(* There is nth_error in the Coq library, but it seems to return something in Type. *)
Fixpoint nth_option(A : Set)(ls : list A)(i : nat) :=
  match ls with 
    | nil => None
    | a :: ls' =>
      match i with
        | O => Some a
        | S i' =>
          nth_option ls' i'
          end
  end.

Theorem nth_option_snd_split : forall (A B : Set)(ls : list (A * B)) n a b, nth_option ls n = Some (a, b) -> nth_option (snd (split ls)) n = Some b.