1 include "logic/equality.ma".
3 (* Inclusion of: BOO032-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : BOO032-1 : TPTP v3.7.0. Released v2.2.0. *)
9 (* Domain : Boolean Algebra *)
11 (* Problem : Independence of a system of Boolean algebra *)
13 (* Version : [MP96] (equality) axioms : Especial. *)
15 (* English : This is part of a proof that a self-dual 3-basis for Boolean *)
17 (* algebra is independent. *)
19 (* Refs : [McC98] McCune (1998), Email to G. Sutcliffe *)
21 (* : [MP96] McCune & Padmanabhan (1996), Automated Deduction in Eq *)
23 (* Source : [McC98] *)
25 (* Names : DUAL-BA-9 [MP96] *)
27 (* Status : Satisfiable *)
29 (* Rating : 0.33 v3.2.0, 0.67 v3.1.0, 0.33 v2.5.0, 0.67 v2.4.0, 0.67 v2.3.0, 1.00 v2.2.1 *)
31 (* Syntax : Number of clauses : 13 ( 0 non-Horn; 13 unit; 1 RR) *)
33 (* Number of atoms : 13 ( 13 equality) *)
35 (* Maximal clause size : 1 ( 1 average) *)
37 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
39 (* Number of functors : 4 ( 1 constant; 0-2 arity) *)
41 (* Number of variables : 28 ( 10 singleton) *)
43 (* Maximal term depth : 4 ( 2 average) *)
45 (* Comments : The smallest model has 5 elements. *)
47 (* -------------------------------------------------------------------------- *)
49 (* ----3 properties of Boolean algebra and the corresponding duals. *)
51 (* ----Majority polynomials: *)
53 (* ----Duals of majority polynomials: *)
55 (* ----A propery of Boolean Algebra fails to hold. *)
56 ntheorem prove_inverse_involution:
57 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
59 ∀add:∀_:Univ.∀_:Univ.Univ.
60 ∀inverse:∀_:Univ.Univ.
61 ∀multiply:∀_:Univ.∀_:Univ.Univ.
62 ∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply (add (multiply X Y) Y) (add X Y)) Y.
63 ∀H1:∀X:Univ.∀Y:Univ.eq Univ (multiply (add (multiply X X) Y) (add X X)) X.
64 ∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply (add (multiply X Y) X) (add X Y)) X.
65 ∀H3:∀X:Univ.∀Y:Univ.eq Univ (add (multiply (add X Y) Y) (multiply X Y)) Y.
66 ∀H4:∀X:Univ.∀Y:Univ.eq Univ (add (multiply (add X X) Y) (multiply X X)) X.
67 ∀H5:∀X:Univ.∀Y:Univ.eq Univ (add (multiply (add X Y) X) (multiply X Y)) X.
68 ∀H6:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X (inverse X)) Y) Y.
69 ∀H7:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply (add X Y) (add Y Z)) Y) Y.
70 ∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y (add X Z))) X.
71 ∀H9:∀X:Univ.∀Y:Univ.eq Univ (multiply (add X (inverse X)) Y) Y.
72 ∀H10:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add (multiply X Y) (multiply Y Z)) Y) Y.
73 ∀H11:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add X (multiply Y (multiply X Z))) X.eq Univ (inverse (inverse a)) a)
95 nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11 ##;
96 ntry (nassumption) ##;
99 (* -------------------------------------------------------------------------- *)