%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET108-7 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:25:15 AM UTC 2025
% Result : Unsatisfiable 0.23s 0.43s
% Output : Proof 0.23s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 9
% Syntax : Number of formulae : 21 ( 8 unt; 0 typ; 0 def)
% Number of atoms : 52 ( 19 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 56 ( 26 ~; 22 |; 0 &)
% ( 8 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 4 ( 2 avg)
% Number of FOOLs : 1 ( 1 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 4 ( 2 usr; 1 prp; 0-2 aty)
% Number of functors : 7 ( 7 usr; 3 con; 0-2 aty)
% Number of variables : 39 ( 36 !; 0 ?; 39 :)
% 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(second_type,type,
second: $i > $i ).
tff(z_type,type,
z: $i ).
tff(y_type,type,
y: $i ).
tff(first_type,type,
first: $i > $i ).
tff(1,plain,
( member(ordered_pair(y,z),cross_product(universal_class,universal_class))
<=> member(ordered_pair(y,z),cross_product(universal_class,universal_class)) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
member(ordered_pair(y,z),cross_product(universal_class,universal_class)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_existence_of_1st_and_2nd_1_1) ).
tff(3,plain,
member(ordered_pair(y,z),cross_product(universal_class,universal_class)),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,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(5,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)],[4]) ).
tff(6,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(7,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(8,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)],[7,6]) ).
tff(9,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) ),
inference(skolemize,[status(sab)],[8]) ).
tff(10,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)],[9,5]) ).
tff(11,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
| ~ member(ordered_pair(y,z),cross_product(universal_class,universal_class))
| ( ordered_pair(first(ordered_pair(y,z)),second(ordered_pair(y,z))) = ordered_pair(y,z) ) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
| ~ member(ordered_pair(y,z),cross_product(universal_class,universal_class))
| ( ordered_pair(first(ordered_pair(y,z)),second(ordered_pair(y,z))) = ordered_pair(y,z) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(12,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
| ~ member(ordered_pair(y,z),cross_product(universal_class,universal_class))
| ( ordered_pair(first(ordered_pair(y,z)),second(ordered_pair(y,z))) = ordered_pair(y,z) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(13,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,cross_product(X,Y))
| ( ordered_pair(first(Z),second(Z)) = Z ) )
| ~ member(ordered_pair(y,z),cross_product(universal_class,universal_class))
| ( ordered_pair(first(ordered_pair(y,z)),second(ordered_pair(y,z))) = ordered_pair(y,z) ) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
ordered_pair(first(ordered_pair(y,z)),second(ordered_pair(y,z))) = ordered_pair(y,z),
inference(unit_resolution,[status(thm)],[13,10,3]) ).
tff(15,plain,
( member(ordered_pair(first(ordered_pair(y,z)),second(ordered_pair(y,z))),cross_product(universal_class,universal_class))
<=> member(ordered_pair(y,z),cross_product(universal_class,universal_class)) ),
inference(monotonicity,[status(thm)],[14]) ).
tff(16,plain,
( member(ordered_pair(y,z),cross_product(universal_class,universal_class))
<=> member(ordered_pair(first(ordered_pair(y,z)),second(ordered_pair(y,z))),cross_product(universal_class,universal_class)) ),
inference(symmetry,[status(thm)],[15]) ).
tff(17,plain,
member(ordered_pair(first(ordered_pair(y,z)),second(ordered_pair(y,z))),cross_product(universal_class,universal_class)),
inference(modus_ponens,[status(thm)],[3,16]) ).
tff(18,plain,
( ~ member(ordered_pair(first(ordered_pair(y,z)),second(ordered_pair(y,z))),cross_product(universal_class,universal_class))
<=> ~ member(ordered_pair(first(ordered_pair(y,z)),second(ordered_pair(y,z))),cross_product(universal_class,universal_class)) ),
inference(rewrite,[status(thm)],[]) ).
tff(19,axiom,
~ member(ordered_pair(first(ordered_pair(y,z)),second(ordered_pair(y,z))),cross_product(universal_class,universal_class)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_existence_of_1st_and_2nd_1_2) ).
tff(20,plain,
~ member(ordered_pair(first(ordered_pair(y,z)),second(ordered_pair(y,z))),cross_product(universal_class,universal_class)),
inference(modus_ponens,[status(thm)],[19,18]) ).
tff(21,plain,
$false,
inference(unit_resolution,[status(thm)],[20,17]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.14 % Problem : SET108-7 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.08/0.14 % Command : run_E %s %d THM
% 0.14/0.35 % Computer : n003.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Jun 20 04:18:54 EDT 2025
% 0.14/0.35 % CPUTime :
% 0.23/0.43 % SZS status Unsatisfiable
% 0.23/0.43 % SZS output start Proof
% See solution above
% 0.23/0.44 % E exiting
%------------------------------------------------------------------------------