(** * Iterated differential *)

(** ** Definition *)

Fixpoint Derive_n (f : R -> R) (n : nat) x :=
  match n with
    | O => f x
    | S n => Derive (Derive_n f n) x
  end.Definition ex_derive_n f n x :=
  match n with
  | O => True
  | S n => ex_derive (Derive_n f n) x
  end.Definition is_derive_n f n x l :=
  match n with
  | O => f x = l
  | S n => is_derive (Derive_n f n) x l
  end.

Lemma is_derive_n_plus (f g : R -> R) (n : nat) (x lf lg : R) : locally x (fun y => forall k, (k <= n)%nat -> ex_derive_n f k y) -> locally x (fun y => forall k, (k <= n)%nat -> ex_derive_n g k y) -> is_derive_n f n x lf -> is_derive_n g n x lg -> is_derive_n (fun x => f x + g x) n x (lf + lg).