(** 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_aux3 :
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 ->
(ulp 0 = x)%R.