Module NZGcdProp
(Import A : NZOrdAxiomsSig')
(Import B : NZGcd' A)
(Import C : NZMulOrderProp A).
(** Results concerning divisibility*)
Instance divide_wd : Proper (eq==>eq==>iff) divide.
Instance divide_reflexive : Reflexive divide | 5 := divide_refl.
Instance divide_transitive : Transitive divide | 5 := divide_trans.
(** Due to sign, no general antisymmetry result *)
(** Basic properties of gcd *)
Instance gcd_wd : Proper (eq==>eq==>eq) gcd.

Lemma gcd_divide_iff : forall n m p,
(p | gcd n m) <-> (p | n) /\ (p | m).