(** On natural numbers, we should use a particular form for the Bezout identity, since we don't have full subtraction. *)
#[global]
Instance Bezout_wd : Proper (eq==>eq==>eq==>iff) Bezout.

(* Specifications: *)
Lemma bezout_1_gcd : forall n m, Bezout n m 1 -> gcd n m == 1.
(** Bezout on natural numbers commutes *)
Theorem bezout_comm : forall a b g,
b ~= 0 -> Bezout a b g -> Bezout b a g.
Lemma gcd_bezout_pos : forall n m, 0 < n -> Bezout n m (gcd n m).
(** For strictly positive numbers, we have Bezout in the two directions. *)
Lemma gcd_bezout_pos_pos : forall n, 0<n -> forall m, 0<m ->
Bezout n m (gcd n m) /\ Bezout m n (gcd n m).