TSTP Solution File: SET984+1 by Leo-III-SAT---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III-SAT---1.7.12
% Problem  : SET984+1 : TPTP v8.2.0. Released v3.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n012.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 03:07:31 EDT 2024

% Result   : Theorem 139.68s 20.88s
% Output   : Refutation 139.80s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :   16
%            Number of leaves      :   15
% Syntax   : Number of formulae    :   72 (  28 unt;   8 typ;   0 def)
%            Number of atoms       :  148 ( 115 equ;   0 cnn)
%            Maximal formula atoms :    6 (   2 avg)
%            Number of connectives :  434 (  46   ~;  59   |;   7   &; 310   @)
%                                         (   1 <=>;  11  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   11 (   5 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    6 (   6   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   10 (   8 usr;   6 con; 0-2 aty)
%            Number of variables   :  127 (   0   ^ 127   !;   0   ?; 127   :)

% Comments : 
%------------------------------------------------------------------------------
thf(subset_type,type,
    subset: $i > $i > $o ).

thf(cartesian_product2_type,type,
    cartesian_product2: $i > $i > $i ).

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

thf(set_intersection2_type,type,
    set_intersection2: $i > $i > $i ).

thf(sk1_type,type,
    sk1: $i ).

thf(sk2_type,type,
    sk2: $i ).

thf(sk3_type,type,
    sk3: $i ).

thf(sk4_type,type,
    sk4: $i ).

thf(1,conjecture,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( subset @ ( cartesian_product2 @ A @ B ) @ ( cartesian_product2 @ C @ D ) )
     => ( ( ( cartesian_product2 @ A @ B )
          = empty_set )
        | ( ( subset @ A @ C )
          & ( subset @ B @ D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',t138_zfmisc_1) ).

thf(2,negated_conjecture,
    ~ ! [A: $i,B: $i,C: $i,D: $i] :
        ( ( subset @ ( cartesian_product2 @ A @ B ) @ ( cartesian_product2 @ C @ D ) )
       => ( ( ( cartesian_product2 @ A @ B )
            = empty_set )
          | ( ( subset @ A @ C )
            & ( subset @ B @ D ) ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(14,plain,
    ~ ! [A: $i,B: $i,C: $i,D: $i] :
        ( ( subset @ ( cartesian_product2 @ A @ B ) @ ( cartesian_product2 @ C @ D ) )
       => ( ( ( cartesian_product2 @ A @ B )
            = empty_set )
          | ( ( subset @ A @ C )
            & ( subset @ B @ D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(17,plain,
    subset @ ( cartesian_product2 @ sk1 @ sk2 ) @ ( cartesian_product2 @ sk3 @ sk4 ),
    inference(cnf,[status(esa)],[14]) ).

thf(13,axiom,
    ! [A: $i,B: $i] :
      ( ( subset @ A @ B )
     => ( ( set_intersection2 @ A @ B )
        = A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',t28_xboole_1) ).

thf(53,plain,
    ! [A: $i,B: $i] :
      ( ( subset @ A @ B )
     => ( ( set_intersection2 @ A @ B )
        = A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(54,plain,
    ! [B: $i,A: $i] :
      ( ~ ( subset @ A @ B )
      | ( ( set_intersection2 @ A @ B )
        = A ) ),
    inference(cnf,[status(esa)],[53]) ).

thf(55,plain,
    ! [B: $i,A: $i] :
      ( ( ( set_intersection2 @ A @ B )
        = A )
      | ~ ( subset @ A @ B ) ),
    inference(lifteq,[status(thm)],[54]) ).

thf(851,plain,
    ! [B: $i,A: $i] :
      ( ( ( set_intersection2 @ A @ B )
        = A )
      | ( ( subset @ ( cartesian_product2 @ sk1 @ sk2 ) @ ( cartesian_product2 @ sk3 @ sk4 ) )
       != ( subset @ A @ B ) ) ),
    inference(paramod_ordered,[status(thm)],[17,55]) ).

thf(852,plain,
    ( ( set_intersection2 @ ( cartesian_product2 @ sk1 @ sk2 ) @ ( cartesian_product2 @ sk3 @ sk4 ) )
    = ( cartesian_product2 @ sk1 @ sk2 ) ),
    inference(pattern_uni,[status(thm)],[851:[bind(A,$thf( cartesian_product2 @ sk1 @ sk2 )),bind(B,$thf( cartesian_product2 @ sk3 @ sk4 ))]]) ).

thf(10,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( cartesian_product2 @ ( set_intersection2 @ A @ B ) @ ( set_intersection2 @ C @ D ) )
      = ( set_intersection2 @ ( cartesian_product2 @ A @ C ) @ ( cartesian_product2 @ B @ D ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',t123_zfmisc_1) ).

thf(43,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( cartesian_product2 @ ( set_intersection2 @ A @ B ) @ ( set_intersection2 @ C @ D ) )
      = ( set_intersection2 @ ( cartesian_product2 @ A @ C ) @ ( cartesian_product2 @ B @ D ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(44,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( cartesian_product2 @ ( set_intersection2 @ A @ B ) @ ( set_intersection2 @ C @ D ) )
      = ( set_intersection2 @ ( cartesian_product2 @ A @ C ) @ ( cartesian_product2 @ B @ D ) ) ),
    inference(cnf,[status(esa)],[43]) ).

thf(45,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( set_intersection2 @ ( cartesian_product2 @ A @ C ) @ ( cartesian_product2 @ B @ D ) )
      = ( cartesian_product2 @ ( set_intersection2 @ A @ B ) @ ( set_intersection2 @ C @ D ) ) ),
    inference(lifteq,[status(thm)],[44]) ).

thf(884,plain,
    ( ( cartesian_product2 @ ( set_intersection2 @ sk1 @ sk3 ) @ ( set_intersection2 @ sk2 @ sk4 ) )
    = ( cartesian_product2 @ sk1 @ sk2 ) ),
    inference(rewrite,[status(thm)],[852,45]) ).

thf(11,axiom,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( cartesian_product2 @ A @ B )
        = ( cartesian_product2 @ C @ D ) )
     => ( ( A = empty_set )
        | ( B = empty_set )
        | ( ( A = C )
          & ( B = D ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',t134_zfmisc_1) ).

thf(46,plain,
    ! [A: $i,B: $i,C: $i,D: $i] :
      ( ( ( cartesian_product2 @ A @ B )
        = ( cartesian_product2 @ C @ D ) )
     => ( ( A = empty_set )
        | ( B = empty_set )
        | ( ( A = C )
          & ( B = D ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(48,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( cartesian_product2 @ A @ B )
       != ( cartesian_product2 @ C @ D ) )
      | ( A = empty_set )
      | ( B = empty_set )
      | ( B = D ) ),
    inference(cnf,[status(esa)],[46]) ).

thf(50,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( cartesian_product2 @ A @ B )
       != ( cartesian_product2 @ C @ D ) )
      | ( A = empty_set )
      | ( B = empty_set )
      | ( B = D ) ),
    inference(lifteq,[status(thm)],[48]) ).

thf(903,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( cartesian_product2 @ sk1 @ sk2 )
       != ( cartesian_product2 @ A @ B ) )
      | ( A = empty_set )
      | ( B = empty_set )
      | ( B = D )
      | ( ( cartesian_product2 @ ( set_intersection2 @ sk1 @ sk3 ) @ ( set_intersection2 @ sk2 @ sk4 ) )
       != ( cartesian_product2 @ C @ D ) ) ),
    inference(paramod_ordered,[status(thm)],[884,50]) ).

thf(904,plain,
    ! [B: $i,A: $i] :
      ( ( ( cartesian_product2 @ sk1 @ sk2 )
       != ( cartesian_product2 @ A @ B ) )
      | ( A = empty_set )
      | ( B = empty_set )
      | ( B
        = ( set_intersection2 @ sk2 @ sk4 ) ) ),
    inference(pattern_uni,[status(thm)],[903:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( set_intersection2 @ sk1 @ sk3 )),bind(D,$thf( set_intersection2 @ sk2 @ sk4 ))]]) ).

thf(40642,plain,
    ( ( ( set_intersection2 @ sk2 @ sk4 )
      = sk2 )
    | ( sk2 = empty_set )
    | ( sk1 = empty_set ) ),
    inference(pattern_uni,[status(thm)],[904:[bind(A,$thf( sk1 )),bind(B,$thf( sk2 ))]]) ).

thf(9,axiom,
    ! [A: $i,B: $i] :
      ( ( ( cartesian_product2 @ A @ B )
        = empty_set )
    <=> ( ( A = empty_set )
        | ( B = empty_set ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',t113_zfmisc_1) ).

thf(33,plain,
    ! [A: $i,B: $i] :
      ( ( ( ( cartesian_product2 @ A @ B )
          = empty_set )
       => ( ( A = empty_set )
          | ( B = empty_set ) ) )
      & ( ( ( A = empty_set )
          | ( B = empty_set ) )
       => ( ( cartesian_product2 @ A @ B )
          = empty_set ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(34,plain,
    ( ! [A: $i,B: $i] :
        ( ( ( cartesian_product2 @ A @ B )
          = empty_set )
       => ( ( A = empty_set )
          | ( B = empty_set ) ) )
    & ! [A: $i,B: $i] :
        ( ( ( A = empty_set )
          | ( B = empty_set ) )
       => ( ( cartesian_product2 @ A @ B )
          = empty_set ) ) ),
    inference(miniscope,[status(thm)],[33]) ).

thf(36,plain,
    ! [B: $i,A: $i] :
      ( ( B != empty_set )
      | ( ( cartesian_product2 @ A @ B )
        = empty_set ) ),
    inference(cnf,[status(esa)],[34]) ).

thf(40,plain,
    ! [B: $i,A: $i] :
      ( ( B != empty_set )
      | ( ( cartesian_product2 @ A @ B )
        = empty_set ) ),
    inference(lifteq,[status(thm)],[36]) ).

thf(41,plain,
    ! [A: $i] :
      ( ( cartesian_product2 @ A @ empty_set )
      = empty_set ),
    inference(simp,[status(thm)],[40]) ).

thf(16,plain,
    ( ( cartesian_product2 @ sk1 @ sk2 )
   != empty_set ),
    inference(cnf,[status(esa)],[14]) ).

thf(18,plain,
    ( ( cartesian_product2 @ sk1 @ sk2 )
   != empty_set ),
    inference(lifteq,[status(thm)],[16]) ).

thf(84,plain,
    ! [A: $i] :
      ( ( cartesian_product2 @ A @ empty_set )
     != ( cartesian_product2 @ sk1 @ sk2 ) ),
    inference(paramod_ordered,[status(thm)],[41,18]) ).

thf(86,plain,
    ! [A: $i] :
      ( ( A != sk1 )
      | ( sk2 != empty_set ) ),
    inference(simp,[status(thm)],[84]) ).

thf(89,plain,
    sk2 != empty_set,
    inference(simp,[status(thm)],[86]) ).

thf(3,axiom,
    ! [A: $i,B: $i] :
      ( ( set_intersection2 @ A @ B )
      = ( set_intersection2 @ B @ A ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_k3_xboole_0) ).

thf(19,plain,
    ! [A: $i,B: $i] :
      ( ( set_intersection2 @ A @ B )
      = ( set_intersection2 @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(20,plain,
    ! [B: $i,A: $i] :
      ( ( set_intersection2 @ A @ B )
      = ( set_intersection2 @ B @ A ) ),
    inference(cnf,[status(esa)],[19]) ).

thf(21,plain,
    ! [B: $i,A: $i] :
      ( ( set_intersection2 @ A @ B )
      = ( set_intersection2 @ B @ A ) ),
    inference(lifteq,[status(thm)],[20]) ).

thf(12,axiom,
    ! [A: $i,B: $i] : ( subset @ ( set_intersection2 @ A @ B ) @ A ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',t17_xboole_1) ).

thf(51,plain,
    ! [A: $i,B: $i] : ( subset @ ( set_intersection2 @ A @ B ) @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(52,plain,
    ! [B: $i,A: $i] : ( subset @ ( set_intersection2 @ A @ B ) @ A ),
    inference(cnf,[status(esa)],[51]) ).

thf(152,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( subset @ ( set_intersection2 @ B @ A ) @ C )
      | ( ( set_intersection2 @ A @ B )
       != ( set_intersection2 @ C @ D ) ) ),
    inference(paramod_ordered,[status(thm)],[21,52]) ).

thf(153,plain,
    ! [B: $i,A: $i] : ( subset @ ( set_intersection2 @ B @ A ) @ A ),
    inference(pattern_uni,[status(thm)],[152:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( A )),bind(D,$thf( B ))]]) ).

thf(47,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( cartesian_product2 @ A @ B )
       != ( cartesian_product2 @ C @ D ) )
      | ( A = empty_set )
      | ( B = empty_set )
      | ( A = C ) ),
    inference(cnf,[status(esa)],[46]) ).

thf(49,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( cartesian_product2 @ A @ B )
       != ( cartesian_product2 @ C @ D ) )
      | ( A = empty_set )
      | ( B = empty_set )
      | ( A = C ) ),
    inference(lifteq,[status(thm)],[47]) ).

thf(899,plain,
    ! [D: $i,C: $i,B: $i,A: $i] :
      ( ( ( cartesian_product2 @ sk1 @ sk2 )
       != ( cartesian_product2 @ A @ B ) )
      | ( A = empty_set )
      | ( B = empty_set )
      | ( A = C )
      | ( ( cartesian_product2 @ ( set_intersection2 @ sk1 @ sk3 ) @ ( set_intersection2 @ sk2 @ sk4 ) )
       != ( cartesian_product2 @ C @ D ) ) ),
    inference(paramod_ordered,[status(thm)],[884,49]) ).

thf(900,plain,
    ! [B: $i,A: $i] :
      ( ( ( cartesian_product2 @ sk1 @ sk2 )
       != ( cartesian_product2 @ A @ B ) )
      | ( A = empty_set )
      | ( B = empty_set )
      | ( A
        = ( set_intersection2 @ sk1 @ sk3 ) ) ),
    inference(pattern_uni,[status(thm)],[899:[bind(A,$thf( A )),bind(B,$thf( B )),bind(C,$thf( set_intersection2 @ sk1 @ sk3 )),bind(D,$thf( set_intersection2 @ sk2 @ sk4 ))]]) ).

thf(37666,plain,
    ( ( ( set_intersection2 @ sk1 @ sk3 )
      = sk1 )
    | ( sk2 = empty_set )
    | ( sk1 = empty_set ) ),
    inference(pattern_uni,[status(thm)],[900:[bind(A,$thf( sk1 )),bind(B,$thf( sk2 ))]]) ).

thf(35,plain,
    ! [B: $i,A: $i] :
      ( ( A != empty_set )
      | ( ( cartesian_product2 @ A @ B )
        = empty_set ) ),
    inference(cnf,[status(esa)],[34]) ).

thf(38,plain,
    ! [B: $i,A: $i] :
      ( ( A != empty_set )
      | ( ( cartesian_product2 @ A @ B )
        = empty_set ) ),
    inference(lifteq,[status(thm)],[35]) ).

thf(39,plain,
    ! [A: $i] :
      ( ( cartesian_product2 @ empty_set @ A )
      = empty_set ),
    inference(simp,[status(thm)],[38]) ).

thf(71,plain,
    ! [A: $i] :
      ( ( cartesian_product2 @ empty_set @ A )
     != ( cartesian_product2 @ sk1 @ sk2 ) ),
    inference(paramod_ordered,[status(thm)],[39,18]) ).

thf(74,plain,
    ! [A: $i] :
      ( ( sk1 != empty_set )
      | ( A != sk2 ) ),
    inference(simp,[status(thm)],[71]) ).

thf(77,plain,
    sk1 != empty_set,
    inference(simp,[status(thm)],[74]) ).

thf(40092,plain,
    ( ( set_intersection2 @ sk1 @ sk3 )
    = sk1 ),
    inference(simplifyReflect,[status(thm)],[37666,89,77]) ).

thf(15,plain,
    ( ~ ( subset @ sk1 @ sk3 )
    | ~ ( subset @ sk2 @ sk4 ) ),
    inference(cnf,[status(esa)],[14]) ).

thf(189,plain,
    ! [B: $i,A: $i] :
      ( ~ ( subset @ sk2 @ sk4 )
      | ( ( subset @ ( set_intersection2 @ B @ A ) @ A )
       != ( subset @ sk1 @ sk3 ) ) ),
    inference(paramod_ordered,[status(thm)],[153,15]) ).

thf(193,plain,
    ! [B: $i,A: $i] :
      ( ~ ( subset @ sk2 @ sk4 )
      | ( ( set_intersection2 @ B @ A )
       != sk1 )
      | ( A != sk3 ) ),
    inference(simp,[status(thm)],[189]) ).

thf(202,plain,
    ! [A: $i] :
      ( ~ ( subset @ sk2 @ sk4 )
      | ( ( set_intersection2 @ A @ sk3 )
       != sk1 ) ),
    inference(simp,[status(thm)],[193]) ).

thf(40118,plain,
    ! [A: $i] :
      ( ~ ( subset @ sk2 @ sk4 )
      | ( ( set_intersection2 @ sk1 @ sk3 )
       != ( set_intersection2 @ A @ sk3 ) ) ),
    inference(paramod_ordered,[status(thm)],[40092,202]) ).

thf(40119,plain,
    ~ ( subset @ sk2 @ sk4 ),
    inference(pattern_uni,[status(thm)],[40118:[bind(A,$thf( sk1 ))]]) ).

thf(40566,plain,
    ! [B: $i,A: $i] :
      ( ( subset @ ( set_intersection2 @ B @ A ) @ A )
     != ( subset @ sk2 @ sk4 ) ),
    inference(paramod_ordered,[status(thm)],[153,40119]) ).

thf(40603,plain,
    ! [B: $i,A: $i] :
      ( ( ( set_intersection2 @ B @ A )
       != sk2 )
      | ( A != sk4 ) ),
    inference(simp,[status(thm)],[40566]) ).

thf(40633,plain,
    ! [A: $i] :
      ( ( set_intersection2 @ A @ sk4 )
     != sk2 ),
    inference(simp,[status(thm)],[40603]) ).

thf(43998,plain,
    $false,
    inference(simplifyReflect,[status(thm)],[40642,89,40633,77]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.09  % Problem  : SET984+1 : TPTP v8.2.0. Released v3.2.0.
% 0.08/0.11  % Command  : run_Leo-III %s %d
% 0.10/0.31  % Computer : n012.cluster.edu
% 0.10/0.31  % Model    : x86_64 x86_64
% 0.10/0.31  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.31  % Memory   : 8042.1875MB
% 0.10/0.31  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.10/0.31  % CPULimit : 300
% 0.10/0.31  % WCLimit  : 300
% 0.10/0.31  % DateTime : Mon May 20 11:57:24 EDT 2024
% 0.15/0.31  % CPUTime  : 
% 0.87/0.90  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 1.28/1.06  % [INFO] 	 Parsing done (157ms). 
% 1.28/1.07  % [INFO] 	 Running in sequential loop mode. 
% 1.70/1.42  % [INFO] 	 nitpick registered as external prover. 
% 1.82/1.42  % [INFO] 	 Scanning for conjecture ... 
% 1.91/1.52  % [INFO] 	 Found a conjecture (or negated_conjecture) and 11 axioms. Running axiom selection ... 
% 2.02/1.55  % [INFO] 	 Axiom selection finished. Selected 11 axioms (removed 0 axioms). 
% 2.17/1.57  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 2.17/1.58  % [INFO] 	 Type checking passed. 
% 2.17/1.58  % [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 ... 
% 139.68/20.83  % [INFO] 	 Killing All external provers ... 
% 139.68/20.84  % Time passed: 20380ms (effective reasoning time: 19755ms)
% 139.68/20.84  % 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)>
% 139.68/20.87  % Axioms used in derivation (6): t17_xboole_1, t113_zfmisc_1, commutativity_k3_xboole_0, t123_zfmisc_1, t28_xboole_1, t134_zfmisc_1
% 139.68/20.87  % No. of inferences in proof: 64
% 139.68/20.88  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 20380 ms resp. 19755 ms w/o parsing
% 139.80/21.03  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 139.80/21.03  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------