Section Floor_Ceil.
(** Zfloor and Zceil *)
Definition Zfloor (x : R) := (up x - 1)%Z.
Definition Zceil (x : R) := (- Zfloor (- x))%Z.

Theorem Zceil_le :
forall x y, (x <= y)%R ->
(Zceil x <= Zceil y)%Z.