(* 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 *)
(** b | a if there is q such that a = b * q*)
Definition divides (a b:nat) := exists q:nat,a = (b*q).
intros.
red.
exists n.
auto with arith.
Qed.
intros.
red.
exists O.
auto with arith.
Qed.
intros.
red.
exists 1.
auto with arith.
Qed.
unfold divides.
intros.
elim H;intro q;intro.
elim H0;intro q';intro.
rewrite H2 in H1.
exists (q' * q).
rewrite H1.
auto with arith.
Qed.

(** the relation of divisibility is antisymmetric *)
Lemma divides_antisym : forall (a b:nat),(divides a b)->(divides b a)->a=b.