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-(************************************************************************)
-(*  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: Rcomplete.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*)
-
-Require Rbase.
-Require Rfunctions.
-Require Rseries.
-Require SeqProp.
-Require Max.
-V7only [ Import nat_scope. Import Z_scope. Import R_scope. ].
-Open Local Scope R_scope.
-
-(****************************************************)
-(*              R is complete :                     *)
-(*        Each sequence which satisfies             *)
-(*       the Cauchy's criterion converges           *)
-(*                                                  *)
-(*    Proof with adjacent sequences (Vn and Wn)     *)
-(****************************************************)
-
-Theorem R_complete : (Un:nat->R) (Cauchy_crit Un) -> (sigTT R [l:R](Un_cv Un l)).
-Intros.
-Pose Vn := (sequence_minorant Un (cauchy_min Un H)).
-Pose Wn := (sequence_majorant Un (cauchy_maj Un H)).
-Assert H0 := (maj_cv Un H).
-Fold Wn in H0.
-Assert H1 := (min_cv Un H).
-Fold Vn in H1.
-Elim H0; Intros.
-Elim H1; Intros.
-Cut x==x0.
-Intros.
-Apply existTT with x.
-Rewrite <- H2 in p0.
-Unfold Un_cv.
-Intros.
-Unfold Un_cv in p; Unfold Un_cv in p0.
-Cut ``0<eps/3``.
-Intro.
-Elim (p ``eps/3`` H4); Intros.
-Elim (p0 ``eps/3`` H4); Intros.
-Exists (max x1 x2).
-Intros.
-Unfold R_dist.
-Apply Rle_lt_trans with ``(Rabsolu ((Un n)-(Vn n)))+(Rabsolu ((Vn n)-x))``.
-Replace ``(Un n)-x`` with ``((Un n)-(Vn n))+((Vn n)-x)``; [Apply Rabsolu_triang | Ring].
-Apply Rle_lt_trans with ``(Rabsolu ((Wn n)-(Vn n)))+(Rabsolu ((Vn n)-x))``.
-Do 2 Rewrite <- (Rplus_sym ``(Rabsolu ((Vn n)-x))``).
-Apply Rle_compatibility.
-Repeat Rewrite Rabsolu_right.
-Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-(Vn n)``); Apply Rle_compatibility.
-Assert H8 := (Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)).
-Fold Vn Wn in H8.
-Elim (H8 n); Intros.
-Assumption.
-Apply Rle_sym1.
-Unfold Rminus; Apply Rle_anti_compatibility with (Vn n).
-Rewrite Rplus_Or.
-Replace ``(Vn n)+((Wn n)+ -(Vn n))`` with (Wn n); [Idtac | Ring].
-Assert H8 := (Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)).
-Fold Vn Wn in H8.
-Elim (H8 n); Intros.
-Apply Rle_trans with (Un n); Assumption.
-Apply Rle_sym1.
-Unfold Rminus; Apply Rle_anti_compatibility with (Vn n).
-Rewrite Rplus_Or.
-Replace ``(Vn n)+((Un n)+ -(Vn n))`` with (Un n); [Idtac | Ring].
-Assert H8 := (Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)).
-Fold Vn Wn in H8.
-Elim (H8 n); Intros.
-Assumption.
-Apply Rle_lt_trans with ``(Rabsolu ((Wn n)-x))+(Rabsolu (x-(Vn n)))+(Rabsolu ((Vn n)-x))``.
-Do 2 Rewrite <- (Rplus_sym ``(Rabsolu ((Vn n)-x))``).
-Apply Rle_compatibility.
-Replace ``(Wn n)-(Vn n)`` with ``((Wn n)-x)+(x-(Vn n))``; [Apply Rabsolu_triang | Ring].
-Apply Rlt_le_trans with ``eps/3+eps/3+eps/3``.
-Repeat Apply Rplus_lt.
-Unfold R_dist in H5.
-Apply H5.
-Unfold ge; Apply le_trans with (max x1 x2).
-Apply le_max_l.
-Assumption.
-Rewrite <- Rabsolu_Ropp.
-Replace ``-(x-(Vn n))`` with ``(Vn n)-x``; [Idtac | Ring].
-Unfold R_dist in H6.
-Apply H6.
-Unfold ge; Apply le_trans with (max x1 x2).
-Apply le_max_r.
-Assumption.
-Unfold R_dist in H6.
-Apply H6.
-Unfold ge; Apply le_trans with (max x1 x2).
-Apply le_max_r.
-Assumption.
-Right.
-Pattern 4 eps; Replace ``eps`` with ``3*eps/3``.
-Ring.
-Unfold Rdiv; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m; DiscrR.
-Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0].
-Apply cond_eq.
-Intros.
-Cut ``0<eps/5``.
-Intro.
-Unfold Un_cv in p; Unfold Un_cv in p0.
-Unfold R_dist in p; Unfold R_dist in p0.
-Elim (p ``eps/5`` H3); Intros N1 H4.
-Elim (p0 ``eps/5`` H3); Intros N2 H5.
-Unfold Cauchy_crit in H.
-Unfold R_dist in H.
-Elim (H ``eps/5`` H3); Intros N3 H6.
-Pose N := (max (max N1 N2) N3).
-Apply Rle_lt_trans with ``(Rabsolu (x-(Wn N)))+(Rabsolu ((Wn N)-x0))``.
-Replace ``x-x0`` with ``(x-(Wn N))+((Wn N)-x0)``; [Apply Rabsolu_triang | Ring].
-Apply Rle_lt_trans with ``(Rabsolu (x-(Wn N)))+(Rabsolu ((Wn N)-(Vn N)))+(Rabsolu (((Vn N)-x0)))``.
-Rewrite Rplus_assoc.
-Apply Rle_compatibility.
-Replace ``(Wn N)-x0`` with ``((Wn N)-(Vn N))+((Vn N)-x0)``; [Apply Rabsolu_triang | Ring].
-Replace ``eps`` with ``eps/5+3*eps/5+eps/5``.
-Repeat Apply Rplus_lt.
-Rewrite <- Rabsolu_Ropp.
-Replace ``-(x-(Wn N))`` with ``(Wn N)-x``; [Apply H4 | Ring].
-Unfold ge N.
-Apply le_trans with (max N1 N2); Apply le_max_l.
-Unfold Wn Vn.
-Unfold sequence_majorant sequence_minorant.
-Assert H7 := (approx_maj [k:nat](Un (plus N k)) (maj_ss Un N (cauchy_maj Un H))).
-Assert H8 := (approx_min [k:nat](Un (plus N k)) (min_ss Un N (cauchy_min Un H))).
-Cut (Wn N)==(majorant ([k:nat](Un (plus N k))) (maj_ss Un N (cauchy_maj Un H))).
-Cut (Vn N)==(minorant ([k:nat](Un (plus N k))) (min_ss Un N (cauchy_min Un H))).
-Intros.
-Rewrite <- H9; Rewrite <- H10.
-Rewrite <- H9 in H8.
-Rewrite <- H10 in H7.
-Elim (H7 ``eps/5`` H3); Intros k2 H11.
-Elim (H8 ``eps/5`` H3); Intros k1 H12.
-Apply Rle_lt_trans with ``(Rabsolu ((Wn N)-(Un (plus N k2))))+(Rabsolu ((Un (plus N k2))-(Vn N)))``.
-Replace ``(Wn N)-(Vn N)`` with ``((Wn N)-(Un (plus N k2)))+((Un (plus N k2))-(Vn N))``; [Apply Rabsolu_triang | Ring].
-Apply Rle_lt_trans with ``(Rabsolu ((Wn N)-(Un (plus N k2))))+(Rabsolu ((Un (plus N k2))-(Un (plus N k1))))+(Rabsolu ((Un (plus N k1))-(Vn N)))``.
-Rewrite Rplus_assoc.
-Apply Rle_compatibility.
-Replace ``(Un (plus N k2))-(Vn N)`` with ``((Un (plus N k2))-(Un (plus N k1)))+((Un (plus N k1))-(Vn N))``; [Apply Rabsolu_triang | Ring].
-Replace ``3*eps/5`` with ``eps/5+eps/5+eps/5``; [Repeat Apply Rplus_lt | Ring].
-Assumption.
-Apply H6.
-Unfold ge.
-Apply le_trans with N.
-Unfold N; Apply le_max_r.
-Apply le_plus_l.
-Unfold ge.
-Apply le_trans with N.
-Unfold N; Apply le_max_r.
-Apply le_plus_l.
-Rewrite <- Rabsolu_Ropp.
-Replace ``-((Un (plus N k1))-(Vn N))`` with ``(Vn N)-(Un (plus N k1))``; [Assumption | Ring].
-Reflexivity.
-Reflexivity.
-Apply H5.
-Unfold ge; Apply le_trans with (max N1 N2).
-Apply le_max_r.
-Unfold N; Apply le_max_l.
-Pattern 4 eps; Replace ``eps`` with ``5*eps/5``.
-Ring.
-Unfold Rdiv; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m.
-DiscrR.
-Unfold Rdiv; Apply Rmult_lt_pos.
-Assumption.
-Apply Rlt_Rinv.
-Sup0; Try Apply lt_O_Sn.
-Qed.