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.
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.