TSTP Solution File: NUM952_5 by Leo-III---1.7.12

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%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : NUM952_5 : TPTP v8.2.0. Released v6.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n015.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 : Tue May 21 01:33:48 EDT 2024

% Result   : Theorem 6.02s 2.34s
% Output   : Refutation 6.30s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :  138
% Syntax   : Number of formulae    :  251 ( 132 unt;  27 typ;   0 def)
%            Number of atoms       :  379 ( 223 equ;   0 cnn)
%            Maximal formula atoms :   10 (   1 avg)
%            Number of connectives : 2571 (  28   ~;   5   |;  29   &;2388   @)
%                                         (  11 <=>; 110  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   4 avg)
%            Number of types       :    4 (   3 usr)
%            Number of type conns  :   18 (  18   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   26 (  24 usr;   5 con; 0-3 aty)
%            Number of variables   :  365 (   0   ^ 350   !;   0   ?; 365   :)
%                                         (  15  !>;   0  ?*;   0  @-;   0  @+)

% Comments : 
%------------------------------------------------------------------------------
thf(bool_type,type,
    bool: $tType ).

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

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

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

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

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

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

thf(ring_char_0_type,type,
    ring_char_0: 
      !>[TA: $tType] : $o ).

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

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

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

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

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

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

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

thf(zprime_type,type,
    zprime: int > $o ).

thf(bit0_type,type,
    bit0: int > int ).

thf(bit1_type,type,
    bit1: int > int ).

thf(min_type,type,
    min: int ).

thf(pls_type,type,
    pls: int ).

thf(number_number_of_type,type,
    number_number_of: 
      !>[TA: $tType] : ( int > TA ) ).

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

thf(legendre_type,type,
    legendre: int > int > int ).

thf(quadRes_type,type,
    quadRes: int > int > $o ).

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

thf(m_type,type,
    m: int ).

thf(s_type,type,
    s: int ).

thf(79,axiom,
    ! [A: int] :
      ( ( minus_minus @ int @ pls @ ( bit1 @ A ) )
      = ( bit1 @ ( minus_minus @ int @ min @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_diff__bin__simps_I4_J) ).

thf(340,plain,
    ! [A: int] :
      ( ( minus_minus @ int @ pls @ ( bit1 @ A ) )
      = ( bit1 @ ( minus_minus @ int @ min @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).

thf(73,axiom,
    ! [A: int,B: int] :
      ( ( bit0 @ B )
     != ( bit1 @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_rel__simps_I49_J) ).

thf(321,plain,
    ! [A: int,B: int] :
      ( ( bit0 @ B )
     != ( bit1 @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).

thf(85,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus @ int @ ( number_number_of @ int @ B ) @ ( number_number_of @ int @ A ) )
      = ( number_number_of @ int @ ( plus_plus @ int @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_plus__numeral__code_I9_J) ).

thf(361,plain,
    ! [A: int,B: int] :
      ( ( plus_plus @ int @ ( number_number_of @ int @ B ) @ ( number_number_of @ int @ A ) )
      = ( number_number_of @ int @ ( plus_plus @ int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).

thf(13,axiom,
    ( ( one_one @ nat )
    = ( number_number_of @ nat @ ( bit1 @ pls ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_Numeral1__eq1__nat) ).

thf(143,plain,
    ( ( one_one @ nat )
    = ( number_number_of @ nat @ ( bit1 @ pls ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(39,axiom,
    semiring @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Osemiring) ).

thf(218,plain,
    semiring @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).

thf(45,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int,B: int] :
          ( ( number_number_of @ TA @ ( plus_plus @ int @ B @ A ) )
          = ( plus_plus @ TA @ ( number_number_of @ TA @ B ) @ ( number_number_of @ TA @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_number__of__add) ).

thf(234,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int,B: int] :
          ( ( number_number_of @ TA @ ( plus_plus @ int @ B @ A ) )
          = ( plus_plus @ TA @ ( number_number_of @ TA @ B ) @ ( number_number_of @ TA @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).

thf(82,axiom,
    ! [TA: $tType] :
      ( ( ( number @ TA )
        & ( semiring @ TA ) )
     => ! [A: int,B: TA,C: TA] :
          ( ( times_times @ TA @ ( plus_plus @ TA @ C @ B ) @ ( number_number_of @ TA @ A ) )
          = ( plus_plus @ TA @ ( times_times @ TA @ C @ ( number_number_of @ TA @ A ) ) @ ( times_times @ TA @ B @ ( number_number_of @ TA @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_left__distrib__number__of) ).

thf(352,plain,
    ! [TA: $tType] :
      ( ( ( number @ TA )
        & ( semiring @ TA ) )
     => ! [A: int,B: TA,C: TA] :
          ( ( times_times @ TA @ ( plus_plus @ TA @ C @ B ) @ ( number_number_of @ TA @ A ) )
          = ( plus_plus @ TA @ ( times_times @ TA @ C @ ( number_number_of @ TA @ A ) ) @ ( times_times @ TA @ B @ ( number_number_of @ TA @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).

thf(21,axiom,
    ! [TA: $tType] :
      ( ( monoid_mult @ TA )
     => ! [A: nat,B: TA] :
          ( ( power_power @ TA @ B @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
          = ( power_power @ TA @ ( power_power @ TA @ B @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_power__even__eq) ).

thf(170,plain,
    ! [TA: $tType] :
      ( ( monoid_mult @ TA )
     => ! [A: nat,B: TA] :
          ( ( power_power @ TA @ B @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
          = ( power_power @ TA @ ( power_power @ TA @ B @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(48,axiom,
    ! [A: int,B: int] :
      ( ( bit1 @ B )
     != ( bit0 @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_rel__simps_I50_J) ).

thf(247,plain,
    ! [A: int,B: int] :
      ( ( bit1 @ B )
     != ( bit0 @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).

thf(32,axiom,
    ( ~ ( quadRes @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) @ ( number_number_of @ int @ min ) )
   => ( ( legendre @ ( number_number_of @ int @ min ) @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) )
     != ( one_one @ int ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_126_061_A1_096) ).

thf(202,plain,
    ( ~ ( quadRes @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) @ ( number_number_of @ int @ min ) )
   => ( ( legendre @ ( number_number_of @ int @ min ) @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) )
     != ( one_one @ int ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(26,axiom,
    ! [A: int,B: int,C: int,D: int,E: int] :
      ( ( dvd_dvd @ int @ E @ D )
     => ( ( dvd_dvd @ int @ E @ ( plus_plus @ int @ C @ B ) )
      <=> ( dvd_dvd @ int @ E @ ( plus_plus @ int @ ( plus_plus @ int @ C @ ( times_times @ int @ A @ D ) ) @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_zdvd__period) ).

thf(185,plain,
    ! [A: int,B: int,C: int,D: int,E: int] :
      ( ( dvd_dvd @ int @ E @ D )
     => ( ( ( dvd_dvd @ int @ E @ ( plus_plus @ int @ C @ B ) )
         => ( dvd_dvd @ int @ E @ ( plus_plus @ int @ ( plus_plus @ int @ C @ ( times_times @ int @ A @ D ) ) @ B ) ) )
        & ( ( dvd_dvd @ int @ E @ ( plus_plus @ int @ ( plus_plus @ int @ C @ ( times_times @ int @ A @ D ) ) @ B ) )
         => ( dvd_dvd @ int @ E @ ( plus_plus @ int @ C @ B ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(100,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: nat] :
          ( ( power_power @ TA @ ( number_number_of @ TA @ min ) @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
          = ( one_one @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_power__m1__even) ).

thf(417,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: nat] :
          ( ( power_power @ TA @ ( number_number_of @ TA @ min ) @ ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
          = ( one_one @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).

thf(107,axiom,
    ( ( minus_minus @ int @ ( power_power @ int @ s @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of @ int @ min ) )
    = ( plus_plus @ int @ ( power_power @ int @ s @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( one_one @ int ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0__096s_A_094_A2_A_N_A_N1_A_061_As_A_094_A2_A_L_A1_096) ).

thf(436,plain,
    ( ( minus_minus @ int @ ( power_power @ int @ s @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of @ int @ min ) )
    = ( plus_plus @ int @ ( power_power @ int @ s @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( one_one @ int ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).

thf(7,axiom,
    number @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Int_Onumber) ).

thf(125,plain,
    number @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(30,axiom,
    ! [A: int] :
      ( ( plus_plus @ int @ A @ pls )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_add__Pls__right) ).

thf(196,plain,
    ! [A: int] :
      ( ( plus_plus @ int @ A @ pls )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(102,axiom,
    ! [A: int] :
      ( ( number_number_of @ int @ A )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_number__of__is__id) ).

thf(421,plain,
    ! [A: int] :
      ( ( number_number_of @ int @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).

thf(74,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int,B: int] :
          ( ( number_number_of @ TA @ ( minus_minus @ int @ B @ A ) )
          = ( minus_minus @ TA @ ( number_number_of @ TA @ B ) @ ( number_number_of @ TA @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_number__of__diff) ).

thf(325,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int,B: int] :
          ( ( number_number_of @ TA @ ( minus_minus @ int @ B @ A ) )
          = ( minus_minus @ TA @ ( number_number_of @ TA @ B ) @ ( number_number_of @ TA @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).

thf(93,axiom,
    min != pls,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_rel__simps_I40_J) ).

thf(397,plain,
    min != pls,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).

thf(57,axiom,
    dvd_dvd @ int @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( power_power @ int @ s @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of @ int @ min ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_N_A_N1_096) ).

thf(280,plain,
    dvd_dvd @ int @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( power_power @ int @ s @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of @ int @ min ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).

thf(61,axiom,
    ! [A: int] :
      ( min
     != ( bit0 @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_rel__simps_I42_J) ).

thf(293,plain,
    ! [A: int] :
      ( min
     != ( bit0 @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).

thf(92,axiom,
    semiring @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Osemiring) ).

thf(396,plain,
    semiring @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).

thf(98,axiom,
    ! [TA: $tType] :
      ( ( number_semiring @ TA )
     => ! [A: TA,B: TA] :
          ( ( power_power @ TA @ ( plus_plus @ TA @ B @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
          = ( plus_plus @ TA @ ( plus_plus @ TA @ ( power_power @ TA @ B @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power @ TA @ A @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times @ TA @ ( times_times @ TA @ ( number_number_of @ TA @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_power2__sum) ).

thf(411,plain,
    ! [TA: $tType] :
      ( ( number_semiring @ TA )
     => ! [A: TA,B: TA] :
          ( ( power_power @ TA @ ( plus_plus @ TA @ B @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
          = ( plus_plus @ TA @ ( plus_plus @ TA @ ( power_power @ TA @ B @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power @ TA @ A @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times @ TA @ ( times_times @ TA @ ( number_number_of @ TA @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).

thf(4,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus @ int @ ( bit0 @ B ) @ ( bit1 @ A ) )
      = ( bit1 @ ( plus_plus @ int @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_add__Bit0__Bit1) ).

thf(116,plain,
    ! [A: int,B: int] :
      ( ( plus_plus @ int @ ( bit0 @ B ) @ ( bit1 @ A ) )
      = ( bit1 @ ( plus_plus @ int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(52,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ A @ ( number_number_of @ TA @ ( bit1 @ pls ) ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_mult__numeral__1__right) ).

thf(265,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ A @ ( number_number_of @ TA @ ( bit1 @ pls ) ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).

thf(70,axiom,
    ! [A: int] :
      ( ( minus_minus @ int @ min @ ( bit0 @ A ) )
      = ( bit1 @ ( minus_minus @ int @ min @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_diff__bin__simps_I5_J) ).

thf(312,plain,
    ! [A: int] :
      ( ( minus_minus @ int @ min @ ( bit0 @ A ) )
      = ( bit1 @ ( minus_minus @ int @ min @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).

thf(34,axiom,
    ! [TA: $tType] :
      ( ( ( number @ TA )
        & ( ring @ TA ) )
     => ! [A: TA,B: TA,C: int] :
          ( ( times_times @ TA @ ( number_number_of @ TA @ C ) @ ( minus_minus @ TA @ B @ A ) )
          = ( minus_minus @ TA @ ( times_times @ TA @ ( number_number_of @ TA @ C ) @ B ) @ ( times_times @ TA @ ( number_number_of @ TA @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_right__diff__distrib__number__of) ).

thf(206,plain,
    ! [TA: $tType] :
      ( ( ( number @ TA )
        & ( ring @ TA ) )
     => ! [A: TA,B: TA,C: int] :
          ( ( times_times @ TA @ ( number_number_of @ TA @ C ) @ ( minus_minus @ TA @ B @ A ) )
          = ( minus_minus @ TA @ ( times_times @ TA @ ( number_number_of @ TA @ C ) @ B ) @ ( times_times @ TA @ ( number_number_of @ TA @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(112,axiom,
    number_semiring @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Int_Onumber__semiring) ).

thf(452,plain,
    number_semiring @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).

thf(109,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times @ int @ ( plus_plus @ int @ C @ B ) @ A )
      = ( plus_plus @ int @ ( times_times @ int @ C @ A ) @ ( times_times @ int @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_int__distrib_I1_J) ).

thf(439,plain,
    ! [A: int,B: int,C: int] :
      ( ( times_times @ int @ ( plus_plus @ int @ C @ B ) @ A )
      = ( plus_plus @ int @ ( times_times @ int @ C @ A ) @ ( times_times @ int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).

thf(87,axiom,
    monoid_mult @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Groups_Omonoid__mult) ).

thf(366,plain,
    monoid_mult @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).

thf(77,axiom,
    ! [A: int] :
      ( ( minus_minus @ int @ pls @ ( bit0 @ A ) )
      = ( bit0 @ ( minus_minus @ int @ pls @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_diff__bin__simps_I3_J) ).

thf(334,plain,
    ! [A: int] :
      ( ( minus_minus @ int @ pls @ ( bit0 @ A ) )
      = ( bit0 @ ( minus_minus @ int @ pls @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).

thf(65,axiom,
    number_semiring @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Int_Onumber__semiring) ).

thf(302,plain,
    number_semiring @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).

thf(62,axiom,
    ring @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Oring) ).

thf(297,plain,
    ring @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).

thf(103,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ ( number_number_of @ TA @ ( bit1 @ pls ) ) @ A )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_mult__numeral__1) ).

thf(424,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA] :
          ( ( times_times @ TA @ ( number_number_of @ TA @ ( bit1 @ pls ) ) @ A )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).

thf(29,axiom,
    ! [A: int] :
      ( pls
     != ( bit1 @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_rel__simps_I39_J) ).

thf(192,plain,
    ! [A: int] :
      ( pls
     != ( bit1 @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(95,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus @ int @ ( bit0 @ B ) @ ( bit0 @ A ) )
      = ( bit0 @ ( minus_minus @ int @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_diff__bin__simps_I7_J) ).

thf(403,plain,
    ! [A: int,B: int] :
      ( ( minus_minus @ int @ ( bit0 @ B ) @ ( bit0 @ A ) )
      = ( bit0 @ ( minus_minus @ int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).

thf(27,axiom,
    ( ( number_number_of @ nat @ ( bit1 @ pls ) )
    = ( one_one @ nat ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_nat__numeral__1__eq__1) ).

thf(188,plain,
    ( ( number_number_of @ nat @ ( bit1 @ pls ) )
    = ( one_one @ nat ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(20,axiom,
    zprime @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_zprime__2) ).

thf(169,plain,
    zprime @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(101,axiom,
    zprime @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_p) ).

thf(420,plain,
    zprime @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).

thf(75,axiom,
    ! [A: nat] :
      ( ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ A )
      = ( plus_plus @ nat @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_nat__mult__2) ).

thf(328,plain,
    ! [A: nat] :
      ( ( times_times @ nat @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ A )
      = ( plus_plus @ nat @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).

thf(42,axiom,
    ! [A: nat,B: int,C: int] :
      ( ( zprime @ C )
     => ( ( dvd_dvd @ int @ C @ ( power_power @ int @ B @ A ) )
       => ( dvd_dvd @ int @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_zprime__zdvd__power) ).

thf(225,plain,
    ! [A: nat,B: int,C: int] :
      ( ( zprime @ C )
     => ( ( dvd_dvd @ int @ C @ ( power_power @ int @ B @ A ) )
       => ( dvd_dvd @ int @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).

thf(36,axiom,
    ! [A: int,B: nat,C: int,D: int] :
      ( ( zprime @ D )
     => ( ~ ( dvd_dvd @ int @ D @ C )
       => ( ( dvd_dvd @ int @ ( power_power @ int @ D @ B ) @ ( times_times @ int @ A @ C ) )
         => ( dvd_dvd @ int @ ( power_power @ int @ D @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_zprime__power__zdvd__cancel__right) ).

thf(212,plain,
    ! [A: int,B: nat,C: int,D: int] :
      ( ( zprime @ D )
     => ( ~ ( dvd_dvd @ int @ D @ C )
       => ( ( dvd_dvd @ int @ ( power_power @ int @ D @ B ) @ ( times_times @ int @ A @ C ) )
         => ( dvd_dvd @ int @ ( power_power @ int @ D @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).

thf(91,axiom,
    ! [A: int,B: int] :
      ( ( times_times @ int @ ( bit0 @ B ) @ A )
      = ( bit0 @ ( times_times @ int @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_mult__Bit0) ).

thf(393,plain,
    ! [A: int,B: int] :
      ( ( times_times @ int @ ( bit0 @ B ) @ A )
      = ( bit0 @ ( times_times @ int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).

thf(22,axiom,
    pls != min,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_rel__simps_I37_J) ).

thf(173,plain,
    pls != min,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(9,axiom,
    ! [A: int] :
      ( ( ( bit0 @ A )
        = pls )
    <=> ( A = pls ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_rel__simps_I44_J) ).

thf(129,plain,
    ! [A: int] :
      ( ( ( ( bit0 @ A )
          = pls )
       => ( A = pls ) )
      & ( ( A = pls )
       => ( ( bit0 @ A )
          = pls ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(47,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int] :
          ( ( plus_plus @ TA @ ( one_one @ TA ) @ ( number_number_of @ TA @ A ) )
          = ( number_number_of @ TA @ ( plus_plus @ int @ ( bit1 @ pls ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_add__special_I2_J) ).

thf(244,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int] :
          ( ( plus_plus @ TA @ ( one_one @ TA ) @ ( number_number_of @ TA @ A ) )
          = ( number_number_of @ TA @ ( plus_plus @ int @ ( bit1 @ pls ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).

thf(33,axiom,
    number @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Int_Onumber) ).

thf(205,plain,
    number @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(80,axiom,
    ! [A: int] :
      ( ( ( bit1 @ A )
        = min )
    <=> ( A = min ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_rel__simps_I47_J) ).

thf(343,plain,
    ! [A: int] :
      ( ( ( ( bit1 @ A )
          = min )
       => ( A = min ) )
      & ( ( A = min )
       => ( ( bit1 @ A )
          = min ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).

thf(46,axiom,
    ! [A: int] :
      ( ( min
        = ( bit1 @ A ) )
    <=> ( min = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_rel__simps_I43_J) ).

thf(237,plain,
    ! [A: int] :
      ( ( ( min
          = ( bit1 @ A ) )
       => ( min = A ) )
      & ( ( min = A )
       => ( min
          = ( bit1 @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).

thf(63,axiom,
    semiring_1 @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Rings_Osemiring__1) ).

thf(298,plain,
    semiring_1 @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).

thf(105,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA,B: int,C: int] :
          ( ( plus_plus @ TA @ ( number_number_of @ TA @ C ) @ ( minus_minus @ TA @ ( number_number_of @ TA @ B ) @ A ) )
          = ( minus_minus @ TA @ ( number_number_of @ TA @ ( plus_plus @ int @ C @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_add__number__of__diff1) ).

thf(430,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA,B: int,C: int] :
          ( ( plus_plus @ TA @ ( number_number_of @ TA @ C ) @ ( minus_minus @ TA @ ( number_number_of @ TA @ B ) @ A ) )
          = ( minus_minus @ TA @ ( number_number_of @ TA @ ( plus_plus @ int @ C @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).

thf(19,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int,B: int] :
          ( ( plus_plus @ TA @ ( number_number_of @ TA @ B ) @ ( number_number_of @ TA @ A ) )
          = ( number_number_of @ TA @ ( plus_plus @ int @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_add__number__of__eq) ).

thf(166,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int,B: int] :
          ( ( plus_plus @ TA @ ( number_number_of @ TA @ B ) @ ( number_number_of @ TA @ A ) )
          = ( number_number_of @ TA @ ( plus_plus @ int @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(15,axiom,
    ! [A: int,B: int] :
      ( ( power_power @ int @ ( plus_plus @ int @ B @ A ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
      = ( plus_plus @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( power_power @ int @ B @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( times_times @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) @ ( power_power @ int @ B @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ A ) ) @ ( times_times @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) @ B ) @ ( power_power @ int @ A @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( power_power @ int @ A @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_zadd__power3) ).

thf(148,plain,
    ! [A: int,B: int] :
      ( ( power_power @ int @ ( plus_plus @ int @ B @ A ) @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
      = ( plus_plus @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( power_power @ int @ B @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( times_times @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) @ ( power_power @ int @ B @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ A ) ) @ ( times_times @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit1 @ ( bit1 @ pls ) ) ) @ B ) @ ( power_power @ int @ A @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( power_power @ int @ A @ ( number_number_of @ nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(23,axiom,
    semiring_1 @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Osemiring__1) ).

thf(176,plain,
    semiring_1 @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(58,axiom,
    ! [TA: $tType] :
      ( ( ( number_ring @ TA )
        & ( ring_char_0 @ TA ) )
     => ! [A: int,B: int] :
          ( ( ( number_number_of @ TA @ B )
            = ( number_number_of @ TA @ A ) )
        <=> ( B = A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_eq__number__of) ).

thf(281,plain,
    ! [TA: $tType] :
      ( ( ( number_ring @ TA )
        & ( ring_char_0 @ TA ) )
     => ! [A: int,B: int] :
          ( ( ( ( number_number_of @ TA @ B )
              = ( number_number_of @ TA @ A ) )
           => ( B = A ) )
          & ( ( B = A )
           => ( ( number_number_of @ TA @ B )
              = ( number_number_of @ TA @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).

thf(50,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ A @ ( number_number_of @ TA @ pls ) )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_add__numeral__0__right) ).

thf(259,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ A @ ( number_number_of @ TA @ pls ) )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).

thf(111,axiom,
    ! [A: int] :
      ( ( pls
        = ( bit0 @ A ) )
    <=> ( pls = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_rel__simps_I38_J) ).

thf(445,plain,
    ! [A: int] :
      ( ( ( pls
          = ( bit0 @ A ) )
       => ( pls = A ) )
      & ( ( pls = A )
       => ( pls
          = ( bit0 @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).

thf(81,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd @ int @ C @ ( minus_minus @ int @ B @ A ) )
     => ( ( dvd_dvd @ int @ C @ A )
       => ( dvd_dvd @ int @ C @ B ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_zdvd__zdiffD) ).

thf(350,plain,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd @ int @ C @ ( minus_minus @ int @ B @ A ) )
     => ( ( dvd_dvd @ int @ C @ A )
       => ( dvd_dvd @ int @ C @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).

thf(96,axiom,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ( ( power_power @ TA @ ( one_one @ TA ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
        = ( one_one @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_one__power2) ).

thf(406,plain,
    ! [TA: $tType] :
      ( ( semiring_1 @ TA )
     => ( ( power_power @ TA @ ( one_one @ TA ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
        = ( one_one @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).

thf(1,conjecture,
    dvd_dvd @ int @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) @ ( plus_plus @ int @ ( power_power @ int @ s @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( one_one @ int ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).

thf(2,negated_conjecture,
    ~ ( dvd_dvd @ int @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) @ ( plus_plus @ int @ ( power_power @ int @ s @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( one_one @ int ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(113,plain,
    ~ ( dvd_dvd @ int @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) @ ( plus_plus @ int @ ( power_power @ int @ s @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( one_one @ int ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(40,axiom,
    ! [A: nat] :
      ( ( times_times @ nat @ A @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
      = ( plus_plus @ nat @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_nat__mult__2__right) ).

thf(219,plain,
    ! [A: nat] :
      ( ( times_times @ nat @ A @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
      = ( plus_plus @ nat @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).

thf(55,axiom,
    ! [TA: $tType] :
      ( ( ( monoid_mult @ TA )
        & ( number @ TA ) )
     => ! [A: int] :
          ( ( power_power @ TA @ ( number_number_of @ TA @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
          = ( times_times @ TA @ ( number_number_of @ TA @ A ) @ ( number_number_of @ TA @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_power2__eq__square__number__of) ).

thf(274,plain,
    ! [TA: $tType] :
      ( ( ( monoid_mult @ TA )
        & ( number @ TA ) )
     => ! [A: int] :
          ( ( power_power @ TA @ ( number_number_of @ TA @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
          = ( times_times @ TA @ ( number_number_of @ TA @ A ) @ ( number_number_of @ TA @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).

thf(44,axiom,
    ! [A: int] :
      ( ( bit1 @ A )
     != pls ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_rel__simps_I46_J) ).

thf(230,plain,
    ! [A: int] :
      ( ( bit1 @ A )
     != pls ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).

thf(72,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times @ int @ C @ ( minus_minus @ int @ B @ A ) )
      = ( minus_minus @ int @ ( times_times @ int @ C @ B ) @ ( times_times @ int @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_int__distrib_I4_J) ).

thf(318,plain,
    ! [A: int,B: int,C: int] :
      ( ( times_times @ int @ C @ ( minus_minus @ int @ B @ A ) )
      = ( minus_minus @ int @ ( times_times @ int @ C @ B ) @ ( times_times @ int @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).

thf(66,axiom,
    quadRes @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) @ ( number_number_of @ int @ min ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096) ).

thf(303,plain,
    quadRes @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) @ ( number_number_of @ int @ min ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).

thf(54,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int] :
          ( ( number_number_of @ TA @ ( bit1 @ A ) )
          = ( plus_plus @ TA @ ( plus_plus @ TA @ ( one_one @ TA ) @ ( number_number_of @ TA @ A ) ) @ ( number_number_of @ TA @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_number__of__Bit1) ).

thf(271,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int] :
          ( ( number_number_of @ TA @ ( bit1 @ A ) )
          = ( plus_plus @ TA @ ( plus_plus @ TA @ ( one_one @ TA ) @ ( number_number_of @ TA @ A ) ) @ ( number_number_of @ TA @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).

thf(88,axiom,
    ! [TA: $tType] :
      ( ( number @ TA )
     => ! [A: TA,B: int] :
          ( ( ( number_number_of @ TA @ B )
            = A )
        <=> ( A
            = ( number_number_of @ TA @ B ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_number__of__reorient) ).

thf(367,plain,
    ! [TA: $tType] :
      ( ( number @ TA )
     => ! [A: TA,B: int] :
          ( ( ( ( number_number_of @ TA @ B )
              = A )
           => ( A
              = ( number_number_of @ TA @ B ) ) )
          & ( ( A
              = ( number_number_of @ TA @ B ) )
           => ( ( number_number_of @ TA @ B )
              = A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).

thf(12,axiom,
    ! [A: int] :
      ( ( zprime @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) @ ( one_one @ int ) ) )
     => ( ( legendre @ ( number_number_of @ int @ min ) @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) @ ( one_one @ int ) ) )
        = ( one_one @ int ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_Legendre__1mod4) ).

thf(140,plain,
    ! [A: int] :
      ( ( zprime @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) @ ( one_one @ int ) ) )
     => ( ( legendre @ ( number_number_of @ int @ min ) @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A ) @ ( one_one @ int ) ) )
        = ( one_one @ int ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(49,axiom,
    ! [A: int,B: int] :
      ( ( ( bit0 @ B )
        = ( bit0 @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_rel__simps_I48_J) ).

thf(251,plain,
    ! [A: int,B: int] :
      ( ( ( ( bit0 @ B )
          = ( bit0 @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( bit0 @ B )
          = ( bit0 @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).

thf(17,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times @ int @ B @ A )
        = ( one_one @ int ) )
     => ( ( B
          = ( one_one @ int ) )
        | ( B
          = ( number_number_of @ int @ min ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_pos__zmult__eq__1__iff__lemma) ).

thf(155,plain,
    ! [A: int,B: int] :
      ( ( ( times_times @ int @ B @ A )
        = ( one_one @ int ) )
     => ( ( B
          = ( one_one @ int ) )
        | ( B
          = ( number_number_of @ int @ min ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(68,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus @ int @ ( bit1 @ B ) @ ( bit0 @ A ) )
      = ( bit1 @ ( plus_plus @ int @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_add__Bit1__Bit0) ).

thf(306,plain,
    ! [A: int,B: int] :
      ( ( plus_plus @ int @ ( bit1 @ B ) @ ( bit0 @ A ) )
      = ( bit1 @ ( plus_plus @ int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).

thf(64,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ( ( one_one @ TA )
        = ( number_number_of @ TA @ ( bit1 @ pls ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_semiring__norm_I110_J) ).

thf(299,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ( ( one_one @ TA )
        = ( number_number_of @ TA @ ( bit1 @ pls ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).

thf(108,axiom,
    ring_char_0 @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Int_Oring__char__0) ).

thf(438,plain,
    ring_char_0 @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).

thf(89,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times @ int @ B @ A )
        = ( one_one @ int ) )
    <=> ( ( ( B
            = ( one_one @ int ) )
          & ( A
            = ( one_one @ int ) ) )
        | ( ( B
            = ( number_number_of @ int @ min ) )
          & ( A
            = ( number_number_of @ int @ min ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_zmult__eq__1__iff) ).

thf(374,plain,
    ! [A: int,B: int] :
      ( ( ( ( times_times @ int @ B @ A )
          = ( one_one @ int ) )
       => ( ( ( B
              = ( one_one @ int ) )
            & ( A
              = ( one_one @ int ) ) )
          | ( ( B
              = ( number_number_of @ int @ min ) )
            & ( A
              = ( number_number_of @ int @ min ) ) ) ) )
      & ( ( ( ( B
              = ( one_one @ int ) )
            & ( A
              = ( one_one @ int ) ) )
          | ( ( B
              = ( number_number_of @ int @ min ) )
            & ( A
              = ( number_number_of @ int @ min ) ) ) )
       => ( ( times_times @ int @ B @ A )
          = ( one_one @ int ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).

thf(35,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( number_number_of @ TA @ pls ) @ A )
          = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_add__numeral__0) ).

thf(209,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA] :
          ( ( plus_plus @ TA @ ( number_number_of @ TA @ pls ) @ A )
          = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).

thf(83,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ( ( number_number_of @ TA @ ( bit1 @ pls ) )
        = ( one_one @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_numeral__1__eq__1) ).

thf(355,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ( ( number_number_of @ TA @ ( bit1 @ pls ) )
        = ( one_one @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).

thf(14,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int,B: int] :
          ( ( times_times @ TA @ ( number_number_of @ TA @ B ) @ ( number_number_of @ TA @ A ) )
          = ( number_number_of @ TA @ ( times_times @ int @ B @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_arith__simps_I32_J) ).

thf(145,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int,B: int] :
          ( ( times_times @ TA @ ( number_number_of @ TA @ B ) @ ( number_number_of @ TA @ A ) )
          = ( number_number_of @ TA @ ( times_times @ int @ B @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(99,axiom,
    ! [A: int] :
      ( ( minus_minus @ int @ min @ ( bit1 @ A ) )
      = ( bit0 @ ( minus_minus @ int @ min @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_diff__bin__simps_I6_J) ).

thf(414,plain,
    ! [A: int] :
      ( ( minus_minus @ int @ min @ ( bit1 @ A ) )
      = ( bit0 @ ( minus_minus @ int @ min @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).

thf(97,axiom,
    ( ( legendre @ ( number_number_of @ int @ min ) @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) )
    = ( one_one @ int ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096) ).

thf(409,plain,
    ( ( legendre @ ( number_number_of @ int @ min ) @ ( plus_plus @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ ( one_one @ int ) ) )
    = ( one_one @ int ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).

thf(31,axiom,
    ! [A: int] :
      ( ( bit0 @ A )
      = ( plus_plus @ int @ A @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_Bit0__def) ).

thf(199,plain,
    ! [A: int] :
      ( ( bit0 @ A )
      = ( plus_plus @ int @ A @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(76,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA,B: int,C: int] :
          ( ( plus_plus @ TA @ ( number_number_of @ TA @ C ) @ ( plus_plus @ TA @ ( number_number_of @ TA @ B ) @ A ) )
          = ( plus_plus @ TA @ ( number_number_of @ TA @ ( plus_plus @ int @ C @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_add__number__of__left) ).

thf(331,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA,B: int,C: int] :
          ( ( plus_plus @ TA @ ( number_number_of @ TA @ C ) @ ( plus_plus @ TA @ ( number_number_of @ TA @ B ) @ A ) )
          = ( plus_plus @ TA @ ( number_number_of @ TA @ ( plus_plus @ int @ C @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).

thf(24,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd @ int @ C @ ( plus_plus @ int @ B @ ( times_times @ int @ C @ A ) ) )
    <=> ( dvd_dvd @ int @ C @ B ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_zdvd__reduce) ).

thf(177,plain,
    ! [A: int,B: int,C: int] :
      ( ( ( dvd_dvd @ int @ C @ ( plus_plus @ int @ B @ ( times_times @ int @ C @ A ) ) )
       => ( dvd_dvd @ int @ C @ B ) )
      & ( ( dvd_dvd @ int @ C @ B )
       => ( dvd_dvd @ int @ C @ ( plus_plus @ int @ B @ ( times_times @ int @ C @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(25,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus @ int @ ( bit0 @ B ) @ ( bit0 @ A ) )
      = ( bit0 @ ( plus_plus @ int @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_add__Bit0__Bit0) ).

thf(182,plain,
    ! [A: int,B: int] :
      ( ( plus_plus @ int @ ( bit0 @ B ) @ ( bit0 @ A ) )
      = ( bit0 @ ( plus_plus @ int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(67,axiom,
    ( ( bit0 @ pls )
    = pls ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_Bit0__Pls) ).

thf(304,plain,
    ( ( bit0 @ pls )
    = pls ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).

thf(18,axiom,
    ! [A: int,B: int] :
      ( ( ( bit1 @ B )
        = ( bit1 @ A ) )
    <=> ( B = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_rel__simps_I51_J) ).

thf(158,plain,
    ! [A: int,B: int] :
      ( ( ( ( bit1 @ B )
          = ( bit1 @ A ) )
       => ( B = A ) )
      & ( ( B = A )
       => ( ( bit1 @ B )
          = ( bit1 @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(90,axiom,
    ! [A: int] :
      ( ( minus_minus @ int @ A @ pls )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_diff__bin__simps_I1_J) ).

thf(390,plain,
    ! [A: int] :
      ( ( minus_minus @ int @ A @ pls )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).

thf(3,axiom,
    ( ( bit1 @ min )
    = min ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_Bit1__Min) ).

thf(114,plain,
    ( ( bit1 @ min )
    = min ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(11,axiom,
    number_ring @ int,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Int_Oint___Int_Onumber__ring) ).

thf(139,plain,
    number_ring @ int,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(106,axiom,
    ! [A: int,B: int] :
      ( ( times_times @ int @ ( bit1 @ B ) @ A )
      = ( plus_plus @ int @ ( bit0 @ ( times_times @ int @ B @ A ) ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_mult__Bit1) ).

thf(433,plain,
    ! [A: int,B: int] :
      ( ( times_times @ int @ ( bit1 @ B ) @ A )
      = ( plus_plus @ int @ ( bit0 @ ( times_times @ int @ B @ A ) ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).

thf(60,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA,B: int,C: int] :
          ( ( times_times @ TA @ ( number_number_of @ TA @ C ) @ ( times_times @ TA @ ( number_number_of @ TA @ B ) @ A ) )
          = ( times_times @ TA @ ( number_number_of @ TA @ ( times_times @ int @ C @ B ) ) @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_mult__number__of__left) ).

thf(290,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: TA,B: int,C: int] :
          ( ( times_times @ TA @ ( number_number_of @ TA @ C ) @ ( times_times @ TA @ ( number_number_of @ TA @ B ) @ A ) )
          = ( times_times @ TA @ ( number_number_of @ TA @ ( times_times @ int @ C @ B ) ) @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).

thf(5,axiom,
    ! [TA: $tType] :
      ( ( number_semiring @ TA )
     => ( ( number_number_of @ TA @ ( bit1 @ pls ) )
        = ( one_one @ TA ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_semiring__numeral__1__eq__1) ).

thf(119,plain,
    ! [TA: $tType] :
      ( ( number_semiring @ TA )
     => ( ( number_number_of @ TA @ ( bit1 @ pls ) )
        = ( one_one @ TA ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(37,axiom,
    monoid_mult @ nat,
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Omonoid__mult) ).

thf(214,plain,
    monoid_mult @ nat,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).

thf(59,axiom,
    ! [A: int,B: int] :
      ( ( times_times @ int @ ( number_number_of @ int @ B ) @ ( number_number_of @ int @ A ) )
      = ( number_number_of @ int @ ( times_times @ int @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_times__numeral__code_I5_J) ).

thf(287,plain,
    ! [A: int,B: int] :
      ( ( times_times @ int @ ( number_number_of @ int @ B ) @ ( number_number_of @ int @ A ) )
      = ( number_number_of @ int @ ( times_times @ int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).

thf(16,axiom,
    ! [A: int] :
      ( ( bit0 @ A )
     != min ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_rel__simps_I45_J) ).

thf(151,plain,
    ! [A: int] :
      ( ( bit0 @ A )
     != min ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(71,axiom,
    ! [A: int,B: int] :
      ( ( power_power @ int @ ( plus_plus @ int @ B @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
      = ( plus_plus @ int @ ( plus_plus @ int @ ( power_power @ int @ B @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( times_times @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) @ A ) ) @ ( power_power @ int @ A @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_zadd__power2) ).

thf(315,plain,
    ! [A: int,B: int] :
      ( ( power_power @ int @ ( plus_plus @ int @ B @ A ) @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
      = ( plus_plus @ int @ ( plus_plus @ int @ ( power_power @ int @ B @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( times_times @ int @ ( times_times @ int @ ( number_number_of @ int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) @ A ) ) @ ( power_power @ int @ A @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).

thf(51,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus @ int @ ( bit1 @ B ) @ ( bit0 @ A ) )
      = ( bit1 @ ( minus_minus @ int @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_48_diff__bin__simps_I9_J) ).

thf(262,plain,
    ! [A: int,B: int] :
      ( ( minus_minus @ int @ ( bit1 @ B ) @ ( bit0 @ A ) )
      = ( bit1 @ ( minus_minus @ int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).

thf(28,axiom,
    ! [A: int,B: nat,C: int,D: int] :
      ( ( zprime @ D )
     => ( ~ ( dvd_dvd @ int @ D @ C )
       => ( ( dvd_dvd @ int @ ( power_power @ int @ D @ B ) @ ( times_times @ int @ C @ A ) )
         => ( dvd_dvd @ int @ ( power_power @ int @ D @ B ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_zprime__power__zdvd__cancel__left) ).

thf(190,plain,
    ! [A: int,B: nat,C: int,D: int] :
      ( ( zprime @ D )
     => ( ~ ( dvd_dvd @ int @ D @ C )
       => ( ( dvd_dvd @ int @ ( power_power @ int @ D @ B ) @ ( times_times @ int @ C @ A ) )
         => ( dvd_dvd @ int @ ( power_power @ int @ D @ B ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(53,axiom,
    ! [A: int] :
      ( ( times_times @ int @ pls @ A )
      = pls ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_mult__Pls) ).

thf(268,plain,
    ! [A: int] :
      ( ( times_times @ int @ pls @ A )
      = pls ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).

thf(84,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ( ( plus_plus @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) )
        = ( number_number_of @ TA @ ( bit0 @ ( bit1 @ pls ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_one__add__one__is__two) ).

thf(358,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ( ( plus_plus @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) )
        = ( number_number_of @ TA @ ( bit0 @ ( bit1 @ pls ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).

thf(94,axiom,
    ! [A: int,B: int] :
      ( ( times_times @ int @ ( plus_plus @ int @ B @ A ) @ ( minus_minus @ int @ B @ A ) )
      = ( minus_minus @ int @ ( power_power @ int @ B @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power @ int @ A @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_zspecial__product) ).

thf(400,plain,
    ! [A: int,B: int] :
      ( ( times_times @ int @ ( plus_plus @ int @ B @ A ) @ ( minus_minus @ int @ B @ A ) )
      = ( minus_minus @ int @ ( power_power @ int @ B @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power @ int @ A @ ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).

thf(8,axiom,
    ! [A: int] :
      ( ( bit1 @ A )
      = ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ A ) @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_Bit1__def) ).

thf(126,plain,
    ! [A: int] :
      ( ( bit1 @ A )
      = ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ A ) @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(10,axiom,
    ! [A: int] :
      ( ( plus_plus @ int @ pls @ A )
      = A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_add__Pls) ).

thf(136,plain,
    ! [A: int] :
      ( ( plus_plus @ int @ pls @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(104,axiom,
    ! [A: nat,B: nat,C: int] :
      ( ( power_power @ int @ ( power_power @ int @ C @ B ) @ A )
      = ( power_power @ int @ C @ ( times_times @ nat @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_zpower__zpower) ).

thf(427,plain,
    ! [A: nat,B: nat,C: int] :
      ( ( power_power @ int @ ( power_power @ int @ C @ B ) @ A )
      = ( power_power @ int @ C @ ( times_times @ nat @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).

thf(56,axiom,
    ! [TA: $tType] :
      ( ( ( number @ TA )
        & ( semiring @ TA ) )
     => ! [A: TA,B: TA,C: int] :
          ( ( times_times @ TA @ ( number_number_of @ TA @ C ) @ ( plus_plus @ TA @ B @ A ) )
          = ( plus_plus @ TA @ ( times_times @ TA @ ( number_number_of @ TA @ C ) @ B ) @ ( times_times @ TA @ ( number_number_of @ TA @ C ) @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_right__distrib__number__of) ).

thf(277,plain,
    ! [TA: $tType] :
      ( ( ( number @ TA )
        & ( semiring @ TA ) )
     => ! [A: TA,B: TA,C: int] :
          ( ( times_times @ TA @ ( number_number_of @ TA @ C ) @ ( plus_plus @ TA @ B @ A ) )
          = ( plus_plus @ TA @ ( times_times @ TA @ ( number_number_of @ TA @ C ) @ B ) @ ( times_times @ TA @ ( number_number_of @ TA @ C ) @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).

thf(110,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times @ int @ C @ ( plus_plus @ int @ B @ A ) )
      = ( plus_plus @ int @ ( times_times @ int @ C @ B ) @ ( times_times @ int @ C @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_int__distrib_I2_J) ).

thf(442,plain,
    ! [A: int,B: int,C: int] :
      ( ( times_times @ int @ C @ ( plus_plus @ int @ B @ A ) )
      = ( plus_plus @ int @ ( times_times @ int @ C @ B ) @ ( times_times @ int @ C @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).

thf(43,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times @ int @ ( minus_minus @ int @ C @ B ) @ A )
      = ( minus_minus @ int @ ( times_times @ int @ C @ A ) @ ( times_times @ int @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_int__distrib_I3_J) ).

thf(227,plain,
    ! [A: int,B: int,C: int] :
      ( ( times_times @ int @ ( minus_minus @ int @ C @ B ) @ A )
      = ( minus_minus @ int @ ( times_times @ int @ C @ A ) @ ( times_times @ int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).

thf(69,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int] :
          ( ( plus_plus @ TA @ ( number_number_of @ TA @ A ) @ ( one_one @ TA ) )
          = ( number_number_of @ TA @ ( plus_plus @ int @ A @ ( bit1 @ pls ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_add__special_I3_J) ).

thf(309,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int] :
          ( ( plus_plus @ TA @ ( number_number_of @ TA @ A ) @ ( one_one @ TA ) )
          = ( number_number_of @ TA @ ( plus_plus @ int @ A @ ( bit1 @ pls ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).

thf(38,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus @ int @ ( bit1 @ B ) @ ( bit1 @ A ) )
      = ( bit0 @ ( minus_minus @ int @ B @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_diff__bin__simps_I10_J) ).

thf(215,plain,
    ! [A: int,B: int] :
      ( ( minus_minus @ int @ ( bit1 @ B ) @ ( bit1 @ A ) )
      = ( bit0 @ ( minus_minus @ int @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).

thf(41,axiom,
    ( ( number_number_of @ int @ pls )
   != ( number_number_of @ int @ min ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_eq__number__of__Pls__Min) ).

thf(222,plain,
    ( ( number_number_of @ int @ pls )
   != ( number_number_of @ int @ min ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).

thf(86,axiom,
    ( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
    = ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_nat__1__add__1) ).

thf(364,plain,
    ( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
    = ( number_number_of @ nat @ ( bit0 @ ( bit1 @ pls ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).

thf(78,axiom,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int,B: int] :
          ( ( number_number_of @ TA @ ( times_times @ int @ B @ A ) )
          = ( times_times @ TA @ ( number_number_of @ TA @ B ) @ ( number_number_of @ TA @ A ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_number__of__mult) ).

thf(337,plain,
    ! [TA: $tType] :
      ( ( number_ring @ TA )
     => ! [A: int,B: int] :
          ( ( number_number_of @ TA @ ( times_times @ int @ B @ A ) )
          = ( times_times @ TA @ ( number_number_of @ TA @ B ) @ ( number_number_of @ TA @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).

thf(6,axiom,
    ! [TA: $tType] :
      ( ( ( number @ TA )
        & ( ring @ TA ) )
     => ! [A: int,B: TA,C: TA] :
          ( ( times_times @ TA @ ( minus_minus @ TA @ C @ B ) @ ( number_number_of @ TA @ A ) )
          = ( minus_minus @ TA @ ( times_times @ TA @ C @ ( number_number_of @ TA @ A ) ) @ ( times_times @ TA @ B @ ( number_number_of @ TA @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_left__diff__distrib__number__of) ).

thf(122,plain,
    ! [TA: $tType] :
      ( ( ( number @ TA )
        & ( ring @ TA ) )
     => ! [A: int,B: TA,C: TA] :
          ( ( times_times @ TA @ ( minus_minus @ TA @ C @ B ) @ ( number_number_of @ TA @ A ) )
          = ( minus_minus @ TA @ ( times_times @ TA @ C @ ( number_number_of @ TA @ A ) ) @ ( times_times @ TA @ B @ ( number_number_of @ TA @ A ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(453,plain,
    $false,
    inference(e,[status(thm)],[340,321,361,143,218,234,352,170,247,202,185,417,436,125,196,421,325,397,280,293,396,411,116,265,312,206,452,439,366,334,302,297,424,192,403,188,169,420,328,225,212,393,173,129,244,205,343,237,298,430,166,148,176,281,259,445,350,406,113,219,274,230,318,303,271,367,140,251,155,306,299,438,374,209,355,145,414,409,199,331,177,182,304,158,390,114,139,433,290,119,214,287,151,315,262,190,268,358,400,126,136,427,277,442,227,309,215,222,364,337,122]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12  % Problem  : NUM952_5 : TPTP v8.2.0. Released v6.0.0.
% 0.07/0.15  % Command  : run_Leo-III %s %d
% 0.16/0.36  % Computer : n015.cluster.edu
% 0.16/0.36  % Model    : x86_64 x86_64
% 0.16/0.36  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.36  % Memory   : 8042.1875MB
% 0.16/0.36  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.16/0.36  % CPULimit : 300
% 0.16/0.36  % WCLimit  : 300
% 0.16/0.36  % DateTime : Mon May 20 03:46:54 EDT 2024
% 0.16/0.36  % CPUTime  : 
% 0.94/0.85  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 1.49/1.00  % [INFO] 	 Parsing done (150ms). 
% 1.49/1.01  % [INFO] 	 Running in sequential loop mode. 
% 1.83/1.21  % [INFO] 	 eprover registered as external prover. 
% 1.83/1.21  % [INFO] 	 cvc4 registered as external prover. 
% 1.83/1.22  % [INFO] 	 Scanning for conjecture ... 
% 2.13/1.31  % [INFO] 	 Found a conjecture (or negated_conjecture) and 112 axioms. Running axiom selection ... 
% 2.43/1.38  % [INFO] 	 Axiom selection finished. Selected 110 axioms (removed 2 axioms). 
% 2.73/1.49  % [INFO] 	 Problem is typed first-order (TPTP TFF). 
% 2.73/1.50  % [INFO] 	 Type checking passed. 
% 2.73/1.50  % [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 ... 
% 6.01/2.33  % External prover 'e' found a proof!
% 6.01/2.33  % [INFO] 	 Killing All external provers ... 
% 6.02/2.34  % Time passed: 1815ms (effective reasoning time: 1325ms)
% 6.02/2.34  % 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)>
% 6.02/2.34  % Axioms used in derivation (110): arity_Nat_Onat___Rings_Osemiring, fact_36_left__distrib__number__of, fact_54_diff__bin__simps_I5_J, fact_89_zdvd__period, fact_31_Bit1__Min, arity_Nat_Onat___Groups_Omonoid__mult, fact_41_arith__simps_I32_J, fact_91_zspecial__product, fact_25_mult__Pls, fact_76_zmult__eq__1__iff, fact_1__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_N_A_N1_096, fact_45_add__Bit1__Bit0, fact_69_number__of__is__id, arity_Nat_Onat___Rings_Osemiring__1, fact_48_diff__bin__simps_I9_J, fact_32_rel__simps_I37_J, arity_Int_Oint___Int_Oring__char__0, fact_72_Legendre__1mod4, fact_4_power2__eq__square__number__of, fact_80_times__numeral__code_I5_J, fact_0__096s_A_094_A2_A_N_A_N1_A_061_As_A_094_A2_A_L_A1_096, fact_11__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096, fact_9_mult__Bit1, fact_58_nat__mult__2, fact_87_number__of__add, fact_2_p, arity_Nat_Onat___Int_Onumber__semiring, fact_42_add__number__of__left, fact_12_zadd__power2, fact_7_add__special_I3_J, fact_75_pos__zmult__eq__1__iff__lemma, fact_47_diff__bin__simps_I10_J, fact_71_power__even__eq, fact_61_int__distrib_I3_J, arity_Int_Oint___Rings_Oring, fact_39_right__diff__distrib__number__of, fact_86_number__of__mult, fact_67_zprime__2, arity_Int_Oint___Int_Onumber__semiring, fact_26_mult__Bit0, arity_Int_Oint___Int_Onumber__ring, fact_43_add__number__of__eq, fact_65_zprime__power__zdvd__cancel__right, fact_66_zprime__power__zdvd__cancel__left, arity_Int_Oint___Rings_Osemiring__1, fact_44_nat__numeral__1__eq__1, fact_96_semiring__norm_I110_J, fact_93_mult__numeral__1, fact_40_mult__number__of__left, fact_6_add__special_I2_J, fact_49_diff__bin__simps_I3_J, fact_95_semiring__numeral__1__eq__1, fact_34_rel__simps_I45_J, fact_5_one__power2, fact_22_rel__simps_I44_J, fact_92_number__of__Bit1, fact_88_Bit1__def, fact_38_left__diff__distrib__number__of, arity_Int_Oint___Groups_Omonoid__mult, fact_28_diff__bin__simps_I7_J, fact_59_nat__mult__2__right, fact_68_number__of__reorient, fact_90_zdvd__reduce, fact_81_int__distrib_I1_J, fact_78_add__Pls, fact_62_int__distrib_I4_J, fact_35_rel__simps_I42_J, fact_17_rel__simps_I48_J, fact_77_add__Pls__right, arity_Int_Oint___Rings_Osemiring, fact_13_zadd__power3, fact_24_Bit0__Pls, fact_83_plus__numeral__code_I9_J, fact_57_zprime__zdvd__power, fact_19_rel__simps_I39_J, fact_37_right__distrib__number__of, fact_85_add__numeral__0__right, fact_63_zdvd__zdiffD, fact_8_one__add__one__is__two, fact_51_nat__1__add__1, fact_94_mult__numeral__1__right, fact_23_rel__simps_I38_J, fact_21_rel__simps_I49_J, fact_60_diff__bin__simps_I1_J, arity_Nat_Onat___Int_Onumber, fact_53_diff__bin__simps_I6_J, fact_64_Numeral1__eq1__nat, fact_50__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_126_061_A1_096, fact_3__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096, fact_14_power2__sum, fact_82_int__distrib_I2_J, fact_73_power__m1__even, fact_46_add__Bit0__Bit1, fact_29_rel__simps_I47_J, fact_20_rel__simps_I50_J, fact_56_number__of__diff, fact_10_numeral__1__eq__1, fact_79_Bit0__def, fact_55_zpower__zpower, arity_Int_Oint___Int_Onumber, fact_74_add__number__of__diff1, fact_33_rel__simps_I40_J, fact_27_add__Bit0__Bit0, fact_16_rel__simps_I51_J, fact_18_rel__simps_I46_J, fact_30_rel__simps_I43_J, fact_84_add__numeral__0, fact_15_eq__number__of, fact_52_diff__bin__simps_I4_J, fact_70_eq__number__of__Pls__Min
% 6.02/2.34  % No. of inferences in proof: 224
% 6.02/2.34  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 1815 ms resp. 1325 ms w/o parsing
% 6.30/2.41  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 6.30/2.41  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------