Library Stdlib.Reals.PSeries_reg

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Ranalysis1.
Require Import MVT.
Require Import Lra.
Local Open Scope R_scope.

Definition Boule (x:R) (r:posreal) (y:R) : Prop := Rabs (y - x) < r.

Lemma Boule_convex : forall c d x y z,
  Boule c d x -> Boule c d y -> x <= z <= y -> Boule c d z.

Definition boule_of_interval x y (h : x < y) :
  {c :R & {r : posreal | c - r = x /\ c + r = y }}.

Definition boule_in_interval x y z (h : x < z < y) :
  {c : R & {r | Boule c r z /\ x < c - r /\ c + r < y }}.

Lemma Ball_in_inter : forall c1 c2 r1 r2 x,
  Boule c1 r1 x -> Boule c2 r2 x ->
  {r3 : posreal | forall y, Boule x r3 y -> Boule c1 r1 y /\ Boule c2 r2 y }.

Lemma Boule_center : forall x r, Boule x r x.

Uniform convergence

Definition CVU (fn:nat -> R -> R) (f:R -> R) (x:R)
  (r:posreal) : Prop :=
  forall eps:R,
    0 < eps ->
    exists N : nat ,
      (forall (n:nat) (y:R),
        (N <= n)%nat -> Boule x r y -> Rabs (f y - fn n y) < eps ).

Normal convergence

Definition CVN_r (fn:nat -> R -> R) (r:posreal) : Type :=
  { An :nat -> R &
    { l :R |
      Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (An k)) n) l /\
      (forall (n:nat) (y:R), Boule 0 r y -> Rabs (fn n y) <= An n ) } }.

Definition CVN_R (fn:nat -> R -> R) : Type := forall r:posreal, CVN_r fn r.

Definition SFL (fn:nat -> R -> R)
  (cv:forall x:R, { l :R | Un_cv (fun N:nat => SP fn N x) l })
  (y:R) : R := let (a,_) := cv y in a.

In a complete space, normal convergence implies uniform convergence

Lemma CVN_CVU :
  forall (fn:nat -> R -> R)
    (cv:forall x:R, {l :R | Un_cv (fun N:nat => SP fn N x) l })
    (r:posreal), CVN_r fn r -> CVU (fun n:nat => SP fn n) (SFL fn cv) 0 r.

Each limit of a sequence of functions which converges uniformly is continue

Lemma CVU_continuity :
  forall (fn:nat -> R -> R) (f:R -> R) (x:R) (r:posreal),
    CVU fn f x r ->
    (forall (n:nat) (y:R), Boule x r y -> continuity_pt (fn n) y ) ->
    forall y:R, Boule x r y -> continuity_pt f y.

Lemma continuity_pt_finite_SF :
  forall (fn:nat -> R -> R) (N:nat) (x:R),
    (forall n:nat, (n <= N)%nat -> continuity_pt (fn n) x ) ->
    continuity_pt (fun y:R => sum_f_R0 (fun k:nat => fn k y) N) x.

Continuity and normal convergence

Lemma SFL_continuity_pt :
  forall (fn:nat -> R -> R)
    (cv:forall x:R, { l :R | Un_cv (fun N:nat => SP fn N x) l })
    (r:posreal),
    CVN_r fn r ->
    (forall (n:nat) (y:R), Boule 0 r y -> continuity_pt (fn n) y ) ->
    forall y:R, Boule 0 r y -> continuity_pt (SFL fn cv) y.

Lemma SFL_continuity :
  forall (fn:nat -> R -> R)
    (cv:forall x:R, { l :R | Un_cv (fun N:nat => SP fn N x) l }),
    CVN_R fn -> (forall n:nat, continuity (fn n)) -> continuity (SFL fn cv).

As R is complete, normal convergence implies that (fn) is simply-uniformly convergent

Lemma CVN_R_CVS :
  forall fn:nat -> R -> R,
    CVN_R fn -> forall x:R, { l :R | Un_cv (fun N:nat => SP fn N x) l }.

Lemma CVU_cv : forall f g c d, CVU f g c d ->
   forall x, Boule c d x -> Un_cv (fun n => f n x) (g x).

Lemma CVU_ext_lim :
  forall f g1 g2 c d, CVU f g1 c d -> (forall x, Boule c d x -> g1 x = g2 x ) ->
    CVU f g2 c d.

Lemma CVU_derivable :
  forall f f' g g' c d,
   CVU f' g' c d ->
   (forall x, Boule c d x -> Un_cv (fun n => f n x) (g x)) ->
   (forall n x, Boule c d x -> derivable_pt_lim (f n) x (f' n x)) ->
   forall x, Boule c d x -> derivable_pt_lim g x (g' x).

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