TSTP Solution File: SET131-6 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET131-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n013.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 05:05:50 EDT 2022
% Result : Unsatisfiable 0.19s 0.43s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 63
% Syntax : Number of formulae : 140 ( 44 unt; 12 typ; 0 def)
% Number of atoms : 419 ( 48 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 500 ( 219 ~; 230 |; 0 &)
% ( 51 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 5 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of FOOLs : 10 ( 10 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 12 ( 7 >; 5 *; 0 +; 0 <<)
% Number of predicates : 5 ( 3 usr; 1 prp; 0-2 aty)
% Number of functors : 11 ( 11 usr; 5 con; 0-2 aty)
% Number of variables : 258 ( 237 !; 0 ?; 258 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(complement_type,type,
complement: $i > $i ).
tff(singleton_type,type,
singleton: $i > $i ).
tff(u_type,type,
u: $i ).
tff(intersection_type,type,
intersection: ( $i * $i ) > $i ).
tff(set_builder_type,type,
set_builder: ( $i * $i ) > $i ).
tff(null_class_type,type,
null_class: $i ).
tff(z_type,type,
z: $i ).
tff(union_type,type,
union: ( $i * $i ) > $i ).
tff(x_type,type,
x: $i ).
tff(universal_class_type,type,
universal_class: $i ).
tff(unordered_pair_type,type,
unordered_pair: ( $i * $i ) > $i ).
tff(1,plain,
^ [Y: $i,X: $i] :
refl(
( ( union(singleton(X),Y) = set_builder(X,Y) )
<=> ( union(singleton(X),Y) = set_builder(X,Y) ) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [Y: $i,X: $i] : ( union(singleton(X),Y) = set_builder(X,Y) )
<=> ! [Y: $i,X: $i] : ( union(singleton(X),Y) = set_builder(X,Y) ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [Y: $i,X: $i] : ( union(singleton(X),Y) = set_builder(X,Y) )
<=> ! [Y: $i,X: $i] : ( union(singleton(X),Y) = set_builder(X,Y) ) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [Y: $i,X: $i] : ( union(singleton(X),Y) = set_builder(X,Y) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',definition_of_set_builder) ).
tff(5,plain,
! [Y: $i,X: $i] : ( union(singleton(X),Y) = set_builder(X,Y) ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [Y: $i,X: $i] : ( union(singleton(X),Y) = set_builder(X,Y) ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [Y: $i,X: $i] : ( union(singleton(X),Y) = set_builder(X,Y) ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [Y: $i,X: $i] : ( union(singleton(X),Y) = set_builder(X,Y) )
| ( union(singleton(u),set_builder(z,null_class)) = set_builder(u,set_builder(z,null_class)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
union(singleton(u),set_builder(z,null_class)) = set_builder(u,set_builder(z,null_class)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [Y: $i,X: $i] :
refl(
( ( complement(intersection(complement(X),complement(Y))) = union(X,Y) )
<=> ( complement(intersection(complement(X),complement(Y))) = union(X,Y) ) )),
inference(bind,[status(th)],]) ).
tff(11,plain,
( ! [Y: $i,X: $i] : ( complement(intersection(complement(X),complement(Y))) = union(X,Y) )
<=> ! [Y: $i,X: $i] : ( complement(intersection(complement(X),complement(Y))) = union(X,Y) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [Y: $i,X: $i] : ( complement(intersection(complement(X),complement(Y))) = union(X,Y) )
<=> ! [Y: $i,X: $i] : ( complement(intersection(complement(X),complement(Y))) = union(X,Y) ) ),
inference(rewrite,[status(thm)],]) ).
tff(13,axiom,
! [Y: $i,X: $i] : ( complement(intersection(complement(X),complement(Y))) = union(X,Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',union) ).
tff(14,plain,
! [Y: $i,X: $i] : ( complement(intersection(complement(X),complement(Y))) = union(X,Y) ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [Y: $i,X: $i] : ( complement(intersection(complement(X),complement(Y))) = union(X,Y) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [Y: $i,X: $i] : ( complement(intersection(complement(X),complement(Y))) = union(X,Y) ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [Y: $i,X: $i] : ( complement(intersection(complement(X),complement(Y))) = union(X,Y) )
| ( complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class)))) = union(singleton(u),set_builder(z,null_class)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(18,plain,
complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class)))) = union(singleton(u),set_builder(z,null_class)),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class)))) = set_builder(u,set_builder(z,null_class)),
inference(transitivity,[status(thm)],[18,9]) ).
tff(20,plain,
( member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class)))))
<=> member(u,set_builder(u,set_builder(z,null_class))) ),
inference(monotonicity,[status(thm)],[19]) ).
tff(21,plain,
( member(u,set_builder(u,set_builder(z,null_class)))
<=> member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) ),
inference(symmetry,[status(thm)],[20]) ).
tff(22,plain,
( ~ member(u,set_builder(u,set_builder(z,null_class)))
<=> ~ member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) ),
inference(monotonicity,[status(thm)],[21]) ).
tff(23,plain,
( ~ ! [Y: $i,X: $i] : ( union(singleton(X),Y) = set_builder(X,Y) )
| ( union(singleton(x),set_builder(u,set_builder(z,null_class))) = set_builder(x,set_builder(u,set_builder(z,null_class))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(24,plain,
union(singleton(x),set_builder(u,set_builder(z,null_class))) = set_builder(x,set_builder(u,set_builder(z,null_class))),
inference(unit_resolution,[status(thm)],[23,7]) ).
tff(25,plain,
( ~ ! [Y: $i,X: $i] : ( complement(intersection(complement(X),complement(Y))) = union(X,Y) )
| ( complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class))))) = union(singleton(x),set_builder(u,set_builder(z,null_class))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(26,plain,
complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class))))) = union(singleton(x),set_builder(u,set_builder(z,null_class))),
inference(unit_resolution,[status(thm)],[25,16]) ).
tff(27,plain,
complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class))))) = set_builder(x,set_builder(u,set_builder(z,null_class))),
inference(transitivity,[status(thm)],[26,24]) ).
tff(28,plain,
( member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class))))))
<=> member(u,set_builder(x,set_builder(u,set_builder(z,null_class)))) ),
inference(monotonicity,[status(thm)],[27]) ).
tff(29,plain,
( member(u,set_builder(x,set_builder(u,set_builder(z,null_class))))
<=> member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) ),
inference(symmetry,[status(thm)],[28]) ).
tff(30,plain,
( ~ member(u,set_builder(x,set_builder(u,set_builder(z,null_class))))
<=> ~ member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) ),
inference(monotonicity,[status(thm)],[29]) ).
tff(31,plain,
( ~ member(u,set_builder(x,set_builder(u,set_builder(z,null_class))))
<=> ~ member(u,set_builder(x,set_builder(u,set_builder(z,null_class)))) ),
inference(rewrite,[status(thm)],]) ).
tff(32,axiom,
~ member(u,set_builder(x,set_builder(u,set_builder(z,null_class)))),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_member_of_triple2_2) ).
tff(33,plain,
~ member(u,set_builder(x,set_builder(u,set_builder(z,null_class)))),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
~ member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))),
inference(modus_ponens,[status(thm)],[33,30]) ).
tff(35,plain,
( member(u,universal_class)
<=> member(u,universal_class) ),
inference(rewrite,[status(thm)],]) ).
tff(36,axiom,
member(u,universal_class),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_member_of_triple2_1) ).
tff(37,plain,
member(u,universal_class),
inference(modus_ponens,[status(thm)],[36,35]) ).
tff(38,plain,
^ [Z: $i,X: $i] :
refl(
( ( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
<=> ( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) ) )),
inference(bind,[status(th)],]) ).
tff(39,plain,
( ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
<=> ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) ) ),
inference(quant_intro,[status(thm)],[38]) ).
tff(40,plain,
( ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
<=> ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(41,plain,
^ [Z: $i,X: $i] :
rewrite(
( ( ~ member(Z,universal_class)
| member(Z,complement(X))
| member(Z,X) )
<=> ( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) ) )),
inference(bind,[status(th)],]) ).
tff(42,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,universal_class)
| member(Z,complement(X))
| member(Z,X) )
<=> ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) ) ),
inference(quant_intro,[status(thm)],[41]) ).
tff(43,axiom,
! [Z: $i,X: $i] :
( ~ member(Z,universal_class)
| member(Z,complement(X))
| member(Z,X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',complement2) ).
tff(44,plain,
! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) ),
inference(modus_ponens,[status(thm)],[43,42]) ).
tff(45,plain,
! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) ),
inference(modus_ponens,[status(thm)],[44,40]) ).
tff(46,plain,
! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) ),
inference(skolemize,[status(sab)],[45]) ).
tff(47,plain,
! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) ),
inference(modus_ponens,[status(thm)],[46,39]) ).
tff(48,plain,
( ( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| ~ member(u,universal_class)
| member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) )
<=> ( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| ~ member(u,universal_class)
| member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(49,plain,
( ( member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| ~ member(u,universal_class)
| member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) )
<=> ( ~ member(u,universal_class)
| member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(50,plain,
( ( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| ~ member(u,universal_class)
| member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) )
<=> ( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| ~ member(u,universal_class)
| member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) ) ),
inference(monotonicity,[status(thm)],[49]) ).
tff(51,plain,
( ( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| ~ member(u,universal_class)
| member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) )
<=> ( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| ~ member(u,universal_class)
| member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) ) ),
inference(transitivity,[status(thm)],[50,48]) ).
tff(52,plain,
( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| ~ member(u,universal_class)
| member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) ),
inference(quant_inst,[status(thm)],]) ).
tff(53,plain,
( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| ~ member(u,universal_class)
| member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) ),
inference(modus_ponens,[status(thm)],[52,51]) ).
tff(54,plain,
( member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| member(u,complement(intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))) ),
inference(unit_resolution,[status(thm)],[53,47,37]) ).
tff(55,plain,
member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class))))),
inference(unit_resolution,[status(thm)],[54,34]) ).
tff(56,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(57,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)],[56]) ).
tff(58,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(59,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',intersection2) ).
tff(60,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(modus_ponens,[status(thm)],[59,58]) ).
tff(61,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(skolemize,[status(sab)],[60]) ).
tff(62,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(modus_ponens,[status(thm)],[61,57]) ).
tff(63,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| member(u,complement(set_builder(u,set_builder(z,null_class)))) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| member(u,complement(set_builder(u,set_builder(z,null_class)))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(64,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| member(u,complement(set_builder(u,set_builder(z,null_class)))) ),
inference(quant_inst,[status(thm)],]) ).
tff(65,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(u,intersection(complement(singleton(x)),complement(set_builder(u,set_builder(z,null_class)))))
| member(u,complement(set_builder(u,set_builder(z,null_class)))) ),
inference(modus_ponens,[status(thm)],[64,63]) ).
tff(66,plain,
member(u,complement(set_builder(u,set_builder(z,null_class)))),
inference(unit_resolution,[status(thm)],[65,62,55]) ).
tff(67,plain,
^ [Z: $i,X: $i] :
refl(
( ( ~ member(Z,X)
| ~ member(Z,complement(X)) )
<=> ( ~ member(Z,X)
| ~ member(Z,complement(X)) ) )),
inference(bind,[status(th)],]) ).
tff(68,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ) ),
inference(quant_intro,[status(thm)],[67]) ).
tff(69,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(70,plain,
^ [Z: $i,X: $i] :
rewrite(
( ( ~ member(Z,complement(X))
| ~ member(Z,X) )
<=> ( ~ member(Z,X)
| ~ member(Z,complement(X)) ) )),
inference(bind,[status(th)],]) ).
tff(71,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,complement(X))
| ~ member(Z,X) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ) ),
inference(quant_intro,[status(thm)],[70]) ).
tff(72,axiom,
! [Z: $i,X: $i] :
( ~ member(Z,complement(X))
| ~ member(Z,X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',complement1) ).
tff(73,plain,
! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ),
inference(modus_ponens,[status(thm)],[72,71]) ).
tff(74,plain,
! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ),
inference(modus_ponens,[status(thm)],[73,69]) ).
tff(75,plain,
! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ),
inference(skolemize,[status(sab)],[74]) ).
tff(76,plain,
! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ),
inference(modus_ponens,[status(thm)],[75,68]) ).
tff(77,plain,
( ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
| ~ member(u,set_builder(u,set_builder(z,null_class)))
| ~ member(u,complement(set_builder(u,set_builder(z,null_class)))) )
<=> ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
| ~ member(u,set_builder(u,set_builder(z,null_class)))
| ~ member(u,complement(set_builder(u,set_builder(z,null_class)))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(78,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
| ~ member(u,set_builder(u,set_builder(z,null_class)))
| ~ member(u,complement(set_builder(u,set_builder(z,null_class)))) ),
inference(quant_inst,[status(thm)],]) ).
tff(79,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
| ~ member(u,set_builder(u,set_builder(z,null_class)))
| ~ member(u,complement(set_builder(u,set_builder(z,null_class)))) ),
inference(modus_ponens,[status(thm)],[78,77]) ).
tff(80,plain,
~ member(u,set_builder(u,set_builder(z,null_class))),
inference(unit_resolution,[status(thm)],[79,76,66]) ).
tff(81,plain,
~ member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))),
inference(modus_ponens,[status(thm)],[80,22]) ).
tff(82,plain,
( ( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| ~ member(u,universal_class)
| member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) )
<=> ( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| ~ member(u,universal_class)
| member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(83,plain,
( ( member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| ~ member(u,universal_class)
| member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) )
<=> ( ~ member(u,universal_class)
| member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(84,plain,
( ( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| ~ member(u,universal_class)
| member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) )
<=> ( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| ~ member(u,universal_class)
| member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) ) ),
inference(monotonicity,[status(thm)],[83]) ).
tff(85,plain,
( ( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| ~ member(u,universal_class)
| member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) )
<=> ( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| ~ member(u,universal_class)
| member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) ) ),
inference(transitivity,[status(thm)],[84,82]) ).
tff(86,plain,
( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| ~ member(u,universal_class)
| member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) ),
inference(quant_inst,[status(thm)],]) ).
tff(87,plain,
( ~ ! [Z: $i,X: $i] :
( member(Z,X)
| ~ member(Z,universal_class)
| member(Z,complement(X)) )
| ~ member(u,universal_class)
| member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) ),
inference(modus_ponens,[status(thm)],[86,85]) ).
tff(88,plain,
( member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| member(u,complement(intersection(complement(singleton(u)),complement(set_builder(z,null_class))))) ),
inference(unit_resolution,[status(thm)],[87,47,37]) ).
tff(89,plain,
member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class)))),
inference(unit_resolution,[status(thm)],[88,81]) ).
tff(90,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(
( ( ~ member(Z,intersection(X,Y))
| member(Z,X) )
<=> ( ~ member(Z,intersection(X,Y))
| member(Z,X) ) )),
inference(bind,[status(th)],]) ).
tff(91,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ) ),
inference(quant_intro,[status(thm)],[90]) ).
tff(92,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ) ),
inference(rewrite,[status(thm)],]) ).
tff(93,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',intersection1) ).
tff(94,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
inference(modus_ponens,[status(thm)],[93,92]) ).
tff(95,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
inference(skolemize,[status(sab)],[94]) ).
tff(96,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
inference(modus_ponens,[status(thm)],[95,91]) ).
tff(97,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| member(u,complement(singleton(u))) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| member(u,complement(singleton(u))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(98,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| member(u,complement(singleton(u))) ),
inference(quant_inst,[status(thm)],]) ).
tff(99,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(u,intersection(complement(singleton(u)),complement(set_builder(z,null_class))))
| member(u,complement(singleton(u))) ),
inference(modus_ponens,[status(thm)],[98,97]) ).
tff(100,plain,
member(u,complement(singleton(u))),
inference(unit_resolution,[status(thm)],[99,96,89]) ).
tff(101,plain,
^ [X: $i] :
refl(
( ( unordered_pair(X,X) = singleton(X) )
<=> ( unordered_pair(X,X) = singleton(X) ) )),
inference(bind,[status(th)],]) ).
tff(102,plain,
( ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
<=> ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ) ),
inference(quant_intro,[status(thm)],[101]) ).
tff(103,plain,
( ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
<=> ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ) ),
inference(rewrite,[status(thm)],]) ).
tff(104,axiom,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',singleton_set) ).
tff(105,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(modus_ponens,[status(thm)],[104,103]) ).
tff(106,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(skolemize,[status(sab)],[105]) ).
tff(107,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(modus_ponens,[status(thm)],[106,102]) ).
tff(108,plain,
( ~ ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
| ( unordered_pair(u,u) = singleton(u) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(109,plain,
unordered_pair(u,u) = singleton(u),
inference(unit_resolution,[status(thm)],[108,107]) ).
tff(110,plain,
singleton(u) = unordered_pair(u,u),
inference(symmetry,[status(thm)],[109]) ).
tff(111,plain,
( member(u,singleton(u))
<=> member(u,unordered_pair(u,u)) ),
inference(monotonicity,[status(thm)],[110]) ).
tff(112,plain,
( member(u,unordered_pair(u,u))
<=> member(u,singleton(u)) ),
inference(symmetry,[status(thm)],[111]) ).
tff(113,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(114,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)],[113]) ).
tff(115,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(116,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(117,plain,
! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[116,115]) ).
tff(118,plain,
! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) ),
inference(skolemize,[status(sab)],[117]) ).
tff(119,plain,
! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[118,114]) ).
tff(120,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) )
| ~ member(u,universal_class)
| member(u,unordered_pair(u,u)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) )
| ~ member(u,universal_class)
| member(u,unordered_pair(u,u)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(121,plain,
( ~ ! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) )
| ~ member(u,universal_class)
| member(u,unordered_pair(u,u)) ),
inference(quant_inst,[status(thm)],]) ).
tff(122,plain,
( ~ ! [Y: $i,X: $i] :
( ~ member(X,universal_class)
| member(X,unordered_pair(X,Y)) )
| ~ member(u,universal_class)
| member(u,unordered_pair(u,u)) ),
inference(modus_ponens,[status(thm)],[121,120]) ).
tff(123,plain,
member(u,unordered_pair(u,u)),
inference(unit_resolution,[status(thm)],[122,119,37]) ).
tff(124,plain,
member(u,singleton(u)),
inference(modus_ponens,[status(thm)],[123,112]) ).
tff(125,plain,
( ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
| ~ member(u,singleton(u))
| ~ member(u,complement(singleton(u))) )
<=> ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
| ~ member(u,singleton(u))
| ~ member(u,complement(singleton(u))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(126,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
| ~ member(u,singleton(u))
| ~ member(u,complement(singleton(u))) ),
inference(quant_inst,[status(thm)],]) ).
tff(127,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
| ~ member(u,singleton(u))
| ~ member(u,complement(singleton(u))) ),
inference(modus_ponens,[status(thm)],[126,125]) ).
tff(128,plain,
$false,
inference(unit_resolution,[status(thm)],[127,76,124,100]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET131-6 : 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 : n013.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 02:42:17 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.13/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.34 Usage: tptp [options] [-file:]file
% 0.13/0.34 -h, -? prints this message.
% 0.13/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.34 -m, -model generate model.
% 0.13/0.34 -p, -proof generate proof.
% 0.13/0.34 -c, -core generate unsat core of named formulas.
% 0.13/0.34 -st, -statistics display statistics.
% 0.13/0.34 -t:timeout set timeout (in second).
% 0.13/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.34 -<param>:<value> configuration parameter and value.
% 0.13/0.34 -o:<output-file> file to place output in.
% 0.19/0.43 % SZS status Unsatisfiable
% 0.19/0.43 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------