Imported Upstream version 8.0pl1upstream/8.0pl1

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