TSTP Solution File: SET579^3 by Leo-III---1.7.12

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%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : SET579^3 : TPTP v8.2.0. Released v8.1.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_Leo-III %s %d

% Computer : n022.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 03:02:57 EDT 2024

% Result   : Theorem 3.12s 1.76s
% Output   : Refutation 3.53s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :   34
% Syntax   : Number of formulae    :   51 (  14 unt;  13 typ;   6 def)
%            Number of atoms       :  180 (  11 equ;   0 cnn)
%            Maximal formula atoms :   13 (   4 avg)
%            Number of connectives :  362 (   5   ~;   0   |;   9   &; 337   @)
%                                         (   0 <=>;  11  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   8 avg)
%            Number of types       :    3 (   1 usr)
%            Number of type conns  :   47 (  47   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   14 (  12 usr;   2 con; 0-3 aty)
%            Number of variables   :   97 (  55   ^  40   !;   2   ?;  97   :)

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

thf(mrel_type,type,
    mrel: mworld > mworld > $o ).

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

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

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

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

thf(16,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] : ( subset @ A @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',reflexivity_of_subset) ).

thf(52,plain,
    ! [A: $i] : ( subset @ A @ A @ mactual ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(3,axiom,
    ! [A: mworld] :
    ? [B: mworld] : ( mrel @ A @ B ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',mrel_serial) ).

thf(21,plain,
    ! [A: mworld] :
    ? [B: mworld] : ( mrel @ A @ B ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(12,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(41,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)],[12]) ).

thf(6,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(27,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)],[6]) ).

thf(15,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_defn) ).

thf(49,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)],[15]) ).

thf(11,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(39,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)],[11]) ).

thf(9,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(35,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)],[9]) ).

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

thf(43,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( member @ C @ ( difference @ A @ B ) @ mactual )
      = ( ( member @ C @ A @ mactual )
        & ~ ( member @ C @ B @ mactual ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(4,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] : ( qmltpeq @ A @ A ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',reflexivity) ).

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

thf(10,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(37,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)],[10]) ).

thf(5,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(25,plain,
    ! [A: $i,B: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ B @ A @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(14,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mequiv @ ( qmltpeq @ A @ B ) @ ( mand @ ( subset @ A @ B ) @ ( subset @ B @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',equal_defn) ).

thf(46,plain,
    ! [A: $i,B: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
      = ( ( subset @ A @ B @ mactual )
        & ( subset @ B @ A @ mactual ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(7,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(29,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)],[7]) ).

thf(8,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(32,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)],[8]) ).

thf(1,conjecture,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] :
                  ( mimplies
                  @ ( mforall_di
                    @ ^ [D: $i] : ( mequiv @ ( member @ D @ A ) @ ( mand @ ( member @ D @ B ) @ ( mnot @ ( member @ D @ C ) ) ) ) )
                  @ ( qmltpeq @ A @ ( difference @ B @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_th20) ).

thf(2,negated_conjecture,
    ~ ( mlocal
      @ ( mforall_di
        @ ^ [A: $i] :
            ( mforall_di
            @ ^ [B: $i] :
                ( mforall_di
                @ ^ [C: $i] :
                    ( mimplies
                    @ ( mforall_di
                      @ ^ [D: $i] : ( mequiv @ ( member @ D @ A ) @ ( mand @ ( member @ D @ B ) @ ( mnot @ ( member @ D @ C ) ) ) ) )
                    @ ( qmltpeq @ A @ ( difference @ B @ C ) ) ) ) ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(17,plain,
    ~ ! [A: $i,B: $i,C: $i] :
        ( ! [D: $i] :
            ( ( member @ D @ A @ mactual )
            = ( ( member @ D @ B @ mactual )
              & ~ ( member @ D @ C @ mactual ) ) )
       => ( qmltpeq @ A @ ( difference @ B @ C ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(54,plain,
    $false,
    inference(cvc4,[status(thm)],[52,21,41,27,49,39,35,43,23,37,25,46,29,32,17]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12  % Problem  : SET579^3 : TPTP v8.2.0. Released v8.1.0.
% 0.13/0.15  % Command  : run_Leo-III %s %d
% 0.15/0.36  % Computer : n022.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 13:24:24 EDT 2024
% 0.15/0.36  % CPUTime  : 
% 0.95/0.87  % [INFO] 	 Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ... 
% 1.20/1.00  % [INFO] 	 Parsing done (130ms). 
% 1.20/1.01  % [INFO] 	 Running in sequential loop mode. 
% 1.77/1.21  % [INFO] 	 eprover registered as external prover. 
% 1.77/1.21  % [INFO] 	 cvc4 registered as external prover. 
% 1.77/1.22  % [INFO] 	 Scanning for conjecture ... 
% 1.88/1.29  % [INFO] 	 Found a conjecture (or negated_conjecture) and 14 axioms. Running axiom selection ... 
% 2.08/1.31  % [INFO] 	 Axiom selection finished. Selected 14 axioms (removed 0 axioms). 
% 2.08/1.33  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 2.08/1.34  % [INFO] 	 Type checking passed. 
% 2.08/1.34  % [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 ... 
% 3.12/1.75  % External prover 'cvc4' found a proof!
% 3.12/1.75  % [INFO] 	 Killing All external provers ... 
% 3.12/1.76  % Time passed: 1224ms (effective reasoning time: 741ms)
% 3.12/1.76  % 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)>
% 3.12/1.76  % Axioms used in derivation (14): member_substitution_1, reflexivity_of_subset, subset_substitution_2, difference_substitution_2, subset_substitution_1, difference_substitution_1, member_substitution_2, symmetry, reflexivity, difference_defn, equal_defn, mrel_serial, transitivity, subset_defn
% 3.12/1.76  % No. of inferences in proof: 32
% 3.12/1.76  % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 1224 ms resp. 741 ms w/o parsing
% 3.53/1.81  % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 3.53/1.81  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------