Section Znearest.
(** Roundings to nearest: when in the middle, use the choice function *)
Variable choice : Z -> bool.
Definition Znearest x :=
match Rcompare (x - IZR (Zfloor x)) (/2) with
| Lt => Zfloor x
| Eq => if choice (Zfloor x) then Zceil x else Zfloor x
| Gt => Zceil x
end.
Global Instance valid_rnd_N : Valid_rnd Znearest.

Lemma round_N_small_pos :
forall x,
forall ex,
(Raux.bpow beta (ex - 1) <= x < Raux.bpow beta ex)%R ->
(ex < fexp ex)%Z ->
(round Znearest x = 0)%R.