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

(* Specifications: *)
Lemma Remainder_equiv : forall a b r,
Remainder a b r <-> Remainder_alt a b r.
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.
Theorem Zquot_unique_full a b q r :
Remainder a b r -> a = b*q + r -> q = a÷b.
Theorem Zrem_unique_full a b q r :
Remainder a b r -> a = b*q + r -> r = Z.rem a b.