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

Theorem Zceil_ub :
forall x : R,
(x <= IZR (Zceil x))%R.