1 include "logic/equality.ma".
3 (* Inclusion of: ROB024-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : ROB024-1 : TPTP v3.7.0. Released v1.0.0. *)
9 (* Domain : Robbins Algebra *)
11 (* Problem : -(a + (a + b)) + -(a + -b) = a => Boolean *)
13 (* Version : [Win90] (equality) axioms. *)
15 (* English : If there exist a and b so that -(a + (a + b)) + -(a + -b) *)
17 (* = a then the algebra is Boolean. *)
19 (* Refs : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
21 (* : [WW+90] Wos et al. (1990), Automated Reasoning Contributes to *)
23 (* Source : [WW+90] *)
25 (* Names : RA-1 [WW+90] *)
27 (* Status : Unknown *)
29 (* Rating : 1.00 v2.0.0 *)
31 (* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
33 (* Number of atoms : 5 ( 5 equality) *)
35 (* Maximal clause size : 1 ( 1 average) *)
37 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
39 (* Number of functors : 4 ( 2 constant; 0-2 arity) *)
41 (* Number of variables : 7 ( 0 singleton) *)
43 (* Maximal term depth : 6 ( 3 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.
103 ∀negate:∀_:Univ.Univ.
104 ∀H0:eq Univ (negate (add (negate (add a (add a b))) (negate (add a (negate b))))) a.
105 ∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
106 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
107 ∀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)
121 nauto by H0,H1,H2,H3 ##;
122 ntry (nassumption) ##;
125 (* -------------------------------------------------------------------------- *)