TSTP Solution File: SET004^4 by Leo-III---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : SET004^4 : TPTP v8.2.0. Released v8.1.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n004.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Tue May 21 03:00:42 EDT 2024

% Result   : Theorem 4.15s 2.01s
% Output   : Refutation 4.48s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :   64
% Syntax   : Number of formulae    :   99 (  22 unt;  23 typ;   8 def)
%            Number of atoms       :  354 (  19 equ;   0 cnn)
%            Maximal formula atoms :   10 (   4 avg)
%            Number of connectives :  779 (   5   ~;   3   |;  12   &; 736   @)
%                                         (   0 <=>;  23  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   14 (   9 avg)
%            Number of types       :    3 (   1 usr)
%            Number of type conns  :   71 (  71   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   24 (  22 usr;   3 con; 0-3 aty)
%            Number of variables   :  195 ( 107   ^  86   !;   2   ?; 195   :)

% Comments : 
%------------------------------------------------------------------------------
thf(mworld_type,type,
    mworld: $tType ).

thf(mactual_type,type,
    mactual: mworld ).

thf(mlocal_type,type,
    mlocal: ( mworld > $o ) > $o ).

thf(mlocal_def,definition,
    ( mlocal
    = ( ^ [A: mworld > $o] : ( A @ mactual ) ) ) ).

thf(mnot_type,type,
    mnot: ( mworld > $o ) > mworld > $o ).

thf(mnot_def,definition,
    ( mnot
    = ( ^ [A: mworld > $o,B: mworld] :
          ~ ( A @ B ) ) ) ).

thf(mand_type,type,
    mand: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).

thf(mand_def,definition,
    ( mand
    = ( ^ [A: mworld > $o,B: mworld > $o,C: mworld] :
          ( ( A @ C )
          & ( B @ C ) ) ) ) ).

thf(mor_type,type,
    mor: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).

thf(mor_def,definition,
    ( mor
    = ( ^ [A: mworld > $o,B: mworld > $o,C: mworld] :
          ( ( A @ C )
          | ( B @ C ) ) ) ) ).

thf(mimplies_type,type,
    mimplies: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).

thf(mimplies_def,definition,
    ( mimplies
    = ( ^ [A: mworld > $o,B: mworld > $o,C: mworld] :
          ( ( A @ C )
         => ( B @ C ) ) ) ) ).

thf(mequiv_type,type,
    mequiv: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).

thf(mequiv_def,definition,
    ( mequiv
    = ( ^ [A: mworld > $o,B: mworld > $o,C: mworld] :
          ( ( A @ C )
          = ( B @ C ) ) ) ) ).

thf(mforall_di_type,type,
    mforall_di: ( $i > mworld > $o ) > mworld > $o ).

thf(mforall_di_def,definition,
    ( mforall_di
    = ( ^ [A: $i > mworld > $o,B: mworld] :
        ! [C: $i] : ( A @ C @ B ) ) ) ).

thf(mexists_di_type,type,
    mexists_di: ( $i > mworld > $o ) > mworld > $o ).

thf(mexists_di_def,definition,
    ( mexists_di
    = ( ^ [A: $i > mworld > $o,B: mworld] :
        ? [C: $i] : ( A @ C @ B ) ) ) ).

thf(empty_set_type,type,
    empty_set: $i ).

thf(qmltpeq_type,type,
    qmltpeq: $i > $i > mworld > $o ).

thf(equal_set_type,type,
    equal_set: $i > $i > mworld > $o ).

thf(member_type,type,
    member: $i > $i > mworld > $o ).

thf(subset_type,type,
    subset: $i > $i > mworld > $o ).

thf(singleton_type,type,
    singleton: $i > $i ).

thf(product_type,type,
    product: $i > $i ).

thf(unordered_pair_type,type,
    unordered_pair: $i > $i > $i ).

thf(intersection_type,type,
    intersection: $i > $i > $i ).

thf(difference_type,type,
    difference: $i > $i > $i ).

thf(sum_type,type,
    sum: $i > $i ).

thf(union_type,type,
    union: $i > $i > $i ).

thf(power_set_type,type,
    power_set: $i > $i ).

thf(33,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( unordered_pair @ A @ C ) @ ( unordered_pair @ B @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_substitution_1) ).

thf(115,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( unordered_pair @ A @ C ) @ ( unordered_pair @ B @ C ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).

thf(31,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subset @ C @ A ) ) @ ( subset @ C @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',subset_substitution_2) ).

thf(110,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( qmltpeq @ A @ B @ mactual )
        & ( subset @ C @ A @ mactual ) )
     => ( subset @ C @ B @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).

thf(9,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ B @ C ) ) @ ( qmltpeq @ A @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',transitivity) ).

thf(53,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( qmltpeq @ A @ B @ mactual )
        & ( qmltpeq @ B @ C @ mactual ) )
     => ( qmltpeq @ A @ C @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(17,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( power_set @ A ) @ ( power_set @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_set_substitution_1) ).

thf(73,plain,
    ! [A: $i,B: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( power_set @ A ) @ ( power_set @ B ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(16,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ C @ A ) ) @ ( member @ C @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_substitution_2) ).

thf(71,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( qmltpeq @ A @ B @ mactual )
        & ( member @ C @ A @ mactual ) )
     => ( member @ C @ B @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(14,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( intersection @ A @ C ) @ ( intersection @ B @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection_substitution_1) ).

thf(66,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( intersection @ A @ C ) @ ( intersection @ B @ C ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(32,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ A @ C ) @ ( union @ B @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',union_substitution_1) ).

thf(112,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( union @ A @ C ) @ ( union @ B @ C ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).

thf(8,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( intersection @ C @ A ) @ ( intersection @ C @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection_substitution_2) ).

thf(50,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( intersection @ C @ A ) @ ( intersection @ C @ B ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(10,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subset @ A @ C ) ) @ ( subset @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',subset_substitution_1) ).

thf(55,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( qmltpeq @ A @ B @ mactual )
        & ( subset @ A @ C @ mactual ) )
     => ( subset @ B @ C @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(30,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ C @ A ) @ ( union @ C @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',union_substitution_2) ).

thf(107,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( union @ C @ A ) @ ( union @ C @ B ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).

thf(7,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mequiv @ ( equal_set @ A @ B ) @ ( mand @ ( subset @ A @ B ) @ ( subset @ B @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_set_0) ).

thf(47,plain,
    ! [A: $i,B: $i] :
      ( ( equal_set @ A @ B @ mactual )
      = ( ( subset @ A @ B @ mactual )
        & ( subset @ B @ A @ mactual ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(24,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mequiv @ ( member @ A @ ( unordered_pair @ B @ C ) ) @ ( mor @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_0) ).

thf(90,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( member @ A @ ( unordered_pair @ B @ C ) @ mactual )
      = ( ( qmltpeq @ A @ B @ mactual )
        | ( qmltpeq @ A @ C @ mactual ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).

thf(15,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ A @ C ) ) @ ( member @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_substitution_1) ).

thf(69,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( qmltpeq @ A @ B @ mactual )
        & ( member @ A @ C @ mactual ) )
     => ( member @ B @ C @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(28,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] : ( mnot @ ( member @ A @ empty_set ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',empty_set_0) ).

thf(101,plain,
    ! [A: $i] :
      ~ ( member @ A @ empty_set @ mactual ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).

thf(5,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( equal_set @ A @ C ) ) @ ( equal_set @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_set_substitution_1) ).

thf(42,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( qmltpeq @ A @ B @ mactual )
        & ( equal_set @ A @ C @ mactual ) )
     => ( equal_set @ B @ C @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(3,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( sum @ A ) @ ( sum @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_substitution_1) ).

thf(37,plain,
    ! [A: $i,B: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( sum @ A ) @ ( sum @ B ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(25,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( product @ A ) @ ( product @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',product_substitution_1) ).

thf(93,plain,
    ! [A: $i,B: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( product @ A ) @ ( product @ B ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).

thf(11,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mequiv @ ( member @ A @ ( singleton @ B ) ) @ ( qmltpeq @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_0) ).

thf(57,plain,
    ! [A: $i,B: $i] :
      ( ( member @ A @ ( singleton @ B ) @ mactual )
      = ( qmltpeq @ A @ B @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(19,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( equal_set @ C @ A ) ) @ ( equal_set @ C @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_set_substitution_2) ).

thf(78,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( qmltpeq @ A @ B @ mactual )
        & ( equal_set @ C @ A @ mactual ) )
     => ( equal_set @ C @ B @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(12,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( difference @ A @ C ) @ ( difference @ B @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',difference_substitution_1) ).

thf(60,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( difference @ A @ C ) @ ( difference @ B @ C ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(22,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( singleton @ A ) @ ( singleton @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_substitution_1) ).

thf(85,plain,
    ! [A: $i,B: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( singleton @ A ) @ ( singleton @ B ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(6,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mequiv @ ( member @ A @ ( difference @ C @ B ) ) @ ( mand @ ( member @ A @ C ) @ ( mnot @ ( member @ A @ B ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',difference_0) ).

thf(44,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( member @ A @ ( difference @ C @ B ) @ mactual )
      = ( ( member @ A @ C @ mactual )
        & ~ ( member @ A @ B @ mactual ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(34,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',symmetry) ).

thf(118,plain,
    ! [A: $i,B: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ B @ A @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).

thf(4,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mequiv @ ( member @ A @ ( product @ B ) )
              @ ( mforall_di
                @ ^ [C: $i] : ( mimplies @ ( member @ C @ B ) @ ( member @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',product_0) ).

thf(39,plain,
    ! [A: $i,B: $i] :
      ( ( member @ A @ ( product @ B ) @ mactual )
      = ( ! [C: $i] :
            ( ( member @ C @ B @ mactual )
           => ( member @ A @ C @ mactual ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(27,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mequiv @ ( member @ A @ ( union @ B @ C ) ) @ ( mor @ ( member @ A @ B ) @ ( member @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',union_0) ).

thf(98,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( member @ A @ ( union @ B @ C ) @ mactual )
      = ( ( member @ A @ B @ mactual )
        | ( member @ A @ C @ mactual ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).

thf(20,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] : ( qmltpeq @ A @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',reflexivity) ).

thf(80,plain,
    ! [A: $i] : ( qmltpeq @ A @ A @ mactual ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(1,conjecture,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( equal_set @ ( intersection @ A @ B ) @ ( intersection @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',thI06) ).

thf(2,negated_conjecture,
    ~ ( mlocal
      @ ( mforall_di
        @ ^ [A: $i] :
            ( mforall_di
            @ ^ [B: $i] : ( equal_set @ ( intersection @ A @ B ) @ ( intersection @ B @ A ) ) ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(35,plain,
    ~ ! [A: $i,B: $i] : ( equal_set @ ( intersection @ A @ B ) @ ( intersection @ B @ A ) @ mactual ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(13,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mequiv @ ( subset @ A @ B )
              @ ( mforall_di
                @ ^ [C: $i] : ( mimplies @ ( member @ C @ A ) @ ( member @ C @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',subset_0) ).

thf(63,plain,
    ! [A: $i,B: $i] :
      ( ( subset @ A @ B @ mactual )
      = ( ! [C: $i] :
            ( ( member @ C @ A @ mactual )
           => ( member @ C @ B @ mactual ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(26,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mequiv @ ( member @ A @ ( intersection @ B @ C ) ) @ ( mand @ ( member @ A @ B ) @ ( member @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',intersection_0) ).

thf(95,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( member @ A @ ( intersection @ B @ C ) @ mactual )
      = ( ( member @ A @ B @ mactual )
        & ( member @ A @ C @ mactual ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).

thf(23,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( difference @ C @ A ) @ ( difference @ C @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',difference_substitution_2) ).

thf(87,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( difference @ C @ A ) @ ( difference @ C @ B ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).

thf(29,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mequiv @ ( member @ A @ ( power_set @ B ) ) @ ( subset @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',power_set_0) ).

thf(104,plain,
    ! [A: $i,B: $i] :
      ( ( member @ A @ ( power_set @ B ) @ mactual )
      = ( subset @ A @ B @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).

thf(18,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( unordered_pair @ C @ A ) @ ( unordered_pair @ C @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',unordered_pair_substitution_2) ).

thf(75,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( unordered_pair @ C @ A ) @ ( unordered_pair @ C @ B ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(21,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mequiv @ ( member @ A @ ( sum @ B ) )
              @ ( mexists_di
                @ ^ [C: $i] : ( mand @ ( member @ C @ B ) @ ( member @ A @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',sum_0) ).

thf(82,plain,
    ! [A: $i,B: $i] :
      ( ( member @ A @ ( sum @ B ) @ mactual )
      = ( ? [C: $i] :
            ( ( member @ C @ B @ mactual )
            & ( member @ A @ C @ mactual ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(125,plain,
    $false,
    inference(cvc4,[status(thm)],[115,110,53,73,71,66,112,50,55,107,47,90,69,101,42,37,93,57,78,60,85,44,118,39,98,80,35,63,95,87,104,75,82]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12  % Problem  : SET004^4 : TPTP v8.2.0. Released v8.1.0.
% 0.11/0.15  % Command  : run_Leo-III %s %d
% 0.15/0.36  % Computer : n004.cluster.edu
% 0.15/0.36  % Model    : x86_64 x86_64
% 0.15/0.36  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36  % Memory   : 8042.1875MB
% 0.15/0.36  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36  % CPULimit : 300
% 0.15/0.36  % WCLimit  : 300
% 0.15/0.36  % DateTime : Mon May 20 11:08:24 EDT 2024
% 0.15/0.36  % CPUTime  : 
% 0.97/0.86  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 1.20/1.01  % [INFO] 	 Parsing done (151ms). 
% 1.20/1.02  % [INFO] 	 Running in sequential loop mode. 
% 1.84/1.22  % [INFO] 	 eprover registered as external prover. 
% 1.84/1.22  % [INFO] 	 cvc4 registered as external prover. 
% 1.84/1.23  % [INFO] 	 Scanning for conjecture ... 
% 2.12/1.31  % [INFO] 	 Found a conjecture (or negated_conjecture) and 34 axioms. Running axiom selection ... 
% 2.29/1.36  % [INFO] 	 Axiom selection finished. Selected 32 axioms (removed 2 axioms). 
% 2.29/1.40  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 2.29/1.40  % [INFO] 	 Type checking passed. 
% 2.29/1.41  % [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 ... 
% 4.15/2.00  % External prover 'cvc4' found a proof!
% 4.15/2.00  % [INFO] 	 Killing All external provers ... 
% 4.15/2.00  % Time passed: 1477ms (effective reasoning time: 980ms)
% 4.15/2.00  % 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)>
% 4.15/2.01  % Axioms used in derivation (32): product_substitution_1, intersection_substitution_1, singleton_0, power_set_substitution_1, power_set_0, unordered_pair_0, difference_substitution_1, transitivity, member_substitution_2, member_substitution_1, sum_substitution_1, empty_set_0, subset_substitution_2, unordered_pair_substitution_2, intersection_0, intersection_substitution_2, sum_0, equal_set_0, union_substitution_2, difference_0, reflexivity, product_0, subset_0, equal_set_substitution_2, union_0, unordered_pair_substitution_1, difference_substitution_2, subset_substitution_1, equal_set_substitution_1, union_substitution_1, singleton_substitution_1, symmetry
% 4.15/2.01  % No. of inferences in proof: 68
% 4.15/2.01  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 1477 ms resp. 980 ms w/o parsing
% 4.48/2.07  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 4.48/2.07  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------