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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Mult.
Require Import Compare_dec.
Require Import Wf_nat.
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.
induction m as (m,H0) using gt_wf_rec.
destruct (le_gt_dec n m) as [Hlebn|Hgtbn].
destruct (H0 (m - n)) as (q,r,Hge0,Heq); auto with arith.
apply divex with (S q) r; trivial.
simpl; rewrite <- plus_assoc, <- Heq; auto with arith.
apply divex with 0 m; simpl; trivial.
Defined.
Lemma quotient :
forall n,
n > 0 ->
forall m:nat, {q : nat | exists r : nat, m = q * n + r /\ n > r}.
Proof.
induction m as (m,H0) using gt_wf_rec.
destruct (le_gt_dec n m) as [Hlebn|Hgtbn].
destruct (H0 (m - n)) as (q & Hq); auto with arith; exists (S q).
destruct Hq as (r & Heq & Hgt); exists r; split; trivial.
simpl; rewrite <- plus_assoc, <- Heq; auto with arith.
exists 0; exists m; 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.
induction m as (m,H0) using gt_wf_rec.
destruct (le_gt_dec n m) as [Hlebn|Hgtbn].
destruct (H0 (m - n)) as (r & Hr); auto with arith; exists r.
destruct Hr as (q & Heq & Hgt); exists (S q); split; trivial.
simpl; rewrite <- plus_assoc, <- Heq; auto with arith.
exists m; exists 0; simpl; auto with arith.
Defined.
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