%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET028-3 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 05:24:54 AM UTC 2025
% Result : Unsatisfiable 0.21s 0.46s
% Output : Proof 0.21s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 97
% Syntax : Number of formulae : 237 ( 57 unt; 0 typ; 0 def)
% Number of atoms : 1132 ( 105 equ)
% Maximal formula atoms : 20 ( 4 avg)
% Number of connectives : 1558 ( 716 ~; 728 |; 0 &)
% ( 114 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of FOOLs : 53 ( 53 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 10 ( 8 usr; 1 prp; 0-3 aty)
% Number of functors : 14 ( 14 usr; 2 con; 0-3 aty)
% Number of variables : 598 ( 545 !; 0 ?; 598 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(image_type,type,
image: ( $i * $i ) > $i ).
tff(a_function_type,type,
a_function: $i ).
tff(singleton_set_type,type,
singleton_set: $i > $i ).
tff(element_type,type,
element: $i ).
tff(f7_type,type,
f7: ( $i * $i ) > $i ).
tff(f28_type,type,
f28: ( $i * $i * $i ) > $i ).
tff(f17_type,type,
f17: ( $i * $i ) > $i ).
tff(sigma_type,type,
sigma: $i > $i ).
tff(apply_type,type,
apply: ( $i * $i ) > $i ).
tff(non_ordered_pair_type,type,
non_ordered_pair: ( $i * $i ) > $i ).
tff(second_type,type,
second: $i > $i ).
tff(ordered_pair_type,type,
ordered_pair: ( $i * $i ) > $i ).
tff(f6_type,type,
f6: ( $i * $i ) > $i ).
tff(subset_type,type,
subset: ( $i * $i ) > $o ).
tff(little_set_type,type,
little_set: $i > $o ).
tff(first_type,type,
first: $i > $i ).
tff(ordered_pair_predicate_type,type,
ordered_pair_predicate: $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/sandbox2/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(element) = non_ordered_pair(element,element) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
singleton_set(element) = non_ordered_pair(element,element),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
non_ordered_pair(element,element) = singleton_set(element),
inference(symmetry,[status(thm)],[9]) ).
tff(11,plain,
image(non_ordered_pair(element,element),a_function) = image(singleton_set(element),a_function),
inference(monotonicity,[status(thm)],[10]) ).
tff(12,plain,
image(singleton_set(element),a_function) = image(non_ordered_pair(element,element),a_function),
inference(symmetry,[status(thm)],[11]) ).
tff(13,plain,
( ~ subset(apply(a_function,element),sigma(image(singleton_set(element),a_function)))
<=> ~ subset(apply(a_function,element),sigma(image(singleton_set(element),a_function))) ),
inference(rewrite,[status(thm)],[]) ).
tff(14,axiom,
~ subset(apply(a_function,element),sigma(image(singleton_set(element),a_function))),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_property_of_image_and_apply1) ).
tff(15,plain,
~ subset(apply(a_function,element),sigma(image(singleton_set(element),a_function))),
inference(modus_ponens,[status(thm)],[14,13]) ).
tff(16,plain,
^ [Y: $i,X: $i] :
refl(( ( subset(X,Y)
| member(f17(X,Y),X) )
<=> ( subset(X,Y)
| member(f17(X,Y),X) ) )),
inference(bind,[status(th)],[]) ).
tff(17,plain,
( ! [Y: $i,X: $i] :
( subset(X,Y)
| member(f17(X,Y),X) )
<=> ! [Y: $i,X: $i] :
( subset(X,Y)
| member(f17(X,Y),X) ) ),
inference(quant_intro,[status(thm)],[16]) ).
tff(18,plain,
( ! [Y: $i,X: $i] :
( subset(X,Y)
| member(f17(X,Y),X) )
<=> ! [Y: $i,X: $i] :
( subset(X,Y)
| member(f17(X,Y),X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(19,axiom,
! [Y: $i,X: $i] :
( subset(X,Y)
| member(f17(X,Y),X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',subset2) ).
tff(20,plain,
! [Y: $i,X: $i] :
( subset(X,Y)
| member(f17(X,Y),X) ),
inference(modus_ponens,[status(thm)],[19,18]) ).
tff(21,plain,
! [Y: $i,X: $i] :
( subset(X,Y)
| member(f17(X,Y),X) ),
inference(skolemize,[status(sab)],[20]) ).
tff(22,plain,
! [Y: $i,X: $i] :
( subset(X,Y)
| member(f17(X,Y),X) ),
inference(modus_ponens,[status(thm)],[21,17]) ).
tff(23,plain,
( ( ~ ! [Y: $i,X: $i] :
( subset(X,Y)
| member(f17(X,Y),X) )
| subset(apply(a_function,element),sigma(image(singleton_set(element),a_function)))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element)) )
<=> ( ~ ! [Y: $i,X: $i] :
( subset(X,Y)
| member(f17(X,Y),X) )
| subset(apply(a_function,element),sigma(image(singleton_set(element),a_function)))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(24,plain,
( ~ ! [Y: $i,X: $i] :
( subset(X,Y)
| member(f17(X,Y),X) )
| subset(apply(a_function,element),sigma(image(singleton_set(element),a_function)))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(25,plain,
( ~ ! [Y: $i,X: $i] :
( subset(X,Y)
| member(f17(X,Y),X) )
| subset(apply(a_function,element),sigma(image(singleton_set(element),a_function)))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element)) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element)),
inference(unit_resolution,[status(thm)],[25,22,15]) ).
tff(27,plain,
^ [Z: $i,Y: $i,Xf: $i] :
refl(( ( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) )
<=> ( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) ) )),
inference(bind,[status(th)],[]) ).
tff(28,plain,
( ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) )
<=> ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) ) ),
inference(quant_intro,[status(thm)],[27]) ).
tff(29,plain,
( ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) )
<=> ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(30,axiom,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',apply4) ).
tff(31,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) ),
inference(modus_ponens,[status(thm)],[30,29]) ).
tff(32,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) ),
inference(skolemize,[status(sab)],[31]) ).
tff(33,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) ),
inference(modus_ponens,[status(thm)],[32,28]) ).
tff(34,plain,
( ( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) )
<=> ( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(35,plain,
( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(36,plain,
( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(Z,second(f28(Z,Xf,Y))) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(modus_ponens,[status(thm)],[35,34]) ).
tff(37,plain,
member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))),
inference(unit_resolution,[status(thm)],[36,33,26]) ).
tff(38,plain,
^ [Z: $i,X: $i] :
refl(( ( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) )
<=> ( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(39,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) ) ),
inference(quant_intro,[status(thm)],[38]) ).
tff(40,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,axiom,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',second3) ).
tff(42,plain,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) ),
inference(modus_ponens,[status(thm)],[41,40]) ).
tff(43,plain,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) ),
inference(skolemize,[status(sab)],[42]) ).
tff(44,plain,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) ),
inference(modus_ponens,[status(thm)],[43,39]) ).
tff(45,plain,
( ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element) = ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) )
<=> ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element) = ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(46,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element) = ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(47,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| ( X = ordered_pair(f6(Z,X),f7(Z,X)) ) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element) = ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ),
inference(modus_ponens,[status(thm)],[46,45]) ).
tff(48,plain,
f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element) = ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))),
inference(unit_resolution,[status(thm)],[47,44,37]) ).
tff(49,plain,
second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) = second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))),
inference(monotonicity,[status(thm)],[48]) ).
tff(50,plain,
second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))) = second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),
inference(symmetry,[status(thm)],[49]) ).
tff(51,plain,
^ [Z: $i,X: $i] :
refl(( ( ~ member(Z,second(X))
| little_set(f7(Z,X)) )
<=> ( ~ member(Z,second(X))
| little_set(f7(Z,X)) ) )),
inference(bind,[status(th)],[]) ).
tff(52,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f7(Z,X)) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f7(Z,X)) ) ),
inference(quant_intro,[status(thm)],[51]) ).
tff(53,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f7(Z,X)) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f7(Z,X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(54,axiom,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f7(Z,X)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',second2) ).
tff(55,plain,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f7(Z,X)) ),
inference(modus_ponens,[status(thm)],[54,53]) ).
tff(56,plain,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f7(Z,X)) ),
inference(skolemize,[status(sab)],[55]) ).
tff(57,plain,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f7(Z,X)) ),
inference(modus_ponens,[status(thm)],[56,52]) ).
tff(58,plain,
( ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f7(Z,X)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) )
<=> ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f7(Z,X)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(59,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f7(Z,X)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(60,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f7(Z,X)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(modus_ponens,[status(thm)],[59,58]) ).
tff(61,plain,
little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))),
inference(unit_resolution,[status(thm)],[60,57,37]) ).
tff(62,plain,
^ [Z: $i,X: $i] :
refl(( ( little_set(f6(Z,X))
| ~ member(Z,second(X)) )
<=> ( little_set(f6(Z,X))
| ~ member(Z,second(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(63,plain,
( ! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) )
<=> ! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) ) ),
inference(quant_intro,[status(thm)],[62]) ).
tff(64,plain,
( ! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) )
<=> ! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(65,plain,
^ [Z: $i,X: $i] :
rewrite(( ( ~ member(Z,second(X))
| little_set(f6(Z,X)) )
<=> ( little_set(f6(Z,X))
| ~ member(Z,second(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(66,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f6(Z,X)) )
<=> ! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) ) ),
inference(quant_intro,[status(thm)],[65]) ).
tff(67,axiom,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| little_set(f6(Z,X)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',second1) ).
tff(68,plain,
! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) ),
inference(modus_ponens,[status(thm)],[67,66]) ).
tff(69,plain,
! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) ),
inference(modus_ponens,[status(thm)],[68,64]) ).
tff(70,plain,
! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) ),
inference(skolemize,[status(sab)],[69]) ).
tff(71,plain,
! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) ),
inference(modus_ponens,[status(thm)],[70,63]) ).
tff(72,plain,
( ( ~ ! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) )
| little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) )
<=> ( ~ ! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) )
| little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(73,plain,
( ~ ! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) )
| little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(74,plain,
( ~ ! [Z: $i,X: $i] :
( little_set(f6(Z,X))
| ~ member(Z,second(X)) )
| little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(modus_ponens,[status(thm)],[73,72]) ).
tff(75,plain,
little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))),
inference(unit_resolution,[status(thm)],[74,71,37]) ).
tff(76,plain,
^ [Y: $i,X: $i] :
refl(( ( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ) )),
inference(bind,[status(th)],[]) ).
tff(77,plain,
( ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ) ),
inference(quant_intro,[status(thm)],[76]) ).
tff(78,plain,
( ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(79,plain,
^ [Y: $i,X: $i] :
rewrite(( ( ~ little_set(X)
| ~ little_set(Y)
| ( second(ordered_pair(X,Y)) = Y ) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ) )),
inference(bind,[status(th)],[]) ).
tff(80,plain,
( ! [Y: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ( second(ordered_pair(X,Y)) = Y ) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ) ),
inference(quant_intro,[status(thm)],[79]) ).
tff(81,axiom,
! [Y: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ( second(ordered_pair(X,Y)) = Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_second) ).
tff(82,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[81,80]) ).
tff(83,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[82,78]) ).
tff(84,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(skolemize,[status(sab)],[83]) ).
tff(85,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[84,77]) ).
tff(86,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))) = f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))) = f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(87,plain,
( ( ~ little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))) = f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) )
<=> ( ~ little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))) = f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(88,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))) = f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))) = f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) ) ),
inference(monotonicity,[status(thm)],[87]) ).
tff(89,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))) = f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))) = f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) ) ),
inference(transitivity,[status(thm)],[88,86]) ).
tff(90,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))) = f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(91,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ little_set(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ( second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))) = f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) ),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))) = f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),
inference(unit_resolution,[status(thm)],[91,85,75,61]) ).
tff(93,plain,
f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) = second(ordered_pair(f6(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))),
inference(symmetry,[status(thm)],[92]) ).
tff(94,plain,
f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) = second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),
inference(transitivity,[status(thm)],[93,50]) ).
tff(95,plain,
( member(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(singleton_set(element),a_function))
<=> member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function)) ),
inference(monotonicity,[status(thm)],[94,12]) ).
tff(96,plain,
( member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function))
<=> member(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(singleton_set(element),a_function)) ),
inference(symmetry,[status(thm)],[95]) ).
tff(97,plain,
^ [Z: $i,Y: $i,Xf: $i] :
refl(( ( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) )
<=> ( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) ) )),
inference(bind,[status(th)],[]) ).
tff(98,plain,
( ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) )
<=> ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) ) ),
inference(quant_intro,[status(thm)],[97]) ).
tff(99,plain,
( ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) )
<=> ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(100,axiom,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',apply3) ).
tff(101,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[100,99]) ).
tff(102,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) ),
inference(skolemize,[status(sab)],[101]) ).
tff(103,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[102,98]) ).
tff(104,plain,
( ( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| ( first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) = element ) )
<=> ( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| ( first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) = element ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(105,plain,
( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| ( first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) = element ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(106,plain,
( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ( first(f28(Z,Xf,Y)) = Y ) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| ( first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) = element ) ),
inference(modus_ponens,[status(thm)],[105,104]) ).
tff(107,plain,
first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) = element,
inference(unit_resolution,[status(thm)],[106,103,26]) ).
tff(108,plain,
( member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element))
<=> member(element,singleton_set(element)) ),
inference(monotonicity,[status(thm)],[107,10]) ).
tff(109,plain,
( member(element,singleton_set(element))
<=> member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element)) ),
inference(symmetry,[status(thm)],[108]) ).
tff(110,plain,
( little_set(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
<=> little_set(element) ),
inference(monotonicity,[status(thm)],[107]) ).
tff(111,plain,
^ [Z: $i,Y: $i,Xf: $i] :
refl(( ( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) )
<=> ( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(112,plain,
( ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) )
<=> ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) ) ),
inference(quant_intro,[status(thm)],[111]) ).
tff(113,plain,
( ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) )
<=> ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(114,plain,
^ [Z: $i,Y: $i,Xf: $i] :
rewrite(( ( ~ member(Z,apply(Xf,Y))
| ordered_pair_predicate(f28(Z,Xf,Y)) )
<=> ( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(115,plain,
( ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ordered_pair_predicate(f28(Z,Xf,Y)) )
<=> ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) ) ),
inference(quant_intro,[status(thm)],[114]) ).
tff(116,axiom,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| ordered_pair_predicate(f28(Z,Xf,Y)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',apply1) ).
tff(117,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) ),
inference(modus_ponens,[status(thm)],[116,115]) ).
tff(118,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) ),
inference(modus_ponens,[status(thm)],[117,113]) ).
tff(119,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) ),
inference(skolemize,[status(sab)],[118]) ).
tff(120,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) ),
inference(modus_ponens,[status(thm)],[119,112]) ).
tff(121,plain,
( ( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) )
<=> ( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(122,plain,
( ( ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element)) )
<=> ( ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(123,plain,
( ( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) )
| ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element)) )
<=> ( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) ),
inference(monotonicity,[status(thm)],[122]) ).
tff(124,plain,
( ( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) )
| ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element)) )
<=> ( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ) ),
inference(transitivity,[status(thm)],[123,121]) ).
tff(125,plain,
( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) )
| ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(126,plain,
( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ordered_pair_predicate(f28(Z,Xf,Y))
| ~ member(Z,apply(Xf,Y)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)) ),
inference(modus_ponens,[status(thm)],[125,124]) ).
tff(127,plain,
ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),
inference(unit_resolution,[status(thm)],[126,120,26]) ).
tff(128,plain,
^ [X: $i] :
refl(( ( ~ ordered_pair_predicate(X)
| little_set(first(X)) )
<=> ( ~ ordered_pair_predicate(X)
| little_set(first(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(129,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(first(X)) )
<=> ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(first(X)) ) ),
inference(quant_intro,[status(thm)],[128]) ).
tff(130,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(first(X)) )
<=> ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(first(X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(131,axiom,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(first(X)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',first_component_is_small) ).
tff(132,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(first(X)) ),
inference(modus_ponens,[status(thm)],[131,130]) ).
tff(133,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(first(X)) ),
inference(skolemize,[status(sab)],[132]) ).
tff(134,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(first(X)) ),
inference(modus_ponens,[status(thm)],[133,129]) ).
tff(135,plain,
( ( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(first(X)) )
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| little_set(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) )
<=> ( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(first(X)) )
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| little_set(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(136,plain,
( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(first(X)) )
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| little_set(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(137,plain,
( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(first(X)) )
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| little_set(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(modus_ponens,[status(thm)],[136,135]) ).
tff(138,plain,
little_set(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))),
inference(unit_resolution,[status(thm)],[137,134,127]) ).
tff(139,plain,
little_set(element),
inference(modus_ponens,[status(thm)],[138,110]) ).
tff(140,plain,
^ [X: $i] :
refl(( ( ~ little_set(X)
| member(X,singleton_set(X)) )
<=> ( ~ little_set(X)
| member(X,singleton_set(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(141,plain,
( ! [X: $i] :
( ~ little_set(X)
| member(X,singleton_set(X)) )
<=> ! [X: $i] :
( ~ little_set(X)
| member(X,singleton_set(X)) ) ),
inference(quant_intro,[status(thm)],[140]) ).
tff(142,plain,
( ! [X: $i] :
( ~ little_set(X)
| member(X,singleton_set(X)) )
<=> ! [X: $i] :
( ~ little_set(X)
| member(X,singleton_set(X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(143,axiom,
! [X: $i] :
( ~ little_set(X)
| member(X,singleton_set(X)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_singleton_sets) ).
tff(144,plain,
! [X: $i] :
( ~ little_set(X)
| member(X,singleton_set(X)) ),
inference(modus_ponens,[status(thm)],[143,142]) ).
tff(145,plain,
! [X: $i] :
( ~ little_set(X)
| member(X,singleton_set(X)) ),
inference(skolemize,[status(sab)],[144]) ).
tff(146,plain,
! [X: $i] :
( ~ little_set(X)
| member(X,singleton_set(X)) ),
inference(modus_ponens,[status(thm)],[145,141]) ).
tff(147,plain,
( ( ~ ! [X: $i] :
( ~ little_set(X)
| member(X,singleton_set(X)) )
| ~ little_set(element)
| member(element,singleton_set(element)) )
<=> ( ~ ! [X: $i] :
( ~ little_set(X)
| member(X,singleton_set(X)) )
| ~ little_set(element)
| member(element,singleton_set(element)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(148,plain,
( ~ ! [X: $i] :
( ~ little_set(X)
| member(X,singleton_set(X)) )
| ~ little_set(element)
| member(element,singleton_set(element)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(149,plain,
( ~ ! [X: $i] :
( ~ little_set(X)
| member(X,singleton_set(X)) )
| ~ little_set(element)
| member(element,singleton_set(element)) ),
inference(modus_ponens,[status(thm)],[148,147]) ).
tff(150,plain,
( ~ little_set(element)
| member(element,singleton_set(element)) ),
inference(unit_resolution,[status(thm)],[149,146]) ).
tff(151,plain,
member(element,singleton_set(element)),
inference(unit_resolution,[status(thm)],[150,139]) ).
tff(152,plain,
member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element)),
inference(modus_ponens,[status(thm)],[151,109]) ).
tff(153,plain,
^ [X: $i] :
refl(( ( ~ ordered_pair_predicate(X)
| little_set(second(X)) )
<=> ( ~ ordered_pair_predicate(X)
| little_set(second(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(154,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(second(X)) )
<=> ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(second(X)) ) ),
inference(quant_intro,[status(thm)],[153]) ).
tff(155,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(second(X)) )
<=> ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(second(X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(156,axiom,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(second(X)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',second_component_is_small) ).
tff(157,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(second(X)) ),
inference(modus_ponens,[status(thm)],[156,155]) ).
tff(158,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(second(X)) ),
inference(skolemize,[status(sab)],[157]) ).
tff(159,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(second(X)) ),
inference(modus_ponens,[status(thm)],[158,154]) ).
tff(160,plain,
( ( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(second(X)) )
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) )
<=> ( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(second(X)) )
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(161,plain,
( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(second(X)) )
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(162,plain,
( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(second(X)) )
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(modus_ponens,[status(thm)],[161,160]) ).
tff(163,plain,
little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))),
inference(unit_resolution,[status(thm)],[162,159,127]) ).
tff(164,plain,
^ [Z: $i,Y: $i,Xf: $i] :
refl(( ( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) )
<=> ( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) ) )),
inference(bind,[status(th)],[]) ).
tff(165,plain,
( ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) )
<=> ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) ) ),
inference(quant_intro,[status(thm)],[164]) ).
tff(166,plain,
( ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) )
<=> ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(167,axiom,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',apply2) ).
tff(168,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) ),
inference(modus_ponens,[status(thm)],[167,166]) ).
tff(169,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) ),
inference(skolemize,[status(sab)],[168]) ).
tff(170,plain,
! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) ),
inference(modus_ponens,[status(thm)],[169,165]) ).
tff(171,plain,
( ( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function) )
<=> ( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(172,plain,
( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function) ),
inference(quant_inst,[status(thm)],[]) ).
tff(173,plain,
( ~ ! [Z: $i,Y: $i,Xf: $i] :
( ~ member(Z,apply(Xf,Y))
| member(f28(Z,Xf,Y),Xf) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),apply(a_function,element))
| member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function) ),
inference(modus_ponens,[status(thm)],[172,171]) ).
tff(174,plain,
member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function),
inference(unit_resolution,[status(thm)],[173,170,26]) ).
tff(175,plain,
^ [Y: $i,Xf: $i,X: $i] :
refl(( ( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) )
<=> ( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(176,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) ) ),
inference(quant_intro,[status(thm)],[175]) ).
tff(177,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(178,plain,
^ [Y: $i,Xf: $i,X: $i] :
rewrite(( ( member(second(Y),image(X,Xf))
| ~ member(Y,Xf)
| ~ little_set(second(Y))
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y) )
<=> ( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(179,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( member(second(Y),image(X,Xf))
| ~ member(Y,Xf)
| ~ little_set(second(Y))
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) ) ),
inference(quant_intro,[status(thm)],[178]) ).
tff(180,plain,
( ! [Z: $i,Y: $i,Xf: $i,X: $i] :
( member(second(Y),image(X,Xf))
| ~ member(Y,Xf)
| ~ little_set(second(Y))
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( member(second(Y),image(X,Xf))
| ~ member(Y,Xf)
| ~ little_set(second(Y))
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y) ) ),
inference(elim_unused_vars,[status(thm)],[]) ).
tff(181,plain,
( ! [Z: $i,Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| ( second(Y) != Z )
| member(Z,image(X,Xf)) )
<=> ! [Z: $i,Y: $i,Xf: $i,X: $i] :
( member(second(Y),image(X,Xf))
| ~ member(Y,Xf)
| ~ little_set(second(Y))
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y) ) ),
inference(destructive_equality_resolution,[status(thm)],[]) ).
tff(182,plain,
( ! [Z: $i,Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| ( second(Y) != Z )
| member(Z,image(X,Xf)) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( member(second(Y),image(X,Xf))
| ~ member(Y,Xf)
| ~ little_set(second(Y))
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y) ) ),
inference(transitivity,[status(thm)],[181,180]) ).
tff(183,plain,
( ! [Z: $i,Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| ( second(Y) != Z )
| member(Z,image(X,Xf)) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) ) ),
inference(transitivity,[status(thm)],[182,179]) ).
tff(184,plain,
^ [Z: $i,Y: $i,Xf: $i,X: $i] :
trans(monotonicity(trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite(( ( member(Z,image(X,Xf))
| ~ little_set(Z) )
<=> ( ~ little_set(Z)
| member(Z,image(X,Xf)) ) )),
( ( member(Z,image(X,Xf))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y) )
<=> ( ~ little_set(Z)
| member(Z,image(X,Xf))
| ~ ordered_pair_predicate(Y) ) )),
rewrite(( ( ~ little_set(Z)
| member(Z,image(X,Xf))
| ~ ordered_pair_predicate(Y) )
<=> ( ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| member(Z,image(X,Xf)) ) )),
( ( member(Z,image(X,Xf))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y) )
<=> ( ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| member(Z,image(X,Xf)) ) )),
( ( member(Z,image(X,Xf))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| ~ member(Y,Xf) )
<=> ( ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| member(Z,image(X,Xf))
| ~ member(Y,Xf) ) )),
rewrite(( ( ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| member(Z,image(X,Xf))
| ~ member(Y,Xf) )
<=> ( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| member(Z,image(X,Xf)) ) )),
( ( member(Z,image(X,Xf))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| ~ member(Y,Xf) )
<=> ( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| member(Z,image(X,Xf)) ) )),
( ( member(Z,image(X,Xf))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| ~ member(Y,Xf)
| ~ member(first(Y),X) )
<=> ( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| member(Z,image(X,Xf))
| ~ member(first(Y),X) ) )),
rewrite(( ( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| member(Z,image(X,Xf))
| ~ member(first(Y),X) )
<=> ( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(Z,image(X,Xf)) ) )),
( ( member(Z,image(X,Xf))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| ~ member(Y,Xf)
| ~ member(first(Y),X) )
<=> ( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(Z,image(X,Xf)) ) )),
( ( member(Z,image(X,Xf))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| ~ member(Y,Xf)
| ~ member(first(Y),X)
| ( second(Y) != Z ) )
<=> ( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(Z,image(X,Xf))
| ( second(Y) != Z ) ) )),
rewrite(( ( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(Z,image(X,Xf))
| ( second(Y) != Z ) )
<=> ( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| ( second(Y) != Z )
| member(Z,image(X,Xf)) ) )),
( ( member(Z,image(X,Xf))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| ~ member(Y,Xf)
| ~ member(first(Y),X)
| ( second(Y) != Z ) )
<=> ( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| ( second(Y) != Z )
| member(Z,image(X,Xf)) ) )),
inference(bind,[status(th)],[]) ).
tff(185,plain,
( ! [Z: $i,Y: $i,Xf: $i,X: $i] :
( member(Z,image(X,Xf))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| ~ member(Y,Xf)
| ~ member(first(Y),X)
| ( second(Y) != Z ) )
<=> ! [Z: $i,Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ little_set(Z)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| ( second(Y) != Z )
| member(Z,image(X,Xf)) ) ),
inference(quant_intro,[status(thm)],[184]) ).
tff(186,plain,
( ! [Z: $i,Y: $i,Xf: $i,X: $i] :
( member(Z,image(X,Xf))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| ~ member(Y,Xf)
| ~ member(first(Y),X)
| ( second(Y) != Z ) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) ) ),
inference(transitivity,[status(thm)],[185,183]) ).
tff(187,axiom,
! [Z: $i,Y: $i,Xf: $i,X: $i] :
( member(Z,image(X,Xf))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Y)
| ~ member(Y,Xf)
| ~ member(first(Y),X)
| ( second(Y) != Z ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',image_and_substitution5) ).
tff(188,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) ),
inference(modus_ponens,[status(thm)],[187,186]) ).
tff(189,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) ),
inference(modus_ponens,[status(thm)],[188,177]) ).
tff(190,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) ),
inference(skolemize,[status(sab)],[189]) ).
tff(191,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) ),
inference(modus_ponens,[status(thm)],[190,176]) ).
tff(192,plain,
( ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) )
| ~ member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function)
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| ~ member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element))
| member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function))
| ~ little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) )
<=> ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) )
| ~ member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function)
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| ~ member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element))
| member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function))
| ~ little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(193,plain,
( ( ~ member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function)
| ~ member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element))
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function))
| ~ little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) )
<=> ( ~ member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function)
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| ~ member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element))
| member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function))
| ~ little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(194,plain,
( ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) )
| ~ member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function)
| ~ member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element))
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function))
| ~ little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) )
<=> ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) )
| ~ member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function)
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| ~ member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element))
| member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function))
| ~ little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ),
inference(monotonicity,[status(thm)],[193]) ).
tff(195,plain,
( ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) )
| ~ member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function)
| ~ member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element))
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function))
| ~ little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) )
<=> ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) )
| ~ member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function)
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| ~ member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element))
| member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function))
| ~ little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ),
inference(transitivity,[status(thm)],[194,192]) ).
tff(196,plain,
( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) )
| ~ member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function)
| ~ member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element))
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function))
| ~ little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(197,plain,
( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ member(Y,Xf)
| ~ member(first(Y),X)
| ~ ordered_pair_predicate(Y)
| member(second(Y),image(X,Xf))
| ~ little_set(second(Y)) )
| ~ member(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element),a_function)
| ~ ordered_pair_predicate(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))
| ~ member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element))
| member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function))
| ~ little_set(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(modus_ponens,[status(thm)],[196,195]) ).
tff(198,plain,
( ~ member(first(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),non_ordered_pair(element,element))
| member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function)) ),
inference(unit_resolution,[status(thm)],[197,191,127,174,163]) ).
tff(199,plain,
member(second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(non_ordered_pair(element,element),a_function)),
inference(unit_resolution,[status(thm)],[198,152]) ).
tff(200,plain,
member(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(singleton_set(element),a_function)),
inference(modus_ponens,[status(thm)],[199,96]) ).
tff(201,plain,
^ [Z: $i,X: $i] :
refl(( ( ~ member(Z,second(X))
| member(Z,f7(Z,X)) )
<=> ( ~ member(Z,second(X))
| member(Z,f7(Z,X)) ) )),
inference(bind,[status(th)],[]) ).
tff(202,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| member(Z,f7(Z,X)) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| member(Z,f7(Z,X)) ) ),
inference(quant_intro,[status(thm)],[201]) ).
tff(203,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| member(Z,f7(Z,X)) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| member(Z,f7(Z,X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(204,axiom,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| member(Z,f7(Z,X)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',second4) ).
tff(205,plain,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| member(Z,f7(Z,X)) ),
inference(modus_ponens,[status(thm)],[204,203]) ).
tff(206,plain,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| member(Z,f7(Z,X)) ),
inference(skolemize,[status(sab)],[205]) ).
tff(207,plain,
! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| member(Z,f7(Z,X)) ),
inference(modus_ponens,[status(thm)],[206,202]) ).
tff(208,plain,
( ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| member(Z,f7(Z,X)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) )
<=> ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| member(Z,f7(Z,X)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(209,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| member(Z,f7(Z,X)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(210,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,second(X))
| member(Z,f7(Z,X)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),second(f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))) ),
inference(modus_ponens,[status(thm)],[209,208]) ).
tff(211,plain,
member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element))),
inference(unit_resolution,[status(thm)],[210,207,37]) ).
tff(212,plain,
^ [Y: $i,X: $i] :
refl(( ( subset(X,Y)
| ~ member(f17(X,Y),Y) )
<=> ( subset(X,Y)
| ~ member(f17(X,Y),Y) ) )),
inference(bind,[status(th)],[]) ).
tff(213,plain,
( ! [Y: $i,X: $i] :
( subset(X,Y)
| ~ member(f17(X,Y),Y) )
<=> ! [Y: $i,X: $i] :
( subset(X,Y)
| ~ member(f17(X,Y),Y) ) ),
inference(quant_intro,[status(thm)],[212]) ).
tff(214,plain,
( ! [Y: $i,X: $i] :
( subset(X,Y)
| ~ member(f17(X,Y),Y) )
<=> ! [Y: $i,X: $i] :
( subset(X,Y)
| ~ member(f17(X,Y),Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(215,axiom,
! [Y: $i,X: $i] :
( subset(X,Y)
| ~ member(f17(X,Y),Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',subset3) ).
tff(216,plain,
! [Y: $i,X: $i] :
( subset(X,Y)
| ~ member(f17(X,Y),Y) ),
inference(modus_ponens,[status(thm)],[215,214]) ).
tff(217,plain,
! [Y: $i,X: $i] :
( subset(X,Y)
| ~ member(f17(X,Y),Y) ),
inference(skolemize,[status(sab)],[216]) ).
tff(218,plain,
! [Y: $i,X: $i] :
( subset(X,Y)
| ~ member(f17(X,Y),Y) ),
inference(modus_ponens,[status(thm)],[217,213]) ).
tff(219,plain,
( ( ~ ! [Y: $i,X: $i] :
( subset(X,Y)
| ~ member(f17(X,Y),Y) )
| subset(apply(a_function,element),sigma(image(singleton_set(element),a_function)))
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),sigma(image(singleton_set(element),a_function))) )
<=> ( ~ ! [Y: $i,X: $i] :
( subset(X,Y)
| ~ member(f17(X,Y),Y) )
| subset(apply(a_function,element),sigma(image(singleton_set(element),a_function)))
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),sigma(image(singleton_set(element),a_function))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(220,plain,
( ~ ! [Y: $i,X: $i] :
( subset(X,Y)
| ~ member(f17(X,Y),Y) )
| subset(apply(a_function,element),sigma(image(singleton_set(element),a_function)))
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),sigma(image(singleton_set(element),a_function))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(221,plain,
( ~ ! [Y: $i,X: $i] :
( subset(X,Y)
| ~ member(f17(X,Y),Y) )
| subset(apply(a_function,element),sigma(image(singleton_set(element),a_function)))
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),sigma(image(singleton_set(element),a_function))) ),
inference(modus_ponens,[status(thm)],[220,219]) ).
tff(222,plain,
~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),sigma(image(singleton_set(element),a_function))),
inference(unit_resolution,[status(thm)],[221,218,15]) ).
tff(223,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) )
<=> ( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(224,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) ) ),
inference(quant_intro,[status(thm)],[223]) ).
tff(225,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(226,plain,
^ [Z: $i,Y: $i,X: $i] :
trans(monotonicity(rewrite(( ( member(Z,sigma(X))
| ~ member(Y,X) )
<=> ( ~ member(Y,X)
| member(Z,sigma(X)) ) )),
( ( member(Z,sigma(X))
| ~ member(Y,X)
| ~ member(Z,Y) )
<=> ( ~ member(Y,X)
| member(Z,sigma(X))
| ~ member(Z,Y) ) )),
rewrite(( ( ~ member(Y,X)
| member(Z,sigma(X))
| ~ member(Z,Y) )
<=> ( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) ) )),
( ( member(Z,sigma(X))
| ~ member(Y,X)
| ~ member(Z,Y) )
<=> ( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(227,plain,
( ! [Z: $i,Y: $i,X: $i] :
( member(Z,sigma(X))
| ~ member(Y,X)
| ~ member(Z,Y) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) ) ),
inference(quant_intro,[status(thm)],[226]) ).
tff(228,axiom,
! [Z: $i,Y: $i,X: $i] :
( member(Z,sigma(X))
| ~ member(Y,X)
| ~ member(Z,Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',sigma3) ).
tff(229,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) ),
inference(modus_ponens,[status(thm)],[228,227]) ).
tff(230,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) ),
inference(modus_ponens,[status(thm)],[229,225]) ).
tff(231,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) ),
inference(skolemize,[status(sab)],[230]) ).
tff(232,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) ),
inference(modus_ponens,[status(thm)],[231,224]) ).
tff(233,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ member(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(singleton_set(element),a_function))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),sigma(image(singleton_set(element),a_function))) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ member(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(singleton_set(element),a_function))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),sigma(image(singleton_set(element),a_function))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(234,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ member(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(singleton_set(element),a_function))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),sigma(image(singleton_set(element),a_function))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(235,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,Y)
| ~ member(Y,X)
| member(Z,sigma(X)) )
| ~ member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)))
| ~ member(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(singleton_set(element),a_function))
| member(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),sigma(image(singleton_set(element),a_function))) ),
inference(modus_ponens,[status(thm)],[234,233]) ).
tff(236,plain,
~ member(f7(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),f28(f17(apply(a_function,element),sigma(image(singleton_set(element),a_function))),a_function,element)),image(singleton_set(element),a_function)),
inference(unit_resolution,[status(thm)],[235,232,222,211]) ).
tff(237,plain,
$false,
inference(unit_resolution,[status(thm)],[236,200]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET028-3 : TPTP v9.0.0. Released v1.0.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n001.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 04:07:19 EDT 2025
% 0.12/0.35 % CPUTime :
% 0.21/0.46 % SZS status Unsatisfiable
% 0.21/0.46 % SZS output start Proof
% See solution above
% 0.21/0.51 % E exiting
%------------------------------------------------------------------------------