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set.txt
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set.txt
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% File : SET010-1 : TPTP v7.3.0. Released v1.0.0.
% Domain : Set Theory
% Problem : X \ Y ^ Z = (X \ Y) U (X \ Z)
% Version : [LS74] axioms.
% English :
% Refs : [LS74] Lawrence & Starkey (1974), Experimental Tests of Resol
% : [WM76] Wilson & Minker (1976), Resolution, Refinements, and S
% Source : [SPRFN]
% Names : ls118 [LS74]
% : ls118 [WM76]
%----Include the member and subset axioms
%include('Axioms/SET001-0.ax').
cnf(subsets_are_set_equal_sets,axiom,
( ~ subset(Set1,Set2)
| ~ subset(Set2,Set1)
| equal_sets(Set2,Set1) )).
cnf(set_equal_sets_are_subsets2,axiom,
( subset(Subset,Superset)
| ~ equal_sets(Superset,Subset) )).
cnf(membership_in_subsets,axiom,
( ~ subset(Subset,Superset)
| member(Element,Superset)
| ~ member(Element,Subset) )).
cnf(set_equal_sets_are_subsets1,axiom,
( subset(Subset,Superset)
| ~ equal_sets(Subset,Superset) )).
cnf(subsets_axiom2,axiom,
( subset(Subset,Superset)
| ~ member(member_of_1_not_of_2(Subset,Superset),Superset) )).
cnf(subsets_axiom1,axiom,
( member(member_of_1_not_of_2(Subset,Superset),Subset)
| subset(Subset,Superset) )).
%----Include the member and union axioms
%include('Axioms/SET001-1.ax').
cnf(member_of_union_is_member_of_one_set,axiom,
( ~ member(Element,Union)
| member(Element,Set2)
| member(Element,Set1)
| ~ union(Set1,Set2,Union) )).
cnf(member_of_set1_is_member_of_union,axiom,
( ~ union(Set1,Set2,Union)
| ~ member(Element,Set1)
| member(Element,Union) )).
cnf(union_axiom2,axiom,
( ~ member(g(Set1,Set2,Union),Union)
| union(Set1,Set2,Union)
| ~ member(g(Set1,Set2,Union),Set1) )).
cnf(union_axiom3,axiom,
( ~ member(g(Set1,Set2,Union),Set2)
| ~ member(g(Set1,Set2,Union),Union)
| union(Set1,Set2,Union) )).
cnf(union_axiom1,axiom,
( member(g(Set1,Set2,Union),Set1)
| member(g(Set1,Set2,Union),Set2)
| member(g(Set1,Set2,Union),Union)
| union(Set1,Set2,Union) )).
cnf(member_of_set2_is_member_of_union,axiom,
( ~ member(Element,Set2)
| member(Element,Union)
| ~ union(Set1,Set2,Union) )).
%----Include the member and intersection axioms
%include('Axioms/SET001-2.ax').
cnf(intersection_axiom1,axiom,
( member(h(Set1,Set2,Intersection),Set1)
| intersection(Set1,Set2,Intersection)
| member(h(Set1,Set2,Intersection),Intersection) )).
cnf(member_of_both_is_member_of_intersection,axiom,
( ~ intersection(Set1,Set2,Intersection)
| ~ member(Element,Set2)
| ~ member(Element,Set1)
| member(Element,Intersection) )).
cnf(member_of_intersection_is_member_of_set2,axiom,
( ~ intersection(Set1,Set2,Intersection)
| member(Element,Set2)
| ~ member(Element,Intersection) )).
cnf(member_of_intersection_is_member_of_set1,axiom,
( ~ member(Element,Intersection)
| member(Element,Set1)
| ~ intersection(Set1,Set2,Intersection) )).
cnf(intersection_axiom2,axiom,
( member(h(Set1,Set2,Intersection),Set2)
| intersection(Set1,Set2,Intersection)
| member(h(Set1,Set2,Intersection),Intersection) )).
cnf(intersection_axiom3,axiom,
( intersection(Set1,Set2,Intersection)
| ~ member(h(Set1,Set2,Intersection),Set1)
| ~ member(h(Set1,Set2,Intersection),Set2)
| ~ member(h(Set1,Set2,Intersection),Intersection) )).
%----Include the member and difference axioms
% include('Axioms/SET001-3.ax').
cnf(difference_axiom1,axiom,
( difference(Set1,Set2,Difference)
| member(k(Set1,Set2,Difference),Difference)
| ~ member(k(Set1,Set2,Difference),Set2) )).
cnf(member_of_difference,axiom,
( ~ member(Element,Difference)
| member(Element,Set1)
| ~ difference(Set1,Set2,Difference) )).
cnf(difference_axiom3,axiom,
( difference(Set1,Set2,Difference)
| member(k(Set1,Set2,Difference),Set2)
| ~ member(k(Set1,Set2,Difference),Set1)
| ~ member(k(Set1,Set2,Difference),Difference) )).
cnf(not_member_of_difference,axiom,
( ~ member(Element,Set2)
| ~ difference(A_set,Set1,Set2)
| ~ member(Element,Set1) )).
cnf(difference_axiom2,axiom,
( member(k(Set1,Set2,Difference),Difference)
| member(k(Set1,Set2,Difference),Set1)
| difference(Set1,Set2,Difference) )).
cnf(member_of_difference_or_set2,axiom,
( ~ member(Element,Set1)
| member(Element,Set2)
| member(Element,Difference)
| ~ difference(Set1,Set2,Difference) )).
%--------------------------------------------------------------------------
cnf(a_intersection_b,hypothesis,
( intersection(a,b,aIb) )).
cnf(c_minus_a,hypothesis,
( difference(c,a,cDa) )).
cnf(c_minus_b,hypothesis,
( difference(c,b,cDb) )).
cnf(c_minus_aIb,hypothesis,
( difference(c,aIb,cD_aIb) )).
cnf(prove_cDa_union_cDb_is_cD_aIb,negated_conjecture,
( ~ union(cDa,cDb,cD_aIb) )).