(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
(*   \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.
