(* 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.
(** definition of a "prime number" *)
Definition is_prime (p:nat) := (p<>1)/\(forall (d:nat),(divides p d)->(d=1)\/(d=p)).
unfold is_prime.
intro.
elim H;intros.
case (H1 2);try (apply zero_max_div);intro;discriminate.
Qed.
unfold is_prime.
split.
intro;discriminate.
intros.
elim H;destruct x;rewrite mult_comm.
intro;discriminate.
simpl.
case d.
simpl.
rewrite mult_comm;simpl;intro;discriminate.
intros.
inversion H0.
symmetry in H2.
case (plus_is_one n (x*(S n)) H2);intro.
elim a;intros.
left;rewrite H1;trivial.
elim a;intros.
right;rewrite H1;trivial.
Qed.
unfold is_prime.
unfold rel_prime.
intros.
unfold is_gcd;unfold is_cd.
split.
split;apply one_min_div.
intros.
elim H;intros.
elim H1;intros.
case (H3 d' H4);intro.
rewrite H6;apply divides_refl.
rewrite H6 in H5;tauto.
Qed.
unfold is_prime.
intros.
elim H;intros.
apply H2.
elim H0;intros.
elim H3;trivial.
Qed.
unfold rel_prime.
intros.
generalize (gcd_is_gcd a p);intros.
case (prime_gcd (gcd a p) p a H H1);intro;rewrite H2 in H1;try tauto.
elim H1;intros.
elim H3;trivial.
Qed.
intros.
generalize (gcd_is_gcd a p);intro.
case (prime_gcd (gcd a p) p a H H1);intro;rewrite H2 in H1.
right;apply gauss with a;trivial.
red in H1;elim H1;intros.
red in H3;tauto.
Qed.

(** corollary: if p is prime and p | a^2 then p | a *)
Lemma prime_square : forall (p a:nat),(is_prime p)->(divides (square a) p)->(divides a p).