%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET031-4 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:24:55 AM UTC 2025
% Result : Unsatisfiable 0.23s 0.43s
% Output : Proof 0.23s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 37
% Syntax : Number of formulae : 88 ( 19 unt; 0 typ; 0 def)
% Number of atoms : 373 ( 32 equ)
% Maximal formula atoms : 12 ( 4 avg)
% Number of connectives : 490 ( 219 ~; 229 |; 0 &)
% ( 42 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 5 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of FOOLs : 14 ( 14 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 : 7 ( 7 usr; 2 con; 0-3 aty)
% Number of variables : 222 ( 203 !; 0 ?; 222 :)
% Comments :
%------------------------------------------------------------------------------
tff(ordered_pair_predicate_type,type,
ordered_pair_predicate: $i > $o ).
tff(ordered_pair_type,type,
ordered_pair: ( $i * $i ) > $i ).
tff(f30_type,type,
f30: ( $i * $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(a_type,type,
a: $i ).
tff(f18_type,type,
f18: $i > $i ).
tff(compose_type,type,
compose: ( $i * $i ) > $i ).
tff(f29_type,type,
f29: ( $i * $i * $i ) > $i ).
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(relation_type,type,
relation: $i > $o ).
tff(little_set_type,type,
little_set: $i > $o ).
tff(1,plain,
( ~ relation(compose(a,b))
<=> ~ relation(compose(a,b)) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
~ relation(compose(a,b)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_composition_is_a_relation) ).
tff(3,plain,
~ relation(compose(a,b)),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [Z: $i] :
refl(( ( relation(Z)
| member(f18(Z),Z) )
<=> ( relation(Z)
| member(f18(Z),Z) ) )),
inference(bind,[status(th)],[]) ).
tff(5,plain,
( ! [Z: $i] :
( relation(Z)
| member(f18(Z),Z) )
<=> ! [Z: $i] :
( relation(Z)
| member(f18(Z),Z) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [Z: $i] :
( relation(Z)
| member(f18(Z),Z) )
<=> ! [Z: $i] :
( relation(Z)
| member(f18(Z),Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(7,axiom,
! [Z: $i] :
( relation(Z)
| member(f18(Z),Z) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',relation2) ).
tff(8,plain,
! [Z: $i] :
( relation(Z)
| member(f18(Z),Z) ),
inference(modus_ponens,[status(thm)],[7,6]) ).
tff(9,plain,
! [Z: $i] :
( relation(Z)
| member(f18(Z),Z) ),
inference(skolemize,[status(sab)],[8]) ).
tff(10,plain,
! [Z: $i] :
( relation(Z)
| member(f18(Z),Z) ),
inference(modus_ponens,[status(thm)],[9,5]) ).
tff(11,plain,
( ( ~ ! [Z: $i] :
( relation(Z)
| member(f18(Z),Z) )
| relation(compose(a,b))
| member(f18(compose(a,b)),compose(a,b)) )
<=> ( ~ ! [Z: $i] :
( relation(Z)
| member(f18(Z),Z) )
| relation(compose(a,b))
| member(f18(compose(a,b)),compose(a,b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(12,plain,
( ~ ! [Z: $i] :
( relation(Z)
| member(f18(Z),Z) )
| relation(compose(a,b))
| member(f18(compose(a,b)),compose(a,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(13,plain,
( ~ ! [Z: $i] :
( relation(Z)
| member(f18(Z),Z) )
| relation(compose(a,b))
| member(f18(compose(a,b)),compose(a,b)) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
member(f18(compose(a,b)),compose(a,b)),
inference(unit_resolution,[status(thm)],[13,10,3]) ).
tff(15,plain,
^ [Xg: $i,Z: $i,Xf: $i] :
refl(( ( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) )
<=> ( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(16,plain,
( ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) )
<=> ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) ) ),
inference(quant_intro,[status(thm)],[15]) ).
tff(17,plain,
( ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) )
<=> ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(18,axiom,
! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',compose4) ).
tff(19,plain,
! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) ),
inference(skolemize,[status(sab)],[19]) ).
tff(21,plain,
! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) ),
inference(modus_ponens,[status(thm)],[20,16]) ).
tff(22,plain,
( ( ~ ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) )
| ~ member(f18(compose(a,b)),compose(a,b))
| ( f18(compose(a,b)) = ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b)) ) )
<=> ( ~ ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) )
| ~ member(f18(compose(a,b)),compose(a,b))
| ( f18(compose(a,b)) = ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(23,plain,
( ~ ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) )
| ~ member(f18(compose(a,b)),compose(a,b))
| ( f18(compose(a,b)) = ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(24,plain,
( ~ ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| ( Z = ordered_pair(f29(Z,Xf,Xg),f30(Z,Xf,Xg)) ) )
| ~ member(f18(compose(a,b)),compose(a,b))
| ( f18(compose(a,b)) = ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b)) ) ),
inference(modus_ponens,[status(thm)],[23,22]) ).
tff(25,plain,
f18(compose(a,b)) = ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b)),
inference(unit_resolution,[status(thm)],[24,21,14]) ).
tff(26,plain,
ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b)) = f18(compose(a,b)),
inference(symmetry,[status(thm)],[25]) ).
tff(27,plain,
( ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b)))
<=> ordered_pair_predicate(f18(compose(a,b))) ),
inference(monotonicity,[status(thm)],[26]) ).
tff(28,plain,
( ordered_pair_predicate(f18(compose(a,b)))
<=> ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))) ),
inference(symmetry,[status(thm)],[27]) ).
tff(29,plain,
( ~ ordered_pair_predicate(f18(compose(a,b)))
<=> ~ ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))) ),
inference(monotonicity,[status(thm)],[28]) ).
tff(30,plain,
^ [Z: $i] :
refl(( ( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) )
<=> ( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) ) )),
inference(bind,[status(th)],[]) ).
tff(31,plain,
( ! [Z: $i] :
( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) )
<=> ! [Z: $i] :
( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) ) ),
inference(quant_intro,[status(thm)],[30]) ).
tff(32,plain,
( ! [Z: $i] :
( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) )
<=> ! [Z: $i] :
( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(33,axiom,
! [Z: $i] :
( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',relation3) ).
tff(34,plain,
! [Z: $i] :
( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) ),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
! [Z: $i] :
( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) ),
inference(skolemize,[status(sab)],[34]) ).
tff(36,plain,
! [Z: $i] :
( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) ),
inference(modus_ponens,[status(thm)],[35,31]) ).
tff(37,plain,
( ( ~ ! [Z: $i] :
( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) )
| relation(compose(a,b))
| ~ ordered_pair_predicate(f18(compose(a,b))) )
<=> ( ~ ! [Z: $i] :
( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) )
| relation(compose(a,b))
| ~ ordered_pair_predicate(f18(compose(a,b))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(38,plain,
( ~ ! [Z: $i] :
( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) )
| relation(compose(a,b))
| ~ ordered_pair_predicate(f18(compose(a,b))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(39,plain,
( ~ ! [Z: $i] :
( relation(Z)
| ~ ordered_pair_predicate(f18(Z)) )
| relation(compose(a,b))
| ~ ordered_pair_predicate(f18(compose(a,b))) ),
inference(modus_ponens,[status(thm)],[38,37]) ).
tff(40,plain,
~ ordered_pair_predicate(f18(compose(a,b))),
inference(unit_resolution,[status(thm)],[39,36,3]) ).
tff(41,plain,
~ ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))),
inference(modus_ponens,[status(thm)],[40,29]) ).
tff(42,plain,
^ [Xg: $i,Z: $i,Xf: $i] :
refl(( ( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) )
<=> ( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) ) )),
inference(bind,[status(th)],[]) ).
tff(43,plain,
( ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) )
<=> ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) ) ),
inference(quant_intro,[status(thm)],[42]) ).
tff(44,plain,
( ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) )
<=> ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(45,axiom,
! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',compose2) ).
tff(46,plain,
! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) ),
inference(modus_ponens,[status(thm)],[45,44]) ).
tff(47,plain,
! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) ),
inference(skolemize,[status(sab)],[46]) ).
tff(48,plain,
! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) ),
inference(modus_ponens,[status(thm)],[47,43]) ).
tff(49,plain,
( ( ~ ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) )
| ~ member(f18(compose(a,b)),compose(a,b))
| little_set(f30(f18(compose(a,b)),a,b)) )
<=> ( ~ ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) )
| ~ member(f18(compose(a,b)),compose(a,b))
| little_set(f30(f18(compose(a,b)),a,b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(50,plain,
( ~ ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) )
| ~ member(f18(compose(a,b)),compose(a,b))
| little_set(f30(f18(compose(a,b)),a,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(51,plain,
( ~ ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f30(Z,Xf,Xg)) )
| ~ member(f18(compose(a,b)),compose(a,b))
| little_set(f30(f18(compose(a,b)),a,b)) ),
inference(modus_ponens,[status(thm)],[50,49]) ).
tff(52,plain,
little_set(f30(f18(compose(a,b)),a,b)),
inference(unit_resolution,[status(thm)],[51,48,14]) ).
tff(53,plain,
^ [Xg: $i,Z: $i,Xf: $i] :
refl(( ( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) )
<=> ( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) ) )),
inference(bind,[status(th)],[]) ).
tff(54,plain,
( ! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) )
<=> ! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) ) ),
inference(quant_intro,[status(thm)],[53]) ).
tff(55,plain,
( ! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) )
<=> ! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(56,plain,
^ [Xg: $i,Z: $i,Xf: $i] :
rewrite(( ( ~ member(Z,compose(Xf,Xg))
| little_set(f29(Z,Xf,Xg)) )
<=> ( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) ) )),
inference(bind,[status(th)],[]) ).
tff(57,plain,
( ! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f29(Z,Xf,Xg)) )
<=> ! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) ) ),
inference(quant_intro,[status(thm)],[56]) ).
tff(58,axiom,
! [Xg: $i,Z: $i,Xf: $i] :
( ~ member(Z,compose(Xf,Xg))
| little_set(f29(Z,Xf,Xg)) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',compose1) ).
tff(59,plain,
! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) ),
inference(modus_ponens,[status(thm)],[58,57]) ).
tff(60,plain,
! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) ),
inference(modus_ponens,[status(thm)],[59,55]) ).
tff(61,plain,
! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) ),
inference(skolemize,[status(sab)],[60]) ).
tff(62,plain,
! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) ),
inference(modus_ponens,[status(thm)],[61,54]) ).
tff(63,plain,
( ( ~ ! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) )
| little_set(f29(f18(compose(a,b)),a,b))
| ~ member(f18(compose(a,b)),compose(a,b)) )
<=> ( ~ ! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) )
| little_set(f29(f18(compose(a,b)),a,b))
| ~ member(f18(compose(a,b)),compose(a,b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(64,plain,
( ~ ! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) )
| little_set(f29(f18(compose(a,b)),a,b))
| ~ member(f18(compose(a,b)),compose(a,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(65,plain,
( ~ ! [Xg: $i,Z: $i,Xf: $i] :
( little_set(f29(Z,Xf,Xg))
| ~ member(Z,compose(Xf,Xg)) )
| little_set(f29(f18(compose(a,b)),a,b))
| ~ member(f18(compose(a,b)),compose(a,b)) ),
inference(modus_ponens,[status(thm)],[64,63]) ).
tff(66,plain,
little_set(f29(f18(compose(a,b)),a,b)),
inference(unit_resolution,[status(thm)],[65,62,14]) ).
tff(67,plain,
^ [Z: $i,Y: $i] :
refl(( ( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
<=> ( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ) )),
inference(bind,[status(th)],[]) ).
tff(68,plain,
( ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
<=> ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ) ),
inference(quant_intro,[status(thm)],[67]) ).
tff(69,plain,
( ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
<=> ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(70,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
<=> ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ) ),
inference(elim_unused_vars,[status(thm)],[]) ).
tff(71,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ) ),
inference(destructive_equality_resolution,[status(thm)],[]) ).
tff(72,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) )
<=> ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ) ),
inference(transitivity,[status(thm)],[71,70]) ).
tff(73,plain,
^ [Z: $i,Y: $i,X: $i] :
trans(monotonicity(rewrite(( ( ordered_pair_predicate(X)
| ~ little_set(Y)
| ~ little_set(Z) )
<=> ( ~ little_set(Y)
| ordered_pair_predicate(X)
| ~ little_set(Z) ) )),
( ( ordered_pair_predicate(X)
| ~ little_set(Y)
| ~ little_set(Z)
| ( X != ordered_pair(Y,Z) ) )
<=> ( ~ little_set(Y)
| ordered_pair_predicate(X)
| ~ little_set(Z)
| ( X != ordered_pair(Y,Z) ) ) )),
rewrite(( ( ~ little_set(Y)
| ordered_pair_predicate(X)
| ~ little_set(Z)
| ( X != ordered_pair(Y,Z) ) )
<=> ( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) ) )),
( ( ordered_pair_predicate(X)
| ~ little_set(Y)
| ~ little_set(Z)
| ( X != ordered_pair(Y,Z) ) )
<=> ( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) ) )),
inference(bind,[status(th)],[]) ).
tff(74,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ordered_pair_predicate(X)
| ~ little_set(Y)
| ~ little_set(Z)
| ( X != ordered_pair(Y,Z) ) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) ) ),
inference(quant_intro,[status(thm)],[73]) ).
tff(75,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ordered_pair_predicate(X)
| ~ little_set(Y)
| ~ little_set(Z)
| ( X != ordered_pair(Y,Z) ) )
<=> ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ) ),
inference(transitivity,[status(thm)],[74,72]) ).
tff(76,axiom,
! [Z: $i,Y: $i,X: $i] :
( ordered_pair_predicate(X)
| ~ little_set(Y)
| ~ little_set(Z)
| ( X != ordered_pair(Y,Z) ) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET003-0.ax',ordered_pair_predicate4) ).
tff(77,plain,
! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ),
inference(modus_ponens,[status(thm)],[76,75]) ).
tff(78,plain,
! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ),
inference(modus_ponens,[status(thm)],[77,69]) ).
tff(79,plain,
! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ),
inference(skolemize,[status(sab)],[78]) ).
tff(80,plain,
! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ),
inference(modus_ponens,[status(thm)],[79,68]) ).
tff(81,plain,
( ( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ~ little_set(f29(f18(compose(a,b)),a,b))
| ~ little_set(f30(f18(compose(a,b)),a,b))
| ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))) )
<=> ( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ~ little_set(f29(f18(compose(a,b)),a,b))
| ~ little_set(f30(f18(compose(a,b)),a,b))
| ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(82,plain,
( ( ~ little_set(f30(f18(compose(a,b)),a,b))
| ~ little_set(f29(f18(compose(a,b)),a,b))
| ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))) )
<=> ( ~ little_set(f29(f18(compose(a,b)),a,b))
| ~ little_set(f30(f18(compose(a,b)),a,b))
| ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(83,plain,
( ( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ~ little_set(f30(f18(compose(a,b)),a,b))
| ~ little_set(f29(f18(compose(a,b)),a,b))
| ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))) )
<=> ( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ~ little_set(f29(f18(compose(a,b)),a,b))
| ~ little_set(f30(f18(compose(a,b)),a,b))
| ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))) ) ),
inference(monotonicity,[status(thm)],[82]) ).
tff(84,plain,
( ( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ~ little_set(f30(f18(compose(a,b)),a,b))
| ~ little_set(f29(f18(compose(a,b)),a,b))
| ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))) )
<=> ( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ~ little_set(f29(f18(compose(a,b)),a,b))
| ~ little_set(f30(f18(compose(a,b)),a,b))
| ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))) ) ),
inference(transitivity,[status(thm)],[83,81]) ).
tff(85,plain,
( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ~ little_set(f30(f18(compose(a,b)),a,b))
| ~ little_set(f29(f18(compose(a,b)),a,b))
| ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(86,plain,
( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ~ little_set(f29(f18(compose(a,b)),a,b))
| ~ little_set(f30(f18(compose(a,b)),a,b))
| ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))) ),
inference(modus_ponens,[status(thm)],[85,84]) ).
tff(87,plain,
ordered_pair_predicate(ordered_pair(f29(f18(compose(a,b)),a,b),f30(f18(compose(a,b)),a,b))),
inference(unit_resolution,[status(thm)],[86,80,66,52]) ).
tff(88,plain,
$false,
inference(unit_resolution,[status(thm)],[87,41]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.14 % Problem : SET031-4 : TPTP v9.0.0. Released v1.0.0.
% 0.08/0.14 % Command : run_E %s %d THM
% 0.14/0.35 % Computer : n011.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Jun 20 04:03:41 EDT 2025
% 0.14/0.35 % CPUTime :
% 0.23/0.43 % SZS status Unsatisfiable
% 0.23/0.43 % SZS output start Proof
% See solution above
% 0.23/0.46 % E exiting
%------------------------------------------------------------------------------