(** Zeven, used for rounding to nearest, ties to even *)
Definition Zeven (n : Z) :=
  match n with
  | Zpos (xO _) => true
  | Zneg (xO _) => true
  | Z0 => true
  | _ => false
  end.

Theorem Zeven_ex : forall x, exists p, x = (2 * p + if Zeven x then 0 else 1)%Z.