1 include "logic/equality.ma".
3 (* Inclusion of: BOO001-1.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : BOO001-1 : TPTP v3.7.0. Released v1.0.0. *)
9 (* Domain : Boolean Algebra (Ternary) *)
11 (* Problem : In B3 algebra, inverse is an involution *)
13 (* Version : [OTTER] (equality) axioms. *)
19 (* Source : [OTTER] *)
21 (* Names : tba_gg.in [OTTER] *)
23 (* Status : Unsatisfiable *)
25 (* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.14 v2.1.0, 0.25 v2.0.0 *)
27 (* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
29 (* Number of atoms : 6 ( 6 equality) *)
31 (* Maximal clause size : 1 ( 1 average) *)
33 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
35 (* Number of functors : 3 ( 1 constant; 0-3 arity) *)
37 (* Number of variables : 13 ( 2 singleton) *)
39 (* Maximal term depth : 3 ( 2 average) *)
43 (* -------------------------------------------------------------------------- *)
45 (* ----Include ternary Boolean algebra axioms *)
47 (* Inclusion of: Axioms/BOO001-0.ax *)
49 (* -------------------------------------------------------------------------- *)
51 (* File : BOO001-0 : TPTP v3.7.0. Released v1.0.0. *)
53 (* Domain : Algebra (Ternary Boolean) *)
55 (* Axioms : Ternary Boolean algebra (equality) axioms *)
57 (* Version : [OTTER] (equality) axioms. *)
61 (* Refs : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
63 (* : [Win82] Winker (1982), Generation and Verification of Finite M *)
65 (* Source : [OTTER] *)
71 (* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 0 RR) *)
73 (* Number of atoms : 5 ( 5 equality) *)
75 (* Maximal clause size : 1 ( 1 average) *)
77 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
79 (* Number of functors : 2 ( 0 constant; 1-3 arity) *)
81 (* Number of variables : 13 ( 2 singleton) *)
83 (* Maximal term depth : 3 ( 2 average) *)
85 (* Comments : These axioms appear in [Win82], in which ternary_multiply_1 is *)
87 (* shown to be independant. *)
89 (* : These axioms are also used in [Wos88], p.222. *)
91 (* -------------------------------------------------------------------------- *)
93 (* -------------------------------------------------------------------------- *)
95 (* -------------------------------------------------------------------------- *)
96 ntheorem prove_inverse_is_self_cancelling:
97 (∀Univ:Type.∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.
99 ∀inverse:∀_:Univ.Univ.
100 ∀multiply:∀_:Univ.∀_:Univ.∀_:Univ.Univ.
101 ∀H0:∀X:Univ.∀Y:Univ.eq Univ (multiply X Y (inverse Y)) X.
102 ∀H1:∀X:Univ.∀Y:Univ.eq Univ (multiply (inverse Y) Y X) X.
103 ∀H2:∀X:Univ.∀Y:Univ.eq Univ (multiply X X Y) X.
104 ∀H3:∀X:Univ.∀Y:Univ.eq Univ (multiply Y X X) X.
105 ∀H4:∀V:Univ.∀W:Univ.∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply V W X) Y (multiply V W Z)) (multiply V W (multiply X Y Z)).eq Univ (inverse (inverse a)) a)
121 nauto by H0,H1,H2,H3,H4 ##;
122 ntry (nassumption) ##;
125 (* -------------------------------------------------------------------------- *)