(* Copyright (C) 2005-2008 Sebastien Briais *)
(* http://lamp.epfl.ch/~sbriais/ *)
(* This library is free software; you can redistribute it and/or modify *)
(* it under the terms of the GNU Lesser General Public License as *)
(* published by the Free Software Foundation; either version 2.1 of the *)
(* License, or (at your option) any later version. *)
(* This library is distributed in the hope that it will be useful, but *)
(* WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU *)
(* Lesser General Public License for more details. *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this library; if not, write to the Free Software *)
(* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
Unset Standard Proposition Elimination Names.
intros.
elim H0;intro q;intro.
replace b with (b*1);try ring.
rewrite H1.
apply mult_le_compat;try omega.
destruct q;omega.
Qed.
intros.
apply (lt_wf_rec a (fun a:nat =>{q : nat &  {r : nat | a = b * q + r /\ r < b}})).
intros.
case (le_lt_dec b n);intro.
elim (H0 (n-b)).
intro q;intro.
elim p;intro r;intro.
exists (q+1);exists r.
split;try tauto.
rewrite (le_plus_minus b n);trivial.
elim p0;intros.
rewrite H1;ring.
omega.
exists 0;exists n.
split;try tauto.
ring.
Qed.
Definition quotient_euclide (a b:nat)(H:(b<>O)) := let (q,_) := (euclide a b H) in q.
Definition remainder_euclide (a b:nat)(H:(b<>O)) := let (_,e0) := (euclide a b H) in let (r,_) := e0 in r.
unfold quotient_euclide;unfold remainder_euclide;intros.
generalize (euclide a b H);intros.
elim s;intro q;intro.
elim p;intro r;intro.
tauto.
Qed.
unfold remainder_euclide;intros.
generalize (euclide a b H);intros.
elim s;intro q;intro.
elim p;intro r;intro.
tauto.
Qed.

(** Euclide division is unique *)
Lemma euclide_unique : forall (a b q r q' r':nat),(b<>O)->a=b*q+r->a=b*q'+r'->r<b->r'<b->(q=q')/\(r=r').