(** An injection from [nat] to [Z] *)
Definition nat_to_Z_i (n:nat) := 
  match even_odd_exists_dec n with
  | inl s => let (k, _) := s in Z_of_nat k
  | inr s => let (k, _) := s in Z.opp (Z_of_nat k)
  end.

Lemma Z_to_nat_to_Z_i : forall (n:nat), Z_to_nat_i (nat_to_Z_i n) = n.