%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET098-7 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n002.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 05:25:13 AM UTC 2025
% Result : Unsatisfiable 0.21s 0.41s
% Output : Proof 0.21s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 17
% Syntax : Number of formulae : 34 ( 11 unt; 0 typ; 0 def)
% Number of atoms : 141 ( 64 equ)
% Maximal formula atoms : 12 ( 4 avg)
% Number of connectives : 136 ( 38 ~; 82 |; 0 &)
% ( 16 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 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 : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-3 aty)
% Number of functors : 6 ( 6 usr; 2 con; 0-2 aty)
% Number of variables : 56 ( 51 !; 0 ?; 56 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(intersection_type,type,
intersection: ( $i * $i ) > $i ).
tff(x_type,type,
x: $i ).
tff(complement_type,type,
complement: $i > $i ).
tff(singleton_type,type,
singleton: $i > $i ).
tff(not_subclass_element_type,type,
not_subclass_element: ( $i * $i ) > $i ).
tff(null_class_type,type,
null_class: $i ).
tff(1,plain,
( ( x != null_class )
<=> ( x != null_class ) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
x != null_class,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_1_to_number_of_elements_in_class_3) ).
tff(3,plain,
x != null_class,
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
( ( singleton(not_subclass_element(x,null_class)) != x )
<=> ( singleton(not_subclass_element(x,null_class)) != x ) ),
inference(rewrite,[status(thm)],[]) ).
tff(5,axiom,
singleton(not_subclass_element(x,null_class)) != x,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_1_to_number_of_elements_in_class_2) ).
tff(6,plain,
singleton(not_subclass_element(x,null_class)) != x,
inference(modus_ponens,[status(thm)],[5,4]) ).
tff(7,plain,
^ [X: $i] :
refl(( ( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
<=> ( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) )),
inference(bind,[status(th)],[]) ).
tff(8,plain,
( ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
<=> ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) ),
inference(quant_intro,[status(thm)],[7]) ).
tff(9,plain,
( ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
<=> ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(10,plain,
^ [X: $i] :
trans(monotonicity(rewrite(( ( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( singleton(not_subclass_element(X,null_class)) = X ) )
<=> ( ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) )),
( ( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( singleton(not_subclass_element(X,null_class)) = X )
| ( X = null_class ) )
<=> ( ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( X = null_class ) ) )),
rewrite(( ( ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( X = null_class ) )
<=> ( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) )),
( ( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( singleton(not_subclass_element(X,null_class)) = X )
| ( X = null_class ) )
<=> ( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [X: $i] :
( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( singleton(not_subclass_element(X,null_class)) = X )
| ( X = null_class ) )
<=> ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,axiom,
! [X: $i] :
( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( singleton(not_subclass_element(X,null_class)) = X )
| ( X = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',number_of_elements_in_class) ).
tff(13,plain,
! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ),
inference(modus_ponens,[status(thm)],[13,9]) ).
tff(15,plain,
! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ),
inference(modus_ponens,[status(thm)],[15,8]) ).
tff(17,plain,
( ( ~ ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
| ( x = null_class )
| ( singleton(not_subclass_element(x,null_class)) = x )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)) )
<=> ( ~ ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
| ( x = null_class )
| ( singleton(not_subclass_element(x,null_class)) = x )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(18,plain,
( ~ ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
| ( x = null_class )
| ( singleton(not_subclass_element(x,null_class)) = x )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(19,plain,
( ~ ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
| ( x = null_class )
| ( singleton(not_subclass_element(x,null_class)) = x )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)) ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)),
inference(unit_resolution,[status(thm)],[19,16,6,3]) ).
tff(21,plain,
( ~ member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),x)
<=> ~ member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),x) ),
inference(rewrite,[status(thm)],[]) ).
tff(22,axiom,
~ member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),x),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_1_to_number_of_elements_in_class_1) ).
tff(23,plain,
~ member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),x),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
<=> ( ~ member(Z,intersection(X,Y))
| member(Z,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(25,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ) ),
inference(quant_intro,[status(thm)],[24]) ).
tff(26,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(27,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',intersection2) ).
tff(28,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(skolemize,[status(sab)],[28]) ).
tff(30,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) ),
inference(modus_ponens,[status(thm)],[29,25]) ).
tff(31,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x))
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),x) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x))
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),x) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(32,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x))
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),x) ),
inference(quant_inst,[status(thm)],[]) ).
tff(33,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,Y) )
| ~ member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x))
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),x) ),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
$false,
inference(unit_resolution,[status(thm)],[33,30,23,20]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.05/0.12 % Problem : SET098-7 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.05/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n002.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 04:18:46 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.21/0.41 % SZS status Unsatisfiable
% 0.21/0.41 % SZS output start Proof
% See solution above
% 0.21/0.42 % E exiting
%------------------------------------------------------------------------------