(** * Iterated differential *)

Fixpoint ex_diff_n f n x y :=
  continuity_2d_pt f x y /\
  match n with
  | O => True
  | S n =>
    ex_derive (fun z => f z y) x /\
    ex_derive (fun z => f x z) y /\
    ex_diff_n (fun u v => Derive (fun z => f z v) u) n x y /\
    ex_diff_n (fun u v => Derive (fun z => f u z) v) n x y
  end.

Lemma ex_diff_n_deriv: forall n p q, (p+q <= n)%nat -> forall f x y, ex_diff_n f n x y-> ex_diff_n (partial_derive p q f) (n -(p+q)) x y.