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(* as published by the Free Software Foundation; either version 2.1   *)
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(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
(** Relative primality *)
Definition rel_prime (a b : Z) : Prop := gcd a b 1.
(** Bezout's theorem: [a] and [b] are relatively prime if and only if there exist [u] and [v] such that [ua+vb = 1]. *)
(** Gauss's theorem: if [a] divides [bc] and if [a] and [b] are relatively prime, then [a] divides [c]. *)
(** If [a] is relatively prime to [b] and [c], then it is to [bc] *)
(** Primality *)
Inductive prime (p : Z) : Prop :=
prime_intro :
(1 < p)%Z -> (forall n : Z, (1 <= n < p)%Z -> rel_prime n p) -> prime p.
(** The sole divisors of a prime number [p] are [-1], [1], [p] and [-p]. *)
(** A prime number is relatively prime with any number it does not divide *)

Lemma prime_rel_prime :
forall p : Z, prime p -> forall a : Z, ~ (p | a)%Z -> rel_prime p a.