Definition ZROUND_inf (p : positive) (x : Z) :=
  let (x', p) := ZROUND_inf_spec p x in x'.

(* Specifications: *)
Lemma ZROUND_sup_spec : forall (p : positive) (x : Z), {y : Z | (Zpred y * two_power_pos p < x <= y * two_power_pos p)%Z}.