%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET042-3 : 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 05:24:57 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.42s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 46
% Syntax : Number of formulae : 109 ( 30 unt; 0 typ; 0 def)
% Number of atoms : 652 ( 60 equ)
% Maximal formula atoms : 20 ( 5 avg)
% Number of connectives : 958 ( 450 ~; 445 |; 0 &)
% ( 63 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 35 ( 35 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-3 aty)
% Number of functors : 8 ( 8 usr; 4 con; 0-2 aty)
% Number of variables : 260 ( 235 !; 0 ?; 260 :)
% Comments :
%------------------------------------------------------------------------------
tff(ordered_pair_predicate_type,type,
ordered_pair_predicate: $i > $o ).
tff(ordered_pair_type,type,
ordered_pair: ( $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(a_type,type,
a: $i ).
tff(little_set_type,type,
little_set: $i > $o ).
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(set_b_type,type,
set_b: $i ).
tff(set_a_type,type,
set_a: $i ).
tff(second_type,type,
second: $i > $i ).
tff(first_type,type,
first: $i > $i ).
tff(cross_product_type,type,
cross_product: ( $i * $i ) > $i ).
tff(1,plain,
( member(b,set_b)
<=> member(b,set_b) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
member(b,set_b),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_set_b) ).
tff(3,plain,
member(b,set_b),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [Y: $i,X: $i] :
refl(( ( little_set(X)
| ~ member(X,Y) )
<=> ( little_set(X)
| ~ member(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(5,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)],[4]) ).
tff(6,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(7,plain,
^ [Y: $i,X: $i] :
rewrite(( ( ~ member(X,Y)
| little_set(X) )
<=> ( little_set(X)
| ~ member(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(8,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)],[7]) ).
tff(9,axiom,
! [Y: $i,X: $i] :
( ~ member(X,Y)
| little_set(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',a2) ).
tff(10,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(modus_ponens,[status(thm)],[9,8]) ).
tff(11,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(modus_ponens,[status(thm)],[10,6]) ).
tff(12,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(skolemize,[status(sab)],[11]) ).
tff(13,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(modus_ponens,[status(thm)],[12,5]) ).
tff(14,plain,
( ( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(b)
| ~ member(b,set_b) )
<=> ( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(b)
| ~ member(b,set_b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(15,plain,
( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(b)
| ~ member(b,set_b) ),
inference(quant_inst,[status(thm)],[]) ).
tff(16,plain,
( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(b)
| ~ member(b,set_b) ),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
little_set(b),
inference(unit_resolution,[status(thm)],[16,13,3]) ).
tff(18,plain,
( member(a,set_a)
<=> member(a,set_a) ),
inference(rewrite,[status(thm)],[]) ).
tff(19,axiom,
member(a,set_a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_set_a) ).
tff(20,plain,
member(a,set_a),
inference(modus_ponens,[status(thm)],[19,18]) ).
tff(21,plain,
( ( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(a)
| ~ member(a,set_a) )
<=> ( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(a)
| ~ member(a,set_a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(22,plain,
( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(a)
| ~ member(a,set_a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(23,plain,
( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(a)
| ~ member(a,set_a) ),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
little_set(a),
inference(unit_resolution,[status(thm)],[23,13,20]) ).
tff(25,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(26,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)],[25]) ).
tff(27,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(28,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(29,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(30,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)],[29,28]) ).
tff(31,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(32,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)],[31]) ).
tff(33,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)],[32,30]) ).
tff(34,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/sandbox2/benchmark/Axioms/SET003-0.ax',ordered_pair_predicate4) ).
tff(35,plain,
! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ),
inference(modus_ponens,[status(thm)],[34,33]) ).
tff(36,plain,
! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ),
inference(modus_ponens,[status(thm)],[35,27]) ).
tff(37,plain,
! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ),
inference(skolemize,[status(sab)],[36]) ).
tff(38,plain,
! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) ),
inference(modus_ponens,[status(thm)],[37,26]) ).
tff(39,plain,
( ( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ordered_pair_predicate(ordered_pair(a,b))
| ~ little_set(b)
| ~ little_set(a) )
<=> ( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ordered_pair_predicate(ordered_pair(a,b))
| ~ little_set(b)
| ~ little_set(a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(40,plain,
( ( ~ little_set(b)
| ~ little_set(a)
| ordered_pair_predicate(ordered_pair(a,b)) )
<=> ( ordered_pair_predicate(ordered_pair(a,b))
| ~ little_set(b)
| ~ little_set(a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,plain,
( ( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ~ little_set(b)
| ~ little_set(a)
| ordered_pair_predicate(ordered_pair(a,b)) )
<=> ( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ordered_pair_predicate(ordered_pair(a,b))
| ~ little_set(b)
| ~ little_set(a) ) ),
inference(monotonicity,[status(thm)],[40]) ).
tff(42,plain,
( ( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ~ little_set(b)
| ~ little_set(a)
| ordered_pair_predicate(ordered_pair(a,b)) )
<=> ( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ordered_pair_predicate(ordered_pair(a,b))
| ~ little_set(b)
| ~ little_set(a) ) ),
inference(transitivity,[status(thm)],[41,39]) ).
tff(43,plain,
( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ~ little_set(b)
| ~ little_set(a)
| ordered_pair_predicate(ordered_pair(a,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(44,plain,
( ~ ! [Z: $i,Y: $i] :
( ~ little_set(Z)
| ~ little_set(Y)
| ordered_pair_predicate(ordered_pair(Y,Z)) )
| ordered_pair_predicate(ordered_pair(a,b))
| ~ little_set(b)
| ~ little_set(a) ),
inference(modus_ponens,[status(thm)],[43,42]) ).
tff(45,plain,
ordered_pair_predicate(ordered_pair(a,b)),
inference(unit_resolution,[status(thm)],[44,38,24,17]) ).
tff(46,plain,
^ [Y: $i,X: $i] :
refl(( little_set(ordered_pair(X,Y))
<=> little_set(ordered_pair(X,Y)) )),
inference(bind,[status(th)],[]) ).
tff(47,plain,
( ! [Y: $i,X: $i] : little_set(ordered_pair(X,Y))
<=> ! [Y: $i,X: $i] : little_set(ordered_pair(X,Y)) ),
inference(quant_intro,[status(thm)],[46]) ).
tff(48,plain,
( ! [Y: $i,X: $i] : little_set(ordered_pair(X,Y))
<=> ! [Y: $i,X: $i] : little_set(ordered_pair(X,Y)) ),
inference(rewrite,[status(thm)],[]) ).
tff(49,axiom,
! [Y: $i,X: $i] : little_set(ordered_pair(X,Y)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pairs_are_small1) ).
tff(50,plain,
! [Y: $i,X: $i] : little_set(ordered_pair(X,Y)),
inference(modus_ponens,[status(thm)],[49,48]) ).
tff(51,plain,
! [Y: $i,X: $i] : little_set(ordered_pair(X,Y)),
inference(skolemize,[status(sab)],[50]) ).
tff(52,plain,
! [Y: $i,X: $i] : little_set(ordered_pair(X,Y)),
inference(modus_ponens,[status(thm)],[51,47]) ).
tff(53,plain,
( ~ ! [Y: $i,X: $i] : little_set(ordered_pair(X,Y))
| little_set(ordered_pair(a,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(54,plain,
little_set(ordered_pair(a,b)),
inference(unit_resolution,[status(thm)],[53,52]) ).
tff(55,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(56,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)],[55]) ).
tff(57,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(58,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(59,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)],[58]) ).
tff(60,axiom,
! [Y: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ( second(ordered_pair(X,Y)) = Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_second) ).
tff(61,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[60,59]) ).
tff(62,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[61,57]) ).
tff(63,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(skolemize,[status(sab)],[62]) ).
tff(64,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[63,56]) ).
tff(65,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(b)
| ~ little_set(a)
| ( second(ordered_pair(a,b)) = b ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(b)
| ~ little_set(a)
| ( second(ordered_pair(a,b)) = b ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(66,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(b)
| ~ little_set(a)
| ( second(ordered_pair(a,b)) = b ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(67,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) )
| ~ little_set(b)
| ~ little_set(a)
| ( second(ordered_pair(a,b)) = b ) ),
inference(modus_ponens,[status(thm)],[66,65]) ).
tff(68,plain,
second(ordered_pair(a,b)) = b,
inference(unit_resolution,[status(thm)],[67,64,24,17]) ).
tff(69,plain,
( member(second(ordered_pair(a,b)),set_b)
<=> member(b,set_b) ),
inference(monotonicity,[status(thm)],[68]) ).
tff(70,plain,
( member(b,set_b)
<=> member(second(ordered_pair(a,b)),set_b) ),
inference(symmetry,[status(thm)],[69]) ).
tff(71,plain,
member(second(ordered_pair(a,b)),set_b),
inference(modus_ponens,[status(thm)],[3,70]) ).
tff(72,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(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(quant_intro,[status(thm)],[72]) ).
tff(74,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(75,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(76,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)],[75]) ).
tff(77,axiom,
! [Y: $i,X: $i] :
( ~ little_set(X)
| ~ little_set(Y)
| ( first(ordered_pair(X,Y)) = X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',property_of_first) ).
tff(78,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(modus_ponens,[status(thm)],[77,76]) ).
tff(79,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(modus_ponens,[status(thm)],[78,74]) ).
tff(80,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(skolemize,[status(sab)],[79]) ).
tff(81,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(modus_ponens,[status(thm)],[80,73]) ).
tff(82,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
| ~ little_set(b)
| ~ little_set(a)
| ( first(ordered_pair(a,b)) = a ) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
| ~ little_set(b)
| ~ little_set(a)
| ( first(ordered_pair(a,b)) = a ) ) ),
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(b)
| ~ little_set(a)
| ( first(ordered_pair(a,b)) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(84,plain,
( ~ ! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) )
| ~ little_set(b)
| ~ little_set(a)
| ( first(ordered_pair(a,b)) = a ) ),
inference(modus_ponens,[status(thm)],[83,82]) ).
tff(85,plain,
first(ordered_pair(a,b)) = a,
inference(unit_resolution,[status(thm)],[84,81,24,17]) ).
tff(86,plain,
( member(first(ordered_pair(a,b)),set_a)
<=> member(a,set_a) ),
inference(monotonicity,[status(thm)],[85]) ).
tff(87,plain,
( member(a,set_a)
<=> member(first(ordered_pair(a,b)),set_a) ),
inference(symmetry,[status(thm)],[86]) ).
tff(88,plain,
member(first(ordered_pair(a,b)),set_a),
inference(modus_ponens,[status(thm)],[20,87]) ).
tff(89,plain,
( ~ member(ordered_pair(a,b),cross_product(set_a,set_b))
<=> ~ member(ordered_pair(a,b),cross_product(set_a,set_b)) ),
inference(rewrite,[status(thm)],[]) ).
tff(90,axiom,
~ member(ordered_pair(a,b),cross_product(set_a,set_b)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_ordered_pair_is_in_cross_product) ).
tff(91,plain,
~ member(ordered_pair(a,b),cross_product(set_a,set_b)),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) )
<=> ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) ) )),
inference(bind,[status(th)],[]) ).
tff(93,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) ) ),
inference(quant_intro,[status(thm)],[92]) ).
tff(94,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(95,plain,
^ [Z: $i,Y: $i,X: $i] :
trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite(( ( member(Z,cross_product(X,Y))
| ~ little_set(Z) )
<=> ( ~ little_set(Z)
| member(Z,cross_product(X,Y)) ) )),
( ( member(Z,cross_product(X,Y))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Z) )
<=> ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ ordered_pair_predicate(Z) ) )),
rewrite(( ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ ordered_pair_predicate(Z) )
<=> ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ ordered_pair_predicate(Z) ) )),
( ( member(Z,cross_product(X,Y))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Z) )
<=> ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ ordered_pair_predicate(Z) ) )),
( ( member(Z,cross_product(X,Y))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Z)
| ~ member(first(Z),X) )
<=> ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ ordered_pair_predicate(Z)
| ~ member(first(Z),X) ) )),
rewrite(( ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ ordered_pair_predicate(Z)
| ~ member(first(Z),X) )
<=> ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) ) )),
( ( member(Z,cross_product(X,Y))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Z)
| ~ member(first(Z),X) )
<=> ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) ) )),
( ( member(Z,cross_product(X,Y))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Z)
| ~ member(first(Z),X)
| ~ member(second(Z),Y) )
<=> ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z)
| ~ member(second(Z),Y) ) )),
rewrite(( ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z)
| ~ member(second(Z),Y) )
<=> ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) ) )),
( ( member(Z,cross_product(X,Y))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Z)
| ~ member(first(Z),X)
| ~ member(second(Z),Y) )
<=> ( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) ) )),
inference(bind,[status(th)],[]) ).
tff(96,plain,
( ! [Z: $i,Y: $i,X: $i] :
( member(Z,cross_product(X,Y))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Z)
| ~ member(first(Z),X)
| ~ member(second(Z),Y) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) ) ),
inference(quant_intro,[status(thm)],[95]) ).
tff(97,axiom,
! [Z: $i,Y: $i,X: $i] :
( member(Z,cross_product(X,Y))
| ~ little_set(Z)
| ~ ordered_pair_predicate(Z)
| ~ member(first(Z),X)
| ~ member(second(Z),Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',cross_product4) ).
tff(98,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) ),
inference(modus_ponens,[status(thm)],[97,96]) ).
tff(99,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) ),
inference(modus_ponens,[status(thm)],[98,94]) ).
tff(100,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) ),
inference(skolemize,[status(sab)],[99]) ).
tff(101,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) ),
inference(modus_ponens,[status(thm)],[100,93]) ).
tff(102,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) )
| member(ordered_pair(a,b),cross_product(set_a,set_b))
| ~ little_set(ordered_pair(a,b))
| ~ member(second(ordered_pair(a,b)),set_b)
| ~ member(first(ordered_pair(a,b)),set_a)
| ~ ordered_pair_predicate(ordered_pair(a,b)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) )
| member(ordered_pair(a,b),cross_product(set_a,set_b))
| ~ little_set(ordered_pair(a,b))
| ~ member(second(ordered_pair(a,b)),set_b)
| ~ member(first(ordered_pair(a,b)),set_a)
| ~ ordered_pair_predicate(ordered_pair(a,b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(103,plain,
( ( ~ little_set(ordered_pair(a,b))
| member(ordered_pair(a,b),cross_product(set_a,set_b))
| ~ member(second(ordered_pair(a,b)),set_b)
| ~ member(first(ordered_pair(a,b)),set_a)
| ~ ordered_pair_predicate(ordered_pair(a,b)) )
<=> ( member(ordered_pair(a,b),cross_product(set_a,set_b))
| ~ little_set(ordered_pair(a,b))
| ~ member(second(ordered_pair(a,b)),set_b)
| ~ member(first(ordered_pair(a,b)),set_a)
| ~ ordered_pair_predicate(ordered_pair(a,b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(104,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) )
| ~ little_set(ordered_pair(a,b))
| member(ordered_pair(a,b),cross_product(set_a,set_b))
| ~ member(second(ordered_pair(a,b)),set_b)
| ~ member(first(ordered_pair(a,b)),set_a)
| ~ ordered_pair_predicate(ordered_pair(a,b)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) )
| member(ordered_pair(a,b),cross_product(set_a,set_b))
| ~ little_set(ordered_pair(a,b))
| ~ member(second(ordered_pair(a,b)),set_b)
| ~ member(first(ordered_pair(a,b)),set_a)
| ~ ordered_pair_predicate(ordered_pair(a,b)) ) ),
inference(monotonicity,[status(thm)],[103]) ).
tff(105,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) )
| ~ little_set(ordered_pair(a,b))
| member(ordered_pair(a,b),cross_product(set_a,set_b))
| ~ member(second(ordered_pair(a,b)),set_b)
| ~ member(first(ordered_pair(a,b)),set_a)
| ~ ordered_pair_predicate(ordered_pair(a,b)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) )
| member(ordered_pair(a,b),cross_product(set_a,set_b))
| ~ little_set(ordered_pair(a,b))
| ~ member(second(ordered_pair(a,b)),set_b)
| ~ member(first(ordered_pair(a,b)),set_a)
| ~ ordered_pair_predicate(ordered_pair(a,b)) ) ),
inference(transitivity,[status(thm)],[104,102]) ).
tff(106,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) )
| ~ little_set(ordered_pair(a,b))
| member(ordered_pair(a,b),cross_product(set_a,set_b))
| ~ member(second(ordered_pair(a,b)),set_b)
| ~ member(first(ordered_pair(a,b)),set_a)
| ~ ordered_pair_predicate(ordered_pair(a,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(107,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Z)
| member(Z,cross_product(X,Y))
| ~ member(second(Z),Y)
| ~ member(first(Z),X)
| ~ ordered_pair_predicate(Z) )
| member(ordered_pair(a,b),cross_product(set_a,set_b))
| ~ little_set(ordered_pair(a,b))
| ~ member(second(ordered_pair(a,b)),set_b)
| ~ member(first(ordered_pair(a,b)),set_a)
| ~ ordered_pair_predicate(ordered_pair(a,b)) ),
inference(modus_ponens,[status(thm)],[106,105]) ).
tff(108,plain,
( ~ little_set(ordered_pair(a,b))
| ~ member(second(ordered_pair(a,b)),set_b)
| ~ member(first(ordered_pair(a,b)),set_a)
| ~ ordered_pair_predicate(ordered_pair(a,b)) ),
inference(unit_resolution,[status(thm)],[107,101,91]) ).
tff(109,plain,
$false,
inference(unit_resolution,[status(thm)],[108,88,71,54,45]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SET042-3 : TPTP v9.0.0. Released v1.0.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n003.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Fri Jun 20 04:06:54 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.19/0.42 % SZS status Unsatisfiable
% 0.19/0.42 % SZS output start Proof
% See solution above
% 0.19/0.44 % E exiting
%------------------------------------------------------------------------------