(** * 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_sum_n n (f : nat -> R -> R) (k: nat) (x : R) : locally x (fun t => forall l j , (l <= n)%nat ->(j <= k)%nat -> ex_derive_n (f l) j t) -> is_derive_n (fun y => sum_n (fun j => f j y) n) k x (sum_n (fun j => Derive_n (f j) k x) n).