TSTP Solution File: SET926^20 by Leo-III---1.7.12

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : SET926^20 : 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 03:04:33 EDT 2024

% Result   : Theorem 4.04s 2.16s
% Output   : Refutation 4.24s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :   45
% Syntax   : Number of formulae    :   66 (  14 unt;  18 typ;   8 def)
%            Number of atoms       :  201 (  11 equ;   0 cnn)
%            Maximal formula atoms :    9 (   4 avg)
%            Number of connectives :  459 (   6   ~;   2   |;  10   &; 398   @)
%                                         (   0 <=>;  43  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   14 (   8 avg)
%            Number of types       :    3 (   1 usr)
%            Number of type conns  :   62 (  62   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   19 (  17 usr;   3 con; 0-3 aty)
%            Number of variables   :   98 (  52   ^  41   !;   5   ?;  98   :)

% 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(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(eiw_di_type,type,
    eiw_di: $i > mworld > $o ).

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] :
          ( ( eiw_di @ C @ B )
         => ( 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] :
          ( ( eiw_di @ C @ B )
          & ( A @ C @ B ) ) ) ) ).

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

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

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

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

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

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

thf(17,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( set_difference @ A @ C ) @ ( set_difference @ B @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_difference_substitution_1) ).

thf(56,plain,
    ! [A: $i] :
      ( ( eiw_di @ A @ mactual )
     => ! [B: $i] :
          ( ( eiw_di @ B @ mactual )
         => ! [C: $i] :
              ( ( eiw_di @ C @ mactual )
             => ( ( qmltpeq @ A @ B @ mactual )
               => ( qmltpeq @ ( set_difference @ A @ C ) @ ( set_difference @ B @ C ) @ mactual ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(8,axiom,
    ( mlocal
    @ ( mexists_di
      @ ^ [A: $i] : ( mnot @ ( empty @ A ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc2_xboole_0) ).

thf(37,plain,
    ? [A: $i] :
      ( ( eiw_di @ A @ mactual )
      & ~ ( empty @ A @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

thf(19,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] : ( qmltpeq @ A @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity) ).

thf(61,plain,
    ! [A: $i] :
      ( ( eiw_di @ A @ mactual )
     => ( qmltpeq @ A @ A @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).

thf(16,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mequiv @ ( qmltpeq @ ( set_difference @ ( singleton @ A ) @ B ) @ ( singleton @ A ) ) @ ( mnot @ ( in @ A @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l34_zfmisc_1) ).

thf(53,plain,
    ! [A: $i] :
      ( ( eiw_di @ A @ mactual )
     => ! [B: $i] :
          ( ( eiw_di @ B @ mactual )
         => ( ( qmltpeq @ ( set_difference @ ( singleton @ A ) @ B ) @ ( singleton @ A ) @ mactual )
            = ( ~ ( in @ A @ B @ mactual ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(10,axiom,
    ! [A: mworld] :
    ? [B: $i] : ( eiw_di @ B @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',eiw_di_nonempty) ).

thf(41,plain,
    ! [A: mworld] :
    ? [B: $i] : ( eiw_di @ B @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

thf(14,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( in @ A @ C ) ) @ ( in @ B @ C ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',in_substitution_1) ).

thf(49,plain,
    ! [A: $i] :
      ( ( eiw_di @ A @ mactual )
     => ! [B: $i] :
          ( ( eiw_di @ B @ mactual )
         => ! [C: $i] :
              ( ( eiw_di @ C @ mactual )
             => ( ( ( qmltpeq @ A @ B @ mactual )
                  & ( in @ A @ C @ mactual ) )
               => ( in @ B @ C @ mactual ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).

thf(7,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( singleton @ A ) @ ( singleton @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_substitution_1) ).

thf(35,plain,
    ! [A: $i] :
      ( ( eiw_di @ A @ mactual )
     => ! [B: $i] :
          ( ( eiw_di @ B @ mactual )
         => ( ( qmltpeq @ A @ B @ mactual )
           => ( qmltpeq @ ( singleton @ A ) @ ( singleton @ B ) @ mactual ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(20,axiom,
    ! [A: mworld,B: mworld,C: $i] :
      ( ( ( eiw_di @ C @ A )
        & ( mrel @ B @ A ) )
     => ( eiw_di @ C @ B ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',eiw_di_decr) ).

thf(63,plain,
    ! [A: mworld,B: mworld,C: $i] :
      ( ( ( eiw_di @ C @ A )
        & ( mrel @ B @ A ) )
     => ( eiw_di @ C @ B ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).

thf(11,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( set_difference @ C @ A ) @ ( set_difference @ C @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_difference_substitution_2) ).

thf(43,plain,
    ! [A: $i] :
      ( ( eiw_di @ A @ mactual )
     => ! [B: $i] :
          ( ( eiw_di @ B @ mactual )
         => ! [C: $i] :
              ( ( eiw_di @ C @ mactual )
             => ( ( qmltpeq @ A @ B @ mactual )
               => ( qmltpeq @ ( set_difference @ C @ A ) @ ( set_difference @ C @ B ) @ mactual ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(18,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mequiv @ ( qmltpeq @ ( set_difference @ ( singleton @ A ) @ B ) @ empty_set ) @ ( in @ A @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l36_zfmisc_1) ).

thf(58,plain,
    ! [A: $i] :
      ( ( eiw_di @ A @ mactual )
     => ! [B: $i] :
          ( ( eiw_di @ B @ mactual )
         => ( ( qmltpeq @ ( set_difference @ ( singleton @ A ) @ B ) @ empty_set @ mactual )
            = ( in @ A @ B @ mactual ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(13,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',symmetry) ).

thf(47,plain,
    ! [A: $i] :
      ( ( eiw_di @ A @ mactual )
     => ! [B: $i] :
          ( ( eiw_di @ B @ mactual )
         => ( ( qmltpeq @ A @ B @ mactual )
           => ( qmltpeq @ B @ A @ mactual ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).

thf(4,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mimplies @ ( in @ A @ B ) @ ( mnot @ ( in @ B @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',antisymmetry_r2_hidden) ).

thf(28,plain,
    ! [A: $i] :
      ( ( eiw_di @ A @ mactual )
     => ! [B: $i] :
          ( ( eiw_di @ B @ mactual )
         => ( ( in @ A @ B @ mactual )
           => ~ ( in @ B @ A @ mactual ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).

thf(1,conjecture,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mor @ ( qmltpeq @ ( set_difference @ ( singleton @ A ) @ B ) @ empty_set ) @ ( qmltpeq @ ( set_difference @ ( singleton @ A ) @ B ) @ ( singleton @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t69_zfmisc_1) ).

thf(2,negated_conjecture,
    ~ ( mlocal
      @ ( mforall_di
        @ ^ [A: $i] :
            ( mforall_di
            @ ^ [B: $i] : ( mor @ ( qmltpeq @ ( set_difference @ ( singleton @ A ) @ B ) @ empty_set ) @ ( qmltpeq @ ( set_difference @ ( singleton @ A ) @ B ) @ ( singleton @ A ) ) ) ) ) ),
    inference(neg_conjecture,[status(cth)],[1]) ).

thf(21,plain,
    ~ ! [A: $i] :
        ( ( eiw_di @ A @ mactual )
       => ! [B: $i] :
            ( ( eiw_di @ B @ mactual )
           => ( ( qmltpeq @ ( set_difference @ ( singleton @ A ) @ B ) @ empty_set @ mactual )
              | ( qmltpeq @ ( set_difference @ ( singleton @ A ) @ B ) @ ( singleton @ A ) @ mactual ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(6,axiom,
    mlocal @ ( mexists_di @ empty ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',rc1_xboole_0) ).

thf(32,plain,
    ? [A: $i] :
      ( ( eiw_di @ A @ mactual )
      & ( empty @ A @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).

thf(12,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( empty @ A ) ) @ ( empty @ B ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',empty_substitution_1) ).

thf(45,plain,
    ! [A: $i] :
      ( ( eiw_di @ A @ mactual )
     => ! [B: $i] :
          ( ( eiw_di @ B @ mactual )
         => ( ( ( qmltpeq @ A @ B @ mactual )
              & ( empty @ A @ mactual ) )
           => ( empty @ B @ mactual ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(9,axiom,
    mlocal @ ( empty @ empty_set ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_xboole_0) ).

thf(40,plain,
    empty @ empty_set @ mactual,
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).

thf(3,axiom,
    ! [A: mworld] : ( mrel @ A @ A ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',mrel_reflexive) ).

thf(26,plain,
    ! [A: mworld] : ( mrel @ A @ A ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).

thf(5,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( in @ C @ A ) ) @ ( in @ C @ B ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',in_substitution_2) ).

thf(30,plain,
    ! [A: $i] :
      ( ( eiw_di @ A @ mactual )
     => ! [B: $i] :
          ( ( eiw_di @ B @ mactual )
         => ! [C: $i] :
              ( ( eiw_di @ C @ mactual )
             => ( ( ( qmltpeq @ A @ B @ mactual )
                  & ( in @ C @ A @ mactual ) )
               => ( in @ C @ B @ mactual ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(15,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/sandbox2/benchmark/theBenchmark.p',transitivity) ).

thf(51,plain,
    ! [A: $i] :
      ( ( eiw_di @ A @ mactual )
     => ! [B: $i] :
          ( ( eiw_di @ B @ mactual )
         => ! [C: $i] :
              ( ( eiw_di @ C @ mactual )
             => ( ( ( qmltpeq @ A @ B @ mactual )
                  & ( qmltpeq @ B @ C @ mactual ) )
               => ( qmltpeq @ A @ C @ mactual ) ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).

thf(65,plain,
    $false,
    inference(e,[status(thm)],[56,37,61,53,41,49,35,63,43,58,47,28,21,32,45,40,26,30,51]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.10  % Problem  : SET926^20 : TPTP v8.2.0. Released v8.1.0.
% 0.04/0.12  % Command  : run_Leo-III %s %d
% 0.10/0.31  % Computer : n022.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 11:05:54 EDT 2024
% 0.10/0.31  % CPUTime  : 
% 0.90/0.90  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 1.27/1.06  % [INFO] 	 Parsing done (158ms). 
% 1.41/1.07  % [INFO] 	 Running in sequential loop mode. 
% 1.94/1.37  % [INFO] 	 eprover registered as external prover. 
% 1.94/1.37  % [INFO] 	 cvc4 registered as external prover. 
% 1.94/1.38  % [INFO] 	 Scanning for conjecture ... 
% 2.13/1.46  % [INFO] 	 Found a conjecture (or negated_conjecture) and 18 axioms. Running axiom selection ... 
% 2.23/1.51  % [INFO] 	 Axiom selection finished. Selected 18 axioms (removed 0 axioms). 
% 2.23/1.53  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 2.23/1.53  % [INFO] 	 Type checking passed. 
% 2.23/1.54  % [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 ... 
% 3.88/2.16  % External prover 'e' found a proof!
% 3.88/2.16  % [INFO] 	 Killing All external provers ... 
% 3.88/2.16  % Time passed: 1692ms (effective reasoning time: 1081ms)
% 3.88/2.16  % 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)>
% 4.04/2.16  % Axioms used in derivation (18): in_substitution_2, set_difference_substitution_2, fc1_xboole_0, empty_substitution_1, mrel_reflexive, l34_zfmisc_1, set_difference_substitution_1, reflexivity, eiw_di_nonempty, in_substitution_1, rc1_xboole_0, transitivity, rc2_xboole_0, l36_zfmisc_1, singleton_substitution_1, antisymmetry_r2_hidden, symmetry, eiw_di_decr
% 4.04/2.16  % No. of inferences in proof: 40
% 4.04/2.16  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 1692 ms resp. 1081 ms w/o parsing
% 4.24/2.26  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 4.24/2.26  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------