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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 *)
Hint Resolve prime_rel_prime.
(** [divide] is decidable *)

Axiom divide_dec : forall a b : Z, {(a | b)%Z} + {~ (a | b)%Z}.