(** * 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 euclid_gcd1 : forall (d : nat) (x y q r : Z), gcd x y d -> x = (q * y + r)%Z -> gcd r y d.