(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

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.
forall n : nat, n > 0 -> forall m : nat, diveucl m n
Proof.
forall n : nat, n > 0 -> forall m : nat, diveucl m n
  induction m as (m,H0) using gt_wf_rec.
n:nat
H:n > 0
m:nat
H0:forall m0 : nat, m > m0 -> diveucl m0 n
diveucl m n
  destruct (le_gt_dec n m) as [Hlebn|Hgtbn].
n:nat
H:n > 0
m:nat
H0:forall m0 : nat, m > m0 -> diveucl m0 n
Hlebn:n <= m
diveucl m n
n:nat
H:n > 0
m:nat
H0:forall m0 : nat, m > m0 -> diveucl m0 n
Hgtbn:n > m
diveucl m n
  destruct (H0 (m - n)) as (q,r,Hge0,Heq); auto with arith.
n:nat
H:n > 0
m:nat
H0:forall m0 : nat, m > m0 -> diveucl m0 n
Hlebn:n <= m
q, r:nat
Hge0:n > r
Heq:m - n = q * n + r
diveucl m n
n:nat
H:n > 0
m:nat
H0:forall m0 : nat, m > m0 -> diveucl m0 n
Hgtbn:n > m
diveucl m n
  apply divex with (S q) r; trivial.
n:nat
H:n > 0
m:nat
H0:forall m0 : nat, m > m0 -> diveucl m0 n
Hlebn:n <= m
q, r:nat
Hge0:n > r
Heq:m - n = q * n + r
m = S q * n + r
n:nat
H:n > 0
m:nat
H0:forall m0 : nat, m > m0 -> diveucl m0 n
Hgtbn:n > m
diveucl m n
  simpl; rewrite <- plus_assoc, <- Heq; auto with arith.
n:nat
H:n > 0
m:nat
H0:forall m0 : nat, m > m0 -> diveucl m0 n
Hgtbn:n > m
diveucl m n
  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}.
forall n : nat,
n > 0 ->
forall m : nat,
{q : nat | exists r : nat, m = q * n + r /\ n > r}
Proof.
forall n : nat,
n > 0 ->
forall m : nat,
{q : nat | exists r : nat, m = q * n + r /\ n > r}
  induction m as (m,H0) using gt_wf_rec.
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{q : nat |
exists r : nat, m0 = q * n + r /\ n > r}
{q : nat | exists r : nat, m = q * n + r /\ n > r}
  destruct (le_gt_dec n m) as [Hlebn|Hgtbn].
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{q : nat |
exists r : nat, m0 = q * n + r /\ n > r}
Hlebn:n <= m
{q : nat | exists r : nat, m = q * n + r /\ n > r}
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{q : nat |
exists r : nat, m0 = q * n + r /\ n > r}
Hgtbn:n > m
{q : nat | exists r : nat, m = q * n + r /\ n > r}
  destruct (H0 (m - n)) as (q & Hq); auto with arith; exists (S q).
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{q0 : nat |
exists r : nat, m0 = q0 * n + r /\ n > r}
Hlebn:n <= m
q:nat
Hq:exists r : nat, m - n = q * n + r /\ n > r
exists r : nat, m = S q * n + r /\ n > r
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{q : nat |
exists r : nat, m0 = q * n + r /\ n > r}
Hgtbn:n > m
{q : nat | exists r : nat, m = q * n + r /\ n > r}
  destruct Hq as (r & Heq & Hgt); exists r; split; trivial.
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{q0 : nat |
exists r0 : nat, m0 = q0 * n + r0 /\ n > r0}
Hlebn:n <= m
q, r:nat
Heq:m - n = q * n + r
Hgt:n > r
m = S q * n + r
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{q : nat |
exists r : nat, m0 = q * n + r /\ n > r}
Hgtbn:n > m
{q : nat | exists r : nat, m = q * n + r /\ n > r}
  simpl; rewrite <- plus_assoc, <- Heq; auto with arith.
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{q : nat |
exists r : nat, m0 = q * n + r /\ n > r}
Hgtbn:n > m
{q : nat | exists r : nat, m = q * n + r /\ n > r}
  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}.
forall n : nat,
n > 0 ->
forall m : nat,
{r : nat | exists q : nat, m = q * n + r /\ n > r}
Proof.
forall n : nat,
n > 0 ->
forall m : nat,
{r : nat | exists q : nat, m = q * n + r /\ n > r}
  induction m as (m,H0) using gt_wf_rec.
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{r : nat |
exists q : nat, m0 = q * n + r /\ n > r}
{r : nat | exists q : nat, m = q * n + r /\ n > r}
  destruct (le_gt_dec n m) as [Hlebn|Hgtbn].
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{r : nat |
exists q : nat, m0 = q * n + r /\ n > r}
Hlebn:n <= m
{r : nat | exists q : nat, m = q * n + r /\ n > r}
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{r : nat |
exists q : nat, m0 = q * n + r /\ n > r}
Hgtbn:n > m
{r : nat | exists q : nat, m = q * n + r /\ n > r}
  destruct (H0 (m - n)) as (r & Hr); auto with arith; exists r.
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{r0 : nat |
exists q : nat, m0 = q * n + r0 /\ n > r0}
Hlebn:n <= m
r:nat
Hr:exists q : nat, m - n = q * n + r /\ n > r
exists q : nat, m = q * n + r /\ n > r
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{r : nat |
exists q : nat, m0 = q * n + r /\ n > r}
Hgtbn:n > m
{r : nat | exists q : nat, m = q * n + r /\ n > r}
  destruct Hr as (q & Heq & Hgt); exists (S q); split; trivial.
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{r0 : nat |
exists q0 : nat, m0 = q0 * n + r0 /\ n > r0}
Hlebn:n <= m
r, q:nat
Heq:m - n = q * n + r
Hgt:n > r
m = S q * n + r
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{r : nat |
exists q : nat, m0 = q * n + r /\ n > r}
Hgtbn:n > m
{r : nat | exists q : nat, m = q * n + r /\ n > r}
  simpl; rewrite <- plus_assoc, <- Heq; auto with arith.
n:nat
H:n > 0
m:nat
H0:forall m0 : nat,
m > m0 ->
{r : nat |
exists q : nat, m0 = q * n + r /\ n > r}
Hgtbn:n > m
{r : nat | exists q : nat, m = q * n + r /\ n > r}
  exists m; exists 0; simpl; auto with arith.
Defined.