Section NullDerivative.
Context {V : NormedModule R_AbsRing}.
End NullDerivative.
(** * 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.
(** Extensionality *)
(** Constant function *)
(** ** Operations *)
(** *** Additive operators *)
(** Opposite *)
(** Addition of functions *)

Lemma is_derive_n_sum_n_m n m (f : nat -> R -> R) (k: nat) (x : R) :
locally x (fun t => forall l j , (n <= l <= m)%nat ->(j <= k)%nat -> ex_derive_n (f l) j t) ->
is_derive_n (fun y => sum_n_m (fun j => f j y) n m) k x
(sum_n_m (fun j => Derive_n (f j) k x) n m).