1 include "logic/equality.ma".
3 (* Inclusion of: GRP023-2.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : GRP023-2 : TPTP v3.7.0. Released v1.0.0. *)
9 (* Domain : Group Theory *)
11 (* Problem : The inverse of the identity is the identity *)
13 (* Version : [MOW76] (equality) axioms : Augmented. *)
17 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
21 (* Names : Established lemma [MOW76] *)
23 (* Status : Unsatisfiable *)
25 (* Rating : 0.00 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; 0-2 arity) *)
37 (* Number of variables : 7 ( 0 singleton) *)
39 (* Maximal term depth : 3 ( 2 average) *)
43 (* -------------------------------------------------------------------------- *)
45 (* ----Include equality group theory axioms *)
47 (* Inclusion of: Axioms/GRP004-0.ax *)
49 (* -------------------------------------------------------------------------- *)
51 (* File : GRP004-0 : TPTP v3.7.0. Released v1.0.0. *)
53 (* Domain : Group Theory *)
55 (* Axioms : Group theory (equality) axioms *)
57 (* Version : [MOW76] (equality) axioms : *)
59 (* Reduced > Complete. *)
63 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
65 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
73 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
75 (* Number of atoms : 3 ( 3 equality) *)
77 (* Maximal clause size : 1 ( 1 average) *)
79 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
81 (* Number of functors : 3 ( 1 constant; 0-2 arity) *)
83 (* Number of variables : 5 ( 0 singleton) *)
85 (* Maximal term depth : 3 ( 2 average) *)
87 (* Comments : [MOW76] also contains redundant right_identity and *)
89 (* right_inverse axioms. *)
91 (* : These axioms are also used in [Wos88] p.186, also with *)
93 (* right_identity and right_inverse. *)
95 (* -------------------------------------------------------------------------- *)
97 (* ----For any x and y in the group x*y is also in the group. No clause *)
99 (* ----is needed here since this is an instance of reflexivity *)
101 (* ----There exists an identity element *)
103 (* ----For any x in the group, there exists an element y such that x*y = y*x *)
105 (* ----= identity. *)
107 (* ----The operation '*' is associative *)
109 (* -------------------------------------------------------------------------- *)
111 (* -------------------------------------------------------------------------- *)
113 (* ----Redundant two axioms *)
114 ntheorem prove_inverse_of_id_is_id:
115 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
117 ∀inverse:∀_:Univ.Univ.
118 ∀multiply:∀_:Univ.∀_:Univ.Univ.
119 ∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) identity.
120 ∀H1:∀X:Univ.eq Univ (multiply X identity) X.
121 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
122 ∀H3:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
123 ∀H4:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (inverse identity) identity)
137 nauto by H0,H1,H2,H3,H4 ##;
138 ntry (nassumption) ##;
141 (* -------------------------------------------------------------------------- *)