(** An injection from [Z] to [nat] *)
Definition Z_to_nat_i (z:Z) :nat :=
   match z with 
   | Z0 => O
   | Zpos p => Div2.double (nat_of_P p)
   | Zneg p => pred (Div2.double (nat_of_P p))
   end.

Lemma even_odd_exists_dec:forall n, {p : nat | n = Div2.double p} + {p : nat | n = pred (Div2.double p)}.