(** Definition and properties of pred and succ *)

Definition pred_pos x :=
  if Req_bool x (bpow (ln_beta beta x - 1)) then
    (x - bpow (fexp (ln_beta beta x - 1)))%R
  else
    (x - ulp x)%R.Definition succ x :=
   if (Rle_bool 0 x) then
          (x+ulp x)%R
   else
     (- pred_pos (-x))%R.Definition pred x := (- succ (-x))%R.

Lemma round_N_eq_UP: forall choice x, let d:=round beta fexp Zfloor x in let u:=round beta fexp Zceil x in ((d+u)/2 < x)%R -> round beta fexp (Znearest choice) x = u.