1 include "logic/equality.ma".
3 (* Inclusion of: GRP002-3.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : GRP002-3 : TPTP v3.2.0. Released v1.0.0. *)
9 (* Domain : Group Theory *)
11 (* Problem : Commutator equals identity in groups of order 3 *)
13 (* Version : [Ove90] (equality) axioms. *)
15 (* English : In a group, if (for all x) the cube of x is the identity *)
17 (* (i.e. a group of order 3), then the equation [[x,y],y]= *)
19 (* identity holds, where [x,y] is the product of x, y, the *)
21 (* inverse of x and the inverse of y (i.e. the commutator *)
25 (* Refs : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal *)
27 (* : [LM93] Lusk & McCune (1993), Uniform Strategies: The CADE-11 *)
29 (* : [Zha93] Zhang (1993), Automated Proofs of Equality Problems in *)
31 (* : [Ove90] Overbeek (1990), ATP competition announced at CADE-10 *)
33 (* : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
35 (* Source : [Ove90] *)
37 (* Names : CADE-11 Competition Eq-1 [Ove90] *)
39 (* : THEOREM EQ-1 [LM93] *)
41 (* : PROBLEM 1 [Zha93] *)
43 (* : comm.in [OTTER] *)
45 (* Status : Unsatisfiable *)
47 (* Rating : 0.00 v3.1.0, 0.11 v2.7.0, 0.00 v2.2.1, 0.33 v2.2.0, 0.43 v2.1.0, 0.25 v2.0.0 *)
49 (* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
51 (* Number of atoms : 6 ( 6 equality) *)
53 (* Maximal clause size : 1 ( 1 average) *)
55 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
57 (* Number of functors : 6 ( 3 constant; 0-2 arity) *)
59 (* Number of variables : 8 ( 0 singleton) *)
61 (* Maximal term depth : 5 ( 2 average) *)
63 (* Comments : Uses an explicit formulation of the commutator. *)
65 (* : Same axioms as [MOW76] (equality) axioms. *)
67 (* -------------------------------------------------------------------------- *)
69 (* ----Include group theory axioms *)
71 (* Inclusion of: Axioms/GRP004-0.ax *)
73 (* -------------------------------------------------------------------------- *)
75 (* File : GRP004-0 : TPTP v3.2.0. Released v1.0.0. *)
77 (* Domain : Group Theory *)
79 (* Axioms : Group theory (equality) axioms *)
81 (* Version : [MOW76] (equality) axioms : *)
83 (* Reduced > Complete. *)
87 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
89 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
97 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
99 (* Number of literals : 3 ( 3 equality) *)
101 (* Maximal clause size : 1 ( 1 average) *)
103 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
105 (* Number of functors : 3 ( 1 constant; 0-2 arity) *)
107 (* Number of variables : 5 ( 0 singleton) *)
109 (* Maximal term depth : 3 ( 2 average) *)
111 (* Comments : [MOW76] also contains redundant right_identity and *)
113 (* right_inverse axioms. *)
115 (* : These axioms are also used in [Wos88] p.186, also with *)
117 (* right_identity and right_inverse. *)
119 (* -------------------------------------------------------------------------- *)
121 (* ----For any x and y in the group x*y is also in the group. No clause *)
123 (* ----is needed here since this is an instance of reflexivity *)
125 (* ----There exists an identity element *)
127 (* ----For any x in the group, there exists an element y such that x*y = y*x *)
129 (* ----= identity. *)
131 (* ----The operation '*' is associative *)
133 (* -------------------------------------------------------------------------- *)
135 (* -------------------------------------------------------------------------- *)
137 (* ----Definition of the commutator *)
138 ntheorem prove_commutator:
139 ∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
142 ∀commutator:∀_:Univ.∀_:Univ.Univ.
144 ∀inverse:∀_:Univ.Univ.
145 ∀multiply:∀_:Univ.∀_:Univ.Univ.
146 ∀H0:∀X:Univ.eq Univ (multiply X (multiply X X)) identity.
147 ∀H1:∀X:Univ.∀Y:Univ.eq Univ (commutator X Y) (multiply X (multiply Y (multiply (inverse X) (inverse Y)))).
148 ∀H2:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
149 ∀H3:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
150 ∀H4:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (commutator (commutator a b) b) identity
167 nauto by H0,H1,H2,H3,H4;
170 (* -------------------------------------------------------------------------- *)