TSTP Solution File: SET042-3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET042-3 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n005.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 : Tue Sep 20 05:05:06 EDT 2022
% Result : Unsatisfiable 0.20s 0.45s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 61
% Syntax : Number of formulae : 132 ( 38 unt; 13 typ; 0 def)
% Number of atoms : 675 ( 99 equ)
% Maximal formula atoms : 20 ( 5 avg)
% Number of connectives : 984 ( 464 ~; 455 |; 0 &)
% ( 65 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 36 ( 36 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 13 ( 9 >; 4 *; 0 +; 0 <<)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-3 aty)
% Number of functors : 10 ( 10 usr; 4 con; 0-2 aty)
% Number of variables : 276 ( 248 !; 0 ?; 276 :)
% Comments :
%------------------------------------------------------------------------------
tff(ordered_pair_type,type,
ordered_pair: ( $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(a_type,type,
a: $i ).
tff(non_ordered_pair_type,type,
non_ordered_pair: ( $i * $i ) > $i ).
tff(singleton_set_type,type,
singleton_set: $i > $i ).
tff(ordered_pair_predicate_type,type,
ordered_pair_predicate: $i > $o ).
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(set_b_type,type,
set_b: $i ).
tff(second_type,type,
second: $i > $i ).
tff(little_set_type,type,
little_set: $i > $o ).
tff(set_a_type,type,
set_a: $i ).
tff(first_type,type,
first: $i > $i ).
tff(cross_product_type,type,
cross_product: ( $i * $i ) > $i ).
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/sandbox2/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(a,b) = non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
ordered_pair(a,b) = non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) = ordered_pair(a,b),
inference(symmetry,[status(thm)],[9]) ).
tff(11,plain,
( ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)))
<=> ordered_pair_predicate(ordered_pair(a,b)) ),
inference(monotonicity,[status(thm)],[10]) ).
tff(12,plain,
( ordered_pair_predicate(ordered_pair(a,b))
<=> ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b))) ),
inference(symmetry,[status(thm)],[11]) ).
tff(13,plain,
( ~ ordered_pair_predicate(ordered_pair(a,b))
<=> ~ ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b))) ),
inference(monotonicity,[status(thm)],[12]) ).
tff(14,plain,
( member(b,set_b)
<=> member(b,set_b) ),
inference(rewrite,[status(thm)],]) ).
tff(15,axiom,
member(b,set_b),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_set_b) ).
tff(16,plain,
member(b,set_b),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
^ [Y: $i,X: $i] :
refl(
( ( little_set(X)
| ~ member(X,Y) )
<=> ( little_set(X)
| ~ member(X,Y) ) )),
inference(bind,[status(th)],]) ).
tff(18,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)],[17]) ).
tff(19,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(20,plain,
^ [Y: $i,X: $i] :
rewrite(
( ( ~ member(X,Y)
| little_set(X) )
<=> ( little_set(X)
| ~ member(X,Y) ) )),
inference(bind,[status(th)],]) ).
tff(21,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)],[20]) ).
tff(22,axiom,
! [Y: $i,X: $i] :
( ~ member(X,Y)
| little_set(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET003-0.ax',a2) ).
tff(23,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(modus_ponens,[status(thm)],[23,19]) ).
tff(25,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(skolemize,[status(sab)],[24]) ).
tff(26,plain,
! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) ),
inference(modus_ponens,[status(thm)],[25,18]) ).
tff(27,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(28,plain,
( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(b)
| ~ member(b,set_b) ),
inference(quant_inst,[status(thm)],]) ).
tff(29,plain,
( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(b)
| ~ member(b,set_b) ),
inference(modus_ponens,[status(thm)],[28,27]) ).
tff(30,plain,
little_set(b),
inference(unit_resolution,[status(thm)],[29,26,16]) ).
tff(31,plain,
( member(a,set_a)
<=> member(a,set_a) ),
inference(rewrite,[status(thm)],]) ).
tff(32,axiom,
member(a,set_a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',member_of_set_a) ).
tff(33,plain,
member(a,set_a),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,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(35,plain,
( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(a)
| ~ member(a,set_a) ),
inference(quant_inst,[status(thm)],]) ).
tff(36,plain,
( ~ ! [Y: $i,X: $i] :
( little_set(X)
| ~ member(X,Y) )
| little_set(a)
| ~ member(a,set_a) ),
inference(modus_ponens,[status(thm)],[35,34]) ).
tff(37,plain,
little_set(a),
inference(unit_resolution,[status(thm)],[36,26,33]) ).
tff(38,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(39,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)],[38]) ).
tff(40,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(41,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(42,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)],[41]) ).
tff(43,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(44,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[43,42]) ).
tff(45,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[44,40]) ).
tff(46,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(skolemize,[status(sab)],[45]) ).
tff(47,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( second(ordered_pair(X,Y)) = Y ) ),
inference(modus_ponens,[status(thm)],[46,39]) ).
tff(48,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(49,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(50,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)],[49,48]) ).
tff(51,plain,
second(ordered_pair(a,b)) = b,
inference(unit_resolution,[status(thm)],[50,47,37,30]) ).
tff(52,plain,
( member(second(ordered_pair(a,b)),set_b)
<=> member(b,set_b) ),
inference(monotonicity,[status(thm)],[51]) ).
tff(53,plain,
( member(b,set_b)
<=> member(second(ordered_pair(a,b)),set_b) ),
inference(symmetry,[status(thm)],[52]) ).
tff(54,plain,
member(second(ordered_pair(a,b)),set_b),
inference(modus_ponens,[status(thm)],[16,53]) ).
tff(55,plain,
^ [Y: $i,X: $i] :
refl(
( little_set(ordered_pair(X,Y))
<=> little_set(ordered_pair(X,Y)) )),
inference(bind,[status(th)],]) ).
tff(56,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)],[55]) ).
tff(57,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(58,axiom,
! [Y: $i,X: $i] : little_set(ordered_pair(X,Y)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ordered_pairs_are_small1) ).
tff(59,plain,
! [Y: $i,X: $i] : little_set(ordered_pair(X,Y)),
inference(modus_ponens,[status(thm)],[58,57]) ).
tff(60,plain,
! [Y: $i,X: $i] : little_set(ordered_pair(X,Y)),
inference(skolemize,[status(sab)],[59]) ).
tff(61,plain,
! [Y: $i,X: $i] : little_set(ordered_pair(X,Y)),
inference(modus_ponens,[status(thm)],[60,56]) ).
tff(62,plain,
( ~ ! [Y: $i,X: $i] : little_set(ordered_pair(X,Y))
| little_set(ordered_pair(a,b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(63,plain,
little_set(ordered_pair(a,b)),
inference(unit_resolution,[status(thm)],[62,61]) ).
tff(64,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(65,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)],[64]) ).
tff(66,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(67,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(68,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)],[67]) ).
tff(69,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(70,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(modus_ponens,[status(thm)],[69,68]) ).
tff(71,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(modus_ponens,[status(thm)],[70,66]) ).
tff(72,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(skolemize,[status(sab)],[71]) ).
tff(73,plain,
! [Y: $i,X: $i] :
( ~ little_set(Y)
| ~ little_set(X)
| ( first(ordered_pair(X,Y)) = X ) ),
inference(modus_ponens,[status(thm)],[72,65]) ).
tff(74,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(75,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(76,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)],[75,74]) ).
tff(77,plain,
first(ordered_pair(a,b)) = a,
inference(unit_resolution,[status(thm)],[76,73,37,30]) ).
tff(78,plain,
( member(first(ordered_pair(a,b)),set_a)
<=> member(a,set_a) ),
inference(monotonicity,[status(thm)],[77]) ).
tff(79,plain,
( member(a,set_a)
<=> member(first(ordered_pair(a,b)),set_a) ),
inference(symmetry,[status(thm)],[78]) ).
tff(80,plain,
member(first(ordered_pair(a,b)),set_a),
inference(modus_ponens,[status(thm)],[33,79]) ).
tff(81,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(82,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(83,plain,
~ member(ordered_pair(a,b),cross_product(set_a,set_b)),
inference(modus_ponens,[status(thm)],[82,81]) ).
tff(84,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(85,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)],[84]) ).
tff(86,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(87,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(88,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)],[87]) ).
tff(89,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(90,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)],[89,88]) ).
tff(91,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)],[90,86]) ).
tff(92,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)],[91]) ).
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) ),
inference(modus_ponens,[status(thm)],[92,85]) ).
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) )
| 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(95,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(96,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)],[95]) ).
tff(97,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)],[96,94]) ).
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) )
| ~ 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(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) )
| 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)],[98,97]) ).
tff(100,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)],[99,93,83]) ).
tff(101,plain,
~ ordered_pair_predicate(ordered_pair(a,b)),
inference(unit_resolution,[status(thm)],[100,80,63,54]) ).
tff(102,plain,
~ ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b))),
inference(modus_ponens,[status(thm)],[101,13]) ).
tff(103,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(
( ( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) )
<=> ( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) ) )),
inference(bind,[status(th)],]) ).
tff(104,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(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) ) ),
inference(quant_intro,[status(thm)],[103]) ).
tff(105,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(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) ) ),
inference(rewrite,[status(thm)],]) ).
tff(106,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(107,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)],[106]) ).
tff(108,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(109,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) ),
inference(modus_ponens,[status(thm)],[108,107]) ).
tff(110,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) ),
inference(modus_ponens,[status(thm)],[109,105]) ).
tff(111,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) ),
inference(skolemize,[status(sab)],[110]) ).
tff(112,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) ),
inference(modus_ponens,[status(thm)],[111,104]) ).
tff(113,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) )
| ~ little_set(b)
| ~ little_set(a)
| ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)))
| ( non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) != ordered_pair(a,b) ) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) )
| ~ little_set(b)
| ~ little_set(a)
| ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)))
| ( non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) != ordered_pair(a,b) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(114,plain,
( ( ~ little_set(a)
| ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)))
| ( non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) != ordered_pair(a,b) )
| ~ little_set(b) )
<=> ( ~ little_set(b)
| ~ little_set(a)
| ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)))
| ( non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) != ordered_pair(a,b) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(115,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) )
| ~ little_set(a)
| ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)))
| ( non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) != ordered_pair(a,b) )
| ~ little_set(b) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) )
| ~ little_set(b)
| ~ little_set(a)
| ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)))
| ( non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) != ordered_pair(a,b) ) ) ),
inference(monotonicity,[status(thm)],[114]) ).
tff(116,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) )
| ~ little_set(a)
| ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)))
| ( non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) != ordered_pair(a,b) )
| ~ little_set(b) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) )
| ~ little_set(b)
| ~ little_set(a)
| ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)))
| ( non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) != ordered_pair(a,b) ) ) ),
inference(transitivity,[status(thm)],[115,113]) ).
tff(117,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) )
| ~ little_set(a)
| ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)))
| ( non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) != ordered_pair(a,b) )
| ~ little_set(b) ),
inference(quant_inst,[status(thm)],]) ).
tff(118,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ little_set(Y)
| ordered_pair_predicate(X)
| ( X != ordered_pair(Y,Z) )
| ~ little_set(Z) )
| ~ little_set(b)
| ~ little_set(a)
| ordered_pair_predicate(non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)))
| ( non_ordered_pair(singleton_set(a),non_ordered_pair(a,b)) != ordered_pair(a,b) ) ),
inference(modus_ponens,[status(thm)],[117,116]) ).
tff(119,plain,
$false,
inference(unit_resolution,[status(thm)],[118,112,37,30,102,10]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : SET042-3 : TPTP v8.1.0. Released v1.0.0.
% 0.04/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n005.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Sat Sep 3 01:44:20 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.13/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35 Usage: tptp [options] [-file:]file
% 0.13/0.35 -h, -? prints this message.
% 0.13/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.35 -m, -model generate model.
% 0.13/0.35 -p, -proof generate proof.
% 0.13/0.35 -c, -core generate unsat core of named formulas.
% 0.13/0.35 -st, -statistics display statistics.
% 0.13/0.35 -t:timeout set timeout (in second).
% 0.13/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35 -<param>:<value> configuration parameter and value.
% 0.13/0.35 -o:<output-file> file to place output in.
% 0.20/0.45 % SZS status Unsatisfiable
% 0.20/0.45 % SZS output start Proof
% See solution above
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