src/HOL/Transcendental.thy
 author huffman Sun Mar 25 20:15:39 2012 +0200 (2012-03-25) changeset 47108 2a1953f0d20d parent 46240 933f35c4e126 child 47489 04e7d09ade7a permissions -rw-r--r--
merged fork with new numeral representation (see NEWS)
```     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 proof (rule IVT)
```
```  1069   assume "1 \<le> y"
```
```  1070   hence "0 \<le> y - 1" by simp
```
```  1071   hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
```
```  1072   thus "y \<le> exp (y - 1)" by simp
```
```  1073 qed (simp_all add: le_diff_eq)
```
```  1074
```
```  1075 lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
```
```  1076 proof (rule linorder_le_cases [of 1 y])
```
```  1077   assume "1 \<le> y" thus "\<exists>x. exp x = y"
```
```  1078     by (fast dest: lemma_exp_total)
```
```  1079 next
```
```  1080   assume "0 < y" and "y \<le> 1"
```
```  1081   hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
```
```  1082   then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
```
```  1083   hence "exp (- x) = y" by (simp add: exp_minus)
```
```  1084   thus "\<exists>x. exp x = y" ..
```
```  1085 qed
```
```  1086
```
```  1087
```
```  1088 subsection {* Natural Logarithm *}
```
```  1089
```
```  1090 definition ln :: "real \<Rightarrow> real" where
```
```  1091   "ln x = (THE u. exp u = x)"
```
```  1092
```
```  1093 lemma ln_exp [simp]: "ln (exp x) = x"
```
```  1094   by (simp add: ln_def)
```
```  1095
```
```  1096 lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
```
```  1097   by (auto dest: exp_total)
```
```  1098
```
```  1099 lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
```
```  1100   by (metis exp_gt_zero exp_ln)
```
```  1101
```
```  1102 lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
```
```  1103   by (erule subst, rule ln_exp)
```
```  1104
```
```  1105 lemma ln_one [simp]: "ln 1 = 0"
```
```  1106   by (rule ln_unique, simp)
```
```  1107
```
```  1108 lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"
```
```  1109   by (rule ln_unique, simp add: exp_add)
```
```  1110
```
```  1111 lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
```
```  1112   by (rule ln_unique, simp add: exp_minus)
```
```  1113
```
```  1114 lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"
```
```  1115   by (rule ln_unique, simp add: exp_diff)
```
```  1116
```
```  1117 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
```
```  1118   by (rule ln_unique, simp add: exp_real_of_nat_mult)
```
```  1119
```
```  1120 lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
```
```  1121   by (subst exp_less_cancel_iff [symmetric], simp)
```
```  1122
```
```  1123 lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
```
```  1124   by (simp add: linorder_not_less [symmetric])
```
```  1125
```
```  1126 lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
```
```  1127   by (simp add: order_eq_iff)
```
```  1128
```
```  1129 lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
```
```  1130   apply (rule exp_le_cancel_iff [THEN iffD1])
```
```  1131   apply (simp add: exp_ge_add_one_self_aux)
```
```  1132   done
```
```  1133
```
```  1134 lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
```
```  1135   by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
```
```  1136
```
```  1137 lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
```
```  1138   using ln_le_cancel_iff [of 1 x] by simp
```
```  1139
```
```  1140 lemma ln_ge_zero_imp_ge_one: "\<lbrakk>0 \<le> ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 \<le> x"
```
```  1141   using ln_le_cancel_iff [of 1 x] by simp
```
```  1142
```
```  1143 lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> (0 \<le> ln x) = (1 \<le> x)"
```
```  1144   using ln_le_cancel_iff [of 1 x] by simp
```
```  1145
```
```  1146 lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x < 0) = (x < 1)"
```
```  1147   using ln_less_cancel_iff [of x 1] by simp
```
```  1148
```
```  1149 lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
```
```  1150   using ln_less_cancel_iff [of 1 x] by simp
```
```  1151
```
```  1152 lemma ln_gt_zero_imp_gt_one: "\<lbrakk>0 < ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 < x"
```
```  1153   using ln_less_cancel_iff [of 1 x] by simp
```
```  1154
```
```  1155 lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> (0 < ln x) = (1 < x)"
```
```  1156   using ln_less_cancel_iff [of 1 x] by simp
```
```  1157
```
```  1158 lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x = 0) = (x = 1)"
```
```  1159   using ln_inj_iff [of x 1] by simp
```
```  1160
```
```  1161 lemma ln_less_zero: "\<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> ln x < 0"
```
```  1162   by simp
```
```  1163
```
```  1164 lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
```
```  1165   apply (subgoal_tac "isCont ln (exp (ln x))", simp)
```
```  1166   apply (rule isCont_inverse_function [where f=exp], simp_all)
```
```  1167   done
```
```  1168
```
```  1169 lemma tendsto_ln [tendsto_intros]:
```
```  1170   "\<lbrakk>(f ---> a) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
```
```  1171   by (rule isCont_tendsto_compose [OF isCont_ln])
```
```  1172
```
```  1173 lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
```
```  1174   apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
```
```  1175   apply (erule DERIV_cong [OF DERIV_exp exp_ln])
```
```  1176   apply (simp_all add: abs_if isCont_ln)
```
```  1177   done
```
```  1178
```
```  1179 lemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x"
```
```  1180   by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
```
```  1181
```
```  1182 lemma ln_series: assumes "0 < x" and "x < 2"
```
```  1183   shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))")
```
```  1184 proof -
```
```  1185   let "?f' x n" = "(-1)^n * (x - 1)^n"
```
```  1186
```
```  1187   have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
```
```  1188   proof (rule DERIV_isconst3[where x=x])
```
```  1189     fix x :: real assume "x \<in> {0 <..< 2}" hence "0 < x" and "x < 2" by auto
```
```  1190     have "norm (1 - x) < 1" using `0 < x` and `x < 2` by auto
```
```  1191     have "1 / x = 1 / (1 - (1 - x))" by auto
```
```  1192     also have "\<dots> = (\<Sum> n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
```
```  1193     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)
```
```  1194     finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
```
```  1195     moreover
```
```  1196     have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
```
```  1197     have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
```
```  1198     proof (rule DERIV_power_series')
```
```  1199       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto
```
```  1200       { fix x :: real assume "x \<in> {- 1<..<1}" hence "norm (-x) < 1" by auto
```
```  1201         show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
```
```  1202           unfolding One_nat_def
```
```  1203           by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
```
```  1204       }
```
```  1205     qed
```
```  1206     hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" unfolding One_nat_def by auto
```
```  1207     hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos .
```
```  1208     ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
```
```  1209       by (rule DERIV_diff)
```
```  1210     thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
```
```  1211   qed (auto simp add: assms)
```
```  1212   thus ?thesis by auto
```
```  1213 qed
```
```  1214
```
```  1215 subsection {* Sine and Cosine *}
```
```  1216
```
```  1217 definition sin_coeff :: "nat \<Rightarrow> real" where
```
```  1218   "sin_coeff = (\<lambda>n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"
```
```  1219
```
```  1220 definition cos_coeff :: "nat \<Rightarrow> real" where
```
```  1221   "cos_coeff = (\<lambda>n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"
```
```  1222
```
```  1223 definition sin :: "real \<Rightarrow> real" where
```
```  1224   "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
```
```  1225
```
```  1226 definition cos :: "real \<Rightarrow> real" where
```
```  1227   "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
```
```  1228
```
```  1229 lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
```
```  1230   unfolding sin_coeff_def by simp
```
```  1231
```
```  1232 lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
```
```  1233   unfolding cos_coeff_def by simp
```
```  1234
```
```  1235 lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
```
```  1236   unfolding cos_coeff_def sin_coeff_def
```
```  1237   by (simp del: mult_Suc)
```
```  1238
```
```  1239 lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
```
```  1240   unfolding cos_coeff_def sin_coeff_def
```
```  1241   by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)
```
```  1242
```
```  1243 lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
```
```  1244 unfolding sin_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 summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"
```
```  1250 unfolding cos_coeff_def
```
```  1251 apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
```
```  1252 apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  1253 done
```
```  1254
```
```  1255 lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
```
```  1256 unfolding sin_def by (rule summable_sin [THEN summable_sums])
```
```  1257
```
```  1258 lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
```
```  1259 unfolding cos_def by (rule summable_cos [THEN summable_sums])
```
```  1260
```
```  1261 lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
```
```  1262   by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  1263
```
```  1264 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
```
```  1265   by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  1266
```
```  1267 text{*Now at last we can get the derivatives of exp, sin and cos*}
```
```  1268
```
```  1269 lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
```
```  1270   unfolding sin_def cos_def
```
```  1271   apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
```
```  1272   apply (simp_all add: diffs_sin_coeff diffs_cos_coeff
```
```  1273     summable_minus summable_sin summable_cos)
```
```  1274   done
```
```  1275
```
```  1276 lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
```
```  1277   unfolding cos_def sin_def
```
```  1278   apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
```
```  1279   apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus
```
```  1280     summable_minus summable_sin summable_cos suminf_minus)
```
```  1281   done
```
```  1282
```
```  1283 lemma isCont_sin: "isCont sin x"
```
```  1284   by (rule DERIV_sin [THEN DERIV_isCont])
```
```  1285
```
```  1286 lemma isCont_cos: "isCont cos x"
```
```  1287   by (rule DERIV_cos [THEN DERIV_isCont])
```
```  1288
```
```  1289 lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
```
```  1290   by (rule isCont_o2 [OF _ isCont_sin])
```
```  1291
```
```  1292 lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
```
```  1293   by (rule isCont_o2 [OF _ isCont_cos])
```
```  1294
```
```  1295 lemma tendsto_sin [tendsto_intros]:
```
```  1296   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
```
```  1297   by (rule isCont_tendsto_compose [OF isCont_sin])
```
```  1298
```
```  1299 lemma tendsto_cos [tendsto_intros]:
```
```  1300   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
```
```  1301   by (rule isCont_tendsto_compose [OF isCont_cos])
```
```  1302
```
```  1303 declare
```
```  1304   DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1305   DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1306   DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1307   DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1308
```
```  1309 subsection {* Properties of Sine and Cosine *}
```
```  1310
```
```  1311 lemma sin_zero [simp]: "sin 0 = 0"
```
```  1312   unfolding sin_def sin_coeff_def by (simp add: powser_zero)
```
```  1313
```
```  1314 lemma cos_zero [simp]: "cos 0 = 1"
```
```  1315   unfolding cos_def cos_coeff_def by (simp add: powser_zero)
```
```  1316
```
```  1317 lemma sin_cos_squared_add [simp]: "(sin x)\<twosuperior> + (cos x)\<twosuperior> = 1"
```
```  1318 proof -
```
```  1319   have "\<forall>x. DERIV (\<lambda>x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
```
```  1320     by (auto intro!: DERIV_intros)
```
```  1321   hence "(sin x)\<twosuperior> + (cos x)\<twosuperior> = (sin 0)\<twosuperior> + (cos 0)\<twosuperior>"
```
```  1322     by (rule DERIV_isconst_all)
```
```  1323   thus "(sin x)\<twosuperior> + (cos x)\<twosuperior> = 1" by simp
```
```  1324 qed
```
```  1325
```
```  1326 lemma sin_cos_squared_add2 [simp]: "(cos x)\<twosuperior> + (sin x)\<twosuperior> = 1"
```
```  1327   by (subst add_commute, rule sin_cos_squared_add)
```
```  1328
```
```  1329 lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
```
```  1330   using sin_cos_squared_add2 [unfolded power2_eq_square] .
```
```  1331
```
```  1332 lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
```
```  1333   unfolding eq_diff_eq by (rule sin_cos_squared_add)
```
```  1334
```
```  1335 lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
```
```  1336   unfolding eq_diff_eq by (rule sin_cos_squared_add2)
```
```  1337
```
```  1338 lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
```
```  1339   by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
```
```  1340
```
```  1341 lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
```
```  1342   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  1343
```
```  1344 lemma sin_le_one [simp]: "sin x \<le> 1"
```
```  1345   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  1346
```
```  1347 lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
```
```  1348   by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
```
```  1349
```
```  1350 lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
```
```  1351   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  1352
```
```  1353 lemma cos_le_one [simp]: "cos x \<le> 1"
```
```  1354   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  1355
```
```  1356 lemma DERIV_fun_pow: "DERIV g x :> m ==>
```
```  1357       DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
```
```  1358   by (auto intro!: DERIV_intros)
```
```  1359
```
```  1360 lemma DERIV_fun_exp:
```
```  1361      "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
```
```  1362   by (auto intro!: DERIV_intros)
```
```  1363
```
```  1364 lemma DERIV_fun_sin:
```
```  1365      "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
```
```  1366   by (auto intro!: DERIV_intros)
```
```  1367
```
```  1368 lemma DERIV_fun_cos:
```
```  1369      "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
```
```  1370   by (auto intro!: DERIV_intros)
```
```  1371
```
```  1372 lemma sin_cos_add_lemma:
```
```  1373      "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
```
```  1374       (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
```
```  1375   (is "?f x = 0")
```
```  1376 proof -
```
```  1377   have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
```
```  1378     by (auto intro!: DERIV_intros simp add: algebra_simps)
```
```  1379   hence "?f x = ?f 0"
```
```  1380     by (rule DERIV_isconst_all)
```
```  1381   thus ?thesis by simp
```
```  1382 qed
```
```  1383
```
```  1384 lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
```
```  1385   using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
```
```  1386
```
```  1387 lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
```
```  1388   using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
```
```  1389
```
```  1390 lemma sin_cos_minus_lemma:
```
```  1391   "(sin(-x) + sin(x))\<twosuperior> + (cos(-x) - cos(x))\<twosuperior> = 0" (is "?f x = 0")
```
```  1392 proof -
```
```  1393   have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
```
```  1394     by (auto intro!: DERIV_intros simp add: algebra_simps)
```
```  1395   hence "?f x = ?f 0"
```
```  1396     by (rule DERIV_isconst_all)
```
```  1397   thus ?thesis by simp
```
```  1398 qed
```
```  1399
```
```  1400 lemma sin_minus [simp]: "sin (-x) = -sin(x)"
```
```  1401   using sin_cos_minus_lemma [where x=x] by simp
```
```  1402
```
```  1403 lemma cos_minus [simp]: "cos (-x) = cos(x)"
```
```  1404   using sin_cos_minus_lemma [where x=x] by simp
```
```  1405
```
```  1406 lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
```
```  1407   by (simp add: diff_minus sin_add)
```
```  1408
```
```  1409 lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
```
```  1410   by (simp add: sin_diff mult_commute)
```
```  1411
```
```  1412 lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
```
```  1413   by (simp add: diff_minus cos_add)
```
```  1414
```
```  1415 lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
```
```  1416   by (simp add: cos_diff mult_commute)
```
```  1417
```
```  1418 lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
```
```  1419   using sin_add [where x=x and y=x] by simp
```
```  1420
```
```  1421 lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
```
```  1422   using cos_add [where x=x and y=x]
```
```  1423   by (simp add: power2_eq_square)
```
```  1424
```
```  1425
```
```  1426 subsection {* The Constant Pi *}
```
```  1427
```
```  1428 definition pi :: "real" where
```
```  1429   "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
```
```  1430
```
```  1431 text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
```
```  1432    hence define pi.*}
```
```  1433
```
```  1434 lemma sin_paired:
```
```  1435      "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1))
```
```  1436       sums  sin x"
```
```  1437 proof -
```
```  1438   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
```
```  1439     by (rule sin_converges [THEN sums_group], simp)
```
```  1440   thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)
```
```  1441 qed
```
```  1442
```
```  1443 lemma sin_gt_zero:
```
```  1444   assumes "0 < x" and "x < 2" shows "0 < sin x"
```
```  1445 proof -
```
```  1446   let ?f = "\<lambda>n. \<Sum>k = n*2..<n*2+2. -1 ^ k / real (fact (2*k+1)) * x^(2*k+1)"
```
```  1447   have pos: "\<forall>n. 0 < ?f n"
```
```  1448   proof
```
```  1449     fix n :: nat
```
```  1450     let ?k2 = "real (Suc (Suc (4 * n)))"
```
```  1451     let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
```
```  1452     have "x * x < ?k2 * ?k3"
```
```  1453       using assms by (intro mult_strict_mono', simp_all)
```
```  1454     hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
```
```  1455       by (intro mult_strict_right_mono zero_less_power `0 < x`)
```
```  1456     thus "0 < ?f n"
```
```  1457       by (simp del: mult_Suc,
```
```  1458         simp add: less_divide_eq mult_pos_pos field_simps del: mult_Suc)
```
```  1459   qed
```
```  1460   have sums: "?f sums sin x"
```
```  1461     by (rule sin_paired [THEN sums_group], simp)
```
```  1462   show "0 < sin x"
```
```  1463     unfolding sums_unique [OF sums]
```
```  1464     using sums_summable [OF sums] pos
```
```  1465     by (rule suminf_gt_zero)
```
```  1466 qed
```
```  1467
```
```  1468 lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
```
```  1469 apply (cut_tac x = x in sin_gt_zero)
```
```  1470 apply (auto simp add: cos_squared_eq cos_double)
```
```  1471 done
```
```  1472
```
```  1473 lemma cos_paired:
```
```  1474      "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
```
```  1475 proof -
```
```  1476   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
```
```  1477     by (rule cos_converges [THEN sums_group], simp)
```
```  1478   thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)
```
```  1479 qed
```
```  1480
```
```  1481 lemma real_mult_inverse_cancel:
```
```  1482      "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
```
```  1483       ==> inverse x * y < inverse x1 * u"
```
```  1484 apply (rule_tac c=x in mult_less_imp_less_left)
```
```  1485 apply (auto simp add: mult_assoc [symmetric])
```
```  1486 apply (simp (no_asm) add: mult_ac)
```
```  1487 apply (rule_tac c=x1 in mult_less_imp_less_right)
```
```  1488 apply (auto simp add: mult_ac)
```
```  1489 done
```
```  1490
```
```  1491 lemma real_mult_inverse_cancel2:
```
```  1492      "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
```
```  1493 apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)
```
```  1494 done
```
```  1495
```
```  1496 lemma realpow_num_eq_if:
```
```  1497   fixes m :: "'a::power"
```
```  1498   shows "m ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
```
```  1499 by (cases n, auto)
```
```  1500
```
```  1501 lemma cos_two_less_zero [simp]: "cos (2) < 0"
```
```  1502 apply (cut_tac x = 2 in cos_paired)
```
```  1503 apply (drule sums_minus)
```
```  1504 apply (rule neg_less_iff_less [THEN iffD1])
```
```  1505 apply (frule sums_unique, auto)
```
```  1506 apply (rule_tac y =
```
```  1507  "\<Sum>n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
```
```  1508        in order_less_trans)
```
```  1509 apply (simp (no_asm) add: fact_num_eq_if_nat realpow_num_eq_if del: fact_Suc)
```
```  1510 apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
```
```  1511 apply (rule sumr_pos_lt_pair)
```
```  1512 apply (erule sums_summable, safe)
```
```  1513 unfolding One_nat_def
```
```  1514 apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
```
```  1515             del: fact_Suc)
```
```  1516 apply (simp add: inverse_eq_divide less_divide_eq del: fact_Suc)
```
```  1517 apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
```
```  1518 apply (simp only: real_of_nat_mult)
```
```  1519 apply (rule mult_strict_mono, force)
```
```  1520   apply (rule_tac [3] real_of_nat_ge_zero)
```
```  1521  prefer 2 apply force
```
```  1522 apply (rule real_of_nat_less_iff [THEN iffD2])
```
```  1523 apply (rule fact_less_mono_nat, auto)
```
```  1524 done
```
```  1525
```
```  1526 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
```
```  1527 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
```
```  1528
```
```  1529 lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
```
```  1530 proof (rule ex_ex1I)
```
```  1531   show "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0"
```
```  1532     by (rule IVT2, simp_all)
```
```  1533 next
```
```  1534   fix x y
```
```  1535   assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
```
```  1536   assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
```
```  1537   have [simp]: "\<forall>x. cos differentiable x"
```
```  1538     unfolding differentiable_def by (auto intro: DERIV_cos)
```
```  1539   from x y show "x = y"
```
```  1540     apply (cut_tac less_linear [of x y], auto)
```
```  1541     apply (drule_tac f = cos in Rolle)
```
```  1542     apply (drule_tac [5] f = cos in Rolle)
```
```  1543     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  1544     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  1545     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  1546     done
```
```  1547 qed
```
```  1548
```
```  1549 lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
```
```  1550 by (simp add: pi_def)
```
```  1551
```
```  1552 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
```
```  1553 by (simp add: pi_half cos_is_zero [THEN theI'])
```
```  1554
```
```  1555 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
```
```  1556 apply (rule order_le_neq_trans)
```
```  1557 apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  1558 apply (rule notI, drule arg_cong [where f=cos], simp)
```
```  1559 done
```
```  1560
```
```  1561 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
```
```  1562 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
```
```  1563
```
```  1564 lemma pi_half_less_two [simp]: "pi / 2 < 2"
```
```  1565 apply (rule order_le_neq_trans)
```
```  1566 apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  1567 apply (rule notI, drule arg_cong [where f=cos], simp)
```
```  1568 done
```
```  1569
```
```  1570 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
```
```  1571 lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
```
```  1572
```
```  1573 lemma pi_gt_zero [simp]: "0 < pi"
```
```  1574 by (insert pi_half_gt_zero, simp)
```
```  1575
```
```  1576 lemma pi_ge_zero [simp]: "0 \<le> pi"
```
```  1577 by (rule pi_gt_zero [THEN order_less_imp_le])
```
```  1578
```
```  1579 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
```
```  1580 by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym])
```
```  1581
```
```  1582 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
```
```  1583 by (simp add: linorder_not_less)
```
```  1584
```
```  1585 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
```
```  1586 by simp
```
```  1587
```
```  1588 lemma m2pi_less_pi: "- (2 * pi) < pi"
```
```  1589 by simp
```
```  1590
```
```  1591 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
```
```  1592 apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
```
```  1593 apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
```
```  1594 apply (simp add: power2_eq_1_iff)
```
```  1595 done
```
```  1596
```
```  1597 lemma cos_pi [simp]: "cos pi = -1"
```
```  1598 by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
```
```  1599
```
```  1600 lemma sin_pi [simp]: "sin pi = 0"
```
```  1601 by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
```
```  1602
```
```  1603 lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
```
```  1604 by (simp add: cos_diff)
```
```  1605
```
```  1606 lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
```
```  1607 by (simp add: cos_add)
```
```  1608
```
```  1609 lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
```
```  1610 by (simp add: sin_diff)
```
```  1611
```
```  1612 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
```
```  1613 by (simp add: sin_add)
```
```  1614
```
```  1615 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
```
```  1616 by (simp add: sin_add)
```
```  1617
```
```  1618 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
```
```  1619 by (simp add: cos_add)
```
```  1620
```
```  1621 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
```
```  1622 by (simp add: sin_add cos_double)
```
```  1623
```
```  1624 lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
```
```  1625 by (simp add: cos_add cos_double)
```
```  1626
```
```  1627 lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
```
```  1628 apply (induct "n")
```
```  1629 apply (auto simp add: real_of_nat_Suc left_distrib)
```
```  1630 done
```
```  1631
```
```  1632 lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
```
```  1633 proof -
```
```  1634   have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
```
```  1635   also have "... = -1 ^ n" by (rule cos_npi)
```
```  1636   finally show ?thesis .
```
```  1637 qed
```
```  1638
```
```  1639 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
```
```  1640 apply (induct "n")
```
```  1641 apply (auto simp add: real_of_nat_Suc left_distrib)
```
```  1642 done
```
```  1643
```
```  1644 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
```
```  1645 by (simp add: mult_commute [of pi])
```
```  1646
```
```  1647 lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
```
```  1648 by (simp add: cos_double)
```
```  1649
```
```  1650 lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
```
```  1651 by simp
```
```  1652
```
```  1653 lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
```
```  1654 apply (rule sin_gt_zero, assumption)
```
```  1655 apply (rule order_less_trans, assumption)
```
```  1656 apply (rule pi_half_less_two)
```
```  1657 done
```
```  1658
```
```  1659 lemma sin_less_zero:
```
```  1660   assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
```
```  1661 proof -
```
```  1662   have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
```
```  1663   thus ?thesis by simp
```
```  1664 qed
```
```  1665
```
```  1666 lemma pi_less_4: "pi < 4"
```
```  1667 by (cut_tac pi_half_less_two, auto)
```
```  1668
```
```  1669 lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
```
```  1670 apply (cut_tac pi_less_4)
```
```  1671 apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
```
```  1672 apply (cut_tac cos_is_zero, safe)
```
```  1673 apply (rename_tac y z)
```
```  1674 apply (drule_tac x = y in spec)
```
```  1675 apply (drule_tac x = "pi/2" in spec, simp)
```
```  1676 done
```
```  1677
```
```  1678 lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
```
```  1679 apply (rule_tac x = x and y = 0 in linorder_cases)
```
```  1680 apply (rule cos_minus [THEN subst])
```
```  1681 apply (rule cos_gt_zero)
```
```  1682 apply (auto intro: cos_gt_zero)
```
```  1683 done
```
```  1684
```
```  1685 lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
```
```  1686 apply (auto simp add: order_le_less cos_gt_zero_pi)
```
```  1687 apply (subgoal_tac "x = pi/2", auto)
```
```  1688 done
```
```  1689
```
```  1690 lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
```
```  1691 by (simp add: sin_cos_eq cos_gt_zero_pi)
```
```  1692
```
```  1693 lemma pi_ge_two: "2 \<le> pi"
```
```  1694 proof (rule ccontr)
```
```  1695   assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
```
```  1696   have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
```
```  1697   proof (cases "2 < 2 * pi")
```
```  1698     case True with dense[OF `pi < 2`] show ?thesis by auto
```
```  1699   next
```
```  1700     case False have "pi < 2 * pi" by auto
```
```  1701     from dense[OF this] and False show ?thesis by auto
```
```  1702   qed
```
```  1703   then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
```
```  1704   hence "0 < sin y" using sin_gt_zero by auto
```
```  1705   moreover
```
```  1706   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
```
```  1707   ultimately show False by auto
```
```  1708 qed
```
```  1709
```
```  1710 lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
```
```  1711 by (auto simp add: order_le_less sin_gt_zero_pi)
```
```  1712
```
```  1713 text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}.
```
```  1714   It should be possible to factor out some of the common parts. *}
```
```  1715
```
```  1716 lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
```
```  1717 proof (rule ex_ex1I)
```
```  1718   assume y: "-1 \<le> y" "y \<le> 1"
```
```  1719   show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
```
```  1720     by (rule IVT2, simp_all add: y)
```
```  1721 next
```
```  1722   fix a b
```
```  1723   assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
```
```  1724   assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
```
```  1725   have [simp]: "\<forall>x. cos differentiable x"
```
```  1726     unfolding differentiable_def by (auto intro: DERIV_cos)
```
```  1727   from a b show "a = b"
```
```  1728     apply (cut_tac less_linear [of a b], auto)
```
```  1729     apply (drule_tac f = cos in Rolle)
```
```  1730     apply (drule_tac [5] f = cos in Rolle)
```
```  1731     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  1732     apply (metis order_less_le_trans less_le sin_gt_zero_pi)
```
```  1733     apply (metis order_less_le_trans less_le sin_gt_zero_pi)
```
```  1734     done
```
```  1735 qed
```
```  1736
```
```  1737 lemma sin_total:
```
```  1738      "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
```
```  1739 apply (rule ccontr)
```
```  1740 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))")
```
```  1741 apply (erule contrapos_np)
```
```  1742 apply simp
```
```  1743 apply (cut_tac y="-y" in cos_total, simp) apply simp
```
```  1744 apply (erule ex1E)
```
```  1745 apply (rule_tac a = "x - (pi/2)" in ex1I)
```
```  1746 apply (simp (no_asm) add: add_assoc)
```
```  1747 apply (rotate_tac 3)
```
```  1748 apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all add: cos_add)
```
```  1749 done
```
```  1750
```
```  1751 lemma reals_Archimedean4:
```
```  1752      "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
```
```  1753 apply (auto dest!: reals_Archimedean3)
```
```  1754 apply (drule_tac x = x in spec, clarify)
```
```  1755 apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
```
```  1756  prefer 2 apply (erule LeastI)
```
```  1757 apply (case_tac "LEAST m::nat. x < real m * y", simp)
```
```  1758 apply (subgoal_tac "~ x < real nat * y")
```
```  1759  prefer 2 apply (rule not_less_Least, simp, force)
```
```  1760 done
```
```  1761
```
```  1762 (* Pre Isabelle99-2 proof was simpler- numerals arithmetic
```
```  1763    now causes some unwanted re-arrangements of literals!   *)
```
```  1764 lemma cos_zero_lemma:
```
```  1765      "[| 0 \<le> x; cos x = 0 |] ==>
```
```  1766       \<exists>n::nat. ~even n & x = real n * (pi/2)"
```
```  1767 apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
```
```  1768 apply (subgoal_tac "0 \<le> x - real n * pi &
```
```  1769                     (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
```
```  1770 apply (auto simp add: algebra_simps real_of_nat_Suc)
```
```  1771  prefer 2 apply (simp add: cos_diff)
```
```  1772 apply (simp add: cos_diff)
```
```  1773 apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
```
```  1774 apply (rule_tac [2] cos_total, safe)
```
```  1775 apply (drule_tac x = "x - real n * pi" in spec)
```
```  1776 apply (drule_tac x = "pi/2" in spec)
```
```  1777 apply (simp add: cos_diff)
```
```  1778 apply (rule_tac x = "Suc (2 * n)" in exI)
```
```  1779 apply (simp add: real_of_nat_Suc algebra_simps, auto)
```
```  1780 done
```
```  1781
```
```  1782 lemma sin_zero_lemma:
```
```  1783      "[| 0 \<le> x; sin x = 0 |] ==>
```
```  1784       \<exists>n::nat. even n & x = real n * (pi/2)"
```
```  1785 apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
```
```  1786  apply (clarify, rule_tac x = "n - 1" in exI)
```
```  1787  apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
```
```  1788 apply (rule cos_zero_lemma)
```
```  1789 apply (simp_all add: cos_add)
```
```  1790 done
```
```  1791
```
```  1792
```
```  1793 lemma cos_zero_iff:
```
```  1794      "(cos x = 0) =
```
```  1795       ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
```
```  1796        (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
```
```  1797 apply (rule iffI)
```
```  1798 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  1799 apply (drule cos_zero_lemma, assumption+)
```
```  1800 apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
```
```  1801 apply (force simp add: minus_equation_iff [of x])
```
```  1802 apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
```
```  1803 apply (auto simp add: cos_add)
```
```  1804 done
```
```  1805
```
```  1806 (* ditto: but to a lesser extent *)
```
```  1807 lemma sin_zero_iff:
```
```  1808      "(sin x = 0) =
```
```  1809       ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
```
```  1810        (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
```
```  1811 apply (rule iffI)
```
```  1812 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  1813 apply (drule sin_zero_lemma, assumption+)
```
```  1814 apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
```
```  1815 apply (force simp add: minus_equation_iff [of x])
```
```  1816 apply (auto simp add: even_mult_two_ex)
```
```  1817 done
```
```  1818
```
```  1819 lemma cos_monotone_0_pi: assumes "0 \<le> y" and "y < x" and "x \<le> pi"
```
```  1820   shows "cos x < cos y"
```
```  1821 proof -
```
```  1822   have "- (x - y) < 0" using assms by auto
```
```  1823
```
```  1824   from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
```
```  1825   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" by auto
```
```  1826   hence "0 < z" and "z < pi" using assms by auto
```
```  1827   hence "0 < sin z" using sin_gt_zero_pi by auto
```
```  1828   hence "cos x - cos y < 0" unfolding cos_diff minus_mult_commute[symmetric] using `- (x - y) < 0` by (rule mult_pos_neg2)
```
```  1829   thus ?thesis by auto
```
```  1830 qed
```
```  1831
```
```  1832 lemma cos_monotone_0_pi': assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" shows "cos x \<le> cos y"
```
```  1833 proof (cases "y < x")
```
```  1834   case True show ?thesis using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
```
```  1835 next
```
```  1836   case False hence "y = x" using `y \<le> x` by auto
```
```  1837   thus ?thesis by auto
```
```  1838 qed
```
```  1839
```
```  1840 lemma cos_monotone_minus_pi_0: assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
```
```  1841   shows "cos y < cos x"
```
```  1842 proof -
```
```  1843   have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" using assms by auto
```
```  1844   from cos_monotone_0_pi[OF this]
```
```  1845   show ?thesis unfolding cos_minus .
```
```  1846 qed
```
```  1847
```
```  1848 lemma cos_monotone_minus_pi_0': assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0" shows "cos y \<le> cos x"
```
```  1849 proof (cases "y < x")
```
```  1850   case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] by auto
```
```  1851 next
```
```  1852   case False hence "y = x" using `y \<le> x` by auto
```
```  1853   thus ?thesis by auto
```
```  1854 qed
```
```  1855
```
```  1856 lemma sin_monotone_2pi': assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2" shows "sin y \<le> sin x"
```
```  1857 proof -
```
```  1858   have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi"
```
```  1859     using pi_ge_two and assms by auto
```
```  1860   from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] by auto
```
```  1861 qed
```
```  1862
```
```  1863 subsection {* Tangent *}
```
```  1864
```
```  1865 definition tan :: "real \<Rightarrow> real" where
```
```  1866   "tan = (\<lambda>x. sin x / cos x)"
```
```  1867
```
```  1868 lemma tan_zero [simp]: "tan 0 = 0"
```
```  1869   by (simp add: tan_def)
```
```  1870
```
```  1871 lemma tan_pi [simp]: "tan pi = 0"
```
```  1872   by (simp add: tan_def)
```
```  1873
```
```  1874 lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
```
```  1875   by (simp add: tan_def)
```
```  1876
```
```  1877 lemma tan_minus [simp]: "tan (-x) = - tan x"
```
```  1878   by (simp add: tan_def)
```
```  1879
```
```  1880 lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
```
```  1881   by (simp add: tan_def)
```
```  1882
```
```  1883 lemma lemma_tan_add1:
```
```  1884   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
```
```  1885   by (simp add: tan_def cos_add field_simps)
```
```  1886
```
```  1887 lemma add_tan_eq:
```
```  1888   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
```
```  1889   by (simp add: tan_def sin_add field_simps)
```
```  1890
```
```  1891 lemma tan_add:
```
```  1892      "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
```
```  1893       ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
```
```  1894   by (simp add: add_tan_eq lemma_tan_add1, simp add: tan_def)
```
```  1895
```
```  1896 lemma tan_double:
```
```  1897      "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
```
```  1898       ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
```
```  1899   using tan_add [of x x] by (simp add: power2_eq_square)
```
```  1900
```
```  1901 lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
```
```  1902 by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
```
```  1903
```
```  1904 lemma tan_less_zero:
```
```  1905   assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
```
```  1906 proof -
```
```  1907   have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
```
```  1908   thus ?thesis by simp
```
```  1909 qed
```
```  1910
```
```  1911 lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"
```
```  1912   unfolding tan_def sin_double cos_double sin_squared_eq
```
```  1913   by (simp add: power2_eq_square)
```
```  1914
```
```  1915 lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<twosuperior>)"
```
```  1916   unfolding tan_def
```
```  1917   by (auto intro!: DERIV_intros, simp add: divide_inverse power2_eq_square)
```
```  1918
```
```  1919 lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
```
```  1920   by (rule DERIV_tan [THEN DERIV_isCont])
```
```  1921
```
```  1922 lemma isCont_tan' [simp]:
```
```  1923   "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
```
```  1924   by (rule isCont_o2 [OF _ isCont_tan])
```
```  1925
```
```  1926 lemma tendsto_tan [tendsto_intros]:
```
```  1927   "\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F"
```
```  1928   by (rule isCont_tendsto_compose [OF isCont_tan])
```
```  1929
```
```  1930 lemma LIM_cos_div_sin: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
```
```  1931   by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
```
```  1932
```
```  1933 lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
```
```  1934 apply (cut_tac LIM_cos_div_sin)
```
```  1935 apply (simp only: LIM_eq)
```
```  1936 apply (drule_tac x = "inverse y" in spec, safe, force)
```
```  1937 apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
```
```  1938 apply (rule_tac x = "(pi/2) - e" in exI)
```
```  1939 apply (simp (no_asm_simp))
```
```  1940 apply (drule_tac x = "(pi/2) - e" in spec)
```
```  1941 apply (auto simp add: tan_def sin_diff cos_diff)
```
```  1942 apply (rule inverse_less_iff_less [THEN iffD1])
```
```  1943 apply (auto simp add: divide_inverse)
```
```  1944 apply (rule mult_pos_pos)
```
```  1945 apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
```
```  1946 apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
```
```  1947 done
```
```  1948
```
```  1949 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
```
```  1950 apply (frule order_le_imp_less_or_eq, safe)
```
```  1951  prefer 2 apply force
```
```  1952 apply (drule lemma_tan_total, safe)
```
```  1953 apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
```
```  1954 apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
```
```  1955 apply (drule_tac y = xa in order_le_imp_less_or_eq)
```
```  1956 apply (auto dest: cos_gt_zero)
```
```  1957 done
```
```  1958
```
```  1959 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  1960 apply (cut_tac linorder_linear [of 0 y], safe)
```
```  1961 apply (drule tan_total_pos)
```
```  1962 apply (cut_tac [2] y="-y" in tan_total_pos, safe)
```
```  1963 apply (rule_tac [3] x = "-x" in exI)
```
```  1964 apply (auto del: exI intro!: exI)
```
```  1965 done
```
```  1966
```
```  1967 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  1968 apply (cut_tac y = y in lemma_tan_total1, auto)
```
```  1969 apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
```
```  1970 apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
```
```  1971 apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
```
```  1972 apply (rule_tac [4] Rolle)
```
```  1973 apply (rule_tac [2] Rolle)
```
```  1974 apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
```
```  1975             simp add: differentiable_def)
```
```  1976 txt{*Now, simulate TRYALL*}
```
```  1977 apply (rule_tac [!] DERIV_tan asm_rl)
```
```  1978 apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
```
```  1979             simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
```
```  1980 done
```
```  1981
```
```  1982 lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
```
```  1983   shows "tan y < tan x"
```
```  1984 proof -
```
```  1985   have "\<forall> x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse (cos x'^2)"
```
```  1986   proof (rule allI, rule impI)
```
```  1987     fix x' :: real assume "y \<le> x' \<and> x' \<le> x"
```
```  1988     hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
```
```  1989     from cos_gt_zero_pi[OF this]
```
```  1990     have "cos x' \<noteq> 0" by auto
```
```  1991     thus "DERIV tan x' :> inverse (cos x'^2)" by (rule DERIV_tan)
```
```  1992   qed
```
```  1993   from MVT2[OF `y < x` this]
```
```  1994   obtain z where "y < z" and "z < x" and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<twosuperior>)" by auto
```
```  1995   hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
```
```  1996   hence "0 < cos z" using cos_gt_zero_pi by auto
```
```  1997   hence inv_pos: "0 < inverse ((cos z)\<twosuperior>)" by auto
```
```  1998   have "0 < x - y" using `y < x` by auto
```
```  1999   from mult_pos_pos [OF this inv_pos]
```
```  2000   have "0 < tan x - tan y" unfolding tan_diff by auto
```
```  2001   thus ?thesis by auto
```
```  2002 qed
```
```  2003
```
```  2004 lemma tan_monotone': assumes "- (pi / 2) < y" and "y < pi / 2" and "- (pi / 2) < x" and "x < pi / 2"
```
```  2005   shows "(y < x) = (tan y < tan x)"
```
```  2006 proof
```
```  2007   assume "y < x" thus "tan y < tan x" using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
```
```  2008 next
```
```  2009   assume "tan y < tan x"
```
```  2010   show "y < x"
```
```  2011   proof (rule ccontr)
```
```  2012     assume "\<not> y < x" hence "x \<le> y" by auto
```
```  2013     hence "tan x \<le> tan y"
```
```  2014     proof (cases "x = y")
```
```  2015       case True thus ?thesis by auto
```
```  2016     next
```
```  2017       case False hence "x < y" using `x \<le> y` by auto
```
```  2018       from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
```
```  2019     qed
```
```  2020     thus False using `tan y < tan x` by auto
```
```  2021   qed
```
```  2022 qed
```
```  2023
```
```  2024 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
```
```  2025
```
```  2026 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
```
```  2027   by (simp add: tan_def)
```
```  2028
```
```  2029 lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x"
```
```  2030 proof (induct n arbitrary: x)
```
```  2031   case (Suc n)
```
```  2032   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
```
```  2033   show ?case unfolding split_pi_off using Suc by auto
```
```  2034 qed auto
```
```  2035
```
```  2036 lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
```
```  2037 proof (cases "0 \<le> i")
```
```  2038   case True hence i_nat: "real i = real (nat i)" by auto
```
```  2039   show ?thesis unfolding i_nat by auto
```
```  2040 next
```
```  2041   case False hence i_nat: "real i = - real (nat (-i))" by auto
```
```  2042   have "tan x = tan (x + real i * pi - real i * pi)" by auto
```
```  2043   also have "\<dots> = tan (x + real i * pi)" unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
```
```  2044   finally show ?thesis by auto
```
```  2045 qed
```
```  2046
```
```  2047 lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
```
```  2048   using tan_periodic_int[of _ "numeral n" ] unfolding real_numeral .
```
```  2049
```
```  2050 subsection {* Inverse Trigonometric Functions *}
```
```  2051
```
```  2052 definition
```
```  2053   arcsin :: "real => real" where
```
```  2054   "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
```
```  2055
```
```  2056 definition
```
```  2057   arccos :: "real => real" where
```
```  2058   "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
```
```  2059
```
```  2060 definition
```
```  2061   arctan :: "real => real" where
```
```  2062   "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
```
```  2063
```
```  2064 lemma arcsin:
```
```  2065      "[| -1 \<le> y; y \<le> 1 |]
```
```  2066       ==> -(pi/2) \<le> arcsin y &
```
```  2067            arcsin y \<le> pi/2 & sin(arcsin y) = y"
```
```  2068 unfolding arcsin_def by (rule theI' [OF sin_total])
```
```  2069
```
```  2070 lemma arcsin_pi:
```
```  2071      "[| -1 \<le> y; y \<le> 1 |]
```
```  2072       ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
```
```  2073 apply (drule (1) arcsin)
```
```  2074 apply (force intro: order_trans)
```
```  2075 done
```
```  2076
```
```  2077 lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
```
```  2078 by (blast dest: arcsin)
```
```  2079
```
```  2080 lemma arcsin_bounded:
```
```  2081      "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
```
```  2082 by (blast dest: arcsin)
```
```  2083
```
```  2084 lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
```
```  2085 by (blast dest: arcsin)
```
```  2086
```
```  2087 lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
```
```  2088 by (blast dest: arcsin)
```
```  2089
```
```  2090 lemma arcsin_lt_bounded:
```
```  2091      "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
```
```  2092 apply (frule order_less_imp_le)
```
```  2093 apply (frule_tac y = y in order_less_imp_le)
```
```  2094 apply (frule arcsin_bounded)
```
```  2095 apply (safe, simp)
```
```  2096 apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
```
```  2097 apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
```
```  2098 apply (drule_tac [!] f = sin in arg_cong, auto)
```
```  2099 done
```
```  2100
```
```  2101 lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
```
```  2102 apply (unfold arcsin_def)
```
```  2103 apply (rule the1_equality)
```
```  2104 apply (rule sin_total, auto)
```
```  2105 done
```
```  2106
```
```  2107 lemma arccos:
```
```  2108      "[| -1 \<le> y; y \<le> 1 |]
```
```  2109       ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
```
```  2110 unfolding arccos_def by (rule theI' [OF cos_total])
```
```  2111
```
```  2112 lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
```
```  2113 by (blast dest: arccos)
```
```  2114
```
```  2115 lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
```
```  2116 by (blast dest: arccos)
```
```  2117
```
```  2118 lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
```
```  2119 by (blast dest: arccos)
```
```  2120
```
```  2121 lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
```
```  2122 by (blast dest: arccos)
```
```  2123
```
```  2124 lemma arccos_lt_bounded:
```
```  2125      "[| -1 < y; y < 1 |]
```
```  2126       ==> 0 < arccos y & arccos y < pi"
```
```  2127 apply (frule order_less_imp_le)
```
```  2128 apply (frule_tac y = y in order_less_imp_le)
```
```  2129 apply (frule arccos_bounded, auto)
```
```  2130 apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
```
```  2131 apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
```
```  2132 apply (drule_tac [!] f = cos in arg_cong, auto)
```
```  2133 done
```
```  2134
```
```  2135 lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
```
```  2136 apply (simp add: arccos_def)
```
```  2137 apply (auto intro!: the1_equality cos_total)
```
```  2138 done
```
```  2139
```
```  2140 lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
```
```  2141 apply (simp add: arccos_def)
```
```  2142 apply (auto intro!: the1_equality cos_total)
```
```  2143 done
```
```  2144
```
```  2145 lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<twosuperior>)"
```
```  2146 apply (subgoal_tac "x\<twosuperior> \<le> 1")
```
```  2147 apply (rule power2_eq_imp_eq)
```
```  2148 apply (simp add: cos_squared_eq)
```
```  2149 apply (rule cos_ge_zero)
```
```  2150 apply (erule (1) arcsin_lbound)
```
```  2151 apply (erule (1) arcsin_ubound)
```
```  2152 apply simp
```
```  2153 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
```
```  2154 apply (rule power_mono, simp, simp)
```
```  2155 done
```
```  2156
```
```  2157 lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<twosuperior>)"
```
```  2158 apply (subgoal_tac "x\<twosuperior> \<le> 1")
```
```  2159 apply (rule power2_eq_imp_eq)
```
```  2160 apply (simp add: sin_squared_eq)
```
```  2161 apply (rule sin_ge_zero)
```
```  2162 apply (erule (1) arccos_lbound)
```
```  2163 apply (erule (1) arccos_ubound)
```
```  2164 apply simp
```
```  2165 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
```
```  2166 apply (rule power_mono, simp, simp)
```
```  2167 done
```
```  2168
```
```  2169 lemma arctan [simp]:
```
```  2170      "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
```
```  2171 unfolding arctan_def by (rule theI' [OF tan_total])
```
```  2172
```
```  2173 lemma tan_arctan: "tan(arctan y) = y"
```
```  2174 by auto
```
```  2175
```
```  2176 lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
```
```  2177 by (auto simp only: arctan)
```
```  2178
```
```  2179 lemma arctan_lbound: "- (pi/2) < arctan y"
```
```  2180 by auto
```
```  2181
```
```  2182 lemma arctan_ubound: "arctan y < pi/2"
```
```  2183 by (auto simp only: arctan)
```
```  2184
```
```  2185 lemma arctan_unique:
```
```  2186   assumes "-(pi/2) < x" and "x < pi/2" and "tan x = y"
```
```  2187   shows "arctan y = x"
```
```  2188   using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
```
```  2189
```
```  2190 lemma arctan_tan:
```
```  2191       "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
```
```  2192   by (rule arctan_unique, simp_all)
```
```  2193
```
```  2194 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
```
```  2195   by (rule arctan_unique, simp_all)
```
```  2196
```
```  2197 lemma arctan_minus: "arctan (- x) = - arctan x"
```
```  2198   apply (rule arctan_unique)
```
```  2199   apply (simp only: neg_less_iff_less arctan_ubound)
```
```  2200   apply (metis minus_less_iff arctan_lbound)
```
```  2201   apply simp
```
```  2202   done
```
```  2203
```
```  2204 lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
```
```  2205   by (intro less_imp_neq [symmetric] cos_gt_zero_pi
```
```  2206     arctan_lbound arctan_ubound)
```
```  2207
```
```  2208 lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<twosuperior>)"
```
```  2209 proof (rule power2_eq_imp_eq)
```
```  2210   have "0 < 1 + x\<twosuperior>" by (simp add: add_pos_nonneg)
```
```  2211   show "0 \<le> 1 / sqrt (1 + x\<twosuperior>)" by simp
```
```  2212   show "0 \<le> cos (arctan x)"
```
```  2213     by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
```
```  2214   have "(cos (arctan x))\<twosuperior> * (1 + (tan (arctan x))\<twosuperior>) = 1"
```
```  2215     unfolding tan_def by (simp add: right_distrib power_divide)
```
```  2216   thus "(cos (arctan x))\<twosuperior> = (1 / sqrt (1 + x\<twosuperior>))\<twosuperior>"
```
```  2217     using `0 < 1 + x\<twosuperior>` by (simp add: power_divide eq_divide_eq)
```
```  2218 qed
```
```  2219
```
```  2220 lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<twosuperior>)"
```
```  2221   using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
```
```  2222   using tan_arctan [of x] unfolding tan_def cos_arctan
```
```  2223   by (simp add: eq_divide_eq)
```
```  2224
```
```  2225 lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
```
```  2226 apply (rule power_inverse [THEN subst])
```
```  2227 apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
```
```  2228 apply (auto dest: field_power_not_zero
```
```  2229         simp add: power_mult_distrib left_distrib power_divide tan_def
```
```  2230                   mult_assoc power_inverse [symmetric])
```
```  2231 done
```
```  2232
```
```  2233 lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
```
```  2234   by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
```
```  2235
```
```  2236 lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
```
```  2237   by (simp only: not_less [symmetric] arctan_less_iff)
```
```  2238
```
```  2239 lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
```
```  2240   by (simp only: eq_iff [where 'a=real] arctan_le_iff)
```
```  2241
```
```  2242 lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
```
```  2243   using arctan_less_iff [of 0 x] by simp
```
```  2244
```
```  2245 lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
```
```  2246   using arctan_less_iff [of x 0] by simp
```
```  2247
```
```  2248 lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
```
```  2249   using arctan_le_iff [of 0 x] by simp
```
```  2250
```
```  2251 lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```  2252   using arctan_le_iff [of x 0] by simp
```
```  2253
```
```  2254 lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
```
```  2255   using arctan_eq_iff [of x 0] by simp
```
```  2256
```
```  2257 lemma isCont_inverse_function2:
```
```  2258   fixes f g :: "real \<Rightarrow> real" shows
```
```  2259   "\<lbrakk>a < x; x < b;
```
```  2260     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
```
```  2261     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
```
```  2262    \<Longrightarrow> isCont g (f x)"
```
```  2263 apply (rule isCont_inverse_function
```
```  2264        [where f=f and d="min (x - a) (b - x)"])
```
```  2265 apply (simp_all add: abs_le_iff)
```
```  2266 done
```
```  2267
```
```  2268 lemma isCont_arcsin: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arcsin x"
```
```  2269 apply (subgoal_tac "isCont arcsin (sin (arcsin x))", simp)
```
```  2270 apply (rule isCont_inverse_function2 [where f=sin])
```
```  2271 apply (erule (1) arcsin_lt_bounded [THEN conjunct1])
```
```  2272 apply (erule (1) arcsin_lt_bounded [THEN conjunct2])
```
```  2273 apply (fast intro: arcsin_sin, simp)
```
```  2274 done
```
```  2275
```
```  2276 lemma isCont_arccos: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arccos x"
```
```  2277 apply (subgoal_tac "isCont arccos (cos (arccos x))", simp)
```
```  2278 apply (rule isCont_inverse_function2 [where f=cos])
```
```  2279 apply (erule (1) arccos_lt_bounded [THEN conjunct1])
```
```  2280 apply (erule (1) arccos_lt_bounded [THEN conjunct2])
```
```  2281 apply (fast intro: arccos_cos, simp)
```
```  2282 done
```
```  2283
```
```  2284 lemma isCont_arctan: "isCont arctan x"
```
```  2285 apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
```
```  2286 apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
```
```  2287 apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
```
```  2288 apply (erule (1) isCont_inverse_function2 [where f=tan])
```
```  2289 apply (metis arctan_tan order_le_less_trans order_less_le_trans)
```
```  2290 apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
```
```  2291 done
```
```  2292
```
```  2293 lemma DERIV_arcsin:
```
```  2294   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<twosuperior>))"
```
```  2295 apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
```
```  2296 apply (rule DERIV_cong [OF DERIV_sin])
```
```  2297 apply (simp add: cos_arcsin)
```
```  2298 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
```
```  2299 apply (rule power_strict_mono, simp, simp, simp)
```
```  2300 apply assumption
```
```  2301 apply assumption
```
```  2302 apply simp
```
```  2303 apply (erule (1) isCont_arcsin)
```
```  2304 done
```
```  2305
```
```  2306 lemma DERIV_arccos:
```
```  2307   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<twosuperior>))"
```
```  2308 apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
```
```  2309 apply (rule DERIV_cong [OF DERIV_cos])
```
```  2310 apply (simp add: sin_arccos)
```
```  2311 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
```
```  2312 apply (rule power_strict_mono, simp, simp, simp)
```
```  2313 apply assumption
```
```  2314 apply assumption
```
```  2315 apply simp
```
```  2316 apply (erule (1) isCont_arccos)
```
```  2317 done
```
```  2318
```
```  2319 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<twosuperior>)"
```
```  2320 apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
```
```  2321 apply (rule DERIV_cong [OF DERIV_tan])
```
```  2322 apply (rule cos_arctan_not_zero)
```
```  2323 apply (simp add: power_inverse tan_sec [symmetric])
```
```  2324 apply (subgoal_tac "0 < 1 + x\<twosuperior>", simp)
```
```  2325 apply (simp add: add_pos_nonneg)
```
```  2326 apply (simp, simp, simp, rule isCont_arctan)
```
```  2327 done
```
```  2328
```
```  2329 declare
```
```  2330   DERIV_arcsin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  2331   DERIV_arccos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  2332   DERIV_arctan[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  2333
```
```  2334 subsection {* More Theorems about Sin and Cos *}
```
```  2335
```
```  2336 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
```
```  2337 proof -
```
```  2338   let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
```
```  2339   have nonneg: "0 \<le> ?c"
```
```  2340     by (simp add: cos_ge_zero)
```
```  2341   have "0 = cos (pi / 4 + pi / 4)"
```
```  2342     by simp
```
```  2343   also have "cos (pi / 4 + pi / 4) = ?c\<twosuperior> - ?s\<twosuperior>"
```
```  2344     by (simp only: cos_add power2_eq_square)
```
```  2345   also have "\<dots> = 2 * ?c\<twosuperior> - 1"
```
```  2346     by (simp add: sin_squared_eq)
```
```  2347   finally have "?c\<twosuperior> = (sqrt 2 / 2)\<twosuperior>"
```
```  2348     by (simp add: power_divide)
```
```  2349   thus ?thesis
```
```  2350     using nonneg by (rule power2_eq_imp_eq) simp
```
```  2351 qed
```
```  2352
```
```  2353 lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"
```
```  2354 proof -
```
```  2355   let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
```
```  2356   have pos_c: "0 < ?c"
```
```  2357     by (rule cos_gt_zero, simp, simp)
```
```  2358   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
```
```  2359     by simp
```
```  2360   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
```
```  2361     by (simp only: cos_add sin_add)
```
```  2362   also have "\<dots> = ?c * (?c\<twosuperior> - 3 * ?s\<twosuperior>)"
```
```  2363     by (simp add: algebra_simps power2_eq_square)
```
```  2364   finally have "?c\<twosuperior> = (sqrt 3 / 2)\<twosuperior>"
```
```  2365     using pos_c by (simp add: sin_squared_eq power_divide)
```
```  2366   thus ?thesis
```
```  2367     using pos_c [THEN order_less_imp_le]
```
```  2368     by (rule power2_eq_imp_eq) simp
```
```  2369 qed
```
```  2370
```
```  2371 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
```
```  2372 by (simp add: sin_cos_eq cos_45)
```
```  2373
```
```  2374 lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
```
```  2375 by (simp add: sin_cos_eq cos_30)
```
```  2376
```
```  2377 lemma cos_60: "cos (pi / 3) = 1 / 2"
```
```  2378 apply (rule power2_eq_imp_eq)
```
```  2379 apply (simp add: cos_squared_eq sin_60 power_divide)
```
```  2380 apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
```
```  2381 done
```
```  2382
```
```  2383 lemma sin_30: "sin (pi / 6) = 1 / 2"
```
```  2384 by (simp add: sin_cos_eq cos_60)
```
```  2385
```
```  2386 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
```
```  2387 unfolding tan_def by (simp add: sin_30 cos_30)
```
```  2388
```
```  2389 lemma tan_45: "tan (pi / 4) = 1"
```
```  2390 unfolding tan_def by (simp add: sin_45 cos_45)
```
```  2391
```
```  2392 lemma tan_60: "tan (pi / 3) = sqrt 3"
```
```  2393 unfolding tan_def by (simp add: sin_60 cos_60)
```
```  2394
```
```  2395 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
```
```  2396 proof -
```
```  2397   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
```
```  2398     by (auto simp add: algebra_simps sin_add)
```
```  2399   thus ?thesis
```
```  2400     by (simp add: real_of_nat_Suc left_distrib add_divide_distrib
```
```  2401                   mult_commute [of pi])
```
```  2402 qed
```
```  2403
```
```  2404 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
```
```  2405 by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
```
```  2406
```
```  2407 lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
```
```  2408 apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
```
```  2409 apply (subst cos_add, simp)
```
```  2410 done
```
```  2411
```
```  2412 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
```
```  2413 by (auto simp add: mult_assoc)
```
```  2414
```
```  2415 lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
```
```  2416 apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
```
```  2417 apply (subst sin_add, simp)
```
```  2418 done
```
```  2419
```
```  2420 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
```
```  2421 by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto)
```
```  2422
```
```  2423 lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
```
```  2424   by (auto intro!: DERIV_intros)
```
```  2425
```
```  2426 lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
```
```  2427 by (auto simp add: sin_zero_iff even_mult_two_ex)
```
```  2428
```
```  2429 lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
```
```  2430 by (cut_tac x = x in sin_cos_squared_add3, auto)
```
```  2431
```
```  2432 subsection {* Machins formula *}
```
```  2433
```
```  2434 lemma arctan_one: "arctan 1 = pi / 4"
```
```  2435   by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi)
```
```  2436
```
```  2437 lemma tan_total_pi4: assumes "\<bar>x\<bar> < 1"
```
```  2438   shows "\<exists> z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
```
```  2439 proof
```
```  2440   show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
```
```  2441     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  2442     unfolding arctan_less_iff using assms by auto
```
```  2443 qed
```
```  2444
```
```  2445 lemma arctan_add: assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
```
```  2446   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
```
```  2447 proof (rule arctan_unique [symmetric])
```
```  2448   have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y"
```
```  2449     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  2450     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  2451   from add_le_less_mono [OF this]
```
```  2452   show 1: "- (pi / 2) < arctan x + arctan y" by simp
```
```  2453   have "arctan x \<le> pi / 4" and "arctan y < pi / 4"
```
```  2454     unfolding arctan_one [symmetric]
```
```  2455     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  2456   from add_le_less_mono [OF this]
```
```  2457   show 2: "arctan x + arctan y < pi / 2" by simp
```
```  2458   show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
```
```  2459     using cos_gt_zero_pi [OF 1 2] by (simp add: tan_add)
```
```  2460 qed
```
```  2461
```
```  2462 theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
```
```  2463 proof -
```
```  2464   have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
```
```  2465   from arctan_add[OF less_imp_le[OF this] this]
```
```  2466   have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
```
```  2467   moreover
```
```  2468   have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
```
```  2469   from arctan_add[OF less_imp_le[OF this] this]
```
```  2470   have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
```
```  2471   moreover
```
```  2472   have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
```
```  2473   from arctan_add[OF this]
```
```  2474   have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
```
```  2475   ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
```
```  2476   thus ?thesis unfolding arctan_one by algebra
```
```  2477 qed
```
```  2478
```
```  2479 subsection {* Introducing the arcus tangens power series *}
```
```  2480
```
```  2481 lemma monoseq_arctan_series: fixes x :: real
```
```  2482   assumes "\<bar>x\<bar> \<le> 1" shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
```
```  2483 proof (cases "x = 0") case True thus ?thesis unfolding monoseq_def One_nat_def by auto
```
```  2484 next
```
```  2485   case False
```
```  2486   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  2487   show "monoseq ?a"
```
```  2488   proof -
```
```  2489     { fix n fix x :: real assume "0 \<le> x" and "x \<le> 1"
```
```  2490       have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le> 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
```
```  2491       proof (rule mult_mono)
```
```  2492         show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))" by (rule frac_le) simp_all
```
```  2493         show "0 \<le> 1 / real (Suc (n * 2))" by auto
```
```  2494         show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)" by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`)
```
```  2495         show "0 \<le> x ^ Suc (Suc n * 2)" by (rule zero_le_power) (simp add: `0 \<le> x`)
```
```  2496       qed
```
```  2497     } note mono = this
```
```  2498
```
```  2499     show ?thesis
```
```  2500     proof (cases "0 \<le> x")
```
```  2501       case True from mono[OF this `x \<le> 1`, THEN allI]
```
```  2502       show ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI2)
```
```  2503     next
```
```  2504       case False hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto
```
```  2505       from mono[OF this]
```
```  2506       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
```
```  2507       thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
```
```  2508     qed
```
```  2509   qed
```
```  2510 qed
```
```  2511
```
```  2512 lemma zeroseq_arctan_series: fixes x :: real
```
```  2513   assumes "\<bar>x\<bar> \<le> 1" shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
```
```  2514 proof (cases "x = 0") case True thus ?thesis unfolding One_nat_def by (auto simp add: tendsto_const)
```
```  2515 next
```
```  2516   case False
```
```  2517   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  2518   show "?a ----> 0"
```
```  2519   proof (cases "\<bar>x\<bar> < 1")
```
```  2520     case True hence "norm x < 1" by auto
```
```  2521     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]]
```
```  2522     have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"
```
```  2523       unfolding inverse_eq_divide Suc_eq_plus1 by simp
```
```  2524     then show ?thesis using pos2 by (rule LIMSEQ_linear)
```
```  2525   next
```
```  2526     case False hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  2527     hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x" unfolding One_nat_def by auto
```
```  2528     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
```
```  2529     show ?thesis unfolding n_eq Suc_eq_plus1 by auto
```
```  2530   qed
```
```  2531 qed
```
```  2532
```
```  2533 lemma summable_arctan_series: fixes x :: real and n :: nat
```
```  2534   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)")
```
```  2535   by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
```
```  2536
```
```  2537 lemma less_one_imp_sqr_less_one: fixes x :: real assumes "\<bar>x\<bar> < 1" shows "x^2 < 1"
```
```  2538 proof -
```
```  2539   from mult_left_mono[OF less_imp_le[OF `\<bar>x\<bar> < 1`] abs_ge_zero[of x]]
```
```  2540   have "\<bar> x^2 \<bar> < 1" using `\<bar> x \<bar> < 1` unfolding numeral_2_eq_2 power_Suc2 by auto
```
```  2541   thus ?thesis using zero_le_power2 by auto
```
```  2542 qed
```
```  2543
```
```  2544 lemma DERIV_arctan_series: assumes "\<bar> x \<bar> < 1"
```
```  2545   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")
```
```  2546 proof -
```
```  2547   let "?f n" = "if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
```
```  2548
```
```  2549   { fix n :: nat assume "even n" hence "2 * (n div 2) = n" by presburger } note n_even=this
```
```  2550   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
```
```  2551
```
```  2552   { fix x :: real assume "\<bar>x\<bar> < 1" hence "x^2 < 1" by (rule less_one_imp_sqr_less_one)
```
```  2553     have "summable (\<lambda> n. -1 ^ n * (x^2) ^n)"
```
```  2554       by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x^2 < 1` order_less_imp_le[OF `x^2 < 1`])
```
```  2555     hence "summable (\<lambda> n. -1 ^ n * x^(2*n))" unfolding power_mult .
```
```  2556   } note summable_Integral = this
```
```  2557
```
```  2558   { fix f :: "nat \<Rightarrow> real"
```
```  2559     have "\<And> x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  2560     proof
```
```  2561       fix x :: real assume "f sums x"
```
```  2562       from sums_if[OF sums_zero this]
```
```  2563       show "(\<lambda> n. if even n then f (n div 2) else 0) sums x" by auto
```
```  2564     next
```
```  2565       fix x :: real assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  2566       from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult_commute]]
```
```  2567       show "f sums x" unfolding sums_def by auto
```
```  2568     qed
```
```  2569     hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
```
```  2570   } note sums_even = this
```
```  2571
```
```  2572   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]
```
```  2573     by auto
```
```  2574
```
```  2575   { fix x :: real
```
```  2576     have if_eq': "\<And> n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
```
```  2577       (if even n then -1 ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
```
```  2578       using n_even by auto
```
```  2579     have idx_eq: "\<And> n. n * 2 + 1 = Suc (2 * n)" by auto
```
```  2580     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]
```
```  2581       by auto
```
```  2582   } note arctan_eq = this
```
```  2583
```
```  2584   have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
```
```  2585   proof (rule DERIV_power_series')
```
```  2586     show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto
```
```  2587     { fix x' :: real assume x'_bounds: "x' \<in> {- 1 <..< 1}"
```
```  2588       hence "\<bar>x'\<bar> < 1" by auto
```
```  2589
```
```  2590       let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
```
```  2591       show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
```
```  2592         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`])
```
```  2593     }
```
```  2594   qed auto
```
```  2595   thus ?thesis unfolding Int_eq arctan_eq .
```
```  2596 qed
```
```  2597
```
```  2598 lemma arctan_series: assumes "\<bar> x \<bar> \<le> 1"
```
```  2599   shows "arctan x = (\<Sum> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "_ = suminf (\<lambda> n. ?c x n)")
```
```  2600 proof -
```
```  2601   let "?c' x n" = "(-1)^n * x^(n*2)"
```
```  2602
```
```  2603   { fix r x :: real assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
```
```  2604     have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto
```
```  2605     from DERIV_arctan_series[OF this]
```
```  2606     have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
```
```  2607   } note DERIV_arctan_suminf = this
```
```  2608
```
```  2609   { fix x :: real assume "\<bar>x\<bar> \<le> 1" note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] }
```
```  2610   note arctan_series_borders = this
```
```  2611
```
```  2612   { fix x :: real assume "\<bar>x\<bar> < 1" have "arctan x = (\<Sum> k. ?c x k)"
```
```  2613   proof -
```
```  2614     obtain r where "\<bar>x\<bar> < r" and "r < 1" using dense[OF `\<bar>x\<bar> < 1`] by blast
```
```  2615     hence "0 < r" and "-r < x" and "x < r" by auto
```
```  2616
```
```  2617     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"
```
```  2618     proof -
```
```  2619       fix x a b assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
```
```  2620       hence "\<bar>x\<bar> < r" by auto
```
```  2621       show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  2622       proof (rule DERIV_isconst2[of "a" "b"])
```
```  2623         show "a < b" and "a \<le> x" and "x \<le> b" using `a < b` `a \<le> x` `x \<le> b` by auto
```
```  2624         have "\<forall> x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  2625         proof (rule allI, rule impI)
```
```  2626           fix x assume "-r < x \<and> x < r" hence "\<bar>x\<bar> < r" by auto
```
```  2627           hence "\<bar>x\<bar> < 1" using `r < 1` by auto
```
```  2628           have "\<bar> - (x^2) \<bar> < 1" using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
```
```  2629           hence "(\<lambda> n. (- (x^2)) ^ n) sums (1 / (1 - (- (x^2))))" unfolding real_norm_def[symmetric] by (rule geometric_sums)
```
```  2630           hence "(?c' x) sums (1 / (1 - (- (x^2))))" unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto
```
```  2631           hence suminf_c'_eq_geom: "inverse (1 + x^2) = suminf (?c' x)" using sums_unique unfolding inverse_eq_divide by auto
```
```  2632           have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x^2))" unfolding suminf_c'_eq_geom
```
```  2633             by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
```
```  2634           from DERIV_add_minus[OF this DERIV_arctan]
```
```  2635           show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0" unfolding diff_minus by auto
```
```  2636         qed
```
```  2637         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
```
```  2638         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
```
```  2639         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
```
```  2640       qed
```
```  2641     qed
```
```  2642
```
```  2643     have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
```
```  2644       unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero by auto
```
```  2645
```
```  2646     have "suminf (?c x) - arctan x = 0"
```
```  2647     proof (cases "x = 0")
```
```  2648       case True thus ?thesis using suminf_arctan_zero by auto
```
```  2649     next
```
```  2650       case False hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
```
```  2651       have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
```
```  2652         by (rule suminf_eq_arctan_bounded[where x="0" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>", symmetric])
```
```  2653           (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  2654       moreover
```
```  2655       have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
```
```  2656         by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"])
```
```  2657           (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  2658       ultimately
```
```  2659       show ?thesis using suminf_arctan_zero by auto
```
```  2660     qed
```
```  2661     thus ?thesis by auto
```
```  2662   qed } note when_less_one = this
```
```  2663
```
```  2664   show "arctan x = suminf (\<lambda> n. ?c x n)"
```
```  2665   proof (cases "\<bar>x\<bar> < 1")
```
```  2666     case True thus ?thesis by (rule when_less_one)
```
```  2667   next case False hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  2668     let "?a x n" = "\<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
```
```  2669     let "?diff x n" = "\<bar> arctan x - (\<Sum> i = 0..<n. ?c x i)\<bar>"
```
```  2670     { fix n :: nat
```
```  2671       have "0 < (1 :: real)" by auto
```
```  2672       moreover
```
```  2673       { fix x :: real assume "0 < x" and "x < 1" hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
```
```  2674         from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)" by auto
```
```  2675         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]
```
```  2676         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)
```
```  2677         hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)" by (rule abs_of_pos)
```
```  2678         have "?diff x n \<le> ?a x n"
```
```  2679         proof (cases "even n")
```
```  2680           case True hence sgn_pos: "(-1)^n = (1::real)" by auto
```
```  2681           from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
```
```  2682           from bounds[of m, unfolded this atLeastAtMost_iff]
```
```  2683           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
```
```  2684           also have "\<dots> = ?c x n" unfolding One_nat_def by auto
```
```  2685           also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
```
```  2686           finally show ?thesis .
```
```  2687         next
```
```  2688           case False hence sgn_neg: "(-1)^n = (-1::real)" by auto
```
```  2689           from `odd n` obtain m where m_def: "2 * m + 1 = n" unfolding odd_Suc_mult_two_ex by auto
```
```  2690           hence m_plus: "2 * (m + 1) = n + 1" by auto
```
```  2691           from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
```
```  2692           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
```
```  2693           also have "\<dots> = - ?c x n" unfolding One_nat_def by auto
```
```  2694           also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
```
```  2695           finally show ?thesis .
```
```  2696         qed
```
```  2697         hence "0 \<le> ?a x n - ?diff x n" by auto
```
```  2698       }
```
```  2699       hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
```
```  2700       moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
```
```  2701         unfolding diff_minus divide_inverse
```
```  2702         by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum)
```
```  2703       ultimately have "0 \<le> ?a 1 n - ?diff 1 n" by (rule LIM_less_bound)
```
```  2704       hence "?diff 1 n \<le> ?a 1 n" by auto
```
```  2705     }
```
```  2706     have "?a 1 ----> 0"
```
```  2707       unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
```
```  2708       by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)
```
```  2709     have "?diff 1 ----> 0"
```
```  2710     proof (rule LIMSEQ_I)
```
```  2711       fix r :: real assume "0 < r"
```
```  2712       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
```
```  2713       { fix n assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this]
```
```  2714         have "norm (?diff 1 n - 0) < r" by auto }
```
```  2715       thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
```
```  2716     qed
```
```  2717     from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
```
```  2718     have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
```
```  2719     hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
```
```  2720
```
```  2721     show ?thesis
```
```  2722     proof (cases "x = 1", simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
```
```  2723       assume "x \<noteq> 1" hence "x = -1" using `\<bar>x\<bar> = 1` by auto
```
```  2724
```
```  2725       have "- (pi / 2) < 0" using pi_gt_zero by auto
```
```  2726       have "- (2 * pi) < 0" using pi_gt_zero by auto
```
```  2727
```
```  2728       have c_minus_minus: "\<And> i. ?c (- 1) i = - ?c 1 i" unfolding One_nat_def by auto
```
```  2729
```
```  2730       have "arctan (- 1) = arctan (tan (-(pi / 4)))" unfolding tan_45 tan_minus ..
```
```  2731       also have "\<dots> = - (pi / 4)" by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
```
```  2732       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])
```
```  2733       also have "\<dots> = - (arctan 1)" unfolding tan_45 ..
```
```  2734       also have "\<dots> = - (\<Sum> i. ?c 1 i)" using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto
```
```  2735       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
```
```  2736       finally show ?thesis using `x = -1` by auto
```
```  2737     qed
```
```  2738   qed
```
```  2739 qed
```
```  2740
```
```  2741 lemma arctan_half: fixes x :: real
```
```  2742   shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x^2)))"
```
```  2743 proof -
```
```  2744   obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x" using tan_total by blast
```
```  2745   hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2" by auto
```
```  2746
```
```  2747   have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> A / B = (A / C) / (B / C)" by auto
```
```  2748
```
```  2749   have "0 < cos y" using cos_gt_zero_pi[OF low high] .
```
```  2750   hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y) ^ 2) = cos y" by auto
```
```  2751
```
```  2752   have "1 + (tan y)^2 = 1 + sin y^2 / cos y^2" unfolding tan_def power_divide ..
```
```  2753   also have "\<dots> = cos y^2 / cos y^2 + sin y^2 / cos y^2" using `cos y \<noteq> 0` by auto
```
```  2754   also have "\<dots> = 1 / cos y^2" unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
```
```  2755   finally have "1 + (tan y)^2 = 1 / cos y^2" .
```
```  2756
```
```  2757   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] ..
```
```  2758   also have "\<dots> = tan y / (1 + 1 / cos y)" using `cos y \<noteq> 0` unfolding add_divide_distrib by auto
```
```  2759   also have "\<dots> = tan y / (1 + 1 / sqrt(cos y^2))" unfolding cos_sqrt ..
```
```  2760   also have "\<dots> = tan y / (1 + sqrt(1 / cos y^2))" unfolding real_sqrt_divide by auto
```
```  2761   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` .
```
```  2762
```
```  2763   have "arctan x = y" using arctan_tan low high y_eq by auto
```
```  2764   also have "\<dots> = 2 * (arctan (tan (y/2)))" using arctan_tan[OF low2 high2] by auto
```
```  2765   also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" unfolding tan_half by auto
```
```  2766   finally show ?thesis unfolding eq `tan y = x` .
```
```  2767 qed
```
```  2768
```
```  2769 lemma arctan_monotone: assumes "x < y"
```
```  2770   shows "arctan x < arctan y"
```
```  2771   using assms by (simp only: arctan_less_iff)
```
```  2772
```
```  2773 lemma arctan_monotone': assumes "x \<le> y" shows "arctan x \<le> arctan y"
```
```  2774   using assms by (simp only: arctan_le_iff)
```
```  2775
```
```  2776 lemma arctan_inverse:
```
```  2777   assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
```
```  2778 proof (rule arctan_unique)
```
```  2779   show "- (pi / 2) < sgn x * pi / 2 - arctan x"
```
```  2780     using arctan_bounded [of x] assms
```
```  2781     unfolding sgn_real_def
```
```  2782     apply (auto simp add: algebra_simps)
```
```  2783     apply (drule zero_less_arctan_iff [THEN iffD2])
```
```  2784     apply arith
```
```  2785     done
```
```  2786   show "sgn x * pi / 2 - arctan x < pi / 2"
```
```  2787     using arctan_bounded [of "- x"] assms
```
```  2788     unfolding sgn_real_def arctan_minus
```
```  2789     by auto
```
```  2790   show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
```
```  2791     unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
```
```  2792     unfolding sgn_real_def
```
```  2793     by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
```
```  2794 qed
```
```  2795
```
```  2796 theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
```
```  2797 proof -
```
```  2798   have "pi / 4 = arctan 1" using arctan_one by auto
```
```  2799   also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
```
```  2800   finally show ?thesis by auto
```
```  2801 qed
```
```  2802
```
```  2803 subsection {* Existence of Polar Coordinates *}
```
```  2804
```
```  2805 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<twosuperior> + y\<twosuperior>)\<bar> \<le> 1"
```
```  2806 apply (rule power2_le_imp_le [OF _ zero_le_one])
```
```  2807 apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
```
```  2808 done
```
```  2809
```
```  2810 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
```
```  2811 by (simp add: abs_le_iff)
```
```  2812
```
```  2813 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<twosuperior>)"
```
```  2814 by (simp add: sin_arccos abs_le_iff)
```
```  2815
```
```  2816 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
```
```  2817
```
```  2818 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
```
```  2819
```
```  2820 lemma polar_ex1:
```
```  2821      "0 < y ==> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  2822 apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>)" in exI)
```
```  2823 apply (rule_tac x = "arccos (x / sqrt (x\<twosuperior> + y\<twosuperior>))" in exI)
```
```  2824 apply (simp add: cos_arccos_lemma1)
```
```  2825 apply (simp add: sin_arccos_lemma1)
```
```  2826 apply (simp add: power_divide)
```
```  2827 apply (simp add: real_sqrt_mult [symmetric])
```
```  2828 apply (simp add: right_diff_distrib)
```
```  2829 done
```
```  2830
```
```  2831 lemma polar_ex2:
```
```  2832      "y < 0 ==> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  2833 apply (insert polar_ex1 [where x=x and y="-y"], simp, clarify)
```
```  2834 apply (metis cos_minus minus_minus minus_mult_right sin_minus)
```
```  2835 done
```
```  2836
```
```  2837 lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
```
```  2838 apply (rule_tac x=0 and y=y in linorder_cases)
```
```  2839 apply (erule polar_ex1)
```
```  2840 apply (rule_tac x=x in exI, rule_tac x=0 in exI, simp)
```
```  2841 apply (erule polar_ex2)
```
```  2842 done
```
```  2843
```
```  2844 end
```