diff options
Diffstat (limited to 'theories/Reals/Cos_plus.v')
| -rw-r--r-- | theories/Reals/Cos_plus.v | 1061 |
1 files changed, 1061 insertions, 0 deletions
diff --git a/theories/Reals/Cos_plus.v b/theories/Reals/Cos_plus.v new file mode 100644 index 00000000..422eb4a4 --- /dev/null +++ b/theories/Reals/Cos_plus.v @@ -0,0 +1,1061 @@ +(************************************************************************) +(* 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: Cos_plus.v,v 1.11.2.1 2004/07/16 19:31:10 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo_def. +Require Import Cos_rel. +Require Import Max. Open Local Scope nat_scope. Open Local Scope R_scope. + +Definition Majxy (x y:R) (n:nat) : R := + Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n) / INR (fact n). + +Lemma Majxy_cv_R0 : forall x y:R, Un_cv (Majxy x y) 0. +intros. +set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))). +set (C0 := C ^ 4). +cut (0 < C). +intro. +cut (0 < C0). +intro. +assert (H1 := cv_speed_pow_fact C0). +unfold Un_cv in H1; unfold R_dist in H1. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +cut (0 < eps / C0); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; assumption ] ]. +elim (H1 (eps / C0) H3); intros N0 H4. +exists N0; intros. +replace (Majxy x y n) with (C0 ^ S n / INR (fact n)). +simpl in |- *. +apply Rmult_lt_reg_l with (Rabs (/ C0)). +apply Rabs_pos_lt. +apply Rinv_neq_0_compat. +red in |- *; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0). +rewrite <- Rabs_mult. +unfold Rminus in |- *; rewrite Rmult_plus_distr_l. +rewrite Ropp_0; rewrite Rmult_0_r. +unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +rewrite (Rabs_right (/ C0)). +rewrite <- (Rmult_comm eps). +replace (C0 ^ n * / INR (fact n) + 0) with (C0 ^ n * / INR (fact n) - 0); + [ idtac | ring ]. +unfold Rdiv in H4; apply H4; assumption. +apply Rle_ge; left; apply Rinv_0_lt_compat; assumption. +red in |- *; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0). +unfold Majxy in |- *. +unfold C0 in |- *. +rewrite pow_mult. +unfold C in |- *; reflexivity. +unfold C0 in |- *; apply pow_lt; assumption. +apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *. +apply RmaxLess1. +Qed. + +Lemma reste1_maj : + forall (x y:R) (N:nat), + (0 < N)%nat -> Rabs (Reste1 x y N) <= Majxy x y (pred N). +intros. +set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))). +unfold Reste1 in |- *. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + Rabs + (sum_f_R0 + (fun l:nat => + (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) * + x ^ (2 * S (l + k)) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) * + y ^ (2 * (N - l))) (pred (N - k)))) ( + pred N)). +apply + (Rsum_abs + (fun k:nat => + sum_f_R0 + (fun l:nat => + (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) * + x ^ (2 * S (l + k)) * ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) * + y ^ (2 * (N - l))) (pred (N - k))) (pred N)). +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + Rabs + ((-1) ^ S (l + k) / INR (fact (2 * S (l + k))) * + x ^ (2 * S (l + k)) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) * + y ^ (2 * (N - l)))) (pred (N - k))) ( + pred N)). +apply sum_Rle. +intros. +apply + (Rsum_abs + (fun l:nat => + (-1) ^ S (l + n) / INR (fact (2 * S (l + n))) * x ^ (2 * S (l + n)) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) * + y ^ (2 * (N - l))) (pred (N - n))). +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + / INR (fact (2 * S (l + k)) * fact (2 * (N - l))) * + C ^ (2 * S (N + k))) (pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +unfold Rdiv in |- *; repeat rewrite Rabs_mult. +do 2 rewrite pow_1_abs. +do 2 rewrite Rmult_1_l. +rewrite (Rabs_right (/ INR (fact (2 * S (n0 + n))))). +rewrite (Rabs_right (/ INR (fact (2 * (N - n0))))). +rewrite mult_INR. +rewrite Rinv_mult_distr. +repeat rewrite Rmult_assoc. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +rewrite <- Rmult_assoc. +rewrite <- (Rmult_comm (/ INR (fact (2 * (N - n0))))). +rewrite Rmult_assoc. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +do 2 rewrite <- RPow_abs. +apply Rle_trans with (Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0))). +apply Rmult_le_compat_l. +apply pow_le; apply Rabs_pos. +apply pow_incr. +split. +apply Rabs_pos. +unfold C in |- *. +apply Rle_trans with (Rmax (Rabs x) (Rabs y)); apply RmaxLess2. +apply Rle_trans with (C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0))). +do 2 rewrite <- (Rmult_comm (C ^ (2 * (N - n0)))). +apply Rmult_le_compat_l. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +apply pow_incr. +split. +apply Rabs_pos. +unfold C in |- *; apply Rle_trans with (Rmax (Rabs x) (Rabs y)). +apply RmaxLess1. +apply RmaxLess2. +right. +replace (2 * S (N + n))%nat with (2 * (N - n0) + 2 * S (n0 + n))%nat. +rewrite pow_add. +apply Rmult_comm. +apply INR_eq; rewrite plus_INR; do 3 rewrite mult_INR. +rewrite minus_INR. +repeat rewrite S_INR; do 2 rewrite plus_INR; ring. +apply le_trans with (pred (N - n)). +exact H1. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply Rle_ge; left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_ge; left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + / INR (fact (2 * S (l + k)) * fact (2 * (N - l))) * C ^ (4 * N)) + (pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat. +rewrite mult_INR; apply Rmult_lt_0_compat; apply INR_fact_lt_0. +apply Rle_pow. +unfold C in |- *; apply RmaxLess1. +replace (4 * N)%nat with (2 * (2 * N))%nat; [ idtac | ring ]. +apply (fun m n p:nat => mult_le_compat_l p n m). +replace (2 * N)%nat with (S (N + pred N)). +apply le_n_S. +apply plus_le_compat_l; assumption. +rewrite pred_of_minus. +apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR; + rewrite minus_INR. +repeat rewrite S_INR; ring. +apply lt_le_S; assumption. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => C ^ (4 * N) * Rsqr (/ INR (fact (S (N + k))))) + (pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +rewrite <- (Rmult_comm (C ^ (4 * N))). +apply Rmult_le_compat_l. +apply pow_le. +left; apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +replace (/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))) with + (Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) / INR (fact (2 * S (N + n)))). +apply Rle_trans with + (Binomial.C (2 * S (N + n)) (S (N + n)) / INR (fact (2 * S (N + n)))). +unfold Rdiv in |- *; + do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (N + n))))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply C_maj. +apply (fun m n p:nat => mult_le_compat_l p n m). +apply le_n_S. +apply plus_le_compat_r. +apply le_trans with (pred (N - n)). +assumption. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +right. +unfold Rdiv in |- *; rewrite Rmult_comm. +unfold Binomial.C in |- *. +unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +replace (2 * S (N + n) - S (N + n))%nat with (S (N + n)). +rewrite Rinv_mult_distr. +unfold Rsqr in |- *; reflexivity. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_eq; rewrite S_INR; rewrite minus_INR. +rewrite mult_INR; repeat rewrite S_INR; rewrite plus_INR; ring. +apply le_n_2n. +apply INR_fact_neq_0. +unfold Rdiv in |- *; rewrite Rmult_comm. +unfold Binomial.C in |- *. +unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +replace (2 * S (N + n) - 2 * S (n0 + n))%nat with (2 * (N - n0))%nat. +rewrite mult_INR. +reflexivity. +apply INR_eq; rewrite minus_INR. +do 3 rewrite mult_INR; repeat rewrite S_INR; do 2 rewrite plus_INR; + rewrite minus_INR. +ring. +apply le_trans with (pred (N - n)). +assumption. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply (fun m n p:nat => mult_le_compat_l p n m). +apply le_n_S. +apply plus_le_compat_r. +apply le_trans with (pred (N - n)). +assumption. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply INR_fact_neq_0. +apply Rle_trans with + (sum_f_R0 (fun k:nat => INR N / INR (fact (S N)) * C ^ (4 * N)) (pred N)). +apply sum_Rle; intros. +rewrite <- + (scal_sum (fun _:nat => C ^ (4 * N)) (pred (N - n)) + (Rsqr (/ INR (fact (S (N + n)))))). +rewrite sum_cte. +rewrite <- Rmult_assoc. +do 2 rewrite <- (Rmult_comm (C ^ (4 * N))). +rewrite Rmult_assoc. +apply Rmult_le_compat_l. +apply pow_le. +left; apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +apply Rle_trans with (Rsqr (/ INR (fact (S (N + n)))) * INR N). +apply Rmult_le_compat_l. +apply Rle_0_sqr. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_INR. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +rewrite Rmult_comm; unfold Rdiv in |- *; apply Rmult_le_compat_l. +apply pos_INR. +apply Rle_trans with (/ INR (fact (S (N + n)))). +pattern (/ INR (fact (S (N + n)))) at 2 in |- *; rewrite <- Rmult_1_r. +unfold Rsqr in |- *. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rmult_le_reg_l with (INR (fact (S (N + n)))). +apply INR_fact_lt_0. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r. +replace 1 with (INR 1). +apply le_INR. +apply lt_le_S. +apply INR_lt; apply INR_fact_lt_0. +reflexivity. +apply INR_fact_neq_0. +apply Rmult_le_reg_l with (INR (fact (S (N + n)))). +apply INR_fact_lt_0. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with (INR (fact (S N))). +apply INR_fact_lt_0. +rewrite Rmult_1_r. +rewrite (Rmult_comm (INR (fact (S N)))). +rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +apply le_INR. +apply fact_le. +apply le_n_S. +apply le_plus_l. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +rewrite sum_cte. +apply Rle_trans with (C ^ (4 * N) / INR (fact (pred N))). +rewrite <- (Rmult_comm (C ^ (4 * N))). +unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l. +apply pow_le. +left; apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +cut (S (pred N) = N). +intro; rewrite H0. +pattern N at 2 in |- *; rewrite <- H0. +do 2 rewrite fact_simpl. +rewrite H0. +repeat rewrite mult_INR. +repeat rewrite Rinv_mult_distr. +rewrite (Rmult_comm (/ INR (S N))). +repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_l. +pattern (/ INR (fact (pred N))) at 2 in |- *; rewrite <- Rmult_1_r. +rewrite Rmult_assoc. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rmult_le_reg_l with (INR (S N)). +apply lt_INR_0; apply lt_O_Sn. +rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; rewrite Rmult_1_l. +apply le_INR; apply le_n_Sn. +apply not_O_INR; discriminate. +apply not_O_INR. +red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H). +apply not_O_INR. +red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H). +apply INR_fact_neq_0. +apply not_O_INR; discriminate. +apply prod_neq_R0. +apply not_O_INR. +red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H). +apply INR_fact_neq_0. +symmetry in |- *; apply S_pred with 0%nat; assumption. +right. +unfold Majxy in |- *. +unfold C in |- *. +replace (S (pred N)) with N. +reflexivity. +apply S_pred with 0%nat; assumption. +Qed. + +Lemma reste2_maj : + forall (x y:R) (N:nat), (0 < N)%nat -> Rabs (Reste2 x y N) <= Majxy x y N. +intros. +set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))). +unfold Reste2 in |- *. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + Rabs + (sum_f_R0 + (fun l:nat => + (-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) * + x ^ (2 * S (l + k) + 1) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) * + y ^ (2 * (N - l) + 1)) (pred (N - k)))) ( + pred N)). +apply + (Rsum_abs + (fun k:nat => + sum_f_R0 + (fun l:nat => + (-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) * + x ^ (2 * S (l + k) + 1) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) * + y ^ (2 * (N - l) + 1)) (pred (N - k))) ( + pred N)). +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + Rabs + ((-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) * + x ^ (2 * S (l + k) + 1) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) * + y ^ (2 * (N - l) + 1))) (pred (N - k))) ( + pred N)). +apply sum_Rle. +intros. +apply + (Rsum_abs + (fun l:nat => + (-1) ^ S (l + n) / INR (fact (2 * S (l + n) + 1)) * + x ^ (2 * S (l + n) + 1) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) * + y ^ (2 * (N - l) + 1)) (pred (N - n))). +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + / INR (fact (2 * S (l + k) + 1) * fact (2 * (N - l) + 1)) * + C ^ (2 * S (S (N + k)))) (pred (N - k))) ( + pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +unfold Rdiv in |- *; repeat rewrite Rabs_mult. +do 2 rewrite pow_1_abs. +do 2 rewrite Rmult_1_l. +rewrite (Rabs_right (/ INR (fact (2 * S (n0 + n) + 1)))). +rewrite (Rabs_right (/ INR (fact (2 * (N - n0) + 1)))). +rewrite mult_INR. +rewrite Rinv_mult_distr. +repeat rewrite Rmult_assoc. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +rewrite <- Rmult_assoc. +rewrite <- (Rmult_comm (/ INR (fact (2 * (N - n0) + 1)))). +rewrite Rmult_assoc. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +do 2 rewrite <- RPow_abs. +apply Rle_trans with (Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1)). +apply Rmult_le_compat_l. +apply pow_le; apply Rabs_pos. +apply pow_incr. +split. +apply Rabs_pos. +unfold C in |- *. +apply Rle_trans with (Rmax (Rabs x) (Rabs y)); apply RmaxLess2. +apply Rle_trans with (C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1)). +do 2 rewrite <- (Rmult_comm (C ^ (2 * (N - n0) + 1))). +apply Rmult_le_compat_l. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +apply pow_incr. +split. +apply Rabs_pos. +unfold C in |- *; apply Rle_trans with (Rmax (Rabs x) (Rabs y)). +apply RmaxLess1. +apply RmaxLess2. +right. +replace (2 * S (S (N + n)))%nat with + (2 * (N - n0) + 1 + (2 * S (n0 + n) + 1))%nat. +repeat rewrite pow_add. +ring. +apply INR_eq; repeat rewrite plus_INR; do 3 rewrite mult_INR. +rewrite minus_INR. +repeat rewrite S_INR; do 2 rewrite plus_INR; ring. +apply le_trans with (pred (N - n)). +exact H1. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply Rle_ge; left; apply Rinv_0_lt_compat. +apply INR_fact_lt_0. +apply Rle_ge; left; apply Rinv_0_lt_compat. +apply INR_fact_lt_0. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + / INR (fact (2 * S (l + k) + 1) * fact (2 * (N - l) + 1)) * + C ^ (4 * S N)) (pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat. +rewrite mult_INR; apply Rmult_lt_0_compat; apply INR_fact_lt_0. +apply Rle_pow. +unfold C in |- *; apply RmaxLess1. +replace (4 * S N)%nat with (2 * (2 * S N))%nat; [ idtac | ring ]. +apply (fun m n p:nat => mult_le_compat_l p n m). +replace (2 * S N)%nat with (S (S (N + N))). +repeat apply le_n_S. +apply plus_le_compat_l. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply INR_eq; do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR. +repeat rewrite S_INR; ring. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => C ^ (4 * S N) * Rsqr (/ INR (fact (S (S (N + k)))))) + (pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +rewrite <- (Rmult_comm (C ^ (4 * S N))). +apply Rmult_le_compat_l. +apply pow_le. +left; apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +replace (/ INR (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))) with + (Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) / + INR (fact (2 * S (S (N + n))))). +apply Rle_trans with + (Binomial.C (2 * S (S (N + n))) (S (S (N + n))) / + INR (fact (2 * S (S (N + n))))). +unfold Rdiv in |- *; + do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (S (N + n)))))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply C_maj. +apply le_trans with (2 * S (S (n0 + n)))%nat. +replace (2 * S (S (n0 + n)))%nat with (S (2 * S (n0 + n) + 1)). +apply le_n_Sn. +apply INR_eq; rewrite S_INR; rewrite plus_INR; do 2 rewrite mult_INR; + repeat rewrite S_INR; rewrite plus_INR; ring. +apply (fun m n p:nat => mult_le_compat_l p n m). +repeat apply le_n_S. +apply plus_le_compat_r. +apply le_trans with (pred (N - n)). +assumption. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +right. +unfold Rdiv in |- *; rewrite Rmult_comm. +unfold Binomial.C in |- *. +unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +replace (2 * S (S (N + n)) - S (S (N + n)))%nat with (S (S (N + n))). +rewrite Rinv_mult_distr. +unfold Rsqr in |- *; reflexivity. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_eq; do 2 rewrite S_INR; rewrite minus_INR. +rewrite mult_INR; repeat rewrite S_INR; rewrite plus_INR; ring. +apply le_n_2n. +apply INR_fact_neq_0. +unfold Rdiv in |- *; rewrite Rmult_comm. +unfold Binomial.C in |- *. +unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +replace (2 * S (S (N + n)) - (2 * S (n0 + n) + 1))%nat with + (2 * (N - n0) + 1)%nat. +rewrite mult_INR. +reflexivity. +apply INR_eq; rewrite minus_INR. +do 2 rewrite plus_INR; do 3 rewrite mult_INR; repeat rewrite S_INR; + do 2 rewrite plus_INR; rewrite minus_INR. +ring. +apply le_trans with (pred (N - n)). +assumption. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_trans with (2 * S (S (n0 + n)))%nat. +replace (2 * S (S (n0 + n)))%nat with (S (2 * S (n0 + n) + 1)). +apply le_n_Sn. +apply INR_eq; rewrite S_INR; rewrite plus_INR; do 2 rewrite mult_INR; + repeat rewrite S_INR; rewrite plus_INR; ring. +apply (fun m n p:nat => mult_le_compat_l p n m). +repeat apply le_n_S. +apply plus_le_compat_r. +apply le_trans with (pred (N - n)). +assumption. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply INR_fact_neq_0. +apply Rle_trans with + (sum_f_R0 (fun k:nat => INR N / INR (fact (S (S N))) * C ^ (4 * S N)) + (pred N)). +apply sum_Rle; intros. +rewrite <- + (scal_sum (fun _:nat => C ^ (4 * S N)) (pred (N - n)) + (Rsqr (/ INR (fact (S (S (N + n))))))). +rewrite sum_cte. +rewrite <- Rmult_assoc. +do 2 rewrite <- (Rmult_comm (C ^ (4 * S N))). +rewrite Rmult_assoc. +apply Rmult_le_compat_l. +apply pow_le. +left; apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +apply Rle_trans with (Rsqr (/ INR (fact (S (S (N + n))))) * INR N). +apply Rmult_le_compat_l. +apply Rle_0_sqr. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_INR. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +rewrite Rmult_comm; unfold Rdiv in |- *; apply Rmult_le_compat_l. +apply pos_INR. +apply Rle_trans with (/ INR (fact (S (S (N + n))))). +pattern (/ INR (fact (S (S (N + n))))) at 2 in |- *; rewrite <- Rmult_1_r. +unfold Rsqr in |- *. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rmult_le_reg_l with (INR (fact (S (S (N + n))))). +apply INR_fact_lt_0. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r. +replace 1 with (INR 1). +apply le_INR. +apply lt_le_S. +apply INR_lt; apply INR_fact_lt_0. +reflexivity. +apply INR_fact_neq_0. +apply Rmult_le_reg_l with (INR (fact (S (S (N + n))))). +apply INR_fact_lt_0. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with (INR (fact (S (S N)))). +apply INR_fact_lt_0. +rewrite Rmult_1_r. +rewrite (Rmult_comm (INR (fact (S (S N))))). +rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +apply le_INR. +apply fact_le. +repeat apply le_n_S. +apply le_plus_l. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +rewrite sum_cte. +apply Rle_trans with (C ^ (4 * S N) / INR (fact N)). +rewrite <- (Rmult_comm (C ^ (4 * S N))). +unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l. +apply pow_le. +left; apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +cut (S (pred N) = N). +intro; rewrite H0. +do 2 rewrite fact_simpl. +repeat rewrite mult_INR. +repeat rewrite Rinv_mult_distr. +apply Rle_trans with + (INR (S (S N)) * (/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) * INR N). +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (INR N)). +rewrite (Rmult_comm (INR (S (S N)))). +apply Rmult_le_compat_l. +repeat apply Rmult_le_pos. +left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. +left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. +left; apply Rinv_0_lt_compat. +apply INR_fact_lt_0. +apply pos_INR. +apply le_INR. +apply le_trans with (S N); apply le_n_Sn. +repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_l. +apply Rle_trans with (/ INR (S N) * / INR (fact N) * INR (S N)). +repeat rewrite Rmult_assoc. +repeat apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply le_INR; apply le_n_Sn. +rewrite (Rmult_comm (/ INR (S N))). +rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; right; reflexivity. +apply not_O_INR; discriminate. +apply not_O_INR; discriminate. +apply not_O_INR; discriminate. +apply INR_fact_neq_0. +apply not_O_INR; discriminate. +apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ]. +symmetry in |- *; apply S_pred with 0%nat; assumption. +right. +unfold Majxy in |- *. +unfold C in |- *. +reflexivity. +Qed. + +Lemma reste1_cv_R0 : forall x y:R, Un_cv (Reste1 x y) 0. +intros. +assert (H := Majxy_cv_R0 x y). +unfold Un_cv in H; unfold R_dist in H. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H eps H0); intros N0 H1. +exists (S N0); intros. +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r. +apply Rle_lt_trans with (Rabs (Majxy x y (pred n))). +rewrite (Rabs_right (Majxy x y (pred n))). +apply reste1_maj. +apply lt_le_trans with (S N0). +apply lt_O_Sn. +assumption. +apply Rle_ge. +unfold Majxy in |- *. +unfold Rdiv in |- *; apply Rmult_le_pos. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +apply RmaxLess1. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +replace (Majxy x y (pred n)) with (Majxy x y (pred n) - 0); [ idtac | ring ]. +apply H1. +unfold ge in |- *; apply le_S_n. +replace (S (pred n)) with n. +assumption. +apply S_pred with 0%nat. +apply lt_le_trans with (S N0); [ apply lt_O_Sn | assumption ]. +Qed. + +Lemma reste2_cv_R0 : forall x y:R, Un_cv (Reste2 x y) 0. +intros. +assert (H := Majxy_cv_R0 x y). +unfold Un_cv in H; unfold R_dist in H. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H eps H0); intros N0 H1. +exists (S N0); intros. +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r. +apply Rle_lt_trans with (Rabs (Majxy x y n)). +rewrite (Rabs_right (Majxy x y n)). +apply reste2_maj. +apply lt_le_trans with (S N0). +apply lt_O_Sn. +assumption. +apply Rle_ge. +unfold Majxy in |- *. +unfold Rdiv in |- *; apply Rmult_le_pos. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +apply RmaxLess1. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +replace (Majxy x y n) with (Majxy x y n - 0); [ idtac | ring ]. +apply H1. +unfold ge in |- *; apply le_trans with (S N0). +apply le_n_Sn. +exact H2. +Qed. + +Lemma reste_cv_R0 : forall x y:R, Un_cv (Reste x y) 0. +intros. +unfold Reste in |- *. +set (An := fun n:nat => Reste2 x y n). +set (Bn := fun n:nat => Reste1 x y (S n)). +cut + (Un_cv (fun n:nat => An n - Bn n) (0 - 0) -> + Un_cv (fun N:nat => Reste2 x y N - Reste1 x y (S N)) 0). +intro. +apply H. +apply CV_minus. +unfold An in |- *. +replace (fun n:nat => Reste2 x y n) with (Reste2 x y). +apply reste2_cv_R0. +reflexivity. +unfold Bn in |- *. +assert (H0 := reste1_cv_R0 x y). +unfold Un_cv in H0; unfold R_dist in H0. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H0 eps H1); intros N0 H2. +exists N0; intros. +apply H2. +unfold ge in |- *; apply le_trans with (S N0). +apply le_n_Sn. +apply le_n_S; assumption. +unfold An, Bn in |- *. +intro. +replace 0 with (0 - 0); [ idtac | ring ]. +exact H. +Qed. + +Theorem cos_plus : forall x y:R, cos (x + y) = cos x * cos y - sin x * sin y. +intros. +cut (Un_cv (C1 x y) (cos x * cos y - sin x * sin y)). +cut (Un_cv (C1 x y) (cos (x + y))). +intros. +apply UL_sequence with (C1 x y); assumption. +apply C1_cvg. +unfold Un_cv in |- *; unfold R_dist in |- *. +intros. +assert (H0 := A1_cvg x). +assert (H1 := A1_cvg y). +assert (H2 := B1_cvg x). +assert (H3 := B1_cvg y). +assert (H4 := CV_mult _ _ _ _ H0 H1). +assert (H5 := CV_mult _ _ _ _ H2 H3). +assert (H6 := reste_cv_R0 x y). +unfold Un_cv in H4; unfold Un_cv in H5; unfold Un_cv in H6. +unfold R_dist in H4; unfold R_dist in H5; unfold R_dist in H6. +cut (0 < eps / 3); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H4 (eps / 3) H7); intros N1 H8. +elim (H5 (eps / 3) H7); intros N2 H9. +elim (H6 (eps / 3) H7); intros N3 H10. +set (N := S (S (max (max N1 N2) N3))). +exists N. +intros. +cut (n = S (pred n)). +intro; rewrite H12. +rewrite <- cos_plus_form. +rewrite <- H12. +apply Rle_lt_trans with + (Rabs (A1 x n * A1 y n - cos x * cos y) + + Rabs (sin x * sin y - B1 x (pred n) * B1 y (pred n) + Reste x y (pred n))). +replace + (A1 x n * A1 y n - B1 x (pred n) * B1 y (pred n) + Reste x y (pred n) - + (cos x * cos y - sin x * sin y)) with + (A1 x n * A1 y n - cos x * cos y + + (sin x * sin y - B1 x (pred n) * B1 y (pred n) + Reste x y (pred n))); + [ apply Rabs_triang | ring ]. +replace eps with (eps / 3 + (eps / 3 + eps / 3)). +apply Rplus_lt_compat. +apply H8. +unfold ge in |- *; apply le_trans with N. +unfold N in |- *. +apply le_trans with (max N1 N2). +apply le_max_l. +apply le_trans with (max (max N1 N2) N3). +apply le_max_l. +apply le_trans with (S (max (max N1 N2) N3)); apply le_n_Sn. +assumption. +apply Rle_lt_trans with + (Rabs (sin x * sin y - B1 x (pred n) * B1 y (pred n)) + + Rabs (Reste x y (pred n))). +apply Rabs_triang. +apply Rplus_lt_compat. +rewrite <- Rabs_Ropp. +rewrite Ropp_minus_distr. +apply H9. +unfold ge in |- *; apply le_trans with (max N1 N2). +apply le_max_r. +apply le_S_n. +rewrite <- H12. +apply le_trans with N. +unfold N in |- *. +apply le_n_S. +apply le_trans with (max (max N1 N2) N3). +apply le_max_l. +apply le_n_Sn. +assumption. +replace (Reste x y (pred n)) with (Reste x y (pred n) - 0). +apply H10. +unfold ge in |- *. +apply le_S_n. +rewrite <- H12. +apply le_trans with N. +unfold N in |- *. +apply le_n_S. +apply le_trans with (max (max N1 N2) N3). +apply le_max_r. +apply le_n_Sn. +assumption. +ring. +pattern eps at 4 in |- *; replace eps with (3 * (eps / 3)). +ring. +unfold Rdiv in |- *. +rewrite <- Rmult_assoc. +apply Rinv_r_simpl_m. +discrR. +apply lt_le_trans with (pred N). +unfold N in |- *; simpl in |- *; apply lt_O_Sn. +apply le_S_n. +rewrite <- H12. +replace (S (pred N)) with N. +assumption. +unfold N in |- *; simpl in |- *; reflexivity. +cut (0 < N)%nat. +intro. +cut (0 < n)%nat. +intro. +apply S_pred with 0%nat; assumption. +apply lt_le_trans with N; assumption. +unfold N in |- *; apply lt_O_Sn. +Qed.
\ No newline at end of file |