Fixpoint nth_error (l:list A) (n:nat) {struct n} : option A :=
    match n, l with
      | O, x :: _ => Some x
      | S n, _ :: l' => nth_error l' n
      | _, _ => None
    end.Definition nth_default (default:A) (l:list A) (n:nat) : A :=
    match nth_error l n with
      | Some x => x
      | None => default
    end.

Lemma nth_error_app l l' n : nth_error (l ++ l') n = if Nat.ltb n (length l) then nth_error l n else nth_error l' (n - length l).