diff options
Diffstat (limited to 'theories/Reals/PSeries_reg.v')
| -rw-r--r-- | theories/Reals/PSeries_reg.v | 259 |
1 files changed, 259 insertions, 0 deletions
diff --git a/theories/Reals/PSeries_reg.v b/theories/Reals/PSeries_reg.v new file mode 100644 index 00000000..0c19c8da --- /dev/null +++ b/theories/Reals/PSeries_reg.v @@ -0,0 +1,259 @@ +(************************************************************************) +(* 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: PSeries_reg.v,v 1.12.2.1 2004/07/16 19:31:10 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Ranalysis1. +Require Import Max. +Require Import Even. Open Local Scope R_scope. + +Definition Boule (x:R) (r:posreal) (y:R) : Prop := Rabs (y - x) < r. + +(* Uniform convergence *) +Definition CVU (fn:nat -> R -> R) (f:R -> R) (x:R) + (r:posreal) : Prop := + forall eps:R, + 0 < eps -> + exists N : nat, + (forall (n:nat) (y:R), + (N <= n)%nat -> Boule x r y -> Rabs (f y - fn n y) < eps). + +(* Normal convergence *) +Definition CVN_r (fn:nat -> R -> R) (r:posreal) : Type := + sigT + (fun An:nat -> R => + sigT + (fun 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))). + +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. + +(* 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)) + (r:posreal), CVN_r fn r -> CVU (fun n:nat => SP fn n) (SFL fn cv) 0 r. +intros; unfold CVU in |- *; intros. +unfold CVN_r in X. +elim X; intros An X0. +elim X0; intros s H0. +elim H0; intros. +cut (Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (An k)) n - s) 0). +intro; unfold Un_cv in H3. +elim (H3 eps H); intros N0 H4. +exists N0; intros. +apply Rle_lt_trans with (Rabs (sum_f_R0 (fun k:nat => Rabs (An k)) n - s)). +rewrite <- (Rabs_Ropp (sum_f_R0 (fun k:nat => Rabs (An k)) n - s)); + rewrite Ropp_minus_distr'; + rewrite (Rabs_right (s - sum_f_R0 (fun k:nat => Rabs (An k)) n)). +eapply sum_maj1. +unfold SFL in |- *; case (cv y); intro. +trivial. +apply H1. +intro; elim H0; intros. +rewrite (Rabs_right (An n0)). +apply H8; apply H6. +apply Rle_ge; apply Rle_trans with (Rabs (fn n0 y)). +apply Rabs_pos. +apply H8; apply H6. +apply Rle_ge; + apply Rplus_le_reg_l with (sum_f_R0 (fun k:nat => Rabs (An k)) n). +rewrite Rplus_0_r; unfold Rminus in |- *; rewrite (Rplus_comm s); + rewrite <- Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_l; + apply sum_incr. +apply H1. +intro; apply Rabs_pos. +unfold R_dist in H4; unfold Rminus in H4; rewrite Ropp_0 in H4. +assert (H7 := H4 n H5). +rewrite Rplus_0_r in H7; apply H7. +unfold Un_cv in H1; unfold Un_cv in |- *; intros. +elim (H1 _ H3); intros. +exists x; intros. +unfold R_dist in |- *; unfold R_dist in H4. +rewrite Rminus_0_r; apply H4; assumption. +Qed. + +(* Each limit of a sequence of functions which converges uniformly is continue *) +Lemma CVU_continuity : + forall (fn:nat -> R -> R) (f:R -> R) (x:R) (r:posreal), + CVU fn f x r -> + (forall (n:nat) (y:R), Boule x r y -> continuity_pt (fn n) y) -> + forall y:R, Boule x r y -> continuity_pt f y. +intros; unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +unfold CVU in H. +cut (0 < eps / 3); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H _ H3); intros N0 H4. +assert (H5 := H0 N0 y H1). +cut (exists del : posreal, (forall h:R, Rabs h < del -> Boule x r (y + h))). +intro. +elim H6; intros del1 H7. +unfold continuity_pt in H5; unfold continue_in in H5; unfold limit1_in in H5; + unfold limit_in in H5; simpl in H5; unfold R_dist in H5. +elim (H5 _ H3); intros del2 H8. +set (del := Rmin del1 del2). +exists del; intros. +split. +unfold del in |- *; unfold Rmin in |- *; case (Rle_dec del1 del2); intro. +apply (cond_pos del1). +elim H8; intros; assumption. +intros; + apply Rle_lt_trans with (Rabs (f x0 - fn N0 x0) + Rabs (fn N0 x0 - f y)). +replace (f x0 - f y) with (f x0 - fn N0 x0 + (fn N0 x0 - f y)); + [ apply Rabs_triang | ring ]. +apply Rle_lt_trans with + (Rabs (f x0 - fn N0 x0) + Rabs (fn N0 x0 - fn N0 y) + Rabs (fn N0 y - f y)). +rewrite Rplus_assoc; apply Rplus_le_compat_l. +replace (fn N0 x0 - f y) with (fn N0 x0 - fn N0 y + (fn N0 y - f y)); + [ apply Rabs_triang | ring ]. +replace eps with (eps / 3 + eps / 3 + eps / 3). +repeat apply Rplus_lt_compat. +apply H4. +apply le_n. +replace x0 with (y + (x0 - y)); [ idtac | ring ]; apply H7. +elim H9; intros. +apply Rlt_le_trans with del. +assumption. +unfold del in |- *; apply Rmin_l. +elim H8; intros. +apply H11. +split. +elim H9; intros; assumption. +elim H9; intros; apply Rlt_le_trans with del. +assumption. +unfold del in |- *; apply Rmin_r. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply H4. +apply le_n. +assumption. +apply Rmult_eq_reg_l with 3. +do 2 rewrite Rmult_plus_distr_l; unfold Rdiv in |- *; rewrite <- Rmult_assoc; + rewrite Rinv_r_simpl_m. +ring. +discrR. +discrR. +cut (0 < r - Rabs (x - y)). +intro; exists (mkposreal _ H6). +simpl in |- *; intros. +unfold Boule in |- *; replace (y + h - x) with (h + (y - x)); + [ idtac | ring ]; apply Rle_lt_trans with (Rabs h + Rabs (y - x)). +apply Rabs_triang. +apply Rplus_lt_reg_r with (- Rabs (x - y)). +rewrite <- (Rabs_Ropp (y - x)); rewrite Ropp_minus_distr'. +replace (- Rabs (x - y) + r) with (r - Rabs (x - y)). +replace (- Rabs (x - y) + (Rabs h + Rabs (x - y))) with (Rabs h). +apply H7. +ring. +ring. +unfold Boule in H1; rewrite <- (Rabs_Ropp (x - y)); rewrite Ropp_minus_distr'; + apply Rplus_lt_reg_r with (Rabs (y - x)). +rewrite Rplus_0_r; replace (Rabs (y - x) + (r - Rabs (y - x))) with (pos r); + [ apply H1 | ring ]. +Qed. + +(**********) +Lemma continuity_pt_finite_SF : + forall (fn:nat -> R -> R) (N:nat) (x:R), + (forall n:nat, (n <= N)%nat -> continuity_pt (fn n) x) -> + continuity_pt (fun y:R => sum_f_R0 (fun k:nat => fn k y) N) x. +intros; induction N as [| N HrecN]. +simpl in |- *; apply (H 0%nat); apply le_n. +simpl in |- *; + replace (fun y:R => sum_f_R0 (fun k:nat => fn k y) N + fn (S N) y) with + ((fun y:R => sum_f_R0 (fun k:nat => fn k y) N) + (fun y:R => fn (S N) y))%F; + [ idtac | reflexivity ]. +apply continuity_pt_plus. +apply HrecN. +intros; apply H. +apply le_trans with N; [ assumption | apply le_n_Sn ]. +apply (H (S N)); apply le_n. +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)) + (r:posreal), + CVN_r fn r -> + (forall (n:nat) (y:R), Boule 0 r y -> continuity_pt (fn n) y) -> + forall y:R, Boule 0 r y -> continuity_pt (SFL fn cv) y. +intros; eapply CVU_continuity. +apply CVN_CVU. +apply X. +intros; unfold SP in |- *; apply continuity_pt_finite_SF. +intros; apply H. +apply H1. +apply H0. +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)), + CVN_R fn -> (forall n:nat, continuity (fn n)) -> continuity (SFL fn cv). +intros; unfold continuity in |- *; intro. +cut (0 < Rabs x + 1); + [ intro | apply Rplus_le_lt_0_compat; [ apply Rabs_pos | apply Rlt_0_1 ] ]. +cut (Boule 0 (mkposreal _ H0) x). +intro; eapply SFL_continuity_pt with (mkposreal _ H0). +apply X. +intros; apply (H n y). +apply H1. +unfold Boule in |- *; simpl in |- *; rewrite Rminus_0_r; + pattern (Rabs x) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_lt_compat_l; apply Rlt_0_1. +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). +intros; apply R_complete. +unfold SP in |- *; set (An := fun N:nat => fn N x). +change (Cauchy_crit_series An) in |- *. +apply cauchy_abs. +unfold Cauchy_crit_series in |- *; apply CV_Cauchy. +unfold CVN_R in X; cut (0 < Rabs x + 1). +intro; assert (H0 := X (mkposreal _ H)). +unfold CVN_r in H0; elim H0; intros Bn H1. +elim H1; intros l H2. +elim H2; intros. +apply Rseries_CV_comp with Bn. +intro; split. +apply Rabs_pos. +unfold An in |- *; apply H4; unfold Boule in |- *; simpl in |- *; + 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. +cut (forall n:nat, 0 <= Bn n). +intro; unfold Un_cv in H3; unfold Un_cv in |- *; intros. +elim (H3 _ H6); intros. +exists x0; intros. +replace (sum_f_R0 Bn n) with (sum_f_R0 (fun k:nat => Rabs (Bn k)) n). +apply H7; assumption. +apply sum_eq; intros; apply Rabs_right; apply Rle_ge; apply H5. +intro; apply Rle_trans with (Rabs (An n)). +apply Rabs_pos. +unfold An in |- *; apply H4; unfold Boule in |- *; simpl in |- *; + rewrite Rminus_0_r; pattern (Rabs x) at 1 in |- *; + rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rlt_0_1. +apply Rplus_le_lt_0_compat; [ apply Rabs_pos | apply Rlt_0_1 ]. +Qed. |