diff options
Diffstat (limited to 'theories/Reals/Ranalysis4.v')
| -rw-r--r-- | theories/Reals/Ranalysis4.v | 21 |
1 files changed, 11 insertions, 10 deletions
diff --git a/theories/Reals/Ranalysis4.v b/theories/Reals/Ranalysis4.v index 205c06b4..95f6d27e 100644 --- a/theories/Reals/Ranalysis4.v +++ b/theories/Reals/Ranalysis4.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Ranalysis4.v 9245 2006-10-17 12:53:34Z notin $ i*) +(*i $Id: Ranalysis4.v 10710 2008-03-23 09:24:09Z herbelin $ i*) Require Import Rbase. Require Import Rfunctions. @@ -14,7 +14,8 @@ Require Import SeqSeries. Require Import Rtrigo. Require Import Ranalysis1. Require Import Ranalysis3. -Require Import Exp_prop. Open Local Scope R_scope. +Require Import Exp_prop. +Open Local Scope R_scope. (**********) Lemma derivable_pt_inv : @@ -28,7 +29,7 @@ Proof. assumption. assumption. unfold div_fct, inv_fct, fct_cte in |- *; intro X0; elim X0; intros; - unfold derivable_pt in |- *; apply existT with x0; + unfold derivable_pt in |- *; exists x0; unfold derivable_pt_abs in |- *; unfold derivable_pt_lim in |- *; unfold derivable_pt_abs in p; unfold derivable_pt_lim in p; intros; elim (p eps H0); intros; exists x1; intros; @@ -164,10 +165,10 @@ Proof. intros. case (total_order_T x 0); intro. elim s; intro. - unfold derivable_pt in |- *; apply existT with (-1). + unfold derivable_pt in |- *; exists (-1). apply (Rabs_derive_2 x a). elim H; exact b. - unfold derivable_pt in |- *; apply existT with 1. + unfold derivable_pt in |- *; exists 1. apply (Rabs_derive_1 x r). Qed. @@ -294,8 +295,8 @@ Proof. unfold derivable_pt in |- *. assert (H := derivable_pt_lim_finite_sum An x N). induction N as [| N HrecN]. - apply existT with 0; apply H. - apply existT with + exists 0; apply H. + exists (sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred (S N))); apply H. Qed. @@ -352,7 +353,7 @@ Lemma derivable_pt_exp : forall x:R, derivable_pt exp x. Proof. intro. unfold derivable_pt in |- *. - apply existT with (exp x). + exists (exp x). apply derivable_pt_lim_exp. Qed. @@ -360,7 +361,7 @@ Lemma derivable_pt_cosh : forall x:R, derivable_pt cosh x. Proof. intro. unfold derivable_pt in |- *. - apply existT with (sinh x). + exists (sinh x). apply derivable_pt_lim_cosh. Qed. @@ -368,7 +369,7 @@ Lemma derivable_pt_sinh : forall x:R, derivable_pt sinh x. Proof. intro. unfold derivable_pt in |- *. - apply existT with (cosh x). + exists (cosh x). apply derivable_pt_lim_sinh. Qed. |