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Diffstat (limited to 'theories/Reals/Rtrigo_calc.v')
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diff --git a/theories/Reals/Rtrigo_calc.v b/theories/Reals/Rtrigo_calc.v new file mode 100644 index 00000000..0ef87322 --- /dev/null +++ b/theories/Reals/Rtrigo_calc.v @@ -0,0 +1,434 @@ +(************************************************************************) +(* 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_calc.v,v 1.15.2.1 2004/07/16 19:31:14 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import R_sqrt. +Open Local Scope R_scope. + +Lemma tan_PI : tan PI = 0. +unfold tan in |- *; rewrite sin_PI; rewrite cos_PI; unfold Rdiv in |- *; + apply Rmult_0_l. +Qed. + +Lemma sin_3PI2 : sin (3 * (PI / 2)) = -1. +replace (3 * (PI / 2)) with (PI + PI / 2). +rewrite sin_plus; rewrite sin_PI; rewrite cos_PI; rewrite sin_PI2; ring. +pattern PI at 1 in |- *; rewrite (double_var PI); ring. +Qed. + +Lemma tan_2PI : tan (2 * PI) = 0. +unfold tan in |- *; rewrite sin_2PI; unfold Rdiv in |- *; apply Rmult_0_l. +Qed. + +Lemma sin_cos_PI4 : sin (PI / 4) = cos (PI / 4). +Proof with trivial. +rewrite cos_sin... +replace (PI / 2 + PI / 4) with (- (PI / 4) + PI)... +rewrite neg_sin; rewrite sin_neg; ring... +cut (PI = PI / 2 + PI / 2); [ intro | apply double_var ]... +pattern PI at 2 3 in |- *; rewrite H; pattern PI at 2 3 in |- *; rewrite H... +assert (H0 : 2 <> 0); + [ discrR | unfold Rdiv in |- *; rewrite Rinv_mult_distr; try ring ]... +Qed. + +Lemma sin_PI3_cos_PI6 : sin (PI / 3) = cos (PI / 6). +Proof with trivial. +replace (PI / 6) with (PI / 2 - PI / 3)... +rewrite cos_shift... +assert (H0 : 6 <> 0); [ discrR | idtac ]... +assert (H1 : 3 <> 0); [ discrR | idtac ]... +assert (H2 : 2 <> 0); [ discrR | idtac ]... +apply Rmult_eq_reg_l with 6... +rewrite Rmult_minus_distr_l; repeat rewrite (Rmult_comm 6)... +unfold Rdiv in |- *; repeat rewrite Rmult_assoc... +rewrite <- Rinv_l_sym... +rewrite (Rmult_comm (/ 3)); repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym... +pattern PI at 2 in |- *; rewrite (Rmult_comm PI); repeat rewrite Rmult_1_r; + repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym... +ring... +Qed. + +Lemma sin_PI6_cos_PI3 : cos (PI / 3) = sin (PI / 6). +Proof with trivial. +replace (PI / 6) with (PI / 2 - PI / 3)... +rewrite sin_shift... +assert (H0 : 6 <> 0); [ discrR | idtac ]... +assert (H1 : 3 <> 0); [ discrR | idtac ]... +assert (H2 : 2 <> 0); [ discrR | idtac ]... +apply Rmult_eq_reg_l with 6... +rewrite Rmult_minus_distr_l; repeat rewrite (Rmult_comm 6)... +unfold Rdiv in |- *; repeat rewrite Rmult_assoc... +rewrite <- Rinv_l_sym... +rewrite (Rmult_comm (/ 3)); repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym... +pattern PI at 2 in |- *; rewrite (Rmult_comm PI); repeat rewrite Rmult_1_r; + repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym... +ring... +Qed. + +Lemma PI6_RGT_0 : 0 < PI / 6. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup0 ]. +Qed. + +Lemma PI6_RLT_PI2 : PI / 6 < PI / 2. +unfold Rdiv in |- *; apply Rmult_lt_compat_l. +apply PI_RGT_0. +apply Rinv_lt_contravar; prove_sup. +Qed. + +Lemma sin_PI6 : sin (PI / 6) = 1 / 2. +Proof with trivial. +assert (H : 2 <> 0); [ discrR | idtac ]... +apply Rmult_eq_reg_l with (2 * cos (PI / 6))... +replace (2 * cos (PI / 6) * sin (PI / 6)) with + (2 * sin (PI / 6) * cos (PI / 6))... +rewrite <- sin_2a; replace (2 * (PI / 6)) with (PI / 3)... +rewrite sin_PI3_cos_PI6... +unfold Rdiv in |- *; rewrite Rmult_1_l; rewrite Rmult_assoc; + pattern 2 at 2 in |- *; rewrite (Rmult_comm 2); rewrite Rmult_assoc; + rewrite <- Rinv_l_sym... +rewrite Rmult_1_r... +unfold Rdiv in |- *; rewrite Rinv_mult_distr... +rewrite (Rmult_comm (/ 2)); rewrite (Rmult_comm 2); + repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... +rewrite Rmult_1_r... +discrR... +ring... +apply prod_neq_R0... +cut (0 < cos (PI / 6)); + [ intro H1; auto with real + | apply cos_gt_0; + [ apply (Rlt_trans (- (PI / 2)) 0 (PI / 6) _PI2_RLT_0 PI6_RGT_0) + | apply PI6_RLT_PI2 ] ]... +Qed. + +Lemma sqrt2_neq_0 : sqrt 2 <> 0. +assert (Hyp : 0 < 2); + [ prove_sup0 + | generalize (Rlt_le 0 2 Hyp); intro H1; red in |- *; intro H2; + generalize (sqrt_eq_0 2 H1 H2); intro H; absurd (2 = 0); + [ discrR | assumption ] ]. +Qed. + +Lemma R1_sqrt2_neq_0 : 1 / sqrt 2 <> 0. +generalize (Rinv_neq_0_compat (sqrt 2) sqrt2_neq_0); intro H; + generalize (prod_neq_R0 1 (/ sqrt 2) R1_neq_R0 H); + intro H0; assumption. +Qed. + +Lemma sqrt3_2_neq_0 : 2 * sqrt 3 <> 0. +apply prod_neq_R0; + [ discrR + | assert (Hyp : 0 < 3); + [ prove_sup0 + | generalize (Rlt_le 0 3 Hyp); intro H1; red in |- *; intro H2; + generalize (sqrt_eq_0 3 H1 H2); intro H; absurd (3 = 0); + [ discrR | assumption ] ] ]. +Qed. + +Lemma Rlt_sqrt2_0 : 0 < sqrt 2. +assert (Hyp : 0 < 2); + [ prove_sup0 + | generalize (sqrt_positivity 2 (Rlt_le 0 2 Hyp)); intro H1; elim H1; + intro H2; + [ assumption + | absurd (0 = sqrt 2); + [ apply (sym_not_eq (A:=R)); apply sqrt2_neq_0 | assumption ] ] ]. +Qed. + +Lemma Rlt_sqrt3_0 : 0 < sqrt 3. +cut (0%nat <> 1%nat); + [ intro H0; assert (Hyp : 0 < 2); + [ prove_sup0 + | generalize (Rlt_le 0 2 Hyp); intro H1; assert (Hyp2 : 0 < 3); + [ prove_sup0 + | generalize (Rlt_le 0 3 Hyp2); intro H2; + generalize (lt_INR_0 1 (neq_O_lt 1 H0)); + unfold INR in |- *; intro H3; + generalize (Rplus_lt_compat_l 2 0 1 H3); + rewrite Rplus_comm; rewrite Rplus_0_l; replace (2 + 1) with 3; + [ intro H4; generalize (sqrt_lt_1 2 3 H1 H2 H4); clear H3; intro H3; + apply (Rlt_trans 0 (sqrt 2) (sqrt 3) Rlt_sqrt2_0 H3) + | ring ] ] ] + | discriminate ]. +Qed. + +Lemma PI4_RGT_0 : 0 < PI / 4. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup0 ]. +Qed. + +Lemma cos_PI4 : cos (PI / 4) = 1 / sqrt 2. +Proof with trivial. +apply Rsqr_inj... +apply cos_ge_0... +left; apply (Rlt_trans (- (PI / 2)) 0 (PI / 4) _PI2_RLT_0 PI4_RGT_0)... +left; apply PI4_RLT_PI2... +left; apply (Rmult_lt_0_compat 1 (/ sqrt 2))... +prove_sup... +apply Rinv_0_lt_compat; apply Rlt_sqrt2_0... +rewrite Rsqr_div... +rewrite Rsqr_1; rewrite Rsqr_sqrt... +assert (H : 2 <> 0); [ discrR | idtac ]... +unfold Rsqr in |- *; pattern (cos (PI / 4)) at 1 in |- *; + rewrite <- sin_cos_PI4; + replace (sin (PI / 4) * cos (PI / 4)) with + (1 / 2 * (2 * sin (PI / 4) * cos (PI / 4)))... +rewrite <- sin_2a; replace (2 * (PI / 4)) with (PI / 2)... +rewrite sin_PI2... +apply Rmult_1_r... +unfold Rdiv in |- *; rewrite (Rmult_comm 2); rewrite Rinv_mult_distr... +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... +rewrite Rmult_1_r... +unfold Rdiv in |- *; rewrite Rmult_1_l; repeat rewrite <- Rmult_assoc... +rewrite <- Rinv_l_sym... +rewrite Rmult_1_l... +left; prove_sup... +apply sqrt2_neq_0... +Qed. + +Lemma sin_PI4 : sin (PI / 4) = 1 / sqrt 2. +rewrite sin_cos_PI4; apply cos_PI4. +Qed. + +Lemma tan_PI4 : tan (PI / 4) = 1. +unfold tan in |- *; rewrite sin_cos_PI4. +unfold Rdiv in |- *; apply Rinv_r. +change (cos (PI / 4) <> 0) in |- *; rewrite cos_PI4; apply R1_sqrt2_neq_0. +Qed. + +Lemma cos3PI4 : cos (3 * (PI / 4)) = -1 / sqrt 2. +Proof with trivial. +replace (3 * (PI / 4)) with (PI / 2 - - (PI / 4))... +rewrite cos_shift; rewrite sin_neg; rewrite sin_PI4... +unfold Rdiv in |- *; rewrite Ropp_mult_distr_l_reverse... +unfold Rminus in |- *; rewrite Ropp_involutive; pattern PI at 1 in |- *; + rewrite double_var; unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; + repeat rewrite Rmult_assoc; rewrite <- Rinv_mult_distr; + [ ring | discrR | discrR ]... +Qed. + +Lemma sin3PI4 : sin (3 * (PI / 4)) = 1 / sqrt 2. +Proof with trivial. +replace (3 * (PI / 4)) with (PI / 2 - - (PI / 4))... +rewrite sin_shift; rewrite cos_neg; rewrite cos_PI4... +unfold Rminus in |- *; rewrite Ropp_involutive; pattern PI at 1 in |- *; + rewrite double_var; unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; + repeat rewrite Rmult_assoc; rewrite <- Rinv_mult_distr; + [ ring | discrR | discrR ]... +Qed. + +Lemma cos_PI6 : cos (PI / 6) = sqrt 3 / 2. +Proof with trivial. +apply Rsqr_inj... +apply cos_ge_0... +left; apply (Rlt_trans (- (PI / 2)) 0 (PI / 6) _PI2_RLT_0 PI6_RGT_0)... +left; apply PI6_RLT_PI2... +left; apply (Rmult_lt_0_compat (sqrt 3) (/ 2))... +apply Rlt_sqrt3_0... +apply Rinv_0_lt_compat; prove_sup0... +assert (H : 2 <> 0); [ discrR | idtac ]... +assert (H1 : 4 <> 0); [ apply prod_neq_R0 | idtac ]... +rewrite Rsqr_div... +rewrite cos2; unfold Rsqr in |- *; rewrite sin_PI6; rewrite sqrt_def... +unfold Rdiv in |- *; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4... +rewrite Rmult_minus_distr_l; rewrite (Rmult_comm 3); + repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym... +rewrite Rmult_1_l; rewrite Rmult_1_r... +rewrite <- (Rmult_comm (/ 2)); repeat rewrite <- Rmult_assoc... +rewrite <- Rinv_l_sym... +rewrite Rmult_1_l; rewrite <- Rinv_r_sym... +ring... +left; prove_sup0... +Qed. + +Lemma tan_PI6 : tan (PI / 6) = 1 / sqrt 3. +unfold tan in |- *; rewrite sin_PI6; rewrite cos_PI6; unfold Rdiv in |- *; + repeat rewrite Rmult_1_l; rewrite Rinv_mult_distr. +rewrite Rinv_involutive. +rewrite (Rmult_comm (/ 2)); rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +apply Rmult_1_r. +discrR. +discrR. +red in |- *; intro; assert (H1 := Rlt_sqrt3_0); rewrite H in H1; + elim (Rlt_irrefl 0 H1). +apply Rinv_neq_0_compat; discrR. +Qed. + +Lemma sin_PI3 : sin (PI / 3) = sqrt 3 / 2. +rewrite sin_PI3_cos_PI6; apply cos_PI6. +Qed. + +Lemma cos_PI3 : cos (PI / 3) = 1 / 2. +rewrite sin_PI6_cos_PI3; apply sin_PI6. +Qed. + +Lemma tan_PI3 : tan (PI / 3) = sqrt 3. +unfold tan in |- *; rewrite sin_PI3; rewrite cos_PI3; unfold Rdiv in |- *; + rewrite Rmult_1_l; rewrite Rinv_involutive. +rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +apply Rmult_1_r. +discrR. +discrR. +Qed. + +Lemma sin_2PI3 : sin (2 * (PI / 3)) = sqrt 3 / 2. +rewrite double; rewrite sin_plus; rewrite sin_PI3; rewrite cos_PI3; + unfold Rdiv in |- *; repeat rewrite Rmult_1_l; rewrite (Rmult_comm (/ 2)); + repeat rewrite <- Rmult_assoc; rewrite double_var; + reflexivity. +Qed. + +Lemma cos_2PI3 : cos (2 * (PI / 3)) = -1 / 2. +Proof with trivial. +assert (H : 2 <> 0); [ discrR | idtac ]... +assert (H0 : 4 <> 0); [ apply prod_neq_R0 | idtac ]... +rewrite double; rewrite cos_plus; rewrite sin_PI3; rewrite cos_PI3; + unfold Rdiv in |- *; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4... +rewrite Rmult_minus_distr_l; repeat rewrite Rmult_assoc; + rewrite (Rmult_comm 2)... +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... +rewrite Rmult_1_r; rewrite <- Rinv_r_sym... +pattern 2 at 4 in |- *; rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym... +rewrite Rmult_1_r; rewrite Ropp_mult_distr_r_reverse; rewrite Rmult_1_r... +rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... +rewrite Rmult_1_r; rewrite (Rmult_comm 2); rewrite (Rmult_comm (/ 2))... +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... +rewrite Rmult_1_r; rewrite sqrt_def... +ring... +left; prove_sup... +Qed. + +Lemma tan_2PI3 : tan (2 * (PI / 3)) = - sqrt 3. +Proof with trivial. +assert (H : 2 <> 0); [ discrR | idtac ]... +unfold tan in |- *; rewrite sin_2PI3; rewrite cos_2PI3; unfold Rdiv in |- *; + rewrite Ropp_mult_distr_l_reverse; rewrite Rmult_1_l; + rewrite <- Ropp_inv_permute... +rewrite Rinv_involutive... +rewrite Rmult_assoc; rewrite Ropp_mult_distr_r_reverse; rewrite <- Rinv_l_sym... +ring... +apply Rinv_neq_0_compat... +Qed. + +Lemma cos_5PI4 : cos (5 * (PI / 4)) = -1 / sqrt 2. +Proof with trivial. +replace (5 * (PI / 4)) with (PI / 4 + PI)... +rewrite neg_cos; rewrite cos_PI4; unfold Rdiv in |- *; + rewrite Ropp_mult_distr_l_reverse... +pattern PI at 2 in |- *; rewrite double_var; pattern PI at 2 3 in |- *; + rewrite double_var; assert (H : 2 <> 0); + [ discrR | unfold Rdiv in |- *; repeat rewrite Rinv_mult_distr; try ring ]... +Qed. + +Lemma sin_5PI4 : sin (5 * (PI / 4)) = -1 / sqrt 2. +Proof with trivial. +replace (5 * (PI / 4)) with (PI / 4 + PI)... +rewrite neg_sin; rewrite sin_PI4; unfold Rdiv in |- *; + rewrite Ropp_mult_distr_l_reverse... +pattern PI at 2 in |- *; rewrite double_var; pattern PI at 2 3 in |- *; + rewrite double_var; assert (H : 2 <> 0); + [ discrR | unfold Rdiv in |- *; repeat rewrite Rinv_mult_distr; try ring ]... +Qed. + +Lemma sin_cos5PI4 : cos (5 * (PI / 4)) = sin (5 * (PI / 4)). +rewrite cos_5PI4; rewrite sin_5PI4; reflexivity. +Qed. + +Lemma Rgt_3PI2_0 : 0 < 3 * (PI / 2). +apply Rmult_lt_0_compat; + [ prove_sup0 + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup0 ] ]. +Qed. + +Lemma Rgt_2PI_0 : 0 < 2 * PI. +apply Rmult_lt_0_compat; [ prove_sup0 | apply PI_RGT_0 ]. +Qed. + +Lemma Rlt_PI_3PI2 : PI < 3 * (PI / 2). +generalize PI2_RGT_0; intro H1; + generalize (Rplus_lt_compat_l PI 0 (PI / 2) H1); + replace (PI + PI / 2) with (3 * (PI / 2)). +rewrite Rplus_0_r; intro H2; assumption. +pattern PI at 2 in |- *; rewrite double_var; ring. +Qed. + +Lemma Rlt_3PI2_2PI : 3 * (PI / 2) < 2 * PI. +generalize PI2_RGT_0; intro H1; + generalize (Rplus_lt_compat_l (3 * (PI / 2)) 0 (PI / 2) H1); + replace (3 * (PI / 2) + PI / 2) with (2 * PI). +rewrite Rplus_0_r; intro H2; assumption. +rewrite double; pattern PI at 1 2 in |- *; rewrite double_var; ring. +Qed. + +(***************************************************************) +(* Radian -> Degree | Degree -> Radian *) +(***************************************************************) + +Definition plat : R := 180. +Definition toRad (x:R) : R := x * PI * / plat. +Definition toDeg (x:R) : R := x * plat * / PI. + +Lemma rad_deg : forall x:R, toRad (toDeg x) = x. +intro; unfold toRad, toDeg in |- *; + replace (x * plat * / PI * PI * / plat) with + (x * (plat * / plat) * (PI * / PI)); [ idtac | ring ]. +repeat rewrite <- Rinv_r_sym. +ring. +apply PI_neq0. +unfold plat in |- *; discrR. +Qed. + +Lemma toRad_inj : forall x y:R, toRad x = toRad y -> x = y. +intros; unfold toRad in H; apply Rmult_eq_reg_l with PI. +rewrite <- (Rmult_comm x); rewrite <- (Rmult_comm y). +apply Rmult_eq_reg_l with (/ plat). +rewrite <- (Rmult_comm (x * PI)); rewrite <- (Rmult_comm (y * PI)); + assumption. +apply Rinv_neq_0_compat; unfold plat in |- *; discrR. +apply PI_neq0. +Qed. + +Lemma deg_rad : forall x:R, toDeg (toRad x) = x. +intro x; apply toRad_inj; rewrite (rad_deg (toRad x)); reflexivity. +Qed. + +Definition sind (x:R) : R := sin (toRad x). +Definition cosd (x:R) : R := cos (toRad x). +Definition tand (x:R) : R := tan (toRad x). + +Lemma Rsqr_sin_cos_d_one : forall x:R, Rsqr (sind x) + Rsqr (cosd x) = 1. +intro x; unfold sind in |- *; unfold cosd in |- *; apply sin2_cos2. +Qed. + +(***************************************************) +(* Other properties *) +(***************************************************) + +Lemma sin_lb_ge_0 : forall a:R, 0 <= a -> a <= PI / 2 -> 0 <= sin_lb a. +intros; case (Rtotal_order 0 a); intro. +left; apply sin_lb_gt_0; assumption. +elim H1; intro. +rewrite <- H2; unfold sin_lb in |- *; unfold sin_approx in |- *; + unfold sum_f_R0 in |- *; unfold sin_term in |- *; + repeat rewrite pow_ne_zero. +unfold Rdiv in |- *; repeat rewrite Rmult_0_l; repeat rewrite Rmult_0_r; + repeat rewrite Rplus_0_r; right; reflexivity. +discriminate. +discriminate. +discriminate. +discriminate. +elim (Rlt_irrefl 0 (Rle_lt_trans 0 a 0 H H2)). +Qed.
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