diff options
Diffstat (limited to 'theories/QArith/Qcanon.v')
| -rw-r--r-- | theories/QArith/Qcanon.v | 36 |
1 files changed, 16 insertions, 20 deletions
diff --git a/theories/QArith/Qcanon.v b/theories/QArith/Qcanon.v index 71a3b474..d1160cbe 100644 --- a/theories/QArith/Qcanon.v +++ b/theories/QArith/Qcanon.v @@ -1,13 +1,11 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Qcanon.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Import Field. Require Import QArith. Require Import Znumtheory. @@ -20,43 +18,43 @@ Record Qc : Set := Qcmake { this :> Q ; canon : Qred this = this }. Delimit Scope Qc_scope with Qc. Bind Scope Qc_scope with Qc. -Arguments Scope Qcmake [Q_scope]. +Arguments Qcmake this%Q _. Open Scope Qc_scope. Lemma Qred_identity : - forall q:Q, Zgcd (Qnum q) (QDen q) = 1%Z -> Qred q = q. + forall q:Q, Z.gcd (Qnum q) (QDen q) = 1%Z -> Qred q = q. Proof. unfold Qred; intros (a,b); simpl. - generalize (Zggcd_gcd a ('b)) (Zggcd_correct_divisors a ('b)). + generalize (Z.ggcd_gcd a ('b)) (Z.ggcd_correct_divisors a ('b)). intros. rewrite H1 in H; clear H1. - destruct (Zggcd a ('b)) as (g,(aa,bb)); simpl in *; subst. + destruct (Z.ggcd a ('b)) as (g,(aa,bb)); simpl in *; subst. destruct H0. - rewrite Zmult_1_l in H, H0. + rewrite Z.mul_1_l in H, H0. subst; simpl; auto. Qed. Lemma Qred_identity2 : - forall q:Q, Qred q = q -> Zgcd (Qnum q) (QDen q) = 1%Z. + forall q:Q, Qred q = q -> Z.gcd (Qnum q) (QDen q) = 1%Z. Proof. unfold Qred; intros (a,b); simpl. - generalize (Zggcd_gcd a ('b)) (Zggcd_correct_divisors a ('b)) (Zgcd_is_pos a ('b)). + generalize (Z.ggcd_gcd a ('b)) (Z.ggcd_correct_divisors a ('b)) (Z.gcd_nonneg a ('b)). intros. rewrite <- H; rewrite <- H in H1; clear H. - destruct (Zggcd a ('b)) as (g,(aa,bb)); simpl in *; subst. + destruct (Z.ggcd a ('b)) as (g,(aa,bb)); simpl in *; subst. injection H2; intros; clear H2. destruct H0. clear H0 H3. destruct g as [|g|g]; destruct bb as [|bb|bb]; simpl in *; try discriminate. f_equal. - apply Pmult_reg_r with bb. + apply Pos.mul_reg_r with bb. injection H2; intros. rewrite <- H0. rewrite H; simpl; auto. elim H1; auto. Qed. -Lemma Qred_iff : forall q:Q, Qred q = q <-> Zgcd (Qnum q) (QDen q) = 1%Z. +Lemma Qred_iff : forall q:Q, Qred q = q <-> Z.gcd (Qnum q) (QDen q) = 1%Z. Proof. split; intros. apply Qred_identity2; auto. @@ -71,7 +69,7 @@ Proof. Qed. Definition Q2Qc (q:Q) : Qc := Qcmake (Qred q) (Qred_involutive q). -Arguments Scope Q2Qc [Q_scope]. +Arguments Q2Qc q%Q. Notation " !! " := Q2Qc : Qc_scope. Lemma Qc_is_canon : forall q q' : Qc, q == q' -> q = q'. @@ -468,18 +466,16 @@ Proof. destruct n; simpl. destruct 1; auto. intros. - apply Qc_is_canon. - simpl. - compute; auto. + now apply Qc_is_canon. Qed. Lemma Qcpower_pos : forall p n, 0 <= p -> 0 <= p^n. Proof. induction n; simpl; auto with qarith. - intros; compute; intro; discriminate. + easy. intros. apply Qcle_trans with (0*(p^n)). - compute; intro; discriminate. + easy. apply Qcmult_le_compat_r; auto. Qed. @@ -492,7 +488,7 @@ Definition Qc_eq_bool (x y : Qc) := Lemma Qc_eq_bool_correct : forall x y : Qc, Qc_eq_bool x y = true -> x=y. Proof. - intros x y; unfold Qc_eq_bool in |- *; case (Qc_eq_dec x y); simpl in |- *; auto. + intros x y; unfold Qc_eq_bool; case (Qc_eq_dec x y); simpl; auto. intros _ H; inversion H. Qed. |