TSTP Solution File: SET952+1 by Leo-III---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : SET952+1 : TPTP v8.2.0. Released v3.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n007.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 03:04:37 EDT 2024

% Result   : Theorem 4.47s 2.48s
% Output   : Refutation 4.76s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :   30
% Syntax   : Number of formulae    :   53 (  22 unt;   9 typ;   0 def)
%            Number of atoms       :  136 (  35 equ;   0 cnn)
%            Maximal formula atoms :   18 (   3 avg)
%            Number of connectives :  402 (  16   ~;  10   |;  30   &; 294   @)
%                                         (  11 <=>;  41  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   15 (   7 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :   15 (  15   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   11 (   9 usr;   1 con; 0-2 aty)
%            Number of variables   :  115 (   0   ^ 101   !;  14   ?; 115   :)

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

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

thf(powerset_type,type,
    powerset: $i > $i ).

thf(set_union2_type,type,
    set_union2: $i > $i > $i ).

thf(empty_type,type,
    empty: $i > $o ).

thf(ordered_pair_type,type,
    ordered_pair: $i > $i > $i ).

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

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

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

thf(15,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( empty @ A )
     => ~ ( empty @ ( set_union2 @ A @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc2_xboole_0) ).

thf(120,plain,
    ! [A: $i,B: $i] :
      ( ~ ( empty @ A )
     => ~ ( empty @ ( set_union2 @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(3,axiom,
    ? [A: $i] : ( empty @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_xboole_0) ).

thf(25,plain,
    ? [A: $i] : ( empty @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(19,axiom,
    ! [A: $i,B: $i] :
      ( ( subset @ A @ B )
    <=> ! [C: $i] :
          ( ( in @ C @ A )
         => ( in @ C @ B ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_tarski) ).

thf(164,plain,
    ! [A: $i,B: $i] :
      ( ( ( subset @ A @ B )
       => ! [C: $i] :
            ( ( in @ C @ A )
           => ( in @ C @ B ) ) )
      & ( ! [C: $i] :
            ( ( in @ C @ A )
           => ( in @ C @ B ) )
       => ( subset @ A @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(12,axiom,
    ! [A: $i,B: $i] : ( subset @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_r1_tarski) ).

thf(96,plain,
    ! [A: $i] : ( subset @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(9,axiom,
    ! [A: $i,B: $i] :
      ~ ( empty @ ( ordered_pair @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_zfmisc_1) ).

thf(64,plain,
    ! [A: $i,B: $i] :
      ~ ( empty @ ( ordered_pair @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(13,axiom,
    ! [A: $i,B: $i] :
      ( ( B
        = ( powerset @ A ) )
    <=> ! [C: $i] :
          ( ( in @ C @ B )
        <=> ( subset @ C @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d1_zfmisc_1) ).

thf(98,plain,
    ! [A: $i,B: $i] :
      ( ( ( B
          = ( powerset @ A ) )
       => ! [C: $i] :
            ( ( ( in @ C @ B )
             => ( subset @ C @ A ) )
            & ( ( subset @ C @ A )
             => ( in @ C @ B ) ) ) )
      & ( ! [C: $i] :
            ( ( ( in @ C @ B )
             => ( subset @ C @ A ) )
            & ( ( subset @ C @ A )
             => ( in @ C @ B ) ) )
       => ( B
          = ( powerset @ A ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(7,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( unordered_pair @ A @ B ) )
    <=> ! [D: $i] :
          ( ( in @ D @ C )
        <=> ( ( D = A )
            | ( D = B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_tarski) ).

thf(35,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( unordered_pair @ A @ B ) )
       => ! [D: $i] :
            ( ( ( in @ D @ C )
             => ( ( D = A )
                | ( D = B ) ) )
            & ( ( ( D = A )
                | ( D = B ) )
             => ( in @ D @ C ) ) ) )
      & ( ! [D: $i] :
            ( ( ( in @ D @ C )
             => ( ( D = A )
                | ( D = B ) ) )
            & ( ( ( D = A )
                | ( D = B ) )
             => ( in @ D @ C ) ) )
       => ( C
          = ( unordered_pair @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(16,axiom,
    ! [A: $i,B: $i] :
      ( ~ ( empty @ A )
     => ~ ( empty @ ( set_union2 @ B @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc3_xboole_0) ).

thf(123,plain,
    ! [A: $i,B: $i] :
      ( ~ ( empty @ A )
     => ~ ( empty @ ( set_union2 @ B @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(10,axiom,
    ! [A: $i,B: $i] :
      ( ( set_union2 @ A @ A )
      = A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k2_xboole_0) ).

thf(67,plain,
    ! [A: $i] :
      ( ( set_union2 @ A @ A )
      = A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(5,axiom,
    ! [A: $i,B: $i] :
      ( ( ordered_pair @ A @ B )
      = ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d5_tarski) ).

thf(30,plain,
    ! [A: $i,B: $i] :
      ( ( ordered_pair @ A @ B )
      = ( unordered_pair @ ( unordered_pair @ A @ B ) @ ( singleton @ A ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(17,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( subset @ ( unordered_pair @ A @ B ) @ C )
    <=> ( ( in @ A @ C )
        & ( in @ B @ C ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t38_zfmisc_1) ).

thf(126,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( subset @ ( unordered_pair @ A @ B ) @ C )
       => ( ( in @ A @ C )
          & ( in @ B @ C ) ) )
      & ( ( ( in @ A @ C )
          & ( in @ B @ C ) )
       => ( subset @ ( unordered_pair @ A @ B ) @ C ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(22,axiom,
    ! [A: $i,B: $i] :
      ( ( set_union2 @ A @ B )
      = ( set_union2 @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k2_xboole_0) ).

thf(178,plain,
    ! [A: $i,B: $i] :
      ( ( set_union2 @ A @ B )
      = ( set_union2 @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).

thf(8,axiom,
    ! [A: $i,B: $i] :
      ( ( unordered_pair @ A @ B )
      = ( unordered_pair @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k2_tarski) ).

thf(61,plain,
    ! [A: $i,B: $i] :
      ( ( unordered_pair @ A @ B )
      = ( unordered_pair @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(18,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( cartesian_product2 @ A @ B ) )
    <=> ! [D: $i] :
          ( ( in @ D @ C )
        <=> ? [E: $i,F: $i] :
              ( ( in @ E @ A )
              & ( in @ F @ B )
              & ( D
                = ( ordered_pair @ E @ F ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_zfmisc_1) ).

thf(132,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( cartesian_product2 @ A @ B ) )
       => ! [D: $i] :
            ( ( ( in @ D @ C )
             => ? [E: $i,F: $i] :
                  ( ( in @ E @ A )
                  & ( in @ F @ B )
                  & ( D
                    = ( ordered_pair @ E @ F ) ) ) )
            & ( ? [E: $i,F: $i] :
                  ( ( in @ E @ A )
                  & ( in @ F @ B )
                  & ( D
                    = ( ordered_pair @ E @ F ) ) )
             => ( in @ D @ C ) ) ) )
      & ( ! [D: $i] :
            ( ( ( in @ D @ C )
             => ? [E: $i,F: $i] :
                  ( ( in @ E @ A )
                  & ( in @ F @ B )
                  & ( D
                    = ( ordered_pair @ E @ F ) ) ) )
            & ( ? [E: $i,F: $i] :
                  ( ( in @ E @ A )
                  & ( in @ F @ B )
                  & ( D
                    = ( ordered_pair @ E @ F ) ) )
             => ( in @ D @ C ) ) )
       => ( C
          = ( cartesian_product2 @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(11,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( C
        = ( set_union2 @ A @ B ) )
    <=> ! [D: $i] :
          ( ( in @ D @ C )
        <=> ( ( in @ D @ A )
            | ( in @ D @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d2_xboole_0) ).

thf(70,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( C
          = ( set_union2 @ A @ B ) )
       => ! [D: $i] :
            ( ( ( in @ D @ C )
             => ( ( in @ D @ A )
                | ( in @ D @ B ) ) )
            & ( ( ( in @ D @ A )
                | ( in @ D @ B ) )
             => ( in @ D @ C ) ) ) )
      & ( ! [D: $i] :
            ( ( ( in @ D @ C )
             => ( ( in @ D @ A )
                | ( in @ D @ B ) ) )
            & ( ( ( in @ D @ A )
                | ( in @ D @ B ) )
             => ( in @ D @ C ) ) )
       => ( C
          = ( set_union2 @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(6,axiom,
    ! [A: $i,B: $i] :
      ( ( in @ A @ B )
     => ~ ( in @ B @ A ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',antisymmetry_r2_hidden) ).

thf(33,plain,
    ! [A: $i,B: $i] :
      ( ( in @ A @ B )
     => ~ ( in @ B @ A ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(21,axiom,
    ! [A: $i,B: $i] : ( subset @ A @ ( set_union2 @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_xboole_1) ).

thf(176,plain,
    ! [A: $i,B: $i] : ( subset @ A @ ( set_union2 @ A @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).

thf(14,axiom,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( subset @ A @ B )
        & ( subset @ B @ C ) )
     => ( subset @ A @ C ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_xboole_1) ).

thf(118,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( ( subset @ A @ B )
        & ( subset @ B @ C ) )
     => ( subset @ A @ C ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(4,axiom,
    ? [A: $i] :
      ~ ( empty @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_xboole_0) ).

thf(27,plain,
    ? [A: $i] :
      ~ ( empty @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(20,axiom,
    ! [A: $i,B: $i] :
      ( ( subset @ ( singleton @ A ) @ B )
    <=> ( in @ A @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l2_zfmisc_1) ).

thf(171,plain,
    ! [A: $i,B: $i] :
      ( ( ( subset @ ( singleton @ A ) @ B )
       => ( in @ A @ B ) )
      & ( ( in @ A @ B )
       => ( subset @ ( singleton @ A ) @ B ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(1,conjecture,
    ! [A: $i,B: $i] : ( subset @ ( cartesian_product2 @ A @ B ) @ ( powerset @ ( powerset @ ( set_union2 @ A @ B ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t105_zfmisc_1) ).

thf(2,negated_conjecture,
    ~ ! [A: $i,B: $i] : ( subset @ ( cartesian_product2 @ A @ B ) @ ( powerset @ ( powerset @ ( set_union2 @ A @ B ) ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(23,plain,
    ~ ! [A: $i,B: $i] : ( subset @ ( cartesian_product2 @ A @ B ) @ ( powerset @ ( powerset @ ( set_union2 @ A @ B ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(192,plain,
    $false,
    inference(cvc4,[status(thm)],[120,25,164,96,64,98,35,123,67,30,126,178,61,132,70,33,176,118,27,171,23]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.09  % Problem  : SET952+1 : TPTP v8.2.0. Released v3.2.0.
% 0.09/0.11  % Command  : run_Leo-III %s %d
% 0.12/0.31  % Computer : n007.cluster.edu
% 0.12/0.31  % Model    : x86_64 x86_64
% 0.12/0.31  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.31  % Memory   : 8042.1875MB
% 0.12/0.31  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.12/0.31  % CPULimit : 300
% 0.12/0.31  % WCLimit  : 300
% 0.12/0.31  % DateTime : Mon May 20 11:40:39 EDT 2024
% 0.16/0.31  % CPUTime  : 
% 0.87/0.90  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 1.30/1.07  % [INFO] 	 Parsing done (157ms). 
% 1.33/1.09  % [INFO] 	 Running in sequential loop mode. 
% 1.86/1.43  % [INFO] 	 eprover registered as external prover. 
% 1.86/1.44  % [INFO] 	 cvc4 registered as external prover. 
% 1.86/1.44  % [INFO] 	 Scanning for conjecture ... 
% 1.94/1.51  % [INFO] 	 Found a conjecture (or negated_conjecture) and 20 axioms. Running axiom selection ... 
% 2.09/1.57  % [INFO] 	 Axiom selection finished. Selected 20 axioms (removed 0 axioms). 
% 2.21/1.62  % [INFO] 	 Problem is first-order (TPTP FOF). 
% 2.21/1.63  % [INFO] 	 Type checking passed. 
% 2.21/1.63  % [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 ... 
% 4.47/2.48  % External prover 'cvc4' found a proof!
% 4.47/2.48  % [INFO] 	 Killing All external provers ... 
% 4.47/2.48  % Time passed: 2040ms (effective reasoning time: 1378ms)
% 4.47/2.48  % 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)>
% 4.47/2.48  % Axioms used in derivation (20): fc1_zfmisc_1, d5_tarski, l2_zfmisc_1, rc1_xboole_0, rc2_xboole_0, antisymmetry_r2_hidden, d2_tarski, fc2_xboole_0, t1_xboole_1, commutativity_k2_xboole_0, fc3_xboole_0, d2_xboole_0, d2_zfmisc_1, commutativity_k2_tarski, d1_zfmisc_1, t38_zfmisc_1, d3_tarski, reflexivity_r1_tarski, idempotence_k2_xboole_0, t7_xboole_1
% 4.47/2.48  % No. of inferences in proof: 44
% 4.47/2.48  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 2040 ms resp. 1378 ms w/o parsing
% 4.76/2.54  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 4.76/2.55  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------