Definition Z_to_K (z: Z) :=
  match z with
  | Z0 => 0
  | Zpos _ => n_to_K (Z.abs_nat z)
  | Zneg _ => - (n_to_K (Z.abs_nat z))
  end.

Lemma Z_to_K_opp (z: Z): Z_to_K (-z)%Z = - Z_to_K z.