(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \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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Require Export Zpower.
Local Open Scope Z_scope.
(** Properties of the power function over [Z] *)
(** Nota: the usual properties of [Z.pow] are now already provided by [BinInt.Z]. Only remain here some compatibility elements, as well as more specific results about power and modulo and/or primality. *)
Proof (Z.pow_1_r x).
Notation Zpower_nat_Zpower := Zpower_nat_Zpower (only parsing).
(** * Z.pow and modulo *)
(** A direct way to compute Z.pow modulo **)
Fixpoint Zpow_mod_pos (a: Z)(m: positive)(n : Z) : Z :=
match m with
| xH => a mod n
| xO m' =>
let z := Zpow_mod_pos a m' n in
match z with
| 0 => 0
| _ => (z * z) mod n
end
| xI m' =>
let z := Zpow_mod_pos a m' n in
match z with
| 0 => 0
| _ => (z * z * a) mod n
end
end.
Definition Zpow_mod a m n :=
match m with
| 0 => 1 mod n
| Zpos p => Zpow_mod_pos a p n
| Zneg p => 0
end.
(* Complements about power and number theory. *)

Theorem Zdivide_power_2 x p n :
0 <= n -> 0 <= x -> prime p -> (x | p^n) -> exists m, x = p^m.