%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET024-3 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n011.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 05:24:52 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.42s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 16
% Syntax : Number of formulae : 39 ( 16 unt; 0 typ; 0 def)
% Number of atoms : 123 ( 26 equ)
% Maximal formula atoms : 8 ( 3 avg)
% Number of connectives : 136 ( 61 ~; 54 |; 0 &)
% ( 21 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Maximal term depth : 2 ( 1 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 : 8 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 3 ( 3 usr; 1 con; 0-2 aty)
% Number of variables : 69 ( 63 !; 0 ?; 69 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(non_ordered_pair_type,type,
non_ordered_pair: ( $i * $i ) > $i ).
tff(a_type,type,
a: $i ).
tff(singleton_set_type,type,
singleton_set: $i > $i ).
tff(little_set_type,type,
little_set: $i > $o ).
tff(1,plain,
^ [X: $i] :
refl(( ( singleton_set(X) = non_ordered_pair(X,X) )
<=> ( singleton_set(X) = non_ordered_pair(X,X) ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : ( singleton_set(X) = non_ordered_pair(X,X) )
<=> ! [X: $i] : ( singleton_set(X) = non_ordered_pair(X,X) ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : ( singleton_set(X) = non_ordered_pair(X,X) )
<=> ! [X: $i] : ( singleton_set(X) = non_ordered_pair(X,X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : ( singleton_set(X) = non_ordered_pair(X,X) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',singleton_set) ).
tff(5,plain,
! [X: $i] : ( singleton_set(X) = non_ordered_pair(X,X) ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : ( singleton_set(X) = non_ordered_pair(X,X) ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : ( singleton_set(X) = non_ordered_pair(X,X) ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : ( singleton_set(X) = non_ordered_pair(X,X) )
| ( singleton_set(a) = non_ordered_pair(a,a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
singleton_set(a) = non_ordered_pair(a,a),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
non_ordered_pair(a,a) = singleton_set(a),
inference(symmetry,[status(thm)],[9]) ).
tff(11,plain,
( member(a,non_ordered_pair(a,a))
<=> member(a,singleton_set(a)) ),
inference(monotonicity,[status(thm)],[10]) ).
tff(12,plain,
( member(a,singleton_set(a))
<=> member(a,non_ordered_pair(a,a)) ),
inference(symmetry,[status(thm)],[11]) ).
tff(13,plain,
( ~ member(a,singleton_set(a))
<=> ~ member(a,non_ordered_pair(a,a)) ),
inference(monotonicity,[status(thm)],[12]) ).
tff(14,plain,
( ~ member(a,singleton_set(a))
<=> ~ member(a,singleton_set(a)) ),
inference(rewrite,[status(thm)],[]) ).
tff(15,axiom,
~ member(a,singleton_set(a)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_membership_of_singleton_set) ).
tff(16,plain,
~ member(a,singleton_set(a)),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
~ member(a,non_ordered_pair(a,a)),
inference(modus_ponens,[status(thm)],[16,13]) ).
tff(18,plain,
( little_set(a)
<=> little_set(a) ),
inference(rewrite,[status(thm)],[]) ).
tff(19,axiom,
little_set(a),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_little_set) ).
tff(20,plain,
little_set(a),
inference(modus_ponens,[status(thm)],[19,18]) ).
tff(21,plain,
^ [Y: $i,X: $i] :
refl(( ( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) )
<=> ( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(22,plain,
( ! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) ) ),
inference(quant_intro,[status(thm)],[21]) ).
tff(23,plain,
( ! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(24,plain,
( ! [Y: $i,U: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) ) ),
inference(elim_unused_vars,[status(thm)],[]) ).
tff(25,plain,
( ! [Y: $i,U: $i,X: $i] :
( member(U,non_ordered_pair(X,Y))
| ( U != X )
| ~ little_set(U) )
<=> ! [Y: $i,U: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) ) ),
inference(destructive_equality_resolution,[status(thm)],[]) ).
tff(26,plain,
( ! [Y: $i,U: $i,X: $i] :
( member(U,non_ordered_pair(X,Y))
| ( U != X )
| ~ little_set(U) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) ) ),
inference(transitivity,[status(thm)],[25,24]) ).
tff(27,plain,
^ [Y: $i,U: $i,X: $i] :
trans(monotonicity(rewrite(( ( member(U,non_ordered_pair(X,Y))
| ~ little_set(U) )
<=> ( member(U,non_ordered_pair(X,Y))
| ~ little_set(U) ) )),
( ( member(U,non_ordered_pair(X,Y))
| ~ little_set(U)
| ( U != X ) )
<=> ( member(U,non_ordered_pair(X,Y))
| ~ little_set(U)
| ( U != X ) ) )),
rewrite(( ( member(U,non_ordered_pair(X,Y))
| ~ little_set(U)
| ( U != X ) )
<=> ( member(U,non_ordered_pair(X,Y))
| ( U != X )
| ~ little_set(U) ) )),
( ( member(U,non_ordered_pair(X,Y))
| ~ little_set(U)
| ( U != X ) )
<=> ( member(U,non_ordered_pair(X,Y))
| ( U != X )
| ~ little_set(U) ) )),
inference(bind,[status(th)],[]) ).
tff(28,plain,
( ! [Y: $i,U: $i,X: $i] :
( member(U,non_ordered_pair(X,Y))
| ~ little_set(U)
| ( U != X ) )
<=> ! [Y: $i,U: $i,X: $i] :
( member(U,non_ordered_pair(X,Y))
| ( U != X )
| ~ little_set(U) ) ),
inference(quant_intro,[status(thm)],[27]) ).
tff(29,plain,
( ! [Y: $i,U: $i,X: $i] :
( member(U,non_ordered_pair(X,Y))
| ~ little_set(U)
| ( U != X ) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) ) ),
inference(transitivity,[status(thm)],[28,26]) ).
tff(30,axiom,
! [Y: $i,U: $i,X: $i] :
( member(U,non_ordered_pair(X,Y))
| ~ little_set(U)
| ( U != X ) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',non_ordered_pair2) ).
tff(31,plain,
! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[30,29]) ).
tff(32,plain,
! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[31,23]) ).
tff(33,plain,
! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) ),
inference(skolemize,[status(sab)],[32]) ).
tff(34,plain,
! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[33,22]) ).
tff(35,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) )
| ~ little_set(a)
| member(a,non_ordered_pair(a,a)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) )
| ~ little_set(a)
| member(a,non_ordered_pair(a,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(36,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) )
| ~ little_set(a)
| member(a,non_ordered_pair(a,a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(37,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(X)
| member(X,non_ordered_pair(X,Y)) )
| ~ little_set(a)
| member(a,non_ordered_pair(a,a)) ),
inference(modus_ponens,[status(thm)],[36,35]) ).
tff(38,plain,
member(a,non_ordered_pair(a,a)),
inference(unit_resolution,[status(thm)],[37,34,20]) ).
tff(39,plain,
$false,
inference(unit_resolution,[status(thm)],[38,17]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : SET024-3 : TPTP v9.0.0. Released v1.0.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n011.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 04:03:11 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.20/0.42 % SZS status Unsatisfiable
% 0.20/0.42 % SZS output start Proof
% See solution above
% 0.20/0.43 % E exiting
%------------------------------------------------------------------------------