Definition Zrnd_odd x :=  match Req_EM_T x (Z2R (Zfloor x))  with
  | left _   => Zfloor x
  | right _  => match (Zeven (Zfloor x)) with
      | true => Zceil x
      | false => Zfloor x
     end
  end.Global Instance valid_rnd_odd : Valid_rnd Zrnd_odd.

Lemma Zrnd_odd_Zodd: forall x, x <> (Z2R (Zfloor x)) -> (Zeven (Zrnd_odd x)) = false.