Proof with auto with typeclass_instances.
intros rnd Hr x ex He.
apply round_abs_abs...
clear rnd Hr x.
intros rnd' Hr x _.
apply round_bounded_large_pos...
Qed.

Theorem exp_small_round_0 :
forall rnd {Hr : Valid_rnd rnd} x ex,
(bpow (ex - 1) <= Rabs x < bpow ex)%R ->
round rnd x = 0%R -> (ex <= fexp ex)%Z .