diff options
Diffstat (limited to 'theories/Reals/Rlimit.v')
| -rw-r--r-- | theories/Reals/Rlimit.v | 557 |
1 files changed, 557 insertions, 0 deletions
diff --git a/theories/Reals/Rlimit.v b/theories/Reals/Rlimit.v new file mode 100644 index 00000000..0fbb17c6 --- /dev/null +++ b/theories/Reals/Rlimit.v @@ -0,0 +1,557 @@ +(************************************************************************) +(* 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: Rlimit.v,v 1.23.2.1 2004/07/16 19:31:13 herbelin Exp $ i*) + +(*********************************************************) +(* Definition of the limit *) +(* *) +(*********************************************************) + +Require Import Rbase. +Require Import Rfunctions. +Require Import Classical_Prop. +Require Import Fourier. Open Local Scope R_scope. + +(*******************************) +(* Calculus *) +(*******************************) +(*********) +Lemma eps2_Rgt_R0 : forall eps:R, eps > 0 -> eps * / 2 > 0. +intros; fourier. +Qed. + +(*********) +Lemma eps2 : forall eps:R, eps * / 2 + eps * / 2 = eps. +intro esp. +assert (H := double_var esp). +unfold Rdiv in H. +symmetry in |- *; exact H. +Qed. + +(*********) +Lemma eps4 : forall eps:R, eps * / (2 + 2) + eps * / (2 + 2) = eps * / 2. +intro eps. +replace (2 + 2) with 4. +pattern eps at 3 in |- *; rewrite double_var. +rewrite (Rmult_plus_distr_r (eps / 2) (eps / 2) (/ 2)). +unfold Rdiv in |- *. +repeat rewrite Rmult_assoc. +rewrite <- Rinv_mult_distr. +reflexivity. +discrR. +discrR. +ring. +Qed. + +(*********) +Lemma Rlt_eps2_eps : forall eps:R, eps > 0 -> eps * / 2 < eps. +intros. +pattern eps at 2 in |- *; rewrite <- Rmult_1_r. +repeat rewrite (Rmult_comm eps). +apply Rmult_lt_compat_r. +exact H. +apply Rmult_lt_reg_l with 2. +fourier. +rewrite Rmult_1_r; rewrite <- Rinv_r_sym. +fourier. +discrR. +Qed. + +(*********) +Lemma Rlt_eps4_eps : forall eps:R, eps > 0 -> eps * / (2 + 2) < eps. +intros. +replace (2 + 2) with 4. +pattern eps at 2 in |- *; rewrite <- Rmult_1_r. +repeat rewrite (Rmult_comm eps). +apply Rmult_lt_compat_r. +exact H. +apply Rmult_lt_reg_l with 4. +replace 4 with 4. +apply Rmult_lt_0_compat; fourier. +ring. +rewrite Rmult_1_r; rewrite <- Rinv_r_sym. +fourier. +discrR. +ring. +Qed. + +(*********) +Lemma prop_eps : forall r:R, (forall eps:R, eps > 0 -> r < eps) -> r <= 0. +intros; elim (Rtotal_order r 0); intro. +apply Rlt_le; assumption. +elim H0; intro. +apply Req_le; assumption. +clear H0; generalize (H r H1); intro; generalize (Rlt_irrefl r); intro; + elimtype False; auto. +Qed. + +(*********) +Definition mul_factor (l l':R) := / (1 + (Rabs l + Rabs l')). + +(*********) +Lemma mul_factor_wd : forall l l':R, 1 + (Rabs l + Rabs l') <> 0. +intros; rewrite (Rplus_comm 1 (Rabs l + Rabs l')); apply tech_Rplus. +cut (Rabs (l + l') <= Rabs l + Rabs l'). +cut (0 <= Rabs (l + l')). +exact (Rle_trans _ _ _). +exact (Rabs_pos (l + l')). +exact (Rabs_triang _ _). +exact Rlt_0_1. +Qed. + +(*********) +Lemma mul_factor_gt : forall eps l l':R, eps > 0 -> eps * mul_factor l l' > 0. +intros; unfold Rgt in |- *; rewrite <- (Rmult_0_r eps); + apply Rmult_lt_compat_l. +assumption. +unfold mul_factor in |- *; apply Rinv_0_lt_compat; + cut (1 <= 1 + (Rabs l + Rabs l')). +cut (0 < 1). +exact (Rlt_le_trans _ _ _). +exact Rlt_0_1. +replace (1 <= 1 + (Rabs l + Rabs l')) with (1 + 0 <= 1 + (Rabs l + Rabs l')). +apply Rplus_le_compat_l. +cut (Rabs (l + l') <= Rabs l + Rabs l'). +cut (0 <= Rabs (l + l')). +exact (Rle_trans _ _ _). +exact (Rabs_pos _). +exact (Rabs_triang _ _). +rewrite (proj1 (Rplus_ne 1)); trivial. +Qed. + +(*********) +Lemma mul_factor_gt_f : + forall eps l l':R, eps > 0 -> Rmin 1 (eps * mul_factor l l') > 0. +intros; apply Rmin_Rgt_r; split. +exact Rlt_0_1. +exact (mul_factor_gt eps l l' H). +Qed. + + +(*******************************) +(* Metric space *) +(*******************************) + +(*********) +Record Metric_Space : Type := + {Base : Type; + dist : Base -> Base -> R; + dist_pos : forall x y:Base, dist x y >= 0; + dist_sym : forall x y:Base, dist x y = dist y x; + dist_refl : forall x y:Base, dist x y = 0 <-> x = y; + dist_tri : forall x y z:Base, dist x y <= dist x z + dist z y}. + +(*******************************) +(* Limit in Metric space *) +(*******************************) + +(*********) +Definition limit_in (X X':Metric_Space) (f:Base X -> Base X') + (D:Base X -> Prop) (x0:Base X) (l:Base X') := + forall eps:R, + eps > 0 -> + exists alp : R, + alp > 0 /\ + (forall x:Base X, D x /\ dist X x x0 < alp -> dist X' (f x) l < eps). + +(*******************************) +(* R is a metric space *) +(*******************************) + +(*********) +Definition R_met : Metric_Space := + Build_Metric_Space R R_dist R_dist_pos R_dist_sym R_dist_refl R_dist_tri. + +(*******************************) +(* Limit 1 arg *) +(*******************************) +(*********) +Definition Dgf (Df Dg:R -> Prop) (f:R -> R) (x:R) := Df x /\ Dg (f x). + +(*********) +Definition limit1_in (f:R -> R) (D:R -> Prop) (l x0:R) : Prop := + limit_in R_met R_met f D x0 l. + +(*********) +Lemma tech_limit : + forall (f:R -> R) (D:R -> Prop) (l x0:R), + D x0 -> limit1_in f D l x0 -> l = f x0. +intros f D l x0 H H0. +case (Rabs_pos (f x0 - l)); intros H1. +absurd (dist R_met (f x0) l < dist R_met (f x0) l). +apply Rlt_irrefl. +case (H0 (dist R_met (f x0) l)); auto. +intros alpha1 [H2 H3]; apply H3; auto; split; auto. +case (dist_refl R_met x0 x0); intros Hr1 Hr2; rewrite Hr2; auto. +case (dist_refl R_met (f x0) l); intros Hr1 Hr2; apply sym_eq; auto. +Qed. + +(*********) +Lemma tech_limit_contr : + forall (f:R -> R) (D:R -> Prop) (l x0:R), + D x0 -> l <> f x0 -> ~ limit1_in f D l x0. +intros; generalize (tech_limit f D l x0); tauto. +Qed. + +(*********) +Lemma lim_x : forall (D:R -> Prop) (x0:R), limit1_in (fun x:R => x) D x0 x0. +unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros; + split with eps; split; auto; intros; elim H0; intros; + auto. +Qed. + +(*********) +Lemma limit_plus : + forall (f g:R -> R) (D:R -> Prop) (l l' x0:R), + limit1_in f D l x0 -> + limit1_in g D l' x0 -> limit1_in (fun x:R => f x + g x) D (l + l') x0. +intros; unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; + intros; elim (H (eps * / 2) (eps2_Rgt_R0 eps H1)); + elim (H0 (eps * / 2) (eps2_Rgt_R0 eps H1)); simpl in |- *; + clear H H0; intros; elim H; elim H0; clear H H0; intros; + split with (Rmin x1 x); split. +exact (Rmin_Rgt_r x1 x 0 (conj H H2)). +intros; elim H4; clear H4; intros; + cut (R_dist (f x2) l + R_dist (g x2) l' < eps). + cut (R_dist (f x2 + g x2) (l + l') <= R_dist (f x2) l + R_dist (g x2) l'). +exact (Rle_lt_trans _ _ _). +exact (R_dist_plus _ _ _ _). +elim (Rmin_Rgt_l x1 x (R_dist x2 x0) H5); clear H5; intros. +generalize (H3 x2 (conj H4 H6)); generalize (H0 x2 (conj H4 H5)); intros; + replace eps with (eps * / 2 + eps * / 2). +exact (Rplus_lt_compat _ _ _ _ H7 H8). +exact (eps2 eps). +Qed. + +(*********) +Lemma limit_Ropp : + forall (f:R -> R) (D:R -> Prop) (l x0:R), + limit1_in f D l x0 -> limit1_in (fun x:R => - f x) D (- l) x0. +unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros; + elim (H eps H0); clear H; intros; elim H; clear H; + intros; split with x; split; auto; intros; generalize (H1 x1 H2); + clear H1; intro; unfold R_dist in |- *; unfold Rminus in |- *; + rewrite (Ropp_involutive l); rewrite (Rplus_comm (- f x1) l); + fold (l - f x1) in |- *; fold (R_dist l (f x1)) in |- *; + rewrite R_dist_sym; assumption. +Qed. + +(*********) +Lemma limit_minus : + forall (f g:R -> R) (D:R -> Prop) (l l' x0:R), + limit1_in f D l x0 -> + limit1_in g D l' x0 -> limit1_in (fun x:R => f x - g x) D (l - l') x0. +intros; unfold Rminus in |- *; generalize (limit_Ropp g D l' x0 H0); intro; + exact (limit_plus f (fun x:R => - g x) D l (- l') x0 H H1). +Qed. + +(*********) +Lemma limit_free : + forall (f:R -> R) (D:R -> Prop) (x x0:R), + limit1_in (fun h:R => f x) D (f x) x0. +unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros; + split with eps; split; auto; intros; elim (R_dist_refl (f x) (f x)); + intros a b; rewrite (b (refl_equal (f x))); unfold Rgt in H; + assumption. +Qed. + +(*********) +Lemma limit_mul : + forall (f g:R -> R) (D:R -> Prop) (l l' x0:R), + limit1_in f D l x0 -> + limit1_in g D l' x0 -> limit1_in (fun x:R => f x * g x) D (l * l') x0. +intros; unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; + intros; + elim (H (Rmin 1 (eps * mul_factor l l')) (mul_factor_gt_f eps l l' H1)); + elim (H0 (eps * mul_factor l l') (mul_factor_gt eps l l' H1)); + clear H H0; simpl in |- *; intros; elim H; elim H0; + clear H H0; intros; split with (Rmin x1 x); split. +exact (Rmin_Rgt_r x1 x 0 (conj H H2)). +intros; elim H4; clear H4; intros; unfold R_dist in |- *; + replace (f x2 * g x2 - l * l') with (f x2 * (g x2 - l') + l' * (f x2 - l)). +cut (Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l)) < eps). +cut + (Rabs (f x2 * (g x2 - l') + l' * (f x2 - l)) <= + Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l))). +exact (Rle_lt_trans _ _ _). +exact (Rabs_triang _ _). +rewrite (Rabs_mult (f x2) (g x2 - l')); rewrite (Rabs_mult l' (f x2 - l)); + cut + ((1 + Rabs l) * (eps * mul_factor l l') + Rabs l' * (eps * mul_factor l l') <= + eps). +cut + (Rabs (f x2) * Rabs (g x2 - l') + Rabs l' * Rabs (f x2 - l) < + (1 + Rabs l) * (eps * mul_factor l l') + Rabs l' * (eps * mul_factor l l')). +exact (Rlt_le_trans _ _ _). +elim (Rmin_Rgt_l x1 x (R_dist x2 x0) H5); clear H5; intros; + generalize (H0 x2 (conj H4 H5)); intro; generalize (Rmin_Rgt_l _ _ _ H7); + intro; elim H8; intros; clear H0 H8; apply Rplus_lt_le_compat. +apply Rmult_ge_0_gt_0_lt_compat. +apply Rle_ge. +exact (Rabs_pos (g x2 - l')). +rewrite (Rplus_comm 1 (Rabs l)); unfold Rgt in |- *; apply Rle_lt_0_plus_1; + exact (Rabs_pos l). +unfold R_dist in H9; + apply (Rplus_lt_reg_r (- Rabs l) (Rabs (f x2)) (1 + Rabs l)). +rewrite <- (Rplus_assoc (- Rabs l) 1 (Rabs l)); + rewrite (Rplus_comm (- Rabs l) 1); + rewrite (Rplus_assoc 1 (- Rabs l) (Rabs l)); rewrite (Rplus_opp_l (Rabs l)); + rewrite (proj1 (Rplus_ne 1)); rewrite (Rplus_comm (- Rabs l) (Rabs (f x2))); + generalize H9; cut (Rabs (f x2) - Rabs l <= Rabs (f x2 - l)). +exact (Rle_lt_trans _ _ _). +exact (Rabs_triang_inv _ _). +generalize (H3 x2 (conj H4 H6)); trivial. +apply Rmult_le_compat_l. +exact (Rabs_pos l'). +unfold Rle in |- *; left; assumption. +rewrite (Rmult_comm (1 + Rabs l) (eps * mul_factor l l')); + rewrite (Rmult_comm (Rabs l') (eps * mul_factor l l')); + rewrite <- + (Rmult_plus_distr_l (eps * mul_factor l l') (1 + Rabs l) (Rabs l')) + ; rewrite (Rmult_assoc eps (mul_factor l l') (1 + Rabs l + Rabs l')); + rewrite (Rplus_assoc 1 (Rabs l) (Rabs l')); unfold mul_factor in |- *; + rewrite (Rinv_l (1 + (Rabs l + Rabs l')) (mul_factor_wd l l')); + rewrite (proj1 (Rmult_ne eps)); apply Req_le; trivial. +ring. +Qed. + +(*********) +Definition adhDa (D:R -> Prop) (a:R) : Prop := + forall alp:R, alp > 0 -> exists x : R, D x /\ R_dist x a < alp. + +(*********) +Lemma single_limit : + forall (f:R -> R) (D:R -> Prop) (l l' x0:R), + adhDa D x0 -> limit1_in f D l x0 -> limit1_in f D l' x0 -> l = l'. +unfold limit1_in in |- *; unfold limit_in in |- *; intros. +cut (forall eps:R, eps > 0 -> dist R_met l l' < 2 * eps). +clear H0 H1; unfold dist in |- *; unfold R_met in |- *; unfold R_dist in |- *; + unfold Rabs in |- *; case (Rcase_abs (l - l')); intros. +cut (forall eps:R, eps > 0 -> - (l - l') < eps). +intro; generalize (prop_eps (- (l - l')) H1); intro; + generalize (Ropp_gt_lt_0_contravar (l - l') r); intro; + unfold Rgt in H3; generalize (Rgt_not_le (- (l - l')) 0 H3); + intro; elimtype False; auto. +intros; cut (eps * / 2 > 0). +intro; generalize (H0 (eps * / 2) H2); rewrite (Rmult_comm eps (/ 2)); + rewrite <- (Rmult_assoc 2 (/ 2) eps); rewrite (Rinv_r 2). +elim (Rmult_ne eps); intros a b; rewrite b; clear a b; trivial. +apply (Rlt_dichotomy_converse 2 0); right; generalize Rlt_0_1; intro; + unfold Rgt in |- *; generalize (Rplus_lt_compat_l 1 0 1 H3); + intro; elim (Rplus_ne 1); intros a b; rewrite a in H4; + clear a b; apply (Rlt_trans 0 1 2 H3 H4). +unfold Rgt in |- *; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2)); + rewrite <- (Rmult_0_r (/ 2)); apply (Rmult_lt_compat_l (/ 2) 0 eps); + auto. +apply (Rinv_0_lt_compat 2); cut (1 < 2). +intro; apply (Rlt_trans 0 1 2 Rlt_0_1 H2). +generalize (Rplus_lt_compat_l 1 0 1 Rlt_0_1); elim (Rplus_ne 1); intros a b; + rewrite a; clear a b; trivial. +(**) +cut (forall eps:R, eps > 0 -> l - l' < eps). +intro; generalize (prop_eps (l - l') H1); intro; elim (Rle_le_eq (l - l') 0); + intros a b; clear b; apply (Rminus_diag_uniq l l'); + apply a; split. +assumption. +apply (Rge_le (l - l') 0 r). +intros; cut (eps * / 2 > 0). +intro; generalize (H0 (eps * / 2) H2); rewrite (Rmult_comm eps (/ 2)); + rewrite <- (Rmult_assoc 2 (/ 2) eps); rewrite (Rinv_r 2). +elim (Rmult_ne eps); intros a b; rewrite b; clear a b; trivial. +apply (Rlt_dichotomy_converse 2 0); right; generalize Rlt_0_1; intro; + unfold Rgt in |- *; generalize (Rplus_lt_compat_l 1 0 1 H3); + intro; elim (Rplus_ne 1); intros a b; rewrite a in H4; + clear a b; apply (Rlt_trans 0 1 2 H3 H4). +unfold Rgt in |- *; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2)); + rewrite <- (Rmult_0_r (/ 2)); apply (Rmult_lt_compat_l (/ 2) 0 eps); + auto. +apply (Rinv_0_lt_compat 2); cut (1 < 2). +intro; apply (Rlt_trans 0 1 2 Rlt_0_1 H2). +generalize (Rplus_lt_compat_l 1 0 1 Rlt_0_1); elim (Rplus_ne 1); intros a b; + rewrite a; clear a b; trivial. +(**) +intros; unfold adhDa in H; elim (H0 eps H2); intros; elim (H1 eps H2); intros; + clear H0 H1; elim H3; elim H4; clear H3 H4; intros; + simpl in |- *; simpl in H1, H4; generalize (Rmin_Rgt x x1 0); + intro; elim H5; intros; clear H5; elim (H (Rmin x x1) (H7 (conj H3 H0))); + intros; elim H5; intros; clear H5 H H6 H7; + generalize (Rmin_Rgt x x1 (R_dist x2 x0)); intro; + elim H; intros; clear H H6; unfold Rgt in H5; elim (H5 H9); + intros; clear H5 H9; generalize (H1 x2 (conj H8 H6)); + generalize (H4 x2 (conj H8 H)); clear H8 H H6 H1 H4 H0 H3; + intros; + generalize + (Rplus_lt_compat (R_dist (f x2) l) eps (R_dist (f x2) l') eps H H0); + unfold R_dist in |- *; intros; rewrite (Rabs_minus_sym (f x2) l) in H1; + rewrite (Rmult_comm 2 eps); rewrite (Rmult_plus_distr_l eps 1 1); + elim (Rmult_ne eps); intros a b; rewrite a; clear a b; + generalize (R_dist_tri l l' (f x2)); unfold R_dist in |- *; + intros; + apply + (Rle_lt_trans (Rabs (l - l')) (Rabs (l - f x2) + Rabs (f x2 - l')) + (eps + eps) H3 H1). +Qed. + +(*********) +Lemma limit_comp : + forall (f g:R -> R) (Df Dg:R -> Prop) (l l' x0:R), + limit1_in f Df l x0 -> + limit1_in g Dg l' l -> limit1_in (fun x:R => g (f x)) (Dgf Df Dg f) l' x0. +unfold limit1_in, limit_in, Dgf in |- *; simpl in |- *. +intros f g Df Dg l l' x0 Hf Hg eps eps_pos. +elim (Hg eps eps_pos). +intros alpg lg. +elim (Hf alpg). +2: tauto. +intros alpf lf. +exists alpf. +intuition. +Qed. + +(*********) + +Lemma limit_inv : + forall (f:R -> R) (D:R -> Prop) (l x0:R), + limit1_in f D l x0 -> l <> 0 -> limit1_in (fun x:R => / f x) D (/ l) x0. +unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; + unfold R_dist in |- *; intros; elim (H (Rabs l / 2)). +intros delta1 H2; elim (H (eps * (Rsqr l / 2))). +intros delta2 H3; elim H2; elim H3; intros; exists (Rmin delta1 delta2); + split. +unfold Rmin in |- *; case (Rle_dec delta1 delta2); intro; assumption. +intro; generalize (H5 x); clear H5; intro H5; generalize (H7 x); clear H7; + intro H7; intro H10; elim H10; intros; cut (D x /\ Rabs (x - x0) < delta1). +cut (D x /\ Rabs (x - x0) < delta2). +intros; generalize (H5 H11); clear H5; intro H5; generalize (H7 H12); + clear H7; intro H7; generalize (Rabs_triang_inv l (f x)); + intro; rewrite Rabs_minus_sym in H7; + generalize + (Rle_lt_trans (Rabs l - Rabs (f x)) (Rabs (l - f x)) (Rabs l / 2) H13 H7); + intro; + generalize + (Rplus_lt_compat_l (Rabs (f x) - Rabs l / 2) (Rabs l - Rabs (f x)) + (Rabs l / 2) H14); + replace (Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))) with (Rabs l / 2). +unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_r; intro; cut (f x <> 0). +intro; replace (/ f x + - / l) with ((l - f x) * / (l * f x)). +rewrite Rabs_mult; rewrite Rabs_Rinv. +cut (/ Rabs (l * f x) < 2 / Rsqr l). +intro; rewrite Rabs_minus_sym in H5; cut (0 <= / Rabs (l * f x)). +intro; + generalize + (Rmult_le_0_lt_compat (Rabs (l - f x)) (eps * (Rsqr l / 2)) + (/ Rabs (l * f x)) (2 / Rsqr l) (Rabs_pos (l - f x)) H18 H5 H17); + replace (eps * (Rsqr l / 2) * (2 / Rsqr l)) with eps. +intro; assumption. +unfold Rdiv in |- *; unfold Rsqr in |- *; rewrite Rinv_mult_distr. +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm l). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite (Rmult_comm l). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; reflexivity. +discrR. +exact H0. +exact H0. +exact H0. +exact H0. +left; apply Rinv_0_lt_compat; apply Rabs_pos_lt; apply prod_neq_R0; + assumption. +rewrite Rmult_comm; rewrite Rabs_mult; rewrite Rinv_mult_distr. +rewrite (Rsqr_abs l); unfold Rsqr in |- *; unfold Rdiv in |- *; + rewrite Rinv_mult_distr. +repeat rewrite <- Rmult_assoc; apply Rmult_lt_compat_r. +apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. +apply Rmult_lt_reg_l with (Rabs (f x) * Rabs l * / 2). +repeat apply Rmult_lt_0_compat. +apply Rabs_pos_lt; assumption. +apply Rabs_pos_lt; assumption. +apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); + [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR in |- *; + intro H18; assumption + | discriminate ]. +replace (Rabs (f x) * Rabs l * / 2 * / Rabs (f x)) with (Rabs l / 2). +replace (Rabs (f x) * Rabs l * / 2 * (2 * / Rabs l)) with (Rabs (f x)). +assumption. +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (Rabs l)). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; reflexivity. +discrR. +apply Rabs_no_R0. +assumption. +unfold Rdiv in |- *. +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (Rabs (f x))). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +reflexivity. +apply Rabs_no_R0; assumption. +apply Rabs_no_R0; assumption. +apply Rabs_no_R0; assumption. +apply Rabs_no_R0; assumption. +apply Rabs_no_R0; assumption. +apply prod_neq_R0; assumption. +rewrite (Rinv_mult_distr _ _ H0 H16). +unfold Rminus in |- *; rewrite Rmult_plus_distr_r. +rewrite <- Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_l. +rewrite Ropp_mult_distr_l_reverse. +rewrite (Rmult_comm (f x)). +rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +reflexivity. +assumption. +assumption. +red in |- *; intro; rewrite H16 in H15; rewrite Rabs_R0 in H15; + cut (0 < Rabs l / 2). +intro; elim (Rlt_irrefl 0 (Rlt_trans 0 (Rabs l / 2) 0 H17 H15)). +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply Rabs_pos_lt; assumption. +apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); + [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR in |- *; + intro; assumption + | discriminate ]. +pattern (Rabs l) at 3 in |- *; rewrite double_var. +ring. +split; + [ assumption + | apply Rlt_le_trans with (Rmin delta1 delta2); + [ assumption | apply Rmin_r ] ]. +split; + [ assumption + | apply Rlt_le_trans with (Rmin delta1 delta2); + [ assumption | apply Rmin_l ] ]. +change (0 < eps * (Rsqr l / 2)) in |- *; unfold Rdiv in |- *; + repeat rewrite Rmult_assoc; repeat apply Rmult_lt_0_compat. +assumption. +apply Rsqr_pos_lt; assumption. +apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); + [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR in |- *; + intro; assumption + | discriminate ]. +change (0 < Rabs l / 2) in |- *; unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply Rabs_pos_lt; assumption + | apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); + [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR in |- *; + intro; assumption + | discriminate ] ]. +Qed. |