Timings for Alembert.v
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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 Rfunctions.
Local Open Scope R_scope.
(***************************************************)
(* Various versions of the criterion of D'Alembert *)
(***************************************************)
Lemma Alembert_C1 :
forall An:nat -> R,
(forall n:nat, 0 < An n) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 ->
{ l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.
assert
(X:{ l:R | is_lub (EUn (fun N:nat => sum_f_R0 An N)) l } ->
{ l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }).
exists x; apply Un_cv_crit_lub;
[ unfold Un_growing; intro; rewrite tech5;
pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r;
apply Rplus_le_compat_l; left; apply H
| apply H1 ].
exists (sum_f_R0 An 0); unfold EUn; exists 0%nat; reflexivity.
unfold Un_cv in H0; unfold bound; cut (0 < / 2);
[ intro | apply Rinv_0_lt_compat; prove_sup0 ].
elim (H0 (/ 2) H1); intros.
exists (sum_f_R0 An x + 2 * An (S x)).
unfold is_upper_bound; intros; unfold EUn in H3; destruct H3 as (x1,->).
destruct (lt_eq_lt_dec x1 x) as [[| -> ]|].
replace (sum_f_R0 An x) with
(sum_f_R0 An x1 + sum_f_R0 (fun i:nat => An (S x1 + i)%nat) (x - S x1)).
symmetry; apply tech2; assumption.
pattern (sum_f_R0 An x1) at 1; rewrite <- Rplus_0_r;
rewrite Rplus_assoc; apply Rplus_le_compat_l.
left; apply Rplus_lt_0_compat.
apply tech1; intros; apply H.
apply Rmult_lt_0_compat; [ prove_sup0 | apply H ].
pattern (sum_f_R0 An x) at 1; rewrite <- Rplus_0_r;
apply Rplus_le_compat_l.
left; apply Rmult_lt_0_compat; [ prove_sup0 | apply H ].
replace (sum_f_R0 An x1) with
(sum_f_R0 An x + sum_f_R0 (fun i:nat => An (S x + i)%nat) (x1 - S x)).
symmetry; apply tech2; assumption.
cut
(sum_f_R0 (fun i:nat => An (S x + i)%nat) (x1 - S x) <=
An (S x) * sum_f_R0 (fun i:nat => (/ 2) ^ i) (x1 - S x)).
intro;
apply Rle_trans with
(An (S x) * sum_f_R0 (fun i:nat => (/ 2) ^ i) (x1 - S x)).
rewrite <- (Rmult_comm (An (S x))); apply Rmult_le_compat_l.
replace (1 - / 2) with (/ 2).
unfold Rdiv; rewrite Rinv_inv.
pattern 2 at 3; rewrite <- Rmult_1_r; rewrite <- (Rmult_comm 2);
apply Rmult_le_compat_l.
left; apply Rplus_lt_reg_l with ((/ 2) ^ S (x1 - S x)).
replace ((/ 2) ^ S (x1 - S x) + (1 - (/ 2) ^ S (x1 - S x))) with 1;
[ idtac | ring ].
rewrite <- (Rplus_comm 1); pattern 1 at 1; rewrite <- Rplus_0_r;
apply Rplus_lt_compat_l.
apply pow_lt; apply Rinv_0_lt_compat; prove_sup0.
replace 1 with (/ 1);
[ apply tech7; discrR | apply Rinv_1 ].
replace (An (S x)) with (An (S x + 0)%nat) by (f_equal; ring).
apply (tech6 (fun i:nat => An (S x + i)%nat) (/ 2)).
left; apply Rinv_0_lt_compat; prove_sup0.
intro; cut (forall n:nat, (n >= x)%nat -> An (S n) < / 2 * An n).
intro H4; replace (S x + S i)%nat with (S (S x + i)) by auto with zarith.
apply H4; unfold ge; apply tech8.
intros; unfold Rdist in H2; apply Rmult_lt_reg_l with (/ An n).
apply Rinv_0_lt_compat; apply H.
do 2 rewrite (Rmult_comm (/ An n)); rewrite Rmult_assoc;
rewrite Rinv_r.
intro H5; assert (H8 := H n); rewrite H5 in H8;
elim (Rlt_irrefl _ H8).
rewrite Rmult_1_r;
replace (An (S n) * / An n) with (Rabs (Rabs (An (S n) / An n) - 0)).
unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r;
rewrite Rabs_Rabsolu; rewrite Rabs_right.
unfold Rdiv; reflexivity.
left; unfold Rdiv; change (0 < An (S n) * / An n);
apply Rmult_lt_0_compat; [ apply H | apply Rinv_0_lt_compat; apply H ].
Lemma Alembert_C2_aux_positivity :
forall Xn : nat -> R,
let Yn i := (2 * Rabs (Xn i) + Xn i) / 2 in
(forall n, Xn n <> 0) ->
forall n, 0 < Yn n.
intros Xn Yn H n; unfold Yn; unfold Rdiv; rewrite <- (Rmult_0_r (/ 2));
rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l.
apply Rinv_0_lt_compat; prove_sup0.
apply Rplus_lt_reg_l with (- Xn n); rewrite Rplus_0_r; unfold Rminus;
rewrite (Rplus_comm (- Xn n)); rewrite Rplus_assoc;
rewrite Rplus_opp_r; rewrite Rplus_0_r;
apply Rle_lt_trans with (Rabs (Xn n)).
rewrite <- Rabs_Ropp; apply RRle_abs.
rewrite <-Rplus_diag; pattern (Rabs (Xn n)) at 1; rewrite <- Rplus_0_r;
apply Rplus_lt_compat_l; apply Rabs_pos_lt; apply H.
Lemma Alembert_C2_aux_Un_cv :
forall Xn : nat -> R,
let Yn i := (2 * Rabs (Xn i) + Xn i) / 2 in
(forall n, Xn n <> 0) ->
Un_cv (fun n:nat => Rabs (Xn (S n) / Xn n)) 0 ->
Un_cv (fun n => Rabs (Yn (S n) / Yn n)) 0.
pose proof (Alembert_C2_aux_positivity An H); fold Vn in H1.
(* <- stupid name compat hack *)
cut (forall n:nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)).
1:intro; cut (forall n:nat, / Vn n <= 2 * / Rabs (An n)).
1:intro; cut (forall n:nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)).
intro; unfold Un_cv; intros; unfold Un_cv in H1; assert (0 < eps / 3).
unfold Rdiv; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim (H0 (eps / 3) H7); intros.
unfold Rdist; unfold Rminus; rewrite Ropp_0;
rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold Rdist in H10;
unfold Rminus in H10; rewrite Ropp_0 in H10; rewrite Rplus_0_r in H10;
rewrite Rabs_Rabsolu in H10; rewrite Rabs_right.
left; change (0 < Vn (S n) / Vn n); unfold Rdiv;
apply Rmult_lt_0_compat.
apply Rinv_0_lt_compat; apply H1.
apply Rle_lt_trans with (3 * Rabs (An (S n) / An n)).
apply Rmult_lt_reg_l with (/ 3).
apply Rinv_0_lt_compat; prove_sup0.
rewrite <- Rmult_assoc; rewrite Rinv_l; [ idtac | discrR ];
rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H10;
exact H10.
intro; unfold Rdiv; rewrite Rabs_mult; rewrite <- Rmult_assoc;
replace 3 with (2 * (3 * / 2));
[ idtac | rewrite <- Rmult_assoc; apply Rmult_inv_r_id_m; discrR ];
apply Rle_trans with (Vn (S n) * 2 * / Rabs (An n)).
rewrite Rmult_assoc; apply Rmult_le_compat_l.
replace (Vn (S n) * 2 * / Rabs (An n)) with (2 * / Rabs (An n) * Vn (S n)) by ring;
replace (2 * (3 * / 2) * Rabs (An (S n)) * / Rabs (An n)) with
(2 * / Rabs (An n) * (3 * / 2 * Rabs (An (S n)))) by ring;
apply Rmult_le_compat_l.
left; apply Rmult_lt_0_compat.
apply Rinv_0_lt_compat; apply Rabs_pos_lt; apply H.
elim (H3 (S n)); intros; assumption.
intro; apply Rmult_le_reg_l with (Vn n).
red; intro; assert (H5 := H1 n); rewrite H4 in H5;
elim (Rlt_irrefl _ H5).
apply Rmult_le_reg_l with (Rabs (An n)).
apply Rabs_pos_lt; apply H.
rewrite Rmult_1_r;
replace (Rabs (An n) * (Vn n * (2 * / Rabs (An n)))) with
(2 * Vn n * (Rabs (An n) * / Rabs (An n))); [ idtac | ring ];
rewrite Rinv_r.
apply Rabs_no_R0; apply H.
rewrite Rmult_1_r; apply Rmult_le_reg_l with (/ 2).
apply Rinv_0_lt_compat; prove_sup0.
rewrite <- Rmult_assoc; rewrite Rinv_l.
rewrite Rmult_1_l; elim (H3 n); intros; assumption.
unfold Vn; unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; prove_sup0.
pattern (Rabs (An n)) at 1; rewrite <- Rplus_0_r; rewrite <-Rplus_diag;
rewrite Rplus_assoc; apply Rplus_le_compat_l.
apply Rplus_le_reg_l with (- An n); rewrite Rplus_0_r;
rewrite <- (Rplus_comm (An n)); rewrite <- Rplus_assoc;
rewrite Rplus_opp_l; rewrite Rplus_0_l; rewrite <- Rabs_Ropp;
apply RRle_abs.
unfold Vn; unfold Rdiv; repeat rewrite <- (Rmult_comm (/ 2));
repeat rewrite Rmult_assoc; apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; prove_sup0.
unfold Rminus; rewrite <-Rplus_diag;
replace (3 * Rabs (An n)) with (Rabs (An n) + Rabs (An n) + Rabs (An n));
[ idtac | ring ]; repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l;
apply RRle_abs.
Lemma Alembert_C2 :
forall An:nat -> R,
(forall n:nat, An n <> 0) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 ->
{ l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.
set (Vn := fun i:nat => (2 * Rabs (An i) + An i) / 2).
set (Wn := fun i:nat => (2 * Rabs (An i) - An i) / 2).
assert (forall n:nat, 0 < Vn n).
apply Alembert_C2_aux_positivity;assumption.
assert (Wn_aux : Wn = fun i => (2 * Rabs (- An i) + (- An i)) / 2).
apply FunctionalExtensionality.functional_extensionality.
symmetry;apply Rabs_Ropp.
assert (forall n:nat, 0 < Wn n).
apply Alembert_C2_aux_positivity.
intros;apply Ropp_neq_0_compat, H.
assert (Un_cv (fun n:nat => Rabs (Vn (S n) / Vn n)) 0).
apply Alembert_C2_aux_Un_cv;assumption.
assert (Un_cv (fun n:nat => Rabs (Wn (S n) / Wn n)) 0).
apply (Alembert_C2_aux_Un_cv (fun n => - An n)).
intros;apply Ropp_neq_0_compat, H.
replace (fun n : nat => Rabs (- An (S n) / - An n)) with
(fun n : nat => Rabs (An (S n) / An n));[assumption|].
apply FunctionalExtensionality.functional_extensionality.
pose proof (Alembert_C1 Vn H1 H3) as (x,p).
pose proof (Alembert_C1 Wn H2 H4) as (x0,p0).
exists (x - x0); unfold Un_cv; unfold Un_cv in p;
unfold Un_cv in p0; intros; assert (H6:0 < eps / 2).
unfold Rdiv; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
destruct (p (eps / 2) H6) as (x1,H8).
destruct (p0 (eps / 2) H6) as (x2,H9).
exists N; intros;
replace (sum_f_R0 An n) with (sum_f_R0 Vn n - sum_f_R0 Wn n).
symmetry ; apply tech11; intro; unfold Vn, Wn;
unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ 2));
apply Rmult_eq_reg_l with 2.
rewrite Rmult_minus_distr_l; repeat rewrite <- Rmult_assoc;
rewrite Rinv_r.
unfold Rdist;
replace (sum_f_R0 Vn n - sum_f_R0 Wn n - (x - x0)) with
(sum_f_R0 Vn n - x + - (sum_f_R0 Wn n - x0)) by ring;
apply Rle_lt_trans with
(Rabs (sum_f_R0 Vn n - x) + Rabs (- (sum_f_R0 Wn n - x0))).
rewrite Rabs_Ropp; apply Rlt_le_trans with (eps / 2 + eps / 2).
unfold Rdist in H8; apply H8; unfold ge; apply Nat.le_trans with N;
[ unfold N; apply Nat.le_max_l | assumption ].
unfold Rdist in H9; apply H9; unfold ge; apply Nat.le_trans with N;
[ unfold N; apply Nat.le_max_r | assumption ].
right; apply Rplus_half_diag.
Lemma AlembertC3_step1 :
forall (An:nat -> R) (x:R),
x <> 0 ->
(forall n:nat, An n <> 0) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 ->
{ l:R | Pser An x l }.
intros; set (Bn := fun i:nat => An i * x ^ i).
assert (forall n:nat, Bn n <> 0).
intro; unfold Bn; apply prod_neq_R0;
[ apply H0 | apply pow_nonzero; assumption ].
cut (Un_cv (fun n:nat => Rabs (Bn (S n) / Bn n)) 0).
intro; destruct (Alembert_C2 Bn H2 H3) as (x0,H4).
exists x0; unfold Bn in H4; apply tech12; assumption.
unfold Un_cv; intros; unfold Un_cv in H1; cut (0 < eps / Rabs x).
unfold Rdiv; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ].
intro; elim (H1 (eps / Rabs x) H4); intros.
exists x0; intros; unfold Rdist; unfold Rminus;
rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu;
unfold Bn;
replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x).
replace (S n) with (n + 1)%nat by ring; rewrite pow_add;
unfold Rdiv; rewrite Rinv_mult.
replace (An (n + 1)%nat * (x ^ n * x ^ 1) * (/ An n * / x ^ n)) with
(An (n + 1)%nat * x ^ 1 * / An n * (x ^ n * / x ^ n)) by ring;
rewrite Rinv_r.
apply pow_nonzero; assumption.
rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs x).
apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
rewrite <- (Rmult_comm (Rabs x)); rewrite <- Rmult_assoc;
rewrite Rinv_l.
apply Rabs_no_R0; assumption.
rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H5;
replace (Rabs (An (S n) / An n)) with (Rdist (Rabs (An (S n) * / An n)) 0).
unfold Rdist; unfold Rminus; rewrite Ropp_0;
rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold Rdiv;
reflexivity.
Lemma AlembertC3_step2 :
forall (An:nat -> R) (x:R), x = 0 -> { l:R | Pser An x l }.
intros; exists (An 0%nat).
unfold Pser; unfold infinite_sum; intros; exists 0%nat; intros;
replace (sum_f_R0 (fun n0:nat => An n0 * x ^ n0) n) with (An 0%nat).
unfold Rdist; unfold Rminus; rewrite Rplus_opp_r;
rewrite Rabs_R0; assumption.
induction n as [| n Hrecn].
rewrite tech5; rewrite Hrecn;
[ rewrite H; simpl; ring | unfold ge; apply Nat.le_0_l ].
(** A useful criterion of convergence for power series *)
Theorem Alembert_C3 :
forall (An:nat -> R) (x:R),
(forall n:nat, An n <> 0) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 ->
{ l:R | Pser An x l }.
intros; destruct (total_order_T x 0) as [[Hlt|Heq]|Hgt].
intro; apply AlembertC3_step1; assumption.
red; intro; rewrite H1 in Hlt; elim (Rlt_irrefl _ Hlt).
apply AlembertC3_step2; assumption.
intro; apply AlembertC3_step1; assumption.
red; intro; rewrite H1 in Hgt; elim (Rlt_irrefl _ Hgt).
Lemma Alembert_C4 :
forall (An:nat -> R) (k:R),
0 <= k < 1 ->
(forall n:nat, 0 < An n) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) k ->
{ l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.
assert
(X:{ l:R | is_lub (EUn (fun N:nat => sum_f_R0 An N)) l } ->
{ l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }).
exists x; apply Un_cv_crit_lub;
[ unfold Un_growing; intro; rewrite tech5;
pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r;
apply Rplus_le_compat_l; left; apply H
| apply H1].
exists (sum_f_R0 An 0); unfold EUn; exists 0%nat; reflexivity.
assert (H1 := tech13 _ _ Hyp H0).
unfold bound; exists (sum_f_R0 An x0 + / (1 - x) * An (S x0)).
unfold is_upper_bound; intros; unfold EUn in H6.
destruct (lt_eq_lt_dec x2 x0) as [[| -> ]|].
replace (sum_f_R0 An x0) with
(sum_f_R0 An x2 + sum_f_R0 (fun i:nat => An (S x2 + i)%nat) (x0 - S x2)).
symmetry ; apply tech2; assumption.
pattern (sum_f_R0 An x2) at 1; rewrite <- Rplus_0_r.
rewrite Rplus_assoc; apply Rplus_le_compat_l.
left; apply Rplus_lt_0_compat.
apply Rinv_0_lt_compat; apply Rplus_lt_reg_l with x; rewrite Rplus_0_r;
replace (x + (1 - x)) with 1 by ring; elim H3; intros; assumption.
pattern (sum_f_R0 An x0) at 1; rewrite <- Rplus_0_r;
apply Rplus_le_compat_l.
left; apply Rmult_lt_0_compat.
apply Rinv_0_lt_compat; apply Rplus_lt_reg_l with x; rewrite Rplus_0_r;
replace (x + (1 - x)) with 1 by ring; elim H3; intros; assumption.
replace (sum_f_R0 An x2) with
(sum_f_R0 An x0 + sum_f_R0 (fun i:nat => An (S x0 + i)%nat) (x2 - S x0)).
symmetry ; apply tech2; assumption.
cut
(sum_f_R0 (fun i:nat => An (S x0 + i)%nat) (x2 - S x0) <=
An (S x0) * sum_f_R0 (fun i:nat => x ^ i) (x2 - S x0)).
intro;
apply Rle_trans with (An (S x0) * sum_f_R0 (fun i:nat => x ^ i) (x2 - S x0)).
rewrite <- (Rmult_comm (An (S x0))); apply Rmult_le_compat_l.
unfold Rdiv; apply Rmult_le_reg_l with (1 - x).
do 2 rewrite (Rmult_comm (1 - x)).
rewrite Rmult_assoc; rewrite Rinv_l.
rewrite Rmult_1_r; apply Rplus_le_reg_l with (x ^ S (x2 - S x0)).
replace (x ^ S (x2 - S x0) + (1 - x ^ S (x2 - S x0))) with 1 by ring.
rewrite <- (Rplus_comm 1); pattern 1 at 1; rewrite <- Rplus_0_r;
apply Rplus_le_compat_l.
replace (An (S x0)) with (An (S x0 + 0)%nat) by (f_equal;ring).
apply (tech6 (fun i:nat => An (S x0 + i)%nat) x).
cut (forall n:nat, (n >= x0)%nat -> An (S n) < x * An n).
replace (S x0 + S i)%nat with (S (S x0 + i)) by ring.
apply Rmult_lt_reg_l with (/ An n).
apply Rinv_0_lt_compat; apply H.
do 2 rewrite (Rmult_comm (/ An n)).
replace (An (S n) * / An n) with (Rabs (An (S n) / An n)).
unfold Rdiv; reflexivity.
left; unfold Rdiv; change (0 < An (S n) * / An n);
apply Rmult_lt_0_compat.
apply Rinv_0_lt_compat; apply H.
Lemma Alembert_C5 :
forall (An:nat -> R) (k:R),
0 <= k < 1 ->
(forall n:nat, An n <> 0) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) k ->
{ l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }.
assert
(Hyp0:{ l:R | Un_cv (fun N:nat => sum_f_R0 An N) l } ->
{ l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }).
assert
(Hyp:{ l:R | Un_cv (fun N:nat => sum_f_R0 (fun i:nat => Rabs (An i)) N) l } ->
{ l:R | Un_cv (fun N:nat => sum_f_R0 (fun i:nat => Rabs (An i)) N) l }).
apply (Alembert_C4 (fun i:nat => Rabs (An i)) k).
intro; apply Rabs_pos_lt; apply H0.
elim (H1 eps H2); intros.
unfold Rdiv in H3; apply H3; assumption.
(** Convergence of power series in D(O,1/k)
k=0 is described in Alembert_C3 *)
Lemma Alembert_C6 :
forall (An:nat -> R) (x k:R),
0 < k ->
(forall n:nat, An n <> 0) ->
Un_cv (fun n:nat => Rabs (An (S n) / An n)) k ->
Rabs x < / k -> { l:R | Pser An x l }.
cut { l:R | Un_cv (fun N:nat => sum_f_R0 (fun i:nat => An i * x ^ i) N) l }.
apply tech12; assumption.
destruct (total_order_T x 0) as [[Hlt|Heq]|Hgt].
eapply Alembert_C5 with (k * Rabs x).
unfold Rdiv; apply Rmult_le_pos.
apply Rmult_lt_reg_l with (/ k).
apply Rinv_0_lt_compat; assumption.
rewrite Rmult_1_r; assumption.
intro; apply prod_neq_R0.
unfold Un_cv; unfold Un_cv in H1.
assert (0 < eps / Rabs x).
unfold Rdiv; apply Rmult_lt_0_compat.
apply Rinv_0_lt_compat; apply Rabs_pos_lt.
elim (H1 (eps / Rabs x) H4); intros.
replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x).
unfold Rdiv; replace (S n) with (n + 1)%nat by ring.
replace (An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)) with
(An (n + 1)%nat * / An n * x * (x ^ n * / x ^ n)) by ring.
replace (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) with
(Rabs x * (Rabs (An (S n) / An n) - k)) by ring.
apply Rmult_lt_reg_l with (/ Rabs x).
apply Rinv_0_lt_compat; apply Rabs_pos_lt.
rewrite <- (Rmult_comm eps).
unfold Rdiv; unfold Rdiv in H5; apply H5; assumption.
replace (sum_f_R0 (fun i:nat => An i * x ^ i) n) with (An 0%nat).
unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
induction n as [| n Hrecn].
rewrite <- Hrecn,Heq;simpl.
unfold ge; apply Nat.le_0_l.
eapply Alembert_C5 with (k * Rabs x).
unfold Rdiv; apply Rmult_le_pos.
apply Rmult_lt_reg_l with (/ k).
apply Rinv_0_lt_compat; assumption.
rewrite Rmult_1_r; assumption.
intro; apply prod_neq_R0.
unfold Un_cv; unfold Un_cv in H1.
assert (0 < eps / Rabs x).
unfold Rdiv; apply Rmult_lt_0_compat.
apply Rinv_0_lt_compat; apply Rabs_pos_lt.
elim (H1 (eps / Rabs x) H4); intros.
replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x).
unfold Rdiv; replace (S n) with (n + 1)%nat by ring.
replace (An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)) with
(An (n + 1)%nat * / An n * x * (x ^ n * / x ^ n)) by ring.
replace (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) with
(Rabs x * (Rabs (An (S n) / An n) - k)); [ idtac | ring ].
apply Rmult_lt_reg_l with (/ Rabs x).
apply Rinv_0_lt_compat; apply Rabs_pos_lt.
rewrite <- (Rmult_comm eps).
unfold Rdiv; unfold Rdiv in H5; apply H5; assumption.