(** * Relative primality *)
Definition rel_prime (a b:Z) : Prop := Zis_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]. *)

(* Specifications: *)
Lemma rel_prime_bezout : forall a b:Z, rel_prime a b -> Bezout a b 1.
Lemma bezout_rel_prime : forall a b:Z, Bezout a b 1 -> rel_prime a b.
(** Gauss's theorem: if [a] divides [bc] and if [a] and [b] are relatively prime, then [a] divides [c]. *)
Theorem Gauss : forall a b c:Z, (a | b * c) -> rel_prime a b -> (a | c).
(** If [a] is relatively prime to [b] and [c], then it is to [bc] *)
Lemma rel_prime_mult :
forall a b c:Z, rel_prime a b -> rel_prime a c -> rel_prime a (b * c).
Lemma rel_prime_cross_prod :
forall a b c d:Z,
rel_prime a b ->
rel_prime c d -> b > 0 -> d > 0 -> a * d = b * c -> a = c /\ b = d.