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 *)

Lemma is_derive_Sn (f : R -> R) (n : nat) (x l : R) :
locally x (ex_derive f) ->
(is_derive_n f (S n) x l <-> is_derive_n (Derive f) n x l).