Proof with auto with typeclass_instances.
intros P HP rnd Hr x.
destruct (Rle_or_lt 0 x) as [Hx|Hx].
(* . *)
rewrite 2!Rabs_pos_eq.
now apply HP.
rewrite <- (round_0 rnd).
now apply round_le.
exact Hx.
(* . *)
rewrite (Rabs_left _ Hx).
rewrite Rabs_left1.
pattern x at 2 ; rewrite <- Ropp_involutive.
rewrite round_opp.
rewrite Ropp_involutive.
apply HP...
rewrite <- Ropp_0.
apply Ropp_le_contravar.
now apply Rlt_le.
rewrite <- (round_0 rnd).
apply round_le...
now apply Rlt_le.
Qed.

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