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| author |  desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2003-01-21 16:37:05 +0000 |
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| committer | desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2003-01-21 16:37:05 +0000 |
| commit | 1a121610f8bc6761fea9dd4c41ed5255e37db657 (patch) | |
| tree | 6fa2a871749190c65c75df5a0e86151408a087ab /theories/Reals/Exp_prop.v | |
| parent | d9a43db80ff4151406a0ed6e3b38c781a06ba8e7 (diff) | |
MAJ dans Exp_prop
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@3563 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Reals/Exp_prop.v')
| -rw-r--r-- | theories/Reals/Exp_prop.v | 3 |
1 files changed, 0 insertions, 3 deletions
diff --git a/theories/Reals/Exp_prop.v b/theories/Reals/Exp_prop.v index 936f943d3..750c0571f 100644 --- a/theories/Reals/Exp_prop.v +++ b/theories/Reals/Exp_prop.v @@ -26,7 +26,6 @@ Case (exist_exp x); Intro. Unfold exp_in Un_cv; Unfold infinit_sum E1; Trivial. Qed. -(* Le reste du produit de Cauchy des sommes partielles de l'exponentielle *) Definition Reste_E [x,y:R] : nat->R := [N:nat](sum_f_R0 [k:nat](sum_f_R0 [l:nat]``/(INR (fact (S (plus l k))))*(pow x (S (plus l k)))*(/(INR (fact (minus N l)))*(pow y (minus N l)))`` (pred (minus N k))) (pred N)). Lemma exp_form : (x,y:R;n:nat) (lt O n) -> ``(E1 x n)*(E1 y n)-(Reste_E x y n)==(E1 (x+y) n)``. @@ -44,7 +43,6 @@ Apply INR_fact_neq_0. Apply H. Qed. -(* Un majorant du reste *) Definition maj_Reste_E [x,y:R] : nat->R := [N:nat]``4*(pow (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S O)) N))/(Rsqr (INR (fact (div2 (pred N)))))``. Lemma Rle_Rinv : (x,y:R) ``0<x`` -> ``0<y`` -> ``x<=y`` -> ``/y<=/x``. @@ -91,7 +89,6 @@ Right; Reflexivity. Left; Apply lt_le_trans with (2); [Apply lt_n_Sn | Apply H1]. Qed. -(* Majoration du reste par une suite convergeant vers 0 *) Lemma Reste_E_maj : (x,y:R;N:nat) (lt O N) -> ``(Rabsolu (Reste_E x y N))<=(maj_Reste_E x y N)``. Intros; Pose M := (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))). Apply Rle_trans with (Rmult (pow M (mult (2) N)) (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``/(Rsqr (INR (fact (div2 (S N)))))`` (pred (minus N k))) (pred N))). |