(** * 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).

Theorem gcd_lincomb : forall (x y : Z) (d : nat), x <> 0%Z -> gcd x y d -> LinComb (Z_of_nat d) x y.