(** 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.

(* Specifications: *)
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.
Lemma Piter_op_square : forall p a, Pos.iter_op Z.mul p (a*a) = (Pos.iter_op Z.mul p a)*(Pos.iter_op Z.mul p a).
Lemma Zpower_equiv a b : a^^b = a^b.
Lemma Zpower_alt_0_r n : n^^0 = 1.
Lemma Zpower_alt_succ_r a b : 0<=b -> a^^(Z.succ b) = a * a^^b.
Lemma Zpower_alt_neg_r a b : b<0 -> a^^b = 0.
Lemma Zpower_alt_Ppow p q : (Zpos p)^^(Zpos q) = Zpos (p^q).