src/HOL/Transcendental.thy
 author huffman Mon Sep 05 18:06:02 2011 -0700 (2011-09-05) changeset 44745 b068207a7400 parent 44730 11a1290fd0ac child 44746 9e4f7d3b5376 permissions -rw-r--r--
convert lemma cos_total to Isar-style proof
```     1 (*  Title:      HOL/Transcendental.thy
```
```     2     Author:     Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
```
```     3     Author:     Lawrence C Paulson
```
```     4 *)
```
```     5
```
```     6 header{*Power Series, Transcendental Functions etc.*}
```
```     7
```
```     8 theory Transcendental
```
```     9 imports Fact Series Deriv NthRoot
```
```    10 begin
```
```    11
```
```    12 subsection {* Properties of Power Series *}
```
```    13
```
```    14 lemma lemma_realpow_diff:
```
```    15   fixes y :: "'a::monoid_mult"
```
```    16   shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
```
```    17 proof -
```
```    18   assume "p \<le> n"
```
```    19   hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
```
```    20   thus ?thesis by (simp add: power_commutes)
```
```    21 qed
```
```    22
```
```    23 lemma lemma_realpow_diff_sumr:
```
```    24   fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows
```
```    25      "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
```
```    26       y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
```
```    27 by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac
```
```    28          del: setsum_op_ivl_Suc)
```
```    29
```
```    30 lemma lemma_realpow_diff_sumr2:
```
```    31   fixes y :: "'a::{comm_ring,monoid_mult}" shows
```
```    32      "x ^ (Suc n) - y ^ (Suc n) =
```
```    33       (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
```
```    34 apply (induct n, simp)
```
```    35 apply (simp del: setsum_op_ivl_Suc)
```
```    36 apply (subst setsum_op_ivl_Suc)
```
```    37 apply (subst lemma_realpow_diff_sumr)
```
```    38 apply (simp add: right_distrib del: setsum_op_ivl_Suc)
```
```    39 apply (subst mult_left_commute [of "x - y"])
```
```    40 apply (erule subst)
```
```    41 apply (simp add: algebra_simps)
```
```    42 done
```
```    43
```
```    44 lemma lemma_realpow_rev_sumr:
```
```    45      "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
```
```    46       (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
```
```    47 apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
```
```    48 apply (rule inj_onI, simp)
```
```    49 apply auto
```
```    50 apply (rule_tac x="n - x" in image_eqI, simp, simp)
```
```    51 done
```
```    52
```
```    53 text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
```
```    54 x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
```
```    55
```
```    56 lemma powser_insidea:
```
```    57   fixes x z :: "'a::real_normed_field"
```
```    58   assumes 1: "summable (\<lambda>n. f n * x ^ n)"
```
```    59   assumes 2: "norm z < norm x"
```
```    60   shows "summable (\<lambda>n. norm (f n * z ^ n))"
```
```    61 proof -
```
```    62   from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
```
```    63   from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
```
```    64     by (rule summable_LIMSEQ_zero)
```
```    65   hence "convergent (\<lambda>n. f n * x ^ n)"
```
```    66     by (rule convergentI)
```
```    67   hence "Cauchy (\<lambda>n. f n * x ^ n)"
```
```    68     by (rule convergent_Cauchy)
```
```    69   hence "Bseq (\<lambda>n. f n * x ^ n)"
```
```    70     by (rule Cauchy_Bseq)
```
```    71   then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
```
```    72     by (simp add: Bseq_def, safe)
```
```    73   have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
```
```    74                    K * norm (z ^ n) * inverse (norm (x ^ n))"
```
```    75   proof (intro exI allI impI)
```
```    76     fix n::nat assume "0 \<le> n"
```
```    77     have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
```
```    78           norm (f n * x ^ n) * norm (z ^ n)"
```
```    79       by (simp add: norm_mult abs_mult)
```
```    80     also have "\<dots> \<le> K * norm (z ^ n)"
```
```    81       by (simp only: mult_right_mono 4 norm_ge_zero)
```
```    82     also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
```
```    83       by (simp add: x_neq_0)
```
```    84     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
```
```    85       by (simp only: mult_assoc)
```
```    86     finally show "norm (norm (f n * z ^ n)) \<le>
```
```    87                   K * norm (z ^ n) * inverse (norm (x ^ n))"
```
```    88       by (simp add: mult_le_cancel_right x_neq_0)
```
```    89   qed
```
```    90   moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
```
```    91   proof -
```
```    92     from 2 have "norm (norm (z * inverse x)) < 1"
```
```    93       using x_neq_0
```
```    94       by (simp add: nonzero_norm_divide divide_inverse [symmetric])
```
```    95     hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
```
```    96       by (rule summable_geometric)
```
```    97     hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
```
```    98       by (rule summable_mult)
```
```    99     thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
```
```   100       using x_neq_0
```
```   101       by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
```
```   102                     power_inverse norm_power mult_assoc)
```
```   103   qed
```
```   104   ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   105     by (rule summable_comparison_test)
```
```   106 qed
```
```   107
```
```   108 lemma powser_inside:
```
```   109   fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" shows
```
```   110      "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]
```
```   111       ==> summable (%n. f(n) * (z ^ n))"
```
```   112 by (rule powser_insidea [THEN summable_norm_cancel])
```
```   113
```
```   114 lemma sum_split_even_odd: fixes f :: "nat \<Rightarrow> real" shows
```
```   115   "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
```
```   116    (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
```
```   117 proof (induct n)
```
```   118   case (Suc n)
```
```   119   have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =
```
```   120         (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
```
```   121     using Suc.hyps unfolding One_nat_def by auto
```
```   122   also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))" by auto
```
```   123   finally show ?case .
```
```   124 qed auto
```
```   125
```
```   126 lemma sums_if': fixes g :: "nat \<Rightarrow> real" assumes "g sums x"
```
```   127   shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   128   unfolding sums_def
```
```   129 proof (rule LIMSEQ_I)
```
```   130   fix r :: real assume "0 < r"
```
```   131   from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
```
```   132   obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast
```
```   133
```
```   134   let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
```
```   135   { fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> no" by auto
```
```   136     have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
```
```   137       using sum_split_even_odd by auto
```
```   138     hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
```
```   139     moreover
```
```   140     have "?SUM (2 * (m div 2)) = ?SUM m"
```
```   141     proof (cases "even m")
```
```   142       case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
```
```   143     next
```
```   144       case False hence "even (Suc m)" by auto
```
```   145       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]]
```
```   146       have eq: "Suc (2 * (m div 2)) = m" by auto
```
```   147       hence "even (2 * (m div 2))" using `odd m` by auto
```
```   148       have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
```
```   149       also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
```
```   150       finally show ?thesis by auto
```
```   151     qed
```
```   152     ultimately have "(norm (?SUM m - x) < r)" by auto
```
```   153   }
```
```   154   thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
```
```   155 qed
```
```   156
```
```   157 lemma sums_if: fixes g :: "nat \<Rightarrow> real" assumes "g sums x" and "f sums y"
```
```   158   shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
```
```   159 proof -
```
```   160   let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
```
```   161   { fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
```
```   162       by (cases B) auto } note if_sum = this
```
```   163   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`] .
```
```   164   {
```
```   165     have "?s 0 = 0" by auto
```
```   166     have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
```
```   167     have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
```
```   168
```
```   169     have "?s sums y" using sums_if'[OF `f sums y`] .
```
```   170     from this[unfolded sums_def, THEN LIMSEQ_Suc]
```
```   171     have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
```
```   172       unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
```
```   173                 image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
```
```   174                 even_Suc Suc_m1 if_eq .
```
```   175   } from sums_add[OF g_sums this]
```
```   176   show ?thesis unfolding if_sum .
```
```   177 qed
```
```   178
```
```   179 subsection {* Alternating series test / Leibniz formula *}
```
```   180
```
```   181 lemma sums_alternating_upper_lower:
```
```   182   fixes a :: "nat \<Rightarrow> real"
```
```   183   assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
```
```   184   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>
```
```   185              ((\<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)"
```
```   186   (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
```
```   187 proof -
```
```   188   have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
```
```   189
```
```   190   have "\<forall> n. ?f n \<le> ?f (Suc n)"
```
```   191   proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed
```
```   192   moreover
```
```   193   have "\<forall> n. ?g (Suc n) \<le> ?g n"
```
```   194   proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
```
```   195     unfolding One_nat_def by auto qed
```
```   196   moreover
```
```   197   have "\<forall> n. ?f n \<le> ?g n"
```
```   198   proof fix n show "?f n \<le> ?g n" using fg_diff a_pos
```
```   199     unfolding One_nat_def by auto qed
```
```   200   moreover
```
```   201   have "(\<lambda> n. ?f n - ?g n) ----> 0" unfolding fg_diff
```
```   202   proof (rule LIMSEQ_I)
```
```   203     fix r :: real assume "0 < r"
```
```   204     with `a ----> 0`[THEN LIMSEQ_D]
```
```   205     obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" by auto
```
```   206     hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   207     thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   208   qed
```
```   209   ultimately
```
```   210   show ?thesis by (rule lemma_nest_unique)
```
```   211 qed
```
```   212
```
```   213 lemma summable_Leibniz': fixes a :: "nat \<Rightarrow> real"
```
```   214   assumes a_zero: "a ----> 0" and a_pos: "\<And> n. 0 \<le> a n"
```
```   215   and a_monotone: "\<And> n. a (Suc n) \<le> a n"
```
```   216   shows summable: "summable (\<lambda> n. (-1)^n * a n)"
```
```   217   and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
```
```   218   and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   219   and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
```
```   220   and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   221 proof -
```
```   222   let "?S n" = "(-1)^n * a n"
```
```   223   let "?P n" = "\<Sum>i=0..<n. ?S i"
```
```   224   let "?f n" = "?P (2 * n)"
```
```   225   let "?g n" = "?P (2 * n + 1)"
```
```   226   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"
```
```   227     using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
```
```   228
```
```   229   let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n"
```
```   230   have "?Sa ----> l"
```
```   231   proof (rule LIMSEQ_I)
```
```   232     fix r :: real assume "0 < r"
```
```   233
```
```   234     with `?f ----> l`[THEN LIMSEQ_D]
```
```   235     obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
```
```   236
```
```   237     from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
```
```   238     obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
```
```   239
```
```   240     { fix n :: nat
```
```   241       assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
```
```   242       have "norm (?Sa n - l) < r"
```
```   243       proof (cases "even n")
```
```   244         case True from even_nat_div_two_times_two[OF this]
```
```   245         have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto
```
```   246         with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" by auto
```
```   247         from f[OF this]
```
```   248         show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
```
```   249       next
```
```   250         case False hence "even (n - 1)" by simp
```
```   251         from even_nat_div_two_times_two[OF this]
```
```   252         have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto
```
```   253         hence range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto
```
```   254
```
```   255         from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" by auto
```
```   256         from g[OF this]
```
```   257         show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
```
```   258       qed
```
```   259     }
```
```   260     thus "\<exists> no. \<forall> n \<ge> no. norm (?Sa n - l) < r" by blast
```
```   261   qed
```
```   262   hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
```
```   263   thus "summable ?S" using summable_def by auto
```
```   264
```
```   265   have "l = suminf ?S" using sums_unique[OF sums_l] .
```
```   266
```
```   267   { fix n show "suminf ?S \<le> ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto }
```
```   268   { fix n show "?f n \<le> suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_l by auto }
```
```   269   show "?g ----> suminf ?S" using `?g ----> l` `l = suminf ?S` by auto
```
```   270   show "?f ----> suminf ?S" using `?f ----> l` `l = suminf ?S` by auto
```
```   271 qed
```
```   272
```
```   273 theorem summable_Leibniz: fixes a :: "nat \<Rightarrow> real"
```
```   274   assumes a_zero: "a ----> 0" and "monoseq a"
```
```   275   shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
```
```   276   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")
```
```   277   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")
```
```   278   and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
```
```   279   and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
```
```   280 proof -
```
```   281   have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
```
```   282   proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
```
```   283     case True
```
```   284     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" by auto
```
```   285     { fix n have "a (Suc n) \<le> a n" using ord[where n="Suc n" and m=n] by auto }
```
```   286     note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] and mono = this
```
```   287     from leibniz[OF mono]
```
```   288     show ?thesis using `0 \<le> a 0` by auto
```
```   289   next
```
```   290     let ?a = "\<lambda> n. - a n"
```
```   291     case False
```
```   292     with monoseq_le[OF `monoseq a` `a ----> 0`]
```
```   293     have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
```
```   294     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" by auto
```
```   295     { fix n have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] by auto }
```
```   296     note monotone = this
```
```   297     note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
```
```   298     have "summable (\<lambda> n. (-1)^n * ?a n)" using leibniz(1) by auto
```
```   299     then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" unfolding summable_def by auto
```
```   300     from this[THEN sums_minus]
```
```   301     have "(\<lambda> n. (-1)^n * a n) sums -l" by auto
```
```   302     hence ?summable unfolding summable_def by auto
```
```   303     moreover
```
```   304     have "\<And> a b :: real. \<bar> - a - - b \<bar> = \<bar>a - b\<bar>" unfolding minus_diff_minus by auto
```
```   305
```
```   306     from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
```
```   307     have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)" by auto
```
```   308
```
```   309     have ?pos using `0 \<le> ?a 0` by auto
```
```   310     moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto
```
```   311     moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel] by auto
```
```   312     ultimately show ?thesis by auto
```
```   313   qed
```
```   314   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]
```
```   315        this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]
```
```   316   show ?summable and ?pos and ?neg and ?f and ?g .
```
```   317 qed
```
```   318
```
```   319 subsection {* Term-by-Term Differentiability of Power Series *}
```
```   320
```
```   321 definition
```
```   322   diffs :: "(nat => 'a::ring_1) => nat => 'a" where
```
```   323   "diffs c = (%n. of_nat (Suc n) * c(Suc n))"
```
```   324
```
```   325 text{*Lemma about distributing negation over it*}
```
```   326 lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
```
```   327 by (simp add: diffs_def)
```
```   328
```
```   329 lemma sums_Suc_imp:
```
```   330   assumes f: "f 0 = 0"
```
```   331   shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
```
```   332 unfolding sums_def
```
```   333 apply (rule LIMSEQ_imp_Suc)
```
```   334 apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
```
```   335 apply (simp only: setsum_shift_bounds_Suc_ivl)
```
```   336 done
```
```   337
```
```   338 lemma diffs_equiv:
```
```   339   fixes x :: "'a::{real_normed_vector, ring_1}"
```
```   340   shows "summable (%n. (diffs c)(n) * (x ^ n)) ==>
```
```   341       (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
```
```   342          (\<Sum>n. (diffs c)(n) * (x ^ n))"
```
```   343 unfolding diffs_def
```
```   344 apply (drule summable_sums)
```
```   345 apply (rule sums_Suc_imp, simp_all)
```
```   346 done
```
```   347
```
```   348 lemma lemma_termdiff1:
```
```   349   fixes z :: "'a :: {monoid_mult,comm_ring}" shows
```
```   350   "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
```
```   351    (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
```
```   352 by(auto simp add: algebra_simps power_add [symmetric])
```
```   353
```
```   354 lemma sumr_diff_mult_const2:
```
```   355   "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
```
```   356 by (simp add: setsum_subtractf)
```
```   357
```
```   358 lemma lemma_termdiff2:
```
```   359   fixes h :: "'a :: {field}"
```
```   360   assumes h: "h \<noteq> 0" shows
```
```   361   "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
```
```   362    h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
```
```   363         (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
```
```   364 apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
```
```   365 apply (simp add: right_diff_distrib diff_divide_distrib h)
```
```   366 apply (simp add: mult_assoc [symmetric])
```
```   367 apply (cases "n", simp)
```
```   368 apply (simp add: lemma_realpow_diff_sumr2 h
```
```   369                  right_diff_distrib [symmetric] mult_assoc
```
```   370             del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
```
```   371 apply (subst lemma_realpow_rev_sumr)
```
```   372 apply (subst sumr_diff_mult_const2)
```
```   373 apply simp
```
```   374 apply (simp only: lemma_termdiff1 setsum_right_distrib)
```
```   375 apply (rule setsum_cong [OF refl])
```
```   376 apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
```
```   377 apply (clarify)
```
```   378 apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
```
```   379             del: setsum_op_ivl_Suc power_Suc)
```
```   380 apply (subst mult_assoc [symmetric], subst power_add [symmetric])
```
```   381 apply (simp add: mult_ac)
```
```   382 done
```
```   383
```
```   384 lemma real_setsum_nat_ivl_bounded2:
```
```   385   fixes K :: "'a::linordered_semidom"
```
```   386   assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
```
```   387   assumes K: "0 \<le> K"
```
```   388   shows "setsum f {0..<n-k} \<le> of_nat n * K"
```
```   389 apply (rule order_trans [OF setsum_mono])
```
```   390 apply (rule f, simp)
```
```   391 apply (simp add: mult_right_mono K)
```
```   392 done
```
```   393
```
```   394 lemma lemma_termdiff3:
```
```   395   fixes h z :: "'a::{real_normed_field}"
```
```   396   assumes 1: "h \<noteq> 0"
```
```   397   assumes 2: "norm z \<le> K"
```
```   398   assumes 3: "norm (z + h) \<le> K"
```
```   399   shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
```
```   400           \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   401 proof -
```
```   402   have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
```
```   403         norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
```
```   404           (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
```
```   405     apply (subst lemma_termdiff2 [OF 1])
```
```   406     apply (subst norm_mult)
```
```   407     apply (rule mult_commute)
```
```   408     done
```
```   409   also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
```
```   410   proof (rule mult_right_mono [OF _ norm_ge_zero])
```
```   411     from norm_ge_zero 2 have K: "0 \<le> K" by (rule order_trans)
```
```   412     have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
```
```   413       apply (erule subst)
```
```   414       apply (simp only: norm_mult norm_power power_add)
```
```   415       apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
```
```   416       done
```
```   417     show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
```
```   418               (z + h) ^ q * z ^ (n - 2 - q))
```
```   419           \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
```
```   420       apply (intro
```
```   421          order_trans [OF norm_setsum]
```
```   422          real_setsum_nat_ivl_bounded2
```
```   423          mult_nonneg_nonneg
```
```   424          zero_le_imp_of_nat
```
```   425          zero_le_power K)
```
```   426       apply (rule le_Kn, simp)
```
```   427       done
```
```   428   qed
```
```   429   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   430     by (simp only: mult_assoc)
```
```   431   finally show ?thesis .
```
```   432 qed
```
```   433
```
```   434 lemma lemma_termdiff4:
```
```   435   fixes f :: "'a::{real_normed_field} \<Rightarrow>
```
```   436               'b::real_normed_vector"
```
```   437   assumes k: "0 < (k::real)"
```
```   438   assumes le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
```
```   439   shows "f -- 0 --> 0"
```
```   440 unfolding LIM_eq diff_0_right
```
```   441 proof (safe)
```
```   442   let ?h = "of_real (k / 2)::'a"
```
```   443   have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
```
```   444   hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
```
```   445   hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
```
```   446   hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
```
```   447
```
```   448   fix r::real assume r: "0 < r"
```
```   449   show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
```
```   450   proof (cases)
```
```   451     assume "K = 0"
```
```   452     with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
```
```   453       by simp
```
```   454     thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
```
```   455   next
```
```   456     assume K_neq_zero: "K \<noteq> 0"
```
```   457     with zero_le_K have K: "0 < K" by simp
```
```   458     show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
```
```   459     proof (rule exI, safe)
```
```   460       from k r K show "0 < min k (r * inverse K / 2)"
```
```   461         by (simp add: mult_pos_pos positive_imp_inverse_positive)
```
```   462     next
```
```   463       fix x::'a
```
```   464       assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
```
```   465       from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
```
```   466         by simp_all
```
```   467       from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
```
```   468       also from x4 K have "K * norm x < K * (r * inverse K / 2)"
```
```   469         by (rule mult_strict_left_mono)
```
```   470       also have "\<dots> = r / 2"
```
```   471         using K_neq_zero by simp
```
```   472       also have "r / 2 < r"
```
```   473         using r by simp
```
```   474       finally show "norm (f x) < r" .
```
```   475     qed
```
```   476   qed
```
```   477 qed
```
```   478
```
```   479 lemma lemma_termdiff5:
```
```   480   fixes g :: "'a::{real_normed_field} \<Rightarrow>
```
```   481               nat \<Rightarrow> 'b::banach"
```
```   482   assumes k: "0 < (k::real)"
```
```   483   assumes f: "summable f"
```
```   484   assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
```
```   485   shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
```
```   486 proof (rule lemma_termdiff4 [OF k])
```
```   487   fix h::'a assume "h \<noteq> 0" and "norm h < k"
```
```   488   hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
```
```   489     by (simp add: le)
```
```   490   hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
```
```   491     by simp
```
```   492   moreover from f have B: "summable (\<lambda>n. f n * norm h)"
```
```   493     by (rule summable_mult2)
```
```   494   ultimately have C: "summable (\<lambda>n. norm (g h n))"
```
```   495     by (rule summable_comparison_test)
```
```   496   hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
```
```   497     by (rule summable_norm)
```
```   498   also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
```
```   499     by (rule summable_le)
```
```   500   also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
```
```   501     by (rule suminf_mult2 [symmetric])
```
```   502   finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
```
```   503 qed
```
```   504
```
```   505
```
```   506 text{* FIXME: Long proofs*}
```
```   507
```
```   508 lemma termdiffs_aux:
```
```   509   fixes x :: "'a::{real_normed_field,banach}"
```
```   510   assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
```
```   511   assumes 2: "norm x < norm K"
```
```   512   shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
```
```   513              - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   514 proof -
```
```   515   from dense [OF 2]
```
```   516   obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
```
```   517   from norm_ge_zero r1 have r: "0 < r"
```
```   518     by (rule order_le_less_trans)
```
```   519   hence r_neq_0: "r \<noteq> 0" by simp
```
```   520   show ?thesis
```
```   521   proof (rule lemma_termdiff5)
```
```   522     show "0 < r - norm x" using r1 by simp
```
```   523   next
```
```   524     from r r2 have "norm (of_real r::'a) < norm K"
```
```   525       by simp
```
```   526     with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
```
```   527       by (rule powser_insidea)
```
```   528     hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
```
```   529       using r
```
```   530       by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
```
```   531     hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
```
```   532       by (rule diffs_equiv [THEN sums_summable])
```
```   533     also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))
```
```   534       = (\<lambda>n. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
```
```   535       apply (rule ext)
```
```   536       apply (simp add: diffs_def)
```
```   537       apply (case_tac n, simp_all add: r_neq_0)
```
```   538       done
```
```   539     finally have "summable
```
```   540       (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
```
```   541       by (rule diffs_equiv [THEN sums_summable])
```
```   542     also have
```
```   543       "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
```
```   544            r ^ (n - Suc 0)) =
```
```   545        (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
```
```   546       apply (rule ext)
```
```   547       apply (case_tac "n", simp)
```
```   548       apply (case_tac "nat", simp)
```
```   549       apply (simp add: r_neq_0)
```
```   550       done
```
```   551     finally show
```
```   552       "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
```
```   553   next
```
```   554     fix h::'a and n::nat
```
```   555     assume h: "h \<noteq> 0"
```
```   556     assume "norm h < r - norm x"
```
```   557     hence "norm x + norm h < r" by simp
```
```   558     with norm_triangle_ineq have xh: "norm (x + h) < r"
```
```   559       by (rule order_le_less_trans)
```
```   560     show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
```
```   561           \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
```
```   562       apply (simp only: norm_mult mult_assoc)
```
```   563       apply (rule mult_left_mono [OF _ norm_ge_zero])
```
```   564       apply (simp (no_asm) add: mult_assoc [symmetric])
```
```   565       apply (rule lemma_termdiff3)
```
```   566       apply (rule h)
```
```   567       apply (rule r1 [THEN order_less_imp_le])
```
```   568       apply (rule xh [THEN order_less_imp_le])
```
```   569       done
```
```   570   qed
```
```   571 qed
```
```   572
```
```   573 lemma termdiffs:
```
```   574   fixes K x :: "'a::{real_normed_field,banach}"
```
```   575   assumes 1: "summable (\<lambda>n. c n * K ^ n)"
```
```   576   assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
```
```   577   assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
```
```   578   assumes 4: "norm x < norm K"
```
```   579   shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
```
```   580 unfolding deriv_def
```
```   581 proof (rule LIM_zero_cancel)
```
```   582   show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
```
```   583             - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
```
```   584   proof (rule LIM_equal2)
```
```   585     show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
```
```   586   next
```
```   587     fix h :: 'a
```
```   588     assume "h \<noteq> 0"
```
```   589     assume "norm (h - 0) < norm K - norm x"
```
```   590     hence "norm x + norm h < norm K" by simp
```
```   591     hence 5: "norm (x + h) < norm K"
```
```   592       by (rule norm_triangle_ineq [THEN order_le_less_trans])
```
```   593     have A: "summable (\<lambda>n. c n * x ^ n)"
```
```   594       by (rule powser_inside [OF 1 4])
```
```   595     have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
```
```   596       by (rule powser_inside [OF 1 5])
```
```   597     have C: "summable (\<lambda>n. diffs c n * x ^ n)"
```
```   598       by (rule powser_inside [OF 2 4])
```
```   599     show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
```
```   600              - (\<Sum>n. diffs c n * x ^ n) =
```
```   601           (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
```
```   602       apply (subst sums_unique [OF diffs_equiv [OF C]])
```
```   603       apply (subst suminf_diff [OF B A])
```
```   604       apply (subst suminf_divide [symmetric])
```
```   605       apply (rule summable_diff [OF B A])
```
```   606       apply (subst suminf_diff)
```
```   607       apply (rule summable_divide)
```
```   608       apply (rule summable_diff [OF B A])
```
```   609       apply (rule sums_summable [OF diffs_equiv [OF C]])
```
```   610       apply (rule arg_cong [where f="suminf"], rule ext)
```
```   611       apply (simp add: algebra_simps)
```
```   612       done
```
```   613   next
```
```   614     show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h -
```
```   615                of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   616         by (rule termdiffs_aux [OF 3 4])
```
```   617   qed
```
```   618 qed
```
```   619
```
```   620
```
```   621 subsection {* Derivability of power series *}
```
```   622
```
```   623 lemma DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
```
```   624   assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
```
```   625   and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
```
```   626   and "summable (f' x0)"
```
```   627   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>"
```
```   628   shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
```
```   629   unfolding deriv_def
```
```   630 proof (rule LIM_I)
```
```   631   fix r :: real assume "0 < r" hence "0 < r/3" by auto
```
```   632
```
```   633   obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
```
```   634     using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
```
```   635
```
```   636   obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
```
```   637     using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
```
```   638
```
```   639   let ?N = "Suc (max N_L N_f')"
```
```   640   have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
```
```   641     L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
```
```   642
```
```   643   let "?diff i x" = "(f (x0 + x) i - f x0 i) / x"
```
```   644
```
```   645   let ?r = "r / (3 * real ?N)"
```
```   646   have "0 < 3 * real ?N" by auto
```
```   647   from divide_pos_pos[OF `0 < r` this]
```
```   648   have "0 < ?r" .
```
```   649
```
```   650   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)"
```
```   651   def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
```
```   652
```
```   653   have "0 < S'" unfolding S'_def
```
```   654   proof (rule iffD2[OF Min_gr_iff])
```
```   655     show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x"
```
```   656     proof (rule ballI)
```
```   657       fix x assume "x \<in> ?s ` {0..<?N}"
```
```   658       then obtain n where "x = ?s n" and "n \<in> {0..<?N}" using image_iff[THEN iffD1] by blast
```
```   659       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
```
```   660       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
```
```   661       have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)
```
```   662       thus "0 < x" unfolding `x = ?s n` .
```
```   663     qed
```
```   664   qed auto
```
```   665
```
```   666   def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
```
```   667   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'`
```
```   668     by auto
```
```   669
```
```   670   { fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
```
```   671     hence x_in_I: "x0 + x \<in> { a <..< b }" using S_a S_b by auto
```
```   672
```
```   673     note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   674     note div_smbl = summable_divide[OF diff_smbl]
```
```   675     note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
```
```   676     note ign = summable_ignore_initial_segment[where k="?N"]
```
```   677     note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
```
```   678     note div_shft_smbl = summable_divide[OF diff_shft_smbl]
```
```   679     note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
```
```   680
```
```   681     { fix n
```
```   682       have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
```
```   683         using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .
```
```   684       hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 0` by auto
```
```   685     } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
```
```   686     from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
```
```   687     have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
```
```   688     hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") using L_estimate by auto
```
```   689
```
```   690     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>)" ..
```
```   691     also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
```
```   692     proof (rule setsum_strict_mono)
```
```   693       fix n assume "n \<in> { 0 ..< ?N}"
```
```   694       have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .
```
```   695       also have "S \<le> S'" using `S \<le> S'` .
```
```   696       also have "S' \<le> ?s n" unfolding S'_def
```
```   697       proof (rule Min_le_iff[THEN iffD2])
```
```   698         have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto
```
```   699         thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
```
```   700       qed auto
```
```   701       finally have "\<bar> x \<bar> < ?s n" .
```
```   702
```
```   703       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]
```
```   704       have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
```
```   705       with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n`
```
```   706       show "\<bar>?diff n x - f' x0 n\<bar> < ?r" by blast
```
```   707     qed auto
```
```   708     also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)
```
```   709     also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto
```
```   710     also have "\<dots> = r/3" by auto
```
```   711     finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
```
```   712
```
```   713     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   714     have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> =
```
```   715                     \<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
```
```   716     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)
```
```   717     also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto
```
```   718     also have "\<dots> < r /3 + r/3 + r/3"
```
```   719       using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
```
```   720       by (rule add_strict_mono [OF add_less_le_mono])
```
```   721     finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"
```
```   722       by auto
```
```   723   } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
```
```   724       norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`
```
```   725     unfolding real_norm_def diff_0_right by blast
```
```   726 qed
```
```   727
```
```   728 lemma DERIV_power_series': fixes f :: "nat \<Rightarrow> real"
```
```   729   assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
```
```   730   and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
```
```   731   shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
```
```   732   (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
```
```   733 proof -
```
```   734   { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
```
```   735     hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" by auto
```
```   736     have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
```
```   737     proof (rule DERIV_series')
```
```   738       show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
```
```   739       proof -
```
```   740         have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto
```
```   741         hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" using `R' < R` by auto
```
```   742         have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto
```
```   743         from powser_insidea[OF converges[OF in_Rball] this] show ?thesis by auto
```
```   744       qed
```
```   745       { fix n x y assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
```
```   746         show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
```
```   747         proof -
```
```   748           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>"
```
```   749             unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto
```
```   750           also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
```
```   751           proof (rule mult_left_mono)
```
```   752             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)
```
```   753             also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
```
```   754             proof (rule setsum_mono)
```
```   755               fix p assume "p \<in> {0..<Suc n}" hence "p \<le> n" by auto
```
```   756               { fix n fix x :: real assume "x \<in> {-R'<..<R'}"
```
```   757                 hence "\<bar>x\<bar> \<le> R'"  by auto
```
```   758                 hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto)
```
```   759               } from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
```
```   760               have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" unfolding abs_mult by auto
```
```   761               thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> n` by auto
```
```   762             qed
```
```   763             also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
```
```   764             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'`]]] .
```
```   765             show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult[symmetric] by auto
```
```   766           qed
```
```   767           also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult mult_assoc[symmetric] by algebra
```
```   768           finally show ?thesis .
```
```   769         qed }
```
```   770       { fix n show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
```
```   771           by (auto intro!: DERIV_intros simp del: power_Suc) }
```
```   772       { fix x assume "x \<in> {-R' <..< R'}" hence "R' \<in> {-R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto
```
```   773         have "summable (\<lambda> n. f n * x^n)"
```
```   774         proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
```
```   775           fix n
```
```   776           have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto)
```
```   777           show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult
```
```   778             by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right])
```
```   779         qed
```
```   780         from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
```
```   781         show "summable (?f x)" by auto }
```
```   782       show "summable (?f' x0)" using converges[OF `x0 \<in> {-R <..< R}`] .
```
```   783       show "x0 \<in> {-R' <..< R'}" using `x0 \<in> {-R' <..< R'}` .
```
```   784     qed
```
```   785   } note for_subinterval = this
```
```   786   let ?R = "(R + \<bar>x0\<bar>) / 2"
```
```   787   have "\<bar>x0\<bar> < ?R" using assms by auto
```
```   788   hence "- ?R < x0"
```
```   789   proof (cases "x0 < 0")
```
```   790     case True
```
```   791     hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
```
```   792     thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
```
```   793   next
```
```   794     case False
```
```   795     have "- ?R < 0" using assms by auto
```
```   796     also have "\<dots> \<le> x0" using False by auto
```
```   797     finally show ?thesis .
```
```   798   qed
```
```   799   hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto
```
```   800   from for_subinterval[OF this]
```
```   801   show ?thesis .
```
```   802 qed
```
```   803
```
```   804 subsection {* Exponential Function *}
```
```   805
```
```   806 definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" where
```
```   807   "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
```
```   808
```
```   809 lemma summable_exp_generic:
```
```   810   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   811   defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
```
```   812   shows "summable S"
```
```   813 proof -
```
```   814   have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
```
```   815     unfolding S_def by (simp del: mult_Suc)
```
```   816   obtain r :: real where r0: "0 < r" and r1: "r < 1"
```
```   817     using dense [OF zero_less_one] by fast
```
```   818   obtain N :: nat where N: "norm x < real N * r"
```
```   819     using reals_Archimedean3 [OF r0] by fast
```
```   820   from r1 show ?thesis
```
```   821   proof (rule ratio_test [rule_format])
```
```   822     fix n :: nat
```
```   823     assume n: "N \<le> n"
```
```   824     have "norm x \<le> real N * r"
```
```   825       using N by (rule order_less_imp_le)
```
```   826     also have "real N * r \<le> real (Suc n) * r"
```
```   827       using r0 n by (simp add: mult_right_mono)
```
```   828     finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
```
```   829       using norm_ge_zero by (rule mult_right_mono)
```
```   830     hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
```
```   831       by (rule order_trans [OF norm_mult_ineq])
```
```   832     hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
```
```   833       by (simp add: pos_divide_le_eq mult_ac)
```
```   834     thus "norm (S (Suc n)) \<le> r * norm (S n)"
```
```   835       by (simp add: S_Suc inverse_eq_divide)
```
```   836   qed
```
```   837 qed
```
```   838
```
```   839 lemma summable_norm_exp:
```
```   840   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   841   shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
```
```   842 proof (rule summable_norm_comparison_test [OF exI, rule_format])
```
```   843   show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
```
```   844     by (rule summable_exp_generic)
```
```   845 next
```
```   846   fix n show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
```
```   847     by (simp add: norm_power_ineq)
```
```   848 qed
```
```   849
```
```   850 lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
```
```   851 by (insert summable_exp_generic [where x=x], simp)
```
```   852
```
```   853 lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
```
```   854 unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
```
```   855
```
```   856
```
```   857 lemma exp_fdiffs:
```
```   858       "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
```
```   859 by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
```
```   860          del: mult_Suc of_nat_Suc)
```
```   861
```
```   862 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
```
```   863 by (simp add: diffs_def)
```
```   864
```
```   865 lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
```
```   866 unfolding exp_def scaleR_conv_of_real
```
```   867 apply (rule DERIV_cong)
```
```   868 apply (rule termdiffs [where K="of_real (1 + norm x)"])
```
```   869 apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
```
```   870 apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
```
```   871 apply (simp del: of_real_add)
```
```   872 done
```
```   873
```
```   874 lemma isCont_exp: "isCont exp x"
```
```   875   by (rule DERIV_exp [THEN DERIV_isCont])
```
```   876
```
```   877 lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
```
```   878   by (rule isCont_o2 [OF _ isCont_exp])
```
```   879
```
```   880 lemma tendsto_exp [tendsto_intros]:
```
```   881   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
```
```   882   by (rule isCont_tendsto_compose [OF isCont_exp])
```
```   883
```
```   884
```
```   885 subsubsection {* Properties of the Exponential Function *}
```
```   886
```
```   887 lemma powser_zero:
```
```   888   fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
```
```   889   shows "(\<Sum>n. f n * 0 ^ n) = f 0"
```
```   890 proof -
```
```   891   have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
```
```   892     by (rule sums_unique [OF series_zero], simp add: power_0_left)
```
```   893   thus ?thesis unfolding One_nat_def by simp
```
```   894 qed
```
```   895
```
```   896 lemma exp_zero [simp]: "exp 0 = 1"
```
```   897 unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
```
```   898
```
```   899 lemma setsum_cl_ivl_Suc2:
```
```   900   "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
```
```   901 by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
```
```   902          del: setsum_cl_ivl_Suc)
```
```   903
```
```   904 lemma exp_series_add:
```
```   905   fixes x y :: "'a::{real_field}"
```
```   906   defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
```
```   907   shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
```
```   908 proof (induct n)
```
```   909   case 0
```
```   910   show ?case
```
```   911     unfolding S_def by simp
```
```   912 next
```
```   913   case (Suc n)
```
```   914   have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
```
```   915     unfolding S_def by (simp del: mult_Suc)
```
```   916   hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
```
```   917     by simp
```
```   918
```
```   919   have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
```
```   920     by (simp only: times_S)
```
```   921   also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
```
```   922     by (simp only: Suc)
```
```   923   also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
```
```   924                 + y * (\<Sum>i=0..n. S x i * S y (n-i))"
```
```   925     by (rule left_distrib)
```
```   926   also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
```
```   927                 + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
```
```   928     by (simp only: setsum_right_distrib mult_ac)
```
```   929   also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
```
```   930                 + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```   931     by (simp add: times_S Suc_diff_le)
```
```   932   also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
```
```   933              (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
```
```   934     by (subst setsum_cl_ivl_Suc2, simp)
```
```   935   also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```   936              (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```   937     by (subst setsum_cl_ivl_Suc, simp)
```
```   938   also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
```
```   939              (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```   940              (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```   941     by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
```
```   942               real_of_nat_add [symmetric], simp)
```
```   943   also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
```
```   944     by (simp only: scaleR_right.setsum)
```
```   945   finally show
```
```   946     "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
```
```   947     by (simp del: setsum_cl_ivl_Suc)
```
```   948 qed
```
```   949
```
```   950 lemma exp_add: "exp (x + y) = exp x * exp y"
```
```   951 unfolding exp_def
```
```   952 by (simp only: Cauchy_product summable_norm_exp exp_series_add)
```
```   953
```
```   954 lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
```
```   955 by (rule exp_add [symmetric])
```
```   956
```
```   957 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
```
```   958 unfolding exp_def
```
```   959 apply (subst suminf_of_real)
```
```   960 apply (rule summable_exp_generic)
```
```   961 apply (simp add: scaleR_conv_of_real)
```
```   962 done
```
```   963
```
```   964 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
```
```   965 proof
```
```   966   have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
```
```   967   also assume "exp x = 0"
```
```   968   finally show "False" by simp
```
```   969 qed
```
```   970
```
```   971 lemma exp_minus: "exp (- x) = inverse (exp x)"
```
```   972 by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
```
```   973
```
```   974 lemma exp_diff: "exp (x - y) = exp x / exp y"
```
```   975   unfolding diff_minus divide_inverse
```
```   976   by (simp add: exp_add exp_minus)
```
```   977
```
```   978
```
```   979 subsubsection {* Properties of the Exponential Function on Reals *}
```
```   980
```
```   981 text {* Comparisons of @{term "exp x"} with zero. *}
```
```   982
```
```   983 text{*Proof: because every exponential can be seen as a square.*}
```
```   984 lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
```
```   985 proof -
```
```   986   have "0 \<le> exp (x/2) * exp (x/2)" by simp
```
```   987   thus ?thesis by (simp add: exp_add [symmetric])
```
```   988 qed
```
```   989
```
```   990 lemma exp_gt_zero [simp]: "0 < exp (x::real)"
```
```   991 by (simp add: order_less_le)
```
```   992
```
```   993 lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
```
```   994 by (simp add: not_less)
```
```   995
```
```   996 lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
```
```   997 by (simp add: not_le)
```
```   998
```
```   999 lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
```
```  1000 by simp
```
```  1001
```
```  1002 lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
```
```  1003 apply (induct "n")
```
```  1004 apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
```
```  1005 done
```
```  1006
```
```  1007 text {* Strict monotonicity of exponential. *}
```
```  1008
```
```  1009 lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
```
```  1010 apply (drule order_le_imp_less_or_eq, auto)
```
```  1011 apply (simp add: exp_def)
```
```  1012 apply (rule order_trans)
```
```  1013 apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
```
```  1014 apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
```
```  1015 done
```
```  1016
```
```  1017 lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
```
```  1018 proof -
```
```  1019   assume x: "0 < x"
```
```  1020   hence "1 < 1 + x" by simp
```
```  1021   also from x have "1 + x \<le> exp x"
```
```  1022     by (simp add: exp_ge_add_one_self_aux)
```
```  1023   finally show ?thesis .
```
```  1024 qed
```
```  1025
```
```  1026 lemma exp_less_mono:
```
```  1027   fixes x y :: real
```
```  1028   assumes "x < y" shows "exp x < exp y"
```
```  1029 proof -
```
```  1030   from `x < y` have "0 < y - x" by simp
```
```  1031   hence "1 < exp (y - x)" by (rule exp_gt_one)
```
```  1032   hence "1 < exp y / exp x" by (simp only: exp_diff)
```
```  1033   thus "exp x < exp y" by simp
```
```  1034 qed
```
```  1035
```
```  1036 lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
```
```  1037 apply (simp add: linorder_not_le [symmetric])
```
```  1038 apply (auto simp add: order_le_less exp_less_mono)
```
```  1039 done
```
```  1040
```
```  1041 lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
```
```  1042 by (auto intro: exp_less_mono exp_less_cancel)
```
```  1043
```
```  1044 lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
```
```  1045 by (auto simp add: linorder_not_less [symmetric])
```
```  1046
```
```  1047 lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
```
```  1048 by (simp add: order_eq_iff)
```
```  1049
```
```  1050 text {* Comparisons of @{term "exp x"} with one. *}
```
```  1051
```
```  1052 lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
```
```  1053   using exp_less_cancel_iff [where x=0 and y=x] by simp
```
```  1054
```
```  1055 lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
```
```  1056   using exp_less_cancel_iff [where x=x and y=0] by simp
```
```  1057
```
```  1058 lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
```
```  1059   using exp_le_cancel_iff [where x=0 and y=x] by simp
```
```  1060
```
```  1061 lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
```
```  1062   using exp_le_cancel_iff [where x=x and y=0] by simp
```
```  1063
```
```  1064 lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
```
```  1065   using exp_inj_iff [where x=x and y=0] by simp
```
```  1066
```
```  1067 lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
```
```  1068 apply (rule IVT)
```
```  1069 apply (auto intro: isCont_exp simp add: le_diff_eq)
```
```  1070 apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)")
```
```  1071 apply simp
```
```  1072 apply (rule exp_ge_add_one_self_aux, simp)
```
```  1073 done
```
```  1074
```
```  1075 lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
```
```  1076 apply (rule_tac x = 1 and y = y in linorder_cases)
```
```  1077 apply (drule order_less_imp_le [THEN lemma_exp_total])
```
```  1078 apply (rule_tac [2] x = 0 in exI)
```
```  1079 apply (frule_tac [3] one_less_inverse)
```
```  1080 apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
```
```  1081 apply (rule_tac x = "-x" in exI)
```
```  1082 apply (simp add: exp_minus)
```
```  1083 done
```
```  1084
```
```  1085
```
```  1086 subsection {* Natural Logarithm *}
```
```  1087
```
```  1088 definition ln :: "real \<Rightarrow> real" where
```
```  1089   "ln x = (THE u. exp u = x)"
```
```  1090
```
```  1091 lemma ln_exp [simp]: "ln (exp x) = x"
```
```  1092   by (simp add: ln_def)
```
```  1093
```
```  1094 lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
```
```  1095   by (auto dest: exp_total)
```
```  1096
```
```  1097 lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
```
```  1098   by (metis exp_gt_zero exp_ln)
```
```  1099
```
```  1100 lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
```
```  1101   by (erule subst, rule ln_exp)
```
```  1102
```
```  1103 lemma ln_one [simp]: "ln 1 = 0"
```
```  1104   by (rule ln_unique, simp)
```
```  1105
```
```  1106 lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"
```
```  1107   by (rule ln_unique, simp add: exp_add)
```
```  1108
```
```  1109 lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
```
```  1110   by (rule ln_unique, simp add: exp_minus)
```
```  1111
```
```  1112 lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"
```
```  1113   by (rule ln_unique, simp add: exp_diff)
```
```  1114
```
```  1115 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
```
```  1116   by (rule ln_unique, simp add: exp_real_of_nat_mult)
```
```  1117
```
```  1118 lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
```
```  1119   by (subst exp_less_cancel_iff [symmetric], simp)
```
```  1120
```
```  1121 lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
```
```  1122   by (simp add: linorder_not_less [symmetric])
```
```  1123
```
```  1124 lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
```
```  1125   by (simp add: order_eq_iff)
```
```  1126
```
```  1127 lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
```
```  1128   apply (rule exp_le_cancel_iff [THEN iffD1])
```
```  1129   apply (simp add: exp_ge_add_one_self_aux)
```
```  1130   done
```
```  1131
```
```  1132 lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
```
```  1133   by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
```
```  1134
```
```  1135 lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
```
```  1136   using ln_le_cancel_iff [of 1 x] by simp
```
```  1137
```
```  1138 lemma ln_ge_zero_imp_ge_one: "\<lbrakk>0 \<le> ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 \<le> x"
```
```  1139   using ln_le_cancel_iff [of 1 x] by simp
```
```  1140
```
```  1141 lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> (0 \<le> ln x) = (1 \<le> x)"
```
```  1142   using ln_le_cancel_iff [of 1 x] by simp
```
```  1143
```
```  1144 lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x < 0) = (x < 1)"
```
```  1145   using ln_less_cancel_iff [of x 1] by simp
```
```  1146
```
```  1147 lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
```
```  1148   using ln_less_cancel_iff [of 1 x] by simp
```
```  1149
```
```  1150 lemma ln_gt_zero_imp_gt_one: "\<lbrakk>0 < ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 < x"
```
```  1151   using ln_less_cancel_iff [of 1 x] by simp
```
```  1152
```
```  1153 lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> (0 < ln x) = (1 < x)"
```
```  1154   using ln_less_cancel_iff [of 1 x] by simp
```
```  1155
```
```  1156 lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x = 0) = (x = 1)"
```
```  1157   using ln_inj_iff [of x 1] by simp
```
```  1158
```
```  1159 lemma ln_less_zero: "\<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> ln x < 0"
```
```  1160   by simp
```
```  1161
```
```  1162 lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
```
```  1163   apply (subgoal_tac "isCont ln (exp (ln x))", simp)
```
```  1164   apply (rule isCont_inverse_function [where f=exp], simp_all)
```
```  1165   done
```
```  1166
```
```  1167 lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
```
```  1168   apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
```
```  1169   apply (erule DERIV_cong [OF DERIV_exp exp_ln])
```
```  1170   apply (simp_all add: abs_if isCont_ln)
```
```  1171   done
```
```  1172
```
```  1173 lemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x"
```
```  1174   by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
```
```  1175
```
```  1176 lemma ln_series: assumes "0 < x" and "x < 2"
```
```  1177   shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))")
```
```  1178 proof -
```
```  1179   let "?f' x n" = "(-1)^n * (x - 1)^n"
```
```  1180
```
```  1181   have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
```
```  1182   proof (rule DERIV_isconst3[where x=x])
```
```  1183     fix x :: real assume "x \<in> {0 <..< 2}" hence "0 < x" and "x < 2" by auto
```
```  1184     have "norm (1 - x) < 1" using `0 < x` and `x < 2` by auto
```
```  1185     have "1 / x = 1 / (1 - (1 - x))" by auto
```
```  1186     also have "\<dots> = (\<Sum> n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
```
```  1187     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)
```
```  1188     finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
```
```  1189     moreover
```
```  1190     have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
```
```  1191     have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
```
```  1192     proof (rule DERIV_power_series')
```
```  1193       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto
```
```  1194       { fix x :: real assume "x \<in> {- 1<..<1}" hence "norm (-x) < 1" by auto
```
```  1195         show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
```
```  1196           unfolding One_nat_def
```
```  1197           by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
```
```  1198       }
```
```  1199     qed
```
```  1200     hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" unfolding One_nat_def by auto
```
```  1201     hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos .
```
```  1202     ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
```
```  1203       by (rule DERIV_diff)
```
```  1204     thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
```
```  1205   qed (auto simp add: assms)
```
```  1206   thus ?thesis by auto
```
```  1207 qed
```
```  1208
```
```  1209 subsection {* Sine and Cosine *}
```
```  1210
```
```  1211 definition sin_coeff :: "nat \<Rightarrow> real" where
```
```  1212   "sin_coeff = (\<lambda>n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"
```
```  1213
```
```  1214 definition cos_coeff :: "nat \<Rightarrow> real" where
```
```  1215   "cos_coeff = (\<lambda>n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"
```
```  1216
```
```  1217 definition sin :: "real \<Rightarrow> real" where
```
```  1218   "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
```
```  1219
```
```  1220 definition cos :: "real \<Rightarrow> real" where
```
```  1221   "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
```
```  1222
```
```  1223 lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
```
```  1224   unfolding sin_coeff_def by simp
```
```  1225
```
```  1226 lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
```
```  1227   unfolding cos_coeff_def by simp
```
```  1228
```
```  1229 lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
```
```  1230   unfolding cos_coeff_def sin_coeff_def
```
```  1231   by (simp del: mult_Suc)
```
```  1232
```
```  1233 lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
```
```  1234   unfolding cos_coeff_def sin_coeff_def
```
```  1235   by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)
```
```  1236
```
```  1237 lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
```
```  1238 unfolding sin_coeff_def
```
```  1239 apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
```
```  1240 apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  1241 done
```
```  1242
```
```  1243 lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"
```
```  1244 unfolding cos_coeff_def
```
```  1245 apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
```
```  1246 apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  1247 done
```
```  1248
```
```  1249 lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
```
```  1250 unfolding sin_def by (rule summable_sin [THEN summable_sums])
```
```  1251
```
```  1252 lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
```
```  1253 unfolding cos_def by (rule summable_cos [THEN summable_sums])
```
```  1254
```
```  1255 lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
```
```  1256   by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  1257
```
```  1258 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
```
```  1259   by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  1260
```
```  1261 text{*Now at last we can get the derivatives of exp, sin and cos*}
```
```  1262
```
```  1263 lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
```
```  1264   unfolding sin_def cos_def
```
```  1265   apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
```
```  1266   apply (simp_all add: diffs_sin_coeff diffs_cos_coeff
```
```  1267     summable_minus summable_sin summable_cos)
```
```  1268   done
```
```  1269
```
```  1270 lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
```
```  1271   unfolding cos_def sin_def
```
```  1272   apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
```
```  1273   apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus
```
```  1274     summable_minus summable_sin summable_cos suminf_minus)
```
```  1275   done
```
```  1276
```
```  1277 lemma isCont_sin: "isCont sin x"
```
```  1278   by (rule DERIV_sin [THEN DERIV_isCont])
```
```  1279
```
```  1280 lemma isCont_cos: "isCont cos x"
```
```  1281   by (rule DERIV_cos [THEN DERIV_isCont])
```
```  1282
```
```  1283 lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
```
```  1284   by (rule isCont_o2 [OF _ isCont_sin])
```
```  1285
```
```  1286 lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
```
```  1287   by (rule isCont_o2 [OF _ isCont_cos])
```
```  1288
```
```  1289 lemma tendsto_sin [tendsto_intros]:
```
```  1290   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
```
```  1291   by (rule isCont_tendsto_compose [OF isCont_sin])
```
```  1292
```
```  1293 lemma tendsto_cos [tendsto_intros]:
```
```  1294   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
```
```  1295   by (rule isCont_tendsto_compose [OF isCont_cos])
```
```  1296
```
```  1297 declare
```
```  1298   DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1299   DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1300   DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1301   DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1302
```
```  1303 subsection {* Properties of Sine and Cosine *}
```
```  1304
```
```  1305 lemma sin_zero [simp]: "sin 0 = 0"
```
```  1306   unfolding sin_def sin_coeff_def by (simp add: powser_zero)
```
```  1307
```
```  1308 lemma cos_zero [simp]: "cos 0 = 1"
```
```  1309   unfolding cos_def cos_coeff_def by (simp add: powser_zero)
```
```  1310
```
```  1311 lemma sin_cos_squared_add [simp]: "(sin x)\<twosuperior> + (cos x)\<twosuperior> = 1"
```
```  1312 proof -
```
```  1313   have "\<forall>x. DERIV (\<lambda>x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
```
```  1314     by (auto intro!: DERIV_intros)
```
```  1315   hence "(sin x)\<twosuperior> + (cos x)\<twosuperior> = (sin 0)\<twosuperior> + (cos 0)\<twosuperior>"
```
```  1316     by (rule DERIV_isconst_all)
```
```  1317   thus "(sin x)\<twosuperior> + (cos x)\<twosuperior> = 1" by simp
```
```  1318 qed
```
```  1319
```
```  1320 lemma sin_cos_squared_add2 [simp]: "(cos x)\<twosuperior> + (sin x)\<twosuperior> = 1"
```
```  1321   by (subst add_commute, rule sin_cos_squared_add)
```
```  1322
```
```  1323 lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
```
```  1324   using sin_cos_squared_add2 [unfolded power2_eq_square] .
```
```  1325
```
```  1326 lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
```
```  1327   unfolding eq_diff_eq by (rule sin_cos_squared_add)
```
```  1328
```
```  1329 lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
```
```  1330   unfolding eq_diff_eq by (rule sin_cos_squared_add2)
```
```  1331
```
```  1332 lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
```
```  1333   by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
```
```  1334
```
```  1335 lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
```
```  1336   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  1337
```
```  1338 lemma sin_le_one [simp]: "sin x \<le> 1"
```
```  1339   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  1340
```
```  1341 lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
```
```  1342   by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
```
```  1343
```
```  1344 lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
```
```  1345   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  1346
```
```  1347 lemma cos_le_one [simp]: "cos x \<le> 1"
```
```  1348   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  1349
```
```  1350 lemma DERIV_fun_pow: "DERIV g x :> m ==>
```
```  1351       DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
```
```  1352   by (auto intro!: DERIV_intros)
```
```  1353
```
```  1354 lemma DERIV_fun_exp:
```
```  1355      "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
```
```  1356   by (auto intro!: DERIV_intros)
```
```  1357
```
```  1358 lemma DERIV_fun_sin:
```
```  1359      "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
```
```  1360   by (auto intro!: DERIV_intros)
```
```  1361
```
```  1362 lemma DERIV_fun_cos:
```
```  1363      "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
```
```  1364   by (auto intro!: DERIV_intros)
```
```  1365
```
```  1366 lemma sin_cos_add_lemma:
```
```  1367      "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
```
```  1368       (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
```
```  1369   (is "?f x = 0")
```
```  1370 proof -
```
```  1371   have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
```
```  1372     by (auto intro!: DERIV_intros simp add: algebra_simps)
```
```  1373   hence "?f x = ?f 0"
```
```  1374     by (rule DERIV_isconst_all)
```
```  1375   thus ?thesis by simp
```
```  1376 qed
```
```  1377
```
```  1378 lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
```
```  1379   using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
```
```  1380
```
```  1381 lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
```
```  1382   using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
```
```  1383
```
```  1384 lemma sin_cos_minus_lemma:
```
```  1385   "(sin(-x) + sin(x))\<twosuperior> + (cos(-x) - cos(x))\<twosuperior> = 0" (is "?f x = 0")
```
```  1386 proof -
```
```  1387   have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
```
```  1388     by (auto intro!: DERIV_intros simp add: algebra_simps)
```
```  1389   hence "?f x = ?f 0"
```
```  1390     by (rule DERIV_isconst_all)
```
```  1391   thus ?thesis by simp
```
```  1392 qed
```
```  1393
```
```  1394 lemma sin_minus [simp]: "sin (-x) = -sin(x)"
```
```  1395   using sin_cos_minus_lemma [where x=x] by simp
```
```  1396
```
```  1397 lemma cos_minus [simp]: "cos (-x) = cos(x)"
```
```  1398   using sin_cos_minus_lemma [where x=x] by simp
```
```  1399
```
```  1400 lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
```
```  1401   by (simp add: diff_minus sin_add)
```
```  1402
```
```  1403 lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
```
```  1404   by (simp add: sin_diff mult_commute)
```
```  1405
```
```  1406 lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
```
```  1407   by (simp add: diff_minus cos_add)
```
```  1408
```
```  1409 lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
```
```  1410   by (simp add: cos_diff mult_commute)
```
```  1411
```
```  1412 lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
```
```  1413   using sin_add [where x=x and y=x] by simp
```
```  1414
```
```  1415 lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
```
```  1416   using cos_add [where x=x and y=x]
```
```  1417   by (simp add: power2_eq_square)
```
```  1418
```
```  1419
```
```  1420 subsection {* The Constant Pi *}
```
```  1421
```
```  1422 definition pi :: "real" where
```
```  1423   "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
```
```  1424
```
```  1425 text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
```
```  1426    hence define pi.*}
```
```  1427
```
```  1428 lemma sin_paired:
```
```  1429      "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1))
```
```  1430       sums  sin x"
```
```  1431 proof -
```
```  1432   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
```
```  1433     by (rule sin_converges [THEN sums_group], simp)
```
```  1434   thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)
```
```  1435 qed
```
```  1436
```
```  1437 lemma sin_gt_zero:
```
```  1438   assumes "0 < x" and "x < 2" shows "0 < sin x"
```
```  1439 proof -
```
```  1440   let ?f = "\<lambda>n. \<Sum>k = n*2..<n*2+2. -1 ^ k / real (fact (2*k+1)) * x^(2*k+1)"
```
```  1441   have pos: "\<forall>n. 0 < ?f n"
```
```  1442   proof
```
```  1443     fix n :: nat
```
```  1444     let ?k2 = "real (Suc (Suc (4 * n)))"
```
```  1445     let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
```
```  1446     have "x * x < ?k2 * ?k3"
```
```  1447       using assms by (intro mult_strict_mono', simp_all)
```
```  1448     hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
```
```  1449       by (intro mult_strict_right_mono zero_less_power `0 < x`)
```
```  1450     thus "0 < ?f n"
```
```  1451       by (simp del: mult_Suc,
```
```  1452         simp add: less_divide_eq mult_pos_pos field_simps del: mult_Suc)
```
```  1453   qed
```
```  1454   have sums: "?f sums sin x"
```
```  1455     by (rule sin_paired [THEN sums_group], simp)
```
```  1456   show "0 < sin x"
```
```  1457     unfolding sums_unique [OF sums]
```
```  1458     using sums_summable [OF sums] pos
```
```  1459     by (rule suminf_gt_zero)
```
```  1460 qed
```
```  1461
```
```  1462 lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
```
```  1463 apply (cut_tac x = x in sin_gt_zero)
```
```  1464 apply (auto simp add: cos_squared_eq cos_double)
```
```  1465 done
```
```  1466
```
```  1467 lemma cos_paired:
```
```  1468      "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
```
```  1469 proof -
```
```  1470   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
```
```  1471     by (rule cos_converges [THEN sums_group], simp)
```
```  1472   thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)
```
```  1473 qed
```
```  1474
```
```  1475 lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
```
```  1476 by simp
```
```  1477
```
```  1478 lemma real_mult_inverse_cancel:
```
```  1479      "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
```
```  1480       ==> inverse x * y < inverse x1 * u"
```
```  1481 apply (rule_tac c=x in mult_less_imp_less_left)
```
```  1482 apply (auto simp add: mult_assoc [symmetric])
```
```  1483 apply (simp (no_asm) add: mult_ac)
```
```  1484 apply (rule_tac c=x1 in mult_less_imp_less_right)
```
```  1485 apply (auto simp add: mult_ac)
```
```  1486 done
```
```  1487
```
```  1488 lemma real_mult_inverse_cancel2:
```
```  1489      "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
```
```  1490 apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)
```
```  1491 done
```
```  1492
```
```  1493 lemma realpow_num_eq_if:
```
```  1494   fixes m :: "'a::power"
```
```  1495   shows "m ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
```
```  1496 by (cases n, auto)
```
```  1497
```
```  1498 lemma cos_two_less_zero [simp]: "cos (2) < 0"
```
```  1499 apply (cut_tac x = 2 in cos_paired)
```
```  1500 apply (drule sums_minus)
```
```  1501 apply (rule neg_less_iff_less [THEN iffD1])
```
```  1502 apply (frule sums_unique, auto)
```
```  1503 apply (rule_tac y =
```
```  1504  "\<Sum>n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
```
```  1505        in order_less_trans)
```
```  1506 apply (simp (no_asm) add: fact_num_eq_if_nat realpow_num_eq_if del: fact_Suc)
```
```  1507 apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
```
```  1508 apply (rule sumr_pos_lt_pair)
```
```  1509 apply (erule sums_summable, safe)
```
```  1510 unfolding One_nat_def
```
```  1511 apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
```
```  1512             del: fact_Suc)
```
```  1513 apply (rule real_mult_inverse_cancel2)
```
```  1514 apply (simp del: fact_Suc)
```
```  1515 apply (simp del: fact_Suc)
```
```  1516 apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
```
```  1517 apply (subst fact_lemma)
```
```  1518 apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
```
```  1519 apply (simp only: real_of_nat_mult)
```
```  1520 apply (rule mult_strict_mono, force)
```
```  1521   apply (rule_tac [3] real_of_nat_ge_zero)
```
```  1522  prefer 2 apply force
```
```  1523 apply (rule real_of_nat_less_iff [THEN iffD2])
```
```  1524 apply (rule fact_less_mono_nat, auto)
```
```  1525 done
```
```  1526
```
```  1527 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
```
```  1528 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
```
```  1529
```
```  1530 lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
```
```  1531 proof (rule ex_ex1I)
```
```  1532   show "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0"
```
```  1533     by (rule IVT2, simp_all)
```
```  1534 next
```
```  1535   fix x y
```
```  1536   assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
```
```  1537   assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
```
```  1538   have [simp]: "\<forall>x. cos differentiable x"
```
```  1539     unfolding differentiable_def by (auto intro: DERIV_cos)
```
```  1540   from x y show "x = y"
```
```  1541     apply (cut_tac less_linear [of x y], auto)
```
```  1542     apply (drule_tac f = cos in Rolle)
```
```  1543     apply (drule_tac [5] f = cos in Rolle)
```
```  1544     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  1545     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  1546     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  1547     done
```
```  1548 qed
```
```  1549
```
```  1550 lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
```
```  1551 by (simp add: pi_def)
```
```  1552
```
```  1553 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
```
```  1554 by (simp add: pi_half cos_is_zero [THEN theI'])
```
```  1555
```
```  1556 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
```
```  1557 apply (rule order_le_neq_trans)
```
```  1558 apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  1559 apply (rule notI, drule arg_cong [where f=cos], simp)
```
```  1560 done
```
```  1561
```
```  1562 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
```
```  1563 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
```
```  1564
```
```  1565 lemma pi_half_less_two [simp]: "pi / 2 < 2"
```
```  1566 apply (rule order_le_neq_trans)
```
```  1567 apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  1568 apply (rule notI, drule arg_cong [where f=cos], simp)
```
```  1569 done
```
```  1570
```
```  1571 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
```
```  1572 lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
```
```  1573
```
```  1574 lemma pi_gt_zero [simp]: "0 < pi"
```
```  1575 by (insert pi_half_gt_zero, simp)
```
```  1576
```
```  1577 lemma pi_ge_zero [simp]: "0 \<le> pi"
```
```  1578 by (rule pi_gt_zero [THEN order_less_imp_le])
```
```  1579
```
```  1580 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
```
```  1581 by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym])
```
```  1582
```
```  1583 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
```
```  1584 by (simp add: linorder_not_less)
```
```  1585
```
```  1586 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
```
```  1587 by simp
```
```  1588
```
```  1589 lemma m2pi_less_pi: "- (2 * pi) < pi"
```
```  1590 proof -
```
```  1591   have "- (2 * pi) < 0" and "0 < pi" by auto
```
```  1592   from order_less_trans[OF this] show ?thesis .
```
```  1593 qed
```
```  1594
```
```  1595 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
```
```  1596 apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
```
```  1597 apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
```
```  1598 apply (simp add: power2_eq_1_iff)
```
```  1599 done
```
```  1600
```
```  1601 lemma cos_pi [simp]: "cos pi = -1"
```
```  1602 by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
```
```  1603
```
```  1604 lemma sin_pi [simp]: "sin pi = 0"
```
```  1605 by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
```
```  1606
```
```  1607 lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
```
```  1608 by (simp add: diff_minus cos_add)
```
```  1609 declare sin_cos_eq [symmetric, simp]
```
```  1610
```
```  1611 lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
```
```  1612 by (simp add: cos_add)
```
```  1613 declare minus_sin_cos_eq [symmetric, simp]
```
```  1614
```
```  1615 lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
```
```  1616 by (simp add: diff_minus sin_add)
```
```  1617 declare cos_sin_eq [symmetric, simp]
```
```  1618
```
```  1619 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
```
```  1620 by (simp add: sin_add)
```
```  1621
```
```  1622 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
```
```  1623 by (simp add: sin_add)
```
```  1624
```
```  1625 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
```
```  1626 by (simp add: cos_add)
```
```  1627
```
```  1628 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
```
```  1629 by (simp add: sin_add cos_double)
```
```  1630
```
```  1631 lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
```
```  1632 by (simp add: cos_add cos_double)
```
```  1633
```
```  1634 lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
```
```  1635 apply (induct "n")
```
```  1636 apply (auto simp add: real_of_nat_Suc left_distrib)
```
```  1637 done
```
```  1638
```
```  1639 lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
```
```  1640 proof -
```
```  1641   have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
```
```  1642   also have "... = -1 ^ n" by (rule cos_npi)
```
```  1643   finally show ?thesis .
```
```  1644 qed
```
```  1645
```
```  1646 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
```
```  1647 apply (induct "n")
```
```  1648 apply (auto simp add: real_of_nat_Suc left_distrib)
```
```  1649 done
```
```  1650
```
```  1651 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
```
```  1652 by (simp add: mult_commute [of pi])
```
```  1653
```
```  1654 lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
```
```  1655 by (simp add: cos_double)
```
```  1656
```
```  1657 lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
```
```  1658 by simp
```
```  1659
```
```  1660 lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
```
```  1661 apply (rule sin_gt_zero, assumption)
```
```  1662 apply (rule order_less_trans, assumption)
```
```  1663 apply (rule pi_half_less_two)
```
```  1664 done
```
```  1665
```
```  1666 lemma sin_less_zero:
```
```  1667   assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
```
```  1668 proof -
```
```  1669   have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
```
```  1670   thus ?thesis by simp
```
```  1671 qed
```
```  1672
```
```  1673 lemma pi_less_4: "pi < 4"
```
```  1674 by (cut_tac pi_half_less_two, auto)
```
```  1675
```
```  1676 lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
```
```  1677 apply (cut_tac pi_less_4)
```
```  1678 apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
```
```  1679 apply (cut_tac cos_is_zero, safe)
```
```  1680 apply (rename_tac y z)
```
```  1681 apply (drule_tac x = y in spec)
```
```  1682 apply (drule_tac x = "pi/2" in spec, simp)
```
```  1683 done
```
```  1684
```
```  1685 lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
```
```  1686 apply (rule_tac x = x and y = 0 in linorder_cases)
```
```  1687 apply (rule cos_minus [THEN subst])
```
```  1688 apply (rule cos_gt_zero)
```
```  1689 apply (auto intro: cos_gt_zero)
```
```  1690 done
```
```  1691
```
```  1692 lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
```
```  1693 apply (auto simp add: order_le_less cos_gt_zero_pi)
```
```  1694 apply (subgoal_tac "x = pi/2", auto)
```
```  1695 done
```
```  1696
```
```  1697 lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
```
```  1698 apply (subst sin_cos_eq)
```
```  1699 apply (rotate_tac 1)
```
```  1700 apply (drule real_sum_of_halves [THEN ssubst])
```
```  1701 apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
```
```  1702 done
```
```  1703
```
```  1704
```
```  1705 lemma pi_ge_two: "2 \<le> pi"
```
```  1706 proof (rule ccontr)
```
```  1707   assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
```
```  1708   have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
```
```  1709   proof (cases "2 < 2 * pi")
```
```  1710     case True with dense[OF `pi < 2`] show ?thesis by auto
```
```  1711   next
```
```  1712     case False have "pi < 2 * pi" by auto
```
```  1713     from dense[OF this] and False show ?thesis by auto
```
```  1714   qed
```
```  1715   then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
```
```  1716   hence "0 < sin y" using sin_gt_zero by auto
```
```  1717   moreover
```
```  1718   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
```
```  1719   ultimately show False by auto
```
```  1720 qed
```
```  1721
```
```  1722 lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
```
```  1723 by (auto simp add: order_le_less sin_gt_zero_pi)
```
```  1724
```
```  1725 text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}.
```
```  1726   It should be possible to factor out some of the common parts. *}
```
```  1727
```
```  1728 lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
```
```  1729 proof (rule ex_ex1I)
```
```  1730   assume y: "-1 \<le> y" "y \<le> 1"
```
```  1731   show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
```
```  1732     by (rule IVT2, simp_all add: y)
```
```  1733 next
```
```  1734   fix a b
```
```  1735   assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
```
```  1736   assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
```
```  1737   have [simp]: "\<forall>x. cos differentiable x"
```
```  1738     unfolding differentiable_def by (auto intro: DERIV_cos)
```
```  1739   from a b show "a = b"
```
```  1740     apply (cut_tac less_linear [of a b], auto)
```
```  1741     apply (drule_tac f = cos in Rolle)
```
```  1742     apply (drule_tac [5] f = cos in Rolle)
```
```  1743     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  1744     apply (metis order_less_le_trans less_le sin_gt_zero_pi)
```
```  1745     apply (metis order_less_le_trans less_le sin_gt_zero_pi)
```
```  1746     done
```
```  1747 qed
```
```  1748
```
```  1749 lemma sin_total:
```
```  1750      "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
```
```  1751 apply (rule ccontr)
```
```  1752 apply (subgoal_tac "\<forall>x. (- (pi/2) \<le> x & x \<le> pi/2 & (sin x = y)) = (0 \<le> (x + pi/2) & (x + pi/2) \<le> pi & (cos (x + pi/2) = -y))")
```
```  1753 apply (erule contrapos_np)
```
```  1754 apply (simp del: minus_sin_cos_eq [symmetric])
```
```  1755 apply (cut_tac y="-y" in cos_total, simp) apply simp
```
```  1756 apply (erule ex1E)
```
```  1757 apply (rule_tac a = "x - (pi/2)" in ex1I)
```
```  1758 apply (simp (no_asm) add: add_assoc)
```
```  1759 apply (rotate_tac 3)
```
```  1760 apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all)
```
```  1761 done
```
```  1762
```
```  1763 lemma reals_Archimedean4:
```
```  1764      "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
```
```  1765 apply (auto dest!: reals_Archimedean3)
```
```  1766 apply (drule_tac x = x in spec, clarify)
```
```  1767 apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
```
```  1768  prefer 2 apply (erule LeastI)
```
```  1769 apply (case_tac "LEAST m::nat. x < real m * y", simp)
```
```  1770 apply (subgoal_tac "~ x < real nat * y")
```
```  1771  prefer 2 apply (rule not_less_Least, simp, force)
```
```  1772 done
```
```  1773
```
```  1774 (* Pre Isabelle99-2 proof was simpler- numerals arithmetic
```
```  1775    now causes some unwanted re-arrangements of literals!   *)
```
```  1776 lemma cos_zero_lemma:
```
```  1777      "[| 0 \<le> x; cos x = 0 |] ==>
```
```  1778       \<exists>n::nat. ~even n & x = real n * (pi/2)"
```
```  1779 apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
```
```  1780 apply (subgoal_tac "0 \<le> x - real n * pi &
```
```  1781                     (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
```
```  1782 apply (auto simp add: algebra_simps real_of_nat_Suc)
```
```  1783  prefer 2 apply (simp add: cos_diff)
```
```  1784 apply (simp add: cos_diff)
```
```  1785 apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
```
```  1786 apply (rule_tac [2] cos_total, safe)
```
```  1787 apply (drule_tac x = "x - real n * pi" in spec)
```
```  1788 apply (drule_tac x = "pi/2" in spec)
```
```  1789 apply (simp add: cos_diff)
```
```  1790 apply (rule_tac x = "Suc (2 * n)" in exI)
```
```  1791 apply (simp add: real_of_nat_Suc algebra_simps, auto)
```
```  1792 done
```
```  1793
```
```  1794 lemma sin_zero_lemma:
```
```  1795      "[| 0 \<le> x; sin x = 0 |] ==>
```
```  1796       \<exists>n::nat. even n & x = real n * (pi/2)"
```
```  1797 apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
```
```  1798  apply (clarify, rule_tac x = "n - 1" in exI)
```
```  1799  apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
```
```  1800 apply (rule cos_zero_lemma)
```
```  1801 apply (simp_all add: add_increasing)
```
```  1802 done
```
```  1803
```
```  1804
```
```  1805 lemma cos_zero_iff:
```
```  1806      "(cos x = 0) =
```
```  1807       ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
```
```  1808        (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
```
```  1809 apply (rule iffI)
```
```  1810 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  1811 apply (drule cos_zero_lemma, assumption+)
```
```  1812 apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
```
```  1813 apply (force simp add: minus_equation_iff [of x])
```
```  1814 apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
```
```  1815 apply (auto simp add: cos_add)
```
```  1816 done
```
```  1817
```
```  1818 (* ditto: but to a lesser extent *)
```
```  1819 lemma sin_zero_iff:
```
```  1820      "(sin x = 0) =
```
```  1821       ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
```
```  1822        (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
```
```  1823 apply (rule iffI)
```
```  1824 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  1825 apply (drule sin_zero_lemma, assumption+)
```
```  1826 apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
```
```  1827 apply (force simp add: minus_equation_iff [of x])
```
```  1828 apply (auto simp add: even_mult_two_ex)
```
```  1829 done
```
```  1830
```
```  1831 lemma cos_monotone_0_pi: assumes "0 \<le> y" and "y < x" and "x \<le> pi"
```
```  1832   shows "cos x < cos y"
```
```  1833 proof -
```
```  1834   have "- (x - y) < 0" using assms by auto
```
```  1835
```
```  1836   from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
```
```  1837   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" by auto
```
```  1838   hence "0 < z" and "z < pi" using assms by auto
```
```  1839   hence "0 < sin z" using sin_gt_zero_pi by auto
```
```  1840   hence "cos x - cos y < 0" unfolding cos_diff minus_mult_commute[symmetric] using `- (x - y) < 0` by (rule mult_pos_neg2)
```
```  1841   thus ?thesis by auto
```
```  1842 qed
```
```  1843
```
```  1844 lemma cos_monotone_0_pi': assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" shows "cos x \<le> cos y"
```
```  1845 proof (cases "y < x")
```
```  1846   case True show ?thesis using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
```
```  1847 next
```
```  1848   case False hence "y = x" using `y \<le> x` by auto
```
```  1849   thus ?thesis by auto
```
```  1850 qed
```
```  1851
```
```  1852 lemma cos_monotone_minus_pi_0: assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
```
```  1853   shows "cos y < cos x"
```
```  1854 proof -
```
```  1855   have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" using assms by auto
```
```  1856   from cos_monotone_0_pi[OF this]
```
```  1857   show ?thesis unfolding cos_minus .
```
```  1858 qed
```
```  1859
```
```  1860 lemma cos_monotone_minus_pi_0': assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0" shows "cos y \<le> cos x"
```
```  1861 proof (cases "y < x")
```
```  1862   case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] by auto
```
```  1863 next
```
```  1864   case False hence "y = x" using `y \<le> x` by auto
```
```  1865   thus ?thesis by auto
```
```  1866 qed
```
```  1867
```
```  1868 lemma sin_monotone_2pi': assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2" shows "sin y \<le> sin x"
```
```  1869 proof -
```
```  1870   have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi"
```
```  1871     using pi_ge_two and assms by auto
```
```  1872   from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] by auto
```
```  1873 qed
```
```  1874
```
```  1875 subsection {* Tangent *}
```
```  1876
```
```  1877 definition tan :: "real \<Rightarrow> real" where
```
```  1878   "tan = (\<lambda>x. sin x / cos x)"
```
```  1879
```
```  1880 lemma tan_zero [simp]: "tan 0 = 0"
```
```  1881   by (simp add: tan_def)
```
```  1882
```
```  1883 lemma tan_pi [simp]: "tan pi = 0"
```
```  1884   by (simp add: tan_def)
```
```  1885
```
```  1886 lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
```
```  1887   by (simp add: tan_def)
```
```  1888
```
```  1889 lemma tan_minus [simp]: "tan (-x) = - tan x"
```
```  1890   by (simp add: tan_def)
```
```  1891
```
```  1892 lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
```
```  1893   by (simp add: tan_def)
```
```  1894
```
```  1895 lemma lemma_tan_add1:
```
```  1896   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
```
```  1897   by (simp add: tan_def cos_add field_simps)
```
```  1898
```
```  1899 lemma add_tan_eq:
```
```  1900   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
```
```  1901   by (simp add: tan_def sin_add field_simps)
```
```  1902
```
```  1903 lemma tan_add:
```
```  1904      "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
```
```  1905       ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
```
```  1906   by (simp add: add_tan_eq lemma_tan_add1, simp add: tan_def)
```
```  1907
```
```  1908 lemma tan_double:
```
```  1909      "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
```
```  1910       ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
```
```  1911   using tan_add [of x x] by (simp add: power2_eq_square)
```
```  1912
```
```  1913 lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
```
```  1914 by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
```
```  1915
```
```  1916 lemma tan_less_zero:
```
```  1917   assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
```
```  1918 proof -
```
```  1919   have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
```
```  1920   thus ?thesis by simp
```
```  1921 qed
```
```  1922
```
```  1923 lemma tan_half: fixes x :: real assumes "- (pi / 2) < x" and "x < pi / 2"
```
```  1924   shows "tan x = sin (2 * x) / (cos (2 * x) + 1)"
```
```  1925 proof -
```
```  1926   from cos_gt_zero_pi[OF `- (pi / 2) < x` `x < pi / 2`]
```
```  1927   have "cos x \<noteq> 0" by auto
```
```  1928
```
```  1929   have minus_cos_2x: "\<And>X. X - cos (2*x) = X - (cos x) ^ 2 + (sin x) ^ 2" unfolding cos_double by algebra
```
```  1930
```
```  1931   have "tan x = (tan x + tan x) / 2" by auto
```
```  1932   also have "\<dots> = sin (x + x) / (cos x * cos x) / 2" unfolding add_tan_eq[OF `cos x \<noteq> 0` `cos x \<noteq> 0`] ..
```
```  1933   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
```
```  1934   also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + cos (2*x) + (sin x)^2)" unfolding minus_cos_2x by auto
```
```  1935   also have "\<dots> = sin (2 * x) / (cos (2*x) + 1)" by auto
```
```  1936   finally show ?thesis .
```
```  1937 qed
```
```  1938
```
```  1939 lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<twosuperior>)"
```
```  1940   unfolding tan_def
```
```  1941   by (auto intro!: DERIV_intros, simp add: divide_inverse power2_eq_square)
```
```  1942
```
```  1943 lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
```
```  1944   by (rule DERIV_tan [THEN DERIV_isCont])
```
```  1945
```
```  1946 lemma isCont_tan' [simp]:
```
```  1947   "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
```
```  1948   by (rule isCont_o2 [OF _ isCont_tan])
```
```  1949
```
```  1950 lemma tendsto_tan [tendsto_intros]:
```
```  1951   "\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F"
```
```  1952   by (rule isCont_tendsto_compose [OF isCont_tan])
```
```  1953
```
```  1954 lemma LIM_cos_div_sin: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
```
```  1955   by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
```
```  1956
```
```  1957 lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
```
```  1958 apply (cut_tac LIM_cos_div_sin)
```
```  1959 apply (simp only: LIM_eq)
```
```  1960 apply (drule_tac x = "inverse y" in spec, safe, force)
```
```  1961 apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
```
```  1962 apply (rule_tac x = "(pi/2) - e" in exI)
```
```  1963 apply (simp (no_asm_simp))
```
```  1964 apply (drule_tac x = "(pi/2) - e" in spec)
```
```  1965 apply (auto simp add: tan_def)
```
```  1966 apply (rule inverse_less_iff_less [THEN iffD1])
```
```  1967 apply (auto simp add: divide_inverse)
```
```  1968 apply (rule mult_pos_pos)
```
```  1969 apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
```
```  1970 apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
```
```  1971 done
```
```  1972
```
```  1973 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
```
```  1974 apply (frule order_le_imp_less_or_eq, safe)
```
```  1975  prefer 2 apply force
```
```  1976 apply (drule lemma_tan_total, safe)
```
```  1977 apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
```
```  1978 apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
```
```  1979 apply (drule_tac y = xa in order_le_imp_less_or_eq)
```
```  1980 apply (auto dest: cos_gt_zero)
```
```  1981 done
```
```  1982
```
```  1983 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  1984 apply (cut_tac linorder_linear [of 0 y], safe)
```
```  1985 apply (drule tan_total_pos)
```
```  1986 apply (cut_tac [2] y="-y" in tan_total_pos, safe)
```
```  1987 apply (rule_tac [3] x = "-x" in exI)
```
```  1988 apply (auto del: exI intro!: exI)
```
```  1989 done
```
```  1990
```
```  1991 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  1992 apply (cut_tac y = y in lemma_tan_total1, auto)
```
```  1993 apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
```
```  1994 apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
```
```  1995 apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
```
```  1996 apply (rule_tac [4] Rolle)
```
```  1997 apply (rule_tac [2] Rolle)
```
```  1998 apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
```
```  1999             simp add: differentiable_def)
```
```  2000 txt{*Now, simulate TRYALL*}
```
```  2001 apply (rule_tac [!] DERIV_tan asm_rl)
```
```  2002 apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
```
```  2003             simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
```
```  2004 done
```
```  2005
```
```  2006 lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
```
```  2007   shows "tan y < tan x"
```
```  2008 proof -
```
```  2009   have "\<forall> x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse (cos x'^2)"
```
```  2010   proof (rule allI, rule impI)
```
```  2011     fix x' :: real assume "y \<le> x' \<and> x' \<le> x"
```
```  2012     hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
```
```  2013     from cos_gt_zero_pi[OF this]
```
```  2014     have "cos x' \<noteq> 0" by auto
```
```  2015     thus "DERIV tan x' :> inverse (cos x'^2)" by (rule DERIV_tan)
```
```  2016   qed
```
```  2017   from MVT2[OF `y < x` this]
```
```  2018   obtain z where "y < z" and "z < x" and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<twosuperior>)" by auto
```
```  2019   hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
```
```  2020   hence "0 < cos z" using cos_gt_zero_pi by auto
```
```  2021   hence inv_pos: "0 < inverse ((cos z)\<twosuperior>)" by auto
```
```  2022   have "0 < x - y" using `y < x` by auto
```
```  2023   from mult_pos_pos [OF this inv_pos]
```
```  2024   have "0 < tan x - tan y" unfolding tan_diff by auto
```
```  2025   thus ?thesis by auto
```
```  2026 qed
```
```  2027
```
```  2028 lemma tan_monotone': assumes "- (pi / 2) < y" and "y < pi / 2" and "- (pi / 2) < x" and "x < pi / 2"
```
```  2029   shows "(y < x) = (tan y < tan x)"
```
```  2030 proof
```
```  2031   assume "y < x" thus "tan y < tan x" using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
```
```  2032 next
```
```  2033   assume "tan y < tan x"
```
```  2034   show "y < x"
```
```  2035   proof (rule ccontr)
```
```  2036     assume "\<not> y < x" hence "x \<le> y" by auto
```
```  2037     hence "tan x \<le> tan y"
```
```  2038     proof (cases "x = y")
```
```  2039       case True thus ?thesis by auto
```
```  2040     next
```
```  2041       case False hence "x < y" using `x \<le> y` by auto
```
```  2042       from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
```
```  2043     qed
```
```  2044     thus False using `tan y < tan x` by auto
```
```  2045   qed
```
```  2046 qed
```
```  2047
```
```  2048 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
```
```  2049
```
```  2050 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
```
```  2051   by (simp add: tan_def)
```
```  2052
```
```  2053 lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x"
```
```  2054 proof (induct n arbitrary: x)
```
```  2055   case (Suc n)
```
```  2056   have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" unfolding Suc_eq_plus1 real_of_nat_add real_of_one left_distrib by auto
```
```  2057   show ?case unfolding split_pi_off using Suc by auto
```
```  2058 qed auto
```
```  2059
```
```  2060 lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
```
```  2061 proof (cases "0 \<le> i")
```
```  2062   case True hence i_nat: "real i = real (nat i)" by auto
```
```  2063   show ?thesis unfolding i_nat by auto
```
```  2064 next
```
```  2065   case False hence i_nat: "real i = - real (nat (-i))" by auto
```
```  2066   have "tan x = tan (x + real i * pi - real i * pi)" by auto
```
```  2067   also have "\<dots> = tan (x + real i * pi)" unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
```
```  2068   finally show ?thesis by auto
```
```  2069 qed
```
```  2070
```
```  2071 lemma tan_periodic_n[simp]: "tan (x + number_of n * pi) = tan x"
```
```  2072   using tan_periodic_int[of _ "number_of n" ] unfolding real_number_of .
```
```  2073
```
```  2074 subsection {* Inverse Trigonometric Functions *}
```
```  2075
```
```  2076 definition
```
```  2077   arcsin :: "real => real" where
```
```  2078   "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
```
```  2079
```
```  2080 definition
```
```  2081   arccos :: "real => real" where
```
```  2082   "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
```
```  2083
```
```  2084 definition
```
```  2085   arctan :: "real => real" where
```
```  2086   "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
```
```  2087
```
```  2088 lemma arcsin:
```
```  2089      "[| -1 \<le> y; y \<le> 1 |]
```
```  2090       ==> -(pi/2) \<le> arcsin y &
```
```  2091            arcsin y \<le> pi/2 & sin(arcsin y) = y"
```
```  2092 unfolding arcsin_def by (rule theI' [OF sin_total])
```
```  2093
```
```  2094 lemma arcsin_pi:
```
```  2095      "[| -1 \<le> y; y \<le> 1 |]
```
```  2096       ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
```
```  2097 apply (drule (1) arcsin)
```
```  2098 apply (force intro: order_trans)
```
```  2099 done
```
```  2100
```
```  2101 lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
```
```  2102 by (blast dest: arcsin)
```
```  2103
```
```  2104 lemma arcsin_bounded:
```
```  2105      "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
```
```  2106 by (blast dest: arcsin)
```
```  2107
```
```  2108 lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
```
```  2109 by (blast dest: arcsin)
```
```  2110
```
```  2111 lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
```
```  2112 by (blast dest: arcsin)
```
```  2113
```
```  2114 lemma arcsin_lt_bounded:
```
```  2115      "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
```
```  2116 apply (frule order_less_imp_le)
```
```  2117 apply (frule_tac y = y in order_less_imp_le)
```
```  2118 apply (frule arcsin_bounded)
```
```  2119 apply (safe, simp)
```
```  2120 apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
```
```  2121 apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
```
```  2122 apply (drule_tac [!] f = sin in arg_cong, auto)
```
```  2123 done
```
```  2124
```
```  2125 lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
```
```  2126 apply (unfold arcsin_def)
```
```  2127 apply (rule the1_equality)
```
```  2128 apply (rule sin_total, auto)
```
```  2129 done
```
```  2130
```
```  2131 lemma arccos:
```
```  2132      "[| -1 \<le> y; y \<le> 1 |]
```
```  2133       ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
```
```  2134 unfolding arccos_def by (rule theI' [OF cos_total])
```
```  2135
```
```  2136 lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
```
```  2137 by (blast dest: arccos)
```
```  2138
```
```  2139 lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
```
```  2140 by (blast dest: arccos)
```
```  2141
```
```  2142 lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
```
```  2143 by (blast dest: arccos)
```
```  2144
```
```  2145 lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
```
```  2146 by (blast dest: arccos)
```
```  2147
```
```  2148 lemma arccos_lt_bounded:
```
```  2149      "[| -1 < y; y < 1 |]
```
```  2150       ==> 0 < arccos y & arccos y < pi"
```
```  2151 apply (frule order_less_imp_le)
```
```  2152 apply (frule_tac y = y in order_less_imp_le)
```
```  2153 apply (frule arccos_bounded, auto)
```
```  2154 apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
```
```  2155 apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
```
```  2156 apply (drule_tac [!] f = cos in arg_cong, auto)
```
```  2157 done
```
```  2158
```
```  2159 lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
```
```  2160 apply (simp add: arccos_def)
```
```  2161 apply (auto intro!: the1_equality cos_total)
```
```  2162 done
```
```  2163
```
```  2164 lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
```
```  2165 apply (simp add: arccos_def)
```
```  2166 apply (auto intro!: the1_equality cos_total)
```
```  2167 done
```
```  2168
```
```  2169 lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<twosuperior>)"
```
```  2170 apply (subgoal_tac "x\<twosuperior> \<le> 1")
```
```  2171 apply (rule power2_eq_imp_eq)
```
```  2172 apply (simp add: cos_squared_eq)
```
```  2173 apply (rule cos_ge_zero)
```
```  2174 apply (erule (1) arcsin_lbound)
```
```  2175 apply (erule (1) arcsin_ubound)
```
```  2176 apply simp
```
```  2177 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
```
```  2178 apply (rule power_mono, simp, simp)
```
```  2179 done
```
```  2180
```
```  2181 lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<twosuperior>)"
```
```  2182 apply (subgoal_tac "x\<twosuperior> \<le> 1")
```
```  2183 apply (rule power2_eq_imp_eq)
```
```  2184 apply (simp add: sin_squared_eq)
```
```  2185 apply (rule sin_ge_zero)
```
```  2186 apply (erule (1) arccos_lbound)
```
```  2187 apply (erule (1) arccos_ubound)
```
```  2188 apply simp
```
```  2189 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
```
```  2190 apply (rule power_mono, simp, simp)
```
```  2191 done
```
```  2192
```
```  2193 lemma arctan [simp]:
```
```  2194      "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
```
```  2195 unfolding arctan_def by (rule theI' [OF tan_total])
```
```  2196
```
```  2197 lemma tan_arctan: "tan(arctan y) = y"
```
```  2198 by auto
```
```  2199
```
```  2200 lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
```
```  2201 by (auto simp only: arctan)
```
```  2202
```
```  2203 lemma arctan_lbound: "- (pi/2) < arctan y"
```
```  2204 by auto
```
```  2205
```
```  2206 lemma arctan_ubound: "arctan y < pi/2"
```
```  2207 by (auto simp only: arctan)
```
```  2208
```
```  2209 lemma arctan_tan:
```
```  2210       "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
```
```  2211 apply (unfold arctan_def)
```
```  2212 apply (rule the1_equality)
```
```  2213 apply (rule tan_total, auto)
```
```  2214 done
```
```  2215
```
```  2216 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
```
```  2217 by (insert arctan_tan [of 0], simp)
```
```  2218
```
```  2219 lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
```
```  2220   by (intro less_imp_neq [symmetric] cos_gt_zero_pi
```
```  2221     arctan_lbound arctan_ubound)
```
```  2222
```
```  2223 lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<twosuperior>)"
```
```  2224 proof (rule power2_eq_imp_eq)
```
```  2225   have "0 < 1 + x\<twosuperior>" by (simp add: add_pos_nonneg)
```
```  2226   show "0 \<le> 1 / sqrt (1 + x\<twosuperior>)" by simp
```
```  2227   show "0 \<le> cos (arctan x)"
```
```  2228     by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
```
```  2229   have "(cos (arctan x))\<twosuperior> * (1 + (tan (arctan x))\<twosuperior>) = 1"
```
```  2230     unfolding tan_def by (simp add: right_distrib power_divide)
```
```  2231   thus "(cos (arctan x))\<twosuperior> = (1 / sqrt (1 + x\<twosuperior>))\<twosuperior>"
```
```  2232     using `0 < 1 + x\<twosuperior>` by (simp add: power_divide eq_divide_eq)
```
```  2233 qed
```
```  2234
```
```  2235 lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<twosuperior>)"
```
```  2236   using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
```
```  2237   using tan_arctan [of x] unfolding tan_def cos_arctan
```
```  2238   by (simp add: eq_divide_eq)
```
```  2239
```
```  2240 lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
```
```  2241 apply (rule power_inverse [THEN subst])
```
```  2242 apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
```
```  2243 apply (auto dest: field_power_not_zero
```
```  2244         simp add: power_mult_distrib left_distrib power_divide tan_def
```
```  2245                   mult_assoc power_inverse [symmetric])
```
```  2246 done
```
```  2247
```
```  2248 lemma isCont_inverse_function2:
```
```  2249   fixes f g :: "real \<Rightarrow> real" shows
```
```  2250   "\<lbrakk>a < x; x < b;
```
```  2251     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
```
```  2252     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
```
```  2253    \<Longrightarrow> isCont g (f x)"
```
```  2254 apply (rule isCont_inverse_function
```
```  2255        [where f=f and d="min (x - a) (b - x)"])
```
```  2256 apply (simp_all add: abs_le_iff)
```
```  2257 done
```
```  2258
```
```  2259 lemma isCont_arcsin: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arcsin x"
```
```  2260 apply (subgoal_tac "isCont arcsin (sin (arcsin x))", simp)
```
```  2261 apply (rule isCont_inverse_function2 [where f=sin])
```
```  2262 apply (erule (1) arcsin_lt_bounded [THEN conjunct1])
```
```  2263 apply (erule (1) arcsin_lt_bounded [THEN conjunct2])
```
```  2264 apply (fast intro: arcsin_sin, simp)
```
```  2265 done
```
```  2266
```
```  2267 lemma isCont_arccos: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arccos x"
```
```  2268 apply (subgoal_tac "isCont arccos (cos (arccos x))", simp)
```
```  2269 apply (rule isCont_inverse_function2 [where f=cos])
```
```  2270 apply (erule (1) arccos_lt_bounded [THEN conjunct1])
```
```  2271 apply (erule (1) arccos_lt_bounded [THEN conjunct2])
```
```  2272 apply (fast intro: arccos_cos, simp)
```
```  2273 done
```
```  2274
```
```  2275 lemma isCont_arctan: "isCont arctan x"
```
```  2276 apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
```
```  2277 apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
```
```  2278 apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
```
```  2279 apply (erule (1) isCont_inverse_function2 [where f=tan])
```
```  2280 apply (metis arctan_tan order_le_less_trans order_less_le_trans)
```
```  2281 apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
```
```  2282 done
```
```  2283
```
```  2284 lemma DERIV_arcsin:
```
```  2285   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<twosuperior>))"
```
```  2286 apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
```
```  2287 apply (rule DERIV_cong [OF DERIV_sin])
```
```  2288 apply (simp add: cos_arcsin)
```
```  2289 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
```
```  2290 apply (rule power_strict_mono, simp, simp, simp)
```
```  2291 apply assumption
```
```  2292 apply assumption
```
```  2293 apply simp
```
```  2294 apply (erule (1) isCont_arcsin)
```
```  2295 done
```
```  2296
```
```  2297 lemma DERIV_arccos:
```
```  2298   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<twosuperior>))"
```
```  2299 apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
```
```  2300 apply (rule DERIV_cong [OF DERIV_cos])
```
```  2301 apply (simp add: sin_arccos)
```
```  2302 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
```
```  2303 apply (rule power_strict_mono, simp, simp, simp)
```
```  2304 apply assumption
```
```  2305 apply assumption
```
```  2306 apply simp
```
```  2307 apply (erule (1) isCont_arccos)
```
```  2308 done
```
```  2309
```
```  2310 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<twosuperior>)"
```
```  2311 apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
```
```  2312 apply (rule DERIV_cong [OF DERIV_tan])
```
```  2313 apply (rule cos_arctan_not_zero)
```
```  2314 apply (simp add: power_inverse tan_sec [symmetric])
```
```  2315 apply (subgoal_tac "0 < 1 + x\<twosuperior>", simp)
```
```  2316 apply (simp add: add_pos_nonneg)
```
```  2317 apply (simp, simp, simp, rule isCont_arctan)
```
```  2318 done
```
```  2319
```
```  2320 declare
```
```  2321   DERIV_arcsin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  2322   DERIV_arccos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  2323   DERIV_arctan[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  2324
```
```  2325 subsection {* More Theorems about Sin and Cos *}
```
```  2326
```
```  2327 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
```
```  2328 proof -
```
```  2329   let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
```
```  2330   have nonneg: "0 \<le> ?c"
```
```  2331     by (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
```
```  2332   have "0 = cos (pi / 4 + pi / 4)"
```
```  2333     by simp
```
```  2334   also have "cos (pi / 4 + pi / 4) = ?c\<twosuperior> - ?s\<twosuperior>"
```
```  2335     by (simp only: cos_add power2_eq_square)
```
```  2336   also have "\<dots> = 2 * ?c\<twosuperior> - 1"
```
```  2337     by (simp add: sin_squared_eq)
```
```  2338   finally have "?c\<twosuperior> = (sqrt 2 / 2)\<twosuperior>"
```
```  2339     by (simp add: power_divide)
```
```  2340   thus ?thesis
```
```  2341     using nonneg by (rule power2_eq_imp_eq) simp
```
```  2342 qed
```
```  2343
```
```  2344 lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"
```
```  2345 proof -
```
```  2346   let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
```
```  2347   have pos_c: "0 < ?c"
```
```  2348     by (rule cos_gt_zero, simp, simp)
```
```  2349   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
```
```  2350     by simp
```
```  2351   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
```
```  2352     by (simp only: cos_add sin_add)
```
```  2353   also have "\<dots> = ?c * (?c\<twosuperior> - 3 * ?s\<twosuperior>)"
```
```  2354     by (simp add: algebra_simps power2_eq_square)
```
```  2355   finally have "?c\<twosuperior> = (sqrt 3 / 2)\<twosuperior>"
```
```  2356     using pos_c by (simp add: sin_squared_eq power_divide)
```
```  2357   thus ?thesis
```
```  2358     using pos_c [THEN order_less_imp_le]
```
```  2359     by (rule power2_eq_imp_eq) simp
```
```  2360 qed
```
```  2361
```
```  2362 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
```
```  2363 proof -
```
```  2364   have "sin (pi / 4) = cos (pi / 2 - pi / 4)" by (rule sin_cos_eq)
```
```  2365   also have "pi / 2 - pi / 4 = pi / 4" by simp
```
```  2366   also have "cos (pi / 4) = sqrt 2 / 2" by (rule cos_45)
```
```  2367   finally show ?thesis .
```
```  2368 qed
```
```  2369
```
```  2370 lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
```
```  2371 proof -
```
```  2372   have "sin (pi / 3) = cos (pi / 2 - pi / 3)" by (rule sin_cos_eq)
```
```  2373   also have "pi / 2 - pi / 3 = pi / 6" by simp
```
```  2374   also have "cos (pi / 6) = sqrt 3 / 2" by (rule cos_30)
```
```  2375   finally show ?thesis .
```
```  2376 qed
```
```  2377
```
```  2378 lemma cos_60: "cos (pi / 3) = 1 / 2"
```
```  2379 apply (rule power2_eq_imp_eq)
```
```  2380 apply (simp add: cos_squared_eq sin_60 power_divide)
```
```  2381 apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
```
```  2382 done
```
```  2383
```
```  2384 lemma sin_30: "sin (pi / 6) = 1 / 2"
```
```  2385 proof -
```
```  2386   have "sin (pi / 6) = cos (pi / 2 - pi / 6)" by (rule sin_cos_eq)
```
```  2387   also have "pi / 2 - pi / 6 = pi / 3" by simp
```
```  2388   also have "cos (pi / 3) = 1 / 2" by (rule cos_60)
```
```  2389   finally show ?thesis .
```
```  2390 qed
```
```  2391
```
```  2392 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
```
```  2393 unfolding tan_def by (simp add: sin_30 cos_30)
```
```  2394
```
```  2395 lemma tan_45: "tan (pi / 4) = 1"
```
```  2396 unfolding tan_def by (simp add: sin_45 cos_45)
```
```  2397
```
```  2398 lemma tan_60: "tan (pi / 3) = sqrt 3"
```
```  2399 unfolding tan_def by (simp add: sin_60 cos_60)
```
```  2400
```
```  2401 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
```
```  2402 proof -
```
```  2403   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
```
```  2404     by (auto simp add: algebra_simps sin_add)
```
```  2405   thus ?thesis
```
```  2406     by (simp add: real_of_nat_Suc left_distrib add_divide_distrib
```
```  2407                   mult_commute [of pi])
```
```  2408 qed
```
```  2409
```
```  2410 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
```
```  2411 by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
```
```  2412
```
```  2413 lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
```
```  2414 apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
```
```  2415 apply (subst cos_add, simp)
```
```  2416 done
```
```  2417
```
```  2418 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
```
```  2419 by (auto simp add: mult_assoc)
```
```  2420
```
```  2421 lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
```
```  2422 apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
```
```  2423 apply (subst sin_add, simp)
```
```  2424 done
```
```  2425
```
```  2426 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
```
```  2427 by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto)
```
```  2428
```
```  2429 lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
```
```  2430   by (auto intro!: DERIV_intros)
```
```  2431
```
```  2432 lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
```
```  2433 by (auto simp add: sin_zero_iff even_mult_two_ex)
```
```  2434
```
```  2435 lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
```
```  2436 by (cut_tac x = x in sin_cos_squared_add3, auto)
```
```  2437
```
```  2438 subsection {* Machins formula *}
```
```  2439
```
```  2440 lemma tan_total_pi4: assumes "\<bar>x\<bar> < 1"
```
```  2441   shows "\<exists> z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
```
```  2442 proof -
```
```  2443   obtain z where "- (pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast
```
```  2444   have "tan (pi / 4) = 1" and "tan (- (pi / 4)) = - 1" using tan_45 tan_minus by auto
```
```  2445   have "z \<noteq> pi / 4"
```
```  2446   proof (rule ccontr)
```
```  2447     assume "\<not> (z \<noteq> pi / 4)" hence "z = pi / 4" by auto
```
```  2448     have "tan z = 1" unfolding `z = pi / 4` `tan (pi / 4) = 1` ..
```
```  2449     thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto
```
```  2450   qed
```
```  2451   have "z \<noteq> - (pi / 4)"
```
```  2452   proof (rule ccontr)
```
```  2453     assume "\<not> (z \<noteq> - (pi / 4))" hence "z = - (pi / 4)" by auto
```
```  2454     have "tan z = - 1" unfolding `z = - (pi / 4)` `tan (- (pi / 4)) = - 1` ..
```
```  2455     thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto
```
```  2456   qed
```
```  2457
```
```  2458   have "z < pi / 4"
```
```  2459   proof (rule ccontr)
```
```  2460     assume "\<not> (z < pi / 4)" hence "pi / 4 < z" using `z \<noteq> pi / 4` by auto
```
```  2461     have "- (pi / 2) < pi / 4" using m2pi_less_pi by auto
```
```  2462     from tan_monotone[OF this `pi / 4 < z` `z < pi / 2`]
```
```  2463     have "1 < x" unfolding `tan z = x` `tan (pi / 4) = 1` .
```
```  2464     thus False using `\<bar>x\<bar> < 1` by auto
```
```  2465   qed
```
```  2466   moreover
```
```  2467   have "-(pi / 4) < z"
```
```  2468   proof (rule ccontr)
```
```  2469     assume "\<not> (-(pi / 4) < z)" hence "z < - (pi / 4)" using `z \<noteq> - (pi / 4)` by auto
```
```  2470     have "-(pi / 4) < pi / 2" using m2pi_less_pi by auto
```
```  2471     from tan_monotone[OF `-(pi / 2) < z` `z < -(pi / 4)` this]
```
```  2472     have "x < - 1" unfolding `tan z = x` `tan (-(pi / 4)) = - 1` .
```
```  2473     thus False using `\<bar>x\<bar> < 1` by auto
```
```  2474   qed
```
```  2475   ultimately show ?thesis using `tan z = x` by auto
```
```  2476 qed
```
```  2477
```
```  2478 lemma arctan_add: assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
```
```  2479   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
```
```  2480 proof -
```
```  2481   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
```
```  2482
```
```  2483   have "pi / 4 < pi / 2" by auto
```
```  2484
```
```  2485   have "\<exists> x'. -(pi/4) \<le> x' \<and> x' \<le> pi/4 \<and> tan x' = x"
```
```  2486   proof (cases "\<bar>x\<bar> < 1")
```
```  2487     case True from tan_total_pi4[OF this] obtain x' where "-(pi/4) < x'" and "x' < pi/4" and "tan x' = x" by blast
```
```  2488     hence "-(pi/4) \<le> x'" and "x' \<le> pi/4" and "tan x' = x" by auto
```
```  2489     thus ?thesis by auto
```
```  2490   next
```
```  2491     case False
```
```  2492     show ?thesis
```
```  2493     proof (cases "x = 1")
```
```  2494       case True hence "tan (pi/4) = x" using tan_45 by auto
```
```  2495       moreover
```
```  2496       have "- pi \<le> pi" unfolding minus_le_self_iff by auto
```
```  2497       hence "-(pi/4) \<le> pi/4" and "pi/4 \<le> pi/4" by auto
```
```  2498       ultimately show ?thesis by blast
```
```  2499     next
```
```  2500       case False hence "x = -1" using `\<not> \<bar>x\<bar> < 1` and `\<bar>x\<bar> \<le> 1` by auto
```
```  2501       hence "tan (-(pi/4)) = x" using tan_45 tan_minus by auto
```
```  2502       moreover
```
```  2503       have "- pi \<le> pi" unfolding minus_le_self_iff by auto
```
```  2504       hence "-(pi/4) \<le> pi/4" and "-(pi/4) \<le> -(pi/4)" by auto
```
```  2505       ultimately show ?thesis by blast
```
```  2506     qed
```
```  2507   qed
```
```  2508   then obtain x' where "-(pi/4) \<le> x'" and "x' \<le> pi/4" and "tan x' = x" by blast
```
```  2509   hence "-(pi/2) < x'" and "x' < pi/2" using order_le_less_trans[OF `x' \<le> pi/4` `pi / 4 < pi / 2`] by auto
```
```  2510
```
```  2511   have "cos x' \<noteq> 0" using cos_gt_zero_pi[THEN less_imp_neq] and `-(pi/2) < x'` and `x' < pi/2` by auto
```
```  2512   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
```
```  2513   ultimately have "cos x' * cos y' \<noteq> 0" by auto
```
```  2514
```
```  2515   have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> (A / C) / (B / C) = A / B" by auto
```
```  2516   have divide_mult_commute: "\<And> A B C D :: real. A * B / (C * D) = (A / C) * (B / D)" by auto
```
```  2517
```
```  2518   have "tan (x' + y') = sin (x' + y') / (cos x' * cos y' - sin x' * sin y')" unfolding tan_def cos_add ..
```
```  2519   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`] ..
```
```  2520   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 ..
```
```  2521   finally have tan_eq: "tan (x' + y') = (x + y) / (1 - x * y)" unfolding `tan y' = y` `tan x' = x` .
```
```  2522
```
```  2523   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)
```
```  2524   moreover have "arctan (tan (x')) = x'" using `-(pi/2) < x'` and `x' < pi/2` by (auto intro!: arctan_tan)
```
```  2525   moreover have "arctan (tan (y')) = y'" using `-(pi/4) < y'` and `y' < pi/4` by (auto intro!: arctan_tan)
```
```  2526   ultimately have "arctan x + arctan y = arctan (tan (x' + y'))" unfolding `tan y' = y` [symmetric] `tan x' = x`[symmetric] by auto
```
```  2527   thus "arctan x + arctan y = arctan ((x + y) / (1 - x * y))" unfolding tan_eq .
```
```  2528 qed
```
```  2529
```
```  2530 lemma arctan1_eq_pi4: "arctan 1 = pi / 4" unfolding tan_45[symmetric] by (rule arctan_tan, auto simp add: m2pi_less_pi)
```
```  2531
```
```  2532 theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
```
```  2533 proof -
```
```  2534   have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
```
```  2535   from arctan_add[OF less_imp_le[OF this] this]
```
```  2536   have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
```
```  2537   moreover
```
```  2538   have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
```
```  2539   from arctan_add[OF less_imp_le[OF this] this]
```
```  2540   have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
```
```  2541   moreover
```
```  2542   have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
```
```  2543   from arctan_add[OF this]
```
```  2544   have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
```
```  2545   ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
```
```  2546   thus ?thesis unfolding arctan1_eq_pi4 by algebra
```
```  2547 qed
```
```  2548 subsection {* Introducing the arcus tangens power series *}
```
```  2549
```
```  2550 lemma monoseq_arctan_series: fixes x :: real
```
```  2551   assumes "\<bar>x\<bar> \<le> 1" shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
```
```  2552 proof (cases "x = 0") case True thus ?thesis unfolding monoseq_def One_nat_def by auto
```
```  2553 next
```
```  2554   case False
```
```  2555   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  2556   show "monoseq ?a"
```
```  2557   proof -
```
```  2558     { fix n fix x :: real assume "0 \<le> x" and "x \<le> 1"
```
```  2559       have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le> 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
```
```  2560       proof (rule mult_mono)
```
```  2561         show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))" by (rule frac_le) simp_all
```
```  2562         show "0 \<le> 1 / real (Suc (n * 2))" by auto
```
```  2563         show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)" by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`)
```
```  2564         show "0 \<le> x ^ Suc (Suc n * 2)" by (rule zero_le_power) (simp add: `0 \<le> x`)
```
```  2565       qed
```
```  2566     } note mono = this
```
```  2567
```
```  2568     show ?thesis
```
```  2569     proof (cases "0 \<le> x")
```
```  2570       case True from mono[OF this `x \<le> 1`, THEN allI]
```
```  2571       show ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI2)
```
```  2572     next
```
```  2573       case False hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto
```
```  2574       from mono[OF this]
```
```  2575       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
```
```  2576       thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
```
```  2577     qed
```
```  2578   qed
```
```  2579 qed
```
```  2580
```
```  2581 lemma zeroseq_arctan_series: fixes x :: real
```
```  2582   assumes "\<bar>x\<bar> \<le> 1" shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
```
```  2583 proof (cases "x = 0") case True thus ?thesis unfolding One_nat_def by (auto simp add: tendsto_const)
```
```  2584 next
```
```  2585   case False
```
```  2586   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  2587   show "?a ----> 0"
```
```  2588   proof (cases "\<bar>x\<bar> < 1")
```
```  2589     case True hence "norm x < 1" by auto
```
```  2590     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]]
```
```  2591     have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"
```
```  2592       unfolding inverse_eq_divide Suc_eq_plus1 by simp
```
```  2593     then show ?thesis using pos2 by (rule LIMSEQ_linear)
```
```  2594   next
```
```  2595     case False hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  2596     hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x" unfolding One_nat_def by auto
```
```  2597     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
```
```  2598     show ?thesis unfolding n_eq Suc_eq_plus1 by auto
```
```  2599   qed
```
```  2600 qed
```
```  2601
```
```  2602 lemma summable_arctan_series: fixes x :: real and n :: nat
```
```  2603   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)")
```
```  2604   by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
```
```  2605
```
```  2606 lemma less_one_imp_sqr_less_one: fixes x :: real assumes "\<bar>x\<bar> < 1" shows "x^2 < 1"
```
```  2607 proof -
```
```  2608   from mult_left_mono[OF less_imp_le[OF `\<bar>x\<bar> < 1`] abs_ge_zero[of x]]
```
```  2609   have "\<bar> x^2 \<bar> < 1" using `\<bar> x \<bar> < 1` unfolding numeral_2_eq_2 power_Suc2 by auto
```
```  2610   thus ?thesis using zero_le_power2 by auto
```
```  2611 qed
```
```  2612
```
```  2613 lemma DERIV_arctan_series: assumes "\<bar> x \<bar> < 1"
```
```  2614   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")
```
```  2615 proof -
```
```  2616   let "?f n" = "if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
```
```  2617
```
```  2618   { fix n :: nat assume "even n" hence "2 * (n div 2) = n" by presburger } note n_even=this
```
```  2619   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
```
```  2620
```
```  2621   { fix x :: real assume "\<bar>x\<bar> < 1" hence "x^2 < 1" by (rule less_one_imp_sqr_less_one)
```
```  2622     have "summable (\<lambda> n. -1 ^ n * (x^2) ^n)"
```
```  2623       by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x^2 < 1` order_less_imp_le[OF `x^2 < 1`])
```
```  2624     hence "summable (\<lambda> n. -1 ^ n * x^(2*n))" unfolding power_mult .
```
```  2625   } note summable_Integral = this
```
```  2626
```
```  2627   { fix f :: "nat \<Rightarrow> real"
```
```  2628     have "\<And> x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  2629     proof
```
```  2630       fix x :: real assume "f sums x"
```
```  2631       from sums_if[OF sums_zero this]
```
```  2632       show "(\<lambda> n. if even n then f (n div 2) else 0) sums x" by auto
```
```  2633     next
```
```  2634       fix x :: real assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  2635       from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult_commute]]
```
```  2636       show "f sums x" unfolding sums_def by auto
```
```  2637     qed
```
```  2638     hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
```
```  2639   } note sums_even = this
```
```  2640
```
```  2641   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]
```
```  2642     by auto
```
```  2643
```
```  2644   { fix x :: real
```
```  2645     have if_eq': "\<And> n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
```
```  2646       (if even n then -1 ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
```
```  2647       using n_even by auto
```
```  2648     have idx_eq: "\<And> n. n * 2 + 1 = Suc (2 * n)" by auto
```
```  2649     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]
```
```  2650       by auto
```
```  2651   } note arctan_eq = this
```
```  2652
```
```  2653   have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
```
```  2654   proof (rule DERIV_power_series')
```
```  2655     show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto
```
```  2656     { fix x' :: real assume x'_bounds: "x' \<in> {- 1 <..< 1}"
```
```  2657       hence "\<bar>x'\<bar> < 1" by auto
```
```  2658
```
```  2659       let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
```
```  2660       show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
```
```  2661         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`])
```
```  2662     }
```
```  2663   qed auto
```
```  2664   thus ?thesis unfolding Int_eq arctan_eq .
```
```  2665 qed
```
```  2666
```
```  2667 lemma arctan_series: assumes "\<bar> x \<bar> \<le> 1"
```
```  2668   shows "arctan x = (\<Sum> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "_ = suminf (\<lambda> n. ?c x n)")
```
```  2669 proof -
```
```  2670   let "?c' x n" = "(-1)^n * x^(n*2)"
```
```  2671
```
```  2672   { fix r x :: real assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
```
```  2673     have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto
```
```  2674     from DERIV_arctan_series[OF this]
```
```  2675     have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
```
```  2676   } note DERIV_arctan_suminf = this
```
```  2677
```
```  2678   { fix x :: real assume "\<bar>x\<bar> \<le> 1" note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] }
```
```  2679   note arctan_series_borders = this
```
```  2680
```
```  2681   { fix x :: real assume "\<bar>x\<bar> < 1" have "arctan x = (\<Sum> k. ?c x k)"
```
```  2682   proof -
```
```  2683     obtain r where "\<bar>x\<bar> < r" and "r < 1" using dense[OF `\<bar>x\<bar> < 1`] by blast
```
```  2684     hence "0 < r" and "-r < x" and "x < r" by auto
```
```  2685
```
```  2686     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"
```
```  2687     proof -
```
```  2688       fix x a b assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
```
```  2689       hence "\<bar>x\<bar> < r" by auto
```
```  2690       show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  2691       proof (rule DERIV_isconst2[of "a" "b"])
```
```  2692         show "a < b" and "a \<le> x" and "x \<le> b" using `a < b` `a \<le> x` `x \<le> b` by auto
```
```  2693         have "\<forall> x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  2694         proof (rule allI, rule impI)
```
```  2695           fix x assume "-r < x \<and> x < r" hence "\<bar>x\<bar> < r" by auto
```
```  2696           hence "\<bar>x\<bar> < 1" using `r < 1` by auto
```
```  2697           have "\<bar> - (x^2) \<bar> < 1" using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
```
```  2698           hence "(\<lambda> n. (- (x^2)) ^ n) sums (1 / (1 - (- (x^2))))" unfolding real_norm_def[symmetric] by (rule geometric_sums)
```
```  2699           hence "(?c' x) sums (1 / (1 - (- (x^2))))" unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto
```
```  2700           hence suminf_c'_eq_geom: "inverse (1 + x^2) = suminf (?c' x)" using sums_unique unfolding inverse_eq_divide by auto
```
```  2701           have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x^2))" unfolding suminf_c'_eq_geom
```
```  2702             by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
```
```  2703           from DERIV_add_minus[OF this DERIV_arctan]
```
```  2704           show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0" unfolding diff_minus by auto
```
```  2705         qed
```
```  2706         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
```
```  2707         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
```
```  2708         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
```
```  2709       qed
```
```  2710     qed
```
```  2711
```
```  2712     have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
```
```  2713       unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero by auto
```
```  2714
```
```  2715     have "suminf (?c x) - arctan x = 0"
```
```  2716     proof (cases "x = 0")
```
```  2717       case True thus ?thesis using suminf_arctan_zero by auto
```
```  2718     next
```
```  2719       case False hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
```
```  2720       have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
```
```  2721         by (rule suminf_eq_arctan_bounded[where x="0" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>", symmetric])
```
```  2722           (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  2723       moreover
```
```  2724       have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
```
```  2725         by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"])
```
```  2726           (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  2727       ultimately
```
```  2728       show ?thesis using suminf_arctan_zero by auto
```
```  2729     qed
```
```  2730     thus ?thesis by auto
```
```  2731   qed } note when_less_one = this
```
```  2732
```
```  2733   show "arctan x = suminf (\<lambda> n. ?c x n)"
```
```  2734   proof (cases "\<bar>x\<bar> < 1")
```
```  2735     case True thus ?thesis by (rule when_less_one)
```
```  2736   next case False hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  2737     let "?a x n" = "\<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
```
```  2738     let "?diff x n" = "\<bar> arctan x - (\<Sum> i = 0..<n. ?c x i)\<bar>"
```
```  2739     { fix n :: nat
```
```  2740       have "0 < (1 :: real)" by auto
```
```  2741       moreover
```
```  2742       { fix x :: real assume "0 < x" and "x < 1" hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
```
```  2743         from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)" by auto
```
```  2744         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]
```
```  2745         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)
```
```  2746         hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)" by (rule abs_of_pos)
```
```  2747         have "?diff x n \<le> ?a x n"
```
```  2748         proof (cases "even n")
```
```  2749           case True hence sgn_pos: "(-1)^n = (1::real)" by auto
```
```  2750           from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
```
```  2751           from bounds[of m, unfolded this atLeastAtMost_iff]
```
```  2752           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
```
```  2753           also have "\<dots> = ?c x n" unfolding One_nat_def by auto
```
```  2754           also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
```
```  2755           finally show ?thesis .
```
```  2756         next
```
```  2757           case False hence sgn_neg: "(-1)^n = (-1::real)" by auto
```
```  2758           from `odd n` obtain m where m_def: "2 * m + 1 = n" unfolding odd_Suc_mult_two_ex by auto
```
```  2759           hence m_plus: "2 * (m + 1) = n + 1" by auto
```
```  2760           from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
```
```  2761           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
```
```  2762           also have "\<dots> = - ?c x n" unfolding One_nat_def by auto
```
```  2763           also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
```
```  2764           finally show ?thesis .
```
```  2765         qed
```
```  2766         hence "0 \<le> ?a x n - ?diff x n" by auto
```
```  2767       }
```
```  2768       hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
```
```  2769       moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
```
```  2770         unfolding diff_minus divide_inverse
```
```  2771         by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum)
```
```  2772       ultimately have "0 \<le> ?a 1 n - ?diff 1 n" by (rule LIM_less_bound)
```
```  2773       hence "?diff 1 n \<le> ?a 1 n" by auto
```
```  2774     }
```
```  2775     have "?a 1 ----> 0"
```
```  2776       unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
```
```  2777       by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)
```
```  2778     have "?diff 1 ----> 0"
```
```  2779     proof (rule LIMSEQ_I)
```
```  2780       fix r :: real assume "0 < r"
```
```  2781       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
```
```  2782       { fix n assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this]
```
```  2783         have "norm (?diff 1 n - 0) < r" by auto }
```
```  2784       thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
```
```  2785     qed
```
```  2786     from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
```
```  2787     have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
```
```  2788     hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
```
```  2789
```
```  2790     show ?thesis
```
```  2791     proof (cases "x = 1", simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
```
```  2792       assume "x \<noteq> 1" hence "x = -1" using `\<bar>x\<bar> = 1` by auto
```
```  2793
```
```  2794       have "- (pi / 2) < 0" using pi_gt_zero by auto
```
```  2795       have "- (2 * pi) < 0" using pi_gt_zero by auto
```
```  2796
```
```  2797       have c_minus_minus: "\<And> i. ?c (- 1) i = - ?c 1 i" unfolding One_nat_def by auto
```
```  2798
```
```  2799       have "arctan (- 1) = arctan (tan (-(pi / 4)))" unfolding tan_45 tan_minus ..
```
```  2800       also have "\<dots> = - (pi / 4)" by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
```
```  2801       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])
```
```  2802       also have "\<dots> = - (arctan 1)" unfolding tan_45 ..
```
```  2803       also have "\<dots> = - (\<Sum> i. ?c 1 i)" using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto
```
```  2804       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
```
```  2805       finally show ?thesis using `x = -1` by auto
```
```  2806     qed
```
```  2807   qed
```
```  2808 qed
```
```  2809
```
```  2810 lemma arctan_half: fixes x :: real
```
```  2811   shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x^2)))"
```
```  2812 proof -
```
```  2813   obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x" using tan_total by blast
```
```  2814   hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2" by auto
```
```  2815
```
```  2816   have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> A / B = (A / C) / (B / C)" by auto
```
```  2817
```
```  2818   have "0 < cos y" using cos_gt_zero_pi[OF low high] .
```
```  2819   hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y) ^ 2) = cos y" by auto
```
```  2820
```
```  2821   have "1 + (tan y)^2 = 1 + sin y^2 / cos y^2" unfolding tan_def power_divide ..
```
```  2822   also have "\<dots> = cos y^2 / cos y^2 + sin y^2 / cos y^2" using `cos y \<noteq> 0` by auto
```
```  2823   also have "\<dots> = 1 / cos y^2" unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
```
```  2824   finally have "1 + (tan y)^2 = 1 / cos y^2" .
```
```  2825
```
```  2826   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] ..
```
```  2827   also have "\<dots> = tan y / (1 + 1 / cos y)" using `cos y \<noteq> 0` unfolding add_divide_distrib by auto
```
```  2828   also have "\<dots> = tan y / (1 + 1 / sqrt(cos y^2))" unfolding cos_sqrt ..
```
```  2829   also have "\<dots> = tan y / (1 + sqrt(1 / cos y^2))" unfolding real_sqrt_divide by auto
```
```  2830   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` .
```
```  2831
```
```  2832   have "arctan x = y" using arctan_tan low high y_eq by auto
```
```  2833   also have "\<dots> = 2 * (arctan (tan (y/2)))" using arctan_tan[OF low2 high2] by auto
```
```  2834   also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" unfolding tan_half[OF low2 high2] by auto
```
```  2835   finally show ?thesis unfolding eq `tan y = x` .
```
```  2836 qed
```
```  2837
```
```  2838 lemma arctan_monotone: assumes "x < y"
```
```  2839   shows "arctan x < arctan y"
```
```  2840 proof -
```
```  2841   obtain z where "-(pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast
```
```  2842   obtain w where "-(pi / 2) < w" and "w < pi / 2" and "tan w = y" using tan_total by blast
```
```  2843   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` .
```
```  2844   thus ?thesis
```
```  2845     unfolding `tan z = x`[symmetric] arctan_tan[OF `-(pi / 2) < z` `z < pi / 2`]
```
```  2846     unfolding `tan w = y`[symmetric] arctan_tan[OF `-(pi / 2) < w` `w < pi / 2`] .
```
```  2847 qed
```
```  2848
```
```  2849 lemma arctan_monotone': assumes "x \<le> y" shows "arctan x \<le> arctan y"
```
```  2850 proof (cases "x = y")
```
```  2851   case False hence "x < y" using `x \<le> y` by auto from arctan_monotone[OF this] show ?thesis by auto
```
```  2852 qed auto
```
```  2853
```
```  2854 lemma arctan_minus: "arctan (- x) = - arctan x"
```
```  2855 proof -
```
```  2856   obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast
```
```  2857   thus ?thesis unfolding `tan y = x`[symmetric] tan_minus[symmetric] using arctan_tan[of y] arctan_tan[of "-y"] by auto
```
```  2858 qed
```
```  2859
```
```  2860 lemma arctan_inverse: assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
```
```  2861 proof -
```
```  2862   obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast
```
```  2863   hence "y = arctan x" unfolding `tan y = x`[symmetric] using arctan_tan by auto
```
```  2864
```
```  2865   { 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
```
```  2866     have "tan y > 0" using tan_monotone'[OF _ _ `- (pi / 2) < y` `y < pi / 2`, of 0] tan_zero `0 < y` by auto
```
```  2867     hence "x > 0" using `tan y = x` by auto
```
```  2868
```
```  2869     have "- (pi / 2) < pi / 2 - y" using `y > 0` `y < pi / 2` by auto
```
```  2870     moreover have "pi / 2 - y < pi / 2" using `y > 0` `y < pi / 2` by auto
```
```  2871     ultimately have "arctan (1 / x) = pi / 2 - y" unfolding `tan y = x`[symmetric] tan_inverse using arctan_tan by auto
```
```  2872     hence "arctan (1 / x) = sgn x * pi / 2 - arctan x" unfolding `y = arctan x` real_sgn_pos[OF `x > 0`] by auto
```
```  2873   } note pos_y = this
```
```  2874
```
```  2875   show ?thesis
```
```  2876   proof (cases "y > 0")
```
```  2877     case True from pos_y[OF this `y < pi / 2` `y = arctan x` `tan y = x`] show ?thesis .
```
```  2878   next
```
```  2879     case False hence "y \<le> 0" by auto
```
```  2880     moreover have "y \<noteq> 0"
```
```  2881     proof (rule ccontr)
```
```  2882       assume "\<not> y \<noteq> 0" hence "y = 0" by auto
```
```  2883       have "x = 0" unfolding `tan y = x`[symmetric] `y = 0` tan_zero ..
```
```  2884       thus False using `x \<noteq> 0` by auto
```
```  2885     qed
```
```  2886     ultimately have "y < 0" by auto
```
```  2887     hence "0 < - y" and "-y < pi / 2" using `- (pi / 2) < y` by auto
```
```  2888     moreover have "-y = arctan (-x)" unfolding arctan_minus `y = arctan x` ..
```
```  2889     moreover have "tan (-y) = -x" unfolding tan_minus `tan y = x` ..
```
```  2890     ultimately have "arctan (1 / -x) = sgn (-x) * pi / 2 - arctan (-x)" using pos_y by blast
```
```  2891     hence "arctan (- (1 / x)) = - (sgn x * pi / 2 - arctan x)" unfolding arctan_minus[of x] divide_minus_right sgn_minus by auto
```
```  2892     thus ?thesis unfolding arctan_minus neg_equal_iff_equal .
```
```  2893   qed
```
```  2894 qed
```
```  2895
```
```  2896 theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
```
```  2897 proof -
```
```  2898   have "pi / 4 = arctan 1" using arctan1_eq_pi4 by auto
```
```  2899   also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
```
```  2900   finally show ?thesis by auto
```
```  2901 qed
```
```  2902
```
```  2903 subsection {* Existence of Polar Coordinates *}
```
```  2904
```
```  2905 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<twosuperior> + y\<twosuperior>)\<bar> \<le> 1"
```
```  2906 apply (rule power2_le_imp_le [OF _ zero_le_one])
```
```  2907 apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
```
```  2908 done
```
```  2909
```
```  2910 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
```
```  2911 by (simp add: abs_le_iff)
```
```  2912
```
```  2913 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<twosuperior>)"
```
```  2914 by (simp add: sin_arccos abs_le_iff)
```
```  2915
```
```  2916 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
```
```  2917
```
```  2918 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
```
```  2919
```
```  2920 lemma polar_ex1:
```
```  2921      "0 < y ==> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  2922 apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>)" in exI)
```
```  2923 apply (rule_tac x = "arccos (x / sqrt (x\<twosuperior> + y\<twosuperior>))" in exI)
```
```  2924 apply (simp add: cos_arccos_lemma1)
```
```  2925 apply (simp add: sin_arccos_lemma1)
```
```  2926 apply (simp add: power_divide)
```
```  2927 apply (simp add: real_sqrt_mult [symmetric])
```
```  2928 apply (simp add: right_diff_distrib)
```
```  2929 done
```
```  2930
```
```  2931 lemma polar_ex2:
```
```  2932      "y < 0 ==> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  2933 apply (insert polar_ex1 [where x=x and y="-y"], simp, clarify)
```
```  2934 apply (metis cos_minus minus_minus minus_mult_right sin_minus)
```
```  2935 done
```
```  2936
```
```  2937 lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
```
```  2938 apply (rule_tac x=0 and y=y in linorder_cases)
```
```  2939 apply (erule polar_ex1)
```
```  2940 apply (rule_tac x=x in exI, rule_tac x=0 in exI, simp)
```
```  2941 apply (erule polar_ex2)
```
```  2942 done
```
```  2943
```
```  2944 end
```