(* 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_app_None : forall (A : Set)(ls1 ls2 : list A) i, nth_option ls1 i = None -> nth_option (ls1 ++ ls2) i = nth_option ls2 (i - length ls1).