1 include "logic/equality.ma".
3 (* Inclusion of: ROB004-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : ROB004-1 : TPTP v3.7.0. Released v1.0.0. *)
9 (* Domain : Robbins Algebra *)
11 (* Problem : c = -d, c + d=d, and c + c=c => Boolean *)
13 (* Version : [Win90] (equality) axioms. *)
15 (* English : If there exist c, d such that c = -d, c+d=d, and c+c=c, then *)
17 (* the algebra is Boolean. *)
19 (* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
21 (* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
23 (* Source : [Win90] *)
25 (* Names : Lemma 2.3 [Win90] *)
27 (* Status : Unsatisfiable *)
29 (* Rating : 0.00 v3.4.0, 0.12 v3.3.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
31 (* Syntax : Number of clauses : 7 ( 0 non-Horn; 7 unit; 4 RR) *)
33 (* Number of atoms : 7 ( 7 equality) *)
35 (* Maximal clause size : 1 ( 1 average) *)
37 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
39 (* Number of functors : 6 ( 4 constant; 0-2 arity) *)
41 (* Number of variables : 7 ( 0 singleton) *)
43 (* Maximal term depth : 6 ( 2 average) *)
45 (* Comments : Commutativity, associativity, and Huntington's axiom *)
47 (* axiomatize Boolean algebra. *)
49 (* -------------------------------------------------------------------------- *)
51 (* ----Include axioms for Robbins algebra *)
53 (* Inclusion of: Axioms/ROB001-0.ax *)
55 (* -------------------------------------------------------------------------- *)
57 (* File : ROB001-0 : TPTP v3.7.0. Released v1.0.0. *)
59 (* Domain : Robbins algebra *)
61 (* Axioms : Robbins algebra axioms *)
63 (* Version : [Win90] (equality) axioms. *)
67 (* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
69 (* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
71 (* Source : [OTTER] *)
73 (* Names : Lemma 2.2 [Win90] *)
77 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
79 (* Number of atoms : 3 ( 3 equality) *)
81 (* Maximal clause size : 1 ( 1 average) *)
83 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
85 (* Number of functors : 2 ( 0 constant; 1-2 arity) *)
87 (* Number of variables : 7 ( 0 singleton) *)
89 (* Maximal term depth : 6 ( 3 average) *)
93 (* -------------------------------------------------------------------------- *)
95 (* -------------------------------------------------------------------------- *)
97 (* -------------------------------------------------------------------------- *)
98 ntheorem prove_huntingtons_axiom:
99 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
101 ∀add:∀_:Univ.∀_:Univ.Univ.
105 ∀negate:∀_:Univ.Univ.
106 ∀H0:eq Univ (add c c) c.
107 ∀H1:eq Univ (add c d) d.
108 ∀H2:eq Univ (negate d) c.
109 ∀H3:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
110 ∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
111 ∀H5:∀X:Univ.∀Y:Univ.eq Univ (add X Y) (add Y X).eq Univ (add (negate (add a (negate b))) (negate (add (negate a) (negate b)))) b)
129 nauto by H0,H1,H2,H3,H4,H5 ##;
130 ntry (nassumption) ##;
133 (* -------------------------------------------------------------------------- *)