(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

Require Import Rbase.
Require Import Rfunctions.
Require Import Rseries.
Require Import SeqProp.
Require Import PartSum.
Require Import Lra.
Local Open Scope R_scope.

(**********)
(** * Formalization of alternated series *)
Definition tg_alt (Un:nat -> R) (i:nat) : R := (-1) ^ i * Un i.
Definition positivity_seq (Un:nat -> R) : Prop := forall n:nat, 0 <= Un n.

Lemma CV_ALT_step0 :
  forall Un:nat -> R,
    Un_decreasing Un ->
    Un_growing (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))).
Proof.
  intros; unfold Un_growing; intro.
  cut ((2 * S n)%nat = S (S (2 * n))).
  - intro; rewrite H0.
    do 4 rewrite tech5; repeat rewrite Rplus_assoc; apply Rplus_le_compat_l.
    pattern (tg_alt Un (S (2 * n))) at 1; rewrite <- Rplus_0_r.
    apply Rplus_le_compat_l.
    unfold tg_alt; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even;
      rewrite Rmult_1_l.
    apply Rplus_le_reg_l with (Un (S (2 * S n))).
    rewrite Rplus_0_r;
      replace (Un (S (2 * S n)) + (Un (2 * S n)%nat + -1 * Un (S (2 * S n)))) with
      (Un (2 * S n)%nat); [ idtac | ring ].
    apply H.
  - cut (forall n:nat, S n = (n + 1)%nat); [ intro | intro; ring ].
    rewrite (H0 n); rewrite (H0 (S (2 * n))); rewrite (H0 (2 * n)%nat); ring.
Qed.

Lemma CV_ALT_step1 :
  forall Un:nat -> R,
    Un_decreasing Un ->
    Un_decreasing (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N)).
Proof.
  intros; unfold Un_decreasing; intro.
  cut ((2 * S n)%nat = S (S (2 * n))).
  - intro; rewrite H0; do 2 rewrite tech5; repeat rewrite Rplus_assoc.
    pattern (sum_f_R0 (tg_alt Un) (2 * n)) at 2; rewrite <- Rplus_0_r.
    apply Rplus_le_compat_l.
    unfold tg_alt; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even;
      rewrite Rmult_1_l.
    apply Rplus_le_reg_l with (Un (S (2 * n))).
    rewrite Rplus_0_r;
      replace (Un (S (2 * n)) + (-1 * Un (S (2 * n)) + Un (2 * S n)%nat)) with
      (Un (2 * S n)%nat); [ idtac | ring ].
    rewrite H0; apply H.
  - cut (forall n:nat, S n = (n + 1)%nat); [ intro | intro; ring ].
    rewrite (H0 n); rewrite (H0 (S (2 * n))); rewrite (H0 (2 * n)%nat); ring.
Qed.

(**********)
Lemma CV_ALT_step2 :
  forall (Un:nat -> R) (N:nat),
    Un_decreasing Un ->
    positivity_seq Un ->
    sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N)) <= 0.
Proof.
  intros; induction  N as [| N HrecN].
  - simpl; unfold tg_alt; simpl; rewrite Rmult_1_r.
    replace (-1 * -1 * Un 2%nat) with (Un 2%nat); [ idtac | ring ].
    apply Rplus_le_reg_l with (Un 1%nat); rewrite Rplus_0_r.
    replace (Un 1%nat + (-1 * Un 1%nat + Un 2%nat)) with (Un 2%nat);
      [ apply H | ring ].
  - cut (S (2 * S N) = S (S (S (2 * N)))).
    + intro; rewrite H1; do 2 rewrite tech5.
      apply Rle_trans with (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N))).
      * pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N))) at 2;
          rewrite <- Rplus_0_r.
        rewrite Rplus_assoc; apply Rplus_le_compat_l.
        unfold tg_alt; rewrite <- H1.
        rewrite pow_1_odd.
        cut (S (S (2 * S N)) = (2 * S (S N))%nat).
        -- intro; rewrite H2; rewrite pow_1_even; rewrite Rmult_1_l; rewrite <- H2.
           apply Rplus_le_reg_l with (Un (S (2 * S N))).
           rewrite Rplus_0_r;
             replace (Un (S (2 * S N)) + (-1 * Un (S (2 * S N)) + Un (S (S (2 * S N)))))
             with (Un (S (S (2 * S N)))); [ idtac | ring ].
           apply H.
        -- ring.
      * apply HrecN.
    + ring.
Qed.

(** A more general inequality *)
Lemma CV_ALT_step3 :
  forall (Un:nat -> R) (N:nat),
    Un_decreasing Un ->
    positivity_seq Un -> sum_f_R0 (fun i:nat => tg_alt Un (S i)) N <= 0.
Proof.
  intros; induction  N as [| N HrecN].
  - simpl; unfold tg_alt; simpl; rewrite Rmult_1_r.
    apply Rplus_le_reg_l with (Un 1%nat).
    rewrite Rplus_0_r; replace (Un 1%nat + -1 * Un 1%nat) with 0;
      [ apply H0 | ring ].
  - assert (H1 := even_odd_cor N).
    elim H1; intros.
    elim H2; intro.
    + rewrite H3; apply CV_ALT_step2; assumption.
    + rewrite H3; rewrite tech5.
      apply Rle_trans with (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * x))).
      * pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * x))) at 2;
          rewrite <- Rplus_0_r.
        apply Rplus_le_compat_l.
        unfold tg_alt; simpl.
        replace (x + (x + 0))%nat with (2 * x)%nat; [ idtac | ring ].
        rewrite pow_1_even.
        replace (-1 * (-1 * (-1 * 1)) * Un (S (S (S (2 * x))))) with
          (- Un (S (S (S (2 * x))))); [ idtac | ring ].
        apply Rplus_le_reg_l with (Un (S (S (S (2 * x))))).
        rewrite Rplus_0_r; rewrite Rplus_opp_r.
        apply H0.
      * apply CV_ALT_step2; assumption.
Qed.

  (**********)
Lemma CV_ALT_step4 :
  forall Un:nat -> R,
    Un_decreasing Un ->
    positivity_seq Un ->
    has_ub (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))).
Proof.
  intros; unfold has_ub; unfold bound.
  exists (Un 0%nat).
  unfold is_upper_bound; intros; elim H1; intros.
  rewrite H2; rewrite decomp_sum.
  - replace (tg_alt Un 0) with (Un 0%nat).
    + pattern (Un 0%nat) at 2; rewrite <- Rplus_0_r.
      apply Rplus_le_compat_l.
      apply CV_ALT_step3; assumption.
    + unfold tg_alt; simpl; ring.
  - apply Nat.lt_0_succ.
Qed.

(** This lemma gives an interesting result about alternated series *)
Lemma CV_ALT :
  forall Un:nat -> R,
    Un_decreasing Un ->
    positivity_seq Un ->
    Un_cv Un 0 ->
    { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l }.
Proof.
  intros.
  assert (H2 := CV_ALT_step0 _ H).
  assert (H3 := CV_ALT_step4 _ H H0).
  destruct (growing_cv _ H2 H3) as (x,p).
  exists x.
  unfold Un_cv; unfold Rdist; unfold Un_cv in H1;
    unfold Rdist in H1; unfold Un_cv in p; unfold Rdist in p.
  intros; cut (0 < eps / 2);
    [ intro
      | unfold Rdiv; apply Rmult_lt_0_compat;
	[ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
  elim (H1 (eps / 2) H5); intros N2 H6.
  elim (p (eps / 2) H5); intros N1 H7.
  set (N := max (S (2 * N1)) N2).
  exists N; intros.
  assert (H9 := even_odd_cor n).
  elim H9; intros P H10.
  assert (N1 <= P)%nat. {
    elim H10; intro; apply le_double.
    - rewrite <- H11; apply Nat.le_trans with N.
      + unfold N; apply Nat.le_trans with (S (2 * N1));
          [ apply Nat.le_succ_diag_r | apply Nat.le_max_l ].
      + assumption.
    - apply Nat.lt_succ_r.
      rewrite <- H11.
      apply Nat.lt_le_trans with N.
      + unfold N; apply Nat.lt_le_trans with (S (2 * N1)).
        * apply Nat.lt_succ_diag_r.
        * apply Nat.le_max_l.
      + assumption.
  }
  elim H10; intro.
  - replace (sum_f_R0 (tg_alt Un) n - x) with
      (sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n)).
    + apply Rle_lt_trans with
        (Rabs (sum_f_R0 (tg_alt Un) (S n) - x) + Rabs (- tg_alt Un (S n))).
      * apply Rabs_triang.
      * rewrite <-(Rplus_half_diag eps); apply Rplus_lt_compat.
        -- rewrite H12; apply H7; assumption.
        -- rewrite Rabs_Ropp; unfold tg_alt; rewrite Rabs_mult;
             rewrite pow_1_abs; rewrite Rmult_1_l; unfold Rminus in H6;
             rewrite Ropp_0 in H6; rewrite <- (Rplus_0_r (Un (S n)));
             apply H6.
           unfold ge; apply Nat.le_trans with n.
           ++ apply Nat.le_trans with N; [ unfold N; apply Nat.le_max_r | assumption ].
           ++ apply Nat.le_succ_diag_r.
    + rewrite tech5; ring.
  - rewrite H12; apply Rlt_trans with (eps / 2).
    + apply H7; assumption.
    + unfold Rdiv; apply Rmult_lt_reg_l with 2.
      * prove_sup0.
      * rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite Rinv_l;
          [ rewrite Rmult_1_r | discrR ].
        rewrite <-Rplus_diag.
        pattern eps at 1; rewrite <- (Rplus_0_r eps); apply Rplus_lt_compat_l;
          assumption.

Qed.


(*************************************************)
(** *       Convergence of alternated series     *)
Theorem alternated_series :
  forall Un:nat -> R,
    Un_decreasing Un ->
    Un_cv Un 0 ->
    { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l }.
Proof.
  intros; apply CV_ALT.
  - assumption.
  - unfold positivity_seq; apply decreasing_ineq; assumption.
  - assumption.
Qed.

Theorem alternated_series_ineq :
  forall (Un:nat -> R) (l:R) (N:nat),
    Un_decreasing Un ->
    Un_cv Un 0 ->
    Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l ->
    sum_f_R0 (tg_alt Un) (S (2 * N)) <= l <= sum_f_R0 (tg_alt Un) (2 * N).
Proof.
  intros.
  assert (Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N)) l). {
    unfold Un_cv; unfold Rdist; unfold Un_cv in H1;
      unfold Rdist in H1; intros.
    elim (H1 eps H2); intros.
    exists x; intros.
    apply H3.
    apply Nat.le_trans with n; [ assumption | ].
    rewrite <- Nat.double_twice; apply Nat.le_add_r.
  }
  assert (Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))) l). {
    unfold Un_cv; unfold Rdist; unfold Un_cv in H1;
      unfold Rdist in H1; intros.
    elim (H1 eps H3); intros.
    exists x; intros.
    apply H4.
    apply Nat.le_trans with n; [ assumption | ].
    apply Nat.le_le_succ_r.
    rewrite <- Nat.double_twice; apply Nat.le_add_r.
  }
  intros; split.
  - apply (growing_ineq (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N)))).
    + apply CV_ALT_step0; assumption.
    + assumption.
  - apply (decreasing_ineq (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N))).
    + apply CV_ALT_step1; assumption.
    + assumption.
Qed.

(***************************************)
(** * Application : construction of PI *)
(***************************************)

Definition PI_tg (n:nat) := / INR (2 * n + 1).

Lemma PI_tg_pos : forall n:nat, 0 <= PI_tg n.
Proof.
  intro; unfold PI_tg; left; apply Rinv_0_lt_compat; apply lt_INR_0;
    replace (2 * n + 1)%nat with (S (2 * n)); [ apply Nat.lt_0_succ | ring ].
Qed.

Lemma PI_tg_decreasing : Un_decreasing PI_tg.
Proof.
  unfold PI_tg, Un_decreasing; intro.
  apply Rmult_le_reg_l with (INR (2 * n + 1)).
  - apply lt_INR_0.
    replace (2 * n + 1)%nat with (S (2 * n)); [ apply Nat.lt_0_succ | ring ].
  - rewrite Rinv_r.
    + apply Rmult_le_reg_l with (INR (2 * S n + 1)).
      * apply lt_INR_0.
        replace (2 * S n + 1)%nat with (S (2 * S n)); [ apply Nat.lt_0_succ | ring ].
      * rewrite (Rmult_comm (INR (2 * S n + 1))); rewrite Rmult_assoc;
          rewrite Rinv_l.
        -- do 2 rewrite Rmult_1_r; apply le_INR.
           replace (2 * S n + 1)%nat with (S (S (2 * n + 1))).
           ++ apply Nat.le_trans with (S (2 * n + 1)); apply Nat.le_succ_diag_r.
           ++ ring.
        -- apply not_O_INR; discriminate.
    + apply not_O_INR; replace (2 * n + 1)%nat with (S (2 * n));
        [ discriminate | ring ].
Qed.

Lemma PI_tg_cv : Un_cv PI_tg 0.
Proof.
  unfold Un_cv; unfold Rdist; intros.
  assert (0 < 2 * eps) by lra.
  assert (H1 := archimed (/ (2 * eps))).
  assert (0 <= up (/ (2 * eps)))%Z. {
    apply le_IZR.
    left; apply Rlt_trans with (/ (2 * eps)).
    - apply Rinv_0_lt_compat; assumption.
    - elim H1; intros; assumption.
  }
  assert (H3 := IZN (up (/ (2 * eps))) H2).
  elim H3; intros N H4.
  assert (0 < N)%nat. {
    elim H1; intros H5 _.
    destruct (lt_eq_lt_dec 0 N) as [[| <- ]|H6].
    - assumption.
    - rewrite H4 in H5.
      simpl in H5.
      cut (0 < / (2 * eps)); [ intro | apply Rinv_0_lt_compat; assumption ].
      elim (Rlt_irrefl _ (Rlt_trans _ _ _ H6 H5)).
    - elim (Nat.nlt_0_r _ H6).
  }
  exists N; intros.
  assert (0 < n)%nat by (apply Nat.lt_le_trans with N; assumption).
  unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r;
    rewrite Rabs_right.
  2:{ apply Rle_ge; apply PI_tg_pos. }
  unfold PI_tg; apply Rlt_trans with (/ INR (2 * n)).
  - apply Rmult_lt_reg_l with (INR (2 * n)).
    { apply lt_INR_0. auto with zarith. }
    rewrite Rinv_r.
    2:{ replace n with (S (pred n)).
        - apply not_O_INR; discriminate.
        - apply Nat.lt_succ_pred with 0%nat.
          assumption. }
    apply Rmult_lt_reg_l with (INR (2 * n + 1)).
    { apply lt_INR_0.
      replace (2 * n + 1)%nat with (S (2 * n)) by ring; apply Nat.lt_0_succ. }
    rewrite (Rmult_comm (INR (2 * n + 1))).
    rewrite Rmult_assoc; rewrite Rinv_l.
    2:{ apply not_O_INR; replace (2 * n + 1)%nat with (S (2 * n));
        [ discriminate | ring ]. }
    do 2 rewrite Rmult_1_r; apply lt_INR.
    replace (2 * n + 1)%nat with (S (2 * n)) by ring; apply Nat.lt_succ_diag_r.

  - apply Rle_lt_trans with (/ INR (2 * N)).
    + apply Rinv_le_contravar.
      * rewrite mult_INR; apply Rmult_lt_0_compat;
          [ simpl; prove_sup0 | apply lt_INR_0; assumption ].
      * apply le_INR.
        now apply Nat.mul_le_mono_nonneg_l; [ apply Nat.le_0_l | ].
    + rewrite mult_INR.
      apply Rmult_lt_reg_l with (INR N / eps).
      * apply Rdiv_lt_0_compat with (2 := H).
        now apply (lt_INR 0).
      * replace (_ */ _) with (/(2 * eps)).
        -- replace (_ / _ * _) with (INR N).
           ++ rewrite INR_IZR_INZ.
              now rewrite <- H4.
           ++ field.
              now apply Rgt_not_eq.
        -- simpl (INR 2); field; split.
           ++ now apply Rgt_not_eq, (lt_INR 0).
           ++ now apply Rgt_not_eq.

Qed.

Lemma exist_PI :
  { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt PI_tg) N) l }.
Proof.
  apply alternated_series.
  - apply PI_tg_decreasing.
  - apply PI_tg_cv.
Qed.

(** Now, PI is defined *)
Definition Alt_PI : R := 4 * (let (a,_) := exist_PI in a).

(** We can get an approximation of PI with the following inequality *)
Lemma Alt_PI_ineq :
  forall N:nat,
    sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= Alt_PI / 4 <=
    sum_f_R0 (tg_alt PI_tg) (2 * N).
Proof.
  intro; apply alternated_series_ineq.
  - apply PI_tg_decreasing.
  - apply PI_tg_cv.
  - unfold Alt_PI; case exist_PI; intro.
    replace (4 * x / 4) with x.
    + trivial.
    + unfold Rdiv; rewrite (Rmult_comm 4); rewrite Rmult_assoc;
        rewrite Rinv_r; [ rewrite Rmult_1_r; reflexivity | discrR ].
Qed.

Lemma Alt_PI_RGT_0 : 0 < Alt_PI.
Proof.
  assert (H := Alt_PI_ineq 0).
  apply Rmult_lt_reg_l with (/ 4).
  - apply Rinv_0_lt_compat; prove_sup0.
  - rewrite Rmult_0_r; rewrite Rmult_comm.
    elim H; clear H; intros H _.
    unfold Rdiv in H;
      apply Rlt_le_trans with (sum_f_R0 (tg_alt PI_tg) (S (2 * 0))).
    + simpl; unfold tg_alt; simpl; rewrite Rmult_1_l;
        rewrite Rmult_1_r; apply Rplus_lt_reg_l with (PI_tg 1).
      rewrite Rplus_0_r;
        replace (PI_tg 1 + (PI_tg 0 + -1 * PI_tg 1)) with (PI_tg 0);
        [ unfold PI_tg | ring ].
      simpl; apply Rinv_lt_contravar.
      * rewrite Rmult_1_l; replace (2 + 1) with 3; [ prove_sup0 | ring ].
      * rewrite Rplus_comm; pattern 1 at 1; rewrite <- Rplus_0_r;
          apply Rplus_lt_compat_l; prove_sup0.
    + assumption.
Qed.
