(** * Greatest common divisor. *)

Definition common_div (x y : Z) (d : nat) :=
  Divides d (Zabs_nat x) /\ Divides d (Zabs_nat y).Definition gcd (x y : Z) (d : nat) :=
  common_div x y d /\ (forall e : nat, common_div x y e -> e <= d).

Lemma gcd_lincomb_pos : forall (x y : Z) (d : nat), (x > 0)%Z -> gcd x y d -> LinComb (Z_of_nat d) x y.