path: root/theories/Reals/Rlimit.v

(************************************************************************)
(*  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 $Id$ 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.