(** * 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 Zquot_mod_unique_full a b q r : Remainder a b r -> a = b*q + r -> q = a÷b /\ r = Z.rem a b.