1 include "logic/equality.ma".
3 (* Inclusion of: ROB027-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : ROB027-1 : TPTP v3.7.0. Released v1.2.0. *)
9 (* Domain : Robbins Algebra *)
11 (* Problem : -(-c) = c => Boolean *)
13 (* Version : [Win90] (equality) axioms. *)
15 (* Theorem formulation : Denies Huntington's axiom. *)
17 (* English : If there are elements c and d such that c+d=d, then the *)
19 (* algebra is Boolean. *)
21 (* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
23 (* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
25 (* : [Wos94] Wos (1994), Two Challenge Problems *)
27 (* Source : [Wos94] *)
29 (* Names : - [Wos94] *)
33 (* Rating : 1.00 v2.0.0 *)
35 (* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
37 (* Number of atoms : 5 ( 5 equality) *)
39 (* Maximal clause size : 1 ( 1 average) *)
41 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
43 (* Number of functors : 5 ( 3 constant; 0-2 arity) *)
45 (* Number of variables : 7 ( 0 singleton) *)
47 (* Maximal term depth : 6 ( 3 average) *)
49 (* Comments : Commutativity, associativity, and Huntington's axiom *)
51 (* axiomatize Boolean algebra. *)
53 (* -------------------------------------------------------------------------- *)
55 (* ----Include axioms for Robbins algebra *)
57 (* Inclusion of: Axioms/ROB001-0.ax *)
59 (* -------------------------------------------------------------------------- *)
61 (* File : ROB001-0 : TPTP v3.7.0. Released v1.0.0. *)
63 (* Domain : Robbins algebra *)
65 (* Axioms : Robbins algebra axioms *)
67 (* Version : [Win90] (equality) axioms. *)
71 (* Refs : [HMT71] Henkin et al. (1971), Cylindrical Algebras *)
73 (* : [Win90] Winker (1990), Robbins Algebra: Conditions that make a *)
75 (* Source : [OTTER] *)
77 (* Names : Lemma 2.2 [Win90] *)
81 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
83 (* Number of atoms : 3 ( 3 equality) *)
85 (* Maximal clause size : 1 ( 1 average) *)
87 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
89 (* Number of functors : 2 ( 0 constant; 1-2 arity) *)
91 (* Number of variables : 7 ( 0 singleton) *)
93 (* Maximal term depth : 6 ( 3 average) *)
97 (* -------------------------------------------------------------------------- *)
99 (* -------------------------------------------------------------------------- *)
101 (* -------------------------------------------------------------------------- *)
102 ntheorem prove_huntingtons_axiom:
103 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
105 ∀add:∀_:Univ.∀_:Univ.Univ.
108 ∀negate:∀_:Univ.Univ.
109 ∀H0:eq Univ (negate (negate c)) c.
110 ∀H1:∀X:Univ.∀Y:Univ.eq Univ (negate (add (negate (add X Y)) (negate (add X (negate Y))))) X.
111 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (add (add X Y) Z) (add X (add Y Z)).
112 ∀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)
127 nauto by H0,H1,H2,H3 ##;
128 ntry (nassumption) ##;
131 (* -------------------------------------------------------------------------- *)