Definition modulo a b := Z.modulo a (Z.abs b).Definition div a b := (Z.sgn b) * (Z.div a (Z.abs b)).

(* Specifications: *)
Theorem div_mod a b : b<>0 -> a = b*(div a b) + modulo a b.
Lemma mod_always_pos a b : b<>0 -> 0 <= modulo a b < Z.abs b.
Lemma mod_bound_pos a b : 0<=a -> 0<b -> 0 <= modulo a b < b.