1 include "logic/equality.ma".
3 (* Inclusion of: GRP002-4.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : GRP002-4 : TPTP v3.7.0. Released v1.0.0. *)
9 (* Domain : Group Theory *)
11 (* Problem : Commutator equals identity in groups of order 3 *)
13 (* Version : [MOW76] (equality) axioms. *)
15 (* Theorem formulation : Explicit formulation of the commutator. *)
17 (* English : In a group, if (for all x) the cube of x is the identity *)
19 (* (i.e. a group of order 3), then the equation [[x,y],y]= *)
21 (* identity holds, where [x,y] is the product of x, y, the *)
23 (* inverse of x and the inverse of y (i.e. the commutator *)
27 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
29 (* : [LO85] Lusk & Overbeek (1985), Reasoning about Equality *)
31 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
33 (* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
37 (* Names : Problem 4 [LO85] *)
39 (* : Test Problem 2 [Wos88] *)
41 (* : Commutator Theorem [Wos88] *)
45 (* Status : Unsatisfiable *)
47 (* Rating : 0.11 v3.4.0, 0.12 v3.3.0, 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 : 8 ( 0 non-Horn; 8 unit; 1 RR) *)
51 (* Number of atoms : 8 ( 8 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 : 10 ( 0 singleton) *)
61 (* Maximal term depth : 5 ( 2 average) *)
65 (* -------------------------------------------------------------------------- *)
67 (* ----Include group theory axioms *)
69 (* Inclusion of: Axioms/GRP004-0.ax *)
71 (* -------------------------------------------------------------------------- *)
73 (* File : GRP004-0 : TPTP v3.7.0. Released v1.0.0. *)
75 (* Domain : Group Theory *)
77 (* Axioms : Group theory (equality) axioms *)
79 (* Version : [MOW76] (equality) axioms : *)
81 (* Reduced > Complete. *)
85 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
87 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
95 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
97 (* Number of atoms : 3 ( 3 equality) *)
99 (* Maximal clause size : 1 ( 1 average) *)
101 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
103 (* Number of functors : 3 ( 1 constant; 0-2 arity) *)
105 (* Number of variables : 5 ( 0 singleton) *)
107 (* Maximal term depth : 3 ( 2 average) *)
109 (* Comments : [MOW76] also contains redundant right_identity and *)
111 (* right_inverse axioms. *)
113 (* : These axioms are also used in [Wos88] p.186, also with *)
115 (* right_identity and right_inverse. *)
117 (* -------------------------------------------------------------------------- *)
119 (* ----For any x and y in the group x*y is also in the group. No clause *)
121 (* ----is needed here since this is an instance of reflexivity *)
123 (* ----There exists an identity element *)
125 (* ----For any x in the group, there exists an element y such that x*y = y*x *)
127 (* ----= identity. *)
129 (* ----The operation '*' is associative *)
131 (* -------------------------------------------------------------------------- *)
133 (* -------------------------------------------------------------------------- *)
135 (* ----Redundant two axioms, but used in established axiomatizations. *)
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.eq Univ (multiply X (inverse X)) identity.
149 ∀H3:∀X:Univ.eq Univ (multiply X identity) X.
150 ∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
151 ∀H5:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
152 ∀H6:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (commutator (commutator a b) b) identity)
171 nauto by H0,H1,H2,H3,H4,H5,H6 ##;
172 ntry (nassumption) ##;
175 (* -------------------------------------------------------------------------- *)