TSTP Solution File: SET094-7 by Z3---4.8.9.0

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Z3---4.8.9.0
% Problem  : SET094-7 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm  : none
% Format   : tptp
% Command  : z3_tptp -proof -model -t:%d -file:%s

% Computer : n029.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 05:05:36 EDT 2022

% Result   : Unsatisfiable 0.20s 0.45s
% Output   : Proof 0.20s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    5
%            Number of leaves      :   17
% Syntax   : Number of formulae    :   33 (  10 unt;   5 typ;   0 def)
%            Number of atoms       :   64 (  34 equ)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives :   65 (  30   ~;  23   |;   0   &)
%                                         (  12 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    8 (   3 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of FOOLs       :    1 (   1 fml;   0 var)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    4 (   3   >;   1   *;   0   +;   0  <<)
%            Number of predicates  :    4 (   2 usr;   1 prp; 0-2 aty)
%            Number of functors    :    4 (   4 usr;   2 con; 0-1 aty)
%            Number of variables   :   26 (  24   !;   0   ?;  26   :)

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

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

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

tff(x_type,type,
    x: $i ).

tff(y_type,type,
    y: $i ).

tff(1,plain,
    ( ( singleton(member_of(x)) = x )
  <=> ( singleton(member_of(x)) = x ) ),
    inference(rewrite,[status(thm)],]) ).

tff(2,axiom,
    singleton(member_of(x)) = x,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_property_of_singletons1_1) ).

tff(3,plain,
    singleton(member_of(x)) = x,
    inference(modus_ponens,[status(thm)],[2,1]) ).

tff(4,plain,
    ( member(y,singleton(member_of(x)))
  <=> member(y,x) ),
    inference(monotonicity,[status(thm)],[3]) ).

tff(5,plain,
    ( member(y,x)
  <=> member(y,singleton(member_of(x))) ),
    inference(symmetry,[status(thm)],[4]) ).

tff(6,plain,
    ( member(y,x)
  <=> member(y,x) ),
    inference(rewrite,[status(thm)],]) ).

tff(7,axiom,
    member(y,x),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_property_of_singletons1_2) ).

tff(8,plain,
    member(y,x),
    inference(modus_ponens,[status(thm)],[7,6]) ).

tff(9,plain,
    member(y,singleton(member_of(x))),
    inference(modus_ponens,[status(thm)],[8,5]) ).

tff(10,plain,
    ( ( y = member_of(x) )
  <=> ( member_of(x) = y ) ),
    inference(commutativity,[status(thm)],]) ).

tff(11,plain,
    ( ( member_of(x) = y )
  <=> ( y = member_of(x) ) ),
    inference(symmetry,[status(thm)],[10]) ).

tff(12,plain,
    ( ( member_of(x) != y )
  <=> ( y != member_of(x) ) ),
    inference(monotonicity,[status(thm)],[11]) ).

tff(13,plain,
    ( ( member_of(x) != y )
  <=> ( member_of(x) != y ) ),
    inference(rewrite,[status(thm)],]) ).

tff(14,axiom,
    member_of(x) != y,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_property_of_singletons1_3) ).

tff(15,plain,
    member_of(x) != y,
    inference(modus_ponens,[status(thm)],[14,13]) ).

tff(16,plain,
    y != member_of(x),
    inference(modus_ponens,[status(thm)],[15,12]) ).

tff(17,plain,
    ^ [Y: $i,X: $i] :
      refl(
        ( ( ~ member(Y,singleton(X))
          | ( Y = X ) )
      <=> ( ~ member(Y,singleton(X))
          | ( Y = X ) ) )),
    inference(bind,[status(th)],]) ).

tff(18,plain,
    ( ! [Y: $i,X: $i] :
        ( ~ member(Y,singleton(X))
        | ( Y = X ) )
  <=> ! [Y: $i,X: $i] :
        ( ~ member(Y,singleton(X))
        | ( Y = X ) ) ),
    inference(quant_intro,[status(thm)],[17]) ).

tff(19,plain,
    ( ! [Y: $i,X: $i] :
        ( ~ member(Y,singleton(X))
        | ( Y = X ) )
  <=> ! [Y: $i,X: $i] :
        ( ~ member(Y,singleton(X))
        | ( Y = X ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(20,axiom,
    ! [Y: $i,X: $i] :
      ( ~ member(Y,singleton(X))
      | ( Y = X ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',only_member_in_singleton) ).

tff(21,plain,
    ! [Y: $i,X: $i] :
      ( ~ member(Y,singleton(X))
      | ( Y = X ) ),
    inference(modus_ponens,[status(thm)],[20,19]) ).

tff(22,plain,
    ! [Y: $i,X: $i] :
      ( ~ member(Y,singleton(X))
      | ( Y = X ) ),
    inference(skolemize,[status(sab)],[21]) ).

tff(23,plain,
    ! [Y: $i,X: $i] :
      ( ~ member(Y,singleton(X))
      | ( Y = X ) ),
    inference(modus_ponens,[status(thm)],[22,18]) ).

tff(24,plain,
    ( ( ~ ! [Y: $i,X: $i] :
            ( ~ member(Y,singleton(X))
            | ( Y = X ) )
      | ~ member(y,singleton(member_of(x)))
      | ( y = member_of(x) ) )
  <=> ( ~ ! [Y: $i,X: $i] :
            ( ~ member(Y,singleton(X))
            | ( Y = X ) )
      | ~ member(y,singleton(member_of(x)))
      | ( y = member_of(x) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(25,plain,
    ( ~ ! [Y: $i,X: $i] :
          ( ~ member(Y,singleton(X))
          | ( Y = X ) )
    | ~ member(y,singleton(member_of(x)))
    | ( y = member_of(x) ) ),
    inference(quant_inst,[status(thm)],]) ).

tff(26,plain,
    ( ~ ! [Y: $i,X: $i] :
          ( ~ member(Y,singleton(X))
          | ( Y = X ) )
    | ~ member(y,singleton(member_of(x)))
    | ( y = member_of(x) ) ),
    inference(modus_ponens,[status(thm)],[25,24]) ).

tff(27,plain,
    ( ~ member(y,singleton(member_of(x)))
    | ( y = member_of(x) ) ),
    inference(unit_resolution,[status(thm)],[26,23]) ).

tff(28,plain,
    $false,
    inference(unit_resolution,[status(thm)],[27,16,9]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12  % Problem  : SET094-7 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.07/0.13  % Command  : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34  % Computer : n029.cluster.edu
% 0.13/0.34  % Model    : x86_64 x86_64
% 0.13/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34  % Memory   : 8042.1875MB
% 0.13/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34  % CPULimit : 300
% 0.13/0.34  % WCLimit  : 300
% 0.13/0.34  % DateTime : Sat Sep  3 02:28:00 EDT 2022
% 0.13/0.35  % CPUTime  : 
% 0.13/0.35  Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35  Usage: tptp [options] [-file:]file
% 0.13/0.35    -h, -?       prints this message.
% 0.13/0.35    -smt2        print SMT-LIB2 benchmark.
% 0.13/0.35    -m, -model   generate model.
% 0.13/0.35    -p, -proof   generate proof.
% 0.13/0.35    -c, -core    generate unsat core of named formulas.
% 0.13/0.35    -st, -statistics display statistics.
% 0.13/0.35    -t:timeout   set timeout (in second).
% 0.13/0.35    -smt2status  display status in smt2 format instead of SZS.
% 0.13/0.35    -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35    -<param>:<value> configuration parameter and value.
% 0.13/0.35    -o:<output-file> file to place output in.
% 0.20/0.45  % SZS status Unsatisfiable
% 0.20/0.45  % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------