TSTP Solution File: ITP132^2 by Leo-III---1.7.12

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%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : ITP132^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n014.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 22:23:14 EDT 2024

% Result   : Theorem 58.75s 41.68s
% Output   : Refutation 59.77s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  356
% Syntax   : Number of formulae    :  654 ( 182 unt;  60 typ;   0 def)
%            Number of atoms       : 1786 ( 555 equ;   8 cnn)
%            Maximal formula atoms :   13 (   3 avg)
%            Number of connectives : 6726 ( 134   ~;  50   |;  92   &;5576   @)
%                                         (   0 <=>; 874  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   7 avg)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :  224 ( 224   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   64 (  61 usr;   6 con; 0-6 aty)
%            Number of variables   : 1860 ( 111   ^1676   !;  16   ?;1860   :)
%                                         (  57  !>;   0  ?*;   0  @-;   0  @+)

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

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

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

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

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

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

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

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

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

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

thf(mult_zero_type,type,
    mult_zero: 
      !>[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_0_type,type,
    semiring_0: 
      !>[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(ordered_ring_type,type,
    ordered_ring: 
      !>[TA: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

thf(comm_s1003936772cancel_type,type,
    comm_s1003936772cancel: 
      !>[TA: $tType] : $o ).

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

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

thf(ring_11004092258visors_type,type,
    ring_11004092258visors: 
      !>[TA: $tType] : $o ).

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

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

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

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

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

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

thf(fun_upd_type,type,
    fun_upd: 
      !>[TA: $tType,TB: $tType] : ( ( TB > TA ) > TB > TA > TB > TA ) ).

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

thf(one_one_type,type,
    one_one: 
      !>[TA: $tType] : TA ).

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

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

thf(groups1340683514dd_sum_type,type,
    groups1340683514dd_sum: 
      !>[TA: $tType,TB: $tType] : ( ( TB > TA ) > ( set @ TB ) > TA ) ).

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

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

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

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

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

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

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

thf(k_type,type,
    k: nat ).

thf(n_type,type,
    n: nat ).

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

thf(28,axiom,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_zero__le) ).

thf(365,plain,
    ! [TA: $tType] :
      ( ( canoni770627133id_add @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(23,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_159_ord__le__eq__subst) ).

thf(347,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)],[23]) ).

thf(17,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ~ ( ord_less @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_less__numeral__extra_I4_J) ).

thf(333,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ~ ( ord_less @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(70,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( member @ TA @ C @ B ) )
     => ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_subsetI) ).

thf(518,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ! [C: TA] :
          ( ( member @ TA @ C @ A )
         => ( member @ TA @ C @ B ) )
     => ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(226,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_mult @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ A @ ( one_one @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_mult_Ocomm__neutral) ).

thf(962,plain,
    ! [TA: $tType] :
      ( ( comm_monoid_mult @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ A @ ( one_one @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(219,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( one_one @ TA ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ( ord_less_eq @ TA @ B @ ( one_one @ TA ) )
             => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( one_one @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_mult__le__one) ).

thf(941,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( one_one @ TA ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ( ord_less_eq @ TA @ B @ ( one_one @ TA ) )
             => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( one_one @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(85,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( preorder @ TA )
     => ( preorder @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Opreorder) ).

thf(555,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( preorder @ TA )
     => ( preorder @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(118,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ C @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
       => ( ord_less_eq @ nat @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_mult__le__cancel2) ).

thf(666,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ C @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
       => ( ord_less_eq @ nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(279,axiom,
    one @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oone) ).

thf(1110,plain,
    one @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(291,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_le__0__eq) ).

thf(1142,plain,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).

thf(67,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_141_dual__order_Oantisym) ).

thf(511,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)],[67]) ).

thf(72,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_dual__order_Orefl) ).

thf(523,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(297,axiom,
    canoni770627133id_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add) ).

thf(1158,plain,
    canoni770627133id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).

thf(144,axiom,
    ! [A: nat,B: nat] :
      ( ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
     => ( ( ( minus_minus @ nat @ B @ A )
          = ( zero_zero @ nat ) )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_diffs0__imp__equal) ).

thf(741,plain,
    ! [A: nat,B: nat] :
      ( ( ( minus_minus @ nat @ A @ B )
        = ( zero_zero @ nat ) )
     => ( ( ( minus_minus @ nat @ B @ A )
          = ( zero_zero @ nat ) )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(264,axiom,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ( ( ^ [A: TA] : ( zero_zero @ TA ) )
        = ( times_times @ TA @ ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_lambda__zero) ).

thf(1073,plain,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ( ( ^ [A: TA] : ( zero_zero @ TA ) )
        = ( times_times @ TA @ ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(249,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_le__trans) ).

thf(1041,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ C )
       => ( ord_less_eq @ nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(189,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_248_infinite__descent__measure) ).

thf(861,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)],[189]) ).

thf(266,axiom,
    number2016821345itions @ p @ n,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_partitions) ).

thf(1078,plain,
    number2016821345itions @ p @ n,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(35,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B != A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_less__imp__not__eq2) ).

thf(384,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(77,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_148_ord__le__eq__trans) ).

thf(533,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)],[77]) ).

thf(167,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ordere216010020id_add @ TB )
     => ! [A: set @ TA,B: TA > TB,C: TA > TB] :
          ( ! [D: TA] :
              ( ( member @ TA @ D @ A )
             => ( ord_less_eq @ TB @ ( B @ D ) @ ( C @ D ) ) )
         => ( ord_less_eq @ TB @ ( groups1340683514dd_sum @ TA @ TB @ B @ A ) @ ( groups1340683514dd_sum @ TA @ TB @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_sum__mono) ).

thf(797,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ordere216010020id_add @ TB )
     => ! [A: set @ TA,B: TA > TB,C: TA > TB] :
          ( ! [D: TA] :
              ( ( member @ TA @ D @ A )
             => ( ord_less_eq @ TB @ ( B @ D ) @ ( C @ D ) ) )
         => ( ord_less_eq @ TB @ ( groups1340683514dd_sum @ TA @ TB @ B @ A ) @ ( groups1340683514dd_sum @ TA @ TB @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(60,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( fun_upd @ TA @ TB )
      = ( ^ [A: TA > TB,B: TA,C: TB,D: TA] : ( if @ TB @ ( D = B ) @ C @ ( A @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_fun__upd__def) ).

thf(472,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( fun_upd @ TA @ TB )
      = ( ^ [A: TA > TB,B: TA,C: TB,D: TA] : ( if @ TB @ ( D = B ) @ C @ ( A @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(259,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_bot__nat__0_Oextremum__unique) ).

thf(1063,plain,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(146,axiom,
    ab_semigroup_mult @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oab__semigroup__mult) ).

thf(747,plain,
    ab_semigroup_mult @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(207,axiom,
    ord @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oord_3) ).

thf(913,plain,
    ord @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(42,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_227_dual__order_Oasym) ).

thf(404,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)],[42]) ).

thf(202,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType] :
      ( ( comm_monoid_add @ TC )
     => ! [A: TB > TA > TC,B: set @ TA,C: set @ TB] :
          ( ( groups1340683514dd_sum @ TB @ TC
            @ ^ [D: TB] : ( groups1340683514dd_sum @ TA @ TC @ ( A @ D ) @ B )
            @ C )
          = ( groups1340683514dd_sum @ TA @ TC
            @ ^ [D: TA] :
                ( groups1340683514dd_sum @ TB @ TC
                @ ^ [E: TB] : ( A @ E @ D )
                @ C )
            @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_sum_Oswap) ).

thf(898,plain,
    ! [TA: $tType,TB: $tType,TC: $tType] :
      ( ( comm_monoid_add @ TC )
     => ! [A: TB > TA > TC,B: set @ TA,C: set @ TB] :
          ( ( groups1340683514dd_sum @ TB @ TC
            @ ^ [D: TB] : ( groups1340683514dd_sum @ TA @ TC @ ( A @ D ) @ B )
            @ C )
          = ( groups1340683514dd_sum @ TA @ TC
            @ ^ [D: TA] :
                ( groups1340683514dd_sum @ TB @ TC
                @ ^ [E: TB] : ( A @ E @ D )
                @ C )
            @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(89,axiom,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: TA,C: TB] :
      ( ( ( A @ B )
        = C )
     => ( ( fun_upd @ TA @ TB @ A @ B @ C )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_fun__upd__idem) ).

thf(565,plain,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: TA,C: TB] :
      ( ( ( A @ B )
        = C )
     => ( ( fun_upd @ TA @ TB @ A @ B @ C )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(53,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_161_order__subst2) ).

thf(436,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)],[53]) ).

thf(112,axiom,
    ! [A: nat > nat,B: nat,C: nat] :
      ( ( number2016821345itions @ A @ B )
     => ( ord_less_eq @ nat @ ( A @ C ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_partitions__bounds) ).

thf(651,plain,
    ! [A: nat > nat,B: nat,C: nat] :
      ( ( number2016821345itions @ A @ B )
     => ( ord_less_eq @ nat @ ( A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(137,axiom,
    ! [TA: $tType] :
      ( ( semiri1193490041visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( zero_zero @ TA ) )
         => ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_divisors__zero) ).

thf(724,plain,
    ! [TA: $tType] :
      ( ( semiri1193490041visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( zero_zero @ TA ) )
         => ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(135,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_le0) ).

thf(719,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(161,axiom,
    mult_zero @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Omult__zero) ).

thf(785,plain,
    mult_zero @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(66,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_160_ord__eq__le__subst) ).

thf(504,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)],[66]) ).

thf(22,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_163_le__fun__def) ).

thf(344,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)],[22]) ).

thf(97,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_236_dense) ).

thf(587,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)],[97]) ).

thf(108,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_251_infinite__descent) ).

thf(619,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)],[108]) ).

thf(129,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
          = ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_zero__le__mult__iff) ).

thf(702,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
          = ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(148,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_mult__right__mono) ).

thf(751,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(47,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_order__refl) ).

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

thf(103,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_232_less__trans) ).

thf(606,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)],[103]) ).

thf(13,axiom,
    minus @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Groups_Ominus_13) ).

thf(325,plain,
    minus @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(10,axiom,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: TA] :
      ( ( fun_upd @ TA @ TB @ A @ B @ ( A @ B ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_fun__upd__triv) ).

thf(316,plain,
    ! [TA: $tType,TB: $tType,A: TA > TB,B: TA] :
      ( ( fun_upd @ TA @ TB @ A @ B @ ( A @ B ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(275,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( semiring_0 @ TB )
     => ! [A: TA > TB,B: set @ TA,C: TB] :
          ( ( times_times @ TB @ ( groups1340683514dd_sum @ TA @ TB @ A @ B ) @ C )
          = ( groups1340683514dd_sum @ TA @ TB
            @ ^ [D: TA] : ( times_times @ TB @ ( A @ D ) @ C )
            @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_sum__distrib__right) ).

thf(1100,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( semiring_0 @ TB )
     => ! [A: TA > TB,B: set @ TA,C: TB] :
          ( ( times_times @ TB @ ( groups1340683514dd_sum @ TA @ TB @ A @ B ) @ C )
          = ( groups1340683514dd_sum @ TA @ TB
            @ ^ [D: TA] : ( times_times @ TB @ ( A @ D ) @ C )
            @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

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_152_le__cases3) ).

thf(475,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(63,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_158_eq__iff) ).

thf(492,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)],[63]) ).

thf(150,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_le__refl) ).

thf(756,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(163,axiom,
    ordered_semiring_0 @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Oordered__semiring__0) ).

thf(789,plain,
    ordered_semiring_0 @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(55,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B != A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_dual__order_Ostrict__implies__not__eq) ).

thf(443,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( B != A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(68,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ( ord_less_eq @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_le__numeral__extra_I4_J) ).

thf(514,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ( ord_less_eq @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(170,axiom,
    no_top @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Ono__top) ).

thf(806,plain,
    no_top @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(176,axiom,
    ! [A: nat > nat] :
      ( ( number2016821345itions @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( ^ [B: nat] : ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_partitions__zero) ).

thf(821,plain,
    ! [A: nat > nat] :
      ( ( number2016821345itions @ A @ ( zero_zero @ nat ) )
      = ( A
        = ( ^ [B: nat] : ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(289,axiom,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ A @ ( zero_zero @ TA ) )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_mult__zero__right) ).

thf(1136,plain,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ A @ ( zero_zero @ TA ) )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).

thf(179,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ ( times_times @ nat @ A @ ( times_times @ nat @ A @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_le__cube) ).

thf(834,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ ( times_times @ nat @ A @ ( times_times @ nat @ A @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(102,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_less__imp__not__eq) ).

thf(602,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(111,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType] :
      ( ( comm_monoid_add @ TB )
     => ! [A: set @ TC,B: TA > TC,C: TC > TA,D: set @ TA,E: TA > TB,F: TC > TB] :
          ( ! [G: TC] :
              ( ( member @ TC @ G @ A )
             => ( ( B @ ( C @ G ) )
                = G ) )
         => ( ! [G: TC] :
                ( ( member @ TC @ G @ A )
               => ( member @ TA @ ( C @ G ) @ D ) )
           => ( ! [G: TA] :
                  ( ( member @ TA @ G @ D )
                 => ( ( C @ ( B @ G ) )
                    = G ) )
             => ( ! [G: TA] :
                    ( ( member @ TA @ G @ D )
                   => ( member @ TC @ ( B @ G ) @ A ) )
               => ( ! [G: TC] :
                      ( ( member @ TC @ G @ A )
                     => ( ( E @ ( C @ G ) )
                        = ( F @ G ) ) )
                 => ( ( groups1340683514dd_sum @ TC @ TB @ F @ A )
                    = ( groups1340683514dd_sum @ TA @ TB @ E @ D ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_sum_Oreindex__bij__witness) ).

thf(634,plain,
    ! [TA: $tType,TB: $tType,TC: $tType] :
      ( ( comm_monoid_add @ TB )
     => ! [A: set @ TC,B: TA > TC,C: TC > TA,D: set @ TA,E: TA > TB,F: TC > TB] :
          ( ! [G: TC] :
              ( ( member @ TC @ G @ A )
             => ( ( B @ ( C @ G ) )
                = G ) )
         => ( ! [G: TC] :
                ( ( member @ TC @ G @ A )
               => ( member @ TA @ ( C @ G ) @ D ) )
           => ( ! [G: TA] :
                  ( ( member @ TA @ G @ D )
                 => ( ( C @ ( B @ G ) )
                    = G ) )
             => ( ! [G: TA] :
                    ( ( member @ TA @ G @ D )
                   => ( member @ TC @ ( B @ G ) @ A ) )
               => ( ! [G: TC] :
                      ( ( member @ TC @ G @ A )
                     => ( ( E @ ( C @ G ) )
                        = ( F @ G ) ) )
                 => ( ( groups1340683514dd_sum @ TC @ TB @ F @ A )
                    = ( groups1340683514dd_sum @ TA @ TB @ E @ D ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(140,axiom,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
     => ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_bot__nat__0_Oextremum__uniqueI) ).

thf(734,plain,
    ! [A: nat] :
      ( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
     => ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(224,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_cancel__comm__monoid__add__class_Odiff__cancel) ).

thf(956,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(283,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_53_diff__commute) ).

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

thf(116,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times @ nat @ A @ ( minus_minus @ nat @ B @ C ) )
      = ( minus_minus @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_diff__mult__distrib2) ).

thf(660,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times @ nat @ A @ ( minus_minus @ nat @ B @ C ) )
      = ( minus_minus @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(50,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_66_le__zero__eq) ).

thf(428,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)],[50]) ).

thf(263,axiom,
    semiri1193490041visors @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors) ).

thf(1072,plain,
    semiri1193490041visors @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(58,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( one_one @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_zero__le__one) ).

thf(453,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( one_one @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(6,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_234_less__asym) ).

thf(307,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)],[6]) ).

thf(211,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
             => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
               => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_mult__mono) ).

thf(920,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
             => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
               => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(54,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_order_Ostrict__implies__not__eq) ).

thf(439,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(81,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( order @ TA )
     => ( order @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder) ).

thf(546,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( order @ TA )
     => ( order @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

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

thf(925,plain,
    zero_neq_one @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(152,axiom,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ C @ B ) @ ( times_times @ TA @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_mult__left__mono__neg) ).

thf(761,plain,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ C @ B ) @ ( times_times @ TA @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(172,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_diff__le__self) ).

thf(810,plain,
    ! [A: nat,B: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(34,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_239_neqE) ).

thf(381,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)],[34]) ).

thf(261,axiom,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) )
          = ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_mult__le__0__iff) ).

thf(1067,plain,
    ! [TA: $tType] :
      ( ( linord581940658strict @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) )
          = ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(251,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_mult__nonneg__nonpos2) ).

thf(1047,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(257,axiom,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ A )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_diff__self__eq__0) ).

thf(1057,plain,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ A )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(206,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_184_diff__eq__diff__less__eq) ).

thf(910,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)],[206]) ).

thf(126,axiom,
    ! [TA: $tType] :
      ( ( ring_11004092258visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = B )
          = ( ( B
              = ( zero_zero @ TA ) )
            | ( A
              = ( one_one @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_mult__cancel__right2) ).

thf(693,plain,
    ! [TA: $tType] :
      ( ( ring_11004092258visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = B )
          = ( ( B
              = ( zero_zero @ TA ) )
            | ( A
              = ( one_one @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(174,axiom,
    ! [A: nat,B: nat] :
      ( ( ( one_one @ nat )
        = ( times_times @ nat @ A @ B ) )
      = ( ( A
          = ( one_one @ nat ) )
        & ( B
          = ( one_one @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_nat__1__eq__mult__iff) ).

thf(815,plain,
    ! [A: nat,B: nat] :
      ( ( ( one_one @ nat )
        = ( times_times @ nat @ A @ B ) )
      = ( ( A
          = ( one_one @ nat ) )
        & ( B
          = ( one_one @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(209,axiom,
    ! [TA: $tType] :
      ( ( ring_11004092258visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
            = ( times_times @ TA @ A @ B ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( one_one @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_mult__cancel__left1) ).

thf(916,plain,
    ! [TA: $tType] :
      ( ( ring_11004092258visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
            = ( times_times @ TA @ A @ B ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( one_one @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(196,axiom,
    ! [A: nat] :
      ( ( times_times @ nat @ A @ ( zero_zero @ nat ) )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_mult__0__right) ).

thf(879,plain,
    ! [A: nat] :
      ( ( times_times @ nat @ A @ ( zero_zero @ nat ) )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(200,axiom,
    ! [A: nat > nat,B: nat] :
      ( ( number2016821345itions @ A @ B )
     => ~ ( ! [C: nat] :
              ( ( ( A @ C )
               != ( zero_zero @ nat ) )
             => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ C )
                & ( ord_less_eq @ nat @ C @ B ) ) )
         => ( ( groups1340683514dd_sum @ nat @ nat
              @ ^ [C: nat] : ( times_times @ nat @ ( A @ C ) @ C )
              @ ( set_ord_atMost @ nat @ B ) )
           != B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_partitionsE) ).

thf(889,plain,
    ! [A: nat > nat,B: nat] :
      ( ( number2016821345itions @ A @ B )
     => ~ ( ! [C: nat] :
              ( ( ( A @ C )
               != ( zero_zero @ nat ) )
             => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ C )
                & ( ord_less_eq @ nat @ C @ B ) ) )
         => ( ( groups1340683514dd_sum @ nat @ nat
              @ ^ [C: nat] : ( times_times @ nat @ ( A @ C ) @ C )
              @ ( set_ord_atMost @ nat @ B ) )
           != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(267,axiom,
    ! [TA: $tType] :
      ( ( comm_s1003936772cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( minus_minus @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_left__diff__distrib_H) ).

thf(1079,plain,
    ! [TA: $tType] :
      ( ( comm_s1003936772cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( minus_minus @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(122,axiom,
    ! [A: nat] :
      ( ( times_times @ nat @ ( one_one @ nat ) @ A )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_nat__mult__1) ).

thf(682,plain,
    ! [A: nat] :
      ( ( times_times @ nat @ ( one_one @ nat ) @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(83,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_233_less__asym_H) ).

thf(551,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)],[83]) ).

thf(277,axiom,
    semiring_0 @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__0) ).

thf(1106,plain,
    semiring_0 @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(93,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ~ ( ord_less_eq @ TA @ ( one_one @ TA ) @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_not__one__le__zero) ).

thf(578,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ~ ( ord_less_eq @ TA @ ( one_one @ TA ) @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(194,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ( ord_less_eq @ TA @ B @ ( one_one @ TA ) )
             => ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_mult__left__le__one__le) ).

thf(874,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ( ord_less_eq @ TA @ B @ ( one_one @ TA ) )
             => ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(242,axiom,
    ! [A: nat > nat,B: nat,C: nat,D: nat] :
      ( ( number2016821345itions @ A @ B )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ C ) )
       => ( ( ( fun_upd @ nat @ nat @ A @ C @ ( minus_minus @ nat @ ( A @ C ) @ ( one_one @ nat ) ) @ D )
           != ( zero_zero @ nat ) )
         => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ D )
            & ( ord_less_eq @ nat @ D @ ( minus_minus @ nat @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_partitions__remove1__bounds) ).

thf(1017,plain,
    ! [A: nat > nat,B: nat,C: nat,D: nat] :
      ( ( number2016821345itions @ A @ B )
     => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ C ) )
       => ( ( ( fun_upd @ nat @ nat @ A @ C @ ( minus_minus @ nat @ ( A @ C ) @ ( one_one @ nat ) ) @ D )
           != ( zero_zero @ nat ) )
         => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ D )
            & ( ord_less_eq @ nat @ D @ ( minus_minus @ nat @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(106,axiom,
    ! [TA: $tType] :
      ( ( no_bot @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_lt__ex) ).

thf(615,plain,
    ! [TA: $tType] :
      ( ( no_bot @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(280,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_less__eq__nat_Osimps_I1_J) ).

thf(1111,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(229,axiom,
    ( ( groups1340683514dd_sum @ nat @ nat
      @ ^ [A: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ A ) @ A )
      @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
    = ( minus_minus @ nat
      @ ( groups1340683514dd_sum @ nat @ nat
        @ ^ [A: nat] : ( times_times @ nat @ ( p @ A ) @ A )
        @ ( set_ord_atMost @ nat @ n ) )
      @ k ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_calculation) ).

thf(980,plain,
    ( ( groups1340683514dd_sum @ nat @ nat
      @ ^ [A: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ A ) @ A )
      @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
    = ( minus_minus @ nat
      @ ( groups1340683514dd_sum @ nat @ nat
        @ ^ [A: nat] : ( times_times @ nat @ ( p @ A ) @ A )
        @ ( set_ord_atMost @ nat @ n ) )
      @ k ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(239,axiom,
    ! [TA: $tType] :
      ( ( monoid_mult @ TA )
     => ( ( ^ [A: TA] : A )
        = ( times_times @ TA @ ( one_one @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_lambda__one) ).

thf(1006,plain,
    ! [TA: $tType] :
      ( ( monoid_mult @ TA )
     => ( ( ^ [A: TA] : A )
        = ( times_times @ TA @ ( one_one @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(36,axiom,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = A )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_zero__reorient) ).

thf(388,plain,
    ! [TA: $tType] :
      ( ( zero @ TA )
     => ! [A: TA] :
          ( ( ( zero_zero @ TA )
            = A )
          = ( A
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(25,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_224_less__induct) ).

thf(356,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)],[25]) ).

thf(15,axiom,
    ! [TA: $tType,TB: $tType,A: TA,B: TA,C: TA > TB,D: TB] :
      ( ( A != B )
     => ( ( fun_upd @ TA @ TB @ C @ B @ D @ A )
        = ( C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_fun__upd__other) ).

thf(328,plain,
    ! [TA: $tType,TB: $tType,A: TA,B: TA,C: TA > TB,D: TB] :
      ( ( A != B )
     => ( ( fun_upd @ TA @ TB @ C @ B @ D @ A )
        = ( C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(268,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
        & ( ord_less @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_nat__mult__less__cancel__disj) ).

thf(1082,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
        & ( ord_less @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(20,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_151_antisym__conv) ).

thf(339,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)],[20]) ).

thf(149,axiom,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ ( zero_zero @ nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_minus__nat_Odiff__0) ).

thf(753,plain,
    ! [A: nat] :
      ( ( minus_minus @ nat @ A @ ( zero_zero @ nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(78,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_229_ord__eq__less__trans) ).

thf(537,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)],[78]) ).

thf(119,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ ( times_times @ nat @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_le__square) ).

thf(669,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ A @ ( times_times @ nat @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(203,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( times_times @ nat @ A @ C ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ( B = C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_nat__mult__eq__cancel__disj) ).

thf(901,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( times_times @ nat @ A @ C ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ( B = C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(62,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_162_order__subst1) ).

thf(489,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)],[62]) ).

thf(98,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_218_order_Ostrict__trans) ).

thf(590,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)],[98]) ).

thf(296,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( semiring_0 @ TB )
     => ! [A: TB,B: TA > TB,C: set @ TA] :
          ( ( times_times @ TB @ A @ ( groups1340683514dd_sum @ TA @ TB @ B @ C ) )
          = ( groups1340683514dd_sum @ TA @ TB
            @ ^ [D: TA] : ( times_times @ TB @ A @ ( B @ D ) )
            @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_sum__distrib__left) ).

thf(1155,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( semiring_0 @ TB )
     => ! [A: TB,B: TA > TB,C: set @ TA] :
          ( ( times_times @ TB @ A @ ( groups1340683514dd_sum @ TA @ TB @ B @ C ) )
          = ( groups1340683514dd_sum @ TA @ TB
            @ ^ [D: TA] : ( times_times @ TB @ A @ ( B @ D ) )
            @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).

thf(38,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_214_dual__order_Ostrict__trans) ).

thf(393,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)],[38]) ).

thf(236,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_171_diff__eq__diff__eq) ).

thf(997,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)],[236]) ).

thf(231,axiom,
    ord_less @ nat @ ( zero_zero @ nat ) @ ( p @ k ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_gr0) ).

thf(985,plain,
    ord_less @ nat @ ( zero_zero @ nat ) @ ( p @ k ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(142,axiom,
    ordered_semiring @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Oordered__semiring) ).

thf(738,plain,
    ordered_semiring @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(136,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( one_one @ nat ) )
      = ( ( A
          = ( one_one @ nat ) )
        & ( B
          = ( one_one @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_nat__mult__eq__1__iff) ).

thf(721,plain,
    ! [A: nat,B: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( one_one @ nat ) )
      = ( ( A
          = ( one_one @ nat ) )
        & ( B
          = ( one_one @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(90,axiom,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ! [C: TA] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect @ TA @ A )
        = ( collect @ TA @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_Collect__cong) ).

thf(569,plain,
    ! [TA: $tType,A: TA > $o,B: TA > $o] :
      ( ! [C: TA] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect @ TA @ A )
        = ( collect @ TA @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(159,axiom,
    ordere1490568538miring @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Oordered__comm__semiring) ).

thf(779,plain,
    ordere1490568538miring @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(287,axiom,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_bot__nat__0_Oextremum) ).

thf(1131,plain,
    ! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).

thf(113,axiom,
    ! [TA: $tType] :
      ( ( ( comm_ring @ TA )
        & ( dvd @ TA ) )
     => ! [A: TA > $o,B: TA,C: TA > $o] :
          ( ! [D: TA,E: TA] :
              ( ( A @ D )
              = ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
         => ( ! [D: TA,E: TA] :
                ( ( C @ D )
                = ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
           => ! [D: TA,E: TA] :
                ( ( ( A @ D )
                  & ( C @ D ) )
                = ( ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) )
                  & ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_inf__period_I1_J) ).

thf(654,plain,
    ! [TA: $tType] :
      ( ( ( comm_ring @ TA )
        & ( dvd @ TA ) )
     => ! [A: TA > $o,B: TA,C: TA > $o] :
          ( ! [D: TA,E: TA] :
              ( ( A @ D )
              = ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
         => ( ! [D: TA,E: TA] :
                ( ( C @ D )
                = ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
           => ! [D: TA,E: TA] :
                ( ( ( A @ D )
                  & ( C @ D ) )
                = ( ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) )
                  & ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(258,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_mult @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( times_times @ TA @ A @ B ) @ C )
          = ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_ab__semigroup__mult__class_Omult__ac_I1_J) ).

thf(1060,plain,
    ! [TA: $tType] :
      ( ( ab_semigroup_mult @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( times_times @ TA @ A @ B ) @ C )
          = ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(253,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_mult__left__mono) ).

thf(1050,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(218,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_15_diff__diff__cancel) ).

thf(938,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)],[218]) ).

thf(27,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_144_linorder__wlog) ).

thf(360,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)],[27]) ).

thf(8,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_146_order__trans) ).

thf(311,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)],[8]) ).

thf(79,axiom,
    ! [TA: $tType] : ( order @ ( set @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oorder_6) ).

thf(541,plain,
    ! [TA: $tType] : ( order @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(114,axiom,
    ordere216010020id_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__comm__monoid__add) ).

thf(657,plain,
    ordere216010020id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(256,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ordere216010020id_add @ TA )
     => ! [A: set @ TB,B: TB > TA] :
          ( ! [C: TB] :
              ( ( member @ TB @ C @ A )
             => ( ord_less_eq @ TA @ ( B @ C ) @ ( zero_zero @ TA ) ) )
         => ( ord_less_eq @ TA @ ( groups1340683514dd_sum @ TB @ TA @ B @ A ) @ ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_sum__nonpos) ).

thf(1054,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ordere216010020id_add @ TA )
     => ! [A: set @ TB,B: TB > TA] :
          ( ! [C: TB] :
              ( ( member @ TB @ C @ A )
             => ( ord_less_eq @ TA @ ( B @ C ) @ ( zero_zero @ TA ) ) )
         => ( ord_less_eq @ TA @ ( groups1340683514dd_sum @ TB @ TA @ B @ A ) @ ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(121,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType] :
      ( ( comm_monoid_add @ TB )
     => ! [A: set @ TA,B: TA > TC,C: set @ TC,D: TC > TA,E: TA > TB,F: TC > TB] :
          ( ! [G: TA] :
              ( ( member @ TA @ G @ A )
             => ( ( member @ TC @ ( B @ G ) @ C )
                & ( ( D @ ( B @ G ) )
                  = G ) ) )
         => ( ! [G: TC] :
                ( ( member @ TC @ G @ C )
               => ( ( member @ TA @ ( D @ G ) @ A )
                  & ( ( B @ ( D @ G ) )
                    = G )
                  & ( ( E @ ( D @ G ) )
                    = ( F @ G ) ) ) )
           => ( ( groups1340683514dd_sum @ TC @ TB @ F @ C )
              = ( groups1340683514dd_sum @ TA @ TB @ E @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_sum_Oeq__general__inverses) ).

thf(674,plain,
    ! [TA: $tType,TB: $tType,TC: $tType] :
      ( ( comm_monoid_add @ TB )
     => ! [A: set @ TA,B: TA > TC,C: set @ TC,D: TC > TA,E: TA > TB,F: TC > TB] :
          ( ! [G: TA] :
              ( ( member @ TA @ G @ A )
             => ( ( member @ TC @ ( B @ G ) @ C )
                & ( ( D @ ( B @ G ) )
                  = G ) ) )
         => ( ! [G: TC] :
                ( ( member @ TC @ G @ C )
               => ( ( member @ TA @ ( D @ G ) @ A )
                  & ( ( B @ ( D @ G ) )
                    = G )
                  & ( ( E @ ( D @ G ) )
                    = ( F @ G ) ) ) )
           => ( ( groups1340683514dd_sum @ TC @ TB @ F @ C )
              = ( groups1340683514dd_sum @ TA @ TB @ E @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(244,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_32_eq__diff__iff) ).

thf(1026,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)],[244]) ).

thf(223,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_202_diff__gt__0__iff__gt) ).

thf(953,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)],[223]) ).

thf(214,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ D )
       => ( ord_less_eq @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_mult__le__mono) ).

thf(926,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ C @ D )
       => ( ord_less_eq @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(40,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_149_ord__eq__le__trans) ).

thf(398,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)],[40]) ).

thf(247,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ab_group_add @ TB )
     => ! [A: TA > TB,B: TA > TB,C: set @ TA] :
          ( ( groups1340683514dd_sum @ TA @ TB
            @ ^ [D: TA] : ( minus_minus @ TB @ ( A @ D ) @ ( B @ D ) )
            @ C )
          = ( minus_minus @ TB @ ( groups1340683514dd_sum @ TA @ TB @ A @ C ) @ ( groups1340683514dd_sum @ TA @ TB @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_sum__subtractf) ).

thf(1035,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ab_group_add @ TB )
     => ! [A: TA > TB,B: TA > TB,C: set @ TA] :
          ( ( groups1340683514dd_sum @ TA @ TB
            @ ^ [D: TA] : ( minus_minus @ TB @ ( A @ D ) @ ( B @ D ) )
            @ C )
          = ( minus_minus @ TB @ ( groups1340683514dd_sum @ TA @ TB @ A @ C ) @ ( groups1340683514dd_sum @ TA @ TB @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(156,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( one_one @ TA ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_mult__left__le) ).

thf(773,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( one_one @ TA ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(166,axiom,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( ( times_times @ TA @ A @ B )
              = ( times_times @ TA @ A @ C ) )
            = ( B = C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_mult__left__cancel) ).

thf(794,plain,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( ( times_times @ TA @ A @ B )
              = ( times_times @ TA @ A @ C ) )
            = ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(1,conjecture,
    ( ( minus_minus @ nat
      @ ( groups1340683514dd_sum @ nat @ nat
        @ ^ [A: nat] : ( times_times @ nat @ ( p @ A ) @ A )
        @ ( set_ord_atMost @ nat @ n ) )
      @ k )
    = ( minus_minus @ nat @ n @ k ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).

thf(2,negated_conjecture,
    ( ( minus_minus @ nat
      @ ( groups1340683514dd_sum @ nat @ nat
        @ ^ [A: nat] : ( times_times @ nat @ ( p @ A ) @ A )
        @ ( set_ord_atMost @ nat @ n ) )
      @ k )
   != ( minus_minus @ nat @ n @ k ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(298,plain,
    ( ( minus_minus @ nat
      @ ( groups1340683514dd_sum @ nat @ nat
        @ ^ [A: nat] : ( times_times @ nat @ ( p @ A ) @ A )
        @ ( set_ord_atMost @ nat @ n ) )
      @ k )
   != ( minus_minus @ nat @ n @ k ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(45,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_217_less__imp__not__less) ).

thf(412,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)],[45]) ).

thf(227,axiom,
    ! [TA: $tType] :
      ( ( ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( minus_minus @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_left__diff__distrib) ).

thf(965,plain,
    ! [TA: $tType] :
      ( ( ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
          = ( minus_minus @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(293,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_203_zero__less__diff) ).

thf(1146,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)],[293]) ).

thf(276,axiom,
    ! [TA: $tType] :
      ( ( neg_numeral @ TA )
     => ( ( minus_minus @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) )
        = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_diff__numeral__special_I9_J) ).

thf(1103,plain,
    ! [TA: $tType] :
      ( ( neg_numeral @ TA )
     => ( ( minus_minus @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) )
        = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(177,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( times_times @ nat @ C @ A ) @ ( times_times @ nat @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_mult__le__mono2) ).

thf(824,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( times_times @ nat @ C @ A ) @ ( times_times @ nat @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(187,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_26_diff__le__mono2) ).

thf(856,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)],[187]) ).

thf(32,axiom,
    ! [TA: $tType] :
      ( ( zero_neq_one @ TA )
     => ( ( zero_zero @ TA )
       != ( one_one @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_zero__neq__one) ).

thf(375,plain,
    ! [TA: $tType] :
      ( ( zero_neq_one @ TA )
     => ( ( zero_zero @ TA )
       != ( one_one @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(96,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_220_linorder__cases) ).

thf(584,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)],[96]) ).

thf(204,axiom,
    ! [TA: $tType] :
      ( ( monoid_mult @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ ( one_one @ TA ) @ A )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_mult_Oleft__neutral) ).

thf(904,plain,
    ! [TA: $tType] :
      ( ( monoid_mult @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ ( one_one @ TA ) @ A )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(48,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_243_order__less__subst1) ).

thf(423,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)],[48]) ).

thf(41,axiom,
    ! [TA: $tType] : ( preorder @ ( set @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Opreorder_5) ).

thf(402,plain,
    ! [TA: $tType] : ( preorder @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

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

thf(772,plain,
    comm_monoid_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(132,axiom,
    minus @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ominus_4) ).

thf(711,plain,
    minus @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(285,axiom,
    ! [A: nat > nat,B: nat] :
      ( ! [C: nat] :
          ( ( ( A @ C )
           != ( zero_zero @ nat ) )
         => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ C )
            & ( ord_less_eq @ nat @ C @ B ) ) )
     => ( ( ( groups1340683514dd_sum @ nat @ nat
            @ ^ [C: nat] : ( times_times @ nat @ ( A @ C ) @ C )
            @ ( set_ord_atMost @ nat @ B ) )
          = B )
       => ( number2016821345itions @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_partitionsI) ).

thf(1124,plain,
    ! [A: nat > nat,B: nat] :
      ( ! [C: nat] :
          ( ( ( A @ C )
           != ( zero_zero @ nat ) )
         => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ C )
            & ( ord_less_eq @ nat @ C @ B ) ) )
     => ( ( ( groups1340683514dd_sum @ nat @ nat
            @ ^ [C: nat] : ( times_times @ nat @ ( A @ C ) @ C )
            @ ( set_ord_atMost @ nat @ B ) )
          = B )
       => ( number2016821345itions @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).

thf(91,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_228_ord__less__eq__trans) ).

thf(572,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)],[91]) ).

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_231_less__linear) ).

thf(313,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(105,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_less__imp__neq) ).

thf(611,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less @ TA @ A @ B )
         => ( A != B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(278,axiom,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) )
         => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_split__mult__pos__le) ).

thf(1107,plain,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) )
         => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(18,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_153_order_Otrans) ).

thf(335,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)],[18]) ).

thf(271,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ A )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_le__antisym) ).

thf(1090,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ( ord_less_eq @ nat @ B @ A )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(210,axiom,
    dvd @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Odvd) ).

thf(919,plain,
    dvd @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(37,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_165_le__funE) ).

thf(391,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)],[37]) ).

thf(290,axiom,
    ! [A: nat] :
      ( ( times_times @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_mult__0) ).

thf(1139,plain,
    ! [A: nat] :
      ( ( times_times @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).

thf(43,axiom,
    ! [TA: $tType,TB: $tType,A: TB,B: TB,C: TB > TA,D: TA,E: TA] :
      ( ( A != B )
     => ( ( fun_upd @ TB @ TA @ ( fun_upd @ TB @ TA @ C @ A @ D ) @ B @ E )
        = ( fun_upd @ TB @ TA @ ( fun_upd @ TB @ TA @ C @ B @ E ) @ A @ D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_fun__upd__twist) ).

thf(406,plain,
    ! [TA: $tType,TB: $tType,A: TB,B: TB,C: TB > TA,D: TA,E: TA] :
      ( ( A != B )
     => ( ( fun_upd @ TB @ TA @ ( fun_upd @ TB @ TA @ C @ A @ D ) @ B @ E )
        = ( fun_upd @ TB @ TA @ ( fun_upd @ TB @ TA @ C @ B @ E ) @ A @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(141,axiom,
    cancel146912293up_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add) ).

thf(737,plain,
    cancel146912293up_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(101,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_216_exists__least__iff) ).

thf(599,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)],[101]) ).

thf(233,axiom,
    ! [TA: $tType] :
      ( ( comm_monoid_mult @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ ( one_one @ TA ) @ A )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_comm__monoid__mult__class_Omult__1) ).

thf(990,plain,
    ! [TA: $tType] :
      ( ( comm_monoid_mult @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ ( one_one @ TA ) @ A )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(235,axiom,
    comm_monoid_mult @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__mult) ).

thf(996,plain,
    comm_monoid_mult @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(4,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_164_le__funI) ).

thf(303,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)],[4]) ).

thf(284,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
      | ( ord_less_eq @ nat @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_nat__le__linear) ).

thf(1122,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
      | ( ord_less_eq @ nat @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(110,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( minus_minus @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_diff__mult__distrib) ).

thf(631,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
      = ( minus_minus @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(151,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ C @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
        & ( ord_less @ nat @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_mult__less__cancel2) ).

thf(758,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ C @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
        & ( ord_less @ nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(29,axiom,
    ! [TA: $tType] :
      ( ( one @ TA )
     => ! [A: TA] :
          ( ( ( one_one @ TA )
            = A )
          = ( A
            = ( one_one @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_one__reorient) ).

thf(367,plain,
    ! [TA: $tType] :
      ( ( one @ TA )
     => ! [A: TA] :
          ( ( ( one_one @ TA )
            = A )
          = ( A
            = ( one_one @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(294,axiom,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( ( times_times @ TA @ B @ A )
              = ( times_times @ TA @ C @ A ) )
            = ( B = C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_mult__right__cancel) ).

thf(1149,plain,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( ( times_times @ TA @ B @ A )
              = ( times_times @ TA @ C @ A ) )
            = ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).

thf(220,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) )
         => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_split__mult__neg__le) ).

thf(943,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
              & ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
            | ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
              & ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) )
         => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(216,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType] :
      ( ( semiring_0 @ TC )
     => ! [A: TA > TC,B: set @ TA,C: TB > TC,D: set @ TB] :
          ( ( times_times @ TC @ ( groups1340683514dd_sum @ TA @ TC @ A @ B ) @ ( groups1340683514dd_sum @ TB @ TC @ C @ D ) )
          = ( groups1340683514dd_sum @ TA @ TC
            @ ^ [E: TA] :
                ( groups1340683514dd_sum @ TB @ TC
                @ ^ [F: TB] : ( times_times @ TC @ ( A @ E ) @ ( C @ F ) )
                @ D )
            @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_sum__product) ).

thf(932,plain,
    ! [TA: $tType,TB: $tType,TC: $tType] :
      ( ( semiring_0 @ TC )
     => ! [A: TA > TC,B: set @ TA,C: TB > TC,D: set @ TB] :
          ( ( times_times @ TC @ ( groups1340683514dd_sum @ TA @ TC @ A @ B ) @ ( groups1340683514dd_sum @ TB @ TC @ C @ D ) )
          = ( groups1340683514dd_sum @ TA @ TC
            @ ^ [E: TA] :
                ( groups1340683514dd_sum @ TB @ TC
                @ ^ [F: TB] : ( times_times @ TC @ ( A @ E ) @ ( C @ F ) )
                @ D )
            @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(184,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_130_le__iff__diff__le__0) ).

thf(848,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)],[184]) ).

thf(128,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_181_diff__mono) ).

thf(700,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)],[128]) ).

thf(225,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_92_diff__is__0__eq) ).

thf(959,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)],[225]) ).

thf(250,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( minus @ TB )
     => ( ( minus_minus @ ( TA > TB ) )
        = ( ^ [A: TA > TB,B: TA > TB,C: TA] : ( minus_minus @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_minus__apply) ).

thf(1044,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( minus @ TB )
     => ( ( minus_minus @ ( TA > TB ) )
        = ( ^ [A: TA > TB,B: TA > TB,C: TA] : ( minus_minus @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(160,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TB > TA,B: set @ TB] :
          ( ( ( groups1340683514dd_sum @ TB @ TA @ A @ B )
           != ( zero_zero @ TA ) )
         => ~ ! [C: TB] :
                ( ( member @ TB @ C @ B )
               => ( ( A @ C )
                  = ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_sum_Onot__neutral__contains__not__neutral) ).

thf(780,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: TB > TA,B: set @ TB] :
          ( ( ( groups1340683514dd_sum @ TB @ TA @ A @ B )
           != ( zero_zero @ TA ) )
         => ~ ! [C: TB] :
                ( ( member @ TB @ C @ B )
               => ( ( A @ C )
                  = ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(104,axiom,
    ! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA] :
      ( ( fun_upd @ TB @ TA @ A @ B @ C @ B )
      = C ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_fun__upd__same) ).

thf(608,plain,
    ! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA] :
      ( ( fun_upd @ TB @ TA @ A @ B @ C @ B )
      = C ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(232,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_eq__imp__le) ).

thf(986,plain,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ord_less_eq @ nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(237,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_diff__self) ).

thf(1000,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(185,axiom,
    ! [TA: $tType] :
      ( ( ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
          = ( minus_minus @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_right__diff__distrib) ).

thf(851,plain,
    ! [TA: $tType] :
      ( ( ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
          = ( minus_minus @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(51,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_less__irrefl) ).

thf(431,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(212,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ A @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
        & ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_nat__0__less__mult__iff) ).

thf(922,plain,
    ! [A: nat,B: nat] :
      ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ A @ B ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
        & ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(115,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_mult__nonneg__nonpos) ).

thf(658,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(123,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: set @ TB] :
          ( ( groups1340683514dd_sum @ TB @ TA
            @ ^ [B: TB] : ( zero_zero @ TA )
            @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_sum_Oneutral__const) ).

thf(685,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: set @ TB] :
          ( ( groups1340683514dd_sum @ TB @ TA
            @ ^ [B: TB] : ( zero_zero @ TA )
            @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(71,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_166_le__funD) ).

thf(521,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)],[71]) ).

thf(154,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_31_le__diff__iff) ).

thf(768,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)],[154]) ).

thf(178,axiom,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
           != ( zero_zero @ TA ) )
         => ( ( A
             != ( zero_zero @ TA ) )
            & ( B
             != ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_mult__not__zero) ).

thf(827,plain,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
           != ( zero_zero @ TA ) )
         => ( ( A
             != ( zero_zero @ TA ) )
            & ( B
             != ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(260,axiom,
    comm_s1003936772cancel @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel) ).

thf(1066,plain,
    comm_s1003936772cancel @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(192,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_bot__nat__0_Onot__eq__extremum) ).

thf(868,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(59,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_215_linorder__less__wlog) ).

thf(455,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)],[59]) ).

thf(198,axiom,
    ! [TA: $tType] :
      ( ( comm_s1003936772cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
          = ( minus_minus @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_right__diff__distrib_H) ).

thf(883,plain,
    ! [TA: $tType] :
      ( ( comm_s1003936772cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
          = ( minus_minus @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(281,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_29_diff__le__mono) ).

thf(1113,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)],[281]) ).

thf(88,axiom,
    ! [TA: $tType] : ( minus @ ( set @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Groups_Ominus_8) ).

thf(563,plain,
    ! [TA: $tType] : ( minus @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(265,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_mult__nonneg__nonneg) ).

thf(1076,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(49,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_154_le__cases) ).

thf(426,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)],[49]) ).

thf(75,axiom,
    linorder @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Olinorder_10) ).

thf(531,plain,
    linorder @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(173,axiom,
    ! [TA: $tType] :
      ( ( ring_11004092258visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
            = ( times_times @ TA @ B @ A ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( one_one @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_mult__cancel__right1) ).

thf(812,plain,
    ! [TA: $tType] :
      ( ( ring_11004092258visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
            = ( times_times @ TA @ B @ A ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( one_one @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(21,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_less__numeral__extra_I3_J) ).

thf(342,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(292,axiom,
    order @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oorder_2) ).

thf(1145,plain,
    order @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).

thf(84,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_156_linear) ).

thf(553,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)],[84]) ).

thf(228,axiom,
    ! [TA: $tType,TB: $tType,TC: $tType] :
      ( ( comm_monoid_add @ TB )
     => ! [A: set @ TA,B: set @ TC,C: TC > TA,D: TA > TB,E: TC > TB] :
          ( ! [F: TA] :
              ( ( member @ TA @ F @ A )
             => ? [G: TC] :
                  ( ( member @ TC @ G @ B )
                  & ( ( C @ G )
                    = F )
                  & ! [H: TC] :
                      ( ( ( member @ TC @ H @ B )
                        & ( ( C @ H )
                          = F ) )
                     => ( H = G ) ) ) )
         => ( ! [F: TC] :
                ( ( member @ TC @ F @ B )
               => ( ( member @ TA @ ( C @ F ) @ A )
                  & ( ( D @ ( C @ F ) )
                    = ( E @ F ) ) ) )
           => ( ( groups1340683514dd_sum @ TC @ TB @ E @ B )
              = ( groups1340683514dd_sum @ TA @ TB @ D @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_sum_Oeq__general) ).

thf(968,plain,
    ! [TA: $tType,TB: $tType,TC: $tType] :
      ( ( comm_monoid_add @ TB )
     => ! [A: set @ TA,B: set @ TC,C: TC > TA,D: TA > TB,E: TC > TB] :
          ( ! [F: TA] :
              ( ( member @ TA @ F @ A )
             => ? [G: TC] :
                  ( ( member @ TC @ G @ B )
                  & ( ( C @ G )
                    = F )
                  & ! [H: TC] :
                      ( ( ( member @ TC @ H @ B )
                        & ( ( C @ H )
                          = F ) )
                     => ( H = G ) ) ) )
         => ( ! [F: TC] :
                ( ( member @ TC @ F @ B )
               => ( ( member @ TA @ ( C @ F ) @ A )
                  & ( ( D @ ( C @ F ) )
                    = ( E @ F ) ) ) )
           => ( ( groups1340683514dd_sum @ TC @ TB @ E @ B )
              = ( groups1340683514dd_sum @ TA @ TB @ D @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(180,axiom,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( times_times @ TA @ A @ C ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B = C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_mult__cancel__left) ).

thf(836,plain,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( times_times @ TA @ A @ C ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(252,axiom,
    comm_monoid_diff @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__diff) ).

thf(1049,plain,
    comm_monoid_diff @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(117,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
       => ( ord_less_eq @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_nat__mult__le__cancel__disj) ).

thf(663,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
      = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
       => ( ord_less_eq @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(82,axiom,
    ! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA,D: TA] :
      ( ( fun_upd @ TB @ TA @ ( fun_upd @ TB @ TA @ A @ B @ C ) @ B @ D )
      = ( fun_upd @ TB @ TA @ A @ B @ D ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_fun__upd__upd) ).

thf(548,plain,
    ! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA,D: TA] :
      ( ( fun_upd @ TB @ TA @ ( fun_upd @ TB @ TA @ A @ B @ C ) @ B @ D )
      = ( fun_upd @ TB @ TA @ A @ B @ D ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(107,axiom,
    ( number2016821345itions
    = ( ^ [A: nat > nat,B: nat] :
          ( ! [C: nat] :
              ( ( ( A @ C )
               != ( zero_zero @ nat ) )
             => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ C )
                & ( ord_less_eq @ nat @ C @ B ) ) )
          & ( ( groups1340683514dd_sum @ nat @ nat
              @ ^ [C: nat] : ( times_times @ nat @ ( A @ C ) @ C )
              @ ( set_ord_atMost @ nat @ B ) )
            = B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_partitions__def) ).

thf(617,plain,
    ( number2016821345itions
    = ( ^ [A: nat > nat,B: nat] :
          ( ! [C: nat] :
              ( ( ( A @ C )
               != ( zero_zero @ nat ) )
             => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ C )
                & ( ord_less_eq @ nat @ C @ B ) ) )
          & ( ( groups1340683514dd_sum @ nat @ nat
              @ ^ [C: nat] : ( times_times @ nat @ ( A @ C ) @ C )
              @ ( set_ord_atMost @ nat @ B ) )
            = B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(69,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TA )
     => ( ord @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oord) ).

thf(516,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ord @ TA )
     => ( ord @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(158,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( times_times @ nat @ A @ B ) )
     => ( ( B
          = ( one_one @ nat ) )
        | ( A
          = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_mult__eq__self__implies__10) ).

thf(776,plain,
    ! [A: nat,B: nat] :
      ( ( A
        = ( times_times @ nat @ A @ B ) )
     => ( ( B
          = ( one_one @ nat ) )
        | ( A
          = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(73,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_249_measure__induct__rule) ).

thf(525,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)],[73]) ).

thf(94,axiom,
    ord @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oord_12) ).

thf(580,plain,
    ord @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(125,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( ordere216010020id_add @ TB )
     => ! [A: set @ TA,B: TA > TB] :
          ( ! [C: TA] :
              ( ( member @ TA @ C @ A )
             => ( ord_less_eq @ TB @ ( zero_zero @ TB ) @ ( B @ C ) ) )
         => ( ord_less_eq @ TB @ ( zero_zero @ TB ) @ ( groups1340683514dd_sum @ TA @ TB @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_sum__nonneg) ).

thf(690,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( ordere216010020id_add @ TB )
     => ! [A: set @ TA,B: TA > TB] :
          ( ! [C: TA] :
              ( ( member @ TA @ C @ A )
             => ( ord_less_eq @ TB @ ( zero_zero @ TB ) @ ( B @ C ) ) )
         => ( ord_less_eq @ TB @ ( zero_zero @ TB ) @ ( groups1340683514dd_sum @ TA @ TB @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(262,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_183_diff__right__mono) ).

thf(1070,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)],[262]) ).

thf(130,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( times_times @ nat @ C @ B ) )
      = ( ( A = C )
        | ( B
          = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_mult__cancel2) ).

thf(705,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( times_times @ nat @ C @ B ) )
      = ( ( A = C )
        | ( B
          = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(46,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_244_ord__less__eq__subst) ).

thf(414,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)],[46]) ).

thf(143,axiom,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ B @ C ) @ ( times_times @ TA @ A @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_mult__right__mono__neg) ).

thf(739,plain,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ B @ C ) @ ( times_times @ TA @ A @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(44,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_238_neq__iff) ).

thf(409,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)],[44]) ).

thf(234,axiom,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_less__nat__zero__code) ).

thf(993,plain,
    ! [A: nat] :
      ~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(133,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_30_Nat_Odiff__diff__eq) ).

thf(712,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)],[133]) ).

thf(86,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_157_antisym) ).

thf(557,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)],[86]) ).

thf(238,axiom,
    ! [TA: $tType] :
      ( ( semiri1193490041visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( B
             != ( zero_zero @ TA ) )
           => ( ( times_times @ TA @ A @ B )
             != ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_no__zero__divisors) ).

thf(1003,plain,
    ! [TA: $tType] :
      ( ( semiri1193490041visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( A
           != ( zero_zero @ TA ) )
         => ( ( B
             != ( zero_zero @ TA ) )
           => ( ( times_times @ TA @ A @ B )
             != ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(255,axiom,
    ord_less_eq @ nat @ k @ n,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1__092_060open_062k_A_092_060le_062_An_092_060close_062) ).

thf(1053,plain,
    ord_less_eq @ nat @ k @ n,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(165,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_mult__le__mono1) ).

thf(791,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq @ nat @ A @ B )
     => ( ord_less_eq @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(16,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_237_order_Oasym) ).

thf(331,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)],[16]) ).

thf(222,axiom,
    ! [A: nat > nat,B: nat] :
      ( ( number2016821345itions @ A @ B )
     => ( ord_less_eq @ nat @ ( groups1340683514dd_sum @ nat @ nat @ A @ ( set_ord_atMost @ nat @ B ) ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_partitions__parts__bounded) ).

thf(951,plain,
    ! [A: nat > nat,B: nat] :
      ( ( number2016821345itions @ A @ B )
     => ( ord_less_eq @ nat @ ( groups1340683514dd_sum @ nat @ nat @ A @ ( set_ord_atMost @ nat @ B ) ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(26,axiom,
    preorder @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Opreorder_9) ).

thf(359,plain,
    preorder @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(205,axiom,
    ! [TA: $tType] :
      ( ( ring_11004092258visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = A )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( one_one @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_mult__cancel__left2) ).

thf(907,plain,
    ! [TA: $tType] :
      ( ( ring_11004092258visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = A )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( one_one @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(248,axiom,
    ! [TA: $tType] :
      ( ( semiri1193490041visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( zero_zero @ TA ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_mult__eq__0__iff) ).

thf(1038,plain,
    ! [TA: $tType] :
      ( ( semiri1193490041visors @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( zero_zero @ TA ) )
          = ( ( A
              = ( zero_zero @ TA ) )
            | ( B
              = ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(30,axiom,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
       => ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_subset__antisym) ).

thf(370,plain,
    ! [TA: $tType,A: set @ TA,B: set @ TA] :
      ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
     => ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
       => ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(269,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_86_diff__ge__0__iff__ge) ).

thf(1085,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)],[269]) ).

thf(92,axiom,
    ! [TA: $tType] :
      ( ( no_top @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_gt__ex) ).

thf(576,plain,
    ! [TA: $tType] :
      ( ( no_top @ TA )
     => ! [A: TA] :
        ? [B: TA] : ( ord_less @ TA @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(145,axiom,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_diff__zero) ).

thf(744,plain,
    ! [TA: $tType] :
      ( ( cancel1352612707id_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(243,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ( B
          = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_mult__is__0) ).

thf(1023,plain,
    ! [A: nat,B: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( zero_zero @ nat ) )
      = ( ( A
          = ( zero_zero @ nat ) )
        | ( B
          = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(254,axiom,
    preorder @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Opreorder_1) ).

thf(1052,plain,
    preorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(190,axiom,
    cancel1352612707id_add @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add) ).

thf(864,plain,
    cancel1352612707id_add @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(138,axiom,
    zero @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ozero) ).

thf(727,plain,
    zero @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(273,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( times_times @ nat @ A @ C ) )
      = ( ( B = C )
        | ( A
          = ( zero_zero @ nat ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_mult__cancel1) ).

thf(1094,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( times_times @ nat @ A @ B )
        = ( times_times @ nat @ A @ C ) )
      = ( ( B = C )
        | ( A
          = ( zero_zero @ nat ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(188,axiom,
    ! [TA: $tType] :
      ( ( ordere1490568538miring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_ordered__comm__semiring__class_Ocomm__mult__left__mono) ).

thf(859,plain,
    ! [TA: $tType] :
      ( ( ordere1490568538miring @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(65,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_245_ord__eq__less__subst) ).

thf(497,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)],[65]) ).

thf(195,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_172_cancel__ab__semigroup__add__class_Odiff__right__commute) ).

thf(876,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)],[195]) ).

thf(183,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ( ( set_ord_atMost @ TA )
        = ( ^ [A: TA] :
              ( collect @ TA
              @ ^ [B: TA] : ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_atMost__def) ).

thf(845,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ( ( set_ord_atMost @ TA )
        = ( ^ [A: TA] :
              ( collect @ TA
              @ ^ [B: TA] : ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(57,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_242_order__less__subst2) ).

thf(450,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)],[57]) ).

thf(120,axiom,
    ! [A: nat] :
      ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_diff__0__eq__0) ).

thf(671,plain,
    ! [A: nat] :
      ( ( minus_minus @ nat @ ( zero_zero @ nat ) @ A )
      = ( zero_zero @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(124,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_182_diff__left__mono) ).

thf(688,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)],[124]) ).

thf(95,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_147_order__class_Oorder_Oantisym) ).

thf(581,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)],[95]) ).

thf(64,axiom,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_le__numeral__extra_I3_J) ).

thf(495,plain,
    ! [TA: $tType] :
      ( ( linord1659791738miring @ TA )
     => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(241,axiom,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ D )
           => ( ord_less_eq @ nat @ D @ C ) )
       => ~ ! [D: nat] :
              ( ( A @ D )
             => ~ ! [E: nat] :
                    ( ( A @ E )
                   => ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_bounded__Max__nat) ).

thf(1010,plain,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ D )
           => ( ord_less_eq @ nat @ D @ C ) )
       => ~ ! [D: nat] :
              ( ( A @ D )
             => ~ ! [E: nat] :
                    ( ( A @ E )
                   => ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(182,axiom,
    monoid_mult @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Omonoid__mult) ).

thf(844,plain,
    monoid_mult @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(11,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( fun_upd @ TA @ TB )
      = ( ^ [A: TA > TB,B: TA,C: TB,D: TA] : ( if @ TB @ ( D = B ) @ C @ ( A @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_fun__upd__apply) ).

thf(319,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( fun_upd @ TA @ TB )
      = ( ^ [A: TA > TB,B: TA,C: TB,D: TA] : ( if @ TB @ ( D = B ) @ C @ ( A @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(221,axiom,
    ! [TA: $tType,A: set @ TA,B: TA > nat,C: TA > nat] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( ord_less_eq @ nat @ ( B @ D ) @ ( C @ D ) ) )
     => ( ( groups1340683514dd_sum @ TA @ nat
          @ ^ [D: TA] : ( minus_minus @ nat @ ( C @ D ) @ ( B @ D ) )
          @ A )
        = ( minus_minus @ nat @ ( groups1340683514dd_sum @ TA @ nat @ C @ A ) @ ( groups1340683514dd_sum @ TA @ nat @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_sum__subtractf__nat) ).

thf(946,plain,
    ! [TA: $tType,A: set @ TA,B: TA > nat,C: TA > nat] :
      ( ! [D: TA] :
          ( ( member @ TA @ D @ A )
         => ( ord_less_eq @ nat @ ( B @ D ) @ ( C @ D ) ) )
     => ( ( groups1340683514dd_sum @ TA @ nat
          @ ^ [D: TA] : ( minus_minus @ nat @ ( C @ D ) @ ( B @ D ) )
          @ A )
        = ( minus_minus @ nat @ ( groups1340683514dd_sum @ TA @ nat @ C @ A ) @ ( groups1340683514dd_sum @ TA @ nat @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(162,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_250_linorder__neqE__nat) ).

thf(786,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)],[162]) ).

thf(201,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_mult__nonpos__nonneg) ).

thf(896,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring_0 @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(147,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_91_diff__is__0__eq_H) ).

thf(748,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)],[147]) ).

thf(245,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( minus @ TB )
     => ( ( minus_minus @ ( TA > TB ) )
        = ( ^ [A: TA > TB,B: TA > TB,C: TA] : ( minus_minus @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_fun__diff__def) ).

thf(1030,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( minus @ TB )
     => ( ( minus_minus @ ( TA > TB ) )
        = ( ^ [A: TA > TB,B: TA > TB,C: TA] : ( minus_minus @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(52,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_198_not__gr__zero) ).

thf(433,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)],[52]) ).

thf(215,axiom,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( times_times @ TA @ C @ B ) )
          = ( ( B
              = ( zero_zero @ TA ) )
            | ( A = C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_mult__cancel__right) ).

thf(929,plain,
    ! [TA: $tType] :
      ( ( semiri1923998003cancel @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( ( times_times @ TA @ A @ B )
            = ( times_times @ TA @ C @ B ) )
          = ( ( B
              = ( zero_zero @ TA ) )
            | ( A = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(153,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: set @ TB,B: TB > TA] :
          ( ! [C: TB] :
              ( ( member @ TB @ C @ A )
             => ( ( B @ C )
                = ( zero_zero @ TA ) ) )
         => ( ( groups1340683514dd_sum @ TB @ TA @ B @ A )
            = ( zero_zero @ TA ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_sum_Oneutral) ).

thf(763,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( comm_monoid_add @ TA )
     => ! [A: set @ TB,B: TB > TA] :
          ( ! [C: TB] :
              ( ( member @ TB @ C @ A )
             => ( ( B @ C )
                = ( zero_zero @ TA ) ) )
         => ( ( groups1340683514dd_sum @ TB @ TA @ B @ A )
            = ( zero_zero @ TA ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(208,axiom,
    ! [TA: $tType] :
      ( ( linordered_ring @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_zero__le__square) ).

thf(914,plain,
    ! [TA: $tType] :
      ( ( linordered_ring @ TA )
     => ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(33,axiom,
    ! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA] :
      ( ( ( fun_upd @ TB @ TA @ A @ B @ C )
        = A )
      = ( ( A @ B )
        = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_fun__upd__idem__iff) ).

thf(378,plain,
    ! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA] :
      ( ( ( fun_upd @ TB @ TA @ A @ B @ C )
        = A )
      = ( ( A @ B )
        = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(134,axiom,
    ! [A: nat] :
      ( ( ord_less @ nat @ A @ ( one_one @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_less__one) ).

thf(716,plain,
    ! [A: nat] :
      ( ( ord_less @ nat @ A @ ( one_one @ nat ) )
      = ( A
        = ( zero_zero @ nat ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(168,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_74_zero__diff) ).

thf(800,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)],[168]) ).

thf(282,axiom,
    ! [A: nat] :
      ( ( times_times @ nat @ A @ ( one_one @ nat ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_nat__mult__1__right) ).

thf(1116,plain,
    ! [A: nat] :
      ( ( times_times @ nat @ A @ ( one_one @ nat ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(288,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_101_eq__iff__diff__eq__0) ).

thf(1133,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)],[288]) ).

thf(3,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_213_not__less__iff__gr__or__eq) ).

thf(300,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)],[3]) ).

thf(56,axiom,
    ! [TA: $tType,A: set @ TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( member @ TA @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_Collect__mem__eq) ).

thf(447,plain,
    ! [TA: $tType,A: set @ TA] :
      ( ( collect @ TA
        @ ^ [B: TA] : ( member @ TA @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(274,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_mult @ TA )
     => ( ( times_times @ TA )
        = ( ^ [A: TA,B: TA] : ( times_times @ TA @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_mult_Ocommute) ).

thf(1097,plain,
    ! [TA: $tType] :
      ( ( ab_semigroup_mult @ TA )
     => ( ( times_times @ TA )
        = ( ^ [A: TA,B: TA] : ( times_times @ TA @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(175,axiom,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ ( set @ TA ) @ ( set_ord_atMost @ TA @ A ) @ ( set_ord_atMost @ TA @ B ) )
          = ( ord_less_eq @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_atMost__subset__iff) ).

thf(818,plain,
    ! [TA: $tType] :
      ( ( preorder @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ ( set @ TA ) @ ( set_ord_atMost @ TA @ A ) @ ( set_ord_atMost @ TA @ B ) )
          = ( ord_less_eq @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(186,axiom,
    ( ( groups1340683514dd_sum @ nat @ nat
      @ ^ [A: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ A ) @ A )
      @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
    = ( groups1340683514dd_sum @ nat @ nat
      @ ^ [A: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ A ) @ A )
      @ ( set_ord_atMost @ nat @ n ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4__092_060open_062_I_092_060Sum_062i_092_060le_062n_A_N_Ak_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_A_061_A_I_092_060Sum_062i_092_060le_062n_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_092_060close_062) ).

thf(854,plain,
    ( ( groups1340683514dd_sum @ nat @ nat
      @ ^ [A: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ A ) @ A )
      @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
    = ( groups1340683514dd_sum @ nat @ nat
      @ ^ [A: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ A ) @ A )
      @ ( set_ord_atMost @ nat @ n ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(164,axiom,
    linorder @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Olinorder) ).

thf(790,plain,
    linorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(191,axiom,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA] :
          ( ( member @ TA @ A @ ( set_ord_atMost @ TA @ B ) )
          = ( ord_less_eq @ TA @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_atMost__iff) ).

thf(865,plain,
    ! [TA: $tType] :
      ( ( ord @ TA )
     => ! [A: TA,B: TA] :
          ( ( member @ TA @ A @ ( set_ord_atMost @ TA @ B ) )
          = ( ord_less_eq @ TA @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(31,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_dual__order_Oirrefl) ).

thf(373,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA] :
          ~ ( ord_less @ TA @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(5,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_221_less__imp__triv) ).

thf(305,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)],[5]) ).

thf(14,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_143_dual__order_Otrans) ).

thf(326,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)],[14]) ).

thf(171,axiom,
    ! [TA: $tType] :
      ( ( monoid_mult @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ A @ ( one_one @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_mult_Oright__neutral) ).

thf(807,plain,
    ! [TA: $tType] :
      ( ( monoid_mult @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ A @ ( one_one @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(246,axiom,
    ! [TA: $tType] :
      ( ( ordered_semiring @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
             => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
               => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_mult__mono_H) ).

thf(1033,plain,
    ! [TA: $tType] :
      ( ( ordered_semiring @ TA )
     => ! [A: TA,B: TA,C: TA,D: TA] :
          ( ( ord_less_eq @ TA @ A @ B )
         => ( ( ord_less_eq @ TA @ C @ D )
           => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
             => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
               => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(76,axiom,
    order @ $o,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oorder_11) ).

thf(532,plain,
    order @ $o,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(127,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_27_le__diff__iff_H) ).

thf(696,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)],[127]) ).

thf(199,axiom,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_neq0__conv) ).

thf(886,plain,
    ! [A: nat] :
      ( ( A
       != ( zero_zero @ nat ) )
      = ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(139,axiom,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ D )
           => ( ord_less_eq @ nat @ D @ C ) )
       => ? [D: nat] :
            ( ( A @ D )
            & ! [E: nat] :
                ( ( A @ E )
               => ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_Nat_Oex__has__greatest__nat) ).

thf(728,plain,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ D )
           => ( ord_less_eq @ nat @ D @ C ) )
       => ? [D: nat] :
            ( ( A @ D )
            & ! [E: nat] :
                ( ( A @ E )
               => ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(286,axiom,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ( ord_less_eq @ TA @ B @ ( one_one @ TA ) )
             => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_mult__right__le__one__le) ).

thf(1129,plain,
    ! [TA: $tType] :
      ( ( linordered_idom @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
         => ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
           => ( ( ord_less_eq @ TA @ B @ ( one_one @ TA ) )
             => ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).

thf(295,axiom,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( set_ord_atMost @ TA @ A )
            = ( set_ord_atMost @ TA @ B ) )
          = ( A = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_atMost__eq__iff) ).

thf(1152,plain,
    ! [TA: $tType] :
      ( ( order @ TA )
     => ! [A: TA,B: TA] :
          ( ( ( set_ord_atMost @ TA @ A )
            = ( set_ord_atMost @ TA @ B ) )
          = ( A = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).

thf(100,axiom,
    ! [TA: $tType,A: TA,B: TA > $o] :
      ( ( member @ TA @ A @ ( collect @ TA @ B ) )
      = ( B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_mem__Collect__eq) ).

thf(596,plain,
    ! [TA: $tType,A: TA,B: TA > $o] :
      ( ( member @ TA @ A @ ( collect @ TA @ B ) )
      = ( B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(74,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_210_linorder__neqE__linordered__idom) ).

thf(528,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)],[74]) ).

thf(230,axiom,
    ! [TA: $tType] :
      ( ( ab_semigroup_mult @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) )
          = ( times_times @ TA @ B @ ( times_times @ TA @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_mult_Oleft__commute) ).

thf(982,plain,
    ! [TA: $tType] :
      ( ( ab_semigroup_mult @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) )
          = ( times_times @ TA @ B @ ( times_times @ TA @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(193,axiom,
    ! [TA: $tType] :
      ( ( semigroup_mult @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( times_times @ TA @ A @ B ) @ C )
          = ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_mult_Oassoc) ).

thf(871,plain,
    ! [TA: $tType] :
      ( ( semigroup_mult @ TA )
     => ! [A: TA,B: TA,C: TA] :
          ( ( times_times @ TA @ ( times_times @ TA @ A @ B ) @ C )
          = ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(272,axiom,
    semigroup_mult @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Osemigroup__mult) ).

thf(1093,plain,
    semigroup_mult @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(39,axiom,
    ! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA,D: TB > TA,E: TA] :
      ( ( ( fun_upd @ TB @ TA @ A @ B @ C )
        = ( fun_upd @ TB @ TA @ D @ B @ E ) )
     => ( C = E ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_fun__upd__eqD) ).

thf(395,plain,
    ! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA,D: TB > TA,E: TA] :
      ( ( ( fun_upd @ TB @ TA @ A @ B @ C )
        = ( fun_upd @ TB @ TA @ D @ B @ E ) )
     => ( C = E ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(109,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( comm_monoid_add @ TB )
     => ! [A: set @ TA,B: set @ TA,C: TA > TB,D: TA > TB] :
          ( ( A = B )
         => ( ! [E: TA] :
                ( ( member @ TA @ E @ B )
               => ( ( C @ E )
                  = ( D @ E ) ) )
           => ( ( groups1340683514dd_sum @ TA @ TB @ C @ A )
              = ( groups1340683514dd_sum @ TA @ TB @ D @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_sum_Ocong) ).

thf(624,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( comm_monoid_add @ TB )
     => ! [A: set @ TA,B: set @ TA,C: TA > TB,D: TA > TB] :
          ( ( A = B )
         => ( ! [E: TA] :
                ( ( member @ TA @ E @ B )
               => ( ( C @ E )
                  = ( D @ E ) ) )
           => ( ( groups1340683514dd_sum @ TA @ TB @ C @ A )
              = ( groups1340683514dd_sum @ TA @ TB @ D @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(181,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_252_nat__less__induct) ).

thf(839,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)],[181]) ).

thf(12,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_150_order__class_Oorder_Oeq__iff) ).

thf(322,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)],[12]) ).

thf(169,axiom,
    ! [TA: $tType] :
      ( ( ( comm_ring @ TA )
        & ( dvd @ TA ) )
     => ! [A: TA > $o,B: TA,C: TA > $o] :
          ( ! [D: TA,E: TA] :
              ( ( A @ D )
              = ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
         => ( ! [D: TA,E: TA] :
                ( ( C @ D )
                = ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
           => ! [D: TA,E: TA] :
                ( ( ( A @ D )
                  | ( C @ D ) )
                = ( ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) )
                  | ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_inf__period_I2_J) ).

thf(803,plain,
    ! [TA: $tType] :
      ( ( ( comm_ring @ TA )
        & ( dvd @ TA ) )
     => ! [A: TA > $o,B: TA,C: TA > $o] :
          ( ! [D: TA,E: TA] :
              ( ( A @ D )
              = ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
         => ( ! [D: TA,E: TA] :
                ( ( C @ D )
                = ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
           => ! [D: TA,E: TA] :
                ( ( ( A @ D )
                  | ( C @ D ) )
                = ( ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) )
                  | ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(19,axiom,
    ! [TA: $tType] : ( ord @ ( set @ TA ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oord_7) ).

thf(337,plain,
    ! [TA: $tType] : ( ord @ ( set @ TA ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(24,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_225_less__not__sym) ).

thf(354,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)],[24]) ).

thf(7,axiom,
    ! [TA: $tType,TB: $tType] :
      ( ( minus @ TA )
     => ( minus @ ( TB > TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Groups_Ominus) ).

thf(309,plain,
    ! [TA: $tType,TB: $tType] :
      ( ( minus @ TA )
     => ( minus @ ( TB > TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(80,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_142_dual__order_Oeq__iff) ).

thf(543,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)],[80]) ).

thf(270,axiom,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_mult__nonpos__nonpos) ).

thf(1088,plain,
    ! [TA: $tType] :
      ( ( ordered_ring @ TA )
     => ! [A: TA,B: TA] :
          ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
         => ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
           => ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(99,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_155_eq__refl) ).

thf(592,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)],[99]) ).

thf(131,axiom,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_mult__zero__left) ).

thf(708,plain,
    ! [TA: $tType] :
      ( ( mult_zero @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ ( zero_zero @ TA ) @ A )
          = ( zero_zero @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(197,axiom,
    linord1659791738miring @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring) ).

thf(882,plain,
    linord1659791738miring @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(217,axiom,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_diff__0__right) ).

thf(935,plain,
    ! [TA: $tType] :
      ( ( group_add @ TA )
     => ! [A: TA] :
          ( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(240,axiom,
    wellorder @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Owellorder) ).

thf(1009,plain,
    wellorder @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(157,axiom,
    semiri1923998003cancel @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel) ).

thf(775,plain,
    semiri1923998003cancel @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(87,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_223_antisym__conv3) ).

thf(560,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)],[87]) ).

thf(1159,plain,
    $false,
    inference(e,[status(thm)],[365,347,333,518,962,941,555,666,1110,1142,511,523,1158,741,1073,1041,861,1078,384,533,797,472,1063,747,913,404,898,565,436,651,724,719,785,504,344,587,619,702,751,421,606,325,316,1100,475,492,756,789,443,514,806,821,1136,834,602,634,734,956,1119,660,428,1072,453,307,920,439,546,925,761,810,381,1067,1047,1057,910,693,815,916,879,889,1079,682,551,1106,578,874,1017,615,1111,980,1006,388,356,328,1082,339,753,537,669,901,489,590,1155,393,997,985,738,721,569,779,1131,654,1060,1050,938,360,311,541,657,1054,674,1026,953,926,398,1035,773,794,298,412,965,1146,1103,824,856,375,584,904,423,402,772,711,1124,572,313,611,1107,335,1090,919,391,1139,406,737,599,990,996,303,1122,631,758,367,1149,943,932,848,700,959,1044,780,608,986,1000,851,431,922,658,685,521,768,827,1066,868,455,883,1113,563,1076,426,531,812,342,1145,553,968,836,1049,663,548,617,516,776,525,580,690,1070,705,414,739,409,993,712,557,1003,1053,791,331,951,359,907,1038,370,1085,576,744,1023,1052,864,727,1094,859,497,876,845,450,671,688,581,495,1010,844,319,946,786,896,748,1030,433,929,763,914,378,716,800,1116,1133,300,447,1097,818,854,790,865,373,305,326,807,1033,532,696,886,728,1129,1152,596,528,982,871,1093,395,624,839,322,803,337,354,309,543,1088,592,708,882,935,1009,775,560]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.11  % Problem  : ITP132^2 : TPTP v8.2.0. Released v7.5.0.
% 0.03/0.14  % Command  : run_Leo-III %s %d
% 0.14/0.35  % Computer : n014.cluster.edu
% 0.14/0.35  % Model    : x86_64 x86_64
% 0.14/0.35  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35  % Memory   : 8042.1875MB
% 0.14/0.35  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35  % CPULimit : 300
% 0.14/0.35  % WCLimit  : 300
% 0.14/0.35  % DateTime : Sat May 18 17:03:09 EDT 2024
% 0.14/0.35  % CPUTime  : 
% 0.92/0.85  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 1.69/1.08  % [INFO] 	 Parsing done (225ms). 
% 1.89/1.09  % [INFO] 	 Running in sequential loop mode. 
% 2.54/1.30  % [INFO] 	 eprover registered as external prover. 
% 2.54/1.30  % [INFO] 	 cvc4 registered as external prover. 
% 2.54/1.30  % [INFO] 	 Scanning for conjecture ... 
% 2.78/1.39  % [INFO] 	 Found a conjecture (or negated_conjecture) and 299 axioms. Running axiom selection ... 
% 3.14/1.48  % [INFO] 	 Axiom selection finished. Selected 295 axioms (removed 4 axioms). 
% 3.70/1.65  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 3.70/1.66  % [INFO] 	 Type checking passed. 
% 3.70/1.67  % [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 ... 
% 58.75/41.65  % External prover 'e' found a proof!
% 58.75/41.65  % [INFO] 	 Killing All external provers ... 
% 58.75/41.66  % Time passed: 41138ms (effective reasoning time: 40565ms)
% 58.75/41.66  % 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)>
% 58.75/41.67  % Axioms used in derivation (295): fact_116_mult__nonpos__nonneg, fact_55_inf__period_I1_J, fact_180_fun__upd__def, fact_62_lambda__one, fact_40_sum__mono, tcon_fun___Groups_Ominus, fact_186_right__diff__distrib, fact_179_fun__upd__eqD, fact_61_order__refl, fact_242_order__less__subst2, fact_70_mult__cancel__left, fact_206_nat__0__less__mult__iff, tcon_Nat_Onat___Orderings_Oorder_2, fact_204_nat__mult__less__cancel__disj, fact_16_nat__mult__eq__1__iff, fact_189_comm__monoid__mult__class_Omult__1, tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel, fact_127_zero__le__square, tcon_Nat_Onat___Orderings_Ono__top, fact_132_zero__le__one, tcon_Nat_Onat___Groups_Ocomm__monoid__diff, fact_14_atMost__iff, tcon_Nat_Onat___Orderings_Olinorder, fact_217_less__imp__not__less, fact_58_fun__upd__upd, fact_213_not__less__iff__gr__or__eq, fact_125_mult__left__mono__neg, fact_210_linorder__neqE__linordered__idom, tcon_Nat_Onat___Rings_Odvd, tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, fact_64_partitionsI, fact_246_less__numeral__extra_I3_J, fact_25_partitions__bounds, fact_28_diff__le__self, fact_221_less__imp__triv, fact_57_mult_Oleft__neutral, fact_233_less__asym_H, fact_149_ord__eq__le__trans, fact_244_ord__less__eq__subst, tcon_Nat_Onat___Rings_Ozero__neq__one, fact_100_mult__right__cancel, fact_220_linorder__cases, fact_112_mult__0, tcon_fun___Orderings_Oord, fact_23_Nat_Oex__has__greatest__nat, fact_224_less__induct, fact_227_dual__order_Oasym, fact_216_exists__least__iff, fact_218_order_Ostrict__trans, fact_102_zero__neq__one, fact_103_lambda__zero, tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors, fact_10_diff__mult__distrib, fact_75_diff__zero, fact_176_fun__upd__other, fact_208_nat__mult__le__cancel__disj, fact_197_le__numeral__extra_I3_J, fact_89_mult__cancel__left2, fact_191_diff__numeral__special_I9_J, fact_188_right__diff__distrib_H, fact_5_calculation, fact_158_eq__iff, tcon_Nat_Onat___Groups_Ocomm__monoid__mult, fact_3_minus__apply, fact_237_order_Oasym, fact_142_dual__order_Oeq__iff, fact_41_atMost__def, tcon_HOL_Obool___Groups_Ominus_13, tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, fact_137_mult__left__le__one__le, fact_241_lt__ex, fact_113_ordered__comm__semiring__class_Ocomm__mult__left__mono, tcon_Nat_Onat___Rings_Oordered__semiring, fact_130_le__iff__diff__le__0, fact_129_mult__mono, tcon_Nat_Onat___Groups_Ocomm__monoid__add, fact_161_order__subst2, fact_195_nat__mult__eq__cancel__disj, fact_81_mult__is__0, fact_231_less__linear, tcon_Nat_Onat___Groups_Oone, fact_173_one__reorient, fact_167_ab__semigroup__mult__class_Omult__ac_I1_J, fact_56_mult_Oright__neutral, fact_247_less__numeral__extra_I4_J, fact_131_not__one__le__zero, fact_11_diff__mult__distrib2, fact_13_atMost__subset__iff, fact_17_nat__1__eq__mult__iff, fact_12_inf__period_I2_J, tcon_Nat_Onat___Groups_Oab__semigroup__mult, fact_46_Collect__mem__eq, fact_166_le__funD, fact_141_dual__order_Oantisym, fact_177_fun__upd__same, fact_209_mult__le__cancel2, fact_63_partitions__def, fact_24_bounded__Max__nat, fact_120_mult__le__0__iff, tcon_HOL_Obool___Orderings_Oorder_11, fact_147_order__class_Oorder_Oantisym, tcon_HOL_Obool___Orderings_Olinorder_10, tcon_Nat_Onat___Groups_Ominus_4, fact_140_mult__eq__self__implies__10, fact_108_le__0__eq, fact_199_bot__nat__0_Onot__eq__extremum, fact_37_le__cube, fact_19_le__trans, fact_22_nat__le__linear, fact_182_diff__left__mono, fact_136_mult__right__le__one__le, fact_225_less__not__sym, fact_150_order__class_Oorder_Oeq__iff, fact_226_less__imp__not__eq, tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel, fact_76_cancel__comm__monoid__add__class_Odiff__cancel, fact_164_le__funI, fact_15_diff__diff__cancel, fact_27_le__diff__iff_H, fact_155_eq__refl, fact_36_le__square, fact_172_cancel__ab__semigroup__add__class_Odiff__right__commute, fact_198_not__gr__zero, fact_53_diff__commute, fact_169_mult_Ocommute, fact_98_no__zero__divisors, fact_187_left__diff__distrib_H, fact_92_diff__is__0__eq, fact_79_diff__0__eq__0, fact_235_less__imp__neq, tcon_Set_Oset___Orderings_Oorder_6, fact_67_mult__zero__left, fact_110_diffs0__imp__equal, fact_2_atMost__eq__iff, tcon_Nat_Onat___Groups_Ozero, fact_230_less__irrefl, fact_107_bot__nat__0_Oextremum__unique, tcon_Nat_Onat___Orderings_Owellorder, fact_86_diff__ge__0__iff__ge, fact_65_partitionsE, fact_160_ord__eq__le__subst, fact_145_dual__order_Orefl, fact_215_linorder__less__wlog, fact_59_fun__upd__triv, fact_51_sum_Oeq__general, tcon_Nat_Onat___Groups_Omonoid__mult, fact_45_mem__Collect__eq, fact_123_mult__left__mono, fact_146_order__trans, tcon_Set_Oset___Orderings_Oord_7, fact_202_diff__gt__0__iff__gt, fact_60_fun__upd__apply, fact_72_diff__self, fact_135_mult__le__one, fact_29_diff__le__mono, fact_229_ord__eq__less__trans, fact_99_mult__left__cancel, fact_243_order__less__subst1, fact_30_Nat_Odiff__diff__eq, fact_0_partitions, fact_71_mult__cancel__right, fact_239_neqE, fact_168_mult_Oassoc, fact_96_mult__not__zero, fact_184_diff__eq__diff__less__eq, fact_91_diff__is__0__eq_H, fact_26_diff__le__mono2, fact_32_eq__diff__iff, tcon_Nat_Onat___Orderings_Oord_3, fact_49_sum_Oreindex__bij__witness, fact_104_sum_Oneutral, fact_232_less__trans, tcon_HOL_Obool___Orderings_Opreorder_9, fact_185_left__diff__distrib, fact_97_divisors__zero, fact_66_le__zero__eq, fact_174_fun__upd__idem__iff, fact_69_mult__eq__0__iff, fact_47_Collect__cong, fact_33_mult__le__mono2, fact_193_subset__antisym, fact_207_less__one, tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add, tcon_Nat_Onat___Rings_Omult__zero, tcon_Nat_Onat___Rings_Oordered__semiring__0, fact_133_partitions__zero, fact_156_linear, fact_152_le__cases3, fact_196_le__numeral__extra_I4_J, fact_119_split__mult__neg__le, fact_205_mult__less__cancel2, fact_95_zero__le, fact_38_nat__mult__1__right, fact_163_le__fun__def, fact_159_ord__le__eq__subst, fact_122_mult__right__mono__neg, fact_170_mult_Oleft__commute, fact_228_ord__less__eq__trans, fact_214_dual__order_Ostrict__trans, fact_73_diff__0__right, fact_138_sum__nonpos, fact_236_dense, fact_148_ord__le__eq__trans, fact_80_diff__self__eq__0, fact_128_mult__mono_H, fact_109_less__eq__nat_Osimps_I1_J, fact_54_sum_Oswap, fact_175_fun__upd__twist, fact_124_mult__nonpos__nonpos, fact_21_le__antisym, tcon_fun___Orderings_Opreorder, fact_200_less__nat__zero__code, fact_245_ord__eq__less__subst, fact_134_mult__left__le, fact_35_mult__le__mono, fact_83_mult__cancel1, fact_203_zero__less__diff, fact_157_antisym, tcon_Nat_Onat___Orderings_Opreorder_1, fact_126_split__mult__pos__le, fact_181_diff__mono, fact_201_neq0__conv, fact_252_nat__less__induct, fact_18_le__refl, fact_42_sum__subtractf__nat, fact_101_eq__iff__diff__eq__0, fact_248_infinite__descent__measure, fact_117_mult__nonneg__nonpos, fact_139_sum__nonneg, fact_219_dual__order_Oirrefl, fact_238_neq__iff, fact_106_bot__nat__0_Oextremum__uniqueI, fact_78_le0, fact_8_sum__distrib__left, tcon_Nat_Onat___Rings_Osemiring__0, fact_153_order_Otrans, fact_118_mult__nonneg__nonneg, fact_90_mult__cancel__left1, fact_77_bot__nat__0_Oextremum, fact_9_sum__distrib__right, fact_105_sum_Onot__neutral__contains__not__neutral, fact_20_eq__imp__le, tcon_Nat_Onat___Rings_Oordered__comm__semiring, tcon_fun___Orderings_Oorder, fact_74_zero__diff, fact_115_mult__nonneg__nonpos2, fact_4__092_060open_062_I_092_060Sum_062i_092_060le_062n_A_N_Ak_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_A_061_A_I_092_060Sum_062i_092_060le_062n_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_092_060close_062, fact_7_sum__product, fact_143_dual__order_Otrans, fact_111_minus__nat_Odiff__0, tcon_Nat_Onat___Groups_Osemigroup__mult, fact_165_le__funE, fact_43_partitions__parts__bounded, fact_194_partitions__remove1__bounds, fact_68_mult__zero__right, tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, fact_6_sum__subtractf, fact_1__092_060open_062k_A_092_060le_062_An_092_060close_062, fact_82_mult__0__right, fact_94_zero__reorient, fact_84_mult__cancel2, tcon_HOL_Obool___Orderings_Oord_12, fact_251_infinite__descent, tcon_Set_Oset___Orderings_Opreorder_5, fact_151_antisym__conv, fact_249_measure__induct__rule, fact_52_sum_Ocong, fact_212_order_Ostrict__implies__not__eq, fact_178_fun__upd__idem, fact_34_mult__le__mono1, fact_183_diff__right__mono, fact_223_antisym__conv3, fact_31_le__diff__iff, fact_44_fun__diff__def, fact_222_less__imp__not__eq2, fact_93_gr0, fact_88_mult__cancel__right1, fact_162_order__subst1, fact_87_mult__cancel__right2, fact_50_sum_Oeq__general__inverses, fact_250_linorder__neqE__nat, fact_171_diff__eq__diff__eq, fact_190_mult_Ocomm__neutral, fact_144_linorder__wlog, fact_240_gt__ex, fact_234_less__asym, fact_211_dual__order_Ostrict__implies__not__eq, tcon_Nat_Onat___Groups_Oordered__comm__monoid__add, fact_121_mult__right__mono, fact_114_zero__le__mult__iff, fact_154_le__cases, fact_39_nat__mult__1, fact_192_subsetI, fact_85_sum_Oneutral__const, tcon_Set_Oset___Groups_Ominus_8
% 58.75/41.67  % No. of inferences in proof: 594
% 58.75/41.68  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 41138 ms resp. 40565 ms w/o parsing
% 59.77/41.91  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 59.77/41.91  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------