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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo_def.
Local Open Scope R_scope.
(***************************************************************)
(** Using series definitions of cos and sin *)
(***************************************************************)
Definition sin_term (a:R) (i:nat) : R :=
(-1) ^ i * (a ^ (2 * i + 1) / INR (fact (2 * i + 1))).
Definition cos_term (a:R) (i:nat) : R :=
(-1) ^ i * (a ^ (2 * i) / INR (fact (2 * i))).
Definition sin_approx (a:R) (n:nat) : R := sum_f_R0 (sin_term a) n.
Definition cos_approx (a:R) (n:nat) : R := sum_f_R0 (cos_term a) n.
(**********)
(*
Lemma Alt_PI_4 : Alt_PI <= 4.
Proof.
assert (H0 := PI_ineq 0).
elim H0; clear H0; intros _ H0.
unfold tg_alt, PI_tg in H0; simpl in H0.
rewrite Rinv_1 in H0; rewrite Rmult_1_r in H0; unfold Rdiv in H0.
apply Rmult_le_reg_l with (/ 4).
apply Rinv_0_lt_compat; prove_sup0.
rewrite <- Rinv_l_sym; [ rewrite Rmult_comm; assumption | discrR ].
Qed.
*)
(**********)
Theorem pre_sin_bound :
forall (a:R) (n:nat),
0 <= a ->
a <= 4 -> sin_approx a (2 * n + 1) <= sin a <= sin_approx a (2 * (n + 1)).
Proof.
intros; case (Req_dec a 0); intro Hyp_a.
rewrite Hyp_a; rewrite sin_0; split; right; unfold sin_approx;
apply sum_eq_R0 || (symmetry ; apply sum_eq_R0);
intros; unfold sin_term; rewrite pow_add;
simpl; unfold Rdiv; rewrite Rmult_0_l;
ring.
unfold sin_approx; cut (0 < a).
intro Hyp_a_pos.
rewrite (decomp_sum (sin_term a) (2 * n + 1)).
rewrite (decomp_sum (sin_term a) (2 * (n + 1))).
replace (sin_term a 0) with a.
cut
(sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * n + 1)) <= sin a - a /\
sin a - a <= sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * (n + 1))) ->
a + sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * n + 1)) <= sin a /\
sin a <= a + sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * (n + 1)))).
intro; apply H1.
set (Un := fun n:nat => a ^ (2 * S n + 1) / INR (fact (2 * S n + 1))).
replace (pred (2 * n + 1)) with (2 * n)%nat.
replace (pred (2 * (n + 1))) with (S (2 * n)).
replace (sum_f_R0 (fun i:nat => sin_term a (S i)) (2 * n)) with
(- sum_f_R0 (tg_alt Un) (2 * n)).
replace (sum_f_R0 (fun i:nat => sin_term a (S i)) (S (2 * n))) with
(- sum_f_R0 (tg_alt Un) (S (2 * n))).
cut
(sum_f_R0 (tg_alt Un) (S (2 * n)) <= a - sin a <=
sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= sin a - a <=
- sum_f_R0 (tg_alt Un) (S (2 * n))).
intro; apply H2.
apply alternated_series_ineq.
unfold Un_decreasing, Un; intro;
cut ((2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))).
intro; rewrite H3.
replace (a ^ S (S (2 * S n0 + 1))) with (a ^ (2 * S n0 + 1) * (a * a)).
unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l.
left; apply pow_lt; assumption.
apply Rmult_le_reg_l with (INR (fact (S (S (2 * S n0 + 1))))).
rewrite <- H3; apply lt_INR_0; apply neq_O_lt; red; intro;
assert (H5 := eq_sym H4); elim (fact_neq_0 _ H5).
rewrite <- H3; rewrite (Rmult_comm (INR (fact (2 * S (S n0) + 1))));
rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; rewrite H3; do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
rewrite Rmult_1_r.
do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR;
simpl;
replace
(((0 + 1 + 1) * (INR n0 + 1) + (0 + 1) + 1 + 1) *
((0 + 1 + 1) * (INR n0 + 1) + (0 + 1) + 1)) with
(4 * INR n0 * INR n0 + 18 * INR n0 + 20); [ idtac | ring ].
apply Rle_trans with 20.
apply Rle_trans with 16.
replace 16 with (Rsqr 4); [ idtac | ring_Rsqr ].
apply Rsqr_incr_1.
assumption.
assumption.
now apply IZR_le.
now apply IZR_le.
rewrite <- (Rplus_0_l 20) at 1;
apply Rplus_le_compat_r.
apply Rplus_le_le_0_compat.
apply Rmult_le_pos.
apply Rmult_le_pos.
now apply IZR_le.
apply pos_INR.
apply pos_INR.
apply Rmult_le_pos.
now apply IZR_le.
apply pos_INR.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
simpl; ring.
ring.
assert (H3 := cv_speed_pow_fact a); unfold Un; unfold Un_cv in H3;
unfold R_dist in H3; unfold Un_cv; unfold R_dist;
intros; elim (H3 eps H4); intros N H5.
exists N; intros; apply H5.
replace (2 * S n0 + 1)%nat with (S (2 * S n0)).
unfold ge; apply le_trans with (2 * S n0)%nat.
apply le_trans with (2 * S N)%nat.
apply le_trans with (2 * N)%nat.
apply le_n_2n.
apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_Sn.
apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_S; assumption.
apply le_n_Sn.
ring.
unfold sin.
destruct (exist_sin (Rsqr a)) as (x,p).
unfold sin_in, infinite_sum, R_dist in p;
unfold Un_cv, R_dist;
intros.
cut (0 < eps / Rabs a).
intro H4; destruct (p _ H4) as (N,H6).
exists N; intros.
replace (sum_f_R0 (tg_alt Un) n0) with
(a * (1 - sum_f_R0 (fun i:nat => sin_n i * Rsqr a ^ i) (S n0))).
unfold Rminus; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r;
rewrite Ropp_plus_distr; rewrite Ropp_involutive;
repeat rewrite Rplus_assoc; rewrite (Rplus_comm a);
rewrite (Rplus_comm (- a)); repeat rewrite Rplus_assoc;
rewrite Rplus_opp_l; rewrite Rplus_0_r; apply Rmult_lt_reg_l with (/ Rabs a).
apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
pattern (/ Rabs a) at 1; rewrite <- (Rabs_Rinv a Hyp_a).
rewrite <- Rabs_mult, Rmult_plus_distr_l, <- 2!Rmult_assoc, <- Rinv_l_sym;
[ rewrite Rmult_1_l | assumption ];
rewrite (Rmult_comm (/ Rabs a)),
<- Rabs_Ropp, Ropp_plus_distr, Ropp_involutive, Rmult_1_l.
unfold Rminus, Rdiv in H6. apply H6; unfold ge;
apply le_trans with n0; [ exact H5 | apply le_n_Sn ].
rewrite (decomp_sum (fun i:nat => sin_n i * Rsqr a ^ i) (S n0)).
replace (sin_n 0) with 1.
simpl; rewrite Rmult_1_r; unfold Rminus;
rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r;
rewrite Rplus_0_l; rewrite Ropp_mult_distr_r_reverse;
rewrite <- Ropp_mult_distr_l_reverse; rewrite scal_sum;
apply sum_eq.
intros; unfold sin_n, Un, tg_alt;
replace ((-1) ^ S i) with (- (-1) ^ i).
replace (a ^ (2 * S i + 1)) with (Rsqr a * Rsqr a ^ i * a).
unfold Rdiv; ring.
rewrite pow_add; rewrite pow_Rsqr; simpl; ring.
simpl; ring.
unfold sin_n; unfold Rdiv; simpl; rewrite Rinv_1;
rewrite Rmult_1_r; reflexivity.
apply lt_O_Sn.
unfold Rdiv; apply Rmult_lt_0_compat.
assumption.
apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
intros; elim H2; intros.
replace (sin a - a) with (- (a - sin a)); [ idtac | ring ].
split; apply Ropp_le_contravar; assumption.
replace (- sum_f_R0 (tg_alt Un) (S (2 * n))) with
(-1 * sum_f_R0 (tg_alt Un) (S (2 * n))); [ rewrite scal_sum | ring ].
apply sum_eq; intros; unfold sin_term, Un, tg_alt;
change ((-1) ^ S i) with (-1 * (-1) ^ i).
unfold Rdiv; ring.
replace (- sum_f_R0 (tg_alt Un) (2 * n)) with
(-1 * sum_f_R0 (tg_alt Un) (2 * n)); [ rewrite scal_sum | ring ].
apply sum_eq; intros.
unfold sin_term, Un, tg_alt;
change ((-1) ^ S i) with (-1 * (-1) ^ i).
unfold Rdiv; ring.
replace (2 * (n + 1))%nat with (S (S (2 * n))).
reflexivity.
ring.
replace (2 * n + 1)%nat with (S (2 * n)).
reflexivity.
ring.
intro; elim H1; intros.
split.
apply Rplus_le_reg_l with (- a).
rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
rewrite (Rplus_comm (- a)); apply H2.
apply Rplus_le_reg_l with (- a).
rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
rewrite (Rplus_comm (- a)); apply H3.
unfold sin_term; simpl; unfold Rdiv; rewrite Rinv_1;
ring.
replace (2 * (n + 1))%nat with (S (S (2 * n))).
apply lt_O_Sn.
ring.
replace (2 * n + 1)%nat with (S (2 * n)).
apply lt_O_Sn.
ring.
inversion H; [ assumption | elim Hyp_a; symmetry ; assumption ].
Qed.
(**********)
Lemma pre_cos_bound :
forall (a:R) (n:nat),
- 2 <= a -> a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1)).
Proof.
cut
((forall (a:R) (n:nat),
0 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1))) ->
forall (a:R) (n:nat),
- 2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1))).
intros H a n; apply H.
intros; unfold cos_approx.
rewrite (decomp_sum (cos_term a0) (2 * n0 + 1)).
rewrite (decomp_sum (cos_term a0) (2 * (n0 + 1))).
replace (cos_term a0 0) with 1.
cut
(sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * n0 + 1)) <= cos a0 - 1 /\
cos a0 - 1 <=
sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * (n0 + 1))) ->
1 + sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * n0 + 1)) <= cos a0 /\
cos a0 <=
1 + sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * (n0 + 1)))).
intro; apply H2.
set (Un := fun n:nat => a0 ^ (2 * S n) / INR (fact (2 * S n))).
replace (pred (2 * n0 + 1)) with (2 * n0)%nat.
replace (pred (2 * (n0 + 1))) with (S (2 * n0)).
replace (sum_f_R0 (fun i:nat => cos_term a0 (S i)) (2 * n0)) with
(- sum_f_R0 (tg_alt Un) (2 * n0)).
replace (sum_f_R0 (fun i:nat => cos_term a0 (S i)) (S (2 * n0))) with
(- sum_f_R0 (tg_alt Un) (S (2 * n0))).
cut
(sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))).
intro; apply H3.
apply alternated_series_ineq.
unfold Un_decreasing; intro; unfold Un.
cut ((2 * S (S n1))%nat = S (S (2 * S n1))).
intro; rewrite H4;
replace (a0 ^ S (S (2 * S n1))) with (a0 ^ (2 * S n1) * (a0 * a0)).
unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l.
apply pow_le; assumption.
apply Rmult_le_reg_l with (INR (fact (S (S (2 * S n1))))).
rewrite <- H4; apply lt_INR_0; apply neq_O_lt; red; intro;
assert (H6 := eq_sym H5); elim (fact_neq_0 _ H6).
rewrite <- H4; rewrite (Rmult_comm (INR (fact (2 * S (S n1)))));
rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; rewrite H4; do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
rewrite Rmult_1_r; do 2 rewrite S_INR; rewrite mult_INR; repeat rewrite S_INR;
simpl;
replace
(((0 + 1 + 1) * (INR n1 + 1) + 1 + 1) * ((0 + 1 + 1) * (INR n1 + 1) + 1))
with (4 * INR n1 * INR n1 + 14 * INR n1 + 12); [ idtac | ring ].
apply Rle_trans with 12.
apply Rle_trans with 4.
change 4 with (Rsqr 2).
apply Rsqr_incr_1.
assumption.
assumption.
now apply IZR_le.
now apply IZR_le.
rewrite <- (Rplus_0_l 12) at 1;
apply Rplus_le_compat_r.
apply Rplus_le_le_0_compat.
apply Rmult_le_pos.
apply Rmult_le_pos.
now apply IZR_le.
apply pos_INR.
apply pos_INR.
apply Rmult_le_pos.
now apply IZR_le.
apply pos_INR.
apply INR_fact_neq_0.
apply INR_fact_neq_0.
simpl; ring.
ring.
assert (H4 := cv_speed_pow_fact a0); unfold Un; unfold Un_cv in H4;
unfold R_dist in H4; unfold Un_cv; unfold R_dist;
intros; elim (H4 eps H5); intros N H6; exists N; intros.
apply H6; unfold ge; apply le_trans with (2 * S N)%nat.
apply le_trans with (2 * N)%nat.
apply le_n_2n.
apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_Sn.
apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_S; assumption.
unfold cos. destruct (exist_cos (Rsqr a0)) as (x,p).
unfold cos_in, infinite_sum, R_dist in p;
unfold Un_cv, R_dist; intros.
destruct (p _ H4) as (N,H6).
exists N; intros.
replace (sum_f_R0 (tg_alt Un) n1) with
(1 - sum_f_R0 (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)).
unfold Rminus; rewrite Ropp_plus_distr; rewrite Ropp_involutive;
repeat rewrite Rplus_assoc; rewrite (Rplus_comm 1);
rewrite (Rplus_comm (-(1))); repeat rewrite Rplus_assoc;
rewrite Rplus_opp_l; rewrite Rplus_0_r; rewrite <- Rabs_Ropp;
rewrite Ropp_plus_distr; rewrite Ropp_involutive;
unfold Rminus in H6; apply H6.
unfold ge; apply le_trans with n1.
exact H5.
apply le_n_Sn.
rewrite (decomp_sum (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)).
replace (cos_n 0) with 1.
simpl; rewrite Rmult_1_r; unfold Rminus;
rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r;
rewrite Rplus_0_l;
replace (- sum_f_R0 (fun i:nat => cos_n (S i) * (Rsqr a0 * Rsqr a0 ^ i)) n1)
with
(-1 * sum_f_R0 (fun i:nat => cos_n (S i) * (Rsqr a0 * Rsqr a0 ^ i)) n1);
[ idtac | ring ]; rewrite scal_sum; apply sum_eq;
intros; unfold cos_n, Un, tg_alt.
replace ((-1) ^ S i) with (- (-1) ^ i).
replace (a0 ^ (2 * S i)) with (Rsqr a0 * Rsqr a0 ^ i).
unfold Rdiv; ring.
rewrite pow_Rsqr; reflexivity.
simpl; ring.
unfold cos_n; unfold Rdiv; simpl; rewrite Rinv_1;
rewrite Rmult_1_r; reflexivity.
apply lt_O_Sn.
intros; elim H3; intros; replace (cos a0 - 1) with (- (1 - cos a0));
[ idtac | ring ].
split; apply Ropp_le_contravar; assumption.
replace (- sum_f_R0 (tg_alt Un) (S (2 * n0))) with
(-1 * sum_f_R0 (tg_alt Un) (S (2 * n0))); [ rewrite scal_sum | ring ].
apply sum_eq; intros; unfold cos_term, Un, tg_alt;
change ((-1) ^ S i) with (-1 * (-1) ^ i).
unfold Rdiv; ring.
replace (- sum_f_R0 (tg_alt Un) (2 * n0)) with
(-1 * sum_f_R0 (tg_alt Un) (2 * n0)); [ rewrite scal_sum | ring ];
apply sum_eq; intros; unfold cos_term, Un, tg_alt;
change ((-1) ^ S i) with (-1 * (-1) ^ i).
unfold Rdiv; ring.
replace (2 * (n0 + 1))%nat with (S (S (2 * n0))).
reflexivity.
ring.
replace (2 * n0 + 1)%nat with (S (2 * n0)).
reflexivity.
ring.
intro; elim H2; intros; split.
apply Rplus_le_reg_l with (-(1)).
rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
rewrite (Rplus_comm (-1)); apply H3.
apply Rplus_le_reg_l with (-(1)).
rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
rewrite (Rplus_comm (-1)); apply H4.
unfold cos_term; simpl; unfold Rdiv; rewrite Rinv_1;
ring.
replace (2 * (n0 + 1))%nat with (S (S (2 * n0))).
apply lt_O_Sn.
ring.
replace (2 * n0 + 1)%nat with (S (2 * n0)).
apply lt_O_Sn.
ring.
intros; destruct (total_order_T 0 a) as [[Hlt|Heq]|Hgt].
apply H; [ left; assumption | assumption ].
apply H; [ right; assumption | assumption ].
cut (0 < - a).
intro; cut (forall (x:R) (n:nat), cos_approx x n = cos_approx (- x) n).
intro; rewrite H3; rewrite (H3 a (2 * (n + 1))%nat); rewrite cos_sym; apply H.
left; assumption.
rewrite <- (Ropp_involutive 2); apply Ropp_le_contravar; exact H0.
intros; unfold cos_approx; apply sum_eq; intros;
unfold cos_term; do 2 rewrite pow_Rsqr; rewrite Rsqr_neg;
unfold Rdiv; reflexivity.
apply Ropp_0_gt_lt_contravar; assumption.
Qed.
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