(* NB: Z.modulo and Z.div are not morphisms with respect to eqm. For instance, let (==) be (eqm 2). Then we have (3 == 1) but: ~ (3 mod 3 == 1 mod 3) ~ (1 mod 3 == 1 mod 1) ~ (3/3 == 1/3) ~ (1/3 == 1/1) *)

(* Specifications: *)
Lemma Zdiv_Zdiv : forall a b c, 0<=b -> 0<=c -> (a/b)/c = a/(b*c).
(** Unfortunately, the previous result isn't always true on negative numbers. For instance: 3/(-2)/(-2) = 1 <> 0 = 3 / (-2*-2) *)
Lemma Zmod_div : forall a b, a mod b / b = 0.
(** A last inequality: *)
Theorem Zdiv_mult_le:
forall a b c, 0<=a -> 0<=b -> 0<=c -> c*(a/b) <= (c*a)/b.