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_add (z1 z2: Z): Z_to_K (z1 + z2)%Z = Z_to_K z1 + Z_to_K z2.