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Diffstat (limited to 'theories/Reals/Rtrigo.v')
| -rw-r--r-- | theories/Reals/Rtrigo.v | 1796 |
1 files changed, 9 insertions, 1787 deletions
diff --git a/theories/Reals/Rtrigo.v b/theories/Reals/Rtrigo.v index e45353b5..32c4d7d3 100644 --- a/theories/Reals/Rtrigo.v +++ b/theories/Reals/Rtrigo.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -16,1789 +16,11 @@ Require Export Cos_rel. Require Export Cos_plus. Require Import ZArith_base. Require Import Zcomplements. -Local Open Scope nat_scope. -Local Open Scope R_scope. - -(** sin_PI2 is the only remaining axiom **) -Axiom sin_PI2 : sin (PI / 2) = 1. - -(**********) -Lemma PI_neq0 : PI <> 0. -Proof. - 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. -Proof. - 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. -Proof. - 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). -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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). -Proof. - 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. -Proof. - 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. -Proof. - 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). -Proof. - 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). -Proof. - 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. -Proof. - 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. -Proof. - intro x; rewrite double; apply cos_plus. -Qed. - -Lemma cos_2a_cos : forall x:R, cos (2 * x) = 2 * cos x * cos x - 1. -Proof. - 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. -Proof. - 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). -Proof. - 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. -Proof. - apply sin_antisym. -Qed. - -Lemma cos_neg : forall x:R, cos (- x) = cos x. -Proof. - intro; symmetry in |- *; apply cos_sym. -Qed. - -Lemma tan_0 : tan 0 = 0. -Proof. - 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. -Proof. - 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). -Proof. - 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. -Proof. - 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. -Proof. - rewrite sin_2a; rewrite sin_PI; ring. -Qed. - -Lemma cos_2PI : cos (2 * PI) = 1. -Proof. - rewrite cos_2a; rewrite sin_PI; rewrite cos_PI; ring. -Qed. - -Lemma neg_sin : forall x:R, sin (x + PI) = - sin x. -Proof. - 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. -Proof. - 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. -Proof. - intros x k; induction k as [| k Hreck]. - simpl in |- *; ring_simplify (x + 2 * 0 * PI). - trivial. - - replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI). - rewrite sin_plus in |- *; rewrite sin_2PI in |- *; rewrite cos_2PI in |- *. - ring_simplify; trivial. - rewrite S_INR in |- *; ring. -Qed. - -Lemma cos_period : forall (x:R) (k:nat), cos (x + 2 * INR k * PI) = cos x. -Proof. - intros x k; induction k as [| k Hreck]. - simpl in |- *; ring_simplify (x + 2 * 0 * PI). - trivial. - - replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI). - rewrite cos_plus in |- *; rewrite sin_2PI in |- *; rewrite cos_2PI in |- *. - ring_simplify; trivial. - rewrite S_INR in |- *; ring. -Qed. - -Lemma sin_shift : forall x:R, sin (PI / 2 - x) = cos x. -Proof. - 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. -Proof. - 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). -Proof. - intro x; rewrite sin_plus; rewrite sin_PI2; rewrite cos_PI2; ring. -Qed. - -Lemma PI2_RGT_0 : 0 < PI / 2. -Proof. - 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. -Proof. - 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. -Proof. - intro; rewrite <- sin_shift; apply SIN_bound. -Qed. - -Lemma cos_sin_0 : forall x:R, ~ (cos x = 0 /\ sin x = 0). -Proof. - 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. -Proof. - intros x. - destruct (Req_dec (cos x) 0). 2: now left. - right. intros H'. - apply (cos_sin_0 x). - now split. -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. -Proof. - 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. - ring. - apply INR_fact_neq_0. - apply INR_fact_neq_0. - 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. -Proof. - rewrite <- Ropp_0; apply Ropp_lt_contravar; apply PI2_RGT_0. -Qed. - -Lemma PI4_RLT_PI2 : PI / 4 < PI / 2. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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). -Proof. - 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). -Proof. - 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). -Proof. - 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). -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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). -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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. -Proof. - 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; simpl. -unfold INR in H3. field_simplify [(sym_eq H3)]. field. -(** - ring_simplify. - (* rewrite (Rmult_comm PI);*) (* old ring compat *) - rewrite <- H3; simpl; - field;repeat split; discrR. -*) -Qed. - -Lemma cos_eq_0_1 : - forall x:R, (exists k : Z, x = IZR k * PI + PI / 2) -> cos x = 0. -Proof. - 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. -Proof. - 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. -Proof. - 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). -Proof. - 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. -Proof. - intros x H1 H2 H3; elim H3; intro H4; - [ rewrite H4; rewrite cos_PI2; reflexivity - | rewrite H4; rewrite cos_3PI2; reflexivity ]. -Qed. +Require Import Classical_Prop. +Require Import Fourier. +Require Import Ranalysis1. +Require Import Rsqrt_def. +Require Import PSeries_reg. +Require Export Rtrigo1. +Require Export Ratan. +Require Export Machin.
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