--- a/src/HOL/Transcendental.thy Wed Feb 04 18:10:07 2009 +0100
+++ b/src/HOL/Transcendental.thy Thu Feb 05 11:34:42 2009 +0100
@@ -113,6 +113,208 @@
==> summable (%n. f(n) * (z ^ n))"
by (rule powser_insidea [THEN summable_norm_cancel])
+lemma sum_split_even_odd: fixes f :: "nat \<Rightarrow> real" shows
+ "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
+ (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
+proof (induct n)
+ case (Suc n)
+ have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =
+ (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
+ using Suc.hyps by auto
+ also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))" by auto
+ finally show ?case .
+qed auto
+
+lemma sums_if': fixes g :: "nat \<Rightarrow> real" assumes "g sums x"
+ shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
+ unfolding sums_def
+proof (rule LIMSEQ_I)
+ fix r :: real assume "0 < r"
+ from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
+ obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast
+
+ let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
+ { fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> no" by auto
+ have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
+ using sum_split_even_odd by auto
+ hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
+ moreover
+ have "?SUM (2 * (m div 2)) = ?SUM m"
+ proof (cases "even m")
+ case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
+ next
+ case False hence "even (Suc m)" by auto
+ from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]] odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]
+ have eq: "Suc (2 * (m div 2)) = m" by auto
+ hence "even (2 * (m div 2))" using `odd m` by auto
+ have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
+ also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
+ finally show ?thesis by auto
+ qed
+ ultimately have "(norm (?SUM m - x) < r)" by auto
+ }
+ thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
+qed
+
+lemma sums_if: fixes g :: "nat \<Rightarrow> real" assumes "g sums x" and "f sums y"
+ shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
+proof -
+ let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
+ { fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
+ by (cases B) auto } note if_sum = this
+ have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" using sums_if'[OF `g sums x`] .
+ {
+ have "?s 0 = 0" by auto
+ have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
+ { fix B T E have "(if \<not> B then T else E) = (if B then E else T)" by auto } note if_eq = this
+
+ have "?s sums y" using sums_if'[OF `f sums y`] .
+ from this[unfolded sums_def, THEN LIMSEQ_Suc]
+ have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
+ unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
+ image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
+ even_nat_Suc Suc_m1 if_eq .
+ } from sums_add[OF g_sums this]
+ show ?thesis unfolding if_sum .
+qed
+
+subsection {* Alternating series test / Leibniz formula *}
+
+lemma sums_alternating_upper_lower:
+ fixes a :: "nat \<Rightarrow> real"
+ assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
+ shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and>
+ ((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. -1^i*a i) ----> l)"
+ (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
+proof -
+ have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" by auto
+
+ have "\<forall> n. ?f n \<le> ?f (Suc n)"
+ proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed
+ moreover
+ have "\<forall> n. ?g (Suc n) \<le> ?g n"
+ proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"] by auto qed
+ moreover
+ have "\<forall> n. ?f n \<le> ?g n"
+ proof fix n show "?f n \<le> ?g n" using fg_diff a_pos by auto qed
+ moreover
+ have "(\<lambda> n. ?f n - ?g n) ----> 0" unfolding fg_diff
+ proof (rule LIMSEQ_I)
+ fix r :: real assume "0 < r"
+ with `a ----> 0`[THEN LIMSEQ_D]
+ obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" by auto
+ hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
+ thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
+ qed
+ ultimately
+ show ?thesis by (rule lemma_nest_unique)
+qed
+
+lemma summable_Leibniz': fixes a :: "nat \<Rightarrow> real"
+ assumes a_zero: "a ----> 0" and a_pos: "\<And> n. 0 \<le> a n"
+ and a_monotone: "\<And> n. a (Suc n) \<le> a n"
+ shows summable: "summable (\<lambda> n. (-1)^n * a n)"
+ and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
+ and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
+ and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
+ and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
+proof -
+ let "?S n" = "(-1)^n * a n"
+ let "?P n" = "\<Sum>i=0..<n. ?S i"
+ let "?f n" = "?P (2 * n)"
+ let "?g n" = "?P (2 * n + 1)"
+ obtain l :: real where below_l: "\<forall> n. ?f n \<le> l" and "?f ----> l" and above_l: "\<forall> n. l \<le> ?g n" and "?g ----> l"
+ using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
+
+ let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n"
+ have "?Sa ----> l"
+ proof (rule LIMSEQ_I)
+ fix r :: real assume "0 < r"
+
+ with `?f ----> l`[THEN LIMSEQ_D]
+ obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
+
+ from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
+ obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
+
+ { fix n :: nat
+ assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
+ have "norm (?Sa n - l) < r"
+ proof (cases "even n")
+ case True from even_nat_div_two_times_two[OF this]
+ have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto
+ with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" by auto
+ from f[OF this]
+ show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
+ next
+ case False hence "even (n - 1)" using even_num_iff odd_pos by auto
+ from even_nat_div_two_times_two[OF this]
+ have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto
+ hence range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto
+
+ from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" by auto
+ from g[OF this]
+ show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
+ qed
+ }
+ thus "\<exists> no. \<forall> n \<ge> no. norm (?Sa n - l) < r" by blast
+ qed
+ hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
+ thus "summable ?S" using summable_def by auto
+
+ have "l = suminf ?S" using sums_unique[OF sums_l] .
+
+ { fix n show "suminf ?S \<le> ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto }
+ { fix n show "?f n \<le> suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_l by auto }
+ show "?g ----> suminf ?S" using `?g ----> l` `l = suminf ?S` by auto
+ show "?f ----> suminf ?S" using `?f ----> l` `l = suminf ?S` by auto
+qed
+
+theorem summable_Leibniz: fixes a :: "nat \<Rightarrow> real"
+ assumes a_zero: "a ----> 0" and "monoseq a"
+ shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
+ and "0 < a 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n. -1^i * a i .. \<Sum>i=0..<2*n+1. -1^i * a i})" (is "?pos")
+ and "a 0 < 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n+1. -1^i * a i .. \<Sum>i=0..<2*n. -1^i * a i})" (is "?neg")
+ and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
+ and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
+proof -
+ have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
+ proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
+ case True
+ hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" by auto
+ { fix n have "a (Suc n) \<le> a n" using ord[where n="Suc n" and m=n] by auto }
+ note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] and mono = this
+ from leibniz[OF mono]
+ show ?thesis using `0 \<le> a 0` by auto
+ next
+ let ?a = "\<lambda> n. - a n"
+ case False
+ with monoseq_le[OF `monoseq a` `a ----> 0`]
+ have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
+ hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" by auto
+ { fix n have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] by auto }
+ note monotone = this
+ note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF LIMSEQ_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
+ have "summable (\<lambda> n. (-1)^n * ?a n)" using leibniz(1) by auto
+ then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" unfolding summable_def by auto
+ from this[THEN sums_minus]
+ have "(\<lambda> n. (-1)^n * a n) sums -l" by auto
+ hence ?summable unfolding summable_def by auto
+ moreover
+ have "\<And> a b :: real. \<bar> - a - - b \<bar> = \<bar>a - b\<bar>" unfolding minus_diff_minus by auto
+
+ from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
+ have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)" by auto
+
+ have ?pos using `0 \<le> ?a 0` by auto
+ moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto
+ moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN LIMSEQ_minus_cancel] by auto
+ ultimately show ?thesis by auto
+ qed
+ from this[THEN conjunct1] this[THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1]
+ this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]
+ show ?summable and ?pos and ?neg and ?f and ?g .
+qed
subsection {* Term-by-Term Differentiability of Power Series *}
@@ -432,6 +634,188 @@
lemma inv_real_of_nat_fact_ge_zero [simp]: "0 \<le> inverse (real (fact n))"
by (auto intro: order_less_imp_le)
+subsection {* Derivability of power series *}
+
+lemma DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
+ assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
+ and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
+ and "summable (f' x0)"
+ and "summable L" and L_def: "\<And> n x y. \<lbrakk> x \<in> { a <..< b} ; y \<in> { a <..< b} \<rbrakk> \<Longrightarrow> \<bar> f x n - f y n \<bar> \<le> L n * \<bar> x - y \<bar>"
+ shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
+ unfolding deriv_def
+proof (rule LIM_I)
+ fix r :: real assume "0 < r" hence "0 < r/3" by auto
+
+ obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
+ using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
+
+ obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
+ using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
+
+ let ?N = "Suc (max N_L N_f')"
+ have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
+ L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
+
+ let "?diff i x" = "(f (x0 + x) i - f x0 i) / x"
+
+ let ?r = "r / (3 * real ?N)"
+ have "0 < 3 * real ?N" by auto
+ from divide_pos_pos[OF `0 < r` this]
+ have "0 < ?r" .
+
+ let "?s n" = "SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"
+ def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
+
+ have "0 < S'" unfolding S'_def
+ proof (rule iffD2[OF Min_gr_iff])
+ show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x"
+ proof (rule ballI)
+ fix x assume "x \<in> ?s ` {0..<?N}"
+ then obtain n where "x = ?s n" and "n \<in> {0..<?N}" using image_iff[THEN iffD1] by blast
+ from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
+ obtain s where s_bound: "0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r)" by auto
+ have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)
+ thus "0 < x" unfolding `x = ?s n` .
+ qed
+ qed auto
+
+ def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
+ hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0" and "S \<le> S'" using x0_in_I and `0 < S'`
+ by auto
+
+ { fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
+ hence x_in_I: "x0 + x \<in> { a <..< b }" using S_a S_b by auto
+
+ note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
+ note div_smbl = summable_divide[OF diff_smbl]
+ note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
+ note ign = summable_ignore_initial_segment[where k="?N"]
+ note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
+ note div_shft_smbl = summable_divide[OF diff_shft_smbl]
+ note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
+
+ { fix n
+ have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
+ using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .
+ hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 0` by auto
+ } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
+ from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
+ have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
+ hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") using L_estimate by auto
+
+ have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..
+ also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
+ proof (rule setsum_strict_mono)
+ fix n assume "n \<in> { 0 ..< ?N}"
+ have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .
+ also have "S \<le> S'" using `S \<le> S'` .
+ also have "S' \<le> ?s n" unfolding S'_def
+ proof (rule Min_le_iff[THEN iffD2])
+ have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto
+ thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
+ qed auto
+ finally have "\<bar> x \<bar> < ?s n" .
+
+ from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
+ have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
+ with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n`
+ show "\<bar>?diff n x - f' x0 n\<bar> < ?r" by blast
+ qed auto
+ also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)
+ also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto
+ also have "\<dots> = r/3" by auto
+ finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
+
+ from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
+ have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> =
+ \<bar> \<Sum>n. ?diff n x - f' x0 n \<bar>" unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] using suminf_divide[OF diff_smbl, symmetric] by auto
+ also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>" unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]] by (rule abs_triangle_ineq)
+ also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto
+ also have "\<dots> < r /3 + r/3 + r/3"
+ using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3` by auto
+ finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"
+ by auto
+ } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
+ norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`
+ unfolding real_norm_def diff_0_right by blast
+qed
+
+lemma DERIV_power_series': fixes f :: "nat \<Rightarrow> real"
+ assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
+ and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
+ shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
+ (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
+proof -
+ { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
+ hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" by auto
+ have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
+ proof (rule DERIV_series')
+ show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
+ proof -
+ have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto
+ hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" using `R' < R` by auto
+ have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto
+ from powser_insidea[OF converges[OF in_Rball] this] show ?thesis by auto
+ qed
+ { fix n x y assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
+ show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
+ proof -
+ have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> = (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
+ unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto
+ also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
+ proof (rule mult_left_mono)
+ have "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p = 0..<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)" by (rule setsum_abs)
+ also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
+ proof (rule setsum_mono)
+ fix p assume "p \<in> {0..<Suc n}" hence "p \<le> n" by auto
+ { fix n fix x :: real assume "x \<in> {-R'<..<R'}"
+ hence "\<bar>x\<bar> \<le> R'" by auto
+ hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto)
+ } from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
+ have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" unfolding abs_mult by auto
+ thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> n` by auto
+ qed
+ also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
+ finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>" unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
+ show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult[symmetric] by auto
+ qed
+ also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult real_mult_assoc[symmetric] by algebra
+ finally show ?thesis .
+ qed }
+ { fix n
+ from DERIV_pow[of "Suc n" x0, THEN DERIV_cmult[where c="f n"]]
+ show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)" unfolding real_mult_assoc by auto }
+ { fix x assume "x \<in> {-R' <..< R'}" hence "R' \<in> {-R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto
+ have "summable (\<lambda> n. f n * x^n)"
+ proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
+ fix n
+ have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto)
+ show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult
+ by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right])
+ qed
+ from this[THEN summable_mult2[where c=x], unfolded real_mult_assoc, unfolded real_mult_commute]
+ show "summable (?f x)" by auto }
+ show "summable (?f' x0)" using converges[OF `x0 \<in> {-R <..< R}`] .
+ show "x0 \<in> {-R' <..< R'}" using `x0 \<in> {-R' <..< R'}` .
+ qed
+ } note for_subinterval = this
+ let ?R = "(R + \<bar>x0\<bar>) / 2"
+ have "\<bar>x0\<bar> < ?R" using assms by auto
+ hence "- ?R < x0"
+ proof (cases "x0 < 0")
+ case True
+ hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
+ thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
+ next
+ case False
+ have "- ?R < 0" using assms by auto
+ also have "\<dots> \<le> x0" using False by auto
+ finally show ?thesis .
+ qed
+ hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto
+ from for_subinterval[OF this]
+ show ?thesis .
+qed
subsection {* Exponential Function *}
@@ -830,6 +1214,37 @@
apply (simp_all add: abs_if isCont_ln)
done
+lemma ln_series: assumes "0 < x" and "x < 2"
+ shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))")
+proof -
+ let "?f' x n" = "(-1)^n * (x - 1)^n"
+
+ have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
+ proof (rule DERIV_isconst3[where x=x])
+ fix x :: real assume "x \<in> {0 <..< 2}" hence "0 < x" and "x < 2" by auto
+ have "norm (1 - x) < 1" using `0 < x` and `x < 2` by auto
+ have "1 / x = 1 / (1 - (1 - x))" by auto
+ also have "\<dots> = (\<Sum> n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
+ also have "\<dots> = suminf (?f' x)" unfolding power_mult_distrib[symmetric] by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
+ finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding real_divide_def by auto
+ moreover
+ have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
+ have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
+ proof (rule DERIV_power_series')
+ show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto
+ { fix x :: real assume "x \<in> {- 1<..<1}" hence "norm (-x) < 1" by auto
+ show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
+ by (auto simp del: power_mult_distrib simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
+ }
+ qed
+ hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" by auto
+ hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos .
+ ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
+ by (rule DERIV_diff)
+ thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
+ qed (auto simp add: assms)
+ thus ?thesis by (auto simp add: suminf_zero)
+qed
subsection {* Sine and Cosine *}
@@ -1378,6 +1793,12 @@
lemma minus_pi_half_less_zero: "-(pi/2) < 0"
by simp
+lemma m2pi_less_pi: "- (2 * pi) < pi"
+proof -
+ have "- (2 * pi) < 0" and "0 < pi" by auto
+ from order_less_trans[OF this] show ?thesis .
+qed
+
lemma sin_pi_half [simp]: "sin(pi/2) = 1"
apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
@@ -1487,6 +1908,24 @@
apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
done
+
+lemma pi_ge_two: "2 \<le> pi"
+proof (rule ccontr)
+ assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
+ have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
+ proof (cases "2 < 2 * pi")
+ case True with dense[OF `pi < 2`] show ?thesis by auto
+ next
+ case False have "pi < 2 * pi" by auto
+ from dense[OF this] and False show ?thesis by auto
+ qed
+ then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
+ hence "0 < sin y" using sin_gt_zero by auto
+ moreover
+ have "sin y < 0" using sin_gt_zero_pi[of "y - pi"] `pi < y` and `y < 2 * pi` sin_periodic_pi[of "y - pi"] by auto
+ ultimately show False by auto
+qed
+
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
by (auto simp add: order_le_less sin_gt_zero_pi)
@@ -1586,6 +2025,48 @@
apply (auto simp add: even_mult_two_ex)
done
+lemma cos_monotone_0_pi: assumes "0 \<le> y" and "y < x" and "x \<le> pi"
+ shows "cos x < cos y"
+proof -
+ have "- (x - y) < 0" by (auto!)
+
+ from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
+ obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" by auto
+ hence "0 < z" and "z < pi" by (auto!)
+ hence "0 < sin z" using sin_gt_zero_pi by auto
+ hence "cos x - cos y < 0" unfolding cos_diff minus_mult_commute[symmetric] using `- (x - y) < 0` by (rule mult_pos_neg2)
+ thus ?thesis by auto
+qed
+
+lemma cos_monotone_0_pi': assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" shows "cos x \<le> cos y"
+proof (cases "y < x")
+ case True show ?thesis using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
+next
+ case False hence "y = x" using `y \<le> x` by auto
+ thus ?thesis by auto
+qed
+
+lemma cos_monotone_minus_pi_0: assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
+ shows "cos y < cos x"
+proof -
+ have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" by (auto!)
+ from cos_monotone_0_pi[OF this]
+ show ?thesis unfolding cos_minus .
+qed
+
+lemma cos_monotone_minus_pi_0': assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0" shows "cos y \<le> cos x"
+proof (cases "y < x")
+ case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] by auto
+next
+ case False hence "y = x" using `y \<le> x` by auto
+ thus ?thesis by auto
+qed
+
+lemma sin_monotone_2pi': assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2" shows "sin y \<le> sin x"
+proof -
+ have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi" using pi_ge_two by (auto!)
+ from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] by auto
+qed
subsection {* Tangent *}
@@ -1653,6 +2134,22 @@
thus ?thesis by simp
qed
+lemma tan_half: fixes x :: real assumes "- (pi / 2) < x" and "x < pi / 2"
+ shows "tan x = sin (2 * x) / (cos (2 * x) + 1)"
+proof -
+ from cos_gt_zero_pi[OF `- (pi / 2) < x` `x < pi / 2`]
+ have "cos x \<noteq> 0" by auto
+
+ have minus_cos_2x: "\<And>X. X - cos (2*x) = X - (cos x) ^ 2 + (sin x) ^ 2" unfolding cos_double by algebra
+
+ have "tan x = (tan x + tan x) / 2" by auto
+ also have "\<dots> = sin (x + x) / (cos x * cos x) / 2" unfolding add_tan_eq[OF `cos x \<noteq> 0` `cos x \<noteq> 0`] ..
+ also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + (cos x) ^ 2 + cos (2*x) - cos (2*x))" unfolding divide_divide_eq_left numeral_2_eq_2 by auto
+ also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + cos (2*x) + (sin x)^2)" unfolding minus_cos_2x by auto
+ also have "\<dots> = sin (2 * x) / (cos (2*x) + 1)" by auto
+ finally show ?thesis .
+qed
+
lemma lemma_DERIV_tan:
"cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"
apply (rule lemma_DERIV_subst)
@@ -1726,6 +2223,73 @@
simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
done
+lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
+ shows "tan y < tan x"
+proof -
+ have "\<forall> x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse (cos x'^2)"
+ proof (rule allI, rule impI)
+ fix x' :: real assume "y \<le> x' \<and> x' \<le> x"
+ hence "-(pi/2) < x'" and "x' < pi/2" by (auto!)
+ from cos_gt_zero_pi[OF this]
+ have "cos x' \<noteq> 0" by auto
+ thus "DERIV tan x' :> inverse (cos x'^2)" by (rule DERIV_tan)
+ qed
+ from MVT2[OF `y < x` this]
+ obtain z where "y < z" and "z < x" and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<twosuperior>)" by auto
+ hence "- (pi / 2) < z" and "z < pi / 2" by (auto!)
+ hence "0 < cos z" using cos_gt_zero_pi by auto
+ hence inv_pos: "0 < inverse ((cos z)\<twosuperior>)" by auto
+ have "0 < x - y" using `y < x` by auto
+ from real_mult_order[OF this inv_pos]
+ have "0 < tan x - tan y" unfolding tan_diff by auto
+ thus ?thesis by auto
+qed
+
+lemma tan_monotone': assumes "- (pi / 2) < y" and "y < pi / 2" and "- (pi / 2) < x" and "x < pi / 2"
+ shows "(y < x) = (tan y < tan x)"
+proof
+ assume "y < x" thus "tan y < tan x" using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
+next
+ assume "tan y < tan x"
+ show "y < x"
+ proof (rule ccontr)
+ assume "\<not> y < x" hence "x \<le> y" by auto
+ hence "tan x \<le> tan y"
+ proof (cases "x = y")
+ case True thus ?thesis by auto
+ next
+ case False hence "x < y" using `x \<le> y` by auto
+ from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
+ qed
+ thus False using `tan y < tan x` by auto
+ qed
+qed
+
+lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)" unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
+
+lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
+ by (simp add: tan_def)
+
+lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x"
+proof (induct n arbitrary: x)
+ case (Suc n)
+ have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" unfolding Suc_plus1 real_of_nat_add real_of_one real_add_mult_distrib by auto
+ show ?case unfolding split_pi_off using Suc by auto
+qed auto
+
+lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
+proof (cases "0 \<le> i")
+ case True hence i_nat: "real i = real (nat i)" by auto
+ show ?thesis unfolding i_nat by auto
+next
+ case False hence i_nat: "real i = - real (nat (-i))" by auto
+ have "tan x = tan (x + real i * pi - real i * pi)" by auto
+ also have "\<dots> = tan (x + real i * pi)" unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
+ finally show ?thesis by auto
+qed
+
+lemma tan_periodic_n[simp]: "tan (x + number_of n * pi) = tan x"
+ using tan_periodic_int[of _ "number_of n" ] unfolding real_number_of .
subsection {* Inverse Trigonometric Functions *}
@@ -1968,7 +2532,6 @@
apply (simp, simp, simp, rule isCont_arctan)
done
-
subsection {* More Theorems about Sin and Cos *}
lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
@@ -2115,6 +2678,464 @@
lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
by (cut_tac x = x in sin_cos_squared_add3, auto)
+subsection {* Machins formula *}
+
+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
+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 .
+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
+ from arctan_add[OF less_imp_le[OF this] this]
+ have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
+ moreover
+ have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
+ from arctan_add[OF less_imp_le[OF this] this]
+ have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
+ moreover
+ have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
+ 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
+qed
+subsection {* Introducing the arcus tangens power series *}
+
+lemma monoseq_arctan_series: fixes x :: real
+ assumes "\<bar>x\<bar> \<le> 1" shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
+proof (cases "x = 0") case True thus ?thesis unfolding monoseq_def by auto
+next
+ case False
+ have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
+ show "monoseq ?a"
+ proof -
+ { fix n fix x :: real assume "0 \<le> x" and "x \<le> 1"
+ have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le> 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
+ proof (rule mult_mono)
+ show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))" by (rule frac_le) simp_all
+ show "0 \<le> 1 / real (Suc (n * 2))" by auto
+ show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)" by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`)
+ show "0 \<le> x ^ Suc (Suc n * 2)" by (rule zero_le_power) (simp add: `0 \<le> x`)
+ qed
+ } note mono = this
+
+ show ?thesis
+ proof (cases "0 \<le> x")
+ case True from mono[OF this `x \<le> 1`, THEN allI]
+ show ?thesis unfolding Suc_plus1[symmetric] by (rule mono_SucI2)
+ next
+ case False hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto
+ from mono[OF this]
+ have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge> 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using `0 \<le> -x` by auto
+ thus ?thesis unfolding Suc_plus1[symmetric] by (rule mono_SucI1[OF allI])
+ qed
+ qed
+qed
+
+lemma zeroseq_arctan_series: fixes x :: real
+ assumes "\<bar>x\<bar> \<le> 1" shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
+proof (cases "x = 0") case True thus ?thesis by (auto simp add: LIMSEQ_const)
+next
+ case False
+ have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
+ show "?a ----> 0"
+ proof (cases "\<bar>x\<bar> < 1")
+ case True hence "norm x < 1" by auto
+ from LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]]
+ show ?thesis unfolding inverse_eq_divide Suc_plus1 using LIMSEQ_linear[OF _ pos2] by auto
+ next
+ case False hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto
+ hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x" by auto
+ from LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] LIMSEQ_const[of x]]
+ show ?thesis unfolding n_eq by auto
+ qed
+qed
+
+lemma summable_arctan_series: fixes x :: real and n :: nat
+ assumes "\<bar>x\<bar> \<le> 1" shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "summable (?c x)")
+ by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
+
+lemma less_one_imp_sqr_less_one: fixes x :: real assumes "\<bar>x\<bar> < 1" shows "x^2 < 1"
+proof -
+ from mult_mono1[OF less_imp_le[OF `\<bar>x\<bar> < 1`] abs_ge_zero[of x]]
+ have "\<bar> x^2 \<bar> < 1" using `\<bar> x \<bar> < 1` unfolding numeral_2_eq_2 power_Suc2 by auto
+ thus ?thesis using zero_le_power2 by auto
+qed
+
+lemma DERIV_arctan_series: assumes "\<bar> x \<bar> < 1"
+ shows "DERIV (\<lambda> x'. \<Sum> k. (-1)^k * (1 / real (k*2+1) * x' ^ (k*2+1))) x :> (\<Sum> k. (-1)^k * x^(k*2))" (is "DERIV ?arctan _ :> ?Int")
+proof -
+ let "?f n" = "if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
+
+ { fix n :: nat assume "even n" hence "2 * (n div 2) = n" by presburger } note n_even=this
+ have if_eq: "\<And> n x'. ?f n * real (Suc n) * x'^n = (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)" using n_even by auto
+
+ { fix x :: real assume "\<bar>x\<bar> < 1" hence "x^2 < 1" by (rule less_one_imp_sqr_less_one)
+ have "summable (\<lambda> n. -1 ^ n * (x^2) ^n)"
+ by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x^2 < 1` order_less_imp_le[OF `x^2 < 1`])
+ hence "summable (\<lambda> n. -1 ^ n * x^(2*n))" unfolding power_mult .
+ } note summable_Integral = this
+
+ { fix f :: "nat \<Rightarrow> real"
+ have "\<And> x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
+ proof
+ fix x :: real assume "f sums x"
+ from sums_if[OF sums_zero this]
+ show "(\<lambda> n. if even n then f (n div 2) else 0) sums x" by auto
+ next
+ fix x :: real assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
+ from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult_commute]]
+ show "f sums x" unfolding sums_def by auto
+ qed
+ hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
+ } note sums_even = this
+
+ have Int_eq: "(\<Sum> n. ?f n * real (Suc n) * x^n) = ?Int" unfolding if_eq mult_commute[of _ 2] suminf_def sums_even[of "\<lambda> n. -1 ^ n * x ^ (2 * n)", symmetric]
+ by auto
+
+ { fix x :: real
+ have if_eq': "\<And> n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
+ (if even n then -1 ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
+ using n_even by auto
+ have idx_eq: "\<And> n. n * 2 + 1 = Suc (2 * n)" by auto
+ have "(\<Sum> n. ?f n * x^(Suc n)) = ?arctan x" unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. -1 ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
+ by auto
+ } note arctan_eq = this
+
+ have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
+ proof (rule DERIV_power_series')
+ show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto
+ { fix x' :: real assume x'_bounds: "x' \<in> {- 1 <..< 1}"
+ hence "\<bar>x'\<bar> < 1" by auto
+
+ let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
+ show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
+ by (rule sums_summable[where l="0 + ?S"], rule sums_if, rule sums_zero, rule summable_sums, rule summable_Integral[OF `\<bar>x'\<bar> < 1`])
+ }
+ qed auto
+ thus ?thesis unfolding Int_eq arctan_eq .
+qed
+
+lemma arctan_series: assumes "\<bar> x \<bar> \<le> 1"
+ shows "arctan x = (\<Sum> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "_ = suminf (\<lambda> n. ?c x n)")
+proof -
+ let "?c' x n" = "(-1)^n * x^(n*2)"
+
+ { fix r x :: real assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
+ have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto
+ from DERIV_arctan_series[OF this]
+ have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
+ } note DERIV_arctan_suminf = this
+
+ { fix x :: real assume "\<bar>x\<bar> \<le> 1" note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] }
+ note arctan_series_borders = this
+
+ { fix x :: real assume "\<bar>x\<bar> < 1" have "arctan x = (\<Sum> k. ?c x k)"
+ proof -
+ obtain r where "\<bar>x\<bar> < r" and "r < 1" using dense[OF `\<bar>x\<bar> < 1`] by blast
+ hence "0 < r" and "-r < x" and "x < r" by auto
+
+ have suminf_eq_arctan_bounded: "\<And> x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow> suminf (?c x) - arctan x = suminf (?c a) - arctan a"
+ proof -
+ fix x a b assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
+ hence "\<bar>x\<bar> < r" by auto
+ show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
+ proof (rule DERIV_isconst2[of "a" "b"])
+ show "a < b" and "a \<le> x" and "x \<le> b" using `a < b` `a \<le> x` `x \<le> b` by auto
+ have "\<forall> x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
+ proof (rule allI, rule impI)
+ fix x assume "-r < x \<and> x < r" hence "\<bar>x\<bar> < r" by auto
+ hence "\<bar>x\<bar> < 1" using `r < 1` by auto
+ have "\<bar> - (x^2) \<bar> < 1" using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
+ hence "(\<lambda> n. (- (x^2)) ^ n) sums (1 / (1 - (- (x^2))))" unfolding real_norm_def[symmetric] by (rule geometric_sums)
+ hence "(?c' x) sums (1 / (1 - (- (x^2))))" unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto
+ hence suminf_c'_eq_geom: "inverse (1 + x^2) = suminf (?c' x)" using sums_unique unfolding inverse_eq_divide by auto
+ have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x^2))" unfolding suminf_c'_eq_geom
+ by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
+ from DERIV_add_minus[OF this DERIV_arctan]
+ show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0" unfolding diff_minus by auto
+ qed
+ hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0" using `-r < a` `b < r` by auto
+ thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0" using `\<bar>x\<bar> < r` by auto
+ show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y" using DERIV_in_rball DERIV_isCont by auto
+ qed
+ qed
+
+ have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
+ unfolding Suc_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero by auto
+
+ have "suminf (?c x) - arctan x = 0"
+ proof (cases "x = 0")
+ case True thus ?thesis using suminf_arctan_zero by auto
+ next
+ case False hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
+ have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
+ by (rule suminf_eq_arctan_bounded[where x="0" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>", symmetric], auto simp add: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>`)
+ moreover
+ have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
+ by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"], auto simp add: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>`)
+ ultimately
+ show ?thesis using suminf_arctan_zero by auto
+ qed
+ thus ?thesis by auto
+ qed } note when_less_one = this
+
+ show "arctan x = suminf (\<lambda> n. ?c x n)"
+ proof (cases "\<bar>x\<bar> < 1")
+ case True thus ?thesis by (rule when_less_one)
+ next case False hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto
+ let "?a x n" = "\<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
+ let "?diff x n" = "\<bar> arctan x - (\<Sum> i = 0..<n. ?c x i)\<bar>"
+ { fix n :: nat
+ have "0 < (1 :: real)" by auto
+ moreover
+ { fix x :: real assume "0 < x" and "x < 1" hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
+ from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)" by auto
+ note bounds = mp[OF arctan_series_borders(2)[OF `\<bar>x\<bar> \<le> 1`] this, unfolded when_less_one[OF `\<bar>x\<bar> < 1`, symmetric], THEN spec]
+ have "0 < 1 / real (n*2+1) * x^(n*2+1)" by (rule mult_pos_pos, auto simp only: zero_less_power[OF `0 < x`], auto)
+ hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)" by (rule abs_of_pos)
+ have "?diff x n \<le> ?a x n"
+ proof (cases "even n")
+ case True hence sgn_pos: "(-1)^n = (1::real)" by auto
+ from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
+ from bounds[of m, unfolded this atLeastAtMost_iff]
+ have "\<bar>arctan x - (\<Sum>i = 0..<n. (?c x i))\<bar> \<le> (\<Sum>i = 0..<n + 1. (?c x i)) - (\<Sum>i = 0..<n. (?c x i))" by auto
+ also have "\<dots> = ?c x n" by auto
+ also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
+ finally show ?thesis .
+ next
+ case False hence sgn_neg: "(-1)^n = (-1::real)" by auto
+ from `odd n` obtain m where m_def: "2 * m + 1 = n" unfolding odd_Suc_mult_two_ex by auto
+ hence m_plus: "2 * (m + 1) = n + 1" by auto
+ from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
+ have "\<bar>arctan x - (\<Sum>i = 0..<n. (?c x i))\<bar> \<le> (\<Sum>i = 0..<n. (?c x i)) - (\<Sum>i = 0..<n+1. (?c x i))" by auto
+ also have "\<dots> = - ?c x n" by auto
+ also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
+ finally show ?thesis .
+ qed
+ hence "0 \<le> ?a x n - ?diff x n" by auto
+ }
+ hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
+ moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
+ unfolding real_diff_def divide_inverse
+ by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum)
+ ultimately have "0 \<le> ?a 1 n - ?diff 1 n" by (rule LIM_less_bound)
+ hence "?diff 1 n \<le> ?a 1 n" by auto
+ }
+ have "?a 1 ----> 0" unfolding LIMSEQ_rabs_zero power_one divide_inverse by (auto intro!: LIMSEQ_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)
+ have "?diff 1 ----> 0"
+ proof (rule LIMSEQ_I)
+ fix r :: real assume "0 < r"
+ obtain N :: nat where N_I: "\<And> n. N \<le> n \<Longrightarrow> ?a 1 n < r" using LIMSEQ_D[OF `?a 1 ----> 0` `0 < r`] by auto
+ { fix n assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this]
+ have "norm (?diff 1 n - 0) < r" by auto }
+ thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
+ qed
+ from this[unfolded LIMSEQ_rabs_zero real_diff_def add_commute[of "arctan 1"], THEN LIMSEQ_add_const, of "- arctan 1", THEN LIMSEQ_minus]
+ have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
+ hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
+
+ show ?thesis
+ proof (cases "x = 1", simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
+ assume "x \<noteq> 1" hence "x = -1" using `\<bar>x\<bar> = 1` by auto
+
+ have "- (pi / 2) < 0" using pi_gt_zero by auto
+ have "- (2 * pi) < 0" using pi_gt_zero by auto
+
+ have c_minus_minus: "\<And> i. ?c (- 1) i = - ?c 1 i" by auto
+
+ have "arctan (- 1) = arctan (tan (-(pi / 4)))" unfolding tan_45 tan_minus ..
+ also have "\<dots> = - (pi / 4)" by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
+ also have "\<dots> = - (arctan (tan (pi / 4)))" unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero])
+ also have "\<dots> = - (arctan 1)" unfolding tan_45 ..
+ also have "\<dots> = - (\<Sum> i. ?c 1 i)" using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto
+ also have "\<dots> = (\<Sum> i. ?c (- 1) i)" using suminf_minus[OF sums_summable[OF `(?c 1) sums (arctan 1)`]] unfolding c_minus_minus by auto
+ finally show ?thesis using `x = -1` by auto
+ qed
+ qed
+qed
+
+lemma arctan_half: fixes x :: real
+ shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x^2)))"
+proof -
+ obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x" using tan_total by blast
+ hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2" by auto
+
+ have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> A / B = (A / C) / (B / C)" by auto
+
+ have "0 < cos y" using cos_gt_zero_pi[OF low high] .
+ hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y) ^ 2) = cos y" by auto
+
+ have "1 + (tan y)^2 = 1 + sin y^2 / cos y^2" unfolding tan_def power_divide ..
+ also have "\<dots> = cos y^2 / cos y^2 + sin y^2 / cos y^2" using `cos y \<noteq> 0` by auto
+ also have "\<dots> = 1 / cos y^2" unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
+ finally have "1 + (tan y)^2 = 1 / cos y^2" .
+
+ have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)" unfolding tan_def divide_nonzero_divide[OF `cos y \<noteq> 0`, symmetric] ..
+ also have "\<dots> = tan y / (1 + 1 / cos y)" using `cos y \<noteq> 0` unfolding add_divide_distrib by auto
+ also have "\<dots> = tan y / (1 + 1 / sqrt(cos y^2))" unfolding cos_sqrt ..
+ also have "\<dots> = tan y / (1 + sqrt(1 / cos y^2))" unfolding real_sqrt_divide by auto
+ finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)^2))" unfolding `1 + (tan y)^2 = 1 / cos y^2` .
+
+ have "arctan x = y" using arctan_tan low high y_eq by auto
+ also have "\<dots> = 2 * (arctan (tan (y/2)))" using arctan_tan[OF low2 high2] by auto
+ also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" unfolding tan_half[OF low2 high2] by auto
+ finally show ?thesis unfolding eq `tan y = x` .
+qed
+
+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
+
+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
+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
+ also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
+ finally show ?thesis by auto
+qed
subsection {* Existence of Polar Coordinates *}