diff options
Diffstat (limited to 'theories/Reals/Rseries.v')
| -rw-r--r-- | theories/Reals/Rseries.v | 44 |
1 files changed, 22 insertions, 22 deletions
diff --git a/theories/Reals/Rseries.v b/theories/Reals/Rseries.v index 479d381d..3c10725b 100644 --- a/theories/Reals/Rseries.v +++ b/theories/Reals/Rseries.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -9,7 +9,7 @@ Require Import Rbase. Require Import Rfunctions. Require Import Compare. -Open Local Scope R_scope. +Local Open Scope R_scope. Implicit Type r : R. @@ -54,20 +54,20 @@ Section sequence. (*********) Lemma EUn_noempty : exists r : R, EUn r. Proof. - unfold EUn in |- *; split with (Un 0); split with 0%nat; trivial. + unfold EUn; split with (Un 0); split with 0%nat; trivial. Qed. (*********) Lemma Un_in_EUn : forall n:nat, EUn (Un n). Proof. - intro; unfold EUn in |- *; split with n; trivial. + intro; unfold EUn; split with n; trivial. Qed. (*********) Lemma Un_bound_imp : forall x:R, (forall n:nat, Un n <= x) -> is_upper_bound EUn x. Proof. - intros; unfold is_upper_bound in |- *; intros; unfold EUn in H0; elim H0; + intros; unfold is_upper_bound; intros; unfold EUn in H0; elim H0; clear H0; intros; generalize (H x1); intro; rewrite <- H0 in H1; trivial. Qed. @@ -77,7 +77,7 @@ Section sequence. forall n m:nat, Un_growing -> (n >= m)%nat -> Un n >= Un m. Proof. double induction n m; intros. - unfold Rge in |- *; right; trivial. + unfold Rge; right; trivial. exfalso; unfold ge in H1; generalize (le_Sn_O n0); intro; auto. cut (n0 >= 0)%nat. generalize H0; intros; unfold Un_growing in H0; @@ -89,7 +89,7 @@ Section sequence. elim y; clear y; intro y. unfold ge in H2; generalize (le_not_lt n0 n1 (le_S_n n0 n1 H2)); intro; exfalso; auto. - rewrite y; unfold Rge in |- *; right; trivial. + rewrite y; unfold Rge; right; trivial. unfold ge in H0; generalize (H0 (S n0) H1 (lt_le_S n0 n1 y)); intro; unfold Un_growing in H1; apply @@ -182,13 +182,13 @@ Section sequence. assert (Hs0: forall n, sum n = 0). intros n. - specialize (Hm1 (sum n) (ex_intro _ _ (refl_equal _))). + specialize (Hm1 (sum n) (ex_intro _ _ (eq_refl _))). apply Rle_antisym with (2 := proj1 (Hsum n)). now rewrite <- Hm. assert (Hub: forall n, Un n <= l - eps). intros n. - generalize (refl_equal (sum (S n))). + generalize (eq_refl (sum (S n))). simpl sum at 1. rewrite 2!Hs0, Rplus_0_l. unfold test. @@ -238,7 +238,7 @@ Section sequence. rewrite (IHN H6), Rplus_0_l. unfold test. destruct Rle_lt_dec. - apply refl_equal. + apply eq_refl. now elim Rlt_not_le with (1 := r). destruct (le_or_lt N n) as [Hn|Hn]. @@ -272,20 +272,20 @@ Section sequence. Proof. 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))). + apply (Req_le (Un n) (Un n) (eq_refl (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))))). + (H2 (or_introl (Un n <= x) (Req_le (Un n) (Un n) (eq_refl (Un n))))). apply (H2 (or_intror (Un n <= Un (S N)) (H n H3))). Qed. (*********) Lemma cauchy_bound : Cauchy_crit -> bound EUn. Proof. - unfold Cauchy_crit, bound in |- *; intros; unfold is_upper_bound in |- *; + unfold Cauchy_crit, bound; intros; unfold is_upper_bound; 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)); @@ -324,12 +324,12 @@ End Isequence. Lemma GP_infinite : forall x:R, Rabs x < 1 -> Pser (fun n:nat => 1) x (/ (1 - x)). Proof. - intros; unfold Pser in |- *; unfold infinite_sum in |- *; intros; + intros; unfold Pser; unfold infinite_sum; 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 |- *. + elim n; simpl. ring. intros; rewrite H3; ring. intro; cut (0 < eps * (Rabs (1 - x) * Rabs (/ x))). @@ -344,11 +344,11 @@ Proof. apply Rabs_pos_lt. apply Rminus_eq_contra. apply Rlt_dichotomy_converse. - right; unfold Rgt in |- *. + right; unfold Rgt. apply (Rle_lt_trans x (Rabs x) 1). apply RRle_abs. assumption. - unfold R_dist in |- *; rewrite <- Rabs_mult. + unfold R_dist; rewrite <- Rabs_mult. rewrite Rmult_minus_distr_l. cut ((1 - x) * sum_f_R0 (fun n0:nat => x ^ n0) n = @@ -359,7 +359,7 @@ Proof. 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; + intro H8; rewrite H8; simpl; rewrite Rabs_mult; apply (Rlt_le_trans (Rabs x * Rabs (x ^ n)) (Rabs x * (eps * (Rabs (1 - x) * Rabs (/ x)))) ( @@ -373,7 +373,7 @@ Proof. 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. + intros; rewrite H9; unfold Rle; right; reflexivity. ring. assumption. ring. @@ -381,12 +381,12 @@ Proof. ring. apply Rminus_eq_contra. apply Rlt_dichotomy_converse. - right; unfold Rgt in |- *. + right; unfold Rgt. apply (Rle_lt_trans x (Rabs x) 1). apply RRle_abs. assumption. ring; ring. - elim n; simpl in |- *. + elim n; simpl. ring. intros; rewrite H5. ring. @@ -396,7 +396,7 @@ Proof. apply Rabs_pos_lt. apply Rminus_eq_contra. apply Rlt_dichotomy_converse. - right; unfold Rgt in |- *. + right; unfold Rgt. apply (Rle_lt_trans x (Rabs x) 1). apply RRle_abs. assumption. |