Section Even_Odd.
(** 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.