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authorGravatar Samuel Mimram <smimram@debian.org>2006-11-21 21:38:49 +0000
committerGravatar Samuel Mimram <smimram@debian.org>2006-11-21 21:38:49 +0000
commit208a0f7bfa5249f9795e6e225f309cbe715c0fad (patch)
tree591e9e512063e34099782e2518573f15ffeac003 /theories/Reals/Ranalysis2.v
parentde0085539583f59dc7c4bf4e272e18711d565466 (diff)
Imported Upstream version 8.1~gammaupstream/8.1.gamma
Diffstat (limited to 'theories/Reals/Ranalysis2.v')
-rw-r--r--theories/Reals/Ranalysis2.v775
1 files changed, 394 insertions, 381 deletions
diff --git a/theories/Reals/Ranalysis2.v b/theories/Reals/Ranalysis2.v
index 0627e22c..fb89da67 100644
--- a/theories/Reals/Ranalysis2.v
+++ b/theories/Reals/Ranalysis2.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Ranalysis2.v 5920 2004-07-16 20:01:26Z herbelin $ i*)
+(*i $Id: Ranalysis2.v 9245 2006-10-17 12:53:34Z notin $ i*)
Require Import Rbase.
Require Import Rfunctions.
@@ -14,437 +14,450 @@ Require Import Ranalysis1. Open Local Scope R_scope.
(**********)
Lemma formule :
- forall (x h l1 l2:R) (f1 f2:R -> R),
- h <> 0 ->
- f2 x <> 0 ->
- f2 (x + h) <> 0 ->
- (f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
- (l1 * f2 x - l2 * f1 x) / Rsqr (f2 x) =
- / f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
- l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
- f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2) +
- l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x).
-intros; unfold Rdiv, Rminus, Rsqr in |- *.
-repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l;
- repeat rewrite Rinv_mult_distr; try assumption.
-replace (l1 * f2 x * (/ f2 x * / f2 x)) with (l1 * / f2 x * (f2 x * / f2 x));
- [ idtac | ring ].
-replace (l1 * (/ f2 x * / f2 (x + h)) * f2 x) with
- (l1 * / f2 (x + h) * (f2 x * / f2 x)); [ idtac | ring ].
-replace (l1 * (/ f2 x * / f2 (x + h)) * - f2 (x + h)) with
- (- (l1 * / f2 x * (f2 (x + h) * / f2 (x + h)))); [ idtac | ring ].
-replace (f1 x * (/ f2 x * / f2 (x + h)) * (f2 (x + h) * / h)) with
- (f1 x * / f2 x * / h * (f2 (x + h) * / f2 (x + h)));
- [ idtac | ring ].
-replace (f1 x * (/ f2 x * / f2 (x + h)) * (- f2 x * / h)) with
- (- (f1 x * / f2 (x + h) * / h * (f2 x * / f2 x)));
- [ idtac | ring ].
-replace (l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * f2 (x + h)) with
- (l2 * f1 x * / f2 x * / f2 x * (f2 (x + h) * / f2 (x + h)));
- [ idtac | ring ].
-replace (l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x) with
- (- (l2 * f1 x * / f2 x * / f2 (x + h) * (f2 x * / f2 x)));
- [ idtac | ring ].
-repeat rewrite <- Rinv_r_sym; try assumption || ring.
-apply prod_neq_R0; assumption.
+ forall (x h l1 l2:R) (f1 f2:R -> R),
+ h <> 0 ->
+ f2 x <> 0 ->
+ f2 (x + h) <> 0 ->
+ (f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
+ (l1 * f2 x - l2 * f1 x) / Rsqr (f2 x) =
+ / f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
+ l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
+ f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2) +
+ l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x).
+Proof.
+ intros; unfold Rdiv, Rminus, Rsqr in |- *.
+ repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l;
+ repeat rewrite Rinv_mult_distr; try assumption.
+ replace (l1 * f2 x * (/ f2 x * / f2 x)) with (l1 * / f2 x * (f2 x * / f2 x));
+ [ idtac | ring ].
+ replace (l1 * (/ f2 x * / f2 (x + h)) * f2 x) with
+ (l1 * / f2 (x + h) * (f2 x * / f2 x)); [ idtac | ring ].
+ replace (l1 * (/ f2 x * / f2 (x + h)) * - f2 (x + h)) with
+ (- (l1 * / f2 x * (f2 (x + h) * / f2 (x + h)))); [ idtac | ring ].
+ replace (f1 x * (/ f2 x * / f2 (x + h)) * (f2 (x + h) * / h)) with
+ (f1 x * / f2 x * / h * (f2 (x + h) * / f2 (x + h)));
+ [ idtac | ring ].
+ replace (f1 x * (/ f2 x * / f2 (x + h)) * (- f2 x * / h)) with
+ (- (f1 x * / f2 (x + h) * / h * (f2 x * / f2 x)));
+ [ idtac | ring ].
+ replace (l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * f2 (x + h)) with
+ (l2 * f1 x * / f2 x * / f2 x * (f2 (x + h) * / f2 (x + h)));
+ [ idtac | ring ].
+ replace (l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x) with
+ (- (l2 * f1 x * / f2 x * / f2 (x + h) * (f2 x * / f2 x)));
+ [ idtac | ring ].
+ repeat rewrite <- Rinv_r_sym; try assumption || ring.
+ apply prod_neq_R0; assumption.
Qed.
Lemma Rmin_pos : forall x y:R, 0 < x -> 0 < y -> 0 < Rmin x y.
-intros; unfold Rmin in |- *.
-case (Rle_dec x y); intro; assumption.
+Proof.
+ intros; unfold Rmin in |- *.
+ case (Rle_dec x y); intro; assumption.
Qed.
Lemma maj_term1 :
- forall (x h eps l1 alp_f2:R) (eps_f2 alp_f1d:posreal)
- (f1 f2:R -> R),
- 0 < eps ->
- f2 x <> 0 ->
- f2 (x + h) <> 0 ->
- (forall h:R,
+ forall (x h eps l1 alp_f2:R) (eps_f2 alp_f1d:posreal)
+ (f1 f2:R -> R),
+ 0 < eps ->
+ f2 x <> 0 ->
+ f2 (x + h) <> 0 ->
+ (forall h:R,
h <> 0 ->
Rabs h < alp_f1d ->
Rabs ((f1 (x + h) - f1 x) / h - l1) < Rabs (eps * f2 x / 8)) ->
- (forall a:R,
+ (forall a:R,
Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
- h <> 0 ->
- Rabs h < alp_f1d ->
- Rabs h < Rmin eps_f2 alp_f2 ->
- Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) < eps / 4.
-intros.
-assert (H7 := H3 h H6).
-assert (H8 := H2 h H4 H5).
-apply Rle_lt_trans with
- (2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1)).
-rewrite Rabs_mult.
-apply Rmult_le_compat_r.
-apply Rabs_pos.
-rewrite Rabs_Rinv; [ left; exact H7 | assumption ].
-apply Rlt_le_trans with (2 / Rabs (f2 x) * Rabs (eps * f2 x / 8)).
-apply Rmult_lt_compat_l.
-unfold Rdiv in |- *; apply Rmult_lt_0_compat;
- [ prove_sup0 | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ].
-exact H8.
-right; unfold Rdiv in |- *.
-repeat rewrite Rabs_mult.
-rewrite Rabs_Rinv; discrR.
-replace (Rabs 8) with 8.
-replace 8 with 8; [ idtac | ring ].
-rewrite Rinv_mult_distr; [ idtac | discrR | discrR ].
-replace (2 * / Rabs (f2 x) * (Rabs eps * Rabs (f2 x) * (/ 2 * / 4))) with
- (Rabs eps * / 4 * (2 * / 2) * (Rabs (f2 x) * / Rabs (f2 x)));
- [ idtac | ring ].
-replace (Rabs eps) with eps.
-repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).
-ring.
-symmetry in |- *; apply Rabs_right; left; assumption.
-symmetry in |- *; apply Rabs_right; left; prove_sup.
+ h <> 0 ->
+ Rabs h < alp_f1d ->
+ Rabs h < Rmin eps_f2 alp_f2 ->
+ Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) < eps / 4.
+Proof.
+ intros.
+ assert (H7 := H3 h H6).
+ assert (H8 := H2 h H4 H5).
+ apply Rle_lt_trans with
+ (2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1)).
+ rewrite Rabs_mult.
+ apply Rmult_le_compat_r.
+ apply Rabs_pos.
+ rewrite Rabs_Rinv; [ left; exact H7 | assumption ].
+ apply Rlt_le_trans with (2 / Rabs (f2 x) * Rabs (eps * f2 x / 8)).
+ apply Rmult_lt_compat_l.
+ unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+ [ prove_sup0 | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ].
+ exact H8.
+ right; unfold Rdiv in |- *.
+ repeat rewrite Rabs_mult.
+ rewrite Rabs_Rinv; discrR.
+ replace (Rabs 8) with 8.
+ replace 8 with 8; [ idtac | ring ].
+ rewrite Rinv_mult_distr; [ idtac | discrR | discrR ].
+ replace (2 * / Rabs (f2 x) * (Rabs eps * Rabs (f2 x) * (/ 2 * / 4))) with
+ (Rabs eps * / 4 * (2 * / 2) * (Rabs (f2 x) * / Rabs (f2 x)));
+ [ idtac | ring ].
+ replace (Rabs eps) with eps.
+ repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).
+ ring.
+ symmetry in |- *; apply Rabs_right; left; assumption.
+ symmetry in |- *; apply Rabs_right; left; prove_sup.
Qed.
Lemma maj_term2 :
- forall (x h eps l1 alp_f2 alp_f2t2:R) (eps_f2:posreal)
- (f2:R -> R),
- 0 < eps ->
- f2 x <> 0 ->
- f2 (x + h) <> 0 ->
- (forall a:R,
+ forall (x h eps l1 alp_f2 alp_f2t2:R) (eps_f2:posreal)
+ (f2:R -> R),
+ 0 < eps ->
+ f2 x <> 0 ->
+ f2 (x + h) <> 0 ->
+ (forall a:R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))) ->
- (forall a:R,
+ (forall a:R,
Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
- h <> 0 ->
- Rabs h < alp_f2t2 ->
- Rabs h < Rmin eps_f2 alp_f2 ->
- l1 <> 0 -> Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) < eps / 4.
-intros.
-assert (H8 := H3 h H6).
-assert (H9 := H2 h H5).
-apply Rle_lt_trans with
- (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))).
-rewrite Rabs_mult; apply Rmult_le_compat_l.
-apply Rabs_pos.
-rewrite <- (Rabs_Ropp (f2 x - f2 (x + h))); rewrite Ropp_minus_distr.
-left; apply H9.
-apply Rlt_le_trans with
- (Rabs (2 * (l1 / (f2 x * f2 x))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))).
-apply Rmult_lt_compat_r.
-apply Rabs_pos_lt.
-unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0;
- try assumption || discrR.
-red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).
-apply Rinv_neq_0_compat; apply prod_neq_R0; try assumption || discrR.
-unfold Rdiv in |- *.
-repeat rewrite Rinv_mult_distr; try assumption.
-repeat rewrite Rabs_mult.
-replace (Rabs 2) with 2.
-rewrite (Rmult_comm 2).
-replace (Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with
- (Rabs l1 * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2)));
- [ idtac | ring ].
-repeat apply Rmult_lt_compat_l.
-apply Rabs_pos_lt; assumption.
-apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption.
-repeat rewrite Rabs_Rinv; try assumption.
-rewrite <- (Rmult_comm 2).
-unfold Rdiv in H8; exact H8.
-symmetry in |- *; apply Rabs_right; left; prove_sup0.
-right.
-unfold Rsqr, Rdiv in |- *.
-do 1 rewrite Rinv_mult_distr; try assumption || discrR.
-do 1 rewrite Rinv_mult_distr; try assumption || discrR.
-repeat rewrite Rabs_mult.
-repeat rewrite Rabs_Rinv; try assumption || discrR.
-replace (Rabs eps) with eps.
-replace (Rabs 8) with 8.
-replace (Rabs 2) with 2.
-replace 8 with (4 * 2); [ idtac | ring ].
-rewrite Rinv_mult_distr; discrR.
-replace
- (2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) *
- (eps * (Rabs (f2 x) * Rabs (f2 x)) * (/ 4 * / 2 * / Rabs l1))) with
- (eps * / 4 * (Rabs l1 * / Rabs l1) * (Rabs (f2 x) * / Rabs (f2 x)) *
- (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ].
-repeat rewrite <- Rinv_r_sym; try (apply Rabs_no_R0; assumption) || discrR.
-ring.
-symmetry in |- *; apply Rabs_right; left; prove_sup0.
-symmetry in |- *; apply Rabs_right; left; prove_sup.
-symmetry in |- *; apply Rabs_right; left; assumption.
+ h <> 0 ->
+ Rabs h < alp_f2t2 ->
+ Rabs h < Rmin eps_f2 alp_f2 ->
+ l1 <> 0 -> Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) < eps / 4.
+Proof.
+ intros.
+ assert (H8 := H3 h H6).
+ assert (H9 := H2 h H5).
+ apply Rle_lt_trans with
+ (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))).
+ rewrite Rabs_mult; apply Rmult_le_compat_l.
+ apply Rabs_pos.
+ rewrite <- (Rabs_Ropp (f2 x - f2 (x + h))); rewrite Ropp_minus_distr.
+ left; apply H9.
+ apply Rlt_le_trans with
+ (Rabs (2 * (l1 / (f2 x * f2 x))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))).
+ apply Rmult_lt_compat_r.
+ apply Rabs_pos_lt.
+ unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0;
+ try assumption || discrR.
+ red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).
+ apply Rinv_neq_0_compat; apply prod_neq_R0; try assumption || discrR.
+ unfold Rdiv in |- *.
+ repeat rewrite Rinv_mult_distr; try assumption.
+ repeat rewrite Rabs_mult.
+ replace (Rabs 2) with 2.
+ rewrite (Rmult_comm 2).
+ replace (Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with
+ (Rabs l1 * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2)));
+ [ idtac | ring ].
+ repeat apply Rmult_lt_compat_l.
+ apply Rabs_pos_lt; assumption.
+ apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption.
+ repeat rewrite Rabs_Rinv; try assumption.
+ rewrite <- (Rmult_comm 2).
+ unfold Rdiv in H8; exact H8.
+ symmetry in |- *; apply Rabs_right; left; prove_sup0.
+ right.
+ unfold Rsqr, Rdiv in |- *.
+ do 1 rewrite Rinv_mult_distr; try assumption || discrR.
+ do 1 rewrite Rinv_mult_distr; try assumption || discrR.
+ repeat rewrite Rabs_mult.
+ repeat rewrite Rabs_Rinv; try assumption || discrR.
+ replace (Rabs eps) with eps.
+ replace (Rabs 8) with 8.
+ replace (Rabs 2) with 2.
+ replace 8 with (4 * 2); [ idtac | ring ].
+ rewrite Rinv_mult_distr; discrR.
+ replace
+ (2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) *
+ (eps * (Rabs (f2 x) * Rabs (f2 x)) * (/ 4 * / 2 * / Rabs l1))) with
+ (eps * / 4 * (Rabs l1 * / Rabs l1) * (Rabs (f2 x) * / Rabs (f2 x)) *
+ (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ].
+ repeat rewrite <- Rinv_r_sym; try (apply Rabs_no_R0; assumption) || discrR.
+ ring.
+ symmetry in |- *; apply Rabs_right; left; prove_sup0.
+ symmetry in |- *; apply Rabs_right; left; prove_sup.
+ symmetry in |- *; apply Rabs_right; left; assumption.
Qed.
Lemma maj_term3 :
- forall (x h eps l2 alp_f2:R) (eps_f2 alp_f2d:posreal)
- (f1 f2:R -> R),
- 0 < eps ->
- f2 x <> 0 ->
- f2 (x + h) <> 0 ->
- (forall h:R,
+ forall (x h eps l2 alp_f2:R) (eps_f2 alp_f2d:posreal)
+ (f1 f2:R -> R),
+ 0 < eps ->
+ f2 x <> 0 ->
+ f2 (x + h) <> 0 ->
+ (forall h:R,
h <> 0 ->
Rabs h < alp_f2d ->
Rabs ((f2 (x + h) - f2 x) / h - l2) <
Rabs (Rsqr (f2 x) * eps / (8 * f1 x))) ->
- (forall a:R,
+ (forall a:R,
Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
- h <> 0 ->
- Rabs h < alp_f2d ->
- Rabs h < Rmin eps_f2 alp_f2 ->
- f1 x <> 0 ->
- Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) <
- eps / 4.
-intros.
-assert (H8 := H2 h H4 H5).
-assert (H9 := H3 h H6).
-apply Rle_lt_trans with
- (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))).
-rewrite Rabs_mult.
-apply Rmult_le_compat_l.
-apply Rabs_pos.
-left; apply H8.
-apply Rlt_le_trans with
- (Rabs (2 * (f1 x / (f2 x * f2 x))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))).
-apply Rmult_lt_compat_r.
-apply Rabs_pos_lt.
-unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0;
- try assumption.
-red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).
-apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption.
-unfold Rdiv in |- *.
-repeat rewrite Rinv_mult_distr; try assumption.
-repeat rewrite Rabs_mult.
-replace (Rabs 2) with 2.
-rewrite (Rmult_comm 2).
-replace (Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with
- (Rabs (f1 x) * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2)));
- [ idtac | ring ].
-repeat apply Rmult_lt_compat_l.
-apply Rabs_pos_lt; assumption.
-apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption.
-repeat rewrite Rabs_Rinv; assumption || idtac.
-rewrite <- (Rmult_comm 2).
-unfold Rdiv in H9; exact H9.
-symmetry in |- *; apply Rabs_right; left; prove_sup0.
-right.
-unfold Rsqr, Rdiv in |- *.
-rewrite Rinv_mult_distr; try assumption || discrR.
-rewrite Rinv_mult_distr; try assumption || discrR.
-repeat rewrite Rabs_mult.
-repeat rewrite Rabs_Rinv; try assumption || discrR.
-replace (Rabs eps) with eps.
-replace (Rabs 8) with 8.
-replace (Rabs 2) with 2.
-replace 8 with (4 * 2); [ idtac | ring ].
-rewrite Rinv_mult_distr; discrR.
-replace
- (2 * (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x))) *
- (Rabs (f2 x) * Rabs (f2 x) * eps * (/ 4 * / 2 * / Rabs (f1 x)))) with
- (eps * / 4 * (Rabs (f2 x) * / Rabs (f2 x)) * (Rabs (f2 x) * / Rabs (f2 x)) *
- (Rabs (f1 x) * / Rabs (f1 x)) * (2 * / 2)); [ idtac | ring ].
-repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).
-ring.
-symmetry in |- *; apply Rabs_right; left; prove_sup0.
-symmetry in |- *; apply Rabs_right; left; prove_sup.
-symmetry in |- *; apply Rabs_right; left; assumption.
+ h <> 0 ->
+ Rabs h < alp_f2d ->
+ Rabs h < Rmin eps_f2 alp_f2 ->
+ f1 x <> 0 ->
+ Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) <
+ eps / 4.
+Proof.
+ intros.
+ assert (H8 := H2 h H4 H5).
+ assert (H9 := H3 h H6).
+ apply Rle_lt_trans with
+ (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))).
+ rewrite Rabs_mult.
+ apply Rmult_le_compat_l.
+ apply Rabs_pos.
+ left; apply H8.
+ apply Rlt_le_trans with
+ (Rabs (2 * (f1 x / (f2 x * f2 x))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))).
+ apply Rmult_lt_compat_r.
+ apply Rabs_pos_lt.
+ unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0;
+ try assumption.
+ red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).
+ apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption.
+ unfold Rdiv in |- *.
+ repeat rewrite Rinv_mult_distr; try assumption.
+ repeat rewrite Rabs_mult.
+ replace (Rabs 2) with 2.
+ rewrite (Rmult_comm 2).
+ replace (Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with
+ (Rabs (f1 x) * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2)));
+ [ idtac | ring ].
+ repeat apply Rmult_lt_compat_l.
+ apply Rabs_pos_lt; assumption.
+ apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption.
+ repeat rewrite Rabs_Rinv; assumption || idtac.
+ rewrite <- (Rmult_comm 2).
+ unfold Rdiv in H9; exact H9.
+ symmetry in |- *; apply Rabs_right; left; prove_sup0.
+ right.
+ unfold Rsqr, Rdiv in |- *.
+ rewrite Rinv_mult_distr; try assumption || discrR.
+ rewrite Rinv_mult_distr; try assumption || discrR.
+ repeat rewrite Rabs_mult.
+ repeat rewrite Rabs_Rinv; try assumption || discrR.
+ replace (Rabs eps) with eps.
+ replace (Rabs 8) with 8.
+ replace (Rabs 2) with 2.
+ replace 8 with (4 * 2); [ idtac | ring ].
+ rewrite Rinv_mult_distr; discrR.
+ replace
+ (2 * (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x))) *
+ (Rabs (f2 x) * Rabs (f2 x) * eps * (/ 4 * / 2 * / Rabs (f1 x)))) with
+ (eps * / 4 * (Rabs (f2 x) * / Rabs (f2 x)) * (Rabs (f2 x) * / Rabs (f2 x)) *
+ (Rabs (f1 x) * / Rabs (f1 x)) * (2 * / 2)); [ idtac | ring ].
+ repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).
+ ring.
+ symmetry in |- *; apply Rabs_right; left; prove_sup0.
+ symmetry in |- *; apply Rabs_right; left; prove_sup.
+ symmetry in |- *; apply Rabs_right; left; assumption.
Qed.
Lemma maj_term4 :
- forall (x h eps l2 alp_f2 alp_f2c:R) (eps_f2:posreal)
- (f1 f2:R -> R),
- 0 < eps ->
- f2 x <> 0 ->
- f2 (x + h) <> 0 ->
- (forall a:R,
+ forall (x h eps l2 alp_f2 alp_f2c:R) (eps_f2:posreal)
+ (f1 f2:R -> R),
+ 0 < eps ->
+ f2 x <> 0 ->
+ f2 (x + h) <> 0 ->
+ (forall a:R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) ->
- (forall a:R,
+ (forall a:R,
Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
- h <> 0 ->
- Rabs h < alp_f2c ->
- Rabs h < Rmin eps_f2 alp_f2 ->
- f1 x <> 0 ->
- l2 <> 0 ->
- Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x)) <
- eps / 4.
-intros.
-assert (H9 := H2 h H5).
-assert (H10 := H3 h H6).
-apply Rle_lt_trans with
- (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) *
- Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).
-rewrite Rabs_mult.
-apply Rmult_le_compat_l.
-apply Rabs_pos.
-left; apply H9.
-apply Rlt_le_trans with
- (Rabs (2 * l2 * (f1 x / (Rsqr (f2 x) * f2 x))) *
- Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).
-apply Rmult_lt_compat_r.
-apply Rabs_pos_lt.
-unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0;
- assumption || idtac.
-red in |- *; intro H11; rewrite H11 in H; elim (Rlt_irrefl _ H).
-apply Rinv_neq_0_compat; apply prod_neq_R0.
-apply prod_neq_R0.
-discrR.
-assumption.
-assumption.
-unfold Rdiv in |- *.
-repeat rewrite Rinv_mult_distr;
- try assumption || (unfold Rsqr in |- *; apply prod_neq_R0; assumption).
-repeat rewrite Rabs_mult.
-replace (Rabs 2) with 2.
-replace
- (2 * Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 x)))) with
- (Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * (Rabs (/ f2 x) * 2))));
- [ idtac | ring ].
-replace
- (Rabs l2 * Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 (x + h)))) with
- (Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 (x + h)))));
- [ idtac | ring ].
-repeat apply Rmult_lt_compat_l.
-apply Rabs_pos_lt; assumption.
-apply Rabs_pos_lt; assumption.
-apply Rabs_pos_lt; apply Rinv_neq_0_compat; unfold Rsqr in |- *;
- apply prod_neq_R0; assumption.
-repeat rewrite Rabs_Rinv; [ idtac | assumption | assumption ].
-rewrite <- (Rmult_comm 2).
-unfold Rdiv in H10; exact H10.
-symmetry in |- *; apply Rabs_right; left; prove_sup0.
-right; unfold Rsqr, Rdiv in |- *.
-rewrite Rinv_mult_distr; try assumption || discrR.
-rewrite Rinv_mult_distr; try assumption || discrR.
-rewrite Rinv_mult_distr; try assumption || discrR.
-rewrite Rinv_mult_distr; try assumption || discrR.
-repeat rewrite Rabs_mult.
-repeat rewrite Rabs_Rinv; try assumption || discrR.
-replace (Rabs eps) with eps.
-replace (Rabs 8) with 8.
-replace (Rabs 2) with 2.
-replace 8 with (4 * 2); [ idtac | ring ].
-rewrite Rinv_mult_distr; discrR.
-replace
- (2 * Rabs l2 *
- (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x) * / Rabs (f2 x))) *
- (Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * eps *
- (/ 4 * / 2 * / Rabs (f1 x) * / Rabs l2))) with
- (eps * / 4 * (Rabs l2 * / Rabs l2) * (Rabs (f1 x) * / Rabs (f1 x)) *
- (Rabs (f2 x) * / Rabs (f2 x)) * (Rabs (f2 x) * / Rabs (f2 x)) *
- (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ].
-repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).
-ring.
-symmetry in |- *; apply Rabs_right; left; prove_sup0.
-symmetry in |- *; apply Rabs_right; left; prove_sup.
-symmetry in |- *; apply Rabs_right; left; assumption.
-apply prod_neq_R0; assumption || discrR.
-apply prod_neq_R0; assumption.
+ h <> 0 ->
+ Rabs h < alp_f2c ->
+ Rabs h < Rmin eps_f2 alp_f2 ->
+ f1 x <> 0 ->
+ l2 <> 0 ->
+ Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x)) <
+ eps / 4.
+Proof.
+ intros.
+ assert (H9 := H2 h H5).
+ assert (H10 := H3 h H6).
+ apply Rle_lt_trans with
+ (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) *
+ Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).
+ rewrite Rabs_mult.
+ apply Rmult_le_compat_l.
+ apply Rabs_pos.
+ left; apply H9.
+ apply Rlt_le_trans with
+ (Rabs (2 * l2 * (f1 x / (Rsqr (f2 x) * f2 x))) *
+ Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).
+ apply Rmult_lt_compat_r.
+ apply Rabs_pos_lt.
+ unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0;
+ assumption || idtac.
+ red in |- *; intro H11; rewrite H11 in H; elim (Rlt_irrefl _ H).
+ apply Rinv_neq_0_compat; apply prod_neq_R0.
+ apply prod_neq_R0.
+ discrR.
+ assumption.
+ assumption.
+ unfold Rdiv in |- *.
+ repeat rewrite Rinv_mult_distr;
+ try assumption || (unfold Rsqr in |- *; apply prod_neq_R0; assumption).
+ repeat rewrite Rabs_mult.
+ replace (Rabs 2) with 2.
+ replace
+ (2 * Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 x)))) with
+ (Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * (Rabs (/ f2 x) * 2))));
+ [ idtac | ring ].
+ replace
+ (Rabs l2 * Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 (x + h)))) with
+ (Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 (x + h)))));
+ [ idtac | ring ].
+ repeat apply Rmult_lt_compat_l.
+ apply Rabs_pos_lt; assumption.
+ apply Rabs_pos_lt; assumption.
+ apply Rabs_pos_lt; apply Rinv_neq_0_compat; unfold Rsqr in |- *;
+ apply prod_neq_R0; assumption.
+ repeat rewrite Rabs_Rinv; [ idtac | assumption | assumption ].
+ rewrite <- (Rmult_comm 2).
+ unfold Rdiv in H10; exact H10.
+ symmetry in |- *; apply Rabs_right; left; prove_sup0.
+ right; unfold Rsqr, Rdiv in |- *.
+ rewrite Rinv_mult_distr; try assumption || discrR.
+ rewrite Rinv_mult_distr; try assumption || discrR.
+ rewrite Rinv_mult_distr; try assumption || discrR.
+ rewrite Rinv_mult_distr; try assumption || discrR.
+ repeat rewrite Rabs_mult.
+ repeat rewrite Rabs_Rinv; try assumption || discrR.
+ replace (Rabs eps) with eps.
+ replace (Rabs 8) with 8.
+ replace (Rabs 2) with 2.
+ replace 8 with (4 * 2); [ idtac | ring ].
+ rewrite Rinv_mult_distr; discrR.
+ replace
+ (2 * Rabs l2 *
+ (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x) * / Rabs (f2 x))) *
+ (Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * eps *
+ (/ 4 * / 2 * / Rabs (f1 x) * / Rabs l2))) with
+ (eps * / 4 * (Rabs l2 * / Rabs l2) * (Rabs (f1 x) * / Rabs (f1 x)) *
+ (Rabs (f2 x) * / Rabs (f2 x)) * (Rabs (f2 x) * / Rabs (f2 x)) *
+ (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ].
+ repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).
+ ring.
+ symmetry in |- *; apply Rabs_right; left; prove_sup0.
+ symmetry in |- *; apply Rabs_right; left; prove_sup.
+ symmetry in |- *; apply Rabs_right; left; assumption.
+ apply prod_neq_R0; assumption || discrR.
+ apply prod_neq_R0; assumption.
Qed.
Lemma D_x_no_cond : forall x a:R, a <> 0 -> D_x no_cond x (x + a).
-intros.
-unfold D_x, no_cond in |- *.
-split.
-trivial.
-apply Rminus_not_eq.
-unfold Rminus in |- *.
-rewrite Ropp_plus_distr.
-rewrite <- Rplus_assoc.
-rewrite Rplus_opp_r.
-rewrite Rplus_0_l.
-apply Ropp_neq_0_compat; assumption.
+Proof.
+ intros.
+ unfold D_x, no_cond in |- *.
+ split.
+ trivial.
+ apply Rminus_not_eq.
+ unfold Rminus in |- *.
+ rewrite Ropp_plus_distr.
+ rewrite <- Rplus_assoc.
+ rewrite Rplus_opp_r.
+ rewrite Rplus_0_l.
+ apply Ropp_neq_0_compat; assumption.
Qed.
Lemma Rabs_4 :
- forall a b c d:R, Rabs (a + b + c + d) <= Rabs a + Rabs b + Rabs c + Rabs d.
-intros.
-apply Rle_trans with (Rabs (a + b) + Rabs (c + d)).
-replace (a + b + c + d) with (a + b + (c + d)); [ apply Rabs_triang | ring ].
-apply Rle_trans with (Rabs a + Rabs b + Rabs (c + d)).
-apply Rplus_le_compat_r.
-apply Rabs_triang.
-repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l.
-apply Rabs_triang.
+ forall a b c d:R, Rabs (a + b + c + d) <= Rabs a + Rabs b + Rabs c + Rabs d.
+Proof.
+ intros.
+ apply Rle_trans with (Rabs (a + b) + Rabs (c + d)).
+ replace (a + b + c + d) with (a + b + (c + d)); [ apply Rabs_triang | ring ].
+ apply Rle_trans with (Rabs a + Rabs b + Rabs (c + d)).
+ apply Rplus_le_compat_r.
+ apply Rabs_triang.
+ repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l.
+ apply Rabs_triang.
Qed.
Lemma Rlt_4 :
- forall a b c d e f g h:R,
- a < b -> c < d -> e < f -> g < h -> a + c + e + g < b + d + f + h.
-intros; apply Rlt_trans with (b + c + e + g).
-repeat apply Rplus_lt_compat_r; assumption.
-repeat rewrite Rplus_assoc; apply Rplus_lt_compat_l.
-apply Rlt_trans with (d + e + g).
-rewrite Rplus_assoc; apply Rplus_lt_compat_r; assumption.
-rewrite Rplus_assoc; apply Rplus_lt_compat_l; apply Rlt_trans with (f + g).
-apply Rplus_lt_compat_r; assumption.
-apply Rplus_lt_compat_l; assumption.
+ forall a b c d e f g h:R,
+ a < b -> c < d -> e < f -> g < h -> a + c + e + g < b + d + f + h.
+Proof.
+ intros; apply Rlt_trans with (b + c + e + g).
+ repeat apply Rplus_lt_compat_r; assumption.
+ repeat rewrite Rplus_assoc; apply Rplus_lt_compat_l.
+ apply Rlt_trans with (d + e + g).
+ rewrite Rplus_assoc; apply Rplus_lt_compat_r; assumption.
+ rewrite Rplus_assoc; apply Rplus_lt_compat_l; apply Rlt_trans with (f + g).
+ apply Rplus_lt_compat_r; assumption.
+ apply Rplus_lt_compat_l; assumption.
Qed.
Lemma Rmin_2 : forall a b c:R, a < b -> a < c -> a < Rmin b c.
-intros; unfold Rmin in |- *; case (Rle_dec b c); intro; assumption.
+Proof.
+ intros; unfold Rmin in |- *; case (Rle_dec b c); intro; assumption.
Qed.
Lemma quadruple : forall x:R, 4 * x = x + x + x + x.
-intro; ring.
+Proof.
+ intro; ring.
Qed.
Lemma quadruple_var : forall x:R, x = x / 4 + x / 4 + x / 4 + x / 4.
-intro; rewrite <- quadruple.
-unfold Rdiv in |- *; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; discrR.
-reflexivity.
+Proof.
+ intro; rewrite <- quadruple.
+ unfold Rdiv in |- *; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; discrR.
+ reflexivity.
Qed.
(**********)
Lemma continuous_neq_0 :
- forall (f:R -> R) (x0:R),
- continuity_pt f x0 ->
- f x0 <> 0 ->
+ forall (f:R -> R) (x0:R),
+ continuity_pt f x0 ->
+ f x0 <> 0 ->
exists eps : posreal, (forall h:R, Rabs h < eps -> f (x0 + h) <> 0).
-intros; unfold continuity_pt in H; unfold continue_in in H;
- unfold limit1_in in H; unfold limit_in in H; elim (H (Rabs (f x0 / 2))).
-intros; elim H1; intros.
-exists (mkposreal x H2).
-intros; assert (H5 := H3 (x0 + h)).
-cut
- (dist R_met (x0 + h) x0 < x ->
- dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)).
-unfold dist in |- *; simpl in |- *; unfold R_dist in |- *;
- replace (x0 + h - x0) with h.
-intros; assert (H7 := H6 H4).
-red in |- *; intro.
-rewrite H8 in H7; unfold Rminus in H7; rewrite Rplus_0_l in H7;
- rewrite Rabs_Ropp in H7; unfold Rdiv in H7; rewrite Rabs_mult in H7;
- pattern (Rabs (f x0)) at 1 in H7; rewrite <- Rmult_1_r in H7.
-cut (0 < Rabs (f x0)).
-intro; assert (H10 := Rmult_lt_reg_l _ _ _ H9 H7).
-cut (Rabs (/ 2) = / 2).
-assert (Hyp : 0 < 2).
-prove_sup0.
-intro; rewrite H11 in H10; assert (H12 := Rmult_lt_compat_l 2 _ _ Hyp H10);
- rewrite Rmult_1_r in H12; rewrite <- Rinv_r_sym in H12;
- [ idtac | discrR ].
-cut (IZR 1 < IZR 2).
-unfold IZR in |- *; unfold INR, nat_of_P in |- *; simpl in |- *; intro;
- elim (Rlt_irrefl 1 (Rlt_trans _ _ _ H13 H12)).
-apply IZR_lt; omega.
-unfold Rabs in |- *; case (Rcase_abs (/ 2)); intro.
-assert (Hyp : 0 < 2).
-prove_sup0.
-assert (H11 := Rmult_lt_compat_l 2 _ _ Hyp r); rewrite Rmult_0_r in H11;
- rewrite <- Rinv_r_sym in H11; [ idtac | discrR ].
-elim (Rlt_irrefl 0 (Rlt_trans _ _ _ Rlt_0_1 H11)).
-reflexivity.
-apply (Rabs_pos_lt _ H0).
-ring.
-assert (H6 := Req_dec x0 (x0 + h)); elim H6; intro.
-intro; rewrite <- H7; unfold dist, R_met in |- *; unfold R_dist in |- *;
- unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
- apply Rabs_pos_lt.
-unfold Rdiv in |- *; apply prod_neq_R0;
- [ assumption | apply Rinv_neq_0_compat; discrR ].
-intro; apply H5.
-split.
-unfold D_x, no_cond in |- *.
-split; trivial || assumption.
-assumption.
-change (0 < Rabs (f x0 / 2)) in |- *.
-apply Rabs_pos_lt; unfold Rdiv in |- *; apply prod_neq_R0.
-assumption.
-apply Rinv_neq_0_compat; discrR.
-Qed. \ No newline at end of file
+Proof.
+ intros; unfold continuity_pt in H; unfold continue_in in H;
+ unfold limit1_in in H; unfold limit_in in H; elim (H (Rabs (f x0 / 2))).
+ intros; elim H1; intros.
+ exists (mkposreal x H2).
+ intros; assert (H5 := H3 (x0 + h)).
+ cut
+ (dist R_met (x0 + h) x0 < x ->
+ dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)).
+ unfold dist in |- *; simpl in |- *; unfold R_dist in |- *;
+ replace (x0 + h - x0) with h.
+ intros; assert (H7 := H6 H4).
+ red in |- *; intro.
+ rewrite H8 in H7; unfold Rminus in H7; rewrite Rplus_0_l in H7;
+ rewrite Rabs_Ropp in H7; unfold Rdiv in H7; rewrite Rabs_mult in H7;
+ pattern (Rabs (f x0)) at 1 in H7; rewrite <- Rmult_1_r in H7.
+ cut (0 < Rabs (f x0)).
+ intro; assert (H10 := Rmult_lt_reg_l _ _ _ H9 H7).
+ cut (Rabs (/ 2) = / 2).
+ assert (Hyp : 0 < 2).
+ prove_sup0.
+ intro; rewrite H11 in H10; assert (H12 := Rmult_lt_compat_l 2 _ _ Hyp H10);
+ rewrite Rmult_1_r in H12; rewrite <- Rinv_r_sym in H12;
+ [ idtac | discrR ].
+ cut (IZR 1 < IZR 2).
+ unfold IZR in |- *; unfold INR, nat_of_P in |- *; simpl in |- *; intro;
+ elim (Rlt_irrefl 1 (Rlt_trans _ _ _ H13 H12)).
+ apply IZR_lt; omega.
+ unfold Rabs in |- *; case (Rcase_abs (/ 2)); intro.
+ assert (Hyp : 0 < 2).
+ prove_sup0.
+ assert (H11 := Rmult_lt_compat_l 2 _ _ Hyp r); rewrite Rmult_0_r in H11;
+ rewrite <- Rinv_r_sym in H11; [ idtac | discrR ].
+ elim (Rlt_irrefl 0 (Rlt_trans _ _ _ Rlt_0_1 H11)).
+ reflexivity.
+ apply (Rabs_pos_lt _ H0).
+ ring.
+ assert (H6 := Req_dec x0 (x0 + h)); elim H6; intro.
+ intro; rewrite <- H7; unfold dist, R_met in |- *; unfold R_dist in |- *;
+ unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+ apply Rabs_pos_lt.
+ unfold Rdiv in |- *; apply prod_neq_R0;
+ [ assumption | apply Rinv_neq_0_compat; discrR ].
+ intro; apply H5.
+ split.
+ unfold D_x, no_cond in |- *.
+ split; trivial || assumption.
+ assumption.
+ change (0 < Rabs (f x0 / 2)) in |- *.
+ apply Rabs_pos_lt; unfold Rdiv in |- *; apply prod_neq_R0.
+ assumption.
+ apply Rinv_neq_0_compat; discrR.
+Qed.