TSTP Solution File: SET080-6 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET080-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n009.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 05:05:27 EDT 2022
% Result : Unsatisfiable 0.19s 0.41s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 36
% Syntax : Number of formulae : 73 ( 24 unt; 8 typ; 0 def)
% Number of atoms : 212 ( 13 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 253 ( 118 ~; 107 |; 0 &)
% ( 28 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 12 ( 12 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 8 ( 5 >; 3 *; 0 +; 0 <<)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-3 aty)
% Number of functors : 5 ( 5 usr; 3 con; 0-2 aty)
% Number of variables : 110 ( 99 !; 0 ?; 110 :)
% Comments :
%------------------------------------------------------------------------------
tff(subclass_type,type,
subclass: ( $i * $i ) > $o ).
tff(universal_class_type,type,
universal_class: $i ).
tff(omega_type,type,
omega: $i ).
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(null_class_type,type,
null_class: $i ).
tff(unordered_pair_type,type,
unordered_pair: ( $i * $i ) > $i ).
tff(singleton_type,type,
singleton: $i > $i ).
tff(inductive_type,type,
inductive: $i > $o ).
tff(1,plain,
^ [X: $i] :
refl(
( ( unordered_pair(X,X) = singleton(X) )
<=> ( unordered_pair(X,X) = singleton(X) ) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
<=> ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
<=> ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',singleton_set) ).
tff(5,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
| ( unordered_pair(null_class,null_class) = singleton(null_class) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
unordered_pair(null_class,null_class) = singleton(null_class),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
( member(null_class,unordered_pair(null_class,null_class))
<=> member(null_class,singleton(null_class)) ),
inference(monotonicity,[status(thm)],[9]) ).
tff(11,plain,
( member(null_class,singleton(null_class))
<=> member(null_class,unordered_pair(null_class,null_class)) ),
inference(symmetry,[status(thm)],[10]) ).
tff(12,plain,
( ~ member(null_class,singleton(null_class))
<=> ~ member(null_class,unordered_pair(null_class,null_class)) ),
inference(monotonicity,[status(thm)],[11]) ).
tff(13,plain,
( ~ member(null_class,singleton(null_class))
<=> ~ member(null_class,singleton(null_class)) ),
inference(rewrite,[status(thm)],]) ).
tff(14,axiom,
~ member(null_class,singleton(null_class)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_null_class_in_its_singleton_1) ).
tff(15,plain,
~ member(null_class,singleton(null_class)),
inference(modus_ponens,[status(thm)],[14,13]) ).
tff(16,plain,
~ member(null_class,unordered_pair(null_class,null_class)),
inference(modus_ponens,[status(thm)],[15,12]) ).
tff(17,plain,
^ [Y: $i,X: $i] :
refl(
( ( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) )
<=> ( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) ) )),
inference(bind,[status(th)],]) ).
tff(18,plain,
( ! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) ) ),
inference(quant_intro,[status(thm)],[17]) ).
tff(19,plain,
( ! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(20,axiom,
! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',unordered_pair2) ).
tff(21,plain,
! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[20,19]) ).
tff(22,plain,
! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) ),
inference(skolemize,[status(sab)],[21]) ).
tff(23,plain,
! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[22,18]) ).
tff(24,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) )
| ~ member(null_class,universal_class)
| member(null_class,unordered_pair(null_class,null_class)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) )
| ~ member(null_class,universal_class)
| member(null_class,unordered_pair(null_class,null_class)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(25,plain,
( ~ ! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) )
| ~ member(null_class,universal_class)
| member(null_class,unordered_pair(null_class,null_class)) ),
inference(quant_inst,[status(thm)],]) ).
tff(26,plain,
( ~ ! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) )
| ~ member(null_class,universal_class)
| member(null_class,unordered_pair(null_class,null_class)) ),
inference(modus_ponens,[status(thm)],[25,24]) ).
tff(27,plain,
( ~ member(null_class,universal_class)
| member(null_class,unordered_pair(null_class,null_class)) ),
inference(unit_resolution,[status(thm)],[26,23]) ).
tff(28,plain,
~ member(null_class,universal_class),
inference(unit_resolution,[status(thm)],[27,16]) ).
tff(29,plain,
( inductive(omega)
<=> inductive(omega) ),
inference(rewrite,[status(thm)],]) ).
tff(30,axiom,
inductive(omega),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',omega_is_inductive1) ).
tff(31,plain,
inductive(omega),
inference(modus_ponens,[status(thm)],[30,29]) ).
tff(32,plain,
^ [X: $i] :
refl(
( ( ~ inductive(X)
| member(null_class,X) )
<=> ( ~ inductive(X)
| member(null_class,X) ) )),
inference(bind,[status(th)],]) ).
tff(33,plain,
( ! [X: $i] :
( ~ inductive(X)
| member(null_class,X) )
<=> ! [X: $i] :
( ~ inductive(X)
| member(null_class,X) ) ),
inference(quant_intro,[status(thm)],[32]) ).
tff(34,plain,
( ! [X: $i] :
( ~ inductive(X)
| member(null_class,X) )
<=> ! [X: $i] :
( ~ inductive(X)
| member(null_class,X) ) ),
inference(rewrite,[status(thm)],]) ).
tff(35,axiom,
! [X: $i] :
( ~ inductive(X)
| member(null_class,X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',inductive1) ).
tff(36,plain,
! [X: $i] :
( ~ inductive(X)
| member(null_class,X) ),
inference(modus_ponens,[status(thm)],[35,34]) ).
tff(37,plain,
! [X: $i] :
( ~ inductive(X)
| member(null_class,X) ),
inference(skolemize,[status(sab)],[36]) ).
tff(38,plain,
! [X: $i] :
( ~ inductive(X)
| member(null_class,X) ),
inference(modus_ponens,[status(thm)],[37,33]) ).
tff(39,plain,
( ( ~ ! [X: $i] :
( ~ inductive(X)
| member(null_class,X) )
| ~ inductive(omega)
| member(null_class,omega) )
<=> ( ~ ! [X: $i] :
( ~ inductive(X)
| member(null_class,X) )
| ~ inductive(omega)
| member(null_class,omega) ) ),
inference(rewrite,[status(thm)],]) ).
tff(40,plain,
( ~ ! [X: $i] :
( ~ inductive(X)
| member(null_class,X) )
| ~ inductive(omega)
| member(null_class,omega) ),
inference(quant_inst,[status(thm)],]) ).
tff(41,plain,
( ~ ! [X: $i] :
( ~ inductive(X)
| member(null_class,X) )
| ~ inductive(omega)
| member(null_class,omega) ),
inference(modus_ponens,[status(thm)],[40,39]) ).
tff(42,plain,
member(null_class,omega),
inference(unit_resolution,[status(thm)],[41,38,31]) ).
tff(43,plain,
^ [Y: $i,U: $i,X: $i] :
refl(
( ( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
<=> ( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ) )),
inference(bind,[status(th)],]) ).
tff(44,plain,
( ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
<=> ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ) ),
inference(quant_intro,[status(thm)],[43]) ).
tff(45,plain,
( ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
<=> ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ) ),
inference(rewrite,[status(thm)],]) ).
tff(46,plain,
^ [Y: $i,U: $i,X: $i] :
trans(
monotonicity(
rewrite(
( ( ~ subclass(X,Y)
| ~ member(U,X) )
<=> ( ~ member(U,X)
| ~ subclass(X,Y) ) )),
( ( ~ subclass(X,Y)
| ~ member(U,X)
| member(U,Y) )
<=> ( ~ member(U,X)
| ~ subclass(X,Y)
| member(U,Y) ) )),
rewrite(
( ( ~ member(U,X)
| ~ subclass(X,Y)
| member(U,Y) )
<=> ( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ) )),
( ( ~ subclass(X,Y)
| ~ member(U,X)
| member(U,Y) )
<=> ( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ) )),
inference(bind,[status(th)],]) ).
tff(47,plain,
( ! [Y: $i,U: $i,X: $i] :
( ~ subclass(X,Y)
| ~ member(U,X)
| member(U,Y) )
<=> ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ) ),
inference(quant_intro,[status(thm)],[46]) ).
tff(48,axiom,
! [Y: $i,U: $i,X: $i] :
( ~ subclass(X,Y)
| ~ member(U,X)
| member(U,Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',subclass_members) ).
tff(49,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[48,47]) ).
tff(50,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[49,45]) ).
tff(51,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(skolemize,[status(sab)],[50]) ).
tff(52,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[51,44]) ).
tff(53,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(null_class,universal_class)
| ~ member(null_class,omega)
| ~ subclass(omega,universal_class) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(null_class,universal_class)
| ~ member(null_class,omega)
| ~ subclass(omega,universal_class) ) ),
inference(rewrite,[status(thm)],]) ).
tff(54,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(null_class,universal_class)
| ~ member(null_class,omega)
| ~ subclass(omega,universal_class) ),
inference(quant_inst,[status(thm)],]) ).
tff(55,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(null_class,universal_class)
| ~ member(null_class,omega)
| ~ subclass(omega,universal_class) ),
inference(modus_ponens,[status(thm)],[54,53]) ).
tff(56,plain,
~ subclass(omega,universal_class),
inference(unit_resolution,[status(thm)],[55,52,42,28]) ).
tff(57,plain,
^ [X: $i] :
refl(
( subclass(X,universal_class)
<=> subclass(X,universal_class) )),
inference(bind,[status(th)],]) ).
tff(58,plain,
( ! [X: $i] : subclass(X,universal_class)
<=> ! [X: $i] : subclass(X,universal_class) ),
inference(quant_intro,[status(thm)],[57]) ).
tff(59,plain,
( ! [X: $i] : subclass(X,universal_class)
<=> ! [X: $i] : subclass(X,universal_class) ),
inference(rewrite,[status(thm)],]) ).
tff(60,axiom,
! [X: $i] : subclass(X,universal_class),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',class_elements_are_sets) ).
tff(61,plain,
! [X: $i] : subclass(X,universal_class),
inference(modus_ponens,[status(thm)],[60,59]) ).
tff(62,plain,
! [X: $i] : subclass(X,universal_class),
inference(skolemize,[status(sab)],[61]) ).
tff(63,plain,
! [X: $i] : subclass(X,universal_class),
inference(modus_ponens,[status(thm)],[62,58]) ).
tff(64,plain,
( ~ ! [X: $i] : subclass(X,universal_class)
| subclass(omega,universal_class) ),
inference(quant_inst,[status(thm)],]) ).
tff(65,plain,
$false,
inference(unit_resolution,[status(thm)],[64,63,56]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11 % Problem : SET080-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.06/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n009.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Sat Sep 3 02:04:20 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.12/0.33 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.33 Usage: tptp [options] [-file:]file
% 0.12/0.33 -h, -? prints this message.
% 0.12/0.33 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.33 -m, -model generate model.
% 0.12/0.33 -p, -proof generate proof.
% 0.12/0.33 -c, -core generate unsat core of named formulas.
% 0.12/0.33 -st, -statistics display statistics.
% 0.12/0.33 -t:timeout set timeout (in second).
% 0.12/0.33 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.33 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.33 -<param>:<value> configuration parameter and value.
% 0.12/0.33 -o:<output-file> file to place output in.
% 0.19/0.41 % SZS status Unsatisfiable
% 0.19/0.41 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------