grp_pow and related things

[jdchristensen]

Based on a comment I made in in #2000 (comment)_

I think issemigrouppreserving_mult_rng_int_mult can be proven by combining some pieces that will be useful on their own. See below for a sketch.

Group.v:

(** [grp_pow g n] commutes with [g].  Actually, for the inductive step below, we might need to know that if [g] commutes with [h], then [g] commutes with powers of [h]? *)
Definition grp_pow_commutes {G : Group} (n : Int) (g : G)
  : (grp_pow g n) * g = g * (grp_pow g n).

(** If [g] and [h] commute, then [grp_pow (g * h) n] = (grp_pow g n) * (grp_pow h n)]. *)
(* Note that part of the proof that [ab_mul] is a homomorphism will be covered by this. *)
Definition grp_pow_mul {G : Group} (n : Int) (g h : G)
  (c : g * h = h * g)
  : grp_pow (g * h) n] = (grp_pow g n) * (grp_pow h n).

(** [grp_pow] satisfies a multiplicative law of exponents. *)
Definition grp_pow_int_mul {G : Group} (m n : Int) (g : G)
  : grp_pow g (m * n)%int = grp_pow (grp_pow g m) n.
(* This will follow from the previous two. *)

Rings/Z.v:

Definition rng_int_mult_foo {R : Ring} (r : R) (n : Int)
  : rng_int_mult r n = (rng_int_mult 1 n) * r.

(* I think issemigrouppreserving_mult_rng_int_mult will follow from the previous item and grp_pow_int_mul. *)

cc: @ThomatoTomato