%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET411-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% 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 : Sat Jun 21 05:25:43 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.40s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 6
% Number of leaves : 25
% Syntax : Number of formulae : 55 ( 16 unt; 0 typ; 0 def)
% Number of atoms : 220 ( 4 equ)
% Maximal formula atoms : 12 ( 4 avg)
% Number of connectives : 291 ( 136 ~; 129 |; 0 &)
% ( 26 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of FOOLs : 10 ( 10 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 11 ( 11 usr; 6 con; 0-2 aty)
% Number of variables : 154 ( 141 !; 0 ?; 154 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(cross_product_type,type,
cross_product: ( $i * $i ) > $i ).
tff(universal_class_type,type,
universal_class: $i ).
tff(ordered_pair_type,type,
ordered_pair: ( $i * $i ) > $i ).
tff(y_type,type,
y: $i ).
tff(x_type,type,
x: $i ).
tff(element_relation_type,type,
element_relation: $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(identity_relation_type,type,
identity_relation: $i ).
tff(singleton_relation_type,type,
singleton_relation: $i ).
tff(subclass_type,type,
subclass: ( $i * $i ) > $o ).
tff(1,plain,
( ~ member(x,universal_class)
<=> ~ member(x,universal_class) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
~ member(x,universal_class),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_compose_condition_for_singleton_membership1_2) ).
tff(3,plain,
~ member(x,universal_class),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
refl(( ( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) )
<=> ( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) ) )),
inference(bind,[status(th)],[]) ).
tff(5,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(7,axiom,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',cartesian_product1) ).
tff(8,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) ),
inference(modus_ponens,[status(thm)],[7,6]) ).
tff(9,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) ),
inference(skolemize,[status(sab)],[8]) ).
tff(10,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) ),
inference(modus_ponens,[status(thm)],[9,5]) ).
tff(11,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) )
| ~ member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| member(x,universal_class) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) )
| ~ member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| member(x,universal_class) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(12,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) )
| ~ member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| member(x,universal_class) ),
inference(quant_inst,[status(thm)],[]) ).
tff(13,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) )
| ~ member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| member(x,universal_class) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
~ member(ordered_pair(x,y),cross_product(universal_class,universal_class)),
inference(unit_resolution,[status(thm)],[13,10,3]) ).
tff(15,plain,
( ( intersection(complement(compose(element_relation,complement(identity_relation))),element_relation) = singleton_relation )
<=> ( intersection(complement(compose(element_relation,complement(identity_relation))),element_relation) = singleton_relation ) ),
inference(rewrite,[status(thm)],[]) ).
tff(16,axiom,
intersection(complement(compose(element_relation,complement(identity_relation))),element_relation) = singleton_relation,
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-1.ax',compose_can_define_singleton) ).
tff(17,plain,
intersection(complement(compose(element_relation,complement(identity_relation))),element_relation) = singleton_relation,
inference(modus_ponens,[status(thm)],[16,15]) ).
tff(18,plain,
( member(ordered_pair(x,y),intersection(complement(compose(element_relation,complement(identity_relation))),element_relation))
<=> member(ordered_pair(x,y),singleton_relation) ),
inference(monotonicity,[status(thm)],[17]) ).
tff(19,plain,
( member(ordered_pair(x,y),singleton_relation)
<=> member(ordered_pair(x,y),intersection(complement(compose(element_relation,complement(identity_relation))),element_relation)) ),
inference(symmetry,[status(thm)],[18]) ).
tff(20,plain,
( member(ordered_pair(x,y),singleton_relation)
<=> member(ordered_pair(x,y),singleton_relation) ),
inference(rewrite,[status(thm)],[]) ).
tff(21,axiom,
member(ordered_pair(x,y),singleton_relation),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_compose_condition_for_singleton_membership1_1) ).
tff(22,plain,
member(ordered_pair(x,y),singleton_relation),
inference(modus_ponens,[status(thm)],[21,20]) ).
tff(23,plain,
member(ordered_pair(x,y),intersection(complement(compose(element_relation,complement(identity_relation))),element_relation)),
inference(modus_ponens,[status(thm)],[22,19]) ).
tff(24,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(25,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)],[24]) ).
tff(26,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(27,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(28,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(skolemize,[status(sab)],[28]) ).
tff(30,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(modus_ponens,[status(thm)],[29,25]) ).
tff(31,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(identity_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(identity_relation))),element_relation))
| member(ordered_pair(x,y),element_relation) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(32,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(identity_relation))),element_relation))
| member(ordered_pair(x,y),element_relation) ),
inference(quant_inst,[status(thm)],[]) ).
tff(33,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(identity_relation))),element_relation))
| member(ordered_pair(x,y),element_relation) ),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
( ~ member(ordered_pair(x,y),intersection(complement(compose(element_relation,complement(identity_relation))),element_relation))
| member(ordered_pair(x,y),element_relation) ),
inference(unit_resolution,[status(thm)],[33,30]) ).
tff(35,plain,
member(ordered_pair(x,y),element_relation),
inference(unit_resolution,[status(thm)],[34,23]) ).
tff(36,plain,
( subclass(element_relation,cross_product(universal_class,universal_class))
<=> subclass(element_relation,cross_product(universal_class,universal_class)) ),
inference(rewrite,[status(thm)],[]) ).
tff(37,axiom,
subclass(element_relation,cross_product(universal_class,universal_class)),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',element_relation1) ).
tff(38,plain,
subclass(element_relation,cross_product(universal_class,universal_class)),
inference(modus_ponens,[status(thm)],[37,36]) ).
tff(39,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(40,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)],[39]) ).
tff(41,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(42,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(43,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)],[42]) ).
tff(44,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(45,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[45,41]) ).
tff(47,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(skolemize,[status(sab)],[46]) ).
tff(48,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[47,40]) ).
tff(49,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| ~ subclass(element_relation,cross_product(universal_class,universal_class))
| member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| ~ member(ordered_pair(x,y),element_relation) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| ~ subclass(element_relation,cross_product(universal_class,universal_class))
| member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| ~ member(ordered_pair(x,y),element_relation) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(50,plain,
( ( member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| ~ member(ordered_pair(x,y),element_relation)
| ~ subclass(element_relation,cross_product(universal_class,universal_class)) )
<=> ( ~ subclass(element_relation,cross_product(universal_class,universal_class))
| member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| ~ member(ordered_pair(x,y),element_relation) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(51,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| ~ member(ordered_pair(x,y),element_relation)
| ~ subclass(element_relation,cross_product(universal_class,universal_class)) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| ~ subclass(element_relation,cross_product(universal_class,universal_class))
| member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| ~ member(ordered_pair(x,y),element_relation) ) ),
inference(monotonicity,[status(thm)],[50]) ).
tff(52,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| ~ member(ordered_pair(x,y),element_relation)
| ~ subclass(element_relation,cross_product(universal_class,universal_class)) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| ~ subclass(element_relation,cross_product(universal_class,universal_class))
| member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| ~ member(ordered_pair(x,y),element_relation) ) ),
inference(transitivity,[status(thm)],[51,49]) ).
tff(53,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| ~ member(ordered_pair(x,y),element_relation)
| ~ subclass(element_relation,cross_product(universal_class,universal_class)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(54,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| ~ subclass(element_relation,cross_product(universal_class,universal_class))
| member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| ~ member(ordered_pair(x,y),element_relation) ),
inference(modus_ponens,[status(thm)],[53,52]) ).
tff(55,plain,
$false,
inference(unit_resolution,[status(thm)],[54,48,38,35,14]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET411-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n010.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Fri Jun 20 05:06:27 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.19/0.40 % SZS status Unsatisfiable
% 0.19/0.40 % SZS output start Proof
% See solution above
% 0.19/0.42 % E exiting
%------------------------------------------------------------------------------