(** An alternative power function for Z *)

(** This [Zpower_alt] is extensionally equal to [Z.pow],
    but not convertible with it. The number of
    multiplications is logarithmic instead of linear, but
    these multiplications are bigger. Experimentally, it seems
    that [Zpower_alt] is slightly quicker than [Z.pow] on average,
    but can be quite slower on powers of 2.
*)

Definition Zpower_alt n m :=
  match m with
    | Z0 => 1
    | Zpos p => Pos.iter_op Z.mul p n
    | Zneg p => 0
  end.Infix "^^" := Zpower_alt (at level 30, right associativity) : Z_scope.

Lemma Piter_mul_acc : forall f, (forall x y:Z, (f x)*y = f (x*y)) -> forall p k, Pos.iter f k p = (Pos.iter f 1 p)*k.