(** 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.

Theorem le_pred_lt : forall x y, F x -> F y -> (x < y)%R -> (x <= pred y)%R.