TSTP Solution File: SET216-6 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET216-6 : 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 05:06:12 EDT 2022
% Result : Unsatisfiable 0.20s 0.43s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 50
% Syntax : Number of formulae : 107 ( 20 unt; 11 typ; 0 def)
% Number of atoms : 334 ( 19 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 406 ( 177 ~; 191 |; 0 &)
% ( 38 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 6 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of FOOLs : 9 ( 9 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 14 ( 8 >; 6 *; 0 +; 0 <<)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 9 ( 9 usr; 3 con; 0-2 aty)
% Number of variables : 310 ( 282 !; 0 ?; 310 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(y_type,type,
y: $i ).
tff(second_type,type,
second: $i > $i ).
tff(not_subclass_element_type,type,
not_subclass_element: ( $i * $i ) > $i ).
tff(cross_product_type,type,
cross_product: ( $i * $i ) > $i ).
tff(x_type,type,
x: $i ).
tff(intersection_type,type,
intersection: ( $i * $i ) > $i ).
tff(z_type,type,
z: $i ).
tff(ordered_pair_type,type,
ordered_pair: ( $i * $i ) > $i ).
tff(first_type,type,
first: $i > $i ).
tff(subclass_type,type,
subclass: ( $i * $i ) > $o ).
tff(1,plain,
( ~ subclass(cross_product(x,intersection(y,z)),cross_product(x,y))
<=> ~ subclass(cross_product(x,intersection(y,z)),cross_product(x,y)) ),
inference(rewrite,[status(thm)],]) ).
tff(2,axiom,
~ subclass(cross_product(x,intersection(y,z)),cross_product(x,y)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_corollary_7_to_X_product_monotonicity_1) ).
tff(3,plain,
~ subclass(cross_product(x,intersection(y,z)),cross_product(x,y)),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [Y: $i,X: $i] :
refl(
( ( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
<=> ( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ) )),
inference(bind,[status(th)],]) ).
tff(5,plain,
( ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
<=> ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
<=> ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ) ),
inference(rewrite,[status(thm)],]) ).
tff(7,plain,
^ [Y: $i,X: $i] :
rewrite(
( ( member(not_subclass_element(X,Y),X)
| subclass(X,Y) )
<=> ( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ) )),
inference(bind,[status(th)],]) ).
tff(8,plain,
( ! [Y: $i,X: $i] :
( member(not_subclass_element(X,Y),X)
| subclass(X,Y) )
<=> ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ) ),
inference(quant_intro,[status(thm)],[7]) ).
tff(9,axiom,
! [Y: $i,X: $i] :
( member(not_subclass_element(X,Y),X)
| subclass(X,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',not_subclass_members1) ).
tff(10,plain,
! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ),
inference(modus_ponens,[status(thm)],[9,8]) ).
tff(11,plain,
! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ),
inference(modus_ponens,[status(thm)],[10,6]) ).
tff(12,plain,
! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ),
inference(skolemize,[status(sab)],[11]) ).
tff(13,plain,
! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ),
inference(modus_ponens,[status(thm)],[12,5]) ).
tff(14,plain,
( ( ~ ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
| subclass(cross_product(x,intersection(y,z)),cross_product(x,y))
| member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,intersection(y,z))) )
<=> ( ~ ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
| subclass(cross_product(x,intersection(y,z)),cross_product(x,y))
| member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,intersection(y,z))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(15,plain,
( ~ ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
| subclass(cross_product(x,intersection(y,z)),cross_product(x,y))
| member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,intersection(y,z))) ),
inference(quant_inst,[status(thm)],]) ).
tff(16,plain,
( ~ ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
| subclass(cross_product(x,intersection(y,z)),cross_product(x,y))
| member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,intersection(y,z))) ),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,intersection(y,z))),
inference(unit_resolution,[status(thm)],[16,13,3]) ).
tff(18,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(
( ( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
<=> ( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) ) )),
inference(bind,[status(th)],]) ).
tff(19,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) ) ),
inference(quant_intro,[status(thm)],[18]) ).
tff(20,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(21,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',cartesian_product4) ).
tff(22,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) ),
inference(modus_ponens,[status(thm)],[21,20]) ).
tff(23,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) ),
inference(skolemize,[status(sab)],[22]) ).
tff(24,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) ),
inference(modus_ponens,[status(thm)],[23,19]) ).
tff(25,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
| ~ member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,intersection(y,z)))
| ( ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))) = not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)) ) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
| ~ member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,intersection(y,z)))
| ( ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))) = not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(26,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
| ~ member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,intersection(y,z)))
| ( ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))) = not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(27,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
| ~ member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,intersection(y,z)))
| ( ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))) = not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)) ) ),
inference(modus_ponens,[status(thm)],[26,25]) ).
tff(28,plain,
ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))) = not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),
inference(unit_resolution,[status(thm)],[27,24,17]) ).
tff(29,plain,
( member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,y))
<=> member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,y)) ),
inference(monotonicity,[status(thm)],[28]) ).
tff(30,plain,
( member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,y))
<=> member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,y)) ),
inference(symmetry,[status(thm)],[29]) ).
tff(31,plain,
( ~ member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,y))
<=> ~ member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,y)) ),
inference(monotonicity,[status(thm)],[30]) ).
tff(32,plain,
^ [Y: $i,X: $i] :
refl(
( ( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
<=> ( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ) )),
inference(bind,[status(th)],]) ).
tff(33,plain,
( ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
<=> ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ) ),
inference(quant_intro,[status(thm)],[32]) ).
tff(34,plain,
( ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
<=> ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ) ),
inference(rewrite,[status(thm)],]) ).
tff(35,axiom,
! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',not_subclass_members2) ).
tff(36,plain,
! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[35,34]) ).
tff(37,plain,
! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ),
inference(skolemize,[status(sab)],[36]) ).
tff(38,plain,
! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[37,33]) ).
tff(39,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
| ~ member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,y))
| subclass(cross_product(x,intersection(y,z)),cross_product(x,y)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
| ~ member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,y))
| subclass(cross_product(x,intersection(y,z)),cross_product(x,y)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(40,plain,
( ~ ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
| ~ member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,y))
| subclass(cross_product(x,intersection(y,z)),cross_product(x,y)) ),
inference(quant_inst,[status(thm)],]) ).
tff(41,plain,
( ~ ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
| ~ member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,y))
| subclass(cross_product(x,intersection(y,z)),cross_product(x,y)) ),
inference(modus_ponens,[status(thm)],[40,39]) ).
tff(42,plain,
~ member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,y)),
inference(unit_resolution,[status(thm)],[41,38,3]) ).
tff(43,plain,
~ member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,y)),
inference(modus_ponens,[status(thm)],[42,31]) ).
tff(44,plain,
( member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z)))
<=> member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,intersection(y,z))) ),
inference(monotonicity,[status(thm)],[28]) ).
tff(45,plain,
( member(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)),cross_product(x,intersection(y,z)))
<=> member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z))) ),
inference(symmetry,[status(thm)],[44]) ).
tff(46,plain,
member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z))),
inference(modus_ponens,[status(thm)],[17,45]) ).
tff(47,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(48,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)],[47]) ).
tff(49,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(50,axiom,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',cartesian_product1) ).
tff(51,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)],[50,49]) ).
tff(52,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)],[51]) ).
tff(53,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)],[52,48]) ).
tff(54,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) )
| ~ member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z)))
| member(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),x) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) )
| ~ member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z)))
| member(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),x) ) ),
inference(rewrite,[status(thm)],]) ).
tff(55,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) )
| ~ member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z)))
| member(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),x) ),
inference(quant_inst,[status(thm)],]) ).
tff(56,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,X) )
| ~ member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z)))
| member(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),x) ),
inference(modus_ponens,[status(thm)],[55,54]) ).
tff(57,plain,
( ~ member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z)))
| member(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),x) ),
inference(unit_resolution,[status(thm)],[56,53]) ).
tff(58,plain,
member(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),x),
inference(unit_resolution,[status(thm)],[57,46]) ).
tff(59,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
refl(
( ( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) )
<=> ( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) ) )),
inference(bind,[status(th)],]) ).
tff(60,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) ) ),
inference(quant_intro,[status(thm)],[59]) ).
tff(61,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) ) ),
inference(rewrite,[status(thm)],]) ).
tff(62,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
rewrite(
( ( ~ member(U,X)
| ~ member(V,Y)
| member(ordered_pair(U,V),cross_product(X,Y)) )
<=> ( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) ) )),
inference(bind,[status(th)],]) ).
tff(63,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| ~ member(V,Y)
| member(ordered_pair(U,V),cross_product(X,Y)) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) ) ),
inference(quant_intro,[status(thm)],[62]) ).
tff(64,axiom,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| ~ member(V,Y)
| member(ordered_pair(U,V),cross_product(X,Y)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',cartesian_product3) ).
tff(65,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) ),
inference(modus_ponens,[status(thm)],[64,63]) ).
tff(66,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) ),
inference(modus_ponens,[status(thm)],[65,61]) ).
tff(67,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) ),
inference(skolemize,[status(sab)],[66]) ).
tff(68,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) ),
inference(modus_ponens,[status(thm)],[67,60]) ).
tff(69,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) )
| ~ member(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),x)
| member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,y))
| ~ member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),y) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) )
| ~ member(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),x)
| member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,y))
| ~ member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),y) ) ),
inference(rewrite,[status(thm)],]) ).
tff(70,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) )
| ~ member(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),x)
| member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,y))
| ~ member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),y) ),
inference(quant_inst,[status(thm)],]) ).
tff(71,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(ordered_pair(U,V),cross_product(X,Y))
| ~ member(V,Y) )
| ~ member(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),x)
| member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,y))
| ~ member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),y) ),
inference(modus_ponens,[status(thm)],[70,69]) ).
tff(72,plain,
( ~ member(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),x)
| member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,y))
| ~ member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),y) ),
inference(unit_resolution,[status(thm)],[71,68]) ).
tff(73,plain,
~ member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),y),
inference(unit_resolution,[status(thm)],[72,58,43]) ).
tff(74,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
refl(
( ( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) )
<=> ( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) ) )),
inference(bind,[status(th)],]) ).
tff(75,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) ) ),
inference(quant_intro,[status(thm)],[74]) ).
tff(76,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) ) ),
inference(rewrite,[status(thm)],]) ).
tff(77,axiom,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',cartesian_product2) ).
tff(78,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) ),
inference(modus_ponens,[status(thm)],[77,76]) ).
tff(79,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) ),
inference(skolemize,[status(sab)],[78]) ).
tff(80,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) ),
inference(modus_ponens,[status(thm)],[79,75]) ).
tff(81,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) )
| ~ member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z)))
| member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),intersection(y,z)) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) )
| ~ member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z)))
| member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),intersection(y,z)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(82,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) )
| ~ member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z)))
| member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),intersection(y,z)) ),
inference(quant_inst,[status(thm)],]) ).
tff(83,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,Y) )
| ~ member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z)))
| member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),intersection(y,z)) ),
inference(modus_ponens,[status(thm)],[82,81]) ).
tff(84,plain,
( ~ member(ordered_pair(first(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y)))),cross_product(x,intersection(y,z)))
| member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),intersection(y,z)) ),
inference(unit_resolution,[status(thm)],[83,80]) ).
tff(85,plain,
member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),intersection(y,z)),
inference(unit_resolution,[status(thm)],[84,46]) ).
tff(86,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(87,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)],[86]) ).
tff(88,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(89,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',intersection1) ).
tff(90,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
inference(modus_ponens,[status(thm)],[89,88]) ).
tff(91,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
inference(skolemize,[status(sab)],[90]) ).
tff(92,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
inference(modus_ponens,[status(thm)],[91,87]) ).
tff(93,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),intersection(y,z))
| member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),y) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),intersection(y,z))
| member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),y) ) ),
inference(rewrite,[status(thm)],]) ).
tff(94,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),intersection(y,z))
| member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),y) ),
inference(quant_inst,[status(thm)],]) ).
tff(95,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),intersection(y,z))
| member(second(not_subclass_element(cross_product(x,intersection(y,z)),cross_product(x,y))),y) ),
inference(modus_ponens,[status(thm)],[94,93]) ).
tff(96,plain,
$false,
inference(unit_resolution,[status(thm)],[95,92,85,73]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET216-6 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.07/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.33 % Computer : n009.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Sat Sep 3 03:25:20 EDT 2022
% 0.13/0.33 % 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.
% 0.20/0.43 % SZS status Unsatisfiable
% 0.20/0.43 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------