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Diffstat (limited to 'theories/Reals/Alembert.v')
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diff --git a/theories/Reals/Alembert.v b/theories/Reals/Alembert.v new file mode 100644 index 00000000..a691b189 --- /dev/null +++ b/theories/Reals/Alembert.v @@ -0,0 +1,726 @@ +(************************************************************************) +(* 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: Alembert.v,v 1.14.2.1 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. + +(***************************************************) +(* Various versions of the criterion of D'Alembert *) +(***************************************************) + +Lemma Alembert_C1 : + forall An:nat -> R, + (forall n:nat, 0 < An n) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l). +intros An H H0. +cut + (sigT (fun l:R => is_lub (EUn (fun N:nat => sum_f_R0 An N)) l) -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l)). +intro; apply X. +apply completeness. +unfold Un_cv in H0; unfold bound in |- *; cut (0 < / 2); + [ intro | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (H0 (/ 2) H1); intros. +exists (sum_f_R0 An x + 2 * An (S x)). +unfold is_upper_bound in |- *; intros; unfold EUn in H3; elim H3; intros. +rewrite H4; assert (H5 := lt_eq_lt_dec x1 x). +elim H5; intros. +elim a; intro. +replace (sum_f_R0 An x) with + (sum_f_R0 An x1 + sum_f_R0 (fun i:nat => An (S x1 + i)%nat) (x - S x1)). +pattern (sum_f_R0 An x1) at 1 in |- *; rewrite <- Rplus_0_r; + rewrite Rplus_assoc; apply Rplus_le_compat_l. +left; apply Rplus_lt_0_compat. +apply tech1; intros; apply H. +apply Rmult_lt_0_compat; [ prove_sup0 | apply H ]. +symmetry in |- *; apply tech2; assumption. +rewrite b; pattern (sum_f_R0 An x) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l. +left; apply Rmult_lt_0_compat; [ prove_sup0 | apply H ]. +replace (sum_f_R0 An x1) with + (sum_f_R0 An x + sum_f_R0 (fun i:nat => An (S x + i)%nat) (x1 - S x)). +apply Rplus_le_compat_l. +cut + (sum_f_R0 (fun i:nat => An (S x + i)%nat) (x1 - S x) <= + An (S x) * sum_f_R0 (fun i:nat => (/ 2) ^ i) (x1 - S x)). +intro; + apply Rle_trans with + (An (S x) * sum_f_R0 (fun i:nat => (/ 2) ^ i) (x1 - S x)). +assumption. +rewrite <- (Rmult_comm (An (S x))); apply Rmult_le_compat_l. +left; apply H. +rewrite tech3. +replace (1 - / 2) with (/ 2). +unfold Rdiv in |- *; rewrite Rinv_involutive. +pattern 2 at 3 in |- *; rewrite <- Rmult_1_r; rewrite <- (Rmult_comm 2); + apply Rmult_le_compat_l. +left; prove_sup0. +left; apply Rplus_lt_reg_r with ((/ 2) ^ S (x1 - S x)). +replace ((/ 2) ^ S (x1 - S x) + (1 - (/ 2) ^ S (x1 - S x))) with 1; + [ idtac | ring ]. +rewrite <- (Rplus_comm 1); pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_lt_compat_l. +apply pow_lt; apply Rinv_0_lt_compat; prove_sup0. +discrR. +apply Rmult_eq_reg_l with 2. +rewrite Rmult_minus_distr_l; rewrite <- Rinv_r_sym. +ring. +discrR. +discrR. +pattern 1 at 3 in |- *; replace 1 with (/ 1); + [ apply tech7; discrR | apply Rinv_1 ]. +replace (An (S x)) with (An (S x + 0)%nat). +apply (tech6 (fun i:nat => An (S x + i)%nat) (/ 2)). +left; apply Rinv_0_lt_compat; prove_sup0. +intro; cut (forall n:nat, (n >= x)%nat -> An (S n) < / 2 * An n). +intro; replace (S x + S i)%nat with (S (S x + i)). +apply H6; unfold ge in |- *; apply tech8. +apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; do 2 rewrite S_INR; ring. +intros; unfold R_dist in H2; apply Rmult_lt_reg_l with (/ An n). +apply Rinv_0_lt_compat; apply H. +do 2 rewrite (Rmult_comm (/ An n)); rewrite Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; + replace (An (S n) * / An n) with (Rabs (Rabs (An (S n) / An n) - 0)). +apply H2; assumption. +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + rewrite Rabs_Rabsolu; rewrite Rabs_right. +unfold Rdiv in |- *; reflexivity. +left; unfold Rdiv in |- *; change (0 < An (S n) * / An n) in |- *; + apply Rmult_lt_0_compat; [ apply H | apply Rinv_0_lt_compat; apply H ]. +red in |- *; intro; assert (H8 := H n); rewrite H7 in H8; + elim (Rlt_irrefl _ H8). +replace (S x + 0)%nat with (S x); [ reflexivity | ring ]. +symmetry in |- *; apply tech2; assumption. +exists (sum_f_R0 An 0); unfold EUn in |- *; exists 0%nat; reflexivity. +intro; elim X; intros. +apply existT with x; apply tech10; + [ unfold Un_growing in |- *; intro; rewrite tech5; + pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l; left; apply H + | apply p ]. +Qed. + +Lemma Alembert_C2 : + forall An:nat -> R, + (forall n:nat, An n <> 0) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l). +intros. +set (Vn := fun i:nat => (2 * Rabs (An i) + An i) / 2). +set (Wn := fun i:nat => (2 * Rabs (An i) - An i) / 2). +cut (forall n:nat, 0 < Vn n). +intro; cut (forall n:nat, 0 < Wn n). +intro; cut (Un_cv (fun n:nat => Rabs (Vn (S n) / Vn n)) 0). +intro; cut (Un_cv (fun n:nat => Rabs (Wn (S n) / Wn n)) 0). +intro; assert (H5 := Alembert_C1 Vn H1 H3). +assert (H6 := Alembert_C1 Wn H2 H4). +elim H5; intros. +elim H6; intros. +apply existT with (x - x0); unfold Un_cv in |- *; unfold Un_cv in p; + unfold Un_cv in p0; intros; cut (0 < eps / 2). +intro; elim (p (eps / 2) H8); clear p; intros. +elim (p0 (eps / 2) H8); clear p0; intros. +set (N := max x1 x2). +exists N; intros; + replace (sum_f_R0 An n) with (sum_f_R0 Vn n - sum_f_R0 Wn n). +unfold R_dist in |- *; + replace (sum_f_R0 Vn n - sum_f_R0 Wn n - (x - x0)) with + (sum_f_R0 Vn n - x + - (sum_f_R0 Wn n - x0)); [ idtac | ring ]; + apply Rle_lt_trans with + (Rabs (sum_f_R0 Vn n - x) + Rabs (- (sum_f_R0 Wn n - x0))). +apply Rabs_triang. +rewrite Rabs_Ropp; apply Rlt_le_trans with (eps / 2 + eps / 2). +apply Rplus_lt_compat. +unfold R_dist in H9; apply H9; unfold ge in |- *; apply le_trans with N; + [ unfold N in |- *; apply le_max_l | assumption ]. +unfold R_dist in H10; apply H10; unfold ge in |- *; apply le_trans with N; + [ unfold N in |- *; apply le_max_r | assumption ]. +right; symmetry in |- *; apply double_var. +symmetry in |- *; apply tech11; intro; unfold Vn, Wn in |- *; + unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ 2)); + apply Rmult_eq_reg_l with 2. +rewrite Rmult_minus_distr_l; repeat rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +ring. +discrR. +discrR. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +cut (forall n:nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)). +intro; cut (forall n:nat, / Wn n <= 2 * / Rabs (An n)). +intro; cut (forall n:nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)). +intro; unfold Un_cv in |- *; intros; unfold Un_cv in H0; cut (0 < eps / 3). +intro; elim (H0 (eps / 3) H8); intros. +exists x; intros. +assert (H11 := H9 n H10). +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold R_dist in H11; + unfold Rminus in H11; rewrite Ropp_0 in H11; rewrite Rplus_0_r in H11; + rewrite Rabs_Rabsolu in H11; rewrite Rabs_right. +apply Rle_lt_trans with (3 * Rabs (An (S n) / An n)). +apply H6. +apply Rmult_lt_reg_l with (/ 3). +apply Rinv_0_lt_compat; prove_sup0. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]; + rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H11; + exact H11. +left; change (0 < Wn (S n) / Wn n) in |- *; unfold Rdiv in |- *; + apply Rmult_lt_0_compat. +apply H2. +apply Rinv_0_lt_compat; apply H2. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +intro; unfold Rdiv in |- *; rewrite Rabs_mult; rewrite <- Rmult_assoc; + replace 3 with (2 * (3 * / 2)); + [ idtac | rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR ]; + apply Rle_trans with (Wn (S n) * 2 * / Rabs (An n)). +rewrite Rmult_assoc; apply Rmult_le_compat_l. +left; apply H2. +apply H5. +rewrite Rabs_Rinv. +replace (Wn (S n) * 2 * / Rabs (An n)) with (2 * / Rabs (An n) * Wn (S n)); + [ idtac | ring ]; + replace (2 * (3 * / 2) * Rabs (An (S n)) * / Rabs (An n)) with + (2 * / Rabs (An n) * (3 * / 2 * Rabs (An (S n)))); + [ idtac | ring ]; apply Rmult_le_compat_l. +left; apply Rmult_lt_0_compat. +prove_sup0. +apply Rinv_0_lt_compat; apply Rabs_pos_lt; apply H. +elim (H4 (S n)); intros; assumption. +apply H. +intro; apply Rmult_le_reg_l with (Wn n). +apply H2. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with (Rabs (An n)). +apply Rabs_pos_lt; apply H. +rewrite Rmult_1_r; + replace (Rabs (An n) * (Wn n * (2 * / Rabs (An n)))) with + (2 * Wn n * (Rabs (An n) * / Rabs (An n))); [ idtac | ring ]; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; apply Rmult_le_reg_l with (/ 2). +apply Rinv_0_lt_compat; prove_sup0. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; elim (H4 n); intros; assumption. +discrR. +apply Rabs_no_R0; apply H. +red in |- *; intro; assert (H6 := H2 n); rewrite H5 in H6; + elim (Rlt_irrefl _ H6). +intro; split. +unfold Wn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; prove_sup0. +pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; rewrite double; + unfold Rminus in |- *; rewrite Rplus_assoc; apply Rplus_le_compat_l. +apply Rplus_le_reg_l with (An n). +rewrite Rplus_0_r; rewrite (Rplus_comm (An n)); rewrite Rplus_assoc; + rewrite Rplus_opp_l; rewrite Rplus_0_r; apply RRle_abs. +unfold Wn in |- *; unfold Rdiv in |- *; repeat rewrite <- (Rmult_comm (/ 2)); + repeat rewrite Rmult_assoc; apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; prove_sup0. +unfold Rminus in |- *; rewrite double; + replace (3 * Rabs (An n)) with (Rabs (An n) + Rabs (An n) + Rabs (An n)); + [ idtac | ring ]; repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l. +rewrite <- Rabs_Ropp; apply RRle_abs. +cut (forall n:nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)). +intro; cut (forall n:nat, / Vn n <= 2 * / Rabs (An n)). +intro; cut (forall n:nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)). +intro; unfold Un_cv in |- *; intros; unfold Un_cv in H1; cut (0 < eps / 3). +intro; elim (H0 (eps / 3) H7); intros. +exists x; intros. +assert (H10 := H8 n H9). +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold R_dist in H10; + unfold Rminus in H10; rewrite Ropp_0 in H10; rewrite Rplus_0_r in H10; + rewrite Rabs_Rabsolu in H10; rewrite Rabs_right. +apply Rle_lt_trans with (3 * Rabs (An (S n) / An n)). +apply H5. +apply Rmult_lt_reg_l with (/ 3). +apply Rinv_0_lt_compat; prove_sup0. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]; + rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H10; + exact H10. +left; change (0 < Vn (S n) / Vn n) in |- *; unfold Rdiv in |- *; + apply Rmult_lt_0_compat. +apply H1. +apply Rinv_0_lt_compat; apply H1. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +intro; unfold Rdiv in |- *; rewrite Rabs_mult; rewrite <- Rmult_assoc; + replace 3 with (2 * (3 * / 2)); + [ idtac | rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR ]; + apply Rle_trans with (Vn (S n) * 2 * / Rabs (An n)). +rewrite Rmult_assoc; apply Rmult_le_compat_l. +left; apply H1. +apply H4. +rewrite Rabs_Rinv. +replace (Vn (S n) * 2 * / Rabs (An n)) with (2 * / Rabs (An n) * Vn (S n)); + [ idtac | ring ]; + replace (2 * (3 * / 2) * Rabs (An (S n)) * / Rabs (An n)) with + (2 * / Rabs (An n) * (3 * / 2 * Rabs (An (S n)))); + [ idtac | ring ]; apply Rmult_le_compat_l. +left; apply Rmult_lt_0_compat. +prove_sup0. +apply Rinv_0_lt_compat; apply Rabs_pos_lt; apply H. +elim (H3 (S n)); intros; assumption. +apply H. +intro; apply Rmult_le_reg_l with (Vn n). +apply H1. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with (Rabs (An n)). +apply Rabs_pos_lt; apply H. +rewrite Rmult_1_r; + replace (Rabs (An n) * (Vn n * (2 * / Rabs (An n)))) with + (2 * Vn n * (Rabs (An n) * / Rabs (An n))); [ idtac | ring ]; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; apply Rmult_le_reg_l with (/ 2). +apply Rinv_0_lt_compat; prove_sup0. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; elim (H3 n); intros; assumption. +discrR. +apply Rabs_no_R0; apply H. +red in |- *; intro; assert (H5 := H1 n); rewrite H4 in H5; + elim (Rlt_irrefl _ H5). +intro; split. +unfold Vn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; prove_sup0. +pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; rewrite double; + rewrite Rplus_assoc; apply Rplus_le_compat_l. +apply Rplus_le_reg_l with (- An n); rewrite Rplus_0_r; + rewrite <- (Rplus_comm (An n)); rewrite <- Rplus_assoc; + rewrite Rplus_opp_l; rewrite Rplus_0_l; rewrite <- Rabs_Ropp; + apply RRle_abs. +unfold Vn in |- *; unfold Rdiv in |- *; repeat rewrite <- (Rmult_comm (/ 2)); + repeat rewrite Rmult_assoc; apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; prove_sup0. +unfold Rminus in |- *; rewrite double; + replace (3 * Rabs (An n)) with (Rabs (An n) + Rabs (An n) + Rabs (An n)); + [ idtac | ring ]; repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l; + apply RRle_abs. +intro; unfold Wn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_0_r (/ 2)); + rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l. +apply Rinv_0_lt_compat; prove_sup0. +apply Rplus_lt_reg_r with (An n); rewrite Rplus_0_r; unfold Rminus in |- *; + rewrite (Rplus_comm (An n)); rewrite Rplus_assoc; + rewrite Rplus_opp_l; rewrite Rplus_0_r; + apply Rle_lt_trans with (Rabs (An n)). +apply RRle_abs. +rewrite double; pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_lt_compat_l; apply Rabs_pos_lt; apply H. +intro; unfold Vn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_0_r (/ 2)); + rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l. +apply Rinv_0_lt_compat; prove_sup0. +apply Rplus_lt_reg_r with (- An n); rewrite Rplus_0_r; unfold Rminus in |- *; + rewrite (Rplus_comm (- An n)); rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; + apply Rle_lt_trans with (Rabs (An n)). +rewrite <- Rabs_Ropp; apply RRle_abs. +rewrite double; pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_lt_compat_l; apply Rabs_pos_lt; apply H. +Qed. + +Lemma AlembertC3_step1 : + forall (An:nat -> R) (x:R), + x <> 0 -> + (forall n:nat, An n <> 0) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 -> + sigT (fun l:R => Pser An x l). +intros; set (Bn := fun i:nat => An i * x ^ i). +cut (forall n:nat, Bn n <> 0). +intro; cut (Un_cv (fun n:nat => Rabs (Bn (S n) / Bn n)) 0). +intro; assert (H4 := Alembert_C2 Bn H2 H3). +elim H4; intros. +apply existT with x0; unfold Bn in p; apply tech12; assumption. +unfold Un_cv in |- *; intros; unfold Un_cv in H1; cut (0 < eps / Rabs x). +intro; elim (H1 (eps / Rabs x) H4); intros. +exists x0; intros; unfold R_dist in |- *; unfold Rminus in |- *; + rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; + unfold Bn in |- *; + replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x). +rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs x). +apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. +rewrite <- (Rmult_comm (Rabs x)); rewrite <- Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H5; + replace (Rabs (An (S n) / An n)) with (R_dist (Rabs (An (S n) * / An n)) 0). +apply H5; assumption. +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold Rdiv in |- *; + reflexivity. +apply Rabs_no_R0; assumption. +replace (S n) with (n + 1)%nat; [ idtac | ring ]; rewrite pow_add; + unfold Rdiv in |- *; rewrite Rinv_mult_distr. +replace (An (n + 1)%nat * (x ^ n * x ^ 1) * (/ An n * / x ^ n)) with + (An (n + 1)%nat * x ^ 1 * / An n * (x ^ n * / x ^ n)); + [ idtac | ring ]; rewrite <- Rinv_r_sym. +simpl in |- *; ring. +apply pow_nonzero; assumption. +apply H0. +apply pow_nonzero; assumption. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ]. +intro; unfold Bn in |- *; apply prod_neq_R0; + [ apply H0 | apply pow_nonzero; assumption ]. +Qed. + +Lemma AlembertC3_step2 : + forall (An:nat -> R) (x:R), x = 0 -> sigT (fun l:R => Pser An x l). +intros; apply existT with (An 0%nat). +unfold Pser in |- *; unfold infinit_sum in |- *; intros; exists 0%nat; intros; + replace (sum_f_R0 (fun n0:nat => An n0 * x ^ n0) n) with (An 0%nat). +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; assumption. +induction n as [| n Hrecn]. +simpl in |- *; ring. +rewrite tech5; rewrite Hrecn; + [ rewrite H; simpl in |- *; ring | unfold ge in |- *; apply le_O_n ]. +Qed. + +(* An useful criterion of convergence for power series *) +Theorem Alembert_C3 : + forall (An:nat -> R) (x:R), + (forall n:nat, An n <> 0) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 -> + sigT (fun l:R => Pser An x l). +intros; case (total_order_T x 0); intro. +elim s; intro. +cut (x <> 0). +intro; apply AlembertC3_step1; assumption. +red in |- *; intro; rewrite H1 in a; elim (Rlt_irrefl _ a). +apply AlembertC3_step2; assumption. +cut (x <> 0). +intro; apply AlembertC3_step1; assumption. +red in |- *; intro; rewrite H1 in r; elim (Rlt_irrefl _ r). +Qed. + +Lemma Alembert_C4 : + forall (An:nat -> R) (k:R), + 0 <= k < 1 -> + (forall n:nat, 0 < An n) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) k -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l). +intros An k Hyp H H0. +cut + (sigT (fun l:R => is_lub (EUn (fun N:nat => sum_f_R0 An N)) l) -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l)). +intro; apply X. +apply completeness. +assert (H1 := tech13 _ _ Hyp H0). +elim H1; intros. +elim H2; intros. +elim H4; intros. +unfold bound in |- *; exists (sum_f_R0 An x0 + / (1 - x) * An (S x0)). +unfold is_upper_bound in |- *; intros; unfold EUn in H6. +elim H6; intros. +rewrite H7. +assert (H8 := lt_eq_lt_dec x2 x0). +elim H8; intros. +elim a; intro. +replace (sum_f_R0 An x0) with + (sum_f_R0 An x2 + sum_f_R0 (fun i:nat => An (S x2 + i)%nat) (x0 - S x2)). +pattern (sum_f_R0 An x2) at 1 in |- *; rewrite <- Rplus_0_r. +rewrite Rplus_assoc; apply Rplus_le_compat_l. +left; apply Rplus_lt_0_compat. +apply tech1. +intros; apply H. +apply Rmult_lt_0_compat. +apply Rinv_0_lt_compat; apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; + replace (x + (1 - x)) with 1; [ elim H3; intros; assumption | ring ]. +apply H. +symmetry in |- *; apply tech2; assumption. +rewrite b; pattern (sum_f_R0 An x0) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l. +left; apply Rmult_lt_0_compat. +apply Rinv_0_lt_compat; apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; + replace (x + (1 - x)) with 1; [ elim H3; intros; assumption | ring ]. +apply H. +replace (sum_f_R0 An x2) with + (sum_f_R0 An x0 + sum_f_R0 (fun i:nat => An (S x0 + i)%nat) (x2 - S x0)). +apply Rplus_le_compat_l. +cut + (sum_f_R0 (fun i:nat => An (S x0 + i)%nat) (x2 - S x0) <= + An (S x0) * sum_f_R0 (fun i:nat => x ^ i) (x2 - S x0)). +intro; + apply Rle_trans with (An (S x0) * sum_f_R0 (fun i:nat => x ^ i) (x2 - S x0)). +assumption. +rewrite <- (Rmult_comm (An (S x0))); apply Rmult_le_compat_l. +left; apply H. +rewrite tech3. +unfold Rdiv in |- *; apply Rmult_le_reg_l with (1 - x). +apply Rplus_lt_reg_r with x; rewrite Rplus_0_r. +replace (x + (1 - x)) with 1; [ elim H3; intros; assumption | ring ]. +do 2 rewrite (Rmult_comm (1 - x)). +rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; apply Rplus_le_reg_l with (x ^ S (x2 - S x0)). +replace (x ^ S (x2 - S x0) + (1 - x ^ S (x2 - S x0))) with 1; + [ idtac | ring ]. +rewrite <- (Rplus_comm 1); pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l. +left; apply pow_lt. +apply Rle_lt_trans with k. +elim Hyp; intros; assumption. +elim H3; intros; assumption. +apply Rminus_eq_contra. +red in |- *; intro. +elim H3; intros. +rewrite H10 in H12; elim (Rlt_irrefl _ H12). +red in |- *; intro. +elim H3; intros. +rewrite H10 in H12; elim (Rlt_irrefl _ H12). +replace (An (S x0)) with (An (S x0 + 0)%nat). +apply (tech6 (fun i:nat => An (S x0 + i)%nat) x). +left; apply Rle_lt_trans with k. +elim Hyp; intros; assumption. +elim H3; intros; assumption. +intro. +cut (forall n:nat, (n >= x0)%nat -> An (S n) < x * An n). +intro. +replace (S x0 + S i)%nat with (S (S x0 + i)). +apply H9. +unfold ge in |- *. +apply tech8. + apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; do 2 rewrite S_INR; + ring. +intros. +apply Rmult_lt_reg_l with (/ An n). +apply Rinv_0_lt_compat; apply H. +do 2 rewrite (Rmult_comm (/ An n)). +rewrite Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r. +replace (An (S n) * / An n) with (Rabs (An (S n) / An n)). +apply H5; assumption. +rewrite Rabs_right. +unfold Rdiv in |- *; reflexivity. +left; unfold Rdiv in |- *; change (0 < An (S n) * / An n) in |- *; + apply Rmult_lt_0_compat. +apply H. +apply Rinv_0_lt_compat; apply H. +red in |- *; intro. +assert (H11 := H n). +rewrite H10 in H11; elim (Rlt_irrefl _ H11). +replace (S x0 + 0)%nat with (S x0); [ reflexivity | ring ]. +symmetry in |- *; apply tech2; assumption. +exists (sum_f_R0 An 0); unfold EUn in |- *; exists 0%nat; reflexivity. +intro; elim X; intros. +apply existT with x; apply tech10; + [ unfold Un_growing in |- *; intro; rewrite tech5; + pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l; left; apply H + | apply p ]. +Qed. + +Lemma Alembert_C5 : + forall (An:nat -> R) (k:R), + 0 <= k < 1 -> + (forall n:nat, An n <> 0) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) k -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l). +intros. +cut + (sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l) -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l)). +intro Hyp0; apply Hyp0. +apply cv_cauchy_2. +apply cauchy_abs. +apply cv_cauchy_1. +cut + (sigT + (fun l:R => Un_cv (fun N:nat => sum_f_R0 (fun i:nat => Rabs (An i)) N) l) -> + sigT + (fun l:R => Un_cv (fun N:nat => sum_f_R0 (fun i:nat => Rabs (An i)) N) l)). +intro Hyp; apply Hyp. +apply (Alembert_C4 (fun i:nat => Rabs (An i)) k). +assumption. +intro; apply Rabs_pos_lt; apply H0. +unfold Un_cv in |- *. +unfold Un_cv in H1. +unfold Rdiv in |- *. +intros. +elim (H1 eps H2); intros. +exists x; intros. +rewrite <- Rabs_Rinv. +rewrite <- Rabs_mult. +rewrite Rabs_Rabsolu. +unfold Rdiv in H3; apply H3; assumption. +apply H0. +intro. +elim X; intros. +apply existT with x. +assumption. +intro. +elim X; intros. +apply existT with x. +assumption. +Qed. + +(* Convergence of power series in D(O,1/k) *) +(* k=0 is described in Alembert_C3 *) +Lemma Alembert_C6 : + forall (An:nat -> R) (x k:R), + 0 < k -> + (forall n:nat, An n <> 0) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) k -> + Rabs x < / k -> sigT (fun l:R => Pser An x l). +intros. +cut + (sigT + (fun l:R => Un_cv (fun N:nat => sum_f_R0 (fun i:nat => An i * x ^ i) N) l)). +intro. +elim X; intros. +apply existT with x0. +apply tech12; assumption. +case (total_order_T x 0); intro. +elim s; intro. +eapply Alembert_C5 with (k * Rabs x). +split. +unfold Rdiv in |- *; apply Rmult_le_pos. +left; assumption. +left; apply Rabs_pos_lt. +red in |- *; intro; rewrite H3 in a; elim (Rlt_irrefl _ a). +apply Rmult_lt_reg_l with (/ k). +apply Rinv_0_lt_compat; assumption. +rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +rewrite Rmult_1_r; assumption. +red in |- *; intro; rewrite H3 in H; elim (Rlt_irrefl _ H). +intro; apply prod_neq_R0. +apply H0. +apply pow_nonzero. +red in |- *; intro; rewrite H3 in a; elim (Rlt_irrefl _ a). +unfold Un_cv in |- *; unfold Un_cv in H1. +intros. +cut (0 < eps / Rabs x). +intro. +elim (H1 (eps / Rabs x) H4); intros. +exists x0. +intros. +replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x). +unfold R_dist in |- *. +rewrite Rabs_mult. +replace (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) with + (Rabs x * (Rabs (An (S n) / An n) - k)); [ idtac | ring ]. +rewrite Rabs_mult. +rewrite Rabs_Rabsolu. +apply Rmult_lt_reg_l with (/ Rabs x). +apply Rinv_0_lt_compat; apply Rabs_pos_lt. +red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). +rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +rewrite <- (Rmult_comm eps). +unfold R_dist in H5. +unfold Rdiv in |- *; unfold Rdiv in H5; apply H5; assumption. +apply Rabs_no_R0. +red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). +unfold Rdiv in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. +rewrite pow_add. +simpl in |- *. +rewrite Rmult_1_r. +rewrite Rinv_mult_distr. +replace (An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)) with + (An (n + 1)%nat * / An n * x * (x ^ n * / x ^ n)); + [ idtac | ring ]. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; reflexivity. +apply pow_nonzero. +red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). +apply H0. +apply pow_nonzero. +red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +assumption. +apply Rinv_0_lt_compat; apply Rabs_pos_lt. +red in |- *; intro H7; rewrite H7 in a; elim (Rlt_irrefl _ a). +apply existT with (An 0%nat). +unfold Un_cv in |- *. +intros. +exists 0%nat. +intros. +unfold R_dist in |- *. +replace (sum_f_R0 (fun i:nat => An i * x ^ i) n) with (An 0%nat). +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. +induction n as [| n Hrecn]. +simpl in |- *; ring. +rewrite tech5. +rewrite <- Hrecn. +rewrite b; simpl in |- *; ring. +unfold ge in |- *; apply le_O_n. +eapply Alembert_C5 with (k * Rabs x). +split. +unfold Rdiv in |- *; apply Rmult_le_pos. +left; assumption. +left; apply Rabs_pos_lt. +red in |- *; intro; rewrite H3 in r; elim (Rlt_irrefl _ r). +apply Rmult_lt_reg_l with (/ k). +apply Rinv_0_lt_compat; assumption. +rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +rewrite Rmult_1_r; assumption. +red in |- *; intro; rewrite H3 in H; elim (Rlt_irrefl _ H). +intro; apply prod_neq_R0. +apply H0. +apply pow_nonzero. +red in |- *; intro; rewrite H3 in r; elim (Rlt_irrefl _ r). +unfold Un_cv in |- *; unfold Un_cv in H1. +intros. +cut (0 < eps / Rabs x). +intro. +elim (H1 (eps / Rabs x) H4); intros. +exists x0. +intros. +replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x). +unfold R_dist in |- *. +rewrite Rabs_mult. +replace (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) with + (Rabs x * (Rabs (An (S n) / An n) - k)); [ idtac | ring ]. +rewrite Rabs_mult. +rewrite Rabs_Rabsolu. +apply Rmult_lt_reg_l with (/ Rabs x). +apply Rinv_0_lt_compat; apply Rabs_pos_lt. +red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). +rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +rewrite <- (Rmult_comm eps). +unfold R_dist in H5. +unfold Rdiv in |- *; unfold Rdiv in H5; apply H5; assumption. +apply Rabs_no_R0. +red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). +unfold Rdiv in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. +rewrite pow_add. +simpl in |- *. +rewrite Rmult_1_r. +rewrite Rinv_mult_distr. +replace (An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)) with + (An (n + 1)%nat * / An n * x * (x ^ n * / x ^ n)); + [ idtac | ring ]. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; reflexivity. +apply pow_nonzero. +red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). +apply H0. +apply pow_nonzero. +red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +assumption. +apply Rinv_0_lt_compat; apply Rabs_pos_lt. +red in |- *; intro H7; rewrite H7 in r; elim (Rlt_irrefl _ r). +Qed.
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