(** * 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.

Lemma Z_quot_plus : forall a b c:Z, 0 <= (a+b*c) * a -> c<>0 -> (a + b * c) ÷ c = a ÷ c + b.