(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Local Open Scope Z_scope.
(** 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_neg_r a b : b<0 -> a^^b = 0.
Lemma Zpower_alt_Ppow p q : (Zpos p)^^(Zpos q) = Zpos (p^q).