diff options
Diffstat (limited to 'theories/Reals/AltSeries.v')
| -rw-r--r-- | theories/Reals/AltSeries.v | 14 |
1 files changed, 6 insertions, 8 deletions
diff --git a/theories/Reals/AltSeries.v b/theories/Reals/AltSeries.v index 581c181f..5c4bbd6a 100644 --- a/theories/Reals/AltSeries.v +++ b/theories/Reals/AltSeries.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) - (*i $Id: AltSeries.v 9551 2007-01-29 15:13:35Z bgregoir $ i*) + (*i $Id: AltSeries.v 10710 2008-03-23 09:24:09Z herbelin $ i*) Require Import Rbase. Require Import Rfunctions. @@ -153,14 +153,14 @@ Lemma CV_ALT : Un_decreasing Un -> positivity_seq Un -> Un_cv Un 0 -> - sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l). + { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l }. Proof. intros. assert (H2 := CV_ALT_step0 _ H). assert (H3 := CV_ALT_step4 _ H H0). assert (X := growing_cv _ H2 H3). elim X; intros. - apply existT with x. + exists x. unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1; unfold R_dist in H1; unfold Un_cv in p; unfold R_dist in p. intros; cut (0 < eps / 2); @@ -220,7 +220,7 @@ Theorem alternated_series : forall Un:nat -> R, Un_decreasing Un -> Un_cv Un 0 -> - sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l). + { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l }. Proof. intros; apply CV_ALT. assumption. @@ -408,7 +408,7 @@ Proof. Qed. Lemma exist_PI : - sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 (tg_alt PI_tg) N) l). + { l:R | Un_cv (fun N:nat => sum_f_R0 (tg_alt PI_tg) N) l }. Proof. apply alternated_series. apply PI_tg_decreasing. @@ -416,9 +416,7 @@ Proof. Qed. (** Now, PI is defined *) -Definition PI : R := 4 * match exist_PI with - | existT a b => a - end. +Definition PI : R := 4 * (let (a,_) := exist_PI in a). (** We can get an approximation of PI with the following inequality *) Lemma PI_ineq : |