Module NZQuot := Nop <+ NZDivProp A Quot2Div B.
End Private_Div.
Ltac pos_or_neg a :=
let LT := fresh "LT" in
let LE := fresh "LE" in
destruct (le_gt_cases 0 a) as [LE|LT]; [|rewrite <- opp_pos_neg in LT].
(** Another formulation of the main equation *)
(** A few sign rules (simple ones) *)

Lemma rem_opp_opp : forall a b, b ~= 0 -> (-a) rem (-b) == - (a rem b).