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|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Rderiv.v 13323 2010-07-24 15:57:30Z herbelin $ i*)
(*********************************************************)
(** Definition of the derivative,continuity *)
(* *)
(*********************************************************)
Require Import Rbase.
Require Import Rfunctions.
Require Import Rlimit.
Require Import Fourier.
Require Import Classical_Prop.
Require Import Classical_Pred_Type.
Require Import Omega.
Open Local Scope R_scope.
(*********)
Definition D_x (D:R -> Prop) (y x:R) : Prop := D x /\ y <> x.
(*********)
Definition continue_in (f:R -> R) (D:R -> Prop) (x0:R) : Prop :=
limit1_in f (D_x D x0) (f x0) x0.
(*********)
Definition D_in (f d:R -> R) (D:R -> Prop) (x0:R) : Prop :=
limit1_in (fun x:R => (f x - f x0) / (x - x0)) (D_x D x0) (d x0) x0.
(*********)
Lemma cont_deriv :
forall (f d:R -> R) (D:R -> Prop) (x0:R),
D_in f d D x0 -> continue_in f D x0.
Proof.
unfold continue_in in |- *; unfold D_in in |- *; unfold limit1_in in |- *;
unfold limit_in in |- *; unfold Rdiv in |- *; simpl in |- *;
intros; elim (H eps H0); clear H; intros; elim H;
clear H; intros; elim (Req_dec (d x0) 0); intro.
split with (Rmin 1 x); split.
elim (Rmin_Rgt 1 x 0); intros a b; apply (b (conj Rlt_0_1 H)).
intros; elim H3; clear H3; intros;
generalize (let (H1, H2) := Rmin_Rgt 1 x (R_dist x1 x0) in H1);
unfold Rgt in |- *; intro; elim (H5 H4); clear H5;
intros; generalize (H1 x1 (conj H3 H6)); clear H1;
intro; unfold D_x in H3; elim H3; intros.
rewrite H2 in H1; unfold R_dist in |- *; unfold R_dist in H1;
cut (Rabs (f x1 - f x0) < eps * Rabs (x1 - x0)).
intro; unfold R_dist in H5;
generalize (Rmult_lt_compat_l eps (Rabs (x1 - x0)) 1 H0 H5);
rewrite Rmult_1_r; intro; apply Rlt_trans with (r2 := eps * Rabs (x1 - x0));
assumption.
rewrite (Rminus_0_r ((f x1 - f x0) * / (x1 - x0))) in H1;
rewrite Rabs_mult in H1; cut (x1 - x0 <> 0).
intro; rewrite (Rabs_Rinv (x1 - x0) H9) in H1;
generalize
(Rmult_lt_compat_l (Rabs (x1 - x0)) (Rabs (f x1 - f x0) * / Rabs (x1 - x0))
eps (Rabs_pos_lt (x1 - x0) H9) H1); intro; rewrite Rmult_comm in H10;
rewrite Rmult_assoc in H10; rewrite Rinv_l in H10.
rewrite Rmult_1_r in H10; rewrite Rmult_comm; assumption.
apply Rabs_no_R0; auto.
apply Rminus_eq_contra; auto.
(**)
split with (Rmin (Rmin (/ 2) x) (eps * / Rabs (2 * d x0))); split.
cut (Rmin (/ 2) x > 0).
cut (eps * / Rabs (2 * d x0) > 0).
intros; elim (Rmin_Rgt (Rmin (/ 2) x) (eps * / Rabs (2 * d x0)) 0);
intros a b; apply (b (conj H4 H3)).
apply Rmult_gt_0_compat; auto.
unfold Rgt in |- *; apply Rinv_0_lt_compat; apply Rabs_pos_lt;
apply Rmult_integral_contrapositive; split.
discrR.
assumption.
elim (Rmin_Rgt (/ 2) x 0); intros a b; cut (0 < 2).
intro; generalize (Rinv_0_lt_compat 2 H3); intro; fold (/ 2 > 0) in H4;
apply (b (conj H4 H)).
fourier.
intros; elim H3; clear H3; intros;
generalize
(let (H1, H2) :=
Rmin_Rgt (Rmin (/ 2) x) (eps * / Rabs (2 * d x0)) (R_dist x1 x0) in
H1); unfold Rgt in |- *; intro; elim (H5 H4); clear H5;
intros; generalize (let (H1, H2) := Rmin_Rgt (/ 2) x (R_dist x1 x0) in H1);
unfold Rgt in |- *; intro; elim (H7 H5); clear H7;
intros; clear H4 H5; generalize (H1 x1 (conj H3 H8));
clear H1; intro; unfold D_x in H3; elim H3; intros;
generalize (sym_not_eq H5); clear H5; intro H5;
generalize (Rminus_eq_contra x1 x0 H5); intro; generalize H1;
pattern (d x0) at 1 in |- *;
rewrite <- (let (H1, H2) := Rmult_ne (d x0) in H2);
rewrite <- (Rinv_l (x1 - x0) H9); unfold R_dist in |- *;
unfold Rminus at 1 in |- *; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0)));
rewrite (Rmult_comm (/ (x1 - x0) * (x1 - x0)) (d x0));
rewrite <- (Ropp_mult_distr_l_reverse (d x0) (/ (x1 - x0) * (x1 - x0)));
rewrite (Rmult_comm (- d x0) (/ (x1 - x0) * (x1 - x0)));
rewrite (Rmult_assoc (/ (x1 - x0)) (x1 - x0) (- d x0));
rewrite <-
(Rmult_plus_distr_l (/ (x1 - x0)) (f x1 - f x0) ((x1 - x0) * - d x0))
; rewrite (Rabs_mult (/ (x1 - x0)) (f x1 - f x0 + (x1 - x0) * - d x0));
clear H1; intro;
generalize
(Rmult_lt_compat_l (Rabs (x1 - x0))
(Rabs (/ (x1 - x0)) * Rabs (f x1 - f x0 + (x1 - x0) * - d x0)) eps
(Rabs_pos_lt (x1 - x0) H9) H1);
rewrite <-
(Rmult_assoc (Rabs (x1 - x0)) (Rabs (/ (x1 - x0)))
(Rabs (f x1 - f x0 + (x1 - x0) * - d x0)));
rewrite (Rabs_Rinv (x1 - x0) H9);
rewrite (Rinv_r (Rabs (x1 - x0)) (Rabs_no_R0 (x1 - x0) H9));
rewrite
(let (H1, H2) := Rmult_ne (Rabs (f x1 - f x0 + (x1 - x0) * - d x0)) in H2)
; generalize (Rabs_triang_inv (f x1 - f x0) ((x1 - x0) * d x0));
intro; rewrite (Rmult_comm (x1 - x0) (- d x0));
rewrite (Ropp_mult_distr_l_reverse (d x0) (x1 - x0));
fold (f x1 - f x0 - d x0 * (x1 - x0)) in |- *;
rewrite (Rmult_comm (x1 - x0) (d x0)) in H10; clear H1;
intro;
generalize
(Rle_lt_trans (Rabs (f x1 - f x0) - Rabs (d x0 * (x1 - x0)))
(Rabs (f x1 - f x0 - d x0 * (x1 - x0))) (Rabs (x1 - x0) * eps) H10 H1);
clear H1; intro;
generalize
(Rplus_lt_compat_l (Rabs (d x0 * (x1 - x0)))
(Rabs (f x1 - f x0) - Rabs (d x0 * (x1 - x0))) (
Rabs (x1 - x0) * eps) H1); unfold Rminus at 2 in |- *;
rewrite (Rplus_comm (Rabs (f x1 - f x0)) (- Rabs (d x0 * (x1 - x0))));
rewrite <-
(Rplus_assoc (Rabs (d x0 * (x1 - x0))) (- Rabs (d x0 * (x1 - x0)))
(Rabs (f x1 - f x0))); rewrite (Rplus_opp_r (Rabs (d x0 * (x1 - x0))));
rewrite (let (H1, H2) := Rplus_ne (Rabs (f x1 - f x0)) in H2);
clear H1; intro; cut (Rabs (d x0 * (x1 - x0)) + Rabs (x1 - x0) * eps < eps).
intro;
apply
(Rlt_trans (Rabs (f x1 - f x0))
(Rabs (d x0 * (x1 - x0)) + Rabs (x1 - x0) * eps) eps H1 H11).
clear H1 H5 H3 H10; generalize (Rabs_pos_lt (d x0) H2); intro;
unfold Rgt in H0;
generalize (Rmult_lt_compat_l eps (R_dist x1 x0) (/ 2) H0 H7);
clear H7; intro;
generalize
(Rmult_lt_compat_l (Rabs (d x0)) (R_dist x1 x0) (
eps * / Rabs (2 * d x0)) H1 H6); clear H6; intro;
rewrite (Rmult_comm eps (R_dist x1 x0)) in H3; unfold R_dist in H3, H5;
rewrite <- (Rabs_mult (d x0) (x1 - x0)) in H5;
rewrite (Rabs_mult 2 (d x0)) in H5; cut (Rabs 2 <> 0).
intro; fold (Rabs (d x0) > 0) in H1;
rewrite
(Rinv_mult_distr (Rabs 2) (Rabs (d x0)) H6
(Rlt_dichotomy_converse (Rabs (d x0)) 0 (or_intror (Rabs (d x0) < 0) H1)))
in H5;
rewrite (Rmult_comm (Rabs (d x0)) (eps * (/ Rabs 2 * / Rabs (d x0)))) in H5;
rewrite <- (Rmult_assoc eps (/ Rabs 2) (/ Rabs (d x0))) in H5;
rewrite (Rmult_assoc (eps * / Rabs 2) (/ Rabs (d x0)) (Rabs (d x0))) in H5;
rewrite
(Rinv_l (Rabs (d x0))
(Rlt_dichotomy_converse (Rabs (d x0)) 0 (or_intror (Rabs (d x0) < 0) H1)))
in H5; rewrite (let (H1, H2) := Rmult_ne (eps * / Rabs 2) in H1) in H5;
cut (Rabs 2 = 2).
intro; rewrite H7 in H5;
generalize
(Rplus_lt_compat (Rabs (d x0 * (x1 - x0))) (eps * / 2)
(Rabs (x1 - x0) * eps) (eps * / 2) H5 H3); intro;
rewrite eps2 in H10; assumption.
unfold Rabs in |- *; case (Rcase_abs 2); auto.
intro; cut (0 < 2).
intro; generalize (Rlt_asym 0 2 H7); intro; exfalso; auto.
fourier.
apply Rabs_no_R0.
discrR.
Qed.
(*********)
Lemma Dconst :
forall (D:R -> Prop) (y x0:R), D_in (fun x:R => y) (fun x:R => 0) D x0.
Proof.
unfold D_in in |- *; intros; unfold limit1_in in |- *;
unfold limit_in in |- *; unfold Rdiv in |- *; intros;
simpl in |- *; split with eps; split; auto.
intros; rewrite (Rminus_diag_eq y y (refl_equal y)); rewrite Rmult_0_l;
unfold R_dist in |- *; rewrite (Rminus_diag_eq 0 0 (refl_equal 0));
unfold Rabs in |- *; case (Rcase_abs 0); intro.
absurd (0 < 0); auto.
red in |- *; intro; apply (Rlt_irrefl 0 H1).
unfold Rgt in H0; assumption.
Qed.
(*********)
Lemma Dx :
forall (D:R -> Prop) (x0:R), D_in (fun x:R => x) (fun x:R => 1) D x0.
Proof.
unfold D_in in |- *; unfold Rdiv in |- *; intros; unfold limit1_in in |- *;
unfold limit_in in |- *; intros; simpl in |- *; split with eps;
split; auto.
intros; elim H0; clear H0; intros; unfold D_x in H0; elim H0; intros;
rewrite (Rinv_r (x - x0) (Rminus_eq_contra x x0 (sym_not_eq H3)));
unfold R_dist in |- *; rewrite (Rminus_diag_eq 1 1 (refl_equal 1));
unfold Rabs in |- *; case (Rcase_abs 0); intro.
absurd (0 < 0); auto.
red in |- *; intro; apply (Rlt_irrefl 0 r).
unfold Rgt in H; assumption.
Qed.
(*********)
Lemma Dadd :
forall (D:R -> Prop) (df dg f g:R -> R) (x0:R),
D_in f df D x0 ->
D_in g dg D x0 ->
D_in (fun x:R => f x + g x) (fun x:R => df x + dg x) D x0.
Proof.
unfold D_in in |- *; intros;
generalize
(limit_plus (fun x:R => (f x - f x0) * / (x - x0))
(fun x:R => (g x - g x0) * / (x - x0)) (D_x D x0) (
df x0) (dg x0) x0 H H0); clear H H0; 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; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))) in H1;
rewrite (Rmult_comm (g x1 - g x0) (/ (x1 - x0))) in H1;
rewrite <- (Rmult_plus_distr_l (/ (x1 - x0)) (f x1 - f x0) (g x1 - g x0))
in H1;
rewrite (Rmult_comm (/ (x1 - x0)) (f x1 - f x0 + (g x1 - g x0))) in H1;
cut (f x1 - f x0 + (g x1 - g x0) = f x1 + g x1 - (f x0 + g x0)).
intro; rewrite H3 in H1; assumption.
ring.
Qed.
(*********)
Lemma Dmult :
forall (D:R -> Prop) (df dg f g:R -> R) (x0:R),
D_in f df D x0 ->
D_in g dg D x0 ->
D_in (fun x:R => f x * g x) (fun x:R => df x * g x + f x * dg x) D x0.
Proof.
intros; unfold D_in in |- *; generalize H H0; intros; unfold D_in in H, H0;
generalize (cont_deriv f df D x0 H1); unfold continue_in in |- *;
intro;
generalize
(limit_mul (fun x:R => (g x - g x0) * / (x - x0)) (
fun x:R => f x) (D_x D x0) (dg x0) (f x0) x0 H0 H3);
intro; cut (limit1_in (fun x:R => g x0) (D_x D x0) (g x0) x0).
intro;
generalize
(limit_mul (fun x:R => (f x - f x0) * / (x - x0)) (
fun _:R => g x0) (D_x D x0) (df x0) (g x0) x0 H H5);
clear H H0 H1 H2 H3 H5; intro;
generalize
(limit_plus (fun x:R => (f x - f x0) * / (x - x0) * g x0)
(fun x:R => (g x - g x0) * / (x - x0) * f x) (
D_x D x0) (df x0 * g x0) (dg x0 * f x0) x0 H H4);
clear H4 H; intro; unfold limit1_in in H; unfold limit_in in H;
simpl in H; 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;
rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))) in H1;
rewrite (Rmult_comm (g x1 - g x0) (/ (x1 - x0))) in H1;
rewrite (Rmult_assoc (/ (x1 - x0)) (f x1 - f x0) (g x0)) in H1;
rewrite (Rmult_assoc (/ (x1 - x0)) (g x1 - g x0) (f x1)) in H1;
rewrite <-
(Rmult_plus_distr_l (/ (x1 - x0)) ((f x1 - f x0) * g x0)
((g x1 - g x0) * f x1)) in H1;
rewrite
(Rmult_comm (/ (x1 - x0)) ((f x1 - f x0) * g x0 + (g x1 - g x0) * f x1))
in H1; rewrite (Rmult_comm (dg x0) (f x0)) in H1;
cut
((f x1 - f x0) * g x0 + (g x1 - g x0) * f x1 = f x1 * g x1 - f x0 * g x0).
intro; rewrite H3 in H1; assumption.
ring.
unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros;
split with eps; split; auto; intros; elim (R_dist_refl (g x0) (g x0));
intros a b; rewrite (b (refl_equal (g x0))); unfold Rgt in H;
assumption.
Qed.
(*********)
Lemma Dmult_const :
forall (D:R -> Prop) (f df:R -> R) (x0 a:R),
D_in f df D x0 -> D_in (fun x:R => a * f x) (fun x:R => a * df x) D x0.
Proof.
intros;
generalize (Dmult D (fun _:R => 0) df (fun _:R => a) f x0 (Dconst D a x0) H);
unfold D_in in |- *; intros; rewrite (Rmult_0_l (f x0)) in H0;
rewrite (let (H1, H2) := Rplus_ne (a * df x0) in H2) in H0;
assumption.
Qed.
(*********)
Lemma Dopp :
forall (D:R -> Prop) (f df:R -> R) (x0:R),
D_in f df D x0 -> D_in (fun x:R => - f x) (fun x:R => - df x) D x0.
Proof.
intros; generalize (Dmult_const D f df x0 (-1) H); unfold D_in in |- *;
unfold limit1_in in |- *; unfold limit_in in |- *;
intros; generalize (H0 eps H1); clear H0; intro; elim H0;
clear H0; intros; elim H0; clear H0; simpl in |- *;
intros; split with x; split; auto.
intros; generalize (H2 x1 H3); clear H2; intro;
rewrite Ropp_mult_distr_l_reverse in H2;
rewrite Ropp_mult_distr_l_reverse in H2;
rewrite Ropp_mult_distr_l_reverse in H2;
rewrite (let (H1, H2) := Rmult_ne (f x1) in H2) in H2;
rewrite (let (H1, H2) := Rmult_ne (f x0) in H2) in H2;
rewrite (let (H1, H2) := Rmult_ne (df x0) in H2) in H2;
assumption.
Qed.
(*********)
Lemma Dminus :
forall (D:R -> Prop) (df dg f g:R -> R) (x0:R),
D_in f df D x0 ->
D_in g dg D x0 ->
D_in (fun x:R => f x - g x) (fun x:R => df x - dg x) D x0.
Proof.
unfold Rminus in |- *; intros; generalize (Dopp D g dg x0 H0); intro;
apply (Dadd D df (fun x:R => - dg x) f (fun x:R => - g x) x0);
assumption.
Qed.
(*********)
Lemma Dx_pow_n :
forall (n:nat) (D:R -> Prop) (x0:R),
D_in (fun x:R => x ^ n) (fun x:R => INR n * x ^ (n - 1)) D x0.
Proof.
simple induction n; intros.
simpl in |- *; rewrite Rmult_0_l; apply Dconst.
intros; cut (n0 = (S n0 - 1)%nat);
[ intro a; rewrite <- a; clear a | simpl in |- *; apply minus_n_O ].
generalize
(Dmult D (fun _:R => 1) (fun x:R => INR n0 * x ^ (n0 - 1)) (
fun x:R => x) (fun x:R => x ^ n0) x0 (Dx D x0) (
H D x0)); unfold D_in in |- *; unfold limit1_in in |- *;
unfold limit_in in |- *; simpl in |- *; intros; elim (H0 eps H1);
clear H0; intros; elim H0; clear H0; intros; split with x;
split; auto.
intros; generalize (H2 x1 H3); clear H2 H3; intro;
rewrite (let (H1, H2) := Rmult_ne (x0 ^ n0) in H2) in H2;
rewrite (tech_pow_Rmult x1 n0) in H2; rewrite (tech_pow_Rmult x0 n0) in H2;
rewrite (Rmult_comm (INR n0) (x0 ^ (n0 - 1))) in H2;
rewrite <- (Rmult_assoc x0 (x0 ^ (n0 - 1)) (INR n0)) in H2;
rewrite (tech_pow_Rmult x0 (n0 - 1)) in H2; elim (classic (n0 = 0%nat));
intro cond.
rewrite cond in H2; rewrite cond; simpl in H2; simpl in |- *;
cut (1 + x0 * 1 * 0 = 1 * 1);
[ intro A; rewrite A in H2; assumption | ring ].
cut (n0 <> 0%nat -> S (n0 - 1) = n0); [ intro | omega ];
rewrite (H3 cond) in H2; rewrite (Rmult_comm (x0 ^ n0) (INR n0)) in H2;
rewrite (tech_pow_Rplus x0 n0 n0) in H2; assumption.
Qed.
(*********)
Lemma Dcomp :
forall (Df Dg:R -> Prop) (df dg f g:R -> R) (x0:R),
D_in f df Df x0 ->
D_in g dg Dg (f x0) ->
D_in (fun x:R => g (f x)) (fun x:R => df x * dg (f x)) (Dgf Df Dg f) x0.
Proof.
intros Df Dg df dg f g x0 H H0; generalize H H0; unfold D_in in |- *;
unfold Rdiv in |- *; intros;
generalize
(limit_comp f (fun x:R => (g x - g (f x0)) * / (x - f x0)) (
D_x Df x0) (D_x Dg (f x0)) (f x0) (dg (f x0)) x0);
intro; generalize (cont_deriv f df Df x0 H); intro;
unfold continue_in in H4; generalize (H3 H4 H2); clear H3;
intro;
generalize
(limit_mul (fun x:R => (g (f x) - g (f x0)) * / (f x - f x0))
(fun x:R => (f x - f x0) * / (x - x0))
(Dgf (D_x Df x0) (D_x Dg (f x0)) f) (dg (f x0)) (
df x0) x0 H3); intro;
cut
(limit1_in (fun x:R => (f x - f x0) * / (x - x0))
(Dgf (D_x Df x0) (D_x Dg (f x0)) f) (df x0) x0).
intro; generalize (H5 H6); clear H5; intro;
generalize
(limit_mul (fun x:R => (f x - f x0) * / (x - x0)) (
fun x:R => dg (f x0)) (D_x Df x0) (df x0) (dg (f x0)) x0 H1
(limit_free (fun x:R => dg (f x0)) (D_x Df x0) x0 x0));
intro; unfold limit1_in in |- *; unfold limit_in in |- *;
simpl in |- *; unfold limit1_in in H5, H7; unfold limit_in in H5, H7;
simpl in H5, H7; intros; elim (H5 eps H8); elim (H7 eps H8);
clear H5 H7; intros; elim H5; elim H7; clear H5 H7;
intros; split with (Rmin x x1); split.
elim (Rmin_Rgt x x1 0); intros a b; apply (b (conj H9 H5)); clear a b.
intros; elim H11; clear H11; intros; elim (Rmin_Rgt x x1 (R_dist x2 x0));
intros a b; clear b; unfold Rgt in a; elim (a H12);
clear H5 a; intros; unfold D_x, Dgf in H11, H7, H10;
clear H12; elim (classic (f x2 = f x0)); intro.
elim H11; clear H11; intros; elim H11; clear H11; intros;
generalize (H10 x2 (conj (conj H11 H14) H5)); intro;
rewrite (Rminus_diag_eq (f x2) (f x0) H12) in H16;
rewrite (Rmult_0_l (/ (x2 - x0))) in H16;
rewrite (Rmult_0_l (dg (f x0))) in H16; rewrite H12;
rewrite (Rminus_diag_eq (g (f x0)) (g (f x0)) (refl_equal (g (f x0))));
rewrite (Rmult_0_l (/ (x2 - x0))); assumption.
clear H10 H5; elim H11; clear H11; intros; elim H5; clear H5; intros;
cut
(((Df x2 /\ x0 <> x2) /\ Dg (f x2) /\ f x0 <> f x2) /\ R_dist x2 x0 < x1);
auto; intro; generalize (H7 x2 H14); intro;
generalize (Rminus_eq_contra (f x2) (f x0) H12); intro;
rewrite
(Rmult_assoc (g (f x2) - g (f x0)) (/ (f x2 - f x0))
((f x2 - f x0) * / (x2 - x0))) in H15;
rewrite <- (Rmult_assoc (/ (f x2 - f x0)) (f x2 - f x0) (/ (x2 - x0)))
in H15; rewrite (Rinv_l (f x2 - f x0) H16) in H15;
rewrite (let (H1, H2) := Rmult_ne (/ (x2 - x0)) in H2) in H15;
rewrite (Rmult_comm (df x0) (dg (f x0))); assumption.
clear H5 H3 H4 H2; unfold limit1_in in |- *; unfold limit_in in |- *;
simpl in |- *; unfold limit1_in in H1; unfold limit_in in H1;
simpl in H1; intros; elim (H1 eps H2); clear H1; intros;
elim H1; clear H1; intros; split with x; split; auto;
intros; unfold D_x, Dgf in H4, H3; elim H4; clear H4;
intros; elim H4; clear H4; intros; exact (H3 x1 (conj H4 H5)).
Qed.
(*********)
Lemma D_pow_n :
forall (n:nat) (D:R -> Prop) (x0:R) (expr dexpr:R -> R),
D_in expr dexpr D x0 ->
D_in (fun x:R => expr x ^ n)
(fun x:R => INR n * expr x ^ (n - 1) * dexpr x) (
Dgf D D expr) x0.
Proof.
intros n D x0 expr dexpr H;
generalize
(Dcomp D D dexpr (fun x:R => INR n * x ^ (n - 1)) expr (
fun x:R => x ^ n) x0 H (Dx_pow_n n D (expr x0)));
intro; unfold D_in in |- *; unfold limit1_in in |- *;
unfold limit_in in |- *; simpl in |- *; intros; unfold D_in in H0;
unfold limit1_in in H0; unfold limit_in in H0; simpl in H0;
elim (H0 eps H1); clear H0; intros; elim H0; clear H0;
intros; split with x; split; intros; auto.
cut
(dexpr x0 * (INR n * expr x0 ^ (n - 1)) =
INR n * expr x0 ^ (n - 1) * dexpr x0);
[ intro Rew; rewrite <- Rew; exact (H2 x1 H3) | ring ].
Qed.
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