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(************************************************************************)
(* * The Coq Proof Assistant / The Coq 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) *)
(************************************************************************)
Require Import BinInt Ring_theory.
Local Open Scope Z_scope.
(** * Power functions over [Z] *)
(** Nota : this file is mostly deprecated. The definition of [Z.pow]
and its usual properties are now provided by module [BinInt.Z]. *)
Notation Zpower_pos := Z.pow_pos (only parsing).
Notation Zpower := Z.pow (only parsing).
Notation Zpower_0_r := Z.pow_0_r (only parsing).
Notation Zpower_succ_r := Z.pow_succ_r (only parsing).
Notation Zpower_neg_r := Z.pow_neg_r (only parsing).
Notation Zpower_Ppow := Pos2Z.inj_pow (only parsing).
Lemma Zpower_theory : power_theory 1 Z.mul (@eq Z) Z.of_N Z.pow.
Proof.
constructor. intros z n.
destruct n as [|p];simpl;trivial.
unfold Z.pow_pos.
rewrite <- (Z.mul_1_r (pow_pos _ _ _)). generalize 1.
induction p as [p IHp|p IHp|]; simpl; intros; rewrite ?IHp, ?Z.mul_assoc; trivial.
Qed.
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