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|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Cos_plus.v,v 1.11.2.1 2004/07/16 19:31:10 herbelin Exp $ i*)
Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo_def.
Require Import Cos_rel.
Require Import Max. Open Local Scope nat_scope. Open Local Scope R_scope.
Definition Majxy (x y:R) (n:nat) : R :=
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n) / INR (fact n).
Lemma Majxy_cv_R0 : forall x y:R, Un_cv (Majxy x y) 0.
intros.
set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))).
set (C0 := C ^ 4).
cut (0 < C).
intro.
cut (0 < C0).
intro.
assert (H1 := cv_speed_pow_fact C0).
unfold Un_cv in H1; unfold R_dist in H1.
unfold Un_cv in |- *; unfold R_dist in |- *; intros.
cut (0 < eps / C0);
[ intro
| unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; assumption ] ].
elim (H1 (eps / C0) H3); intros N0 H4.
exists N0; intros.
replace (Majxy x y n) with (C0 ^ S n / INR (fact n)).
simpl in |- *.
apply Rmult_lt_reg_l with (Rabs (/ C0)).
apply Rabs_pos_lt.
apply Rinv_neq_0_compat.
red in |- *; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0).
rewrite <- Rabs_mult.
unfold Rminus in |- *; rewrite Rmult_plus_distr_l.
rewrite Ropp_0; rewrite Rmult_0_r.
unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_l.
rewrite (Rabs_right (/ C0)).
rewrite <- (Rmult_comm eps).
replace (C0 ^ n * / INR (fact n) + 0) with (C0 ^ n * / INR (fact n) - 0);
[ idtac | ring ].
unfold Rdiv in H4; apply H4; assumption.
apply Rle_ge; left; apply Rinv_0_lt_compat; assumption.
red in |- *; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0).
unfold Majxy in |- *.
unfold C0 in |- *.
rewrite pow_mult.
unfold C in |- *; reflexivity.
unfold C0 in |- *; apply pow_lt; assumption.
apply Rlt_le_trans with 1.
apply Rlt_0_1.
unfold C in |- *.
apply RmaxLess1.
Qed.
Lemma reste1_maj :
forall (x y:R) (N:nat),
(0 < N)%nat -> Rabs (Reste1 x y N) <= Majxy x y (pred N).
intros.
set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))).
unfold Reste1 in |- *.
apply Rle_trans with
(sum_f_R0
(fun k:nat =>
Rabs
(sum_f_R0
(fun l:nat =>
(-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
x ^ (2 * S (l + k)) *
((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
y ^ (2 * (N - l))) (pred (N - k)))) (
pred N)).
apply
(Rsum_abs
(fun k:nat =>
sum_f_R0
(fun l:nat =>
(-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
x ^ (2 * S (l + k)) * ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
y ^ (2 * (N - l))) (pred (N - k))) (pred N)).
apply Rle_trans with
(sum_f_R0
(fun k:nat =>
sum_f_R0
(fun l:nat =>
Rabs
((-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
x ^ (2 * S (l + k)) *
((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
y ^ (2 * (N - l)))) (pred (N - k))) (
pred N)).
apply sum_Rle.
intros.
apply
(Rsum_abs
(fun l:nat =>
(-1) ^ S (l + n) / INR (fact (2 * S (l + n))) * x ^ (2 * S (l + n)) *
((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
y ^ (2 * (N - l))) (pred (N - n))).
apply Rle_trans with
(sum_f_R0
(fun k:nat =>
sum_f_R0
(fun l:nat =>
/ INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
C ^ (2 * S (N + k))) (pred (N - k))) (pred N)).
apply sum_Rle; intros.
apply sum_Rle; intros.
unfold Rdiv in |- *; repeat rewrite Rabs_mult.
do 2 rewrite pow_1_abs.
do 2 rewrite Rmult_1_l.
rewrite (Rabs_right (/ INR (fact (2 * S (n0 + n))))).
rewrite (Rabs_right (/ INR (fact (2 * (N - n0))))).
rewrite mult_INR.
rewrite Rinv_mult_distr.
repeat rewrite Rmult_assoc.
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
rewrite <- Rmult_assoc.
rewrite <- (Rmult_comm (/ INR (fact (2 * (N - n0))))).
rewrite Rmult_assoc.
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
do 2 rewrite <- RPow_abs.
apply Rle_trans with (Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0))).
apply Rmult_le_compat_l.
apply pow_le; apply Rabs_pos.
apply pow_incr.
split.
apply Rabs_pos.
unfold C in |- *.
apply Rle_trans with (Rmax (Rabs x) (Rabs y)); apply RmaxLess2.
apply Rle_trans with (C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0))).
do 2 rewrite <- (Rmult_comm (C ^ (2 * (N - n0)))).
apply Rmult_le_compat_l.
apply pow_le.
apply Rle_trans with 1.
left; apply Rlt_0_1.
unfold C in |- *; apply RmaxLess1.
apply pow_incr.
split.
apply Rabs_pos.
unfold C in |- *; apply Rle_trans with (Rmax (Rabs x) (Rabs y)).
apply RmaxLess1.
apply RmaxLess2.
right.
replace (2 * S (N + n))%nat with (2 * (N - n0) + 2 * S (n0 + n))%nat.
rewrite pow_add.
apply Rmult_comm.
apply INR_eq; rewrite plus_INR; do 3 rewrite mult_INR.
rewrite minus_INR.
repeat rewrite S_INR; do 2 rewrite plus_INR; ring.
apply le_trans with (pred (N - n)).
exact H1.
apply le_S_n.
replace (S (pred (N - n))) with (N - n)%nat.
apply le_trans with N.
apply (fun p n m:nat => plus_le_reg_l n m p) with n.
rewrite <- le_plus_minus.
apply le_plus_r.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply le_n_Sn.
apply S_pred with 0%nat.
apply plus_lt_reg_l with n.
rewrite <- le_plus_minus.
replace (n + 0)%nat with n; [ idtac | ring ].
apply le_lt_trans with (pred N).
assumption.
apply lt_pred_n_n; assumption.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply Rle_ge; left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
apply Rle_ge; left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
apply Rle_trans with
(sum_f_R0
(fun k:nat =>
sum_f_R0
(fun l:nat =>
/ INR (fact (2 * S (l + k)) * fact (2 * (N - l))) * C ^ (4 * N))
(pred (N - k))) (pred N)).
apply sum_Rle; intros.
apply sum_Rle; intros.
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat.
rewrite mult_INR; apply Rmult_lt_0_compat; apply INR_fact_lt_0.
apply Rle_pow.
unfold C in |- *; apply RmaxLess1.
replace (4 * N)%nat with (2 * (2 * N))%nat; [ idtac | ring ].
apply (fun m n p:nat => mult_le_compat_l p n m).
replace (2 * N)%nat with (S (N + pred N)).
apply le_n_S.
apply plus_le_compat_l; assumption.
rewrite pred_of_minus.
apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR;
rewrite minus_INR.
repeat rewrite S_INR; ring.
apply lt_le_S; assumption.
apply Rle_trans with
(sum_f_R0
(fun k:nat =>
sum_f_R0 (fun l:nat => C ^ (4 * N) * Rsqr (/ INR (fact (S (N + k)))))
(pred (N - k))) (pred N)).
apply sum_Rle; intros.
apply sum_Rle; intros.
rewrite <- (Rmult_comm (C ^ (4 * N))).
apply Rmult_le_compat_l.
apply pow_le.
left; apply Rlt_le_trans with 1.
apply Rlt_0_1.
unfold C in |- *; apply RmaxLess1.
replace (/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))) with
(Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) / INR (fact (2 * S (N + n)))).
apply Rle_trans with
(Binomial.C (2 * S (N + n)) (S (N + n)) / INR (fact (2 * S (N + n)))).
unfold Rdiv in |- *;
do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (N + n))))).
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
apply C_maj.
apply (fun m n p:nat => mult_le_compat_l p n m).
apply le_n_S.
apply plus_le_compat_r.
apply le_trans with (pred (N - n)).
assumption.
apply le_S_n.
replace (S (pred (N - n))) with (N - n)%nat.
apply le_trans with N.
apply (fun p n m:nat => plus_le_reg_l n m p) with n.
rewrite <- le_plus_minus.
apply le_plus_r.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply le_n_Sn.
apply S_pred with 0%nat.
apply plus_lt_reg_l with n.
rewrite <- le_plus_minus.
replace (n + 0)%nat with n; [ idtac | ring ].
apply le_lt_trans with (pred N).
assumption.
apply lt_pred_n_n; assumption.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
right.
unfold Rdiv in |- *; rewrite Rmult_comm.
unfold Binomial.C in |- *.
unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_l.
replace (2 * S (N + n) - S (N + n))%nat with (S (N + n)).
rewrite Rinv_mult_distr.
unfold Rsqr in |- *; reflexivity.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply INR_eq; rewrite S_INR; rewrite minus_INR.
rewrite mult_INR; repeat rewrite S_INR; rewrite plus_INR; ring.
apply le_n_2n.
apply INR_fact_neq_0.
unfold Rdiv in |- *; rewrite Rmult_comm.
unfold Binomial.C in |- *.
unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_l.
replace (2 * S (N + n) - 2 * S (n0 + n))%nat with (2 * (N - n0))%nat.
rewrite mult_INR.
reflexivity.
apply INR_eq; rewrite minus_INR.
do 3 rewrite mult_INR; repeat rewrite S_INR; do 2 rewrite plus_INR;
rewrite minus_INR.
ring.
apply le_trans with (pred (N - n)).
assumption.
apply le_S_n.
replace (S (pred (N - n))) with (N - n)%nat.
apply le_trans with N.
apply (fun p n m:nat => plus_le_reg_l n m p) with n.
rewrite <- le_plus_minus.
apply le_plus_r.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply le_n_Sn.
apply S_pred with 0%nat.
apply plus_lt_reg_l with n.
rewrite <- le_plus_minus.
replace (n + 0)%nat with n; [ idtac | ring ].
apply le_lt_trans with (pred N).
assumption.
apply lt_pred_n_n; assumption.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply (fun m n p:nat => mult_le_compat_l p n m).
apply le_n_S.
apply plus_le_compat_r.
apply le_trans with (pred (N - n)).
assumption.
apply le_S_n.
replace (S (pred (N - n))) with (N - n)%nat.
apply le_trans with N.
apply (fun p n m:nat => plus_le_reg_l n m p) with n.
rewrite <- le_plus_minus.
apply le_plus_r.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply le_n_Sn.
apply S_pred with 0%nat.
apply plus_lt_reg_l with n.
rewrite <- le_plus_minus.
replace (n + 0)%nat with n; [ idtac | ring ].
apply le_lt_trans with (pred N).
assumption.
apply lt_pred_n_n; assumption.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply INR_fact_neq_0.
apply Rle_trans with
(sum_f_R0 (fun k:nat => INR N / INR (fact (S N)) * C ^ (4 * N)) (pred N)).
apply sum_Rle; intros.
rewrite <-
(scal_sum (fun _:nat => C ^ (4 * N)) (pred (N - n))
(Rsqr (/ INR (fact (S (N + n)))))).
rewrite sum_cte.
rewrite <- Rmult_assoc.
do 2 rewrite <- (Rmult_comm (C ^ (4 * N))).
rewrite Rmult_assoc.
apply Rmult_le_compat_l.
apply pow_le.
left; apply Rlt_le_trans with 1.
apply Rlt_0_1.
unfold C in |- *; apply RmaxLess1.
apply Rle_trans with (Rsqr (/ INR (fact (S (N + n)))) * INR N).
apply Rmult_le_compat_l.
apply Rle_0_sqr.
replace (S (pred (N - n))) with (N - n)%nat.
apply le_INR.
apply (fun p n m:nat => plus_le_reg_l n m p) with n.
rewrite <- le_plus_minus.
apply le_plus_r.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply S_pred with 0%nat.
apply plus_lt_reg_l with n.
rewrite <- le_plus_minus.
replace (n + 0)%nat with n; [ idtac | ring ].
apply le_lt_trans with (pred N).
assumption.
apply lt_pred_n_n; assumption.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
rewrite Rmult_comm; unfold Rdiv in |- *; apply Rmult_le_compat_l.
apply pos_INR.
apply Rle_trans with (/ INR (fact (S (N + n)))).
pattern (/ INR (fact (S (N + n)))) at 2 in |- *; rewrite <- Rmult_1_r.
unfold Rsqr in |- *.
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
apply Rmult_le_reg_l with (INR (fact (S (N + n)))).
apply INR_fact_lt_0.
rewrite <- Rinv_r_sym.
rewrite Rmult_1_r.
replace 1 with (INR 1).
apply le_INR.
apply lt_le_S.
apply INR_lt; apply INR_fact_lt_0.
reflexivity.
apply INR_fact_neq_0.
apply Rmult_le_reg_l with (INR (fact (S (N + n)))).
apply INR_fact_lt_0.
rewrite <- Rinv_r_sym.
apply Rmult_le_reg_l with (INR (fact (S N))).
apply INR_fact_lt_0.
rewrite Rmult_1_r.
rewrite (Rmult_comm (INR (fact (S N)))).
rewrite Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_r.
apply le_INR.
apply fact_le.
apply le_n_S.
apply le_plus_l.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
rewrite sum_cte.
apply Rle_trans with (C ^ (4 * N) / INR (fact (pred N))).
rewrite <- (Rmult_comm (C ^ (4 * N))).
unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l.
apply pow_le.
left; apply Rlt_le_trans with 1.
apply Rlt_0_1.
unfold C in |- *; apply RmaxLess1.
cut (S (pred N) = N).
intro; rewrite H0.
pattern N at 2 in |- *; rewrite <- H0.
do 2 rewrite fact_simpl.
rewrite H0.
repeat rewrite mult_INR.
repeat rewrite Rinv_mult_distr.
rewrite (Rmult_comm (/ INR (S N))).
repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_r_sym.
rewrite Rmult_1_l.
pattern (/ INR (fact (pred N))) at 2 in |- *; rewrite <- Rmult_1_r.
rewrite Rmult_assoc.
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
apply Rmult_le_reg_l with (INR (S N)).
apply lt_INR_0; apply lt_O_Sn.
rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
rewrite Rmult_1_r; rewrite Rmult_1_l.
apply le_INR; apply le_n_Sn.
apply not_O_INR; discriminate.
apply not_O_INR.
red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H).
apply not_O_INR.
red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H).
apply INR_fact_neq_0.
apply not_O_INR; discriminate.
apply prod_neq_R0.
apply not_O_INR.
red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H).
apply INR_fact_neq_0.
symmetry in |- *; apply S_pred with 0%nat; assumption.
right.
unfold Majxy in |- *.
unfold C in |- *.
replace (S (pred N)) with N.
reflexivity.
apply S_pred with 0%nat; assumption.
Qed.
Lemma reste2_maj :
forall (x y:R) (N:nat), (0 < N)%nat -> Rabs (Reste2 x y N) <= Majxy x y N.
intros.
set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))).
unfold Reste2 in |- *.
apply Rle_trans with
(sum_f_R0
(fun k:nat =>
Rabs
(sum_f_R0
(fun l:nat =>
(-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) *
x ^ (2 * S (l + k) + 1) *
((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
y ^ (2 * (N - l) + 1)) (pred (N - k)))) (
pred N)).
apply
(Rsum_abs
(fun k:nat =>
sum_f_R0
(fun l:nat =>
(-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) *
x ^ (2 * S (l + k) + 1) *
((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
y ^ (2 * (N - l) + 1)) (pred (N - k))) (
pred N)).
apply Rle_trans with
(sum_f_R0
(fun k:nat =>
sum_f_R0
(fun l:nat =>
Rabs
((-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) *
x ^ (2 * S (l + k) + 1) *
((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
y ^ (2 * (N - l) + 1))) (pred (N - k))) (
pred N)).
apply sum_Rle.
intros.
apply
(Rsum_abs
(fun l:nat =>
(-1) ^ S (l + n) / INR (fact (2 * S (l + n) + 1)) *
x ^ (2 * S (l + n) + 1) *
((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
y ^ (2 * (N - l) + 1)) (pred (N - n))).
apply Rle_trans with
(sum_f_R0
(fun k:nat =>
sum_f_R0
(fun l:nat =>
/ INR (fact (2 * S (l + k) + 1) * fact (2 * (N - l) + 1)) *
C ^ (2 * S (S (N + k)))) (pred (N - k))) (
pred N)).
apply sum_Rle; intros.
apply sum_Rle; intros.
unfold Rdiv in |- *; repeat rewrite Rabs_mult.
do 2 rewrite pow_1_abs.
do 2 rewrite Rmult_1_l.
rewrite (Rabs_right (/ INR (fact (2 * S (n0 + n) + 1)))).
rewrite (Rabs_right (/ INR (fact (2 * (N - n0) + 1)))).
rewrite mult_INR.
rewrite Rinv_mult_distr.
repeat rewrite Rmult_assoc.
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
rewrite <- Rmult_assoc.
rewrite <- (Rmult_comm (/ INR (fact (2 * (N - n0) + 1)))).
rewrite Rmult_assoc.
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
do 2 rewrite <- RPow_abs.
apply Rle_trans with (Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1)).
apply Rmult_le_compat_l.
apply pow_le; apply Rabs_pos.
apply pow_incr.
split.
apply Rabs_pos.
unfold C in |- *.
apply Rle_trans with (Rmax (Rabs x) (Rabs y)); apply RmaxLess2.
apply Rle_trans with (C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1)).
do 2 rewrite <- (Rmult_comm (C ^ (2 * (N - n0) + 1))).
apply Rmult_le_compat_l.
apply pow_le.
apply Rle_trans with 1.
left; apply Rlt_0_1.
unfold C in |- *; apply RmaxLess1.
apply pow_incr.
split.
apply Rabs_pos.
unfold C in |- *; apply Rle_trans with (Rmax (Rabs x) (Rabs y)).
apply RmaxLess1.
apply RmaxLess2.
right.
replace (2 * S (S (N + n)))%nat with
(2 * (N - n0) + 1 + (2 * S (n0 + n) + 1))%nat.
repeat rewrite pow_add.
ring.
apply INR_eq; repeat rewrite plus_INR; do 3 rewrite mult_INR.
rewrite minus_INR.
repeat rewrite S_INR; do 2 rewrite plus_INR; ring.
apply le_trans with (pred (N - n)).
exact H1.
apply le_S_n.
replace (S (pred (N - n))) with (N - n)%nat.
apply le_trans with N.
apply (fun p n m:nat => plus_le_reg_l n m p) with n.
rewrite <- le_plus_minus.
apply le_plus_r.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply le_n_Sn.
apply S_pred with 0%nat.
apply plus_lt_reg_l with n.
rewrite <- le_plus_minus.
replace (n + 0)%nat with n; [ idtac | ring ].
apply le_lt_trans with (pred N).
assumption.
apply lt_pred_n_n; assumption.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply Rle_ge; left; apply Rinv_0_lt_compat.
apply INR_fact_lt_0.
apply Rle_ge; left; apply Rinv_0_lt_compat.
apply INR_fact_lt_0.
apply Rle_trans with
(sum_f_R0
(fun k:nat =>
sum_f_R0
(fun l:nat =>
/ INR (fact (2 * S (l + k) + 1) * fact (2 * (N - l) + 1)) *
C ^ (4 * S N)) (pred (N - k))) (pred N)).
apply sum_Rle; intros.
apply sum_Rle; intros.
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat.
rewrite mult_INR; apply Rmult_lt_0_compat; apply INR_fact_lt_0.
apply Rle_pow.
unfold C in |- *; apply RmaxLess1.
replace (4 * S N)%nat with (2 * (2 * S N))%nat; [ idtac | ring ].
apply (fun m n p:nat => mult_le_compat_l p n m).
replace (2 * S N)%nat with (S (S (N + N))).
repeat apply le_n_S.
apply plus_le_compat_l.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply INR_eq; do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR.
repeat rewrite S_INR; ring.
apply Rle_trans with
(sum_f_R0
(fun k:nat =>
sum_f_R0
(fun l:nat => C ^ (4 * S N) * Rsqr (/ INR (fact (S (S (N + k))))))
(pred (N - k))) (pred N)).
apply sum_Rle; intros.
apply sum_Rle; intros.
rewrite <- (Rmult_comm (C ^ (4 * S N))).
apply Rmult_le_compat_l.
apply pow_le.
left; apply Rlt_le_trans with 1.
apply Rlt_0_1.
unfold C in |- *; apply RmaxLess1.
replace (/ INR (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))) with
(Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n))))).
apply Rle_trans with
(Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n))))).
unfold Rdiv in |- *;
do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (S (N + n)))))).
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
apply C_maj.
apply le_trans with (2 * S (S (n0 + n)))%nat.
replace (2 * S (S (n0 + n)))%nat with (S (2 * S (n0 + n) + 1)).
apply le_n_Sn.
apply INR_eq; rewrite S_INR; rewrite plus_INR; do 2 rewrite mult_INR;
repeat rewrite S_INR; rewrite plus_INR; ring.
apply (fun m n p:nat => mult_le_compat_l p n m).
repeat apply le_n_S.
apply plus_le_compat_r.
apply le_trans with (pred (N - n)).
assumption.
apply le_S_n.
replace (S (pred (N - n))) with (N - n)%nat.
apply le_trans with N.
apply (fun p n m:nat => plus_le_reg_l n m p) with n.
rewrite <- le_plus_minus.
apply le_plus_r.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply le_n_Sn.
apply S_pred with 0%nat.
apply plus_lt_reg_l with n.
rewrite <- le_plus_minus.
replace (n + 0)%nat with n; [ idtac | ring ].
apply le_lt_trans with (pred N).
assumption.
apply lt_pred_n_n; assumption.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
right.
unfold Rdiv in |- *; rewrite Rmult_comm.
unfold Binomial.C in |- *.
unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_l.
replace (2 * S (S (N + n)) - S (S (N + n)))%nat with (S (S (N + n))).
rewrite Rinv_mult_distr.
unfold Rsqr in |- *; reflexivity.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
apply INR_eq; do 2 rewrite S_INR; rewrite minus_INR.
rewrite mult_INR; repeat rewrite S_INR; rewrite plus_INR; ring.
apply le_n_2n.
apply INR_fact_neq_0.
unfold Rdiv in |- *; rewrite Rmult_comm.
unfold Binomial.C in |- *.
unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_l.
replace (2 * S (S (N + n)) - (2 * S (n0 + n) + 1))%nat with
(2 * (N - n0) + 1)%nat.
rewrite mult_INR.
reflexivity.
apply INR_eq; rewrite minus_INR.
do 2 rewrite plus_INR; do 3 rewrite mult_INR; repeat rewrite S_INR;
do 2 rewrite plus_INR; rewrite minus_INR.
ring.
apply le_trans with (pred (N - n)).
assumption.
apply le_S_n.
replace (S (pred (N - n))) with (N - n)%nat.
apply le_trans with N.
apply (fun p n m:nat => plus_le_reg_l n m p) with n.
rewrite <- le_plus_minus.
apply le_plus_r.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply le_n_Sn.
apply S_pred with 0%nat.
apply plus_lt_reg_l with n.
rewrite <- le_plus_minus.
replace (n + 0)%nat with n; [ idtac | ring ].
apply le_lt_trans with (pred N).
assumption.
apply lt_pred_n_n; assumption.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply le_trans with (2 * S (S (n0 + n)))%nat.
replace (2 * S (S (n0 + n)))%nat with (S (2 * S (n0 + n) + 1)).
apply le_n_Sn.
apply INR_eq; rewrite S_INR; rewrite plus_INR; do 2 rewrite mult_INR;
repeat rewrite S_INR; rewrite plus_INR; ring.
apply (fun m n p:nat => mult_le_compat_l p n m).
repeat apply le_n_S.
apply plus_le_compat_r.
apply le_trans with (pred (N - n)).
assumption.
apply le_S_n.
replace (S (pred (N - n))) with (N - n)%nat.
apply le_trans with N.
apply (fun p n m:nat => plus_le_reg_l n m p) with n.
rewrite <- le_plus_minus.
apply le_plus_r.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply le_n_Sn.
apply S_pred with 0%nat.
apply plus_lt_reg_l with n.
rewrite <- le_plus_minus.
replace (n + 0)%nat with n; [ idtac | ring ].
apply le_lt_trans with (pred N).
assumption.
apply lt_pred_n_n; assumption.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply INR_fact_neq_0.
apply Rle_trans with
(sum_f_R0 (fun k:nat => INR N / INR (fact (S (S N))) * C ^ (4 * S N))
(pred N)).
apply sum_Rle; intros.
rewrite <-
(scal_sum (fun _:nat => C ^ (4 * S N)) (pred (N - n))
(Rsqr (/ INR (fact (S (S (N + n))))))).
rewrite sum_cte.
rewrite <- Rmult_assoc.
do 2 rewrite <- (Rmult_comm (C ^ (4 * S N))).
rewrite Rmult_assoc.
apply Rmult_le_compat_l.
apply pow_le.
left; apply Rlt_le_trans with 1.
apply Rlt_0_1.
unfold C in |- *; apply RmaxLess1.
apply Rle_trans with (Rsqr (/ INR (fact (S (S (N + n))))) * INR N).
apply Rmult_le_compat_l.
apply Rle_0_sqr.
replace (S (pred (N - n))) with (N - n)%nat.
apply le_INR.
apply (fun p n m:nat => plus_le_reg_l n m p) with n.
rewrite <- le_plus_minus.
apply le_plus_r.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
apply S_pred with 0%nat.
apply plus_lt_reg_l with n.
rewrite <- le_plus_minus.
replace (n + 0)%nat with n; [ idtac | ring ].
apply le_lt_trans with (pred N).
assumption.
apply lt_pred_n_n; assumption.
apply le_trans with (pred N).
assumption.
apply le_pred_n.
rewrite Rmult_comm; unfold Rdiv in |- *; apply Rmult_le_compat_l.
apply pos_INR.
apply Rle_trans with (/ INR (fact (S (S (N + n))))).
pattern (/ INR (fact (S (S (N + n))))) at 2 in |- *; rewrite <- Rmult_1_r.
unfold Rsqr in |- *.
apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
apply Rmult_le_reg_l with (INR (fact (S (S (N + n))))).
apply INR_fact_lt_0.
rewrite <- Rinv_r_sym.
rewrite Rmult_1_r.
replace 1 with (INR 1).
apply le_INR.
apply lt_le_S.
apply INR_lt; apply INR_fact_lt_0.
reflexivity.
apply INR_fact_neq_0.
apply Rmult_le_reg_l with (INR (fact (S (S (N + n))))).
apply INR_fact_lt_0.
rewrite <- Rinv_r_sym.
apply Rmult_le_reg_l with (INR (fact (S (S N)))).
apply INR_fact_lt_0.
rewrite Rmult_1_r.
rewrite (Rmult_comm (INR (fact (S (S N))))).
rewrite Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_r.
apply le_INR.
apply fact_le.
repeat apply le_n_S.
apply le_plus_l.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
rewrite sum_cte.
apply Rle_trans with (C ^ (4 * S N) / INR (fact N)).
rewrite <- (Rmult_comm (C ^ (4 * S N))).
unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l.
apply pow_le.
left; apply Rlt_le_trans with 1.
apply Rlt_0_1.
unfold C in |- *; apply RmaxLess1.
cut (S (pred N) = N).
intro; rewrite H0.
do 2 rewrite fact_simpl.
repeat rewrite mult_INR.
repeat rewrite Rinv_mult_distr.
apply Rle_trans with
(INR (S (S N)) * (/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) * INR N).
repeat rewrite Rmult_assoc.
rewrite (Rmult_comm (INR N)).
rewrite (Rmult_comm (INR (S (S N)))).
apply Rmult_le_compat_l.
repeat apply Rmult_le_pos.
left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn.
left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn.
left; apply Rinv_0_lt_compat.
apply INR_fact_lt_0.
apply pos_INR.
apply le_INR.
apply le_trans with (S N); apply le_n_Sn.
repeat rewrite <- Rmult_assoc.
rewrite <- Rinv_r_sym.
rewrite Rmult_1_l.
apply Rle_trans with (/ INR (S N) * / INR (fact N) * INR (S N)).
repeat rewrite Rmult_assoc.
repeat apply Rmult_le_compat_l.
left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
apply le_INR; apply le_n_Sn.
rewrite (Rmult_comm (/ INR (S N))).
rewrite Rmult_assoc.
rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; right; reflexivity.
apply not_O_INR; discriminate.
apply not_O_INR; discriminate.
apply not_O_INR; discriminate.
apply INR_fact_neq_0.
apply not_O_INR; discriminate.
apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ].
symmetry in |- *; apply S_pred with 0%nat; assumption.
right.
unfold Majxy in |- *.
unfold C in |- *.
reflexivity.
Qed.
Lemma reste1_cv_R0 : forall x y:R, Un_cv (Reste1 x y) 0.
intros.
assert (H := Majxy_cv_R0 x y).
unfold Un_cv in H; unfold R_dist in H.
unfold Un_cv in |- *; unfold R_dist in |- *; intros.
elim (H eps H0); intros N0 H1.
exists (S N0); intros.
unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r.
apply Rle_lt_trans with (Rabs (Majxy x y (pred n))).
rewrite (Rabs_right (Majxy x y (pred n))).
apply reste1_maj.
apply lt_le_trans with (S N0).
apply lt_O_Sn.
assumption.
apply Rle_ge.
unfold Majxy in |- *.
unfold Rdiv in |- *; apply Rmult_le_pos.
apply pow_le.
apply Rle_trans with 1.
left; apply Rlt_0_1.
apply RmaxLess1.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
replace (Majxy x y (pred n)) with (Majxy x y (pred n) - 0); [ idtac | ring ].
apply H1.
unfold ge in |- *; apply le_S_n.
replace (S (pred n)) with n.
assumption.
apply S_pred with 0%nat.
apply lt_le_trans with (S N0); [ apply lt_O_Sn | assumption ].
Qed.
Lemma reste2_cv_R0 : forall x y:R, Un_cv (Reste2 x y) 0.
intros.
assert (H := Majxy_cv_R0 x y).
unfold Un_cv in H; unfold R_dist in H.
unfold Un_cv in |- *; unfold R_dist in |- *; intros.
elim (H eps H0); intros N0 H1.
exists (S N0); intros.
unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r.
apply Rle_lt_trans with (Rabs (Majxy x y n)).
rewrite (Rabs_right (Majxy x y n)).
apply reste2_maj.
apply lt_le_trans with (S N0).
apply lt_O_Sn.
assumption.
apply Rle_ge.
unfold Majxy in |- *.
unfold Rdiv in |- *; apply Rmult_le_pos.
apply pow_le.
apply Rle_trans with 1.
left; apply Rlt_0_1.
apply RmaxLess1.
left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
replace (Majxy x y n) with (Majxy x y n - 0); [ idtac | ring ].
apply H1.
unfold ge in |- *; apply le_trans with (S N0).
apply le_n_Sn.
exact H2.
Qed.
Lemma reste_cv_R0 : forall x y:R, Un_cv (Reste x y) 0.
intros.
unfold Reste in |- *.
set (An := fun n:nat => Reste2 x y n).
set (Bn := fun n:nat => Reste1 x y (S n)).
cut
(Un_cv (fun n:nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N:nat => Reste2 x y N - Reste1 x y (S N)) 0).
intro.
apply H.
apply CV_minus.
unfold An in |- *.
replace (fun n:nat => Reste2 x y n) with (Reste2 x y).
apply reste2_cv_R0.
reflexivity.
unfold Bn in |- *.
assert (H0 := reste1_cv_R0 x y).
unfold Un_cv in H0; unfold R_dist in H0.
unfold Un_cv in |- *; unfold R_dist in |- *; intros.
elim (H0 eps H1); intros N0 H2.
exists N0; intros.
apply H2.
unfold ge in |- *; apply le_trans with (S N0).
apply le_n_Sn.
apply le_n_S; assumption.
unfold An, Bn in |- *.
intro.
replace 0 with (0 - 0); [ idtac | ring ].
exact H.
Qed.
Theorem cos_plus : forall x y:R, cos (x + y) = cos x * cos y - sin x * sin y.
intros.
cut (Un_cv (C1 x y) (cos x * cos y - sin x * sin y)).
cut (Un_cv (C1 x y) (cos (x + y))).
intros.
apply UL_sequence with (C1 x y); assumption.
apply C1_cvg.
unfold Un_cv in |- *; unfold R_dist in |- *.
intros.
assert (H0 := A1_cvg x).
assert (H1 := A1_cvg y).
assert (H2 := B1_cvg x).
assert (H3 := B1_cvg y).
assert (H4 := CV_mult _ _ _ _ H0 H1).
assert (H5 := CV_mult _ _ _ _ H2 H3).
assert (H6 := reste_cv_R0 x y).
unfold Un_cv in H4; unfold Un_cv in H5; unfold Un_cv in H6.
unfold R_dist in H4; unfold R_dist in H5; unfold R_dist in H6.
cut (0 < eps / 3);
[ intro
| unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
elim (H4 (eps / 3) H7); intros N1 H8.
elim (H5 (eps / 3) H7); intros N2 H9.
elim (H6 (eps / 3) H7); intros N3 H10.
set (N := S (S (max (max N1 N2) N3))).
exists N.
intros.
cut (n = S (pred n)).
intro; rewrite H12.
rewrite <- cos_plus_form.
rewrite <- H12.
apply Rle_lt_trans with
(Rabs (A1 x n * A1 y n - cos x * cos y) +
Rabs (sin x * sin y - B1 x (pred n) * B1 y (pred n) + Reste x y (pred n))).
replace
(A1 x n * A1 y n - B1 x (pred n) * B1 y (pred n) + Reste x y (pred n) -
(cos x * cos y - sin x * sin y)) with
(A1 x n * A1 y n - cos x * cos y +
(sin x * sin y - B1 x (pred n) * B1 y (pred n) + Reste x y (pred n)));
[ apply Rabs_triang | ring ].
replace eps with (eps / 3 + (eps / 3 + eps / 3)).
apply Rplus_lt_compat.
apply H8.
unfold ge in |- *; apply le_trans with N.
unfold N in |- *.
apply le_trans with (max N1 N2).
apply le_max_l.
apply le_trans with (max (max N1 N2) N3).
apply le_max_l.
apply le_trans with (S (max (max N1 N2) N3)); apply le_n_Sn.
assumption.
apply Rle_lt_trans with
(Rabs (sin x * sin y - B1 x (pred n) * B1 y (pred n)) +
Rabs (Reste x y (pred n))).
apply Rabs_triang.
apply Rplus_lt_compat.
rewrite <- Rabs_Ropp.
rewrite Ropp_minus_distr.
apply H9.
unfold ge in |- *; apply le_trans with (max N1 N2).
apply le_max_r.
apply le_S_n.
rewrite <- H12.
apply le_trans with N.
unfold N in |- *.
apply le_n_S.
apply le_trans with (max (max N1 N2) N3).
apply le_max_l.
apply le_n_Sn.
assumption.
replace (Reste x y (pred n)) with (Reste x y (pred n) - 0).
apply H10.
unfold ge in |- *.
apply le_S_n.
rewrite <- H12.
apply le_trans with N.
unfold N in |- *.
apply le_n_S.
apply le_trans with (max (max N1 N2) N3).
apply le_max_r.
apply le_n_Sn.
assumption.
ring.
pattern eps at 4 in |- *; replace eps with (3 * (eps / 3)).
ring.
unfold Rdiv in |- *.
rewrite <- Rmult_assoc.
apply Rinv_r_simpl_m.
discrR.
apply lt_le_trans with (pred N).
unfold N in |- *; simpl in |- *; apply lt_O_Sn.
apply le_S_n.
rewrite <- H12.
replace (S (pred N)) with N.
assumption.
unfold N in |- *; simpl in |- *; reflexivity.
cut (0 < N)%nat.
intro.
cut (0 < n)%nat.
intro.
apply S_pred with 0%nat; assumption.
apply lt_le_trans with N; assumption.
unfold N in |- *; apply lt_O_Sn.
Qed.
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