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

Theorem Zceil_floor_neq :
forall x : R,
(IZR (Zfloor x) <> x)%R ->
(Zceil x = Zfloor x + 1)%Z.