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(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(*********************************************************)
(** Definition of the limit *)
(* *)
(*********************************************************)
Require Import Rbase.
Require Import Rfunctions.
Require Import Lra.
Local Open Scope R_scope.
(*******************************)
(** * Calculus *)
(*******************************)
(*********)
Lemma eps2_Rgt_R0 : forall eps:R, eps > 0 -> eps * / 2 > 0.
Proof.
intros; lra.
Qed.
(*********)
Lemma eps2 : forall eps:R, eps * / 2 + eps * / 2 = eps.
Proof.
intro esp.
apply eq_sym, double_var.
Qed.
(*********)
Lemma eps4 : forall eps:R, eps * / (2 + 2) + eps * / (2 + 2) = eps * / 2.
Proof.
intro eps.
field.
Qed.
(*********)
Lemma Rlt_eps2_eps : forall eps:R, eps > 0 -> eps * / 2 < eps.
Proof.
intros.
lra.
Qed.
(*********)
Lemma Rlt_eps4_eps : forall eps:R, eps > 0 -> eps * / (2 + 2) < eps.
Proof.
intros.
lra.
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;
exfalso; 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; rewrite <- (Rmult_0_r eps);
apply Rmult_lt_compat_l.
assumption.
unfold mul_factor; 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.
Declare Equivalent Keys dist R_dist.
(*******************************)
(** * 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; symmetry; 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; unfold limit_in; simpl; 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; unfold limit_in; simpl;
intros; elim (H (eps * / 2) (eps2_Rgt_R0 eps H1));
elim (H0 (eps * / 2) (eps2_Rgt_R0 eps H1)); simpl;
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; unfold limit_in; simpl; 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; unfold Rminus;
rewrite (Ropp_involutive l); rewrite (Rplus_comm (- f x1) l);
fold (l - f x1); fold (R_dist l (f x1));
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; 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; unfold limit_in; simpl; intros;
split with eps; split; auto; intros; elim (R_dist_refl (f x) (f x));
intros a b; rewrite (b (eq_refl (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; unfold limit_in; simpl;
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; 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;
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; apply Rle_lt_0_plus_1;
exact (Rabs_pos l).
unfold R_dist in H9;
apply (Rplus_lt_reg_l (- 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; 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;
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; unfold limit_in; intros.
simpl in *.
cut (forall eps:R, eps > 0 -> dist R_met l l' < 2 * eps).
clear H0 H1; unfold dist in |- *; unfold R_met; unfold R_dist in |- *;
unfold Rabs; case (Rcase_abs (l - l')) as [Hlt|Hge]; 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') Hlt); intro;
unfold Rgt in H3; generalize (Rgt_not_le (- (l - l')) 0 H3);
intro; exfalso; 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; 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; 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 Hge).
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; 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; 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; 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; intros; rewrite (Rabs_minus_sym (f x2) l) in H1;
rewrite (Rmult_comm 2 eps); replace (eps *2) with (eps + eps) by ring;
generalize (R_dist_tri l l' (f x2)); unfold R_dist;
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; simpl.
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; unfold limit_in; simpl;
unfold R_dist; 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; 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; 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; unfold Rsqr; 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; unfold Rdiv;
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;
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.
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; 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; 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; 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;
intro; assumption
| discriminate ].
pattern (Rabs l) at 3; 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)); unfold Rdiv;
repeat rewrite Rmult_assoc; apply Rmult_lt_0_compat.
assumption.
apply Rmult_lt_0_compat. 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;
intro; assumption
| discriminate ].
change (0 < Rabs l / 2); unfold Rdiv; 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;
intro; assumption
| discriminate ] ].
Qed.
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