Definition Zceil (x : R) := (- Zfloor (- x))%Z.

(* Specifications: *)
Theorem Zceil_ub :
forall x : R,
(x <= Z2R (Zceil x))%R.
Theorem Zceil_glb :
forall n x,
(x <= Z2R n)%R ->
(Zceil x <= n)%Z.
Theorem Zceil_imp :
forall n x,
(Z2R (n - 1) < x <= Z2R n)%R ->
Zceil x = n.
Theorem Zceil_Z2R :
forall n,
Zceil (Z2R n) = n.
Theorem Zceil_le :
forall x y, (x <= y)%R ->
(Zceil x <= Zceil y)%Z.
Theorem Zceil_floor_neq :
forall x : R,
(Z2R (Zfloor x) <> x)%R ->
(Zceil x = Zfloor x + 1)%Z.