Library Coq.Reals.Rlimit

``` ```
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. Proof.   intros; fourier. Qed. Lemma eps2 : forall eps:R, eps * / 2 + eps * / 2 = eps. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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. Proof.   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'. Proof.   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. Proof.   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. Proof.   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. ```