(** 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.

Theorem Z_div_mod_full a b : b <> 0 -> let (q, r) := Z.div_eucl a b in a = b * q + r /\ Remainder r b.