1 include "logic/equality.ma".
3 (* Inclusion of: GRP010-4.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : GRP010-4 : TPTP v3.7.0. Released v1.0.0. *)
9 (* Domain : Group Theory *)
11 (* Problem : Inverse is a symmetric relationship *)
13 (* Version : [Wos65] (equality) axioms : Incomplete. *)
15 (* English : If a is an inverse of b then b is an inverse of a. *)
17 (* Refs : [Wos65] Wos (1965), Unpublished Note *)
19 (* : [Pel86] Pelletier (1986), Seventy-five Problems for Testing Au *)
21 (* Source : [Pel86] *)
23 (* Names : Pelletier 64 [Pel86] *)
25 (* Status : Unsatisfiable *)
27 (* Rating : 0.00 v2.1.0, 0.13 v2.0.0 *)
29 (* Syntax : Number of clauses : 5 ( 0 non-Horn; 5 unit; 2 RR) *)
31 (* Number of atoms : 5 ( 5 equality) *)
33 (* Maximal clause size : 1 ( 1 average) *)
35 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
37 (* Number of functors : 5 ( 3 constant; 0-2 arity) *)
39 (* Number of variables : 5 ( 0 singleton) *)
41 (* Maximal term depth : 3 ( 2 average) *)
43 (* Comments : [Pel86] says "... problems, published I think, by Larry Wos *)
45 (* (but I cannot locate where)." *)
47 (* -------------------------------------------------------------------------- *)
49 (* ----The operation '*' is associative *)
51 (* ----There exists an identity element 'e' defined below. *)
52 ntheorem prove_b_times_c_is_e:
53 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
57 ∀inverse:∀_:Univ.Univ.
58 ∀multiply:∀_:Univ.∀_:Univ.Univ.
59 ∀H0:eq Univ (multiply c b) identity.
60 ∀H1:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
61 ∀H2:∀X:Univ.eq Univ (multiply identity X) X.
62 ∀H3:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).eq Univ (multiply b c) identity)
77 nauto by H0,H1,H2,H3 ##;
78 ntry (nassumption) ##;
81 (* -------------------------------------------------------------------------- *)