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Diffstat (limited to 'theories/Reals/Rtrigo_fun.v')
| -rw-r--r-- | theories/Reals/Rtrigo_fun.v | 109 |
1 files changed, 109 insertions, 0 deletions
diff --git a/theories/Reals/Rtrigo_fun.v b/theories/Reals/Rtrigo_fun.v new file mode 100644 index 00000000..b0f29e5c --- /dev/null +++ b/theories/Reals/Rtrigo_fun.v @@ -0,0 +1,109 @@ +(************************************************************************) +(* 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: Rtrigo_fun.v,v 1.7.2.1 2004/07/16 19:31:15 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Open Local Scope R_scope. + +(*****************************************************************) +(* 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. +(**) +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 ((fun (n m:nat) (H:(m >= n)%nat) => H) 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. + + + + + |