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diff --git a/theories/Reals/Rtrigo.v b/theories/Reals/Rtrigo.v new file mode 100644 index 00000000..e4cae6c6 --- /dev/null +++ b/theories/Reals/Rtrigo.v @@ -0,0 +1,1707 @@ +(************************************************************************) +(* 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: Rtrigo.v,v 1.40.2.1 2004/07/16 19:31:14 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Export Rtrigo_fun. +Require Export Rtrigo_def. +Require Export Rtrigo_alt. +Require Export Cos_rel. +Require Export Cos_plus. +Require Import ZArith_base. +Require Import Zcomplements. +Require Import Classical_Prop. +Open Local Scope nat_scope. +Open Local Scope R_scope. + +(** sin_PI2 is the only remaining axiom **) +Axiom sin_PI2 : sin (PI / 2) = 1. + +(**********) +Lemma PI_neq0 : PI <> 0. +red in |- *; intro; assert (H0 := PI_RGT_0); rewrite H in H0; + elim (Rlt_irrefl _ H0). +Qed. + +(**********) +Lemma cos_minus : forall x y:R, cos (x - y) = cos x * cos y + sin x * sin y. +intros; unfold Rminus in |- *; rewrite cos_plus. +rewrite <- cos_sym; rewrite sin_antisym; ring. +Qed. + +(**********) +Lemma sin2_cos2 : forall x:R, Rsqr (sin x) + Rsqr (cos x) = 1. +intro; unfold Rsqr in |- *; rewrite Rplus_comm; rewrite <- (cos_minus x x); + unfold Rminus in |- *; rewrite Rplus_opp_r; apply cos_0. +Qed. + +Lemma cos2 : forall x:R, Rsqr (cos x) = 1 - Rsqr (sin x). +intro x; generalize (sin2_cos2 x); intro H1; rewrite <- H1; + unfold Rminus in |- *; rewrite <- (Rplus_comm (Rsqr (cos x))); + rewrite Rplus_assoc; rewrite Rplus_opp_r; symmetry in |- *; + apply Rplus_0_r. +Qed. + +(**********) +Lemma cos_PI2 : cos (PI / 2) = 0. +apply Rsqr_eq_0; rewrite cos2; rewrite sin_PI2; rewrite Rsqr_1; + unfold Rminus in |- *; apply Rplus_opp_r. +Qed. + +(**********) +Lemma cos_PI : cos PI = -1. +replace PI with (PI / 2 + PI / 2). +rewrite cos_plus. +rewrite sin_PI2; rewrite cos_PI2. +ring. +symmetry in |- *; apply double_var. +Qed. + +Lemma sin_PI : sin PI = 0. +assert (H := sin2_cos2 PI). +rewrite cos_PI in H. +rewrite <- Rsqr_neg in H. +rewrite Rsqr_1 in H. +cut (Rsqr (sin PI) = 0). +intro; apply (Rsqr_eq_0 _ H0). +apply Rplus_eq_reg_l with 1. +rewrite Rplus_0_r; rewrite Rplus_comm; exact H. +Qed. + +(**********) +Lemma neg_cos : forall x:R, cos (x + PI) = - cos x. +intro x; rewrite cos_plus; rewrite sin_PI; rewrite cos_PI; ring. +Qed. + +(**********) +Lemma sin_cos : forall x:R, sin x = - cos (PI / 2 + x). +intro x; rewrite cos_plus; rewrite sin_PI2; rewrite cos_PI2; ring. +Qed. + +(**********) +Lemma sin_plus : forall x y:R, sin (x + y) = sin x * cos y + cos x * sin y. +intros. +rewrite (sin_cos (x + y)). +replace (PI / 2 + (x + y)) with (PI / 2 + x + y); [ rewrite cos_plus | ring ]. +rewrite (sin_cos (PI / 2 + x)). +replace (PI / 2 + (PI / 2 + x)) with (x + PI). +rewrite neg_cos. +replace (cos (PI / 2 + x)) with (- sin x). +ring. +rewrite sin_cos; rewrite Ropp_involutive; reflexivity. +pattern PI at 1 in |- *; rewrite (double_var PI); ring. +Qed. + +Lemma sin_minus : forall x y:R, sin (x - y) = sin x * cos y - cos x * sin y. +intros; unfold Rminus in |- *; rewrite sin_plus. +rewrite <- cos_sym; rewrite sin_antisym; ring. +Qed. + +(**********) +Definition tan (x:R) : R := sin x / cos x. + +Lemma tan_plus : + forall x y:R, + cos x <> 0 -> + cos y <> 0 -> + cos (x + y) <> 0 -> + 1 - tan x * tan y <> 0 -> + tan (x + y) = (tan x + tan y) / (1 - tan x * tan y). +intros; unfold tan in |- *; rewrite sin_plus; rewrite cos_plus; + unfold Rdiv in |- *; + replace (cos x * cos y - sin x * sin y) with + (cos x * cos y * (1 - sin x * / cos x * (sin y * / cos y))). +rewrite Rinv_mult_distr. +repeat rewrite <- Rmult_assoc; + replace ((sin x * cos y + cos x * sin y) * / (cos x * cos y)) with + (sin x * / cos x + sin y * / cos y). +reflexivity. +rewrite Rmult_plus_distr_r; rewrite Rinv_mult_distr. +repeat rewrite Rmult_assoc; repeat rewrite (Rmult_comm (sin x)); + repeat rewrite <- Rmult_assoc. +repeat rewrite Rinv_r_simpl_m; [ reflexivity | assumption | assumption ]. +assumption. +assumption. +apply prod_neq_R0; assumption. +assumption. +unfold Rminus in |- *; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r; + apply Rplus_eq_compat_l; repeat rewrite Rmult_assoc; + rewrite (Rmult_comm (sin x)); rewrite (Rmult_comm (cos y)); + rewrite <- Ropp_mult_distr_r_reverse; repeat rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite (Rmult_comm (sin x)); + rewrite <- Ropp_mult_distr_r_reverse; repeat rewrite Rmult_assoc; + apply Rmult_eq_compat_l; rewrite (Rmult_comm (/ cos y)); + rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +apply Rmult_1_r. +assumption. +assumption. +Qed. + +(*******************************************************) +(* Some properties of cos, sin and tan *) +(*******************************************************) + +Lemma sin2 : forall x:R, Rsqr (sin x) = 1 - Rsqr (cos x). +intro x; generalize (cos2 x); intro H1; rewrite H1. +unfold Rminus in |- *; rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_l; symmetry in |- *; + apply Ropp_involutive. +Qed. + +Lemma sin_2a : forall x:R, sin (2 * x) = 2 * sin x * cos x. +intro x; rewrite double; rewrite sin_plus. +rewrite <- (Rmult_comm (sin x)); symmetry in |- *; rewrite Rmult_assoc; + apply double. +Qed. + +Lemma cos_2a : forall x:R, cos (2 * x) = cos x * cos x - sin x * sin x. +intro x; rewrite double; apply cos_plus. +Qed. + +Lemma cos_2a_cos : forall x:R, cos (2 * x) = 2 * cos x * cos x - 1. +intro x; rewrite double; unfold Rminus in |- *; rewrite Rmult_assoc; + rewrite cos_plus; generalize (sin2_cos2 x); rewrite double; + intro H1; rewrite <- H1; ring_Rsqr. +Qed. + +Lemma cos_2a_sin : forall x:R, cos (2 * x) = 1 - 2 * sin x * sin x. +intro x; rewrite Rmult_assoc; unfold Rminus in |- *; repeat rewrite double. +generalize (sin2_cos2 x); intro H1; rewrite <- H1; rewrite cos_plus; + ring_Rsqr. +Qed. + +Lemma tan_2a : + forall x:R, + cos x <> 0 -> + cos (2 * x) <> 0 -> + 1 - tan x * tan x <> 0 -> tan (2 * x) = 2 * tan x / (1 - tan x * tan x). +repeat rewrite double; intros; repeat rewrite double; rewrite double in H0; + apply tan_plus; assumption. +Qed. + +Lemma sin_neg : forall x:R, sin (- x) = - sin x. +apply sin_antisym. +Qed. + +Lemma cos_neg : forall x:R, cos (- x) = cos x. +intro; symmetry in |- *; apply cos_sym. +Qed. + +Lemma tan_0 : tan 0 = 0. +unfold tan in |- *; rewrite sin_0; rewrite cos_0. +unfold Rdiv in |- *; apply Rmult_0_l. +Qed. + +Lemma tan_neg : forall x:R, tan (- x) = - tan x. +intros x; unfold tan in |- *; rewrite sin_neg; rewrite cos_neg; + unfold Rdiv in |- *. +apply Ropp_mult_distr_l_reverse. +Qed. + +Lemma tan_minus : + forall x y:R, + cos x <> 0 -> + cos y <> 0 -> + cos (x - y) <> 0 -> + 1 + tan x * tan y <> 0 -> + tan (x - y) = (tan x - tan y) / (1 + tan x * tan y). +intros; unfold Rminus in |- *; rewrite tan_plus. +rewrite tan_neg; unfold Rminus in |- *; rewrite <- Ropp_mult_distr_l_reverse; + rewrite Rmult_opp_opp; reflexivity. +assumption. +rewrite cos_neg; assumption. +assumption. +rewrite tan_neg; unfold Rminus in |- *; rewrite <- Ropp_mult_distr_l_reverse; + rewrite Rmult_opp_opp; assumption. +Qed. + +Lemma cos_3PI2 : cos (3 * (PI / 2)) = 0. +replace (3 * (PI / 2)) with (PI + PI / 2). +rewrite cos_plus; rewrite sin_PI; rewrite cos_PI2; ring. +pattern PI at 1 in |- *; rewrite (double_var PI). +ring. +Qed. + +Lemma sin_2PI : sin (2 * PI) = 0. +rewrite sin_2a; rewrite sin_PI; ring. +Qed. + +Lemma cos_2PI : cos (2 * PI) = 1. +rewrite cos_2a; rewrite sin_PI; rewrite cos_PI; ring. +Qed. + +Lemma neg_sin : forall x:R, sin (x + PI) = - sin x. +intro x; rewrite sin_plus; rewrite sin_PI; rewrite cos_PI; ring. +Qed. + +Lemma sin_PI_x : forall x:R, sin (PI - x) = sin x. +intro x; rewrite sin_minus; rewrite sin_PI; rewrite cos_PI; rewrite Rmult_0_l; + unfold Rminus in |- *; rewrite Rplus_0_l; rewrite Ropp_mult_distr_l_reverse; + rewrite Ropp_involutive; apply Rmult_1_l. +Qed. + +Lemma sin_period : forall (x:R) (k:nat), sin (x + 2 * INR k * PI) = sin x. +intros x k; induction k as [| k Hreck]. +cut (x + 2 * INR 0 * PI = x); [ intro; rewrite H; reflexivity | ring ]. +replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI); + [ rewrite sin_plus; rewrite sin_2PI; rewrite cos_2PI; ring; apply Hreck + | rewrite S_INR; ring ]. +Qed. + +Lemma cos_period : forall (x:R) (k:nat), cos (x + 2 * INR k * PI) = cos x. +intros x k; induction k as [| k Hreck]. +cut (x + 2 * INR 0 * PI = x); [ intro; rewrite H; reflexivity | ring ]. +replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI); + [ rewrite cos_plus; rewrite sin_2PI; rewrite cos_2PI; ring; apply Hreck + | rewrite S_INR; ring ]. +Qed. + +Lemma sin_shift : forall x:R, sin (PI / 2 - x) = cos x. +intro x; rewrite sin_minus; rewrite sin_PI2; rewrite cos_PI2; ring. +Qed. + +Lemma cos_shift : forall x:R, cos (PI / 2 - x) = sin x. +intro x; rewrite cos_minus; rewrite sin_PI2; rewrite cos_PI2; ring. +Qed. + +Lemma cos_sin : forall x:R, cos x = sin (PI / 2 + x). +intro x; rewrite sin_plus; rewrite sin_PI2; rewrite cos_PI2; ring. +Qed. + +Lemma PI2_RGT_0 : 0 < PI / 2. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup ]. +Qed. + +Lemma SIN_bound : forall x:R, -1 <= sin x <= 1. +intro; case (Rle_dec (-1) (sin x)); intro. +case (Rle_dec (sin x) 1); intro. +split; assumption. +cut (1 < sin x). +intro; + generalize + (Rsqr_incrst_1 1 (sin x) H (Rlt_le 0 1 Rlt_0_1) + (Rlt_le 0 (sin x) (Rlt_trans 0 1 (sin x) Rlt_0_1 H))); + rewrite Rsqr_1; intro; rewrite sin2 in H0; unfold Rminus in H0; + generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0); + repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l; + rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1; + generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1); + repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x)); + intro; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rsqr (cos x)) 0 H3 H2)). +auto with real. +cut (sin x < -1). +intro; generalize (Ropp_lt_gt_contravar (sin x) (-1) H); + rewrite Ropp_involutive; clear H; intro; + generalize + (Rsqr_incrst_1 1 (- sin x) H (Rlt_le 0 1 Rlt_0_1) + (Rlt_le 0 (- sin x) (Rlt_trans 0 1 (- sin x) Rlt_0_1 H))); + rewrite Rsqr_1; intro; rewrite <- Rsqr_neg in H0; + rewrite sin2 in H0; unfold Rminus in H0; + generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0); + repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l; + rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1; + generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1); + repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x)); + intro; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rsqr (cos x)) 0 H3 H2)). +auto with real. +Qed. + +Lemma COS_bound : forall x:R, -1 <= cos x <= 1. +intro; rewrite <- sin_shift; apply SIN_bound. +Qed. + +Lemma cos_sin_0 : forall x:R, ~ (cos x = 0 /\ sin x = 0). +intro; red in |- *; intro; elim H; intros; generalize (sin2_cos2 x); intro; + rewrite H0 in H2; rewrite H1 in H2; repeat rewrite Rsqr_0 in H2; + rewrite Rplus_0_r in H2; generalize Rlt_0_1; intro; + rewrite <- H2 in H3; elim (Rlt_irrefl 0 H3). +Qed. + +Lemma cos_sin_0_var : forall x:R, cos x <> 0 \/ sin x <> 0. +intro; apply not_and_or; apply cos_sin_0. +Qed. + +(*****************************************************************) +(* Using series definitions of cos and sin *) +(*****************************************************************) + +Definition sin_lb (a:R) : R := sin_approx a 3. +Definition sin_ub (a:R) : R := sin_approx a 4. +Definition cos_lb (a:R) : R := cos_approx a 3. +Definition cos_ub (a:R) : R := cos_approx a 4. + +Lemma sin_lb_gt_0 : forall a:R, 0 < a -> a <= PI / 2 -> 0 < sin_lb a. +intros. +unfold sin_lb in |- *; unfold sin_approx in |- *; unfold sin_term in |- *. +set (Un := fun i:nat => a ^ (2 * i + 1) / INR (fact (2 * i + 1))). +replace + (sum_f_R0 + (fun i:nat => (-1) ^ i * (a ^ (2 * i + 1) / INR (fact (2 * i + 1)))) 3) + with (sum_f_R0 (fun i:nat => (-1) ^ i * Un i) 3); + [ idtac | apply sum_eq; intros; unfold Un in |- *; reflexivity ]. +cut (forall n:nat, Un (S n) < Un n). +intro; simpl in |- *. +repeat rewrite Rmult_1_l; repeat rewrite Rmult_1_r; + replace (-1 * Un 1%nat) with (- Un 1%nat); [ idtac | ring ]; + replace (-1 * -1 * Un 2%nat) with (Un 2%nat); [ idtac | ring ]; + replace (-1 * (-1 * -1) * Un 3%nat) with (- Un 3%nat); + [ idtac | ring ]; + replace (Un 0%nat + - Un 1%nat + Un 2%nat + - Un 3%nat) with + (Un 0%nat - Un 1%nat + (Un 2%nat - Un 3%nat)); [ idtac | ring ]. +apply Rplus_lt_0_compat. +unfold Rminus in |- *; apply Rplus_lt_reg_r with (Un 1%nat); + rewrite Rplus_0_r; rewrite (Rplus_comm (Un 1%nat)); + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; + apply H1. +unfold Rminus in |- *; apply Rplus_lt_reg_r with (Un 3%nat); + rewrite Rplus_0_r; rewrite (Rplus_comm (Un 3%nat)); + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; + apply H1. +intro; unfold Un in |- *. +cut ((2 * S n + 1)%nat = (2 * n + 1 + 2)%nat). +intro; rewrite H1. +rewrite pow_add; unfold Rdiv in |- *; rewrite Rmult_assoc; + apply Rmult_lt_compat_l. +apply pow_lt; assumption. +rewrite <- H1; apply Rmult_lt_reg_l with (INR (fact (2 * n + 1))). +apply lt_INR_0; apply neq_O_lt. +assert (H2 := fact_neq_0 (2 * n + 1)). +red in |- *; intro; elim H2; symmetry in |- *; assumption. +rewrite <- Rinv_r_sym. +apply Rmult_lt_reg_l with (INR (fact (2 * S n + 1))). +apply lt_INR_0; apply neq_O_lt. +assert (H2 := fact_neq_0 (2 * S n + 1)). +red in |- *; intro; elim H2; symmetry in |- *; assumption. +rewrite (Rmult_comm (INR (fact (2 * S n + 1)))); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +do 2 rewrite Rmult_1_r; apply Rle_lt_trans with (INR (fact (2 * n + 1)) * 4). +apply Rmult_le_compat_l. +replace 0 with (INR 0); [ idtac | reflexivity ]; apply le_INR; apply le_O_n. +simpl in |- *; rewrite Rmult_1_r; replace 4 with (Rsqr 2); + [ idtac | ring_Rsqr ]; replace (a * a) with (Rsqr a); + [ idtac | reflexivity ]; apply Rsqr_incr_1. +apply Rle_trans with (PI / 2); + [ assumption + | unfold Rdiv in |- *; apply Rmult_le_reg_l with 2; + [ prove_sup0 + | rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; + [ replace 4 with 4; [ apply PI_4 | ring ] | discrR ] ] ]. +left; assumption. +left; prove_sup0. +rewrite H1; replace (2 * n + 1 + 2)%nat with (S (S (2 * n + 1))). +do 2 rewrite fact_simpl; do 2 rewrite mult_INR. +repeat rewrite <- Rmult_assoc. +rewrite <- (Rmult_comm (INR (fact (2 * n + 1)))). +rewrite Rmult_assoc. +apply Rmult_lt_compat_l. +apply lt_INR_0; apply neq_O_lt. +assert (H2 := fact_neq_0 (2 * n + 1)). +red in |- *; intro; elim H2; symmetry in |- *; assumption. +do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR; set (x := INR n); + unfold INR in |- *. +replace ((2 * x + 1 + 1 + 1) * (2 * x + 1 + 1)) with (4 * x * x + 10 * x + 6); + [ idtac | ring ]. +apply Rplus_lt_reg_r with (-4); rewrite Rplus_opp_l; + replace (-4 + (4 * x * x + 10 * x + 6)) with (4 * x * x + 10 * x + 2); + [ idtac | ring ]. +apply Rplus_le_lt_0_compat. +cut (0 <= x). +intro; apply Rplus_le_le_0_compat; repeat apply Rmult_le_pos; + assumption || left; prove_sup. +unfold x in |- *; replace 0 with (INR 0); + [ apply le_INR; apply le_O_n | reflexivity ]. +prove_sup0. +apply INR_eq; do 2 rewrite S_INR; do 3 rewrite plus_INR; rewrite mult_INR; + repeat rewrite S_INR; ring. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_eq; do 3 rewrite plus_INR; do 2 rewrite mult_INR; + repeat rewrite S_INR; ring. +Qed. + +Lemma SIN : forall a:R, 0 <= a -> a <= PI -> sin_lb a <= sin a <= sin_ub a. +intros; unfold sin_lb, sin_ub in |- *; apply (sin_bound a 1 H H0). +Qed. + +Lemma COS : + forall a:R, - PI / 2 <= a -> a <= PI / 2 -> cos_lb a <= cos a <= cos_ub a. +intros; unfold cos_lb, cos_ub in |- *; apply (cos_bound a 1 H H0). +Qed. + +(**********) +Lemma _PI2_RLT_0 : - (PI / 2) < 0. +rewrite <- Ropp_0; apply Ropp_lt_contravar; apply PI2_RGT_0. +Qed. + +Lemma PI4_RLT_PI2 : PI / 4 < PI / 2. +unfold Rdiv in |- *; apply Rmult_lt_compat_l. +apply PI_RGT_0. +apply Rinv_lt_contravar. +apply Rmult_lt_0_compat; prove_sup0. +pattern 2 at 1 in |- *; rewrite <- Rplus_0_r. +replace 4 with (2 + 2); [ apply Rplus_lt_compat_l; prove_sup0 | ring ]. +Qed. + +Lemma PI2_Rlt_PI : PI / 2 < PI. +unfold Rdiv in |- *; pattern PI at 2 in |- *; rewrite <- Rmult_1_r. +apply Rmult_lt_compat_l. +apply PI_RGT_0. +pattern 1 at 3 in |- *; rewrite <- Rinv_1; apply Rinv_lt_contravar. +rewrite Rmult_1_l; prove_sup0. +pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + apply Rlt_0_1. +Qed. + +(********************************************) +(* Increasing and decreasing of COS and SIN *) +(********************************************) +Theorem sin_gt_0 : forall x:R, 0 < x -> x < PI -> 0 < sin x. +intros; elim (SIN x (Rlt_le 0 x H) (Rlt_le x PI H0)); intros H1 _; + case (Rtotal_order x (PI / 2)); intro H2. +apply Rlt_le_trans with (sin_lb x). +apply sin_lb_gt_0; [ assumption | left; assumption ]. +assumption. +elim H2; intro H3. +rewrite H3; rewrite sin_PI2; apply Rlt_0_1. +rewrite <- sin_PI_x; generalize (Ropp_gt_lt_contravar x (PI / 2) H3); + intro H4; generalize (Rplus_lt_compat_l PI (- x) (- (PI / 2)) H4). +replace (PI + - x) with (PI - x). +replace (PI + - (PI / 2)) with (PI / 2). +intro H5; generalize (Ropp_lt_gt_contravar x PI H0); intro H6; + change (- PI < - x) in H6; generalize (Rplus_lt_compat_l PI (- PI) (- x) H6). +rewrite Rplus_opp_r. +replace (PI + - x) with (PI - x). +intro H7; + elim + (SIN (PI - x) (Rlt_le 0 (PI - x) H7) + (Rlt_le (PI - x) PI (Rlt_trans (PI - x) (PI / 2) PI H5 PI2_Rlt_PI))); + intros H8 _; + generalize (sin_lb_gt_0 (PI - x) H7 (Rlt_le (PI - x) (PI / 2) H5)); + intro H9; apply (Rlt_le_trans 0 (sin_lb (PI - x)) (sin (PI - x)) H9 H8). +reflexivity. +pattern PI at 2 in |- *; rewrite double_var; ring. +reflexivity. +Qed. + +Theorem cos_gt_0 : forall x:R, - (PI / 2) < x -> x < PI / 2 -> 0 < cos x. +intros; rewrite cos_sin; + generalize (Rplus_lt_compat_l (PI / 2) (- (PI / 2)) x H). +rewrite Rplus_opp_r; intro H1; + generalize (Rplus_lt_compat_l (PI / 2) x (PI / 2) H0); + rewrite <- double_var; intro H2; apply (sin_gt_0 (PI / 2 + x) H1 H2). +Qed. + +Lemma sin_ge_0 : forall x:R, 0 <= x -> x <= PI -> 0 <= sin x. +intros x H1 H2; elim H1; intro H3; + [ elim H2; intro H4; + [ left; apply (sin_gt_0 x H3 H4) + | rewrite H4; right; symmetry in |- *; apply sin_PI ] + | rewrite <- H3; right; symmetry in |- *; apply sin_0 ]. +Qed. + +Lemma cos_ge_0 : forall x:R, - (PI / 2) <= x -> x <= PI / 2 -> 0 <= cos x. +intros x H1 H2; elim H1; intro H3; + [ elim H2; intro H4; + [ left; apply (cos_gt_0 x H3 H4) + | rewrite H4; right; symmetry in |- *; apply cos_PI2 ] + | rewrite <- H3; rewrite cos_neg; right; symmetry in |- *; apply cos_PI2 ]. +Qed. + +Lemma sin_le_0 : forall x:R, PI <= x -> x <= 2 * PI -> sin x <= 0. +intros x H1 H2; apply Rge_le; rewrite <- Ropp_0; + rewrite <- (Ropp_involutive (sin x)); apply Ropp_le_ge_contravar; + rewrite <- neg_sin; replace (x + PI) with (x - PI + 2 * INR 1 * PI); + [ rewrite (sin_period (x - PI) 1); apply sin_ge_0; + [ replace (x - PI) with (x + - PI); + [ rewrite Rplus_comm; replace 0 with (- PI + PI); + [ apply Rplus_le_compat_l; assumption | ring ] + | ring ] + | replace (x - PI) with (x + - PI); rewrite Rplus_comm; + [ pattern PI at 2 in |- *; replace PI with (- PI + 2 * PI); + [ apply Rplus_le_compat_l; assumption | ring ] + | ring ] ] + | unfold INR in |- *; ring ]. +Qed. + +Lemma cos_le_0 : forall x:R, PI / 2 <= x -> x <= 3 * (PI / 2) -> cos x <= 0. +intros x H1 H2; apply Rge_le; rewrite <- Ropp_0; + rewrite <- (Ropp_involutive (cos x)); apply Ropp_le_ge_contravar; + rewrite <- neg_cos; replace (x + PI) with (x - PI + 2 * INR 1 * PI). +rewrite cos_period; apply cos_ge_0. +replace (- (PI / 2)) with (- PI + PI / 2). +unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_le_compat_l; + assumption. +pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr; + ring. +unfold Rminus in |- *; rewrite Rplus_comm; + replace (PI / 2) with (- PI + 3 * (PI / 2)). +apply Rplus_le_compat_l; assumption. +pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr; + ring. +unfold INR in |- *; ring. +Qed. + +Lemma sin_lt_0 : forall x:R, PI < x -> x < 2 * PI -> sin x < 0. +intros x H1 H2; rewrite <- Ropp_0; rewrite <- (Ropp_involutive (sin x)); + apply Ropp_lt_gt_contravar; rewrite <- neg_sin; + replace (x + PI) with (x - PI + 2 * INR 1 * PI); + [ rewrite (sin_period (x - PI) 1); apply sin_gt_0; + [ replace (x - PI) with (x + - PI); + [ rewrite Rplus_comm; replace 0 with (- PI + PI); + [ apply Rplus_lt_compat_l; assumption | ring ] + | ring ] + | replace (x - PI) with (x + - PI); rewrite Rplus_comm; + [ pattern PI at 2 in |- *; replace PI with (- PI + 2 * PI); + [ apply Rplus_lt_compat_l; assumption | ring ] + | ring ] ] + | unfold INR in |- *; ring ]. +Qed. + +Lemma sin_lt_0_var : forall x:R, - PI < x -> x < 0 -> sin x < 0. +intros; generalize (Rplus_lt_compat_l (2 * PI) (- PI) x H); + replace (2 * PI + - PI) with PI; + [ intro H1; rewrite Rplus_comm in H1; + generalize (Rplus_lt_compat_l (2 * PI) x 0 H0); + intro H2; rewrite (Rplus_comm (2 * PI)) in H2; + rewrite <- (Rplus_comm 0) in H2; rewrite Rplus_0_l in H2; + rewrite <- (sin_period x 1); unfold INR in |- *; + replace (2 * 1 * PI) with (2 * PI); + [ apply (sin_lt_0 (x + 2 * PI) H1 H2) | ring ] + | ring ]. +Qed. + +Lemma cos_lt_0 : forall x:R, PI / 2 < x -> x < 3 * (PI / 2) -> cos x < 0. +intros x H1 H2; rewrite <- Ropp_0; rewrite <- (Ropp_involutive (cos x)); + apply Ropp_lt_gt_contravar; rewrite <- neg_cos; + replace (x + PI) with (x - PI + 2 * INR 1 * PI). +rewrite cos_period; apply cos_gt_0. +replace (- (PI / 2)) with (- PI + PI / 2). +unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_lt_compat_l; + assumption. +pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr; + ring. +unfold Rminus in |- *; rewrite Rplus_comm; + replace (PI / 2) with (- PI + 3 * (PI / 2)). +apply Rplus_lt_compat_l; assumption. +pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr; + ring. +unfold INR in |- *; ring. +Qed. + +Lemma tan_gt_0 : forall x:R, 0 < x -> x < PI / 2 -> 0 < tan x. +intros x H1 H2; unfold tan in |- *; generalize _PI2_RLT_0; + generalize (Rlt_trans 0 x (PI / 2) H1 H2); intros; + generalize (Rlt_trans (- (PI / 2)) 0 x H0 H1); intro H5; + generalize (Rlt_trans x (PI / 2) PI H2 PI2_Rlt_PI); + intro H7; unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply sin_gt_0; assumption. +apply Rinv_0_lt_compat; apply cos_gt_0; assumption. +Qed. + +Lemma tan_lt_0 : forall x:R, - (PI / 2) < x -> x < 0 -> tan x < 0. +intros x H1 H2; unfold tan in |- *; + generalize (cos_gt_0 x H1 (Rlt_trans x 0 (PI / 2) H2 PI2_RGT_0)); + intro H3; rewrite <- Ropp_0; + replace (sin x / cos x) with (- (- sin x / cos x)). +rewrite <- sin_neg; apply Ropp_gt_lt_contravar; + change (0 < sin (- x) / cos x) in |- *; unfold Rdiv in |- *; + apply Rmult_lt_0_compat. +apply sin_gt_0. +rewrite <- Ropp_0; apply Ropp_gt_lt_contravar; assumption. +apply Rlt_trans with (PI / 2). +rewrite <- (Ropp_involutive (PI / 2)); apply Ropp_gt_lt_contravar; assumption. +apply PI2_Rlt_PI. +apply Rinv_0_lt_compat; assumption. +unfold Rdiv in |- *; ring. +Qed. + +Lemma cos_ge_0_3PI2 : + forall x:R, 3 * (PI / 2) <= x -> x <= 2 * PI -> 0 <= cos x. +intros; rewrite <- cos_neg; rewrite <- (cos_period (- x) 1); + unfold INR in |- *; replace (- x + 2 * 1 * PI) with (2 * PI - x). +generalize (Ropp_le_ge_contravar x (2 * PI) H0); intro H1; + generalize (Rge_le (- x) (- (2 * PI)) H1); clear H1; + intro H1; generalize (Rplus_le_compat_l (2 * PI) (- (2 * PI)) (- x) H1). +rewrite Rplus_opp_r. +intro H2; generalize (Ropp_le_ge_contravar (3 * (PI / 2)) x H); intro H3; + generalize (Rge_le (- (3 * (PI / 2))) (- x) H3); clear H3; + intro H3; + generalize (Rplus_le_compat_l (2 * PI) (- x) (- (3 * (PI / 2))) H3). +replace (2 * PI + - (3 * (PI / 2))) with (PI / 2). +intro H4; + apply + (cos_ge_0 (2 * PI - x) + (Rlt_le (- (PI / 2)) (2 * PI - x) + (Rlt_le_trans (- (PI / 2)) 0 (2 * PI - x) _PI2_RLT_0 H2)) H4). +rewrite double; pattern PI at 2 3 in |- *; rewrite double_var; ring. +ring. +Qed. + +Lemma form1 : + forall p q:R, cos p + cos q = 2 * cos ((p - q) / 2) * cos ((p + q) / 2). +intros p q; pattern p at 1 in |- *; + replace p with ((p - q) / 2 + (p + q) / 2). +rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2). +rewrite cos_plus; rewrite cos_minus; ring. +pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +Qed. + +Lemma form2 : + forall p q:R, cos p - cos q = -2 * sin ((p - q) / 2) * sin ((p + q) / 2). +intros p q; pattern p at 1 in |- *; + replace p with ((p - q) / 2 + (p + q) / 2). +rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2). +rewrite cos_plus; rewrite cos_minus; ring. +pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +Qed. + +Lemma form3 : + forall p q:R, sin p + sin q = 2 * cos ((p - q) / 2) * sin ((p + q) / 2). +intros p q; pattern p at 1 in |- *; + replace p with ((p - q) / 2 + (p + q) / 2). +pattern q at 3 in |- *; replace q with ((p + q) / 2 - (p - q) / 2). +rewrite sin_plus; rewrite sin_minus; ring. +pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +Qed. + +Lemma form4 : + forall p q:R, sin p - sin q = 2 * cos ((p + q) / 2) * sin ((p - q) / 2). +intros p q; pattern p at 1 in |- *; + replace p with ((p - q) / 2 + (p + q) / 2). +pattern q at 3 in |- *; replace q with ((p + q) / 2 - (p - q) / 2). +rewrite sin_plus; rewrite sin_minus; ring. +pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. + +Qed. + +Lemma sin_increasing_0 : + forall x y:R, + - (PI / 2) <= x -> + x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> sin x < sin y -> x < y. +intros; cut (sin ((x - y) / 2) < 0). +intro H4; case (Rtotal_order ((x - y) / 2) 0); intro H5. +assert (Hyp : 0 < 2). +prove_sup0. +generalize (Rmult_lt_compat_l 2 ((x - y) / 2) 0 Hyp H5). +unfold Rdiv in |- *. +rewrite <- Rmult_assoc. +rewrite Rinv_r_simpl_m. +rewrite Rmult_0_r. +clear H5; intro H5; apply Rminus_lt; assumption. +discrR. +elim H5; intro H6. +rewrite H6 in H4; rewrite sin_0 in H4; elim (Rlt_irrefl 0 H4). +change (0 < (x - y) / 2) in H6; + generalize (Ropp_le_ge_contravar (- (PI / 2)) y H1). +rewrite Ropp_involutive. +intro H7; generalize (Rge_le (PI / 2) (- y) H7); clear H7; intro H7; + generalize (Rplus_le_compat x (PI / 2) (- y) (PI / 2) H0 H7). +rewrite <- double_var. +intro H8. +assert (Hyp : 0 < 2). +prove_sup0. +generalize + (Rmult_le_compat_l (/ 2) (x - y) PI + (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H8). +repeat rewrite (Rmult_comm (/ 2)). +intro H9; + generalize + (sin_gt_0 ((x - y) / 2) H6 + (Rle_lt_trans ((x - y) / 2) (PI / 2) PI H9 PI2_Rlt_PI)); + intro H10; + elim + (Rlt_irrefl (sin ((x - y) / 2)) + (Rlt_trans (sin ((x - y) / 2)) 0 (sin ((x - y) / 2)) H4 H10)). +generalize (Rlt_minus (sin x) (sin y) H3); clear H3; intro H3; + rewrite form4 in H3; + generalize (Rplus_le_compat x (PI / 2) y (PI / 2) H0 H2). +rewrite <- double_var. +assert (Hyp : 0 < 2). +prove_sup0. +intro H4; + generalize + (Rmult_le_compat_l (/ 2) (x + y) PI + (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H4). +repeat rewrite (Rmult_comm (/ 2)). +clear H4; intro H4; + generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) y H H1); + replace (- (PI / 2) + - (PI / 2)) with (- PI). +intro H5; + generalize + (Rmult_le_compat_l (/ 2) (- PI) (x + y) + (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H5). +replace (/ 2 * (x + y)) with ((x + y) / 2). +replace (/ 2 * - PI) with (- (PI / 2)). +clear H5; intro H5; elim H4; intro H40. +elim H5; intro H50. +generalize (cos_gt_0 ((x + y) / 2) H50 H40); intro H6; + generalize (Rmult_lt_compat_l 2 0 (cos ((x + y) / 2)) Hyp H6). +rewrite Rmult_0_r. +clear H6; intro H6; case (Rcase_abs (sin ((x - y) / 2))); intro H7. +assumption. +generalize (Rge_le (sin ((x - y) / 2)) 0 H7); clear H7; intro H7; + generalize + (Rmult_le_pos (2 * cos ((x + y) / 2)) (sin ((x - y) / 2)) + (Rlt_le 0 (2 * cos ((x + y) / 2)) H6) H7); intro H8; + generalize + (Rle_lt_trans 0 (2 * cos ((x + y) / 2) * sin ((x - y) / 2)) 0 H8 H3); + intro H9; elim (Rlt_irrefl 0 H9). +rewrite <- H50 in H3; rewrite cos_neg in H3; rewrite cos_PI2 in H3; + rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3; + elim (Rlt_irrefl 0 H3). +unfold Rdiv in H3. +rewrite H40 in H3; assert (H50 := cos_PI2); unfold Rdiv in H50; + rewrite H50 in H3; rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3; + elim (Rlt_irrefl 0 H3). +unfold Rdiv in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +apply Rmult_comm. +unfold Rdiv in |- *; apply Rmult_comm. +pattern PI at 1 in |- *; rewrite double_var. +rewrite Ropp_plus_distr. +reflexivity. +Qed. + +Lemma sin_increasing_1 : + forall x y:R, + - (PI / 2) <= x -> + x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> x < y -> sin x < sin y. +intros; generalize (Rplus_lt_compat_l x x y H3); intro H4; + generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) x H H); + replace (- (PI / 2) + - (PI / 2)) with (- PI). +assert (Hyp : 0 < 2). +prove_sup0. +intro H5; generalize (Rle_lt_trans (- PI) (x + x) (x + y) H5 H4); intro H6; + generalize + (Rmult_lt_compat_l (/ 2) (- PI) (x + y) (Rinv_0_lt_compat 2 Hyp) H6); + replace (/ 2 * - PI) with (- (PI / 2)). +replace (/ 2 * (x + y)) with ((x + y) / 2). +clear H4 H5 H6; intro H4; generalize (Rplus_lt_compat_l y x y H3); intro H5; + rewrite Rplus_comm in H5; + generalize (Rplus_le_compat y (PI / 2) y (PI / 2) H2 H2). +rewrite <- double_var. +intro H6; generalize (Rlt_le_trans (x + y) (y + y) PI H5 H6); intro H7; + generalize (Rmult_lt_compat_l (/ 2) (x + y) PI (Rinv_0_lt_compat 2 Hyp) H7); + replace (/ 2 * PI) with (PI / 2). +replace (/ 2 * (x + y)) with ((x + y) / 2). +clear H5 H6 H7; intro H5; generalize (Ropp_le_ge_contravar (- (PI / 2)) y H1); + rewrite Ropp_involutive; clear H1; intro H1; + generalize (Rge_le (PI / 2) (- y) H1); clear H1; intro H1; + generalize (Ropp_le_ge_contravar y (PI / 2) H2); clear H2; + intro H2; generalize (Rge_le (- y) (- (PI / 2)) H2); + clear H2; intro H2; generalize (Rplus_lt_compat_l (- y) x y H3); + replace (- y + x) with (x - y). +rewrite Rplus_opp_l. +intro H6; + generalize (Rmult_lt_compat_l (/ 2) (x - y) 0 (Rinv_0_lt_compat 2 Hyp) H6); + rewrite Rmult_0_r; replace (/ 2 * (x - y)) with ((x - y) / 2). +clear H6; intro H6; + generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) (- y) H H2); + replace (- (PI / 2) + - (PI / 2)) with (- PI). +replace (x + - y) with (x - y). +intro H7; + generalize + (Rmult_le_compat_l (/ 2) (- PI) (x - y) + (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H7); + replace (/ 2 * - PI) with (- (PI / 2)). +replace (/ 2 * (x - y)) with ((x - y) / 2). +clear H7; intro H7; clear H H0 H1 H2; apply Rminus_lt; rewrite form4; + generalize (cos_gt_0 ((x + y) / 2) H4 H5); intro H8; + generalize (Rmult_lt_0_compat 2 (cos ((x + y) / 2)) Hyp H8); + clear H8; intro H8; cut (- PI < - (PI / 2)). +intro H9; + generalize + (sin_lt_0_var ((x - y) / 2) + (Rlt_le_trans (- PI) (- (PI / 2)) ((x - y) / 2) H9 H7) H6); + intro H10; + generalize + (Rmult_lt_gt_compat_neg_l (sin ((x - y) / 2)) 0 ( + 2 * cos ((x + y) / 2)) H10 H8); intro H11; rewrite Rmult_0_r in H11; + rewrite Rmult_comm; assumption. +apply Ropp_lt_gt_contravar; apply PI2_Rlt_PI. +unfold Rdiv in |- *; apply Rmult_comm. +unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse; apply Rmult_comm. +reflexivity. +pattern PI at 1 in |- *; rewrite double_var. +rewrite Ropp_plus_distr. +reflexivity. +unfold Rdiv in |- *; apply Rmult_comm. +unfold Rminus in |- *; apply Rplus_comm. +unfold Rdiv in |- *; apply Rmult_comm. +unfold Rdiv in |- *; apply Rmult_comm. +unfold Rdiv in |- *; apply Rmult_comm. +unfold Rdiv in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +apply Rmult_comm. +pattern PI at 1 in |- *; rewrite double_var. +rewrite Ropp_plus_distr. +reflexivity. +Qed. + +Lemma sin_decreasing_0 : + forall x y:R, + x <= 3 * (PI / 2) -> + PI / 2 <= x -> y <= 3 * (PI / 2) -> PI / 2 <= y -> sin x < sin y -> y < x. +intros; rewrite <- (sin_PI_x x) in H3; rewrite <- (sin_PI_x y) in H3; + generalize (Ropp_lt_gt_contravar (sin (PI - x)) (sin (PI - y)) H3); + repeat rewrite <- sin_neg; + generalize (Rplus_le_compat_l (- PI) x (3 * (PI / 2)) H); + generalize (Rplus_le_compat_l (- PI) (PI / 2) x H0); + generalize (Rplus_le_compat_l (- PI) y (3 * (PI / 2)) H1); + generalize (Rplus_le_compat_l (- PI) (PI / 2) y H2); + replace (- PI + x) with (x - PI). +replace (- PI + PI / 2) with (- (PI / 2)). +replace (- PI + y) with (y - PI). +replace (- PI + 3 * (PI / 2)) with (PI / 2). +replace (- (PI - x)) with (x - PI). +replace (- (PI - y)) with (y - PI). +intros; change (sin (y - PI) < sin (x - PI)) in H8; + apply Rplus_lt_reg_r with (- PI); rewrite Rplus_comm; + replace (y + - PI) with (y - PI). +rewrite Rplus_comm; replace (x + - PI) with (x - PI). +apply (sin_increasing_0 (y - PI) (x - PI) H4 H5 H6 H7 H8). +reflexivity. +reflexivity. +unfold Rminus in |- *; rewrite Ropp_plus_distr. +rewrite Ropp_involutive. +apply Rplus_comm. +unfold Rminus in |- *; rewrite Ropp_plus_distr. +rewrite Ropp_involutive. +apply Rplus_comm. +pattern PI at 2 in |- *; rewrite double_var. +rewrite Ropp_plus_distr. +ring. +unfold Rminus in |- *; apply Rplus_comm. +pattern PI at 2 in |- *; rewrite double_var. +rewrite Ropp_plus_distr. +ring. +unfold Rminus in |- *; apply Rplus_comm. +Qed. + +Lemma sin_decreasing_1 : + forall x y:R, + x <= 3 * (PI / 2) -> + PI / 2 <= x -> y <= 3 * (PI / 2) -> PI / 2 <= y -> x < y -> sin y < sin x. +intros; rewrite <- (sin_PI_x x); rewrite <- (sin_PI_x y); + generalize (Rplus_le_compat_l (- PI) x (3 * (PI / 2)) H); + generalize (Rplus_le_compat_l (- PI) (PI / 2) x H0); + generalize (Rplus_le_compat_l (- PI) y (3 * (PI / 2)) H1); + generalize (Rplus_le_compat_l (- PI) (PI / 2) y H2); + generalize (Rplus_lt_compat_l (- PI) x y H3); + replace (- PI + PI / 2) with (- (PI / 2)). +replace (- PI + y) with (y - PI). +replace (- PI + 3 * (PI / 2)) with (PI / 2). +replace (- PI + x) with (x - PI). +intros; apply Ropp_lt_cancel; repeat rewrite <- sin_neg; + replace (- (PI - x)) with (x - PI). +replace (- (PI - y)) with (y - PI). +apply (sin_increasing_1 (x - PI) (y - PI) H7 H8 H5 H6 H4). +unfold Rminus in |- *; rewrite Ropp_plus_distr. +rewrite Ropp_involutive. +apply Rplus_comm. +unfold Rminus in |- *; rewrite Ropp_plus_distr. +rewrite Ropp_involutive. +apply Rplus_comm. +unfold Rminus in |- *; apply Rplus_comm. +pattern PI at 2 in |- *; rewrite double_var; ring. +unfold Rminus in |- *; apply Rplus_comm. +pattern PI at 2 in |- *; rewrite double_var; ring. +Qed. + +Lemma cos_increasing_0 : + forall x y:R, + PI <= x -> x <= 2 * PI -> PI <= y -> y <= 2 * PI -> cos x < cos y -> x < y. +intros x y H1 H2 H3 H4; rewrite <- (cos_neg x); rewrite <- (cos_neg y); + rewrite <- (cos_period (- x) 1); rewrite <- (cos_period (- y) 1); + unfold INR in |- *; + replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))). +replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))). +repeat rewrite cos_shift; intro H5; + generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI x H1); + generalize (Rplus_le_compat_l (-3 * (PI / 2)) x (2 * PI) H2); + generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI y H3); + generalize (Rplus_le_compat_l (-3 * (PI / 2)) y (2 * PI) H4). +replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)). +replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)). +replace (-3 * (PI / 2) + 2 * PI) with (PI / 2). +replace (-3 * (PI / 2) + PI) with (- (PI / 2)). +clear H1 H2 H3 H4; intros H1 H2 H3 H4; + apply Rplus_lt_reg_r with (-3 * (PI / 2)); + replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)). +replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)). +apply (sin_increasing_0 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H4 H3 H2 H1 H5). +unfold Rminus in |- *. +rewrite Ropp_mult_distr_l_reverse. +apply Rplus_comm. +unfold Rminus in |- *. +rewrite Ropp_mult_distr_l_reverse. +apply Rplus_comm. +pattern PI at 3 in |- *; rewrite double_var. +ring. +rewrite double; pattern PI at 3 4 in |- *; rewrite double_var. +ring. +unfold Rminus in |- *. +rewrite Ropp_mult_distr_l_reverse. +apply Rplus_comm. +unfold Rminus in |- *. +rewrite Ropp_mult_distr_l_reverse. +apply Rplus_comm. +rewrite Rmult_1_r. +rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. +ring. +rewrite Rmult_1_r. +rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. +ring. +Qed. + +Lemma cos_increasing_1 : + forall x y:R, + PI <= x -> x <= 2 * PI -> PI <= y -> y <= 2 * PI -> x < y -> cos x < cos y. +intros x y H1 H2 H3 H4 H5; + generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI x H1); + generalize (Rplus_le_compat_l (-3 * (PI / 2)) x (2 * PI) H2); + generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI y H3); + generalize (Rplus_le_compat_l (-3 * (PI / 2)) y (2 * PI) H4); + generalize (Rplus_lt_compat_l (-3 * (PI / 2)) x y H5); + rewrite <- (cos_neg x); rewrite <- (cos_neg y); + rewrite <- (cos_period (- x) 1); rewrite <- (cos_period (- y) 1); + unfold INR in |- *; replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)). +replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)). +replace (-3 * (PI / 2) + PI) with (- (PI / 2)). +replace (-3 * (PI / 2) + 2 * PI) with (PI / 2). +clear H1 H2 H3 H4 H5; intros H1 H2 H3 H4 H5; + replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))). +replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))). +repeat rewrite cos_shift; + apply + (sin_increasing_1 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H5 H4 H3 H2 H1). +rewrite Rmult_1_r. +rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. +ring. +rewrite Rmult_1_r. +rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. +ring. +rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. +ring. +pattern PI at 3 in |- *; rewrite double_var; ring. +unfold Rminus in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +apply Rplus_comm. +unfold Rminus in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +apply Rplus_comm. +Qed. + +Lemma cos_decreasing_0 : + forall x y:R, + 0 <= x -> x <= PI -> 0 <= y -> y <= PI -> cos x < cos y -> y < x. +intros; generalize (Ropp_lt_gt_contravar (cos x) (cos y) H3); + repeat rewrite <- neg_cos; intro H4; + change (cos (y + PI) < cos (x + PI)) in H4; rewrite (Rplus_comm x) in H4; + rewrite (Rplus_comm y) in H4; generalize (Rplus_le_compat_l PI 0 x H); + generalize (Rplus_le_compat_l PI x PI H0); + generalize (Rplus_le_compat_l PI 0 y H1); + generalize (Rplus_le_compat_l PI y PI H2); rewrite Rplus_0_r. +rewrite <- double. +clear H H0 H1 H2 H3; intros; apply Rplus_lt_reg_r with PI; + apply (cos_increasing_0 (PI + y) (PI + x) H0 H H2 H1 H4). +Qed. + +Lemma cos_decreasing_1 : + forall x y:R, + 0 <= x -> x <= PI -> 0 <= y -> y <= PI -> x < y -> cos y < cos x. +intros; apply Ropp_lt_cancel; repeat rewrite <- neg_cos; + rewrite (Rplus_comm x); rewrite (Rplus_comm y); + generalize (Rplus_le_compat_l PI 0 x H); + generalize (Rplus_le_compat_l PI x PI H0); + generalize (Rplus_le_compat_l PI 0 y H1); + generalize (Rplus_le_compat_l PI y PI H2); rewrite Rplus_0_r. +rewrite <- double. +generalize (Rplus_lt_compat_l PI x y H3); clear H H0 H1 H2 H3; intros; + apply (cos_increasing_1 (PI + x) (PI + y) H3 H2 H1 H0 H). +Qed. + +Lemma tan_diff : + forall x y:R, + cos x <> 0 -> cos y <> 0 -> tan x - tan y = sin (x - y) / (cos x * cos y). +intros; unfold tan in |- *; rewrite sin_minus. +unfold Rdiv in |- *. +unfold Rminus in |- *. +rewrite Rmult_plus_distr_r. +rewrite Rinv_mult_distr. +repeat rewrite (Rmult_comm (sin x)). +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (cos y)). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite (Rmult_comm (sin x)). +apply Rplus_eq_compat_l. +rewrite <- Ropp_mult_distr_l_reverse. +rewrite <- Ropp_mult_distr_r_reverse. +rewrite (Rmult_comm (/ cos x)). +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (cos x)). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +reflexivity. +assumption. +assumption. +assumption. +assumption. +Qed. + +Lemma tan_increasing_0 : + forall x y:R, + - (PI / 4) <= x -> + x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> tan x < tan y -> x < y. +intros; generalize PI4_RLT_PI2; intro H4; + generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4); + intro H5; change (- (PI / 2) < - (PI / 4)) in H5; + generalize + (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H) + (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)); intro HP1; + generalize + (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1) + (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)); intro HP2; + generalize + (sym_not_eq + (Rlt_not_eq 0 (cos x) + (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H) + (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)))); + intro H6; + generalize + (sym_not_eq + (Rlt_not_eq 0 (cos y) + (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1) + (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)))); + intro H7; generalize (tan_diff x y H6 H7); intro H8; + generalize (Rlt_minus (tan x) (tan y) H3); clear H3; + intro H3; rewrite H8 in H3; cut (sin (x - y) < 0). +intro H9; generalize (Ropp_le_ge_contravar (- (PI / 4)) y H1); + rewrite Ropp_involutive; intro H10; generalize (Rge_le (PI / 4) (- y) H10); + clear H10; intro H10; generalize (Ropp_le_ge_contravar y (PI / 4) H2); + intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11); + clear H11; intro H11; + generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11); + generalize (Rplus_le_compat x (PI / 4) (- y) (PI / 4) H0 H10); + replace (x + - y) with (x - y). +replace (PI / 4 + PI / 4) with (PI / 2). +replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)). +intros; case (Rtotal_order 0 (x - y)); intro H14. +generalize + (sin_gt_0 (x - y) H14 (Rle_lt_trans (x - y) (PI / 2) PI H12 PI2_Rlt_PI)); + intro H15; elim (Rlt_irrefl 0 (Rlt_trans 0 (sin (x - y)) 0 H15 H9)). +elim H14; intro H15. +rewrite <- H15 in H9; rewrite sin_0 in H9; elim (Rlt_irrefl 0 H9). +apply Rminus_lt; assumption. +pattern PI at 1 in |- *; rewrite double_var. +unfold Rdiv in |- *. +rewrite Rmult_plus_distr_r. +repeat rewrite Rmult_assoc. +rewrite <- Rinv_mult_distr. +rewrite Ropp_plus_distr. +replace 4 with 4. +reflexivity. +ring. +discrR. +discrR. +pattern PI at 1 in |- *; rewrite double_var. +unfold Rdiv in |- *. +rewrite Rmult_plus_distr_r. +repeat rewrite Rmult_assoc. +rewrite <- Rinv_mult_distr. +replace 4 with 4. +reflexivity. +ring. +discrR. +discrR. +reflexivity. +case (Rcase_abs (sin (x - y))); intro H9. +assumption. +generalize (Rge_le (sin (x - y)) 0 H9); clear H9; intro H9; + generalize (Rinv_0_lt_compat (cos x) HP1); intro H10; + generalize (Rinv_0_lt_compat (cos y) HP2); intro H11; + generalize (Rmult_lt_0_compat (/ cos x) (/ cos y) H10 H11); + replace (/ cos x * / cos y) with (/ (cos x * cos y)). +intro H12; + generalize + (Rmult_le_pos (sin (x - y)) (/ (cos x * cos y)) H9 + (Rlt_le 0 (/ (cos x * cos y)) H12)); intro H13; + elim + (Rlt_irrefl 0 (Rle_lt_trans 0 (sin (x - y) * / (cos x * cos y)) 0 H13 H3)). +rewrite Rinv_mult_distr. +reflexivity. +assumption. +assumption. +Qed. + +Lemma tan_increasing_1 : + forall x y:R, + - (PI / 4) <= x -> + x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> x < y -> tan x < tan y. +intros; apply Rminus_lt; generalize PI4_RLT_PI2; intro H4; + generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4); + intro H5; change (- (PI / 2) < - (PI / 4)) in H5; + generalize + (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H) + (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)); intro HP1; + generalize + (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1) + (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)); intro HP2; + generalize + (sym_not_eq + (Rlt_not_eq 0 (cos x) + (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H) + (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)))); + intro H6; + generalize + (sym_not_eq + (Rlt_not_eq 0 (cos y) + (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1) + (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)))); + intro H7; rewrite (tan_diff x y H6 H7); + generalize (Rinv_0_lt_compat (cos x) HP1); intro H10; + generalize (Rinv_0_lt_compat (cos y) HP2); intro H11; + generalize (Rmult_lt_0_compat (/ cos x) (/ cos y) H10 H11); + replace (/ cos x * / cos y) with (/ (cos x * cos y)). +clear H10 H11; intro H8; generalize (Ropp_le_ge_contravar y (PI / 4) H2); + intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11); + clear H11; intro H11; + generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11); + replace (x + - y) with (x - y). +replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)). +clear H11; intro H9; generalize (Rlt_minus x y H3); clear H3; intro H3; + clear H H0 H1 H2 H4 H5 HP1 HP2; generalize PI2_Rlt_PI; + intro H1; generalize (Ropp_lt_gt_contravar (PI / 2) PI H1); + clear H1; intro H1; + generalize + (sin_lt_0_var (x - y) (Rlt_le_trans (- PI) (- (PI / 2)) (x - y) H1 H9) H3); + intro H2; + generalize + (Rmult_lt_gt_compat_neg_l (sin (x - y)) 0 (/ (cos x * cos y)) H2 H8); + rewrite Rmult_0_r; intro H4; assumption. +pattern PI at 1 in |- *; rewrite double_var. +unfold Rdiv in |- *. +rewrite Rmult_plus_distr_r. +repeat rewrite Rmult_assoc. +rewrite <- Rinv_mult_distr. +replace 4 with 4. +rewrite Ropp_plus_distr. +reflexivity. +ring. +discrR. +discrR. +reflexivity. +apply Rinv_mult_distr; assumption. +Qed. + +Lemma sin_incr_0 : + forall x y:R, + - (PI / 2) <= x -> + x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> sin x <= sin y -> x <= y. +intros; case (Rtotal_order (sin x) (sin y)); intro H4; + [ left; apply (sin_increasing_0 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order x y); intro H6; + [ left; assumption + | elim H6; intro H7; + [ right; assumption + | generalize (sin_increasing_1 y x H1 H2 H H0 H7); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl (sin y) H8) ] ] + | elim (Rlt_irrefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5)) ] ]. +Qed. + +Lemma sin_incr_1 : + forall x y:R, + - (PI / 2) <= x -> + x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> x <= y -> sin x <= sin y. +intros; case (Rtotal_order x y); intro H4; + [ left; apply (sin_increasing_1 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order (sin x) (sin y)); intro H6; + [ left; assumption + | elim H6; intro H7; + [ right; assumption + | generalize (sin_increasing_0 y x H1 H2 H H0 H7); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ] + | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ]. +Qed. + +Lemma sin_decr_0 : + forall x y:R, + x <= 3 * (PI / 2) -> + PI / 2 <= x -> + y <= 3 * (PI / 2) -> PI / 2 <= y -> sin x <= sin y -> y <= x. +intros; case (Rtotal_order (sin x) (sin y)); intro H4; + [ left; apply (sin_decreasing_0 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order x y); intro H6; + [ generalize (sin_decreasing_1 x y H H0 H1 H2 H6); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl (sin y) H8) + | elim H6; intro H7; + [ right; symmetry in |- *; assumption | left; assumption ] ] + | elim (Rlt_irrefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5)) ] ]. +Qed. + +Lemma sin_decr_1 : + forall x y:R, + x <= 3 * (PI / 2) -> + PI / 2 <= x -> + y <= 3 * (PI / 2) -> PI / 2 <= y -> x <= y -> sin y <= sin x. +intros; case (Rtotal_order x y); intro H4; + [ left; apply (sin_decreasing_1 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order (sin x) (sin y)); intro H6; + [ generalize (sin_decreasing_0 x y H H0 H1 H2 H6); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl y H8) + | elim H6; intro H7; + [ right; symmetry in |- *; assumption | left; assumption ] ] + | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ]. +Qed. + +Lemma cos_incr_0 : + forall x y:R, + PI <= x -> + x <= 2 * PI -> PI <= y -> y <= 2 * PI -> cos x <= cos y -> x <= y. +intros; case (Rtotal_order (cos x) (cos y)); intro H4; + [ left; apply (cos_increasing_0 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order x y); intro H6; + [ left; assumption + | elim H6; intro H7; + [ right; assumption + | generalize (cos_increasing_1 y x H1 H2 H H0 H7); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl (cos y) H8) ] ] + | elim (Rlt_irrefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5)) ] ]. +Qed. + +Lemma cos_incr_1 : + forall x y:R, + PI <= x -> + x <= 2 * PI -> PI <= y -> y <= 2 * PI -> x <= y -> cos x <= cos y. +intros; case (Rtotal_order x y); intro H4; + [ left; apply (cos_increasing_1 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order (cos x) (cos y)); intro H6; + [ left; assumption + | elim H6; intro H7; + [ right; assumption + | generalize (cos_increasing_0 y x H1 H2 H H0 H7); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ] + | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ]. +Qed. + +Lemma cos_decr_0 : + forall x y:R, + 0 <= x -> x <= PI -> 0 <= y -> y <= PI -> cos x <= cos y -> y <= x. +intros; case (Rtotal_order (cos x) (cos y)); intro H4; + [ left; apply (cos_decreasing_0 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order x y); intro H6; + [ generalize (cos_decreasing_1 x y H H0 H1 H2 H6); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl (cos y) H8) + | elim H6; intro H7; + [ right; symmetry in |- *; assumption | left; assumption ] ] + | elim (Rlt_irrefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5)) ] ]. +Qed. + +Lemma cos_decr_1 : + forall x y:R, + 0 <= x -> x <= PI -> 0 <= y -> y <= PI -> x <= y -> cos y <= cos x. +intros; case (Rtotal_order x y); intro H4; + [ left; apply (cos_decreasing_1 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order (cos x) (cos y)); intro H6; + [ generalize (cos_decreasing_0 x y H H0 H1 H2 H6); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl y H8) + | elim H6; intro H7; + [ right; symmetry in |- *; assumption | left; assumption ] ] + | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ]. +Qed. + +Lemma tan_incr_0 : + forall x y:R, + - (PI / 4) <= x -> + x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> tan x <= tan y -> x <= y. +intros; case (Rtotal_order (tan x) (tan y)); intro H4; + [ left; apply (tan_increasing_0 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order x y); intro H6; + [ left; assumption + | elim H6; intro H7; + [ right; assumption + | generalize (tan_increasing_1 y x H1 H2 H H0 H7); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl (tan y) H8) ] ] + | elim (Rlt_irrefl (tan x) (Rle_lt_trans (tan x) (tan y) (tan x) H3 H5)) ] ]. +Qed. + +Lemma tan_incr_1 : + forall x y:R, + - (PI / 4) <= x -> + x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> x <= y -> tan x <= tan y. +intros; case (Rtotal_order x y); intro H4; + [ left; apply (tan_increasing_1 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order (tan x) (tan y)); intro H6; + [ left; assumption + | elim H6; intro H7; + [ right; assumption + | generalize (tan_increasing_0 y x H1 H2 H H0 H7); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ] + | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ]. +Qed. + +(**********) +Lemma sin_eq_0_1 : forall x:R, (exists k : Z, x = IZR k * PI) -> sin x = 0. +intros. +elim H; intros. +apply (Zcase_sign x0). +intro. +rewrite H1 in H0. +simpl in H0. +rewrite H0; rewrite Rmult_0_l; apply sin_0. +intro. +cut (0 <= x0)%Z. +intro. +elim (IZN x0 H2); intros. +rewrite H3 in H0. +rewrite <- INR_IZR_INZ in H0. +rewrite H0. +elim (even_odd_cor x1); intros. +elim H4; intro. +rewrite H5. +rewrite mult_INR. +simpl in |- *. +rewrite <- (Rplus_0_l (2 * INR x2 * PI)). +rewrite sin_period. +apply sin_0. +rewrite H5. +rewrite S_INR; rewrite mult_INR. +simpl in |- *. +rewrite Rmult_plus_distr_r. +rewrite Rmult_1_l; rewrite sin_plus. +rewrite sin_PI. +rewrite Rmult_0_r. +rewrite <- (Rplus_0_l (2 * INR x2 * PI)). +rewrite sin_period. +rewrite sin_0; ring. +apply le_IZR. +left; apply IZR_lt. +assert (H2 := Zorder.Zgt_iff_lt). +elim (H2 x0 0%Z); intros. +apply H3; assumption. +intro. +rewrite H0. +replace (sin (IZR x0 * PI)) with (- sin (- IZR x0 * PI)). +cut (0 <= - x0)%Z. +intro. +rewrite <- Ropp_Ropp_IZR. +elim (IZN (- x0) H2); intros. +rewrite H3. +rewrite <- INR_IZR_INZ. +elim (even_odd_cor x1); intros. +elim H4; intro. +rewrite H5. +rewrite mult_INR. +simpl in |- *. +rewrite <- (Rplus_0_l (2 * INR x2 * PI)). +rewrite sin_period. +rewrite sin_0; ring. +rewrite H5. +rewrite S_INR; rewrite mult_INR. +simpl in |- *. +rewrite Rmult_plus_distr_r. +rewrite Rmult_1_l; rewrite sin_plus. +rewrite sin_PI. +rewrite Rmult_0_r. +rewrite <- (Rplus_0_l (2 * INR x2 * PI)). +rewrite sin_period. +rewrite sin_0; ring. +apply le_IZR. +apply Rplus_le_reg_l with (IZR x0). +rewrite Rplus_0_r. +rewrite Ropp_Ropp_IZR. +rewrite Rplus_opp_r. +left; replace 0 with (IZR 0); [ apply IZR_lt | reflexivity ]. +assumption. +rewrite <- sin_neg. +rewrite Ropp_mult_distr_l_reverse. +rewrite Ropp_involutive. +reflexivity. +Qed. + +Lemma sin_eq_0_0 : forall x:R, sin x = 0 -> exists k : Z, x = IZR k * PI. +intros. +assert (H0 := euclidian_division x PI PI_neq0). +elim H0; intros q H1. +elim H1; intros r H2. +exists q. +cut (r = 0). +intro. +elim H2; intros H4 _; rewrite H4; rewrite H3. +apply Rplus_0_r. +elim H2; intros. +rewrite H3 in H. +rewrite sin_plus in H. +cut (sin (IZR q * PI) = 0). +intro. +rewrite H5 in H. +rewrite Rmult_0_l in H. +rewrite Rplus_0_l in H. +assert (H6 := Rmult_integral _ _ H). +elim H6; intro. +assert (H8 := sin2_cos2 (IZR q * PI)). +rewrite H5 in H8; rewrite H7 in H8. +rewrite Rsqr_0 in H8. +rewrite Rplus_0_r in H8. +elim R1_neq_R0; symmetry in |- *; assumption. +cut (r = 0 \/ 0 < r < PI). +intro; elim H8; intro. +assumption. +elim H9; intros. +assert (H12 := sin_gt_0 _ H10 H11). +rewrite H7 in H12; elim (Rlt_irrefl _ H12). +rewrite Rabs_right in H4. +elim H4; intros. +case (Rtotal_order 0 r); intro. +right; split; assumption. +elim H10; intro. +left; symmetry in |- *; assumption. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H8 H11)). +apply Rle_ge. +left; apply PI_RGT_0. +apply sin_eq_0_1. +exists q; reflexivity. +Qed. + +Lemma cos_eq_0_0 : + forall x:R, cos x = 0 -> exists k : Z, x = IZR k * PI + PI / 2. +intros x H; rewrite cos_sin in H; generalize (sin_eq_0_0 (PI / INR 2 + x) H); + intro H2; elim H2; intros x0 H3; exists (x0 - Z_of_nat 1)%Z; + rewrite <- Z_R_minus; ring; rewrite Rmult_comm; rewrite <- H3; + unfold INR in |- *. +rewrite (double_var (- PI)); unfold Rdiv in |- *; ring. +Qed. + +Lemma cos_eq_0_1 : + forall x:R, (exists k : Z, x = IZR k * PI + PI / 2) -> cos x = 0. +intros x H1; rewrite cos_sin; elim H1; intros x0 H2; rewrite H2; + replace (PI / 2 + (IZR x0 * PI + PI / 2)) with (IZR x0 * PI + PI). +rewrite neg_sin; rewrite <- Ropp_0. +apply Ropp_eq_compat; apply sin_eq_0_1; exists x0; reflexivity. +pattern PI at 2 in |- *; rewrite (double_var PI); ring. +Qed. + +Lemma sin_eq_O_2PI_0 : + forall x:R, + 0 <= x -> x <= 2 * PI -> sin x = 0 -> x = 0 \/ x = PI \/ x = 2 * PI. +intros; generalize (sin_eq_0_0 x H1); intro. +elim H2; intros k0 H3. +case (Rtotal_order PI x); intro. +rewrite H3 in H4; rewrite H3 in H0. +right; right. +generalize + (Rmult_lt_compat_r (/ PI) PI (IZR k0 * PI) (Rinv_0_lt_compat PI PI_RGT_0) H4); + rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; intro; + generalize + (Rmult_le_compat_r (/ PI) (IZR k0 * PI) (2 * PI) + (Rlt_le 0 (/ PI) (Rinv_0_lt_compat PI PI_RGT_0)) H0); + repeat rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym. +repeat rewrite Rmult_1_r; intro; + generalize (Rplus_lt_compat_l (IZR (-2)) 1 (IZR k0) H5); + rewrite <- plus_IZR. +replace (IZR (-2) + 1) with (-1). +intro; generalize (Rplus_le_compat_l (IZR (-2)) (IZR k0) 2 H6); + rewrite <- plus_IZR. +replace (IZR (-2) + 2) with 0. +intro; cut (-1 < IZR (-2 + k0) < 1). +intro; generalize (one_IZR_lt1 (-2 + k0) H9); intro. +cut (k0 = 2%Z). +intro; rewrite H11 in H3; rewrite H3; simpl in |- *. +reflexivity. +rewrite <- (Zplus_opp_l 2) in H10; generalize (Zplus_reg_l (-2) k0 2 H10); + intro; assumption. +split. +assumption. +apply Rle_lt_trans with 0. +assumption. +apply Rlt_0_1. +simpl in |- *; ring. +simpl in |- *; ring. +apply PI_neq0. +apply PI_neq0. +elim H4; intro. +right; left. +symmetry in |- *; assumption. +left. +rewrite H3 in H5; rewrite H3 in H; + generalize + (Rmult_lt_compat_r (/ PI) (IZR k0 * PI) PI (Rinv_0_lt_compat PI PI_RGT_0) + H5); rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; intro; + generalize + (Rmult_le_compat_r (/ PI) 0 (IZR k0 * PI) + (Rlt_le 0 (/ PI) (Rinv_0_lt_compat PI PI_RGT_0)) H); + repeat rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; rewrite Rmult_0_l; intro. +cut (-1 < IZR k0 < 1). +intro; generalize (one_IZR_lt1 k0 H8); intro; rewrite H9 in H3; rewrite H3; + simpl in |- *; apply Rmult_0_l. +split. +apply Rlt_le_trans with 0. +rewrite <- Ropp_0; apply Ropp_gt_lt_contravar; apply Rlt_0_1. +assumption. +assumption. +apply PI_neq0. +apply PI_neq0. +Qed. + +Lemma sin_eq_O_2PI_1 : + forall x:R, + 0 <= x -> x <= 2 * PI -> x = 0 \/ x = PI \/ x = 2 * PI -> sin x = 0. +intros x H1 H2 H3; elim H3; intro H4; + [ rewrite H4; rewrite sin_0; reflexivity + | elim H4; intro H5; + [ rewrite H5; rewrite sin_PI; reflexivity + | rewrite H5; rewrite sin_2PI; reflexivity ] ]. +Qed. + +Lemma cos_eq_0_2PI_0 : + forall x:R, + 0 <= x -> x <= 2 * PI -> cos x = 0 -> x = PI / 2 \/ x = 3 * (PI / 2). +intros; case (Rtotal_order x (3 * (PI / 2))); intro. +rewrite cos_sin in H1. +cut (0 <= PI / 2 + x). +cut (PI / 2 + x <= 2 * PI). +intros; generalize (sin_eq_O_2PI_0 (PI / 2 + x) H4 H3 H1); intros. +decompose [or] H5. +generalize (Rplus_le_compat_l (PI / 2) 0 x H); rewrite Rplus_0_r; rewrite H6; + intro. +elim (Rlt_irrefl 0 (Rlt_le_trans 0 (PI / 2) 0 PI2_RGT_0 H7)). +left. +generalize (Rplus_eq_compat_l (- (PI / 2)) (PI / 2 + x) PI H7). +replace (- (PI / 2) + (PI / 2 + x)) with x. +replace (- (PI / 2) + PI) with (PI / 2). +intro; assumption. +pattern PI at 3 in |- *; rewrite (double_var PI); ring. +ring. +right. +generalize (Rplus_eq_compat_l (- (PI / 2)) (PI / 2 + x) (2 * PI) H7). +replace (- (PI / 2) + (PI / 2 + x)) with x. +replace (- (PI / 2) + 2 * PI) with (3 * (PI / 2)). +intro; assumption. +rewrite double; pattern PI at 3 4 in |- *; rewrite (double_var PI); ring. +ring. +left; replace (2 * PI) with (PI / 2 + 3 * (PI / 2)). +apply Rplus_lt_compat_l; assumption. +rewrite (double PI); pattern PI at 3 4 in |- *; rewrite (double_var PI); ring. +apply Rplus_le_le_0_compat. +left; unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply PI_RGT_0. +apply Rinv_0_lt_compat; prove_sup0. +assumption. +elim H2; intro. +right; assumption. +generalize (cos_eq_0_0 x H1); intro; elim H4; intros k0 H5. +rewrite H5 in H3; rewrite H5 in H0; + generalize + (Rplus_lt_compat_l (- (PI / 2)) (3 * (PI / 2)) (IZR k0 * PI + PI / 2) H3); + generalize + (Rplus_le_compat_l (- (PI / 2)) (IZR k0 * PI + PI / 2) (2 * PI) H0). +replace (- (PI / 2) + 3 * (PI / 2)) with PI. +replace (- (PI / 2) + (IZR k0 * PI + PI / 2)) with (IZR k0 * PI). +replace (- (PI / 2) + 2 * PI) with (3 * (PI / 2)). +intros; + generalize + (Rmult_lt_compat_l (/ PI) PI (IZR k0 * PI) (Rinv_0_lt_compat PI PI_RGT_0) + H7); + generalize + (Rmult_le_compat_l (/ PI) (IZR k0 * PI) (3 * (PI / 2)) + (Rlt_le 0 (/ PI) (Rinv_0_lt_compat PI PI_RGT_0)) H6). +replace (/ PI * (IZR k0 * PI)) with (IZR k0). +replace (/ PI * (3 * (PI / 2))) with (3 * / 2). +rewrite <- Rinv_l_sym. +intros; generalize (Rplus_lt_compat_l (IZR (-2)) 1 (IZR k0) H9); + rewrite <- plus_IZR. +replace (IZR (-2) + 1) with (-1). +intro; generalize (Rplus_le_compat_l (IZR (-2)) (IZR k0) (3 * / 2) H8); + rewrite <- plus_IZR. +replace (IZR (-2) + 2) with 0. +intro; cut (-1 < IZR (-2 + k0) < 1). +intro; generalize (one_IZR_lt1 (-2 + k0) H12); intro. +cut (k0 = 2%Z). +intro; rewrite H14 in H8. +assert (Hyp : 0 < 2). +prove_sup0. +generalize (Rmult_le_compat_l 2 (IZR 2) (3 * / 2) (Rlt_le 0 2 Hyp) H8); + simpl in |- *. +replace 4 with 4. +replace (2 * (3 * / 2)) with 3. +intro; cut (3 < 4). +intro; elim (Rlt_irrefl 3 (Rlt_le_trans 3 4 3 H16 H15)). +generalize (Rplus_lt_compat_l 3 0 1 Rlt_0_1); rewrite Rplus_0_r. +replace (3 + 1) with 4. +intro; assumption. +ring. +symmetry in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. +discrR. +ring. +rewrite <- (Zplus_opp_l 2) in H13; generalize (Zplus_reg_l (-2) k0 2 H13); + intro; assumption. +split. +assumption. +apply Rle_lt_trans with (IZR (-2) + 3 * / 2). +assumption. +simpl in |- *; replace (-2 + 3 * / 2) with (- (1 * / 2)). +apply Rlt_trans with 0. +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar. +apply Rmult_lt_0_compat; + [ apply Rlt_0_1 | apply Rinv_0_lt_compat; prove_sup0 ]. +apply Rlt_0_1. +rewrite Rmult_1_l; apply Rmult_eq_reg_l with 2. +rewrite Ropp_mult_distr_r_reverse; rewrite <- Rinv_r_sym. +rewrite Rmult_plus_distr_l; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m. +ring. +discrR. +discrR. +discrR. +simpl in |- *; ring. +simpl in |- *; ring. +apply PI_neq0. +unfold Rdiv in |- *; pattern 3 at 1 in |- *; rewrite (Rmult_comm 3); + repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; apply Rmult_comm. +apply PI_neq0. +symmetry in |- *; rewrite (Rmult_comm (/ PI)); rewrite Rmult_assoc; + rewrite <- Rinv_r_sym. +apply Rmult_1_r. +apply PI_neq0. +rewrite double; pattern PI at 3 4 in |- *; rewrite double_var; ring. +ring. +pattern PI at 1 in |- *; rewrite double_var; ring. +Qed. + +Lemma cos_eq_0_2PI_1 : + forall x:R, + 0 <= x -> x <= 2 * PI -> x = PI / 2 \/ x = 3 * (PI / 2) -> cos x = 0. +intros x H1 H2 H3; elim H3; intro H4; + [ rewrite H4; rewrite cos_PI2; reflexivity + | rewrite H4; rewrite cos_3PI2; reflexivity ]. +Qed.
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