diff options
Diffstat (limited to 'theories/Reals/PSeries_reg.v')
| -rw-r--r-- | theories/Reals/PSeries_reg.v | 29 |
1 files changed, 13 insertions, 16 deletions
diff --git a/theories/Reals/PSeries_reg.v b/theories/Reals/PSeries_reg.v index 64b8e0af..e122a26a 100644 --- a/theories/Reals/PSeries_reg.v +++ b/theories/Reals/PSeries_reg.v @@ -6,14 +6,15 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: PSeries_reg.v 9245 2006-10-17 12:53:34Z notin $ i*) +(*i $Id: PSeries_reg.v 10710 2008-03-23 09:24:09Z herbelin $ i*) Require Import Rbase. Require Import Rfunctions. Require Import SeqSeries. Require Import Ranalysis1. Require Import Max. -Require Import Even. Open Local Scope R_scope. +Require Import Even. +Open Local Scope R_scope. Definition Boule (x:R) (r:posreal) (y:R) : Prop := Rabs (y - x) < r. @@ -28,25 +29,21 @@ Definition CVU (fn:nat -> R -> R) (f:R -> R) (x:R) (** Normal convergence *) Definition CVN_r (fn:nat -> R -> R) (r:posreal) : Type := - sigT - (fun An:nat -> R => - sigT - (fun l:R => + { An:nat -> R & + { l:R | Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (An k)) n) l /\ - (forall (n:nat) (y:R), Boule 0 r y -> Rabs (fn n y) <= An n))). + (forall (n:nat) (y:R), Boule 0 r y -> Rabs (fn n y) <= An n) } }. Definition CVN_R (fn:nat -> R -> R) : Type := forall r:posreal, CVN_r fn r. Definition SFL (fn:nat -> R -> R) - (cv:forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)) - (y:R) : R := match cv y with - | existT a b => a - end. + (cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }) + (y:R) : R := let (a,_) := cv y in a. (** In a complete space, normal convergence implies uniform convergence *) Lemma CVN_CVU : forall (fn:nat -> R -> R) - (cv:forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)) + (cv:forall x:R, {l:R | Un_cv (fun N:nat => SP fn N x) l }) (r:posreal), CVN_r fn r -> CVU (fun n:nat => SP fn n) (SFL fn cv) 0 r. Proof. intros; unfold CVU in |- *; intros. @@ -193,7 +190,7 @@ Qed. (** Continuity and normal convergence *) Lemma SFL_continuity_pt : forall (fn:nat -> R -> R) - (cv:forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)) + (cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }) (r:posreal), CVN_r fn r -> (forall (n:nat) (y:R), Boule 0 r y -> continuity_pt (fn n) y) -> @@ -210,7 +207,7 @@ Qed. Lemma SFL_continuity : forall (fn:nat -> R -> R) - (cv:forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)), + (cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }), CVN_R fn -> (forall n:nat, continuity (fn n)) -> continuity (SFL fn cv). Proof. intros; unfold continuity in |- *; intro. @@ -229,7 +226,7 @@ Qed. (** As R is complete, normal convergence implies that (fn) is simply-uniformly convergent *) Lemma CVN_R_CVS : forall fn:nat -> R -> R, - CVN_R fn -> forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l). + CVN_R fn -> forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }. Proof. intros; apply R_complete. unfold SP in |- *; set (An := fun N:nat => fn N x). @@ -248,7 +245,7 @@ Proof. rewrite Rminus_0_r. pattern (Rabs x) at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rlt_0_1. - apply existT with l. + exists l. cut (forall n:nat, 0 <= Bn n). intro; unfold Un_cv in H3; unfold Un_cv in |- *; intros. elim (H3 _ H6); intros. |