Notation bpow e := (bpow beta e).Notation format := (generic_format beta fexp).Notation canonic := (canonic beta fexp).Notation cexp := (canonic_exp beta fexp).Definition Rnd_odd_pt (x f : R) :=
  format f /\ ((f = x)%R \/
    ((Rnd_DN_pt format x f \/ Rnd_UP_pt format x f) /\
    exists g : float beta, f = F2R g /\ canonic g /\ Zeven (Fnum g) = false)).Definition Rnd_odd (rnd : R -> R) :=
  forall x : R, Rnd_odd_pt x (rnd x).

Lemma DN_odd_d_aux: forall z, (F2R d<= z< F2R u)%R -> Rnd_DN_pt (generic_format beta fexp) z (F2R d).