1 set "baseuri" "cic:/matita/TPTP/GRP022-2".
2 include "logic/equality.ma".
3 (* Inclusion of: GRP022-2.p *)
4 (* -------------------------------------------------------------------------- *)
5 (* File : GRP022-2 : TPTP v3.1.1. Released v1.0.0. *)
6 (* Domain : Group Theory *)
7 (* Problem : Inverse is an involution *)
8 (* Version : [MOW76] (equality) axioms : Augmented. *)
10 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
11 (* : [LO85] Lusk & Overbeek (1985), Reasoning about Equality *)
13 (* Names : Established lemma [MOW76] *)
14 (* : Problem 2 [LO85] *)
15 (* Status : Unsatisfiable *)
16 (* Rating : 0.00 v2.0.0 *)
17 (* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
18 (* Number of atoms : 6 ( 6 equality) *)
19 (* Maximal clause size : 1 ( 1 average) *)
20 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
21 (* Number of functors : 4 ( 2 constant; 0-2 arity) *)
22 (* Number of variables : 7 ( 0 singleton) *)
23 (* Maximal term depth : 3 ( 2 average) *)
25 (* -------------------------------------------------------------------------- *)
26 (* ----Include equality group theory axioms *)
27 (* Inclusion of: Axioms/GRP004-0.ax *)
28 (* -------------------------------------------------------------------------- *)
29 (* File : GRP004-0 : TPTP v3.1.1. Released v1.0.0. *)
30 (* Domain : Group Theory *)
31 (* Axioms : Group theory (equality) axioms *)
32 (* Version : [MOW76] (equality) axioms : *)
33 (* Reduced > Complete. *)
35 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
36 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
40 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
41 (* Number of literals : 3 ( 3 equality) *)
42 (* Maximal clause size : 1 ( 1 average) *)
43 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
44 (* Number of functors : 3 ( 1 constant; 0-2 arity) *)
45 (* Number of variables : 5 ( 0 singleton) *)
46 (* Maximal term depth : 3 ( 2 average) *)
47 (* Comments : [MOW76] also contains redundant right_identity and *)
48 (* right_inverse axioms. *)
49 (* : These axioms are also used in [Wos88] p.186, also with *)
50 (* right_identity and right_inverse. *)
51 (* -------------------------------------------------------------------------- *)
52 (* ----For any x and y in the group x*y is also in the group. No clause *)
53 (* ----is needed here since this is an instance of reflexivity *)
54 (* ----There exists an identity element *)
55 (* ----For any x in the group, there exists an element y such that x*y = y*x *)
57 (* ----The operation '*' is associative *)
58 (* -------------------------------------------------------------------------- *)
59 (* -------------------------------------------------------------------------- *)
60 (* ----Redundant two axioms *)
61 theorem prove_inverse_of_inverse_is_original:
64 \forall identity:Univ.
65 \forall inverse:\forall _:Univ.Univ.
66 \forall multiply:\forall _:Univ.\forall _:Univ.Univ.
67 \forall H0:\forall X:Univ.eq Univ (multiply X (inverse X)) identity.
68 \forall H1:\forall X:Univ.eq Univ (multiply X identity) X.
69 \forall H2:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
70 \forall H3:\forall X:Univ.eq Univ (multiply (inverse X) X) identity.
71 \forall H4:\forall X:Univ.eq Univ (multiply identity X) X.eq Univ (inverse (inverse a)) a
74 autobatch paramodulation timeout=100.
78 (* -------------------------------------------------------------------------- *)