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Diffstat (limited to 'theories/Reals/AltSeries.v')
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diff --git a/theories/Reals/AltSeries.v b/theories/Reals/AltSeries.v new file mode 100644 index 00000000..166a8a46 --- /dev/null +++ b/theories/Reals/AltSeries.v @@ -0,0 +1,448 @@ +(************************************************************************) +(* 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: AltSeries.v,v 1.12.2.1 2004/07/16 19:31:10 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import Rseries. +Require Import SeqProp. +Require Import PartSum. +Require Import Max. +Open Local Scope R_scope. + +(**********) +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))). +intros; unfold Un_growing in |- *; 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 in |- *; rewrite <- Rplus_0_r. +apply Rplus_le_compat_l. +unfold tg_alt in |- *; 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)). +intros; unfold Un_decreasing in |- *; 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 in |- *; rewrite <- Rplus_0_r. +apply Rplus_le_compat_l. +unfold tg_alt in |- *; 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. +intros; induction N as [| N HrecN]. +simpl in |- *; unfold tg_alt in |- *; simpl in |- *; 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 in |- *; + rewrite <- Rplus_0_r. +rewrite Rplus_assoc; apply Rplus_le_compat_l. +unfold tg_alt in |- *; 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. +apply INR_eq; rewrite mult_INR; repeat rewrite S_INR; rewrite mult_INR; + repeat rewrite S_INR; ring. +apply HrecN. +apply INR_eq; repeat rewrite S_INR; do 2 rewrite mult_INR; + repeat rewrite S_INR; 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. +intros; induction N as [| N HrecN]. +simpl in |- *; unfold tg_alt in |- *; simpl in |- *; 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 in |- *; + rewrite <- Rplus_0_r. +apply Rplus_le_compat_l. +unfold tg_alt in |- *; simpl in |- *. +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))). +intros; unfold has_ub in |- *; unfold bound in |- *. +exists (Un 0%nat). +unfold is_upper_bound in |- *; intros; elim H1; intros. +rewrite H2; rewrite decomp_sum. +replace (tg_alt Un 0) with (Un 0%nat). +pattern (Un 0%nat) at 2 in |- *; rewrite <- Rplus_0_r. +apply Rplus_le_compat_l. +apply CV_ALT_step3; assumption. +unfold tg_alt in |- *; simpl in |- *; ring. +apply lt_O_Sn. +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 -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l). +intros. +assert (H2 := CV_ALT_step0 _ H). +assert (H3 := CV_ALT_step4 _ H H0). +assert (X := growing_cv _ H2 H3). +elim X; intros. +apply existT with x. +unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1; + unfold R_dist in H1; unfold Un_cv in p; unfold R_dist in p. +intros; cut (0 < eps / 2); + [ intro + | unfold Rdiv in |- *; 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. +cut (N1 <= P)%nat. +intro; 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 (double_var eps); apply Rplus_lt_compat. +rewrite H12; apply H7; assumption. +rewrite Rabs_Ropp; unfold tg_alt in |- *; 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 in |- *; apply le_trans with n. +apply le_trans with N; [ unfold N in |- *; apply le_max_r | assumption ]. +apply le_n_Sn. +rewrite tech5; ring. +rewrite H12; apply Rlt_trans with (eps / 2). +apply H7; assumption. +unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym; + [ rewrite Rmult_1_r | discrR ]. +rewrite double. +pattern eps at 1 in |- *; rewrite <- (Rplus_0_r eps); apply Rplus_lt_compat_l; + assumption. +elim H10; intro; apply le_double. +rewrite <- H11; apply le_trans with N. +unfold N in |- *; apply le_trans with (S (2 * N1)); + [ apply le_n_Sn | apply le_max_l ]. +assumption. +apply lt_n_Sm_le. +rewrite <- H11. +apply lt_le_trans with N. +unfold N in |- *; apply lt_le_trans with (S (2 * N1)). +apply lt_n_Sn. +apply le_max_l. +assumption. +Qed. + +(************************************************) +(* Convergence of alternated series *) +(* *) +(* Applications: PI, cos, sin *) +(************************************************) +Theorem alternated_series : + forall Un:nat -> R, + Un_decreasing Un -> + Un_cv Un 0 -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l). +intros; apply CV_ALT. +assumption. +unfold positivity_seq in |- *; 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). +intros. +cut (Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N)) l). +cut (Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))) l). +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. +unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1; + unfold R_dist in H1; intros. +elim (H1 eps H2); intros. +exists x; intros. +apply H3. +unfold ge in |- *; apply le_trans with (2 * n)%nat. +apply le_trans with n. +assumption. +assert (H5 := mult_O_le n 2). +elim H5; intro. +cut (0%nat <> 2%nat); + [ intro; elim H7; symmetry in |- *; assumption | discriminate ]. +assumption. +apply le_n_Sn. +unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1; + unfold R_dist in H1; intros. +elim (H1 eps H2); intros. +exists x; intros. +apply H3. +unfold ge in |- *; apply le_trans with n. +assumption. +assert (H5 := mult_O_le n 2). +elim H5; intro. +cut (0%nat <> 2%nat); + [ intro; elim H7; symmetry in |- *; assumption | discriminate ]. +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. +intro; unfold PI_tg in |- *; left; apply Rinv_0_lt_compat; apply lt_INR_0; + replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ]. +Qed. + +Lemma PI_tg_decreasing : Un_decreasing PI_tg. +unfold PI_tg, Un_decreasing in |- *; 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 lt_O_Sn | ring ]. +rewrite <- Rinv_r_sym. +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 lt_O_Sn | ring ]. +rewrite (Rmult_comm (INR (2 * S n + 1))); rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +do 2 rewrite Rmult_1_r; apply le_INR. +replace (2 * S n + 1)%nat with (S (S (2 * n + 1))). +apply le_trans with (S (2 * n + 1)); apply le_n_Sn. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite plus_INR; + do 2 rewrite mult_INR; repeat rewrite S_INR; 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. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +cut (0 < 2 * eps); + [ intro | apply Rmult_lt_0_compat; [ prove_sup0 | assumption ] ]. +assert (H1 := archimed (/ (2 * eps))). +cut (0 <= up (/ (2 * eps)))%Z. +intro; assert (H3 := IZN (up (/ (2 * eps))) H2). +elim H3; intros N H4. +cut (0 < N)%nat. +intro; exists N; intros. +cut (0 < n)%nat. +intro; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + rewrite Rabs_right. +unfold PI_tg in |- *; apply Rlt_trans with (/ INR (2 * n)). +apply Rmult_lt_reg_l with (INR (2 * n)). +apply lt_INR_0. +replace (2 * n)%nat with (n + n)%nat; [ idtac | ring ]. +apply lt_le_trans with n. +assumption. +apply le_plus_l. +rewrite <- Rinv_r_sym. +apply Rmult_lt_reg_l with (INR (2 * n + 1)). +apply lt_INR_0. +replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ]. +rewrite (Rmult_comm (INR (2 * n + 1))). +rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +do 2 rewrite Rmult_1_r; apply lt_INR. +replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_n_Sn | ring ]. +apply not_O_INR; replace (2 * n + 1)%nat with (S (2 * n)); + [ discriminate | ring ]. +replace n with (S (pred n)). +apply not_O_INR; discriminate. +symmetry in |- *; apply S_pred with 0%nat. +assumption. +apply Rle_lt_trans with (/ INR (2 * N)). +apply Rmult_le_reg_l with (INR (2 * N)). +rewrite mult_INR; apply Rmult_lt_0_compat; + [ simpl in |- *; prove_sup0 | apply lt_INR_0; assumption ]. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with (INR (2 * n)). +rewrite mult_INR; apply Rmult_lt_0_compat; + [ simpl in |- *; prove_sup0 | apply lt_INR_0; assumption ]. +rewrite (Rmult_comm (INR (2 * n))); rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +do 2 rewrite Rmult_1_r; apply le_INR. +apply (fun m n p:nat => mult_le_compat_l p n m); assumption. +replace n with (S (pred n)). +apply not_O_INR; discriminate. +symmetry in |- *; apply S_pred with 0%nat. +assumption. +replace N with (S (pred N)). +apply not_O_INR; discriminate. +symmetry in |- *; apply S_pred with 0%nat. +assumption. +rewrite mult_INR. +rewrite Rinv_mult_distr. +replace (INR 2) with 2; [ idtac | reflexivity ]. +apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; [ idtac | discrR ]. +rewrite Rmult_1_l; apply Rmult_lt_reg_l with (INR N). +apply lt_INR_0; assumption. +rewrite <- Rinv_r_sym. +apply Rmult_lt_reg_l with (/ (2 * eps)). +apply Rinv_0_lt_compat; assumption. +rewrite Rmult_1_r; + replace (/ (2 * eps) * (INR N * (2 * eps))) with + (INR N * (2 * eps * / (2 * eps))); [ idtac | ring ]. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; replace (INR N) with (IZR (Z_of_nat N)). +rewrite <- H4. +elim H1; intros; assumption. +symmetry in |- *; apply INR_IZR_INZ. +apply prod_neq_R0; + [ discrR | red in |- *; intro; rewrite H8 in H; elim (Rlt_irrefl _ H) ]. +apply not_O_INR. +red in |- *; intro; rewrite H8 in H5; elim (lt_irrefl _ H5). +replace (INR 2) with 2; [ discrR | reflexivity ]. +apply not_O_INR. +red in |- *; intro; rewrite H8 in H5; elim (lt_irrefl _ H5). +apply Rle_ge; apply PI_tg_pos. +apply lt_le_trans with N; assumption. +elim H1; intros H5 _. +assert (H6 := lt_eq_lt_dec 0 N). +elim H6; intro. +elim a; intro. +assumption. +rewrite <- b in H4. +rewrite H4 in H5. +simpl in H5. +cut (0 < / (2 * eps)); [ intro | apply Rinv_0_lt_compat; assumption ]. +elim (Rlt_irrefl _ (Rlt_trans _ _ _ H7 H5)). +elim (lt_n_O _ b). +apply le_IZR. +simpl in |- *. +left; apply Rlt_trans with (/ (2 * eps)). +apply Rinv_0_lt_compat; assumption. +elim H1; intros; assumption. +Qed. + +Lemma exist_PI : + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 (tg_alt PI_tg) N) l). +apply alternated_series. +apply PI_tg_decreasing. +apply PI_tg_cv. +Qed. + +(* Now, PI is defined *) +Definition PI : R := 4 * match exist_PI with + | existT a b => a + end. + +(* We can get an approximation of PI with the following inequality *) +Lemma PI_ineq : + forall N:nat, + sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= PI / 4 <= + sum_f_R0 (tg_alt PI_tg) (2 * N). +intro; apply alternated_series_ineq. +apply PI_tg_decreasing. +apply PI_tg_cv. +unfold PI in |- *; case exist_PI; intro. +replace (4 * x / 4) with x. +trivial. +unfold Rdiv in |- *; rewrite (Rmult_comm 4); rewrite Rmult_assoc; + rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r; reflexivity | discrR ]. +Qed. + +Lemma PI_RGT_0 : 0 < PI. +assert (H := 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 in |- *; unfold tg_alt in |- *; simpl in |- *; rewrite Rmult_1_l; + rewrite Rmult_1_r; apply Rplus_lt_reg_r 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 in |- * | ring ]. +simpl in |- *; apply Rinv_lt_contravar. +rewrite Rmult_1_l; replace (2 + 1) with 3; [ prove_sup0 | ring ]. +rewrite Rplus_comm; pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_lt_compat_l; prove_sup0. +assumption. +Qed.
\ No newline at end of file |