(** 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_opp: forall x, (pred (-x) = - succ x)%R.