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Definition is_gcd (a b d : S) :=
  divide d a /\
  divide d b /\ (forall q : S, divide q a -> divide q b -> divide q d).

(* Specifications: *)
Lemma gcd_null : forall a b : S, is_gcd a b O -> a = O /\ b = O.
Lemma gcd_null2 : is_ring -> forall d : S, is_gcd O O d -> d = O.
Lemma simplification_integrity : is_unitary_commutative_ring -> integrity -> forall a x : S, A a -> A x -> a <> O -> Mult a x = a -> x = I.
Lemma gcd_unicity_apart_unities : is_unitary_commutative_ring -> integrity -> forall a b d1 d2 : S, is_gcd a b d1 -> is_gcd a b d2 -> exists x : S, inversible S Mult I x /\ A x /\ d2 = Mult d1 x.
Lemma opp_O : is_ring -> forall x : S, A x -> Opp x = O -> x = O.