1 include "logic/equality.ma".
3 (* Inclusion of: LAT009-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : LAT009-1 : TPTP v3.7.0. Released v2.2.0. *)
9 (* Domain : Lattice Theory *)
11 (* Problem : A self-dual form of distributivity for lattice theory. *)
13 (* Version : [MP96] (equality) axioms : Especial. *)
15 (* English : From lattice theory axioms and a self-dual form of *)
17 (* distributivity, we prove ordinary distributivity. *)
19 (* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
21 (* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
23 (* Source : [McC98] *)
25 (* Names : LT-5 [MP96] *)
27 (* Status : Unsatisfiable *)
29 (* Rating : 0.11 v3.4.0, 0.12 v3.3.0, 0.00 v2.2.1 *)
31 (* Syntax : Number of clauses : 10 ( 0 non-Horn; 10 unit; 1 RR) *)
33 (* Number of atoms : 10 ( 10 equality) *)
35 (* Maximal clause size : 1 ( 1 average) *)
37 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
39 (* Number of functors : 5 ( 3 constant; 0-2 arity) *)
41 (* Number of variables : 19 ( 2 singleton) *)
43 (* Maximal term depth : 5 ( 2 average) *)
47 (* -------------------------------------------------------------------------- *)
49 (* ----Include lattice axioms *)
51 (* Inclusion of: Axioms/LAT001-0.ax *)
53 (* -------------------------------------------------------------------------- *)
55 (* File : LAT001-0 : TPTP v3.7.0. Released v1.0.0. *)
57 (* Domain : Lattice Theory *)
59 (* Axioms : Lattice theory (equality) axioms *)
61 (* Version : [McC88] (equality) axioms. *)
65 (* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
67 (* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
69 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
71 (* Source : [McC88] *)
77 (* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
79 (* Number of atoms : 8 ( 8 equality) *)
81 (* Maximal clause size : 1 ( 1 average) *)
83 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
85 (* Number of functors : 2 ( 0 constant; 2-2 arity) *)
87 (* Number of variables : 16 ( 2 singleton) *)
89 (* Maximal term depth : 3 ( 2 average) *)
93 (* -------------------------------------------------------------------------- *)
95 (* ----The following 8 clauses characterise lattices *)
97 (* -------------------------------------------------------------------------- *)
99 (* -------------------------------------------------------------------------- *)
101 (* ----A self-dual form of distributivity for lattice theory. *)
103 (* ----Denial of ordinary distributivity. *)
104 ntheorem prove_distributivity:
105 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
109 ∀join:∀_:Univ.∀_:Univ.Univ.
110 ∀meet:∀_:Univ.∀_:Univ.Univ.
111 ∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet (join (meet X Y) Z) Y) (meet Z X)) (meet (join (meet (join X Y) Z) Y) (join Z X)).
112 ∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
113 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
114 ∀H3:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
115 ∀H4:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
116 ∀H5:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
117 ∀H6:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
118 ∀H7:∀X:Univ.eq Univ (join X X) X.
119 ∀H8:∀X:Univ.eq Univ (meet X X) X.eq Univ (join a (meet b c)) (meet (join a b) (join a c)))
139 nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8 ##;
140 ntry (nassumption) ##;
143 (* -------------------------------------------------------------------------- *)