(*+ Stdlib: some list functions *)
Fixpoint nth_upd {A} (l : list A) i x :=
match l with
| nil => nil
| y :: l =>
match i with
| O => x :: l
| S i => y :: nth_upd l i x
end
end.

Lemma nth_len {A} (l : list A) :
nth l (List.length l) = None.