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Diffstat (limited to 'theories/ZArith/Zpow_def.v')
| -rw-r--r-- | theories/ZArith/Zpow_def.v | 42 |
1 files changed, 23 insertions, 19 deletions
diff --git a/theories/ZArith/Zpow_def.v b/theories/ZArith/Zpow_def.v index 620d6324..6f1ebc06 100644 --- a/theories/ZArith/Zpow_def.v +++ b/theories/ZArith/Zpow_def.v @@ -1,27 +1,31 @@ -Require Import ZArith_base. -Require Import Ring_theory. +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) -Open Local Scope Z_scope. +Require Import BinInt Ring_theory. +Local Open Scope Z_scope. -(** [Zpower_pos z n] is the n-th power of [z] when [n] is an binary - integer (type [positive]) and [z] a signed integer (type [Z]) *) -Definition Zpower_pos (z:Z) (n:positive) := iter_pos n Z (fun x:Z => z * x) 1. +(** * Power functions over [Z] *) -Definition Zpower (x y:Z) := - match y with - | Zpos p => Zpower_pos x p - | Z0 => 1 - | Zneg p => 0 - end. +(** Nota : this file is mostly deprecated. The definition of [Z.pow] + and its usual properties are now provided by module [BinInt.Z]. *) -Lemma Zpower_theory : power_theory 1 Zmult (eq (A:=Z)) Z_of_N Zpower. +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 := Z.pow_Zpos (only parsing). + +Lemma Zpower_theory : power_theory 1 Z.mul (@eq Z) Z.of_N Z.pow. Proof. constructor. intros. destruct n;simpl;trivial. - unfold Zpower_pos. - assert (forall k, iter_pos p Z (fun x : Z => r * x) k = pow_pos Zmult r p*k). - induction p;simpl;intros;repeat rewrite IHp;trivial; - repeat rewrite Zmult_assoc;trivial. - rewrite H;rewrite Zmult_1_r;trivial. + unfold Z.pow_pos. + rewrite <- (Z.mul_1_r (pow_pos _ _ _)). generalize 1. + induction p; simpl; intros; rewrite ?IHp, ?Z.mul_assoc; trivial. Qed. - |