(** * 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).

(* Specifications: *)
Lemma gcd_unq :
forall (d1 d2 : nat) (x y : Z), gcd x y d1 -> gcd x y d2 -> d1 = d2.
Lemma gcd_sym : forall (d : nat) (x y : Z), gcd x y d -> gcd y x d.
Lemma gcd_opp_l : forall (d : nat) (x y : Z), gcd x y d -> gcd (- x) y d.
Lemma gcd_opp_r : forall (d : nat) (x y : Z), gcd x y d -> gcd x (- y) d.
Lemma gcd_0_l : forall d : nat, d > 0 -> gcd 0 (Z_of_nat d) d.
Lemma gcd_0_r : forall d : nat, d > 0 -> gcd (Z_of_nat d) 0 d.
(** * Euclid's theorem. *)
Lemma euclid_gcd1 :
forall (d : nat) (x y q r : Z), gcd x y d -> x = (q * y + r)%Z -> gcd r y 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.