1 include "logic/equality.ma".
3 (* Inclusion of: ROB003-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : ROB003-1 : TPTP v3.2.0. Released v1.0.0. *)
9 (* Domain : Robbins Algebra *)
11 (* Problem : X + c=c => Boolean *)
13 (* Version : [Win90] (equality) axioms. *)
15 (* English : If there exists c such that X+c=c, then the algebra *)
19 (* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
21 (* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
23 (* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
25 (* Source : [OTTER] *)
27 (* Names : Lemma 2.2 [Win90] *)
31 (* : robbins.in [OTTER] *)
33 (* Status : Unsatisfiable *)
35 (* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
37 (* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 1 RR) *)
39 (* Number of atoms : 5 ( 5 equality) *)
41 (* Maximal clause size : 1 ( 1 average) *)
43 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
45 (* Number of functors : 5 ( 3 constant; 0-2 arity) *)
47 (* Number of variables : 8 ( 1 singleton) *)
49 (* Maximal term depth : 6 ( 3 average) *)
51 (* Comments : Commutativity, associativity, and Huntington's axiom *)
53 (* axiomatize Boolean algebra. *)
55 (* : In Overbeek's version, the hypothesis is slightly different : *)
57 (* ...an element c such that c+c=c, then... Mail from McCune says *)
59 (* that this is a simpler problem. *)
61 (* -------------------------------------------------------------------------- *)
63 (* ----Include axioms for Robbins algebra *)
65 (* Inclusion of: Axioms/ROB001-0.ax *)
67 (* -------------------------------------------------------------------------- *)
69 (* File : ROB001-0 : TPTP v3.2.0. Released v1.0.0. *)
71 (* Domain : Robbins algebra *)
73 (* Axioms : Robbins algebra axioms *)
75 (* Version : [Win90] (equality) axioms. *)
79 (* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
81 (* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
83 (* Source : [OTTER] *)
85 (* Names : Lemma 2.2 [Win90] *)
89 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
91 (* Number of literals : 3 ( 3 equality) *)
93 (* Maximal clause size : 1 ( 1 average) *)
95 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
97 (* Number of functors : 2 ( 0 constant; 1-2 arity) *)
99 (* Number of variables : 7 ( 0 singleton) *)
101 (* Maximal term depth : 6 ( 3 average) *)
105 (* -------------------------------------------------------------------------- *)
107 (* -------------------------------------------------------------------------- *)
109 (* -------------------------------------------------------------------------- *)
110 ntheorem prove_huntingtons_axiom:
111 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
113 ∀add:∀_:Univ.∀_:Univ.Univ.
116 ∀negate:∀_:Univ.Univ.
117 ∀H0:∀X:Univ.eq Univ (add X c) c.
118 ∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
119 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
120 ∀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)
135 nauto by H0,H1,H2,H3 ##;
138 (* -------------------------------------------------------------------------- *)