TSTP Solution File: ITP106^1 by Leo-III---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : ITP106^1 : TPTP v8.2.0. Released v7.5.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n016.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Mon May 20 22:22:57 EDT 2024

% Result   : Theorem 12.12s 3.60s
% Output   : Refutation 12.64s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  339
% Syntax   : Number of formulae    :  624 ( 214 unt;  56 typ;   0 def)
%            Number of atoms       : 1604 ( 764 equ;   0 cnn)
%            Maximal formula atoms :   17 (   2 avg)
%            Number of connectives : 7755 (  18   ~;   2   |;  36   &;6837   @)
%                                         (   0 <=>; 862  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   6 avg)
%            Number of types       :   11 (  10 usr)
%            Number of type conns  :   78 (  78   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   48 (  46 usr;   9 con; 0-3 aty)
%            Number of variables   : 1364 (  26   ^1306   !;  32   ?;1364   :)

% Comments : 
%------------------------------------------------------------------------------
thf(partia1993116613t_unit_type,type,
    partia1993116613t_unit: $tType ).

thf(partia1833973666xt_a_b_type,type,
    partia1833973666xt_a_b: $tType ).

thf(partia96731725t_unit_type,type,
    partia96731725t_unit: $tType ).

thf(set_Product_prod_a_a_type,type,
    set_Product_prod_a_a: $tType ).

thf(product_prod_a_a_type,type,
    product_prod_a_a: $tType ).

thf(set_a_type,type,
    set_a: $tType ).

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

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

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

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

thf(partia206007992t_unit_type,type,
    partia206007992t_unit: partia1993116613t_unit > set_Product_prod_a_a ).

thf(partia1955795460t_unit_type,type,
    partia1955795460t_unit: partia96731725t_unit > set_a ).

thf(partia1066395285xt_a_b_type,type,
    partia1066395285xt_a_b: partia1833973666xt_a_b > set_a ).

thf(units_a_Product_unit_type,type,
    units_a_Product_unit: partia96731725t_unit > set_a ).

thf(units_a_ring_ext_a_b_type,type,
    units_a_ring_ext_a_b: partia1833973666xt_a_b > set_a ).

thf(comm_g1684316527t_unit_type,type,
    comm_g1684316527t_unit: partia96731725t_unit > $o ).

thf(comm_g791708116xt_a_b_type,type,
    comm_g791708116xt_a_b: partia1833973666xt_a_b > $o ).

thf(group_a_Product_unit_type,type,
    group_a_Product_unit: partia96731725t_unit > $o ).

thf(group_a_ring_ext_a_b_type,type,
    group_a_ring_ext_a_b: partia1833973666xt_a_b > $o ).

thf(mult_a_Product_unit_type,type,
    mult_a_Product_unit: partia96731725t_unit > a > a > a ).

thf(mult_a_ring_ext_a_b_type,type,
    mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a ).

thf(one_a_Product_unit_type,type,
    one_a_Product_unit: partia96731725t_unit > a ).

thf(one_a_ring_ext_a_b_type,type,
    one_a_ring_ext_a_b: partia1833973666xt_a_b > a ).

thf(units_873712258t_unit_type,type,
    units_873712258t_unit: partia96731725t_unit > partia96731725t_unit ).

thf(units_1411277569xt_a_b_type,type,
    units_1411277569xt_a_b: partia1833973666xt_a_b > partia96731725t_unit ).

thf(minus_minus_int_type,type,
    minus_minus_int: int > int > int ).

thf(minus_minus_nat_type,type,
    minus_minus_nat: nat > nat > nat ).

thf(plus_plus_int_type,type,
    plus_plus_int: int > int > int ).

thf(plus_plus_nat_type,type,
    plus_plus_nat: nat > nat > nat ).

thf(uminus_uminus_int_type,type,
    uminus_uminus_int: int > int ).

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

thf(semiri2019852685at_int_type,type,
    semiri2019852685at_int: nat > int ).

thf(semiri1382578993at_nat_type,type,
    semiri1382578993at_nat: nat > nat ).

thf(product_Pair_a_a_type,type,
    product_Pair_a_a: a > a > product_prod_a_a ).

thf(a_inv_a_b_type,type,
    a_inv_a_b: partia1833973666xt_a_b > a > a ).

thf(a_minus_a_b_type,type,
    a_minus_a_b: partia1833973666xt_a_b > a > a > a ).

thf(abelian_group_a_b_type,type,
    abelian_group_a_b: partia1833973666xt_a_b > $o ).

thf(add_pow_a_b_int_type,type,
    add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).

thf(add_pow_a_b_nat_type,type,
    add_pow_a_b_nat: partia1833973666xt_a_b > nat > a > a ).

thf(field_a_b_type,type,
    field_a_b: partia1833973666xt_a_b > $o ).

thf(add_a_b_type,type,
    add_a_b: partia1833973666xt_a_b > a > a > a ).

thf(more_a_b_type,type,
    more_a_b: partia1833973666xt_a_b > b ).

thf(zero_a_b_type,type,
    zero_a_b: partia1833973666xt_a_b > a ).

thf(semiring_a_b_type,type,
    semiring_a_b: partia1833973666xt_a_b > $o ).

thf(collec645855634od_a_a_type,type,
    collec645855634od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).

thf(collect_a_type,type,
    collect_a: ( a > $o ) > set_a ).

thf(member449909584od_a_a_type,type,
    member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(member_a_type,type,
    member_a: a > set_a > $o ).

thf(r_type,type,
    r: partia1833973666xt_a_b ).

thf(r2_type,type,
    r2: a ).

thf(r3_type,type,
    r3: a ).

thf(r4_type,type,
    r4: a ).

thf(rel_type,type,
    rel: partia1993116613t_unit ).

thf(s_type,type,
    s: a ).

thf(s2_type,type,
    s2: a ).

thf(s3_type,type,
    s3: a ).

thf(103,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
      = ( plus_plus_int @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_260_diff__minus__eq__add) ).

thf(629,plain,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
      = ( plus_plus_int @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(210,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ A ) @ B )
          = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_l__minus) ).

thf(1024,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ A ) @ B )
          = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).

thf(255,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
              = A )
            = ( B
              = ( add_a_b @ r @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_add_Oinv__solve__right_H) ).

thf(1168,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
              = A )
            = ( B
              = ( add_a_b @ r @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).

thf(21,axiom,
    ! [A: partia96731725t_unit,B: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ? [C: a] :
            ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
            & ( ( mult_a_Product_unit @ A @ C @ B )
              = ( one_a_Product_unit @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_group_Ol__inv__ex) ).

thf(347,plain,
    ! [A: partia96731725t_unit,B: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ? [C: a] :
            ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
            & ( ( mult_a_Product_unit @ A @ C @ B )
              = ( one_a_Product_unit @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(163,axiom,
    ! [A: a,B: nat,C: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( ( add_pow_a_b_nat @ r @ B @ A )
          = ( add_pow_a_b_nat @ r @ C @ A ) )
       => ( ( add_pow_a_b_nat @ r @ ( minus_minus_nat @ B @ C ) @ A )
          = ( zero_a_b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_add_Opow__eq__div2) ).

thf(846,plain,
    ! [A: a,B: nat,C: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( ( add_pow_a_b_nat @ r @ B @ A )
          = ( add_pow_a_b_nat @ r @ C @ A ) )
       => ( ( add_pow_a_b_nat @ r @ ( minus_minus_nat @ B @ C ) @ A )
          = ( zero_a_b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).

thf(212,axiom,
    ! [A: a,B: a,C: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ C @ B ) )
          = ( add_pow_a_b_int @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_add__pow__rdistr__int) ).

thf(1031,plain,
    ! [A: a,B: a,C: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ C @ B ) )
          = ( add_pow_a_b_int @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).

thf(227,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( add_a_b @ A @ ( zero_a_b @ A ) @ B )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_semiring_Osemiring__simprules_I6_J) ).

thf(1083,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( add_a_b @ A @ ( zero_a_b @ A ) @ B )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).

thf(114,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( abelian_group_a_b @ A )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
           => ( ( ( a_minus_a_b @ A @ B @ C )
                = ( zero_a_b @ A ) )
             => ( B = C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_abelian__group_Ominus__to__eq) ).

thf(666,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( abelian_group_a_b @ A )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
           => ( ( ( a_minus_a_b @ A @ B @ C )
                = ( zero_a_b @ A ) )
             => ( B = C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).

thf(60,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1382578993at_nat @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( semiri1382578993at_nat @ A ) @ ( semiri1382578993at_nat @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_272_of__nat__add) ).

thf(481,plain,
    ! [A: nat,B: nat] :
      ( ( semiri1382578993at_nat @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( semiri1382578993at_nat @ A ) @ ( semiri1382578993at_nat @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(22,axiom,
    ! [A: partia96731725t_unit,B: a,C: a,D: a] :
      ( ( comm_g1684316527t_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( member_a @ D @ ( partia1955795460t_unit @ A ) )
           => ( ( mult_a_Product_unit @ A @ ( mult_a_Product_unit @ A @ B @ C ) @ D )
              = ( mult_a_Product_unit @ A @ B @ ( mult_a_Product_unit @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_276_comm__groupE_I3_J) ).

thf(352,plain,
    ! [A: partia96731725t_unit,B: a,C: a,D: a] :
      ( ( comm_g1684316527t_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( member_a @ D @ ( partia1955795460t_unit @ A ) )
           => ( ( mult_a_Product_unit @ A @ ( mult_a_Product_unit @ A @ B @ C ) @ D )
              = ( mult_a_Product_unit @ A @ B @ ( mult_a_Product_unit @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(138,axiom,
    ! [A: partia1833973666xt_a_b] :
      ( ( mult_a_Product_unit @ ( units_1411277569xt_a_b @ A ) )
      = ( mult_a_ring_ext_a_b @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_units__of__mult) ).

thf(762,plain,
    ! [A: partia1833973666xt_a_b] :
      ( ( mult_a_Product_unit @ ( units_1411277569xt_a_b @ A ) )
      = ( mult_a_ring_ext_a_b @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).

thf(205,axiom,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_old_Oprod_Oinject) ).

thf(1005,plain,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).

thf(133,axiom,
    ! [A: partia1833973666xt_a_b] :
      ( ! [B: a] :
          ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
         => ! [C: a] :
              ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
             => ( member_a @ ( add_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) )
     => ( ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) )
       => ( ! [B: a] :
              ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
             => ! [C: a] :
                  ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
                 => ! [D: a] :
                      ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
                     => ( ( add_a_b @ A @ ( add_a_b @ A @ B @ C ) @ D )
                        = ( add_a_b @ A @ B @ ( add_a_b @ A @ C @ D ) ) ) ) ) )
         => ( ! [B: a] :
                ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
               => ! [C: a] :
                    ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
                   => ( ( add_a_b @ A @ B @ C )
                      = ( add_a_b @ A @ C @ B ) ) ) )
           => ( ! [B: a] :
                  ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
                 => ( ( add_a_b @ A @ ( zero_a_b @ A ) @ B )
                    = B ) )
             => ( ! [B: a] :
                    ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
                   => ? [C: a] :
                        ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
                        & ( ( add_a_b @ A @ C @ B )
                          = ( zero_a_b @ A ) ) ) )
               => ( abelian_group_a_b @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_abelian__groupI) ).

thf(730,plain,
    ! [A: partia1833973666xt_a_b] :
      ( ! [B: a] :
          ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
         => ! [C: a] :
              ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
             => ( member_a @ ( add_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) )
     => ( ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) )
       => ( ! [B: a] :
              ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
             => ! [C: a] :
                  ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
                 => ! [D: a] :
                      ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
                     => ( ( add_a_b @ A @ ( add_a_b @ A @ B @ C ) @ D )
                        = ( add_a_b @ A @ B @ ( add_a_b @ A @ C @ D ) ) ) ) ) )
         => ( ! [B: a] :
                ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
               => ! [C: a] :
                    ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
                   => ( ( add_a_b @ A @ B @ C )
                      = ( add_a_b @ A @ C @ B ) ) ) )
           => ( ! [B: a] :
                  ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
                 => ( ( add_a_b @ A @ ( zero_a_b @ A ) @ B )
                    = B ) )
             => ( ! [B: a] :
                    ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
                   => ? [C: a] :
                        ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
                        & ( ( add_a_b @ A @ C @ B )
                          = ( zero_a_b @ A ) ) ) )
               => ( abelian_group_a_b @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).

thf(50,axiom,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( B
              = ( mult_a_Product_unit @ A @ C @ B ) )
            = ( C
              = ( one_a_Product_unit @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_group_Or__cancel__one_H) ).

thf(449,plain,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( B
              = ( mult_a_Product_unit @ A @ C @ B ) )
            = ( C
              = ( one_a_Product_unit @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(29,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ A @ ( suc @ B ) )
      = ( suc @ ( plus_plus_nat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_add__Suc__right) ).

thf(379,plain,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ A @ ( suc @ B ) )
      = ( suc @ ( plus_plus_nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(167,axiom,
    ( a_minus_a_b
    = ( ^ [A: partia1833973666xt_a_b,B: a,C: a] : ( add_a_b @ A @ B @ ( a_inv_a_b @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_a__minus__def) ).

thf(861,plain,
    ( a_minus_a_b
    = ( ^ [A: partia1833973666xt_a_b,B: a,C: a] : ( add_a_b @ A @ B @ ( a_inv_a_b @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).

thf(245,axiom,
    group_a_Product_unit @ ( units_1411277569xt_a_b @ r ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_units__group) ).

thf(1137,plain,
    group_a_Product_unit @ ( units_1411277569xt_a_b @ r ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).

thf(148,axiom,
    ! [A: a,B: int,C: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ B @ C ) @ A )
        = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ B @ A ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ C @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_add_Oint__pow__diff) ).

thf(797,plain,
    ! [A: a,B: int,C: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ B @ C ) @ A )
        = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ B @ A ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).

thf(283,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ! [B: a] :
            ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
           => ( ( mult_a_ring_ext_a_b @ r @ A @ B )
              = B ) )
       => ( A
          = ( one_a_ring_ext_a_b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_one__unique) ).

thf(1275,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ! [B: a] :
            ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
           => ( ( mult_a_ring_ext_a_b @ r @ A @ B )
              = B ) )
       => ( A
          = ( one_a_ring_ext_a_b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).

thf(74,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ C @ B ) )
     => ( A = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_add__right__imp__eq) ).

thf(538,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ C @ B ) )
     => ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(188,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( ( mult_a_ring_ext_a_b @ A @ C @ B )
              = B )
            = ( C
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_group_Or__cancel__one) ).

thf(930,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( ( mult_a_ring_ext_a_b @ A @ C @ B )
              = B )
            = ( C
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).

thf(144,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ A @ B ) )
          = ( add_a_b @ r @ ( a_inv_a_b @ r @ A ) @ ( a_inv_a_b @ r @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_local_Ominus__add) ).

thf(782,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ A @ B ) )
          = ( add_a_b @ r @ ( a_inv_a_b @ r @ A ) @ ( a_inv_a_b @ r @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).

thf(127,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ A @ B ) )
          = ( a_minus_a_b @ r @ ( a_inv_a_b @ r @ A ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_inv__add) ).

thf(709,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ A @ B ) )
          = ( a_minus_a_b @ r @ ( a_inv_a_b @ r @ A ) @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).

thf(40,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_add__left__cancel) ).

thf(417,plain,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(46,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_256_add__minus__cancel) ).

thf(436,plain,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(270,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ A @ ( zero_a_b @ r ) )
        = ( zero_a_b @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_r__null) ).

thf(1228,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ A @ ( zero_a_b @ r ) )
        = ( zero_a_b @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).

thf(158,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( B
              = ( mult_a_ring_ext_a_b @ A @ B @ C ) )
            = ( C
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_group_Ol__cancel__one_H) ).

thf(829,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( B
              = ( mult_a_ring_ext_a_b @ A @ B @ C ) )
            = ( C
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).

thf(231,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ A )
        = ( zero_a_b @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_l__null) ).

thf(1095,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ A )
        = ( zero_a_b @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).

thf(274,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ A @ B )
          = ( mult_a_ring_ext_a_b @ r @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_m__comm) ).

thf(1243,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ A @ B )
          = ( mult_a_ring_ext_a_b @ r @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).

thf(242,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( comm_g791708116xt_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( mult_a_ring_ext_a_b @ A @ B @ C )
            = ( mult_a_ring_ext_a_b @ A @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_275_comm__groupE_I4_J) ).

thf(1127,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( comm_g791708116xt_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( mult_a_ring_ext_a_b @ A @ B @ C )
            = ( mult_a_ring_ext_a_b @ A @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).

thf(199,axiom,
    ! [A: product_prod_a_a > $o,B: product_prod_a_a] :
      ( ! [C: a,D: a] : ( A @ ( product_Pair_a_a @ C @ D ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_old_Oprod_Oinducts) ).

thf(973,plain,
    ! [A: product_prod_a_a > $o,B: product_prod_a_a] :
      ( ! [C: a,D: a] : ( A @ ( product_Pair_a_a @ C @ D ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).

thf(175,axiom,
    ! [A: nat] :
      ( ( add_pow_a_b_nat @ r @ A @ ( zero_a_b @ r ) )
      = ( zero_a_b @ r ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_add_Onat__pow__one) ).

thf(885,plain,
    ! [A: nat] :
      ( ( add_pow_a_b_nat @ r @ A @ ( zero_a_b @ r ) )
      = ( zero_a_b @ r ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).

thf(217,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ A @ B )
            = ( zero_a_b @ r ) )
         => ( A
            = ( a_inv_a_b @ r @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_sum__zero__eq__neg) ).

thf(1046,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ A @ B )
            = ( zero_a_b @ r ) )
         => ( A
            = ( a_inv_a_b @ r @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).

thf(32,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_diff__eq__eq) ).

thf(389,plain,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(27,axiom,
    ! [A: partia1833973666xt_a_b] :
      ( ( field_a_b @ A )
     => ( ( one_a_ring_ext_a_b @ A )
       != ( zero_a_b @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_one__not__zero) ).

thf(372,plain,
    ! [A: partia1833973666xt_a_b] :
      ( ( field_a_b @ A )
     => ( ( one_a_ring_ext_a_b @ A )
       != ( zero_a_b @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(67,axiom,
    ! [A: a,B: a > $o] :
      ( ( member_a @ A @ ( collect_a @ B ) )
      = ( B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_mem__Collect__eq) ).

thf(504,plain,
    ! [A: a,B: a > $o] :
      ( ( member_a @ A @ ( collect_a @ B ) )
      = ( B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(135,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ? [C: a] :
            ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
            & ( ( add_a_b @ A @ C @ B )
              = ( zero_a_b @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_abelian__groupE_I6_J) ).

thf(751,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ? [C: a] :
            ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
            & ( ( add_a_b @ A @ C @ B )
              = ( zero_a_b @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).

thf(20,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_diff__add__eq) ).

thf(344,plain,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(189,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( mult_a_ring_ext_a_b @ A @ B @ ( zero_a_b @ A ) )
          = ( zero_a_b @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_semiring_Or__null) ).

thf(934,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( mult_a_ring_ext_a_b @ A @ B @ ( zero_a_b @ A ) )
          = ( zero_a_b @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).

thf(219,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_semiring__simprules_I3_J) ).

thf(1055,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).

thf(131,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( member_a @ ( add_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_abelian__groupE_I1_J) ).

thf(724,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( member_a @ ( add_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).

thf(90,axiom,
    ! [A: product_prod_a_a > $o,B: product_prod_a_a > $o] :
      ( ! [C: product_prod_a_a] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collec645855634od_a_a @ A )
        = ( collec645855634od_a_a @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_Collect__cong) ).

thf(587,plain,
    ! [A: product_prod_a_a > $o,B: product_prod_a_a > $o] :
      ( ! [C: product_prod_a_a] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collec645855634od_a_a @ A )
        = ( collec645855634od_a_a @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(161,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
          = ( one_a_ring_ext_a_b @ A ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
           => ( ( mult_a_ring_ext_a_b @ A @ C @ B )
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_group_Oinv__comm) ).

thf(838,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
          = ( one_a_ring_ext_a_b @ A ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
           => ( ( mult_a_ring_ext_a_b @ A @ C @ B )
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).

thf(75,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_258_minus__add__distrib) ).

thf(542,plain,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(54,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_add__left__imp__eq) ).

thf(460,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(154,axiom,
    ! [A: product_prod_a_a] :
    ? [B: a,C: a] :
      ( A
      = ( product_Pair_a_a @ B @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_surj__pair) ).

thf(817,plain,
    ! [A: product_prod_a_a] :
    ? [B: a,C: a] :
      ( A
      = ( product_Pair_a_a @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).

thf(106,axiom,
    ( minus_minus_int
    = ( ^ [A: int,B: int] : ( plus_plus_int @ A @ ( uminus_uminus_int @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_264_ab__group__add__class_Oab__diff__conv__add__uminus) ).

thf(638,plain,
    ( minus_minus_int
    = ( ^ [A: int,B: int] : ( plus_plus_int @ A @ ( uminus_uminus_int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(228,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( B
              = ( mult_a_ring_ext_a_b @ A @ C @ B ) )
            = ( C
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_group_Or__cancel__one_H) ).

thf(1087,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( B
              = ( mult_a_ring_ext_a_b @ A @ C @ B ) )
            = ( C
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).

thf(275,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( A
            = ( add_a_b @ r @ B @ A ) )
          = ( B
            = ( zero_a_b @ r ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_add_Or__cancel__one_H) ).

thf(1247,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( A
            = ( add_a_b @ r @ B @ A ) )
          = ( B
            = ( zero_a_b @ r ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).

thf(198,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ A @ B ) @ C )
            = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_l__distr) ).

thf(969,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ A @ B ) @ C )
            = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).

thf(174,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ C )
            = ( mult_a_ring_ext_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_m__assoc) ).

thf(881,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ C )
            = ( mult_a_ring_ext_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).

thf(238,axiom,
    ! [A: a,B: int,C: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ B @ C ) @ A )
        = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ B @ A ) @ ( add_pow_a_b_int @ r @ C @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_add_Oint__pow__mult) ).

thf(1115,plain,
    ! [A: a,B: int,C: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ B @ C ) @ A )
        = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ B @ A ) @ ( add_pow_a_b_int @ r @ C @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).

thf(4,axiom,
    ! [A: nat,B: nat] :
      ( ( ( semiri1382578993at_nat @ A )
        = ( semiri1382578993at_nat @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_270_of__nat__eq__iff) ).

thf(289,plain,
    ! [A: nat,B: nat] :
      ( ( ( semiri1382578993at_nat @ A )
        = ( semiri1382578993at_nat @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(11,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_128_diff__commute) ).

thf(316,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)],[11]) ).

thf(186,axiom,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ r4 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_f10) ).

thf(924,plain,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ r4 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).

thf(136,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
     => ( member_a @ A @ ( partia1066395285xt_a_b @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_Units__closed) ).

thf(756,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
     => ( member_a @ A @ ( partia1066395285xt_a_b @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).

thf(209,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ A ) @ ( add_a_b @ r @ A @ B ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_r__neg1) ).

thf(1020,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ A ) @ ( add_a_b @ r @ A @ B ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).

thf(86,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_267_minus__diff__commute) ).

thf(574,plain,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(95,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [B: a] : ( member_a @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_48_Collect__mem__eq) ).

thf(602,plain,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [B: a] : ( member_a @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(121,axiom,
    abelian_group_a_b @ r,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_is__abelian__group) ).

thf(692,plain,
    abelian_group_a_b @ r,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).

thf(34,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A
        = ( minus_minus_int @ B @ C ) )
      = ( ( plus_plus_int @ A @ C )
        = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_eq__diff__eq) ).

thf(396,plain,
    ! [A: int,B: int,C: int] :
      ( ( A
        = ( minus_minus_int @ B @ C ) )
      = ( ( plus_plus_int @ A @ C )
        = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(112,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_diff__diff__add) ).

thf(660,plain,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(263,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( zero_a_b @ r ) @ A )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_l__zero) ).

thf(1201,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( zero_a_b @ r ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).

thf(235,axiom,
    ! [A: product_prod_a_a > $o,B: product_prod_a_a] :
      ( ! [C: a,D: a] : ( A @ ( product_Pair_a_a @ C @ D ) )
     => ( A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_prod__cases) ).

thf(1104,plain,
    ! [A: product_prod_a_a > $o,B: product_prod_a_a] :
      ( ! [C: a,D: a] : ( A @ ( product_Pair_a_a @ C @ D ) )
     => ( A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).

thf(68,axiom,
    ! [A: partia96731725t_unit] :
      ( ! [B: a] :
          ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
         => ! [C: a] :
              ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
             => ( member_a @ ( mult_a_Product_unit @ A @ B @ C ) @ ( partia1955795460t_unit @ A ) ) ) )
     => ( ( member_a @ ( one_a_Product_unit @ A ) @ ( partia1955795460t_unit @ A ) )
       => ( ! [B: a] :
              ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
             => ! [C: a] :
                  ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
                 => ! [D: a] :
                      ( ( member_a @ D @ ( partia1955795460t_unit @ A ) )
                     => ( ( mult_a_Product_unit @ A @ ( mult_a_Product_unit @ A @ B @ C ) @ D )
                        = ( mult_a_Product_unit @ A @ B @ ( mult_a_Product_unit @ A @ C @ D ) ) ) ) ) )
         => ( ! [B: a] :
                ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
               => ( ( mult_a_Product_unit @ A @ ( one_a_Product_unit @ A ) @ B )
                  = B ) )
           => ( ! [B: a] :
                  ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
                 => ? [C: a] :
                      ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
                      & ( ( mult_a_Product_unit @ A @ C @ B )
                        = ( one_a_Product_unit @ A ) ) ) )
             => ( group_a_Product_unit @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_groupI) ).

thf(507,plain,
    ! [A: partia96731725t_unit] :
      ( ! [B: a] :
          ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
         => ! [C: a] :
              ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
             => ( member_a @ ( mult_a_Product_unit @ A @ B @ C ) @ ( partia1955795460t_unit @ A ) ) ) )
     => ( ( member_a @ ( one_a_Product_unit @ A ) @ ( partia1955795460t_unit @ A ) )
       => ( ! [B: a] :
              ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
             => ! [C: a] :
                  ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
                 => ! [D: a] :
                      ( ( member_a @ D @ ( partia1955795460t_unit @ A ) )
                     => ( ( mult_a_Product_unit @ A @ ( mult_a_Product_unit @ A @ B @ C ) @ D )
                        = ( mult_a_Product_unit @ A @ B @ ( mult_a_Product_unit @ A @ C @ D ) ) ) ) ) )
         => ( ! [B: a] :
                ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
               => ( ( mult_a_Product_unit @ A @ ( one_a_Product_unit @ A ) @ B )
                  = B ) )
           => ( ! [B: a] :
                  ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
                 => ? [C: a] :
                      ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
                      & ( ( mult_a_Product_unit @ A @ C @ B )
                        = ( one_a_Product_unit @ A ) ) ) )
             => ( group_a_Product_unit @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(224,axiom,
    ! [A: product_prod_a_a] :
      ~ ! [B: a,C: a] :
          ( A
         != ( product_Pair_a_a @ B @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_old_Oprod_Oexhaust) ).

thf(1072,plain,
    ! [A: product_prod_a_a] :
      ~ ! [B: a,C: a] :
          ( A
         != ( product_Pair_a_a @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).

thf(51,axiom,
    ! [A: partia1833973666xt_a_b] :
      ( ( abelian_group_a_b @ A )
     => ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_abelian__groupE_I2_J) ).

thf(453,plain,
    ! [A: partia1833973666xt_a_b] :
      ( ( abelian_group_a_b @ A )
     => ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(239,axiom,
    member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_zero__closed) ).

thf(1119,plain,
    member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).

thf(142,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( comm_g791708116xt_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
              = ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_277_comm__groupE_I3_J) ).

thf(774,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( comm_g791708116xt_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
              = ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).

thf(8,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_257_minus__add__cancel) ).

thf(307,plain,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(5,axiom,
    ! [A: partia96731725t_unit] :
      ( ( comm_g1684316527t_unit @ A )
     => ( member_a @ ( one_a_Product_unit @ A ) @ ( partia1955795460t_unit @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_280_comm__groupE_I2_J) ).

thf(292,plain,
    ! [A: partia96731725t_unit] :
      ( ( comm_g1684316527t_unit @ A )
     => ( member_a @ ( one_a_Product_unit @ A ) @ ( partia1955795460t_unit @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(185,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ C @ A ) @ B )
            = ( a_minus_a_b @ r @ C @ ( add_a_b @ r @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_right__inv__add) ).

thf(920,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ C @ A ) @ B )
            = ( a_minus_a_b @ r @ C @ ( add_a_b @ r @ A @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).

thf(71,axiom,
    ! [A: partia1833973666xt_a_b] :
      ( ( semiring_a_b @ A )
     => ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_semiring_Osemiring__simprules_I4_J) ).

thf(529,plain,
    ! [A: partia1833973666xt_a_b] :
      ( ( semiring_a_b @ A )
     => ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(73,axiom,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( ( mult_a_Product_unit @ A @ B @ C )
              = B )
            = ( C
              = ( one_a_Product_unit @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_group_Ol__cancel__one) ).

thf(534,plain,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( ( mult_a_Product_unit @ A @ B @ C )
              = B )
            = ( C
              = ( one_a_Product_unit @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(191,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( add_a_b @ A @ B @ ( add_a_b @ A @ ( a_inv_a_b @ A @ B ) @ C ) )
            = C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_abelian__group_Or__neg2) ).

thf(942,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( add_a_b @ A @ B @ ( add_a_b @ A @ ( a_inv_a_b @ A @ B ) @ C ) )
            = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).

thf(47,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( suc @ A ) @ B )
      = ( suc @ ( plus_plus_nat @ A @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_add__Suc) ).

thf(439,plain,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( suc @ A ) @ B )
      = ( suc @ ( plus_plus_nat @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(282,axiom,
    member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) ) @ ( partia206007992t_unit @ rel ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_f12) ).

thf(1274,plain,
    member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) ) @ ( partia206007992t_unit @ rel ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).

thf(178,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ? [C: a] :
            ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
            & ( ( mult_a_ring_ext_a_b @ A @ C @ B )
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_group_Ol__inv__ex) ).

thf(894,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ? [C: a] :
            ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
            & ( ( mult_a_ring_ext_a_b @ A @ C @ B )
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).

thf(229,axiom,
    member449909584od_a_a @ ( product_Pair_a_a @ r4 @ s3 ) @ ( partia206007992t_unit @ rel ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_assms_I3_J) ).

thf(1091,plain,
    member449909584od_a_a @ ( product_Pair_a_a @ r4 @ s3 ) @ ( partia206007992t_unit @ rel ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).

thf(143,axiom,
    ! [A: a,B: a,C: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( add_pow_a_b_nat @ r @ C @ ( add_a_b @ r @ A @ B ) )
          = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ C @ A ) @ ( add_pow_a_b_nat @ r @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_add_Onat__pow__distrib) ).

thf(778,plain,
    ! [A: a,B: a,C: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( add_pow_a_b_nat @ r @ C @ ( add_a_b @ r @ A @ B ) )
          = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ C @ A ) @ ( add_pow_a_b_nat @ r @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).

thf(277,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( A
              = ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
            = ( C
              = ( add_a_b @ r @ B @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_add_Oinv__solve__left) ).

thf(1255,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( A
              = ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
            = ( C
              = ( add_a_b @ r @ B @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).

thf(241,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ A @ B )
            = A )
          = ( B
            = ( zero_a_b @ r ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_add_Ol__cancel__one) ).

thf(1123,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ A @ B )
            = A )
          = ( B
            = ( zero_a_b @ r ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).

thf(155,axiom,
    ! [A: a,B: a] :
      ( ( a_minus_a_b @ r @ A @ B )
      = ( add_a_b @ r @ A @ ( a_inv_a_b @ r @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_minus__eq) ).

thf(820,plain,
    ! [A: a,B: a] :
      ( ( a_minus_a_b @ r @ A @ B )
      = ( add_a_b @ r @ A @ ( a_inv_a_b @ r @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).

thf(33,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = C )
     => ( A
        = ( minus_minus_int @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_add__implies__diff) ).

thf(392,plain,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = C )
     => ( A
        = ( minus_minus_int @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(7,axiom,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ ( suc @ D ) )
           => ( A @ D ) )
       => ( A @ ( minus_minus_nat @ B @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_zero__induct__lemma) ).

thf(297,plain,
    ! [A: nat > $o,B: nat,C: nat] :
      ( ( A @ B )
     => ( ! [D: nat] :
            ( ( A @ ( suc @ D ) )
           => ( A @ D ) )
       => ( A @ ( minus_minus_nat @ B @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(42,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( minus_minus_nat @ A @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_diff__Suc__Suc) ).

thf(424,plain,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( suc @ A ) @ ( suc @ B ) )
      = ( minus_minus_nat @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(122,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ A ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_local_Ominus__minus) ).

thf(693,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ A ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).

thf(268,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( add_a_b @ A @ B @ ( a_inv_a_b @ A @ B ) )
          = ( zero_a_b @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_abelian__group_Or__neg) ).

thf(1217,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( add_a_b @ A @ B @ ( a_inv_a_b @ A @ B ) )
          = ( zero_a_b @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).

thf(94,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
     => ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_Suc__inject) ).

thf(598,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
     => ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(184,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ A @ ( a_inv_a_b @ r @ B ) )
          = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_r__minus) ).

thf(916,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ A @ ( a_inv_a_b @ r @ B ) )
          = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).

thf(24,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_diff__add__eq__diff__diff__swap) ).

thf(361,plain,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(15,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_add__diff__cancel) ).

thf(329,plain,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(69,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_diff__diff__left) ).

thf(522,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(251,axiom,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) )
    = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) @ r4 ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_f7) ).

thf(1159,plain,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) )
    = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) @ r4 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).

thf(118,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ B @ A )
              = ( add_a_b @ r @ C @ A ) )
            = ( B = C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_local_Oadd_Oright__cancel) ).

thf(682,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ B @ A )
              = ( add_a_b @ r @ C @ A ) )
            = ( B = C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).

thf(78,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_add_Oleft__commute) ).

thf(551,plain,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(162,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
     => ? [B: a] :
          ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
          & ( ( mult_a_ring_ext_a_b @ r @ A @ B )
            = ( one_a_ring_ext_a_b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_Units__r__inv__ex) ).

thf(842,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
     => ? [B: a] :
          ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
          & ( ( mult_a_ring_ext_a_b @ r @ A @ B )
            = ( one_a_ring_ext_a_b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).

thf(207,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( ( add_a_b @ A @ B @ C )
          = ( zero_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
           => ( ( a_inv_a_b @ A @ C )
              = B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_abelian__group_Ominus__equality) ).

thf(1012,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( ( add_a_b @ A @ B @ C )
          = ( zero_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
           => ( ( a_inv_a_b @ A @ C )
              = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).

thf(1,conjecture,
    member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).

thf(2,negated_conjecture,
    ~ ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(285,plain,
    ~ ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) ) @ ( partia206007992t_unit @ rel ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(83,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri2019852685at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri2019852685at_int @ A ) @ ( semiri2019852685at_int @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_273_of__nat__add) ).

thf(566,plain,
    ! [A: nat,B: nat] :
      ( ( semiri2019852685at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri2019852685at_int @ A ) @ ( semiri2019852685at_int @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(61,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
      = ( minus_minus_nat @ B @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_Nat_Odiff__cancel) ).

thf(484,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
      = ( minus_minus_nat @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(99,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ A @ C ) )
      = ( minus_minus_int @ B @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_add__diff__cancel__left) ).

thf(615,plain,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ A @ C ) )
      = ( minus_minus_int @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(237,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a,E: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( member_a @ E @ ( partia1066395285xt_a_b @ A ) )
             => ( ( a_minus_a_b @ A @ ( add_a_b @ A @ ( a_minus_a_b @ A @ B @ D ) @ C ) @ E )
                = ( a_minus_a_b @ A @ ( a_minus_a_b @ A @ ( add_a_b @ A @ B @ C ) @ D ) @ E ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_abelian__group_Ofour__elem__comm) ).

thf(1111,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a,E: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( member_a @ E @ ( partia1066395285xt_a_b @ A ) )
             => ( ( a_minus_a_b @ A @ ( add_a_b @ A @ ( a_minus_a_b @ A @ B @ D ) @ C ) @ E )
                = ( a_minus_a_b @ A @ ( a_minus_a_b @ A @ ( add_a_b @ A @ B @ C ) @ D ) @ E ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).

thf(201,axiom,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062) ).

thf(980,plain,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).

thf(92,axiom,
    ( plus_plus_nat
    = ( ^ [A: nat,B: nat] : ( plus_plus_nat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_add_Ocommute) ).

thf(593,plain,
    ( plus_plus_nat
    = ( ^ [A: nat,B: nat] : ( plus_plus_nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(41,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ C @ B ) )
     => ( A = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_add__right__imp__eq) ).

thf(420,plain,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ C @ B ) )
     => ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(281,axiom,
    ! [A: a,B: a,C: int] :
      ( ( ( add_a_b @ r @ A @ B )
        = ( add_a_b @ r @ B @ A ) )
     => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( add_pow_a_b_int @ r @ C @ ( add_a_b @ r @ A @ B ) )
            = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ C @ A ) @ ( add_pow_a_b_int @ r @ C @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_add_Oint__pow__mult__distrib) ).

thf(1270,plain,
    ! [A: a,B: a,C: int] :
      ( ( ( add_a_b @ r @ A @ B )
        = ( add_a_b @ r @ B @ A ) )
     => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( add_pow_a_b_int @ r @ C @ ( add_a_b @ r @ A @ B ) )
            = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ C @ A ) @ ( add_pow_a_b_int @ r @ C @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).

thf(278,axiom,
    ! [A: a,B: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_nat @ r @ ( suc @ B ) @ A )
        = ( add_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_add_Onat__pow__Suc2) ).

thf(1259,plain,
    ! [A: a,B: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_nat @ r @ ( suc @ B ) @ A )
        = ( add_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).

thf(246,axiom,
    ( ( ( zero_a_b @ r )
     != ( one_a_ring_ext_a_b @ r ) )
   => ( ! [A: a] :
          ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
         => ( ( A
             != ( zero_a_b @ r ) )
           => ? [B: a] :
                ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
                & ( ( mult_a_ring_ext_a_b @ r @ A @ B )
                  = ( one_a_ring_ext_a_b @ r ) ) ) ) )
     => ( field_a_b @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_cring__fieldI2) ).

thf(1138,plain,
    ( ( ( zero_a_b @ r )
     != ( one_a_ring_ext_a_b @ r ) )
   => ( ! [A: a] :
          ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
         => ( ( A
             != ( zero_a_b @ r ) )
           => ? [B: a] :
                ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
                & ( ( mult_a_ring_ext_a_b @ r @ A @ B )
                  = ( one_a_ring_ext_a_b @ r ) ) ) ) )
     => ( field_a_b @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).

thf(23,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A
        = ( plus_plus_int @ B @ C ) )
     => ( ( minus_minus_int @ D @ A )
        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( minus_minus_int @ D @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_262_group__cancel_Osub2) ).

thf(356,plain,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A
        = ( plus_plus_int @ B @ C ) )
     => ( ( minus_minus_int @ D @ A )
        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( minus_minus_int @ D @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(57,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_add__diff__cancel__right_H) ).

thf(471,plain,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(211,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( member_a @ ( a_minus_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_minus__closed) ).

thf(1028,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( member_a @ ( a_minus_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).

thf(79,axiom,
    ! [A: partia1833973666xt_a_b] :
      ( ( one_a_Product_unit @ ( units_1411277569xt_a_b @ A ) )
      = ( one_a_ring_ext_a_b @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_units__of__one) ).

thf(554,plain,
    ! [A: partia1833973666xt_a_b] :
      ( ( one_a_Product_unit @ ( units_1411277569xt_a_b @ A ) )
      = ( one_a_ring_ext_a_b @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(119,axiom,
    ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) )
    = ( zero_a_b @ r ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_A_061_A_092_060zero_062_092_060close_062) ).

thf(686,plain,
    ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) )
    = ( zero_a_b @ r ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).

thf(267,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ A ) @ A )
        = ( zero_a_b @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_l__neg) ).

thf(1214,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ A ) @ A )
        = ( zero_a_b @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).

thf(215,axiom,
    ! [A: a,B: a,C: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ C @ B ) )
          = ( add_pow_a_b_nat @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_add__pow__rdistr) ).

thf(1039,plain,
    ! [A: a,B: a,C: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ C @ B ) )
          = ( add_pow_a_b_nat @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).

thf(250,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
     => ? [B: a] :
          ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
          & ( ( mult_a_ring_ext_a_b @ r @ B @ A )
            = ( one_a_ring_ext_a_b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_Units__l__inv__ex) ).

thf(1155,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
     => ? [B: a] :
          ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
          & ( ( mult_a_ring_ext_a_b @ r @ B @ A )
            = ( one_a_ring_ext_a_b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).

thf(208,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( mult_a_ring_ext_a_b @ A @ ( add_a_b @ A @ B @ C ) @ D )
              = ( add_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ D ) @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_semiring_Ol__distr) ).

thf(1016,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( mult_a_ring_ext_a_b @ A @ ( add_a_b @ A @ B @ C ) @ D )
              = ( add_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ D ) @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).

thf(91,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_diff__add__inverse2) ).

thf(590,plain,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(243,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( ( a_inv_a_b @ r @ A )
          = ( zero_a_b @ r ) )
        = ( A
          = ( zero_a_b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_add_Oinv__eq__1__iff) ).

thf(1131,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( ( a_inv_a_b @ r @ A )
          = ( zero_a_b @ r ) )
        = ( A
          = ( zero_a_b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).

thf(218,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ? [C: a] :
            ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
            & ( ( mult_a_ring_ext_a_b @ A @ B @ C )
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_group_Or__inv__ex) ).

thf(1050,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ? [C: a] :
            ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
            & ( ( mult_a_ring_ext_a_b @ A @ B @ C )
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).

thf(182,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( add_a_b @ r @ A @ ( add_a_b @ r @ B @ C ) )
            = ( add_a_b @ r @ B @ ( add_a_b @ r @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_add_Om__lcomm) ).

thf(911,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( add_a_b @ r @ A @ ( add_a_b @ r @ B @ C ) )
            = ( add_a_b @ r @ B @ ( add_a_b @ r @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).

thf(84,axiom,
    ! [A: partia96731725t_unit] :
      ( ( group_a_Product_unit @ A )
     => ( ( units_a_Product_unit @ A )
        = ( partia1955795460t_unit @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_group_OUnits__eq) ).

thf(569,plain,
    ! [A: partia96731725t_unit] :
      ( ( group_a_Product_unit @ A )
     => ( ( units_a_Product_unit @ A )
        = ( partia1955795460t_unit @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(96,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_diff__eq__diff__eq) ).

thf(605,plain,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(109,axiom,
    ! [A: partia96731725t_unit,B: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ? [C: a] :
            ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
            & ( ( mult_a_Product_unit @ A @ B @ C )
              = ( one_a_Product_unit @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_group_Or__inv__ex) ).

thf(647,plain,
    ! [A: partia96731725t_unit,B: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ? [C: a] :
            ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
            & ( ( mult_a_Product_unit @ A @ B @ C )
              = ( one_a_Product_unit @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(279,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
       => ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_Units__m__closed) ).

thf(1263,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
       => ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).

thf(280,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( A
              = ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
            = ( B
              = ( add_a_b @ r @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_add_Oinv__solve__right) ).

thf(1266,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( A
              = ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
            = ( B
              = ( add_a_b @ r @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).

thf(271,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ C @ ( add_a_b @ r @ A @ B ) )
            = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_r__distr) ).

thf(1231,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ C @ ( add_a_b @ r @ A @ B ) )
            = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).

thf(190,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( add_a_b @ r @ A @ B )
          = ( add_a_b @ r @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_add_Om__comm) ).

thf(938,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( add_a_b @ r @ A @ B )
          = ( add_a_b @ r @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).

thf(111,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_diff__diff__eq2) ).

thf(657,plain,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(139,axiom,
    ! [A: a,B: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_int @ r @ B @ ( a_inv_a_b @ r @ A ) )
        = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_add_Oint__pow__inv) ).

thf(765,plain,
    ! [A: a,B: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_int @ r @ B @ ( a_inv_a_b @ r @ A ) )
        = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).

thf(213,axiom,
    ! [A: int] :
      ( ( add_pow_a_b_int @ r @ A @ ( zero_a_b @ r ) )
      = ( zero_a_b @ r ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_add_Oint__pow__one) ).

thf(1035,plain,
    ! [A: int] :
      ( ( add_pow_a_b_int @ r @ A @ ( zero_a_b @ r ) )
      = ( zero_a_b @ r ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).

thf(176,axiom,
    ! [A: a,B: a,C: a,D: a] :
      ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
     => ( ( member449909584od_a_a @ ( product_Pair_a_a @ C @ D ) @ ( partia206007992t_unit @ rel ) )
       => ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ D @ A ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) @ ( mult_a_ring_ext_a_b @ r @ B @ D ) ) @ ( partia206007992t_unit @ rel ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_closed__rel__add) ).

thf(888,plain,
    ! [A: a,B: a,C: a,D: a] :
      ( ( member449909584od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( partia206007992t_unit @ rel ) )
     => ( ( member449909584od_a_a @ ( product_Pair_a_a @ C @ D ) @ ( partia206007992t_unit @ rel ) )
       => ( member449909584od_a_a @ ( product_Pair_a_a @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ D @ A ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) @ ( mult_a_ring_ext_a_b @ r @ B @ D ) ) @ ( partia206007992t_unit @ rel ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).

thf(187,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( add_a_b @ A @ B @ C )
            = ( add_a_b @ A @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_abelian__groupE_I4_J) ).

thf(926,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( add_a_b @ A @ B @ C )
            = ( add_a_b @ A @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).

thf(141,axiom,
    member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_Units__minus__one__closed) ).

thf(773,plain,
    member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).

thf(147,axiom,
    ! [A: a,B: a] :
      ( ( ( add_a_b @ r @ A @ B )
        = ( zero_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
         => ( ( a_inv_a_b @ r @ B )
            = A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_minus__equality) ).

thf(794,plain,
    ! [A: a,B: a] :
      ( ( ( add_a_b @ r @ A @ B )
        = ( zero_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
         => ( ( a_inv_a_b @ r @ B )
            = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).

thf(234,axiom,
    member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) ) @ ( partia206007992t_unit @ rel ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_f4) ).

thf(1103,plain,
    member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) ) @ ( partia206007992t_unit @ rel ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).

thf(28,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ C ) )
     => ( ( suc @ A )
        = ( plus_plus_nat @ B @ ( suc @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_nat__arith_Osuc1) ).

thf(375,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ C ) )
     => ( ( suc @ A )
        = ( plus_plus_nat @ B @ ( suc @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(276,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ B @ A )
            = A )
          = ( B
            = ( zero_a_b @ r ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_add_Or__cancel__one) ).

thf(1251,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ B @ A )
            = A )
          = ( B
            = ( zero_a_b @ r ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).

thf(100,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_add__diff__cancel__right) ).

thf(618,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(89,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_diff__add__inverse) ).

thf(584,plain,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(36,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_nat_Oinject) ).

thf(402,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(3,axiom,
    ! [A: partia96731725t_unit] :
      ( ( one_a_Product_unit @ ( units_873712258t_unit @ A ) )
      = ( one_a_Product_unit @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_units__of__one) ).

thf(286,plain,
    ! [A: partia96731725t_unit] :
      ( ( one_a_Product_unit @ ( units_873712258t_unit @ A ) )
      = ( one_a_Product_unit @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(76,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_266_add_Oinverse__distrib__swap) ).

thf(545,plain,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(171,axiom,
    member449909584od_a_a @ ( product_Pair_a_a @ r3 @ s2 ) @ ( partia206007992t_unit @ rel ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_assms_I2_J) ).

thf(875,plain,
    member449909584od_a_a @ ( product_Pair_a_a @ r3 @ s2 ) @ ( partia206007992t_unit @ rel ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).

thf(157,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_a_b @ r @ A @ ( zero_a_b @ r ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_r__zero) ).

thf(826,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_a_b @ r @ A @ ( zero_a_b @ r ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).

thf(87,axiom,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( comm_g1684316527t_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( mult_a_Product_unit @ A @ B @ C )
            = ( mult_a_Product_unit @ A @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_274_comm__groupE_I4_J) ).

thf(577,plain,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( comm_g1684316527t_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( mult_a_Product_unit @ A @ B @ C )
            = ( mult_a_Product_unit @ A @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(49,axiom,
    ! [A: partia96731725t_unit,B: a,C: a,D: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( member_a @ D @ ( partia1955795460t_unit @ A ) )
           => ( ( ( mult_a_Product_unit @ A @ C @ B )
                = ( mult_a_Product_unit @ A @ D @ B ) )
              = ( C = D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_Group_Ogroup_Oright__cancel) ).

thf(445,plain,
    ! [A: partia96731725t_unit,B: a,C: a,D: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( member_a @ D @ ( partia1955795460t_unit @ A ) )
           => ( ( ( mult_a_Product_unit @ A @ C @ B )
                = ( mult_a_Product_unit @ A @ D @ B ) )
              = ( C = D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(30,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = C )
     => ( A
        = ( minus_minus_nat @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_add__implies__diff) ).

thf(382,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = C )
     => ( A
        = ( minus_minus_nat @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(85,axiom,
    ( plus_plus_int
    = ( ^ [A: int,B: int] : ( plus_plus_int @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_add_Ocommute) ).

thf(572,plain,
    ( plus_plus_int
    = ( ^ [A: int,B: int] : ( plus_plus_int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(10,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_171_diff__right__commute) ).

thf(313,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)],[10]) ).

thf(128,axiom,
    ! [A: a,B: a] :
      ( ( abelian_group_a_b @ r )
     => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( ( a_minus_a_b @ r @ A @ B )
              = ( zero_a_b @ r ) )
           => ( A = B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_minus__to__eq) ).

thf(713,plain,
    ! [A: a,B: a] :
      ( ( abelian_group_a_b @ r )
     => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( ( a_minus_a_b @ r @ A @ B )
              = ( zero_a_b @ r ) )
           => ( A = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).

thf(273,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( add_a_b @ A @ ( add_a_b @ A @ B @ C ) @ D )
              = ( add_a_b @ A @ B @ ( add_a_b @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_abelian__groupE_I3_J) ).

thf(1239,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( add_a_b @ A @ ( add_a_b @ A @ B @ C ) @ D )
              = ( add_a_b @ A @ B @ ( add_a_b @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).

thf(230,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( comm_g791708116xt_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_279_comm__groupE_I1_J) ).

thf(1092,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( comm_g791708116xt_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).

thf(17,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_diff__diff__add) ).

thf(335,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(150,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_a_b @ r @ A @ ( a_inv_a_b @ r @ A ) )
        = ( zero_a_b @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_r__neg) ).

thf(804,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_a_b @ r @ A @ ( a_inv_a_b @ r @ A ) )
        = ( zero_a_b @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).

thf(125,axiom,
    member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia1066395285xt_a_b @ r ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_semiring__simprules_I4_J) ).

thf(704,plain,
    member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia1066395285xt_a_b @ r ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).

thf(220,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ? [B: a] :
          ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
          & ( ( add_a_b @ r @ A @ B )
            = ( zero_a_b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_add_Or__inv__ex) ).

thf(1058,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ? [B: a] :
          ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
          & ( ( add_a_b @ r @ A @ B )
            = ( zero_a_b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).

thf(236,axiom,
    ! [A: a,B: a,C: nat] :
      ( ( ( add_a_b @ r @ A @ B )
        = ( add_a_b @ r @ B @ A ) )
     => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( add_pow_a_b_nat @ r @ C @ ( add_a_b @ r @ A @ B ) )
            = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ C @ A ) @ ( add_pow_a_b_nat @ r @ C @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_add_Opow__mult__distrib) ).

thf(1107,plain,
    ! [A: a,B: a,C: nat] :
      ( ( ( add_a_b @ r @ A @ B )
        = ( add_a_b @ r @ B @ A ) )
     => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( add_pow_a_b_nat @ r @ C @ ( add_a_b @ r @ A @ B ) )
            = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ C @ A ) @ ( add_pow_a_b_nat @ r @ C @ B ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).

thf(204,axiom,
    ! [A: partia1833973666xt_a_b] :
      ( ! [B: a] :
          ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
         => ! [C: a] :
              ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
             => ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) )
     => ( ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) )
       => ( ! [B: a] :
              ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
             => ! [C: a] :
                  ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
                 => ! [D: a] :
                      ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
                     => ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
                        = ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) )
         => ( ! [B: a] :
                ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
               => ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
                  = B ) )
           => ( ! [B: a] :
                  ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
                 => ? [C: a] :
                      ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
                      & ( ( mult_a_ring_ext_a_b @ A @ C @ B )
                        = ( one_a_ring_ext_a_b @ A ) ) ) )
             => ( group_a_ring_ext_a_b @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_groupI) ).

thf(990,plain,
    ! [A: partia1833973666xt_a_b] :
      ( ! [B: a] :
          ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
         => ! [C: a] :
              ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
             => ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) )
     => ( ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) )
       => ( ! [B: a] :
              ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
             => ! [C: a] :
                  ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
                 => ! [D: a] :
                      ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
                     => ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
                        = ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) )
         => ( ! [B: a] :
                ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
               => ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
                  = B ) )
           => ( ! [B: a] :
                  ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
                 => ? [C: a] :
                      ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
                      & ( ( mult_a_ring_ext_a_b @ A @ C @ B )
                        = ( one_a_ring_ext_a_b @ A ) ) ) )
             => ( group_a_ring_ext_a_b @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).

thf(168,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( a_minus_a_b @ A @ ( a_minus_a_b @ A @ D @ B ) @ C )
              = ( a_minus_a_b @ A @ D @ ( add_a_b @ A @ B @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_abelian__group_Oright__inv__add) ).

thf(863,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( a_minus_a_b @ A @ ( a_minus_a_b @ A @ D @ B ) @ C )
              = ( a_minus_a_b @ A @ D @ ( add_a_b @ A @ B @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).

thf(173,axiom,
    member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( partia206007992t_unit @ rel ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_f5) ).

thf(880,plain,
    member449909584od_a_a @ ( product_Pair_a_a @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( partia206007992t_unit @ rel ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).

thf(216,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( member_a @ ( add_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_semiring_Osemiring__simprules_I1_J) ).

thf(1043,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( member_a @ ( add_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).

thf(59,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A
        = ( plus_plus_int @ B @ C ) )
     => ( ( uminus_uminus_int @ A )
        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_265_group__cancel_Oneg1) ).

thf(477,plain,
    ! [A: int,B: int,C: int] :
      ( ( A
        = ( plus_plus_int @ B @ C ) )
     => ( ( uminus_uminus_int @ A )
        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(152,axiom,
    ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
    = ( zero_a_b @ r ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_local_Ominus__zero) ).

thf(811,plain,
    ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
    = ( zero_a_b @ r ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).

thf(137,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( A
            = ( add_a_b @ r @ A @ B ) )
          = ( B
            = ( zero_a_b @ r ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_add_Ol__cancel__one_H) ).

thf(758,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( A
            = ( add_a_b @ r @ A @ B ) )
          = ( B
            = ( zero_a_b @ r ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).

thf(13,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ C @ B ) )
      = ( A = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_add__right__cancel) ).

thf(323,plain,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ C @ B ) )
      = ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(253,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( a_minus_a_b @ A @ B @ C )
        = ( add_a_b @ A @ B @ ( a_inv_a_b @ A @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_abelian__group_Ominus__eq) ).

thf(1162,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( a_minus_a_b @ A @ B @ C )
        = ( add_a_b @ A @ B @ ( a_inv_a_b @ A @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).

thf(101,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A
        = ( plus_plus_int @ B @ C ) )
     => ( ( plus_plus_int @ D @ A )
        = ( plus_plus_int @ B @ ( plus_plus_int @ D @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_group__cancel_Oadd2) ).

thf(621,plain,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A
        = ( plus_plus_int @ B @ C ) )
     => ( ( plus_plus_int @ D @ A )
        = ( plus_plus_int @ B @ ( plus_plus_int @ D @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(124,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: nat] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( mult_a_ring_ext_a_b @ A @ ( add_pow_a_b_nat @ A @ D @ B ) @ C )
            = ( add_pow_a_b_nat @ A @ D @ ( mult_a_ring_ext_a_b @ A @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_semiring_Oadd__pow__ldistr) ).

thf(700,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: nat] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( mult_a_ring_ext_a_b @ A @ ( add_pow_a_b_nat @ A @ D @ B ) @ C )
            = ( add_pow_a_b_nat @ A @ D @ ( mult_a_ring_ext_a_b @ A @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).

thf(248,axiom,
    ! [A: a,B: a] :
      ( ( ( add_a_b @ r @ A @ B )
        = ( zero_a_b @ r ) )
     => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( add_a_b @ r @ B @ A )
            = ( zero_a_b @ r ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_add_Oinv__comm) ).

thf(1149,plain,
    ! [A: a,B: a] :
      ( ( ( add_a_b @ r @ A @ B )
        = ( zero_a_b @ r ) )
     => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( add_a_b @ r @ B @ A )
            = ( zero_a_b @ r ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).

thf(140,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( add_a_b @ A @ ( a_inv_a_b @ A @ B ) @ B )
          = ( zero_a_b @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_abelian__group_Ol__neg) ).

thf(769,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( add_a_b @ A @ ( a_inv_a_b @ A @ B ) @ B )
          = ( zero_a_b @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).

thf(64,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_diff__cancel2) ).

thf(494,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(226,axiom,
    ! [A: set_Product_prod_a_a] :
      ( ( collec645855634od_a_a
        @ ^ [B: product_prod_a_a] : ( member449909584od_a_a @ B @ A ) )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_Collect__mem__eq) ).

thf(1080,plain,
    ! [A: set_Product_prod_a_a] :
      ( ( collec645855634od_a_a
        @ ^ [B: product_prod_a_a] : ( member449909584od_a_a @ B @ A ) )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).

thf(166,axiom,
    ! [A: a,B: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( member_a @ ( add_pow_a_b_int @ r @ B @ A ) @ ( partia1066395285xt_a_b @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_add_Oint__pow__closed) ).

thf(858,plain,
    ! [A: a,B: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( member_a @ ( add_pow_a_b_int @ r @ B @ A ) @ ( partia1066395285xt_a_b @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).

thf(26,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ C ) )
     => ( ( plus_plus_nat @ D @ A )
        = ( plus_plus_nat @ B @ ( plus_plus_nat @ D @ C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_group__cancel_Oadd2) ).

thf(367,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ C ) )
     => ( ( plus_plus_nat @ D @ A )
        = ( plus_plus_nat @ B @ ( plus_plus_nat @ D @ C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(102,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
      = ( minus_minus_nat @ B @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_add__diff__cancel__left) ).

thf(626,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ ( plus_plus_nat @ A @ C ) )
      = ( minus_minus_nat @ B @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(183,axiom,
    member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_Units__one__closed) ).

thf(915,plain,
    member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).

thf(35,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_diff__right__commute) ).

thf(399,plain,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(195,axiom,
    ! [A: a,B: a,C: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ C @ A ) @ B )
          = ( add_pow_a_b_int @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_add__pow__ldistr__int) ).

thf(959,plain,
    ! [A: a,B: a,C: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ C @ A ) @ B )
          = ( add_pow_a_b_int @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).

thf(222,axiom,
    ! [A: a,B: a,C: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( add_pow_a_b_int @ r @ C @ ( add_a_b @ r @ A @ B ) )
          = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ C @ A ) @ ( add_pow_a_b_int @ r @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_add_Oint__pow__distrib) ).

thf(1066,plain,
    ! [A: a,B: a,C: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( add_pow_a_b_int @ r @ C @ ( add_a_b @ r @ A @ B ) )
          = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ C @ A ) @ ( add_pow_a_b_int @ r @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).

thf(52,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( member_a @ ( a_minus_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_abelian__group_Ominus__closed) ).

thf(455,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( member_a @ ( a_minus_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(203,axiom,
    ! [A: a,B: nat,C: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ B @ A ) @ ( add_pow_a_b_nat @ r @ C @ A ) )
        = ( add_pow_a_b_nat @ r @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_add_Onat__pow__mult) ).

thf(986,plain,
    ! [A: a,B: nat,C: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ B @ A ) @ ( add_pow_a_b_nat @ r @ C @ A ) )
        = ( add_pow_a_b_nat @ r @ ( plus_plus_nat @ B @ C ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).

thf(82,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_add__diff__eq) ).

thf(563,plain,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(233,axiom,
    member449909584od_a_a @ ( product_Pair_a_a @ r2 @ s ) @ ( partia206007992t_unit @ rel ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_assms_I1_J) ).

thf(1102,plain,
    member449909584od_a_a @ ( product_Pair_a_a @ r2 @ s ) @ ( partia206007992t_unit @ rel ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).

thf(44,axiom,
    ! [A: partia1833973666xt_a_b] :
      ( ( semiring_a_b @ A )
     => ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_semiring_Osemiring__simprules_I2_J) ).

thf(431,plain,
    ! [A: partia1833973666xt_a_b] :
      ( ( semiring_a_b @ A )
     => ( member_a @ ( zero_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(257,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( ( add_a_b @ A @ B @ C )
          = D )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
           => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
             => ( ( add_a_b @ A @ ( a_inv_a_b @ A @ B ) @ D )
                = C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_254_abelian__group_Oa__transpose__inv) ).

thf(1176,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( ( add_a_b @ A @ B @ C )
          = D )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
           => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
             => ( ( add_a_b @ A @ ( a_inv_a_b @ A @ B ) @ D )
                = C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).

thf(97,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( a_inv_a_b @ A @ ( a_inv_a_b @ A @ B ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_abelian__group_Ominus__minus) ).

thf(608,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( a_inv_a_b @ A @ ( a_inv_a_b @ A @ B ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(225,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( ( mult_a_ring_ext_a_b @ A @ C @ B )
                = ( mult_a_ring_ext_a_b @ A @ D @ B ) )
              = ( C = D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_Group_Ogroup_Oright__cancel) ).

thf(1076,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( ( mult_a_ring_ext_a_b @ A @ C @ B )
                = ( mult_a_ring_ext_a_b @ A @ D @ B ) )
              = ( C = D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).

thf(254,axiom,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) @ r4 ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060otimes_062_Ar_H_H_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062) ).

thf(1166,plain,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ r4 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) @ r4 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).

thf(265,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( mult_a_ring_ext_a_b @ A @ ( zero_a_b @ A ) @ B )
          = ( zero_a_b @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_semiring_Ol__null) ).

thf(1208,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( mult_a_ring_ext_a_b @ A @ ( zero_a_b @ A ) @ B )
          = ( zero_a_b @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).

thf(72,axiom,
    ! [A: nat,B: nat] :
      ( ( ( semiri2019852685at_int @ A )
        = ( semiri2019852685at_int @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_271_of__nat__eq__iff) ).

thf(531,plain,
    ! [A: nat,B: nat] :
      ( ( ( semiri2019852685at_int @ A )
        = ( semiri2019852685at_int @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(117,axiom,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) @ r4 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062) ).

thf(680,plain,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) @ r4 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).

thf(134,axiom,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) )
    = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_f8) ).

thf(749,plain,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) )
    = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).

thf(247,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
              = ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_semiring_Osemiring__simprules_I8_J) ).

thf(1145,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( mult_a_ring_ext_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ D )
              = ( mult_a_ring_ext_a_b @ A @ B @ ( mult_a_ring_ext_a_b @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).

thf(63,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A = B )
        & ( C = D ) )
     => ( ( plus_plus_nat @ A @ C )
        = ( plus_plus_nat @ B @ D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_add__mono__thms__linordered__semiring_I4_J) ).

thf(490,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A = B )
        & ( C = D ) )
     => ( ( plus_plus_nat @ A @ C )
        = ( plus_plus_nat @ B @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(116,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ! [B: a] :
            ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
           => ( ( add_a_b @ r @ A @ B )
              = B ) )
       => ( A
          = ( zero_a_b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_add_Oone__unique) ).

thf(675,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ! [B: a] :
            ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
           => ( ( add_a_b @ r @ A @ B )
              = B ) )
       => ( A
          = ( zero_a_b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).

thf(113,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ C @ B ) )
      = ( A = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_add__right__cancel) ).

thf(663,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ C @ B ) )
      = ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).

thf(77,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_diff__add__cancel) ).

thf(548,plain,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(244,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ A )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_l__one) ).

thf(1134,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ A )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).

thf(62,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_add_Oleft__commute) ).

thf(487,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(53,axiom,
    ( minus_minus_int
    = ( ^ [A: int,B: int] : ( plus_plus_int @ A @ ( uminus_uminus_int @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_263_diff__conv__add__uminus) ).

thf(458,plain,
    ( minus_minus_int
    = ( ^ [A: int,B: int] : ( plus_plus_int @ A @ ( uminus_uminus_int @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(258,axiom,
    ! [A: a,B: a] :
      ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
        = ( one_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
       => ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ B @ A )
            = ( one_a_ring_ext_a_b @ r ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_Units__inv__comm) ).

thf(1181,plain,
    ! [A: a,B: a] :
      ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
        = ( one_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
       => ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ B @ A )
            = ( one_a_ring_ext_a_b @ r ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).

thf(156,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( member_a @ ( add_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_add_Om__closed) ).

thf(823,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( member_a @ ( add_a_b @ r @ A @ B ) @ ( partia1066395285xt_a_b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).

thf(70,axiom,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( ( mult_a_Product_unit @ A @ B @ C )
          = ( one_a_Product_unit @ A ) )
       => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
         => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
           => ( ( mult_a_Product_unit @ A @ C @ B )
              = ( one_a_Product_unit @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_group_Oinv__comm) ).

thf(525,plain,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( ( mult_a_Product_unit @ A @ B @ C )
          = ( one_a_Product_unit @ A ) )
       => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
         => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
           => ( ( mult_a_Product_unit @ A @ C @ B )
              = ( one_a_Product_unit @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(80,axiom,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( comm_g1684316527t_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( member_a @ ( mult_a_Product_unit @ A @ B @ C ) @ ( partia1955795460t_unit @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_278_comm__groupE_I1_J) ).

thf(557,plain,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( comm_g1684316527t_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( member_a @ ( mult_a_Product_unit @ A @ B @ C ) @ ( partia1955795460t_unit @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(107,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_add__diff__cancel__left_H) ).

thf(640,plain,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(223,axiom,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s3 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_f9) ).

thf(1070,plain,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ s3 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).

thf(126,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ A @ B ) )
          = ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ ( a_inv_a_b @ r @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_add_Oinv__mult__group) ).

thf(705,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ A @ B ) )
          = ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ ( a_inv_a_b @ r @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).

thf(39,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_add__diff__cancel__right_H) ).

thf(414,plain,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(232,axiom,
    ! [A: a,B: nat,C: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ B @ A ) @ ( add_pow_a_b_nat @ r @ C @ A ) )
        = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ C @ A ) @ ( add_pow_a_b_nat @ r @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_add_Onat__pow__comm) ).

thf(1098,plain,
    ! [A: a,B: nat,C: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ B @ A ) @ ( add_pow_a_b_nat @ r @ C @ A ) )
        = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ C @ A ) @ ( add_pow_a_b_nat @ r @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).

thf(9,axiom,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_old_Onat_Oinject) ).

thf(310,plain,
    ! [A: nat,B: nat] :
      ( ( ( suc @ A )
        = ( suc @ B ) )
      = ( A = B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(214,axiom,
    comm_g1684316527t_unit @ ( units_1411277569xt_a_b @ r ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_269_units__comm__group) ).

thf(1038,plain,
    comm_g1684316527t_unit @ ( units_1411277569xt_a_b @ r ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).

thf(129,axiom,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_prod_Oinject) ).

thf(717,plain,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ C @ D ) )
      = ( ( A = C )
        & ( B = D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).

thf(272,axiom,
    ! [A: a,B: a,C: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ C @ A ) @ B )
          = ( add_pow_a_b_nat @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_add__pow__ldistr) ).

thf(1235,plain,
    ! [A: a,B: a,C: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ C @ A ) @ B )
          = ( add_pow_a_b_nat @ r @ C @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).

thf(18,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
      = ( minus_minus_int @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_259_minus__diff__eq) ).

thf(338,plain,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
      = ( minus_minus_int @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(108,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A = B )
        & ( C = D ) )
     => ( ( plus_plus_int @ A @ C )
        = ( plus_plus_int @ B @ D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_add__mono__thms__linordered__semiring_I4_J) ).

thf(643,plain,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A = B )
        & ( C = D ) )
     => ( ( plus_plus_int @ A @ C )
        = ( plus_plus_int @ B @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(181,axiom,
    ! [A: a,B: a,C: a] :
      ( ( ( add_a_b @ r @ A @ B )
        = C )
     => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
           => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ A ) @ C )
              = B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_a__transpose__inv) ).

thf(907,plain,
    ! [A: a,B: a,C: a] :
      ( ( ( add_a_b @ r @ A @ B )
        = C )
     => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
           => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ A ) @ C )
              = B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).

thf(269,axiom,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ C @ D ) )
     => ~ ( ( A = C )
         => ( B != D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_Pair__inject) ).

thf(1221,plain,
    ! [A: a,B: a,C: a,D: a] :
      ( ( ( product_Pair_a_a @ A @ B )
        = ( product_Pair_a_a @ C @ D ) )
     => ~ ( ( A = C )
         => ( B != D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).

thf(132,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_semiring_Osemiring__simprules_I3_J) ).

thf(727,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( member_a @ ( mult_a_ring_ext_a_b @ A @ B @ C ) @ ( partia1066395285xt_a_b @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).

thf(31,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ C @ B ) )
      = ( A = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_group__add__class_Oadd_Oright__cancel) ).

thf(386,plain,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ C @ B ) )
      = ( A = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(206,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( a_inv_a_b @ A @ ( add_a_b @ A @ B @ C ) )
            = ( a_minus_a_b @ A @ ( a_inv_a_b @ A @ B ) @ C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_253_abelian__group_Oinv__add) ).

thf(1008,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( a_inv_a_b @ A @ ( add_a_b @ A @ B @ C ) )
            = ( a_minus_a_b @ A @ ( a_inv_a_b @ A @ B ) @ C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).

thf(93,axiom,
    ! [A: partia96731725t_unit] :
      ( ( mult_a_Product_unit @ ( units_873712258t_unit @ A ) )
      = ( mult_a_Product_unit @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_units__of__mult) ).

thf(595,plain,
    ! [A: partia96731725t_unit] :
      ( ( mult_a_Product_unit @ ( units_873712258t_unit @ A ) )
      = ( mult_a_Product_unit @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(98,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_add__diff__cancel__right) ).

thf(612,plain,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(65,axiom,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( B
              = ( mult_a_Product_unit @ A @ B @ C ) )
            = ( C
              = ( one_a_Product_unit @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_group_Ol__cancel__one_H) ).

thf(497,plain,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( B
              = ( mult_a_Product_unit @ A @ B @ C ) )
            = ( C
              = ( one_a_Product_unit @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(172,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( mult_a_ring_ext_a_b @ A @ D @ ( add_a_b @ A @ B @ C ) )
              = ( add_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ D @ B ) @ ( mult_a_ring_ext_a_b @ A @ D @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_semiring_Or__distr) ).

thf(876,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( mult_a_ring_ext_a_b @ A @ D @ ( add_a_b @ A @ B @ C ) )
              = ( add_a_b @ A @ ( mult_a_ring_ext_a_b @ A @ D @ B ) @ ( mult_a_ring_ext_a_b @ A @ D @ C ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).

thf(115,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( add_a_b @ A @ B @ ( zero_a_b @ A ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_semiring_Osemiring__simprules_I11_J) ).

thf(671,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( add_a_b @ A @ B @ ( zero_a_b @ A ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).

thf(88,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_add__left__cancel) ).

thf(581,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(177,axiom,
    ! [A: product_prod_a_a,B: product_prod_a_a > $o] :
      ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ B ) )
      = ( B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_mem__Collect__eq) ).

thf(891,plain,
    ! [A: product_prod_a_a,B: product_prod_a_a > $o] :
      ( ( member449909584od_a_a @ A @ ( collec645855634od_a_a @ B ) )
      = ( B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).

thf(260,axiom,
    ! [A: a,B: a,C: a,D: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ r ) )
           => ( ( a_minus_a_b @ r @ ( add_a_b @ r @ ( a_minus_a_b @ r @ A @ C ) @ B ) @ D )
              = ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ ( add_a_b @ r @ A @ B ) @ C ) @ D ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_four__elem__comm) ).

thf(1189,plain,
    ! [A: a,B: a,C: a,D: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ r ) )
           => ( ( a_minus_a_b @ r @ ( add_a_b @ r @ ( a_minus_a_b @ r @ A @ C ) @ B ) @ D )
              = ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ ( add_a_b @ r @ A @ B ) @ C ) @ D ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).

thf(200,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
              = B )
            = ( C
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_group_Ol__cancel__one) ).

thf(976,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
              = B )
            = ( C
              = ( one_a_ring_ext_a_b @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).

thf(120,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: nat] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( mult_a_ring_ext_a_b @ A @ B @ ( add_pow_a_b_nat @ A @ D @ C ) )
            = ( add_pow_a_b_nat @ A @ D @ ( mult_a_ring_ext_a_b @ A @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_semiring_Oadd__pow__rdistr) ).

thf(688,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: nat] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( mult_a_ring_ext_a_b @ A @ B @ ( add_pow_a_b_nat @ A @ D @ C ) )
            = ( add_pow_a_b_nat @ A @ D @ ( mult_a_ring_ext_a_b @ A @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).

thf(180,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( add_a_b @ r @ ( add_a_b @ r @ A @ B ) @ C )
            = ( add_a_b @ r @ A @ ( add_a_b @ r @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_add_Om__assoc) ).

thf(903,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( add_a_b @ r @ ( add_a_b @ r @ A @ B ) @ C )
            = ( add_a_b @ r @ A @ ( add_a_b @ r @ B @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).

thf(153,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( add_a_b @ A @ ( zero_a_b @ A ) @ B )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_abelian__groupE_I5_J) ).

thf(813,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( add_a_b @ A @ ( zero_a_b @ A ) @ B )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).

thf(196,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( add_a_b @ A @ ( a_inv_a_b @ A @ B ) @ ( add_a_b @ A @ B @ C ) )
            = C ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_abelian__group_Or__neg1) ).

thf(963,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( add_a_b @ A @ ( a_inv_a_b @ A @ B ) @ ( add_a_b @ A @ B @ C ) )
            = C ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).

thf(12,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_add__left__imp__eq) ).

thf(319,plain,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(221,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
              = ( mult_a_ring_ext_a_b @ r @ A @ C ) )
            = ( B = C ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_Units__l__cancel) ).

thf(1062,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
              = ( mult_a_ring_ext_a_b @ r @ A @ C ) )
            = ( B = C ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).

thf(192,axiom,
    ! [A: a,B: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ ( semiri2019852685at_int @ B ) ) @ A )
        = ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_268_add_Oint__pow__neg__int) ).

thf(946,plain,
    ! [A: a,B: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ ( semiri2019852685at_int @ B ) ) @ A )
        = ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).

thf(110,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A
        = ( plus_plus_int @ B @ C ) )
     => ( ( minus_minus_int @ A @ D )
        = ( plus_plus_int @ B @ ( minus_minus_int @ C @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_group__cancel_Osub1) ).

thf(652,plain,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A
        = ( plus_plus_int @ B @ C ) )
     => ( ( minus_minus_int @ A @ D )
        = ( plus_plus_int @ B @ ( minus_minus_int @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(104,axiom,
    ! [A: partia1833973666xt_a_b] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( units_a_ring_ext_a_b @ A )
        = ( partia1066395285xt_a_b @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_group_OUnits__eq) ).

thf(632,plain,
    ! [A: partia1833973666xt_a_b] :
      ( ( group_a_ring_ext_a_b @ A )
     => ( ( units_a_ring_ext_a_b @ A )
        = ( partia1066395285xt_a_b @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(145,axiom,
    ! [A: a,B: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ B ) @ A )
        = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_255_add_Oint__pow__neg) ).

thf(786,plain,
    ! [A: a,B: int] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ B ) @ A )
        = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).

thf(45,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(433,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(261,axiom,
    ! [A: a,B: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_nat @ r @ B @ ( a_inv_a_b @ r @ A ) )
        = ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_add_Onat__pow__inv) ).

thf(1193,plain,
    ! [A: a,B: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( add_pow_a_b_nat @ r @ B @ ( a_inv_a_b @ r @ A ) )
        = ( a_inv_a_b @ r @ ( add_pow_a_b_nat @ r @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).

thf(266,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( member_a @ ( a_inv_a_b @ r @ A ) @ ( partia1066395285xt_a_b @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_add_Oinv__closed) ).

thf(1212,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( member_a @ ( a_inv_a_b @ r @ A ) @ ( partia1066395285xt_a_b @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).

thf(179,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ? [B: a] :
          ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
          & ( ( add_a_b @ r @ B @ A )
            = ( zero_a_b @ r ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_add_Ol__inv__ex) ).

thf(899,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ? [B: a] :
          ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
          & ( ( add_a_b @ r @ B @ A )
            = ( zero_a_b @ r ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).

thf(252,axiom,
    semiring_a_b @ r,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_local_Osemiring__axioms) ).

thf(1161,plain,
    semiring_a_b @ r,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).

thf(284,axiom,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ s ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_H_H_A_092_060otimes_062_As_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062) ).

thf(1280,plain,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ s ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).

thf(37,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A
        = ( plus_plus_int @ B @ C ) )
     => ( ( plus_plus_int @ A @ D )
        = ( plus_plus_int @ B @ ( plus_plus_int @ C @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_group__cancel_Oadd1) ).

thf(405,plain,
    ! [A: int,B: int,C: int,D: int] :
      ( ( A
        = ( plus_plus_int @ B @ C ) )
     => ( ( plus_plus_int @ A @ D )
        = ( plus_plus_int @ B @ ( plus_plus_int @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(165,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( add_a_b @ A @ ( add_a_b @ A @ B @ C ) @ D )
              = ( add_a_b @ A @ B @ ( add_a_b @ A @ C @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_semiring_Osemiring__simprules_I5_J) ).

thf(854,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( add_a_b @ A @ ( add_a_b @ A @ B @ C ) @ D )
              = ( add_a_b @ A @ B @ ( add_a_b @ A @ C @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).

thf(146,axiom,
    ! [A: a,B: a,C: a] :
      ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
        = ( one_a_ring_ext_a_b @ r ) )
     => ( ( ( mult_a_ring_ext_a_b @ r @ B @ C )
          = ( one_a_ring_ext_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
           => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
             => ( A = C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_inv__unique) ).

thf(790,plain,
    ! [A: a,B: a,C: a] :
      ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
        = ( one_a_ring_ext_a_b @ r ) )
     => ( ( ( mult_a_ring_ext_a_b @ r @ B @ C )
          = ( one_a_ring_ext_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
           => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
             => ( A = C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).

thf(169,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( add_a_b @ A @ B @ C )
            = ( add_a_b @ A @ C @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_semiring_Osemiring__simprules_I7_J) ).

thf(867,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( add_a_b @ A @ B @ C )
            = ( add_a_b @ A @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).

thf(159,axiom,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) @ s3 ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_061_As_A_092_060otimes_062_A_Is_H_H_A_092_060otimes_062_As_H_J_A_092_060otimes_062_As_H_H_092_060close_062) ).

thf(833,plain,
    ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) )
    = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ ( mult_a_ring_ext_a_b @ r @ s3 @ s2 ) ) @ s3 ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).

thf(38,axiom,
    ! [A: nat] :
      ( A
     != ( suc @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_n__not__Suc__n) ).

thf(410,plain,
    ! [A: nat] :
      ( A
     != ( suc @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(197,axiom,
    ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ) )
    = ( zero_a_b @ r ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_f11) ).

thf(967,plain,
    ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s @ r3 ) ) @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s2 ) @ s3 ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r2 @ r4 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ r3 @ r4 ) ) ) ) ) )
    = ( zero_a_b @ r ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).

thf(193,axiom,
    ! [A: a,B: a,C: a] :
      ( ( A = B )
     => ( ( a_minus_a_b @ r @ C @ A )
        = ( a_minus_a_b @ r @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_local_Oright__minus__eq) ).

thf(950,plain,
    ! [A: a,B: a,C: a] :
      ( ( A = B )
     => ( ( a_minus_a_b @ r @ C @ A )
        = ( a_minus_a_b @ r @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).

thf(160,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ A @ ( one_a_ring_ext_a_b @ r ) )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_r__one) ).

thf(835,plain,
    ! [A: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ A @ ( one_a_ring_ext_a_b @ r ) )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).

thf(14,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( member_a @ ( a_inv_a_b @ A @ B ) @ ( partia1066395285xt_a_b @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_abelian__group_Oa__inv__closed) ).

thf(326,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( member_a @ ( a_inv_a_b @ A @ B ) @ ( partia1066395285xt_a_b @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(19,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( suc @ A ) @ B )
      = ( plus_plus_nat @ A @ ( suc @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_add__Suc__shift) ).

thf(341,plain,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( suc @ A ) @ B )
      = ( plus_plus_nat @ A @ ( suc @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(151,axiom,
    ! [A: a,B: a,C: a] :
      ( ( ( add_a_b @ r @ A @ B )
        = ( zero_a_b @ r ) )
     => ( ( ( add_a_b @ r @ B @ C )
          = ( zero_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
           => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
             => ( A = C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_local_Ominus__unique) ).

thf(807,plain,
    ! [A: a,B: a,C: a] :
      ( ( ( add_a_b @ r @ A @ B )
        = ( zero_a_b @ r ) )
     => ( ( ( add_a_b @ r @ B @ C )
          = ( zero_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
           => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
             => ( A = C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).

thf(262,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( a_inv_a_b @ A @ ( add_a_b @ A @ B @ C ) )
            = ( add_a_b @ A @ ( a_inv_a_b @ A @ B ) @ ( a_inv_a_b @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_abelian__group_Ominus__add) ).

thf(1197,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( abelian_group_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( a_inv_a_b @ A @ ( add_a_b @ A @ B @ C ) )
            = ( add_a_b @ A @ ( a_inv_a_b @ A @ B ) @ ( a_inv_a_b @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).

thf(105,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_add_Oassoc) ).

thf(635,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(123,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( field_a_b @ A )
     => ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
          = ( zero_a_b @ A ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
           => ( ( B
                = ( zero_a_b @ A ) )
              | ( C
                = ( zero_a_b @ A ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_integral) ).

thf(696,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a] :
      ( ( field_a_b @ A )
     => ( ( ( mult_a_ring_ext_a_b @ A @ B @ C )
          = ( zero_a_b @ A ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
           => ( ( B
                = ( zero_a_b @ A ) )
              | ( C
                = ( zero_a_b @ A ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).

thf(130,axiom,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( add_a_b @ A @ B @ ( add_a_b @ A @ C @ D ) )
              = ( add_a_b @ A @ C @ ( add_a_b @ A @ B @ D ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_semiring_Osemiring__simprules_I12_J) ).

thf(720,plain,
    ! [A: partia1833973666xt_a_b,B: a,C: a,D: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ A ) )
         => ( ( member_a @ D @ ( partia1066395285xt_a_b @ A ) )
           => ( ( add_a_b @ A @ B @ ( add_a_b @ A @ C @ D ) )
              = ( add_a_b @ A @ C @ ( add_a_b @ A @ B @ D ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).

thf(194,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) )
            = ( mult_a_ring_ext_a_b @ r @ B @ ( mult_a_ring_ext_a_b @ r @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_m__lcomm) ).

thf(955,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) )
            = ( mult_a_ring_ext_a_b @ r @ B @ ( mult_a_ring_ext_a_b @ r @ A @ C ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).

thf(6,axiom,
    ! [A: a > $o,B: a > $o] :
      ( ! [C: a] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect_a @ A )
        = ( collect_a @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_Collect__cong) ).

thf(294,plain,
    ! [A: a > $o,B: a > $o] :
      ( ! [C: a] :
          ( ( A @ C )
          = ( B @ C ) )
     => ( ( collect_a @ A )
        = ( collect_a @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(149,axiom,
    ! [A: partia1833973666xt_a_b,B: partia1833973666xt_a_b] :
      ( ( ( partia1066395285xt_a_b @ A )
        = ( partia1066395285xt_a_b @ B ) )
     => ( ( ( mult_a_ring_ext_a_b @ A )
          = ( mult_a_ring_ext_a_b @ B ) )
       => ( ( ( one_a_ring_ext_a_b @ A )
            = ( one_a_ring_ext_a_b @ B ) )
         => ( ( ( zero_a_b @ A )
              = ( zero_a_b @ B ) )
           => ( ( ( add_a_b @ A )
                = ( add_a_b @ B ) )
             => ( ( ( more_a_b @ A )
                  = ( more_a_b @ B ) )
               => ( A = B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_ring_Oequality) ).

thf(801,plain,
    ! [A: partia1833973666xt_a_b,B: partia1833973666xt_a_b] :
      ( ( ( partia1066395285xt_a_b @ A )
        = ( partia1066395285xt_a_b @ B ) )
     => ( ( ( mult_a_ring_ext_a_b @ A )
          = ( mult_a_ring_ext_a_b @ B ) )
       => ( ( ( one_a_ring_ext_a_b @ A )
            = ( one_a_ring_ext_a_b @ B ) )
         => ( ( ( zero_a_b @ A )
              = ( zero_a_b @ B ) )
           => ( ( ( add_a_b @ A )
                = ( add_a_b @ B ) )
             => ( ( ( more_a_b @ A )
                  = ( more_a_b @ B ) )
               => ( A = B ) ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).

thf(256,axiom,
    ! [A: a,B: a,C: nat] :
      ( ( ( add_a_b @ r @ A @ B )
        = ( add_a_b @ r @ B @ A ) )
     => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ C @ A ) @ B )
            = ( add_a_b @ r @ B @ ( add_pow_a_b_nat @ r @ C @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_add_Ogroup__commutes__pow) ).

thf(1172,plain,
    ! [A: a,B: a,C: nat] :
      ( ( ( add_a_b @ r @ A @ B )
        = ( add_a_b @ r @ B @ A ) )
     => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
         => ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ C @ A ) @ B )
            = ( add_a_b @ r @ B @ ( add_pow_a_b_nat @ r @ C @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).

thf(58,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ A ) @ B ) @ ( suc @ C ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_Suc__diff__diff) ).

thf(474,plain,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ A ) @ B ) @ ( suc @ C ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(55,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ C ) )
     => ( ( plus_plus_nat @ A @ D )
        = ( plus_plus_nat @ B @ ( plus_plus_nat @ C @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_group__cancel_Oadd1) ).

thf(464,plain,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ C ) )
     => ( ( plus_plus_nat @ A @ D )
        = ( plus_plus_nat @ B @ ( plus_plus_nat @ C @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(16,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_add__diff__cancel__left_H) ).

thf(332,plain,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(164,axiom,
    ( ! [A: a] :
        ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ? [B: a] :
            ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
            & ( ( mult_a_ring_ext_a_b @ r @ B @ A )
              = ( one_a_ring_ext_a_b @ r ) ) ) )
   => ( group_a_ring_ext_a_b @ r ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_group__l__invI) ).

thf(850,plain,
    ( ! [A: a] :
        ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
       => ? [B: a] :
            ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
            & ( ( mult_a_ring_ext_a_b @ r @ B @ A )
              = ( one_a_ring_ext_a_b @ r ) ) ) )
   => ( group_a_ring_ext_a_b @ r ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).

thf(202,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
              = A )
            = ( C
              = ( add_a_b @ r @ B @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_add_Oinv__solve__left_H) ).

thf(982,plain,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
              = A )
            = ( C
              = ( add_a_b @ r @ B @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).

thf(48,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_add_Oassoc) ).

thf(442,plain,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(170,axiom,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_semiring_Osemiring__simprules_I9_J) ).

thf(871,plain,
    ! [A: partia1833973666xt_a_b,B: a] :
      ( ( semiring_a_b @ A )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ A ) )
       => ( ( mult_a_ring_ext_a_b @ A @ ( one_a_ring_ext_a_b @ A ) @ B )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).

thf(240,axiom,
    ! [A: a,B: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( member_a @ ( add_pow_a_b_nat @ r @ B @ A ) @ ( partia1066395285xt_a_b @ r ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_add_Onat__pow__closed) ).

thf(1120,plain,
    ! [A: a,B: nat] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( member_a @ ( add_pow_a_b_nat @ r @ B @ A ) @ ( partia1066395285xt_a_b @ r ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).

thf(66,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_ab__semigroup__add__class_Oadd__ac_I1_J) ).

thf(501,plain,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(249,axiom,
    ! [A: nat,B: a] :
      ( ( add_pow_a_b_nat @ r @ ( suc @ A ) @ B )
      = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ A @ B ) @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_add_Onat__pow__Suc) ).

thf(1152,plain,
    ! [A: nat,B: a] :
      ( ( add_pow_a_b_nat @ r @ ( suc @ A ) @ B )
      = ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ A @ B ) @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).

thf(43,axiom,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( ( mult_a_Product_unit @ A @ C @ B )
              = B )
            = ( C
              = ( one_a_Product_unit @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_group_Or__cancel__one) ).

thf(427,plain,
    ! [A: partia96731725t_unit,B: a,C: a] :
      ( ( group_a_Product_unit @ A )
     => ( ( member_a @ B @ ( partia1955795460t_unit @ A ) )
       => ( ( member_a @ C @ ( partia1955795460t_unit @ A ) )
         => ( ( ( mult_a_Product_unit @ A @ C @ B )
              = B )
            = ( C
              = ( one_a_Product_unit @ A ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(56,axiom,
    ! [A: partia1833973666xt_a_b] :
      ( ( comm_g791708116xt_a_b @ A )
     => ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_281_comm__groupE_I2_J) ).

thf(469,plain,
    ! [A: partia1833973666xt_a_b] :
      ( ( comm_g791708116xt_a_b @ A )
     => ( member_a @ ( one_a_ring_ext_a_b @ A ) @ ( partia1066395285xt_a_b @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(259,axiom,
    ! [A: a,B: a,C: a] :
      ( ( A = B )
     => ( ( add_a_b @ r @ C @ A )
        = ( add_a_b @ r @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_right__add__eq) ).

thf(1184,plain,
    ! [A: a,B: a,C: a] :
      ( ( A = B )
     => ( ( add_a_b @ r @ C @ A )
        = ( add_a_b @ r @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).

thf(81,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_261_uminus__add__conv__diff) ).

thf(560,plain,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(264,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( add_a_b @ r @ A @ ( add_a_b @ r @ ( a_inv_a_b @ r @ A ) @ B ) )
          = B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_r__neg2) ).

thf(1204,plain,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
       => ( ( add_a_b @ r @ A @ ( add_a_b @ r @ ( a_inv_a_b @ r @ A ) @ B ) )
          = B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).

thf(25,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_add_Oleft__cancel) ).

thf(364,plain,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(1303,plain,
    $false,
    inference(cvc4,[status(thm)],[629,1024,1168,347,846,1031,1083,666,481,352,762,1005,730,449,379,861,1137,797,1275,538,930,782,709,417,436,1228,829,1095,1243,1127,973,885,1046,389,372,504,751,344,934,1055,724,587,838,542,460,817,638,1087,1247,969,881,1115,289,316,924,756,1020,574,602,692,396,660,1201,1104,507,1072,453,1119,774,307,292,920,529,534,942,439,1274,894,1091,778,1255,1123,820,392,297,424,693,1217,598,916,361,329,522,1159,682,551,842,1012,285,566,484,615,1111,980,593,420,1270,1259,1138,356,471,1028,554,686,1214,1039,1155,1016,590,1131,1050,911,569,605,647,1263,1266,1231,938,657,765,1035,888,926,773,794,1103,375,1251,618,584,402,286,545,875,826,577,445,382,572,313,713,1239,1092,335,804,704,1058,1107,990,863,880,1043,477,811,758,323,1162,621,700,1149,769,494,1080,858,367,626,915,399,959,1066,455,986,563,1102,431,1176,608,1076,1166,1208,531,680,749,1145,490,675,663,548,1134,487,458,1181,823,525,557,640,1070,705,414,1098,310,1038,717,1235,338,643,907,1221,727,386,1008,595,612,497,876,671,581,891,1189,976,688,903,813,963,319,1062,946,652,632,786,433,1193,1212,899,1161,1280,405,854,790,867,833,410,967,950,835,326,341,807,1197,635,696,720,955,294,801,1172,474,464,332,850,982,442,871,1120,501,1152,427,469,1184,560,1204,364]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12  % Problem  : ITP106^1 : TPTP v8.2.0. Released v7.5.0.
% 0.13/0.16  % Command  : run_Leo-III %s %d
% 0.17/0.37  % Computer : n016.cluster.edu
% 0.17/0.37  % Model    : x86_64 x86_64
% 0.17/0.37  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.37  % Memory   : 8042.1875MB
% 0.17/0.37  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.17/0.37  % CPULimit : 300
% 0.17/0.37  % WCLimit  : 300
% 0.17/0.37  % DateTime : Sat May 18 18:28:39 EDT 2024
% 0.17/0.37  % CPUTime  : 
% 0.99/0.91  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 1.97/1.17  % [INFO] 	 Parsing done (254ms). 
% 1.97/1.18  % [INFO] 	 Running in sequential loop mode. 
% 2.72/1.40  % [INFO] 	 eprover registered as external prover. 
% 2.72/1.40  % [INFO] 	 cvc4 registered as external prover. 
% 2.72/1.40  % [INFO] 	 Scanning for conjecture ... 
% 3.04/1.51  % [INFO] 	 Found a conjecture (or negated_conjecture) and 282 axioms. Running axiom selection ... 
% 3.20/1.61  % [INFO] 	 Axiom selection finished. Selected 282 axioms (removed 0 axioms). 
% 4.07/1.78  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 4.12/1.79  % [INFO] 	 Type checking passed. 
% 4.12/1.80  % [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 ... 
% 12.12/3.59  % External prover 'cvc4' found a proof!
% 12.12/3.59  % [INFO] 	 Killing All external provers ... 
% 12.12/3.60  % Time passed: 3058ms (effective reasoning time: 2416ms)
% 12.12/3.60  % 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)>
% 12.12/3.60  % Axioms used in derivation (282): fact_257_minus__add__cancel, fact_249_abelian__group_Ominus__eq, fact_68_add__pow__rdistr__int, fact_223_a__transpose__inv, fact_261_uminus__add__conv__diff, fact_259_minus__diff__eq, fact_59_prod__cases, fact_159_group__add__class_Oadd_Oright__cancel, fact_176_eq__diff__eq, fact_133_diff__add__inverse2, fact_48_Collect__mem__eq, fact_130_nat__arith_Osuc1, fact_146_add__right__cancel, fact_35_right__inv__add, fact_56_old_Oprod_Oinducts, fact_279_comm__groupE_I1_J, fact_149_add__diff__cancel, fact_272_of__nat__add, fact_255_add_Oint__pow__neg, fact_165_group__cancel_Oadd1, fact_21_old_Oprod_Oinject, fact_156_add_Oleft__commute, fact_63_semiring_Osemiring__simprules_I6_J, fact_124_nat_Oinject, fact_275_comm__groupE_I4_J, fact_27_add_Om__comm, fact_214_inv__add, fact_264_ab__group__add__class_Oab__diff__conv__add__uminus, fact_198_group_Ol__inv__ex, fact_92_add_Ogroup__commutes__pow, fact_169_ab__semigroup__add__class_Oadd__ac_I1_J, fact_64_semiring_Ol__null, fact_6_f9, fact_188_Units__closed, fact_20_prod_Oinject, fact_127_n__not__Suc__n, fact_278_comm__groupE_I1_J, fact_148_add__left__cancel, fact_191_Units__one__closed, fact_190_Units__m__closed, fact_224_r__minus, fact_136_Nat_Odiff__cancel, fact_242_abelian__group_Ominus__minus, fact_233_add_Oint__pow__diff, fact_168_add__mono__thms__linordered__semiring_I4_J, fact_215_r__neg2, fact_258_minus__add__distrib, fact_28_add_Om__lcomm, fact_54_r__null, fact_201_group_OUnits__eq, fact_182_diff__diff__add, fact_250_abelian__group_Ol__neg, fact_86_semiring_Osemiring__simprules_I12_J, fact_263_diff__conv__add__uminus, fact_85_semiring_Osemiring__simprules_I7_J, fact_222_add_Oinv__mult__group, fact_238_Units__minus__one__closed, fact_144_add__diff__cancel__left, fact_49_Collect__cong, fact_134_diff__add__inverse, fact_218_add_Oinv__solve__right_H, fact_177_add__diff__eq, fact_75_add_Oint__pow__closed, fact_33_add_Ol__inv__ex, fact_252_abelian__group_Ominus__equality, fact_65_semiring_Or__null, fact_112_abelian__groupE_I2_J, fact_268_add_Oint__pow__neg__int, fact_50_Collect__cong, fact_30_local_Ominus__unique, fact_277_comm__groupE_I3_J, fact_83_semiring_Osemiring__simprules_I1_J, fact_139_add__diff__cancel__right, fact_202_group_OUnits__eq, fact_194_groupI, fact_180_diff__add__eq__diff__diff__swap, fact_91_add__pow__rdistr, fact_153_add__left__imp__eq, fact_226_add_Onat__pow__inv, fact_66_semiring_Or__distr, fact_274_comm__groupE_I4_J, fact_118_abelian__group_Oright__inv__add, fact_232_l__neg, fact_18_r__distr, fact_163_group__cancel_Oadd2, fact_89_add_Oint__pow__mult, fact_82_semiring_Osemiring__simprules_I8_J, fact_161_add_Oassoc, fact_178_diff__diff__eq2, fact_138_add__diff__cancel__right_H, fact_96_add_Onat__pow__closed, fact_7__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_H_H_A_092_060otimes_062_As_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062, fact_147_add__left__cancel, fact_8__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062, fact_111_abelian__groupE_I4_J, fact_245_units__of__one, fact_97_add_Onat__pow__one, fact_22_local_Osemiring__axioms, fact_151_add__right__imp__eq, fact_132_zero__induct__lemma, fact_125_old_Onat_Oinject, fact_128_diff__commute, fact_13__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_092_060close_062, fact_229_sum__zero__eq__neg, fact_166_group__cancel_Oadd1, fact_70_inv__unique, fact_204_group_Or__cancel__one_H, fact_126_Suc__inject, fact_114_abelian__groupI, fact_72_add_Oint__pow__distrib, fact_103_add_Onat__pow__Suc2, fact_100_semiring_Oadd__pow__ldistr, fact_186_Units__r__inv__ex, fact_212_group_Or__inv__ex, fact_32_add_Oone__unique, fact_117_abelian__group_Ofour__elem__comm, fact_10_f5, fact_51_add_Or__cancel__one, fact_42_r__zero, fact_25_m__lcomm, fact_256_add__minus__cancel, fact_24_m__comm, fact_31_add_Or__inv__ex, fact_95_add_Onat__pow__comm, fact_246_abelian__group_Ominus__add, fact_44_add_Or__cancel__one_H, fact_81_semiring_Osemiring__simprules_I3_J, fact_73_add_Oint__pow__mult__distrib, fact_154_add__left__imp__eq, fact_38_add_Om__closed, fact_69_add__pow__ldistr__int, fact_192_Units__l__cancel, fact_37_local_Oadd_Oright__cancel, fact_17_l__distr, fact_197_group_Ol__inv__ex, fact_267_minus__diff__commute, fact_46_mem__Collect__eq, fact_172_diff__right__commute, fact_152_add__right__imp__eq, fact_143_add__diff__cancel__left, fact_220_add_Oinv__solve__left_H, fact_122_diff__Suc__Suc, fact_58_Pair__inject, fact_45_mem__Collect__eq, fact_40_semiring__simprules_I3_J, fact_203_group_Or__cancel__one_H, fact_231_minus__equality, fact_164_group__cancel_Oadd2, fact_175_diff__eq__eq, fact_88_cring__fieldI2, fact_79_semiring_Osemiring__simprules_I4_J, fact_41_minus__closed, fact_211_group_Or__inv__ex, fact_189_Units__inv__comm, fact_11_f4, fact_237_add_Oinv__eq__1__iff, fact_171_diff__right__commute, fact_247_abelian__group_Or__neg2, fact_62_semiring_Osemiring__simprules_I11_J, fact_78_r__one, fact_173_diff__eq__diff__eq, fact_181_diff__diff__add, fact_115_abelian__groupE_I5_J, fact_184_add__implies__diff, fact_265_group__cancel_Oneg1, fact_123_add__Suc__right, fact_145_add__right__cancel, fact_131_add__Suc__shift, fact_266_add_Oinverse__distrib__swap, fact_104_add_Opow__eq__div2, fact_53_add_Ol__cancel__one, fact_221_add_Oinv__solve__left, fact_227_add_Oint__pow__inv, fact_67_semiring_Ol__distr, fact_193_groupI, fact_270_of__nat__eq__iff, fact_120_diff__diff__left, fact_43_l__zero, fact_243_a__minus__def, fact_12__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_061_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060otimes_062_Ar_H_H_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_A_092_060otimes_062_Ar_H_H_J_092_060close_062, fact_34_add_Oinv__comm, fact_2_assms_I2_J, fact_240_units__of__mult, fact_260_diff__minus__eq__add, fact_108_add_Onat__pow__Suc, fact_1_assms_I3_J, fact_101_semiring_Oadd__pow__rdistr, fact_23_m__assoc, fact_216_r__neg1, fact_135_diff__cancel2, fact_74_semiring__simprules_I4_J, fact_105_minus__to__eq, fact_16_f12, fact_157_add_Ocommute, fact_235_add_Oinv__closed, fact_9_f10, fact_199_Group_Ogroup_Oright__cancel, fact_208_group_Or__cancel__one, fact_113_abelian__group_Ominus__closed, fact_15_f8, fact_3_assms_I1_J, fact_57_old_Oprod_Oexhaust, fact_217_local_Ominus__add, fact_196_group_Oinv__comm, fact_262_group__cancel_Osub2, fact_200_Group_Ogroup_Oright__cancel, fact_36_four__elem__comm, fact_280_comm__groupE_I2_J, fact_210_group_Ol__cancel__one, fact_236_local_Ominus__zero, fact_47_Collect__mem__eq, fact_179_diff__add__eq, fact_234_local_Ominus__minus, fact_228_minus__eq, fact_225_l__minus, fact_253_abelian__group_Oinv__add, fact_160_add_Oleft__cancel, fact_230_r__neg, fact_219_add_Oinv__solve__right, fact_87_semiring_Osemiring__simprules_I2_J, fact_55_l__null, fact_187_group__l__invI, fact_119_abelian__group_Ominus__to__eq, fact_109_abelian__groupE_I1_J, fact_107_is__abelian__group, fact_140_add__diff__cancel__right, fact_273_of__nat__add, fact_195_group_Oinv__comm, fact_60_surj__pair, fact_213_units__group, fact_93_add_Opow__mult__distrib, fact_137_add__diff__cancel__right_H, fact_206_group_Ol__cancel__one_H, fact_269_units__comm__group, fact_0_right__add__eq, fact_121_Suc__diff__diff, fact_129_add__Suc, fact_248_abelian__group_Or__neg1, fact_110_abelian__groupE_I3_J, fact_185_Units__l__inv__ex, fact_239_units__of__mult, fact_106_ring_Oequality, fact_14_f7, fact_271_of__nat__eq__iff, fact_244_units__of__one, fact_167_add__mono__thms__linordered__semiring_I4_J, fact_174_group__cancel_Osub1, fact_5__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_061_As_A_092_060otimes_062_A_Is_H_H_A_092_060otimes_062_As_H_J_A_092_060otimes_062_As_H_H_092_060close_062, fact_205_group_Ol__cancel__one_H, fact_61_f11, fact_80_semiring_Osemiring__simprules_I9_J, fact_209_group_Ol__cancel__one, fact_155_add_Oleft__commute, fact_158_add_Ocommute, fact_116_abelian__groupE_I6_J, fact_141_add__diff__cancel__left_H, fact_39_zero__closed, fact_254_abelian__group_Oa__transpose__inv, fact_71_one__unique, fact_170_ab__semigroup__add__class_Oadd__ac_I1_J, fact_76_add_Oint__pow__one, fact_26_add_Om__assoc, fact_98_one__not__zero, fact_29_local_Oright__minus__eq, fact_281_comm__groupE_I2_J, fact_90_add__pow__ldistr, fact_251_abelian__group_Or__neg, fact_183_add__implies__diff, fact_77_l__one, fact_99_integral, fact_142_add__diff__cancel__left_H, fact_276_comm__groupE_I3_J, fact_4_closed__rel__add, fact_162_add_Oassoc, fact_94_add_Onat__pow__distrib, fact_241_abelian__group_Oa__inv__closed, fact_52_add_Ol__cancel__one_H, fact_19__092_060open_062s_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_J_A_092_060otimes_062_A_I_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_A_092_060otimes_062_Ar_H_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_A_092_060otimes_062_Ar_H_H_J_A_092_060oplus_062_As_A_092_060otimes_062_As_H_H_A_092_060otimes_062_A_Ir_H_A_092_060otimes_062_Ar_H_H_J_J_A_061_A_092_060zero_062_092_060close_062, fact_102_add_Onat__pow__mult, fact_150_diff__add__cancel, fact_207_group_Or__cancel__one, fact_84_semiring_Osemiring__simprules_I5_J
% 12.12/3.60  % No. of inferences in proof: 568
% 12.12/3.60  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 3058 ms resp. 2416 ms w/o parsing
% 12.45/3.75  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 12.64/3.75  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------