Section Floor_Ceil.
(** Zfloor and Zceil *)
Definition Zfloor (x : R) := (up x - 1)%Z.
Definition Zceil (x : R) := (- Zfloor (- x))%Z.
Definition Ztrunc x := if Rlt_bool x 0 then Zceil x else Zfloor x.

Theorem Ztrunc_lub :
forall n x,
(IZR n <= Rabs x)%R ->
(n <= Z.abs (Ztrunc x))%Z.