%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET121-7 : TPTP v9.0.0. Bugfixed v7.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.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 05:25:18 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.40s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 21
% Syntax : Number of formulae : 50 ( 14 unt; 0 typ; 0 def)
% Number of atoms : 152 ( 74 equ)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 139 ( 42 ~; 76 |; 0 &)
% ( 21 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 4 ( 2 avg)
% Number of FOOLs : 5 ( 5 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 5 ( 3 usr; 1 prp; 0-2 aty)
% Number of functors : 7 ( 7 usr; 3 con; 0-2 aty)
% Number of variables : 86 ( 78 !; 0 ?; 86 :)
% 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(y_type,type,
y: $i ).
tff(x_type,type,
x: $i ).
tff(first_type,type,
first: $i > $i ).
tff(1,plain,
( ( first(ordered_pair(x,y)) != ordered_pair(x,y) )
<=> ( first(ordered_pair(x,y)) != ordered_pair(x,y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
first(ordered_pair(x,y)) != ordered_pair(x,y),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_corollary_1_to_OP_determines_components2_2) ).
tff(3,plain,
first(ordered_pair(x,y)) != ordered_pair(x,y),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [X: $i] :
refl(( ( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) )
<=> ( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) ) )),
inference(bind,[status(th)],[]) ).
tff(5,plain,
( ! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) )
<=> ! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) )
<=> ! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(7,plain,
^ [X: $i] :
rewrite(( ( ( ordered_pair(first(X),second(X)) = X )
| ( first(X) = X ) )
<=> ( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) ) )),
inference(bind,[status(th)],[]) ).
tff(8,plain,
( ! [X: $i] :
( ( ordered_pair(first(X),second(X)) = X )
| ( first(X) = X ) )
<=> ! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) ) ),
inference(quant_intro,[status(thm)],[7]) ).
tff(9,axiom,
! [X: $i] :
( ( ordered_pair(first(X),second(X)) = X )
| ( first(X) = X ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_1st_and_2nd_4) ).
tff(10,plain,
! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) ),
inference(modus_ponens,[status(thm)],[9,8]) ).
tff(11,plain,
! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) ),
inference(modus_ponens,[status(thm)],[10,6]) ).
tff(12,plain,
! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) ),
inference(skolemize,[status(sab)],[11]) ).
tff(13,plain,
! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) ),
inference(modus_ponens,[status(thm)],[12,5]) ).
tff(14,plain,
( ( ~ ! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) )
| ( first(ordered_pair(x,y)) = ordered_pair(x,y) )
| ( ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))) = ordered_pair(x,y) ) )
<=> ( ~ ! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) )
| ( first(ordered_pair(x,y)) = ordered_pair(x,y) )
| ( ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))) = ordered_pair(x,y) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(15,plain,
( ~ ! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) )
| ( first(ordered_pair(x,y)) = ordered_pair(x,y) )
| ( ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))) = ordered_pair(x,y) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(16,plain,
( ~ ! [X: $i] :
( ( first(X) = X )
| ( ordered_pair(first(X),second(X)) = X ) )
| ( first(ordered_pair(x,y)) = ordered_pair(x,y) )
| ( ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))) = ordered_pair(x,y) ) ),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))) = ordered_pair(x,y),
inference(unit_resolution,[status(thm)],[16,13,3]) ).
tff(18,plain,
( member(ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))),cross_product(universal_class,universal_class))
<=> member(ordered_pair(x,y),cross_product(universal_class,universal_class)) ),
inference(monotonicity,[status(thm)],[17]) ).
tff(19,plain,
( member(ordered_pair(x,y),cross_product(universal_class,universal_class))
<=> member(ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))),cross_product(universal_class,universal_class)) ),
inference(symmetry,[status(thm)],[18]) ).
tff(20,plain,
( ~ member(ordered_pair(x,y),cross_product(universal_class,universal_class))
<=> ~ member(ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))),cross_product(universal_class,universal_class)) ),
inference(monotonicity,[status(thm)],[19]) ).
tff(21,plain,
( ~ member(y,universal_class)
<=> ~ member(y,universal_class) ),
inference(rewrite,[status(thm)],[]) ).
tff(22,axiom,
~ member(y,universal_class),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_corollary_1_to_OP_determines_components2_1) ).
tff(23,plain,
~ member(y,universal_class),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
refl(( ( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) )
<=> ( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) ) )),
inference(bind,[status(th)],[]) ).
tff(25,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) ) ),
inference(quant_intro,[status(thm)],[24]) ).
tff(26,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(27,axiom,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',corollary_2_to_cartesian_product) ).
tff(28,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) ),
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) ),
inference(skolemize,[status(sab)],[28]) ).
tff(30,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) ),
inference(modus_ponens,[status(thm)],[29,25]) ).
tff(31,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) )
| ~ member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| member(y,universal_class) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) )
| ~ member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| member(y,universal_class) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(32,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) )
| ~ member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| member(y,universal_class) ),
inference(quant_inst,[status(thm)],[]) ).
tff(33,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) )
| ~ member(ordered_pair(x,y),cross_product(universal_class,universal_class))
| member(y,universal_class) ),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
~ member(ordered_pair(x,y),cross_product(universal_class,universal_class)),
inference(unit_resolution,[status(thm)],[33,30,23]) ).
tff(35,plain,
~ member(ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))),cross_product(universal_class,universal_class)),
inference(modus_ponens,[status(thm)],[34,20]) ).
tff(36,plain,
^ [X: $i] :
refl(( ( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) )
<=> ( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) ) )),
inference(bind,[status(th)],[]) ).
tff(37,plain,
( ! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) )
<=> ! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) ) ),
inference(quant_intro,[status(thm)],[36]) ).
tff(38,plain,
( ! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) )
<=> ! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(39,plain,
^ [X: $i] :
rewrite(( ( member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class))
| ( first(X) = X ) )
<=> ( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) ) )),
inference(bind,[status(th)],[]) ).
tff(40,plain,
( ! [X: $i] :
( member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class))
| ( first(X) = X ) )
<=> ! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) ) ),
inference(quant_intro,[status(thm)],[39]) ).
tff(41,axiom,
! [X: $i] :
( member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class))
| ( first(X) = X ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',existence_of_1st_and_2nd_2) ).
tff(42,plain,
! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) ),
inference(modus_ponens,[status(thm)],[41,40]) ).
tff(43,plain,
! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) ),
inference(modus_ponens,[status(thm)],[42,38]) ).
tff(44,plain,
! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) ),
inference(skolemize,[status(sab)],[43]) ).
tff(45,plain,
! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) ),
inference(modus_ponens,[status(thm)],[44,37]) ).
tff(46,plain,
( ( ~ ! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) )
| ( first(ordered_pair(x,y)) = ordered_pair(x,y) )
| member(ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))),cross_product(universal_class,universal_class)) )
<=> ( ~ ! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) )
| ( first(ordered_pair(x,y)) = ordered_pair(x,y) )
| member(ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))),cross_product(universal_class,universal_class)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(47,plain,
( ~ ! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) )
| ( first(ordered_pair(x,y)) = ordered_pair(x,y) )
| member(ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))),cross_product(universal_class,universal_class)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(48,plain,
( ~ ! [X: $i] :
( ( first(X) = X )
| member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class)) )
| ( first(ordered_pair(x,y)) = ordered_pair(x,y) )
| member(ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))),cross_product(universal_class,universal_class)) ),
inference(modus_ponens,[status(thm)],[47,46]) ).
tff(49,plain,
member(ordered_pair(first(ordered_pair(x,y)),second(ordered_pair(x,y))),cross_product(universal_class,universal_class)),
inference(unit_resolution,[status(thm)],[48,45,3]) ).
tff(50,plain,
$false,
inference(unit_resolution,[status(thm)],[49,35]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SET121-7 : TPTP v9.0.0. Bugfixed v7.3.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n005.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Fri Jun 20 04:21:35 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
%------------------------------------------------------------------------------