%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET038-3 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:24:56 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.46s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 108
% Syntax : Number of formulae : 282 ( 76 unt; 0 typ; 0 def)
% Number of atoms : 1355 ( 213 equ)
% Maximal formula atoms : 20 ( 4 avg)
% Number of connectives : 1896 ( 886 ~; 871 |; 0 &)
% ( 139 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of FOOLs : 63 ( 63 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-3 aty)
% Number of functors : 15 ( 15 usr; 4 con; 0-2 aty)
% Number of variables : 626 ( 567 !; 0 ?; 626 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(range_of_type,type,
range_of: $i > $i ).
tff(a_function_type,type,
a_function: $i ).
tff(apply_type,type,
apply: ( $i * $i ) > $i ).
tff(a_type,type,
a: $i ).
tff(second_type,type,
second: $i > $i ).
tff(non_ordered_pair_type,type,
non_ordered_pair: ( $i * $i ) > $i ).
tff(first_type,type,
first: $i > $i ).
tff(f8_type,type,
f8: ( $i * $i ) > $i ).
tff(singleton_set_type,type,
singleton_set: $i > $i ).
tff(ordered_pair_type,type,
ordered_pair: ( $i * $i ) > $i ).
tff(little_set_type,type,
little_set: $i > $o ).
tff(domain_of_type,type,
domain_of: $i > $i ).
tff(a_domain_type,type,
a_domain: $i ).
tff(maps_type,type,
maps: ( $i * $i * $i ) > $o ).
tff(a_range_type,type,
a_range: $i ).
tff(function_type,type,
function: $i > $o ).
tff(f3_type,type,
f3: $i > $i ).
tff(f2_type,type,
f2: $i > $i ).
tff(ordered_pair_predicate_type,type,
ordered_pair_predicate: $i > $o ).
tff(subset_type,type,
subset: ( $i * $i ) > $o ).
tff(1,plain,
^ [Y: $i,X: $i] :
refl(( ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) )
<=> ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) )
<=> ! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) )
<=> ! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',ordered_pair) ).
tff(5,plain,
! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [Y: $i,X: $i] : ( ordered_pair(X,Y) = non_ordered_pair(singleton_set(X),non_ordered_pair(X,Y)) )
| ( ordered_pair(first(f8(a,a_function)),apply(a_function,a)) = non_ordered_pair(singleton_set(first(f8(a,a_function))),non_ordered_pair(first(f8(a,a_function)),apply(a_function,a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
ordered_pair(first(f8(a,a_function)),apply(a_function,a)) = non_ordered_pair(singleton_set(first(f8(a,a_function))),non_ordered_pair(first(f8(a,a_function)),apply(a_function,a))),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
non_ordered_pair(singleton_set(first(f8(a,a_function))),non_ordered_pair(first(f8(a,a_function)),apply(a_function,a))) = ordered_pair(first(f8(a,a_function)),apply(a_function,a)),
inference(symmetry,[status(thm)],[9]) ).
tff(11,plain,
second(non_ordered_pair(singleton_set(first(f8(a,a_function))),non_ordered_pair(first(f8(a,a_function)),apply(a_function,a)))) = second(ordered_pair(first(f8(a,a_function)),apply(a_function,a))),
inference(monotonicity,[status(thm)],[10]) ).
tff(12,plain,
second(ordered_pair(first(f8(a,a_function)),apply(a_function,a))) = second(non_ordered_pair(singleton_set(first(f8(a,a_function))),non_ordered_pair(first(f8(a,a_function)),apply(a_function,a)))),
inference(symmetry,[status(thm)],[11]) ).
tff(13,plain,
( maps(a_function,a_domain,a_range)
<=> maps(a_function,a_domain,a_range) ),
inference(rewrite,[status(thm)],[]) ).
tff(14,axiom,
maps(a_function,a_domain,a_range),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_mapping) ).
tff(15,plain,
maps(a_function,a_domain,a_range),
inference(modus_ponens,[status(thm)],[14,13]) ).
tff(16,plain,
^ [Y: $i,Xf: $i,X: $i] :
refl(( ( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) )
<=> ( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) ) )),
inference(bind,[status(th)],[]) ).
tff(17,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) ) ),
inference(quant_intro,[status(thm)],[16]) ).
tff(18,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(19,axiom,
! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',maps2) ).
tff(20,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) ),
inference(modus_ponens,[status(thm)],[19,18]) ).
tff(21,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) ),
inference(skolemize,[status(sab)],[20]) ).
tff(22,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) ),
inference(modus_ponens,[status(thm)],[21,17]) ).
tff(23,plain,
( ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) )
| ~ maps(a_function,a_domain,a_range)
| ( domain_of(a_function) = a_domain ) )
<=> ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) )
| ~ maps(a_function,a_domain,a_range)
| ( domain_of(a_function) = a_domain ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(24,plain,
( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) )
| ~ maps(a_function,a_domain,a_range)
| ( domain_of(a_function) = a_domain ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(25,plain,
( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| ( domain_of(Xf) = X ) )
| ~ maps(a_function,a_domain,a_range)
| ( domain_of(a_function) = a_domain ) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
domain_of(a_function) = a_domain,
inference(unit_resolution,[status(thm)],[25,22,15]) ).
tff(27,plain,
( member(a,domain_of(a_function))
<=> member(a,a_domain) ),
inference(monotonicity,[status(thm)],[26]) ).
tff(28,plain,
( member(a,a_domain)
<=> member(a,domain_of(a_function)) ),
inference(symmetry,[status(thm)],[27]) ).
tff(29,plain,
( member(a,a_domain)
<=> member(a,a_domain) ),
inference(rewrite,[status(thm)],[]) ).
tff(30,axiom,
member(a,a_domain),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_domain) ).
tff(31,plain,
member(a,a_domain),
inference(modus_ponens,[status(thm)],[30,29]) ).
tff(32,plain,
member(a,domain_of(a_function)),
inference(modus_ponens,[status(thm)],[31,28]) ).
tff(33,plain,
^ [Z: $i,X: $i] :
refl(( ( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) )
<=> ( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(34,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) ) ),
inference(quant_intro,[status(thm)],[33]) ).
tff(35,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(36,axiom,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',domain3) ).
tff(37,plain,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) ),
inference(modus_ponens,[status(thm)],[36,35]) ).
tff(38,plain,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) ),
inference(skolemize,[status(sab)],[37]) ).
tff(39,plain,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) ),
inference(modus_ponens,[status(thm)],[38,34]) ).
tff(40,plain,
( ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) )
| ~ member(a,domain_of(a_function))
| ( a = first(f8(a,a_function)) ) )
<=> ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) )
| ~ member(a,domain_of(a_function))
| ( a = first(f8(a,a_function)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) )
| ~ member(a,domain_of(a_function))
| ( a = first(f8(a,a_function)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(42,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) )
| ~ member(a,domain_of(a_function))
| ( a = first(f8(a,a_function)) ) ),
inference(modus_ponens,[status(thm)],[41,40]) ).
tff(43,plain,
a = first(f8(a,a_function)),
inference(unit_resolution,[status(thm)],[42,39,32]) ).
tff(44,plain,
first(f8(a,a_function)) = a,
inference(symmetry,[status(thm)],[43]) ).
tff(45,plain,
( little_set(first(f8(a,a_function)))
<=> little_set(a) ),
inference(monotonicity,[status(thm)],[44]) ).
tff(46,plain,
( little_set(a)
<=> little_set(first(f8(a,a_function))) ),
inference(symmetry,[status(thm)],[45]) ).
tff(47,plain,
^ [Y: $i,X: $i] :
refl(( ( little_set(X)
| ~ member(X,Y) )
<=> ( little_set(X)
| ~ member(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(48,plain,
( ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
<=> ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ) ),
inference(quant_intro,[status(thm)],[47]) ).
tff(49,plain,
( ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
<=> ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(50,plain,
^ [Y: $i,X: $i] :
rewrite(( ( ~ member(X,Y)
| little_set(X) )
<=> ( little_set(X)
| ~ member(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(51,plain,
( ! [Y: $i,X: $i] :
( ~ member(X,Y)
| little_set(X) )
<=> ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ) ),
inference(quant_intro,[status(thm)],[50]) ).
tff(52,axiom,
! [Y: $i,X: $i] :
( ~ member(X,Y)
| little_set(X) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',a2) ).
tff(53,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(modus_ponens,[status(thm)],[52,51]) ).
tff(54,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(modus_ponens,[status(thm)],[53,49]) ).
tff(55,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(skolemize,[status(sab)],[54]) ).
tff(56,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(modus_ponens,[status(thm)],[55,48]) ).
tff(57,plain,
( ( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(a)
| ~ member(a,a_domain) )
<=> ( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(a)
| ~ member(a,a_domain) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(58,plain,
( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(a)
| ~ member(a,a_domain) ),
inference(quant_inst,[status(thm)],[]) ).
tff(59,plain,
( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(a)
| ~ member(a,a_domain) ),
inference(modus_ponens,[status(thm)],[58,57]) ).
tff(60,plain,
little_set(a),
inference(unit_resolution,[status(thm)],[59,56,31]) ).
tff(61,plain,
little_set(first(f8(a,a_function))),
inference(modus_ponens,[status(thm)],[60,46]) ).
tff(62,plain,
^ [Y: $i,Xf: $i,X: $i] :
refl(( ( function(Xf)
| ~ maps(Xf,X,Y) )
<=> ( function(Xf)
| ~ maps(Xf,X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(63,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) ) ),
inference(quant_intro,[status(thm)],[62]) ).
tff(64,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(65,plain,
^ [Y: $i,Xf: $i,X: $i] :
rewrite(( ( ~ maps(Xf,X,Y)
| function(Xf) )
<=> ( function(Xf)
| ~ maps(Xf,X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(66,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| function(Xf) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) ) ),
inference(quant_intro,[status(thm)],[65]) ).
tff(67,axiom,
! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| function(Xf) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',maps1) ).
tff(68,plain,
! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) ),
inference(modus_ponens,[status(thm)],[67,66]) ).
tff(69,plain,
! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) ),
inference(modus_ponens,[status(thm)],[68,64]) ).
tff(70,plain,
! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) ),
inference(skolemize,[status(sab)],[69]) ).
tff(71,plain,
! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) ),
inference(modus_ponens,[status(thm)],[70,63]) ).
tff(72,plain,
( ( ~ ! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) )
| function(a_function)
| ~ maps(a_function,a_domain,a_range) )
<=> ( ~ ! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) )
| function(a_function)
| ~ maps(a_function,a_domain,a_range) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(73,plain,
( ~ ! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) )
| function(a_function)
| ~ maps(a_function,a_domain,a_range) ),
inference(quant_inst,[status(thm)],[]) ).
tff(74,plain,
( ~ ! [Y: $i,Xf: $i,X: $i] :
( function(Xf)
| ~ maps(Xf,X,Y) )
| function(a_function)
| ~ maps(a_function,a_domain,a_range) ),
inference(modus_ponens,[status(thm)],[73,72]) ).
tff(75,plain,
function(a_function),
inference(unit_resolution,[status(thm)],[74,71,15]) ).
tff(76,plain,
^ [Y: $i,X: $i] :
refl(( ( ~ function(Y)
| little_set(apply(Y,X)) )
<=> ( ~ function(Y)
| little_set(apply(Y,X)) ) )),
inference(bind,[status(th)],[]) ).
tff(77,plain,
( ! [Y: $i,X: $i] :
( ~ function(Y)
| little_set(apply(Y,X)) )
<=> ! [Y: $i,X: $i] :
( ~ function(Y)
| little_set(apply(Y,X)) ) ),
inference(quant_intro,[status(thm)],[76]) ).
tff(78,plain,
( ! [Y: $i,X: $i] :
( ~ function(Y)
| little_set(apply(Y,X)) )
<=> ! [Y: $i,X: $i] :
( ~ function(Y)
| little_set(apply(Y,X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(79,axiom,
! [Y: $i,X: $i] :
( ~ function(Y)
| little_set(apply(Y,X)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',function_values_are_small) ).
tff(80,plain,
! [Y: $i,X: $i] :
( ~ function(Y)
| little_set(apply(Y,X)) ),
inference(modus_ponens,[status(thm)],[79,78]) ).
tff(81,plain,
! [Y: $i,X: $i] :
( ~ function(Y)
| little_set(apply(Y,X)) ),
inference(skolemize,[status(sab)],[80]) ).
tff(82,plain,
! [Y: $i,X: $i] :
( ~ function(Y)
| little_set(apply(Y,X)) ),
inference(modus_ponens,[status(thm)],[81,77]) ).
tff(83,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ function(Y)
| little_set(apply(Y,X)) )
| ~ function(a_function)
| little_set(apply(a_function,a)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ function(Y)
| little_set(apply(Y,X)) )
| ~ function(a_function)
| little_set(apply(a_function,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(84,plain,
( ~ ! [Y: $i,X: $i] :
( ~ function(Y)
| little_set(apply(Y,X)) )
| ~ function(a_function)
| little_set(apply(a_function,a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(85,plain,
( ~ ! [Y: $i,X: $i] :
( ~ function(Y)
| little_set(apply(Y,X)) )
| ~ function(a_function)
| little_set(apply(a_function,a)) ),
inference(modus_ponens,[status(thm)],[84,83]) ).
tff(86,plain,
( ~ function(a_function)
| little_set(apply(a_function,a)) ),
inference(unit_resolution,[status(thm)],[85,82]) ).
tff(87,plain,
little_set(apply(a_function,a)),
inference(unit_resolution,[status(thm)],[86,75]) ).
tff(88,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(89,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)],[88]) ).
tff(90,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(91,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(92,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)],[91]) ).
tff(93,axiom,
! [Y: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ( second(ordered_pair(X,Y)) = Y ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_of_second) ).
tff(94,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[93,92]) ).
tff(95,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[94,90]) ).
tff(96,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(skolemize,[status(sab)],[95]) ).
tff(97,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[96,89]) ).
tff(98,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(apply(a_function,a))
| ~ little_set(first(f8(a,a_function)))
| ( second(ordered_pair(first(f8(a,a_function)),apply(a_function,a))) = apply(a_function,a) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(apply(a_function,a))
| ~ little_set(first(f8(a,a_function)))
| ( second(ordered_pair(first(f8(a,a_function)),apply(a_function,a))) = apply(a_function,a) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(99,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(apply(a_function,a))
| ~ little_set(first(f8(a,a_function)))
| ( second(ordered_pair(first(f8(a,a_function)),apply(a_function,a))) = apply(a_function,a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(100,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(apply(a_function,a))
| ~ little_set(first(f8(a,a_function)))
| ( second(ordered_pair(first(f8(a,a_function)),apply(a_function,a))) = apply(a_function,a) ) ),
inference(modus_ponens,[status(thm)],[99,98]) ).
tff(101,plain,
( ~ little_set(first(f8(a,a_function)))
| ( second(ordered_pair(first(f8(a,a_function)),apply(a_function,a))) = apply(a_function,a) ) ),
inference(unit_resolution,[status(thm)],[100,97,87]) ).
tff(102,plain,
second(ordered_pair(first(f8(a,a_function)),apply(a_function,a))) = apply(a_function,a),
inference(unit_resolution,[status(thm)],[101,61]) ).
tff(103,plain,
apply(a_function,a) = second(ordered_pair(first(f8(a,a_function)),apply(a_function,a))),
inference(symmetry,[status(thm)],[102]) ).
tff(104,plain,
apply(a_function,a) = second(non_ordered_pair(singleton_set(first(f8(a,a_function))),non_ordered_pair(first(f8(a,a_function)),apply(a_function,a)))),
inference(transitivity,[status(thm)],[103,12]) ).
tff(105,plain,
( member(apply(a_function,a),range_of(a_function))
<=> member(second(non_ordered_pair(singleton_set(first(f8(a,a_function))),non_ordered_pair(first(f8(a,a_function)),apply(a_function,a)))),range_of(a_function)) ),
inference(monotonicity,[status(thm)],[104]) ).
tff(106,plain,
( member(second(non_ordered_pair(singleton_set(first(f8(a,a_function))),non_ordered_pair(first(f8(a,a_function)),apply(a_function,a)))),range_of(a_function))
<=> member(apply(a_function,a),range_of(a_function)) ),
inference(symmetry,[status(thm)],[105]) ).
tff(107,plain,
^ [Z: $i,X: $i] :
refl(( ( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) )
<=> ( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(108,plain,
( ! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) )
<=> ! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ) ),
inference(quant_intro,[status(thm)],[107]) ).
tff(109,plain,
( ! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) )
<=> ! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(110,plain,
^ [Z: $i,X: $i] :
rewrite(( ( ~ member(Z,domain_of(X))
| ordered_pair_predicate(f8(Z,X)) )
<=> ( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(111,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ordered_pair_predicate(f8(Z,X)) )
<=> ! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ) ),
inference(quant_intro,[status(thm)],[110]) ).
tff(112,axiom,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ordered_pair_predicate(f8(Z,X)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',domain1) ).
tff(113,plain,
! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ),
inference(modus_ponens,[status(thm)],[112,111]) ).
tff(114,plain,
! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ),
inference(modus_ponens,[status(thm)],[113,109]) ).
tff(115,plain,
! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ),
inference(skolemize,[status(sab)],[114]) ).
tff(116,plain,
! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ),
inference(modus_ponens,[status(thm)],[115,108]) ).
tff(117,plain,
( ( ~ ! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) )
| ordered_pair_predicate(f8(a,a_function))
| ~ member(a,domain_of(a_function)) )
<=> ( ~ ! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) )
| ordered_pair_predicate(f8(a,a_function))
| ~ member(a,domain_of(a_function)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(118,plain,
( ~ ! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) )
| ordered_pair_predicate(f8(a,a_function))
| ~ member(a,domain_of(a_function)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(119,plain,
( ~ ! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) )
| ordered_pair_predicate(f8(a,a_function))
| ~ member(a,domain_of(a_function)) ),
inference(modus_ponens,[status(thm)],[118,117]) ).
tff(120,plain,
ordered_pair_predicate(f8(a,a_function)),
inference(unit_resolution,[status(thm)],[119,116,32]) ).
tff(121,plain,
^ [X: $i] :
refl(( ( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
<=> ( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(122,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
<=> ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ) ),
inference(quant_intro,[status(thm)],[121]) ).
tff(123,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
<=> ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(124,axiom,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',ordered_pair_predicate3) ).
tff(125,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ),
inference(modus_ponens,[status(thm)],[124,123]) ).
tff(126,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ),
inference(skolemize,[status(sab)],[125]) ).
tff(127,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ),
inference(modus_ponens,[status(thm)],[126,122]) ).
tff(128,plain,
( ( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
| ~ ordered_pair_predicate(f8(a,a_function))
| ( f8(a,a_function) = ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))) ) )
<=> ( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
| ~ ordered_pair_predicate(f8(a,a_function))
| ( f8(a,a_function) = ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(129,plain,
( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
| ~ ordered_pair_predicate(f8(a,a_function))
| ( f8(a,a_function) = ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(130,plain,
( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) )
| ~ ordered_pair_predicate(f8(a,a_function))
| ( f8(a,a_function) = ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))) ) ),
inference(modus_ponens,[status(thm)],[129,128]) ).
tff(131,plain,
f8(a,a_function) = ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),
inference(unit_resolution,[status(thm)],[130,127,120]) ).
tff(132,plain,
ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))) = f8(a,a_function),
inference(symmetry,[status(thm)],[131]) ).
tff(133,plain,
first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = first(f8(a,a_function)),
inference(monotonicity,[status(thm)],[132]) ).
tff(134,plain,
^ [X: $i] :
refl(( ( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
<=> ( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(135,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
<=> ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ) ),
inference(quant_intro,[status(thm)],[134]) ).
tff(136,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
<=> ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(137,axiom,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',ordered_pair_predicate2) ).
tff(138,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ),
inference(modus_ponens,[status(thm)],[137,136]) ).
tff(139,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ),
inference(skolemize,[status(sab)],[138]) ).
tff(140,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ),
inference(modus_ponens,[status(thm)],[139,135]) ).
tff(141,plain,
( ( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
| ~ ordered_pair_predicate(f8(a,a_function))
| little_set(f3(f8(a,a_function))) )
<=> ( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
| ~ ordered_pair_predicate(f8(a,a_function))
| little_set(f3(f8(a,a_function))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(142,plain,
( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
| ~ ordered_pair_predicate(f8(a,a_function))
| little_set(f3(f8(a,a_function))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(143,plain,
( ~ ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) )
| ~ ordered_pair_predicate(f8(a,a_function))
| little_set(f3(f8(a,a_function))) ),
inference(modus_ponens,[status(thm)],[142,141]) ).
tff(144,plain,
little_set(f3(f8(a,a_function))),
inference(unit_resolution,[status(thm)],[143,140,120]) ).
tff(145,plain,
^ [X: $i] :
refl(( ( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
<=> ( little_set(f2(X))
| ~ ordered_pair_predicate(X) ) )),
inference(bind,[status(th)],[]) ).
tff(146,plain,
( ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
<=> ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ) ),
inference(quant_intro,[status(thm)],[145]) ).
tff(147,plain,
( ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
<=> ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(148,plain,
^ [X: $i] :
rewrite(( ( ~ ordered_pair_predicate(X)
| little_set(f2(X)) )
<=> ( little_set(f2(X))
| ~ ordered_pair_predicate(X) ) )),
inference(bind,[status(th)],[]) ).
tff(149,plain,
( ! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f2(X)) )
<=> ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ) ),
inference(quant_intro,[status(thm)],[148]) ).
tff(150,axiom,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f2(X)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',ordered_pair_predicate1) ).
tff(151,plain,
! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(modus_ponens,[status(thm)],[150,149]) ).
tff(152,plain,
! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(modus_ponens,[status(thm)],[151,147]) ).
tff(153,plain,
! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(skolemize,[status(sab)],[152]) ).
tff(154,plain,
! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(modus_ponens,[status(thm)],[153,146]) ).
tff(155,plain,
( ( ~ ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
| little_set(f2(f8(a,a_function)))
| ~ ordered_pair_predicate(f8(a,a_function)) )
<=> ( ~ ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
| little_set(f2(f8(a,a_function)))
| ~ ordered_pair_predicate(f8(a,a_function)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(156,plain,
( ~ ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
| little_set(f2(f8(a,a_function)))
| ~ ordered_pair_predicate(f8(a,a_function)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(157,plain,
( ~ ! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) )
| little_set(f2(f8(a,a_function)))
| ~ ordered_pair_predicate(f8(a,a_function)) ),
inference(modus_ponens,[status(thm)],[156,155]) ).
tff(158,plain,
little_set(f2(f8(a,a_function))),
inference(unit_resolution,[status(thm)],[157,154,120]) ).
tff(159,plain,
^ [Y: $i,X: $i] :
refl(( ( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ) )),
inference(bind,[status(th)],[]) ).
tff(160,plain,
( ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ) ),
inference(quant_intro,[status(thm)],[159]) ).
tff(161,plain,
( ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(162,plain,
^ [Y: $i,X: $i] :
rewrite(( ( ~ little_set(X)
| ~ little_set(Y)
| ( first(ordered_pair(X,Y)) = X ) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ) )),
inference(bind,[status(th)],[]) ).
tff(163,plain,
( ! [Y: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ( first(ordered_pair(X,Y)) = X ) )
<=> ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ) ),
inference(quant_intro,[status(thm)],[162]) ).
tff(164,axiom,
! [Y: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ( first(ordered_pair(X,Y)) = X ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_of_first) ).
tff(165,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(modus_ponens,[status(thm)],[164,163]) ).
tff(166,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(modus_ponens,[status(thm)],[165,161]) ).
tff(167,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(skolemize,[status(sab)],[166]) ).
tff(168,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(modus_ponens,[status(thm)],[167,160]) ).
tff(169,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ( first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f2(f8(a,a_function)) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ( first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f2(f8(a,a_function)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(170,plain,
( ( ~ little_set(f3(f8(a,a_function)))
| ~ little_set(f2(f8(a,a_function)))
| ( first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f2(f8(a,a_function)) ) )
<=> ( ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ( first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f2(f8(a,a_function)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(171,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
| ~ little_set(f3(f8(a,a_function)))
| ~ little_set(f2(f8(a,a_function)))
| ( first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f2(f8(a,a_function)) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ( first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f2(f8(a,a_function)) ) ) ),
inference(monotonicity,[status(thm)],[170]) ).
tff(172,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
| ~ little_set(f3(f8(a,a_function)))
| ~ little_set(f2(f8(a,a_function)))
| ( first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f2(f8(a,a_function)) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ( first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f2(f8(a,a_function)) ) ) ),
inference(transitivity,[status(thm)],[171,169]) ).
tff(173,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
| ~ little_set(f3(f8(a,a_function)))
| ~ little_set(f2(f8(a,a_function)))
| ( first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f2(f8(a,a_function)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(174,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ( first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f2(f8(a,a_function)) ) ),
inference(modus_ponens,[status(thm)],[173,172]) ).
tff(175,plain,
first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f2(f8(a,a_function)),
inference(unit_resolution,[status(thm)],[174,168,158,144]) ).
tff(176,plain,
f2(f8(a,a_function)) = first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))),
inference(symmetry,[status(thm)],[175]) ).
tff(177,plain,
f2(f8(a,a_function)) = a,
inference(transitivity,[status(thm)],[176,133,44]) ).
tff(178,plain,
apply(a_function,f2(f8(a,a_function))) = apply(a_function,a),
inference(monotonicity,[status(thm)],[177]) ).
tff(179,plain,
( member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
<=> member(f8(a,a_function),a_function) ),
inference(monotonicity,[status(thm)],[132]) ).
tff(180,plain,
( member(f8(a,a_function),a_function)
<=> member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function) ),
inference(symmetry,[status(thm)],[179]) ).
tff(181,plain,
^ [Z: $i,X: $i] :
refl(( ( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) )
<=> ( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) ) )),
inference(bind,[status(th)],[]) ).
tff(182,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) ) ),
inference(quant_intro,[status(thm)],[181]) ).
tff(183,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(184,axiom,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',domain2) ).
tff(185,plain,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) ),
inference(modus_ponens,[status(thm)],[184,183]) ).
tff(186,plain,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) ),
inference(skolemize,[status(sab)],[185]) ).
tff(187,plain,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) ),
inference(modus_ponens,[status(thm)],[186,182]) ).
tff(188,plain,
( ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) )
| ~ member(a,domain_of(a_function))
| member(f8(a,a_function),a_function) )
<=> ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) )
| ~ member(a,domain_of(a_function))
| member(f8(a,a_function),a_function) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(189,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) )
| ~ member(a,domain_of(a_function))
| member(f8(a,a_function),a_function) ),
inference(quant_inst,[status(thm)],[]) ).
tff(190,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) )
| ~ member(a,domain_of(a_function))
| member(f8(a,a_function),a_function) ),
inference(modus_ponens,[status(thm)],[189,188]) ).
tff(191,plain,
member(f8(a,a_function),a_function),
inference(unit_resolution,[status(thm)],[190,187,32]) ).
tff(192,plain,
member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function),
inference(modus_ponens,[status(thm)],[191,180]) ).
tff(193,plain,
^ [Y: $i,Xf: $i,X: $i] :
refl(( ( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) ) )),
inference(bind,[status(th)],[]) ).
tff(194,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) ) ),
inference(quant_intro,[status(thm)],[193]) ).
tff(195,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(196,plain,
^ [Y: $i,Xf: $i,X: $i] :
trans(monotonicity(trans(monotonicity(rewrite(( ( ~ little_set(X)
| ~ little_set(Y)
| ~ function(Xf) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf) ) )),
( ( ~ little_set(X)
| ~ little_set(Y)
| ~ function(Xf)
| ~ member(ordered_pair(X,Y),Xf) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ~ member(ordered_pair(X,Y),Xf) ) )),
rewrite(( ( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ~ member(ordered_pair(X,Y),Xf) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ~ member(ordered_pair(X,Y),Xf) ) )),
( ( ~ little_set(X)
| ~ little_set(Y)
| ~ function(Xf)
| ~ member(ordered_pair(X,Y),Xf) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ~ member(ordered_pair(X,Y),Xf) ) )),
( ( ~ little_set(X)
| ~ little_set(Y)
| ~ function(Xf)
| ~ member(ordered_pair(X,Y),Xf)
| ( apply(Xf,X) = Y ) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ~ member(ordered_pair(X,Y),Xf)
| ( apply(Xf,X) = Y ) ) )),
rewrite(( ( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ~ member(ordered_pair(X,Y),Xf)
| ( apply(Xf,X) = Y ) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) ) )),
( ( ~ little_set(X)
| ~ little_set(Y)
| ~ function(Xf)
| ~ member(ordered_pair(X,Y),Xf)
| ( apply(Xf,X) = Y ) )
<=> ( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) ) )),
inference(bind,[status(th)],[]) ).
tff(197,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ~ function(Xf)
| ~ member(ordered_pair(X,Y),Xf)
| ( apply(Xf,X) = Y ) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) ) ),
inference(quant_intro,[status(thm)],[196]) ).
tff(198,axiom,
! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ~ function(Xf)
| ~ member(ordered_pair(X,Y),Xf)
| ( apply(Xf,X) = Y ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',apply_for_functions1) ).
tff(199,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) ),
inference(modus_ponens,[status(thm)],[198,197]) ).
tff(200,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) ),
inference(modus_ponens,[status(thm)],[199,195]) ).
tff(201,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) ),
inference(skolemize,[status(sab)],[200]) ).
tff(202,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) ),
inference(modus_ponens,[status(thm)],[201,194]) ).
tff(203,plain,
( ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) )
| ~ function(a_function)
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
| ( apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)) ) )
<=> ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) )
| ~ function(a_function)
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
| ( apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(204,plain,
( ( ~ little_set(f3(f8(a,a_function)))
| ~ little_set(f2(f8(a,a_function)))
| ~ function(a_function)
| ( apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)) )
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function) )
<=> ( ~ function(a_function)
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
| ( apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(205,plain,
( ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) )
| ~ little_set(f3(f8(a,a_function)))
| ~ little_set(f2(f8(a,a_function)))
| ~ function(a_function)
| ( apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)) )
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function) )
<=> ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) )
| ~ function(a_function)
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
| ( apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)) ) ) ),
inference(monotonicity,[status(thm)],[204]) ).
tff(206,plain,
( ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) )
| ~ little_set(f3(f8(a,a_function)))
| ~ little_set(f2(f8(a,a_function)))
| ~ function(a_function)
| ( apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)) )
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function) )
<=> ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) )
| ~ function(a_function)
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
| ( apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)) ) ) ),
inference(transitivity,[status(thm)],[205,203]) ).
tff(207,plain,
( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) )
| ~ little_set(f3(f8(a,a_function)))
| ~ little_set(f2(f8(a,a_function)))
| ~ function(a_function)
| ( apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)) )
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function) ),
inference(quant_inst,[status(thm)],[]) ).
tff(208,plain,
( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ~ function(Xf)
| ( apply(Xf,X) = Y )
| ~ member(ordered_pair(X,Y),Xf) )
| ~ function(a_function)
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
| ( apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)) ) ),
inference(modus_ponens,[status(thm)],[207,206]) ).
tff(209,plain,
( ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
| ( apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)) ) ),
inference(unit_resolution,[status(thm)],[208,202,75,158,144]) ).
tff(210,plain,
apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)),
inference(unit_resolution,[status(thm)],[209,192]) ).
tff(211,plain,
f3(f8(a,a_function)) = apply(a_function,f2(f8(a,a_function))),
inference(symmetry,[status(thm)],[210]) ).
tff(212,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ( second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f3(f8(a,a_function)) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ( second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f3(f8(a,a_function)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(213,plain,
( ( ~ little_set(f3(f8(a,a_function)))
| ~ little_set(f2(f8(a,a_function)))
| ( second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f3(f8(a,a_function)) ) )
<=> ( ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ( second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f3(f8(a,a_function)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(214,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f3(f8(a,a_function)))
| ~ little_set(f2(f8(a,a_function)))
| ( second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f3(f8(a,a_function)) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ( second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f3(f8(a,a_function)) ) ) ),
inference(monotonicity,[status(thm)],[213]) ).
tff(215,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f3(f8(a,a_function)))
| ~ little_set(f2(f8(a,a_function)))
| ( second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f3(f8(a,a_function)) ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ( second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f3(f8(a,a_function)) ) ) ),
inference(transitivity,[status(thm)],[214,212]) ).
tff(216,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f3(f8(a,a_function)))
| ~ little_set(f2(f8(a,a_function)))
| ( second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f3(f8(a,a_function)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(217,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(f2(f8(a,a_function)))
| ~ little_set(f3(f8(a,a_function)))
| ( second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f3(f8(a,a_function)) ) ),
inference(modus_ponens,[status(thm)],[216,215]) ).
tff(218,plain,
second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f3(f8(a,a_function)),
inference(unit_resolution,[status(thm)],[217,97,158,144]) ).
tff(219,plain,
second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = second(non_ordered_pair(singleton_set(first(f8(a,a_function))),non_ordered_pair(first(f8(a,a_function)),apply(a_function,a)))),
inference(transitivity,[status(thm)],[218,211,178,103,12]) ).
tff(220,plain,
( member(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))),range_of(a_function))
<=> member(second(non_ordered_pair(singleton_set(first(f8(a,a_function))),non_ordered_pair(first(f8(a,a_function)),apply(a_function,a)))),range_of(a_function)) ),
inference(monotonicity,[status(thm)],[219]) ).
tff(221,plain,
( little_set(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))))
<=> little_set(f3(f8(a,a_function))) ),
inference(monotonicity,[status(thm)],[218]) ).
tff(222,plain,
( little_set(f3(f8(a,a_function)))
<=> little_set(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))) ),
inference(symmetry,[status(thm)],[221]) ).
tff(223,plain,
little_set(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))),
inference(modus_ponens,[status(thm)],[144,222]) ).
tff(224,plain,
( ordered_pair_predicate(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))
<=> ordered_pair_predicate(f8(a,a_function)) ),
inference(monotonicity,[status(thm)],[132]) ).
tff(225,plain,
( ordered_pair_predicate(f8(a,a_function))
<=> ordered_pair_predicate(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) ),
inference(symmetry,[status(thm)],[224]) ).
tff(226,plain,
ordered_pair_predicate(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))),
inference(modus_ponens,[status(thm)],[120,225]) ).
tff(227,plain,
^ [Xp: $i,X: $i] :
refl(( ( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) )
<=> ( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) ) )),
inference(bind,[status(th)],[]) ).
tff(228,plain,
( ! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) )
<=> ! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) ) ),
inference(quant_intro,[status(thm)],[227]) ).
tff(229,plain,
( ! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) )
<=> ! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(230,plain,
^ [Xp: $i,X: $i] :
rewrite(( ( member(second(Xp),range_of(X))
| ~ little_set(second(Xp))
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp) )
<=> ( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) ) )),
inference(bind,[status(th)],[]) ).
tff(231,plain,
( ! [Xp: $i,X: $i] :
( member(second(Xp),range_of(X))
| ~ little_set(second(Xp))
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp) )
<=> ! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) ) ),
inference(quant_intro,[status(thm)],[230]) ).
tff(232,plain,
( ! [Z: $i,Xp: $i,X: $i] :
( member(second(Xp),range_of(X))
| ~ little_set(second(Xp))
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp) )
<=> ! [Xp: $i,X: $i] :
( member(second(Xp),range_of(X))
| ~ little_set(second(Xp))
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp) ) ),
inference(elim_unused_vars,[status(thm)],[]) ).
tff(233,plain,
( ! [Z: $i,Xp: $i,X: $i] :
( ~ little_set(Z)
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| ( Z != second(Xp) )
| member(Z,range_of(X)) )
<=> ! [Z: $i,Xp: $i,X: $i] :
( member(second(Xp),range_of(X))
| ~ little_set(second(Xp))
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp) ) ),
inference(destructive_equality_resolution,[status(thm)],[]) ).
tff(234,plain,
( ! [Z: $i,Xp: $i,X: $i] :
( ~ little_set(Z)
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| ( Z != second(Xp) )
| member(Z,range_of(X)) )
<=> ! [Xp: $i,X: $i] :
( member(second(Xp),range_of(X))
| ~ little_set(second(Xp))
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp) ) ),
inference(transitivity,[status(thm)],[233,232]) ).
tff(235,plain,
( ! [Z: $i,Xp: $i,X: $i] :
( ~ little_set(Z)
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| ( Z != second(Xp) )
| member(Z,range_of(X)) )
<=> ! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) ) ),
inference(transitivity,[status(thm)],[234,231]) ).
tff(236,plain,
^ [Z: $i,Xp: $i,X: $i] :
trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite(( ( member(Z,range_of(X))
| ~ little_set(Z) )
<=> ( ~ little_set(Z)
| member(Z,range_of(X)) ) )),
( ( member(Z,range_of(X))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Xp) )
<=> ( ~ little_set(Z)
| member(Z,range_of(X))
| ~ ordered_pair_predicate(Xp) ) )),
rewrite(( ( ~ little_set(Z)
| member(Z,range_of(X))
| ~ ordered_pair_predicate(Xp) )
<=> ( ~ little_set(Z)
| ~ ordered_pair_predicate(Xp)
| member(Z,range_of(X)) ) )),
( ( member(Z,range_of(X))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Xp) )
<=> ( ~ little_set(Z)
| ~ ordered_pair_predicate(Xp)
| member(Z,range_of(X)) ) )),
( ( member(Z,range_of(X))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Xp)
| ~ member(Xp,X) )
<=> ( ~ little_set(Z)
| ~ ordered_pair_predicate(Xp)
| member(Z,range_of(X))
| ~ member(Xp,X) ) )),
rewrite(( ( ~ little_set(Z)
| ~ ordered_pair_predicate(Xp)
| member(Z,range_of(X))
| ~ member(Xp,X) )
<=> ( ~ little_set(Z)
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(Z,range_of(X)) ) )),
( ( member(Z,range_of(X))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Xp)
| ~ member(Xp,X) )
<=> ( ~ little_set(Z)
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(Z,range_of(X)) ) )),
( ( member(Z,range_of(X))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Xp)
| ~ member(Xp,X)
| ( Z != second(Xp) ) )
<=> ( ~ little_set(Z)
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(Z,range_of(X))
| ( Z != second(Xp) ) ) )),
rewrite(( ( ~ little_set(Z)
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(Z,range_of(X))
| ( Z != second(Xp) ) )
<=> ( ~ little_set(Z)
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| ( Z != second(Xp) )
| member(Z,range_of(X)) ) )),
( ( member(Z,range_of(X))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Xp)
| ~ member(Xp,X)
| ( Z != second(Xp) ) )
<=> ( ~ little_set(Z)
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| ( Z != second(Xp) )
| member(Z,range_of(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(237,plain,
( ! [Z: $i,Xp: $i,X: $i] :
( member(Z,range_of(X))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Xp)
| ~ member(Xp,X)
| ( Z != second(Xp) ) )
<=> ! [Z: $i,Xp: $i,X: $i] :
( ~ little_set(Z)
| ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| ( Z != second(Xp) )
| member(Z,range_of(X)) ) ),
inference(quant_intro,[status(thm)],[236]) ).
tff(238,plain,
( ! [Z: $i,Xp: $i,X: $i] :
( member(Z,range_of(X))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Xp)
| ~ member(Xp,X)
| ( Z != second(Xp) ) )
<=> ! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) ) ),
inference(transitivity,[status(thm)],[237,235]) ).
tff(239,axiom,
! [Z: $i,Xp: $i,X: $i] :
( member(Z,range_of(X))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Xp)
| ~ member(Xp,X)
| ( Z != second(Xp) ) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',range_of4) ).
tff(240,plain,
! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) ),
inference(modus_ponens,[status(thm)],[239,238]) ).
tff(241,plain,
! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) ),
inference(modus_ponens,[status(thm)],[240,229]) ).
tff(242,plain,
! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) ),
inference(skolemize,[status(sab)],[241]) ).
tff(243,plain,
! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) ),
inference(modus_ponens,[status(thm)],[242,228]) ).
tff(244,plain,
( ( ~ ! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) )
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
| ~ ordered_pair_predicate(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))
| member(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))),range_of(a_function))
| ~ little_set(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))) )
<=> ( ~ ! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) )
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
| ~ ordered_pair_predicate(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))
| member(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))),range_of(a_function))
| ~ little_set(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(245,plain,
( ~ ! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) )
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
| ~ ordered_pair_predicate(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))
| member(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))),range_of(a_function))
| ~ little_set(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(246,plain,
( ~ ! [Xp: $i,X: $i] :
( ~ member(Xp,X)
| ~ ordered_pair_predicate(Xp)
| member(second(Xp),range_of(X))
| ~ little_set(second(Xp)) )
| ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
| ~ ordered_pair_predicate(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))
| member(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))),range_of(a_function))
| ~ little_set(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))) ),
inference(modus_ponens,[status(thm)],[245,244]) ).
tff(247,plain,
( ~ member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function)
| ~ ordered_pair_predicate(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))
| member(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))),range_of(a_function))
| ~ little_set(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))))) ),
inference(unit_resolution,[status(thm)],[246,243]) ).
tff(248,plain,
member(second(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))),range_of(a_function)),
inference(unit_resolution,[status(thm)],[247,226,192,223]) ).
tff(249,plain,
member(second(non_ordered_pair(singleton_set(first(f8(a,a_function))),non_ordered_pair(first(f8(a,a_function)),apply(a_function,a)))),range_of(a_function)),
inference(modus_ponens,[status(thm)],[248,220]) ).
tff(250,plain,
member(apply(a_function,a),range_of(a_function)),
inference(modus_ponens,[status(thm)],[249,106]) ).
tff(251,plain,
^ [Y: $i,Xf: $i,X: $i] :
refl(( ( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) )
<=> ( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) ) )),
inference(bind,[status(th)],[]) ).
tff(252,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) ) ),
inference(quant_intro,[status(thm)],[251]) ).
tff(253,plain,
( ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) )
<=> ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(254,axiom,
! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',maps3) ).
tff(255,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) ),
inference(modus_ponens,[status(thm)],[254,253]) ).
tff(256,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) ),
inference(skolemize,[status(sab)],[255]) ).
tff(257,plain,
! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) ),
inference(modus_ponens,[status(thm)],[256,252]) ).
tff(258,plain,
( ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) )
| ~ maps(a_function,a_domain,a_range)
| subset(range_of(a_function),a_range) )
<=> ( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) )
| ~ maps(a_function,a_domain,a_range)
| subset(range_of(a_function),a_range) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(259,plain,
( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) )
| ~ maps(a_function,a_domain,a_range)
| subset(range_of(a_function),a_range) ),
inference(quant_inst,[status(thm)],[]) ).
tff(260,plain,
( ~ ! [Y: $i,Xf: $i,X: $i] :
( ~ maps(Xf,X,Y)
| subset(range_of(Xf),Y) )
| ~ maps(a_function,a_domain,a_range)
| subset(range_of(a_function),a_range) ),
inference(modus_ponens,[status(thm)],[259,258]) ).
tff(261,plain,
subset(range_of(a_function),a_range),
inference(unit_resolution,[status(thm)],[260,257,15]) ).
tff(262,plain,
( ~ member(apply(a_function,a),a_range)
<=> ~ member(apply(a_function,a),a_range) ),
inference(rewrite,[status(thm)],[]) ).
tff(263,axiom,
~ member(apply(a_function,a),a_range),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_mapping_in_range) ).
tff(264,plain,
~ member(apply(a_function,a),a_range),
inference(modus_ponens,[status(thm)],[263,262]) ).
tff(265,plain,
^ [Y: $i,U: $i,X: $i] :
refl(( ( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) )
<=> ( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(266,plain,
( ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) )
<=> ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) ) ),
inference(quant_intro,[status(thm)],[265]) ).
tff(267,plain,
( ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) )
<=> ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(268,plain,
^ [Y: $i,U: $i,X: $i] :
trans(monotonicity(rewrite(( ( ~ subset(X,Y)
| ~ member(U,X) )
<=> ( ~ member(U,X)
| ~ subset(X,Y) ) )),
( ( ~ subset(X,Y)
| ~ member(U,X)
| member(U,Y) )
<=> ( ~ member(U,X)
| ~ subset(X,Y)
| member(U,Y) ) )),
rewrite(( ( ~ member(U,X)
| ~ subset(X,Y)
| member(U,Y) )
<=> ( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) ) )),
( ( ~ subset(X,Y)
| ~ member(U,X)
| member(U,Y) )
<=> ( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(269,plain,
( ! [Y: $i,U: $i,X: $i] :
( ~ subset(X,Y)
| ~ member(U,X)
| member(U,Y) )
<=> ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) ) ),
inference(quant_intro,[status(thm)],[268]) ).
tff(270,axiom,
! [Y: $i,U: $i,X: $i] :
( ~ subset(X,Y)
| ~ member(U,X)
| member(U,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',subset1) ).
tff(271,plain,
! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) ),
inference(modus_ponens,[status(thm)],[270,269]) ).
tff(272,plain,
! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) ),
inference(modus_ponens,[status(thm)],[271,267]) ).
tff(273,plain,
! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) ),
inference(skolemize,[status(sab)],[272]) ).
tff(274,plain,
! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) ),
inference(modus_ponens,[status(thm)],[273,266]) ).
tff(275,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) )
| member(apply(a_function,a),a_range)
| ~ subset(range_of(a_function),a_range)
| ~ member(apply(a_function,a),range_of(a_function)) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) )
| member(apply(a_function,a),a_range)
| ~ subset(range_of(a_function),a_range)
| ~ member(apply(a_function,a),range_of(a_function)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(276,plain,
( ( ~ member(apply(a_function,a),range_of(a_function))
| member(apply(a_function,a),a_range)
| ~ subset(range_of(a_function),a_range) )
<=> ( member(apply(a_function,a),a_range)
| ~ subset(range_of(a_function),a_range)
| ~ member(apply(a_function,a),range_of(a_function)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(277,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) )
| ~ member(apply(a_function,a),range_of(a_function))
| member(apply(a_function,a),a_range)
| ~ subset(range_of(a_function),a_range) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) )
| member(apply(a_function,a),a_range)
| ~ subset(range_of(a_function),a_range)
| ~ member(apply(a_function,a),range_of(a_function)) ) ),
inference(monotonicity,[status(thm)],[276]) ).
tff(278,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) )
| ~ member(apply(a_function,a),range_of(a_function))
| member(apply(a_function,a),a_range)
| ~ subset(range_of(a_function),a_range) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) )
| member(apply(a_function,a),a_range)
| ~ subset(range_of(a_function),a_range)
| ~ member(apply(a_function,a),range_of(a_function)) ) ),
inference(transitivity,[status(thm)],[277,275]) ).
tff(279,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) )
| ~ member(apply(a_function,a),range_of(a_function))
| member(apply(a_function,a),a_range)
| ~ subset(range_of(a_function),a_range) ),
inference(quant_inst,[status(thm)],[]) ).
tff(280,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( ~ member(U,X)
| member(U,Y)
| ~ subset(X,Y) )
| member(apply(a_function,a),a_range)
| ~ subset(range_of(a_function),a_range)
| ~ member(apply(a_function,a),range_of(a_function)) ),
inference(modus_ponens,[status(thm)],[279,278]) ).
tff(281,plain,
~ member(apply(a_function,a),range_of(a_function)),
inference(unit_resolution,[status(thm)],[280,274,264,261]) ).
tff(282,plain,
$false,
inference(unit_resolution,[status(thm)],[281,250]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : SET038-3 : TPTP v9.0.0. Released v1.0.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n003.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 04:06:54 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.46 % SZS status Unsatisfiable
% 0.20/0.46 % SZS output start Proof
% See solution above
% 0.20/0.50 % E exiting
%------------------------------------------------------------------------------