TSTP Solution File: SET002^3 by Leo-III---1.7.12

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%------------------------------------------------------------------------------
% File     : Leo-III---1.7.12
% Problem  : SET002^3 : TPTP v8.2.0. Released v8.1.0.
% 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 03:00:40 EDT 2024

% Result   : Theorem 4.93s 2.87s
% Output   : Refutation 5.15s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    3
%            Number of leaves      :   35
% Syntax   : Number of formulae    :   54 (  15 unt;  12 typ;   6 def)
%            Number of atoms       :  180 (  12 equ;   0 cnn)
%            Maximal formula atoms :    9 (   4 avg)
%            Number of connectives :  372 (   2   ~;   2   |;   7   &; 350   @)
%                                         (   0 <=>;  11  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   14 (   8 avg)
%            Number of types       :    3 (   1 usr)
%            Number of type conns  :   48 (  48   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   13 (  11 usr;   2 con; 0-3 aty)
%            Number of variables   :   99 (  57   ^  42   !;   0   ?;  99   :)

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

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(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(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(union_type,type,
    union: $i > $i > $i ).

thf(16,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ A @ C ) @ ( union @ B @ C ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_substitution_1) ).

thf(52,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( union @ A @ C ) @ ( union @ B @ C ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).

thf(8,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mequiv @ ( qmltpeq @ A @ B )
              @ ( mforall_di
                @ ^ [C: $i] : ( mequiv @ ( member @ C @ A ) @ ( member @ C @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_member_defn) ).

thf(34,plain,
    ! [A: $i,B: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
      = ( ! [C: $i] :
            ( ( member @ C @ A @ mactual )
            = ( member @ C @ B @ mactual ) ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).

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

thf(44,plain,
    ! [A: $i,B: $i] :
      ( ( subset @ A @ B @ mactual )
     => ( qmltpeq @ ( union @ A @ B ) @ B @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).

thf(14,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/sandbox2/benchmark/theBenchmark.p',member_substitution_2) ).

thf(48,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)],[14]) ).

thf(1,conjecture,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] : ( qmltpeq @ ( union @ A @ A ) @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_idempotency_of_union) ).

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

thf(19,plain,
    ~ ! [A: $i] : ( qmltpeq @ ( union @ A @ A ) @ A @ mactual ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).

thf(11,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] : ( subset @ A @ A ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',reflexivity_of_subset) ).

thf(42,plain,
    ! [A: $i] : ( subset @ A @ A @ mactual ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).

thf(9,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( mequiv @ ( qmltpeq @ A @ B ) @ ( mand @ ( subset @ A @ B ) @ ( subset @ B @ A ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',equal_defn) ).

thf(37,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)],[9]) ).

thf(5,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mequiv @ ( member @ C @ ( union @ A @ B ) ) @ ( mor @ ( member @ C @ A ) @ ( member @ C @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_defn) ).

thf(25,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( member @ C @ ( union @ A @ B ) @ mactual )
      = ( ( member @ C @ A @ mactual )
        | ( member @ C @ B @ mactual ) ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).

thf(13,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/sandbox2/benchmark/theBenchmark.p',member_substitution_1) ).

thf(46,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)],[13]) ).

thf(18,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(57,plain,
    ! [A: $i,B: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ B @ A @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).

thf(6,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/sandbox2/benchmark/theBenchmark.p',subset_defn) ).

thf(28,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)],[6]) ).

thf(3,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(21,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)],[3]) ).

thf(15,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/sandbox2/benchmark/theBenchmark.p',subset_substitution_2) ).

thf(50,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)],[15]) ).

thf(7,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] :
              ( mforall_di
              @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ C @ A ) @ ( union @ C @ B ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_substitution_2) ).

thf(31,plain,
    ! [A: $i,B: $i,C: $i] :
      ( ( qmltpeq @ A @ B @ mactual )
     => ( qmltpeq @ ( union @ C @ A ) @ ( union @ C @ B ) @ mactual ) ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).

thf(10,axiom,
    ( mlocal
    @ ( mforall_di
      @ ^ [A: $i] :
          ( mforall_di
          @ ^ [B: $i] : ( qmltpeq @ ( union @ A @ B ) @ ( union @ B @ A ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_of_union) ).

thf(40,plain,
    ! [A: $i,B: $i] : ( qmltpeq @ ( union @ A @ B ) @ ( union @ B @ A ) @ mactual ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).

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

thf(55,plain,
    ! [A: $i] : ( qmltpeq @ A @ A @ mactual ),
    inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).

thf(4,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/sandbox2/benchmark/theBenchmark.p',subset_substitution_1) ).

thf(23,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)],[4]) ).

thf(59,plain,
    $false,
    inference(e,[status(thm)],[52,34,44,48,19,42,37,25,46,57,28,21,50,31,40,55,23]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12  % Problem  : SET002^3 : TPTP v8.2.0. Released v8.1.0.
% 0.13/0.15  % Command  : run_Leo-III %s %d
% 0.15/0.36  % Computer : n003.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
% 0.15/0.37  % CPULimit : 300
% 0.15/0.37  % WCLimit  : 300
% 0.15/0.37  % DateTime : Mon May 20 13:06:39 EDT 2024
% 0.15/0.37  % CPUTime  : 
% 1.16/1.13  % [INFO] 	 Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ... 
% 1.69/1.36  % [INFO] 	 Parsing done (227ms). 
% 1.69/1.38  % [INFO] 	 Running in sequential loop mode. 
% 2.51/1.81  % [INFO] 	 eprover registered as external prover. 
% 2.51/1.82  % [INFO] 	 cvc4 registered as external prover. 
% 2.54/1.82  % [INFO] 	 Scanning for conjecture ... 
% 2.74/1.97  % [INFO] 	 Found a conjecture (or negated_conjecture) and 17 axioms. Running axiom selection ... 
% 2.89/2.05  % [INFO] 	 Axiom selection finished. Selected 16 axioms (removed 1 axioms). 
% 3.17/2.08  % [INFO] 	 Problem is higher-order (TPTP THF). 
% 3.17/2.09  % [INFO] 	 Type checking passed. 
% 3.17/2.10  % [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 ... 
% 4.93/2.86  % External prover 'e' found a proof!
% 4.93/2.86  % [INFO] 	 Killing All external provers ... 
% 4.93/2.86  % Time passed: 2331ms (effective reasoning time: 1477ms)
% 4.93/2.86  % 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)>
% 4.93/2.87  % Axioms used in derivation (16): member_substitution_1, reflexivity_of_subset, equal_member_defn, subset_substitution_2, union_substitution_2, subset_union, equal_defn, member_substitution_2, symmetry, reflexivity, commutativity_of_union, subset_substitution_1, union_substitution_1, transitivity, union_defn, subset_defn
% 4.93/2.87  % No. of inferences in proof: 36
% 4.93/2.87  % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 2331 ms resp. 1477 ms w/o parsing
% 5.15/2.97  % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 5.15/2.97  % [INFO] 	 Killing All external provers ... 
%------------------------------------------------------------------------------