TPTP Problem File: SET076-7.p
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%------------------------------------------------------------------------------
% File : SET076-7 : TPTP v9.0.0. Bugfixed v2.1.0.
% Domain : Set Theory
% Problem : Unorderd pair is a subset
% Version : [Qua92] axioms : Augmented.
% English : If both members of an unordered pair belong to a set, the
% pair is a subset.
% Refs : [Qua92] Quaife (1992), Automated Deduction in von Neumann-Bern
% Source : [Quaife]
% Names : UP7 [Qua92]
% Status : Unsatisfiable
% Rating : 0.25 v8.2.0, 0.29 v8.1.0, 0.16 v7.5.0, 0.21 v7.4.0, 0.24 v7.3.0, 0.25 v7.1.0, 0.17 v7.0.0, 0.27 v6.2.0, 0.30 v6.1.0, 0.50 v5.5.0, 0.70 v5.3.0, 0.72 v5.2.0, 0.62 v5.1.0, 0.71 v4.1.0, 0.77 v4.0.1, 0.73 v4.0.0, 0.55 v3.7.0, 0.30 v3.5.0, 0.45 v3.4.0, 0.58 v3.3.0, 0.64 v3.2.0, 0.69 v3.1.0, 0.64 v2.7.0, 0.58 v2.6.0, 0.78 v2.5.0, 0.82 v2.4.0, 0.88 v2.2.1, 0.83 v2.2.0, 0.67 v2.1.0
% Syntax : Number of clauses : 121 ( 40 unt; 15 nHn; 82 RR)
% Number of literals : 238 ( 56 equ; 108 neg)
% Maximal clause size : 5 ( 1 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-3 aty)
% Number of functors : 41 ( 41 usr; 11 con; 0-3 aty)
% Number of variables : 236 ( 44 sgn)
% SPC : CNF_UNS_RFO_SEQ_NHN
% Comments : Preceding lemmas are added.
% Bugfixes : v2.1.0 - Bugfix in SET004-0.ax.
%------------------------------------------------------------------------------
%----Include von Neuman-Bernays-Godel set theory axioms
include('Axioms/SET004-0.ax').
%------------------------------------------------------------------------------
%----Corollaries to Unordered pair axiom. Not in paper, but in email.
cnf(corollary_1_to_unordered_pair,axiom,
( ~ member(ordered_pair(X,Y),cross_product(U,V))
| member(X,unordered_pair(X,Y)) ) ).
cnf(corollary_2_to_unordered_pair,axiom,
( ~ member(ordered_pair(X,Y),cross_product(U,V))
| member(Y,unordered_pair(X,Y)) ) ).
%----Corollaries to Cartesian product axiom.
cnf(corollary_1_to_cartesian_product,axiom,
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(U,universal_class) ) ).
cnf(corollary_2_to_cartesian_product,axiom,
( ~ member(ordered_pair(U,V),cross_product(X,Y))
| member(V,universal_class) ) ).
%---- PARTIAL ORDER.
%----(PO1): reflexive.
cnf(subclass_is_reflexive,axiom,
subclass(X,X) ).
%----(PO2): antisymmetry is part of A-3.
%----(x < y), (y < x) --> (x = y).
%----(PO3): transitivity.
cnf(transitivity_of_subclass,axiom,
( ~ subclass(X,Y)
| ~ subclass(Y,Z)
| subclass(X,Z) ) ).
%---- EQUALITY.
%----(EQ1): equality axiom.
%----a:x:(x = x).
%----This is always an axiom in the TPTP presentation.
%----(EQ2): expanded equality definition.
cnf(equality1,axiom,
( X = Y
| member(not_subclass_element(X,Y),X)
| member(not_subclass_element(Y,X),Y) ) ).
cnf(equality2,axiom,
( ~ member(not_subclass_element(X,Y),Y)
| X = Y
| member(not_subclass_element(Y,X),Y) ) ).
cnf(equality3,axiom,
( ~ member(not_subclass_element(Y,X),X)
| X = Y
| member(not_subclass_element(X,Y),X) ) ).
cnf(equality4,axiom,
( ~ member(not_subclass_element(X,Y),Y)
| ~ member(not_subclass_element(Y,X),X)
| X = Y ) ).
%---- SPECIAL CLASSES.
%----(SP1): lemma.
cnf(special_classes_lemma,axiom,
~ member(Y,intersection(complement(X),X)) ).
%----(SP2): Existence of O (null class).
%----e:x:a:z:(-(z e x)).
cnf(existence_of_null_class,axiom,
~ member(Z,null_class) ).
%----(SP3): O is a subclass of every class.
cnf(null_class_is_subclass,axiom,
subclass(null_class,X) ).
%----corollary.
cnf(corollary_of_null_class_is_subclass,axiom,
( ~ subclass(X,null_class)
| X = null_class ) ).
%----(SP4): uniqueness of null class.
cnf(null_class_is_unique,axiom,
( Z = null_class
| member(not_subclass_element(Z,null_class),Z) ) ).
%----(SP5): O is a set (follows from axiom of infinity).
cnf(null_class_is_a_set,axiom,
member(null_class,universal_class) ).
%---- UNORDERED PAIRS.
%----(UP1): unordered pair is commutative.
cnf(commutativity_of_unordered_pair,axiom,
unordered_pair(X,Y) = unordered_pair(Y,X) ).
%----(UP2): if one argument is a proper class, pair contains only the
%----other. In a slightly different form to the paper
cnf(singleton_in_unordered_pair1,axiom,
subclass(singleton(X),unordered_pair(X,Y)) ).
cnf(singleton_in_unordered_pair2,axiom,
subclass(singleton(Y),unordered_pair(X,Y)) ).
cnf(unordered_pair_equals_singleton1,axiom,
( member(Y,universal_class)
| unordered_pair(X,Y) = singleton(X) ) ).
cnf(unordered_pair_equals_singleton2,axiom,
( member(X,universal_class)
| unordered_pair(X,Y) = singleton(Y) ) ).
%----(UP3): if both arguments are proper classes, pair is null.
cnf(null_unordered_pair,axiom,
( unordered_pair(X,Y) = null_class
| member(X,universal_class)
| member(Y,universal_class) ) ).
%----(UP4): left cancellation for unordered pairs.
cnf(left_cancellation,axiom,
( unordered_pair(X,Y) != unordered_pair(X,Z)
| ~ member(ordered_pair(Y,Z),cross_product(universal_class,universal_class))
| Y = Z ) ).
%----(UP5): right cancellation for unordered pairs.
cnf(right_cancellation,axiom,
( unordered_pair(X,Z) != unordered_pair(Y,Z)
| ~ member(ordered_pair(X,Y),cross_product(universal_class,universal_class))
| X = Y ) ).
%----(UP6): corollary to (A-4).
cnf(corollary_to_unordered_pair_axiom1,axiom,
( ~ member(X,universal_class)
| unordered_pair(X,Y) != null_class ) ).
cnf(corollary_to_unordered_pair_axiom2,axiom,
( ~ member(Y,universal_class)
| unordered_pair(X,Y) != null_class ) ).
%----corollary to instantiate variables.
%----Not in the paper
cnf(corollary_to_unordered_pair_axiom3,axiom,
( ~ member(ordered_pair(X,Y),cross_product(U,V))
| unordered_pair(X,Y) != null_class ) ).
cnf(prove_unordered_pair_is_subset_1,negated_conjecture,
member(x,z) ).
cnf(prove_unordered_pair_is_subset_2,negated_conjecture,
member(y,z) ).
cnf(prove_unordered_pair_is_subset_3,negated_conjecture,
~ subclass(unordered_pair(x,y),z) ).
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