%--------------------------------------------------------------------------
% File     : SET002-0 : TPTP v9.0.0. Released v1.0.0.
% Domain   : Set Theory
% Axioms   : Set theory axioms
% Version  : [MOW76] axioms : Biased.
% English  :

% Refs     : [MOW76] McCharen et al. (1976), Problems and Experiments for a
% Source   : [ANL]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses     :   21 (   3 unt;   3 nHn;  15 RR)
%            Number of literals    :   45 (   0 equ;  23 neg)
%            Maximal clause size   :    3 (   2 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    4 (   4 usr;   0 prp; 2-2 aty)
%            Number of functors    :    5 (   5 usr;   1 con; 0-2 aty)
%            Number of variables   :   48 (   5 sgn)
% SPC      : 

% Comments :
%--------------------------------------------------------------------------
%-----Definition of the empty set.
cnf(empty_set,axiom,
    ~ member(X,empty_set) ).

%-----Subset axioms. These are the same as in SET001-0.ax
cnf(membership_in_subsets,axiom,
    ( ~ member(Element,Subset)
    | ~ subset(Subset,Superset)
    | member(Element,Superset) ) ).

cnf(subsets_axiom1,axiom,
    ( subset(Subset,Superset)
    | member(member_of_1_not_of_2(Subset,Superset),Subset) ) ).

cnf(subsets_axiom2,axiom,
    ( ~ member(member_of_1_not_of_2(Subset,Superset),Superset)
    | subset(Subset,Superset) ) ).

%-----Axioms of complementation.
cnf(member_of_set_or_complement,axiom,
    ( member(X,Xs)
    | member(X,complement(Xs)) ) ).

cnf(not_member_of_set_and_complement,axiom,
    ( ~ member(X,Xs)
    | ~ member(X,complement(Xs)) ) ).

%-----Axioms of union.
cnf(member_of_set1_is_member_of_union,axiom,
    ( ~ member(X,Xs)
    | member(X,union(Xs,Ys)) ) ).

cnf(member_of_set2_is_member_of_union,axiom,
    ( ~ member(X,Ys)
    | member(X,union(Xs,Ys)) ) ).

cnf(member_of_union_is_member_of_one_set,axiom,
    ( ~ member(X,union(Xs,Ys))
    | member(X,Xs)
    | member(X,Ys) ) ).

%-----Axioms of intersection.
cnf(member_of_both_is_member_of_intersection,axiom,
    ( ~ member(X,Xs)
    | ~ member(X,Ys)
    | member(X,intersection(Xs,Ys)) ) ).

cnf(member_of_intersection_is_member_of_set1,axiom,
    ( ~ member(X,intersection(Xs,Ys))
    | member(X,Xs) ) ).

cnf(member_of_intersection_is_member_of_set2,axiom,
    ( ~ member(X,intersection(Xs,Ys))
    | member(X,Ys) ) ).

%-----Set equality axioms.
cnf(set_equal_sets_are_subsets1,axiom,
    ( ~ equal_sets(Subset,Superset)
    | subset(Subset,Superset) ) ).

cnf(set_equal_sets_are_subsets2,axiom,
    ( ~ equal_sets(Superset,Subset)
    | subset(Subset,Superset) ) ).

cnf(subsets_are_set_equal_sets,axiom,
    ( ~ subset(Set1,Set2)
    | ~ subset(Set2,Set1)
    | equal_sets(Set2,Set1) ) ).

%-----Equality axioms.
cnf(reflexivity_for_set_equal,axiom,
    equal_sets(Xs,Xs) ).

cnf(symmetry_for_set_equal,axiom,
    ( ~ equal_sets(Xs,Ys)
    | equal_sets(Ys,Xs) ) ).

cnf(transitivity_for_set_equal,axiom,
    ( ~ equal_sets(Xs,Ys)
    | ~ equal_sets(Ys,Zs)
    | equal_sets(Xs,Zs) ) ).

cnf(reflexivity_for_equal_elements,axiom,
    equal_elements(X,X) ).

cnf(symmetry_for_equal_elements,axiom,
    ( ~ equal_elements(X,Y)
    | equal_elements(Y,X) ) ).

cnf(transitivity_for_equal_elements,axiom,
    ( ~ equal_elements(X,Y)
    | ~ equal_elements(Y,Z)
    | equal_elements(X,Z) ) ).

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