(* This lemma used to be useful since it was mentioned with an unnecessary premise 
   `x>=0` as Z_modulo_2 in ZArith, but the ZArith version has been fixed. *)

Definition Z_modulo_2_always :
  forall x : Z, {y : Z | x = (2 * y)%Z} + {y : Z | x = (2 * y + 1)%Z} :=
  Zeven.Z_modulo_2.

Lemma Z_div_le : forall a b c : Z, (0 < c)%Z -> (b <= a)%Z -> (b / c <= a / c)%Z.