diff options
Diffstat (limited to 'theories/Reals/Rtrigo1.v')
| -rw-r--r-- | theories/Reals/Rtrigo1.v | 81 |
1 files changed, 28 insertions, 53 deletions
diff --git a/theories/Reals/Rtrigo1.v b/theories/Reals/Rtrigo1.v index 4d2418639..17b9677ef 100644 --- a/theories/Reals/Rtrigo1.v +++ b/theories/Reals/Rtrigo1.v @@ -182,13 +182,10 @@ destruct (pre_cos_bound _ 0 lo up) as [_ upper]. apply Rle_lt_trans with (1 := upper). apply Rlt_le_trans with (2 := lower). unfold cos_approx, sin_approx. -simpl sum_f_R0; replace 7 with (IZR 7) by (simpl; field). -replace 8 with (IZR 8) by (simpl; field). +simpl sum_f_R0. unfold cos_term, sin_term; simpl fact; rewrite !INR_IZR_INZ. -simpl plus; simpl mult. -field_simplify; - try (repeat apply conj; apply not_eq_sym, Rlt_not_eq, (IZR_lt 0); reflexivity). -unfold Rminus; rewrite !pow_IZR, <- !mult_IZR, <- !opp_IZR, <- ?plus_IZR. +simpl plus; simpl mult; simpl Z_of_nat. +field_simplify. match goal with |- IZR ?a / ?b < ?c / ?d => apply Rmult_lt_reg_r with d;[apply (IZR_lt 0); reflexivity | @@ -198,7 +195,7 @@ match goal with end. unfold Rdiv; rewrite !Rmult_assoc, Rinv_l, Rmult_1_r; [ | apply not_eq_sym, Rlt_not_eq, (IZR_lt 0); reflexivity]. -repeat (rewrite <- !plus_IZR || rewrite <- !mult_IZR). +rewrite <- !mult_IZR. apply IZR_lt; reflexivity. Qed. @@ -323,6 +320,7 @@ Lemma sin_PI : sin PI = 0. Proof. assert (H := sin2_cos2 PI). rewrite cos_PI in H. + change (-1) with (-(1)) in H. rewrite <- Rsqr_neg in H. rewrite Rsqr_1 in H. cut (Rsqr (sin PI) = 0). @@ -533,9 +531,8 @@ 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. + intro x; rewrite sin_minus; rewrite sin_PI; rewrite cos_PI. + ring. Qed. Lemma sin_period : forall (x:R) (k:nat), sin (x + 2 * INR k * PI) = sin x. @@ -593,9 +590,9 @@ Proof. 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; + 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; + repeat rewrite <- Rplus_assoc; change (-1) with (-(1)); 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)); @@ -603,6 +600,7 @@ Proof. auto with real. cut (sin x < -1). intro; generalize (Ropp_lt_gt_contravar (sin x) (-1) H); + change (-1) with (-(1)); rewrite Ropp_involutive; clear H; intro; generalize (Rsqr_incrst_1 1 (- sin x) H (Rlt_le 0 1 Rlt_0_1) @@ -610,7 +608,7 @@ Proof. 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_assoc; change (-1) with (-(1)); 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)); @@ -712,17 +710,16 @@ Proof. 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); + replace (((1 + 1) * x + 1 + 1 + 1) * ((1 + 1) * x + 1 + 1)) with (4 * x * x + 10 * x + 6); [ idtac | ring ]. - apply Rplus_lt_reg_l with (-4); rewrite Rplus_opp_l; - replace (-4 + (4 * x * x + 10 * x + 6)) with (4 * x * x + 10 * x + 2); + apply Rplus_lt_reg_l 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). @@ -767,7 +764,7 @@ 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 <- 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. @@ -1260,44 +1257,22 @@ 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))). + replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))) by field. + replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))) by field. 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)). + replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)) by ring. + replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)) by ring. + replace (-3 * (PI / 2) + 2 * PI) with (PI / 2) by field. + replace (-3 * (PI / 2) + PI) with (- (PI / 2)) by field. clear H1 H2 H3 H4; intros H1 H2 H3 H4; apply Rplus_lt_reg_l with (-3 * (PI / 2)); - replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)). - replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)). + replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)) by ring. + replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)) by ring. 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 : @@ -1737,7 +1712,7 @@ Proof. rewrite H5. rewrite mult_INR. simpl in |- *. - rewrite <- (Rplus_0_l (2 * INR x2 * PI)). + rewrite <- (Rplus_0_l ((1 + 1) * INR x2 * PI)). rewrite sin_period. apply sin_0. rewrite H5. @@ -1747,7 +1722,7 @@ Proof. rewrite Rmult_1_l; rewrite sin_plus. rewrite sin_PI. rewrite Rmult_0_r. - rewrite <- (Rplus_0_l (2 * INR x2 * PI)). + rewrite <- (Rplus_0_l ((1 + 1) * INR x2 * PI)). rewrite sin_period. rewrite sin_0; ring. apply le_IZR. @@ -1769,7 +1744,7 @@ Proof. rewrite H5. rewrite mult_INR. simpl in |- *. - rewrite <- (Rplus_0_l (2 * INR x2 * PI)). + rewrite <- (Rplus_0_l ((1 + 1) * INR x2 * PI)). rewrite sin_period. rewrite sin_0; ring. rewrite H5. @@ -1779,7 +1754,7 @@ Proof. rewrite Rmult_1_l; rewrite sin_plus. rewrite sin_PI. rewrite Rmult_0_r. - rewrite <- (Rplus_0_l (2 * INR x2 * PI)). + rewrite <- (Rplus_0_l ((1 + 1) * INR x2 * PI)). rewrite sin_period. rewrite sin_0; ring. apply le_IZR. @@ -1858,7 +1833,7 @@ Proof. - right; left; auto. - left. clear Hi. subst. - replace 0 with (IZR 0 * PI) by (simpl; ring). f_equal. f_equal. + replace 0 with (IZR 0 * PI) by apply Rmult_0_l. f_equal. f_equal. apply one_IZR_lt1. split. + apply Rlt_le_trans with 0; |