Proof with auto with typeclass_instances.
intros x y.
apply round_abs_abs...
clear rnd valid_rnd x.
intros rnd' Hrnd x Hx.
apply round_le_generic...
Qed.
End monotone_abs.

Theorem round_DN_opp :
forall x,
round Zfloor (-x) = (- round Zceil x)%R.