1 include "logic/equality.ma".
3 (* Inclusion of: GRP022-2.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : GRP022-2 : TPTP v3.7.0. Released v1.0.0. *)
9 (* Domain : Group Theory *)
11 (* Problem : Inverse is an involution *)
13 (* Version : [MOW76] (equality) axioms : Augmented. *)
17 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
19 (* : [LO85] Lusk & Overbeek (1985), Reasoning about Equality *)
23 (* Names : Established lemma [MOW76] *)
25 (* : Problem 2 [LO85] *)
27 (* Status : Unsatisfiable *)
29 (* Rating : 0.00 v2.0.0 *)
31 (* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
33 (* Number of atoms : 6 ( 6 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 : 3 ( 2 average) *)
47 (* -------------------------------------------------------------------------- *)
49 (* ----Include equality group theory axioms *)
51 (* Inclusion of: Axioms/GRP004-0.ax *)
53 (* -------------------------------------------------------------------------- *)
55 (* File : GRP004-0 : TPTP v3.7.0. Released v1.0.0. *)
57 (* Domain : Group Theory *)
59 (* Axioms : Group theory (equality) axioms *)
61 (* Version : [MOW76] (equality) axioms : *)
63 (* Reduced > Complete. *)
67 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
69 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
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 : 3 ( 1 constant; 0-2 arity) *)
87 (* Number of variables : 5 ( 0 singleton) *)
89 (* Maximal term depth : 3 ( 2 average) *)
91 (* Comments : [MOW76] also contains redundant right_identity and *)
93 (* right_inverse axioms. *)
95 (* : These axioms are also used in [Wos88] p.186, also with *)
97 (* right_identity and right_inverse. *)
99 (* -------------------------------------------------------------------------- *)
101 (* ----For any x and y in the group x*y is also in the group. No clause *)
103 (* ----is needed here since this is an instance of reflexivity *)
105 (* ----There exists an identity element *)
107 (* ----For any x in the group, there exists an element y such that x*y = y*x *)
109 (* ----= identity. *)
111 (* ----The operation '*' is associative *)
113 (* -------------------------------------------------------------------------- *)
115 (* -------------------------------------------------------------------------- *)
117 (* ----Redundant two axioms *)
118 ntheorem prove_inverse_of_inverse_is_original:
119 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
122 ∀inverse:∀_:Univ.Univ.
123 ∀multiply:∀_:Univ.∀_:Univ.Univ.
124 ∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) identity.
125 ∀H1:∀X:Univ.eq Univ (multiply X identity) X.
126 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
127 ∀H3:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
128 ∀H4:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (inverse (inverse a)) a)
143 nauto by H0,H1,H2,H3,H4 ##;
144 ntry (nassumption) ##;
147 (* -------------------------------------------------------------------------- *)