1 include "logic/equality.ma".
3 (* Inclusion of: LAT013-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : LAT013-1 : TPTP v3.7.0. Released v2.2.0. *)
9 (* Domain : Lattice Theory *)
11 (* Problem : McKenzie's 4-basis for lattice theory, part 2 (of 3) *)
13 (* Version : [MP96] (equality) axioms. *)
15 (* English : This is part of a proof that McKenzie's 4-basis axiomatizes *)
17 (* lattice theory. We prove half of the standard basis. *)
19 (* The other half follows by duality. In this part we prove *)
21 (* associativity of meet. *)
23 (* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
25 (* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
27 (* Source : [McC98] *)
29 (* Names : LT-9-b [MP96] *)
31 (* Status : Unsatisfiable *)
33 (* Rating : 0.00 v3.3.0, 0.21 v3.2.0, 0.14 v3.1.0, 0.22 v2.7.0, 0.18 v2.6.0, 0.17 v2.5.0, 0.00 v2.2.1 *)
35 (* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
37 (* Number of atoms : 5 ( 5 equality) *)
39 (* Maximal clause size : 1 ( 1 average) *)
41 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
43 (* Number of functors : 5 ( 3 constant; 0-2 arity) *)
45 (* Number of variables : 12 ( 8 singleton) *)
47 (* Maximal term depth : 4 ( 3 average) *)
51 (* -------------------------------------------------------------------------- *)
53 (* ----McKenzie's self-dual (independent) absorptive 4-basis for lattice theory. *)
55 (* ----Denial of conclusion: *)
56 ntheorem prove_associativity_of_meet:
57 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
61 ∀join:∀_:Univ.∀_:Univ.Univ.
62 ∀meet:∀_:Univ.∀_:Univ.Univ.
63 ∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet (join X Y) (join Y Z)) Y) Y.
64 ∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join (meet X Y) (meet Y Z)) Y) Y.
65 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet X (join Y (join X Z))) X.
66 ∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join X (meet Y (meet X Z))) X.eq Univ (meet (meet a b) c) (meet a (meet b c)))
81 nauto by H0,H1,H2,H3 ##;
82 ntry (nassumption) ##;
85 (* -------------------------------------------------------------------------- *)