(*****************************************************)
(** * Derivative's definition using Landau's kernel  *)
(*****************************************************)

Definition derivable_pt_lim f (x l:R) : Prop :=
  forall eps:R,
    0 < eps ->
    exists delta : posreal,
      (forall h:R,
        h <> 0 -> Rabs h < delta -> Rabs ((f (x + h) - f x) / h - l) < eps).Definition derivable_pt_abs f (x l:R) : Prop := derivable_pt_lim f x l.Definition derivable_pt f (x:R) := { l:R | derivable_pt_abs f x l }.Definition derivable f := forall x:R, derivable_pt f x.Definition derive_pt f (x:R) (pr:derivable_pt f x) := proj1_sig pr.Definition derive f (pr:derivable f) (x:R) := derive_pt f x (pr x).Definition antiderivative f (g:R -> R) (a b:R) : Prop :=
  (forall x:R,
    a <= x <= b ->  exists pr : derivable_pt g x, f x = derive_pt g x pr) /\
  a <= b.(**************************************)
(** * Class of differential functions *)
(**************************************)
Record Differential : Type := mkDifferential
  {d1 :> R -> R; cond_diff : derivable d1}.(**************************************)
(** * Class of differential functions *)
(**************************************)
Record Differential : Type := mkDifferential
  {d1 :> R -> R; cond_diff : derivable d1}.(**************************************)
(** * Class of differential functions *)
(**************************************)
Record Differential : Type := mkDifferential
  {d1 :> R -> R; cond_diff : derivable d1}.(**************************************)
(** * Class of differential functions *)
(**************************************)
Record Differential : Type := mkDifferential
  {d1 :> R -> R; cond_diff : derivable d1}.Record Differential_D2 : Type := mkDifferential_D2
  {d2 :> R -> R;
    cond_D1 : derivable d2;
    cond_D2 : derivable (derive d2 cond_D1)}.Record Differential_D2 : Type := mkDifferential_D2
  {d2 :> R -> R;
    cond_D1 : derivable d2;
    cond_D2 : derivable (derive d2 cond_D1)}.Record Differential_D2 : Type := mkDifferential_D2
  {d2 :> R -> R;
    cond_D1 : derivable d2;
    cond_D2 : derivable (derive d2 cond_D1)}.Record Differential_D2 : Type := mkDifferential_D2
  {d2 :> R -> R;
    cond_D1 : derivable d2;
    cond_D2 : derivable (derive d2 cond_D1)}.Record Differential_D2 : Type := mkDifferential_D2
  {d2 :> R -> R;
    cond_D1 : derivable d2;
    cond_D2 : derivable (derive d2 cond_D1)}.

Lemma uniqueness_step2 : forall f (x l:R), derivable_pt_lim f x l -> limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0.