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author | 2003-02-14 10:04:41 +0000 | |
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committer | 2003-02-14 10:04:41 +0000 | |
commit | c983078ec789ce6f9ac590747b48a7c7f674c5bd (patch) | |
tree | b517b4d8daea344ffdf7ae0809dd1f96e25fe3b4 /theories/Reals/SeqSeries.v | |
parent | 82ff751239d4ed1320ff9a633a8b94a603fe25e0 (diff) |
Ajout du theoreme de Cesaro
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@3681 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Reals/SeqSeries.v')
-rw-r--r-- | theories/Reals/SeqSeries.v | 103 |
1 files changed, 103 insertions, 0 deletions
diff --git a/theories/Reals/SeqSeries.v b/theories/Reals/SeqSeries.v index bd818a0ba..0d4bfbfad 100644 --- a/theories/Reals/SeqSeries.v +++ b/theories/Reals/SeqSeries.v @@ -10,6 +10,7 @@ Require Rbase. Require Rfunctions. +Require Max. Require Export Rseries. Require Export SeqProp. Require Export Rcomplete. @@ -200,3 +201,105 @@ Apply Rle_sym1. Apply cond_pos_sum; Intro. Elim (H (plus (S m) n0)); Intros; Assumption. Qed. + +(* Cesaro's theorem *) +Lemma Cesaro : (An,Bn:nat->R;l:R) (Un_cv Bn l) -> ((n:nat)``0<(An n)``) -> (cv_infty [n:nat](sum_f_R0 An n)) -> (Un_cv [n:nat](Rdiv (sum_f_R0 [k:nat]``(An k)*(Bn k)`` n) (sum_f_R0 An n)) l). +Proof with Trivial. +Unfold Un_cv; Intros; Assert H3 : (n:nat)``0<(sum_f_R0 An n)``. +Intro; Apply tech1. +Assert H4 : (n:nat) ``(sum_f_R0 An n)<>0``. +Intro; Red; Intro; Assert H5 := (H3 n); Rewrite H4 in H5; Elim (Rlt_antirefl ? H5). +Assert H5 := (cv_infty_cv_R0 ? H4 H1); Assert H6 : ``0<eps/2``. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply Rlt_Rinv; Sup. +Elim (H ? H6); Clear H; Intros N1 H; Pose C := (Rabsolu (sum_f_R0 [k:nat]``(An k)*((Bn k)-l)`` N1)); Assert H7 : (EX N:nat | (n:nat) (le N n) -> ``C/(sum_f_R0 An n)<eps/2``). +Case (Req_EM C R0); Intro. +Exists O; Intros. +Rewrite H7; Unfold Rdiv; Rewrite Rmult_Ol; Apply Rmult_lt_pos. +Apply Rlt_Rinv; Sup. +Assert H8 : ``0<eps/(2*(Rabsolu C))``. +Unfold Rdiv; Apply Rmult_lt_pos. +Apply Rlt_Rinv; Apply Rmult_lt_pos. +Sup. +Apply Rabsolu_pos_lt. +Elim (H5 ? H8); Intros; Exists x; Intros; Assert H11 := (H9 ? H10); Unfold R_dist in H11; Unfold Rminus in H11; Rewrite Ropp_O in H11; Rewrite Rplus_Or in H11. +Apply Rle_lt_trans with (Rabsolu ``C/(sum_f_R0 An n)``). +Apply Rle_Rabsolu. +Unfold Rdiv; Rewrite Rabsolu_mult; Apply Rlt_monotony_contra with ``/(Rabsolu C)``. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt. +Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Replace ``/(Rabsolu C)*(eps*/2)`` with ``eps/(2*(Rabsolu C))``. +Unfold Rdiv; Rewrite Rinv_Rmult. +Ring. +DiscrR. +Apply Rabsolu_no_R0. +Apply Rabsolu_no_R0. +Elim H7; Clear H7; Intros N2 H7; Pose N := (max N1 N2); Exists (S N); Intros; Unfold R_dist; Replace (Rminus (Rdiv (sum_f_R0 [k:nat]``(An k)*(Bn k)`` n) (sum_f_R0 An n)) l) with (Rdiv (sum_f_R0 [k:nat]``(An k)*((Bn k)-l)`` n) (sum_f_R0 An n)). +Assert H9 : (lt N1 n). +Apply lt_le_trans with (S N). +Apply le_lt_n_Sm; Unfold N; Apply le_max_l. +Rewrite (tech2 [k:nat]``(An k)*((Bn k)-l)`` ? ? H9); Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Apply Rle_lt_trans with (Rplus (Rabsolu (Rdiv (sum_f_R0 [k:nat]``(An k)*((Bn k)-l)`` N1) (sum_f_R0 An n))) (Rabsolu (Rdiv (sum_f_R0 [i:nat]``(An (plus (S N1) i))*((Bn (plus (S N1) i))-l)`` (minus n (S N1))) (sum_f_R0 An n)))). +Apply Rabsolu_triang. +Rewrite (double_var eps); Apply Rplus_lt. +Unfold Rdiv; Rewrite Rabsolu_mult; Fold C; Rewrite Rabsolu_right. +Apply (H7 n); Apply le_trans with (S N). +Apply le_trans with N; [Unfold N; Apply le_max_r | Apply le_n_Sn]. +Apply Rle_sym1; Left; Apply Rlt_Rinv. + +Unfold R_dist in H; Unfold Rdiv; Rewrite Rabsolu_mult; Rewrite (Rabsolu_right ``/(sum_f_R0 An n)``). +Apply Rle_lt_trans with (Rmult (sum_f_R0 [i:nat](Rabsolu ``(An (plus (S N1) i))*((Bn (plus (S N1) i))-l)``) (minus n (S N1))) ``/(sum_f_R0 An n)``). +Do 2 Rewrite <- (Rmult_sym ``/(sum_f_R0 An n)``); Apply Rle_monotony. +Left; Apply Rlt_Rinv. +Apply (sum_Rabsolu [i:nat]``(An (plus (S N1) i))*((Bn (plus (S N1) i))-l)`` (minus n (S N1))). +Apply Rle_lt_trans with (Rmult (sum_f_R0 [i:nat]``(An (plus (S N1) i))*eps/2`` (minus n (S N1))) ``/(sum_f_R0 An n)``). +Do 2 Rewrite <- (Rmult_sym ``/(sum_f_R0 An n)``); Apply Rle_monotony. +Left; Apply Rlt_Rinv. +Apply sum_Rle; Intros; Rewrite Rabsolu_mult; Pattern 2 (An (plus (S N1) n0)); Rewrite <- (Rabsolu_right (An (plus (S N1) n0))). +Apply Rle_monotony. +Apply Rabsolu_pos. +Left; Apply H; Unfold ge; Apply le_trans with (S N1); [Apply le_n_Sn | Apply le_plus_l]. +Apply Rle_sym1; Left. +Rewrite <- (scal_sum [i:nat](An (plus (S N1) i)) (minus n (S N1)) ``eps/2``); Unfold Rdiv; Repeat Rewrite Rmult_assoc; Apply Rlt_monotony. +Pattern 2 ``/2``; Rewrite <- Rmult_1r; Apply Rlt_monotony. +Apply Rlt_Rinv; Sup. +Rewrite Rmult_sym; Apply Rlt_monotony_contra with (sum_f_R0 An n). +Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Rewrite Rmult_1r; Rewrite (tech2 An N1 n). +Rewrite Rplus_sym; Pattern 1 (sum_f_R0 [i:nat](An (plus (S N1) i)) (minus n (S N1))); Rewrite <- Rplus_Or; Apply Rlt_compatibility. +Apply Rle_sym1; Left; Apply Rlt_Rinv. +Replace (sum_f_R0 [k:nat]``(An k)*((Bn k)-l)`` n) with (Rplus (sum_f_R0 [k:nat]``(An k)*(Bn k)`` n) (sum_f_R0 [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 : (An:nat->R;l:R) (Un_cv An l) -> (Un_cv [n:nat]``(sum_f_R0 An (pred n))/(INR n)`` l). +Proof with Trivial. +Intros Bn l H; Pose An := [_:nat]R1. +Assert H0 : (n:nat) ``0<(An n)``. +Intro; Unfold An; Apply Rlt_R0_R1. +Assert H1 : (n:nat)``0<(sum_f_R0 An n)``. +Intro; Apply tech1. +Assert H2 : (cv_infty [n:nat](sum_f_R0 An n)). +Unfold cv_infty; Intro; Case (total_order_Rle M R0); Intro. +Exists O; Intros; Apply Rle_lt_trans with R0. +Assert H2 : ``0<M``. +Auto with real. +Clear n; Pose m := (up M); Elim (archimed M); Intros; Assert H5 : `0<=m`. +Apply le_IZR; Unfold m; Simpl; Left; Apply Rlt_trans with M. +Elim (IZN ? H5); Intros; Exists x; Intros; Unfold An; Rewrite sum_cte; Rewrite Rmult_1l; Apply Rlt_trans with (IZR (up M)). +Apply Rle_lt_trans with (INR x). +Rewrite INR_IZR_INZ; Fold m; Rewrite <- H6; Right. +Apply lt_INR; Apply le_lt_n_Sm. +Assert H3 := (Cesaro ? ? ? H H0 H2). +Unfold Un_cv; Unfold Un_cv in H3; Intros; Elim (H3 ? H4); Intros; Exists (S x); Intros; Unfold R_dist; Unfold R_dist in H5; Apply Rle_lt_trans with (Rabsolu (Rminus (Rdiv (sum_f_R0 [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 (Rminus (Rdiv (sum_f_R0 [k:nat]``(An k)*(Bn k)`` (pred n)) (sum_f_R0 An (pred n))) l). +Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-l``); Apply Rplus_plus_r. +Unfold An; Replace (sum_f_R0 [k:nat]``1*(Bn k)`` (pred n)) with (sum_f_R0 Bn (pred n)). +Rewrite sum_cte; Rewrite Rmult_1l; Replace (S (pred n)) with n. +Apply S_pred with O; Apply lt_le_trans with (S x). +Apply lt_O_Sn. +Apply sum_eq; Intros; Ring. +Apply H5; Unfold ge; Apply le_S_n; Replace (S (pred n)) with n. +Apply S_pred with O; Apply lt_le_trans with (S x). +Apply lt_O_Sn. +Qed.
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