From f56b37a990555c085f324fbe12da56b2e3a0096c Mon Sep 17 00:00:00 2001
From: desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7>
Date: Tue, 25 Jun 2002 13:27:35 +0000
Subject: Integration de Rcomplet et Alembert_compl

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2806 85f007b7-540e-0410-9357-904b9bb8a0f7
---
 theories/Reals/Alembert_compl.v | 320 ++++++++++++++++++++++++++++++++++++++++
 1 file changed, 320 insertions(+)
 create mode 100644 theories/Reals/Alembert_compl.v

(limited to 'theories/Reals/Alembert_compl.v')

diff --git a/theories/Reals/Alembert_compl.v b/theories/Reals/Alembert_compl.v
new file mode 100644
index 000000000..a0daa7dd7
--- /dev/null
+++ b/theories/Reals/Alembert_compl.v
@@ -0,0 +1,320 @@
+(***********************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
+(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
+(*   \VV/  *************************************************************)
+(*    //   *      This file is distributed under the terms of the      *)
+(*         *       GNU Lesser General Public License Version 2.1       *)
+(***********************************************************************)
+ 
+(*i $Id$ i*)
+
+Require Max.
+Require Raxioms.
+Require DiscrR.
+Require Rbase.
+Require Rseries.
+Require Rcomplet.
+Require Alembert.
+
+(* Le critère de Cauchy pour les séries *)
+Definition Cauchy_crit_series [An:nat->R] : Prop := (Cauchy_crit [N:nat](sum_f_R0 An N)).
+
+(**********)
+Lemma sum_growing : (An,Bn:nat->R;N:nat) ((n:nat)``(An n)<=(Bn n)``)->``(sum_f_R0 An N)<=(sum_f_R0 Bn N)``.
+Intros.
+Induction N.
+Simpl; Apply H.
+Do 2 Rewrite tech5.
+Apply Rle_trans with ``(sum_f_R0 An N)+(Bn (S N))``.
+Apply Rle_compatibility; Apply H.
+Do 2 Rewrite <- (Rplus_sym (Bn (S N))).
+Apply Rle_compatibility; Apply HrecN.
+Qed.
+
+(**********)
+Lemma Rabsolu_triang_gen : (An:nat->R;N:nat) (Rle (Rabsolu (sum_f_R0 An N)) (sum_f_R0 [i:nat](Rabsolu (An i)) N)). 
+Intros.
+Induction N.
+Simpl.
+Right; Reflexivity.
+Do 2 Rewrite tech5.
+Apply Rle_trans with ``(Rabsolu ((sum_f_R0 An N)))+(Rabsolu (An (S N)))``.
+Apply Rabsolu_triang.
+Do 2 Rewrite <- (Rplus_sym (Rabsolu (An (S N)))).
+Apply Rle_compatibility; Apply HrecN.
+Qed.
+
+(**********)
+Lemma cond_pos_sum : (An:nat->R;N:nat) ((n:nat)``0<=(An n)``) -> ``0<=(sum_f_R0 An N)``.
+Intros.
+Induction N.
+Simpl; Apply H.
+Rewrite tech5.
+Apply ge0_plus_ge0_is_ge0.
+Apply HrecN.
+Apply H.
+Qed.
+
+(* Si (|An|) verifie le critere de Cauchy pour les séries , alors (An) aussi *)
+Lemma cauchy_abs : (An:nat->R) (Cauchy_crit_series [i:nat](Rabsolu (An i))) -> (Cauchy_crit_series An).
+Unfold Cauchy_crit_series; Unfold Cauchy_crit.
+Intros.
+Elim (H eps H0); Intros.
+Exists x.
+Intros.
+Cut (Rle (R_dist (sum_f_R0 An n) (sum_f_R0 An m)) (R_dist (sum_f_R0 [i:nat](Rabsolu (An i)) n) (sum_f_R0 [i:nat](Rabsolu (An i)) m))).
+Intro.
+Apply Rle_lt_trans with (R_dist (sum_f_R0 [i:nat](Rabsolu (An i)) n) (sum_f_R0 [i:nat](Rabsolu (An i)) m)).
+Assumption.
+Apply H1; Assumption.
+Assert H4 := (lt_eq_lt_dec n m).
+Elim H4; Intro.
+Elim a; Intro.
+Rewrite (tech2 An n m); [Idtac | Assumption].
+Rewrite (tech2 [i:nat](Rabsolu (An i)) n m); [Idtac | Assumption].
+Unfold R_dist.
+Unfold Rminus.
+Do 2 Rewrite Ropp_distr1.
+Do 2 Rewrite <- Rplus_assoc.
+Do 2 Rewrite Rplus_Ropp_r.
+Do 2 Rewrite Rplus_Ol.
+Do 2 Rewrite Rabsolu_Ropp.
+Rewrite (Rabsolu_right (sum_f_R0 [i:nat](Rabsolu (An (plus (S n) i))) (minus m (S n)))).
+Pose Bn:=[i:nat](An (plus (S n) i)).
+Replace [i:nat](Rabsolu (An (plus (S n) i))) with [i:nat](Rabsolu (Bn i)).
+Apply Rabsolu_triang_gen.
+Unfold Bn; Reflexivity.
+Apply Rle_sym1.
+Apply cond_pos_sum.
+Intro; Apply Rabsolu_pos.
+Rewrite b.
+Unfold R_dist.
+Unfold Rminus; Do 2 Rewrite Rplus_Ropp_r.
+Rewrite Rabsolu_R0; Right; Reflexivity.
+Rewrite (tech2 An m n); [Idtac | Assumption].
+Rewrite (tech2 [i:nat](Rabsolu (An i)) m n); [Idtac | Assumption].
+Unfold R_dist.
+Unfold Rminus.
+Do 2 Rewrite Rplus_assoc.
+Rewrite (Rplus_sym (sum_f_R0 An m)).
+Rewrite (Rplus_sym (sum_f_R0 [i:nat](Rabsolu (An i)) m)).
+Do 2 Rewrite Rplus_assoc.
+Do 2 Rewrite Rplus_Ropp_l.
+Do 2 Rewrite Rplus_Or.
+Rewrite (Rabsolu_right (sum_f_R0 [i:nat](Rabsolu (An (plus (S m) i))) (minus n (S m)))).
+Pose Bn:=[i:nat](An (plus (S m) i)).
+Replace [i:nat](Rabsolu (An (plus (S m) i))) with [i:nat](Rabsolu (Bn i)).
+Apply Rabsolu_triang_gen.
+Unfold Bn; Reflexivity.
+Apply Rle_sym1.
+Apply cond_pos_sum.
+Intro; Apply Rabsolu_pos.
+Qed.
+
+(**********)
+Lemma cv_cauchy_1 : (An:nat->R) (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)) -> (Cauchy_crit_series An).
+Intros.
+Elim X; Intros.
+Unfold Un_cv in p.
+Unfold Cauchy_crit_series; Unfold Cauchy_crit.
+Intros.
+Cut ``0<eps/2``.
+Intro.
+Elim (p ``eps/2`` H0); Intros.
+Exists x0.
+Intros.
+Apply Rle_lt_trans with ``(R_dist (sum_f_R0 An n) x)+(R_dist (sum_f_R0 An m) x)``.
+Unfold R_dist.
+Replace ``(sum_f_R0 An n)-(sum_f_R0 An m)`` with ``((sum_f_R0 An n)-x)+ -((sum_f_R0 An m)-x)``; [Idtac | Ring].
+Rewrite <- (Rabsolu_Ropp ``(sum_f_R0 An m)-x``).
+Apply Rabsolu_triang.
+Apply Rlt_le_trans with ``eps/2+eps/2``.
+Apply Rplus_lt.
+Apply H1; Assumption.
+Apply H1; Assumption.
+Right; Symmetry; Apply double_var.
+Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_2_0].
+Qed.
+
+Lemma cv_cauchy_2 : (An:nat->R) (Cauchy_crit_series An) -> (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)).
+Intros.
+Apply R_complet.
+Unfold Cauchy_crit_series in H.
+Exact H.
+Qed.
+
+Lemma Alembert_strong_general : (An:nat->R;k:R) ``0<=k<1`` -> ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)).
+Intros.
+Cut (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)).
+Intro Hyp0; Apply Hyp0.
+Apply cv_cauchy_2.
+Apply cauchy_abs.
+Apply cv_cauchy_1.
+Cut (SigT R [l:R](Un_cv [N:nat](sum_f_R0 [i:nat](Rabsolu (An i)) N) l)) -> (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 [i:nat](Rabsolu (An i)) N) l)).
+Intro Hyp; Apply Hyp.
+Apply (Alembert_strong_positive [i:nat](Rabsolu (An i)) k).
+Assumption.
+Intro; Apply Rabsolu_pos_lt; Apply H0.
+Unfold Un_cv.
+Unfold Un_cv in H1.
+Unfold Rdiv.
+Intros.
+Elim (H1 eps H2); Intros.
+Exists x; Intros.
+Rewrite <- Rabsolu_Rinv.
+Rewrite <- Rabsolu_mult.
+Rewrite Rabsolu_Rabsolu.
+Unfold Rdiv in H3; Apply H3; Assumption.
+Apply H0.
+Intro.
+Elim X; Intros.
+Apply existTT with x.
+Assumption.
+Intro.
+Elim X; Intros.
+Apply Specif.existT with x.
+Assumption.
+Qed.
+
+(* Convergence de la SE dans le disque D(O,1/k) *)
+(* le cas k=0 est decrit par le theoreme "Alembert" *)
+Lemma Alembert_strong : (An:nat->R;x,k:R) ``0<k`` -> ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> ``(Rabsolu x)</k`` -> (SigT R [l:R](Pser An x l)).
+Intros.
+Cut (SigT R [l:R](Un_cv [N:nat](sum_f_R0 [i:nat]``(An i)*(pow x i)`` N) l)).
+Intro.
+Elim X; Intros.
+Apply Specif.existT with x0.
+Apply tech12; Assumption.
+Case (total_order_T x R0); Intro.
+Elim s; Intro.
+EApply Alembert_strong_general with ``k*(Rabsolu x)``.
+Split.
+Unfold Rdiv; Apply Rmult_le_pos.
+Left; Assumption.
+Left; Apply Rabsolu_pos_lt.
+Red; Intro; Rewrite H3 in a; Elim (Rlt_antirefl ? a).
+Apply Rlt_monotony_contra with ``/k``.
+Apply Rlt_Rinv; Assumption.
+Rewrite <- Rmult_assoc.
+Rewrite <- Rinv_l_sym.
+Rewrite Rmult_1l.
+Rewrite Rmult_1r; Assumption.
+Red; Intro; Rewrite H3 in H; Elim (Rlt_antirefl ? H).
+Intro; Apply prod_neq_R0.
+Apply H0.
+Apply pow_nonzero.
+Red; Intro; Rewrite H3 in a; Elim (Rlt_antirefl ? a).
+Unfold Un_cv; Unfold Un_cv in H1.
+Intros.
+Cut ``0<eps/(Rabsolu x)``.
+Intro.
+Elim (H1 ``eps/(Rabsolu x)`` H4); Intros.
+Exists x0.
+Intros.
+Replace ``((An (S n))*(pow x (S n)))/((An n)*(pow x n))`` with ``(An (S n))/(An n)*x``.
+Unfold R_dist.
+Rewrite Rabsolu_mult.
+Replace ``(Rabsolu ((An (S n))/(An n)))*(Rabsolu x)-k*(Rabsolu x)`` with ``(Rabsolu x)*((Rabsolu ((An (S n))/(An n)))-k)``; [Idtac | Ring].
+Rewrite Rabsolu_mult.
+Rewrite Rabsolu_Rabsolu.
+Apply Rlt_monotony_contra with ``/(Rabsolu x)``.
+Apply Rlt_Rinv; Apply Rabsolu_pos_lt.
+Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a).
+Rewrite <- Rmult_assoc.
+Rewrite <- Rinv_l_sym.
+Rewrite Rmult_1l.
+Rewrite <- (Rmult_sym eps).
+Unfold R_dist in H5.
+Unfold Rdiv; Unfold Rdiv in H5; Apply H5; Assumption.
+Apply Rabsolu_no_R0.
+Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a).
+Unfold Rdiv; Replace (S n) with (plus n (1)); [Idtac | Ring].
+Rewrite pow_add.
+Simpl.
+Rewrite Rmult_1r.
+Rewrite Rinv_Rmult.
+Replace ``(An (plus n (S O)))*((pow x n)*x)*(/(An n)*/(pow x n))`` with ``(An (plus n (S O)))*/(An n)*x*((pow x n)*/(pow x n))``; [Idtac | Ring].
+Rewrite <- Rinv_r_sym.
+Rewrite Rmult_1r; Reflexivity.
+Apply pow_nonzero.
+Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a).
+Apply H0.
+Apply pow_nonzero.
+Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a).
+Unfold Rdiv; Apply Rmult_lt_pos.
+Assumption.
+Apply Rlt_Rinv; Apply Rabsolu_pos_lt.
+Red; Intro H7; Rewrite H7 in a; Elim (Rlt_antirefl ? a).
+Apply Specif.existT with (An O).
+Unfold Un_cv.
+Intros.
+Exists O.
+Intros.
+Unfold R_dist.
+Replace (sum_f_R0 [i:nat]``(An i)*(pow x i)`` n) with (An O).
+Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption.
+Induction n.
+Simpl; Ring.
+Rewrite tech5.
+Rewrite <- Hrecn.
+Rewrite b; Simpl; Ring.
+Unfold ge; Apply le_O_n.
+EApply Alembert_strong_general with ``k*(Rabsolu x)``.
+Split.
+Unfold Rdiv; Apply Rmult_le_pos.
+Left; Assumption.
+Left; Apply Rabsolu_pos_lt.
+Red; Intro; Rewrite H3 in r; Elim (Rlt_antirefl ? r).
+Apply Rlt_monotony_contra with ``/k``.
+Apply Rlt_Rinv; Assumption.
+Rewrite <- Rmult_assoc.
+Rewrite <- Rinv_l_sym.
+Rewrite Rmult_1l.
+Rewrite Rmult_1r; Assumption.
+Red; Intro; Rewrite H3 in H; Elim (Rlt_antirefl ? H).
+Intro; Apply prod_neq_R0.
+Apply H0.
+Apply pow_nonzero.
+Red; Intro; Rewrite H3 in r; Elim (Rlt_antirefl ? r).
+Unfold Un_cv; Unfold Un_cv in H1.
+Intros.
+Cut ``0<eps/(Rabsolu x)``.
+Intro.
+Elim (H1 ``eps/(Rabsolu x)`` H4); Intros.
+Exists x0.
+Intros.
+Replace ``((An (S n))*(pow x (S n)))/((An n)*(pow x n))`` with ``(An (S n))/(An n)*x``.
+Unfold R_dist.
+Rewrite Rabsolu_mult.
+Replace ``(Rabsolu ((An (S n))/(An n)))*(Rabsolu x)-k*(Rabsolu x)`` with ``(Rabsolu x)*((Rabsolu ((An (S n))/(An n)))-k)``; [Idtac | Ring].
+Rewrite Rabsolu_mult.
+Rewrite Rabsolu_Rabsolu.
+Apply Rlt_monotony_contra with ``/(Rabsolu x)``.
+Apply Rlt_Rinv; Apply Rabsolu_pos_lt.
+Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r).
+Rewrite <- Rmult_assoc.
+Rewrite <- Rinv_l_sym.
+Rewrite Rmult_1l.
+Rewrite <- (Rmult_sym eps).
+Unfold R_dist in H5.
+Unfold Rdiv; Unfold Rdiv in H5; Apply H5; Assumption.
+Apply Rabsolu_no_R0.
+Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r).
+Unfold Rdiv; Replace (S n) with (plus n (1)); [Idtac | Ring].
+Rewrite pow_add.
+Simpl.
+Rewrite Rmult_1r.
+Rewrite Rinv_Rmult.
+Replace ``(An (plus n (S O)))*((pow x n)*x)*(/(An n)*/(pow x n))`` with ``(An (plus n (S O)))*/(An n)*x*((pow x n)*/(pow x n))``; [Idtac | Ring].
+Rewrite <- Rinv_r_sym.
+Rewrite Rmult_1r; Reflexivity.
+Apply pow_nonzero.
+Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r).
+Apply H0.
+Apply pow_nonzero.
+Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r).
+Unfold Rdiv; Apply Rmult_lt_pos.
+Assumption.
+Apply Rlt_Rinv; Apply Rabsolu_pos_lt.
+Red; Intro H7; Rewrite H7 in r; Elim (Rlt_antirefl ? r).
+Qed.
\ No newline at end of file
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