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Diffstat (limited to 'theories/Arith/Euclid.v')
| -rw-r--r-- | theories/Arith/Euclid.v | 31 |
1 files changed, 14 insertions, 17 deletions
diff --git a/theories/Arith/Euclid.v b/theories/Arith/Euclid.v index f32e1ad4..3abdff98 100644 --- a/theories/Arith/Euclid.v +++ b/theories/Arith/Euclid.v @@ -1,71 +1,68 @@ (************************************************************************) (* 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: Euclid.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Import Mult. Require Import Compare_dec. Require Import Wf_nat. -Open Local Scope nat_scope. +Local Open Scope nat_scope. Implicit Types a b n q r : nat. Inductive diveucl a b : Set := divex : forall q r, b > r -> a = q * b + r -> diveucl a b. - Lemma eucl_dev : forall n, n > 0 -> forall m:nat, diveucl m n. Proof. - intros b H a; pattern a in |- *; apply gt_wf_rec; intros n H0. + intros b H a; pattern a; apply gt_wf_rec; intros n H0. elim (le_gt_dec b n). intro lebn. elim (H0 (n - b)); auto with arith. intros q r g e. - apply divex with (S q) r; simpl in |- *; auto with arith. + apply divex with (S q) r; simpl; auto with arith. elim plus_assoc. elim e; auto with arith. intros gtbn. - apply divex with 0 n; simpl in |- *; auto with arith. -Qed. + apply divex with 0 n; simpl; auto with arith. +Defined. Lemma quotient : forall n, n > 0 -> forall m:nat, {q : nat | exists r : nat, m = q * n + r /\ n > r}. Proof. - intros b H a; pattern a in |- *; apply gt_wf_rec; intros n H0. + intros b H a; pattern a; apply gt_wf_rec; intros n H0. elim (le_gt_dec b n). intro lebn. elim (H0 (n - b)); auto with arith. intros q Hq; exists (S q). elim Hq; intros r Hr. - exists r; simpl in |- *; elim Hr; intros. + exists r; simpl; elim Hr; intros. elim plus_assoc. elim H1; auto with arith. intros gtbn. - exists 0; exists n; simpl in |- *; auto with arith. -Qed. + exists 0; exists n; simpl; auto with arith. +Defined. Lemma modulo : forall n, n > 0 -> forall m:nat, {r : nat | exists q : nat, m = q * n + r /\ n > r}. Proof. - intros b H a; pattern a in |- *; apply gt_wf_rec; intros n H0. + intros b H a; pattern a; apply gt_wf_rec; intros n H0. elim (le_gt_dec b n). intro lebn. elim (H0 (n - b)); auto with arith. intros r Hr; exists r. elim Hr; intros q Hq. - elim Hq; intros; exists (S q); simpl in |- *. + elim Hq; intros; exists (S q); simpl. elim plus_assoc. elim H1; auto with arith. intros gtbn. - exists n; exists 0; simpl in |- *; auto with arith. -Qed. + exists n; exists 0; simpl; auto with arith. +Defined. |