Réorganisation de la librairie des réels

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@3311 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 3 insertions, 329 deletions

diff --git a/theories/Reals/PSeries_reg.v b/theories/Reals/PSeries_reg.v
index 743e1184d..d056387f4 100644
--- a/theories/Reals/PSeries_reg.v
+++ b/theories/Reals/PSeries_reg.v
@@ -8,18 +8,9 @@
  
 (*i $Id$ i*)
 
-Require Rbase.
-Require DiscrR.
+Require RealsB.
 Require Rfunctions.
-Require Rseries.
-Require Rsigma.
-Require Alembert.
-Require Alembert_compl.
-Require Binome.
-Require Cv_prop.
-Require Rcomplet.
-Require Rtrigo_alt.
-Require Cos_plus.
+Require SeqSeries.
 Require Ranalysis1.
 Require Max.
 Require Even.
@@ -32,286 +23,8 @@ Definition CVN_r [fn:nat->R->R;r:posreal] : Type := (SigT ? [An:nat->R](sigTT R
 
 Definition CVN_R [fn:nat->R->R] : Type := (r:posreal) (CVN_r fn r).
 
-Definition SP [fn:nat->R->R;N:nat] : R->R := [x:R](sum_f_R0 [k:nat]``(fn k x)`` N).
-
 Definition SFL [fn:nat->R->R;cv:(x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l))] : R-> R := [y:R](Cases (cv y) of (existTT a b) => a end).
 
-(**********)
-Lemma sum_incr : (An:nat->R;N:nat;l:R) (Un_cv [n:nat](sum_f_R0 An n) l) -> ((n:nat)``0<=(An n)``) -> ``(sum_f_R0 An N)<=l``.
-Intros; Case (total_order_T (sum_f_R0 An N) l); Intro.
-Elim s; Intro.
-Left; Apply a.
-Right; Apply b.
-Cut (Un_growing [n:nat](sum_f_R0 An n)).
-Intro; Pose l1 := (sum_f_R0 An N).
-Fold l1 in r.
-Unfold Un_cv in H; Cut ``0<l1-l``.
-Intro; Elim (H ? H2); Intros.
-Pose N0 := (max x N); Cut (ge N0 x).
-Intro; Assert H5 := (H3 N0 H4).
-Cut ``l1<=(sum_f_R0 An N0)``.
-Intro; Unfold R_dist in H5; Rewrite Rabsolu_right in H5.
-Cut ``(sum_f_R0 An N0)<l1``.
-Intro; Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H7 H6)).
-Apply Rlt_anti_compatibility with ``-l``.
-Do 2 Rewrite (Rplus_sym ``-l``).
-Apply H5.
-Apply Rle_sym1; Apply Rle_anti_compatibility with l.
-Rewrite Rplus_Or; Replace ``l+((sum_f_R0 An N0)-l)`` with (sum_f_R0 An N0); [Idtac | Ring]; Apply Rle_trans with l1.
-Left; Apply r.
-Apply H6.
-Unfold l1; Apply Rle_sym2; Apply (growing_prop [k:nat](sum_f_R0 An k)).
-Apply H1.
-Unfold ge N0; Apply le_max_r.
-Unfold ge N0; Apply le_max_l.
-Apply Rlt_anti_compatibility with l; Rewrite Rplus_Or; Replace ``l+(l1-l)`` with l1; [Apply r | Ring].
-Unfold Un_growing; Intro; Simpl; Pattern 1 (sum_f_R0 An n); Rewrite <- Rplus_Or; Apply Rle_compatibility; Apply H0.
-Qed.
-
-(**********)
-Lemma sum_cv_maj : (An:nat->R;fn:nat->R->R;x,l1,l2:R) (Un_cv [n:nat](SP fn n x) l1) -> (Un_cv [n:nat](sum_f_R0 An n) l2) -> ((n:nat)``(Rabsolu (fn n x))<=(An n)``) -> ``(Rabsolu l1)<=l2``.
-Intros; Case (total_order_T (Rabsolu l1) l2); Intro.
-Elim s; Intro.
-Left; Apply a.
-Right; Apply b.
-Cut (n0:nat)``(Rabsolu (SP fn n0 x))<=(sum_f_R0 An n0)``.
-Intro; Cut ``0<((Rabsolu l1)-l2)/2``.
-Intro; Unfold Un_cv in H H0.
-Elim (H ? H3); Intros Na H4.
-Elim (H0 ? H3); Intros Nb H5.
-Pose N := (max Na Nb).
-Unfold R_dist in H4 H5.
-Cut ``(Rabsolu ((sum_f_R0 An N)-l2))<((Rabsolu l1)-l2)/2``.
-Intro; Cut ``(Rabsolu ((Rabsolu l1)-(Rabsolu (SP fn N x))))<((Rabsolu l1)-l2)/2``.
-Intro; Cut ``(sum_f_R0 An N)<((Rabsolu l1)+l2)/2``.
-Intro; Cut ``((Rabsolu l1)+l2)/2<(Rabsolu (SP fn N x))``.
-Intro; Cut ``(sum_f_R0 An N)<(Rabsolu (SP fn N x))``.
-Intro; Assert H11 := (H2 N).
-Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H11 H10)).
-Apply Rlt_trans with ``((Rabsolu l1)+l2)/2``; Assumption.
-Case (case_Rabsolu ``(Rabsolu l1)-(Rabsolu (SP fn N x))``); Intro.
-Apply Rlt_trans with (Rabsolu l1).
-Apply Rlt_monotony_contra with ``2``.
-Apply Rgt_2_0.
-Unfold Rdiv; Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym.
-Rewrite Rmult_1r; Rewrite double; Apply Rlt_compatibility; Apply r.
-DiscrR.
-Apply (Rminus_lt ? ? r0).
-Rewrite (Rabsolu_right ? r0) in H7.
-Apply Rlt_anti_compatibility with ``((Rabsolu l1)-l2)/2-(Rabsolu (SP fn N x))``.
-Replace ``((Rabsolu l1)-l2)/2-(Rabsolu (SP fn N x))+((Rabsolu l1)+l2)/2`` with ``(Rabsolu l1)-(Rabsolu (SP fn N x))``.
-Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply H7.
-Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Rewrite <- (Rmult_sym ``/2``); Rewrite Rminus_distr; Repeat Rewrite (Rmult_sym ``/2``); Pattern 1 (Rabsolu l1); Rewrite double_var; Unfold Rdiv; Ring.
-Case (case_Rabsolu ``(sum_f_R0 An N)-l2``); Intro.
-Apply Rlt_trans with l2.
-Apply (Rminus_lt ? ? r0).
-Apply Rlt_monotony_contra with ``2``.
-Apply Rgt_2_0.
-Rewrite (double l2); Unfold Rdiv; Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym.
-Rewrite Rmult_1r; Rewrite (Rplus_sym (Rabsolu l1)); Apply Rlt_compatibility; Apply r.
-DiscrR.
-Rewrite (Rabsolu_right ? r0) in H6; Apply Rlt_anti_compatibility with ``-l2``.
-Replace ``-l2+((Rabsolu l1)+l2)/2`` with ``((Rabsolu l1)-l2)/2``.
-Rewrite Rplus_sym; Apply H6.
-Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite Rminus_distr; Rewrite Rmult_Rplus_distrl; Pattern 2 l2; Rewrite double_var; Repeat Rewrite (Rmult_sym ``/2``); Rewrite Ropp_distr1; Unfold Rdiv; Ring.
-Apply Rle_lt_trans with ``(Rabsolu ((SP fn N x)-l1))``.
-Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr3; Apply Rabsolu_triang_inv2.
-Apply H4; Unfold ge N; Apply le_max_l.
-Apply H5; Unfold ge N; Apply le_max_r.
-Unfold Rdiv; Apply Rmult_lt_pos.
-Apply Rlt_anti_compatibility with l2.
-Rewrite Rplus_Or; Replace ``l2+((Rabsolu l1)-l2)`` with (Rabsolu l1); [Apply r | Ring].
-Apply Rlt_Rinv; Apply Rgt_2_0.
-Intros; Induction n0.
-Unfold SP; Simpl; Apply H1.
-Unfold SP; Simpl.
-Apply Rle_trans with (Rplus (Rabsolu (sum_f_R0 [k:nat](fn k x) n0)) (Rabsolu (fn (S n0) x))).
-Apply Rabsolu_triang.
-Apply Rle_trans with ``(sum_f_R0 An n0)+(Rabsolu (fn (S n0) x))``.
-Do 2 Rewrite <- (Rplus_sym (Rabsolu (fn (S n0) x))).
-Apply Rle_compatibility; Apply Hrecn0.
-Apply Rle_compatibility; Apply H1.
-Qed.
-
-(**********)
-Lemma sum_maj1 : (fn:nat->R->R;An:nat->R;x,l1,l2:R;N:nat) (Un_cv [n:nat](SP fn n x) l1) -> (Un_cv [n:nat](sum_f_R0 An n) l2) -> ((n:nat)``(Rabsolu (fn n x))<=(An n)``) -> ``(Rabsolu (l1-(SP fn N x)))<=l2-(sum_f_R0 An N)``.
-Intros; Cut (sigTT R [l:R](Un_cv [n:nat](sum_f_R0 [l:nat](fn (plus (S N) l) x) n) l)).
-Intro; Cut (sigTT R [l:R](Un_cv [n:nat](sum_f_R0 [l:nat](An (plus (S N) l)) n) l)).
-Intro; Elim X; Intros l1N H2.
-Elim X0; Intros l2N H3.
-Cut ``l1-(SP fn N x)==l1N``.
-Intro; Cut ``l2-(sum_f_R0 An N)==l2N``.
-Intro; Rewrite H4; Rewrite H5.
-Apply sum_cv_maj with [l:nat](An (plus (S N) l)) [l:nat][x:R](fn (plus (S N) l) x) x.
-Unfold SP; Apply H2.
-Apply H3.
-Intros; Apply H1.
-Symmetry; EApply UL_suite.
-Apply H3.
-Unfold Un_cv in H0; Unfold Un_cv; Intros; Elim (H0 eps H5); Intros N0 H6.
-Unfold R_dist in H6; Exists N0; Intros.
-Unfold R_dist; Replace (Rminus (sum_f_R0 [l:nat](An (plus (S N) l)) n) (Rminus l2 (sum_f_R0 An N))) with (Rminus (Rplus (sum_f_R0 An N) (sum_f_R0 [l:nat](An (plus (S N) l)) n)) l2); [Idtac | Ring].
-Replace (Rplus (sum_f_R0 An N) (sum_f_R0 [l:nat](An (plus (S N) l)) n)) with (sum_f_R0 An (S (plus N n))).
-Apply H6; Unfold ge; Apply le_trans with n.
-Apply H7.
-Apply le_trans with (plus N n).
-Apply le_plus_r.
-Apply le_n_Sn.
-Cut (le O N).
-Cut (lt N (S (plus N n))).
-Intros; Assert H10 := (sigma_split An H9 H8).
-Unfold sigma in H10.
-Do 2 Rewrite <- minus_n_O in H10.
-Replace (sum_f_R0 An (S (plus N n))) with (sum_f_R0 [k:nat](An (plus (0) k)) (S (plus N n))).
-Replace (sum_f_R0 An N) with (sum_f_R0 [k:nat](An (plus (0) k)) N).
-Cut (minus (S (plus N n)) (S N))=n.
-Intro; Rewrite H11 in H10.
-Apply H10.
-Apply INR_eq; Rewrite minus_INR.
-Do 2 Rewrite S_INR; Rewrite plus_INR; Ring.
-Apply le_n_S; Apply le_plus_l.
-Apply sum_eq; Intros.
-Reflexivity.
-Apply sum_eq; Intros.
-Reflexivity.
-Apply le_lt_n_Sm; Apply le_plus_l.
-Apply le_O_n.
-Symmetry; EApply UL_suite.
-Apply H2.
-Unfold Un_cv in H; Unfold Un_cv; Intros.
-Elim (H eps H4); Intros N0 H5.
-Unfold R_dist in H5; Exists N0; Intros.
-Unfold R_dist SP; Replace (Rminus (sum_f_R0 [l:nat](fn (plus (S N) l) x) n) (Rminus l1 (sum_f_R0 [k:nat](fn k x) N))) with (Rminus (Rplus (sum_f_R0 [k:nat](fn k x) N) (sum_f_R0 [l:nat](fn (plus (S N) l) x) n)) l1); [Idtac | Ring].
-Replace (Rplus (sum_f_R0 [k:nat](fn k x) N) (sum_f_R0 [l:nat](fn (plus (S N) l) x) n)) with (sum_f_R0 [k:nat](fn k x) (S (plus N n))).
-Unfold SP in H5; Apply H5; Unfold ge; Apply le_trans with n.
-Apply H6.
-Apply le_trans with (plus N n).
-Apply le_plus_r.
-Apply le_n_Sn.
-Cut (le O N).
-Cut (lt N (S (plus N n))).
-Intros; Assert H9 := (sigma_split [k:nat](fn k x) H8 H7).
-Unfold sigma in H9.
-Do 2 Rewrite <- minus_n_O in H9.
-Replace (sum_f_R0 [k:nat](fn k x) (S (plus N n))) with (sum_f_R0 [k:nat](fn (plus (0) k) x) (S (plus N n))).
-Replace (sum_f_R0 [k:nat](fn k x) N) with (sum_f_R0 [k:nat](fn (plus (0) k) x) N).
-Cut (minus (S (plus N n)) (S N))=n.
-Intro; Rewrite H10 in H9.
-Apply H9.
-Apply INR_eq; Rewrite minus_INR.
-Do 2 Rewrite S_INR; Rewrite plus_INR; Ring.
-Apply le_n_S; Apply le_plus_l.
-Apply sum_eq; Intros.
-Reflexivity.
-Apply sum_eq; Intros.
-Reflexivity.
-Apply le_lt_n_Sm.
-Apply le_plus_l.
-Apply le_O_n.
-Apply existTT with ``l2-(sum_f_R0 An N)``.
-Unfold Un_cv in H0; Unfold Un_cv; Intros.
-Elim (H0 eps H2); Intros N0 H3.
-Unfold R_dist in H3; Exists N0; Intros.
-Unfold R_dist; Replace (Rminus (sum_f_R0 [l:nat](An (plus (S N) l)) n) (Rminus l2 (sum_f_R0 An N))) with (Rminus (Rplus (sum_f_R0 An N) (sum_f_R0 [l:nat](An (plus (S N) l)) n)) l2); [Idtac | Ring].
-Replace (Rplus (sum_f_R0 An N) (sum_f_R0 [l:nat](An (plus (S N) l)) n)) with (sum_f_R0 An (S (plus N n))).
-Apply H3; Unfold ge; Apply le_trans with n.
-Apply H4.
-Apply le_trans with (plus N n).
-Apply le_plus_r.
-Apply le_n_Sn.
-Cut (le O N).
-Cut (lt N (S (plus N n))).
-Intros; Assert H7 := (sigma_split An H6 H5).
-Unfold sigma in H7.
-Do 2 Rewrite <- minus_n_O in H7.
-Replace (sum_f_R0 An (S (plus N n))) with (sum_f_R0 [k:nat](An (plus (0) k)) (S (plus N n))).
-Replace (sum_f_R0 An N) with (sum_f_R0 [k:nat](An (plus (0) k)) N).
-Cut (minus (S (plus N n)) (S N))=n.
-Intro; Rewrite H8 in H7.
-Apply H7.
-Apply INR_eq; Rewrite minus_INR.
-Do 2 Rewrite S_INR; Rewrite plus_INR; Ring.
-Apply le_n_S; Apply le_plus_l.
-Apply sum_eq; Intros.
-Reflexivity.
-Apply sum_eq; Intros.
-Reflexivity.
-Apply le_lt_n_Sm.
-Apply le_plus_l.
-Apply le_O_n.
-Apply existTT with ``l1-(SP fn N x)``.
-Unfold Un_cv in H; Unfold Un_cv; Intros.
-Elim (H eps H2); Intros N0 H3.
-Unfold R_dist in H3; Exists N0; Intros.
-Unfold R_dist SP.
-Replace (Rminus (sum_f_R0 [l:nat](fn (plus (S N) l) x) n) (Rminus l1 (sum_f_R0 [k:nat](fn k x) N))) with (Rminus (Rplus (sum_f_R0 [k:nat](fn k x) N) (sum_f_R0 [l:nat](fn (plus (S N) l) x) n)) l1); [Idtac | Ring].
-Replace (Rplus (sum_f_R0 [k:nat](fn k x) N) (sum_f_R0 [l:nat](fn (plus (S N) l) x) n)) with (sum_f_R0 [k:nat](fn k x) (S (plus N n))).
-Unfold SP in H3; Apply H3.
-Unfold ge; Apply le_trans with n.
-Apply H4.
-Apply le_trans with (plus N n).
-Apply le_plus_r.
-Apply le_n_Sn.
-Cut (le O N).
-Cut (lt N (S (plus N n))).
-Intros; Assert H7 := (sigma_split [k:nat](fn k x) H6 H5).
-Unfold sigma in H7.
-Do 2 Rewrite <- minus_n_O in H7.
-Replace (sum_f_R0 [k:nat](fn k x) (S (plus N n))) with (sum_f_R0 [k:nat](fn (plus (0) k) x) (S (plus N n))).
-Replace (sum_f_R0 [k:nat](fn k x) N) with (sum_f_R0 [k:nat](fn (plus (0) k) x) N).
-Cut (minus (S (plus N n)) (S N))=n.
-Intro; Rewrite H8 in H7.
-Apply H7.
-Apply INR_eq; Rewrite minus_INR.
-Do 2 Rewrite S_INR; Rewrite plus_INR; Ring.
-Apply le_n_S; Apply le_plus_l.
-Apply sum_eq; Intros.
-Reflexivity.
-Apply sum_eq; Intros.
-Reflexivity.
-Apply le_lt_n_Sm.
-Apply le_plus_l.
-Apply le_O_n.
-Qed.
-
-Lemma pow1 : (n:nat) (pow R1 n)==R1.
-Intro; Induction n.
-Reflexivity.
-Simpl; Rewrite Hrecn; Rewrite Rmult_1r; Reflexivity.
-Qed.
-
-Lemma pow_Rabs : (x:R;n:nat) ``(pow x n)<=(pow (Rabsolu x) n)``.
-Intros; Induction n.
-Right; Reflexivity.
-Simpl; Case (case_Rabsolu x); Intro.
-Apply Rle_trans with (Rabsolu ``x*(pow x n)``).
-Apply Rle_Rabsolu.
-Rewrite Rabsolu_mult.
-Apply Rle_monotony.
-Apply Rabsolu_pos.
-Right; Symmetry; Apply Pow_Rabsolu.
-Pattern 1 (Rabsolu x); Rewrite (Rabsolu_right x r); Apply Rle_monotony.
-Apply Rle_sym2; Exact r.
-Apply Hrecn.
-Qed.
-
-Lemma pow_maj_Rabs : (x,y:R;n:nat) ``(Rabsolu y)<=x`` -> ``(pow y n)<=(pow x n)``.
-Intros; Cut ``0<=x``.
-Intro; Apply Rle_trans with (pow (Rabsolu y) n).
-Apply pow_Rabs.
-Induction n.
-Right; Reflexivity.
-Simpl; Apply Rle_trans with ``x*(pow (Rabsolu y) n)``.
-Do 2 Rewrite <- (Rmult_sym (pow (Rabsolu y) n)).
-Apply Rle_monotony.
-Apply pow_le; Apply Rabsolu_pos.
-Assumption.
-Apply Rle_monotony.
-Apply H0.
-Apply Hrecn.
-Apply Rle_trans with (Rabsolu y); [Apply Rabsolu_pos | Exact H].
-Qed.
-
 (* Dans un espace complet, la convergence normale implique la 
    convergence uniforme *)
 Lemma CVN_CVU : (fn:nat->R->R;cv:(x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l));r:posreal) (CVN_r fn r) -> (CVU [n:nat](SP fn n) (SFL fn cv) ``0`` r).
@@ -355,7 +68,7 @@ Qed.
 Lemma CVU_continuity : (fn:nat->R->R;f:R->R;x:R;r:posreal) (CVU fn f x r) -> ((n:nat)(y:R) (Boule x r y)->(continuity_pt (fn n) y)) -> ((y:R) (Boule x r y) -> (continuity_pt f y)).
 Intros; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros.
 Unfold CVU in H.
-Cut ``0<eps/3``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_3_0]].
+Cut ``0<eps/3``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]].
 Elim (H ? H3); Intros N0 H4.
 Assert H5 := (H0 N0 y H1).
 Cut (EXT del : posreal | (h:R) ``(Rabsolu h)<del`` -> (Boule x r ``y+h``) ).
@@ -448,45 +161,6 @@ Apply H1.
 Unfold Boule; Simpl; Rewrite minus_R0; Pattern 1 (Rabsolu x); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1.
 Qed.
 
-(* Théorème de comparaison de convergence pour les séries *)
-Lemma Rseries_CV_comp : (An,Bn:nat->R) ((n:nat)``0<=(An n)<=(Bn n)``) -> (sigTT ? [l:R](Un_cv [N:nat](sum_f_R0 Bn N) l)) -> (sigTT ? [l:R](Un_cv [N:nat](sum_f_R0 An N) l)).
-Intros; Apply cv_cauchy_2.
-Assert H0 := (cv_cauchy_1 ? X).
-Unfold Cauchy_crit_series; Unfold Cauchy_crit.
-Intros; Elim (H0 eps H1); Intros.
-Exists x; Intros.
-Cut (Rle (R_dist (sum_f_R0 An n) (sum_f_R0 An m)) (R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m))).
-Intro; Apply Rle_lt_trans with (R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)).
-Assumption.
-Apply H2; Assumption.
-Assert H5 := (lt_eq_lt_dec n m).
-Elim H5; Intro.
-Elim a; Intro.
-Rewrite (tech2 An n m); [Idtac | Assumption].
-Rewrite (tech2 Bn 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; Repeat Rewrite Rabsolu_right.
-Apply sum_Rle; Intros.
-Elim (H (plus (S n) n0)); Intros.
-Apply H8.
-Apply Rle_sym1; Apply cond_pos_sum; Intro.
-Elim (H (plus (S n) n0)); Intros.
-Apply Rle_trans with (An (plus (S n) n0)); Assumption.
-Apply Rle_sym1; Apply cond_pos_sum; Intro.
-Elim (H (plus (S n) n0)); Intros; Assumption.
-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 Bn 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 Bn m)); Do 2 Rewrite Rplus_assoc; Do 2 Rewrite Rplus_Ropp_l; Do 2 Rewrite Rplus_Or; Repeat Rewrite Rabsolu_right.
-Apply sum_Rle; Intros.
-Elim (H (plus (S m) n0)); Intros; Apply H8.
-Apply Rle_sym1; Apply cond_pos_sum; Intro.
-Elim (H (plus (S m) n0)); Intros.
-Apply Rle_trans with (An (plus (S m) n0)); Assumption.
-Apply Rle_sym1.
-Apply cond_pos_sum; Intro.
-Elim (H (plus (S m) n0)); Intros; Assumption.
-Qed.
-
 (* Grace a la completude de R, on a le lemme suivant *)
 Lemma CVN_R_CVS : (fn:nat->R->R) (CVN_R fn) -> ((x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l))).
 Intros; Apply R_complet.