2 include "logic/equality.ma".
3 (* Inclusion of: GRP023-2.p *)
4 (* -------------------------------------------------------------------------- *)
5 (* File : GRP023-2 : TPTP v3.1.1. Released v1.0.0. *)
6 (* Domain : Group Theory *)
7 (* Problem : The inverse of the identity is the identity *)
8 (* Version : [MOW76] (equality) axioms : Augmented. *)
10 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
12 (* Names : Established lemma [MOW76] *)
13 (* Status : Unsatisfiable *)
14 (* Rating : 0.00 v2.0.0 *)
15 (* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
16 (* Number of atoms : 6 ( 6 equality) *)
17 (* Maximal clause size : 1 ( 1 average) *)
18 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
19 (* Number of functors : 3 ( 1 constant; 0-2 arity) *)
20 (* Number of variables : 7 ( 0 singleton) *)
21 (* Maximal term depth : 3 ( 2 average) *)
23 (* -------------------------------------------------------------------------- *)
24 (* ----Include equality group theory axioms *)
25 (* Inclusion of: Axioms/GRP004-0.ax *)
26 (* -------------------------------------------------------------------------- *)
27 (* File : GRP004-0 : TPTP v3.1.1. Released v1.0.0. *)
28 (* Domain : Group Theory *)
29 (* Axioms : Group theory (equality) axioms *)
30 (* Version : [MOW76] (equality) axioms : *)
31 (* Reduced > Complete. *)
33 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
34 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
38 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
39 (* Number of literals : 3 ( 3 equality) *)
40 (* Maximal clause size : 1 ( 1 average) *)
41 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
42 (* Number of functors : 3 ( 1 constant; 0-2 arity) *)
43 (* Number of variables : 5 ( 0 singleton) *)
44 (* Maximal term depth : 3 ( 2 average) *)
45 (* Comments : [MOW76] also contains redundant right_identity and *)
46 (* right_inverse axioms. *)
47 (* : These axioms are also used in [Wos88] p.186, also with *)
48 (* right_identity and right_inverse. *)
49 (* -------------------------------------------------------------------------- *)
50 (* ----For any x and y in the group x*y is also in the group. No clause *)
51 (* ----is needed here since this is an instance of reflexivity *)
52 (* ----There exists an identity element *)
53 (* ----For any x in the group, there exists an element y such that x*y = y*x *)
55 (* ----The operation '*' is associative *)
56 (* -------------------------------------------------------------------------- *)
57 (* -------------------------------------------------------------------------- *)
58 (* ----Redundant two axioms *)
59 theorem prove_inverse_of_id_is_id:
61 \forall identity:Univ.
62 \forall inverse:\forall _:Univ.Univ.
63 \forall multiply:\forall _:Univ.\forall _:Univ.Univ.
64 \forall H0:\forall X:Univ.eq Univ (multiply X (inverse X)) identity.
65 \forall H1:\forall X:Univ.eq Univ (multiply X identity) X.
66 \forall H2:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
67 \forall H3:\forall X:Univ.eq Univ (multiply (inverse X) X) identity.
68 \forall H4:\forall X:Univ.eq Univ (multiply identity X) X.eq Univ (inverse identity) identity
71 autobatch paramodulation timeout=100;
75 (* -------------------------------------------------------------------------- *)