Built with Alectryon, running Coq+SerAPI v8.10.0+0.7.0. Coq sources are in this panel; goals and messages will appear in the other. Bubbles () indicate interactive fragments: hover for details, tap to reveal contents. Use Ctrl+↑ Ctrl+↓ to navigate, Ctrl+🖱️ to focus.

(************************************************************************)
(*         *   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)         *)
(************************************************************************)

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.power_theory 1 Z.mul eq Z.of_N Z.pow
Proof.power_theory 1 Z.mul eq Z.of_N Z.pow
 constructor.forall (r : Z) (n : N),
r ^ Z.of_N n = pow_N 1 Z.mul r n intros.r:Z
n:N
r ^ Z.of_N n = pow_N 1 Z.mul r n
 destruct n;simpl;trivial.r:Z
p:positive
Z.pow_pos r p = pow_pos Z.mul r p
 unfold Z.pow_pos.r:Z
p:positive
Pos.iter (Z.mul r) 1 p = pow_pos Z.mul r p
 rewrite <- (Z.mul_1_r (pow_pos _ _ _)).r:Z
p:positive
Pos.iter (Z.mul r) 1 p = pow_pos Z.mul r p * 1 generalize 1.r:Z
p:positive
forall z : Z,
Pos.iter (Z.mul r) z p = pow_pos Z.mul r p * z
 induction p; simpl; intros; rewrite ?IHp, ?Z.mul_assoc; trivial.
Qed.