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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Rbase.
Require Import Rfunctions.
Require Import Rseries.
Require Import SeqProp.
Require Import Max.
Local Open 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 :
forall Un:nat -> R, Cauchy_crit Un -> { l:R | Un_cv Un l } .
Proof.
intros.
set (Vn := sequence_minorant Un (cauchy_min Un H)).
set (Wn := sequence_majorant Un (cauchy_maj Un H)).
pose proof (maj_cv Un H) as (x,p).
fold Wn in p.
pose proof (min_cv Un H) as (x0,p0).
fold Vn in p0.
cut (x = x0).
intros H2.
exists x.
rewrite <- H2 in p0.
unfold Un_cv.
intros.
unfold Un_cv in p; unfold Un_cv in p0.
cut (0 < eps / 3).
intro H4.
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 (Rabs (Un n - Vn n) + Rabs (Vn n - x)).
replace (Un n - x) with (Un n - Vn n + (Vn n - x));
[ apply Rabs_triang | ring ].
apply Rle_lt_trans with (Rabs (Wn n - Vn n) + Rabs (Vn n - x)).
do 2 rewrite <- (Rplus_comm (Rabs (Vn n - x))).
apply Rplus_le_compat_l.
repeat rewrite Rabs_right.
unfold Rminus; do 2 rewrite <- (Rplus_comm (- Vn n));
apply Rplus_le_compat_l.
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_ge.
unfold Rminus; apply Rplus_le_reg_l with (Vn n).
rewrite Rplus_0_r.
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_ge.
unfold Rminus; apply Rplus_le_reg_l with (Vn n).
rewrite Rplus_0_r.
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 (Rabs (Wn n - x) + Rabs (x - Vn n) + Rabs (Vn n - x)).
do 2 rewrite <- (Rplus_comm (Rabs (Vn n - x))).
apply Rplus_le_compat_l.
replace (Wn n - Vn n) with (Wn n - x + (x - Vn n));
[ apply Rabs_triang | ring ].
apply Rlt_le_trans with (eps / 3 + eps / 3 + eps / 3).
repeat apply Rplus_lt_compat.
unfold R_dist in H1.
apply H1.
unfold ge; apply le_trans with (max x1 x2).
apply le_max_l.
assumption.
rewrite <- Rabs_Ropp.
replace (- (x - Vn n)) with (Vn n - x); [ idtac | ring ].
unfold R_dist in H3.
apply H3.
unfold ge; apply le_trans with (max x1 x2).
apply le_max_r.
assumption.
unfold R_dist in H3.
apply H3.
unfold ge; apply le_trans with (max x1 x2).
apply le_max_r.
assumption.
right.
pattern eps at 4; replace eps with (3 * (eps / 3)).
ring.
unfold Rdiv; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR.
unfold Rdiv; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_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) H1); intros N1 H4.
elim (p0 (eps / 5) H1); intros N2 H5.
unfold Cauchy_crit in H.
unfold R_dist in H.
elim (H (eps / 5) H1); intros N3 H6.
set (N := max (max N1 N2) N3).
apply Rle_lt_trans with (Rabs (x - Wn N) + Rabs (Wn N - x0)).
replace (x - x0) with (x - Wn N + (Wn N - x0)); [ apply Rabs_triang | ring ].
apply Rle_lt_trans with
(Rabs (x - Wn N) + Rabs (Wn N - Vn N) + Rabs (Vn N - x0)).
rewrite Rplus_assoc.
apply Rplus_le_compat_l.
replace (Wn N - x0) with (Wn N - Vn N + (Vn N - x0));
[ apply Rabs_triang | ring ].
replace eps with (eps / 5 + 3 * (eps / 5) + eps / 5).
repeat apply Rplus_lt_compat.
rewrite <- Rabs_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 (fun k:nat => Un (N + k)%nat) (maj_ss Un N (cauchy_maj Un H))).
assert
(H8 :=
approx_min (fun k:nat => Un (N + k)%nat) (min_ss Un N (cauchy_min Un H))).
cut
(Wn N =
majorant (fun k:nat => Un (N + k)%nat) (maj_ss Un N (cauchy_maj Un H))).
cut
(Vn N =
minorant (fun k:nat => Un (N + k)%nat) (min_ss Un N (cauchy_min Un H))).
intros H9 H10.
rewrite <- H9 in H8 |- *.
rewrite <- H10 in H7 |- *.
elim (H7 (eps / 5) H1); intros k2 H11.
elim (H8 (eps / 5) H1); intros k1 H12.
apply Rle_lt_trans with
(Rabs (Wn N - Un (N + k2)%nat) + Rabs (Un (N + k2)%nat - Vn N)).
replace (Wn N - Vn N) with
(Wn N - Un (N + k2)%nat + (Un (N + k2)%nat - Vn N));
[ apply Rabs_triang | ring ].
apply Rle_lt_trans with
(Rabs (Wn N - Un (N + k2)%nat) + Rabs (Un (N + k2)%nat - Un (N + k1)%nat) +
Rabs (Un (N + k1)%nat - Vn N)).
rewrite Rplus_assoc.
apply Rplus_le_compat_l.
replace (Un (N + k2)%nat - Vn N) with
(Un (N + k2)%nat - Un (N + k1)%nat + (Un (N + k1)%nat - Vn N));
[ apply Rabs_triang | ring ].
replace (3 * (eps / 5)) with (eps / 5 + eps / 5 + eps / 5);
[ repeat apply Rplus_lt_compat | 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 <- Rabs_Ropp.
replace (- (Un (N + k1)%nat - Vn N)) with (Vn N - Un (N + k1)%nat);
[ 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 eps at 4; replace eps with (5 * (eps / 5)).
ring.
unfold Rdiv; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m.
discrR.
unfold Rdiv; apply Rmult_lt_0_compat.
assumption.
apply Rinv_0_lt_compat.
prove_sup0; try apply lt_O_Sn.
Qed.
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