TSTP Solution File: SET411-6 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET411-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n028.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 05:06:42 EDT 2022
% Result : Unsatisfiable 0.18s 0.42s
% Output : Proof 0.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 38
% Syntax : Number of formulae : 66 ( 18 unt; 13 typ; 0 def)
% Number of atoms : 192 ( 0 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 240 ( 112 ~; 105 |; 0 &)
% ( 23 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 12 ( 2 avg)
% Number of FOOLs : 11 ( 11 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 14 ( 8 >; 6 *; 0 +; 0 <<)
% Number of predicates : 7 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 11 ( 11 usr; 5 con; 0-2 aty)
% Number of variables : 126 ( 114 !; 0 ?; 126 :)
% Comments :
%------------------------------------------------------------------------------
tff(subclass_type,type,
subclass: ( $i * $i ) > $o ).
tff(universal_class_type,type,
universal_class: $i ).
tff(y_type,type,
y: $i ).
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(x_type,type,
x: $i ).
tff(element_relation_type,type,
element_relation: $i ).
tff(ordered_pair_type,type,
ordered_pair: ( $i * $i ) > $i ).
tff(intersection_type,type,
intersection: ( $i * $i ) > $i ).
tff(complement_type,type,
complement: $i > $i ).
tff(compose_type,type,
compose: ( $i * $i ) > $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(cross_product_type,type,
cross_product: ( $i * $i ) > $i ).
tff(singleton_relation_type,type,
singleton_relation: $i ).
tff(1,plain,
( member(ordered_pair(x,y),singleton_relation)
<=> member(ordered_pair(x,y),intersection(complement(compose(element_relation,complement(intersection(inverse(intersection(cross_product(universal_class,universal_class),intersection(cross_product(universal_class,universal_class),complement(compose(complement(element_relation),inverse(element_relation)))))),intersection(cross_product(universal_class,universal_class),intersection(cross_product(universal_class,universal_class),complement(compose(complement(element_relation),inverse(element_relation))))))))),element_relation)) ),
inference(rewrite,[status(thm)],]) ).
tff(2,plain,
( member(ordered_pair(x,y),singleton_relation)
<=> member(ordered_pair(x,y),singleton_relation) ),
inference(rewrite,[status(thm)],]) ).
tff(3,axiom,
member(ordered_pair(x,y),singleton_relation),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_compose_condition_for_singleton_membership1_1) ).
tff(4,plain,
member(ordered_pair(x,y),singleton_relation),
inference(modus_ponens,[status(thm)],[3,2]) ).
tff(5,plain,
member(ordered_pair(x,y),intersection(complement(compose(element_relation,complement(intersection(inverse(intersection(cross_product(universal_class,universal_class),intersection(cross_product(universal_class,universal_class),complement(compose(complement(element_relation),inverse(element_relation)))))),intersection(cross_product(universal_class,universal_class),intersection(cross_product(universal_class,universal_class),complement(compose(complement(element_relation),inverse(element_relation))))))))),element_relation)),
inference(modus_ponens,[status(thm)],[4,1]) ).
tff(6,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(
( ( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
<=> ( ~ member(Z,intersection(X,Y))
| member(Z,Y) ) )),
inference(bind,[status(th)],]) ).
tff(7,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ) ),
inference(quant_intro,[status(thm)],[6]) ).
tff(8,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ) ),
inference(rewrite,[status(thm)],]) ).
tff(9,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',intersection2) ).
tff(10,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(modus_ponens,[status(thm)],[9,8]) ).
tff(11,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(skolemize,[status(sab)],[10]) ).
tff(12,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(modus_ponens,[status(thm)],[11,7]) ).
tff(13,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(ordered_pair(x,y),intersection(complement(compose(element_relation,complement(intersection(inverse(intersection(cross_product(universal_class,universal_class),intersection(cross_product(universal_class,universal_class),complement(compose(complement(element_relation),inverse(element_relation)))))),intersection(cross_product(universal_class,universal_class),intersection(cross_product(universal_class,universal_class),complement(compose(complement(element_relation),inverse(element_relation))))))))),element_relation))
| member(ordered_pair(x,y),element_relation) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(ordered_pair(x,y),intersection(complement(compose(element_relation,complement(intersection(inverse(intersection(cross_product(universal_class,universal_class),intersection(cross_product(universal_class,universal_class),complement(compose(complement(element_relation),inverse(element_relation)))))),intersection(cross_product(universal_class,universal_class),intersection(cross_product(universal_class,universal_class),complement(compose(complement(element_relation),inverse(element_relation))))))))),element_relation))
| member(ordered_pair(x,y),element_relation) ) ),
inference(rewrite,[status(thm)],]) ).
tff(14,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(ordered_pair(x,y),intersection(complement(compose(element_relation,complement(intersection(inverse(intersection(cross_product(universal_class,universal_class),intersection(cross_product(universal_class,universal_class),complement(compose(complement(element_relation),inverse(element_relation)))))),intersection(cross_product(universal_class,universal_class),intersection(cross_product(universal_class,universal_class),complement(compose(complement(element_relation),inverse(element_relation))))))))),element_relation))
| member(ordered_pair(x,y),element_relation) ),
inference(quant_inst,[status(thm)],]) ).
tff(15,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(ordered_pair(x,y),intersection(complement(compose(element_relation,complement(intersection(inverse(intersection(cross_product(universal_class,universal_class),intersection(cross_product(universal_class,universal_class),complement(compose(complement(element_relation),inverse(element_relation)))))),intersection(cross_product(universal_class,universal_class),intersection(cross_product(universal_class,universal_class),complement(compose(complement(element_relation),inverse(element_relation))))))))),element_relation))
| member(ordered_pair(x,y),element_relation) ),
inference(modus_ponens,[status(thm)],[14,13]) ).
tff(16,plain,
member(ordered_pair(x,y),element_relation),
inference(unit_resolution,[status(thm)],[15,12,5]) ).
tff(17,plain,
^ [Y: $i,X: $i] :
refl(
( ( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) )
<=> ( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) ) )),
inference(bind,[status(th)],]) ).
tff(18,plain,
( ! [Y: $i,X: $i] :
( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) )
<=> ! [Y: $i,X: $i] :
( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) ) ),
inference(quant_intro,[status(thm)],[17]) ).
tff(19,plain,
( ! [Y: $i,X: $i] :
( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) )
<=> ! [Y: $i,X: $i] :
( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) ) ),
inference(rewrite,[status(thm)],]) ).
tff(20,axiom,
! [Y: $i,X: $i] :
( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',element_relation2) ).
tff(21,plain,
! [Y: $i,X: $i] :
( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) ),
inference(modus_ponens,[status(thm)],[20,19]) ).
tff(22,plain,
! [Y: $i,X: $i] :
( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) ),
inference(skolemize,[status(sab)],[21]) ).
tff(23,plain,
! [Y: $i,X: $i] :
( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) ),
inference(modus_ponens,[status(thm)],[22,18]) ).
tff(24,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) )
| ~ member(ordered_pair(x,y),element_relation)
| member(x,y) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) )
| ~ member(ordered_pair(x,y),element_relation)
| member(x,y) ) ),
inference(rewrite,[status(thm)],]) ).
tff(25,plain,
( ~ ! [Y: $i,X: $i] :
( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) )
| ~ member(ordered_pair(x,y),element_relation)
| member(x,y) ),
inference(quant_inst,[status(thm)],]) ).
tff(26,plain,
( ~ ! [Y: $i,X: $i] :
( ~ member(ordered_pair(X,Y),element_relation)
| member(X,Y) )
| ~ member(ordered_pair(x,y),element_relation)
| member(x,y) ),
inference(modus_ponens,[status(thm)],[25,24]) ).
tff(27,plain,
member(x,y),
inference(unit_resolution,[status(thm)],[26,23,16]) ).
tff(28,plain,
( ~ member(x,universal_class)
<=> ~ member(x,universal_class) ),
inference(rewrite,[status(thm)],]) ).
tff(29,axiom,
~ member(x,universal_class),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_compose_condition_for_singleton_membership1_2) ).
tff(30,plain,
~ member(x,universal_class),
inference(modus_ponens,[status(thm)],[29,28]) ).
tff(31,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(32,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)],[31]) ).
tff(33,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(34,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(35,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)],[34]) ).
tff(36,axiom,
! [Y: $i,U: $i,X: $i] :
( ~ subclass(X,Y)
| ~ member(U,X)
| member(U,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',subclass_members) ).
tff(37,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[36,35]) ).
tff(38,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[37,33]) ).
tff(39,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(skolemize,[status(sab)],[38]) ).
tff(40,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[39,32]) ).
tff(41,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(x,universal_class)
| ~ member(x,y)
| ~ subclass(y,universal_class) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(x,universal_class)
| ~ member(x,y)
| ~ subclass(y,universal_class) ) ),
inference(rewrite,[status(thm)],]) ).
tff(42,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(x,universal_class)
| ~ member(x,y)
| ~ subclass(y,universal_class) ),
inference(quant_inst,[status(thm)],]) ).
tff(43,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(x,universal_class)
| ~ member(x,y)
| ~ subclass(y,universal_class) ),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
~ subclass(y,universal_class),
inference(unit_resolution,[status(thm)],[43,40,30,27]) ).
tff(45,plain,
^ [X: $i] :
refl(
( subclass(X,universal_class)
<=> subclass(X,universal_class) )),
inference(bind,[status(th)],]) ).
tff(46,plain,
( ! [X: $i] : subclass(X,universal_class)
<=> ! [X: $i] : subclass(X,universal_class) ),
inference(quant_intro,[status(thm)],[45]) ).
tff(47,plain,
( ! [X: $i] : subclass(X,universal_class)
<=> ! [X: $i] : subclass(X,universal_class) ),
inference(rewrite,[status(thm)],]) ).
tff(48,axiom,
! [X: $i] : subclass(X,universal_class),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',class_elements_are_sets) ).
tff(49,plain,
! [X: $i] : subclass(X,universal_class),
inference(modus_ponens,[status(thm)],[48,47]) ).
tff(50,plain,
! [X: $i] : subclass(X,universal_class),
inference(skolemize,[status(sab)],[49]) ).
tff(51,plain,
! [X: $i] : subclass(X,universal_class),
inference(modus_ponens,[status(thm)],[50,46]) ).
tff(52,plain,
( ~ ! [X: $i] : subclass(X,universal_class)
| subclass(y,universal_class) ),
inference(quant_inst,[status(thm)],]) ).
tff(53,plain,
$false,
inference(unit_resolution,[status(thm)],[52,51,44]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : SET411-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.10/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n028.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 05:25:08 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 0.18/0.42 % SZS status Unsatisfiable
% 0.18/0.42 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------