(** * 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 euclid_gcd :
forall (d1 d2 : nat) (x y q r : Z),
x = (q * y + r)%Z -> gcd x y d1 -> gcd r y d2 -> d1 = d2.