1 include "logic/equality.ma".
3 (* Inclusion of: LAT011-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : LAT011-1 : TPTP v3.7.0. Released v2.2.0. *)
9 (* Domain : Lattice Theory *)
11 (* Problem : Uniqueness of meet (dually join) in lattice theory *)
13 (* Version : [MP96] (equality) axioms : Especial. *)
15 (* English : Let's say we have a lattice with two meet operations, say *)
17 (* meet1 and meet2. In other words, {join,meet1} is a lattice, *)
19 (* and {join,meet2} is a lattice. Then, we can prove that the *)
21 (* two meet operations are really the same. *)
23 (* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
25 (* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
27 (* Source : [McC98] *)
29 (* Names : LT-8 [MP96] *)
31 (* Status : Unsatisfiable *)
33 (* Rating : 0.00 v3.3.0, 0.07 v3.1.0, 0.00 v2.7.0, 0.09 v2.6.0, 0.00 v2.2.1 *)
35 (* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 1 RR) *)
37 (* Number of atoms : 14 ( 14 equality) *)
39 (* Maximal clause size : 1 ( 1 average) *)
41 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
43 (* Number of functors : 5 ( 2 constant; 0-2 arity) *)
45 (* Number of variables : 26 ( 4 singleton) *)
47 (* Maximal term depth : 3 ( 2 average) *)
49 (* Comments : For quasilattice, meet (dually join) is not necessarily unique. *)
51 (* -------------------------------------------------------------------------- *)
53 (* ----Include lattice axioms *)
55 (* Inclusion of: Axioms/LAT001-0.ax *)
57 (* -------------------------------------------------------------------------- *)
59 (* File : LAT001-0 : TPTP v3.7.0. Released v1.0.0. *)
61 (* Domain : Lattice Theory *)
63 (* Axioms : Lattice theory (equality) axioms *)
65 (* Version : [McC88] (equality) axioms. *)
69 (* Refs : [Bum65] Bumcroft (1965), Proceedings of the Glasgow Mathematic *)
71 (* : [McC88] McCune (1988), Challenge Equality Problems in Lattice *)
73 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
75 (* Source : [McC88] *)
81 (* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
83 (* Number of atoms : 8 ( 8 equality) *)
85 (* Maximal clause size : 1 ( 1 average) *)
87 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
89 (* Number of functors : 2 ( 0 constant; 2-2 arity) *)
91 (* Number of variables : 16 ( 2 singleton) *)
93 (* Maximal term depth : 3 ( 2 average) *)
97 (* -------------------------------------------------------------------------- *)
99 (* ----The following 8 clauses characterise lattices *)
101 (* -------------------------------------------------------------------------- *)
103 (* -------------------------------------------------------------------------- *)
105 (* ----{join,meet2} is a lattice: *)
107 (* ----Denial that meet1 and meet2 are the same: *)
108 ntheorem prove_meets_are_same:
109 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
112 ∀join:∀_:Univ.∀_:Univ.Univ.
113 ∀meet:∀_:Univ.∀_:Univ.Univ.
114 ∀meet2:∀_:Univ.∀_:Univ.Univ.
115 ∀H0:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet2 (meet2 X Y) Z) (meet2 X (meet2 Y Z)).
116 ∀H1:∀X:Univ.∀Y:Univ.eq Univ (join X (meet2 X Y)) X.
117 ∀H2:∀X:Univ.∀Y:Univ.eq Univ (meet2 X (join X Y)) X.
118 ∀H3:∀X:Univ.∀Y:Univ.eq Univ (meet2 X Y) (meet2 Y X).
119 ∀H4:∀X:Univ.eq Univ (meet2 X X) X.
120 ∀H5:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
121 ∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
122 ∀H7:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
123 ∀H8:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet Y X).
124 ∀H9:∀X:Univ.∀Y:Univ.eq Univ (join X (meet X Y)) X.
125 ∀H10:∀X:Univ.∀Y:Univ.eq Univ (meet X (join X Y)) X.
126 ∀H11:∀X:Univ.eq Univ (join X X) X.
127 ∀H12:∀X:Univ.eq Univ (meet X X) X.eq Univ (meet a b) (meet2 a b))
151 nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11,H12 ##;
152 ntry (nassumption) ##;
155 (* -------------------------------------------------------------------------- *)