src/HOL/Transcendental.thy
 author paulson Sun Nov 24 13:06:22 2013 +0000 (2013-11-24) changeset 54573 07864001495d parent 54489 03ff4d1e6784 child 54575 0b9ca2c865cb permissions -rw-r--r--
cleaned up some messy proofs
```     1 (*  Title:      HOL/Transcendental.thy
```
```     2     Author:     Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
```
```     3     Author:     Lawrence C Paulson
```
```     4     Author:     Jeremy Avigad
```
```     5 *)
```
```     6
```
```     7 header{*Power Series, Transcendental Functions etc.*}
```
```     8
```
```     9 theory Transcendental
```
```    10 imports Fact Series Deriv NthRoot
```
```    11 begin
```
```    12
```
```    13 subsection {* Properties of Power Series *}
```
```    14
```
```    15 lemma lemma_realpow_diff:
```
```    16   fixes y :: "'a::monoid_mult"
```
```    17   shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
```
```    18 proof -
```
```    19   assume "p \<le> n"
```
```    20   hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
```
```    21   thus ?thesis by (simp add: power_commutes)
```
```    22 qed
```
```    23
```
```    24 lemma lemma_realpow_diff_sumr:
```
```    25   fixes y :: "'a::{comm_semiring_0,monoid_mult}"
```
```    26   shows
```
```    27     "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
```
```    28       y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
```
```    29   by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac del: setsum_op_ivl_Suc)
```
```    30
```
```    31 lemma lemma_realpow_diff_sumr2:
```
```    32   fixes y :: "'a::{comm_ring,monoid_mult}"
```
```    33   shows
```
```    34     "x ^ (Suc n) - y ^ (Suc n) =
```
```    35       (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
```
```    36 proof (induct n)
```
```    37   case 0 show ?case
```
```    38     by simp
```
```    39 next
```
```    40   case (Suc n)
```
```    41   have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x ^ n) - y * (y * y ^ n)"
```
```    42     by simp
```
```    43   also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x ^ n)"
```
```    44     by (simp add: algebra_simps)
```
```    45   also have "... = y * ((x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
```
```    46     by (simp only: Suc)
```
```    47   also have "... = (x - y) * (y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
```
```    48     by (simp only: mult_left_commute)
```
```    49   also have "... = (x - y) * (\<Sum>p = 0..<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
```
```    50     by (simp add: setsum_op_ivl_Suc [where n = "Suc n"] distrib_left lemma_realpow_diff_sumr
```
```    51              del: setsum_op_ivl_Suc)
```
```    52   finally show ?case .
```
```    53 qed
```
```    54
```
```    55 lemma lemma_realpow_rev_sumr:
```
```    56    "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
```
```    57     (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
```
```    58   apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
```
```    59   apply (rule inj_onI, auto)
```
```    60   apply (metis atLeastLessThan_iff diff_diff_cancel diff_less_Suc imageI le0 less_Suc_eq_le)
```
```    61   done
```
```    62
```
```    63 text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
```
```    64   x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
```
```    65
```
```    66 lemma powser_insidea:
```
```    67   fixes x z :: "'a::real_normed_div_algebra"
```
```    68   assumes 1: "summable (\<lambda>n. f n * x ^ n)"
```
```    69     and 2: "norm z < norm x"
```
```    70   shows "summable (\<lambda>n. norm (f n * z ^ n))"
```
```    71 proof -
```
```    72   from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
```
```    73   from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
```
```    74     by (rule summable_LIMSEQ_zero)
```
```    75   hence "convergent (\<lambda>n. f n * x ^ n)"
```
```    76     by (rule convergentI)
```
```    77   hence "Cauchy (\<lambda>n. f n * x ^ n)"
```
```    78     by (rule convergent_Cauchy)
```
```    79   hence "Bseq (\<lambda>n. f n * x ^ n)"
```
```    80     by (rule Cauchy_Bseq)
```
```    81   then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
```
```    82     by (simp add: Bseq_def, safe)
```
```    83   have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
```
```    84                    K * norm (z ^ n) * inverse (norm (x ^ n))"
```
```    85   proof (intro exI allI impI)
```
```    86     fix n::nat
```
```    87     assume "0 \<le> n"
```
```    88     have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
```
```    89           norm (f n * x ^ n) * norm (z ^ n)"
```
```    90       by (simp add: norm_mult abs_mult)
```
```    91     also have "\<dots> \<le> K * norm (z ^ n)"
```
```    92       by (simp only: mult_right_mono 4 norm_ge_zero)
```
```    93     also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
```
```    94       by (simp add: x_neq_0)
```
```    95     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
```
```    96       by (simp only: mult_assoc)
```
```    97     finally show "norm (norm (f n * z ^ n)) \<le>
```
```    98                   K * norm (z ^ n) * inverse (norm (x ^ n))"
```
```    99       by (simp add: mult_le_cancel_right x_neq_0)
```
```   100   qed
```
```   101   moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
```
```   102   proof -
```
```   103     from 2 have "norm (norm (z * inverse x)) < 1"
```
```   104       using x_neq_0
```
```   105       by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
```
```   106     hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
```
```   107       by (rule summable_geometric)
```
```   108     hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
```
```   109       by (rule summable_mult)
```
```   110     thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
```
```   111       using x_neq_0
```
```   112       by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
```
```   113                     power_inverse norm_power mult_assoc)
```
```   114   qed
```
```   115   ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   116     by (rule summable_comparison_test)
```
```   117 qed
```
```   118
```
```   119 lemma powser_inside:
```
```   120   fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
```
```   121   shows
```
```   122     "summable (\<lambda>n. f n * (x ^ n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
```
```   123       summable (\<lambda>n. f n * (z ^ n))"
```
```   124   by (rule powser_insidea [THEN summable_norm_cancel])
```
```   125
```
```   126 lemma sum_split_even_odd:
```
```   127   fixes f :: "nat \<Rightarrow> real"
```
```   128   shows
```
```   129     "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
```
```   130      (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
```
```   131 proof (induct n)
```
```   132   case 0
```
```   133   then show ?case by simp
```
```   134 next
```
```   135   case (Suc n)
```
```   136   have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =
```
```   137     (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
```
```   138     using Suc.hyps unfolding One_nat_def by auto
```
```   139   also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))"
```
```   140     by auto
```
```   141   finally show ?case .
```
```   142 qed
```
```   143
```
```   144 lemma sums_if':
```
```   145   fixes g :: "nat \<Rightarrow> real"
```
```   146   assumes "g sums x"
```
```   147   shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   148   unfolding sums_def
```
```   149 proof (rule LIMSEQ_I)
```
```   150   fix r :: real
```
```   151   assume "0 < r"
```
```   152   from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
```
```   153   obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast
```
```   154
```
```   155   let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
```
```   156   {
```
```   157     fix m
```
```   158     assume "m \<ge> 2 * no"
```
```   159     hence "m div 2 \<ge> no" by auto
```
```   160     have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
```
```   161       using sum_split_even_odd by auto
```
```   162     hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
```
```   163       using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
```
```   164     moreover
```
```   165     have "?SUM (2 * (m div 2)) = ?SUM m"
```
```   166     proof (cases "even m")
```
```   167       case True
```
```   168       show ?thesis
```
```   169         unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
```
```   170     next
```
```   171       case False
```
```   172       hence "even (Suc m)" by auto
```
```   173       from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]]
```
```   174         odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]
```
```   175       have eq: "Suc (2 * (m div 2)) = m" by auto
```
```   176       hence "even (2 * (m div 2))" using `odd m` by auto
```
```   177       have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
```
```   178       also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
```
```   179       finally show ?thesis by auto
```
```   180     qed
```
```   181     ultimately have "(norm (?SUM m - x) < r)" by auto
```
```   182   }
```
```   183   thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
```
```   184 qed
```
```   185
```
```   186 lemma sums_if:
```
```   187   fixes g :: "nat \<Rightarrow> real"
```
```   188   assumes "g sums x" and "f sums y"
```
```   189   shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
```
```   190 proof -
```
```   191   let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
```
```   192   {
```
```   193     fix B T E
```
```   194     have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
```
```   195       by (cases B) auto
```
```   196   } note if_sum = this
```
```   197   have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   198     using sums_if'[OF `g sums x`] .
```
```   199   {
```
```   200     have "?s 0 = 0" by auto
```
```   201     have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
```
```   202     have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
```
```   203
```
```   204     have "?s sums y" using sums_if'[OF `f sums y`] .
```
```   205     from this[unfolded sums_def, THEN LIMSEQ_Suc]
```
```   206     have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
```
```   207       unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
```
```   208                 image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
```
```   209                 even_Suc Suc_m1 if_eq .
```
```   210   }
```
```   211   from sums_add[OF g_sums this] show ?thesis unfolding if_sum .
```
```   212 qed
```
```   213
```
```   214 subsection {* Alternating series test / Leibniz formula *}
```
```   215
```
```   216 lemma sums_alternating_upper_lower:
```
```   217   fixes a :: "nat \<Rightarrow> real"
```
```   218   assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
```
```   219   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>
```
```   220              ((\<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)"
```
```   221   (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
```
```   222 proof (rule nested_sequence_unique)
```
```   223   have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
```
```   224
```
```   225   show "\<forall>n. ?f n \<le> ?f (Suc n)"
```
```   226   proof
```
```   227     fix n
```
```   228     show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
```
```   229   qed
```
```   230   show "\<forall>n. ?g (Suc n) \<le> ?g n"
```
```   231   proof
```
```   232     fix n
```
```   233     show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
```
```   234       unfolding One_nat_def by auto
```
```   235   qed
```
```   236   show "\<forall>n. ?f n \<le> ?g n"
```
```   237   proof
```
```   238     fix n
```
```   239     show "?f n \<le> ?g n" using fg_diff a_pos
```
```   240       unfolding One_nat_def by auto
```
```   241   qed
```
```   242   show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff
```
```   243   proof (rule LIMSEQ_I)
```
```   244     fix r :: real
```
```   245     assume "0 < r"
```
```   246     with `a ----> 0`[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
```
```   247       by auto
```
```   248     hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   249     thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   250   qed
```
```   251 qed
```
```   252
```
```   253 lemma summable_Leibniz':
```
```   254   fixes a :: "nat \<Rightarrow> real"
```
```   255   assumes a_zero: "a ----> 0"
```
```   256     and a_pos: "\<And> n. 0 \<le> a n"
```
```   257     and a_monotone: "\<And> n. a (Suc n) \<le> a n"
```
```   258   shows summable: "summable (\<lambda> n. (-1)^n * a n)"
```
```   259     and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
```
```   260     and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   261     and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
```
```   262     and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   263 proof -
```
```   264   let ?S = "\<lambda>n. (-1)^n * a n"
```
```   265   let ?P = "\<lambda>n. \<Sum>i=0..<n. ?S i"
```
```   266   let ?f = "\<lambda>n. ?P (2 * n)"
```
```   267   let ?g = "\<lambda>n. ?P (2 * n + 1)"
```
```   268   obtain l :: real
```
```   269     where below_l: "\<forall> n. ?f n \<le> l"
```
```   270       and "?f ----> l"
```
```   271       and above_l: "\<forall> n. l \<le> ?g n"
```
```   272       and "?g ----> l"
```
```   273     using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
```
```   274
```
```   275   let ?Sa = "\<lambda>m. \<Sum> n = 0..<m. ?S n"
```
```   276   have "?Sa ----> l"
```
```   277   proof (rule LIMSEQ_I)
```
```   278     fix r :: real
```
```   279     assume "0 < r"
```
```   280     with `?f ----> l`[THEN LIMSEQ_D]
```
```   281     obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
```
```   282
```
```   283     from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
```
```   284     obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
```
```   285
```
```   286     {
```
```   287       fix n :: nat
```
```   288       assume "n \<ge> (max (2 * f_no) (2 * g_no))"
```
```   289       hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
```
```   290       have "norm (?Sa n - l) < r"
```
```   291       proof (cases "even n")
```
```   292         case True
```
```   293         from even_nat_div_two_times_two[OF this]
```
```   294         have n_eq: "2 * (n div 2) = n"
```
```   295           unfolding numeral_2_eq_2[symmetric] by auto
```
```   296         with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no"
```
```   297           by auto
```
```   298         from f[OF this] show ?thesis
```
```   299           unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
```
```   300       next
```
```   301         case False
```
```   302         hence "even (n - 1)" by simp
```
```   303         from even_nat_div_two_times_two[OF this]
```
```   304         have n_eq: "2 * ((n - 1) div 2) = n - 1"
```
```   305           unfolding numeral_2_eq_2[symmetric] by auto
```
```   306         hence range_eq: "n - 1 + 1 = n"
```
```   307           using odd_pos[OF False] by auto
```
```   308
```
```   309         from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no"
```
```   310           by auto
```
```   311         from g[OF this] show ?thesis
```
```   312           unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
```
```   313       qed
```
```   314     }
```
```   315     thus "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
```
```   316   qed
```
```   317   hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
```
```   318     unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
```
```   319   thus "summable ?S" using summable_def by auto
```
```   320
```
```   321   have "l = suminf ?S" using sums_unique[OF sums_l] .
```
```   322
```
```   323   fix n
```
```   324   show "suminf ?S \<le> ?g n"
```
```   325     unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
```
```   326   show "?f n \<le> suminf ?S"
```
```   327     unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
```
```   328   show "?g ----> suminf ?S"
```
```   329     using `?g ----> l` `l = suminf ?S` by auto
```
```   330   show "?f ----> suminf ?S"
```
```   331     using `?f ----> l` `l = suminf ?S` by auto
```
```   332 qed
```
```   333
```
```   334 theorem summable_Leibniz:
```
```   335   fixes a :: "nat \<Rightarrow> real"
```
```   336   assumes a_zero: "a ----> 0" and "monoseq a"
```
```   337   shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
```
```   338     and "0 < a 0 \<longrightarrow>
```
```   339       (\<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")
```
```   340     and "a 0 < 0 \<longrightarrow>
```
```   341       (\<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")
```
```   342     and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
```
```   343     and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
```
```   344 proof -
```
```   345   have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
```
```   346   proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
```
```   347     case True
```
```   348     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n"
```
```   349       by auto
```
```   350     {
```
```   351       fix n
```
```   352       have "a (Suc n) \<le> a n"
```
```   353         using ord[where n="Suc n" and m=n] by auto
```
```   354     } note mono = this
```
```   355     note leibniz = summable_Leibniz'[OF `a ----> 0` ge0]
```
```   356     from leibniz[OF mono]
```
```   357     show ?thesis using `0 \<le> a 0` by auto
```
```   358   next
```
```   359     let ?a = "\<lambda> n. - a n"
```
```   360     case False
```
```   361     with monoseq_le[OF `monoseq a` `a ----> 0`]
```
```   362     have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
```
```   363     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
```
```   364       by auto
```
```   365     {
```
```   366       fix n
```
```   367       have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
```
```   368         by auto
```
```   369     } note monotone = this
```
```   370     note leibniz =
```
```   371       summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
```
```   372         OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
```
```   373     have "summable (\<lambda> n. (-1)^n * ?a n)"
```
```   374       using leibniz(1) by auto
```
```   375     then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
```
```   376       unfolding summable_def by auto
```
```   377     from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
```
```   378       by auto
```
```   379     hence ?summable unfolding summable_def by auto
```
```   380     moreover
```
```   381     have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
```
```   382       unfolding minus_diff_minus by auto
```
```   383
```
```   384     from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
```
```   385     have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)"
```
```   386       by auto
```
```   387
```
```   388     have ?pos using `0 \<le> ?a 0` by auto
```
```   389     moreover have ?neg
```
```   390       using leibniz(2,4)
```
```   391       unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
```
```   392       by auto
```
```   393     moreover have ?f and ?g
```
```   394       using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
```
```   395       by auto
```
```   396     ultimately show ?thesis by auto
```
```   397   qed
```
```   398   then
```
```   399   show ?summable and ?pos and ?neg and ?f and ?g
```
```   400     by safe
```
```   401 qed
```
```   402
```
```   403 subsection {* Term-by-Term Differentiability of Power Series *}
```
```   404
```
```   405 definition diffs :: "(nat => 'a::ring_1) => nat => 'a"
```
```   406   where "diffs c = (\<lambda>n. of_nat (Suc n) * c(Suc n))"
```
```   407
```
```   408 text{*Lemma about distributing negation over it*}
```
```   409 lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
```
```   410   by (simp add: diffs_def)
```
```   411
```
```   412 lemma sums_Suc_imp:
```
```   413   assumes f: "f 0 = 0"
```
```   414   shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
```
```   415   unfolding sums_def
```
```   416   apply (rule LIMSEQ_imp_Suc)
```
```   417   apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
```
```   418   apply (simp only: setsum_shift_bounds_Suc_ivl)
```
```   419   done
```
```   420
```
```   421 lemma diffs_equiv:
```
```   422   fixes x :: "'a::{real_normed_vector, ring_1}"
```
```   423   shows "summable (\<lambda>n. (diffs c)(n) * (x ^ n)) \<Longrightarrow>
```
```   424       (\<lambda>n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
```
```   425          (\<Sum>n. (diffs c)(n) * (x ^ n))"
```
```   426   unfolding diffs_def
```
```   427   by (simp add: summable_sums sums_Suc_imp)
```
```   428
```
```   429 lemma lemma_termdiff1:
```
```   430   fixes z :: "'a :: {monoid_mult,comm_ring}" shows
```
```   431   "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
```
```   432    (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
```
```   433   by (auto simp add: algebra_simps power_add [symmetric])
```
```   434
```
```   435 lemma sumr_diff_mult_const2:
```
```   436   "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
```
```   437   by (simp add: setsum_subtractf)
```
```   438
```
```   439 lemma lemma_termdiff2:
```
```   440   fixes h :: "'a :: {field}"
```
```   441   assumes h: "h \<noteq> 0"
```
```   442   shows
```
```   443     "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
```
```   444      h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
```
```   445           (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
```
```   446   apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
```
```   447   apply (simp add: right_diff_distrib diff_divide_distrib h)
```
```   448   apply (simp add: mult_assoc [symmetric])
```
```   449   apply (cases "n", simp)
```
```   450   apply (simp add: lemma_realpow_diff_sumr2 h
```
```   451                    right_diff_distrib [symmetric] mult_assoc
```
```   452               del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
```
```   453   apply (subst lemma_realpow_rev_sumr)
```
```   454   apply (subst sumr_diff_mult_const2)
```
```   455   apply simp
```
```   456   apply (simp only: lemma_termdiff1 setsum_right_distrib)
```
```   457   apply (rule setsum_cong [OF refl])
```
```   458   apply (simp add: less_iff_Suc_add)
```
```   459   apply (clarify)
```
```   460   apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
```
```   461               del: setsum_op_ivl_Suc power_Suc)
```
```   462   apply (subst mult_assoc [symmetric], subst power_add [symmetric])
```
```   463   apply (simp add: mult_ac)
```
```   464   done
```
```   465
```
```   466 lemma real_setsum_nat_ivl_bounded2:
```
```   467   fixes K :: "'a::linordered_semidom"
```
```   468   assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
```
```   469     and K: "0 \<le> K"
```
```   470   shows "setsum f {0..<n-k} \<le> of_nat n * K"
```
```   471   apply (rule order_trans [OF setsum_mono])
```
```   472   apply (rule f, simp)
```
```   473   apply (simp add: mult_right_mono K)
```
```   474   done
```
```   475
```
```   476 lemma lemma_termdiff3:
```
```   477   fixes h z :: "'a::{real_normed_field}"
```
```   478   assumes 1: "h \<noteq> 0"
```
```   479     and 2: "norm z \<le> K"
```
```   480     and 3: "norm (z + h) \<le> K"
```
```   481   shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
```
```   482           \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   483 proof -
```
```   484   have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
```
```   485         norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
```
```   486           (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
```
```   487     by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult_commute norm_mult)
```
```   488   also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
```
```   489   proof (rule mult_right_mono [OF _ norm_ge_zero])
```
```   490     from norm_ge_zero 2 have K: "0 \<le> K"
```
```   491       by (rule order_trans)
```
```   492     have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
```
```   493       apply (erule subst)
```
```   494       apply (simp only: norm_mult norm_power power_add)
```
```   495       apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
```
```   496       done
```
```   497     show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))
```
```   498           \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
```
```   499       apply (intro
```
```   500          order_trans [OF norm_setsum]
```
```   501          real_setsum_nat_ivl_bounded2
```
```   502          mult_nonneg_nonneg
```
```   503          of_nat_0_le_iff
```
```   504          zero_le_power K)
```
```   505       apply (rule le_Kn, simp)
```
```   506       done
```
```   507   qed
```
```   508   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   509     by (simp only: mult_assoc)
```
```   510   finally show ?thesis .
```
```   511 qed
```
```   512
```
```   513 lemma lemma_termdiff4:
```
```   514   fixes f :: "'a::{real_normed_field} \<Rightarrow>
```
```   515               'b::real_normed_vector"
```
```   516   assumes k: "0 < (k::real)"
```
```   517     and le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
```
```   518   shows "f -- 0 --> 0"
```
```   519   unfolding LIM_eq diff_0_right
```
```   520 proof safe
```
```   521   let ?h = "of_real (k / 2)::'a"
```
```   522   have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
```
```   523   hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
```
```   524   hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
```
```   525   hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
```
```   526
```
```   527   fix r::real
```
```   528   assume r: "0 < r"
```
```   529   show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
```
```   530   proof cases
```
```   531     assume "K = 0"
```
```   532     with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
```
```   533       by simp
```
```   534     thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
```
```   535   next
```
```   536     assume K_neq_zero: "K \<noteq> 0"
```
```   537     with zero_le_K have K: "0 < K" by simp
```
```   538     show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
```
```   539     proof (rule exI, safe)
```
```   540       from k r K
```
```   541       show "0 < min k (r * inverse K / 2)"
```
```   542         by (simp add: mult_pos_pos positive_imp_inverse_positive)
```
```   543     next
```
```   544       fix x::'a
```
```   545       assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
```
```   546       from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
```
```   547         by simp_all
```
```   548       from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
```
```   549       also from x4 K have "K * norm x < K * (r * inverse K / 2)"
```
```   550         by (rule mult_strict_left_mono)
```
```   551       also have "\<dots> = r / 2"
```
```   552         using K_neq_zero by simp
```
```   553       also have "r / 2 < r"
```
```   554         using r by simp
```
```   555       finally show "norm (f x) < r" .
```
```   556     qed
```
```   557   qed
```
```   558 qed
```
```   559
```
```   560 lemma lemma_termdiff5:
```
```   561   fixes g :: "'a::real_normed_field \<Rightarrow> nat \<Rightarrow> 'b::banach"
```
```   562   assumes k: "0 < (k::real)"
```
```   563   assumes f: "summable f"
```
```   564   assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
```
```   565   shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
```
```   566 proof (rule lemma_termdiff4 [OF k])
```
```   567   fix h::'a
```
```   568   assume "h \<noteq> 0" and "norm h < k"
```
```   569   hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
```
```   570     by (simp add: le)
```
```   571   hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
```
```   572     by simp
```
```   573   moreover from f have B: "summable (\<lambda>n. f n * norm h)"
```
```   574     by (rule summable_mult2)
```
```   575   ultimately have C: "summable (\<lambda>n. norm (g h n))"
```
```   576     by (rule summable_comparison_test)
```
```   577   hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
```
```   578     by (rule summable_norm)
```
```   579   also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
```
```   580     by (rule summable_le)
```
```   581   also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
```
```   582     by (rule suminf_mult2 [symmetric])
```
```   583   finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
```
```   584 qed
```
```   585
```
```   586
```
```   587 text{* FIXME: Long proofs*}
```
```   588
```
```   589 lemma termdiffs_aux:
```
```   590   fixes x :: "'a::{real_normed_field,banach}"
```
```   591   assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
```
```   592     and 2: "norm x < norm K"
```
```   593   shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
```
```   594              - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   595 proof -
```
```   596   from dense [OF 2]
```
```   597   obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
```
```   598   from norm_ge_zero r1 have r: "0 < r"
```
```   599     by (rule order_le_less_trans)
```
```   600   hence r_neq_0: "r \<noteq> 0" by simp
```
```   601   show ?thesis
```
```   602   proof (rule lemma_termdiff5)
```
```   603     show "0 < r - norm x" using r1 by simp
```
```   604     from r r2 have "norm (of_real r::'a) < norm K"
```
```   605       by simp
```
```   606     with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
```
```   607       by (rule powser_insidea)
```
```   608     hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
```
```   609       using r
```
```   610       by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
```
```   611     hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
```
```   612       by (rule diffs_equiv [THEN sums_summable])
```
```   613     also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
```
```   614       (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
```
```   615       apply (rule ext)
```
```   616       apply (simp add: diffs_def)
```
```   617       apply (case_tac n, simp_all add: r_neq_0)
```
```   618       done
```
```   619     finally have "summable
```
```   620       (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
```
```   621       by (rule diffs_equiv [THEN sums_summable])
```
```   622     also have
```
```   623       "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
```
```   624            r ^ (n - Suc 0)) =
```
```   625        (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
```
```   626       apply (rule ext)
```
```   627       apply (case_tac "n", simp)
```
```   628       apply (case_tac "nat", simp)
```
```   629       apply (simp add: r_neq_0)
```
```   630       done
```
```   631     finally
```
```   632     show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
```
```   633   next
```
```   634     fix h::'a and n::nat
```
```   635     assume h: "h \<noteq> 0"
```
```   636     assume "norm h < r - norm x"
```
```   637     hence "norm x + norm h < r" by simp
```
```   638     with norm_triangle_ineq have xh: "norm (x + h) < r"
```
```   639       by (rule order_le_less_trans)
```
```   640     show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
```
```   641           \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
```
```   642       apply (simp only: norm_mult mult_assoc)
```
```   643       apply (rule mult_left_mono [OF _ norm_ge_zero])
```
```   644       apply (simp (no_asm) add: mult_assoc [symmetric])
```
```   645       apply (rule lemma_termdiff3)
```
```   646       apply (rule h)
```
```   647       apply (rule r1 [THEN order_less_imp_le])
```
```   648       apply (rule xh [THEN order_less_imp_le])
```
```   649       done
```
```   650   qed
```
```   651 qed
```
```   652
```
```   653 lemma termdiffs:
```
```   654   fixes K x :: "'a::{real_normed_field,banach}"
```
```   655   assumes 1: "summable (\<lambda>n. c n * K ^ n)"
```
```   656     and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
```
```   657     and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
```
```   658     and 4: "norm x < norm K"
```
```   659   shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
```
```   660   unfolding deriv_def
```
```   661 proof (rule LIM_zero_cancel)
```
```   662   show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
```
```   663             - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
```
```   664   proof (rule LIM_equal2)
```
```   665     show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
```
```   666   next
```
```   667     fix h :: 'a
```
```   668     assume "h \<noteq> 0"
```
```   669     assume "norm (h - 0) < norm K - norm x"
```
```   670     hence "norm x + norm h < norm K" by simp
```
```   671     hence 5: "norm (x + h) < norm K"
```
```   672       by (rule norm_triangle_ineq [THEN order_le_less_trans])
```
```   673     have A: "summable (\<lambda>n. c n * x ^ n)"
```
```   674       by (rule powser_inside [OF 1 4])
```
```   675     have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
```
```   676       by (rule powser_inside [OF 1 5])
```
```   677     have C: "summable (\<lambda>n. diffs c n * x ^ n)"
```
```   678       by (rule powser_inside [OF 2 4])
```
```   679     show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
```
```   680              - (\<Sum>n. diffs c n * x ^ n) =
```
```   681           (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
```
```   682       apply (subst sums_unique [OF diffs_equiv [OF C]])
```
```   683       apply (subst suminf_diff [OF B A])
```
```   684       apply (subst suminf_divide [symmetric])
```
```   685       apply (rule summable_diff [OF B A])
```
```   686       apply (subst suminf_diff)
```
```   687       apply (rule summable_divide)
```
```   688       apply (rule summable_diff [OF B A])
```
```   689       apply (rule sums_summable [OF diffs_equiv [OF C]])
```
```   690       apply (rule arg_cong [where f="suminf"], rule ext)
```
```   691       apply (simp add: algebra_simps)
```
```   692       done
```
```   693   next
```
```   694     show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   695       by (rule termdiffs_aux [OF 3 4])
```
```   696   qed
```
```   697 qed
```
```   698
```
```   699
```
```   700 subsection {* Derivability of power series *}
```
```   701
```
```   702 lemma DERIV_series':
```
```   703   fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
```
```   704   assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
```
```   705     and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
```
```   706     and "summable (f' x0)"
```
```   707     and "summable L"
```
```   708     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>"
```
```   709   shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
```
```   710   unfolding deriv_def
```
```   711 proof (rule LIM_I)
```
```   712   fix r :: real
```
```   713   assume "0 < r" hence "0 < r/3" by auto
```
```   714
```
```   715   obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
```
```   716     using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
```
```   717
```
```   718   obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
```
```   719     using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
```
```   720
```
```   721   let ?N = "Suc (max N_L N_f')"
```
```   722   have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
```
```   723     L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
```
```   724
```
```   725   let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
```
```   726
```
```   727   let ?r = "r / (3 * real ?N)"
```
```   728   have "0 < 3 * real ?N" by auto
```
```   729   from divide_pos_pos[OF `0 < r` this]
```
```   730   have "0 < ?r" .
```
```   731
```
```   732   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)"
```
```   733   def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
```
```   734
```
```   735   have "0 < S'" unfolding S'_def
```
```   736   proof (rule iffD2[OF Min_gr_iff])
```
```   737     show "\<forall>x \<in> (?s ` { 0 ..< ?N }). 0 < x"
```
```   738     proof
```
```   739       fix x
```
```   740       assume "x \<in> ?s ` {0..<?N}"
```
```   741       then obtain n where "x = ?s n" and "n \<in> {0..<?N}"
```
```   742         using image_iff[THEN iffD1] by blast
```
```   743       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
```
```   744       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)"
```
```   745         by auto
```
```   746       have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound)
```
```   747       thus "0 < x" unfolding `x = ?s n` .
```
```   748     qed
```
```   749   qed auto
```
```   750
```
```   751   def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
```
```   752   hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
```
```   753     and "S \<le> S'" using x0_in_I and `0 < S'`
```
```   754     by auto
```
```   755
```
```   756   {
```
```   757     fix x
```
```   758     assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
```
```   759     hence x_in_I: "x0 + x \<in> { a <..< b }"
```
```   760       using S_a S_b by auto
```
```   761
```
```   762     note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   763     note div_smbl = summable_divide[OF diff_smbl]
```
```   764     note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
```
```   765     note ign = summable_ignore_initial_segment[where k="?N"]
```
```   766     note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
```
```   767     note div_shft_smbl = summable_divide[OF diff_shft_smbl]
```
```   768     note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
```
```   769
```
```   770     {
```
```   771       fix n
```
```   772       have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
```
```   773         using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
```
```   774         unfolding abs_divide .
```
```   775       hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)"
```
```   776         using `x \<noteq> 0` by auto
```
```   777     } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
```
```   778     from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
```
```   779     have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
```
```   780     hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3")
```
```   781       using L_estimate by auto
```
```   782
```
```   783     have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le>
```
```   784       (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..
```
```   785     also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
```
```   786     proof (rule setsum_strict_mono)
```
```   787       fix n
```
```   788       assume "n \<in> { 0 ..< ?N}"
```
```   789       have "\<bar>x\<bar> < S" using `\<bar>x\<bar> < S` .
```
```   790       also have "S \<le> S'" using `S \<le> S'` .
```
```   791       also have "S' \<le> ?s n" unfolding S'_def
```
```   792       proof (rule Min_le_iff[THEN iffD2])
```
```   793         have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n"
```
```   794           using `n \<in> { 0 ..< ?N}` by auto
```
```   795         thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
```
```   796       qed auto
```
```   797       finally have "\<bar>x\<bar> < ?s n" .
```
```   798
```
```   799       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]
```
```   800       have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
```
```   801       with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n` show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
```
```   802         by blast
```
```   803     qed auto
```
```   804     also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r"
```
```   805       by (rule setsum_constant)
```
```   806     also have "\<dots> = real ?N * ?r"
```
```   807       unfolding real_eq_of_nat by auto
```
```   808     also have "\<dots> = r/3" by auto
```
```   809     finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
```
```   810
```
```   811     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   812     have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
```
```   813         \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
```
```   814       unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric]
```
```   815       using suminf_divide[OF diff_smbl, symmetric] by auto
```
```   816     also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>"
```
```   817       unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
```
```   818       unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]]
```
```   819       by (rule abs_triangle_ineq)
```
```   820     also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
```
```   821       using abs_triangle_ineq4 by auto
```
```   822     also have "\<dots> < r /3 + r/3 + r/3"
```
```   823       using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
```
```   824       by (rule add_strict_mono [OF add_less_le_mono])
```
```   825     finally have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
```
```   826       by auto
```
```   827   }
```
```   828   thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
```
```   829       norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
```
```   830     using `0 < S` unfolding real_norm_def diff_0_right by blast
```
```   831 qed
```
```   832
```
```   833 lemma DERIV_power_series':
```
```   834   fixes f :: "nat \<Rightarrow> real"
```
```   835   assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
```
```   836     and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
```
```   837   shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
```
```   838   (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
```
```   839 proof -
```
```   840   {
```
```   841     fix R'
```
```   842     assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
```
```   843     hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
```
```   844       by auto
```
```   845     have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
```
```   846     proof (rule DERIV_series')
```
```   847       show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
```
```   848       proof -
```
```   849         have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
```
```   850           using `0 < R'` `0 < R` `R' < R` by auto
```
```   851         hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
```
```   852           using `R' < R` by auto
```
```   853         have "norm R' < norm ((R' + R) / 2)"
```
```   854           using `0 < R'` `0 < R` `R' < R` by auto
```
```   855         from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
```
```   856           by auto
```
```   857       qed
```
```   858       {
```
```   859         fix n x y
```
```   860         assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
```
```   861         show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
```
```   862         proof -
```
```   863           have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
```
```   864             (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
```
```   865             unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult
```
```   866             by auto
```
```   867           also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
```
```   868           proof (rule mult_left_mono)
```
```   869             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>)"
```
```   870               by (rule setsum_abs)
```
```   871             also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
```
```   872             proof (rule setsum_mono)
```
```   873               fix p
```
```   874               assume "p \<in> {0..<Suc n}"
```
```   875               hence "p \<le> n" by auto
```
```   876               {
```
```   877                 fix n
```
```   878                 fix x :: real
```
```   879                 assume "x \<in> {-R'<..<R'}"
```
```   880                 hence "\<bar>x\<bar> \<le> R'"  by auto
```
```   881                 hence "\<bar>x^n\<bar> \<le> R'^n"
```
```   882                   unfolding power_abs by (rule power_mono, auto)
```
```   883               }
```
```   884               from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
```
```   885               have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
```
```   886                 unfolding abs_mult by auto
```
```   887               thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
```
```   888                 unfolding power_add[symmetric] using `p \<le> n` by auto
```
```   889             qed
```
```   890             also have "\<dots> = real (Suc n) * R' ^ n"
```
```   891               unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
```
```   892             finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
```
```   893               unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
```
```   894             show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
```
```   895               unfolding abs_mult[symmetric] by auto
```
```   896           qed
```
```   897           also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
```
```   898             unfolding abs_mult mult_assoc[symmetric] by algebra
```
```   899           finally show ?thesis .
```
```   900         qed
```
```   901       }
```
```   902       {
```
```   903         fix n
```
```   904         show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
```
```   905           by (auto intro!: DERIV_intros simp del: power_Suc)
```
```   906       }
```
```   907       {
```
```   908         fix x
```
```   909         assume "x \<in> {-R' <..< R'}"
```
```   910         hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
```
```   911           using assms `R' < R` by auto
```
```   912         have "summable (\<lambda> n. f n * x^n)"
```
```   913         proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
```
```   914           fix n
```
```   915           have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
```
```   916             by (rule mult_left_mono) auto
```
```   917           show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)"
```
```   918             unfolding real_norm_def abs_mult
```
```   919             by (rule mult_right_mono) (auto simp add: le[unfolded mult_1_right])
```
```   920         qed
```
```   921         from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
```
```   922         show "summable (?f x)" by auto
```
```   923       }
```
```   924       show "summable (?f' x0)"
```
```   925         using converges[OF `x0 \<in> {-R <..< R}`] .
```
```   926       show "x0 \<in> {-R' <..< R'}"
```
```   927         using `x0 \<in> {-R' <..< R'}` .
```
```   928     qed
```
```   929   } note for_subinterval = this
```
```   930   let ?R = "(R + \<bar>x0\<bar>) / 2"
```
```   931   have "\<bar>x0\<bar> < ?R" using assms by auto
```
```   932   hence "- ?R < x0"
```
```   933   proof (cases "x0 < 0")
```
```   934     case True
```
```   935     hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
```
```   936     thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
```
```   937   next
```
```   938     case False
```
```   939     have "- ?R < 0" using assms by auto
```
```   940     also have "\<dots> \<le> x0" using False by auto
```
```   941     finally show ?thesis .
```
```   942   qed
```
```   943   hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
```
```   944     using assms by auto
```
```   945   from for_subinterval[OF this]
```
```   946   show ?thesis .
```
```   947 qed
```
```   948
```
```   949
```
```   950 subsection {* Exponential Function *}
```
```   951
```
```   952 definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```   953   where "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
```
```   954
```
```   955 lemma summable_exp_generic:
```
```   956   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   957   defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
```
```   958   shows "summable S"
```
```   959 proof -
```
```   960   have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
```
```   961     unfolding S_def by (simp del: mult_Suc)
```
```   962   obtain r :: real where r0: "0 < r" and r1: "r < 1"
```
```   963     using dense [OF zero_less_one] by fast
```
```   964   obtain N :: nat where N: "norm x < real N * r"
```
```   965     using reals_Archimedean3 [OF r0] by fast
```
```   966   from r1 show ?thesis
```
```   967   proof (rule ratio_test [rule_format])
```
```   968     fix n :: nat
```
```   969     assume n: "N \<le> n"
```
```   970     have "norm x \<le> real N * r"
```
```   971       using N by (rule order_less_imp_le)
```
```   972     also have "real N * r \<le> real (Suc n) * r"
```
```   973       using r0 n by (simp add: mult_right_mono)
```
```   974     finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
```
```   975       using norm_ge_zero by (rule mult_right_mono)
```
```   976     hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
```
```   977       by (rule order_trans [OF norm_mult_ineq])
```
```   978     hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
```
```   979       by (simp add: pos_divide_le_eq mult_ac)
```
```   980     thus "norm (S (Suc n)) \<le> r * norm (S n)"
```
```   981       by (simp add: S_Suc inverse_eq_divide)
```
```   982   qed
```
```   983 qed
```
```   984
```
```   985 lemma summable_norm_exp:
```
```   986   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   987   shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
```
```   988 proof (rule summable_norm_comparison_test [OF exI, rule_format])
```
```   989   show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
```
```   990     by (rule summable_exp_generic)
```
```   991   fix n
```
```   992   show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
```
```   993     by (simp add: norm_power_ineq)
```
```   994 qed
```
```   995
```
```   996 lemma summable_exp: "summable (\<lambda>n. inverse (real (fact n)) * x ^ n)"
```
```   997   using summable_exp_generic [where x=x] by simp
```
```   998
```
```   999 lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
```
```  1000   unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
```
```  1001
```
```  1002
```
```  1003 lemma exp_fdiffs:
```
```  1004       "diffs (\<lambda>n. inverse(real (fact n))) = (\<lambda>n. inverse(real (fact n)))"
```
```  1005   by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
```
```  1006         del: mult_Suc of_nat_Suc)
```
```  1007
```
```  1008 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
```
```  1009   by (simp add: diffs_def)
```
```  1010
```
```  1011 lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
```
```  1012   unfolding exp_def scaleR_conv_of_real
```
```  1013   apply (rule DERIV_cong)
```
```  1014   apply (rule termdiffs [where K="of_real (1 + norm x)"])
```
```  1015   apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
```
```  1016   apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
```
```  1017   apply (simp del: of_real_add)
```
```  1018   done
```
```  1019
```
```  1020 declare DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1021
```
```  1022 lemma isCont_exp: "isCont exp x"
```
```  1023   by (rule DERIV_exp [THEN DERIV_isCont])
```
```  1024
```
```  1025 lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
```
```  1026   by (rule isCont_o2 [OF _ isCont_exp])
```
```  1027
```
```  1028 lemma tendsto_exp [tendsto_intros]:
```
```  1029   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
```
```  1030   by (rule isCont_tendsto_compose [OF isCont_exp])
```
```  1031
```
```  1032 lemma continuous_exp [continuous_intros]:
```
```  1033   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
```
```  1034   unfolding continuous_def by (rule tendsto_exp)
```
```  1035
```
```  1036 lemma continuous_on_exp [continuous_on_intros]:
```
```  1037   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
```
```  1038   unfolding continuous_on_def by (auto intro: tendsto_exp)
```
```  1039
```
```  1040
```
```  1041 subsubsection {* Properties of the Exponential Function *}
```
```  1042
```
```  1043 lemma powser_zero:
```
```  1044   fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
```
```  1045   shows "(\<Sum>n. f n * 0 ^ n) = f 0"
```
```  1046 proof -
```
```  1047   have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
```
```  1048     by (rule sums_unique [OF series_zero], simp add: power_0_left)
```
```  1049   thus ?thesis unfolding One_nat_def by simp
```
```  1050 qed
```
```  1051
```
```  1052 lemma exp_zero [simp]: "exp 0 = 1"
```
```  1053   unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
```
```  1054
```
```  1055 lemma setsum_cl_ivl_Suc2:
```
```  1056   "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
```
```  1057   by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
```
```  1058            del: setsum_cl_ivl_Suc)
```
```  1059
```
```  1060 lemma exp_series_add:
```
```  1061   fixes x y :: "'a::{real_field}"
```
```  1062   defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
```
```  1063   shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
```
```  1064 proof (induct n)
```
```  1065   case 0
```
```  1066   show ?case
```
```  1067     unfolding S_def by simp
```
```  1068 next
```
```  1069   case (Suc n)
```
```  1070   have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
```
```  1071     unfolding S_def by (simp del: mult_Suc)
```
```  1072   hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
```
```  1073     by simp
```
```  1074
```
```  1075   have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
```
```  1076     by (simp only: times_S)
```
```  1077   also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
```
```  1078     by (simp only: Suc)
```
```  1079   also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
```
```  1080                 + y * (\<Sum>i=0..n. S x i * S y (n-i))"
```
```  1081     by (rule distrib_right)
```
```  1082   also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
```
```  1083                 + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
```
```  1084     by (simp only: setsum_right_distrib mult_ac)
```
```  1085   also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
```
```  1086                 + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1087     by (simp add: times_S Suc_diff_le)
```
```  1088   also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
```
```  1089              (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1090     by (subst setsum_cl_ivl_Suc2, simp)
```
```  1091   also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```  1092              (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1093     by (subst setsum_cl_ivl_Suc, simp)
```
```  1094   also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
```
```  1095              (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```  1096              (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1097     by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
```
```  1098               real_of_nat_add [symmetric], simp)
```
```  1099   also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
```
```  1100     by (simp only: scaleR_right.setsum)
```
```  1101   finally show
```
```  1102     "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
```
```  1103     by (simp del: setsum_cl_ivl_Suc)
```
```  1104 qed
```
```  1105
```
```  1106 lemma exp_add: "exp (x + y) = exp x * exp y"
```
```  1107   unfolding exp_def
```
```  1108   by (simp only: Cauchy_product summable_norm_exp exp_series_add)
```
```  1109
```
```  1110 lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
```
```  1111   by (rule exp_add [symmetric])
```
```  1112
```
```  1113 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
```
```  1114   unfolding exp_def
```
```  1115   apply (subst suminf_of_real)
```
```  1116   apply (rule summable_exp_generic)
```
```  1117   apply (simp add: scaleR_conv_of_real)
```
```  1118   done
```
```  1119
```
```  1120 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
```
```  1121 proof
```
```  1122   have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
```
```  1123   also assume "exp x = 0"
```
```  1124   finally show "False" by simp
```
```  1125 qed
```
```  1126
```
```  1127 lemma exp_minus: "exp (- x) = inverse (exp x)"
```
```  1128   by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
```
```  1129
```
```  1130 lemma exp_diff: "exp (x - y) = exp x / exp y"
```
```  1131   using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
```
```  1132
```
```  1133
```
```  1134 subsubsection {* Properties of the Exponential Function on Reals *}
```
```  1135
```
```  1136 text {* Comparisons of @{term "exp x"} with zero. *}
```
```  1137
```
```  1138 text{*Proof: because every exponential can be seen as a square.*}
```
```  1139 lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
```
```  1140 proof -
```
```  1141   have "0 \<le> exp (x/2) * exp (x/2)" by simp
```
```  1142   thus ?thesis by (simp add: exp_add [symmetric])
```
```  1143 qed
```
```  1144
```
```  1145 lemma exp_gt_zero [simp]: "0 < exp (x::real)"
```
```  1146   by (simp add: order_less_le)
```
```  1147
```
```  1148 lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
```
```  1149   by (simp add: not_less)
```
```  1150
```
```  1151 lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
```
```  1152   by (simp add: not_le)
```
```  1153
```
```  1154 lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
```
```  1155   by simp
```
```  1156
```
```  1157 lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
```
```  1158   by (induct n) (auto simp add: real_of_nat_Suc distrib_left exp_add mult_commute)
```
```  1159
```
```  1160 text {* Strict monotonicity of exponential. *}
```
```  1161
```
```  1162 lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) \<Longrightarrow> (1 + x) \<le> exp(x)"
```
```  1163   apply (drule order_le_imp_less_or_eq, auto)
```
```  1164   apply (simp add: exp_def)
```
```  1165   apply (rule order_trans)
```
```  1166   apply (rule_tac [2] n = 2 and f = "(\<lambda>n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
```
```  1167   apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
```
```  1168   done
```
```  1169
```
```  1170 lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
```
```  1171 proof -
```
```  1172   assume x: "0 < x"
```
```  1173   hence "1 < 1 + x" by simp
```
```  1174   also from x have "1 + x \<le> exp x"
```
```  1175     by (simp add: exp_ge_add_one_self_aux)
```
```  1176   finally show ?thesis .
```
```  1177 qed
```
```  1178
```
```  1179 lemma exp_less_mono:
```
```  1180   fixes x y :: real
```
```  1181   assumes "x < y"
```
```  1182   shows "exp x < exp y"
```
```  1183 proof -
```
```  1184   from `x < y` have "0 < y - x" by simp
```
```  1185   hence "1 < exp (y - x)" by (rule exp_gt_one)
```
```  1186   hence "1 < exp y / exp x" by (simp only: exp_diff)
```
```  1187   thus "exp x < exp y" by simp
```
```  1188 qed
```
```  1189
```
```  1190 lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
```
```  1191   apply (simp add: linorder_not_le [symmetric])
```
```  1192   apply (auto simp add: order_le_less exp_less_mono)
```
```  1193   done
```
```  1194
```
```  1195 lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
```
```  1196   by (auto intro: exp_less_mono exp_less_cancel)
```
```  1197
```
```  1198 lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
```
```  1199   by (auto simp add: linorder_not_less [symmetric])
```
```  1200
```
```  1201 lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
```
```  1202   by (simp add: order_eq_iff)
```
```  1203
```
```  1204 text {* Comparisons of @{term "exp x"} with one. *}
```
```  1205
```
```  1206 lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
```
```  1207   using exp_less_cancel_iff [where x=0 and y=x] by simp
```
```  1208
```
```  1209 lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
```
```  1210   using exp_less_cancel_iff [where x=x and y=0] by simp
```
```  1211
```
```  1212 lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
```
```  1213   using exp_le_cancel_iff [where x=0 and y=x] by simp
```
```  1214
```
```  1215 lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
```
```  1216   using exp_le_cancel_iff [where x=x and y=0] by simp
```
```  1217
```
```  1218 lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
```
```  1219   using exp_inj_iff [where x=x and y=0] by simp
```
```  1220
```
```  1221 lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
```
```  1222 proof (rule IVT)
```
```  1223   assume "1 \<le> y"
```
```  1224   hence "0 \<le> y - 1" by simp
```
```  1225   hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
```
```  1226   thus "y \<le> exp (y - 1)" by simp
```
```  1227 qed (simp_all add: le_diff_eq)
```
```  1228
```
```  1229 lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y"
```
```  1230 proof (rule linorder_le_cases [of 1 y])
```
```  1231   assume "1 \<le> y"
```
```  1232   thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total)
```
```  1233 next
```
```  1234   assume "0 < y" and "y \<le> 1"
```
```  1235   hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
```
```  1236   then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
```
```  1237   hence "exp (- x) = y" by (simp add: exp_minus)
```
```  1238   thus "\<exists>x. exp x = y" ..
```
```  1239 qed
```
```  1240
```
```  1241
```
```  1242 subsection {* Natural Logarithm *}
```
```  1243
```
```  1244 definition ln :: "real \<Rightarrow> real"
```
```  1245   where "ln x = (THE u. exp u = x)"
```
```  1246
```
```  1247 lemma ln_exp [simp]: "ln (exp x) = x"
```
```  1248   by (simp add: ln_def)
```
```  1249
```
```  1250 lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
```
```  1251   by (auto dest: exp_total)
```
```  1252
```
```  1253 lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
```
```  1254   by (metis exp_gt_zero exp_ln)
```
```  1255
```
```  1256 lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
```
```  1257   by (erule subst, rule ln_exp)
```
```  1258
```
```  1259 lemma ln_one [simp]: "ln 1 = 0"
```
```  1260   by (rule ln_unique) simp
```
```  1261
```
```  1262 lemma ln_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
```
```  1263   by (rule ln_unique) (simp add: exp_add)
```
```  1264
```
```  1265 lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
```
```  1266   by (rule ln_unique) (simp add: exp_minus)
```
```  1267
```
```  1268 lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
```
```  1269   by (rule ln_unique) (simp add: exp_diff)
```
```  1270
```
```  1271 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
```
```  1272   by (rule ln_unique) (simp add: exp_real_of_nat_mult)
```
```  1273
```
```  1274 lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
```
```  1275   by (subst exp_less_cancel_iff [symmetric]) simp
```
```  1276
```
```  1277 lemma ln_le_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
```
```  1278   by (simp add: linorder_not_less [symmetric])
```
```  1279
```
```  1280 lemma ln_inj_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
```
```  1281   by (simp add: order_eq_iff)
```
```  1282
```
```  1283 lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
```
```  1284   apply (rule exp_le_cancel_iff [THEN iffD1])
```
```  1285   apply (simp add: exp_ge_add_one_self_aux)
```
```  1286   done
```
```  1287
```
```  1288 lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
```
```  1289   by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
```
```  1290
```
```  1291 lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
```
```  1292   using ln_le_cancel_iff [of 1 x] by simp
```
```  1293
```
```  1294 lemma ln_ge_zero_imp_ge_one: "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
```
```  1295   using ln_le_cancel_iff [of 1 x] by simp
```
```  1296
```
```  1297 lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
```
```  1298   using ln_le_cancel_iff [of 1 x] by simp
```
```  1299
```
```  1300 lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
```
```  1301   using ln_less_cancel_iff [of x 1] by simp
```
```  1302
```
```  1303 lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
```
```  1304   using ln_less_cancel_iff [of 1 x] by simp
```
```  1305
```
```  1306 lemma ln_gt_zero_imp_gt_one: "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
```
```  1307   using ln_less_cancel_iff [of 1 x] by simp
```
```  1308
```
```  1309 lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
```
```  1310   using ln_less_cancel_iff [of 1 x] by simp
```
```  1311
```
```  1312 lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
```
```  1313   using ln_inj_iff [of x 1] by simp
```
```  1314
```
```  1315 lemma ln_less_zero: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
```
```  1316   by simp
```
```  1317
```
```  1318 lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
```
```  1319   apply (subgoal_tac "isCont ln (exp (ln x))", simp)
```
```  1320   apply (rule isCont_inverse_function [where f=exp], simp_all)
```
```  1321   done
```
```  1322
```
```  1323 lemma tendsto_ln [tendsto_intros]:
```
```  1324   "(f ---> a) F \<Longrightarrow> 0 < a \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
```
```  1325   by (rule isCont_tendsto_compose [OF isCont_ln])
```
```  1326
```
```  1327 lemma continuous_ln:
```
```  1328   "continuous F f \<Longrightarrow> 0 < f (Lim F (\<lambda>x. x)) \<Longrightarrow> continuous F (\<lambda>x. ln (f x))"
```
```  1329   unfolding continuous_def by (rule tendsto_ln)
```
```  1330
```
```  1331 lemma isCont_ln' [continuous_intros]:
```
```  1332   "continuous (at x) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x))"
```
```  1333   unfolding continuous_at by (rule tendsto_ln)
```
```  1334
```
```  1335 lemma continuous_within_ln [continuous_intros]:
```
```  1336   "continuous (at x within s) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x))"
```
```  1337   unfolding continuous_within by (rule tendsto_ln)
```
```  1338
```
```  1339 lemma continuous_on_ln [continuous_on_intros]:
```
```  1340   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. 0 < f x) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x))"
```
```  1341   unfolding continuous_on_def by (auto intro: tendsto_ln)
```
```  1342
```
```  1343 lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
```
```  1344   apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
```
```  1345   apply (erule DERIV_cong [OF DERIV_exp exp_ln])
```
```  1346   apply (simp_all add: abs_if isCont_ln)
```
```  1347   done
```
```  1348
```
```  1349 lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
```
```  1350   by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
```
```  1351
```
```  1352 declare DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1353
```
```  1354 lemma ln_series:
```
```  1355   assumes "0 < x" and "x < 2"
```
```  1356   shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))"
```
```  1357   (is "ln x = suminf (?f (x - 1))")
```
```  1358 proof -
```
```  1359   let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n"
```
```  1360
```
```  1361   have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
```
```  1362   proof (rule DERIV_isconst3[where x=x])
```
```  1363     fix x :: real
```
```  1364     assume "x \<in> {0 <..< 2}"
```
```  1365     hence "0 < x" and "x < 2" by auto
```
```  1366     have "norm (1 - x) < 1"
```
```  1367       using `0 < x` and `x < 2` by auto
```
```  1368     have "1 / x = 1 / (1 - (1 - x))" by auto
```
```  1369     also have "\<dots> = (\<Sum> n. (1 - x)^n)"
```
```  1370       using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
```
```  1371     also have "\<dots> = suminf (?f' x)"
```
```  1372       unfolding power_mult_distrib[symmetric]
```
```  1373       by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
```
```  1374     finally have "DERIV ln x :> suminf (?f' x)"
```
```  1375       using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
```
```  1376     moreover
```
```  1377     have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
```
```  1378     have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :>
```
```  1379       (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
```
```  1380     proof (rule DERIV_power_series')
```
```  1381       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1"
```
```  1382         using `0 < x` `x < 2` by auto
```
```  1383       fix x :: real
```
```  1384       assume "x \<in> {- 1<..<1}"
```
```  1385       hence "norm (-x) < 1" by auto
```
```  1386       show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
```
```  1387         unfolding One_nat_def
```
```  1388         by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
```
```  1389     qed
```
```  1390     hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
```
```  1391       unfolding One_nat_def by auto
```
```  1392     hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
```
```  1393       unfolding DERIV_iff repos .
```
```  1394     ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
```
```  1395       by (rule DERIV_diff)
```
```  1396     thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
```
```  1397   qed (auto simp add: assms)
```
```  1398   thus ?thesis by auto
```
```  1399 qed
```
```  1400
```
```  1401 lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) * (x ^ (n+2)))"
```
```  1402 proof -
```
```  1403   have "exp x = suminf (\<lambda>n. inverse(fact n) * (x ^ n))"
```
```  1404     by (simp add: exp_def)
```
```  1405   also from summable_exp have "... = (\<Sum> n::nat = 0 ..< 2. inverse(fact n) * (x ^ n)) +
```
```  1406       (\<Sum> n. inverse(fact(n+2)) * (x ^ (n+2)))" (is "_ = ?a + _")
```
```  1407     by (rule suminf_split_initial_segment)
```
```  1408   also have "?a = 1 + x"
```
```  1409     by (simp add: numeral_2_eq_2)
```
```  1410   finally show ?thesis .
```
```  1411 qed
```
```  1412
```
```  1413 lemma exp_bound: "0 <= (x::real) \<Longrightarrow> x <= 1 \<Longrightarrow> exp x <= 1 + x + x\<^sup>2"
```
```  1414 proof -
```
```  1415   assume a: "0 <= x"
```
```  1416   assume b: "x <= 1"
```
```  1417   {
```
```  1418     fix n :: nat
```
```  1419     have "2 * 2 ^ n \<le> fact (n + 2)"
```
```  1420       by (induct n) simp_all
```
```  1421     hence "real ((2::nat) * 2 ^ n) \<le> real (fact (n + 2))"
```
```  1422       by (simp only: real_of_nat_le_iff)
```
```  1423     hence "2 * 2 ^ n \<le> real (fact (n + 2))"
```
```  1424       by simp
```
```  1425     hence "inverse (fact (n + 2)) \<le> inverse (2 * 2 ^ n)"
```
```  1426       by (rule le_imp_inverse_le) simp
```
```  1427     hence "inverse (fact (n + 2)) \<le> 1/2 * (1/2)^n"
```
```  1428       by (simp add: power_inverse)
```
```  1429     hence "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
```
```  1430       by (rule mult_mono)
```
```  1431         (rule mult_mono, simp_all add: power_le_one a b mult_nonneg_nonneg)
```
```  1432     hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)"
```
```  1433       unfolding power_add by (simp add: mult_ac del: fact_Suc) }
```
```  1434   note aux1 = this
```
```  1435   have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
```
```  1436     by (intro sums_mult geometric_sums, simp)
```
```  1437   hence aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
```
```  1438     by simp
```
```  1439   have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2"
```
```  1440   proof -
```
```  1441     have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <=
```
```  1442         suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
```
```  1443       apply (rule summable_le)
```
```  1444       apply (rule allI, rule aux1)
```
```  1445       apply (rule summable_exp [THEN summable_ignore_initial_segment])
```
```  1446       by (rule sums_summable, rule aux2)
```
```  1447     also have "... = x\<^sup>2"
```
```  1448       by (rule sums_unique [THEN sym], rule aux2)
```
```  1449     finally show ?thesis .
```
```  1450   qed
```
```  1451   thus ?thesis unfolding exp_first_two_terms by auto
```
```  1452 qed
```
```  1453
```
```  1454 lemma ln_one_minus_pos_upper_bound: "0 <= x \<Longrightarrow> x < 1 \<Longrightarrow> ln (1 - x) <= - x"
```
```  1455 proof -
```
```  1456   assume a: "0 <= (x::real)" and b: "x < 1"
```
```  1457   have "(1 - x) * (1 + x + x\<^sup>2) = (1 - x^3)"
```
```  1458     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
```
```  1459   also have "... <= 1"
```
```  1460     by (auto simp add: a)
```
```  1461   finally have "(1 - x) * (1 + x + x\<^sup>2) <= 1" .
```
```  1462   moreover have c: "0 < 1 + x + x\<^sup>2"
```
```  1463     by (simp add: add_pos_nonneg a)
```
```  1464   ultimately have "1 - x <= 1 / (1 + x + x\<^sup>2)"
```
```  1465     by (elim mult_imp_le_div_pos)
```
```  1466   also have "... <= 1 / exp x"
```
```  1467     apply (rule divide_left_mono)
```
```  1468     apply (rule exp_bound, rule a)
```
```  1469     apply (rule b [THEN less_imp_le])
```
```  1470     apply simp
```
```  1471     apply (rule mult_pos_pos)
```
```  1472     apply (rule c)
```
```  1473     apply simp
```
```  1474     done
```
```  1475   also have "... = exp (-x)"
```
```  1476     by (auto simp add: exp_minus divide_inverse)
```
```  1477   finally have "1 - x <= exp (- x)" .
```
```  1478   also have "1 - x = exp (ln (1 - x))"
```
```  1479   proof -
```
```  1480     have "0 < 1 - x"
```
```  1481       by (insert b, auto)
```
```  1482     thus ?thesis
```
```  1483       by (auto simp only: exp_ln_iff [THEN sym])
```
```  1484   qed
```
```  1485   finally have "exp (ln (1 - x)) <= exp (- x)" .
```
```  1486   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```  1487 qed
```
```  1488
```
```  1489 lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
```
```  1490   apply (case_tac "0 <= x")
```
```  1491   apply (erule exp_ge_add_one_self_aux)
```
```  1492   apply (case_tac "x <= -1")
```
```  1493   apply (subgoal_tac "1 + x <= 0")
```
```  1494   apply (erule order_trans)
```
```  1495   apply simp
```
```  1496   apply simp
```
```  1497   apply (subgoal_tac "1 + x = exp(ln (1 + x))")
```
```  1498   apply (erule ssubst)
```
```  1499   apply (subst exp_le_cancel_iff)
```
```  1500   apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
```
```  1501   apply simp
```
```  1502   apply (rule ln_one_minus_pos_upper_bound)
```
```  1503   apply auto
```
```  1504 done
```
```  1505
```
```  1506 lemma ln_one_plus_pos_lower_bound: "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> x - x\<^sup>2 <= ln (1 + x)"
```
```  1507 proof -
```
```  1508   assume a: "0 <= x" and b: "x <= 1"
```
```  1509   have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
```
```  1510     by (rule exp_diff)
```
```  1511   also have "... <= (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
```
```  1512     apply (rule divide_right_mono)
```
```  1513     apply (rule exp_bound)
```
```  1514     apply (rule a, rule b)
```
```  1515     apply simp
```
```  1516     done
```
```  1517   also have "... <= (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
```
```  1518     apply (rule divide_left_mono)
```
```  1519     apply (simp add: exp_ge_add_one_self_aux)
```
```  1520     apply (simp add: a)
```
```  1521     apply (simp add: mult_pos_pos add_pos_nonneg)
```
```  1522     done
```
```  1523   also from a have "... <= 1 + x"
```
```  1524     by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
```
```  1525   finally have "exp (x - x\<^sup>2) <= 1 + x" .
```
```  1526   also have "... = exp (ln (1 + x))"
```
```  1527   proof -
```
```  1528     from a have "0 < 1 + x" by auto
```
```  1529     thus ?thesis
```
```  1530       by (auto simp only: exp_ln_iff [THEN sym])
```
```  1531   qed
```
```  1532   finally have "exp (x - x\<^sup>2) <= exp (ln (1 + x))" .
```
```  1533   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```  1534 qed
```
```  1535
```
```  1536 lemma aux5: "x < 1 \<Longrightarrow> ln(1 - x) = - ln(1 + x / (1 - x))"
```
```  1537 proof -
```
```  1538   assume a: "x < 1"
```
```  1539   have "ln(1 - x) = - ln(1 / (1 - x))"
```
```  1540   proof -
```
```  1541     have "ln(1 - x) = - (- ln (1 - x))"
```
```  1542       by auto
```
```  1543     also have "- ln(1 - x) = ln 1 - ln(1 - x)"
```
```  1544       by simp
```
```  1545     also have "... = ln(1 / (1 - x))"
```
```  1546       apply (rule ln_div [THEN sym])
```
```  1547       using a apply auto
```
```  1548       done
```
```  1549     finally show ?thesis .
```
```  1550   qed
```
```  1551   also have " 1 / (1 - x) = 1 + x / (1 - x)" using a by(simp add:field_simps)
```
```  1552   finally show ?thesis .
```
```  1553 qed
```
```  1554
```
```  1555 lemma ln_one_minus_pos_lower_bound:
```
```  1556   "0 <= x \<Longrightarrow> x <= (1 / 2) \<Longrightarrow> - x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1557 proof -
```
```  1558   assume a: "0 <= x" and b: "x <= (1 / 2)"
```
```  1559   from b have c: "x < 1" by auto
```
```  1560   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
```
```  1561     by (rule aux5)
```
```  1562   also have "- (x / (1 - x)) <= ..."
```
```  1563   proof -
```
```  1564     have "ln (1 + x / (1 - x)) <= x / (1 - x)"
```
```  1565       apply (rule ln_add_one_self_le_self)
```
```  1566       apply (rule divide_nonneg_pos)
```
```  1567       using a c apply auto
```
```  1568       done
```
```  1569     thus ?thesis
```
```  1570       by auto
```
```  1571   qed
```
```  1572   also have "- (x / (1 - x)) = -x / (1 - x)"
```
```  1573     by auto
```
```  1574   finally have d: "- x / (1 - x) <= ln (1 - x)" .
```
```  1575   have "0 < 1 - x" using a b by simp
```
```  1576   hence e: "-x - 2 * x\<^sup>2 <= - x / (1 - x)"
```
```  1577     using mult_right_le_one_le[of "x*x" "2*x"] a b
```
```  1578     by (simp add: field_simps power2_eq_square)
```
```  1579   from e d show "- x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1580     by (rule order_trans)
```
```  1581 qed
```
```  1582
```
```  1583 lemma ln_add_one_self_le_self2: "-1 < x \<Longrightarrow> ln(1 + x) <= x"
```
```  1584   apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
```
```  1585   apply (subst ln_le_cancel_iff)
```
```  1586   apply auto
```
```  1587   done
```
```  1588
```
```  1589 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
```
```  1590   "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> abs(ln (1 + x) - x) <= x\<^sup>2"
```
```  1591 proof -
```
```  1592   assume x: "0 <= x"
```
```  1593   assume x1: "x <= 1"
```
```  1594   from x have "ln (1 + x) <= x"
```
```  1595     by (rule ln_add_one_self_le_self)
```
```  1596   then have "ln (1 + x) - x <= 0"
```
```  1597     by simp
```
```  1598   then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
```
```  1599     by (rule abs_of_nonpos)
```
```  1600   also have "... = x - ln (1 + x)"
```
```  1601     by simp
```
```  1602   also have "... <= x\<^sup>2"
```
```  1603   proof -
```
```  1604     from x x1 have "x - x\<^sup>2 <= ln (1 + x)"
```
```  1605       by (intro ln_one_plus_pos_lower_bound)
```
```  1606     thus ?thesis
```
```  1607       by simp
```
```  1608   qed
```
```  1609   finally show ?thesis .
```
```  1610 qed
```
```  1611
```
```  1612 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
```
```  1613   "-(1 / 2) <= x \<Longrightarrow> x <= 0 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
```
```  1614 proof -
```
```  1615   assume a: "-(1 / 2) <= x"
```
```  1616   assume b: "x <= 0"
```
```  1617   have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
```
```  1618     apply (subst abs_of_nonpos)
```
```  1619     apply simp
```
```  1620     apply (rule ln_add_one_self_le_self2)
```
```  1621     using a apply auto
```
```  1622     done
```
```  1623   also have "... <= 2 * x\<^sup>2"
```
```  1624     apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 <= ln (1 - (-x))")
```
```  1625     apply (simp add: algebra_simps)
```
```  1626     apply (rule ln_one_minus_pos_lower_bound)
```
```  1627     using a b apply auto
```
```  1628     done
```
```  1629   finally show ?thesis .
```
```  1630 qed
```
```  1631
```
```  1632 lemma abs_ln_one_plus_x_minus_x_bound:
```
```  1633     "abs x <= 1 / 2 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
```
```  1634   apply (case_tac "0 <= x")
```
```  1635   apply (rule order_trans)
```
```  1636   apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
```
```  1637   apply auto
```
```  1638   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
```
```  1639   apply auto
```
```  1640   done
```
```  1641
```
```  1642 lemma ln_x_over_x_mono: "exp 1 <= x \<Longrightarrow> x <= y \<Longrightarrow> (ln y / y) <= (ln x / x)"
```
```  1643 proof -
```
```  1644   assume x: "exp 1 <= x" "x <= y"
```
```  1645   moreover have "0 < exp (1::real)" by simp
```
```  1646   ultimately have a: "0 < x" and b: "0 < y"
```
```  1647     by (fast intro: less_le_trans order_trans)+
```
```  1648   have "x * ln y - x * ln x = x * (ln y - ln x)"
```
```  1649     by (simp add: algebra_simps)
```
```  1650   also have "... = x * ln(y / x)"
```
```  1651     by (simp only: ln_div a b)
```
```  1652   also have "y / x = (x + (y - x)) / x"
```
```  1653     by simp
```
```  1654   also have "... = 1 + (y - x) / x"
```
```  1655     using x a by (simp add: field_simps)
```
```  1656   also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
```
```  1657     apply (rule mult_left_mono)
```
```  1658     apply (rule ln_add_one_self_le_self)
```
```  1659     apply (rule divide_nonneg_pos)
```
```  1660     using x a apply simp_all
```
```  1661     done
```
```  1662   also have "... = y - x" using a by simp
```
```  1663   also have "... = (y - x) * ln (exp 1)" by simp
```
```  1664   also have "... <= (y - x) * ln x"
```
```  1665     apply (rule mult_left_mono)
```
```  1666     apply (subst ln_le_cancel_iff)
```
```  1667     apply fact
```
```  1668     apply (rule a)
```
```  1669     apply (rule x)
```
```  1670     using x apply simp
```
```  1671     done
```
```  1672   also have "... = y * ln x - x * ln x"
```
```  1673     by (rule left_diff_distrib)
```
```  1674   finally have "x * ln y <= y * ln x"
```
```  1675     by arith
```
```  1676   then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
```
```  1677   also have "... = y * (ln x / x)" by simp
```
```  1678   finally show ?thesis using b by (simp add: field_simps)
```
```  1679 qed
```
```  1680
```
```  1681 lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1"
```
```  1682   using exp_ge_add_one_self[of "ln x"] by simp
```
```  1683
```
```  1684 lemma ln_eq_minus_one:
```
```  1685   assumes "0 < x" "ln x = x - 1"
```
```  1686   shows "x = 1"
```
```  1687 proof -
```
```  1688   let ?l = "\<lambda>y. ln y - y + 1"
```
```  1689   have D: "\<And>x. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
```
```  1690     by (auto intro!: DERIV_intros)
```
```  1691
```
```  1692   show ?thesis
```
```  1693   proof (cases rule: linorder_cases)
```
```  1694     assume "x < 1"
```
```  1695     from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast
```
```  1696     from `x < a` have "?l x < ?l a"
```
```  1697     proof (rule DERIV_pos_imp_increasing, safe)
```
```  1698       fix y
```
```  1699       assume "x \<le> y" "y \<le> a"
```
```  1700       with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y"
```
```  1701         by (auto simp: field_simps)
```
```  1702       with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
```
```  1703         by auto
```
```  1704     qed
```
```  1705     also have "\<dots> \<le> 0"
```
```  1706       using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps)
```
```  1707     finally show "x = 1" using assms by auto
```
```  1708   next
```
```  1709     assume "1 < x"
```
```  1710     from dense[OF this] obtain a where "1 < a" "a < x" by blast
```
```  1711     from `a < x` have "?l x < ?l a"
```
```  1712     proof (rule DERIV_neg_imp_decreasing, safe)
```
```  1713       fix y
```
```  1714       assume "a \<le> y" "y \<le> x"
```
```  1715       with `1 < a` have "1 / y - 1 < 0" "0 < y"
```
```  1716         by (auto simp: field_simps)
```
```  1717       with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
```
```  1718         by blast
```
```  1719     qed
```
```  1720     also have "\<dots> \<le> 0"
```
```  1721       using ln_le_minus_one `1 < a` by (auto simp: field_simps)
```
```  1722     finally show "x = 1" using assms by auto
```
```  1723   next
```
```  1724     assume "x = 1"
```
```  1725     then show ?thesis by simp
```
```  1726   qed
```
```  1727 qed
```
```  1728
```
```  1729 lemma exp_at_bot: "(exp ---> (0::real)) at_bot"
```
```  1730   unfolding tendsto_Zfun_iff
```
```  1731 proof (rule ZfunI, simp add: eventually_at_bot_dense)
```
```  1732   fix r :: real assume "0 < r"
```
```  1733   {
```
```  1734     fix x
```
```  1735     assume "x < ln r"
```
```  1736     then have "exp x < exp (ln r)"
```
```  1737       by simp
```
```  1738     with `0 < r` have "exp x < r"
```
```  1739       by simp
```
```  1740   }
```
```  1741   then show "\<exists>k. \<forall>n<k. exp n < r" by auto
```
```  1742 qed
```
```  1743
```
```  1744 lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
```
```  1745   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"])
```
```  1746      (auto intro: eventually_gt_at_top)
```
```  1747
```
```  1748 lemma ln_at_0: "LIM x at_right 0. ln x :> at_bot"
```
```  1749   by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  1750      (auto simp: eventually_at_filter)
```
```  1751
```
```  1752 lemma ln_at_top: "LIM x at_top. ln x :> at_top"
```
```  1753   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  1754      (auto intro: eventually_gt_at_top)
```
```  1755
```
```  1756 lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) ---> (0::real)) at_top"
```
```  1757 proof (induct k)
```
```  1758   case 0
```
```  1759   show "((\<lambda>x. x ^ 0 / exp x) ---> (0::real)) at_top"
```
```  1760     by (simp add: inverse_eq_divide[symmetric])
```
```  1761        (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
```
```  1762               at_top_le_at_infinity order_refl)
```
```  1763 next
```
```  1764   case (Suc k)
```
```  1765   show ?case
```
```  1766   proof (rule lhospital_at_top_at_top)
```
```  1767     show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"
```
```  1768       by eventually_elim (intro DERIV_intros, simp, simp)
```
```  1769     show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top"
```
```  1770       by eventually_elim (auto intro!: DERIV_intros)
```
```  1771     show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
```
```  1772       by auto
```
```  1773     from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
```
```  1774     show "((\<lambda>x. real (Suc k) * x ^ k / exp x) ---> 0) at_top"
```
```  1775       by simp
```
```  1776   qed (rule exp_at_top)
```
```  1777 qed
```
```  1778
```
```  1779
```
```  1780 definition powr :: "[real,real] => real"  (infixr "powr" 80)
```
```  1781   -- {*exponentation with real exponent*}
```
```  1782   where "x powr a = exp(a * ln x)"
```
```  1783
```
```  1784 definition log :: "[real,real] => real"
```
```  1785   -- {*logarithm of @{term x} to base @{term a}*}
```
```  1786   where "log a x = ln x / ln a"
```
```  1787
```
```  1788
```
```  1789 lemma tendsto_log [tendsto_intros]:
```
```  1790   "\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a; a \<noteq> 1; 0 < b\<rbrakk> \<Longrightarrow> ((\<lambda>x. log (f x) (g x)) ---> log a b) F"
```
```  1791   unfolding log_def by (intro tendsto_intros) auto
```
```  1792
```
```  1793 lemma continuous_log:
```
```  1794   assumes "continuous F f"
```
```  1795     and "continuous F g"
```
```  1796     and "0 < f (Lim F (\<lambda>x. x))"
```
```  1797     and "f (Lim F (\<lambda>x. x)) \<noteq> 1"
```
```  1798     and "0 < g (Lim F (\<lambda>x. x))"
```
```  1799   shows "continuous F (\<lambda>x. log (f x) (g x))"
```
```  1800   using assms unfolding continuous_def by (rule tendsto_log)
```
```  1801
```
```  1802 lemma continuous_at_within_log[continuous_intros]:
```
```  1803   assumes "continuous (at a within s) f"
```
```  1804     and "continuous (at a within s) g"
```
```  1805     and "0 < f a"
```
```  1806     and "f a \<noteq> 1"
```
```  1807     and "0 < g a"
```
```  1808   shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
```
```  1809   using assms unfolding continuous_within by (rule tendsto_log)
```
```  1810
```
```  1811 lemma isCont_log[continuous_intros, simp]:
```
```  1812   assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
```
```  1813   shows "isCont (\<lambda>x. log (f x) (g x)) a"
```
```  1814   using assms unfolding continuous_at by (rule tendsto_log)
```
```  1815
```
```  1816 lemma continuous_on_log[continuous_on_intros]:
```
```  1817   assumes "continuous_on s f" "continuous_on s g"
```
```  1818     and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
```
```  1819   shows "continuous_on s (\<lambda>x. log (f x) (g x))"
```
```  1820   using assms unfolding continuous_on_def by (fast intro: tendsto_log)
```
```  1821
```
```  1822 lemma powr_one_eq_one [simp]: "1 powr a = 1"
```
```  1823   by (simp add: powr_def)
```
```  1824
```
```  1825 lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
```
```  1826   by (simp add: powr_def)
```
```  1827
```
```  1828 lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
```
```  1829   by (simp add: powr_def)
```
```  1830 declare powr_one_gt_zero_iff [THEN iffD2, simp]
```
```  1831
```
```  1832 lemma powr_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
```
```  1833   by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
```
```  1834
```
```  1835 lemma powr_gt_zero [simp]: "0 < x powr a"
```
```  1836   by (simp add: powr_def)
```
```  1837
```
```  1838 lemma powr_ge_pzero [simp]: "0 <= x powr y"
```
```  1839   by (rule order_less_imp_le, rule powr_gt_zero)
```
```  1840
```
```  1841 lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
```
```  1842   by (simp add: powr_def)
```
```  1843
```
```  1844 lemma powr_divide: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
```
```  1845   apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
```
```  1846   apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
```
```  1847   done
```
```  1848
```
```  1849 lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
```
```  1850   apply (simp add: powr_def)
```
```  1851   apply (subst exp_diff [THEN sym])
```
```  1852   apply (simp add: left_diff_distrib)
```
```  1853   done
```
```  1854
```
```  1855 lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
```
```  1856   by (simp add: powr_def exp_add [symmetric] distrib_right)
```
```  1857
```
```  1858 lemma powr_mult_base: "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
```
```  1859   using assms by (auto simp: powr_add)
```
```  1860
```
```  1861 lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
```
```  1862   by (simp add: powr_def)
```
```  1863
```
```  1864 lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
```
```  1865   by (simp add: powr_powr mult_commute)
```
```  1866
```
```  1867 lemma powr_minus: "x powr (-a) = inverse (x powr a)"
```
```  1868   by (simp add: powr_def exp_minus [symmetric])
```
```  1869
```
```  1870 lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
```
```  1871   by (simp add: divide_inverse powr_minus)
```
```  1872
```
```  1873 lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
```
```  1874   by (simp add: powr_def)
```
```  1875
```
```  1876 lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
```
```  1877   by (simp add: powr_def)
```
```  1878
```
```  1879 lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a < x powr b) = (a < b)"
```
```  1880   by (blast intro: powr_less_cancel powr_less_mono)
```
```  1881
```
```  1882 lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a \<le> x powr b) = (a \<le> b)"
```
```  1883   by (simp add: linorder_not_less [symmetric])
```
```  1884
```
```  1885 lemma log_ln: "ln x = log (exp(1)) x"
```
```  1886   by (simp add: log_def)
```
```  1887
```
```  1888 lemma DERIV_log:
```
```  1889   assumes "x > 0"
```
```  1890   shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
```
```  1891 proof -
```
```  1892   def lb \<equiv> "1 / ln b"
```
```  1893   moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
```
```  1894     using `x > 0` by (auto intro!: DERIV_intros)
```
```  1895   ultimately show ?thesis
```
```  1896     by (simp add: log_def)
```
```  1897 qed
```
```  1898
```
```  1899 lemmas DERIV_log[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1900
```
```  1901 lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
```
```  1902   by (simp add: powr_def log_def)
```
```  1903
```
```  1904 lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y"
```
```  1905   by (simp add: log_def powr_def)
```
```  1906
```
```  1907 lemma log_mult:
```
```  1908   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow>
```
```  1909     log a (x * y) = log a x + log a y"
```
```  1910   by (simp add: log_def ln_mult divide_inverse distrib_right)
```
```  1911
```
```  1912 lemma log_eq_div_ln_mult_log:
```
```  1913   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow>
```
```  1914     log a x = (ln b/ln a) * log b x"
```
```  1915   by (simp add: log_def divide_inverse)
```
```  1916
```
```  1917 text{*Base 10 logarithms*}
```
```  1918 lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x"
```
```  1919   by (simp add: log_def)
```
```  1920
```
```  1921 lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x"
```
```  1922   by (simp add: log_def)
```
```  1923
```
```  1924 lemma log_one [simp]: "log a 1 = 0"
```
```  1925   by (simp add: log_def)
```
```  1926
```
```  1927 lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
```
```  1928   by (simp add: log_def)
```
```  1929
```
```  1930 lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x"
```
```  1931   apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
```
```  1932   apply (simp add: log_mult [symmetric])
```
```  1933   done
```
```  1934
```
```  1935 lemma log_divide: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a (x/y) = log a x - log a y"
```
```  1936   by (simp add: log_mult divide_inverse log_inverse)
```
```  1937
```
```  1938 lemma log_less_cancel_iff [simp]:
```
```  1939   "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
```
```  1940   apply safe
```
```  1941   apply (rule_tac [2] powr_less_cancel)
```
```  1942   apply (drule_tac a = "log a x" in powr_less_mono, auto)
```
```  1943   done
```
```  1944
```
```  1945 lemma log_inj:
```
```  1946   assumes "1 < b"
```
```  1947   shows "inj_on (log b) {0 <..}"
```
```  1948 proof (rule inj_onI, simp)
```
```  1949   fix x y
```
```  1950   assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
```
```  1951   show "x = y"
```
```  1952   proof (cases rule: linorder_cases)
```
```  1953     assume "x = y"
```
```  1954     then show ?thesis by simp
```
```  1955   next
```
```  1956     assume "x < y" hence "log b x < log b y"
```
```  1957       using log_less_cancel_iff[OF `1 < b`] pos by simp
```
```  1958     then show ?thesis using * by simp
```
```  1959   next
```
```  1960     assume "y < x" hence "log b y < log b x"
```
```  1961       using log_less_cancel_iff[OF `1 < b`] pos by simp
```
```  1962     then show ?thesis using * by simp
```
```  1963   qed
```
```  1964 qed
```
```  1965
```
```  1966 lemma log_le_cancel_iff [simp]:
```
```  1967   "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (log a x \<le> log a y) = (x \<le> y)"
```
```  1968   by (simp add: linorder_not_less [symmetric])
```
```  1969
```
```  1970 lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
```
```  1971   using log_less_cancel_iff[of a 1 x] by simp
```
```  1972
```
```  1973 lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
```
```  1974   using log_le_cancel_iff[of a 1 x] by simp
```
```  1975
```
```  1976 lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
```
```  1977   using log_less_cancel_iff[of a x 1] by simp
```
```  1978
```
```  1979 lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
```
```  1980   using log_le_cancel_iff[of a x 1] by simp
```
```  1981
```
```  1982 lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
```
```  1983   using log_less_cancel_iff[of a a x] by simp
```
```  1984
```
```  1985 lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
```
```  1986   using log_le_cancel_iff[of a a x] by simp
```
```  1987
```
```  1988 lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
```
```  1989   using log_less_cancel_iff[of a x a] by simp
```
```  1990
```
```  1991 lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
```
```  1992   using log_le_cancel_iff[of a x a] by simp
```
```  1993
```
```  1994 lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
```
```  1995   apply (induct n)
```
```  1996   apply simp
```
```  1997   apply (subgoal_tac "real(Suc n) = real n + 1")
```
```  1998   apply (erule ssubst)
```
```  1999   apply (subst powr_add, simp, simp)
```
```  2000   done
```
```  2001
```
```  2002 lemma powr_realpow_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
```
```  2003   unfolding real_of_nat_numeral [symmetric] by (rule powr_realpow)
```
```  2004
```
```  2005 lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
```
```  2006   apply (case_tac "x = 0", simp, simp)
```
```  2007   apply (rule powr_realpow [THEN sym], simp)
```
```  2008   done
```
```  2009
```
```  2010 lemma powr_int:
```
```  2011   assumes "x > 0"
```
```  2012   shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
```
```  2013 proof (cases "i < 0")
```
```  2014   case True
```
```  2015   have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
```
```  2016   show ?thesis using `i < 0` `x > 0` by (simp add: r field_simps powr_realpow[symmetric])
```
```  2017 next
```
```  2018   case False
```
```  2019   then show ?thesis by (simp add: assms powr_realpow[symmetric])
```
```  2020 qed
```
```  2021
```
```  2022 lemma powr_one: "0 < x \<Longrightarrow> x powr 1 = x"
```
```  2023   using powr_realpow [of x 1] by simp
```
```  2024
```
```  2025 lemma powr_numeral: "0 < x \<Longrightarrow> x powr numeral n = x ^ numeral n"
```
```  2026   by (fact powr_realpow_numeral)
```
```  2027
```
```  2028 lemma powr_neg_one: "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
```
```  2029   using powr_int [of x "- 1"] by simp
```
```  2030
```
```  2031 lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
```
```  2032   using powr_int [of x "- numeral n"] by simp
```
```  2033
```
```  2034 lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
```
```  2035   by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
```
```  2036
```
```  2037 lemma ln_powr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
```
```  2038   unfolding powr_def by simp
```
```  2039
```
```  2040 lemma log_powr: "0 < x ==> 0 \<le> y ==> log b (x powr y) = y * log b x"
```
```  2041   apply (cases "y = 0")
```
```  2042   apply force
```
```  2043   apply (auto simp add: log_def ln_powr field_simps)
```
```  2044   done
```
```  2045
```
```  2046 lemma log_nat_power: "0 < x ==> log b (x^n) = real n * log b x"
```
```  2047   apply (subst powr_realpow [symmetric])
```
```  2048   apply (auto simp add: log_powr)
```
```  2049   done
```
```  2050
```
```  2051 lemma ln_bound: "1 <= x ==> ln x <= x"
```
```  2052   apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
```
```  2053   apply simp
```
```  2054   apply (rule ln_add_one_self_le_self, simp)
```
```  2055   done
```
```  2056
```
```  2057 lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
```
```  2058   apply (cases "x = 1", simp)
```
```  2059   apply (cases "a = b", simp)
```
```  2060   apply (rule order_less_imp_le)
```
```  2061   apply (rule powr_less_mono, auto)
```
```  2062   done
```
```  2063
```
```  2064 lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
```
```  2065   apply (subst powr_zero_eq_one [THEN sym])
```
```  2066   apply (rule powr_mono, assumption+)
```
```  2067   done
```
```  2068
```
```  2069 lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a < y powr a"
```
```  2070   apply (unfold powr_def)
```
```  2071   apply (rule exp_less_mono)
```
```  2072   apply (rule mult_strict_left_mono)
```
```  2073   apply (subst ln_less_cancel_iff, assumption)
```
```  2074   apply (rule order_less_trans)
```
```  2075   prefer 2
```
```  2076   apply assumption+
```
```  2077   done
```
```  2078
```
```  2079 lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a < x powr a"
```
```  2080   apply (unfold powr_def)
```
```  2081   apply (rule exp_less_mono)
```
```  2082   apply (rule mult_strict_left_mono_neg)
```
```  2083   apply (subst ln_less_cancel_iff)
```
```  2084   apply assumption
```
```  2085   apply (rule order_less_trans)
```
```  2086   prefer 2
```
```  2087   apply assumption+
```
```  2088   done
```
```  2089
```
```  2090 lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
```
```  2091   apply (case_tac "a = 0", simp)
```
```  2092   apply (case_tac "x = y", simp)
```
```  2093   apply (rule order_less_imp_le)
```
```  2094   apply (rule powr_less_mono2, auto)
```
```  2095   done
```
```  2096
```
```  2097 lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
```
```  2098   unfolding powr_def exp_inj_iff by simp
```
```  2099
```
```  2100 lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
```
```  2101   apply (rule mult_imp_le_div_pos)
```
```  2102   apply (assumption)
```
```  2103   apply (subst mult_commute)
```
```  2104   apply (subst ln_powr [THEN sym])
```
```  2105   apply auto
```
```  2106   apply (rule ln_bound)
```
```  2107   apply (erule ge_one_powr_ge_zero)
```
```  2108   apply (erule order_less_imp_le)
```
```  2109   done
```
```  2110
```
```  2111 lemma ln_powr_bound2:
```
```  2112   assumes "1 < x" and "0 < a"
```
```  2113   shows "(ln x) powr a <= (a powr a) * x"
```
```  2114 proof -
```
```  2115   from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
```
```  2116     apply (intro ln_powr_bound)
```
```  2117     apply (erule order_less_imp_le)
```
```  2118     apply (rule divide_pos_pos)
```
```  2119     apply simp_all
```
```  2120     done
```
```  2121   also have "... = a * (x powr (1 / a))"
```
```  2122     by simp
```
```  2123   finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
```
```  2124     apply (intro powr_mono2)
```
```  2125     apply (rule order_less_imp_le, rule assms)
```
```  2126     apply (rule ln_gt_zero)
```
```  2127     apply (rule assms)
```
```  2128     apply assumption
```
```  2129     done
```
```  2130   also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
```
```  2131     apply (rule powr_mult)
```
```  2132     apply (rule assms)
```
```  2133     apply (rule powr_gt_zero)
```
```  2134     done
```
```  2135   also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
```
```  2136     by (rule powr_powr)
```
```  2137   also have "... = x"
```
```  2138     apply simp
```
```  2139     apply (subgoal_tac "a ~= 0")
```
```  2140     using assms apply auto
```
```  2141     done
```
```  2142   finally show ?thesis .
```
```  2143 qed
```
```  2144
```
```  2145 lemma tendsto_powr [tendsto_intros]:
```
```  2146   "\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x powr g x) ---> a powr b) F"
```
```  2147   unfolding powr_def by (intro tendsto_intros)
```
```  2148
```
```  2149 lemma continuous_powr:
```
```  2150   assumes "continuous F f"
```
```  2151     and "continuous F g"
```
```  2152     and "0 < f (Lim F (\<lambda>x. x))"
```
```  2153   shows "continuous F (\<lambda>x. (f x) powr (g x))"
```
```  2154   using assms unfolding continuous_def by (rule tendsto_powr)
```
```  2155
```
```  2156 lemma continuous_at_within_powr[continuous_intros]:
```
```  2157   assumes "continuous (at a within s) f"
```
```  2158     and "continuous (at a within s) g"
```
```  2159     and "0 < f a"
```
```  2160   shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
```
```  2161   using assms unfolding continuous_within by (rule tendsto_powr)
```
```  2162
```
```  2163 lemma isCont_powr[continuous_intros, simp]:
```
```  2164   assumes "isCont f a" "isCont g a" "0 < f a"
```
```  2165   shows "isCont (\<lambda>x. (f x) powr g x) a"
```
```  2166   using assms unfolding continuous_at by (rule tendsto_powr)
```
```  2167
```
```  2168 lemma continuous_on_powr[continuous_on_intros]:
```
```  2169   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. 0 < f x"
```
```  2170   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
```
```  2171   using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
```
```  2172
```
```  2173 (* FIXME: generalize by replacing d by with g x and g ---> d? *)
```
```  2174 lemma tendsto_zero_powrI:
```
```  2175   assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> 0) F"
```
```  2176     and "0 < d"
```
```  2177   shows "((\<lambda>x. f x powr d) ---> 0) F"
```
```  2178 proof (rule tendstoI)
```
```  2179   fix e :: real assume "0 < e"
```
```  2180   def Z \<equiv> "e powr (1 / d)"
```
```  2181   with `0 < e` have "0 < Z" by simp
```
```  2182   with assms have "eventually (\<lambda>x. 0 < f x \<and> dist (f x) 0 < Z) F"
```
```  2183     by (intro eventually_conj tendstoD)
```
```  2184   moreover
```
```  2185   from assms have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> x powr d < Z powr d"
```
```  2186     by (intro powr_less_mono2) (auto simp: dist_real_def)
```
```  2187   with assms `0 < e` have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> dist (x powr d) 0 < e"
```
```  2188     unfolding dist_real_def Z_def by (auto simp: powr_powr)
```
```  2189   ultimately
```
```  2190   show "eventually (\<lambda>x. dist (f x powr d) 0 < e) F" by (rule eventually_elim1)
```
```  2191 qed
```
```  2192
```
```  2193 lemma tendsto_neg_powr:
```
```  2194   assumes "s < 0"
```
```  2195     and "LIM x F. f x :> at_top"
```
```  2196   shows "((\<lambda>x. f x powr s) ---> 0) F"
```
```  2197 proof (rule tendstoI)
```
```  2198   fix e :: real assume "0 < e"
```
```  2199   def Z \<equiv> "e powr (1 / s)"
```
```  2200   from assms have "eventually (\<lambda>x. Z < f x) F"
```
```  2201     by (simp add: filterlim_at_top_dense)
```
```  2202   moreover
```
```  2203   from assms have "\<And>x. Z < x \<Longrightarrow> x powr s < Z powr s"
```
```  2204     by (auto simp: Z_def intro!: powr_less_mono2_neg)
```
```  2205   with assms `0 < e` have "\<And>x. Z < x \<Longrightarrow> dist (x powr s) 0 < e"
```
```  2206     by (simp add: powr_powr Z_def dist_real_def)
```
```  2207   ultimately
```
```  2208   show "eventually (\<lambda>x. dist (f x powr s) 0 < e) F" by (rule eventually_elim1)
```
```  2209 qed
```
```  2210
```
```  2211 subsection {* Sine and Cosine *}
```
```  2212
```
```  2213 definition sin_coeff :: "nat \<Rightarrow> real" where
```
```  2214   "sin_coeff = (\<lambda>n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"
```
```  2215
```
```  2216 definition cos_coeff :: "nat \<Rightarrow> real" where
```
```  2217   "cos_coeff = (\<lambda>n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"
```
```  2218
```
```  2219 definition sin :: "real \<Rightarrow> real"
```
```  2220   where "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
```
```  2221
```
```  2222 definition cos :: "real \<Rightarrow> real"
```
```  2223   where "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
```
```  2224
```
```  2225 lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
```
```  2226   unfolding sin_coeff_def by simp
```
```  2227
```
```  2228 lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
```
```  2229   unfolding cos_coeff_def by simp
```
```  2230
```
```  2231 lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
```
```  2232   unfolding cos_coeff_def sin_coeff_def
```
```  2233   by (simp del: mult_Suc)
```
```  2234
```
```  2235 lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
```
```  2236   unfolding cos_coeff_def sin_coeff_def
```
```  2237   by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)
```
```  2238
```
```  2239 lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
```
```  2240   unfolding sin_coeff_def
```
```  2241   apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
```
```  2242   apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2243   done
```
```  2244
```
```  2245 lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"
```
```  2246   unfolding cos_coeff_def
```
```  2247   apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
```
```  2248   apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2249   done
```
```  2250
```
```  2251 lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
```
```  2252   unfolding sin_def by (rule summable_sin [THEN summable_sums])
```
```  2253
```
```  2254 lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
```
```  2255   unfolding cos_def by (rule summable_cos [THEN summable_sums])
```
```  2256
```
```  2257 lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
```
```  2258   by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  2259
```
```  2260 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
```
```  2261   by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  2262
```
```  2263 text{*Now at last we can get the derivatives of exp, sin and cos*}
```
```  2264
```
```  2265 lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
```
```  2266   unfolding sin_def cos_def
```
```  2267   apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
```
```  2268   apply (simp_all add: diffs_sin_coeff diffs_cos_coeff
```
```  2269     summable_minus summable_sin summable_cos)
```
```  2270   done
```
```  2271
```
```  2272 declare DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  2273
```
```  2274 lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
```
```  2275   unfolding cos_def sin_def
```
```  2276   apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
```
```  2277   apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus
```
```  2278     summable_minus summable_sin summable_cos suminf_minus)
```
```  2279   done
```
```  2280
```
```  2281 declare DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  2282
```
```  2283 lemma isCont_sin: "isCont sin x"
```
```  2284   by (rule DERIV_sin [THEN DERIV_isCont])
```
```  2285
```
```  2286 lemma isCont_cos: "isCont cos x"
```
```  2287   by (rule DERIV_cos [THEN DERIV_isCont])
```
```  2288
```
```  2289 lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
```
```  2290   by (rule isCont_o2 [OF _ isCont_sin])
```
```  2291
```
```  2292 lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
```
```  2293   by (rule isCont_o2 [OF _ isCont_cos])
```
```  2294
```
```  2295 lemma tendsto_sin [tendsto_intros]:
```
```  2296   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
```
```  2297   by (rule isCont_tendsto_compose [OF isCont_sin])
```
```  2298
```
```  2299 lemma tendsto_cos [tendsto_intros]:
```
```  2300   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
```
```  2301   by (rule isCont_tendsto_compose [OF isCont_cos])
```
```  2302
```
```  2303 lemma continuous_sin [continuous_intros]:
```
```  2304   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
```
```  2305   unfolding continuous_def by (rule tendsto_sin)
```
```  2306
```
```  2307 lemma continuous_on_sin [continuous_on_intros]:
```
```  2308   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
```
```  2309   unfolding continuous_on_def by (auto intro: tendsto_sin)
```
```  2310
```
```  2311 lemma continuous_cos [continuous_intros]:
```
```  2312   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
```
```  2313   unfolding continuous_def by (rule tendsto_cos)
```
```  2314
```
```  2315 lemma continuous_on_cos [continuous_on_intros]:
```
```  2316   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
```
```  2317   unfolding continuous_on_def by (auto intro: tendsto_cos)
```
```  2318
```
```  2319 subsection {* Properties of Sine and Cosine *}
```
```  2320
```
```  2321 lemma sin_zero [simp]: "sin 0 = 0"
```
```  2322   unfolding sin_def sin_coeff_def by (simp add: powser_zero)
```
```  2323
```
```  2324 lemma cos_zero [simp]: "cos 0 = 1"
```
```  2325   unfolding cos_def cos_coeff_def by (simp add: powser_zero)
```
```  2326
```
```  2327 lemma sin_cos_squared_add [simp]: "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
```
```  2328 proof -
```
```  2329   have "\<forall>x. DERIV (\<lambda>x. (sin x)\<^sup>2 + (cos x)\<^sup>2) x :> 0"
```
```  2330     by (auto intro!: DERIV_intros)
```
```  2331   hence "(sin x)\<^sup>2 + (cos x)\<^sup>2 = (sin 0)\<^sup>2 + (cos 0)\<^sup>2"
```
```  2332     by (rule DERIV_isconst_all)
```
```  2333   thus "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1" by simp
```
```  2334 qed
```
```  2335
```
```  2336 lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
```
```  2337   by (subst add_commute, rule sin_cos_squared_add)
```
```  2338
```
```  2339 lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
```
```  2340   using sin_cos_squared_add2 [unfolded power2_eq_square] .
```
```  2341
```
```  2342 lemma sin_squared_eq: "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
```
```  2343   unfolding eq_diff_eq by (rule sin_cos_squared_add)
```
```  2344
```
```  2345 lemma cos_squared_eq: "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
```
```  2346   unfolding eq_diff_eq by (rule sin_cos_squared_add2)
```
```  2347
```
```  2348 lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
```
```  2349   by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
```
```  2350
```
```  2351 lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
```
```  2352   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  2353
```
```  2354 lemma sin_le_one [simp]: "sin x \<le> 1"
```
```  2355   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  2356
```
```  2357 lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
```
```  2358   by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
```
```  2359
```
```  2360 lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
```
```  2361   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  2362
```
```  2363 lemma cos_le_one [simp]: "cos x \<le> 1"
```
```  2364   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  2365
```
```  2366 lemma DERIV_fun_pow: "DERIV g x :> m ==>
```
```  2367       DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
```
```  2368   by (auto intro!: DERIV_intros)
```
```  2369
```
```  2370 lemma DERIV_fun_exp:
```
```  2371      "DERIV g x :> m ==> DERIV (\<lambda>x. exp(g x)) x :> exp(g x) * m"
```
```  2372   by (auto intro!: DERIV_intros)
```
```  2373
```
```  2374 lemma DERIV_fun_sin:
```
```  2375      "DERIV g x :> m ==> DERIV (\<lambda>x. sin(g x)) x :> cos(g x) * m"
```
```  2376   by (auto intro!: DERIV_intros)
```
```  2377
```
```  2378 lemma DERIV_fun_cos:
```
```  2379      "DERIV g x :> m ==> DERIV (\<lambda>x. cos(g x)) x :> -sin(g x) * m"
```
```  2380   by (auto intro!: DERIV_intros)
```
```  2381
```
```  2382 lemma sin_cos_add_lemma:
```
```  2383   "(sin (x + y) - (sin x * cos y + cos x * sin y))\<^sup>2 +
```
```  2384     (cos (x + y) - (cos x * cos y - sin x * sin y))\<^sup>2 = 0"
```
```  2385   (is "?f x = 0")
```
```  2386 proof -
```
```  2387   have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
```
```  2388     by (auto intro!: DERIV_intros simp add: algebra_simps)
```
```  2389   hence "?f x = ?f 0"
```
```  2390     by (rule DERIV_isconst_all)
```
```  2391   thus ?thesis by simp
```
```  2392 qed
```
```  2393
```
```  2394 lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
```
```  2395   using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
```
```  2396
```
```  2397 lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
```
```  2398   using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
```
```  2399
```
```  2400 lemma sin_cos_minus_lemma:
```
```  2401   "(sin(-x) + sin(x))\<^sup>2 + (cos(-x) - cos(x))\<^sup>2 = 0" (is "?f x = 0")
```
```  2402 proof -
```
```  2403   have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
```
```  2404     by (auto intro!: DERIV_intros simp add: algebra_simps)
```
```  2405   hence "?f x = ?f 0"
```
```  2406     by (rule DERIV_isconst_all)
```
```  2407   thus ?thesis by simp
```
```  2408 qed
```
```  2409
```
```  2410 lemma sin_minus [simp]: "sin (-x) = -sin(x)"
```
```  2411   using sin_cos_minus_lemma [where x=x] by simp
```
```  2412
```
```  2413 lemma cos_minus [simp]: "cos (-x) = cos(x)"
```
```  2414   using sin_cos_minus_lemma [where x=x] by simp
```
```  2415
```
```  2416 lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
```
```  2417   using sin_add [of x "- y"] by simp
```
```  2418
```
```  2419 lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
```
```  2420   by (simp add: sin_diff mult_commute)
```
```  2421
```
```  2422 lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
```
```  2423   using cos_add [of x "- y"] by simp
```
```  2424
```
```  2425 lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
```
```  2426   by (simp add: cos_diff mult_commute)
```
```  2427
```
```  2428 lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
```
```  2429   using sin_add [where x=x and y=x] by simp
```
```  2430
```
```  2431 lemma cos_double: "cos(2* x) = ((cos x)\<^sup>2) - ((sin x)\<^sup>2)"
```
```  2432   using cos_add [where x=x and y=x]
```
```  2433   by (simp add: power2_eq_square)
```
```  2434
```
```  2435
```
```  2436 subsection {* The Constant Pi *}
```
```  2437
```
```  2438 definition pi :: real
```
```  2439   where "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
```
```  2440
```
```  2441 text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
```
```  2442    hence define pi.*}
```
```  2443
```
```  2444 lemma sin_paired:
```
```  2445   "(\<lambda>n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) sums  sin x"
```
```  2446 proof -
```
```  2447   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
```
```  2448     by (rule sin_converges [THEN sums_group], simp)
```
```  2449   thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)
```
```  2450 qed
```
```  2451
```
```  2452 lemma sin_gt_zero:
```
```  2453   assumes "0 < x" and "x < 2"
```
```  2454   shows "0 < sin x"
```
```  2455 proof -
```
```  2456   let ?f = "\<lambda>n. \<Sum>k = n*2..<n*2+2. -1 ^ k / real (fact (2*k+1)) * x^(2*k+1)"
```
```  2457   have pos: "\<forall>n. 0 < ?f n"
```
```  2458   proof
```
```  2459     fix n :: nat
```
```  2460     let ?k2 = "real (Suc (Suc (4 * n)))"
```
```  2461     let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
```
```  2462     have "x * x < ?k2 * ?k3"
```
```  2463       using assms by (intro mult_strict_mono', simp_all)
```
```  2464     hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
```
```  2465       by (intro mult_strict_right_mono zero_less_power `0 < x`)
```
```  2466     thus "0 < ?f n"
```
```  2467       by (simp del: mult_Suc,
```
```  2468         simp add: less_divide_eq mult_pos_pos field_simps del: mult_Suc)
```
```  2469   qed
```
```  2470   have sums: "?f sums sin x"
```
```  2471     by (rule sin_paired [THEN sums_group], simp)
```
```  2472   show "0 < sin x"
```
```  2473     unfolding sums_unique [OF sums]
```
```  2474     using sums_summable [OF sums] pos
```
```  2475     by (rule suminf_gt_zero)
```
```  2476 qed
```
```  2477
```
```  2478 lemma cos_double_less_one: "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
```
```  2479   using sin_gt_zero [where x = x] by (auto simp add: cos_squared_eq cos_double)
```
```  2480
```
```  2481 lemma cos_paired: "(\<lambda>n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
```
```  2482 proof -
```
```  2483   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
```
```  2484     by (rule cos_converges [THEN sums_group], simp)
```
```  2485   thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)
```
```  2486 qed
```
```  2487
```
```  2488 lemma real_mult_inverse_cancel:
```
```  2489      "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
```
```  2490       ==> inverse x * y < inverse x1 * u"
```
```  2491   apply (rule_tac c=x in mult_less_imp_less_left)
```
```  2492   apply (auto simp add: mult_assoc [symmetric])
```
```  2493   apply (simp (no_asm) add: mult_ac)
```
```  2494   apply (rule_tac c=x1 in mult_less_imp_less_right)
```
```  2495   apply (auto simp add: mult_ac)
```
```  2496   done
```
```  2497
```
```  2498 lemma real_mult_inverse_cancel2:
```
```  2499      "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
```
```  2500   by (auto dest: real_mult_inverse_cancel simp add: mult_ac)
```
```  2501
```
```  2502 lemmas realpow_num_eq_if = power_eq_if
```
```  2503
```
```  2504 lemma cos_two_less_zero [simp]:
```
```  2505   "cos 2 < 0"
```
```  2506 proof -
```
```  2507   note fact_Suc [simp del]
```
```  2508   from cos_paired
```
```  2509   have "(\<lambda>n. - (-1 ^ n / real (fact (2 * n)) * 2 ^ (2 * n))) sums - cos 2"
```
```  2510     by (rule sums_minus)
```
```  2511   then have *: "(\<lambda>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n)))) sums - cos 2"
```
```  2512     by simp
```
```  2513   then have **: "summable (\<lambda>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2514     by (rule sums_summable)
```
```  2515   have "0 < (\<Sum>n = 0..<Suc (Suc (Suc 0)). - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2516     by (simp add: fact_num_eq_if_nat realpow_num_eq_if)
```
```  2517   moreover have "(\<Sum>n = 0..<Suc (Suc (Suc 0)). - (-1 ^ n  * 2 ^ (2 * n) / real (fact (2 * n))))
```
```  2518     < (\<Sum>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2519   proof -
```
```  2520     { fix d
```
```  2521       have "4 * real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))
```
```  2522        < real (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))) *
```
```  2523            fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))"
```
```  2524         by (simp only: real_of_nat_mult) (auto intro!: mult_strict_mono fact_less_mono_nat)
```
```  2525       then have "4 * real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))
```
```  2526         < real (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))))"
```
```  2527         by (simp only: fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
```
```  2528       then have "4 * inverse (real (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))))
```
```  2529         < inverse (real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))"
```
```  2530         by (simp add: inverse_eq_divide less_divide_eq)
```
```  2531     }
```
```  2532     note *** = this
```
```  2533     have [simp]: "\<And>x y::real. 0 < x - y \<longleftrightarrow> y < x" by arith
```
```  2534     from ** show ?thesis by (rule sumr_pos_lt_pair)
```
```  2535       (simp add: divide_inverse mult_assoc [symmetric] ***)
```
```  2536   qed
```
```  2537   ultimately have "0 < (\<Sum>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2538     by (rule order_less_trans)
```
```  2539   moreover from * have "- cos 2 = (\<Sum>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2540     by (rule sums_unique)
```
```  2541   ultimately have "0 < - cos 2" by simp
```
```  2542   then show ?thesis by simp
```
```  2543 qed
```
```  2544
```
```  2545 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
```
```  2546 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
```
```  2547
```
```  2548 lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 \<and> cos x = 0"
```
```  2549 proof (rule ex_ex1I)
```
```  2550   show "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0"
```
```  2551     by (rule IVT2, simp_all)
```
```  2552 next
```
```  2553   fix x y
```
```  2554   assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
```
```  2555   assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
```
```  2556   have [simp]: "\<forall>x. cos differentiable x"
```
```  2557     unfolding differentiable_def by (auto intro: DERIV_cos)
```
```  2558   from x y show "x = y"
```
```  2559     apply (cut_tac less_linear [of x y], auto)
```
```  2560     apply (drule_tac f = cos in Rolle)
```
```  2561     apply (drule_tac [5] f = cos in Rolle)
```
```  2562     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  2563     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  2564     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  2565     done
```
```  2566 qed
```
```  2567
```
```  2568 lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
```
```  2569   by (simp add: pi_def)
```
```  2570
```
```  2571 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
```
```  2572   by (simp add: pi_half cos_is_zero [THEN theI'])
```
```  2573
```
```  2574 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
```
```  2575   apply (rule order_le_neq_trans)
```
```  2576   apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  2577   apply (rule notI, drule arg_cong [where f=cos], simp)
```
```  2578   done
```
```  2579
```
```  2580 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
```
```  2581 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
```
```  2582
```
```  2583 lemma pi_half_less_two [simp]: "pi / 2 < 2"
```
```  2584   apply (rule order_le_neq_trans)
```
```  2585   apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  2586   apply (rule notI, drule arg_cong [where f=cos], simp)
```
```  2587   done
```
```  2588
```
```  2589 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
```
```  2590 lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
```
```  2591
```
```  2592 lemma pi_gt_zero [simp]: "0 < pi"
```
```  2593   using pi_half_gt_zero by simp
```
```  2594
```
```  2595 lemma pi_ge_zero [simp]: "0 \<le> pi"
```
```  2596   by (rule pi_gt_zero [THEN order_less_imp_le])
```
```  2597
```
```  2598 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
```
```  2599   by (rule pi_gt_zero [THEN less_imp_neq, symmetric])
```
```  2600
```
```  2601 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
```
```  2602   by (simp add: linorder_not_less)
```
```  2603
```
```  2604 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
```
```  2605   by simp
```
```  2606
```
```  2607 lemma m2pi_less_pi: "- (2 * pi) < pi"
```
```  2608   by simp
```
```  2609
```
```  2610 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
```
```  2611   using sin_cos_squared_add2 [where x = "pi/2"]
```
```  2612   using sin_gt_zero [OF pi_half_gt_zero pi_half_less_two]
```
```  2613   by (simp add: power2_eq_1_iff)
```
```  2614
```
```  2615 lemma cos_pi [simp]: "cos pi = -1"
```
```  2616   using cos_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  2617
```
```  2618 lemma sin_pi [simp]: "sin pi = 0"
```
```  2619   using sin_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  2620
```
```  2621 lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
```
```  2622   by (simp add: cos_diff)
```
```  2623
```
```  2624 lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
```
```  2625   by (simp add: cos_add)
```
```  2626
```
```  2627 lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
```
```  2628   by (simp add: sin_diff)
```
```  2629
```
```  2630 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
```
```  2631   by (simp add: sin_add)
```
```  2632
```
```  2633 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
```
```  2634   by (simp add: sin_add)
```
```  2635
```
```  2636 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
```
```  2637   by (simp add: cos_add)
```
```  2638
```
```  2639 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
```
```  2640   by (simp add: sin_add cos_double)
```
```  2641
```
```  2642 lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
```
```  2643   by (simp add: cos_add cos_double)
```
```  2644
```
```  2645 lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
```
```  2646   by (induct n) (auto simp add: real_of_nat_Suc distrib_right)
```
```  2647
```
```  2648 lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
```
```  2649 proof -
```
```  2650   have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
```
```  2651   also have "... = -1 ^ n" by (rule cos_npi)
```
```  2652   finally show ?thesis .
```
```  2653 qed
```
```  2654
```
```  2655 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
```
```  2656   by (induct n) (auto simp add: real_of_nat_Suc distrib_right)
```
```  2657
```
```  2658 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
```
```  2659   by (simp add: mult_commute [of pi])
```
```  2660
```
```  2661 lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
```
```  2662   by (simp add: cos_double)
```
```  2663
```
```  2664 lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
```
```  2665   by simp
```
```  2666
```
```  2667 lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
```
```  2668   apply (rule sin_gt_zero, assumption)
```
```  2669   apply (rule order_less_trans, assumption)
```
```  2670   apply (rule pi_half_less_two)
```
```  2671   done
```
```  2672
```
```  2673 lemma sin_less_zero:
```
```  2674   assumes "- pi/2 < x" and "x < 0"
```
```  2675   shows "sin x < 0"
```
```  2676 proof -
```
```  2677   have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
```
```  2678   thus ?thesis by simp
```
```  2679 qed
```
```  2680
```
```  2681 lemma pi_less_4: "pi < 4"
```
```  2682   using pi_half_less_two by auto
```
```  2683
```
```  2684 lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
```
```  2685   apply (cut_tac pi_less_4)
```
```  2686   apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
```
```  2687   apply (cut_tac cos_is_zero, safe)
```
```  2688   apply (rename_tac y z)
```
```  2689   apply (drule_tac x = y in spec)
```
```  2690   apply (drule_tac x = "pi/2" in spec, simp)
```
```  2691   done
```
```  2692
```
```  2693 lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
```
```  2694   apply (rule_tac x = x and y = 0 in linorder_cases)
```
```  2695   apply (rule cos_minus [THEN subst])
```
```  2696   apply (rule cos_gt_zero)
```
```  2697   apply (auto intro: cos_gt_zero)
```
```  2698   done
```
```  2699
```
```  2700 lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
```
```  2701   apply (auto simp add: order_le_less cos_gt_zero_pi)
```
```  2702   apply (subgoal_tac "x = pi/2", auto)
```
```  2703   done
```
```  2704
```
```  2705 lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
```
```  2706   by (simp add: sin_cos_eq cos_gt_zero_pi)
```
```  2707
```
```  2708 lemma pi_ge_two: "2 \<le> pi"
```
```  2709 proof (rule ccontr)
```
```  2710   assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
```
```  2711   have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
```
```  2712   proof (cases "2 < 2 * pi")
```
```  2713     case True with dense[OF `pi < 2`] show ?thesis by auto
```
```  2714   next
```
```  2715     case False have "pi < 2 * pi" by auto
```
```  2716     from dense[OF this] and False show ?thesis by auto
```
```  2717   qed
```
```  2718   then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
```
```  2719   hence "0 < sin y" using sin_gt_zero by auto
```
```  2720   moreover
```
```  2721   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
```
```  2722   ultimately show False by auto
```
```  2723 qed
```
```  2724
```
```  2725 lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
```
```  2726   by (auto simp add: order_le_less sin_gt_zero_pi)
```
```  2727
```
```  2728 text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}.
```
```  2729   It should be possible to factor out some of the common parts. *}
```
```  2730
```
```  2731 lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
```
```  2732 proof (rule ex_ex1I)
```
```  2733   assume y: "-1 \<le> y" "y \<le> 1"
```
```  2734   show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
```
```  2735     by (rule IVT2, simp_all add: y)
```
```  2736 next
```
```  2737   fix a b
```
```  2738   assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
```
```  2739   assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
```
```  2740   have [simp]: "\<forall>x. cos differentiable x"
```
```  2741     unfolding differentiable_def by (auto intro: DERIV_cos)
```
```  2742   from a b show "a = b"
```
```  2743     apply (cut_tac less_linear [of a b], auto)
```
```  2744     apply (drule_tac f = cos in Rolle)
```
```  2745     apply (drule_tac [5] f = cos in Rolle)
```
```  2746     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  2747     apply (metis order_less_le_trans less_le sin_gt_zero_pi)
```
```  2748     apply (metis order_less_le_trans less_le sin_gt_zero_pi)
```
```  2749     done
```
```  2750 qed
```
```  2751
```
```  2752 lemma sin_total:
```
```  2753      "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
```
```  2754 apply (rule ccontr)
```
```  2755 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))")
```
```  2756 apply (erule contrapos_np)
```
```  2757 apply simp
```
```  2758 apply (cut_tac y="-y" in cos_total, simp) apply simp
```
```  2759 apply (erule ex1E)
```
```  2760 apply (rule_tac a = "x - (pi/2)" in ex1I)
```
```  2761 apply (simp (no_asm) add: add_assoc)
```
```  2762 apply (rotate_tac 3)
```
```  2763 apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all add: cos_add)
```
```  2764 done
```
```  2765
```
```  2766 lemma reals_Archimedean4:
```
```  2767      "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
```
```  2768 apply (auto dest!: reals_Archimedean3)
```
```  2769 apply (drule_tac x = x in spec, clarify)
```
```  2770 apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
```
```  2771  prefer 2 apply (erule LeastI)
```
```  2772 apply (case_tac "LEAST m::nat. x < real m * y", simp)
```
```  2773 apply (subgoal_tac "~ x < real nat * y")
```
```  2774  prefer 2 apply (rule not_less_Least, simp, force)
```
```  2775 done
```
```  2776
```
```  2777 (* Pre Isabelle99-2 proof was simpler- numerals arithmetic
```
```  2778    now causes some unwanted re-arrangements of literals!   *)
```
```  2779 lemma cos_zero_lemma:
```
```  2780      "[| 0 \<le> x; cos x = 0 |] ==>
```
```  2781       \<exists>n::nat. ~even n & x = real n * (pi/2)"
```
```  2782 apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
```
```  2783 apply (subgoal_tac "0 \<le> x - real n * pi &
```
```  2784                     (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
```
```  2785 apply (auto simp add: algebra_simps real_of_nat_Suc)
```
```  2786  prefer 2 apply (simp add: cos_diff)
```
```  2787 apply (simp add: cos_diff)
```
```  2788 apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
```
```  2789 apply (rule_tac [2] cos_total, safe)
```
```  2790 apply (drule_tac x = "x - real n * pi" in spec)
```
```  2791 apply (drule_tac x = "pi/2" in spec)
```
```  2792 apply (simp add: cos_diff)
```
```  2793 apply (rule_tac x = "Suc (2 * n)" in exI)
```
```  2794 apply (simp add: real_of_nat_Suc algebra_simps, auto)
```
```  2795 done
```
```  2796
```
```  2797 lemma sin_zero_lemma:
```
```  2798      "[| 0 \<le> x; sin x = 0 |] ==>
```
```  2799       \<exists>n::nat. even n & x = real n * (pi/2)"
```
```  2800 apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
```
```  2801  apply (clarify, rule_tac x = "n - 1" in exI)
```
```  2802  apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc distrib_right)
```
```  2803 apply (rule cos_zero_lemma)
```
```  2804 apply (simp_all add: cos_add)
```
```  2805 done
```
```  2806
```
```  2807
```
```  2808 lemma cos_zero_iff:
```
```  2809      "(cos x = 0) =
```
```  2810       ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
```
```  2811        (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
```
```  2812 apply (rule iffI)
```
```  2813 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  2814 apply (drule cos_zero_lemma, assumption+)
```
```  2815 apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
```
```  2816 apply (force simp add: minus_equation_iff [of x])
```
```  2817 apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc distrib_right)
```
```  2818 apply (auto simp add: cos_diff cos_add)
```
```  2819 done
```
```  2820
```
```  2821 (* ditto: but to a lesser extent *)
```
```  2822 lemma sin_zero_iff:
```
```  2823      "(sin x = 0) =
```
```  2824       ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
```
```  2825        (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
```
```  2826 apply (rule iffI)
```
```  2827 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  2828 apply (drule sin_zero_lemma, assumption+)
```
```  2829 apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
```
```  2830 apply (force simp add: minus_equation_iff [of x])
```
```  2831 apply (auto simp add: even_mult_two_ex)
```
```  2832 done
```
```  2833
```
```  2834 lemma cos_monotone_0_pi:
```
```  2835   assumes "0 \<le> y" and "y < x" and "x \<le> pi"
```
```  2836   shows "cos x < cos y"
```
```  2837 proof -
```
```  2838   have "- (x - y) < 0" using assms by auto
```
```  2839
```
```  2840   from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
```
```  2841   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z"
```
```  2842     by auto
```
```  2843   hence "0 < z" and "z < pi" using assms by auto
```
```  2844   hence "0 < sin z" using sin_gt_zero_pi by auto
```
```  2845   hence "cos x - cos y < 0"
```
```  2846     unfolding cos_diff minus_mult_commute[symmetric]
```
```  2847     using `- (x - y) < 0` by (rule mult_pos_neg2)
```
```  2848   thus ?thesis by auto
```
```  2849 qed
```
```  2850
```
```  2851 lemma cos_monotone_0_pi':
```
```  2852   assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi"
```
```  2853   shows "cos x \<le> cos y"
```
```  2854 proof (cases "y < x")
```
```  2855   case True
```
```  2856   show ?thesis
```
```  2857     using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
```
```  2858 next
```
```  2859   case False
```
```  2860   hence "y = x" using `y \<le> x` by auto
```
```  2861   thus ?thesis by auto
```
```  2862 qed
```
```  2863
```
```  2864 lemma cos_monotone_minus_pi_0:
```
```  2865   assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
```
```  2866   shows "cos y < cos x"
```
```  2867 proof -
```
```  2868   have "0 \<le> -x" and "-x < -y" and "-y \<le> pi"
```
```  2869     using assms by auto
```
```  2870   from cos_monotone_0_pi[OF this] show ?thesis
```
```  2871     unfolding cos_minus .
```
```  2872 qed
```
```  2873
```
```  2874 lemma cos_monotone_minus_pi_0':
```
```  2875   assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0"
```
```  2876   shows "cos y \<le> cos x"
```
```  2877 proof (cases "y < x")
```
```  2878   case True
```
```  2879   show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`]
```
```  2880     by auto
```
```  2881 next
```
```  2882   case False
```
```  2883   hence "y = x" using `y \<le> x` by auto
```
```  2884   thus ?thesis by auto
```
```  2885 qed
```
```  2886
```
```  2887 lemma sin_monotone_2pi':
```
```  2888   assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2"
```
```  2889   shows "sin y \<le> sin x"
```
```  2890 proof -
```
```  2891   have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi"
```
```  2892     using pi_ge_two and assms by auto
```
```  2893   from cos_monotone_0_pi'[OF this] show ?thesis
```
```  2894     unfolding minus_sin_cos_eq[symmetric] by auto
```
```  2895 qed
```
```  2896
```
```  2897
```
```  2898 subsection {* Tangent *}
```
```  2899
```
```  2900 definition tan :: "real \<Rightarrow> real"
```
```  2901   where "tan = (\<lambda>x. sin x / cos x)"
```
```  2902
```
```  2903 lemma tan_zero [simp]: "tan 0 = 0"
```
```  2904   by (simp add: tan_def)
```
```  2905
```
```  2906 lemma tan_pi [simp]: "tan pi = 0"
```
```  2907   by (simp add: tan_def)
```
```  2908
```
```  2909 lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
```
```  2910   by (simp add: tan_def)
```
```  2911
```
```  2912 lemma tan_minus [simp]: "tan (-x) = - tan x"
```
```  2913   by (simp add: tan_def)
```
```  2914
```
```  2915 lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
```
```  2916   by (simp add: tan_def)
```
```  2917
```
```  2918 lemma lemma_tan_add1:
```
```  2919   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
```
```  2920   by (simp add: tan_def cos_add field_simps)
```
```  2921
```
```  2922 lemma add_tan_eq:
```
```  2923   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
```
```  2924   by (simp add: tan_def sin_add field_simps)
```
```  2925
```
```  2926 lemma tan_add:
```
```  2927      "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
```
```  2928       ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
```
```  2929   by (simp add: add_tan_eq lemma_tan_add1, simp add: tan_def)
```
```  2930
```
```  2931 lemma tan_double:
```
```  2932      "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
```
```  2933       ==> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)"
```
```  2934   using tan_add [of x x] by (simp add: power2_eq_square)
```
```  2935
```
```  2936 lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
```
```  2937   by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
```
```  2938
```
```  2939 lemma tan_less_zero:
```
```  2940   assumes lb: "- pi/2 < x" and "x < 0"
```
```  2941   shows "tan x < 0"
```
```  2942 proof -
```
```  2943   have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
```
```  2944   thus ?thesis by simp
```
```  2945 qed
```
```  2946
```
```  2947 lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"
```
```  2948   unfolding tan_def sin_double cos_double sin_squared_eq
```
```  2949   by (simp add: power2_eq_square)
```
```  2950
```
```  2951 lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)"
```
```  2952   unfolding tan_def
```
```  2953   by (auto intro!: DERIV_intros, simp add: divide_inverse power2_eq_square)
```
```  2954
```
```  2955 lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
```
```  2956   by (rule DERIV_tan [THEN DERIV_isCont])
```
```  2957
```
```  2958 lemma isCont_tan' [simp]:
```
```  2959   "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
```
```  2960   by (rule isCont_o2 [OF _ isCont_tan])
```
```  2961
```
```  2962 lemma tendsto_tan [tendsto_intros]:
```
```  2963   "\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F"
```
```  2964   by (rule isCont_tendsto_compose [OF isCont_tan])
```
```  2965
```
```  2966 lemma continuous_tan:
```
```  2967   "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))"
```
```  2968   unfolding continuous_def by (rule tendsto_tan)
```
```  2969
```
```  2970 lemma isCont_tan'' [continuous_intros]:
```
```  2971   "continuous (at x) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. tan (f x))"
```
```  2972   unfolding continuous_at by (rule tendsto_tan)
```
```  2973
```
```  2974 lemma continuous_within_tan [continuous_intros]:
```
```  2975   "continuous (at x within s) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
```
```  2976   unfolding continuous_within by (rule tendsto_tan)
```
```  2977
```
```  2978 lemma continuous_on_tan [continuous_on_intros]:
```
```  2979   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))"
```
```  2980   unfolding continuous_on_def by (auto intro: tendsto_tan)
```
```  2981
```
```  2982 lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) -- pi/2 --> 0"
```
```  2983   by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
```
```  2984
```
```  2985 lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
```
```  2986   apply (cut_tac LIM_cos_div_sin)
```
```  2987   apply (simp only: LIM_eq)
```
```  2988   apply (drule_tac x = "inverse y" in spec, safe, force)
```
```  2989   apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
```
```  2990   apply (rule_tac x = "(pi/2) - e" in exI)
```
```  2991   apply (simp (no_asm_simp))
```
```  2992   apply (drule_tac x = "(pi/2) - e" in spec)
```
```  2993   apply (auto simp add: tan_def sin_diff cos_diff)
```
```  2994   apply (rule inverse_less_iff_less [THEN iffD1])
```
```  2995   apply (auto simp add: divide_inverse)
```
```  2996   apply (rule mult_pos_pos)
```
```  2997   apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
```
```  2998   apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
```
```  2999   done
```
```  3000
```
```  3001 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
```
```  3002   apply (frule order_le_imp_less_or_eq, safe)
```
```  3003    prefer 2 apply force
```
```  3004   apply (drule lemma_tan_total, safe)
```
```  3005   apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
```
```  3006   apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
```
```  3007   apply (drule_tac y = xa in order_le_imp_less_or_eq)
```
```  3008   apply (auto dest: cos_gt_zero)
```
```  3009   done
```
```  3010
```
```  3011 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  3012   apply (cut_tac linorder_linear [of 0 y], safe)
```
```  3013   apply (drule tan_total_pos)
```
```  3014   apply (cut_tac [2] y="-y" in tan_total_pos, safe)
```
```  3015   apply (rule_tac [3] x = "-x" in exI)
```
```  3016   apply (auto del: exI intro!: exI)
```
```  3017   done
```
```  3018
```
```  3019 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  3020   apply (cut_tac y = y in lemma_tan_total1, auto)
```
```  3021   apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
```
```  3022   apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
```
```  3023   apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
```
```  3024   apply (rule_tac [4] Rolle)
```
```  3025   apply (rule_tac [2] Rolle)
```
```  3026   apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
```
```  3027               simp add: differentiable_def)
```
```  3028   txt{*Now, simulate TRYALL*}
```
```  3029   apply (rule_tac [!] DERIV_tan asm_rl)
```
```  3030   apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
```
```  3031               simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
```
```  3032   done
```
```  3033
```
```  3034 lemma tan_monotone:
```
```  3035   assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
```
```  3036   shows "tan y < tan x"
```
```  3037 proof -
```
```  3038   have "\<forall>x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse ((cos x')\<^sup>2)"
```
```  3039   proof (rule allI, rule impI)
```
```  3040     fix x' :: real
```
```  3041     assume "y \<le> x' \<and> x' \<le> x"
```
```  3042     hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
```
```  3043     from cos_gt_zero_pi[OF this]
```
```  3044     have "cos x' \<noteq> 0" by auto
```
```  3045     thus "DERIV tan x' :> inverse ((cos x')\<^sup>2)" by (rule DERIV_tan)
```
```  3046   qed
```
```  3047   from MVT2[OF `y < x` this]
```
```  3048   obtain z where "y < z" and "z < x"
```
```  3049     and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
```
```  3050   hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
```
```  3051   hence "0 < cos z" using cos_gt_zero_pi by auto
```
```  3052   hence inv_pos: "0 < inverse ((cos z)\<^sup>2)" by auto
```
```  3053   have "0 < x - y" using `y < x` by auto
```
```  3054   from mult_pos_pos [OF this inv_pos]
```
```  3055   have "0 < tan x - tan y" unfolding tan_diff by auto
```
```  3056   thus ?thesis by auto
```
```  3057 qed
```
```  3058
```
```  3059 lemma tan_monotone':
```
```  3060   assumes "- (pi / 2) < y"
```
```  3061     and "y < pi / 2"
```
```  3062     and "- (pi / 2) < x"
```
```  3063     and "x < pi / 2"
```
```  3064   shows "(y < x) = (tan y < tan x)"
```
```  3065 proof
```
```  3066   assume "y < x"
```
```  3067   thus "tan y < tan x"
```
```  3068     using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
```
```  3069 next
```
```  3070   assume "tan y < tan x"
```
```  3071   show "y < x"
```
```  3072   proof (rule ccontr)
```
```  3073     assume "\<not> y < x" hence "x \<le> y" by auto
```
```  3074     hence "tan x \<le> tan y"
```
```  3075     proof (cases "x = y")
```
```  3076       case True thus ?thesis by auto
```
```  3077     next
```
```  3078       case False hence "x < y" using `x \<le> y` by auto
```
```  3079       from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
```
```  3080     qed
```
```  3081     thus False using `tan y < tan x` by auto
```
```  3082   qed
```
```  3083 qed
```
```  3084
```
```  3085 lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)"
```
```  3086   unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
```
```  3087
```
```  3088 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
```
```  3089   by (simp add: tan_def)
```
```  3090
```
```  3091 lemma tan_periodic_nat[simp]:
```
```  3092   fixes n :: nat
```
```  3093   shows "tan (x + real n * pi) = tan x"
```
```  3094 proof (induct n arbitrary: x)
```
```  3095   case 0
```
```  3096   then show ?case by simp
```
```  3097 next
```
```  3098   case (Suc n)
```
```  3099   have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi"
```
```  3100     unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
```
```  3101   show ?case unfolding split_pi_off using Suc by auto
```
```  3102 qed
```
```  3103
```
```  3104 lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
```
```  3105 proof (cases "0 \<le> i")
```
```  3106   case True
```
```  3107   hence i_nat: "real i = real (nat i)" by auto
```
```  3108   show ?thesis unfolding i_nat by auto
```
```  3109 next
```
```  3110   case False
```
```  3111   hence i_nat: "real i = - real (nat (-i))" by auto
```
```  3112   have "tan x = tan (x + real i * pi - real i * pi)"
```
```  3113     by auto
```
```  3114   also have "\<dots> = tan (x + real i * pi)"
```
```  3115     unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
```
```  3116   finally show ?thesis by auto
```
```  3117 qed
```
```  3118
```
```  3119 lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
```
```  3120   using tan_periodic_int[of _ "numeral n" ] unfolding real_numeral .
```
```  3121
```
```  3122 subsection {* Inverse Trigonometric Functions *}
```
```  3123
```
```  3124 definition arcsin :: "real => real"
```
```  3125   where "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
```
```  3126
```
```  3127 definition arccos :: "real => real"
```
```  3128   where "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
```
```  3129
```
```  3130 definition arctan :: "real => real"
```
```  3131   where "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
```
```  3132
```
```  3133 lemma arcsin:
```
```  3134   "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow>
```
```  3135     -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2 & sin(arcsin y) = y"
```
```  3136   unfolding arcsin_def by (rule theI' [OF sin_total])
```
```  3137
```
```  3138 lemma arcsin_pi:
```
```  3139   "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
```
```  3140   apply (drule (1) arcsin)
```
```  3141   apply (force intro: order_trans)
```
```  3142   done
```
```  3143
```
```  3144 lemma sin_arcsin [simp]: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin(arcsin y) = y"
```
```  3145   by (blast dest: arcsin)
```
```  3146
```
```  3147 lemma arcsin_bounded: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
```
```  3148   by (blast dest: arcsin)
```
```  3149
```
```  3150 lemma arcsin_lbound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y"
```
```  3151   by (blast dest: arcsin)
```
```  3152
```
```  3153 lemma arcsin_ubound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
```
```  3154   by (blast dest: arcsin)
```
```  3155
```
```  3156 lemma arcsin_lt_bounded:
```
```  3157      "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
```
```  3158   apply (frule order_less_imp_le)
```
```  3159   apply (frule_tac y = y in order_less_imp_le)
```
```  3160   apply (frule arcsin_bounded)
```
```  3161   apply (safe, simp)
```
```  3162   apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
```
```  3163   apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
```
```  3164   apply (drule_tac [!] f = sin in arg_cong, auto)
```
```  3165   done
```
```  3166
```
```  3167 lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
```
```  3168   apply (unfold arcsin_def)
```
```  3169   apply (rule the1_equality)
```
```  3170   apply (rule sin_total, auto)
```
```  3171   done
```
```  3172
```
```  3173 lemma arccos:
```
```  3174      "[| -1 \<le> y; y \<le> 1 |]
```
```  3175       ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
```
```  3176   unfolding arccos_def by (rule theI' [OF cos_total])
```
```  3177
```
```  3178 lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
```
```  3179   by (blast dest: arccos)
```
```  3180
```
```  3181 lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
```
```  3182   by (blast dest: arccos)
```
```  3183
```
```  3184 lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
```
```  3185   by (blast dest: arccos)
```
```  3186
```
```  3187 lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
```
```  3188   by (blast dest: arccos)
```
```  3189
```
```  3190 lemma arccos_lt_bounded:
```
```  3191      "[| -1 < y; y < 1 |]
```
```  3192       ==> 0 < arccos y & arccos y < pi"
```
```  3193   apply (frule order_less_imp_le)
```
```  3194   apply (frule_tac y = y in order_less_imp_le)
```
```  3195   apply (frule arccos_bounded, auto)
```
```  3196   apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
```
```  3197   apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
```
```  3198   apply (drule_tac [!] f = cos in arg_cong, auto)
```
```  3199   done
```
```  3200
```
```  3201 lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
```
```  3202   apply (simp add: arccos_def)
```
```  3203   apply (auto intro!: the1_equality cos_total)
```
```  3204   done
```
```  3205
```
```  3206 lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
```
```  3207   apply (simp add: arccos_def)
```
```  3208   apply (auto intro!: the1_equality cos_total)
```
```  3209   done
```
```  3210
```
```  3211 lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
```
```  3212   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  3213   apply (rule power2_eq_imp_eq)
```
```  3214   apply (simp add: cos_squared_eq)
```
```  3215   apply (rule cos_ge_zero)
```
```  3216   apply (erule (1) arcsin_lbound)
```
```  3217   apply (erule (1) arcsin_ubound)
```
```  3218   apply simp
```
```  3219   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  3220   apply (rule power_mono, simp, simp)
```
```  3221   done
```
```  3222
```
```  3223 lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
```
```  3224   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  3225   apply (rule power2_eq_imp_eq)
```
```  3226   apply (simp add: sin_squared_eq)
```
```  3227   apply (rule sin_ge_zero)
```
```  3228   apply (erule (1) arccos_lbound)
```
```  3229   apply (erule (1) arccos_ubound)
```
```  3230   apply simp
```
```  3231   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  3232   apply (rule power_mono, simp, simp)
```
```  3233   done
```
```  3234
```
```  3235 lemma arctan [simp]: "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
```
```  3236   unfolding arctan_def by (rule theI' [OF tan_total])
```
```  3237
```
```  3238 lemma tan_arctan: "tan (arctan y) = y"
```
```  3239   by auto
```
```  3240
```
```  3241 lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
```
```  3242   by (auto simp only: arctan)
```
```  3243
```
```  3244 lemma arctan_lbound: "- (pi/2) < arctan y"
```
```  3245   by auto
```
```  3246
```
```  3247 lemma arctan_ubound: "arctan y < pi/2"
```
```  3248   by (auto simp only: arctan)
```
```  3249
```
```  3250 lemma arctan_unique:
```
```  3251   assumes "-(pi/2) < x"
```
```  3252     and "x < pi/2"
```
```  3253     and "tan x = y"
```
```  3254   shows "arctan y = x"
```
```  3255   using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
```
```  3256
```
```  3257 lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x"
```
```  3258   by (rule arctan_unique) simp_all
```
```  3259
```
```  3260 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
```
```  3261   by (rule arctan_unique) simp_all
```
```  3262
```
```  3263 lemma arctan_minus: "arctan (- x) = - arctan x"
```
```  3264   apply (rule arctan_unique)
```
```  3265   apply (simp only: neg_less_iff_less arctan_ubound)
```
```  3266   apply (metis minus_less_iff arctan_lbound)
```
```  3267   apply simp
```
```  3268   done
```
```  3269
```
```  3270 lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
```
```  3271   by (intro less_imp_neq [symmetric] cos_gt_zero_pi
```
```  3272     arctan_lbound arctan_ubound)
```
```  3273
```
```  3274 lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)"
```
```  3275 proof (rule power2_eq_imp_eq)
```
```  3276   have "0 < 1 + x\<^sup>2" by (simp add: add_pos_nonneg)
```
```  3277   show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp
```
```  3278   show "0 \<le> cos (arctan x)"
```
```  3279     by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
```
```  3280   have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1"
```
```  3281     unfolding tan_def by (simp add: distrib_left power_divide)
```
```  3282   thus "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
```
```  3283     using `0 < 1 + x\<^sup>2` by (simp add: power_divide eq_divide_eq)
```
```  3284 qed
```
```  3285
```
```  3286 lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)"
```
```  3287   using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
```
```  3288   using tan_arctan [of x] unfolding tan_def cos_arctan
```
```  3289   by (simp add: eq_divide_eq)
```
```  3290
```
```  3291 lemma tan_sec: "cos x \<noteq> 0 ==> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
```
```  3292   apply (rule power_inverse [THEN subst])
```
```  3293   apply (rule_tac c1 = "(cos x)\<^sup>2" in real_mult_right_cancel [THEN iffD1])
```
```  3294   apply (auto dest: field_power_not_zero
```
```  3295           simp add: power_mult_distrib distrib_right power_divide tan_def
```
```  3296                     mult_assoc power_inverse [symmetric])
```
```  3297   done
```
```  3298
```
```  3299 lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
```
```  3300   by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
```
```  3301
```
```  3302 lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
```
```  3303   by (simp only: not_less [symmetric] arctan_less_iff)
```
```  3304
```
```  3305 lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
```
```  3306   by (simp only: eq_iff [where 'a=real] arctan_le_iff)
```
```  3307
```
```  3308 lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
```
```  3309   using arctan_less_iff [of 0 x] by simp
```
```  3310
```
```  3311 lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
```
```  3312   using arctan_less_iff [of x 0] by simp
```
```  3313
```
```  3314 lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
```
```  3315   using arctan_le_iff [of 0 x] by simp
```
```  3316
```
```  3317 lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```  3318   using arctan_le_iff [of x 0] by simp
```
```  3319
```
```  3320 lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
```
```  3321   using arctan_eq_iff [of x 0] by simp
```
```  3322
```
```  3323 lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin"
```
```  3324 proof -
```
```  3325   have "continuous_on (sin ` {- pi / 2 .. pi / 2}) arcsin"
```
```  3326     by (rule continuous_on_inv) (auto intro: continuous_on_intros simp: arcsin_sin)
```
```  3327   also have "sin ` {- pi / 2 .. pi / 2} = {-1 .. 1}"
```
```  3328   proof safe
```
```  3329     fix x :: real
```
```  3330     assume "x \<in> {-1..1}"
```
```  3331     then show "x \<in> sin ` {- pi / 2..pi / 2}"
```
```  3332       using arcsin_lbound arcsin_ubound
```
```  3333       by (intro image_eqI[where x="arcsin x"]) auto
```
```  3334   qed simp
```
```  3335   finally show ?thesis .
```
```  3336 qed
```
```  3337
```
```  3338 lemma continuous_on_arcsin [continuous_on_intros]:
```
```  3339   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. -1 \<le> f x \<and> f x \<le> 1) \<Longrightarrow> continuous_on s (\<lambda>x. arcsin (f x))"
```
```  3340   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arcsin']]
```
```  3341   by (auto simp: comp_def subset_eq)
```
```  3342
```
```  3343 lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x"
```
```  3344   using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  3345   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  3346
```
```  3347 lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos"
```
```  3348 proof -
```
```  3349   have "continuous_on (cos ` {0 .. pi}) arccos"
```
```  3350     by (rule continuous_on_inv) (auto intro: continuous_on_intros simp: arccos_cos)
```
```  3351   also have "cos ` {0 .. pi} = {-1 .. 1}"
```
```  3352   proof safe
```
```  3353     fix x :: real
```
```  3354     assume "x \<in> {-1..1}"
```
```  3355     then show "x \<in> cos ` {0..pi}"
```
```  3356       using arccos_lbound arccos_ubound
```
```  3357       by (intro image_eqI[where x="arccos x"]) auto
```
```  3358   qed simp
```
```  3359   finally show ?thesis .
```
```  3360 qed
```
```  3361
```
```  3362 lemma continuous_on_arccos [continuous_on_intros]:
```
```  3363   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. -1 \<le> f x \<and> f x \<le> 1) \<Longrightarrow> continuous_on s (\<lambda>x. arccos (f x))"
```
```  3364   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arccos']]
```
```  3365   by (auto simp: comp_def subset_eq)
```
```  3366
```
```  3367 lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x"
```
```  3368   using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  3369   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  3370
```
```  3371 lemma isCont_arctan: "isCont arctan x"
```
```  3372   apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
```
```  3373   apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
```
```  3374   apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
```
```  3375   apply (erule (1) isCont_inverse_function2 [where f=tan])
```
```  3376   apply (metis arctan_tan order_le_less_trans order_less_le_trans)
```
```  3377   apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
```
```  3378   done
```
```  3379
```
```  3380 lemma tendsto_arctan [tendsto_intros]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) ---> arctan x) F"
```
```  3381   by (rule isCont_tendsto_compose [OF isCont_arctan])
```
```  3382
```
```  3383 lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))"
```
```  3384   unfolding continuous_def by (rule tendsto_arctan)
```
```  3385
```
```  3386 lemma continuous_on_arctan [continuous_on_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
```
```  3387   unfolding continuous_on_def by (auto intro: tendsto_arctan)
```
```  3388
```
```  3389 lemma DERIV_arcsin:
```
```  3390   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
```
```  3391   apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
```
```  3392   apply (rule DERIV_cong [OF DERIV_sin])
```
```  3393   apply (simp add: cos_arcsin)
```
```  3394   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  3395   apply (rule power_strict_mono, simp, simp, simp)
```
```  3396   apply assumption
```
```  3397   apply assumption
```
```  3398   apply simp
```
```  3399   apply (erule (1) isCont_arcsin)
```
```  3400   done
```
```  3401
```
```  3402 lemma DERIV_arccos:
```
```  3403   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
```
```  3404   apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
```
```  3405   apply (rule DERIV_cong [OF DERIV_cos])
```
```  3406   apply (simp add: sin_arccos)
```
```  3407   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  3408   apply (rule power_strict_mono, simp, simp, simp)
```
```  3409   apply assumption
```
```  3410   apply assumption
```
```  3411   apply simp
```
```  3412   apply (erule (1) isCont_arccos)
```
```  3413   done
```
```  3414
```
```  3415 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
```
```  3416   apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
```
```  3417   apply (rule DERIV_cong [OF DERIV_tan])
```
```  3418   apply (rule cos_arctan_not_zero)
```
```  3419   apply (simp add: power_inverse tan_sec [symmetric])
```
```  3420   apply (subgoal_tac "0 < 1 + x\<^sup>2", simp)
```
```  3421   apply (simp add: add_pos_nonneg)
```
```  3422   apply (simp, simp, simp, rule isCont_arctan)
```
```  3423   done
```
```  3424
```
```  3425 declare
```
```  3426   DERIV_arcsin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  3427   DERIV_arccos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  3428   DERIV_arctan[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  3429
```
```  3430 lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- pi/2))"
```
```  3431   by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. - pi/2 < x \<and> x < pi/2" and P="\<lambda>x. True" and g=arctan])
```
```  3432      (auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  3433            intro!: tan_monotone exI[of _ "pi/2"])
```
```  3434
```
```  3435 lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"
```
```  3436   by (rule filterlim_at_top_at_left[where Q="\<lambda>x. - pi/2 < x \<and> x < pi/2" and P="\<lambda>x. True" and g=arctan])
```
```  3437      (auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  3438            intro!: tan_monotone exI[of _ "pi/2"])
```
```  3439
```
```  3440 lemma tendsto_arctan_at_top: "(arctan ---> (pi/2)) at_top"
```
```  3441 proof (rule tendstoI)
```
```  3442   fix e :: real
```
```  3443   assume "0 < e"
```
```  3444   def y \<equiv> "pi/2 - min (pi/2) e"
```
```  3445   then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y"
```
```  3446     using `0 < e` by auto
```
```  3447
```
```  3448   show "eventually (\<lambda>x. dist (arctan x) (pi / 2) < e) at_top"
```
```  3449   proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)
```
```  3450     fix x
```
```  3451     assume "tan y < x"
```
```  3452     then have "arctan (tan y) < arctan x"
```
```  3453       by (simp add: arctan_less_iff)
```
```  3454     with y have "y < arctan x"
```
```  3455       by (subst (asm) arctan_tan) simp_all
```
```  3456     with arctan_ubound[of x, arith] y `0 < e`
```
```  3457     show "dist (arctan x) (pi / 2) < e"
```
```  3458       by (simp add: dist_real_def)
```
```  3459   qed
```
```  3460 qed
```
```  3461
```
```  3462 lemma tendsto_arctan_at_bot: "(arctan ---> - (pi/2)) at_bot"
```
```  3463   unfolding filterlim_at_bot_mirror arctan_minus
```
```  3464   by (intro tendsto_minus tendsto_arctan_at_top)
```
```  3465
```
```  3466
```
```  3467 subsection {* More Theorems about Sin and Cos *}
```
```  3468
```
```  3469 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
```
```  3470 proof -
```
```  3471   let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
```
```  3472   have nonneg: "0 \<le> ?c"
```
```  3473     by (simp add: cos_ge_zero)
```
```  3474   have "0 = cos (pi / 4 + pi / 4)"
```
```  3475     by simp
```
```  3476   also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2"
```
```  3477     by (simp only: cos_add power2_eq_square)
```
```  3478   also have "\<dots> = 2 * ?c\<^sup>2 - 1"
```
```  3479     by (simp add: sin_squared_eq)
```
```  3480   finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2"
```
```  3481     by (simp add: power_divide)
```
```  3482   thus ?thesis
```
```  3483     using nonneg by (rule power2_eq_imp_eq) simp
```
```  3484 qed
```
```  3485
```
```  3486 lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"
```
```  3487 proof -
```
```  3488   let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
```
```  3489   have pos_c: "0 < ?c"
```
```  3490     by (rule cos_gt_zero, simp, simp)
```
```  3491   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
```
```  3492     by simp
```
```  3493   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
```
```  3494     by (simp only: cos_add sin_add)
```
```  3495   also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)"
```
```  3496     by (simp add: algebra_simps power2_eq_square)
```
```  3497   finally have "?c\<^sup>2 = (sqrt 3 / 2)\<^sup>2"
```
```  3498     using pos_c by (simp add: sin_squared_eq power_divide)
```
```  3499   thus ?thesis
```
```  3500     using pos_c [THEN order_less_imp_le]
```
```  3501     by (rule power2_eq_imp_eq) simp
```
```  3502 qed
```
```  3503
```
```  3504 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
```
```  3505   by (simp add: sin_cos_eq cos_45)
```
```  3506
```
```  3507 lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
```
```  3508   by (simp add: sin_cos_eq cos_30)
```
```  3509
```
```  3510 lemma cos_60: "cos (pi / 3) = 1 / 2"
```
```  3511   apply (rule power2_eq_imp_eq)
```
```  3512   apply (simp add: cos_squared_eq sin_60 power_divide)
```
```  3513   apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
```
```  3514   done
```
```  3515
```
```  3516 lemma sin_30: "sin (pi / 6) = 1 / 2"
```
```  3517   by (simp add: sin_cos_eq cos_60)
```
```  3518
```
```  3519 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
```
```  3520   unfolding tan_def by (simp add: sin_30 cos_30)
```
```  3521
```
```  3522 lemma tan_45: "tan (pi / 4) = 1"
```
```  3523   unfolding tan_def by (simp add: sin_45 cos_45)
```
```  3524
```
```  3525 lemma tan_60: "tan (pi / 3) = sqrt 3"
```
```  3526   unfolding tan_def by (simp add: sin_60 cos_60)
```
```  3527
```
```  3528 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
```
```  3529 proof -
```
```  3530   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
```
```  3531     by (auto simp add: algebra_simps sin_add)
```
```  3532   thus ?thesis
```
```  3533     by (simp add: real_of_nat_Suc distrib_right add_divide_distrib
```
```  3534                   mult_commute [of pi])
```
```  3535 qed
```
```  3536
```
```  3537 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
```
```  3538   by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
```
```  3539
```
```  3540 lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
```
```  3541   apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
```
```  3542   apply (subst cos_add, simp)
```
```  3543   done
```
```  3544
```
```  3545 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
```
```  3546   by (auto simp add: mult_assoc)
```
```  3547
```
```  3548 lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
```
```  3549   apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
```
```  3550   apply (subst sin_add, simp)
```
```  3551   done
```
```  3552
```
```  3553 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
```
```  3554   apply (simp only: cos_add sin_add real_of_nat_Suc distrib_right distrib_left add_divide_distrib)
```
```  3555   apply auto
```
```  3556   done
```
```  3557
```
```  3558 lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)"
```
```  3559   by (auto intro!: DERIV_intros)
```
```  3560
```
```  3561 lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
```
```  3562   by (auto simp add: sin_zero_iff even_mult_two_ex)
```
```  3563
```
```  3564 lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
```
```  3565   using sin_cos_squared_add3 [where x = x] by auto
```
```  3566
```
```  3567
```
```  3568 subsection {* Machins formula *}
```
```  3569
```
```  3570 lemma arctan_one: "arctan 1 = pi / 4"
```
```  3571   by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi)
```
```  3572
```
```  3573 lemma tan_total_pi4:
```
```  3574   assumes "\<bar>x\<bar> < 1"
```
```  3575   shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
```
```  3576 proof
```
```  3577   show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
```
```  3578     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  3579     unfolding arctan_less_iff using assms by auto
```
```  3580 qed
```
```  3581
```
```  3582 lemma arctan_add:
```
```  3583   assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
```
```  3584   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
```
```  3585 proof (rule arctan_unique [symmetric])
```
```  3586   have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y"
```
```  3587     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  3588     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  3589   from add_le_less_mono [OF this]
```
```  3590   show 1: "- (pi / 2) < arctan x + arctan y" by simp
```
```  3591   have "arctan x \<le> pi / 4" and "arctan y < pi / 4"
```
```  3592     unfolding arctan_one [symmetric]
```
```  3593     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  3594   from add_le_less_mono [OF this]
```
```  3595   show 2: "arctan x + arctan y < pi / 2" by simp
```
```  3596   show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
```
```  3597     using cos_gt_zero_pi [OF 1 2] by (simp add: tan_add)
```
```  3598 qed
```
```  3599
```
```  3600 theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
```
```  3601 proof -
```
```  3602   have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
```
```  3603   from arctan_add[OF less_imp_le[OF this] this]
```
```  3604   have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
```
```  3605   moreover
```
```  3606   have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
```
```  3607   from arctan_add[OF less_imp_le[OF this] this]
```
```  3608   have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
```
```  3609   moreover
```
```  3610   have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
```
```  3611   from arctan_add[OF this]
```
```  3612   have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
```
```  3613   ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
```
```  3614   thus ?thesis unfolding arctan_one by algebra
```
```  3615 qed
```
```  3616
```
```  3617
```
```  3618 subsection {* Introducing the arcus tangens power series *}
```
```  3619
```
```  3620 lemma monoseq_arctan_series:
```
```  3621   fixes x :: real
```
```  3622   assumes "\<bar>x\<bar> \<le> 1"
```
```  3623   shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
```
```  3624 proof (cases "x = 0")
```
```  3625   case True
```
```  3626   thus ?thesis unfolding monoseq_def One_nat_def by auto
```
```  3627 next
```
```  3628   case False
```
```  3629   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  3630   show "monoseq ?a"
```
```  3631   proof -
```
```  3632     {
```
```  3633       fix n
```
```  3634       fix x :: real
```
```  3635       assume "0 \<le> x" and "x \<le> 1"
```
```  3636       have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le>
```
```  3637         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
```
```  3638       proof (rule mult_mono)
```
```  3639         show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))"
```
```  3640           by (rule frac_le) simp_all
```
```  3641         show "0 \<le> 1 / real (Suc (n * 2))"
```
```  3642           by auto
```
```  3643         show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)"
```
```  3644           by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`)
```
```  3645         show "0 \<le> x ^ Suc (Suc n * 2)"
```
```  3646           by (rule zero_le_power) (simp add: `0 \<le> x`)
```
```  3647       qed
```
```  3648     } note mono = this
```
```  3649
```
```  3650     show ?thesis
```
```  3651     proof (cases "0 \<le> x")
```
```  3652       case True from mono[OF this `x \<le> 1`, THEN allI]
```
```  3653       show ?thesis unfolding Suc_eq_plus1[symmetric]
```
```  3654         by (rule mono_SucI2)
```
```  3655     next
```
```  3656       case False
```
```  3657       hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto
```
```  3658       from mono[OF this]
```
```  3659       have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge>
```
```  3660         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using `0 \<le> -x` by auto
```
```  3661       thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
```
```  3662     qed
```
```  3663   qed
```
```  3664 qed
```
```  3665
```
```  3666 lemma zeroseq_arctan_series:
```
```  3667   fixes x :: real
```
```  3668   assumes "\<bar>x\<bar> \<le> 1"
```
```  3669   shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
```
```  3670 proof (cases "x = 0")
```
```  3671   case True
```
```  3672   thus ?thesis
```
```  3673     unfolding One_nat_def by (auto simp add: tendsto_const)
```
```  3674 next
```
```  3675   case False
```
```  3676   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  3677   show "?a ----> 0"
```
```  3678   proof (cases "\<bar>x\<bar> < 1")
```
```  3679     case True
```
```  3680     hence "norm x < 1" by auto
```
```  3681     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]]
```
```  3682     have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"
```
```  3683       unfolding inverse_eq_divide Suc_eq_plus1 by simp
```
```  3684     then show ?thesis using pos2 by (rule LIMSEQ_linear)
```
```  3685   next
```
```  3686     case False
```
```  3687     hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  3688     hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x"
```
```  3689       unfolding One_nat_def by auto
```
```  3690     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
```
```  3691     show ?thesis unfolding n_eq Suc_eq_plus1 by auto
```
```  3692   qed
```
```  3693 qed
```
```  3694
```
```  3695 lemma summable_arctan_series:
```
```  3696   fixes x :: real and n :: nat
```
```  3697   assumes "\<bar>x\<bar> \<le> 1"
```
```  3698   shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  3699   (is "summable (?c x)")
```
```  3700   by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
```
```  3701
```
```  3702 lemma less_one_imp_sqr_less_one:
```
```  3703   fixes x :: real
```
```  3704   assumes "\<bar>x\<bar> < 1"
```
```  3705   shows "x\<^sup>2 < 1"
```
```  3706 proof -
```
```  3707   have "\<bar>x\<^sup>2\<bar> < 1"
```
```  3708     by (metis abs_power2 assms pos2 power2_abs power_0 power_strict_decreasing zero_eq_power2 zero_less_abs_iff)
```
```  3709   thus ?thesis using zero_le_power2 by auto
```
```  3710 qed
```
```  3711
```
```  3712 lemma DERIV_arctan_series:
```
```  3713   assumes "\<bar> x \<bar> < 1"
```
```  3714   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))"
```
```  3715   (is "DERIV ?arctan _ :> ?Int")
```
```  3716 proof -
```
```  3717   let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
```
```  3718
```
```  3719   have n_even: "\<And>n :: nat. even n \<Longrightarrow> 2 * (n div 2) = n"
```
```  3720     by presburger
```
```  3721   then have if_eq: "\<And>n x'. ?f n * real (Suc n) * x'^n =
```
```  3722     (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)"
```
```  3723     by auto
```
```  3724
```
```  3725   {
```
```  3726     fix x :: real
```
```  3727     assume "\<bar>x\<bar> < 1"
```
```  3728     hence "x\<^sup>2 < 1" by (rule less_one_imp_sqr_less_one)
```
```  3729     have "summable (\<lambda> n. -1 ^ n * (x\<^sup>2) ^n)"
```
```  3730       by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x\<^sup>2 < 1` order_less_imp_le[OF `x\<^sup>2 < 1`])
```
```  3731     hence "summable (\<lambda> n. -1 ^ n * x^(2*n))" unfolding power_mult .
```
```  3732   } note summable_Integral = this
```
```  3733
```
```  3734   {
```
```  3735     fix f :: "nat \<Rightarrow> real"
```
```  3736     have "\<And>x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  3737     proof
```
```  3738       fix x :: real
```
```  3739       assume "f sums x"
```
```  3740       from sums_if[OF sums_zero this]
```
```  3741       show "(\<lambda>n. if even n then f (n div 2) else 0) sums x"
```
```  3742         by auto
```
```  3743     next
```
```  3744       fix x :: real
```
```  3745       assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  3746       from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult_commute]]
```
```  3747       show "f sums x" unfolding sums_def by auto
```
```  3748     qed
```
```  3749     hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
```
```  3750   } note sums_even = this
```
```  3751
```
```  3752   have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int"
```
```  3753     unfolding if_eq mult_commute[of _ 2] suminf_def sums_even[of "\<lambda> n. -1 ^ n * x ^ (2 * n)", symmetric]
```
```  3754     by auto
```
```  3755
```
```  3756   {
```
```  3757     fix x :: real
```
```  3758     have if_eq': "\<And>n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
```
```  3759       (if even n then -1 ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
```
```  3760       using n_even by auto
```
```  3761     have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)" by auto
```
```  3762     have "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x"
```
```  3763       unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. -1 ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
```
```  3764       by auto
```
```  3765   } note arctan_eq = this
```
```  3766
```
```  3767   have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
```
```  3768   proof (rule DERIV_power_series')
```
```  3769     show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto
```
```  3770     {
```
```  3771       fix x' :: real
```
```  3772       assume x'_bounds: "x' \<in> {- 1 <..< 1}"
```
```  3773       hence "\<bar>x'\<bar> < 1" by auto
```
```  3774
```
```  3775       let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
```
```  3776       show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
```
```  3777         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`])
```
```  3778     }
```
```  3779   qed auto
```
```  3780   thus ?thesis unfolding Int_eq arctan_eq .
```
```  3781 qed
```
```  3782
```
```  3783 lemma arctan_series:
```
```  3784   assumes "\<bar> x \<bar> \<le> 1"
```
```  3785   shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  3786   (is "_ = suminf (\<lambda> n. ?c x n)")
```
```  3787 proof -
```
```  3788   let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)"
```
```  3789
```
```  3790   {
```
```  3791     fix r x :: real
```
```  3792     assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
```
```  3793     have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto
```
```  3794     from DERIV_arctan_series[OF this] have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
```
```  3795   } note DERIV_arctan_suminf = this
```
```  3796
```
```  3797   {
```
```  3798     fix x :: real
```
```  3799     assume "\<bar>x\<bar> \<le> 1"
```
```  3800     note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]]
```
```  3801   } note arctan_series_borders = this
```
```  3802
```
```  3803   {
```
```  3804     fix x :: real
```
```  3805     assume "\<bar>x\<bar> < 1"
```
```  3806     have "arctan x = (\<Sum>k. ?c x k)"
```
```  3807     proof -
```
```  3808       obtain r where "\<bar>x\<bar> < r" and "r < 1"
```
```  3809         using dense[OF `\<bar>x\<bar> < 1`] by blast
```
```  3810       hence "0 < r" and "-r < x" and "x < r" by auto
```
```  3811
```
```  3812       have suminf_eq_arctan_bounded: "\<And>x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow>
```
```  3813         suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  3814       proof -
```
```  3815         fix x a b
```
```  3816         assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
```
```  3817         hence "\<bar>x\<bar> < r" by auto
```
```  3818         show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  3819         proof (rule DERIV_isconst2[of "a" "b"])
```
```  3820           show "a < b" and "a \<le> x" and "x \<le> b"
```
```  3821             using `a < b` `a \<le> x` `x \<le> b` by auto
```
```  3822           have "\<forall>x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  3823           proof (rule allI, rule impI)
```
```  3824             fix x
```
```  3825             assume "-r < x \<and> x < r"
```
```  3826             hence "\<bar>x\<bar> < r" by auto
```
```  3827             hence "\<bar>x\<bar> < 1" using `r < 1` by auto
```
```  3828             have "\<bar> - (x\<^sup>2) \<bar> < 1"
```
```  3829               using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
```
```  3830             hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  3831               unfolding real_norm_def[symmetric] by (rule geometric_sums)
```
```  3832             hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  3833               unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto
```
```  3834             hence suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)"
```
```  3835               using sums_unique unfolding inverse_eq_divide by auto
```
```  3836             have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))"
```
```  3837               unfolding suminf_c'_eq_geom
```
```  3838               by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
```
```  3839             from DERIV_add_minus[OF this DERIV_arctan]
```
```  3840             show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  3841               by auto
```
```  3842           qed
```
```  3843           hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
```
```  3844             using `-r < a` `b < r` by auto
```
```  3845           thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
```
```  3846             using `\<bar>x\<bar> < r` by auto
```
```  3847           show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y"
```
```  3848             using DERIV_in_rball DERIV_isCont by auto
```
```  3849         qed
```
```  3850       qed
```
```  3851
```
```  3852       have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
```
```  3853         unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero
```
```  3854         by auto
```
```  3855
```
```  3856       have "suminf (?c x) - arctan x = 0"
```
```  3857       proof (cases "x = 0")
```
```  3858         case True
```
```  3859         thus ?thesis using suminf_arctan_zero by auto
```
```  3860       next
```
```  3861         case False
```
```  3862         hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
```
```  3863         have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
```
```  3864           by (rule suminf_eq_arctan_bounded[where x="0" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>", symmetric])
```
```  3865             (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  3866         moreover
```
```  3867         have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
```
```  3868           by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"])
```
```  3869              (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  3870         ultimately
```
```  3871         show ?thesis using suminf_arctan_zero by auto
```
```  3872       qed
```
```  3873       thus ?thesis by auto
```
```  3874     qed
```
```  3875   } note when_less_one = this
```
```  3876
```
```  3877   show "arctan x = suminf (\<lambda> n. ?c x n)"
```
```  3878   proof (cases "\<bar>x\<bar> < 1")
```
```  3879     case True
```
```  3880     thus ?thesis by (rule when_less_one)
```
```  3881   next
```
```  3882     case False
```
```  3883     hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  3884     let ?a = "\<lambda>x n. \<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
```
```  3885     let ?diff = "\<lambda> x n. \<bar> arctan x - (\<Sum> i = 0..<n. ?c x i)\<bar>"
```
```  3886     {
```
```  3887       fix n :: nat
```
```  3888       have "0 < (1 :: real)" by auto
```
```  3889       moreover
```
```  3890       {
```
```  3891         fix x :: real
```
```  3892         assume "0 < x" and "x < 1"
```
```  3893         hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
```
```  3894         from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)"
```
```  3895           by auto
```
```  3896         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]
```
```  3897         have "0 < 1 / real (n*2+1) * x^(n*2+1)"
```
```  3898           by (rule mult_pos_pos, auto simp only: zero_less_power[OF `0 < x`], auto)
```
```  3899         hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)"
```
```  3900           by (rule abs_of_pos)
```
```  3901         have "?diff x n \<le> ?a x n"
```
```  3902         proof (cases "even n")
```
```  3903           case True
```
```  3904           hence sgn_pos: "(-1)^n = (1::real)" by auto
```
```  3905           from `even n` obtain m where "2 * m = n"
```
```  3906             unfolding even_mult_two_ex by auto
```
```  3907           from bounds[of m, unfolded this atLeastAtMost_iff]
```
```  3908           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))"
```
```  3909             by auto
```
```  3910           also have "\<dots> = ?c x n" unfolding One_nat_def by auto
```
```  3911           also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
```
```  3912           finally show ?thesis .
```
```  3913         next
```
```  3914           case False
```
```  3915           hence sgn_neg: "(-1)^n = (-1::real)" by auto
```
```  3916           from `odd n` obtain m where m_def: "2 * m + 1 = n"
```
```  3917             unfolding odd_Suc_mult_two_ex by auto
```
```  3918           hence m_plus: "2 * (m + 1) = n + 1" by auto
```
```  3919           from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
```
```  3920           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))"
```
```  3921             by auto
```
```  3922           also have "\<dots> = - ?c x n" unfolding One_nat_def by auto
```
```  3923           also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
```
```  3924           finally show ?thesis .
```
```  3925         qed
```
```  3926         hence "0 \<le> ?a x n - ?diff x n" by auto
```
```  3927       }
```
```  3928       hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
```
```  3929       moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
```
```  3930         unfolding diff_conv_add_uminus divide_inverse
```
```  3931         by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan
```
```  3932           isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum
```
```  3933           simp del: add_uminus_conv_diff)
```
```  3934       ultimately have "0 \<le> ?a 1 n - ?diff 1 n"
```
```  3935         by (rule LIM_less_bound)
```
```  3936       hence "?diff 1 n \<le> ?a 1 n" by auto
```
```  3937     }
```
```  3938     have "?a 1 ----> 0"
```
```  3939       unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
```
```  3940       by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)
```
```  3941     have "?diff 1 ----> 0"
```
```  3942     proof (rule LIMSEQ_I)
```
```  3943       fix r :: real
```
```  3944       assume "0 < r"
```
```  3945       obtain N :: nat where N_I: "\<And>n. N \<le> n \<Longrightarrow> ?a 1 n < r"
```
```  3946         using LIMSEQ_D[OF `?a 1 ----> 0` `0 < r`] by auto
```
```  3947       {
```
```  3948         fix n
```
```  3949         assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this]
```
```  3950         have "norm (?diff 1 n - 0) < r" by auto
```
```  3951       }
```
```  3952       thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
```
```  3953     qed
```
```  3954     from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
```
```  3955     have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
```
```  3956     hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
```
```  3957
```
```  3958     show ?thesis
```
```  3959     proof (cases "x = 1")
```
```  3960       case True
```
```  3961       then show ?thesis by (simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
```
```  3962     next
```
```  3963       case False
```
```  3964       hence "x = -1" using `\<bar>x\<bar> = 1` by auto
```
```  3965
```
```  3966       have "- (pi / 2) < 0" using pi_gt_zero by auto
```
```  3967       have "- (2 * pi) < 0" using pi_gt_zero by auto
```
```  3968
```
```  3969       have c_minus_minus: "\<And>i. ?c (- 1) i = - ?c 1 i"
```
```  3970         unfolding One_nat_def by auto
```
```  3971
```
```  3972       have "arctan (- 1) = arctan (tan (-(pi / 4)))"
```
```  3973         unfolding tan_45 tan_minus ..
```
```  3974       also have "\<dots> = - (pi / 4)"
```
```  3975         by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
```
```  3976       also have "\<dots> = - (arctan (tan (pi / 4)))"
```
```  3977         unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero])
```
```  3978       also have "\<dots> = - (arctan 1)"
```
```  3979         unfolding tan_45 ..
```
```  3980       also have "\<dots> = - (\<Sum> i. ?c 1 i)"
```
```  3981         using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto
```
```  3982       also have "\<dots> = (\<Sum> i. ?c (- 1) i)"
```
```  3983         using suminf_minus[OF sums_summable[OF `(?c 1) sums (arctan 1)`]]
```
```  3984         unfolding c_minus_minus by auto
```
```  3985       finally show ?thesis using `x = -1` by auto
```
```  3986     qed
```
```  3987   qed
```
```  3988 qed
```
```  3989
```
```  3990 lemma arctan_half:
```
```  3991   fixes x :: real
```
```  3992   shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))"
```
```  3993 proof -
```
```  3994   obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x"
```
```  3995     using tan_total by blast
```
```  3996   hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2"
```
```  3997     by auto
```
```  3998
```
```  3999   have divide_nonzero_divide: "\<And>A B C :: real. C \<noteq> 0 \<Longrightarrow> A / B = (A / C) / (B / C)"
```
```  4000     by auto
```
```  4001
```
```  4002   have "0 < cos y" using cos_gt_zero_pi[OF low high] .
```
```  4003   hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y"
```
```  4004     by auto
```
```  4005
```
```  4006   have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  4007     unfolding tan_def power_divide ..
```
```  4008   also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  4009     using `cos y \<noteq> 0` by auto
```
```  4010   also have "\<dots> = 1 / (cos y)\<^sup>2"
```
```  4011     unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
```
```  4012   finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" .
```
```  4013
```
```  4014   have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)"
```
```  4015     unfolding tan_def divide_nonzero_divide[OF `cos y \<noteq> 0`, symmetric] ..
```
```  4016   also have "\<dots> = tan y / (1 + 1 / cos y)"
```
```  4017     using `cos y \<noteq> 0` unfolding add_divide_distrib by auto
```
```  4018   also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))"
```
```  4019     unfolding cos_sqrt ..
```
```  4020   also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))"
```
```  4021     unfolding real_sqrt_divide by auto
```
```  4022   finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))"
```
```  4023     unfolding `1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2` .
```
```  4024
```
```  4025   have "arctan x = y"
```
```  4026     using arctan_tan low high y_eq by auto
```
```  4027   also have "\<dots> = 2 * (arctan (tan (y/2)))"
```
```  4028     using arctan_tan[OF low2 high2] by auto
```
```  4029   also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))"
```
```  4030     unfolding tan_half by auto
```
```  4031   finally show ?thesis
```
```  4032     unfolding eq `tan y = x` .
```
```  4033 qed
```
```  4034
```
```  4035 lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y"
```
```  4036   by (simp only: arctan_less_iff)
```
```  4037
```
```  4038 lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y"
```
```  4039   by (simp only: arctan_le_iff)
```
```  4040
```
```  4041 lemma arctan_inverse:
```
```  4042   assumes "x \<noteq> 0"
```
```  4043   shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
```
```  4044 proof (rule arctan_unique)
```
```  4045   show "- (pi / 2) < sgn x * pi / 2 - arctan x"
```
```  4046     using arctan_bounded [of x] assms
```
```  4047     unfolding sgn_real_def
```
```  4048     apply (auto simp add: algebra_simps)
```
```  4049     apply (drule zero_less_arctan_iff [THEN iffD2])
```
```  4050     apply arith
```
```  4051     done
```
```  4052   show "sgn x * pi / 2 - arctan x < pi / 2"
```
```  4053     using arctan_bounded [of "- x"] assms
```
```  4054     unfolding sgn_real_def arctan_minus
```
```  4055     by (auto simp add: algebra_simps)
```
```  4056   show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
```
```  4057     unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
```
```  4058     unfolding sgn_real_def
```
```  4059     by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
```
```  4060 qed
```
```  4061
```
```  4062 theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
```
```  4063 proof -
```
```  4064   have "pi / 4 = arctan 1" using arctan_one by auto
```
```  4065   also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
```
```  4066   finally show ?thesis by auto
```
```  4067 qed
```
```  4068
```
```  4069
```
```  4070 subsection {* Existence of Polar Coordinates *}
```
```  4071
```
```  4072 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1"
```
```  4073   apply (rule power2_le_imp_le [OF _ zero_le_one])
```
```  4074   apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
```
```  4075   done
```
```  4076
```
```  4077 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
```
```  4078   by (simp add: abs_le_iff)
```
```  4079
```
```  4080 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
```
```  4081   by (simp add: sin_arccos abs_le_iff)
```
```  4082
```
```  4083 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
```
```  4084
```
```  4085 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
```
```  4086
```
```  4087 lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
```
```  4088 proof -
```
```  4089   have polar_ex1: "\<And>y. 0 < y \<Longrightarrow> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  4090     apply (rule_tac x = "sqrt (x\<^sup>2 + y\<^sup>2)" in exI)
```
```  4091     apply (rule_tac x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))" in exI)
```
```  4092     apply (simp add: cos_arccos_lemma1 sin_arccos_lemma1 power_divide
```
```  4093                      real_sqrt_mult [symmetric] right_diff_distrib)
```
```  4094     done
```
```  4095   show ?thesis
```
```  4096   proof (cases "0::real" y rule: linorder_cases)
```
```  4097     case less
```
```  4098       then show ?thesis by (rule polar_ex1)
```
```  4099   next
```
```  4100     case equal
```
```  4101       then show ?thesis
```
```  4102         by (force simp add: intro!: cos_zero sin_zero)
```
```  4103   next
```
```  4104     case greater
```
```  4105       then show ?thesis
```
```  4106      using polar_ex1 [where y="-y"]
```
```  4107     by auto (metis cos_minus minus_minus minus_mult_right sin_minus)
```
```  4108   qed
```
```  4109 qed
```
```  4110
```
```  4111 end
```