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.

(* Specifications: *)
Theorem round_DN_opp :
forall x,
round Zfloor (-x) = (- round Zceil x)%R.
Theorem round_UP_opp :
forall x,
round Zceil (-x) = (- round Zfloor x)%R.
Theorem round_ZR_opp :
forall x,
round Ztrunc (- x) = Ropp (round Ztrunc x).
Theorem round_ZR_abs :
forall x,
round Ztrunc (Rabs x) = Rabs (round Ztrunc x).
Theorem round_DN_opp :
forall x,
round Zfloor (-x) = (- round Zceil x)%R.
Theorem round_UP_opp :
forall x,
round Zceil (-x) = (- round Zfloor x)%R.
Theorem round_ZR_opp :
forall x,
round Ztrunc (- x) = Ropp (round Ztrunc x).
Theorem round_ZR_abs :
forall x,
round Ztrunc (Rabs x) = Rabs (round Ztrunc x).