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.
Definition Zaway x := if Rlt_bool x 0 then Zfloor x else Zceil x.

Theorem Zaway_floor :
forall x,
(x <= 0)%R ->
Zaway x = Zfloor x.