TSTP Solution File: SET930+1 by Leo-III-SAT---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III-SAT---1.7.12
% Problem  : SET930+1 : TPTP v8.2.0. Released v3.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n003.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 03:07:25 EDT 2024

% Result   : Theorem 15.61s 3.51s
% Output   : Refutation 15.61s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :   18
%            Number of leaves      :   13
% Syntax   : Number of formulae    :   62 (  18 unt;   8 typ;   0 def)
%            Number of atoms       :  153 (  75 equ;   0 cnn)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives :  652 (  79   ~;  45   |;  33   &; 480   @)
%                                         (   3 <=>;  12  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   6 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    7 (   7   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   11 (   8 usr;   6 con; 0-2 aty)
%            Number of variables   :   97 (   0   ^  94   !;   3   ?;  97   :)

% Comments : 
%------------------------------------------------------------------------------
thf(set_difference_type,type,
    set_difference: $i > $i > $i ).

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

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

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

thf(in_type,type,
    in: $i > $i > $o ).

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

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

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

thf(9,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = ( unordered_pair @ A @ B ) )
    <=> ( ~ ( in @ A @ C )
        & ~ ( in @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t72_zfmisc_1) ).

thf(44,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
          = ( unordered_pair @ A @ B ) )
       => ( ~ ( in @ A @ C )
          & ~ ( in @ B @ C ) ) )
      & ( ( ~ ( in @ A @ C )
          & ~ ( in @ B @ C ) )
       => ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
          = ( unordered_pair @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(45,plain,
    ( ! [A: $i,B: $i,C: $i] :
        ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
          = ( unordered_pair @ A @ B ) )
       => ( ~ ( in @ A @ C )
          & ~ ( in @ B @ C ) ) )
    & ! [A: $i,B: $i,C: $i] :
        ( ( ~ ( in @ A @ C )
          & ~ ( in @ B @ C ) )
       => ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
          = ( unordered_pair @ A @ B ) ) ) ),
    inference(miniscope,[status(thm)],[44]) ).

thf(46,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( in @ A @ C )
      | ( in @ B @ C )
      | ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = ( unordered_pair @ A @ B ) ) ),
    inference(cnf,[status(esa)],[45]) ).

thf(49,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = ( unordered_pair @ A @ B ) )
      | ( in @ A @ C )
      | ( in @ B @ C ) ),
    inference(lifteq,[status(thm)],[46]) ).

thf(50,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = ( unordered_pair @ A @ B ) )
      | ( in @ A @ C )
      | ( in @ B @ C ) ),
    inference(simp,[status(thm)],[49]) ).

thf(1,conjecture,
    ! [A: $i,B: $i,C: $i] :
      ~ ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
         != empty_set )
        & ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
         != ( singleton @ A ) )
        & ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
         != ( singleton @ B ) )
        & ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
         != ( unordered_pair @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t74_zfmisc_1) ).

thf(2,negated_conjecture,
    ~ ! [A: $i,B: $i,C: $i] :
        ~ ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
           != empty_set )
          & ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
           != ( singleton @ A ) )
          & ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
           != ( singleton @ B ) )
          & ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
           != ( unordered_pair @ A @ B ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(11,plain,
    ~ ! [A: $i,B: $i,C: $i] :
        ~ ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
           != empty_set )
          & ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
           != ( singleton @ A ) )
          & ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
           != ( singleton @ B ) )
          & ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
           != ( unordered_pair @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(12,plain,
    ~ ~ ? [A: $i,B: $i,C: $i] :
          ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
           != empty_set )
          & ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
           != ( singleton @ A ) )
          & ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
           != ( singleton @ B ) )
          & ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
           != ( unordered_pair @ A @ B ) ) ),
    inference(miniscope,[status(thm)],[11]) ).

thf(13,plain,
    ( ( set_difference @ ( unordered_pair @ sk1 @ sk2 ) @ sk3 )
   != ( unordered_pair @ sk1 @ sk2 ) ),
    inference(cnf,[status(esa)],[12]) ).

thf(17,plain,
    ( ( set_difference @ ( unordered_pair @ sk1 @ sk2 ) @ sk3 )
   != ( unordered_pair @ sk1 @ sk2 ) ),
    inference(lifteq,[status(thm)],[13]) ).

thf(289,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( in @ A @ C )
      | ( in @ B @ C )
      | ( ( unordered_pair @ A @ B )
       != ( unordered_pair @ sk1 @ sk2 ) )
      | ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
       != ( set_difference @ ( unordered_pair @ sk1 @ sk2 ) @ sk3 ) ) ),
    inference(paramod_ordered,[status(thm)],[50,17]) ).

thf(290,plain,
    ( ( in @ sk1 @ sk3 )
    | ( in @ sk2 @ sk3 )
    | ( ( unordered_pair @ sk1 @ sk2 )
     != ( unordered_pair @ sk1 @ sk2 ) ) ),
    inference(pattern_uni,[status(thm)],[289:[bind(A,$thf( sk1 )),bind(B,$thf( sk2 )),bind(C,$thf( sk3 ))]]) ).

thf(321,plain,
    ( ( in @ sk1 @ sk3 )
    | ( in @ sk2 @ sk3 ) ),
    inference(simp,[status(thm)],[290]) ).

thf(6,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = ( singleton @ A ) )
    <=> ( ~ ( in @ A @ C )
        & ( ( in @ B @ C )
          | ( A = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l39_zfmisc_1) ).

thf(27,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
          = ( singleton @ A ) )
       => ( ~ ( in @ A @ C )
          & ( ( in @ B @ C )
            | ( A = B ) ) ) )
      & ( ( ~ ( in @ A @ C )
          & ( ( in @ B @ C )
            | ( A = B ) ) )
       => ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
          = ( singleton @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(28,plain,
    ( ! [A: $i,B: $i,C: $i] :
        ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
          = ( singleton @ A ) )
       => ( ~ ( in @ A @ C )
          & ( ( in @ B @ C )
            | ( A = B ) ) ) )
    & ! [A: $i,B: $i,C: $i] :
        ( ( ~ ( in @ A @ C )
          & ( ( in @ B @ C )
            | ( A = B ) ) )
       => ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
          = ( singleton @ A ) ) ) ),
    inference(miniscope,[status(thm)],[27]) ).

thf(30,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( in @ A @ C )
      | ~ ( in @ B @ C )
      | ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = ( singleton @ A ) ) ),
    inference(cnf,[status(esa)],[28]) ).

thf(35,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = ( singleton @ A ) )
      | ( in @ A @ C )
      | ~ ( in @ B @ C ) ),
    inference(lifteq,[status(thm)],[30]) ).

thf(36,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = ( singleton @ A ) )
      | ( in @ A @ C )
      | ~ ( in @ B @ C ) ),
    inference(simp,[status(thm)],[35]) ).

thf(15,plain,
    ( ( set_difference @ ( unordered_pair @ sk1 @ sk2 ) @ sk3 )
   != ( singleton @ sk1 ) ),
    inference(cnf,[status(esa)],[12]) ).

thf(19,plain,
    ( ( set_difference @ ( unordered_pair @ sk1 @ sk2 ) @ sk3 )
   != ( singleton @ sk1 ) ),
    inference(lifteq,[status(thm)],[15]) ).

thf(198,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( in @ A @ C )
      | ~ ( in @ B @ C )
      | ( ( singleton @ A )
       != ( singleton @ sk1 ) )
      | ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
       != ( set_difference @ ( unordered_pair @ sk1 @ sk2 ) @ sk3 ) ) ),
    inference(paramod_ordered,[status(thm)],[36,19]) ).

thf(199,plain,
    ( ( in @ sk1 @ sk3 )
    | ~ ( in @ sk2 @ sk3 )
    | ( ( singleton @ sk1 )
     != ( singleton @ sk1 ) ) ),
    inference(pattern_uni,[status(thm)],[198:[bind(A,$thf( sk1 )),bind(B,$thf( sk2 )),bind(C,$thf( sk3 ))]]) ).

thf(203,plain,
    ( ( in @ sk1 @ sk3 )
    | ~ ( in @ sk2 @ sk3 ) ),
    inference(simp,[status(thm)],[199]) ).

thf(495,plain,
    ( ( in @ sk1 @ sk3 )
    | ( ( in @ sk2 @ sk3 )
     != ( in @ sk2 @ sk3 ) ) ),
    inference(paramod_ordered,[status(thm)],[321,203]) ).

thf(496,plain,
    in @ sk1 @ sk3,
    inference(pattern_uni,[status(thm)],[495:[]]) ).

thf(534,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = ( singleton @ A ) )
      | ( in @ A @ C )
      | ( ( in @ sk1 @ sk3 )
       != ( in @ B @ C ) ) ),
    inference(paramod_ordered,[status(thm)],[496,36]) ).

thf(535,plain,
    ! [A: $i] :
      ( ( ( set_difference @ ( unordered_pair @ A @ sk1 ) @ sk3 )
        = ( singleton @ A ) )
      | ( in @ A @ sk3 ) ),
    inference(pattern_uni,[status(thm)],[534:[bind(A,$thf( A )),bind(B,$thf( sk1 )),bind(C,$thf( sk3 ))]]) ).

thf(10,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = empty_set )
    <=> ( ( in @ A @ C )
        & ( in @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t73_zfmisc_1) ).

thf(53,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
          = empty_set )
       => ( ( in @ A @ C )
          & ( in @ B @ C ) ) )
      & ( ( ( in @ A @ C )
          & ( in @ B @ C ) )
       => ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
          = empty_set ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(54,plain,
    ( ! [A: $i,B: $i,C: $i] :
        ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
          = empty_set )
       => ( ( in @ A @ C )
          & ( in @ B @ C ) ) )
    & ! [A: $i,B: $i,C: $i] :
        ( ( ( in @ A @ C )
          & ( in @ B @ C ) )
       => ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
          = empty_set ) ) ),
    inference(miniscope,[status(thm)],[53]) ).

thf(55,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( in @ A @ C )
      | ~ ( in @ B @ C )
      | ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = empty_set ) ),
    inference(cnf,[status(esa)],[54]) ).

thf(58,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = empty_set )
      | ~ ( in @ A @ C )
      | ~ ( in @ B @ C ) ),
    inference(lifteq,[status(thm)],[55]) ).

thf(59,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
        = empty_set )
      | ~ ( in @ A @ C )
      | ~ ( in @ B @ C ) ),
    inference(simp,[status(thm)],[58]) ).

thf(16,plain,
    ( ( set_difference @ ( unordered_pair @ sk1 @ sk2 ) @ sk3 )
   != empty_set ),
    inference(cnf,[status(esa)],[12]) ).

thf(20,plain,
    ( ( set_difference @ ( unordered_pair @ sk1 @ sk2 ) @ sk3 )
   != empty_set ),
    inference(lifteq,[status(thm)],[16]) ).

thf(934,plain,
    ! [C: $i,B: $i,A: $i] :
      ( ~ ( in @ A @ C )
      | ~ ( in @ B @ C )
      | ( ( set_difference @ ( unordered_pair @ A @ B ) @ C )
       != ( set_difference @ ( unordered_pair @ sk1 @ sk2 ) @ sk3 ) ) ),
    inference(paramod_ordered,[status(thm)],[59,20]) ).

thf(935,plain,
    ( ~ ( in @ sk1 @ sk3 )
    | ~ ( in @ sk2 @ sk3 ) ),
    inference(pattern_uni,[status(thm)],[934:[bind(A,$thf( sk1 )),bind(B,$thf( sk2 )),bind(C,$thf( sk3 ))]]) ).

thf(1169,plain,
    ( ~ $true
    | ~ ( in @ sk2 @ sk3 ) ),
    inference(rewrite,[status(thm)],[935,496]) ).

thf(1170,plain,
    ~ ( in @ sk2 @ sk3 ),
    inference(simp,[status(thm)],[1169]) ).

thf(3031,plain,
    ! [A: $i] :
      ( ( ( set_difference @ ( unordered_pair @ A @ sk1 ) @ sk3 )
        = ( singleton @ A ) )
      | ( ( in @ A @ sk3 )
       != ( in @ sk2 @ sk3 ) ) ),
    inference(paramod_ordered,[status(thm)],[535,1170]) ).

thf(3032,plain,
    ( ( set_difference @ ( unordered_pair @ sk2 @ sk1 ) @ sk3 )
    = ( singleton @ sk2 ) ),
    inference(pattern_uni,[status(thm)],[3031:[bind(A,$thf( sk2 ))]]) ).

thf(14,plain,
    ( ( set_difference @ ( unordered_pair @ sk1 @ sk2 ) @ sk3 )
   != ( singleton @ sk2 ) ),
    inference(cnf,[status(esa)],[12]) ).

thf(18,plain,
    ( ( set_difference @ ( unordered_pair @ sk1 @ sk2 ) @ sk3 )
   != ( singleton @ sk2 ) ),
    inference(lifteq,[status(thm)],[14]) ).

thf(3226,plain,
    ( ( set_difference @ ( unordered_pair @ sk2 @ sk1 ) @ sk3 )
   != ( set_difference @ ( unordered_pair @ sk1 @ sk2 ) @ sk3 ) ),
    inference(paramod_ordered,[status(thm)],[3032,18]) ).

thf(3255,plain,
    ( ( ( unordered_pair @ sk2 @ sk1 )
     != ( unordered_pair @ sk1 @ sk2 ) )
    | ( sk3 != sk3 ) ),
    inference(simp,[status(thm)],[3226]) ).

thf(3289,plain,
    ( ( unordered_pair @ sk2 @ sk1 )
   != ( unordered_pair @ sk1 @ sk2 ) ),
    inference(simp,[status(thm)],[3255]) ).

thf(4,axiom,
    ! [A: $i,B: $i] :
      ( ( unordered_pair @ A @ B )
      = ( unordered_pair @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k2_tarski) ).

thf(23,plain,
    ! [A: $i,B: $i] :
      ( ( unordered_pair @ A @ B )
      = ( unordered_pair @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(24,plain,
    ! [B: $i,A: $i] :
      ( ( unordered_pair @ A @ B )
      = ( unordered_pair @ B @ A ) ),
    inference(cnf,[status(esa)],[23]) ).

thf(25,plain,
    ! [B: $i,A: $i] :
      ( ( unordered_pair @ A @ B )
      = ( unordered_pair @ B @ A ) ),
    inference(lifteq,[status(thm)],[24]) ).

thf(3330,plain,
    $false,
    inference(simplifyReflect,[status(thm)],[3289,25]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.02/0.11  % Problem  : SET930+1 : TPTP v8.2.0. Released v3.2.0.
% 0.12/0.13  % Command  : run_Leo-III %s %d
% 0.12/0.33  % Computer : n003.cluster.edu
% 0.12/0.33  % Model    : x86_64 x86_64
% 0.12/0.33  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33  % Memory   : 8042.1875MB
% 0.12/0.33  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33  % CPULimit : 300
% 0.12/0.33  % WCLimit  : 300
% 0.12/0.33  % DateTime : Mon May 20 11:25:09 EDT 2024
% 0.12/0.33  % CPUTime  : 
% 0.70/0.75  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 1.01/0.85  % [INFO] 	 Parsing done (97ms). 
% 1.01/0.86  % [INFO] 	 Running in sequential loop mode. 
% 1.32/1.06  % [INFO] 	 nitpick registered as external prover. 
% 1.32/1.06  % [INFO] 	 Scanning for conjecture ... 
% 1.51/1.15  % [INFO] 	 Found a conjecture (or negated_conjecture) and 8 axioms. Running axiom selection ... 
% 1.67/1.18  % [INFO] 	 Axiom selection finished. Selected 8 axioms (removed 0 axioms). 
% 1.67/1.19  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 1.80/1.20  % [INFO] 	 Type checking passed. 
% 1.80/1.20  % [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 ... 
% 15.61/3.50  % [INFO] 	 Killing All external provers ... 
% 15.61/3.51  % Time passed: 3026ms (effective reasoning time: 2644ms)
% 15.61/3.51  % 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)>
% 15.61/3.51  % Axioms used in derivation (4): t72_zfmisc_1, l39_zfmisc_1, t73_zfmisc_1, commutativity_k2_tarski
% 15.61/3.51  % No. of inferences in proof: 54
% 15.61/3.51  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 3026 ms resp. 2644 ms w/o parsing
% 15.61/3.58  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 15.61/3.58  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------