%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET256-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% 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 : Sat Jun 21 05:25:33 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.55s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 63
% Syntax : Number of formulae : 159 ( 39 unt; 0 typ; 0 def)
% Number of atoms : 575 ( 35 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 716 ( 321 ~; 329 |; 0 &)
% ( 66 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of FOOLs : 21 ( 21 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; 4 con; 0-3 aty)
% Number of variables : 484 ( 438 !; 0 ?; 484 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(z1_type,type,
z1: $i ).
tff(second_type,type,
second: $i > $i ).
tff(not_subclass_element_type,type,
not_subclass_element: ( $i * $i ) > $i ).
tff(restrict_type,type,
restrict: ( $i * $i * $i ) > $i ).
tff(z2_type,type,
z2: $i ).
tff(y_type,type,
y: $i ).
tff(x_type,type,
x: $i ).
tff(cross_product_type,type,
cross_product: ( $i * $i ) > $i ).
tff(ordered_pair_type,type,
ordered_pair: ( $i * $i ) > $i ).
tff(first_type,type,
first: $i > $i ).
tff(intersection_type,type,
intersection: ( $i * $i ) > $i ).
tff(subclass_type,type,
subclass: ( $i * $i ) > $o ).
tff(1,plain,
^ [Y: $i,Xr: $i,X: $i] :
refl(( ( intersection(cross_product(X,Y),Xr) = restrict(Xr,X,Y) )
<=> ( intersection(cross_product(X,Y),Xr) = restrict(Xr,X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [Y: $i,Xr: $i,X: $i] : ( intersection(cross_product(X,Y),Xr) = restrict(Xr,X,Y) )
<=> ! [Y: $i,Xr: $i,X: $i] : ( intersection(cross_product(X,Y),Xr) = restrict(Xr,X,Y) ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [Y: $i,Xr: $i,X: $i] : ( intersection(cross_product(X,Y),Xr) = restrict(Xr,X,Y) )
<=> ! [Y: $i,Xr: $i,X: $i] : ( intersection(cross_product(X,Y),Xr) = restrict(Xr,X,Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [Y: $i,Xr: $i,X: $i] : ( intersection(cross_product(X,Y),Xr) = restrict(Xr,X,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',restriction2) ).
tff(5,plain,
! [Y: $i,Xr: $i,X: $i] : ( intersection(cross_product(X,Y),Xr) = restrict(Xr,X,Y) ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [Y: $i,Xr: $i,X: $i] : ( intersection(cross_product(X,Y),Xr) = restrict(Xr,X,Y) ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [Y: $i,Xr: $i,X: $i] : ( intersection(cross_product(X,Y),Xr) = restrict(Xr,X,Y) ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [Y: $i,Xr: $i,X: $i] : ( intersection(cross_product(X,Y),Xr) = restrict(Xr,X,Y) )
| ( intersection(cross_product(y,z1),x) = restrict(x,y,z1) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
intersection(cross_product(y,z1),x) = restrict(x,y,z1),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
( member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x))
<=> member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z1)) ),
inference(monotonicity,[status(thm)],[9]) ).
tff(11,plain,
( member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z1))
<=> member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x)) ),
inference(symmetry,[status(thm)],[10]) ).
tff(12,plain,
( ~ subclass(restrict(x,y,z1),restrict(x,y,z2))
<=> ~ subclass(restrict(x,y,z1),restrict(x,y,z2)) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
~ subclass(restrict(x,y,z1),restrict(x,y,z2)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_monotonicity_of_restriction3_2) ).
tff(14,plain,
~ subclass(restrict(x,y,z1),restrict(x,y,z2)),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,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(16,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)],[15]) ).
tff(17,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(18,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(19,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)],[18]) ).
tff(20,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(21,plain,
! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ),
inference(modus_ponens,[status(thm)],[20,19]) ).
tff(22,plain,
! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ),
inference(modus_ponens,[status(thm)],[21,17]) ).
tff(23,plain,
! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ),
inference(skolemize,[status(sab)],[22]) ).
tff(24,plain,
! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) ),
inference(modus_ponens,[status(thm)],[23,16]) ).
tff(25,plain,
( ( ~ ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
| subclass(restrict(x,y,z1),restrict(x,y,z2))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z1)) )
<=> ( ~ ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
| subclass(restrict(x,y,z1),restrict(x,y,z2))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z1)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(26,plain,
( ~ ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
| subclass(restrict(x,y,z1),restrict(x,y,z2))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z1)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(27,plain,
( ~ ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
| subclass(restrict(x,y,z1),restrict(x,y,z2))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z1)) ),
inference(modus_ponens,[status(thm)],[26,25]) ).
tff(28,plain,
member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z1)),
inference(unit_resolution,[status(thm)],[27,24,14]) ).
tff(29,plain,
member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x)),
inference(modus_ponens,[status(thm)],[28,11]) ).
tff(30,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(31,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)],[30]) ).
tff(32,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(33,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(34,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
inference(skolemize,[status(sab)],[34]) ).
tff(36,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
inference(modus_ponens,[status(thm)],[35,31]) ).
tff(37,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z1)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z1)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(38,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z1)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(39,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z1)) ),
inference(modus_ponens,[status(thm)],[38,37]) ).
tff(40,plain,
( ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z1)) ),
inference(unit_resolution,[status(thm)],[39,36]) ).
tff(41,plain,
member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z1)),
inference(unit_resolution,[status(thm)],[40,29]) ).
tff(42,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(43,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)],[42]) ).
tff(44,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(45,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(46,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)],[45,44]) ).
tff(47,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) ),
inference(skolemize,[status(sab)],[46]) ).
tff(48,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)],[47,43]) ).
tff(49,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z1))
| ( ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))) = not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)) ) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z1))
| ( ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))) = not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(50,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z1))
| ( ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))) = not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(51,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z1))
| ( ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))) = not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)) ) ),
inference(modus_ponens,[status(thm)],[50,49]) ).
tff(52,plain,
ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))) = not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),
inference(unit_resolution,[status(thm)],[51,48,41]) ).
tff(53,plain,
( member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1))
<=> member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z1)) ),
inference(monotonicity,[status(thm)],[52]) ).
tff(54,plain,
( member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z1))
<=> member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1)) ),
inference(symmetry,[status(thm)],[53]) ).
tff(55,plain,
member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1)),
inference(modus_ponens,[status(thm)],[41,54]) ).
tff(56,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(57,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)],[56]) ).
tff(58,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(59,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(60,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)],[59,58]) ).
tff(61,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)],[60]) ).
tff(62,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)],[61,57]) ).
tff(63,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(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1))
| member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z1) )
<=> ( ~ ! [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(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1))
| member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z1) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(64,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(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1))
| member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z1) ),
inference(quant_inst,[status(thm)],[]) ).
tff(65,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(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1))
| member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z1) ),
inference(modus_ponens,[status(thm)],[64,63]) ).
tff(66,plain,
( ~ member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1))
| member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z1) ),
inference(unit_resolution,[status(thm)],[65,62]) ).
tff(67,plain,
member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z1),
inference(unit_resolution,[status(thm)],[66,55]) ).
tff(68,plain,
( member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z2))
<=> member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z2)) ),
inference(monotonicity,[status(thm)],[52]) ).
tff(69,plain,
( member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z2))
<=> member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z2)) ),
inference(symmetry,[status(thm)],[68]) ).
tff(70,plain,
( ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z2))
<=> ~ member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z2)) ),
inference(monotonicity,[status(thm)],[69]) ).
tff(71,plain,
( ~ ! [Y: $i,Xr: $i,X: $i] : ( intersection(cross_product(X,Y),Xr) = restrict(Xr,X,Y) )
| ( intersection(cross_product(y,z2),x) = restrict(x,y,z2) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(72,plain,
intersection(cross_product(y,z2),x) = restrict(x,y,z2),
inference(unit_resolution,[status(thm)],[71,7]) ).
tff(73,plain,
( member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z2),x))
<=> member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z2)) ),
inference(monotonicity,[status(thm)],[72]) ).
tff(74,plain,
( member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z2))
<=> member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z2),x)) ),
inference(symmetry,[status(thm)],[73]) ).
tff(75,plain,
( ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z2))
<=> ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z2),x)) ),
inference(monotonicity,[status(thm)],[74]) ).
tff(76,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(77,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)],[76]) ).
tff(78,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(79,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(80,plain,
! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[79,78]) ).
tff(81,plain,
! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ),
inference(skolemize,[status(sab)],[80]) ).
tff(82,plain,
! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[81,77]) ).
tff(83,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z2))
| subclass(restrict(x,y,z1),restrict(x,y,z2)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z2))
| subclass(restrict(x,y,z1),restrict(x,y,z2)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(84,plain,
( ~ ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z2))
| subclass(restrict(x,y,z1),restrict(x,y,z2)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(85,plain,
( ~ ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z2))
| subclass(restrict(x,y,z1),restrict(x,y,z2)) ),
inference(modus_ponens,[status(thm)],[84,83]) ).
tff(86,plain,
~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),restrict(x,y,z2)),
inference(unit_resolution,[status(thm)],[85,82,14]) ).
tff(87,plain,
~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z2),x)),
inference(modus_ponens,[status(thm)],[86,75]) ).
tff(88,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(89,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)],[88]) ).
tff(90,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(91,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(92,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(modus_ponens,[status(thm)],[91,90]) ).
tff(93,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(skolemize,[status(sab)],[92]) ).
tff(94,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(modus_ponens,[status(thm)],[93,89]) ).
tff(95,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),x) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),x) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(96,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),x) ),
inference(quant_inst,[status(thm)],[]) ).
tff(97,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),x) ),
inference(modus_ponens,[status(thm)],[96,95]) ).
tff(98,plain,
( ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z1),x))
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),x) ),
inference(unit_resolution,[status(thm)],[97,94]) ).
tff(99,plain,
member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),x),
inference(unit_resolution,[status(thm)],[98,29]) ).
tff(100,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) )
<=> ( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) ) )),
inference(bind,[status(th)],[]) ).
tff(101,plain,
( ! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) )
<=> ! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) ) ),
inference(quant_intro,[status(thm)],[100]) ).
tff(102,plain,
( ! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) )
<=> ! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(103,plain,
^ [Z: $i,Y: $i,X: $i] :
rewrite(( ( ~ member(Z,X)
| ~ member(Z,Y)
| member(Z,intersection(X,Y)) )
<=> ( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) ) )),
inference(bind,[status(th)],[]) ).
tff(104,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,Y)
| member(Z,intersection(X,Y)) )
<=> ! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) ) ),
inference(quant_intro,[status(thm)],[103]) ).
tff(105,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,Y)
| member(Z,intersection(X,Y)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',intersection3) ).
tff(106,plain,
! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) ),
inference(modus_ponens,[status(thm)],[105,104]) ).
tff(107,plain,
! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) ),
inference(modus_ponens,[status(thm)],[106,102]) ).
tff(108,plain,
! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) ),
inference(skolemize,[status(sab)],[107]) ).
tff(109,plain,
! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) ),
inference(modus_ponens,[status(thm)],[108,101]) ).
tff(110,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) )
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z2),x))
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),x)
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z2)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) )
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z2),x))
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),x)
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z2)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(111,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) )
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z2),x))
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),x)
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z2)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(112,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( member(Z,intersection(X,Y))
| ~ member(Z,Y)
| ~ member(Z,X) )
| member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z2),x))
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),x)
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z2)) ),
inference(modus_ponens,[status(thm)],[111,110]) ).
tff(113,plain,
( member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),intersection(cross_product(y,z2),x))
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),x)
| ~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z2)) ),
inference(unit_resolution,[status(thm)],[112,109]) ).
tff(114,plain,
~ member(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)),cross_product(y,z2)),
inference(unit_resolution,[status(thm)],[113,99,87]) ).
tff(115,plain,
~ member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z2)),
inference(modus_ponens,[status(thm)],[114,70]) ).
tff(116,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(117,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)],[116]) ).
tff(118,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(119,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(120,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)],[119,118]) ).
tff(121,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)],[120]) ).
tff(122,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)],[121,117]) ).
tff(123,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(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1))
| member(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),y) )
<=> ( ~ ! [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(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1))
| member(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(124,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(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1))
| member(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),y) ),
inference(quant_inst,[status(thm)],[]) ).
tff(125,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(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1))
| member(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),y) ),
inference(modus_ponens,[status(thm)],[124,123]) ).
tff(126,plain,
( ~ member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z1))
| member(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),y) ),
inference(unit_resolution,[status(thm)],[125,122]) ).
tff(127,plain,
member(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),y),
inference(unit_resolution,[status(thm)],[126,55]) ).
tff(128,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(129,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)],[128]) ).
tff(130,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(131,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(132,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)],[131]) ).
tff(133,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(134,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)],[133,132]) ).
tff(135,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)],[134,130]) ).
tff(136,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)],[135]) ).
tff(137,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)],[136,129]) ).
tff(138,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(restrict(x,y,z1),restrict(x,y,z2))),y)
| member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z2))
| ~ member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z2) )
<=> ( ~ ! [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(restrict(x,y,z1),restrict(x,y,z2))),y)
| member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z2))
| ~ member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z2) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(139,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(restrict(x,y,z1),restrict(x,y,z2))),y)
| member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z2))
| ~ member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z2) ),
inference(quant_inst,[status(thm)],[]) ).
tff(140,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(restrict(x,y,z1),restrict(x,y,z2))),y)
| member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z2))
| ~ member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z2) ),
inference(modus_ponens,[status(thm)],[139,138]) ).
tff(141,plain,
( ~ member(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),y)
| member(ordered_pair(first(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2)))),cross_product(y,z2))
| ~ member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z2) ),
inference(unit_resolution,[status(thm)],[140,137]) ).
tff(142,plain,
~ member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z2),
inference(unit_resolution,[status(thm)],[141,127,115]) ).
tff(143,plain,
( subclass(z1,z2)
<=> subclass(z1,z2) ),
inference(rewrite,[status(thm)],[]) ).
tff(144,axiom,
subclass(z1,z2),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_monotonicity_of_restriction3_1) ).
tff(145,plain,
subclass(z1,z2),
inference(modus_ponens,[status(thm)],[144,143]) ).
tff(146,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(147,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)],[146]) ).
tff(148,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(149,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(150,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)],[149]) ).
tff(151,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(152,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[151,150]) ).
tff(153,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[152,148]) ).
tff(154,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(skolemize,[status(sab)],[153]) ).
tff(155,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[154,147]) ).
tff(156,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z2)
| ~ member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z1)
| ~ subclass(z1,z2) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z2)
| ~ member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z1)
| ~ subclass(z1,z2) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(157,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z2)
| ~ member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z1)
| ~ subclass(z1,z2) ),
inference(quant_inst,[status(thm)],[]) ).
tff(158,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z2)
| ~ member(second(not_subclass_element(restrict(x,y,z1),restrict(x,y,z2))),z1)
| ~ subclass(z1,z2) ),
inference(modus_ponens,[status(thm)],[157,156]) ).
tff(159,plain,
$false,
inference(unit_resolution,[status(thm)],[158,155,145,142,67]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : SET256-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.12/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n009.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 04:39:18 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.55 % SZS status Unsatisfiable
% 0.20/0.55 % SZS output start Proof
% See solution above
% 0.41/0.57 % E exiting
%------------------------------------------------------------------------------