1 include "logic/equality.ma".
3 (* Inclusion of: GRP002-2.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : GRP002-2 : TPTP v3.7.0. Bugfixed v1.2.1. *)
9 (* Domain : Group Theory *)
11 (* Problem : Commutator equals identity in groups of order 3 *)
13 (* Version : [MOW76] (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 : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
29 (* Names : commutator.ver2.in [ANL] *)
31 (* Status : Unsatisfiable *)
33 (* Rating : 0.11 v3.4.0, 0.12 v3.3.0, 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.29 v2.0.0 *)
35 (* Syntax : Number of clauses : 12 ( 0 non-Horn; 12 unit; 6 RR) *)
37 (* Number of atoms : 12 ( 12 equality) *)
39 (* Maximal clause size : 1 ( 1 average) *)
41 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
43 (* Number of functors : 10 ( 8 constant; 0-2 arity) *)
45 (* Number of variables : 8 ( 0 singleton) *)
47 (* Maximal term depth : 3 ( 2 average) *)
51 (* Bugfixes : v1.2.1 - Clause x_cubed_is_identity fixed. *)
53 (* -------------------------------------------------------------------------- *)
55 (* Inclusion of: Axioms/GRP004-0.ax *)
57 (* -------------------------------------------------------------------------- *)
59 (* File : GRP004-0 : TPTP v3.7.0. Released v1.0.0. *)
61 (* Domain : Group Theory *)
63 (* Axioms : Group theory (equality) axioms *)
65 (* Version : [MOW76] (equality) axioms : *)
67 (* Reduced > Complete. *)
71 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
73 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
81 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
83 (* Number of atoms : 3 ( 3 equality) *)
85 (* Maximal clause size : 1 ( 1 average) *)
87 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
89 (* Number of functors : 3 ( 1 constant; 0-2 arity) *)
91 (* Number of variables : 5 ( 0 singleton) *)
93 (* Maximal term depth : 3 ( 2 average) *)
95 (* Comments : [MOW76] also contains redundant right_identity and *)
97 (* right_inverse axioms. *)
99 (* : These axioms are also used in [Wos88] p.186, also with *)
101 (* right_identity and right_inverse. *)
103 (* -------------------------------------------------------------------------- *)
105 (* ----For any x and y in the group x*y is also in the group. No clause *)
107 (* ----is needed here since this is an instance of reflexivity *)
109 (* ----There exists an identity element *)
111 (* ----For any x in the group, there exists an element y such that x*y = y*x *)
113 (* ----= identity. *)
115 (* ----The operation '*' is associative *)
117 (* -------------------------------------------------------------------------- *)
119 (* -------------------------------------------------------------------------- *)
121 (* ----Redundant two axioms, but established in standard axiomatizations. *)
123 (* ----This hypothesis is omitted in the ANL source version *)
124 ntheorem prove_k_times_inverse_b_is_e:
125 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
132 ∀inverse:∀_:Univ.Univ.
135 ∀multiply:∀_:Univ.∀_:Univ.Univ.
136 ∀H0:eq Univ (multiply j (inverse h)) k.
137 ∀H1:eq Univ (multiply h b) j.
138 ∀H2:eq Univ (multiply d (inverse b)) h.
139 ∀H3:eq Univ (multiply c (inverse a)) d.
140 ∀H4:eq Univ (multiply a b) c.
141 ∀H5:∀X:Univ.eq Univ (multiply X (multiply X X)) identity.
142 ∀H6:∀X:Univ.eq Univ (multiply X (inverse X)) identity.
143 ∀H7:∀X:Univ.eq Univ (multiply X identity) X.
144 ∀H8:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
145 ∀H9:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
146 ∀H10:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply k (inverse b)) identity)
173 nauto by H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10 ##;
174 ntry (nassumption) ##;
177 (* -------------------------------------------------------------------------- *)