(** the most useful one: R -> F *)
Definition round x :=
F2R (Float beta (rnd (scaled_mantissa x)) (canonic_exp x)).

Theorem exp_small_round_0_pos :
forall x ex,
(bpow (ex - 1) <= x < bpow ex)%R ->
round x = 0%R -> (ex <= fexp ex)%Z .