(************************************************************************)
(*         *   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 Export Rtrigo_fun.
Require Export Rtrigo_def.
Require Export Rtrigo_alt.
Require Export Cos_rel.
Require Export Cos_plus.
Local Open Scope nat_scope.
Local Open Scope R_scope.
Definition PI_2_aux : {z | 7/8 <= z <= 7/4 /\ -cos z = 0}.
assert (cc : continuity (fun r =>- cos r)).
apply continuity_opp, continuity_cos.
assert (cvp : 0 < cos (7/8)).
assert (int78 : -2 <= 7/8 <= 2) by (split; lra).
destruct int78 as [lower upper].
case (pre_cos_bound _ 0 lower upper).
unfold cos_approx; simpl sum_f_R0; unfold cos_term.
intros cl _; apply Rlt_le_trans with (2 := cl); simpl.
lra.
assert (cun : cos (7/4) < 0).
replace (7/4) with (7/8 + 7/8) by field.
rewrite cos_plus.
apply Rlt_minus; apply Rsqr_incrst_1.
exact sin_gt_cos_7_8.
apply Rlt_le; assumption.
apply Rlt_le; apply Rlt_trans with (1 := cvp); exact sin_gt_cos_7_8.
apply IVT; auto; lra.
Qed.
Definition PI2 := proj1_sig PI_2_aux.
Definition PI := 2 * PI2.
unfold PI2; case PI_2_aux; simpl.
intros x [_ q]; rewrite <- (Ropp_involutive (cos x)), q; apply Ropp_0.
Qed.
unfold PI2; case PI_2_aux; simpl; tauto.
Qed.
intros x [int1 int2].
assert (lo : 0 <= x) by (apply Rlt_le; assumption).
assert (up : x <= 4) by (apply Rlt_le, Rlt_trans with (1:=int2); lra).
destruct (pre_sin_bound _ 0 lo up) as [t _]; clear lo up.
apply Rlt_le_trans with (2:= t); clear t.
unfold sin_approx; simpl sum_f_R0; unfold sin_term; simpl.
match goal with |- _ < ?a =>
replace a with (x * (1 - x^2/6)) by (simpl; field)
end.
assert (t' : x ^ 2 <= 4).
replace 4 with (2 ^ 2) by field.
apply (pow_incr x 2); split; apply Rlt_le; assumption.
apply Rmult_lt_0_compat;[assumption | lra ].
Qed.
replace (PI / 2) with PI2 by (unfold PI; field).
assert (int' : 0 < PI2 < 2).
destruct pi2_int; split; lra.
assert (lo2 := sin_pos_tech PI2 int').
assert (t2 : Rabs (sin PI2) = 1).
rewrite <- Rabs_R1; apply Rsqr_eq_abs_0.
rewrite Rsqr_1, sin2, cos_pi2, Rsqr_0, Rminus_0_r; reflexivity.
revert t2; rewrite Rabs_pos_eq;[| apply Rlt_le]; tauto.
Qed.

Lemma cos_bound : forall (a : R) (n : nat), - PI / 2 <= a -> a <= PI / 2 ->
cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1)).