%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET095-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n006.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 05:25:12 AM UTC 2025
% Result : Unsatisfiable 0.12s 0.39s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 26
% Syntax : Number of formulae : 65 ( 21 unt; 0 typ; 0 def)
% Number of atoms : 218 ( 75 equ)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 221 ( 74 ~; 119 |; 0 &)
% ( 28 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 6 ( 6 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 5 ( 5 usr; 2 con; 0-2 aty)
% Number of variables : 133 ( 120 !; 0 ?; 133 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(y_type,type,
y: $i ).
tff(not_subclass_element_type,type,
not_subclass_element: ( $i * $i ) > $i ).
tff(singleton_type,type,
singleton: $i > $i ).
tff(x_type,type,
x: $i ).
tff(unordered_pair_type,type,
unordered_pair: ( $i * $i ) > $i ).
tff(subclass_type,type,
subclass: ( $i * $i ) > $o ).
tff(1,plain,
^ [X: $i] :
refl(( ( unordered_pair(X,X) = singleton(X) )
<=> ( unordered_pair(X,X) = singleton(X) ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
<=> ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
<=> ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',singleton_set) ).
tff(5,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
| ( unordered_pair(x,x) = singleton(x) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
unordered_pair(x,x) = singleton(x),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
( member(not_subclass_element(singleton(x),y),unordered_pair(x,x))
<=> member(not_subclass_element(singleton(x),y),singleton(x)) ),
inference(monotonicity,[status(thm)],[9]) ).
tff(11,plain,
( member(not_subclass_element(singleton(x),y),singleton(x))
<=> member(not_subclass_element(singleton(x),y),unordered_pair(x,x)) ),
inference(symmetry,[status(thm)],[10]) ).
tff(12,plain,
( ~ subclass(singleton(x),y)
<=> ~ subclass(singleton(x),y) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
~ subclass(singleton(x),y),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_property_of_singletons2_2) ).
tff(14,plain,
~ subclass(singleton(x),y),
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(singleton(x),y)
| member(not_subclass_element(singleton(x),y),singleton(x)) )
<=> ( ~ ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
| subclass(singleton(x),y)
| member(not_subclass_element(singleton(x),y),singleton(x)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(26,plain,
( ~ ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
| subclass(singleton(x),y)
| member(not_subclass_element(singleton(x),y),singleton(x)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(27,plain,
( ~ ! [Y: $i,X: $i] :
( subclass(X,Y)
| member(not_subclass_element(X,Y),X) )
| subclass(singleton(x),y)
| member(not_subclass_element(singleton(x),y),singleton(x)) ),
inference(modus_ponens,[status(thm)],[26,25]) ).
tff(28,plain,
member(not_subclass_element(singleton(x),y),singleton(x)),
inference(unit_resolution,[status(thm)],[27,24,14]) ).
tff(29,plain,
member(not_subclass_element(singleton(x),y),unordered_pair(x,x)),
inference(modus_ponens,[status(thm)],[28,11]) ).
tff(30,plain,
^ [Y: $i,U: $i,X: $i] :
refl(( ( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
<=> ( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(31,plain,
( ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
<=> ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ) ),
inference(quant_intro,[status(thm)],[30]) ).
tff(32,plain,
( ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
<=> ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(33,plain,
^ [Y: $i,U: $i,X: $i] :
rewrite(( ( ~ member(U,unordered_pair(X,Y))
| ( U = X )
| ( U = Y ) )
<=> ( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(34,plain,
( ! [Y: $i,U: $i,X: $i] :
( ~ member(U,unordered_pair(X,Y))
| ( U = X )
| ( U = Y ) )
<=> ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ) ),
inference(quant_intro,[status(thm)],[33]) ).
tff(35,axiom,
! [Y: $i,U: $i,X: $i] :
( ~ member(U,unordered_pair(X,Y))
| ( U = X )
| ( U = Y ) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',unordered_pair_member) ).
tff(36,plain,
! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[35,34]) ).
tff(37,plain,
! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[36,32]) ).
tff(38,plain,
! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ),
inference(skolemize,[status(sab)],[37]) ).
tff(39,plain,
! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[38,31]) ).
tff(40,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( not_subclass_element(singleton(x),y) = x )
| ~ member(not_subclass_element(singleton(x),y),unordered_pair(x,x)) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( not_subclass_element(singleton(x),y) = x )
| ~ member(not_subclass_element(singleton(x),y),unordered_pair(x,x)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,plain,
( ( ( not_subclass_element(singleton(x),y) = x )
| ( not_subclass_element(singleton(x),y) = x )
| ~ member(not_subclass_element(singleton(x),y),unordered_pair(x,x)) )
<=> ( ( not_subclass_element(singleton(x),y) = x )
| ~ member(not_subclass_element(singleton(x),y),unordered_pair(x,x)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( not_subclass_element(singleton(x),y) = x )
| ( not_subclass_element(singleton(x),y) = x )
| ~ member(not_subclass_element(singleton(x),y),unordered_pair(x,x)) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( not_subclass_element(singleton(x),y) = x )
| ~ member(not_subclass_element(singleton(x),y),unordered_pair(x,x)) ) ),
inference(monotonicity,[status(thm)],[41]) ).
tff(43,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( not_subclass_element(singleton(x),y) = x )
| ( not_subclass_element(singleton(x),y) = x )
| ~ member(not_subclass_element(singleton(x),y),unordered_pair(x,x)) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( not_subclass_element(singleton(x),y) = x )
| ~ member(not_subclass_element(singleton(x),y),unordered_pair(x,x)) ) ),
inference(transitivity,[status(thm)],[42,40]) ).
tff(44,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( not_subclass_element(singleton(x),y) = x )
| ( not_subclass_element(singleton(x),y) = x )
| ~ member(not_subclass_element(singleton(x),y),unordered_pair(x,x)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(45,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( not_subclass_element(singleton(x),y) = x )
| ~ member(not_subclass_element(singleton(x),y),unordered_pair(x,x)) ),
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
( ( not_subclass_element(singleton(x),y) = x )
| ~ member(not_subclass_element(singleton(x),y),unordered_pair(x,x)) ),
inference(unit_resolution,[status(thm)],[45,39]) ).
tff(47,plain,
not_subclass_element(singleton(x),y) = x,
inference(unit_resolution,[status(thm)],[46,29]) ).
tff(48,plain,
( member(not_subclass_element(singleton(x),y),y)
<=> member(x,y) ),
inference(monotonicity,[status(thm)],[47]) ).
tff(49,plain,
( member(x,y)
<=> member(not_subclass_element(singleton(x),y),y) ),
inference(symmetry,[status(thm)],[48]) ).
tff(50,plain,
( member(x,y)
<=> member(x,y) ),
inference(rewrite,[status(thm)],[]) ).
tff(51,axiom,
member(x,y),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_property_of_singletons2_1) ).
tff(52,plain,
member(x,y),
inference(modus_ponens,[status(thm)],[51,50]) ).
tff(53,plain,
member(not_subclass_element(singleton(x),y),y),
inference(modus_ponens,[status(thm)],[52,49]) ).
tff(54,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(55,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)],[54]) ).
tff(56,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(57,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(58,plain,
! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[57,56]) ).
tff(59,plain,
! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ),
inference(skolemize,[status(sab)],[58]) ).
tff(60,plain,
! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[59,55]) ).
tff(61,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
| ~ member(not_subclass_element(singleton(x),y),y)
| subclass(singleton(x),y) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
| ~ member(not_subclass_element(singleton(x),y),y)
| subclass(singleton(x),y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(62,plain,
( ~ ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
| ~ member(not_subclass_element(singleton(x),y),y)
| subclass(singleton(x),y) ),
inference(quant_inst,[status(thm)],[]) ).
tff(63,plain,
( ~ ! [Y: $i,X: $i] :
( ~ member(not_subclass_element(X,Y),Y)
| subclass(X,Y) )
| ~ member(not_subclass_element(singleton(x),y),y)
| subclass(singleton(x),y) ),
inference(modus_ponens,[status(thm)],[62,61]) ).
tff(64,plain,
~ member(not_subclass_element(singleton(x),y),y),
inference(unit_resolution,[status(thm)],[63,60,14]) ).
tff(65,plain,
$false,
inference(unit_resolution,[status(thm)],[64,53]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET095-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n006.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 04:16:20 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.12/0.39 % SZS status Unsatisfiable
% 0.12/0.39 % SZS output start Proof
% See solution above
% 0.20/0.40 % E exiting
%------------------------------------------------------------------------------