1 include "logic/equality.ma".
3 (* Inclusion of: ROB023-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : ROB023-1 : TPTP v3.7.0. Released v1.0.0. *)
9 (* Domain : Robbins Algebra *)
11 (* Problem : X + X=X => Boolean *)
13 (* Version : [Win90] (equality) axioms. *)
15 (* English : If for all X X + X = X then the algebra is Boolean. *)
17 (* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
19 (* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
21 (* : [LM92] Lusk & McCune (1992), Experiments with ROO, a Parallel *)
23 (* : [McC92] McCune (1992), Email to G. Sutcliffe *)
25 (* Source : [McC92] *)
27 (* Names : Robbins [LM92] *)
29 (* Status : Unsatisfiable *)
31 (* Rating : 0.11 v3.4.0, 0.12 v3.3.0, 0.00 v2.2.1, 0.44 v2.2.0, 0.43 v2.1.0, 0.38 v2.0.0 *)
33 (* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
35 (* Number of atoms : 5 ( 5 equality) *)
37 (* Maximal clause size : 1 ( 1 average) *)
39 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
41 (* Number of functors : 4 ( 2 constant; 0-2 arity) *)
43 (* Number of variables : 8 ( 0 singleton) *)
45 (* Maximal term depth : 6 ( 3 average) *)
47 (* Comments : Commutativity, associativity, and Huntington's axiom *)
49 (* axiomatize Boolean algebra. *)
51 (* -------------------------------------------------------------------------- *)
53 (* ----Include axioms for Robbins algebra *)
55 (* Inclusion of: Axioms/ROB001-0.ax *)
57 (* -------------------------------------------------------------------------- *)
59 (* File : ROB001-0 : TPTP v3.7.0. Released v1.0.0. *)
61 (* Domain : Robbins algebra *)
63 (* Axioms : Robbins algebra axioms *)
65 (* Version : [Win90] (equality) axioms. *)
69 (* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
71 (* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
73 (* Source : [OTTER] *)
75 (* Names : Lemma 2.2 [Win90] *)
79 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
81 (* Number of atoms : 3 ( 3 equality) *)
83 (* Maximal clause size : 1 ( 1 average) *)
85 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
87 (* Number of functors : 2 ( 0 constant; 1-2 arity) *)
89 (* Number of variables : 7 ( 0 singleton) *)
91 (* Maximal term depth : 6 ( 3 average) *)
95 (* -------------------------------------------------------------------------- *)
97 (* -------------------------------------------------------------------------- *)
99 (* -------------------------------------------------------------------------- *)
100 ntheorem prove_huntingtons_axiom:
101 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
103 ∀add:∀_:Univ.∀_:Univ.Univ.
105 ∀negate:∀_:Univ.Univ.
106 ∀H0:∀X:Univ.eq Univ (add X X) X.
107 ∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
108 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
109 ∀H3:∀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)
123 nauto by H0,H1,H2,H3 ##;
124 ntry (nassumption) ##;
127 (* -------------------------------------------------------------------------- *)