(** For stating the fully general result, let's give a short name
    to the condition on the remainder. *)

Definition Remainder r b :=  0 <= r < b \/ b < r <= 0.(** Another equivalent formulation: *)

Definition Remainder_alt r b :=  Z.abs r < Z.abs b /\ Z.sgn r <> - Z.sgn b.

Lemma Z_mod_remainder a b : b<>0 -> Remainder (a mod b) b.