1 include "logic/equality.ma".
3 (* Inclusion of: LAT024-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : LAT024-1 : TPTP v3.2.0. Released v2.2.0. *)
9 (* Domain : Lattice Theory (Quasilattices) *)
11 (* Problem : Meet (dually join) is not necessarily unique for quasilattices. *)
13 (* Version : [MP96] (equality) axioms. *)
15 (* English : Let's say we have a quasilattice 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 show that the *)
21 (* two meet operations not necessarily 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 : QLT-7 [MP96] *)
31 (* Status : Satisfiable *)
33 (* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.4.0, 0.67 v2.3.0, 1.00 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 : 30 ( 0 singleton) *)
47 (* Maximal term depth : 4 ( 3 average) *)
49 (* Comments : There is a 2-element model. *)
51 (* : For lattices meet (dually join) is unique. *)
53 (* -------------------------------------------------------------------------- *)
55 (* ----Include Quasilattice theory (equality) axioms *)
57 (* Inclusion of: Axioms/LAT004-0.ax *)
59 (* -------------------------------------------------------------------------- *)
61 (* File : LAT004-0 : TPTP v3.2.0. Released v2.2.0. *)
63 (* Domain : Lattice Theory (Quasilattices) *)
65 (* Axioms : Quasilattice theory (equality) axioms *)
67 (* Version : [McC98b] (equality) axioms. *)
71 (* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
73 (* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
75 (* Source : [McC98] *)
81 (* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
83 (* Number of literals : 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 : 18 ( 0 singleton) *)
93 (* Maximal term depth : 4 ( 2 average) *)
97 (* -------------------------------------------------------------------------- *)
99 (* ----Quasilattice theory: *)
101 (* -------------------------------------------------------------------------- *)
103 (* -------------------------------------------------------------------------- *)
105 (* ----{join,meet2} is a quasilattice: *)
107 (* ----Denial that meet1 and meet2 are the same: *)
108 ntheorem prove_meets_equal:
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 (join X (meet2 Y Z)) (join X Y)) (join X (meet2 Y Z)).
116 ∀H1:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet2 X (join Y Z)) (meet2 X Y)) (meet2 X (join Y Z)).
117 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet2 (meet2 X Y) Z) (meet2 X (meet2 Y Z)).
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 (meet (join X (meet Y Z)) (join X Y)) (join X (meet Y Z)).
121 ∀H6:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (meet X (join Y Z)) (meet X Y)) (meet X (join Y Z)).
122 ∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (join (join X Y) Z) (join X (join Y Z)).
123 ∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (meet (meet X Y) Z) (meet X (meet Y Z)).
124 ∀H9:∀X:Univ.∀Y:Univ.eq Univ (join X Y) (join Y X).
125 ∀H10:∀X:Univ.∀Y:Univ.eq Univ (meet X Y) (meet 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;
154 (* -------------------------------------------------------------------------- *)