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Diffstat (limited to 'theories/Reals/SeqSeries.v')
| -rw-r--r-- | theories/Reals/SeqSeries.v | 417 |
1 files changed, 417 insertions, 0 deletions
diff --git a/theories/Reals/SeqSeries.v b/theories/Reals/SeqSeries.v new file mode 100644 index 00000000..deb98492 --- /dev/null +++ b/theories/Reals/SeqSeries.v @@ -0,0 +1,417 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: SeqSeries.v,v 1.14.2.1 2004/07/16 19:31:15 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import Max. +Require Export Rseries. +Require Export SeqProp. +Require Export Rcomplete. +Require Export PartSum. +Require Export AltSeries. +Require Export Binomial. +Require Export Rsigma. +Require Export Rprod. +Require Export Cauchy_prod. +Require Export Alembert. +Open Local Scope R_scope. + +(**********) +Lemma sum_maj1 : + forall (fn:nat -> R -> R) (An:nat -> R) (x l1 l2:R) + (N:nat), + Un_cv (fun n:nat => SP fn n x) l1 -> + Un_cv (fun n:nat => sum_f_R0 An n) l2 -> + (forall n:nat, Rabs (fn n x) <= An n) -> + Rabs (l1 - SP fn N x) <= l2 - sum_f_R0 An N. +intros; + cut + (sigT + (fun l:R => + Un_cv (fun n:nat => sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) l)). +intro; + cut + (sigT + (fun l:R => + Un_cv (fun n:nat => sum_f_R0 (fun l:nat => An (S N + l)%nat) n) l)). +intro; elim X; intros l1N H2. +elim X0; intros l2N H3. +cut (l1 - SP fn N x = l1N). +intro; cut (l2 - sum_f_R0 An N = l2N). +intro; rewrite H4; rewrite H5. +apply sum_cv_maj with + (fun l:nat => An (S N + l)%nat) (fun (l:nat) (x:R) => fn (S N + l)%nat x) x. +unfold SP in |- *; apply H2. +apply H3. +intros; apply H1. +symmetry in |- *; eapply UL_sequence. +apply H3. +unfold Un_cv in H0; unfold Un_cv in |- *; intros; elim (H0 eps H5); + intros N0 H6. +unfold R_dist in H6; exists N0; intros. +unfold R_dist in |- *; + replace (sum_f_R0 (fun l:nat => An (S N + l)%nat) n - (l2 - sum_f_R0 An N)) + with (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n - l2); + [ idtac | ring ]. +replace (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n) with + (sum_f_R0 An (S (N + n))). +apply H6; unfold ge in |- *; apply le_trans with n. +apply H7. +apply le_trans with (N + n)%nat. +apply le_plus_r. +apply le_n_Sn. +cut (0 <= N)%nat. +cut (N < S (N + n))%nat. +intros; assert (H10 := sigma_split An H9 H8). +unfold sigma in H10. +do 2 rewrite <- minus_n_O in H10. +replace (sum_f_R0 An (S (N + n))) with + (sum_f_R0 (fun k:nat => An (0 + k)%nat) (S (N + n))). +replace (sum_f_R0 An N) with (sum_f_R0 (fun k:nat => An (0 + k)%nat) N). +cut ((S (N + n) - S N)%nat = n). +intro; rewrite H11 in H10. +apply H10. +apply INR_eq; rewrite minus_INR. +do 2 rewrite S_INR; rewrite plus_INR; ring. +apply le_n_S; apply le_plus_l. +apply sum_eq; intros. +reflexivity. +apply sum_eq; intros. +reflexivity. +apply le_lt_n_Sm; apply le_plus_l. +apply le_O_n. +symmetry in |- *; eapply UL_sequence. +apply H2. +unfold Un_cv in H; unfold Un_cv in |- *; intros. +elim (H eps H4); intros N0 H5. +unfold R_dist in H5; exists N0; intros. +unfold R_dist, SP in |- *; + replace + (sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n - + (l1 - sum_f_R0 (fun k:nat => fn k x) N)) with + (sum_f_R0 (fun k:nat => fn k x) N + + sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n - l1); + [ idtac | ring ]. +replace + (sum_f_R0 (fun k:nat => fn k x) N + + sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) with + (sum_f_R0 (fun k:nat => fn k x) (S (N + n))). +unfold SP in H5; apply H5; unfold ge in |- *; apply le_trans with n. +apply H6. +apply le_trans with (N + n)%nat. +apply le_plus_r. +apply le_n_Sn. +cut (0 <= N)%nat. +cut (N < S (N + n))%nat. +intros; assert (H9 := sigma_split (fun k:nat => fn k x) H8 H7). +unfold sigma in H9. +do 2 rewrite <- minus_n_O in H9. +replace (sum_f_R0 (fun k:nat => fn k x) (S (N + n))) with + (sum_f_R0 (fun k:nat => fn (0 + k)%nat x) (S (N + n))). +replace (sum_f_R0 (fun k:nat => fn k x) N) with + (sum_f_R0 (fun k:nat => fn (0 + k)%nat x) N). +cut ((S (N + n) - S N)%nat = n). +intro; rewrite H10 in H9. +apply H9. +apply INR_eq; rewrite minus_INR. +do 2 rewrite S_INR; rewrite plus_INR; ring. +apply le_n_S; apply le_plus_l. +apply sum_eq; intros. +reflexivity. +apply sum_eq; intros. +reflexivity. +apply le_lt_n_Sm. +apply le_plus_l. +apply le_O_n. +apply existT with (l2 - sum_f_R0 An N). +unfold Un_cv in H0; unfold Un_cv in |- *; intros. +elim (H0 eps H2); intros N0 H3. +unfold R_dist in H3; exists N0; intros. +unfold R_dist in |- *; + replace (sum_f_R0 (fun l:nat => An (S N + l)%nat) n - (l2 - sum_f_R0 An N)) + with (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n - l2); + [ idtac | ring ]. +replace (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n) with + (sum_f_R0 An (S (N + n))). +apply H3; unfold ge in |- *; apply le_trans with n. +apply H4. +apply le_trans with (N + n)%nat. +apply le_plus_r. +apply le_n_Sn. +cut (0 <= N)%nat. +cut (N < S (N + n))%nat. +intros; assert (H7 := sigma_split An H6 H5). +unfold sigma in H7. +do 2 rewrite <- minus_n_O in H7. +replace (sum_f_R0 An (S (N + n))) with + (sum_f_R0 (fun k:nat => An (0 + k)%nat) (S (N + n))). +replace (sum_f_R0 An N) with (sum_f_R0 (fun k:nat => An (0 + k)%nat) N). +cut ((S (N + n) - S N)%nat = n). +intro; rewrite H8 in H7. +apply H7. +apply INR_eq; rewrite minus_INR. +do 2 rewrite S_INR; rewrite plus_INR; ring. +apply le_n_S; apply le_plus_l. +apply sum_eq; intros. +reflexivity. +apply sum_eq; intros. +reflexivity. +apply le_lt_n_Sm. +apply le_plus_l. +apply le_O_n. +apply existT with (l1 - SP fn N x). +unfold Un_cv in H; unfold Un_cv in |- *; intros. +elim (H eps H2); intros N0 H3. +unfold R_dist in H3; exists N0; intros. +unfold R_dist, SP in |- *. +replace + (sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n - + (l1 - sum_f_R0 (fun k:nat => fn k x) N)) with + (sum_f_R0 (fun k:nat => fn k x) N + + sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n - l1); + [ idtac | ring ]. +replace + (sum_f_R0 (fun k:nat => fn k x) N + + sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) with + (sum_f_R0 (fun k:nat => fn k x) (S (N + n))). +unfold SP in H3; apply H3. +unfold ge in |- *; apply le_trans with n. +apply H4. +apply le_trans with (N + n)%nat. +apply le_plus_r. +apply le_n_Sn. +cut (0 <= N)%nat. +cut (N < S (N + n))%nat. +intros; assert (H7 := sigma_split (fun k:nat => fn k x) H6 H5). +unfold sigma in H7. +do 2 rewrite <- minus_n_O in H7. +replace (sum_f_R0 (fun k:nat => fn k x) (S (N + n))) with + (sum_f_R0 (fun k:nat => fn (0 + k)%nat x) (S (N + n))). +replace (sum_f_R0 (fun k:nat => fn k x) N) with + (sum_f_R0 (fun k:nat => fn (0 + k)%nat x) N). +cut ((S (N + n) - S N)%nat = n). +intro; rewrite H8 in H7. +apply H7. +apply INR_eq; rewrite minus_INR. +do 2 rewrite S_INR; rewrite plus_INR; ring. +apply le_n_S; apply le_plus_l. +apply sum_eq; intros. +reflexivity. +apply sum_eq; intros. +reflexivity. +apply le_lt_n_Sm. +apply le_plus_l. +apply le_O_n. +Qed. + +(* Comparaison of convergence for series *) +Lemma Rseries_CV_comp : + forall An Bn:nat -> R, + (forall n:nat, 0 <= An n <= Bn n) -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 Bn N) l) -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l). +intros; apply cv_cauchy_2. +assert (H0 := cv_cauchy_1 _ X). +unfold Cauchy_crit_series in |- *; unfold Cauchy_crit in |- *. +intros; elim (H0 eps H1); intros. +exists x; intros. +cut + (R_dist (sum_f_R0 An n) (sum_f_R0 An m) <= + R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)). +intro; apply Rle_lt_trans with (R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)). +assumption. +apply H2; assumption. +assert (H5 := lt_eq_lt_dec n m). +elim H5; intro. +elim a; intro. +rewrite (tech2 An n m); [ idtac | assumption ]. +rewrite (tech2 Bn n m); [ idtac | assumption ]. +unfold R_dist in |- *; unfold Rminus in |- *; do 2 rewrite Ropp_plus_distr; + do 2 rewrite <- Rplus_assoc; do 2 rewrite Rplus_opp_r; + do 2 rewrite Rplus_0_l; do 2 rewrite Rabs_Ropp; repeat rewrite Rabs_right. +apply sum_Rle; intros. +elim (H (S n + n0)%nat); intros. +apply H8. +apply Rle_ge; apply cond_pos_sum; intro. +elim (H (S n + n0)%nat); intros. +apply Rle_trans with (An (S n + n0)%nat); assumption. +apply Rle_ge; apply cond_pos_sum; intro. +elim (H (S n + n0)%nat); intros; assumption. +rewrite b; unfold R_dist in |- *; unfold Rminus in |- *; + do 2 rewrite Rplus_opp_r; rewrite Rabs_R0; right; + reflexivity. +rewrite (tech2 An m n); [ idtac | assumption ]. +rewrite (tech2 Bn m n); [ idtac | assumption ]. +unfold R_dist in |- *; unfold Rminus in |- *; do 2 rewrite Rplus_assoc; + rewrite (Rplus_comm (sum_f_R0 An m)); rewrite (Rplus_comm (sum_f_R0 Bn m)); + do 2 rewrite Rplus_assoc; do 2 rewrite Rplus_opp_l; + do 2 rewrite Rplus_0_r; repeat rewrite Rabs_right. +apply sum_Rle; intros. +elim (H (S m + n0)%nat); intros; apply H8. +apply Rle_ge; apply cond_pos_sum; intro. +elim (H (S m + n0)%nat); intros. +apply Rle_trans with (An (S m + n0)%nat); assumption. +apply Rle_ge. +apply cond_pos_sum; intro. +elim (H (S m + n0)%nat); intros; assumption. +Qed. + +(* Cesaro's theorem *) +Lemma Cesaro : + forall (An Bn:nat -> R) (l:R), + Un_cv Bn l -> + (forall n:nat, 0 < An n) -> + cv_infty (fun n:nat => sum_f_R0 An n) -> + Un_cv (fun n:nat => sum_f_R0 (fun k:nat => An k * Bn k) n / sum_f_R0 An n) + l. +Proof with trivial. +unfold Un_cv in |- *; intros; assert (H3 : forall n:nat, 0 < sum_f_R0 An n)... +intro; apply tech1... +assert (H4 : forall n:nat, sum_f_R0 An n <> 0)... +intro; red in |- *; intro; assert (H5 := H3 n); rewrite H4 in H5; + elim (Rlt_irrefl _ H5)... +assert (H5 := cv_infty_cv_R0 _ H4 H1); assert (H6 : 0 < eps / 2)... +unfold Rdiv in |- *; apply Rmult_lt_0_compat... +apply Rinv_0_lt_compat; prove_sup... +elim (H _ H6); clear H; intros N1 H; + set (C := Rabs (sum_f_R0 (fun k:nat => An k * (Bn k - l)) N1)); + assert + (H7 : + exists N : nat, + (forall n:nat, (N <= n)%nat -> C / sum_f_R0 An n < eps / 2))... +case (Req_dec C 0); intro... +exists 0%nat; intros... +rewrite H7; unfold Rdiv in |- *; rewrite Rmult_0_l; apply Rmult_lt_0_compat... +apply Rinv_0_lt_compat; prove_sup... +assert (H8 : 0 < eps / (2 * Rabs C))... +unfold Rdiv in |- *; apply Rmult_lt_0_compat... +apply Rinv_0_lt_compat; apply Rmult_lt_0_compat... +prove_sup... +apply Rabs_pos_lt... +elim (H5 _ H8); intros; exists x; intros; assert (H11 := H9 _ H10); + unfold R_dist in H11; unfold Rminus in H11; rewrite Ropp_0 in H11; + rewrite Rplus_0_r in H11... +apply Rle_lt_trans with (Rabs (C / sum_f_R0 An n))... +apply RRle_abs... +unfold Rdiv in |- *; rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs C)... +apply Rinv_0_lt_compat; apply Rabs_pos_lt... +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym... +rewrite Rmult_1_l; replace (/ Rabs C * (eps * / 2)) with (eps / (2 * Rabs C))... +unfold Rdiv in |- *; rewrite Rinv_mult_distr... +ring... +discrR... +apply Rabs_no_R0... +apply Rabs_no_R0... +elim H7; clear H7; intros N2 H7; set (N := max N1 N2); exists (S N); intros; + unfold R_dist in |- *; + replace (sum_f_R0 (fun k:nat => An k * Bn k) n / sum_f_R0 An n - l) with + (sum_f_R0 (fun k:nat => An k * (Bn k - l)) n / sum_f_R0 An n)... +assert (H9 : (N1 < n)%nat)... +apply lt_le_trans with (S N)... +apply le_lt_n_Sm; unfold N in |- *; apply le_max_l... +rewrite (tech2 (fun k:nat => An k * (Bn k - l)) _ _ H9); unfold Rdiv in |- *; + rewrite Rmult_plus_distr_r; + apply Rle_lt_trans with + (Rabs (sum_f_R0 (fun k:nat => An k * (Bn k - l)) N1 / sum_f_R0 An n) + + Rabs + (sum_f_R0 (fun i:nat => An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)) + (n - S N1) / sum_f_R0 An n))... +apply Rabs_triang... +rewrite (double_var eps); apply Rplus_lt_compat... +unfold Rdiv in |- *; rewrite Rabs_mult; fold C in |- *; rewrite Rabs_right... +apply (H7 n); apply le_trans with (S N)... +apply le_trans with N; [ unfold N in |- *; apply le_max_r | apply le_n_Sn ]... +apply Rle_ge; left; apply Rinv_0_lt_compat... + +unfold R_dist in H; unfold Rdiv in |- *; rewrite Rabs_mult; + rewrite (Rabs_right (/ sum_f_R0 An n))... +apply Rle_lt_trans with + (sum_f_R0 (fun i:nat => Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))) + (n - S N1) * / sum_f_R0 An n)... +do 2 rewrite <- (Rmult_comm (/ sum_f_R0 An n)); apply Rmult_le_compat_l... +left; apply Rinv_0_lt_compat... +apply + (Rsum_abs (fun i:nat => An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)) + (n - S N1))... +apply Rle_lt_trans with + (sum_f_R0 (fun i:nat => An (S N1 + i)%nat * (eps / 2)) (n - S N1) * + / sum_f_R0 An n)... +do 2 rewrite <- (Rmult_comm (/ sum_f_R0 An n)); apply Rmult_le_compat_l... +left; apply Rinv_0_lt_compat... +apply sum_Rle; intros; rewrite Rabs_mult; + pattern (An (S N1 + n0)%nat) at 2 in |- *; + rewrite <- (Rabs_right (An (S N1 + n0)%nat))... +apply Rmult_le_compat_l... +apply Rabs_pos... +left; apply H; unfold ge in |- *; apply le_trans with (S N1); + [ apply le_n_Sn | apply le_plus_l ]... +apply Rle_ge; left... +rewrite <- (scal_sum (fun i:nat => An (S N1 + i)%nat) (n - S N1) (eps / 2)); + unfold Rdiv in |- *; repeat rewrite Rmult_assoc; apply Rmult_lt_compat_l... +pattern (/ 2) at 2 in |- *; rewrite <- Rmult_1_r; apply Rmult_lt_compat_l... +apply Rinv_0_lt_compat; prove_sup... +rewrite Rmult_comm; apply Rmult_lt_reg_l with (sum_f_R0 An n)... +rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym... +rewrite Rmult_1_l; rewrite Rmult_1_r; rewrite (tech2 An N1 n)... +rewrite Rplus_comm; + pattern (sum_f_R0 (fun i:nat => An (S N1 + i)%nat) (n - S N1)) at 1 in |- *; + rewrite <- Rplus_0_r; apply Rplus_lt_compat_l... +apply Rle_ge; left; apply Rinv_0_lt_compat... +replace (sum_f_R0 (fun k:nat => An k * (Bn k - l)) n) with + (sum_f_R0 (fun k:nat => An k * Bn k) n + + sum_f_R0 (fun k:nat => An k * - l) n)... +rewrite <- (scal_sum An n (- l)); field... +rewrite <- plus_sum; apply sum_eq; intros; ring... +Qed. + +Lemma Cesaro_1 : + forall (An:nat -> R) (l:R), + Un_cv An l -> Un_cv (fun n:nat => sum_f_R0 An (pred n) / INR n) l. +Proof with trivial. +intros Bn l H; set (An := fun _:nat => 1)... +assert (H0 : forall n:nat, 0 < An n)... +intro; unfold An in |- *; apply Rlt_0_1... +assert (H1 : forall n:nat, 0 < sum_f_R0 An n)... +intro; apply tech1... +assert (H2 : cv_infty (fun n:nat => sum_f_R0 An n))... +unfold cv_infty in |- *; intro; case (Rle_dec M 0); intro... +exists 0%nat; intros; apply Rle_lt_trans with 0... +assert (H2 : 0 < M)... +auto with real... +clear n; set (m := up M); elim (archimed M); intros; + assert (H5 : (0 <= m)%Z)... +apply le_IZR; unfold m in |- *; simpl in |- *; left; apply Rlt_trans with M... +elim (IZN _ H5); intros; exists x; intros; unfold An in |- *; rewrite sum_cte; + rewrite Rmult_1_l; apply Rlt_trans with (IZR (up M))... +apply Rle_lt_trans with (INR x)... +rewrite INR_IZR_INZ; fold m in |- *; rewrite <- H6; right... +apply lt_INR; apply le_lt_n_Sm... +assert (H3 := Cesaro _ _ _ H H0 H2)... +unfold Un_cv in |- *; unfold Un_cv in H3; intros; elim (H3 _ H4); intros; + exists (S x); intros; unfold R_dist in |- *; unfold R_dist in H5; + apply Rle_lt_trans with + (Rabs + (sum_f_R0 (fun k:nat => An k * Bn k) (pred n) / sum_f_R0 An (pred n) - l))... +right; + replace (sum_f_R0 Bn (pred n) / INR n - l) with + (sum_f_R0 (fun k:nat => An k * Bn k) (pred n) / sum_f_R0 An (pred n) - l)... +unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- l)); + apply Rplus_eq_compat_l... +unfold An in |- *; + replace (sum_f_R0 (fun k:nat => 1 * Bn k) (pred n)) with + (sum_f_R0 Bn (pred n))... +rewrite sum_cte; rewrite Rmult_1_l; replace (S (pred n)) with n... +apply S_pred with 0%nat; apply lt_le_trans with (S x)... +apply lt_O_Sn... +apply sum_eq; intros; ring... +apply H5; unfold ge in |- *; apply le_S_n; replace (S (pred n)) with n... +apply S_pred with 0%nat; apply lt_le_trans with (S x)... +apply lt_O_Sn... +Qed. |