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Diffstat (limited to 'theories/Reals/Rseries.v')
| -rw-r--r-- | theories/Reals/Rseries.v | 275 |
1 files changed, 275 insertions, 0 deletions
diff --git a/theories/Reals/Rseries.v b/theories/Reals/Rseries.v new file mode 100644 index 00000000..cbf93278 --- /dev/null +++ b/theories/Reals/Rseries.v @@ -0,0 +1,275 @@ +(************************************************************************) +(* 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: Rseries.v,v 1.11.2.1 2004/07/16 19:31:13 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import Classical. +Require Import Compare. +Open Local Scope R_scope. + +Implicit Type r : R. + +(* classical is needed for [Un_cv_crit] *) +(*********************************************************) +(* Definition of sequence and properties *) +(* *) +(*********************************************************) + +Section sequence. + +(*********) +Variable Un : nat -> R. + +(*********) +Fixpoint Rmax_N (N:nat) : R := + match N with + | O => Un 0 + | S n => Rmax (Un (S n)) (Rmax_N n) + end. + +(*********) +Definition EUn r : Prop := exists i : nat, r = Un i. + +(*********) +Definition Un_cv (l:R) : Prop := + forall eps:R, + eps > 0 -> + exists N : nat, (forall n:nat, (n >= N)%nat -> R_dist (Un n) l < eps). + +(*********) +Definition Cauchy_crit : Prop := + forall eps:R, + eps > 0 -> + exists N : nat, + (forall n m:nat, + (n >= N)%nat -> (m >= N)%nat -> R_dist (Un n) (Un m) < eps). + +(*********) +Definition Un_growing : Prop := forall n:nat, Un n <= Un (S n). + +(*********) +Lemma EUn_noempty : exists r : R, EUn r. +unfold EUn in |- *; split with (Un 0); split with 0%nat; trivial. +Qed. + +(*********) +Lemma Un_in_EUn : forall n:nat, EUn (Un n). +intro; unfold EUn in |- *; split with n; trivial. +Qed. + +(*********) +Lemma Un_bound_imp : + forall x:R, (forall n:nat, Un n <= x) -> is_upper_bound EUn x. +intros; unfold is_upper_bound in |- *; intros; unfold EUn in H0; elim H0; + clear H0; intros; generalize (H x1); intro; rewrite <- H0 in H1; + trivial. +Qed. + +(*********) +Lemma growing_prop : + forall n m:nat, Un_growing -> (n >= m)%nat -> Un n >= Un m. +double induction n m; intros. +unfold Rge in |- *; right; trivial. +elimtype False; unfold ge in H1; generalize (le_Sn_O n0); intro; auto. +cut (n0 >= 0)%nat. +generalize H0; intros; unfold Un_growing in H0; + apply + (Rge_trans (Un (S n0)) (Un n0) (Un 0) (Rle_ge (Un n0) (Un (S n0)) (H0 n0)) + (H 0%nat H2 H3)). +elim n0; auto. +elim (lt_eq_lt_dec n1 n0); intro y. +elim y; clear y; intro y. +unfold ge in H2; generalize (le_not_lt n0 n1 (le_S_n n0 n1 H2)); intro; + elimtype False; auto. +rewrite y; unfold Rge in |- *; right; trivial. +unfold ge in H0; generalize (H0 (S n0) H1 (lt_le_S n0 n1 y)); intro; + unfold Un_growing in H1; + apply + (Rge_trans (Un (S n1)) (Un n1) (Un (S n0)) + (Rle_ge (Un n1) (Un (S n1)) (H1 n1)) H3). +Qed. + + +(* classical is needed: [not_all_not_ex] *) +(*********) +Lemma Un_cv_crit : Un_growing -> bound EUn -> exists l : R, Un_cv l. +unfold Un_growing, Un_cv in |- *; intros; + generalize (completeness_weak EUn H0 EUn_noempty); + intro; elim H1; clear H1; intros; split with x; intros; + unfold is_lub in H1; unfold bound in H0; unfold is_upper_bound in H0, H1; + elim H0; clear H0; intros; elim H1; clear H1; intros; + generalize (H3 x0 H0); intro; cut (forall n:nat, Un n <= x); + intro. +cut (exists N : nat, x - eps < Un N). +intro; elim H6; clear H6; intros; split with x1. +intros; unfold R_dist in |- *; apply (Rabs_def1 (Un n - x) eps). +unfold Rgt in H2; + apply (Rle_lt_trans (Un n - x) 0 eps (Rle_minus (Un n) x (H5 n)) H2). +fold Un_growing in H; generalize (growing_prop n x1 H H7); intro; + generalize + (Rlt_le_trans (x - eps) (Un x1) (Un n) H6 (Rge_le (Un n) (Un x1) H8)); + intro; generalize (Rplus_lt_compat_l (- x) (x - eps) (Un n) H9); + unfold Rminus in |- *; rewrite <- (Rplus_assoc (- x) x (- eps)); + rewrite (Rplus_comm (- x) (Un n)); fold (Un n - x) in |- *; + rewrite Rplus_opp_l; rewrite (let (H1, H2) := Rplus_ne (- eps) in H2); + trivial. +cut (~ (forall N:nat, x - eps >= Un N)). +intro; apply (not_all_not_ex nat (fun N:nat => x - eps < Un N)); red in |- *; + intro; red in H6; elim H6; clear H6; intro; + apply (Rnot_lt_ge (x - eps) (Un N) (H7 N)). +red in |- *; intro; cut (forall N:nat, Un N <= x - eps). +intro; generalize (Un_bound_imp (x - eps) H7); intro; + unfold is_upper_bound in H8; generalize (H3 (x - eps) H8); + intro; generalize (Rle_minus x (x - eps) H9); unfold Rminus in |- *; + rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r; + rewrite (let (H1, H2) := Rplus_ne (- - eps) in H2); + rewrite Ropp_involutive; intro; unfold Rgt in H2; + generalize (Rgt_not_le eps 0 H2); intro; auto. +intro; elim (H6 N); intro; unfold Rle in |- *. +left; unfold Rgt in H7; assumption. +right; auto. +apply (H1 (Un n) (Un_in_EUn n)). +Qed. + +(*********) +Lemma finite_greater : + forall N:nat, exists M : R, (forall n:nat, (n <= N)%nat -> Un n <= M). +intro; induction N as [| N HrecN]. +split with (Un 0); intros; rewrite (le_n_O_eq n H); + apply (Req_le (Un n) (Un n) (refl_equal (Un n))). +elim HrecN; clear HrecN; intros; split with (Rmax (Un (S N)) x); intros; + elim (Rmax_Rle (Un (S N)) x (Un n)); intros; clear H1; + inversion H0. +rewrite <- H1; rewrite <- H1 in H2; + apply + (H2 (or_introl (Un n <= x) (Req_le (Un n) (Un n) (refl_equal (Un n))))). +apply (H2 (or_intror (Un n <= Un (S N)) (H n H3))). +Qed. + +(*********) +Lemma cauchy_bound : Cauchy_crit -> bound EUn. +unfold Cauchy_crit, bound in |- *; intros; unfold is_upper_bound in |- *; + unfold Rgt in H; elim (H 1 Rlt_0_1); clear H; intros; + generalize (H x); intro; generalize (le_dec x); intro; + elim (finite_greater x); intros; split with (Rmax x0 (Un x + 1)); + clear H; intros; unfold EUn in H; elim H; clear H; + intros; elim (H1 x2); clear H1; intro y. +unfold ge in H0; generalize (H0 x2 (le_n x) y); clear H0; intro; + rewrite <- H in H0; unfold R_dist in H0; elim (Rabs_def2 (Un x - x1) 1 H0); + clear H0; intros; elim (Rmax_Rle x0 (Un x + 1) x1); + intros; apply H4; clear H3 H4; right; clear H H0 y; + apply (Rlt_le x1 (Un x + 1)); generalize (Rlt_minus (-1) (Un x - x1) H1); + clear H1; intro; apply (Rminus_lt x1 (Un x + 1)); + cut (-1 - (Un x - x1) = x1 - (Un x + 1)); + [ intro; rewrite H0 in H; assumption | ring ]. +generalize (H2 x2 y); clear H2 H0; intro; rewrite <- H in H0; + elim (Rmax_Rle x0 (Un x + 1) x1); intros; clear H1; + apply H2; left; assumption. +Qed. + +End sequence. + +(*****************************************************************) +(* Definition of Power Series and properties *) +(* *) +(*****************************************************************) + +Section Isequence. + +(*********) +Variable An : nat -> R. + +(*********) +Definition Pser (x l:R) : Prop := infinit_sum (fun n:nat => An n * x ^ n) l. + +End Isequence. + +Lemma GP_infinite : + forall x:R, Rabs x < 1 -> Pser (fun n:nat => 1) x (/ (1 - x)). +intros; unfold Pser in |- *; unfold infinit_sum in |- *; intros; + elim (Req_dec x 0). +intros; exists 0%nat; intros; rewrite H1; rewrite Rminus_0_r; rewrite Rinv_1; + cut (sum_f_R0 (fun n0:nat => 1 * 0 ^ n0) n = 1). +intros; rewrite H3; rewrite R_dist_eq; auto. +elim n; simpl in |- *. +ring. +intros; rewrite H3; ring. +intro; cut (0 < eps * (Rabs (1 - x) * Rabs (/ x))). +intro; elim (pow_lt_1_zero x H (eps * (Rabs (1 - x) * Rabs (/ x))) H2); + intro N; intros; exists N; intros; + cut + (sum_f_R0 (fun n0:nat => 1 * x ^ n0) n = sum_f_R0 (fun n0:nat => x ^ n0) n). +intros; rewrite H5; + apply + (Rmult_lt_reg_l (Rabs (1 - x)) + (R_dist (sum_f_R0 (fun n0:nat => x ^ n0) n) (/ (1 - x))) eps). +apply Rabs_pos_lt. +apply Rminus_eq_contra. +apply Rlt_dichotomy_converse. +right; unfold Rgt in |- *. +apply (Rle_lt_trans x (Rabs x) 1). +apply RRle_abs. +assumption. +unfold R_dist in |- *; rewrite <- Rabs_mult. +rewrite Rmult_minus_distr_l. +cut + ((1 - x) * sum_f_R0 (fun n0:nat => x ^ n0) n = + - (sum_f_R0 (fun n0:nat => x ^ n0) n * (x - 1))). +intro; rewrite H6. +rewrite GP_finite. +rewrite Rinv_r. +cut (- (x ^ (n + 1) - 1) - 1 = - x ^ (n + 1)). +intro; rewrite H7. +rewrite Rabs_Ropp; cut ((n + 1)%nat = S n); auto. +intro H8; rewrite H8; simpl in |- *; rewrite Rabs_mult; + apply + (Rlt_le_trans (Rabs x * Rabs (x ^ n)) + (Rabs x * (eps * (Rabs (1 - x) * Rabs (/ x)))) ( + Rabs (1 - x) * eps)). +apply Rmult_lt_compat_l. +apply Rabs_pos_lt. +assumption. +auto. +cut + (Rabs x * (eps * (Rabs (1 - x) * Rabs (/ x))) = + Rabs x * Rabs (/ x) * (eps * Rabs (1 - x))). +clear H8; intros; rewrite H8; rewrite <- Rabs_mult; rewrite Rinv_r. +rewrite Rabs_R1; cut (1 * (eps * Rabs (1 - x)) = Rabs (1 - x) * eps). +intros; rewrite H9; unfold Rle in |- *; right; reflexivity. +ring. +assumption. +ring. +ring. +ring. +apply Rminus_eq_contra. +apply Rlt_dichotomy_converse. +right; unfold Rgt in |- *. +apply (Rle_lt_trans x (Rabs x) 1). +apply RRle_abs. +assumption. +ring; ring. +elim n; simpl in |- *. +ring. +intros; rewrite H5. +ring. +apply Rmult_lt_0_compat. +auto. +apply Rmult_lt_0_compat. +apply Rabs_pos_lt. +apply Rminus_eq_contra. +apply Rlt_dichotomy_converse. +right; unfold Rgt in |- *. +apply (Rle_lt_trans x (Rabs x) 1). +apply RRle_abs. +assumption. +apply Rabs_pos_lt. +apply Rinv_neq_0_compat. +assumption. +Qed. |