diff options
Diffstat (limited to 'theories/Reals/Rtrigo_fun.v')
| -rw-r--r-- | theories/Reals/Rtrigo_fun.v | 167 |
1 files changed, 80 insertions, 87 deletions
diff --git a/theories/Reals/Rtrigo_fun.v b/theories/Reals/Rtrigo_fun.v index eaf2121e..78ef847f 100644 --- a/theories/Reals/Rtrigo_fun.v +++ b/theories/Reals/Rtrigo_fun.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Rtrigo_fun.v 8691 2006-04-10 09:23:37Z msozeau $ i*) +(*i $Id: Rtrigo_fun.v 9245 2006-10-17 12:53:34Z notin $ i*) Require Import Rbase. Require Import Rfunctions. @@ -14,96 +14,89 @@ Require Import SeqSeries. Open Local Scope R_scope. (*****************************************************************) -(* To define transcendental functions *) -(* *) -(*****************************************************************) -(*****************************************************************) -(* For exponential function *) +(** To define transcendental functions *) +(** for exponential function *) (* *) (*****************************************************************) (*********) Lemma Alembert_exp : - Un_cv (fun n:nat => Rabs (/ INR (fact (S n)) * / / INR (fact n))) 0. -unfold Un_cv in |- *; intros; elim (Rgt_dec eps 1); intro. -split with 0%nat; intros; rewrite (simpl_fact n); unfold R_dist in |- *; - rewrite (Rminus_0_r (Rabs (/ INR (S n)))); - rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0). -intro; rewrite (Rabs_pos_eq (/ INR (S n))). -cut (/ eps - 1 < 0). -intro; generalize (Rlt_le_trans (/ eps - 1) 0 (INR n) H2 (pos_INR n)); - clear H2; intro; unfold Rminus in H2; - generalize (Rplus_lt_compat_l 1 (/ eps + -1) (INR n) H2); - replace (1 + (/ eps + -1)) with (/ eps); [ clear H2; intro | ring ]. -rewrite (Rplus_comm 1 (INR n)) in H2; rewrite <- (S_INR n) in H2; - generalize (Rmult_gt_0_compat (/ INR (S n)) eps H1 H); - intro; unfold Rgt in H3; - generalize (Rmult_lt_compat_l (/ INR (S n) * eps) (/ eps) (INR (S n)) H3 H2); - intro; rewrite (Rmult_assoc (/ INR (S n)) eps (/ eps)) in H4; - rewrite (Rinv_r eps (Rlt_dichotomy_converse eps 0 (or_intror (eps < 0) H))) - in H4; rewrite (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1) in H4; - rewrite (Rmult_comm (/ INR (S n))) in H4; - rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H4; - rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (sym_not_equal (O_S n)))) in H4; - rewrite (let (H1, H2) := Rmult_ne eps in H1) in H4; - assumption. -apply Rlt_minus; unfold Rgt in a; rewrite <- Rinv_1; - apply (Rinv_lt_contravar 1 eps); auto; - rewrite (let (H1, H2) := Rmult_ne eps in H2); unfold Rgt in H; - assumption. -unfold Rgt in H1; apply Rlt_le; assumption. -unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. + Un_cv (fun n:nat => Rabs (/ INR (fact (S n)) * / / INR (fact n))) 0. +Proof. + unfold Un_cv in |- *; intros; elim (Rgt_dec eps 1); intro. + split with 0%nat; intros; rewrite (simpl_fact n); unfold R_dist in |- *; + rewrite (Rminus_0_r (Rabs (/ INR (S n)))); + rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0). + intro; rewrite (Rabs_pos_eq (/ INR (S n))). + cut (/ eps - 1 < 0). + intro; generalize (Rlt_le_trans (/ eps - 1) 0 (INR n) H2 (pos_INR n)); + clear H2; intro; unfold Rminus in H2; + generalize (Rplus_lt_compat_l 1 (/ eps + -1) (INR n) H2); + replace (1 + (/ eps + -1)) with (/ eps); [ clear H2; intro | ring ]. + rewrite (Rplus_comm 1 (INR n)) in H2; rewrite <- (S_INR n) in H2; + generalize (Rmult_gt_0_compat (/ INR (S n)) eps H1 H); + intro; unfold Rgt in H3; + generalize (Rmult_lt_compat_l (/ INR (S n) * eps) (/ eps) (INR (S n)) H3 H2); + intro; rewrite (Rmult_assoc (/ INR (S n)) eps (/ eps)) in H4; + rewrite (Rinv_r eps (Rlt_dichotomy_converse eps 0 (or_intror (eps < 0) H))) + in H4; rewrite (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1) in H4; + rewrite (Rmult_comm (/ INR (S n))) in H4; + rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H4; + rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (sym_not_equal (O_S n)))) in H4; + rewrite (let (H1, H2) := Rmult_ne eps in H1) in H4; + assumption. + apply Rlt_minus; unfold Rgt in a; rewrite <- Rinv_1; + apply (Rinv_lt_contravar 1 eps); auto; + rewrite (let (H1, H2) := Rmult_ne eps in H2); unfold Rgt in H; + assumption. + unfold Rgt in H1; apply Rlt_le; assumption. + unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. (**) -cut (0 <= up (/ eps - 1))%Z. -intro; elim (IZN (up (/ eps - 1)) H0); intros; split with x; intros; - rewrite (simpl_fact n); unfold R_dist in |- *; - rewrite (Rminus_0_r (Rabs (/ INR (S n)))); - rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0). -intro; rewrite (Rabs_pos_eq (/ INR (S n))). -cut (/ eps - 1 < INR x). -intro ; - generalize - (Rlt_le_trans (/ eps - 1) (INR x) (INR n) H4 - (le_INR x n H2)); - clear H4; intro; unfold Rminus in H4; - generalize (Rplus_lt_compat_l 1 (/ eps + -1) (INR n) H4); - replace (1 + (/ eps + -1)) with (/ eps); [ clear H4; intro | ring ]. -rewrite (Rplus_comm 1 (INR n)) in H4; rewrite <- (S_INR n) in H4; - generalize (Rmult_gt_0_compat (/ INR (S n)) eps H3 H); - intro; unfold Rgt in H5; - generalize (Rmult_lt_compat_l (/ INR (S n) * eps) (/ eps) (INR (S n)) H5 H4); - intro; rewrite (Rmult_assoc (/ INR (S n)) eps (/ eps)) in H6; - rewrite (Rinv_r eps (Rlt_dichotomy_converse eps 0 (or_intror (eps < 0) H))) - in H6; rewrite (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1) in H6; - rewrite (Rmult_comm (/ INR (S n))) in H6; - rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H6; - rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (sym_not_equal (O_S n)))) in H6; - rewrite (let (H1, H2) := Rmult_ne eps in H1) in H6; - assumption. -cut (IZR (up (/ eps - 1)) = IZR (Z_of_nat x)); - [ intro | rewrite H1; trivial ]. -elim (archimed (/ eps - 1)); intros; clear H6; unfold Rgt in H5; - rewrite H4 in H5; rewrite INR_IZR_INZ; assumption. -unfold Rgt in H1; apply Rlt_le; assumption. -unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. -apply (le_O_IZR (up (/ eps - 1))); - apply (Rle_trans 0 (/ eps - 1) (IZR (up (/ eps - 1)))). -generalize (Rnot_gt_le eps 1 b); clear b; unfold Rle in |- *; intro; elim H0; - clear H0; intro. -left; unfold Rgt in H; - generalize (Rmult_lt_compat_l (/ eps) eps 1 (Rinv_0_lt_compat eps H) H0); - rewrite - (Rinv_l eps - (sym_not_eq (Rlt_dichotomy_converse 0 eps (or_introl (0 > eps) H)))) - ; rewrite (let (H1, H2) := Rmult_ne (/ eps) in H1); - intro; fold (/ eps - 1 > 0) in |- *; apply Rgt_minus; - unfold Rgt in |- *; assumption. -right; rewrite H0; rewrite Rinv_1; apply sym_eq; apply Rminus_diag_eq; auto. -elim (archimed (/ eps - 1)); intros; clear H1; unfold Rgt in H0; apply Rlt_le; - assumption. + cut (0 <= up (/ eps - 1))%Z. + intro; elim (IZN (up (/ eps - 1)) H0); intros; split with x; intros; + rewrite (simpl_fact n); unfold R_dist in |- *; + rewrite (Rminus_0_r (Rabs (/ INR (S n)))); + rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0). + intro; rewrite (Rabs_pos_eq (/ INR (S n))). + cut (/ eps - 1 < INR x). + intro ; + generalize + (Rlt_le_trans (/ eps - 1) (INR x) (INR n) H4 + (le_INR x n H2)); + clear H4; intro; unfold Rminus in H4; + generalize (Rplus_lt_compat_l 1 (/ eps + -1) (INR n) H4); + replace (1 + (/ eps + -1)) with (/ eps); [ clear H4; intro | ring ]. + rewrite (Rplus_comm 1 (INR n)) in H4; rewrite <- (S_INR n) in H4; + generalize (Rmult_gt_0_compat (/ INR (S n)) eps H3 H); + intro; unfold Rgt in H5; + generalize (Rmult_lt_compat_l (/ INR (S n) * eps) (/ eps) (INR (S n)) H5 H4); + intro; rewrite (Rmult_assoc (/ INR (S n)) eps (/ eps)) in H6; + rewrite (Rinv_r eps (Rlt_dichotomy_converse eps 0 (or_intror (eps < 0) H))) + in H6; rewrite (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1) in H6; + rewrite (Rmult_comm (/ INR (S n))) in H6; + rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H6; + rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (sym_not_equal (O_S n)))) in H6; + rewrite (let (H1, H2) := Rmult_ne eps in H1) in H6; + assumption. + cut (IZR (up (/ eps - 1)) = IZR (Z_of_nat x)); + [ intro | rewrite H1; trivial ]. + elim (archimed (/ eps - 1)); intros; clear H6; unfold Rgt in H5; + rewrite H4 in H5; rewrite INR_IZR_INZ; assumption. + unfold Rgt in H1; apply Rlt_le; assumption. + unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. + apply (le_O_IZR (up (/ eps - 1))); + apply (Rle_trans 0 (/ eps - 1) (IZR (up (/ eps - 1)))). + generalize (Rnot_gt_le eps 1 b); clear b; unfold Rle in |- *; intro; elim H0; + clear H0; intro. + left; unfold Rgt in H; + generalize (Rmult_lt_compat_l (/ eps) eps 1 (Rinv_0_lt_compat eps H) H0); + rewrite + (Rinv_l eps + (sym_not_eq (Rlt_dichotomy_converse 0 eps (or_introl (0 > eps) H)))) + ; rewrite (let (H1, H2) := Rmult_ne (/ eps) in H1); + intro; fold (/ eps - 1 > 0) in |- *; apply Rgt_minus; + unfold Rgt in |- *; assumption. + right; rewrite H0; rewrite Rinv_1; apply sym_eq; apply Rminus_diag_eq; auto. + elim (archimed (/ eps - 1)); intros; clear H1; unfold Rgt in H0; apply Rlt_le; + assumption. Qed. - - - - - |