Require Import Rbase.
Require Import Rfunctions.
Require Import Compare.
Local Open Scope R_scope.

Implicit Type r : R.

Section sequence.

  Variable Un : nat -> R.

  Fixpoint Rmax_N (N:nat) : R :=
    match N with
      | O => Un 0
      | S n => Rmax (Un (S n)) (Rmax_N n)
    end.

  Definition EUn r : Prop := exists i : nat, r = Un i.

  Definition Un_cv (l:R) : Prop :=
    forall eps:R,
      eps > 0 ->
      exists N : nat, (forall n:nat, (n >= N)%nat -> Rdist (Un n) l < eps).

  Definition Cauchy_crit : Prop :=
    forall eps:R,
      eps > 0 ->
      exists N : nat,
        (forall n m:nat,
          (n >= N)%nat -> (m >= N)%nat -> Rdist (Un n) (Un m) < eps).

  Definition Un_growing : Prop := forall n:nat, Un n <= Un (S n).

  Lemma EUn_noempty : exists r : R, EUn r.

  Lemma Un_in_EUn : forall n:nat, EUn (Un n).

  Lemma Un_bound_imp :
    forall x:R, (forall n:nat, Un n <= x) -> is_upper_bound EUn x.

  Lemma growing_prop :
    forall n m:nat, Un_growing -> (n >= m)%nat -> Un n >= Un m.

  Lemma Un_cv_crit_lub : Un_growing -> forall l, is_lub EUn l -> Un_cv l.

  Lemma Un_cv_crit : Un_growing -> bound EUn -> exists l : R, Un_cv l.

  Lemma finite_greater :
    forall N:nat, exists M : R, (forall n:nat, (n <= N)%nat -> Un n <= M).

  Lemma cauchy_bound : Cauchy_crit -> bound EUn.

End sequence.

Section Isequence.

  Variable An : nat -> R.

  Definition Pser (x l:R) : Prop := infinite_sum (fun n:nat => An n * x ^ n) l.

End Isequence.

Lemma GP_infinite :
  forall x:R, Rabs x < 1 -> Pser (fun n:nat => 1) x (/ (1 - x)).

Lemma CV_shift :
  forall f k l, Un_cv (fun n => f (n + k)%nat) l -> Un_cv f l.

Lemma CV_shift' :
  forall f k l, Un_cv f l -> Un_cv (fun n => f (n + k)%nat) l.


Lemma Un_growing_shift :
   forall k un, Un_growing un -> Un_growing (fun n => un (n + k)%nat).