(** 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 pred_pos_plus_ulp_aux2 : forall x, (0 < x)%R -> F x -> let e := ln_beta_val beta x (ln_beta beta x) in x = bpow (e - 1) -> (x - bpow (fexp (e-1)) <> 0)%R -> ((x - bpow (fexp (e-1))) + ulp (x - bpow (fexp (e-1))) = x)%R.