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Diffstat (limited to 'theories/Reals/Exp_prop.v')
| -rw-r--r-- | theories/Reals/Exp_prop.v | 1011 |
1 files changed, 1011 insertions, 0 deletions
diff --git a/theories/Reals/Exp_prop.v b/theories/Reals/Exp_prop.v new file mode 100644 index 00000000..fcaeb11e --- /dev/null +++ b/theories/Reals/Exp_prop.v @@ -0,0 +1,1011 @@ +(************************************************************************) +(* 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: Exp_prop.v,v 1.16.2.1 2004/07/16 19:31:10 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import Ranalysis1. +Require Import PSeries_reg. +Require Import Div2. +Require Import Even. +Require Import Max. +Open Local Scope nat_scope. +Open Local Scope R_scope. + +Definition E1 (x:R) (N:nat) : R := + sum_f_R0 (fun k:nat => / INR (fact k) * x ^ k) N. + +Lemma E1_cvg : forall x:R, Un_cv (E1 x) (exp x). +intro; unfold exp in |- *; unfold projT1 in |- *. +case (exist_exp x); intro. +unfold exp_in, Un_cv in |- *; unfold infinit_sum, E1 in |- *; trivial. +Qed. + +Definition Reste_E (x y:R) (N:nat) : R := + sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + / INR (fact (S (l + k))) * x ^ S (l + k) * + (/ INR (fact (N - l)) * y ^ (N - l))) ( + pred (N - k))) (pred N). + +Lemma exp_form : + forall (x y:R) (n:nat), + (0 < n)%nat -> E1 x n * E1 y n - Reste_E x y n = E1 (x + y) n. +intros; unfold E1 in |- *. +rewrite cauchy_finite. +unfold Reste_E in |- *; unfold Rminus in |- *; rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; apply sum_eq; + intros. +rewrite binomial. +rewrite scal_sum; apply sum_eq; intros. +unfold C in |- *; unfold Rdiv in |- *; repeat rewrite Rmult_assoc; + rewrite (Rmult_comm (INR (fact i))); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite Rinv_mult_distr. +ring. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply H. +Qed. + +Definition maj_Reste_E (x y:R) (N:nat) : R := + 4 * + (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * N) / + Rsqr (INR (fact (div2 (pred N))))). + +Lemma Rle_Rinv : forall x y:R, 0 < x -> 0 < y -> x <= y -> / y <= / x. +intros; apply Rmult_le_reg_l with x. +apply H. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with y. +apply H0. +rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; apply H1. +red in |- *; intro; rewrite H2 in H0; elim (Rlt_irrefl _ H0). +red in |- *; intro; rewrite H2 in H; elim (Rlt_irrefl _ H). +Qed. + +(**********) +Lemma div2_double : forall N:nat, div2 (2 * N) = N. +intro; induction N as [| N HrecN]. +reflexivity. +replace (2 * S N)%nat with (S (S (2 * N))). +simpl in |- *; simpl in HrecN; rewrite HrecN; reflexivity. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +Qed. + +Lemma div2_S_double : forall N:nat, div2 (S (2 * N)) = N. +intro; induction N as [| N HrecN]. +reflexivity. +replace (2 * S N)%nat with (S (S (2 * N))). +simpl in |- *; simpl in HrecN; rewrite HrecN; reflexivity. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +Qed. + +Lemma div2_not_R0 : forall N:nat, (1 < N)%nat -> (0 < div2 N)%nat. +intros; induction N as [| N HrecN]. +elim (lt_n_O _ H). +cut ((1 < N)%nat \/ N = 1%nat). +intro; elim H0; intro. +assert (H2 := even_odd_dec N). +elim H2; intro. +rewrite <- (even_div2 _ a); apply HrecN; assumption. +rewrite <- (odd_div2 _ b); apply lt_O_Sn. +rewrite H1; simpl in |- *; apply lt_O_Sn. +inversion H. +right; reflexivity. +left; apply lt_le_trans with 2%nat; [ apply lt_n_Sn | apply H1 ]. +Qed. + +Lemma Reste_E_maj : + forall (x y:R) (N:nat), + (0 < N)%nat -> Rabs (Reste_E x y N) <= maj_Reste_E x y N. +intros; set (M := Rmax 1 (Rmax (Rabs x) (Rabs y))). +apply Rle_trans with + (M ^ (2 * N) * + sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => / Rsqr (INR (fact (div2 (S N))))) + (pred (N - k))) (pred N)). +unfold Reste_E in |- *. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + Rabs + (sum_f_R0 + (fun l:nat => + / INR (fact (S (l + k))) * x ^ S (l + k) * + (/ INR (fact (N - l)) * y ^ (N - l))) ( + pred (N - k)))) (pred N)). +apply + (Rsum_abs + (fun k:nat => + sum_f_R0 + (fun l:nat => + / INR (fact (S (l + k))) * x ^ S (l + k) * + (/ INR (fact (N - l)) * y ^ (N - l))) ( + pred (N - k))) (pred N)). +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + Rabs + (/ INR (fact (S (l + k))) * x ^ S (l + k) * + (/ INR (fact (N - l)) * y ^ (N - l)))) ( + pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply + (Rsum_abs + (fun l:nat => + / INR (fact (S (l + n))) * x ^ S (l + n) * + (/ INR (fact (N - l)) * y ^ (N - l)))). +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + M ^ (2 * N) * / INR (fact (S l)) * / INR (fact (N - l))) + (pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +repeat rewrite Rabs_mult. +do 2 rewrite <- RPow_abs. +rewrite (Rabs_right (/ INR (fact (S (n0 + n))))). +rewrite (Rabs_right (/ INR (fact (N - n0)))). +replace + (/ INR (fact (S (n0 + n))) * Rabs x ^ S (n0 + n) * + (/ INR (fact (N - n0)) * Rabs y ^ (N - n0))) with + (/ INR (fact (N - n0)) * / INR (fact (S (n0 + n))) * Rabs x ^ S (n0 + n) * + Rabs y ^ (N - n0)); [ idtac | ring ]. +rewrite <- (Rmult_comm (/ INR (fact (N - n0)))). +repeat rewrite Rmult_assoc. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_trans with + (/ INR (fact (S n0)) * Rabs x ^ S (n0 + n) * Rabs y ^ (N - n0)). +rewrite (Rmult_comm (/ INR (fact (S (n0 + n))))); + rewrite (Rmult_comm (/ INR (fact (S n0)))); repeat rewrite Rmult_assoc; + apply Rmult_le_compat_l. +apply pow_le; apply Rabs_pos. +rewrite (Rmult_comm (/ INR (fact (S n0)))); apply Rmult_le_compat_l. +apply pow_le; apply Rabs_pos. +apply Rle_Rinv. +apply INR_fact_lt_0. +apply INR_fact_lt_0. +apply le_INR; apply fact_le; apply le_n_S. +apply le_plus_l. +rewrite (Rmult_comm (M ^ (2 * N))); rewrite Rmult_assoc; + apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_trans with (M ^ S (n0 + n) * Rabs y ^ (N - n0)). +do 2 rewrite <- (Rmult_comm (Rabs y ^ (N - n0))). +apply Rmult_le_compat_l. +apply pow_le; apply Rabs_pos. +apply pow_incr; split. +apply Rabs_pos. +apply Rle_trans with (Rmax (Rabs x) (Rabs y)). +apply RmaxLess1. +unfold M in |- *; apply RmaxLess2. +apply Rle_trans with (M ^ S (n0 + n) * M ^ (N - n0)). +apply Rmult_le_compat_l. +apply pow_le; apply Rle_trans with 1. +left; apply Rlt_0_1. +unfold M in |- *; apply RmaxLess1. +apply pow_incr; split. +apply Rabs_pos. +apply Rle_trans with (Rmax (Rabs x) (Rabs y)). +apply RmaxLess2. +unfold M in |- *; apply RmaxLess2. +rewrite <- pow_add; replace (S (n0 + n) + (N - n0))%nat with (N + S n)%nat. +apply Rle_pow. +unfold M in |- *; apply RmaxLess1. +replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ]. +apply plus_le_compat_l. +replace N with (S (pred N)). +apply le_n_S; apply H0. +symmetry in |- *; apply S_pred with 0%nat; apply H. +apply INR_eq; do 2 rewrite plus_INR; do 2 rewrite S_INR; rewrite plus_INR; + rewrite minus_INR. +ring. +apply le_trans with (pred (N - n)). +apply 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). +apply H0. +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). +apply H0. +apply lt_pred_n_n. +apply H. +apply le_trans with (pred N). +apply H0. +apply le_pred_n. +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. +rewrite scal_sum. +apply sum_Rle; intros. +rewrite <- Rmult_comm. +rewrite scal_sum. +apply sum_Rle; intros. +rewrite (Rmult_comm (/ Rsqr (INR (fact (div2 (S N)))))). +rewrite Rmult_assoc; apply Rmult_le_compat_l. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +unfold M in |- *; apply RmaxLess1. +assert (H2 := even_odd_cor N). +elim H2; intros N0 H3. +elim H3; intro. +apply Rle_trans with (/ INR (fact n0) * / INR (fact (N - n0))). +do 2 rewrite <- (Rmult_comm (/ INR (fact (N - n0)))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_Rinv. +apply INR_fact_lt_0. +apply INR_fact_lt_0. +apply le_INR. +apply fact_le. +apply le_n_Sn. +replace (/ INR (fact n0) * / INR (fact (N - n0))) with + (C N n0 / INR (fact N)). +pattern N at 1 in |- *; rewrite H4. +apply Rle_trans with (C N N0 / INR (fact N)). +unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ INR (fact N))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +rewrite H4. +apply C_maj. +rewrite <- H4; apply le_trans with (pred (N - n)). +apply 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). +apply H0. +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). +apply H0. +apply lt_pred_n_n. +apply H. +apply le_trans with (pred N). +apply H0. +apply le_pred_n. +replace (C N N0 / INR (fact N)) with (/ Rsqr (INR (fact N0))). +rewrite H4; rewrite div2_S_double; right; reflexivity. +unfold Rsqr, C, Rdiv in |- *. +repeat rewrite Rinv_mult_distr. +rewrite (Rmult_comm (INR (fact N))). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; replace (N - N0)%nat with N0. +ring. +replace N with (N0 + N0)%nat. +symmetry in |- *; apply minus_plus. +rewrite H4. +apply INR_eq; rewrite plus_INR; rewrite mult_INR; do 2 rewrite S_INR; ring. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +unfold C, Rdiv in |- *. +rewrite (Rmult_comm (INR (fact N))). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rinv_mult_distr. +rewrite Rmult_1_r; ring. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +replace (/ INR (fact (S n0)) * / INR (fact (N - n0))) with + (C (S N) (S n0) / INR (fact (S N))). +apply Rle_trans with (C (S N) (S N0) / INR (fact (S N))). +unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ INR (fact (S N)))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +cut (S N = (2 * S N0)%nat). +intro; rewrite H5; apply C_maj. +rewrite <- H5; apply le_n_S. +apply le_trans with (pred (N - n)). +apply 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). +apply H0. +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). +apply H0. +apply lt_pred_n_n. +apply H. +apply le_trans with (pred N). +apply H0. +apply le_pred_n. +apply INR_eq; rewrite H4. +do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; ring. +cut (S N = (2 * S N0)%nat). +intro. +replace (C (S N) (S N0) / INR (fact (S N))) with (/ Rsqr (INR (fact (S N0)))). +rewrite H5; rewrite div2_double. +right; reflexivity. +unfold Rsqr, C, Rdiv in |- *. +repeat rewrite Rinv_mult_distr. +replace (S N - S N0)%nat with (S N0). +rewrite (Rmult_comm (INR (fact (S N)))). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; reflexivity. +apply INR_fact_neq_0. +replace (S N) with (S N0 + S N0)%nat. +symmetry in |- *; apply minus_plus. +rewrite H5; ring. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_eq; rewrite H4; do 2 rewrite S_INR; do 2 rewrite mult_INR; + repeat rewrite S_INR; ring. +unfold C, Rdiv in |- *. +rewrite (Rmult_comm (INR (fact (S N)))). +rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; rewrite Rinv_mult_distr. +reflexivity. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +unfold maj_Reste_E in |- *. +unfold Rdiv in |- *; rewrite (Rmult_comm 4). +rewrite Rmult_assoc. +apply Rmult_le_compat_l. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +apply RmaxLess1. +apply Rle_trans with + (sum_f_R0 (fun k:nat => INR (N - k) * / Rsqr (INR (fact (div2 (S N))))) + (pred N)). +apply sum_Rle; intros. +rewrite sum_cte. +replace (S (pred (N - n))) with (N - n)%nat. +right; apply Rmult_comm. +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). +apply H0. +apply lt_pred_n_n. +apply H. +apply le_trans with (pred N). +apply H0. +apply le_pred_n. +apply Rle_trans with + (sum_f_R0 (fun k:nat => INR N * / Rsqr (INR (fact (div2 (S N))))) (pred N)). +apply sum_Rle; intros. +do 2 rewrite <- (Rmult_comm (/ Rsqr (INR (fact (div2 (S N)))))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt. +apply INR_fact_neq_0. +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). +apply H0. +apply le_pred_n. +rewrite sum_cte; replace (S (pred N)) with N. +cut (div2 (S N) = S (div2 (pred N))). +intro; rewrite H0. +rewrite fact_simpl; rewrite mult_comm; rewrite mult_INR; rewrite Rsqr_mult. +rewrite Rinv_mult_distr. +rewrite (Rmult_comm (INR N)); repeat rewrite Rmult_assoc; + apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt; apply INR_fact_neq_0. +rewrite <- H0. +cut (INR N <= INR (2 * div2 (S N))). +intro; apply Rmult_le_reg_l with (Rsqr (INR (div2 (S N)))). +apply Rsqr_pos_lt. +apply not_O_INR; red in |- *; intro. +cut (1 < S N)%nat. +intro; assert (H4 := div2_not_R0 _ H3). +rewrite H2 in H4; elim (lt_n_O _ H4). +apply lt_n_S; apply H. +repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_l. +replace (INR N * INR N) with (Rsqr (INR N)); [ idtac | reflexivity ]. +rewrite Rmult_assoc. +rewrite Rmult_comm. +replace 4 with (Rsqr 2); [ idtac | ring_Rsqr ]. +rewrite <- Rsqr_mult. +apply Rsqr_incr_1. +replace 2 with (INR 2). +rewrite <- mult_INR; apply H1. +reflexivity. +left; apply lt_INR_0; apply H. +left; apply Rmult_lt_0_compat. +prove_sup0. +apply lt_INR_0; apply div2_not_R0. +apply lt_n_S; apply H. +cut (1 < S N)%nat. +intro; unfold Rsqr in |- *; apply prod_neq_R0; apply not_O_INR; intro; + assert (H4 := div2_not_R0 _ H2); rewrite H3 in H4; + elim (lt_n_O _ H4). +apply lt_n_S; apply H. +assert (H1 := even_odd_cor N). +elim H1; intros N0 H2. +elim H2; intro. +pattern N at 2 in |- *; rewrite H3. +rewrite div2_S_double. +right; rewrite H3; reflexivity. +pattern N at 2 in |- *; rewrite H3. +replace (S (S (2 * N0))) with (2 * S N0)%nat. +rewrite div2_double. +rewrite H3. +rewrite S_INR; do 2 rewrite mult_INR. +rewrite (S_INR N0). +rewrite Rmult_plus_distr_l. +apply Rplus_le_compat_l. +rewrite Rmult_1_r. +simpl in |- *. +pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + apply Rlt_0_1. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +unfold Rsqr in |- *; apply prod_neq_R0; apply INR_fact_neq_0. +unfold Rsqr in |- *; apply prod_neq_R0; apply not_O_INR; discriminate. +assert (H0 := even_odd_cor N). +elim H0; intros N0 H1. +elim H1; intro. +cut (0 < N0)%nat. +intro; rewrite H2. +rewrite div2_S_double. +replace (2 * N0)%nat with (S (S (2 * pred N0))). +replace (pred (S (S (2 * pred N0)))) with (S (2 * pred N0)). +rewrite div2_S_double. +apply S_pred with 0%nat; apply H3. +reflexivity. +replace N0 with (S (pred N0)). +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +symmetry in |- *; apply S_pred with 0%nat; apply H3. +rewrite H2 in H. +apply neq_O_lt. +red in |- *; intro. +rewrite <- H3 in H. +simpl in H. +elim (lt_n_O _ H). +rewrite H2. +replace (pred (S (2 * N0))) with (2 * N0)%nat; [ idtac | reflexivity ]. +replace (S (S (2 * N0))) with (2 * S N0)%nat. +do 2 rewrite div2_double. +reflexivity. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +apply S_pred with 0%nat; apply H. +Qed. + +Lemma maj_Reste_cv_R0 : forall x y:R, Un_cv (maj_Reste_E x y) 0. +intros; assert (H := Majxy_cv_R0 x y). +unfold Un_cv in H; unfold Un_cv in |- *; intros. +cut (0 < eps / 4); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H _ H1); intros N0 H2. +exists (max (2 * S N0) 2); intros. +unfold R_dist in H2; unfold R_dist in |- *; rewrite Rminus_0_r; + unfold Majxy in H2; unfold maj_Reste_E in |- *. +rewrite Rabs_right. +apply Rle_lt_trans with + (4 * + (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))) / + INR (fact (div2 (pred n))))). +apply Rmult_le_compat_l. +left; prove_sup0. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr. +rewrite (Rmult_comm (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n))); + rewrite + (Rmult_comm (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))))) + ; rewrite Rmult_assoc; apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_trans with (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)). +rewrite Rmult_comm; + pattern (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)) at 2 in |- *; + rewrite <- Rmult_1_r; apply Rmult_le_compat_l. +apply pow_le; apply Rle_trans with 1. +left; apply Rlt_0_1. +apply RmaxLess1. +apply Rmult_le_reg_l with (INR (fact (div2 (pred n)))). +apply INR_fact_lt_0. +rewrite Rmult_1_r; rewrite <- Rinv_r_sym. +replace 1 with (INR 1); [ apply le_INR | reflexivity ]. +apply lt_le_S. +apply INR_lt. +apply INR_fact_lt_0. +apply INR_fact_neq_0. +apply Rle_pow. +apply RmaxLess1. +assert (H4 := even_odd_cor n). +elim H4; intros N1 H5. +elim H5; intro. +cut (0 < N1)%nat. +intro. +rewrite H6. +replace (pred (2 * N1)) with (S (2 * pred N1)). +rewrite div2_S_double. +replace (S (pred N1)) with N1. +apply INR_le. +right. +do 3 rewrite mult_INR; repeat rewrite S_INR; ring. +apply S_pred with 0%nat; apply H7. +replace (2 * N1)%nat with (S (S (2 * pred N1))). +reflexivity. +pattern N1 at 2 in |- *; replace N1 with (S (pred N1)). +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +symmetry in |- *; apply S_pred with 0%nat; apply H7. +apply INR_lt. +apply Rmult_lt_reg_l with (INR 2). +simpl in |- *; prove_sup0. +rewrite Rmult_0_r; rewrite <- mult_INR. +apply lt_INR_0. +rewrite <- H6. +apply lt_le_trans with 2%nat. +apply lt_O_Sn. +apply le_trans with (max (2 * S N0) 2). +apply le_max_r. +apply H3. +rewrite H6. +replace (pred (S (2 * N1))) with (2 * N1)%nat. +rewrite div2_double. +replace (4 * S N1)%nat with (2 * (2 * S N1))%nat. +apply (fun m n p:nat => mult_le_compat_l p n m). +replace (2 * S N1)%nat with (S (S (2 * N1))). +apply le_n_Sn. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +ring. +reflexivity. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply Rmult_lt_reg_l with (/ 4). +apply Rinv_0_lt_compat; prove_sup0. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite Rmult_comm. +replace + (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))) / + INR (fact (div2 (pred n)))) with + (Rabs + (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))) / + INR (fact (div2 (pred n))) - 0)). +apply H2; unfold ge in |- *. +cut (2 * S N0 <= n)%nat. +intro; apply le_S_n. +apply INR_le; apply Rmult_le_reg_l with (INR 2). +simpl in |- *; prove_sup0. +do 2 rewrite <- mult_INR; apply le_INR. +apply le_trans with n. +apply H4. +assert (H5 := even_odd_cor n). +elim H5; intros N1 H6. +elim H6; intro. +cut (0 < N1)%nat. +intro. +rewrite H7. +apply (fun m n p:nat => mult_le_compat_l p n m). +replace (pred (2 * N1)) with (S (2 * pred N1)). +rewrite div2_S_double. +replace (S (pred N1)) with N1. +apply le_n. +apply S_pred with 0%nat; apply H8. +replace (2 * N1)%nat with (S (S (2 * pred N1))). +reflexivity. +pattern N1 at 2 in |- *; replace N1 with (S (pred N1)). +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +symmetry in |- *; apply S_pred with 0%nat; apply H8. +apply INR_lt. +apply Rmult_lt_reg_l with (INR 2). +simpl in |- *; prove_sup0. +rewrite Rmult_0_r; rewrite <- mult_INR. +apply lt_INR_0. +rewrite <- H7. +apply lt_le_trans with 2%nat. +apply lt_O_Sn. +apply le_trans with (max (2 * S N0) 2). +apply le_max_r. +apply H3. +rewrite H7. +replace (pred (S (2 * N1))) with (2 * N1)%nat. +rewrite div2_double. +replace (2 * S N1)%nat with (S (S (2 * N1))). +apply le_n_Sn. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +reflexivity. +apply le_trans with (max (2 * S N0) 2). +apply le_max_l. +apply H3. +rewrite Rminus_0_r; apply Rabs_right. +apply Rle_ge. +unfold Rdiv in |- *; repeat 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. +discrR. +apply Rle_ge. +unfold Rdiv in |- *; apply Rmult_le_pos. +left; prove_sup0. +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 Rsqr_pos_lt; apply INR_fact_neq_0. +Qed. + +(**********) +Lemma Reste_E_cv : forall x y:R, Un_cv (Reste_E x y) 0. +intros; assert (H := maj_Reste_cv_R0 x y). +unfold Un_cv in H; unfold Un_cv in |- *; intros; elim (H _ H0); intros. +exists (max x0 1); intros. +unfold R_dist in |- *; rewrite Rminus_0_r. +apply Rle_lt_trans with (maj_Reste_E x y n). +apply Reste_E_maj. +apply lt_le_trans with 1%nat. +apply lt_O_Sn. +apply le_trans with (max x0 1). +apply le_max_r. +apply H2. +replace (maj_Reste_E x y n) with (R_dist (maj_Reste_E x y n) 0). +apply H1. +unfold ge in |- *; apply le_trans with (max x0 1). +apply le_max_l. +apply H2. +unfold R_dist in |- *; rewrite Rminus_0_r; apply Rabs_right. +apply Rle_ge; apply Rle_trans with (Rabs (Reste_E x y n)). +apply Rabs_pos. +apply Reste_E_maj. +apply lt_le_trans with 1%nat. +apply lt_O_Sn. +apply le_trans with (max x0 1). +apply le_max_r. +apply H2. +Qed. + +(**********) +Lemma exp_plus : forall x y:R, exp (x + y) = exp x * exp y. +intros; assert (H0 := E1_cvg x). +assert (H := E1_cvg y). +assert (H1 := E1_cvg (x + y)). +eapply UL_sequence. +apply H1. +assert (H2 := CV_mult _ _ _ _ H0 H). +assert (H3 := CV_minus _ _ _ _ H2 (Reste_E_cv x y)). +unfold Un_cv in |- *; unfold Un_cv in H3; intros. +elim (H3 _ H4); intros. +exists (S x0); intros. +rewrite <- (exp_form x y n). +rewrite Rminus_0_r in H5. +apply H5. +unfold ge in |- *; apply le_trans with (S x0). +apply le_n_Sn. +apply H6. +apply lt_le_trans with (S x0). +apply lt_O_Sn. +apply H6. +Qed. + +(**********) +Lemma exp_pos_pos : forall x:R, 0 < x -> 0 < exp x. +intros; set (An := fun N:nat => / INR (fact N) * x ^ N). +cut (Un_cv (fun n:nat => sum_f_R0 An n) (exp x)). +intro; apply Rlt_le_trans with (sum_f_R0 An 0). +unfold An in |- *; simpl in |- *; rewrite Rinv_1; rewrite Rmult_1_r; + apply Rlt_0_1. +apply sum_incr. +assumption. +intro; unfold An in |- *; left; apply Rmult_lt_0_compat. +apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply (pow_lt _ n H). +unfold exp in |- *; unfold projT1 in |- *; case (exist_exp x); intro. +unfold exp_in in |- *; unfold infinit_sum, Un_cv in |- *; trivial. +Qed. + +(**********) +Lemma exp_pos : forall x:R, 0 < exp x. +intro; case (total_order_T 0 x); intro. +elim s; intro. +apply (exp_pos_pos _ a). +rewrite <- b; rewrite exp_0; apply Rlt_0_1. +replace (exp x) with (1 / exp (- x)). +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply Rlt_0_1. +apply Rinv_0_lt_compat; apply exp_pos_pos. +apply (Ropp_0_gt_lt_contravar _ r). +cut (exp (- x) <> 0). +intro; unfold Rdiv in |- *; apply Rmult_eq_reg_l with (exp (- x)). +rewrite Rmult_1_l; rewrite <- Rinv_r_sym. +rewrite <- exp_plus. +rewrite Rplus_opp_l; rewrite exp_0; reflexivity. +apply H. +apply H. +assert (H := exp_plus x (- x)). +rewrite Rplus_opp_r in H; rewrite exp_0 in H. +red in |- *; intro; rewrite H0 in H. +rewrite Rmult_0_r in H. +elim R1_neq_R0; assumption. +Qed. + +(* ((exp h)-1)/h -> 0 quand h->0 *) +Lemma derivable_pt_lim_exp_0 : derivable_pt_lim exp 0 1. +unfold derivable_pt_lim in |- *; intros. +set (fn := fun (N:nat) (x:R) => x ^ N / INR (fact (S N))). +cut (CVN_R fn). +intro; cut (forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)). +intro cv; cut (forall n:nat, continuity (fn n)). +intro; cut (continuity (SFL fn cv)). +intro; unfold continuity in H1. +assert (H2 := H1 0). +unfold continuity_pt in H2; unfold continue_in in H2; unfold limit1_in in H2; + unfold limit_in in H2; simpl in H2; unfold R_dist in H2. +elim (H2 _ H); intros alp H3. +elim H3; intros. +exists (mkposreal _ H4); intros. +rewrite Rplus_0_l; rewrite exp_0. +replace ((exp h - 1) / h) with (SFL fn cv h). +replace 1 with (SFL fn cv 0). +apply H5. +split. +unfold D_x, no_cond in |- *; split. +trivial. +apply (sym_not_eq H6). +rewrite Rminus_0_r; apply H7. +unfold SFL in |- *. +case (cv 0); intros. +eapply UL_sequence. +apply u. +unfold Un_cv, SP in |- *. +intros; exists 1%nat; intros. +unfold R_dist in |- *; rewrite decomp_sum. +rewrite (Rplus_comm (fn 0%nat 0)). +replace (fn 0%nat 0) with 1. +unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_r; + rewrite Rplus_0_r. +replace (sum_f_R0 (fun i:nat => fn (S i) 0) (pred n)) with 0. +rewrite Rabs_R0; apply H8. +symmetry in |- *; apply sum_eq_R0; intros. +unfold fn in |- *. +simpl in |- *. +unfold Rdiv in |- *; do 2 rewrite Rmult_0_l; reflexivity. +unfold fn in |- *; simpl in |- *. +unfold Rdiv in |- *; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity. +apply lt_le_trans with 1%nat; [ apply lt_n_Sn | apply H9 ]. +unfold SFL, exp in |- *. +unfold projT1 in |- *. +case (cv h); case (exist_exp h); intros. +eapply UL_sequence. +apply u. +unfold Un_cv in |- *; intros. +unfold exp_in in e. +unfold infinit_sum in e. +cut (0 < eps0 * Rabs h). +intro; elim (e _ H9); intros N0 H10. +exists N0; intros. +unfold R_dist in |- *. +apply Rmult_lt_reg_l with (Rabs h). +apply Rabs_pos_lt; assumption. +rewrite <- Rabs_mult. +rewrite Rmult_minus_distr_l. +replace (h * ((x - 1) / h)) with (x - 1). +unfold R_dist in H10. +replace (h * SP fn n h - (x - 1)) with + (sum_f_R0 (fun i:nat => / INR (fact i) * h ^ i) (S n) - x). +rewrite (Rmult_comm (Rabs h)). +apply H10. +unfold ge in |- *. +apply le_trans with (S N0). +apply le_n_Sn. +apply le_n_S; apply H11. +rewrite decomp_sum. +replace (/ INR (fact 0) * h ^ 0) with 1. +unfold Rminus in |- *. +rewrite Ropp_plus_distr. +rewrite Ropp_involutive. +rewrite <- (Rplus_comm (- x)). +rewrite <- (Rplus_comm (- x + 1)). +rewrite Rplus_assoc; repeat apply Rplus_eq_compat_l. +replace (pred (S n)) with n; [ idtac | reflexivity ]. +unfold SP in |- *. +rewrite scal_sum. +apply sum_eq; intros. +unfold fn in |- *. +replace (h ^ S i) with (h * h ^ i). +unfold Rdiv in |- *; ring. +simpl in |- *; ring. +simpl in |- *; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity. +apply lt_O_Sn. +unfold Rdiv in |- *. +rewrite <- Rmult_assoc. +symmetry in |- *; apply Rinv_r_simpl_m. +assumption. +apply Rmult_lt_0_compat. +apply H8. +apply Rabs_pos_lt; assumption. +apply SFL_continuity; assumption. +intro; unfold fn in |- *. +replace (fun x:R => x ^ n / INR (fact (S n))) with + (pow_fct n / fct_cte (INR (fact (S n))))%F; [ idtac | reflexivity ]. +apply continuity_div. +apply derivable_continuous; apply (derivable_pow n). +apply derivable_continuous; apply derivable_const. +intro; unfold fct_cte in |- *; apply INR_fact_neq_0. +apply (CVN_R_CVS _ X). +assert (H0 := Alembert_exp). +unfold CVN_R in |- *. +intro; unfold CVN_r in |- *. +apply existT with (fun N:nat => r ^ N / INR (fact (S N))). +cut + (sigT + (fun l:R => + Un_cv + (fun n:nat => + sum_f_R0 (fun k:nat => Rabs (r ^ k / INR (fact (S k)))) n) l)). +intro. +elim X; intros. +exists x; intros. +split. +apply p. +unfold Boule in |- *; intros. +rewrite Rminus_0_r in H1. +unfold fn in |- *. +unfold Rdiv in |- *; rewrite Rabs_mult. +cut (0 < INR (fact (S n))). +intro. +rewrite (Rabs_right (/ INR (fact (S n)))). +do 2 rewrite <- (Rmult_comm (/ INR (fact (S n)))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply H2. +rewrite <- RPow_abs. +apply pow_maj_Rabs. +rewrite Rabs_Rabsolu; left; apply H1. +apply Rle_ge; left; apply Rinv_0_lt_compat; apply H2. +apply INR_fact_lt_0. +cut ((r:R) <> 0). +intro; apply Alembert_C2. +intro; apply Rabs_no_R0. +unfold Rdiv in |- *; apply prod_neq_R0. +apply pow_nonzero; assumption. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. +unfold Un_cv in H0. +unfold Un_cv in |- *; intros. +cut (0 < eps0 / r); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; apply (cond_pos r) ] ]. +elim (H0 _ H3); intros N0 H4. +exists N0; intros. +cut (S n >= N0)%nat. +intro hyp_sn. +assert (H6 := H4 _ hyp_sn). +unfold R_dist in H6; rewrite Rminus_0_r in H6. +rewrite Rabs_Rabsolu in H6. +unfold R_dist in |- *; rewrite Rminus_0_r. +rewrite Rabs_Rabsolu. +replace + (Rabs (r ^ S n / INR (fact (S (S n)))) / Rabs (r ^ n / INR (fact (S n)))) + with (r * / INR (fact (S (S n))) * / / INR (fact (S n))). +rewrite Rmult_assoc; rewrite Rabs_mult. +rewrite (Rabs_right r). +apply Rmult_lt_reg_l with (/ r). +apply Rinv_0_lt_compat; apply (cond_pos r). +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite <- (Rmult_comm eps0). +apply H6. +assumption. +apply Rle_ge; left; apply (cond_pos r). +unfold Rdiv in |- *. +repeat rewrite Rabs_mult. +repeat rewrite Rabs_Rinv. +rewrite Rinv_mult_distr. +repeat rewrite Rabs_right. +rewrite Rinv_involutive. +rewrite (Rmult_comm r). +rewrite (Rmult_comm (r ^ S n)). +repeat rewrite Rmult_assoc. +apply Rmult_eq_compat_l. +rewrite (Rmult_comm r). +rewrite <- Rmult_assoc; rewrite <- (Rmult_comm (INR (fact (S n)))). +apply Rmult_eq_compat_l. +simpl in |- *. +rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +ring. +apply pow_nonzero; assumption. +apply INR_fact_neq_0. +apply Rle_ge; left; apply INR_fact_lt_0. +apply Rle_ge; left; apply pow_lt; apply (cond_pos r). +apply Rle_ge; left; apply INR_fact_lt_0. +apply Rle_ge; left; apply pow_lt; apply (cond_pos r). +apply Rabs_no_R0; apply pow_nonzero; assumption. +apply Rinv_neq_0_compat; apply Rabs_no_R0; apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +unfold ge in |- *; apply le_trans with n. +apply H5. +apply le_n_Sn. +assert (H1 := cond_pos r); red in |- *; intro; rewrite H2 in H1; + elim (Rlt_irrefl _ H1). +Qed. + +(**********) +Lemma derivable_pt_lim_exp : forall x:R, derivable_pt_lim exp x (exp x). +intro; assert (H0 := derivable_pt_lim_exp_0). +unfold derivable_pt_lim in H0; unfold derivable_pt_lim in |- *; intros. +cut (0 < eps / exp x); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply H | apply Rinv_0_lt_compat; apply exp_pos ] ]. +elim (H0 _ H1); intros del H2. +exists del; intros. +assert (H5 := H2 _ H3 H4). +rewrite Rplus_0_l in H5; rewrite exp_0 in H5. +replace ((exp (x + h) - exp x) / h - exp x) with + (exp x * ((exp h - 1) / h - 1)). +rewrite Rabs_mult; rewrite (Rabs_right (exp x)). +apply Rmult_lt_reg_l with (/ exp x). +apply Rinv_0_lt_compat; apply exp_pos. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite <- (Rmult_comm eps). +apply H5. +assert (H6 := exp_pos x); red in |- *; intro; rewrite H7 in H6; + elim (Rlt_irrefl _ H6). +apply Rle_ge; left; apply exp_pos. +rewrite Rmult_minus_distr_l. +rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite <- Rmult_assoc; + rewrite Rmult_minus_distr_l. +rewrite Rmult_1_r; rewrite exp_plus; reflexivity. +Qed.
\ No newline at end of file |