,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)],[40]) ).

thf(186,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_Zero__not__Suc) ).

thf(867,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(185,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) )
         => ( ord_less @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_of__nat__less__imp__less) ).

thf(865,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) )
         => ( ord_less @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

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

thf(373,plain,
    linordered_idom @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(107,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ A )
           => ( ord_less @ TA @ C @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_dual__order_Ostrict__trans) ).

thf(603,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ A )
           => ( ord_less @ TA @ C @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(153,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_gr0__implies__Suc) ).

thf(767,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
     => ? [B: nat] :
          ( A
          = ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(122,axiom,
    condit1037483654norder @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder) ).

thf(664,plain,
    condit1037483654norder @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(31,axiom,
    ord @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Int_Oint___Orderings_Oord_3) ).

thf(400,plain,
    ord @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(173,axiom,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat,C: nat] :
          ( ( ord_less_eq @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( power_power @ TA @ ( semiring_1_of_nat @ TA @ B ) @ C ) )
          = ( ord_less_eq @ nat @ A @ ( power_power @ nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_of__nat__power__le__of__nat__cancel__iff) ).

thf(828,plain,
    ! [TA: $tType] :
      ( ( linordered_semidom @ TA )
     => ! [A: nat,B: nat,C: nat] :
          ( ( ord_less_eq @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( power_power @ TA @ ( semiring_1_of_nat @ TA @ B ) @ C ) )
          = ( ord_less_eq @ nat @ A @ ( power_power @ nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(77,axiom,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ D )
           => ( ord_less @ TA @ ( minus_minus @ TA @ A @ D ) @ ( minus_minus @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_diff__strict__mono) ).

thf(532,plain,
    ! [TA: $tType] :
      ( ( ordered_ab_group_add @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( ( ord_less @ TA @ C @ D )
           => ( ord_less @ TA @ ( minus_minus @ TA @ A @ D ) @ ( minus_minus @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(284,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_Zero__neq__Suc) ).

thf(1165,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(210,axiom,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ A )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_diff__self__eq__0) ).

thf(955,plain,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ A )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(163,axiom,
    ! [TA: $tType,A: TA > nat,B: TA > $o,C: TA] :
      ( ! [D: TA] :
          ( ( ( A @ D )
            = ( zero_zero @ nat ) )
         => ( B @ D ) )
     => ( ! [D: TA] :
            ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ D ) )
           => ( ~ ( B @ D )
             => ? [E: TA] :
                  ( ( ord_less @ nat @ ( A @ E ) @ ( A @ D ) )
                  & ~ ( B @ E ) ) ) )
       => ( B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_infinite__descent0__measure) ).

thf(796,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)],[163]) ).

thf(38,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA > TA > $o,B: TA,C: TA] :
          ( ! [D: TA,E: TA] :
              ( ( ord_less_eq @ TA @ D @ E )
             => ( A @ D @ E ) )
         => ( ! [D: TA,E: TA] :
                ( ( A @ E @ D )
               => ( A @ D @ E ) )
           => ( A @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_linorder__wlog) ).

thf(415,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA > TA > $o,B: TA,C: TA] :
          ( ! [D: TA,E: TA] :
              ( ( ord_less_eq @ TA @ D @ E )
             => ( A @ D @ E ) )
         => ( ! [D: TA,E: TA] :
                ( ( A @ E @ D )
               => ( A @ D @ E ) )
           => ( A @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(222,axiom,
    ! [A: nat,B: nat > $o] :
      ( ( ? [C: nat] :
            ( ( ord_less @ nat @ C @ ( suc @ A ) )
            & ( B @ C ) ) )
      = ( ( B @ ( zero_zero @ nat ) )
        | ? [C: nat] :
            ( ( ord_less @ nat @ C @ A )
            & ( B @ ( suc @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_Ex__less__Suc2) ).

thf(987,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)],[222]) ).

thf(183,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_less__zeroE) ).

thf(860,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(171,axiom,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat,B: nat,C: nat] :
          ( ( ( power_power @ TA @ ( semiring_1_of_nat @ TA @ A ) @ B )
            = ( semiring_1_of_nat @ TA @ C ) )
          = ( ( power_power @ nat @ A @ B )
            = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_of__nat__eq__of__nat__power__cancel__iff) ).

thf(822,plain,
    ! [TA: $tType] :
      ( ( semiring_char_0 @ TA )
     => ! [A: nat,B: nat,C: nat] :
          ( ( ( power_power @ TA @ ( semiring_1_of_nat @ TA @ A ) @ B )
            = ( semiring_1_of_nat @ TA @ C ) )
          = ( ( power_power @ nat @ A @ B )
            = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(179,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_diff__self) ).

thf(850,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(229,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less_eq @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) )
          = ( ord_less_eq @ nat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_of__nat__le__iff) ).

thf(1004,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat,B: nat] :
          ( ( ord_less_eq @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( semiring_1_of_nat @ TA @ B ) )
          = ( ord_less_eq @ nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(187,axiom,
    ! [TA: $tType,A: nat > TA > $o,B: nat > TA > TA > $o] :
      ( ? [C: TA] : ( A @ ( zero_zero @ nat ) @ C )
     => ( ! [C: TA,D: nat] :
            ( ( A @ D @ C )
           => ? [E: TA] :
                ( ( A @ ( suc @ D ) @ E )
                & ( B @ D @ C @ E ) ) )
       => ? [C: nat > TA] :
          ! [D: nat] :
            ( ( A @ D @ ( C @ D ) )
            & ( B @ D @ ( C @ D ) @ ( C @ ( suc @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_dependent__nat__choice) ).

thf(871,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)],[187]) ).

thf(109,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_old_Onat_Oinject) ).

thf(607,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(7,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA > TA > $o,B: TA,C: TA] :
          ( ! [D: TA,E: TA] :
              ( ( ord_less @ TA @ D @ E )
             => ( A @ D @ E ) )
         => ( ! [D: TA] : ( A @ D @ D )
           => ( ! [D: TA,E: TA] :
                  ( ( A @ E @ D )
                 => ( A @ D @ E ) )
             => ( A @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_linorder__less__wlog) ).

thf(322,plain,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA > TA > $o,B: TA,C: TA] :
          ( ! [D: TA,E: TA] :
              ( ( ord_less @ TA @ D @ E )
             => ( A @ D @ E ) )
         => ( ! [D: TA] : ( A @ D @ D )
           => ( ! [D: TA,E: TA] :
                  ( ( A @ E @ D )
                 => ( A @ D @ E ) )
             => ( A @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(164,axiom,
    ! [A: nat] :
      ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_diff__0__eq__0) ).

thf(803,plain,
    ! [A: nat] :
      ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(204,axiom,
    ! [TA: $tType] :
      ( ( monoid_mult @ TA )
     => ! [A: TA] :
          ( ( power_power @ TA @ A @ ( suc @ ( zero_zero @ nat ) ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_power__Suc0__right) ).

thf(923,plain,
    ! [TA: $tType] :
      ( ( monoid_mult @ TA )
     => ! [A: TA] :
          ( ( power_power @ TA @ A @ ( suc @ ( zero_zero @ nat ) ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(292,axiom,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_old_Onat_Odistinct_I1_J) ).

thf(1184,plain,
    ! [A: nat] :
      ( ( zero_zero @ nat )
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(119,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_order_Otrans) ).

thf(659,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ B @ C )
           => ( ord_less_eq @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(149,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat] :
          ( ( ord_less_eq @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_of__nat__le__0__iff) ).

thf(755,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: nat] :
          ( ( ord_less_eq @ TA @ ( semiring_1_of_nat @ TA @ A ) @ ( zero_zero @ TA ) )
          = ( A
            = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(298,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ ( suc @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_not__less__eq) ).

thf(1204,plain,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less @ nat @ A @ B ) )
      = ( ord_less @ nat @ B @ ( suc @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).

thf(64,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ( ( ord_less_eq @ ( TA > TB ) )
        = ( ^ [A: TA > TB,B: TA > TB] :
            ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_le__fun__def) ).

thf(496,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TB )
     => ( ( ord_less_eq @ ( TA > TB ) )
        = ( ^ [A: TA > TB,B: TA > TB] :
            ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(52,axiom,
    ! [TA: $tType] :
      ( ( linorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( A != B )
          = ( ( ord_less @ TA @ A @ B )
            | ( ord_less @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_neq__iff) ).

thf(454,plain,
    ! [TA: $tType] :
      ( ( linorder @ 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)],[52]) ).

thf(251,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ A @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
        | ( B
          = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_nat__zero__less__power__iff) ).

thf(1069,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ A @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
        | ( B
          = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(233,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ A @ ( suc @ B ) )
      = ( ( ord_less @ nat @ A @ B )
        | ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_less__Suc__eq) ).

thf(1014,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)],[233]) ).

thf(217,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_gr__zeroI) ).

thf(972,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)],[217]) ).

thf(8808,plain,
    $false,
    inference(cvc4,[status(thm)],[809,518,555,666,352,645,892,479,1105,962,468,1211,683,730,994,308,878,1041,1173,1158,762,384,1137,550,698,538,898,1001,1127,709,320,565,389,570,917,1059,1055,724,934,357,1191,542,460,559,853,969,770,881,1115,1100,789,1109,376,646,834,348,806,966,1183,981,1072,411,507,1169,397,428,1201,920,678,529,661,1153,534,1047,512,439,910,1223,746,825,952,1174,1217,561,693,847,1032,435,665,889,1079,710,1106,583,522,493,1210,578,1089,317,650,1017,1064,1111,403,1096,356,593,610,499,857,783,420,884,339,901,642,1214,706,1028,1082,1207,590,997,701,654,984,569,515,1150,1060,1050,457,503,605,792,1086,360,1054,1170,1035,558,674,586,888,926,750,425,513,1103,573,733,584,819,1007,843,772,743,958,545,562,904,975,641,391,445,382,713,1092,704,540,1194,406,831,681,1117,1011,990,451,1144,599,863,477,811,494,758,816,520,895,1022,1177,462,345,367,912,1198,1066,1113,438,431,658,1044,695,394,306,1076,795,663,374,1134,812,342,1145,355,754,553,776,1025,409,993,557,1003,414,971,1098,310,791,1188,441,463,1130,1220,907,1038,370,1085,470,576,613,727,595,386,978,1140,465,1180,931,671,1157,903,581,759,1010,844,517,401,786,378,667,346,549,536,1161,433,510,752,447,967,950,854,1148,422,867,865,373,603,767,664,400,828,532,1165,955,796,415,987,860,822,850,1004,871,607,322,803,923,1184,659,755,1204,496,454,1069,1014,972]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.13  % Problem  : ITP151^2 : TPTP v8.2.0. Released v7.5.0.
% 0.09/0.16  % Command  : run_Leo-III %s %d
% 0.16/0.37  % Computer : n004.cluster.edu
% 0.16/0.37  % Model    : x86_64 x86_64
% 0.16/0.37  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.37  % Memory   : 8042.1875MB
% 0.16/0.37  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.16/0.37  % CPULimit : 300
% 0.16/0.37  % WCLimit  : 300
% 0.16/0.37  % DateTime : Sat May 18 18:02:09 EDT 2024
% 0.16/0.38  % CPUTime  : 
% 1.00/0.88  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 1.64/1.15  % [INFO] 	 Parsing done (260ms). 
% 1.64/1.16  % [INFO] 	 Running in sequential loop mode. 
% 2.70/1.39  % [INFO] 	 eprover registered as external prover. 
% 2.70/1.39  % [INFO] 	 cvc4 registered as external prover. 
% 2.70/1.39  % [INFO] 	 Scanning for conjecture ... 
% 2.93/1.48  % [INFO] 	 Found a conjecture (or negated_conjecture) and 303 axioms. Running axiom selection ... 
% 3.21/1.57  % [INFO] 	 Axiom selection finished. Selected 303 axioms (removed 0 axioms). 
% 3.68/1.75  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 3.68/1.76  % [INFO] 	 Type checking passed. 
% 3.68/1.76  % [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 ... 
% 84.03/24.31  % External prover 'cvc4' found a proof!
% 84.03/24.31  % [INFO] 	 Killing All external provers ... 
% 84.03/24.31  % Time passed: 23764ms (effective reasoning time: 23150ms)
% 84.03/24.31  % 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)>
% 84.03/24.31  % Axioms used in derivation (303): fact_108_less__trans, fact_37_dual__order_Ostrict__implies__not__eq, fact_64_Zero__not__Suc, tcon_Nat_Onat___Orderings_Oord_18, fact_214_less__diff__iff, fact_103_order_Oasym, fact_28_field__lbound__gt__zero, fact_42_lessI, fact_121_dual__order_Oirrefl, fact_111_ord__eq__less__trans, fact_71_Suc__lessI, tcon_Int_Oint___Rings_Olinordered__semidom, fact_24_gr__implies__not__zero, tfree_0, fact_180_power__inject__base, fact_197_le__cases, fact_242_diff__strict__mono, fact_16_infinite__descent, fact_222_diff__mono, fact_134_reals__Archimedean2, fact_52_n__not__Suc__n, fact_131_zero__less__imp__eq__int, fact_104_dense, fact_15_linorder__neqE__nat, tcon_Nat_Onat___Groups_Ocomm__monoid__diff, fact_217_diff__right__commute, fact_168_zero__less__diff, tcon_Int_Oint___Groups_Ozero, fact_11_zero__reorient, fact_189_order__class_Oorder_Oantisym, tcon_Int_Oint___Orderings_Oorder_2, fact_122_order_Ostrict__trans, tcon_Nat_Onat___Rings_Osemiring__1_15, fact_198_eq__refl, fact_123_less__imp__not__less, fact_62_Suc__neq__Zero, fact_56_old_Onat_Odistinct_I2_J, fact_137_Suc__pred, fact_155_nat__zero__less__power__iff, fact_98_order__less__subst2, fact_176_zero__integer_Orsp, fact_144_cancel__comm__monoid__add__class_Odiff__cancel, fact_21_less__not__refl, fact_212_le__iff__diff__le__0, fact_66_old_Onat_Oinducts, fact_41_of__nat__eq__iff, tcon_Int_Oint___Groups_Oordered__ab__group__add, fact_161_diff__Suc__Suc, fact_135_of__nat__zero__less__power__iff, fact_199_linear, tcon_Int_Oint___Groups_Omonoid__mult, tcon_fun___Orderings_Oord, fact_200_antisym, fact_55_nat_Odistinct_I1_J, fact_225_diff__le__mono2, fact_191_ord__eq__le__trans, fact_213_diff__less__mono, fact_92_All__less__Suc2, fact_145_diff__zero, fact_194_le__cases3, fact_112_ord__less__eq__trans, fact_2_bot__nat__0_Onot__eq__extremum, fact_95_ord__eq__less__subst, fact_233_less__eq__int__code_I1_J, fact_73_less__SucI, tcon_Nat_Onat___Orderings_Opreorder_12, fact_110_less__irrefl, fact_219_diff__right__mono, fact_114_less__imp__not__eq, fact_26_gr__zeroI, fact_7_less__zeroE, fact_80_Suc__less__SucD, fact_124_exists__least__iff, fact_48_of__nat__less__iff, fact_133_exists__least__lemma, fact_117_antisym__conv3, tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_8, fact_67_not0__implies__Suc, fact_187_dual__order_Orefl, tcon_Int_Oint___Rings_Osemiring__1__cancel, fact_23_zero__less__iff__neq__zero, fact_157_diff__is__0__eq, fact_221_zero__le__power, fact_160_Suc__diff__diff, fact_170_of__nat__less__of__nat__power__cancel__iff, fact_96_ord__less__eq__subst, fact_72_less__SucE, fact_18_less__irrefl__nat, tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_6, fact_136_of__nat__le__0__iff, fact_125_linorder__less__wlog, fact_185_dual__order_Otrans, fact_220_diff__left__mono, fact_244_le__numeral__extra_I3_J, fact_243_zero__le, tcon_Int_Oint___Orderings_Opreorder_1, fact_159_diff__0__eq__0, fact_39_nat_Oinject, fact_229_Nat_Odiff__diff__eq, fact_181_of__nat__mono, fact_34_zero__natural_Orsp, tcon_HOL_Obool___Orderings_Oorder_22, fact_61_zero__induct, fact_179_of__nat__diff, tcon_Nat_Onat___Orderings_Oorder_17, fact_97_order__less__subst1, fact_14_infinite__descent__measure, fact_252_dense__ge__bounded, tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5, fact_40_old_Onat_Oinject, fact_6_not__less0, fact_224_diff__eq__diff__eq, fact_158_diff__self__eq__0, tcon_HOL_Obool___Orderings_Olinorder_21, fact_1_less__nat__zero__code, fact_74_Ex__less__Suc, fact_218_diff__eq__diff__less__eq, fact_116_less__induct, tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, fact_19_less__not__refl3, fact_251_dense__le__bounded, fact_5_not__gr0, fact_8_gr__implies__not0, fact_236_power__eq__imp__eq__base, fact_47_of__nat__eq__0__iff, fact_106_less__asym, fact_100_gt__ex, fact_127_not__less__iff__gr__or__eq, fact_188_order__trans, fact_20_less__not__refl2, fact_248_dual__order_Ostrict__iff__order, fact_102_neq__iff, fact_46_of__nat__0__eq__iff, tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, fact_207_le__funI, fact_22_nat__neq__iff, fact_201_eq__iff, fact_50_zero__less__Suc, fact_129_linordered__field__no__lb, fact_138_order__refl, tcon_Nat_Onat___Orderings_Ono__top_16, fact_69_Suc__lessD, tcon_Int_Oint___Rings_Olinordered__idom, fact_33_degree__isnpolyh__Suc, tcon_Int_Oint___Orderings_Olinorder, fact_86_less__imp__of__nat__less, fact_183_dual__order_Oantisym, tcon_Nat_Onat___Orderings_Owellorder, fact_115_less__not__sym, fact_63_Zero__neq__Suc, fact_130_pos__int__cases, fact_154_of__nat__power, fact_186_linorder__wlog, fact_210_int__int__eq, fact_49_less__Suc0, fact_234_less__int__code_I1_J, fact_93_gr0__implies__Suc, fact_143_le__zero__eq, fact_44_Suc__less__eq, fact_238_eq__iff__diff__eq__0, fact_84_not__less__less__Suc__eq, fact_17_nat__less__induct, fact_171_of__nat__power__less__of__nat__cancel__iff, tcon_Nat_Onat___Nat_Osemiring__char__0_11, fact_105_less__imp__neq, fact_132_dependent__nat__choice, fact_249_dual__order_Oorder__iff__strict, fact_29_less__numeral__extra_I3_J, fact_89_lift__Suc__mono__less__iff, fact_231_le__diff__iff, tcon_Nat_Onat___Groups_Omonoid__mult_14, fact_109_less__linear, fact_203_ord__eq__le__subst, fact_184_dual__order_Oeq__iff, fact_65_old_Onat_Oexhaust, tcon_Int_Oint___Rings_Osemiring__1, fact_99_lt__ex, fact_57_old_Onat_Odistinct_I1_J, fact_12_measure__induct__rule, fact_32_of__nat__0__less__iff, fact_172_power__mono__iff, fact_206_le__fun__def, tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add, fact_13_measure__induct, fact_178_lift__Suc__antimono__le, fact_10_bot__nat__0_Oextremum__strict, fact_118_less__imp__not__eq2, fact_68_Nat_OlessE, fact_202_ord__le__eq__subst, fact_151_power__Suc__0, fact_0_neq0__conv, fact_190_ord__le__eq__trans, tcon_Int_Oint___Groups_Ogroup__add, fact_88_lift__Suc__mono__less, fact_142_diff__diff__cancel, fact_174_nonneg__int__cases, fact_87_of__nat__less__imp__less, fact_128_order_Ostrict__implies__not__eq, fact_195_real__arch__simple, fact_246_order_Onot__eq__order__implies__strict, fact_227_diff__le__self, fact_239_diff__strict__right__mono, fact_146_zero__diff, fact_79_less__antisym, fact_162_diff__ge__0__iff__ge, fact_30_isnpoly__def, fact_58_nat_OdiscI, fact_107_less__asym_H, fact_94_less__Suc__eq__0__disj, fact_165_of__nat__power__le__of__nat__cancel__iff, fact_148_diff__self, fact_36_linorder__neqE__linordered__idom, subcl_Groups_Ozero___HOL_Otype, fact_153_of__nat__eq__of__nat__power__cancel__iff, fact_54_polypow_Ocases, fact_152_of__nat__power__eq__of__nat__cancel__iff, tcon_Nat_Onat___Rings_Olinordered__semidom_9, fact_90_Ex__less__Suc2, fact_223_power__mono, fact_141_Suc__le__mono, tcon_fun___Orderings_Opreorder, fact_140_le0, fact_208_le__funE, fact_81_less__trans__Suc, fact_216_power__less__imp__less__base, fact_192_order__class_Oorder_Oeq__iff, conj_1, tcon_Nat_Onat___Orderings_Olinorder_13, fact_149_of__nat__le__iff, fact_228_diff__le__mono, fact_232_eq__diff__iff, fact_156_diff__is__0__eq_H, fact_31_degreen__0, fact_70_Suc__lessE, fact_215_power__not__zero, tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_7, fact_120_linorder__cases, fact_196_order_Otrans, fact_209_le__funD, fact_78_Suc__less__eq2, tcon_Int_Oint___Nat_Osemiring__char__0, fact_75_less__Suc__eq, fact_205_order__subst1, fact_27_infinite__descent0__measure, fact_230_diff__commute, fact_4_gr0I, fact_235_nat__one__le__power, fact_193_antisym__conv, tcon_HOL_Obool___Orderings_Opreorder_20, fact_237_power__eq__iff__eq__base, tcon_fun___Orderings_Oorder, fact_113_dual__order_Oasym, fact_240_diff__strict__left__mono, tcon_Int_Oint___Orderings_Ono__bot, fact_241_diff__eq__diff__less, fact_164_power__0__Suc, fact_77_All__less__Suc, fact_226_le__diff__iff_H, fact_250_order_Ostrict__implies__order, fact_3_not__gr__zero, tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_4, fact_119_less__imp__triv, fact_147_diff__0__right, fact_60_diff__induct, fact_51_Suc__inject, fact_150_nat__power__eq__Suc__0__iff, fact_182_power__le__imp__le__base, tcon_Int_Oint___Orderings_Ono__top, fact_163_diff__gt__0__iff__gt, fact_38_linordered__field__no__ub, fact_25_not__less__zero, tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, fact_166_of__nat__le__of__nat__power__cancel__iff, fact_247_dual__order_Ostrict__implies__order, fact_9_infinite__descent0, fact_139_bot__nat__0_Oextremum, fact_167_power__Suc0__right, fact_245_complete__interval, tcon_HOL_Obool___Orderings_Oord_23, fact_35_ex__gt__or__lt, fact_43_Suc__mono, tcon_Int_Oint___Orderings_Oord_3, fact_45_of__nat__0, fact_126_dual__order_Ostrict__trans, fact_177_lift__Suc__mono__le, fact_101_neqE, fact_59_nat__induct, fact_169_power__eq__0__iff, fact_175_zero__le__imp__eq__int, conj_0, fact_85_of__nat__less__0__iff, tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, fact_83_strict__inc__induct, fact_173_Suc__diff__le, fact_204_order__subst2, tcon_Nat_Onat___Groups_Ozero_19, fact_76_not__less__eq, tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors, fact_91_gr0__conv__Suc, tcon_Nat_Onat___Rings_Osemiring__1__cancel_10, fact_53_of__nat__neq__0, fact_82_less__Suc__induct, fact_211_zle__int
% 84.03/24.31  % No. of inferences in proof: 610
% 84.03/24.31  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 23764 ms resp. 23150 ms w/o parsing
% 84.30/24.53  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 84.30/24.54  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------