1 include "logic/equality.ma".
3 (* Inclusion of: GRP012-4.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : GRP012-4 : TPTP v3.7.0. Released v1.0.0. *)
9 (* Domain : Group Theory *)
11 (* Problem : Inverse of products = Product of inverses *)
13 (* Version : [MOW76] (equality) axioms : Augmented. *)
15 (* English : The inverse of products equals the product of the inverse, *)
17 (* in opposite order *)
19 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
25 (* Status : Unsatisfiable *)
27 (* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
29 (* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
31 (* Number of atoms : 6 ( 6 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 : 7 ( 0 singleton) *)
41 (* Maximal term depth : 3 ( 2 average) *)
43 (* Comments : In Lemmas.eq.clauses of [ANL] *)
45 (* -------------------------------------------------------------------------- *)
47 (* ----Include equality group theory axioms *)
49 (* Inclusion of: Axioms/GRP004-0.ax *)
51 (* -------------------------------------------------------------------------- *)
53 (* File : GRP004-0 : TPTP v3.7.0. Released v1.0.0. *)
55 (* Domain : Group Theory *)
57 (* Axioms : Group theory (equality) axioms *)
59 (* Version : [MOW76] (equality) axioms : *)
61 (* Reduced > Complete. *)
65 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
67 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
75 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
77 (* Number of atoms : 3 ( 3 equality) *)
79 (* Maximal clause size : 1 ( 1 average) *)
81 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
83 (* Number of functors : 3 ( 1 constant; 0-2 arity) *)
85 (* Number of variables : 5 ( 0 singleton) *)
87 (* Maximal term depth : 3 ( 2 average) *)
89 (* Comments : [MOW76] also contains redundant right_identity and *)
91 (* right_inverse axioms. *)
93 (* : These axioms are also used in [Wos88] p.186, also with *)
95 (* right_identity and right_inverse. *)
97 (* -------------------------------------------------------------------------- *)
99 (* ----For any x and y in the group x*y is also in the group. No clause *)
101 (* ----is needed here since this is an instance of reflexivity *)
103 (* ----There exists an identity element *)
105 (* ----For any x in the group, there exists an element y such that x*y = y*x *)
107 (* ----= identity. *)
109 (* ----The operation '*' is associative *)
111 (* -------------------------------------------------------------------------- *)
113 (* -------------------------------------------------------------------------- *)
115 (* ----Redundant two axioms *)
116 ntheorem prove_inverse_of_product_is_product_of_inverses:
117 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
121 ∀inverse:∀_:Univ.Univ.
122 ∀multiply:∀_:Univ.∀_:Univ.Univ.
123 ∀H0:∀X:Univ.eq Univ (multiply X (inverse X)) identity.
124 ∀H1:∀X:Univ.eq Univ (multiply X identity) X.
125 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
126 ∀H3:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
127 ∀H4:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (inverse (multiply a b)) (multiply (inverse b) (inverse a)))
143 nauto by H0,H1,H2,H3,H4 ##;
144 ntry (nassumption) ##;
147 (* -------------------------------------------------------------------------- *)