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 *)

Lemma is_derive_n_const n a : forall x, is_derive_n (fun _ => a) (S n) x 0.
Lemma ex_derive_n_const a n x: ex_derive_n (fun _  => a) n x.
Lemma Derive_n_const n a  : forall x, Derive_n (fun _ => a) (S n) x = 0.