Definition Zlog_sup (M : Z) :=
  match M with
  | Zpos p => log_sup p
  | _ => 0
  end.

Lemma approx_le1 : forall a b y x : Z, 0 < a -> 0 < b -> 0 < y -> 0 <= x -> x < a * b + a -> x / a * a / y - x / a * (a * b / y) / b <= 1.