diff options
author | Samuel Mimram <samuel.mimram@ens-lyon.org> | 2004-07-28 21:54:47 +0000 |
---|---|---|
committer | Samuel Mimram <samuel.mimram@ens-lyon.org> | 2004-07-28 21:54:47 +0000 |
commit | 6b649aba925b6f7462da07599fe67ebb12a3460e (patch) | |
tree | 43656bcaa51164548f3fa14e5b10de5ef1088574 /theories/Arith/Euclid.v |
Imported Upstream version 8.0pl1upstream/8.0pl1
Diffstat (limited to 'theories/Arith/Euclid.v')
-rw-r--r-- | theories/Arith/Euclid.v | 68 |
1 files changed, 68 insertions, 0 deletions
diff --git a/theories/Arith/Euclid.v b/theories/Arith/Euclid.v new file mode 100644 index 00000000..e50e3d70 --- /dev/null +++ b/theories/Arith/Euclid.v @@ -0,0 +1,68 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Euclid.v,v 1.7.2.1 2004/07/16 19:31:00 herbelin Exp $ i*) + +Require Import Mult. +Require Import Compare_dec. +Require Import Wf_nat. + +Open Local 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. +intros b H a; pattern a in |- *; 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. +elim plus_assoc. +elim e; auto with arith. +intros gtbn. +apply divex with 0 n; simpl in |- *; auto with arith. +Qed. + +Lemma quotient : + forall n, + n > 0 -> + forall m:nat, {q : nat | exists r : nat, m = q * n + r /\ n > r}. +intros b H a; pattern a in |- *; 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. +elim plus_assoc. +elim H1; auto with arith. +intros gtbn. +exists 0; exists n; simpl in |- *; auto with arith. +Qed. + +Lemma modulo : + forall n, + n > 0 -> + forall m:nat, {r : nat | exists q : nat, m = q * n + r /\ n > r}. +intros b H a; pattern a in |- *; 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 plus_assoc. +elim H1; auto with arith. +intros gtbn. +exists n; exists 0; simpl in |- *; auto with arith. +Qed.
\ No newline at end of file |