(** * Unicity results *)

Definition Remainder a b r :=
  (0 <= a /\ 0 <= r < Z.abs b) \/ (a <= 0 /\ -Z.abs b < r <= 0).Definition Remainder_alt a b r :=
  Z.abs r < Z.abs b /\ 0 <= r * a.

Theorem Zquotrem_Zdiv_eucl_pos : forall a b:Z, 0 <= a -> 0 < b -> a÷b = a/b /\ Z.rem a b = a mod b.