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|
(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \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) *)
(************************************************************************)
Require Import Rbase.
Require Import Rfunctions.
Require Import Ranalysis1.
Require Import Ranalysis2.
Local Open Scope R_scope.
(** Division *)
Theorem derivable_pt_lim_div :
forall (f1 f2:R -> R) (x l1 l2:R),
derivable_pt_lim f1 x l1 ->
derivable_pt_lim f2 x l2 ->
f2 x <> 0 ->
derivable_pt_lim (f1 / f2) x ((l1 * f2 x - l2 * f1 x) / Rsqr (f2 x)).
Proof.
intros f1 f2 x l1 l2 H H0 H1.
cut (derivable_pt f2 x);
[ intro X | unfold derivable_pt; exists l2; exact H0 ].
assert (H2 := continuous_neq_0 _ _ (derivable_continuous_pt _ _ X) H1).
elim H2; clear H2; intros eps_f2 H2.
unfold div_fct.
assert (H3 := derivable_continuous_pt _ _ X).
unfold continuity_pt in H3; unfold continue_in in H3; unfold limit1_in in H3;
unfold limit_in in H3; unfold dist in H3.
simpl in H3; unfold R_dist in H3.
elim (H3 (Rabs (f2 x) / 2));
[ idtac
| unfold Rdiv; change (0 < Rabs (f2 x) * / 2);
apply Rmult_lt_0_compat;
[ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
clear H3; intros alp_f2 H3.
cut
(forall x0:R,
Rabs (x0 - x) < alp_f2 -> Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2).
intro H4.
cut (forall a:R, Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)).
intro H5.
cut
(forall a:R,
Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)).
intro Maj.
unfold derivable_pt_lim; intros.
elim (H (Rabs (eps * f2 x / 8)));
[ idtac
| unfold Rdiv; change (0 < Rabs (eps * f2 x * / 8));
apply Rabs_pos_lt; repeat apply prod_neq_R0;
[ red; intro H7; rewrite H7 in H6; elim (Rlt_irrefl _ H6)
| assumption
| apply Rinv_neq_0_compat; discrR ] ].
intros alp_f1d H7.
case (Req_dec (f1 x) 0); intro.
case (Req_dec l1 0); intro.
(***********************************)
(* First case *)
(* (f1 x)=0 l1 =0 *)
(***********************************)
cut (0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d));
[ intro
| repeat apply Rmin_pos;
[ apply (cond_pos eps_f2)
| elim H3; intros; assumption
| apply (cond_pos alp_f1d) ] ].
exists (mkposreal (Rmin eps_f2 (Rmin alp_f2 alp_f1d)) H10).
simpl; intros.
assert (H13 := Rlt_le_trans _ _ _ H12 (Rmin_r _ _)).
assert (H14 := Rlt_le_trans _ _ _ H12 (Rmin_l _ _)).
assert (H15 := Rlt_le_trans _ _ _ H13 (Rmin_r _ _)).
assert (H16 := Rlt_le_trans _ _ _ H13 (Rmin_l _ _)).
assert (H17 := H7 _ H11 H15).
rewrite formule; [ idtac | assumption | assumption | apply H2; apply H14 ].
apply Rle_lt_trans with
(Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).
unfold Rminus.
rewrite <-
(Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
.
apply Rabs_4.
repeat rewrite Rabs_mult.
apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4).
cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4).
cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4).
cut
(Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) <
eps / 4).
cut
(Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) <
eps / 4).
intros.
apply Rlt_4; assumption.
rewrite H8.
unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
rewrite Rabs_R0; rewrite Rmult_0_l.
apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
rewrite H8.
unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
rewrite Rabs_R0; rewrite Rmult_0_l.
apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
rewrite H9.
unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
rewrite Rabs_R0; rewrite Rmult_0_l.
apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
rewrite <- Rabs_mult.
apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2);
try assumption || apply H2.
apply H14.
apply Rmin_2; assumption.
right; symmetry ; apply quadruple_var.
(***********************************)
(* Second case *)
(* (f1 x)=0 l1<>0 *)
(***********************************)
assert (H10 := derivable_continuous_pt _ _ X).
unfold continuity_pt in H10.
unfold continue_in in H10.
unfold limit1_in in H10.
unfold limit_in in H10.
unfold dist in H10.
simpl in H10.
unfold R_dist in H10.
elim (H10 (Rabs (eps * Rsqr (f2 x) / (8 * l1)))).
clear H10; intros alp_f2t2 H10.
cut
(forall a:R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))).
intro H11.
cut (0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)).
intro.
exists (mkposreal (Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)) H12).
simpl.
intros.
assert (H15 := Rlt_le_trans _ _ _ H14 (Rmin_r _ _)).
assert (H16 := Rlt_le_trans _ _ _ H14 (Rmin_l _ _)).
assert (H17 := Rlt_le_trans _ _ _ H15 (Rmin_l _ _)).
assert (H18 := Rlt_le_trans _ _ _ H15 (Rmin_r _ _)).
assert (H19 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)).
assert (H20 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)).
clear H14 H15 H16.
rewrite formule; try assumption.
apply Rle_lt_trans with
(Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).
unfold Rminus.
rewrite <-
(Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
.
apply Rabs_4.
repeat rewrite Rabs_mult.
apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4).
cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4).
cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4).
cut
(Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) <
eps / 4).
cut
(Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) <
eps / 4).
intros.
apply Rlt_4; assumption.
rewrite H8.
unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
rewrite Rabs_R0; rewrite Rmult_0_l.
apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
rewrite H8.
unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
rewrite Rabs_R0; rewrite Rmult_0_l.
apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
rewrite <- Rabs_mult.
apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
rewrite <- Rabs_mult.
apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
right; symmetry ; apply quadruple_var.
apply H2; assumption.
repeat apply Rmin_pos.
apply (cond_pos eps_f2).
apply (cond_pos alp_f1d).
elim H3; intros; assumption.
elim H10; intros; assumption.
intros.
elim H10; intros.
case (Req_dec a 0); intro.
rewrite H14; rewrite Rplus_0_r.
unfold Rminus; rewrite Rplus_opp_r.
rewrite Rabs_R0.
apply Rabs_pos_lt.
unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc.
repeat apply prod_neq_R0; try assumption.
now apply Rgt_not_eq.
apply Rinv_neq_0_compat; apply prod_neq_R0; [discrR | assumption].
apply H13.
split.
apply D_x_no_cond; assumption.
replace (x + a - x) with a; [ assumption | ring ].
change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))).
apply Rabs_pos_lt; unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc;
repeat apply prod_neq_R0.
red; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6).
assumption.
assumption.
apply Rinv_neq_0_compat; apply prod_neq_R0; [discrR | assumption].
(***********************************)
(* Third case *)
(* (f1 x)<>0 l1=0 l2=0 *)
(***********************************)
case (Req_dec l1 0); intro.
case (Req_dec l2 0); intro.
elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x))));
[ idtac
| apply Rabs_pos_lt; unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc;
repeat apply prod_neq_R0 ;
[ assumption
| assumption
| now apply Rgt_not_eq
| apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption ] ].
intros alp_f2d H12.
cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)).
intro.
exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) H11).
simpl.
intros.
assert (H15 := Rlt_le_trans _ _ _ H14 (Rmin_l _ _)).
assert (H16 := Rlt_le_trans _ _ _ H14 (Rmin_r _ _)).
assert (H17 := Rlt_le_trans _ _ _ H15 (Rmin_l _ _)).
assert (H18 := Rlt_le_trans _ _ _ H15 (Rmin_r _ _)).
assert (H19 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)).
assert (H20 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)).
clear H15 H16.
rewrite formule; try assumption.
apply Rle_lt_trans with
(Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).
unfold Rminus.
rewrite <-
(Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
.
apply Rabs_4.
repeat rewrite Rabs_mult.
apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4).
cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4).
cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4).
cut
(Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) <
eps / 4).
cut
(Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) <
eps / 4).
intros.
apply Rlt_4; assumption.
rewrite H10.
unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
rewrite Rabs_R0; rewrite Rmult_0_l.
apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
rewrite <- Rabs_mult.
apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
rewrite H9.
unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
rewrite Rabs_R0; rewrite Rmult_0_l.
apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
rewrite <- Rabs_mult.
apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); assumption || idtac.
apply H2; assumption.
apply Rmin_2; assumption.
right; symmetry ; apply quadruple_var.
apply H2; assumption.
repeat apply Rmin_pos.
apply (cond_pos eps_f2).
elim H3; intros; assumption.
apply (cond_pos alp_f1d).
apply (cond_pos alp_f2d).
(***********************************)
(* Fourth case *)
(* (f1 x)<>0 l1=0 l2<>0 *)
(***********************************)
elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x))));
[ idtac
| apply Rabs_pos_lt; unfold Rsqr, Rdiv;
repeat apply prod_neq_R0 ;
[ assumption..
| now apply Rgt_not_eq
| apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption ] ].
intros alp_f2d H11.
assert (H12 := derivable_continuous_pt _ _ X).
unfold continuity_pt in H12.
unfold continue_in in H12.
unfold limit1_in in H12.
unfold limit_in in H12.
unfold dist in H12.
simpl in H12.
unfold R_dist in H12.
elim (H12 (Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2)))).
intros alp_f2c H13.
cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c))).
intro.
exists
(mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c)))
H14).
simpl; intros.
assert (H17 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)).
assert (H18 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)).
assert (H19 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)).
assert (H20 := Rlt_le_trans _ _ _ H19 (Rmin_l _ _)).
assert (H21 := Rlt_le_trans _ _ _ H19 (Rmin_r _ _)).
assert (H22 := Rlt_le_trans _ _ _ H18 (Rmin_l _ _)).
assert (H23 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)).
assert (H24 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)).
clear H16 H17 H18 H19.
cut
(forall a:R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).
intro.
rewrite formule; try assumption.
apply Rle_lt_trans with
(Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).
unfold Rminus.
rewrite <-
(Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
.
apply Rabs_4.
repeat rewrite Rabs_mult.
apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4).
cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4).
cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4).
cut
(Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) <
eps / 4).
cut
(Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) <
eps / 4).
intros.
apply Rlt_4; assumption.
rewrite <- Rabs_mult.
apply (maj_term4 x h eps l2 alp_f2 alp_f2c eps_f2 f1 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
rewrite <- Rabs_mult.
apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
rewrite H9.
unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
rewrite Rabs_R0; rewrite Rmult_0_l.
apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
rewrite <- Rabs_mult.
apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
right; symmetry ; apply quadruple_var.
apply H2; assumption.
intros.
case (Req_dec a 0); intro.
rewrite H17; rewrite Rplus_0_r.
unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0.
apply Rabs_pos_lt.
unfold Rdiv, Rsqr.
repeat rewrite Rinv_mult.
repeat apply prod_neq_R0; try assumption.
red; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6).
apply Rinv_neq_0_compat; discrR.
apply Rinv_neq_0_compat; assumption.
apply Rinv_neq_0_compat; assumption.
discrR.
elim H13; intros.
apply H19.
split.
apply D_x_no_cond; assumption.
replace (x + a - x) with a; [ assumption | ring ].
repeat apply Rmin_pos.
apply (cond_pos eps_f2).
elim H3; intros; assumption.
apply (cond_pos alp_f1d).
apply (cond_pos alp_f2d).
elim H13; intros; assumption.
change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).
apply Rabs_pos_lt.
unfold Rsqr, Rdiv.
repeat rewrite Rinv_mult.
repeat apply prod_neq_R0; try assumption.
red; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6).
apply Rinv_neq_0_compat; discrR.
apply Rinv_neq_0_compat; assumption.
apply Rinv_neq_0_compat; assumption.
(***********************************)
(* Fifth case *)
(* (f1 x)<>0 l1<>0 l2=0 *)
(***********************************)
case (Req_dec l2 0); intro.
assert (H11 := derivable_continuous_pt _ _ X).
unfold continuity_pt in H11.
unfold continue_in in H11.
unfold limit1_in in H11.
unfold limit_in in H11.
unfold dist in H11.
simpl in H11.
unfold R_dist in H11.
elim (H11 (Rabs (eps * Rsqr (f2 x) / (8 * l1)))).
clear H11; intros alp_f2t2 H11.
elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))).
intros alp_f2d H12.
cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))).
intro.
exists
(mkposreal
(Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))) H13).
simpl.
intros.
cut
(forall a:R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))).
intro.
assert (H17 := Rlt_le_trans _ _ _ H15 (Rmin_l _ _)).
assert (H18 := Rlt_le_trans _ _ _ H15 (Rmin_r _ _)).
assert (H19 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)).
assert (H20 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)).
assert (H21 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)).
assert (H22 := Rlt_le_trans _ _ _ H18 (Rmin_l _ _)).
assert (H23 := Rlt_le_trans _ _ _ H21 (Rmin_l _ _)).
assert (H24 := Rlt_le_trans _ _ _ H21 (Rmin_r _ _)).
clear H15 H17 H18 H21.
rewrite formule; try assumption.
apply Rle_lt_trans with
(Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).
unfold Rminus.
rewrite <-
(Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
.
apply Rabs_4.
repeat rewrite Rabs_mult.
apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4).
cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4).
cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4).
cut
(Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) <
eps / 4).
cut
(Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) <
eps / 4).
intros.
apply Rlt_4; assumption.
rewrite H10.
unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
rewrite Rabs_R0; rewrite Rmult_0_l.
apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
rewrite <- Rabs_mult.
apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
rewrite <- Rabs_mult.
apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
rewrite <- Rabs_mult.
apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
right; symmetry ; apply quadruple_var.
apply H2; assumption.
intros.
case (Req_dec a 0); intro.
rewrite H17; rewrite Rplus_0_r; unfold Rminus; rewrite Rplus_opp_r;
rewrite Rabs_R0.
apply Rabs_pos_lt.
unfold Rdiv; rewrite Rinv_mult.
unfold Rsqr.
repeat apply prod_neq_R0;
assumption ||
(apply Rinv_neq_0_compat; assumption) ||
(apply Rinv_neq_0_compat; discrR) ||
(red; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6)).
elim H11; intros.
apply H19.
split.
apply D_x_no_cond; assumption.
replace (x + a - x) with a; [ assumption | ring ].
repeat apply Rmin_pos.
apply (cond_pos eps_f2).
elim H3; intros; assumption.
apply (cond_pos alp_f1d).
apply (cond_pos alp_f2d).
elim H11; intros; assumption.
apply Rabs_pos_lt.
unfold Rdiv, Rsqr; rewrite Rinv_mult.
repeat apply prod_neq_R0;
assumption ||
(apply Rinv_neq_0_compat; assumption) ||
(apply Rinv_neq_0_compat; discrR) ||
(red; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)).
change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))).
apply Rabs_pos_lt.
unfold Rdiv, Rsqr; rewrite Rinv_mult.
repeat apply prod_neq_R0;
assumption ||
(apply Rinv_neq_0_compat; assumption) ||
(apply Rinv_neq_0_compat; discrR) ||
(red; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)).
(***********************************)
(* Sixth case *)
(* (f1 x)<>0 l1<>0 l2<>0 *)
(***********************************)
elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))).
intros alp_f2d H11.
assert (H12 := derivable_continuous_pt _ _ X).
unfold continuity_pt in H12.
unfold continue_in in H12.
unfold limit1_in in H12.
unfold limit_in in H12.
unfold dist in H12.
simpl in H12.
unfold R_dist in H12.
elim (H12 (Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2)))).
intros alp_f2c H13.
elim (H12 (Rabs (eps * Rsqr (f2 x) / (8 * l1)))).
intros alp_f2t2 H14.
cut
(0 <
Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
(Rmin alp_f2c alp_f2t2)).
intro.
exists
(mkposreal
(Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
(Rmin alp_f2c alp_f2t2)) H15).
simpl.
intros.
assert (H18 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)).
assert (H19 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)).
assert (H20 := Rlt_le_trans _ _ _ H18 (Rmin_l _ _)).
assert (H21 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)).
assert (H22 := Rlt_le_trans _ _ _ H19 (Rmin_l _ _)).
assert (H23 := Rlt_le_trans _ _ _ H19 (Rmin_r _ _)).
assert (H24 := Rlt_le_trans _ _ _ H20 (Rmin_l _ _)).
assert (H25 := Rlt_le_trans _ _ _ H20 (Rmin_r _ _)).
assert (H26 := Rlt_le_trans _ _ _ H21 (Rmin_l _ _)).
assert (H27 := Rlt_le_trans _ _ _ H21 (Rmin_r _ _)).
clear H17 H18 H19 H20 H21.
cut
(forall a:R,
Rabs a < alp_f2t2 ->
Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))).
cut
(forall a:R,
Rabs a < alp_f2c ->
Rabs (f2 (x + a) - f2 x) <
Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).
intros.
rewrite formule; try assumption.
apply Rle_lt_trans with
(Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) +
Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).
unfold Rminus.
rewrite <-
(Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
.
apply Rabs_4.
repeat rewrite Rabs_mult.
apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4).
cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4).
cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4).
cut
(Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) <
eps / 4).
cut
(Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) <
eps / 4).
intros.
apply Rlt_4; assumption.
rewrite <- Rabs_mult.
apply (maj_term4 x h eps l2 alp_f2 alp_f2c eps_f2 f1 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
rewrite <- Rabs_mult.
apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
rewrite <- Rabs_mult.
apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
rewrite <- Rabs_mult.
apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption.
apply H2; assumption.
apply Rmin_2; assumption.
right; symmetry ; apply quadruple_var.
apply H2; assumption.
intros.
case (Req_dec a 0); intro.
rewrite H18; rewrite Rplus_0_r; unfold Rminus; rewrite Rplus_opp_r;
rewrite Rabs_R0; apply Rabs_pos_lt.
unfold Rdiv, Rsqr; rewrite Rinv_mult.
repeat apply prod_neq_R0;
assumption ||
(apply Rinv_neq_0_compat; assumption) ||
(apply Rinv_neq_0_compat; discrR) ||
(red; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)).
apply prod_neq_R0; [ discrR | assumption ].
elim H13; intros.
apply H20.
split.
apply D_x_no_cond; assumption.
replace (x + a - x) with a; [ assumption | ring ].
intros.
case (Req_dec a 0); intro.
rewrite H18; rewrite Rplus_0_r; unfold Rminus; rewrite Rplus_opp_r;
rewrite Rabs_R0; apply Rabs_pos_lt.
unfold Rdiv, Rsqr; rewrite Rinv_mult.
repeat apply prod_neq_R0;
assumption ||
(apply Rinv_neq_0_compat; assumption) ||
(apply Rinv_neq_0_compat; discrR) ||
(red; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)).
elim H14; intros.
apply H20.
split.
unfold D_x, no_cond; split.
trivial.
apply Rminus_not_eq_right.
replace (x + a - x) with a; [ assumption | ring ].
replace (x + a - x) with a; [ assumption | ring ].
repeat apply Rmin_pos.
apply (cond_pos eps_f2).
elim H3; intros; assumption.
apply (cond_pos alp_f1d).
apply (cond_pos alp_f2d).
elim H13; intros; assumption.
elim H14; intros; assumption.
change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))); apply Rabs_pos_lt.
unfold Rdiv, Rsqr; rewrite Rinv_mult.
repeat apply prod_neq_R0;
assumption ||
(apply Rinv_neq_0_compat; assumption) ||
(apply Rinv_neq_0_compat; discrR) ||
(red; intro H14; rewrite H14 in H6; elim (Rlt_irrefl _ H6)).
change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2)));
apply Rabs_pos_lt.
unfold Rdiv, Rsqr; rewrite Rinv_mult.
repeat apply prod_neq_R0;
assumption ||
(apply Rinv_neq_0_compat; assumption) ||
(apply Rinv_neq_0_compat; discrR) ||
(red; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6)).
apply prod_neq_R0; [ discrR | assumption ].
apply Rabs_pos_lt.
unfold Rdiv, Rsqr; rewrite Rinv_mult.
repeat apply prod_neq_R0;
assumption ||
(apply Rinv_neq_0_compat; assumption) ||
(apply Rinv_neq_0_compat; discrR) ||
(red; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6)).
intros.
unfold Rdiv.
apply Rmult_lt_reg_l with (Rabs (f2 (x + a))).
apply Rabs_pos_lt; apply H2.
apply Rlt_le_trans with (Rmin eps_f2 alp_f2).
assumption.
apply Rmin_l.
rewrite <- Rinv_r_sym.
apply Rmult_lt_reg_l with (Rabs (f2 x)).
apply Rabs_pos_lt; assumption.
rewrite Rmult_1_r.
rewrite (Rmult_comm (Rabs (f2 x))).
repeat rewrite Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_r.
apply Rmult_lt_reg_l with (/ 2).
apply Rinv_0_lt_compat; prove_sup0.
repeat rewrite (Rmult_comm (/ 2)).
repeat rewrite Rmult_assoc.
rewrite <- Rinv_r_sym.
rewrite Rmult_1_r.
unfold Rdiv in H5; apply H5.
replace (x + a - x) with a.
assert (H7 := Rlt_le_trans _ _ _ H6 (Rmin_r _ _)); assumption.
ring.
discrR.
apply Rabs_no_R0; assumption.
apply Rabs_no_R0; apply H2.
assert (H7 := Rlt_le_trans _ _ _ H6 (Rmin_l _ _)); assumption.
intros.
assert (H6 := H4 a H5).
rewrite <- (Rabs_Ropp (f2 a - f2 x)) in H6.
rewrite Ropp_minus_distr in H6.
assert (H7 := Rle_lt_trans _ _ _ (Rabs_triang_inv _ _) H6).
apply Rplus_lt_reg_l with (- Rabs (f2 a) + Rabs (f2 x) / 2).
rewrite Rplus_assoc.
rewrite <- double_var.
do 2 rewrite (Rplus_comm (- Rabs (f2 a))).
rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r.
unfold Rminus in H7; assumption.
intros.
case (Req_dec x x0); intro.
rewrite <- H5; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
unfold Rdiv; apply Rmult_lt_0_compat;
[ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim H3; intros.
apply H7.
split.
unfold D_x, no_cond; split.
trivial.
assumption.
assumption.
Qed.
Lemma derivable_pt_div :
forall (f1 f2:R -> R) (x:R),
derivable_pt f1 x ->
derivable_pt f2 x -> f2 x <> 0 -> derivable_pt (f1 / f2) x.
Proof.
unfold derivable_pt.
intros f1 f2 x X X0 H.
elim X; intros.
elim X0; intros.
exists ((x0 * f2 x - x1 * f1 x) / Rsqr (f2 x)).
apply derivable_pt_lim_div; assumption.
Qed.
Lemma derivable_div :
forall f1 f2:R -> R,
derivable f1 ->
derivable f2 -> (forall x:R, f2 x <> 0) -> derivable (f1 / f2).
Proof.
unfold derivable; intros f1 f2 X X0 H x.
apply (derivable_pt_div _ _ _ (X x) (X0 x) (H x)).
Qed.
Lemma derive_pt_div :
forall (f1 f2:R -> R) (x:R) (pr1:derivable_pt f1 x)
(pr2:derivable_pt f2 x) (na:f2 x <> 0),
derive_pt (f1 / f2) x (derivable_pt_div _ _ _ pr1 pr2 na) =
(derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) / Rsqr (f2 x).
Proof.
intros.
assert (H := derivable_derive f1 x pr1).
assert (H0 := derivable_derive f2 x pr2).
assert
(H1 := derivable_derive (f1 / f2)%F x (derivable_pt_div _ _ _ pr1 pr2 na)).
elim H; clear H; intros l1 H.
elim H0; clear H0; intros l2 H0.
elim H1; clear H1; intros l H1.
rewrite H; rewrite H0; apply derive_pt_eq_0.
assert (H3 := proj2_sig pr1).
unfold derive_pt in H; rewrite H in H3.
assert (H4 := proj2_sig pr2).
unfold derive_pt in H0; rewrite H0 in H4.
apply derivable_pt_lim_div; assumption.
Qed.
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