TPTP Problem File: SET016-7.p
View Solutions
- Solve Problem
%--------------------------------------------------------------------------
% File : SET016-7 : TPTP v8.2.0. Bugfixed v7.3.0.
% Domain : Set Theory
% Problem : First components of equal ordered pairs are equal
% Version : [Qua92] axioms : Augmented.
% English :
% Refs : [Qua92] Quaife (1992), Automated Deduction in von Neumann-Bern
% Source : [Quaife]
% Names : OP4 [Qua92]
% : OP10 [Qua92]
% Status : Unsatisfiable
% Rating : 0.05 v7.4.0, 0.06 v7.3.0
% Syntax : Number of clauses : 168 ( 47 unt; 36 nHn; 105 RR)
% Number of literals : 339 ( 111 equ; 139 neg)
% Maximal clause size : 5 ( 2 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-3 aty)
% Number of functors : 44 ( 44 usr; 12 con; 0-3 aty)
% Number of variables : 309 ( 53 sgn)
% SPC : CNF_UNS_RFO_SEQ_NHN
% Comments : Preceding lemmas are added.
% : OP4 uses an extra antecedent, not used in OP10. This is the
% OP10 version.
% Bugfixes : v2.1.0 - Bugfix in SET004-0.ax.
% : v7.3.0 - Changed first1 and second1 to first and second.
%--------------------------------------------------------------------------
%----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 ) ).
%----(UP7): if both members of a pair belong to a set, the pair
%----is a subset.
cnf(unordered_pair_is_subset,axiom,
( ~ member(X,Z)
| ~ member(Y,Z)
| subclass(unordered_pair(X,Y),Z) ) ).
%---- SINGLETONS.
%----(SS1): every singleton is a set.
cnf(singletons_are_sets,axiom,
member(singleton(X),universal_class) ).
%----corollary, not in the paper.
cnf(corollary_1_to_singletons_are_sets,axiom,
member(singleton(Y),unordered_pair(X,singleton(Y))) ).
%----(SS2): a set belongs to its singleton.
%----(u = x), (u e universal_class) --> (u e {x}).
cnf(set_in_its_singleton,axiom,
( ~ member(X,universal_class)
| member(X,singleton(X)) ) ).
%----corollary
cnf(corollary_to_set_in_its_singleton,axiom,
( ~ member(X,universal_class)
| singleton(X) != null_class ) ).
%----Not in the paper
cnf(null_class_in_its_singleton,axiom,
member(null_class,singleton(null_class)) ).
%----(SS3): only x can belong to {x}.
cnf(only_member_in_singleton,axiom,
( ~ member(Y,singleton(X))
| Y = X ) ).
%----(SS4): if x is not a set, {x} = O.
cnf(singleton_is_null_class,axiom,
( member(X,universal_class)
| singleton(X) = null_class ) ).
%----(SS5): a singleton set is determined by its element.
cnf(singleton_identified_by_element1,axiom,
( singleton(X) != singleton(Y)
| ~ member(X,universal_class)
| X = Y ) ).
cnf(singleton_identified_by_element2,axiom,
( singleton(X) != singleton(Y)
| ~ member(Y,universal_class)
| X = Y ) ).
%----(SS5.5).
%----Not in the paper.
cnf(singleton_in_unordered_pair3,axiom,
( unordered_pair(Y,Z) != singleton(X)
| ~ member(X,universal_class)
| X = Y
| X = Z ) ).
%----(SS6): existence of memb.
%----a:x:e:u:(((u e universal_class) & x = {u}) | (-e:y:((y
%----e universal_class) & x = {y}) & u = x)).
cnf(member_exists1,axiom,
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) ) ).
cnf(member_exists2,axiom,
( ~ member(Y,universal_class)
| singleton(member_of(singleton(Y))) = singleton(Y) ) ).
cnf(member_exists3,axiom,
( member(member_of(X),universal_class)
| member_of(X) = X ) ).
cnf(member_exists4,axiom,
( singleton(member_of(X)) = X
| member_of(X) = X ) ).
%----(SS7): uniqueness of memb of a singleton set.
%----a:x:a:u:(((u e universal_class) & x = {u}) ==> member_of(x) = u)
cnf(member_of_singleton_is_unique,axiom,
( ~ member(U,universal_class)
| member_of(singleton(U)) = U ) ).
%----(SS8): uniqueness of memb when x is not a singleton of a set.
%----a:x:a:u:((e:y:((y e universal_class) & x = {y})
%----& u = x) | member_of(x) = u)
cnf(member_of_non_singleton_unique1,axiom,
( member(member_of1(X),universal_class)
| member_of(X) = X ) ).
cnf(member_of_non_singleton_unique2,axiom,
( singleton(member_of1(X)) = X
| member_of(X) = X ) ).
%----(SS9): corollary to (SS1).
cnf(corollary_2_to_singletons_are_sets,axiom,
( singleton(member_of(X)) != X
| member(X,universal_class) ) ).
%----(SS10).
cnf(property_of_singletons1,axiom,
( singleton(member_of(X)) != X
| ~ member(Y,X)
| member_of(X) = Y ) ).
%----(SS11).
cnf(property_of_singletons2,axiom,
( ~ member(X,Y)
| subclass(singleton(X),Y) ) ).
%----(SS12): there are at most two subsets of a singleton.
cnf(two_subsets_of_singleton,axiom,
( ~ subclass(X,singleton(Y))
| X = null_class
| singleton(Y) = X ) ).
%----(SS13): a class contains 0, 1, or at least 2 members.
cnf(number_of_elements_in_class,axiom,
( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| singleton(not_subclass_element(X,null_class)) = X
| X = null_class ) ).
%----corollaries.
cnf(corollary_2_to_number_of_elements_in_class,axiom,
( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),X)
| singleton(not_subclass_element(X,null_class)) = X
| X = null_class ) ).
cnf(corollary_1_to_number_of_elements_in_class,axiom,
( not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class) != not_subclass_element(X,null_class)
| singleton(not_subclass_element(X,null_class)) = X
| X = null_class ) ).
%----(SS14): relation to ordered pair.
%----It looks like we could simplify Godel's axioms by taking singleton
%----as a primitive and using the next as a definition. Not in the paper.
cnf(unordered_pairs_and_singletons,axiom,
unordered_pair(X,Y) = union(singleton(X),singleton(Y)) ).
%---- ORDERED PAIRS.
%----(OP1): an ordered pair is a set.
cnf(ordered_pair_is_set,axiom,
member(ordered_pair(X,Y),universal_class) ).
%----(OP2): members of ordered pair.
cnf(singleton_member_of_ordered_pair,axiom,
member(singleton(X),ordered_pair(X,Y)) ).
cnf(unordered_pair_member_of_ordered_pair,axiom,
member(unordered_pair(X,singleton(Y)),ordered_pair(X,Y)) ).
%----(OP3): special cases.
cnf(property_1_of_ordered_pair,axiom,
( unordered_pair(singleton(X),unordered_pair(X,null_class)) = ordered_pair(X,Y)
| member(Y,universal_class) ) ).
cnf(property_2_of_ordered_pair,axiom,
( ~ member(Y,universal_class)
| unordered_pair(null_class,singleton(singleton(Y))) = ordered_pair(X,Y)
| member(X,universal_class) ) ).
cnf(property_3_of_ordered_pair,axiom,
( unordered_pair(null_class,singleton(null_class)) = ordered_pair(X,Y)
| member(X,universal_class)
| member(Y,universal_class) ) ).
%----(OP4)-(OP5): an ordered pair uniquely determines its components.
%----(OP4). This OP10 from the paper. OP4 is now omitted.
cnf(ordered_pair_determines_components1,axiom,
( ordered_pair(W,X) != ordered_pair(Y,Z)
| ~ member(W,universal_class)
| W = Y ) ).
%----(OP5). This OP11 from the paper. OP5 is now omitted.
cnf(ordered_pair_determines_components2,axiom,
( ordered_pair(W,X) != ordered_pair(Y,Z)
| ~ member(X,universal_class)
| X = Z ) ).
%----(OP6): existence of 1st and 2nd.
%----a:x:e:u:e:v:((([u,v] e cross_product(universal_class,
%----universal_class)) & x = [u,v]) | (-e:y:e:z:(([y,z] e cross_product(
%----universal_class,universal_class)) & x = [y,z]) & u = x & v = x)).
cnf(existence_of_1st_and_2nd_1,axiom,
( ~ member(ordered_pair(Y,Z),cross_product(universal_class,universal_class))
| member(ordered_pair(first(ordered_pair(Y,Z)),second(ordered_pair(Y,Z))),cross_product(universal_class,universal_class)) ) ).
%----next is subsumed by Axiom B5'-b ([y,z]
%----e cross_product(universal_class,universal_class)) -->
%----([first([y,z]),second([y,z])] = [y,z]).
cnf(existence_of_1st_and_2nd_2,axiom,
( member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class))
| first(X) = X ) ).
cnf(existence_of_1st_and_2nd_3,axiom,
( member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class))
| second(X) = X ) ).
cnf(existence_of_1st_and_2nd_4,axiom,
( ordered_pair(first(X),second(X)) = X
| first(X) = X ) ).
cnf(existence_of_1st_and_2nd_5,axiom,
( ordered_pair(first(X),second(X)) = X
| second(X) = X ) ).
%----(OP7): uniqueness of 1st and 2nd when x is an ordered pair of sets.
%----a:x:a:u:a:v:([u,v] e cross_product(universal_class,universal_class)
%---- & x = [u,v] ==> first(x) = u & second(x) = v)
cnf(unique_1st_and_2nd_in_pair_of_sets1,axiom,
( ~ member(ordered_pair(U,V),cross_product(universal_class,universal_class))
| first(ordered_pair(U,V)) = U ) ).
cnf(unique_1st_and_2nd_in_pair_of_sets2,axiom,
( ~ member(ordered_pair(U,V),cross_product(universal_class,universal_class))
| second(ordered_pair(U,V)) = V ) ).
%----(OP8): uniqueness of 1st and 2nd when x is not an ordered pair
%----of sets. a:x:a:u:a:v:((e:y:e:z:(([y,z]
%----e cross_product(universal_class, universal_class)) & x = [y,z])
%----& u = x & v = x) | first(x) = u & second(x) = v)
cnf(unique_1st_and_2nd_in_pair_of_non_sets1,axiom,
( member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class))
| first(X) = X ) ).
cnf(unique_1st_and_2nd_in_pair_of_non_sets2,axiom,
( member(ordered_pair(first(X),second(X)),cross_product(universal_class,universal_class))
| second(X) = X ) ).
cnf(unique_1st_and_2nd_in_pair_of_non_sets3,axiom,
( ordered_pair(first(X),second(X)) = X
| first(X) = X ) ).
cnf(unique_1st_and_2nd_in_pair_of_non_sets4,axiom,
( ordered_pair(first(X),second(X)) = X
| second(X) = X ) ).
%----(OP9): corollaries to (OP1).
cnf(corollary_1_to_ordered_pairs_are_sets,axiom,
( ordered_pair(first(X),second(X)) != X
| member(X,universal_class) ) ).
cnf(corollary_12to_ordered_pairs_are_sets,axiom,
( ~ member(X,cross_product(universal_class,universal_class))
| member(X,universal_class) ) ).
cnf(prove_ordered_pair_determines_components1_1,negated_conjecture,
ordered_pair(w,x) = ordered_pair(y,z) ).
cnf(prove_ordered_pair_determines_components1_2,negated_conjecture,
member(w,universal_class) ).
%----This is the extra clause from [Qua92] for OP4
% input_clause(prove_ordered_pair_determines_components1_2a,negated_conjecture,
% [++member(y,universal_class)]).
cnf(prove_ordered_pair_determines_components1_3,negated_conjecture,
w != y ).
%--------------------------------------------------------------------------