TSTP Solution File: SET090-7 by Z3---4.8.9.0

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Z3---4.8.9.0
% Problem  : SET090-7 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm  : none
% Format   : tptp
% Command  : z3_tptp -proof -model -t:%d -file:%s

% Computer : n010.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 Sep 20 05:05:33 EDT 2022

% Result   : Unsatisfiable 0.22s 0.44s
% Output   : Proof 0.22s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    7
%            Number of leaves      :   31
% Syntax   : Number of formulae    :   62 (  19 unt;   6 typ;   0 def)
%            Number of atoms       :  273 ( 157 equ)
%            Maximal formula atoms :   15 (   4 avg)
%            Number of connectives :  347 ( 147   ~; 172   |;   0   &)
%                                         (  28 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   11 (   5 avg)
%            Maximal term depth    :    4 (   2 avg)
%            Number of FOOLs       :   17 (  17 fml;   0 var)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    6 (   4   >;   2   *;   0   +;   0  <<)
%            Number of predicates  :    7 (   5 usr;   1 prp; 0-3 aty)
%            Number of functors    :    5 (   5 usr;   2 con; 0-2 aty)
%            Number of variables   :   99 (  90   !;   0   ?;  99   :)

% Comments : 
%------------------------------------------------------------------------------
tff(singleton_type,type,
    singleton: $i > $i ).

tff(member_of_type,type,
    member_of: $i > $i ).

tff(u_type,type,
    u: $i ).

tff(unordered_pair_type,type,
    unordered_pair: ( $i * $i ) > $i ).

tff(member_type,type,
    member: ( $i * $i ) > $o ).

tff(universal_class_type,type,
    universal_class: $i ).

tff(1,plain,
    ( member(u,universal_class)
  <=> member(u,universal_class) ),
    inference(rewrite,[status(thm)],]) ).

tff(2,axiom,
    member(u,universal_class),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_member_of_singleton_is_unique_1) ).

tff(3,plain,
    member(u,universal_class),
    inference(modus_ponens,[status(thm)],[2,1]) ).

tff(4,plain,
    ^ [Y: $i] :
      refl(
        ( ( ~ member(Y,universal_class)
          | ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
      <=> ( ~ member(Y,universal_class)
          | ( singleton(member_of(singleton(Y))) = singleton(Y) ) ) )),
    inference(bind,[status(th)],]) ).

tff(5,plain,
    ( ! [Y: $i] :
        ( ~ member(Y,universal_class)
        | ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
  <=> ! [Y: $i] :
        ( ~ member(Y,universal_class)
        | ( singleton(member_of(singleton(Y))) = singleton(Y) ) ) ),
    inference(quant_intro,[status(thm)],[4]) ).

tff(6,plain,
    ( ! [Y: $i] :
        ( ~ member(Y,universal_class)
        | ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
  <=> ! [Y: $i] :
        ( ~ member(Y,universal_class)
        | ( singleton(member_of(singleton(Y))) = singleton(Y) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(7,axiom,
    ! [Y: $i] :
      ( ~ member(Y,universal_class)
      | ( singleton(member_of(singleton(Y))) = singleton(Y) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists2) ).

tff(8,plain,
    ! [Y: $i] :
      ( ~ member(Y,universal_class)
      | ( singleton(member_of(singleton(Y))) = singleton(Y) ) ),
    inference(modus_ponens,[status(thm)],[7,6]) ).

tff(9,plain,
    ! [Y: $i] :
      ( ~ member(Y,universal_class)
      | ( singleton(member_of(singleton(Y))) = singleton(Y) ) ),
    inference(skolemize,[status(sab)],[8]) ).

tff(10,plain,
    ! [Y: $i] :
      ( ~ member(Y,universal_class)
      | ( singleton(member_of(singleton(Y))) = singleton(Y) ) ),
    inference(modus_ponens,[status(thm)],[9,5]) ).

tff(11,plain,
    ( ( ~ ! [Y: $i] :
            ( ~ member(Y,universal_class)
            | ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
      | ~ member(u,universal_class)
      | ( singleton(member_of(singleton(u))) = singleton(u) ) )
  <=> ( ~ ! [Y: $i] :
            ( ~ member(Y,universal_class)
            | ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
      | ~ member(u,universal_class)
      | ( singleton(member_of(singleton(u))) = singleton(u) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(12,plain,
    ( ~ ! [Y: $i] :
          ( ~ member(Y,universal_class)
          | ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
    | ~ member(u,universal_class)
    | ( singleton(member_of(singleton(u))) = singleton(u) ) ),
    inference(quant_inst,[status(thm)],]) ).

tff(13,plain,
    ( ~ ! [Y: $i] :
          ( ~ member(Y,universal_class)
          | ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
    | ~ member(u,universal_class)
    | ( singleton(member_of(singleton(u))) = singleton(u) ) ),
    inference(modus_ponens,[status(thm)],[12,11]) ).

tff(14,plain,
    singleton(member_of(singleton(u))) = singleton(u),
    inference(unit_resolution,[status(thm)],[13,10,3]) ).

tff(15,plain,
    singleton(u) = singleton(member_of(singleton(u))),
    inference(symmetry,[status(thm)],[14]) ).

tff(16,plain,
    ^ [X: $i] :
      refl(
        ( ( unordered_pair(X,X) = singleton(X) )
      <=> ( unordered_pair(X,X) = singleton(X) ) )),
    inference(bind,[status(th)],]) ).

tff(17,plain,
    ( ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
  <=> ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ) ),
    inference(quant_intro,[status(thm)],[16]) ).

tff(18,plain,
    ( ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
  <=> ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(19,axiom,
    ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
    file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',singleton_set) ).

tff(20,plain,
    ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
    inference(modus_ponens,[status(thm)],[19,18]) ).

tff(21,plain,
    ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
    inference(skolemize,[status(sab)],[20]) ).

tff(22,plain,
    ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
    inference(modus_ponens,[status(thm)],[21,17]) ).

tff(23,plain,
    ( ~ ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
    | ( unordered_pair(u,u) = singleton(u) ) ),
    inference(quant_inst,[status(thm)],]) ).

tff(24,plain,
    unordered_pair(u,u) = singleton(u),
    inference(unit_resolution,[status(thm)],[23,22]) ).

tff(25,plain,
    unordered_pair(u,u) = singleton(member_of(singleton(u))),
    inference(transitivity,[status(thm)],[24,15]) ).

tff(26,plain,
    ^ [Y: $i] :
      refl(
        ( ( ~ member(Y,universal_class)
          | member(member_of(singleton(Y)),universal_class) )
      <=> ( ~ member(Y,universal_class)
          | member(member_of(singleton(Y)),universal_class) ) )),
    inference(bind,[status(th)],]) ).

tff(27,plain,
    ( ! [Y: $i] :
        ( ~ member(Y,universal_class)
        | member(member_of(singleton(Y)),universal_class) )
  <=> ! [Y: $i] :
        ( ~ member(Y,universal_class)
        | member(member_of(singleton(Y)),universal_class) ) ),
    inference(quant_intro,[status(thm)],[26]) ).

tff(28,plain,
    ( ! [Y: $i] :
        ( ~ member(Y,universal_class)
        | member(member_of(singleton(Y)),universal_class) )
  <=> ! [Y: $i] :
        ( ~ member(Y,universal_class)
        | member(member_of(singleton(Y)),universal_class) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(29,axiom,
    ! [Y: $i] :
      ( ~ member(Y,universal_class)
      | member(member_of(singleton(Y)),universal_class) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_exists1) ).

tff(30,plain,
    ! [Y: $i] :
      ( ~ member(Y,universal_class)
      | member(member_of(singleton(Y)),universal_class) ),
    inference(modus_ponens,[status(thm)],[29,28]) ).

tff(31,plain,
    ! [Y: $i] :
      ( ~ member(Y,universal_class)
      | member(member_of(singleton(Y)),universal_class) ),
    inference(skolemize,[status(sab)],[30]) ).

tff(32,plain,
    ! [Y: $i] :
      ( ~ member(Y,universal_class)
      | member(member_of(singleton(Y)),universal_class) ),
    inference(modus_ponens,[status(thm)],[31,27]) ).

tff(33,plain,
    ( ( ~ ! [Y: $i] :
            ( ~ member(Y,universal_class)
            | member(member_of(singleton(Y)),universal_class) )
      | ~ member(u,universal_class)
      | member(member_of(singleton(u)),universal_class) )
  <=> ( ~ ! [Y: $i] :
            ( ~ member(Y,universal_class)
            | member(member_of(singleton(Y)),universal_class) )
      | ~ member(u,universal_class)
      | member(member_of(singleton(u)),universal_class) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(34,plain,
    ( ~ ! [Y: $i] :
          ( ~ member(Y,universal_class)
          | member(member_of(singleton(Y)),universal_class) )
    | ~ member(u,universal_class)
    | member(member_of(singleton(u)),universal_class) ),
    inference(quant_inst,[status(thm)],]) ).

tff(35,plain,
    ( ~ ! [Y: $i] :
          ( ~ member(Y,universal_class)
          | member(member_of(singleton(Y)),universal_class) )
    | ~ member(u,universal_class)
    | member(member_of(singleton(u)),universal_class) ),
    inference(modus_ponens,[status(thm)],[34,33]) ).

tff(36,plain,
    member(member_of(singleton(u)),universal_class),
    inference(unit_resolution,[status(thm)],[35,32,3]) ).

tff(37,plain,
    ( ( member_of(singleton(u)) != u )
  <=> ( member_of(singleton(u)) != u ) ),
    inference(rewrite,[status(thm)],]) ).

tff(38,axiom,
    member_of(singleton(u)) != u,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_member_of_singleton_is_unique_2) ).

tff(39,plain,
    member_of(singleton(u)) != u,
    inference(modus_ponens,[status(thm)],[38,37]) ).

tff(40,plain,
    ^ [Z: $i,Y: $i,X: $i] :
      refl(
        ( ( ( X = Y )
          | ~ member(X,universal_class)
          | ( X = Z )
          | ( unordered_pair(Y,Z) != singleton(X) ) )
      <=> ( ( X = Y )
          | ~ member(X,universal_class)
          | ( X = Z )
          | ( unordered_pair(Y,Z) != singleton(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(41,plain,
    ( ! [Z: $i,Y: $i,X: $i] :
        ( ( X = Y )
        | ~ member(X,universal_class)
        | ( X = Z )
        | ( unordered_pair(Y,Z) != singleton(X) ) )
  <=> ! [Z: $i,Y: $i,X: $i] :
        ( ( X = Y )
        | ~ member(X,universal_class)
        | ( X = Z )
        | ( unordered_pair(Y,Z) != singleton(X) ) ) ),
    inference(quant_intro,[status(thm)],[40]) ).

tff(42,plain,
    ( ! [Z: $i,Y: $i,X: $i] :
        ( ( X = Y )
        | ~ member(X,universal_class)
        | ( X = Z )
        | ( unordered_pair(Y,Z) != singleton(X) ) )
  <=> ! [Z: $i,Y: $i,X: $i] :
        ( ( X = Y )
        | ~ member(X,universal_class)
        | ( X = Z )
        | ( unordered_pair(Y,Z) != singleton(X) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(43,plain,
    ^ [Z: $i,Y: $i,X: $i] :
      trans(
        monotonicity(
          trans(
            monotonicity(
              rewrite(
                ( ( ( unordered_pair(Y,Z) != singleton(X) )
                  | ~ member(X,universal_class) )
              <=> ( ~ member(X,universal_class)
                  | ( unordered_pair(Y,Z) != singleton(X) ) ) )),
              ( ( ( unordered_pair(Y,Z) != singleton(X) )
                | ~ member(X,universal_class)
                | ( X = Y ) )
            <=> ( ~ member(X,universal_class)
                | ( unordered_pair(Y,Z) != singleton(X) )
                | ( X = Y ) ) )),
            rewrite(
              ( ( ~ member(X,universal_class)
                | ( unordered_pair(Y,Z) != singleton(X) )
                | ( X = Y ) )
            <=> ( ( X = Y )
                | ~ member(X,universal_class)
                | ( unordered_pair(Y,Z) != singleton(X) ) ) )),
            ( ( ( unordered_pair(Y,Z) != singleton(X) )
              | ~ member(X,universal_class)
              | ( X = Y ) )
          <=> ( ( X = Y )
              | ~ member(X,universal_class)
              | ( unordered_pair(Y,Z) != singleton(X) ) ) )),
          ( ( ( unordered_pair(Y,Z) != singleton(X) )
            | ~ member(X,universal_class)
            | ( X = Y )
            | ( X = Z ) )
        <=> ( ( X = Y )
            | ~ member(X,universal_class)
            | ( unordered_pair(Y,Z) != singleton(X) )
            | ( X = Z ) ) )),
        rewrite(
          ( ( ( X = Y )
            | ~ member(X,universal_class)
            | ( unordered_pair(Y,Z) != singleton(X) )
            | ( X = Z ) )
        <=> ( ( X = Y )
            | ~ member(X,universal_class)
            | ( X = Z )
            | ( unordered_pair(Y,Z) != singleton(X) ) ) )),
        ( ( ( unordered_pair(Y,Z) != singleton(X) )
          | ~ member(X,universal_class)
          | ( X = Y )
          | ( X = Z ) )
      <=> ( ( X = Y )
          | ~ member(X,universal_class)
          | ( X = Z )
          | ( unordered_pair(Y,Z) != singleton(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(44,plain,
    ( ! [Z: $i,Y: $i,X: $i] :
        ( ( unordered_pair(Y,Z) != singleton(X) )
        | ~ member(X,universal_class)
        | ( X = Y )
        | ( X = Z ) )
  <=> ! [Z: $i,Y: $i,X: $i] :
        ( ( X = Y )
        | ~ member(X,universal_class)
        | ( X = Z )
        | ( unordered_pair(Y,Z) != singleton(X) ) ) ),
    inference(quant_intro,[status(thm)],[43]) ).

tff(45,axiom,
    ! [Z: $i,Y: $i,X: $i] :
      ( ( unordered_pair(Y,Z) != singleton(X) )
      | ~ member(X,universal_class)
      | ( X = Y )
      | ( X = Z ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton_in_unordered_pair3) ).

tff(46,plain,
    ! [Z: $i,Y: $i,X: $i] :
      ( ( X = Y )
      | ~ member(X,universal_class)
      | ( X = Z )
      | ( unordered_pair(Y,Z) != singleton(X) ) ),
    inference(modus_ponens,[status(thm)],[45,44]) ).

tff(47,plain,
    ! [Z: $i,Y: $i,X: $i] :
      ( ( X = Y )
      | ~ member(X,universal_class)
      | ( X = Z )
      | ( unordered_pair(Y,Z) != singleton(X) ) ),
    inference(modus_ponens,[status(thm)],[46,42]) ).

tff(48,plain,
    ! [Z: $i,Y: $i,X: $i] :
      ( ( X = Y )
      | ~ member(X,universal_class)
      | ( X = Z )
      | ( unordered_pair(Y,Z) != singleton(X) ) ),
    inference(skolemize,[status(sab)],[47]) ).

tff(49,plain,
    ! [Z: $i,Y: $i,X: $i] :
      ( ( X = Y )
      | ~ member(X,universal_class)
      | ( X = Z )
      | ( unordered_pair(Y,Z) != singleton(X) ) ),
    inference(modus_ponens,[status(thm)],[48,41]) ).

tff(50,plain,
    ( ( ~ ! [Z: $i,Y: $i,X: $i] :
            ( ( X = Y )
            | ~ member(X,universal_class)
            | ( X = Z )
            | ( unordered_pair(Y,Z) != singleton(X) ) )
      | ( member_of(singleton(u)) = u )
      | ~ member(member_of(singleton(u)),universal_class)
      | ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) )
  <=> ( ~ ! [Z: $i,Y: $i,X: $i] :
            ( ( X = Y )
            | ~ member(X,universal_class)
            | ( X = Z )
            | ( unordered_pair(Y,Z) != singleton(X) ) )
      | ( member_of(singleton(u)) = u )
      | ~ member(member_of(singleton(u)),universal_class)
      | ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(51,plain,
    ( ( ( member_of(singleton(u)) = u )
      | ~ member(member_of(singleton(u)),universal_class)
      | ( member_of(singleton(u)) = u )
      | ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) )
  <=> ( ( member_of(singleton(u)) = u )
      | ~ member(member_of(singleton(u)),universal_class)
      | ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(52,plain,
    ( ( ~ ! [Z: $i,Y: $i,X: $i] :
            ( ( X = Y )
            | ~ member(X,universal_class)
            | ( X = Z )
            | ( unordered_pair(Y,Z) != singleton(X) ) )
      | ( member_of(singleton(u)) = u )
      | ~ member(member_of(singleton(u)),universal_class)
      | ( member_of(singleton(u)) = u )
      | ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) )
  <=> ( ~ ! [Z: $i,Y: $i,X: $i] :
            ( ( X = Y )
            | ~ member(X,universal_class)
            | ( X = Z )
            | ( unordered_pair(Y,Z) != singleton(X) ) )
      | ( member_of(singleton(u)) = u )
      | ~ member(member_of(singleton(u)),universal_class)
      | ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) ) ),
    inference(monotonicity,[status(thm)],[51]) ).

tff(53,plain,
    ( ( ~ ! [Z: $i,Y: $i,X: $i] :
            ( ( X = Y )
            | ~ member(X,universal_class)
            | ( X = Z )
            | ( unordered_pair(Y,Z) != singleton(X) ) )
      | ( member_of(singleton(u)) = u )
      | ~ member(member_of(singleton(u)),universal_class)
      | ( member_of(singleton(u)) = u )
      | ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) )
  <=> ( ~ ! [Z: $i,Y: $i,X: $i] :
            ( ( X = Y )
            | ~ member(X,universal_class)
            | ( X = Z )
            | ( unordered_pair(Y,Z) != singleton(X) ) )
      | ( member_of(singleton(u)) = u )
      | ~ member(member_of(singleton(u)),universal_class)
      | ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) ) ),
    inference(transitivity,[status(thm)],[52,50]) ).

tff(54,plain,
    ( ~ ! [Z: $i,Y: $i,X: $i] :
          ( ( X = Y )
          | ~ member(X,universal_class)
          | ( X = Z )
          | ( unordered_pair(Y,Z) != singleton(X) ) )
    | ( member_of(singleton(u)) = u )
    | ~ member(member_of(singleton(u)),universal_class)
    | ( member_of(singleton(u)) = u )
    | ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) ),
    inference(quant_inst,[status(thm)],]) ).

tff(55,plain,
    ( ~ ! [Z: $i,Y: $i,X: $i] :
          ( ( X = Y )
          | ~ member(X,universal_class)
          | ( X = Z )
          | ( unordered_pair(Y,Z) != singleton(X) ) )
    | ( member_of(singleton(u)) = u )
    | ~ member(member_of(singleton(u)),universal_class)
    | ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) ),
    inference(modus_ponens,[status(thm)],[54,53]) ).

tff(56,plain,
    $false,
    inference(unit_resolution,[status(thm)],[55,49,39,36,25]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12  % Problem  : SET090-7 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.07/0.13  % Command  : z3_tptp -proof -model -t:%d -file:%s
% 0.15/0.35  % Computer : n010.cluster.edu
% 0.15/0.35  % Model    : x86_64 x86_64
% 0.15/0.35  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35  % Memory   : 8042.1875MB
% 0.15/0.35  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35  % CPULimit : 300
% 0.15/0.35  % WCLimit  : 300
% 0.15/0.35  % DateTime : Sat Sep  3 02:07:57 EDT 2022
% 0.15/0.35  % CPUTime  : 
% 0.15/0.35  Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.15/0.35  Usage: tptp [options] [-file:]file
% 0.15/0.35    -h, -?       prints this message.
% 0.15/0.35    -smt2        print SMT-LIB2 benchmark.
% 0.15/0.35    -m, -model   generate model.
% 0.15/0.35    -p, -proof   generate proof.
% 0.15/0.35    -c, -core    generate unsat core of named formulas.
% 0.15/0.35    -st, -statistics display statistics.
% 0.15/0.35    -t:timeout   set timeout (in second).
% 0.15/0.35    -smt2status  display status in smt2 format instead of SZS.
% 0.15/0.35    -check_status check the status produced by Z3 against annotation in benchmark.
% 0.15/0.35    -<param>:<value> configuration parameter and value.
% 0.15/0.35    -o:<output-file> file to place output in.
% 0.22/0.44  % SZS status Unsatisfiable
% 0.22/0.44  % SZS output start Proof
% See solution above
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