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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Local Open Scope R_scope.
(*****************************************************************)
(** To define transcendental functions *)
(** and exponential function *)
(*****************************************************************)
(*********)
Lemma Alembert_exp :
Un_cv (fun n:nat => Rabs (/ INR (fact (S n)) * / / INR (fact n))) 0.
Proof.
unfold Un_cv; intros; destruct (Rgt_dec eps 1) as [Hgt|Hnotgt].
- split with 0%nat; intros; rewrite (simpl_fact n); unfold R_dist;
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 H2; 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) (not_eq_sym (O_S n)))) in H4;
rewrite (let (H1, H2) := Rmult_ne eps in H1) in H4;
assumption.
apply Rlt_minus; unfold Rgt in Hgt; 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; 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;
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) (not_eq_sym (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; 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 Hnotgt); clear Hnotgt; unfold Rle; 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
(not_eq_sym (Rlt_dichotomy_converse 0 eps (or_introl (0 > eps) H))))
; rewrite (let (H1, H2) := Rmult_ne (/ eps) in H1);
intro; fold (/ eps - 1 > 0); apply Rgt_minus;
unfold Rgt; assumption.
right; rewrite H0; rewrite Rinv_1; symmetry; apply Rminus_diag_eq; auto.
elim (archimed (/ eps - 1)); intros; clear H1; unfold Rgt in H0; apply Rlt_le;
assumption.
Qed.
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