Fact opp_mod_bound_or : forall a b, b~=0 ->
0 <= -(a mod b) < -b \/ -b < -(a mod b) <= 0.

(** With the current conventions, the other sign rules are rather complex. *)
Lemma div_opp_l_z :
forall a b, b~=0 -> a mod b == 0 -> (-a)/b == -(a/b).