(** Extract the n-th element of a list, as [List.nth_error] does,
    but the index [n] is of type [Z]. *)

Fixpoint list_nth_z (A: Type) (l: list A) (n: Z) {struct l}: option A :=
  match l with
  | nil => None
  | hd :: tl => if zeq n 0 then Some hd else list_nth_z tl (Z.pred n)
  end.

Lemma list_in_map_inv: forall (A B: Type) (f: A -> B) (l: list A) (y: B), In y (List.map f l) -> exists x:A, y = f x /\ In x l.