1 include "logic/equality.ma".
3 (* Inclusion of: BOO027-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : BOO027-1 : TPTP v3.7.0. Released v2.2.0. *)
9 (* Domain : Boolean Algebra *)
11 (* Problem : Independence of self-dual 2-basis. *)
13 (* Version : [MP96] (eqiality) axioms : Especial. *)
15 (* English : Show that half of the self-dual 2-basis in DUAL-BA-3 is not *)
17 (* a basis for Boolean Algebra. *)
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-4 [MP96] *)
27 (* Status : Satisfiable *)
29 (* Rating : 0.00 v3.2.0, 0.33 v3.1.0, 0.00 v2.2.1 *)
31 (* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
33 (* Number of atoms : 6 ( 6 equality) *)
35 (* Maximal clause size : 1 ( 1 average) *)
37 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
39 (* Number of functors : 5 ( 2 constant; 0-2 arity) *)
41 (* Number of variables : 10 ( 0 singleton) *)
43 (* Maximal term depth : 5 ( 3 average) *)
45 (* Comments : There is a 2-element model. *)
47 (* -------------------------------------------------------------------------- *)
49 (* ----Two properties of Boolean algebra: *)
51 (* ----Pixley properties: *)
53 (* ----Denial of a property of Boolean Algebra: *)
54 ntheorem prove_idempotence:
55 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
57 ∀add:∀_:Univ.∀_:Univ.Univ.
58 ∀inverse:∀_:Univ.Univ.
59 ∀multiply:∀_:Univ.∀_:Univ.Univ.
61 ∀H0:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X (inverse Y)) (add (multiply X X) (multiply (inverse Y) X))) X.
62 ∀H1:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X (inverse Y)) (add (multiply X Y) (multiply (inverse Y) Y))) X.
63 ∀H2:∀X:Univ.∀Y:Univ.eq Univ (add (multiply X (inverse X)) (add (multiply X Y) (multiply (inverse X) Y))) Y.
64 ∀H3:∀X:Univ.eq Univ (add X (inverse X)) one.
65 ∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply Y X) (multiply Z X)).eq Univ (add a a) a)
81 nauto by H0,H1,H2,H3,H4 ##;
82 ntry (nassumption) ##;
85 (* -------------------------------------------------------------------------- *)