%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET085-7 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.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 05:25:10 AM UTC 2025
% Result : Unsatisfiable 0.16s 0.38s
% Output : Proof 0.16s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 23
% Syntax : Number of formulae : 51 ( 22 unt; 0 typ; 0 def)
% Number of atoms : 147 ( 72 equ)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 151 ( 59 ~; 73 |; 0 &)
% ( 19 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 4 ( 4 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 : 6 ( 6 usr; 4 con; 0-2 aty)
% Number of variables : 84 ( 75 !; 0 ?; 84 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(unordered_pair_type,type,
unordered_pair: ( $i * $i ) > $i ).
tff(y_type,type,
y: $i ).
tff(z_type,type,
z: $i ).
tff(x_type,type,
x: $i ).
tff(singleton_type,type,
singleton: $i > $i ).
tff(universal_class_type,type,
universal_class: $i ).
tff(1,plain,
( ( unordered_pair(y,z) = singleton(x) )
<=> ( unordered_pair(y,z) = singleton(x) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
unordered_pair(y,z) = singleton(x),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_singleton_in_unordered_pair3_1) ).
tff(3,plain,
unordered_pair(y,z) = singleton(x),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [Y: $i,X: $i] :
refl(( ( unordered_pair(X,Y) = unordered_pair(Y,X) )
<=> ( unordered_pair(X,Y) = unordered_pair(Y,X) ) )),
inference(bind,[status(th)],[]) ).
tff(5,plain,
( ! [Y: $i,X: $i] : ( unordered_pair(X,Y) = unordered_pair(Y,X) )
<=> ! [Y: $i,X: $i] : ( unordered_pair(X,Y) = unordered_pair(Y,X) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [Y: $i,X: $i] : ( unordered_pair(X,Y) = unordered_pair(Y,X) )
<=> ! [Y: $i,X: $i] : ( unordered_pair(X,Y) = unordered_pair(Y,X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(7,axiom,
! [Y: $i,X: $i] : ( unordered_pair(X,Y) = unordered_pair(Y,X) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_of_unordered_pair) ).
tff(8,plain,
! [Y: $i,X: $i] : ( unordered_pair(X,Y) = unordered_pair(Y,X) ),
inference(modus_ponens,[status(thm)],[7,6]) ).
tff(9,plain,
! [Y: $i,X: $i] : ( unordered_pair(X,Y) = unordered_pair(Y,X) ),
inference(skolemize,[status(sab)],[8]) ).
tff(10,plain,
! [Y: $i,X: $i] : ( unordered_pair(X,Y) = unordered_pair(Y,X) ),
inference(modus_ponens,[status(thm)],[9,5]) ).
tff(11,plain,
( ~ ! [Y: $i,X: $i] : ( unordered_pair(X,Y) = unordered_pair(Y,X) )
| ( unordered_pair(z,y) = unordered_pair(y,z) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(12,plain,
unordered_pair(z,y) = unordered_pair(y,z),
inference(unit_resolution,[status(thm)],[11,10]) ).
tff(13,plain,
unordered_pair(z,y) = singleton(x),
inference(transitivity,[status(thm)],[12,3]) ).
tff(14,plain,
( member(x,unordered_pair(z,y))
<=> member(x,singleton(x)) ),
inference(monotonicity,[status(thm)],[13]) ).
tff(15,plain,
( member(x,singleton(x))
<=> member(x,unordered_pair(z,y)) ),
inference(symmetry,[status(thm)],[14]) ).
tff(16,plain,
( member(x,universal_class)
<=> member(x,universal_class) ),
inference(rewrite,[status(thm)],[]) ).
tff(17,axiom,
member(x,universal_class),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_singleton_in_unordered_pair3_2) ).
tff(18,plain,
member(x,universal_class),
inference(modus_ponens,[status(thm)],[17,16]) ).
tff(19,plain,
^ [X: $i] :
refl(( ( ~ member(X,universal_class)
| member(X,singleton(X)) )
<=> ( ~ member(X,universal_class)
| member(X,singleton(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(20,plain,
( ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) )
<=> ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) ) ),
inference(quant_intro,[status(thm)],[19]) ).
tff(21,plain,
( ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) )
<=> ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(22,axiom,
! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',set_in_its_singleton) ).
tff(23,plain,
! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) ),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) ),
inference(skolemize,[status(sab)],[23]) ).
tff(25,plain,
! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) ),
inference(modus_ponens,[status(thm)],[24,20]) ).
tff(26,plain,
( ( ~ ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) )
| ~ member(x,universal_class)
| member(x,singleton(x)) )
<=> ( ~ ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) )
| ~ member(x,universal_class)
| member(x,singleton(x)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(27,plain,
( ~ ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) )
| ~ member(x,universal_class)
| member(x,singleton(x)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(28,plain,
( ~ ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) )
| ~ member(x,universal_class)
| member(x,singleton(x)) ),
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
member(x,singleton(x)),
inference(unit_resolution,[status(thm)],[28,25,18]) ).
tff(30,plain,
member(x,unordered_pair(z,y)),
inference(modus_ponens,[status(thm)],[29,15]) ).
tff(31,plain,
( ( x != z )
<=> ( x != z ) ),
inference(rewrite,[status(thm)],[]) ).
tff(32,axiom,
x != z,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_singleton_in_unordered_pair3_4) ).
tff(33,plain,
x != z,
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
( ( x != y )
<=> ( x != y ) ),
inference(rewrite,[status(thm)],[]) ).
tff(35,axiom,
x != y,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_singleton_in_unordered_pair3_3) ).
tff(36,plain,
x != y,
inference(modus_ponens,[status(thm)],[35,34]) ).
tff(37,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(38,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)],[37]) ).
tff(39,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(40,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(41,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)],[40]) ).
tff(42,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(43,plain,
! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[43,39]) ).
tff(45,plain,
! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ),
inference(skolemize,[status(sab)],[44]) ).
tff(46,plain,
! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[45,38]) ).
tff(47,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( x = y )
| ( x = z )
| ~ member(x,unordered_pair(z,y)) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( x = y )
| ( x = z )
| ~ member(x,unordered_pair(z,y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(48,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( x = y )
| ( x = z )
| ~ member(x,unordered_pair(z,y)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(49,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( x = y )
| ( x = z )
| ~ member(x,unordered_pair(z,y)) ),
inference(modus_ponens,[status(thm)],[48,47]) ).
tff(50,plain,
~ member(x,unordered_pair(z,y)),
inference(unit_resolution,[status(thm)],[49,46,36,33]) ).
tff(51,plain,
$false,
inference(unit_resolution,[status(thm)],[50,30]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.10 % Problem : SET085-7 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.10/0.10 % Command : run_E %s %d THM
% 0.10/0.31 % Computer : n007.cluster.edu
% 0.10/0.31 % Model : x86_64 x86_64
% 0.10/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.31 % Memory : 8042.1875MB
% 0.10/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.31 % CPULimit : 300
% 0.10/0.31 % WCLimit : 300
% 0.10/0.31 % DateTime : Fri Jun 20 04:15:20 EDT 2025
% 0.10/0.31 % CPUTime :
% 0.16/0.38 % SZS status Unsatisfiable
% 0.16/0.38 % SZS output start Proof
% See solution above
% 0.16/0.39 % E exiting
%------------------------------------------------------------------------------