(*+ 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_nth_upd {A} (l : list A) i x y j :
nth l i = Some x ->
nth (nth_upd l i y) j = if eq_nat_dec i j then Some y else nth l j.