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author | huffman |

Mon, 05 Sep 2011 22:30:25 -0700 | |

changeset 44746 | 9e4f7d3b5376 |

parent 44745 | b068207a7400 |

child 44747 | ab7522fbe1a2 |

add lemmas about arctan;
simplify some proofs about arctan;

--- a/src/HOL/Transcendental.thy Mon Sep 05 18:06:02 2011 -0700 +++ b/src/HOL/Transcendental.thy Mon Sep 05 22:30:25 2011 -0700 @@ -2206,15 +2206,24 @@ lemma arctan_ubound: "arctan y < pi/2" by (auto simp only: arctan) +lemma arctan_unique: + assumes "-(pi/2) < x" and "x < pi/2" and "tan x = y" + shows "arctan y = x" + using assms arctan [of y] tan_total [of y] by (fast elim: ex1E) + lemma arctan_tan: "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x" -apply (unfold arctan_def) -apply (rule the1_equality) -apply (rule tan_total, auto) -done + by (rule arctan_unique, simp_all) lemma arctan_zero_zero [simp]: "arctan 0 = 0" -by (insert arctan_tan [of 0], simp) + by (rule arctan_unique, simp_all) + +lemma arctan_minus: "arctan (- x) = - arctan x" + apply (rule arctan_unique) + apply (simp only: neg_less_iff_less arctan_ubound) + apply (metis minus_less_iff arctan_lbound) + apply simp + done lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0" by (intro less_imp_neq [symmetric] cos_gt_zero_pi @@ -2245,6 +2254,30 @@ mult_assoc power_inverse [symmetric]) done +lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y" + by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan) + +lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y" + by (simp only: not_less [symmetric] arctan_less_iff) + +lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y" + by (simp only: eq_iff [where 'a=real] arctan_le_iff) + +lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x" + using arctan_less_iff [of 0 x] by simp + +lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0" + using arctan_less_iff [of x 0] by simp + +lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x" + using arctan_le_iff [of 0 x] by simp + +lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0" + using arctan_le_iff [of x 0] by simp + +lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0" + using arctan_eq_iff [of x 0] by simp + lemma isCont_inverse_function2: fixes f g :: "real \<Rightarrow> real" shows "\<lbrakk>a < x; x < b; @@ -2437,98 +2470,34 @@ subsection {* Machins formula *} +lemma arctan_one: "arctan 1 = pi / 4" + by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi) + lemma tan_total_pi4: assumes "\<bar>x\<bar> < 1" shows "\<exists> z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x" -proof - - obtain z where "- (pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast - have "tan (pi / 4) = 1" and "tan (- (pi / 4)) = - 1" using tan_45 tan_minus by auto - have "z \<noteq> pi / 4" - proof (rule ccontr) - assume "\<not> (z \<noteq> pi / 4)" hence "z = pi / 4" by auto - have "tan z = 1" unfolding `z = pi / 4` `tan (pi / 4) = 1` .. - thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto - qed - have "z \<noteq> - (pi / 4)" - proof (rule ccontr) - assume "\<not> (z \<noteq> - (pi / 4))" hence "z = - (pi / 4)" by auto - have "tan z = - 1" unfolding `z = - (pi / 4)` `tan (- (pi / 4)) = - 1` .. - thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto - qed - - have "z < pi / 4" - proof (rule ccontr) - assume "\<not> (z < pi / 4)" hence "pi / 4 < z" using `z \<noteq> pi / 4` by auto - have "- (pi / 2) < pi / 4" using m2pi_less_pi by auto - from tan_monotone[OF this `pi / 4 < z` `z < pi / 2`] - have "1 < x" unfolding `tan z = x` `tan (pi / 4) = 1` . - thus False using `\<bar>x\<bar> < 1` by auto - qed - moreover - have "-(pi / 4) < z" - proof (rule ccontr) - assume "\<not> (-(pi / 4) < z)" hence "z < - (pi / 4)" using `z \<noteq> - (pi / 4)` by auto - have "-(pi / 4) < pi / 2" using m2pi_less_pi by auto - from tan_monotone[OF `-(pi / 2) < z` `z < -(pi / 4)` this] - have "x < - 1" unfolding `tan z = x` `tan (-(pi / 4)) = - 1` . - thus False using `\<bar>x\<bar> < 1` by auto - qed - ultimately show ?thesis using `tan z = x` by auto +proof + show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x" + unfolding arctan_one [symmetric] arctan_minus [symmetric] + unfolding arctan_less_iff using assms by auto qed lemma arctan_add: assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1" shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))" -proof - - obtain y' where "-(pi/4) < y'" and "y' < pi/4" and "tan y' = y" using tan_total_pi4[OF `\<bar>y\<bar> < 1`] by blast - - have "pi / 4 < pi / 2" by auto - - have "\<exists> x'. -(pi/4) \<le> x' \<and> x' \<le> pi/4 \<and> tan x' = x" - proof (cases "\<bar>x\<bar> < 1") - case True from tan_total_pi4[OF this] obtain x' where "-(pi/4) < x'" and "x' < pi/4" and "tan x' = x" by blast - hence "-(pi/4) \<le> x'" and "x' \<le> pi/4" and "tan x' = x" by auto - thus ?thesis by auto - next - case False - show ?thesis - proof (cases "x = 1") - case True hence "tan (pi/4) = x" using tan_45 by auto - moreover - have "- pi \<le> pi" unfolding minus_le_self_iff by auto - hence "-(pi/4) \<le> pi/4" and "pi/4 \<le> pi/4" by auto - ultimately show ?thesis by blast - next - case False hence "x = -1" using `\<not> \<bar>x\<bar> < 1` and `\<bar>x\<bar> \<le> 1` by auto - hence "tan (-(pi/4)) = x" using tan_45 tan_minus by auto - moreover - have "- pi \<le> pi" unfolding minus_le_self_iff by auto - hence "-(pi/4) \<le> pi/4" and "-(pi/4) \<le> -(pi/4)" by auto - ultimately show ?thesis by blast - qed - qed - then obtain x' where "-(pi/4) \<le> x'" and "x' \<le> pi/4" and "tan x' = x" by blast - hence "-(pi/2) < x'" and "x' < pi/2" using order_le_less_trans[OF `x' \<le> pi/4` `pi / 4 < pi / 2`] by auto - - have "cos x' \<noteq> 0" using cos_gt_zero_pi[THEN less_imp_neq] and `-(pi/2) < x'` and `x' < pi/2` by auto - moreover have "cos y' \<noteq> 0" using cos_gt_zero_pi[THEN less_imp_neq] and `-(pi/4) < y'` and `y' < pi/4` by auto - ultimately have "cos x' * cos y' \<noteq> 0" by auto - - have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> (A / C) / (B / C) = A / B" by auto - have divide_mult_commute: "\<And> A B C D :: real. A * B / (C * D) = (A / C) * (B / D)" by auto - - have "tan (x' + y') = sin (x' + y') / (cos x' * cos y' - sin x' * sin y')" unfolding tan_def cos_add .. - also have "\<dots> = (tan x' + tan y') / ((cos x' * cos y' - sin x' * sin y') / (cos x' * cos y'))" unfolding add_tan_eq[OF `cos x' \<noteq> 0` `cos y' \<noteq> 0`] divide_nonzero_divide[OF `cos x' * cos y' \<noteq> 0`] .. - also have "\<dots> = (tan x' + tan y') / (1 - tan x' * tan y')" unfolding tan_def diff_divide_distrib divide_self[OF `cos x' * cos y' \<noteq> 0`] unfolding divide_mult_commute .. - finally have tan_eq: "tan (x' + y') = (x + y) / (1 - x * y)" unfolding `tan y' = y` `tan x' = x` . - - have "arctan (tan (x' + y')) = x' + y'" using `-(pi/4) < y'` `-(pi/4) \<le> x'` `y' < pi/4` and `x' \<le> pi/4` by (auto intro!: arctan_tan) - moreover have "arctan (tan (x')) = x'" using `-(pi/2) < x'` and `x' < pi/2` by (auto intro!: arctan_tan) - moreover have "arctan (tan (y')) = y'" using `-(pi/4) < y'` and `y' < pi/4` by (auto intro!: arctan_tan) - ultimately have "arctan x + arctan y = arctan (tan (x' + y'))" unfolding `tan y' = y` [symmetric] `tan x' = x`[symmetric] by auto - thus "arctan x + arctan y = arctan ((x + y) / (1 - x * y))" unfolding tan_eq . +proof (rule arctan_unique [symmetric]) + have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y" + unfolding arctan_one [symmetric] arctan_minus [symmetric] + unfolding arctan_le_iff arctan_less_iff using assms by auto + from add_le_less_mono [OF this] + show 1: "- (pi / 2) < arctan x + arctan y" by simp + have "arctan x \<le> pi / 4" and "arctan y < pi / 4" + unfolding arctan_one [symmetric] + unfolding arctan_le_iff arctan_less_iff using assms by auto + from add_le_less_mono [OF this] + show 2: "arctan x + arctan y < pi / 2" by simp + show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)" + using cos_gt_zero_pi [OF 1 2] by (simp add: tan_add) qed -lemma arctan1_eq_pi4: "arctan 1 = pi / 4" unfolding tan_45[symmetric] by (rule arctan_tan, auto simp add: m2pi_less_pi) - theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)" proof - have "\<bar>1 / 5\<bar> < (1 :: real)" by auto @@ -2543,8 +2512,9 @@ from arctan_add[OF this] have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto - thus ?thesis unfolding arctan1_eq_pi4 by algebra + thus ?thesis unfolding arctan_one by algebra qed + subsection {* Introducing the arcus tangens power series *} lemma monoseq_arctan_series: fixes x :: real @@ -2837,65 +2807,34 @@ lemma arctan_monotone: assumes "x < y" shows "arctan x < arctan y" -proof - - obtain z where "-(pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast - obtain w where "-(pi / 2) < w" and "w < pi / 2" and "tan w = y" using tan_total by blast - have "z < w" unfolding tan_monotone'[OF `-(pi / 2) < z` `z < pi / 2` `-(pi / 2) < w` `w < pi / 2`] `tan z = x` `tan w = y` using `x < y` . - thus ?thesis - unfolding `tan z = x`[symmetric] arctan_tan[OF `-(pi / 2) < z` `z < pi / 2`] - unfolding `tan w = y`[symmetric] arctan_tan[OF `-(pi / 2) < w` `w < pi / 2`] . -qed + using assms by (simp only: arctan_less_iff) lemma arctan_monotone': assumes "x \<le> y" shows "arctan x \<le> arctan y" -proof (cases "x = y") - case False hence "x < y" using `x \<le> y` by auto from arctan_monotone[OF this] show ?thesis by auto -qed auto - -lemma arctan_minus: "arctan (- x) = - arctan x" -proof - - obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast - thus ?thesis unfolding `tan y = x`[symmetric] tan_minus[symmetric] using arctan_tan[of y] arctan_tan[of "-y"] by auto -qed - -lemma arctan_inverse: assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x" -proof - - obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast - hence "y = arctan x" unfolding `tan y = x`[symmetric] using arctan_tan by auto - - { fix y x :: real assume "0 < y" and "y < pi /2" and "y = arctan x" and "tan y = x" hence "- (pi / 2) < y" by auto - have "tan y > 0" using tan_monotone'[OF _ _ `- (pi / 2) < y` `y < pi / 2`, of 0] tan_zero `0 < y` by auto - hence "x > 0" using `tan y = x` by auto - - have "- (pi / 2) < pi / 2 - y" using `y > 0` `y < pi / 2` by auto - moreover have "pi / 2 - y < pi / 2" using `y > 0` `y < pi / 2` by auto - ultimately have "arctan (1 / x) = pi / 2 - y" unfolding `tan y = x`[symmetric] tan_inverse using arctan_tan by auto - hence "arctan (1 / x) = sgn x * pi / 2 - arctan x" unfolding `y = arctan x` real_sgn_pos[OF `x > 0`] by auto - } note pos_y = this - - show ?thesis - proof (cases "y > 0") - case True from pos_y[OF this `y < pi / 2` `y = arctan x` `tan y = x`] show ?thesis . - next - case False hence "y \<le> 0" by auto - moreover have "y \<noteq> 0" - proof (rule ccontr) - assume "\<not> y \<noteq> 0" hence "y = 0" by auto - have "x = 0" unfolding `tan y = x`[symmetric] `y = 0` tan_zero .. - thus False using `x \<noteq> 0` by auto - qed - ultimately have "y < 0" by auto - hence "0 < - y" and "-y < pi / 2" using `- (pi / 2) < y` by auto - moreover have "-y = arctan (-x)" unfolding arctan_minus `y = arctan x` .. - moreover have "tan (-y) = -x" unfolding tan_minus `tan y = x` .. - ultimately have "arctan (1 / -x) = sgn (-x) * pi / 2 - arctan (-x)" using pos_y by blast - hence "arctan (- (1 / x)) = - (sgn x * pi / 2 - arctan x)" unfolding arctan_minus[of x] divide_minus_right sgn_minus by auto - thus ?thesis unfolding arctan_minus neg_equal_iff_equal . - qed + using assms by (simp only: arctan_le_iff) + +lemma arctan_inverse: + assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x" +proof (rule arctan_unique) + show "- (pi / 2) < sgn x * pi / 2 - arctan x" + using arctan_bounded [of x] assms + unfolding sgn_real_def + apply (auto simp add: algebra_simps) + apply (drule zero_less_arctan_iff [THEN iffD2]) + apply arith + done + show "sgn x * pi / 2 - arctan x < pi / 2" + using arctan_bounded [of "- x"] assms + unfolding sgn_real_def arctan_minus + by auto + show "tan (sgn x * pi / 2 - arctan x) = 1 / x" + unfolding tan_inverse [of "arctan x", unfolded tan_arctan] + unfolding sgn_real_def + by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff) qed theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM") proof - - have "pi / 4 = arctan 1" using arctan1_eq_pi4 by auto + have "pi / 4 = arctan 1" using arctan_one by auto also have "\<dots> = ?SUM" using arctan_series[of 1] by auto finally show ?thesis by auto qed