%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET037-3 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:24:56 AM UTC 2025
% Result : Unsatisfiable 0.18s 0.42s
% Output : Proof 0.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 54
% Syntax : Number of formulae : 137 ( 41 unt; 0 typ; 0 def)
% Number of atoms : 617 ( 125 equ)
% Maximal formula atoms : 20 ( 4 avg)
% Number of connectives : 866 ( 410 ~; 395 |; 0 &)
% ( 61 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of FOOLs : 24 ( 24 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 : 10 ( 10 usr; 3 con; 0-2 aty)
% Number of variables : 234 ( 212 !; 0 ?; 234 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(a_function_type,type,
a_function: $i ).
tff(f8_type,type,
f8: ( $i * $i ) > $i ).
tff(a_type,type,
a: $i ).
tff(ordered_pair_type,type,
ordered_pair: ( $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(f3_type,type,
f3: $i > $i ).
tff(f2_type,type,
f2: $i > $i ).
tff(apply_type,type,
apply: ( $i * $i ) > $i ).
tff(first_type,type,
first: $i > $i ).
tff(domain_of_type,type,
domain_of: $i > $i ).
tff(ordered_pair_predicate_type,type,
ordered_pair_predicate: $i > $o ).
tff(little_set_type,type,
little_set: $i > $o ).
tff(function_type,type,
function: $i > $o ).
tff(1,plain,
( ( apply(a_function,a) = b )
<=> ( apply(a_function,a) = b ) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
apply(a_function,a) = b,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',applying_the_function) ).
tff(3,plain,
apply(a_function,a) = b,
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
( member(a,domain_of(a_function))
<=> member(a,domain_of(a_function)) ),
inference(rewrite,[status(thm)],[]) ).
tff(5,axiom,
member(a,domain_of(a_function)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_of_function_domain) ).
tff(6,plain,
member(a,domain_of(a_function)),
inference(modus_ponens,[status(thm)],[5,4]) ).
tff(7,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(8,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)],[7]) ).
tff(9,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(10,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(11,plain,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) ),
inference(modus_ponens,[status(thm)],[10,9]) ).
tff(12,plain,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) ),
inference(skolemize,[status(sab)],[11]) ).
tff(13,plain,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| ( Z = first(f8(Z,X)) ) ),
inference(modus_ponens,[status(thm)],[12,8]) ).
tff(14,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(15,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(16,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)],[15,14]) ).
tff(17,plain,
a = first(f8(a,a_function)),
inference(unit_resolution,[status(thm)],[16,13,6]) ).
tff(18,plain,
first(f8(a,a_function)) = a,
inference(symmetry,[status(thm)],[17]) ).
tff(19,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(20,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)],[19]) ).
tff(21,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(22,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(23,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)],[22]) ).
tff(24,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(25,plain,
! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ),
inference(modus_ponens,[status(thm)],[25,21]) ).
tff(27,plain,
! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ),
inference(skolemize,[status(sab)],[26]) ).
tff(28,plain,
! [Z: $i,X: $i] :
( ordered_pair_predicate(f8(Z,X))
| ~ member(Z,domain_of(X)) ),
inference(modus_ponens,[status(thm)],[27,20]) ).
tff(29,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(30,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(31,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)],[30,29]) ).
tff(32,plain,
ordered_pair_predicate(f8(a,a_function)),
inference(unit_resolution,[status(thm)],[31,28,6]) ).
tff(33,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(34,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)],[33]) ).
tff(35,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(36,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(37,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ),
inference(modus_ponens,[status(thm)],[36,35]) ).
tff(38,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ),
inference(skolemize,[status(sab)],[37]) ).
tff(39,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| ( X = ordered_pair(f2(X),f3(X)) ) ),
inference(modus_ponens,[status(thm)],[38,34]) ).
tff(40,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(41,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(42,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)],[41,40]) ).
tff(43,plain,
f8(a,a_function) = ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),
inference(unit_resolution,[status(thm)],[42,39,32]) ).
tff(44,plain,
ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))) = f8(a,a_function),
inference(symmetry,[status(thm)],[43]) ).
tff(45,plain,
first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = first(f8(a,a_function)),
inference(monotonicity,[status(thm)],[44]) ).
tff(46,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(47,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)],[46]) ).
tff(48,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(49,axiom,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',ordered_pair_predicate2) ).
tff(50,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ),
inference(modus_ponens,[status(thm)],[49,48]) ).
tff(51,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ),
inference(skolemize,[status(sab)],[50]) ).
tff(52,plain,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f3(X)) ),
inference(modus_ponens,[status(thm)],[51,47]) ).
tff(53,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(54,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(55,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)],[54,53]) ).
tff(56,plain,
little_set(f3(f8(a,a_function))),
inference(unit_resolution,[status(thm)],[55,52,32]) ).
tff(57,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(58,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)],[57]) ).
tff(59,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(60,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(61,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)],[60]) ).
tff(62,axiom,
! [X: $i] :
( ~ ordered_pair_predicate(X)
| little_set(f2(X)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',ordered_pair_predicate1) ).
tff(63,plain,
! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(modus_ponens,[status(thm)],[62,61]) ).
tff(64,plain,
! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(modus_ponens,[status(thm)],[63,59]) ).
tff(65,plain,
! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(skolemize,[status(sab)],[64]) ).
tff(66,plain,
! [X: $i] :
( little_set(f2(X))
| ~ ordered_pair_predicate(X) ),
inference(modus_ponens,[status(thm)],[65,58]) ).
tff(67,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(68,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(69,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)],[68,67]) ).
tff(70,plain,
little_set(f2(f8(a,a_function))),
inference(unit_resolution,[status(thm)],[69,66,32]) ).
tff(71,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(72,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)],[71]) ).
tff(73,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(74,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(75,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)],[74]) ).
tff(76,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(77,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(modus_ponens,[status(thm)],[76,75]) ).
tff(78,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(modus_ponens,[status(thm)],[77,73]) ).
tff(79,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(skolemize,[status(sab)],[78]) ).
tff(80,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(modus_ponens,[status(thm)],[79,72]) ).
tff(81,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(82,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(83,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)],[82]) ).
tff(84,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)],[83,81]) ).
tff(85,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(86,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)],[85,84]) ).
tff(87,plain,
first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))) = f2(f8(a,a_function)),
inference(unit_resolution,[status(thm)],[86,80,70,56]) ).
tff(88,plain,
f2(f8(a,a_function)) = first(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function)))),
inference(symmetry,[status(thm)],[87]) ).
tff(89,plain,
f2(f8(a,a_function)) = a,
inference(transitivity,[status(thm)],[88,45,18]) ).
tff(90,plain,
apply(a_function,f2(f8(a,a_function))) = apply(a_function,a),
inference(monotonicity,[status(thm)],[89]) ).
tff(91,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)],[44]) ).
tff(92,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)],[91]) ).
tff(93,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(94,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)],[93]) ).
tff(95,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(96,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(97,plain,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) ),
inference(modus_ponens,[status(thm)],[96,95]) ).
tff(98,plain,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) ),
inference(skolemize,[status(sab)],[97]) ).
tff(99,plain,
! [Z: $i,X: $i] :
( ~ member(Z,domain_of(X))
| member(f8(Z,X),X) ),
inference(modus_ponens,[status(thm)],[98,94]) ).
tff(100,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(101,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(102,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)],[101,100]) ).
tff(103,plain,
member(f8(a,a_function),a_function),
inference(unit_resolution,[status(thm)],[102,99,6]) ).
tff(104,plain,
member(ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))),a_function),
inference(modus_ponens,[status(thm)],[103,92]) ).
tff(105,plain,
( function(a_function)
<=> function(a_function) ),
inference(rewrite,[status(thm)],[]) ).
tff(106,axiom,
function(a_function),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_function) ).
tff(107,plain,
function(a_function),
inference(modus_ponens,[status(thm)],[106,105]) ).
tff(108,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(109,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)],[108]) ).
tff(110,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(111,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(112,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)],[111]) ).
tff(113,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(114,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)],[113,112]) ).
tff(115,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)],[114,110]) ).
tff(116,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)],[115]) ).
tff(117,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)],[116,109]) ).
tff(118,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(119,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(120,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)],[119]) ).
tff(121,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)],[120,118]) ).
tff(122,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(123,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)],[122,121]) ).
tff(124,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)],[123,117,107,70,56]) ).
tff(125,plain,
apply(a_function,f2(f8(a,a_function))) = f3(f8(a,a_function)),
inference(unit_resolution,[status(thm)],[124,104]) ).
tff(126,plain,
f3(f8(a,a_function)) = apply(a_function,f2(f8(a,a_function))),
inference(symmetry,[status(thm)],[125]) ).
tff(127,plain,
f3(f8(a,a_function)) = b,
inference(transitivity,[status(thm)],[126,90,3]) ).
tff(128,plain,
ordered_pair(f2(f8(a,a_function)),f3(f8(a,a_function))) = ordered_pair(a,b),
inference(monotonicity,[status(thm)],[89,127]) ).
tff(129,plain,
f8(a,a_function) = ordered_pair(a,b),
inference(transitivity,[status(thm)],[43,128]) ).
tff(130,plain,
( member(f8(a,a_function),a_function)
<=> member(ordered_pair(a,b),a_function) ),
inference(monotonicity,[status(thm)],[129]) ).
tff(131,plain,
( member(ordered_pair(a,b),a_function)
<=> member(f8(a,a_function),a_function) ),
inference(symmetry,[status(thm)],[130]) ).
tff(132,plain,
( ~ member(ordered_pair(a,b),a_function)
<=> ~ member(f8(a,a_function),a_function) ),
inference(monotonicity,[status(thm)],[131]) ).
tff(133,plain,
( ~ member(ordered_pair(a,b),a_function)
<=> ~ member(ordered_pair(a,b),a_function) ),
inference(rewrite,[status(thm)],[]) ).
tff(134,axiom,
~ member(ordered_pair(a,b),a_function),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_ordered_pair_in_function) ).
tff(135,plain,
~ member(ordered_pair(a,b),a_function),
inference(modus_ponens,[status(thm)],[134,133]) ).
tff(136,plain,
~ member(f8(a,a_function),a_function),
inference(modus_ponens,[status(thm)],[135,132]) ).
tff(137,plain,
$false,
inference(unit_resolution,[status(thm)],[103,136]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.11 % Problem : SET037-3 : TPTP v9.0.0. Released v1.0.0.
% 0.11/0.11 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n007.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Fri Jun 20 04:08:50 EDT 2025
% 0.11/0.32 % CPUTime :
% 0.18/0.42 % SZS status Unsatisfiable
% 0.18/0.42 % SZS output start Proof
% See solution above
% 0.18/0.44 % E exiting
%------------------------------------------------------------------------------