src/HOL/Transcendental.thy
 author haftmann Tue Nov 19 10:05:53 2013 +0100 (2013-11-19) changeset 54489 03ff4d1e6784 parent 54230 b1d955791529 child 54573 07864001495d permissions -rw-r--r--
eliminiated neg_numeral in favour of - (numeral _)
```     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   apply (induct n)
```
```    37   apply simp
```
```    38   apply (simp del: setsum_op_ivl_Suc)
```
```    39   apply (subst setsum_op_ivl_Suc)
```
```    40   apply (subst lemma_realpow_diff_sumr)
```
```    41   apply (simp add: distrib_left del: setsum_op_ivl_Suc)
```
```    42   apply (subst mult_left_commute [of "x - y"])
```
```    43   apply (erule subst)
```
```    44   apply (simp add: algebra_simps)
```
```    45   done
```
```    46
```
```    47 lemma lemma_realpow_rev_sumr:
```
```    48   "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
```
```    49     (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
```
```    50   apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
```
```    51   apply (rule inj_onI, simp)
```
```    52   apply auto
```
```    53   apply (rule_tac x="n - x" in image_eqI, simp, simp)
```
```    54   done
```
```    55
```
```    56 text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
```
```    57   x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
```
```    58
```
```    59 lemma powser_insidea:
```
```    60   fixes x z :: "'a::real_normed_div_algebra"
```
```    61   assumes 1: "summable (\<lambda>n. f n * x ^ n)"
```
```    62     and 2: "norm z < norm x"
```
```    63   shows "summable (\<lambda>n. norm (f n * z ^ n))"
```
```    64 proof -
```
```    65   from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
```
```    66   from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
```
```    67     by (rule summable_LIMSEQ_zero)
```
```    68   hence "convergent (\<lambda>n. f n * x ^ n)"
```
```    69     by (rule convergentI)
```
```    70   hence "Cauchy (\<lambda>n. f n * x ^ n)"
```
```    71     by (rule convergent_Cauchy)
```
```    72   hence "Bseq (\<lambda>n. f n * x ^ n)"
```
```    73     by (rule Cauchy_Bseq)
```
```    74   then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
```
```    75     by (simp add: Bseq_def, safe)
```
```    76   have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
```
```    77                    K * norm (z ^ n) * inverse (norm (x ^ n))"
```
```    78   proof (intro exI allI impI)
```
```    79     fix n::nat
```
```    80     assume "0 \<le> n"
```
```    81     have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
```
```    82           norm (f n * x ^ n) * norm (z ^ n)"
```
```    83       by (simp add: norm_mult abs_mult)
```
```    84     also have "\<dots> \<le> K * norm (z ^ n)"
```
```    85       by (simp only: mult_right_mono 4 norm_ge_zero)
```
```    86     also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
```
```    87       by (simp add: x_neq_0)
```
```    88     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
```
```    89       by (simp only: mult_assoc)
```
```    90     finally show "norm (norm (f n * z ^ n)) \<le>
```
```    91                   K * norm (z ^ n) * inverse (norm (x ^ n))"
```
```    92       by (simp add: mult_le_cancel_right x_neq_0)
```
```    93   qed
```
```    94   moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
```
```    95   proof -
```
```    96     from 2 have "norm (norm (z * inverse x)) < 1"
```
```    97       using x_neq_0
```
```    98       by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
```
```    99     hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
```
```   100       by (rule summable_geometric)
```
```   101     hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
```
```   102       by (rule summable_mult)
```
```   103     thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
```
```   104       using x_neq_0
```
```   105       by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
```
```   106                     power_inverse norm_power mult_assoc)
```
```   107   qed
```
```   108   ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   109     by (rule summable_comparison_test)
```
```   110 qed
```
```   111
```
```   112 lemma powser_inside:
```
```   113   fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
```
```   114   shows
```
```   115     "summable (\<lambda>n. f n * (x ^ n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
```
```   116       summable (\<lambda>n. f n * (z ^ n))"
```
```   117   by (rule powser_insidea [THEN summable_norm_cancel])
```
```   118
```
```   119 lemma sum_split_even_odd:
```
```   120   fixes f :: "nat \<Rightarrow> real"
```
```   121   shows
```
```   122     "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
```
```   123      (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
```
```   124 proof (induct n)
```
```   125   case 0
```
```   126   then show ?case by simp
```
```   127 next
```
```   128   case (Suc n)
```
```   129   have "(\<Sum> i = 0 ..< 2 * Suc 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)) + (f (2 * n) + g (2 * n + 1))"
```
```   131     using Suc.hyps unfolding One_nat_def by auto
```
```   132   also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))"
```
```   133     by auto
```
```   134   finally show ?case .
```
```   135 qed
```
```   136
```
```   137 lemma sums_if':
```
```   138   fixes g :: "nat \<Rightarrow> real"
```
```   139   assumes "g sums x"
```
```   140   shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   141   unfolding sums_def
```
```   142 proof (rule LIMSEQ_I)
```
```   143   fix r :: real
```
```   144   assume "0 < r"
```
```   145   from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
```
```   146   obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast
```
```   147
```
```   148   let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
```
```   149   {
```
```   150     fix m
```
```   151     assume "m \<ge> 2 * no"
```
```   152     hence "m div 2 \<ge> no" by auto
```
```   153     have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
```
```   154       using sum_split_even_odd by auto
```
```   155     hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
```
```   156       using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
```
```   157     moreover
```
```   158     have "?SUM (2 * (m div 2)) = ?SUM m"
```
```   159     proof (cases "even m")
```
```   160       case True
```
```   161       show ?thesis
```
```   162         unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
```
```   163     next
```
```   164       case False
```
```   165       hence "even (Suc m)" by auto
```
```   166       from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]]
```
```   167         odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]
```
```   168       have eq: "Suc (2 * (m div 2)) = m" by auto
```
```   169       hence "even (2 * (m div 2))" using `odd m` by auto
```
```   170       have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
```
```   171       also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
```
```   172       finally show ?thesis by auto
```
```   173     qed
```
```   174     ultimately have "(norm (?SUM m - x) < r)" by auto
```
```   175   }
```
```   176   thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
```
```   177 qed
```
```   178
```
```   179 lemma sums_if:
```
```   180   fixes g :: "nat \<Rightarrow> real"
```
```   181   assumes "g sums x" and "f sums y"
```
```   182   shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
```
```   183 proof -
```
```   184   let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
```
```   185   {
```
```   186     fix B T E
```
```   187     have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
```
```   188       by (cases B) auto
```
```   189   } note if_sum = this
```
```   190   have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   191     using sums_if'[OF `g sums x`] .
```
```   192   {
```
```   193     have "?s 0 = 0" by auto
```
```   194     have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
```
```   195     have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
```
```   196
```
```   197     have "?s sums y" using sums_if'[OF `f sums y`] .
```
```   198     from this[unfolded sums_def, THEN LIMSEQ_Suc]
```
```   199     have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
```
```   200       unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
```
```   201                 image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
```
```   202                 even_Suc Suc_m1 if_eq .
```
```   203   }
```
```   204   from sums_add[OF g_sums this] show ?thesis unfolding if_sum .
```
```   205 qed
```
```   206
```
```   207 subsection {* Alternating series test / Leibniz formula *}
```
```   208
```
```   209 lemma sums_alternating_upper_lower:
```
```   210   fixes a :: "nat \<Rightarrow> real"
```
```   211   assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
```
```   212   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>
```
```   213              ((\<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)"
```
```   214   (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
```
```   215 proof (rule nested_sequence_unique)
```
```   216   have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
```
```   217
```
```   218   show "\<forall>n. ?f n \<le> ?f (Suc n)"
```
```   219   proof
```
```   220     fix n
```
```   221     show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
```
```   222   qed
```
```   223   show "\<forall>n. ?g (Suc n) \<le> ?g n"
```
```   224   proof
```
```   225     fix n
```
```   226     show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
```
```   227       unfolding One_nat_def by auto
```
```   228   qed
```
```   229   show "\<forall>n. ?f n \<le> ?g n"
```
```   230   proof
```
```   231     fix n
```
```   232     show "?f n \<le> ?g n" using fg_diff a_pos
```
```   233       unfolding One_nat_def by auto
```
```   234   qed
```
```   235   show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff
```
```   236   proof (rule LIMSEQ_I)
```
```   237     fix r :: real
```
```   238     assume "0 < r"
```
```   239     with `a ----> 0`[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
```
```   240       by auto
```
```   241     hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   242     thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   243   qed
```
```   244 qed
```
```   245
```
```   246 lemma summable_Leibniz':
```
```   247   fixes a :: "nat \<Rightarrow> real"
```
```   248   assumes a_zero: "a ----> 0"
```
```   249     and a_pos: "\<And> n. 0 \<le> a n"
```
```   250     and a_monotone: "\<And> n. a (Suc n) \<le> a n"
```
```   251   shows summable: "summable (\<lambda> n. (-1)^n * a n)"
```
```   252     and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
```
```   253     and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   254     and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
```
```   255     and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   256 proof -
```
```   257   let ?S = "\<lambda>n. (-1)^n * a n"
```
```   258   let ?P = "\<lambda>n. \<Sum>i=0..<n. ?S i"
```
```   259   let ?f = "\<lambda>n. ?P (2 * n)"
```
```   260   let ?g = "\<lambda>n. ?P (2 * n + 1)"
```
```   261   obtain l :: real
```
```   262     where below_l: "\<forall> n. ?f n \<le> l"
```
```   263       and "?f ----> l"
```
```   264       and above_l: "\<forall> n. l \<le> ?g n"
```
```   265       and "?g ----> l"
```
```   266     using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
```
```   267
```
```   268   let ?Sa = "\<lambda>m. \<Sum> n = 0..<m. ?S n"
```
```   269   have "?Sa ----> l"
```
```   270   proof (rule LIMSEQ_I)
```
```   271     fix r :: real
```
```   272     assume "0 < r"
```
```   273     with `?f ----> l`[THEN LIMSEQ_D]
```
```   274     obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
```
```   275
```
```   276     from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
```
```   277     obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
```
```   278
```
```   279     {
```
```   280       fix n :: nat
```
```   281       assume "n \<ge> (max (2 * f_no) (2 * g_no))"
```
```   282       hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
```
```   283       have "norm (?Sa n - l) < r"
```
```   284       proof (cases "even n")
```
```   285         case True
```
```   286         from even_nat_div_two_times_two[OF this]
```
```   287         have n_eq: "2 * (n div 2) = n"
```
```   288           unfolding numeral_2_eq_2[symmetric] by auto
```
```   289         with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no"
```
```   290           by auto
```
```   291         from f[OF this] show ?thesis
```
```   292           unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
```
```   293       next
```
```   294         case False
```
```   295         hence "even (n - 1)" by simp
```
```   296         from even_nat_div_two_times_two[OF this]
```
```   297         have n_eq: "2 * ((n - 1) div 2) = n - 1"
```
```   298           unfolding numeral_2_eq_2[symmetric] by auto
```
```   299         hence range_eq: "n - 1 + 1 = n"
```
```   300           using odd_pos[OF False] by auto
```
```   301
```
```   302         from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no"
```
```   303           by auto
```
```   304         from g[OF this] show ?thesis
```
```   305           unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
```
```   306       qed
```
```   307     }
```
```   308     thus "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
```
```   309   qed
```
```   310   hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
```
```   311     unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
```
```   312   thus "summable ?S" using summable_def by auto
```
```   313
```
```   314   have "l = suminf ?S" using sums_unique[OF sums_l] .
```
```   315
```
```   316   fix n
```
```   317   show "suminf ?S \<le> ?g n"
```
```   318     unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
```
```   319   show "?f n \<le> suminf ?S"
```
```   320     unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
```
```   321   show "?g ----> suminf ?S"
```
```   322     using `?g ----> l` `l = suminf ?S` by auto
```
```   323   show "?f ----> suminf ?S"
```
```   324     using `?f ----> l` `l = suminf ?S` by auto
```
```   325 qed
```
```   326
```
```   327 theorem summable_Leibniz:
```
```   328   fixes a :: "nat \<Rightarrow> real"
```
```   329   assumes a_zero: "a ----> 0" and "monoseq a"
```
```   330   shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
```
```   331     and "0 < a 0 \<longrightarrow>
```
```   332       (\<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")
```
```   333     and "a 0 < 0 \<longrightarrow>
```
```   334       (\<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")
```
```   335     and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
```
```   336     and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
```
```   337 proof -
```
```   338   have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
```
```   339   proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
```
```   340     case True
```
```   341     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n"
```
```   342       by auto
```
```   343     {
```
```   344       fix n
```
```   345       have "a (Suc n) \<le> a n"
```
```   346         using ord[where n="Suc n" and m=n] by auto
```
```   347     } note mono = this
```
```   348     note leibniz = summable_Leibniz'[OF `a ----> 0` ge0]
```
```   349     from leibniz[OF mono]
```
```   350     show ?thesis using `0 \<le> a 0` by auto
```
```   351   next
```
```   352     let ?a = "\<lambda> n. - a n"
```
```   353     case False
```
```   354     with monoseq_le[OF `monoseq a` `a ----> 0`]
```
```   355     have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
```
```   356     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
```
```   357       by auto
```
```   358     {
```
```   359       fix n
```
```   360       have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
```
```   361         by auto
```
```   362     } note monotone = this
```
```   363     note leibniz =
```
```   364       summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
```
```   365         OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
```
```   366     have "summable (\<lambda> n. (-1)^n * ?a n)"
```
```   367       using leibniz(1) by auto
```
```   368     then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
```
```   369       unfolding summable_def by auto
```
```   370     from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
```
```   371       by auto
```
```   372     hence ?summable unfolding summable_def by auto
```
```   373     moreover
```
```   374     have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
```
```   375       unfolding minus_diff_minus by auto
```
```   376
```
```   377     from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
```
```   378     have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)"
```
```   379       by auto
```
```   380
```
```   381     have ?pos using `0 \<le> ?a 0` by auto
```
```   382     moreover have ?neg
```
```   383       using leibniz(2,4)
```
```   384       unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
```
```   385       by auto
```
```   386     moreover have ?f and ?g
```
```   387       using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
```
```   388       by auto
```
```   389     ultimately show ?thesis by auto
```
```   390   qed
```
```   391   from this[THEN conjunct1]
```
```   392     this[THEN conjunct2, THEN conjunct1]
```
```   393     this[THEN conjunct2, THEN conjunct2, THEN conjunct1]
```
```   394     this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1]
```
```   395     this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]
```
```   396   show ?summable and ?pos and ?neg and ?f and ?g .
```
```   397 qed
```
```   398
```
```   399 subsection {* Term-by-Term Differentiability of Power Series *}
```
```   400
```
```   401 definition diffs :: "(nat => 'a::ring_1) => nat => 'a"
```
```   402   where "diffs c = (\<lambda>n. of_nat (Suc n) * c(Suc n))"
```
```   403
```
```   404 text{*Lemma about distributing negation over it*}
```
```   405 lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
```
```   406   by (simp add: diffs_def)
```
```   407
```
```   408 lemma sums_Suc_imp:
```
```   409   assumes f: "f 0 = 0"
```
```   410   shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
```
```   411   unfolding sums_def
```
```   412   apply (rule LIMSEQ_imp_Suc)
```
```   413   apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
```
```   414   apply (simp only: setsum_shift_bounds_Suc_ivl)
```
```   415   done
```
```   416
```
```   417 lemma diffs_equiv:
```
```   418   fixes x :: "'a::{real_normed_vector, ring_1}"
```
```   419   shows "summable (\<lambda>n. (diffs c)(n) * (x ^ n)) \<Longrightarrow>
```
```   420       (\<lambda>n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
```
```   421          (\<Sum>n. (diffs c)(n) * (x ^ n))"
```
```   422   unfolding diffs_def
```
```   423   apply (drule summable_sums)
```
```   424   apply (rule sums_Suc_imp, simp_all)
```
```   425   done
```
```   426
```
```   427 lemma lemma_termdiff1:
```
```   428   fixes z :: "'a :: {monoid_mult,comm_ring}" shows
```
```   429   "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
```
```   430    (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
```
```   431   by (auto simp add: algebra_simps power_add [symmetric])
```
```   432
```
```   433 lemma sumr_diff_mult_const2:
```
```   434   "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
```
```   435   by (simp add: setsum_subtractf)
```
```   436
```
```   437 lemma lemma_termdiff2:
```
```   438   fixes h :: "'a :: {field}"
```
```   439   assumes h: "h \<noteq> 0"
```
```   440   shows
```
```   441     "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
```
```   442      h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
```
```   443           (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
```
```   444   apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
```
```   445   apply (simp add: right_diff_distrib diff_divide_distrib h)
```
```   446   apply (simp add: mult_assoc [symmetric])
```
```   447   apply (cases "n", simp)
```
```   448   apply (simp add: lemma_realpow_diff_sumr2 h
```
```   449                    right_diff_distrib [symmetric] mult_assoc
```
```   450               del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
```
```   451   apply (subst lemma_realpow_rev_sumr)
```
```   452   apply (subst sumr_diff_mult_const2)
```
```   453   apply simp
```
```   454   apply (simp only: lemma_termdiff1 setsum_right_distrib)
```
```   455   apply (rule setsum_cong [OF refl])
```
```   456   apply (simp add: less_iff_Suc_add)
```
```   457   apply (clarify)
```
```   458   apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
```
```   459               del: setsum_op_ivl_Suc power_Suc)
```
```   460   apply (subst mult_assoc [symmetric], subst power_add [symmetric])
```
```   461   apply (simp add: mult_ac)
```
```   462   done
```
```   463
```
```   464 lemma real_setsum_nat_ivl_bounded2:
```
```   465   fixes K :: "'a::linordered_semidom"
```
```   466   assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
```
```   467     and K: "0 \<le> K"
```
```   468   shows "setsum f {0..<n-k} \<le> of_nat n * K"
```
```   469   apply (rule order_trans [OF setsum_mono])
```
```   470   apply (rule f, simp)
```
```   471   apply (simp add: mult_right_mono K)
```
```   472   done
```
```   473
```
```   474 lemma lemma_termdiff3:
```
```   475   fixes h z :: "'a::{real_normed_field}"
```
```   476   assumes 1: "h \<noteq> 0"
```
```   477     and 2: "norm z \<le> K"
```
```   478     and 3: "norm (z + h) \<le> K"
```
```   479   shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
```
```   480           \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   481 proof -
```
```   482   have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
```
```   483         norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
```
```   484           (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
```
```   485     apply (subst lemma_termdiff2 [OF 1])
```
```   486     apply (subst norm_mult)
```
```   487     apply (rule mult_commute)
```
```   488     done
```
```   489   also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
```
```   490   proof (rule mult_right_mono [OF _ norm_ge_zero])
```
```   491     from norm_ge_zero 2 have K: "0 \<le> K"
```
```   492       by (rule order_trans)
```
```   493     have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
```
```   494       apply (erule subst)
```
```   495       apply (simp only: norm_mult norm_power power_add)
```
```   496       apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
```
```   497       done
```
```   498     show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))
```
```   499           \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
```
```   500       apply (intro
```
```   501          order_trans [OF norm_setsum]
```
```   502          real_setsum_nat_ivl_bounded2
```
```   503          mult_nonneg_nonneg
```
```   504          of_nat_0_le_iff
```
```   505          zero_le_power K)
```
```   506       apply (rule le_Kn, simp)
```
```   507       done
```
```   508   qed
```
```   509   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   510     by (simp only: mult_assoc)
```
```   511   finally show ?thesis .
```
```   512 qed
```
```   513
```
```   514 lemma lemma_termdiff4:
```
```   515   fixes f :: "'a::{real_normed_field} \<Rightarrow>
```
```   516               'b::real_normed_vector"
```
```   517   assumes k: "0 < (k::real)"
```
```   518     and le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
```
```   519   shows "f -- 0 --> 0"
```
```   520   unfolding LIM_eq diff_0_right
```
```   521 proof safe
```
```   522   let ?h = "of_real (k / 2)::'a"
```
```   523   have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
```
```   524   hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
```
```   525   hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
```
```   526   hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
```
```   527
```
```   528   fix r::real
```
```   529   assume r: "0 < r"
```
```   530   show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
```
```   531   proof cases
```
```   532     assume "K = 0"
```
```   533     with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
```
```   534       by simp
```
```   535     thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
```
```   536   next
```
```   537     assume K_neq_zero: "K \<noteq> 0"
```
```   538     with zero_le_K have K: "0 < K" by simp
```
```   539     show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
```
```   540     proof (rule exI, safe)
```
```   541       from k r K
```
```   542       show "0 < min k (r * inverse K / 2)"
```
```   543         by (simp add: mult_pos_pos positive_imp_inverse_positive)
```
```   544     next
```
```   545       fix x::'a
```
```   546       assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
```
```   547       from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
```
```   548         by simp_all
```
```   549       from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
```
```   550       also from x4 K have "K * norm x < K * (r * inverse K / 2)"
```
```   551         by (rule mult_strict_left_mono)
```
```   552       also have "\<dots> = r / 2"
```
```   553         using K_neq_zero by simp
```
```   554       also have "r / 2 < r"
```
```   555         using r by simp
```
```   556       finally show "norm (f x) < r" .
```
```   557     qed
```
```   558   qed
```
```   559 qed
```
```   560
```
```   561 lemma lemma_termdiff5:
```
```   562   fixes g :: "'a::real_normed_field \<Rightarrow> nat \<Rightarrow> 'b::banach"
```
```   563   assumes k: "0 < (k::real)"
```
```   564   assumes f: "summable f"
```
```   565   assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
```
```   566   shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
```
```   567 proof (rule lemma_termdiff4 [OF k])
```
```   568   fix h::'a
```
```   569   assume "h \<noteq> 0" and "norm h < k"
```
```   570   hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
```
```   571     by (simp add: le)
```
```   572   hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
```
```   573     by simp
```
```   574   moreover from f have B: "summable (\<lambda>n. f n * norm h)"
```
```   575     by (rule summable_mult2)
```
```   576   ultimately have C: "summable (\<lambda>n. norm (g h n))"
```
```   577     by (rule summable_comparison_test)
```
```   578   hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
```
```   579     by (rule summable_norm)
```
```   580   also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
```
```   581     by (rule summable_le)
```
```   582   also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
```
```   583     by (rule suminf_mult2 [symmetric])
```
```   584   finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
```
```   585 qed
```
```   586
```
```   587
```
```   588 text{* FIXME: Long proofs*}
```
```   589
```
```   590 lemma termdiffs_aux:
```
```   591   fixes x :: "'a::{real_normed_field,banach}"
```
```   592   assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
```
```   593     and 2: "norm x < norm K"
```
```   594   shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
```
```   595              - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   596 proof -
```
```   597   from dense [OF 2]
```
```   598   obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
```
```   599   from norm_ge_zero r1 have r: "0 < r"
```
```   600     by (rule order_le_less_trans)
```
```   601   hence r_neq_0: "r \<noteq> 0" by simp
```
```   602   show ?thesis
```
```   603   proof (rule lemma_termdiff5)
```
```   604     show "0 < r - norm x" using r1 by simp
```
```   605     from r r2 have "norm (of_real r::'a) < norm K"
```
```   606       by simp
```
```   607     with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
```
```   608       by (rule powser_insidea)
```
```   609     hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
```
```   610       using r
```
```   611       by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
```
```   612     hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
```
```   613       by (rule diffs_equiv [THEN sums_summable])
```
```   614     also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
```
```   615       (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
```
```   616       apply (rule ext)
```
```   617       apply (simp add: diffs_def)
```
```   618       apply (case_tac n, simp_all add: r_neq_0)
```
```   619       done
```
```   620     finally have "summable
```
```   621       (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
```
```   622       by (rule diffs_equiv [THEN sums_summable])
```
```   623     also have
```
```   624       "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
```
```   625            r ^ (n - Suc 0)) =
```
```   626        (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
```
```   627       apply (rule ext)
```
```   628       apply (case_tac "n", simp)
```
```   629       apply (case_tac "nat", simp)
```
```   630       apply (simp add: r_neq_0)
```
```   631       done
```
```   632     finally
```
```   633     show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
```
```   634   next
```
```   635     fix h::'a and n::nat
```
```   636     assume h: "h \<noteq> 0"
```
```   637     assume "norm h < r - norm x"
```
```   638     hence "norm x + norm h < r" by simp
```
```   639     with norm_triangle_ineq have xh: "norm (x + h) < r"
```
```   640       by (rule order_le_less_trans)
```
```   641     show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
```
```   642           \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
```
```   643       apply (simp only: norm_mult mult_assoc)
```
```   644       apply (rule mult_left_mono [OF _ norm_ge_zero])
```
```   645       apply (simp (no_asm) add: mult_assoc [symmetric])
```
```   646       apply (rule lemma_termdiff3)
```
```   647       apply (rule h)
```
```   648       apply (rule r1 [THEN order_less_imp_le])
```
```   649       apply (rule xh [THEN order_less_imp_le])
```
```   650       done
```
```   651   qed
```
```   652 qed
```
```   653
```
```   654 lemma termdiffs:
```
```   655   fixes K x :: "'a::{real_normed_field,banach}"
```
```   656   assumes 1: "summable (\<lambda>n. c n * K ^ n)"
```
```   657     and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
```
```   658     and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
```
```   659     and 4: "norm x < norm K"
```
```   660   shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
```
```   661   unfolding deriv_def
```
```   662 proof (rule LIM_zero_cancel)
```
```   663   show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
```
```   664             - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
```
```   665   proof (rule LIM_equal2)
```
```   666     show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
```
```   667   next
```
```   668     fix h :: 'a
```
```   669     assume "h \<noteq> 0"
```
```   670     assume "norm (h - 0) < norm K - norm x"
```
```   671     hence "norm x + norm h < norm K" by simp
```
```   672     hence 5: "norm (x + h) < norm K"
```
```   673       by (rule norm_triangle_ineq [THEN order_le_less_trans])
```
```   674     have A: "summable (\<lambda>n. c n * x ^ n)"
```
```   675       by (rule powser_inside [OF 1 4])
```
```   676     have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
```
```   677       by (rule powser_inside [OF 1 5])
```
```   678     have C: "summable (\<lambda>n. diffs c n * x ^ n)"
```
```   679       by (rule powser_inside [OF 2 4])
```
```   680     show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
```
```   681              - (\<Sum>n. diffs c n * x ^ n) =
```
```   682           (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
```
```   683       apply (subst sums_unique [OF diffs_equiv [OF C]])
```
```   684       apply (subst suminf_diff [OF B A])
```
```   685       apply (subst suminf_divide [symmetric])
```
```   686       apply (rule summable_diff [OF B A])
```
```   687       apply (subst suminf_diff)
```
```   688       apply (rule summable_divide)
```
```   689       apply (rule summable_diff [OF B A])
```
```   690       apply (rule sums_summable [OF diffs_equiv [OF C]])
```
```   691       apply (rule arg_cong [where f="suminf"], rule ext)
```
```   692       apply (simp add: algebra_simps)
```
```   693       done
```
```   694   next
```
```   695     show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   696       by (rule termdiffs_aux [OF 3 4])
```
```   697   qed
```
```   698 qed
```
```   699
```
```   700
```
```   701 subsection {* Derivability of power series *}
```
```   702
```
```   703 lemma DERIV_series':
```
```   704   fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
```
```   705   assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
```
```   706     and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
```
```   707     and "summable (f' x0)"
```
```   708     and "summable L"
```
```   709     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>"
```
```   710   shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
```
```   711   unfolding deriv_def
```
```   712 proof (rule LIM_I)
```
```   713   fix r :: real
```
```   714   assume "0 < r" hence "0 < r/3" by auto
```
```   715
```
```   716   obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
```
```   717     using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
```
```   718
```
```   719   obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
```
```   720     using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
```
```   721
```
```   722   let ?N = "Suc (max N_L N_f')"
```
```   723   have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
```
```   724     L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
```
```   725
```
```   726   let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
```
```   727
```
```   728   let ?r = "r / (3 * real ?N)"
```
```   729   have "0 < 3 * real ?N" by auto
```
```   730   from divide_pos_pos[OF `0 < r` this]
```
```   731   have "0 < ?r" .
```
```   732
```
```   733   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)"
```
```   734   def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
```
```   735
```
```   736   have "0 < S'" unfolding S'_def
```
```   737   proof (rule iffD2[OF Min_gr_iff])
```
```   738     show "\<forall>x \<in> (?s ` { 0 ..< ?N }). 0 < x"
```
```   739     proof
```
```   740       fix x
```
```   741       assume "x \<in> ?s ` {0..<?N}"
```
```   742       then obtain n where "x = ?s n" and "n \<in> {0..<?N}"
```
```   743         using image_iff[THEN iffD1] by blast
```
```   744       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
```
```   745       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)"
```
```   746         by auto
```
```   747       have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound)
```
```   748       thus "0 < x" unfolding `x = ?s n` .
```
```   749     qed
```
```   750   qed auto
```
```   751
```
```   752   def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
```
```   753   hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
```
```   754     and "S \<le> S'" using x0_in_I and `0 < S'`
```
```   755     by auto
```
```   756
```
```   757   {
```
```   758     fix x
```
```   759     assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
```
```   760     hence x_in_I: "x0 + x \<in> { a <..< b }"
```
```   761       using S_a S_b by auto
```
```   762
```
```   763     note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   764     note div_smbl = summable_divide[OF diff_smbl]
```
```   765     note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
```
```   766     note ign = summable_ignore_initial_segment[where k="?N"]
```
```   767     note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
```
```   768     note div_shft_smbl = summable_divide[OF diff_shft_smbl]
```
```   769     note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
```
```   770
```
```   771     {
```
```   772       fix n
```
```   773       have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
```
```   774         using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
```
```   775         unfolding abs_divide .
```
```   776       hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)"
```
```   777         using `x \<noteq> 0` by auto
```
```   778     } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
```
```   779     from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
```
```   780     have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
```
```   781     hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3")
```
```   782       using L_estimate by auto
```
```   783
```
```   784     have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le>
```
```   785       (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..
```
```   786     also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
```
```   787     proof (rule setsum_strict_mono)
```
```   788       fix n
```
```   789       assume "n \<in> { 0 ..< ?N}"
```
```   790       have "\<bar>x\<bar> < S" using `\<bar>x\<bar> < S` .
```
```   791       also have "S \<le> S'" using `S \<le> S'` .
```
```   792       also have "S' \<le> ?s n" unfolding S'_def
```
```   793       proof (rule Min_le_iff[THEN iffD2])
```
```   794         have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n"
```
```   795           using `n \<in> { 0 ..< ?N}` by auto
```
```   796         thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
```
```   797       qed auto
```
```   798       finally have "\<bar>x\<bar> < ?s n" .
```
```   799
```
```   800       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]
```
```   801       have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
```
```   802       with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n` show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
```
```   803         by blast
```
```   804     qed auto
```
```   805     also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r"
```
```   806       by (rule setsum_constant)
```
```   807     also have "\<dots> = real ?N * ?r"
```
```   808       unfolding real_eq_of_nat by auto
```
```   809     also have "\<dots> = r/3" by auto
```
```   810     finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
```
```   811
```
```   812     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   813     have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
```
```   814         \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
```
```   815       unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric]
```
```   816       using suminf_divide[OF diff_smbl, symmetric] by auto
```
```   817     also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>"
```
```   818       unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
```
```   819       unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]]
```
```   820       by (rule abs_triangle_ineq)
```
```   821     also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
```
```   822       using abs_triangle_ineq4 by auto
```
```   823     also have "\<dots> < r /3 + r/3 + r/3"
```
```   824       using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
```
```   825       by (rule add_strict_mono [OF add_less_le_mono])
```
```   826     finally have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
```
```   827       by auto
```
```   828   }
```
```   829   thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
```
```   830       norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
```
```   831     using `0 < S` unfolding real_norm_def diff_0_right by blast
```
```   832 qed
```
```   833
```
```   834 lemma DERIV_power_series':
```
```   835   fixes f :: "nat \<Rightarrow> real"
```
```   836   assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
```
```   837     and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
```
```   838   shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
```
```   839   (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
```
```   840 proof -
```
```   841   {
```
```   842     fix R'
```
```   843     assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
```
```   844     hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
```
```   845       by auto
```
```   846     have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
```
```   847     proof (rule DERIV_series')
```
```   848       show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
```
```   849       proof -
```
```   850         have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
```
```   851           using `0 < R'` `0 < R` `R' < R` by auto
```
```   852         hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
```
```   853           using `R' < R` by auto
```
```   854         have "norm R' < norm ((R' + R) / 2)"
```
```   855           using `0 < R'` `0 < R` `R' < R` by auto
```
```   856         from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
```
```   857           by auto
```
```   858       qed
```
```   859       {
```
```   860         fix n x y
```
```   861         assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
```
```   862         show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
```
```   863         proof -
```
```   864           have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
```
```   865             (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
```
```   866             unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult
```
```   867             by auto
```
```   868           also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
```
```   869           proof (rule mult_left_mono)
```
```   870             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>)"
```
```   871               by (rule setsum_abs)
```
```   872             also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
```
```   873             proof (rule setsum_mono)
```
```   874               fix p
```
```   875               assume "p \<in> {0..<Suc n}"
```
```   876               hence "p \<le> n" by auto
```
```   877               {
```
```   878                 fix n
```
```   879                 fix x :: real
```
```   880                 assume "x \<in> {-R'<..<R'}"
```
```   881                 hence "\<bar>x\<bar> \<le> R'"  by auto
```
```   882                 hence "\<bar>x^n\<bar> \<le> R'^n"
```
```   883                   unfolding power_abs by (rule power_mono, auto)
```
```   884               }
```
```   885               from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
```
```   886               have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
```
```   887                 unfolding abs_mult by auto
```
```   888               thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
```
```   889                 unfolding power_add[symmetric] using `p \<le> n` by auto
```
```   890             qed
```
```   891             also have "\<dots> = real (Suc n) * R' ^ n"
```
```   892               unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
```
```   893             finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
```
```   894               unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
```
```   895             show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
```
```   896               unfolding abs_mult[symmetric] by auto
```
```   897           qed
```
```   898           also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
```
```   899             unfolding abs_mult mult_assoc[symmetric] by algebra
```
```   900           finally show ?thesis .
```
```   901         qed
```
```   902       }
```
```   903       {
```
```   904         fix n
```
```   905         show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
```
```   906           by (auto intro!: DERIV_intros simp del: power_Suc)
```
```   907       }
```
```   908       {
```
```   909         fix x
```
```   910         assume "x \<in> {-R' <..< R'}"
```
```   911         hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
```
```   912           using assms `R' < R` by auto
```
```   913         have "summable (\<lambda> n. f n * x^n)"
```
```   914         proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
```
```   915           fix n
```
```   916           have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
```
```   917             by (rule mult_left_mono) auto
```
```   918           show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)"
```
```   919             unfolding real_norm_def abs_mult
```
```   920             by (rule mult_right_mono) (auto simp add: le[unfolded mult_1_right])
```
```   921         qed
```
```   922         from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
```
```   923         show "summable (?f x)" by auto
```
```   924       }
```
```   925       show "summable (?f' x0)"
```
```   926         using converges[OF `x0 \<in> {-R <..< R}`] .
```
```   927       show "x0 \<in> {-R' <..< R'}"
```
```   928         using `x0 \<in> {-R' <..< R'}` .
```
```   929     qed
```
```   930   } note for_subinterval = this
```
```   931   let ?R = "(R + \<bar>x0\<bar>) / 2"
```
```   932   have "\<bar>x0\<bar> < ?R" using assms by auto
```
```   933   hence "- ?R < x0"
```
```   934   proof (cases "x0 < 0")
```
```   935     case True
```
```   936     hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
```
```   937     thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
```
```   938   next
```
```   939     case False
```
```   940     have "- ?R < 0" using assms by auto
```
```   941     also have "\<dots> \<le> x0" using False by auto
```
```   942     finally show ?thesis .
```
```   943   qed
```
```   944   hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
```
```   945     using assms by auto
```
```   946   from for_subinterval[OF this]
```
```   947   show ?thesis .
```
```   948 qed
```
```   949
```
```   950
```
```   951 subsection {* Exponential Function *}
```
```   952
```
```   953 definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```   954   where "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
```
```   955
```
```   956 lemma summable_exp_generic:
```
```   957   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   958   defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
```
```   959   shows "summable S"
```
```   960 proof -
```
```   961   have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
```
```   962     unfolding S_def by (simp del: mult_Suc)
```
```   963   obtain r :: real where r0: "0 < r" and r1: "r < 1"
```
```   964     using dense [OF zero_less_one] by fast
```
```   965   obtain N :: nat where N: "norm x < real N * r"
```
```   966     using reals_Archimedean3 [OF r0] by fast
```
```   967   from r1 show ?thesis
```
```   968   proof (rule ratio_test [rule_format])
```
```   969     fix n :: nat
```
```   970     assume n: "N \<le> n"
```
```   971     have "norm x \<le> real N * r"
```
```   972       using N by (rule order_less_imp_le)
```
```   973     also have "real N * r \<le> real (Suc n) * r"
```
```   974       using r0 n by (simp add: mult_right_mono)
```
```   975     finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
```
```   976       using norm_ge_zero by (rule mult_right_mono)
```
```   977     hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
```
```   978       by (rule order_trans [OF norm_mult_ineq])
```
```   979     hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
```
```   980       by (simp add: pos_divide_le_eq mult_ac)
```
```   981     thus "norm (S (Suc n)) \<le> r * norm (S n)"
```
```   982       by (simp add: S_Suc inverse_eq_divide)
```
```   983   qed
```
```   984 qed
```
```   985
```
```   986 lemma summable_norm_exp:
```
```   987   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   988   shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
```
```   989 proof (rule summable_norm_comparison_test [OF exI, rule_format])
```
```   990   show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
```
```   991     by (rule summable_exp_generic)
```
```   992   fix n
```
```   993   show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
```
```   994     by (simp add: norm_power_ineq)
```
```   995 qed
```
```   996
```
```   997 lemma summable_exp: "summable (\<lambda>n. inverse (real (fact n)) * x ^ n)"
```
```   998   using summable_exp_generic [where x=x] by simp
```
```   999
```
```  1000 lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
```
```  1001   unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
```
```  1002
```
```  1003
```
```  1004 lemma exp_fdiffs:
```
```  1005       "diffs (\<lambda>n. inverse(real (fact n))) = (\<lambda>n. inverse(real (fact n)))"
```
```  1006   by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
```
```  1007         del: mult_Suc of_nat_Suc)
```
```  1008
```
```  1009 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
```
```  1010   by (simp add: diffs_def)
```
```  1011
```
```  1012 lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
```
```  1013   unfolding exp_def scaleR_conv_of_real
```
```  1014   apply (rule DERIV_cong)
```
```  1015   apply (rule termdiffs [where K="of_real (1 + norm x)"])
```
```  1016   apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
```
```  1017   apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
```
```  1018   apply (simp del: of_real_add)
```
```  1019   done
```
```  1020
```
```  1021 declare DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1022
```
```  1023 lemma isCont_exp: "isCont exp x"
```
```  1024   by (rule DERIV_exp [THEN DERIV_isCont])
```
```  1025
```
```  1026 lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
```
```  1027   by (rule isCont_o2 [OF _ isCont_exp])
```
```  1028
```
```  1029 lemma tendsto_exp [tendsto_intros]:
```
```  1030   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
```
```  1031   by (rule isCont_tendsto_compose [OF isCont_exp])
```
```  1032
```
```  1033 lemma continuous_exp [continuous_intros]:
```
```  1034   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
```
```  1035   unfolding continuous_def by (rule tendsto_exp)
```
```  1036
```
```  1037 lemma continuous_on_exp [continuous_on_intros]:
```
```  1038   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
```
```  1039   unfolding continuous_on_def by (auto intro: tendsto_exp)
```
```  1040
```
```  1041
```
```  1042 subsubsection {* Properties of the Exponential Function *}
```
```  1043
```
```  1044 lemma powser_zero:
```
```  1045   fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
```
```  1046   shows "(\<Sum>n. f n * 0 ^ n) = f 0"
```
```  1047 proof -
```
```  1048   have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
```
```  1049     by (rule sums_unique [OF series_zero], simp add: power_0_left)
```
```  1050   thus ?thesis unfolding One_nat_def by simp
```
```  1051 qed
```
```  1052
```
```  1053 lemma exp_zero [simp]: "exp 0 = 1"
```
```  1054   unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
```
```  1055
```
```  1056 lemma setsum_cl_ivl_Suc2:
```
```  1057   "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
```
```  1058   by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
```
```  1059            del: setsum_cl_ivl_Suc)
```
```  1060
```
```  1061 lemma exp_series_add:
```
```  1062   fixes x y :: "'a::{real_field}"
```
```  1063   defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
```
```  1064   shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
```
```  1065 proof (induct n)
```
```  1066   case 0
```
```  1067   show ?case
```
```  1068     unfolding S_def by simp
```
```  1069 next
```
```  1070   case (Suc n)
```
```  1071   have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
```
```  1072     unfolding S_def by (simp del: mult_Suc)
```
```  1073   hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
```
```  1074     by simp
```
```  1075
```
```  1076   have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
```
```  1077     by (simp only: times_S)
```
```  1078   also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
```
```  1079     by (simp only: Suc)
```
```  1080   also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
```
```  1081                 + y * (\<Sum>i=0..n. S x i * S y (n-i))"
```
```  1082     by (rule distrib_right)
```
```  1083   also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
```
```  1084                 + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
```
```  1085     by (simp only: setsum_right_distrib mult_ac)
```
```  1086   also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
```
```  1087                 + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1088     by (simp add: times_S Suc_diff_le)
```
```  1089   also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
```
```  1090              (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1091     by (subst setsum_cl_ivl_Suc2, simp)
```
```  1092   also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```  1093              (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1094     by (subst setsum_cl_ivl_Suc, simp)
```
```  1095   also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
```
```  1096              (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```  1097              (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1098     by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
```
```  1099               real_of_nat_add [symmetric], simp)
```
```  1100   also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
```
```  1101     by (simp only: scaleR_right.setsum)
```
```  1102   finally show
```
```  1103     "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
```
```  1104     by (simp del: setsum_cl_ivl_Suc)
```
```  1105 qed
```
```  1106
```
```  1107 lemma exp_add: "exp (x + y) = exp x * exp y"
```
```  1108   unfolding exp_def
```
```  1109   by (simp only: Cauchy_product summable_norm_exp exp_series_add)
```
```  1110
```
```  1111 lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
```
```  1112   by (rule exp_add [symmetric])
```
```  1113
```
```  1114 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
```
```  1115   unfolding exp_def
```
```  1116   apply (subst suminf_of_real)
```
```  1117   apply (rule summable_exp_generic)
```
```  1118   apply (simp add: scaleR_conv_of_real)
```
```  1119   done
```
```  1120
```
```  1121 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
```
```  1122 proof
```
```  1123   have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
```
```  1124   also assume "exp x = 0"
```
```  1125   finally show "False" by simp
```
```  1126 qed
```
```  1127
```
```  1128 lemma exp_minus: "exp (- x) = inverse (exp x)"
```
```  1129   by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
```
```  1130
```
```  1131 lemma exp_diff: "exp (x - y) = exp x / exp y"
```
```  1132   using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
```
```  1133
```
```  1134
```
```  1135 subsubsection {* Properties of the Exponential Function on Reals *}
```
```  1136
```
```  1137 text {* Comparisons of @{term "exp x"} with zero. *}
```
```  1138
```
```  1139 text{*Proof: because every exponential can be seen as a square.*}
```
```  1140 lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
```
```  1141 proof -
```
```  1142   have "0 \<le> exp (x/2) * exp (x/2)" by simp
```
```  1143   thus ?thesis by (simp add: exp_add [symmetric])
```
```  1144 qed
```
```  1145
```
```  1146 lemma exp_gt_zero [simp]: "0 < exp (x::real)"
```
```  1147   by (simp add: order_less_le)
```
```  1148
```
```  1149 lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
```
```  1150   by (simp add: not_less)
```
```  1151
```
```  1152 lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
```
```  1153   by (simp add: not_le)
```
```  1154
```
```  1155 lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
```
```  1156   by simp
```
```  1157
```
```  1158 lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
```
```  1159   by (induct n) (auto simp add: real_of_nat_Suc distrib_left exp_add mult_commute)
```
```  1160
```
```  1161 text {* Strict monotonicity of exponential. *}
```
```  1162
```
```  1163 lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) \<Longrightarrow> (1 + x) \<le> exp(x)"
```
```  1164   apply (drule order_le_imp_less_or_eq, auto)
```
```  1165   apply (simp add: exp_def)
```
```  1166   apply (rule order_trans)
```
```  1167   apply (rule_tac [2] n = 2 and f = "(\<lambda>n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
```
```  1168   apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
```
```  1169   done
```
```  1170
```
```  1171 lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
```
```  1172 proof -
```
```  1173   assume x: "0 < x"
```
```  1174   hence "1 < 1 + x" by simp
```
```  1175   also from x have "1 + x \<le> exp x"
```
```  1176     by (simp add: exp_ge_add_one_self_aux)
```
```  1177   finally show ?thesis .
```
```  1178 qed
```
```  1179
```
```  1180 lemma exp_less_mono:
```
```  1181   fixes x y :: real
```
```  1182   assumes "x < y"
```
```  1183   shows "exp x < exp y"
```
```  1184 proof -
```
```  1185   from `x < y` have "0 < y - x" by simp
```
```  1186   hence "1 < exp (y - x)" by (rule exp_gt_one)
```
```  1187   hence "1 < exp y / exp x" by (simp only: exp_diff)
```
```  1188   thus "exp x < exp y" by simp
```
```  1189 qed
```
```  1190
```
```  1191 lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
```
```  1192   apply (simp add: linorder_not_le [symmetric])
```
```  1193   apply (auto simp add: order_le_less exp_less_mono)
```
```  1194   done
```
```  1195
```
```  1196 lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
```
```  1197   by (auto intro: exp_less_mono exp_less_cancel)
```
```  1198
```
```  1199 lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
```
```  1200   by (auto simp add: linorder_not_less [symmetric])
```
```  1201
```
```  1202 lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
```
```  1203   by (simp add: order_eq_iff)
```
```  1204
```
```  1205 text {* Comparisons of @{term "exp x"} with one. *}
```
```  1206
```
```  1207 lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
```
```  1208   using exp_less_cancel_iff [where x=0 and y=x] by simp
```
```  1209
```
```  1210 lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
```
```  1211   using exp_less_cancel_iff [where x=x and y=0] by simp
```
```  1212
```
```  1213 lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
```
```  1214   using exp_le_cancel_iff [where x=0 and y=x] by simp
```
```  1215
```
```  1216 lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
```
```  1217   using exp_le_cancel_iff [where x=x and y=0] by simp
```
```  1218
```
```  1219 lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
```
```  1220   using exp_inj_iff [where x=x and y=0] by simp
```
```  1221
```
```  1222 lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
```
```  1223 proof (rule IVT)
```
```  1224   assume "1 \<le> y"
```
```  1225   hence "0 \<le> y - 1" by simp
```
```  1226   hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
```
```  1227   thus "y \<le> exp (y - 1)" by simp
```
```  1228 qed (simp_all add: le_diff_eq)
```
```  1229
```
```  1230 lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y"
```
```  1231 proof (rule linorder_le_cases [of 1 y])
```
```  1232   assume "1 \<le> y"
```
```  1233   thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total)
```
```  1234 next
```
```  1235   assume "0 < y" and "y \<le> 1"
```
```  1236   hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
```
```  1237   then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
```
```  1238   hence "exp (- x) = y" by (simp add: exp_minus)
```
```  1239   thus "\<exists>x. exp x = y" ..
```
```  1240 qed
```
```  1241
```
```  1242
```
```  1243 subsection {* Natural Logarithm *}
```
```  1244
```
```  1245 definition ln :: "real \<Rightarrow> real"
```
```  1246   where "ln x = (THE u. exp u = x)"
```
```  1247
```
```  1248 lemma ln_exp [simp]: "ln (exp x) = x"
```
```  1249   by (simp add: ln_def)
```
```  1250
```
```  1251 lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
```
```  1252   by (auto dest: exp_total)
```
```  1253
```
```  1254 lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
```
```  1255   by (metis exp_gt_zero exp_ln)
```
```  1256
```
```  1257 lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
```
```  1258   by (erule subst, rule ln_exp)
```
```  1259
```
```  1260 lemma ln_one [simp]: "ln 1 = 0"
```
```  1261   by (rule ln_unique) simp
```
```  1262
```
```  1263 lemma ln_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
```
```  1264   by (rule ln_unique) (simp add: exp_add)
```
```  1265
```
```  1266 lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
```
```  1267   by (rule ln_unique) (simp add: exp_minus)
```
```  1268
```
```  1269 lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
```
```  1270   by (rule ln_unique) (simp add: exp_diff)
```
```  1271
```
```  1272 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
```
```  1273   by (rule ln_unique) (simp add: exp_real_of_nat_mult)
```
```  1274
```
```  1275 lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
```
```  1276   by (subst exp_less_cancel_iff [symmetric]) simp
```
```  1277
```
```  1278 lemma ln_le_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
```
```  1279   by (simp add: linorder_not_less [symmetric])
```
```  1280
```
```  1281 lemma ln_inj_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
```
```  1282   by (simp add: order_eq_iff)
```
```  1283
```
```  1284 lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
```
```  1285   apply (rule exp_le_cancel_iff [THEN iffD1])
```
```  1286   apply (simp add: exp_ge_add_one_self_aux)
```
```  1287   done
```
```  1288
```
```  1289 lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
```
```  1290   by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
```
```  1291
```
```  1292 lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
```
```  1293   using ln_le_cancel_iff [of 1 x] by simp
```
```  1294
```
```  1295 lemma ln_ge_zero_imp_ge_one: "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
```
```  1296   using ln_le_cancel_iff [of 1 x] by simp
```
```  1297
```
```  1298 lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
```
```  1299   using ln_le_cancel_iff [of 1 x] by simp
```
```  1300
```
```  1301 lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
```
```  1302   using ln_less_cancel_iff [of x 1] by simp
```
```  1303
```
```  1304 lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
```
```  1305   using ln_less_cancel_iff [of 1 x] by simp
```
```  1306
```
```  1307 lemma ln_gt_zero_imp_gt_one: "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
```
```  1308   using ln_less_cancel_iff [of 1 x] by simp
```
```  1309
```
```  1310 lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
```
```  1311   using ln_less_cancel_iff [of 1 x] by simp
```
```  1312
```
```  1313 lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
```
```  1314   using ln_inj_iff [of x 1] by simp
```
```  1315
```
```  1316 lemma ln_less_zero: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
```
```  1317   by simp
```
```  1318
```
```  1319 lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
```
```  1320   apply (subgoal_tac "isCont ln (exp (ln x))", simp)
```
```  1321   apply (rule isCont_inverse_function [where f=exp], simp_all)
```
```  1322   done
```
```  1323
```
```  1324 lemma tendsto_ln [tendsto_intros]:
```
```  1325   "(f ---> a) F \<Longrightarrow> 0 < a \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
```
```  1326   by (rule isCont_tendsto_compose [OF isCont_ln])
```
```  1327
```
```  1328 lemma continuous_ln:
```
```  1329   "continuous F f \<Longrightarrow> 0 < f (Lim F (\<lambda>x. x)) \<Longrightarrow> continuous F (\<lambda>x. ln (f x))"
```
```  1330   unfolding continuous_def by (rule tendsto_ln)
```
```  1331
```
```  1332 lemma isCont_ln' [continuous_intros]:
```
```  1333   "continuous (at x) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x))"
```
```  1334   unfolding continuous_at by (rule tendsto_ln)
```
```  1335
```
```  1336 lemma continuous_within_ln [continuous_intros]:
```
```  1337   "continuous (at x within s) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x))"
```
```  1338   unfolding continuous_within by (rule tendsto_ln)
```
```  1339
```
```  1340 lemma continuous_on_ln [continuous_on_intros]:
```
```  1341   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. 0 < f x) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x))"
```
```  1342   unfolding continuous_on_def by (auto intro: tendsto_ln)
```
```  1343
```
```  1344 lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
```
```  1345   apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
```
```  1346   apply (erule DERIV_cong [OF DERIV_exp exp_ln])
```
```  1347   apply (simp_all add: abs_if isCont_ln)
```
```  1348   done
```
```  1349
```
```  1350 lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
```
```  1351   by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
```
```  1352
```
```  1353 declare DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1354
```
```  1355 lemma ln_series:
```
```  1356   assumes "0 < x" and "x < 2"
```
```  1357   shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))"
```
```  1358   (is "ln x = suminf (?f (x - 1))")
```
```  1359 proof -
```
```  1360   let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n"
```
```  1361
```
```  1362   have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
```
```  1363   proof (rule DERIV_isconst3[where x=x])
```
```  1364     fix x :: real
```
```  1365     assume "x \<in> {0 <..< 2}"
```
```  1366     hence "0 < x" and "x < 2" by auto
```
```  1367     have "norm (1 - x) < 1"
```
```  1368       using `0 < x` and `x < 2` by auto
```
```  1369     have "1 / x = 1 / (1 - (1 - x))" by auto
```
```  1370     also have "\<dots> = (\<Sum> n. (1 - x)^n)"
```
```  1371       using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
```
```  1372     also have "\<dots> = suminf (?f' x)"
```
```  1373       unfolding power_mult_distrib[symmetric]
```
```  1374       by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
```
```  1375     finally have "DERIV ln x :> suminf (?f' x)"
```
```  1376       using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
```
```  1377     moreover
```
```  1378     have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
```
```  1379     have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :>
```
```  1380       (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
```
```  1381     proof (rule DERIV_power_series')
```
```  1382       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1"
```
```  1383         using `0 < x` `x < 2` by auto
```
```  1384       fix x :: real
```
```  1385       assume "x \<in> {- 1<..<1}"
```
```  1386       hence "norm (-x) < 1" by auto
```
```  1387       show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
```
```  1388         unfolding One_nat_def
```
```  1389         by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
```
```  1390     qed
```
```  1391     hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
```
```  1392       unfolding One_nat_def by auto
```
```  1393     hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
```
```  1394       unfolding DERIV_iff repos .
```
```  1395     ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
```
```  1396       by (rule DERIV_diff)
```
```  1397     thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
```
```  1398   qed (auto simp add: assms)
```
```  1399   thus ?thesis by auto
```
```  1400 qed
```
```  1401
```
```  1402 lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) * (x ^ (n+2)))"
```
```  1403 proof -
```
```  1404   have "exp x = suminf (\<lambda>n. inverse(fact n) * (x ^ n))"
```
```  1405     by (simp add: exp_def)
```
```  1406   also from summable_exp have "... = (\<Sum> n::nat = 0 ..< 2. inverse(fact n) * (x ^ n)) +
```
```  1407       (\<Sum> n. inverse(fact(n+2)) * (x ^ (n+2)))" (is "_ = ?a + _")
```
```  1408     by (rule suminf_split_initial_segment)
```
```  1409   also have "?a = 1 + x"
```
```  1410     by (simp add: numeral_2_eq_2)
```
```  1411   finally show ?thesis .
```
```  1412 qed
```
```  1413
```
```  1414 lemma exp_bound: "0 <= (x::real) \<Longrightarrow> x <= 1 \<Longrightarrow> exp x <= 1 + x + x\<^sup>2"
```
```  1415 proof -
```
```  1416   assume a: "0 <= x"
```
```  1417   assume b: "x <= 1"
```
```  1418   {
```
```  1419     fix n :: nat
```
```  1420     have "2 * 2 ^ n \<le> fact (n + 2)"
```
```  1421       by (induct n) simp_all
```
```  1422     hence "real ((2::nat) * 2 ^ n) \<le> real (fact (n + 2))"
```
```  1423       by (simp only: real_of_nat_le_iff)
```
```  1424     hence "2 * 2 ^ n \<le> real (fact (n + 2))"
```
```  1425       by simp
```
```  1426     hence "inverse (fact (n + 2)) \<le> inverse (2 * 2 ^ n)"
```
```  1427       by (rule le_imp_inverse_le) simp
```
```  1428     hence "inverse (fact (n + 2)) \<le> 1/2 * (1/2)^n"
```
```  1429       by (simp add: power_inverse)
```
```  1430     hence "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
```
```  1431       by (rule mult_mono)
```
```  1432         (rule mult_mono, simp_all add: power_le_one a b mult_nonneg_nonneg)
```
```  1433     hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)"
```
```  1434       unfolding power_add by (simp add: mult_ac del: fact_Suc) }
```
```  1435   note aux1 = this
```
```  1436   have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
```
```  1437     by (intro sums_mult geometric_sums, simp)
```
```  1438   hence aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
```
```  1439     by simp
```
```  1440   have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2"
```
```  1441   proof -
```
```  1442     have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <=
```
```  1443         suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
```
```  1444       apply (rule summable_le)
```
```  1445       apply (rule allI, rule aux1)
```
```  1446       apply (rule summable_exp [THEN summable_ignore_initial_segment])
```
```  1447       by (rule sums_summable, rule aux2)
```
```  1448     also have "... = x\<^sup>2"
```
```  1449       by (rule sums_unique [THEN sym], rule aux2)
```
```  1450     finally show ?thesis .
```
```  1451   qed
```
```  1452   thus ?thesis unfolding exp_first_two_terms by auto
```
```  1453 qed
```
```  1454
```
```  1455 lemma ln_one_minus_pos_upper_bound: "0 <= x \<Longrightarrow> x < 1 \<Longrightarrow> ln (1 - x) <= - x"
```
```  1456 proof -
```
```  1457   assume a: "0 <= (x::real)" and b: "x < 1"
```
```  1458   have "(1 - x) * (1 + x + x\<^sup>2) = (1 - x^3)"
```
```  1459     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
```
```  1460   also have "... <= 1"
```
```  1461     by (auto simp add: a)
```
```  1462   finally have "(1 - x) * (1 + x + x\<^sup>2) <= 1" .
```
```  1463   moreover have c: "0 < 1 + x + x\<^sup>2"
```
```  1464     by (simp add: add_pos_nonneg a)
```
```  1465   ultimately have "1 - x <= 1 / (1 + x + x\<^sup>2)"
```
```  1466     by (elim mult_imp_le_div_pos)
```
```  1467   also have "... <= 1 / exp x"
```
```  1468     apply (rule divide_left_mono)
```
```  1469     apply (rule exp_bound, rule a)
```
```  1470     apply (rule b [THEN less_imp_le])
```
```  1471     apply simp
```
```  1472     apply (rule mult_pos_pos)
```
```  1473     apply (rule c)
```
```  1474     apply simp
```
```  1475     done
```
```  1476   also have "... = exp (-x)"
```
```  1477     by (auto simp add: exp_minus divide_inverse)
```
```  1478   finally have "1 - x <= exp (- x)" .
```
```  1479   also have "1 - x = exp (ln (1 - x))"
```
```  1480   proof -
```
```  1481     have "0 < 1 - x"
```
```  1482       by (insert b, auto)
```
```  1483     thus ?thesis
```
```  1484       by (auto simp only: exp_ln_iff [THEN sym])
```
```  1485   qed
```
```  1486   finally have "exp (ln (1 - x)) <= exp (- x)" .
```
```  1487   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```  1488 qed
```
```  1489
```
```  1490 lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
```
```  1491   apply (case_tac "0 <= x")
```
```  1492   apply (erule exp_ge_add_one_self_aux)
```
```  1493   apply (case_tac "x <= -1")
```
```  1494   apply (subgoal_tac "1 + x <= 0")
```
```  1495   apply (erule order_trans)
```
```  1496   apply simp
```
```  1497   apply simp
```
```  1498   apply (subgoal_tac "1 + x = exp(ln (1 + x))")
```
```  1499   apply (erule ssubst)
```
```  1500   apply (subst exp_le_cancel_iff)
```
```  1501   apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
```
```  1502   apply simp
```
```  1503   apply (rule ln_one_minus_pos_upper_bound)
```
```  1504   apply auto
```
```  1505 done
```
```  1506
```
```  1507 lemma ln_one_plus_pos_lower_bound: "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> x - x\<^sup>2 <= ln (1 + x)"
```
```  1508 proof -
```
```  1509   assume a: "0 <= x" and b: "x <= 1"
```
```  1510   have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
```
```  1511     by (rule exp_diff)
```
```  1512   also have "... <= (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
```
```  1513     apply (rule divide_right_mono)
```
```  1514     apply (rule exp_bound)
```
```  1515     apply (rule a, rule b)
```
```  1516     apply simp
```
```  1517     done
```
```  1518   also have "... <= (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
```
```  1519     apply (rule divide_left_mono)
```
```  1520     apply (simp add: exp_ge_add_one_self_aux)
```
```  1521     apply (simp add: a)
```
```  1522     apply (simp add: mult_pos_pos add_pos_nonneg)
```
```  1523     done
```
```  1524   also from a have "... <= 1 + x"
```
```  1525     by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
```
```  1526   finally have "exp (x - x\<^sup>2) <= 1 + x" .
```
```  1527   also have "... = exp (ln (1 + x))"
```
```  1528   proof -
```
```  1529     from a have "0 < 1 + x" by auto
```
```  1530     thus ?thesis
```
```  1531       by (auto simp only: exp_ln_iff [THEN sym])
```
```  1532   qed
```
```  1533   finally have "exp (x - x\<^sup>2) <= exp (ln (1 + x))" .
```
```  1534   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```  1535 qed
```
```  1536
```
```  1537 lemma aux5: "x < 1 \<Longrightarrow> ln(1 - x) = - ln(1 + x / (1 - x))"
```
```  1538 proof -
```
```  1539   assume a: "x < 1"
```
```  1540   have "ln(1 - x) = - ln(1 / (1 - x))"
```
```  1541   proof -
```
```  1542     have "ln(1 - x) = - (- ln (1 - x))"
```
```  1543       by auto
```
```  1544     also have "- ln(1 - x) = ln 1 - ln(1 - x)"
```
```  1545       by simp
```
```  1546     also have "... = ln(1 / (1 - x))"
```
```  1547       apply (rule ln_div [THEN sym])
```
```  1548       using a apply auto
```
```  1549       done
```
```  1550     finally show ?thesis .
```
```  1551   qed
```
```  1552   also have " 1 / (1 - x) = 1 + x / (1 - x)" using a by(simp add:field_simps)
```
```  1553   finally show ?thesis .
```
```  1554 qed
```
```  1555
```
```  1556 lemma ln_one_minus_pos_lower_bound:
```
```  1557   "0 <= x \<Longrightarrow> x <= (1 / 2) \<Longrightarrow> - x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1558 proof -
```
```  1559   assume a: "0 <= x" and b: "x <= (1 / 2)"
```
```  1560   from b have c: "x < 1" by auto
```
```  1561   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
```
```  1562     by (rule aux5)
```
```  1563   also have "- (x / (1 - x)) <= ..."
```
```  1564   proof -
```
```  1565     have "ln (1 + x / (1 - x)) <= x / (1 - x)"
```
```  1566       apply (rule ln_add_one_self_le_self)
```
```  1567       apply (rule divide_nonneg_pos)
```
```  1568       using a c apply auto
```
```  1569       done
```
```  1570     thus ?thesis
```
```  1571       by auto
```
```  1572   qed
```
```  1573   also have "- (x / (1 - x)) = -x / (1 - x)"
```
```  1574     by auto
```
```  1575   finally have d: "- x / (1 - x) <= ln (1 - x)" .
```
```  1576   have "0 < 1 - x" using a b by simp
```
```  1577   hence e: "-x - 2 * x\<^sup>2 <= - x / (1 - x)"
```
```  1578     using mult_right_le_one_le[of "x*x" "2*x"] a b
```
```  1579     by (simp add: field_simps power2_eq_square)
```
```  1580   from e d show "- x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1581     by (rule order_trans)
```
```  1582 qed
```
```  1583
```
```  1584 lemma ln_add_one_self_le_self2: "-1 < x \<Longrightarrow> ln(1 + x) <= x"
```
```  1585   apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
```
```  1586   apply (subst ln_le_cancel_iff)
```
```  1587   apply auto
```
```  1588   done
```
```  1589
```
```  1590 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
```
```  1591   "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> abs(ln (1 + x) - x) <= x\<^sup>2"
```
```  1592 proof -
```
```  1593   assume x: "0 <= x"
```
```  1594   assume x1: "x <= 1"
```
```  1595   from x have "ln (1 + x) <= x"
```
```  1596     by (rule ln_add_one_self_le_self)
```
```  1597   then have "ln (1 + x) - x <= 0"
```
```  1598     by simp
```
```  1599   then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
```
```  1600     by (rule abs_of_nonpos)
```
```  1601   also have "... = x - ln (1 + x)"
```
```  1602     by simp
```
```  1603   also have "... <= x\<^sup>2"
```
```  1604   proof -
```
```  1605     from x x1 have "x - x\<^sup>2 <= ln (1 + x)"
```
```  1606       by (intro ln_one_plus_pos_lower_bound)
```
```  1607     thus ?thesis
```
```  1608       by simp
```
```  1609   qed
```
```  1610   finally show ?thesis .
```
```  1611 qed
```
```  1612
```
```  1613 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
```
```  1614   "-(1 / 2) <= x \<Longrightarrow> x <= 0 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
```
```  1615 proof -
```
```  1616   assume a: "-(1 / 2) <= x"
```
```  1617   assume b: "x <= 0"
```
```  1618   have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
```
```  1619     apply (subst abs_of_nonpos)
```
```  1620     apply simp
```
```  1621     apply (rule ln_add_one_self_le_self2)
```
```  1622     using a apply auto
```
```  1623     done
```
```  1624   also have "... <= 2 * x\<^sup>2"
```
```  1625     apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 <= ln (1 - (-x))")
```
```  1626     apply (simp add: algebra_simps)
```
```  1627     apply (rule ln_one_minus_pos_lower_bound)
```
```  1628     using a b apply auto
```
```  1629     done
```
```  1630   finally show ?thesis .
```
```  1631 qed
```
```  1632
```
```  1633 lemma abs_ln_one_plus_x_minus_x_bound:
```
```  1634     "abs x <= 1 / 2 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
```
```  1635   apply (case_tac "0 <= x")
```
```  1636   apply (rule order_trans)
```
```  1637   apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
```
```  1638   apply auto
```
```  1639   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
```
```  1640   apply auto
```
```  1641   done
```
```  1642
```
```  1643 lemma ln_x_over_x_mono: "exp 1 <= x \<Longrightarrow> x <= y \<Longrightarrow> (ln y / y) <= (ln x / x)"
```
```  1644 proof -
```
```  1645   assume x: "exp 1 <= x" "x <= y"
```
```  1646   moreover have "0 < exp (1::real)" by simp
```
```  1647   ultimately have a: "0 < x" and b: "0 < y"
```
```  1648     by (fast intro: less_le_trans order_trans)+
```
```  1649   have "x * ln y - x * ln x = x * (ln y - ln x)"
```
```  1650     by (simp add: algebra_simps)
```
```  1651   also have "... = x * ln(y / x)"
```
```  1652     by (simp only: ln_div a b)
```
```  1653   also have "y / x = (x + (y - x)) / x"
```
```  1654     by simp
```
```  1655   also have "... = 1 + (y - x) / x"
```
```  1656     using x a by (simp add: field_simps)
```
```  1657   also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
```
```  1658     apply (rule mult_left_mono)
```
```  1659     apply (rule ln_add_one_self_le_self)
```
```  1660     apply (rule divide_nonneg_pos)
```
```  1661     using x a apply simp_all
```
```  1662     done
```
```  1663   also have "... = y - x" using a by simp
```
```  1664   also have "... = (y - x) * ln (exp 1)" by simp
```
```  1665   also have "... <= (y - x) * ln x"
```
```  1666     apply (rule mult_left_mono)
```
```  1667     apply (subst ln_le_cancel_iff)
```
```  1668     apply fact
```
```  1669     apply (rule a)
```
```  1670     apply (rule x)
```
```  1671     using x apply simp
```
```  1672     done
```
```  1673   also have "... = y * ln x - x * ln x"
```
```  1674     by (rule left_diff_distrib)
```
```  1675   finally have "x * ln y <= y * ln x"
```
```  1676     by arith
```
```  1677   then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
```
```  1678   also have "... = y * (ln x / x)" by simp
```
```  1679   finally show ?thesis using b by (simp add: field_simps)
```
```  1680 qed
```
```  1681
```
```  1682 lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1"
```
```  1683   using exp_ge_add_one_self[of "ln x"] by simp
```
```  1684
```
```  1685 lemma ln_eq_minus_one:
```
```  1686   assumes "0 < x" "ln x = x - 1"
```
```  1687   shows "x = 1"
```
```  1688 proof -
```
```  1689   let ?l = "\<lambda>y. ln y - y + 1"
```
```  1690   have D: "\<And>x. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
```
```  1691     by (auto intro!: DERIV_intros)
```
```  1692
```
```  1693   show ?thesis
```
```  1694   proof (cases rule: linorder_cases)
```
```  1695     assume "x < 1"
```
```  1696     from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast
```
```  1697     from `x < a` have "?l x < ?l a"
```
```  1698     proof (rule DERIV_pos_imp_increasing, safe)
```
```  1699       fix y
```
```  1700       assume "x \<le> y" "y \<le> a"
```
```  1701       with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y"
```
```  1702         by (auto simp: field_simps)
```
```  1703       with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
```
```  1704         by auto
```
```  1705     qed
```
```  1706     also have "\<dots> \<le> 0"
```
```  1707       using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps)
```
```  1708     finally show "x = 1" using assms by auto
```
```  1709   next
```
```  1710     assume "1 < x"
```
```  1711     from dense[OF this] obtain a where "1 < a" "a < x" by blast
```
```  1712     from `a < x` have "?l x < ?l a"
```
```  1713     proof (rule DERIV_neg_imp_decreasing, safe)
```
```  1714       fix y
```
```  1715       assume "a \<le> y" "y \<le> x"
```
```  1716       with `1 < a` have "1 / y - 1 < 0" "0 < y"
```
```  1717         by (auto simp: field_simps)
```
```  1718       with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
```
```  1719         by blast
```
```  1720     qed
```
```  1721     also have "\<dots> \<le> 0"
```
```  1722       using ln_le_minus_one `1 < a` by (auto simp: field_simps)
```
```  1723     finally show "x = 1" using assms by auto
```
```  1724   next
```
```  1725     assume "x = 1"
```
```  1726     then show ?thesis by simp
```
```  1727   qed
```
```  1728 qed
```
```  1729
```
```  1730 lemma exp_at_bot: "(exp ---> (0::real)) at_bot"
```
```  1731   unfolding tendsto_Zfun_iff
```
```  1732 proof (rule ZfunI, simp add: eventually_at_bot_dense)
```
```  1733   fix r :: real assume "0 < r"
```
```  1734   {
```
```  1735     fix x
```
```  1736     assume "x < ln r"
```
```  1737     then have "exp x < exp (ln r)"
```
```  1738       by simp
```
```  1739     with `0 < r` have "exp x < r"
```
```  1740       by simp
```
```  1741   }
```
```  1742   then show "\<exists>k. \<forall>n<k. exp n < r" by auto
```
```  1743 qed
```
```  1744
```
```  1745 lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
```
```  1746   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"])
```
```  1747      (auto intro: eventually_gt_at_top)
```
```  1748
```
```  1749 lemma ln_at_0: "LIM x at_right 0. ln x :> at_bot"
```
```  1750   by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  1751      (auto simp: eventually_at_filter)
```
```  1752
```
```  1753 lemma ln_at_top: "LIM x at_top. ln x :> at_top"
```
```  1754   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  1755      (auto intro: eventually_gt_at_top)
```
```  1756
```
```  1757 lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) ---> (0::real)) at_top"
```
```  1758 proof (induct k)
```
```  1759   case 0
```
```  1760   show "((\<lambda>x. x ^ 0 / exp x) ---> (0::real)) at_top"
```
```  1761     by (simp add: inverse_eq_divide[symmetric])
```
```  1762        (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
```
```  1763               at_top_le_at_infinity order_refl)
```
```  1764 next
```
```  1765   case (Suc k)
```
```  1766   show ?case
```
```  1767   proof (rule lhospital_at_top_at_top)
```
```  1768     show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"
```
```  1769       by eventually_elim (intro DERIV_intros, simp, simp)
```
```  1770     show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top"
```
```  1771       by eventually_elim (auto intro!: DERIV_intros)
```
```  1772     show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
```
```  1773       by auto
```
```  1774     from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
```
```  1775     show "((\<lambda>x. real (Suc k) * x ^ k / exp x) ---> 0) at_top"
```
```  1776       by simp
```
```  1777   qed (rule exp_at_top)
```
```  1778 qed
```
```  1779
```
```  1780
```
```  1781 definition powr :: "[real,real] => real"  (infixr "powr" 80)
```
```  1782   -- {*exponentation with real exponent*}
```
```  1783   where "x powr a = exp(a * ln x)"
```
```  1784
```
```  1785 definition log :: "[real,real] => real"
```
```  1786   -- {*logarithm of @{term x} to base @{term a}*}
```
```  1787   where "log a x = ln x / ln a"
```
```  1788
```
```  1789
```
```  1790 lemma tendsto_log [tendsto_intros]:
```
```  1791   "\<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"
```
```  1792   unfolding log_def by (intro tendsto_intros) auto
```
```  1793
```
```  1794 lemma continuous_log:
```
```  1795   assumes "continuous F f"
```
```  1796     and "continuous F g"
```
```  1797     and "0 < f (Lim F (\<lambda>x. x))"
```
```  1798     and "f (Lim F (\<lambda>x. x)) \<noteq> 1"
```
```  1799     and "0 < g (Lim F (\<lambda>x. x))"
```
```  1800   shows "continuous F (\<lambda>x. log (f x) (g x))"
```
```  1801   using assms unfolding continuous_def by (rule tendsto_log)
```
```  1802
```
```  1803 lemma continuous_at_within_log[continuous_intros]:
```
```  1804   assumes "continuous (at a within s) f"
```
```  1805     and "continuous (at a within s) g"
```
```  1806     and "0 < f a"
```
```  1807     and "f a \<noteq> 1"
```
```  1808     and "0 < g a"
```
```  1809   shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
```
```  1810   using assms unfolding continuous_within by (rule tendsto_log)
```
```  1811
```
```  1812 lemma isCont_log[continuous_intros, simp]:
```
```  1813   assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
```
```  1814   shows "isCont (\<lambda>x. log (f x) (g x)) a"
```
```  1815   using assms unfolding continuous_at by (rule tendsto_log)
```
```  1816
```
```  1817 lemma continuous_on_log[continuous_on_intros]:
```
```  1818   assumes "continuous_on s f" "continuous_on s g"
```
```  1819     and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
```
```  1820   shows "continuous_on s (\<lambda>x. log (f x) (g x))"
```
```  1821   using assms unfolding continuous_on_def by (fast intro: tendsto_log)
```
```  1822
```
```  1823 lemma powr_one_eq_one [simp]: "1 powr a = 1"
```
```  1824   by (simp add: powr_def)
```
```  1825
```
```  1826 lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
```
```  1827   by (simp add: powr_def)
```
```  1828
```
```  1829 lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
```
```  1830   by (simp add: powr_def)
```
```  1831 declare powr_one_gt_zero_iff [THEN iffD2, simp]
```
```  1832
```
```  1833 lemma powr_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
```
```  1834   by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
```
```  1835
```
```  1836 lemma powr_gt_zero [simp]: "0 < x powr a"
```
```  1837   by (simp add: powr_def)
```
```  1838
```
```  1839 lemma powr_ge_pzero [simp]: "0 <= x powr y"
```
```  1840   by (rule order_less_imp_le, rule powr_gt_zero)
```
```  1841
```
```  1842 lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
```
```  1843   by (simp add: powr_def)
```
```  1844
```
```  1845 lemma powr_divide: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
```
```  1846   apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
```
```  1847   apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
```
```  1848   done
```
```  1849
```
```  1850 lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
```
```  1851   apply (simp add: powr_def)
```
```  1852   apply (subst exp_diff [THEN sym])
```
```  1853   apply (simp add: left_diff_distrib)
```
```  1854   done
```
```  1855
```
```  1856 lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
```
```  1857   by (simp add: powr_def exp_add [symmetric] distrib_right)
```
```  1858
```
```  1859 lemma powr_mult_base: "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
```
```  1860   using assms by (auto simp: powr_add)
```
```  1861
```
```  1862 lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
```
```  1863   by (simp add: powr_def)
```
```  1864
```
```  1865 lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
```
```  1866   by (simp add: powr_powr mult_commute)
```
```  1867
```
```  1868 lemma powr_minus: "x powr (-a) = inverse (x powr a)"
```
```  1869   by (simp add: powr_def exp_minus [symmetric])
```
```  1870
```
```  1871 lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
```
```  1872   by (simp add: divide_inverse powr_minus)
```
```  1873
```
```  1874 lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
```
```  1875   by (simp add: powr_def)
```
```  1876
```
```  1877 lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
```
```  1878   by (simp add: powr_def)
```
```  1879
```
```  1880 lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a < x powr b) = (a < b)"
```
```  1881   by (blast intro: powr_less_cancel powr_less_mono)
```
```  1882
```
```  1883 lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a \<le> x powr b) = (a \<le> b)"
```
```  1884   by (simp add: linorder_not_less [symmetric])
```
```  1885
```
```  1886 lemma log_ln: "ln x = log (exp(1)) x"
```
```  1887   by (simp add: log_def)
```
```  1888
```
```  1889 lemma DERIV_log:
```
```  1890   assumes "x > 0"
```
```  1891   shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
```
```  1892 proof -
```
```  1893   def lb \<equiv> "1 / ln b"
```
```  1894   moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
```
```  1895     using `x > 0` by (auto intro!: DERIV_intros)
```
```  1896   ultimately show ?thesis
```
```  1897     by (simp add: log_def)
```
```  1898 qed
```
```  1899
```
```  1900 lemmas DERIV_log[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1901
```
```  1902 lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
```
```  1903   by (simp add: powr_def log_def)
```
```  1904
```
```  1905 lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y"
```
```  1906   by (simp add: log_def powr_def)
```
```  1907
```
```  1908 lemma log_mult:
```
```  1909   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow>
```
```  1910     log a (x * y) = log a x + log a y"
```
```  1911   by (simp add: log_def ln_mult divide_inverse distrib_right)
```
```  1912
```
```  1913 lemma log_eq_div_ln_mult_log:
```
```  1914   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow>
```
```  1915     log a x = (ln b/ln a) * log b x"
```
```  1916   by (simp add: log_def divide_inverse)
```
```  1917
```
```  1918 text{*Base 10 logarithms*}
```
```  1919 lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x"
```
```  1920   by (simp add: log_def)
```
```  1921
```
```  1922 lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x"
```
```  1923   by (simp add: log_def)
```
```  1924
```
```  1925 lemma log_one [simp]: "log a 1 = 0"
```
```  1926   by (simp add: log_def)
```
```  1927
```
```  1928 lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
```
```  1929   by (simp add: log_def)
```
```  1930
```
```  1931 lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x"
```
```  1932   apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
```
```  1933   apply (simp add: log_mult [symmetric])
```
```  1934   done
```
```  1935
```
```  1936 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"
```
```  1937   by (simp add: log_mult divide_inverse log_inverse)
```
```  1938
```
```  1939 lemma log_less_cancel_iff [simp]:
```
```  1940   "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
```
```  1941   apply safe
```
```  1942   apply (rule_tac [2] powr_less_cancel)
```
```  1943   apply (drule_tac a = "log a x" in powr_less_mono, auto)
```
```  1944   done
```
```  1945
```
```  1946 lemma log_inj:
```
```  1947   assumes "1 < b"
```
```  1948   shows "inj_on (log b) {0 <..}"
```
```  1949 proof (rule inj_onI, simp)
```
```  1950   fix x y
```
```  1951   assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
```
```  1952   show "x = y"
```
```  1953   proof (cases rule: linorder_cases)
```
```  1954     assume "x = y"
```
```  1955     then show ?thesis by simp
```
```  1956   next
```
```  1957     assume "x < y" hence "log b x < log b y"
```
```  1958       using log_less_cancel_iff[OF `1 < b`] pos by simp
```
```  1959     then show ?thesis using * by simp
```
```  1960   next
```
```  1961     assume "y < x" hence "log b y < log b x"
```
```  1962       using log_less_cancel_iff[OF `1 < b`] pos by simp
```
```  1963     then show ?thesis using * by simp
```
```  1964   qed
```
```  1965 qed
```
```  1966
```
```  1967 lemma log_le_cancel_iff [simp]:
```
```  1968   "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (log a x \<le> log a y) = (x \<le> y)"
```
```  1969   by (simp add: linorder_not_less [symmetric])
```
```  1970
```
```  1971 lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
```
```  1972   using log_less_cancel_iff[of a 1 x] by simp
```
```  1973
```
```  1974 lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
```
```  1975   using log_le_cancel_iff[of a 1 x] by simp
```
```  1976
```
```  1977 lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
```
```  1978   using log_less_cancel_iff[of a x 1] by simp
```
```  1979
```
```  1980 lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
```
```  1981   using log_le_cancel_iff[of a x 1] by simp
```
```  1982
```
```  1983 lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
```
```  1984   using log_less_cancel_iff[of a a x] by simp
```
```  1985
```
```  1986 lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
```
```  1987   using log_le_cancel_iff[of a a x] by simp
```
```  1988
```
```  1989 lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
```
```  1990   using log_less_cancel_iff[of a x a] by simp
```
```  1991
```
```  1992 lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
```
```  1993   using log_le_cancel_iff[of a x a] by simp
```
```  1994
```
```  1995 lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
```
```  1996   apply (induct n)
```
```  1997   apply simp
```
```  1998   apply (subgoal_tac "real(Suc n) = real n + 1")
```
```  1999   apply (erule ssubst)
```
```  2000   apply (subst powr_add, simp, simp)
```
```  2001   done
```
```  2002
```
```  2003 lemma powr_realpow_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
```
```  2004   unfolding real_of_nat_numeral [symmetric] by (rule powr_realpow)
```
```  2005
```
```  2006 lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
```
```  2007   apply (case_tac "x = 0", simp, simp)
```
```  2008   apply (rule powr_realpow [THEN sym], simp)
```
```  2009   done
```
```  2010
```
```  2011 lemma powr_int:
```
```  2012   assumes "x > 0"
```
```  2013   shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
```
```  2014 proof (cases "i < 0")
```
```  2015   case True
```
```  2016   have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
```
```  2017   show ?thesis using `i < 0` `x > 0` by (simp add: r field_simps powr_realpow[symmetric])
```
```  2018 next
```
```  2019   case False
```
```  2020   then show ?thesis by (simp add: assms powr_realpow[symmetric])
```
```  2021 qed
```
```  2022
```
```  2023 lemma powr_one: "0 < x \<Longrightarrow> x powr 1 = x"
```
```  2024   using powr_realpow [of x 1] by simp
```
```  2025
```
```  2026 lemma powr_numeral: "0 < x \<Longrightarrow> x powr numeral n = x ^ numeral n"
```
```  2027   by (fact powr_realpow_numeral)
```
```  2028
```
```  2029 lemma powr_neg_one: "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
```
```  2030   using powr_int [of x "- 1"] by simp
```
```  2031
```
```  2032 lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
```
```  2033   using powr_int [of x "- numeral n"] by simp
```
```  2034
```
```  2035 lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
```
```  2036   by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
```
```  2037
```
```  2038 lemma ln_powr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
```
```  2039   unfolding powr_def by simp
```
```  2040
```
```  2041 lemma log_powr: "0 < x ==> 0 \<le> y ==> log b (x powr y) = y * log b x"
```
```  2042   apply (cases "y = 0")
```
```  2043   apply force
```
```  2044   apply (auto simp add: log_def ln_powr field_simps)
```
```  2045   done
```
```  2046
```
```  2047 lemma log_nat_power: "0 < x ==> log b (x^n) = real n * log b x"
```
```  2048   apply (subst powr_realpow [symmetric])
```
```  2049   apply (auto simp add: log_powr)
```
```  2050   done
```
```  2051
```
```  2052 lemma ln_bound: "1 <= x ==> ln x <= x"
```
```  2053   apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
```
```  2054   apply simp
```
```  2055   apply (rule ln_add_one_self_le_self, simp)
```
```  2056   done
```
```  2057
```
```  2058 lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
```
```  2059   apply (cases "x = 1", simp)
```
```  2060   apply (cases "a = b", simp)
```
```  2061   apply (rule order_less_imp_le)
```
```  2062   apply (rule powr_less_mono, auto)
```
```  2063   done
```
```  2064
```
```  2065 lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
```
```  2066   apply (subst powr_zero_eq_one [THEN sym])
```
```  2067   apply (rule powr_mono, assumption+)
```
```  2068   done
```
```  2069
```
```  2070 lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a < y powr a"
```
```  2071   apply (unfold powr_def)
```
```  2072   apply (rule exp_less_mono)
```
```  2073   apply (rule mult_strict_left_mono)
```
```  2074   apply (subst ln_less_cancel_iff, assumption)
```
```  2075   apply (rule order_less_trans)
```
```  2076   prefer 2
```
```  2077   apply assumption+
```
```  2078   done
```
```  2079
```
```  2080 lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a < x powr a"
```
```  2081   apply (unfold powr_def)
```
```  2082   apply (rule exp_less_mono)
```
```  2083   apply (rule mult_strict_left_mono_neg)
```
```  2084   apply (subst ln_less_cancel_iff)
```
```  2085   apply assumption
```
```  2086   apply (rule order_less_trans)
```
```  2087   prefer 2
```
```  2088   apply assumption+
```
```  2089   done
```
```  2090
```
```  2091 lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
```
```  2092   apply (case_tac "a = 0", simp)
```
```  2093   apply (case_tac "x = y", simp)
```
```  2094   apply (rule order_less_imp_le)
```
```  2095   apply (rule powr_less_mono2, auto)
```
```  2096   done
```
```  2097
```
```  2098 lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
```
```  2099   unfolding powr_def exp_inj_iff by simp
```
```  2100
```
```  2101 lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
```
```  2102   apply (rule mult_imp_le_div_pos)
```
```  2103   apply (assumption)
```
```  2104   apply (subst mult_commute)
```
```  2105   apply (subst ln_powr [THEN sym])
```
```  2106   apply auto
```
```  2107   apply (rule ln_bound)
```
```  2108   apply (erule ge_one_powr_ge_zero)
```
```  2109   apply (erule order_less_imp_le)
```
```  2110   done
```
```  2111
```
```  2112 lemma ln_powr_bound2:
```
```  2113   assumes "1 < x" and "0 < a"
```
```  2114   shows "(ln x) powr a <= (a powr a) * x"
```
```  2115 proof -
```
```  2116   from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
```
```  2117     apply (intro ln_powr_bound)
```
```  2118     apply (erule order_less_imp_le)
```
```  2119     apply (rule divide_pos_pos)
```
```  2120     apply simp_all
```
```  2121     done
```
```  2122   also have "... = a * (x powr (1 / a))"
```
```  2123     by simp
```
```  2124   finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
```
```  2125     apply (intro powr_mono2)
```
```  2126     apply (rule order_less_imp_le, rule assms)
```
```  2127     apply (rule ln_gt_zero)
```
```  2128     apply (rule assms)
```
```  2129     apply assumption
```
```  2130     done
```
```  2131   also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
```
```  2132     apply (rule powr_mult)
```
```  2133     apply (rule assms)
```
```  2134     apply (rule powr_gt_zero)
```
```  2135     done
```
```  2136   also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
```
```  2137     by (rule powr_powr)
```
```  2138   also have "... = x"
```
```  2139     apply simp
```
```  2140     apply (subgoal_tac "a ~= 0")
```
```  2141     using assms apply auto
```
```  2142     done
```
```  2143   finally show ?thesis .
```
```  2144 qed
```
```  2145
```
```  2146 lemma tendsto_powr [tendsto_intros]:
```
```  2147   "\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x powr g x) ---> a powr b) F"
```
```  2148   unfolding powr_def by (intro tendsto_intros)
```
```  2149
```
```  2150 lemma continuous_powr:
```
```  2151   assumes "continuous F f"
```
```  2152     and "continuous F g"
```
```  2153     and "0 < f (Lim F (\<lambda>x. x))"
```
```  2154   shows "continuous F (\<lambda>x. (f x) powr (g x))"
```
```  2155   using assms unfolding continuous_def by (rule tendsto_powr)
```
```  2156
```
```  2157 lemma continuous_at_within_powr[continuous_intros]:
```
```  2158   assumes "continuous (at a within s) f"
```
```  2159     and "continuous (at a within s) g"
```
```  2160     and "0 < f a"
```
```  2161   shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
```
```  2162   using assms unfolding continuous_within by (rule tendsto_powr)
```
```  2163
```
```  2164 lemma isCont_powr[continuous_intros, simp]:
```
```  2165   assumes "isCont f a" "isCont g a" "0 < f a"
```
```  2166   shows "isCont (\<lambda>x. (f x) powr g x) a"
```
```  2167   using assms unfolding continuous_at by (rule tendsto_powr)
```
```  2168
```
```  2169 lemma continuous_on_powr[continuous_on_intros]:
```
```  2170   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. 0 < f x"
```
```  2171   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
```
```  2172   using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
```
```  2173
```
```  2174 (* FIXME: generalize by replacing d by with g x and g ---> d? *)
```
```  2175 lemma tendsto_zero_powrI:
```
```  2176   assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> 0) F"
```
```  2177     and "0 < d"
```
```  2178   shows "((\<lambda>x. f x powr d) ---> 0) F"
```
```  2179 proof (rule tendstoI)
```
```  2180   fix e :: real assume "0 < e"
```
```  2181   def Z \<equiv> "e powr (1 / d)"
```
```  2182   with `0 < e` have "0 < Z" by simp
```
```  2183   with assms have "eventually (\<lambda>x. 0 < f x \<and> dist (f x) 0 < Z) F"
```
```  2184     by (intro eventually_conj tendstoD)
```
```  2185   moreover
```
```  2186   from assms have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> x powr d < Z powr d"
```
```  2187     by (intro powr_less_mono2) (auto simp: dist_real_def)
```
```  2188   with assms `0 < e` have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> dist (x powr d) 0 < e"
```
```  2189     unfolding dist_real_def Z_def by (auto simp: powr_powr)
```
```  2190   ultimately
```
```  2191   show "eventually (\<lambda>x. dist (f x powr d) 0 < e) F" by (rule eventually_elim1)
```
```  2192 qed
```
```  2193
```
```  2194 lemma tendsto_neg_powr:
```
```  2195   assumes "s < 0"
```
```  2196     and "LIM x F. f x :> at_top"
```
```  2197   shows "((\<lambda>x. f x powr s) ---> 0) F"
```
```  2198 proof (rule tendstoI)
```
```  2199   fix e :: real assume "0 < e"
```
```  2200   def Z \<equiv> "e powr (1 / s)"
```
```  2201   from assms have "eventually (\<lambda>x. Z < f x) F"
```
```  2202     by (simp add: filterlim_at_top_dense)
```
```  2203   moreover
```
```  2204   from assms have "\<And>x. Z < x \<Longrightarrow> x powr s < Z powr s"
```
```  2205     by (auto simp: Z_def intro!: powr_less_mono2_neg)
```
```  2206   with assms `0 < e` have "\<And>x. Z < x \<Longrightarrow> dist (x powr s) 0 < e"
```
```  2207     by (simp add: powr_powr Z_def dist_real_def)
```
```  2208   ultimately
```
```  2209   show "eventually (\<lambda>x. dist (f x powr s) 0 < e) F" by (rule eventually_elim1)
```
```  2210 qed
```
```  2211
```
```  2212 subsection {* Sine and Cosine *}
```
```  2213
```
```  2214 definition sin_coeff :: "nat \<Rightarrow> real" where
```
```  2215   "sin_coeff = (\<lambda>n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"
```
```  2216
```
```  2217 definition cos_coeff :: "nat \<Rightarrow> real" where
```
```  2218   "cos_coeff = (\<lambda>n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"
```
```  2219
```
```  2220 definition sin :: "real \<Rightarrow> real"
```
```  2221   where "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
```
```  2222
```
```  2223 definition cos :: "real \<Rightarrow> real"
```
```  2224   where "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
```
```  2225
```
```  2226 lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
```
```  2227   unfolding sin_coeff_def by simp
```
```  2228
```
```  2229 lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
```
```  2230   unfolding cos_coeff_def by simp
```
```  2231
```
```  2232 lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
```
```  2233   unfolding cos_coeff_def sin_coeff_def
```
```  2234   by (simp del: mult_Suc)
```
```  2235
```
```  2236 lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
```
```  2237   unfolding cos_coeff_def sin_coeff_def
```
```  2238   by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)
```
```  2239
```
```  2240 lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
```
```  2241   unfolding sin_coeff_def
```
```  2242   apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
```
```  2243   apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2244   done
```
```  2245
```
```  2246 lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"
```
```  2247   unfolding cos_coeff_def
```
```  2248   apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
```
```  2249   apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2250   done
```
```  2251
```
```  2252 lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
```
```  2253   unfolding sin_def by (rule summable_sin [THEN summable_sums])
```
```  2254
```
```  2255 lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
```
```  2256   unfolding cos_def by (rule summable_cos [THEN summable_sums])
```
```  2257
```
```  2258 lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
```
```  2259   by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  2260
```
```  2261 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
```
```  2262   by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  2263
```
```  2264 text{*Now at last we can get the derivatives of exp, sin and cos*}
```
```  2265
```
```  2266 lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
```
```  2267   unfolding sin_def cos_def
```
```  2268   apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
```
```  2269   apply (simp_all add: diffs_sin_coeff diffs_cos_coeff
```
```  2270     summable_minus summable_sin summable_cos)
```
```  2271   done
```
```  2272
```
```  2273 declare DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  2274
```
```  2275 lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
```
```  2276   unfolding cos_def sin_def
```
```  2277   apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
```
```  2278   apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus
```
```  2279     summable_minus summable_sin summable_cos suminf_minus)
```
```  2280   done
```
```  2281
```
```  2282 declare DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  2283
```
```  2284 lemma isCont_sin: "isCont sin x"
```
```  2285   by (rule DERIV_sin [THEN DERIV_isCont])
```
```  2286
```
```  2287 lemma isCont_cos: "isCont cos x"
```
```  2288   by (rule DERIV_cos [THEN DERIV_isCont])
```
```  2289
```
```  2290 lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
```
```  2291   by (rule isCont_o2 [OF _ isCont_sin])
```
```  2292
```
```  2293 lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
```
```  2294   by (rule isCont_o2 [OF _ isCont_cos])
```
```  2295
```
```  2296 lemma tendsto_sin [tendsto_intros]:
```
```  2297   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
```
```  2298   by (rule isCont_tendsto_compose [OF isCont_sin])
```
```  2299
```
```  2300 lemma tendsto_cos [tendsto_intros]:
```
```  2301   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
```
```  2302   by (rule isCont_tendsto_compose [OF isCont_cos])
```
```  2303
```
```  2304 lemma continuous_sin [continuous_intros]:
```
```  2305   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
```
```  2306   unfolding continuous_def by (rule tendsto_sin)
```
```  2307
```
```  2308 lemma continuous_on_sin [continuous_on_intros]:
```
```  2309   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
```
```  2310   unfolding continuous_on_def by (auto intro: tendsto_sin)
```
```  2311
```
```  2312 lemma continuous_cos [continuous_intros]:
```
```  2313   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
```
```  2314   unfolding continuous_def by (rule tendsto_cos)
```
```  2315
```
```  2316 lemma continuous_on_cos [continuous_on_intros]:
```
```  2317   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
```
```  2318   unfolding continuous_on_def by (auto intro: tendsto_cos)
```
```  2319
```
```  2320 subsection {* Properties of Sine and Cosine *}
```
```  2321
```
```  2322 lemma sin_zero [simp]: "sin 0 = 0"
```
```  2323   unfolding sin_def sin_coeff_def by (simp add: powser_zero)
```
```  2324
```
```  2325 lemma cos_zero [simp]: "cos 0 = 1"
```
```  2326   unfolding cos_def cos_coeff_def by (simp add: powser_zero)
```
```  2327
```
```  2328 lemma sin_cos_squared_add [simp]: "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
```
```  2329 proof -
```
```  2330   have "\<forall>x. DERIV (\<lambda>x. (sin x)\<^sup>2 + (cos x)\<^sup>2) x :> 0"
```
```  2331     by (auto intro!: DERIV_intros)
```
```  2332   hence "(sin x)\<^sup>2 + (cos x)\<^sup>2 = (sin 0)\<^sup>2 + (cos 0)\<^sup>2"
```
```  2333     by (rule DERIV_isconst_all)
```
```  2334   thus "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1" by simp
```
```  2335 qed
```
```  2336
```
```  2337 lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
```
```  2338   by (subst add_commute, rule sin_cos_squared_add)
```
```  2339
```
```  2340 lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
```
```  2341   using sin_cos_squared_add2 [unfolded power2_eq_square] .
```
```  2342
```
```  2343 lemma sin_squared_eq: "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
```
```  2344   unfolding eq_diff_eq by (rule sin_cos_squared_add)
```
```  2345
```
```  2346 lemma cos_squared_eq: "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
```
```  2347   unfolding eq_diff_eq by (rule sin_cos_squared_add2)
```
```  2348
```
```  2349 lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
```
```  2350   by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
```
```  2351
```
```  2352 lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
```
```  2353   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  2354
```
```  2355 lemma sin_le_one [simp]: "sin x \<le> 1"
```
```  2356   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  2357
```
```  2358 lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
```
```  2359   by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
```
```  2360
```
```  2361 lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
```
```  2362   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  2363
```
```  2364 lemma cos_le_one [simp]: "cos x \<le> 1"
```
```  2365   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  2366
```
```  2367 lemma DERIV_fun_pow: "DERIV g x :> m ==>
```
```  2368       DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
```
```  2369   by (auto intro!: DERIV_intros)
```
```  2370
```
```  2371 lemma DERIV_fun_exp:
```
```  2372      "DERIV g x :> m ==> DERIV (\<lambda>x. exp(g x)) x :> exp(g x) * m"
```
```  2373   by (auto intro!: DERIV_intros)
```
```  2374
```
```  2375 lemma DERIV_fun_sin:
```
```  2376      "DERIV g x :> m ==> DERIV (\<lambda>x. sin(g x)) x :> cos(g x) * m"
```
```  2377   by (auto intro!: DERIV_intros)
```
```  2378
```
```  2379 lemma DERIV_fun_cos:
```
```  2380      "DERIV g x :> m ==> DERIV (\<lambda>x. cos(g x)) x :> -sin(g x) * m"
```
```  2381   by (auto intro!: DERIV_intros)
```
```  2382
```
```  2383 lemma sin_cos_add_lemma:
```
```  2384   "(sin (x + y) - (sin x * cos y + cos x * sin y))\<^sup>2 +
```
```  2385     (cos (x + y) - (cos x * cos y - sin x * sin y))\<^sup>2 = 0"
```
```  2386   (is "?f x = 0")
```
```  2387 proof -
```
```  2388   have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
```
```  2389     by (auto intro!: DERIV_intros simp add: algebra_simps)
```
```  2390   hence "?f x = ?f 0"
```
```  2391     by (rule DERIV_isconst_all)
```
```  2392   thus ?thesis by simp
```
```  2393 qed
```
```  2394
```
```  2395 lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
```
```  2396   using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
```
```  2397
```
```  2398 lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
```
```  2399   using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
```
```  2400
```
```  2401 lemma sin_cos_minus_lemma:
```
```  2402   "(sin(-x) + sin(x))\<^sup>2 + (cos(-x) - cos(x))\<^sup>2 = 0" (is "?f x = 0")
```
```  2403 proof -
```
```  2404   have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
```
```  2405     by (auto intro!: DERIV_intros simp add: algebra_simps)
```
```  2406   hence "?f x = ?f 0"
```
```  2407     by (rule DERIV_isconst_all)
```
```  2408   thus ?thesis by simp
```
```  2409 qed
```
```  2410
```
```  2411 lemma sin_minus [simp]: "sin (-x) = -sin(x)"
```
```  2412   using sin_cos_minus_lemma [where x=x] by simp
```
```  2413
```
```  2414 lemma cos_minus [simp]: "cos (-x) = cos(x)"
```
```  2415   using sin_cos_minus_lemma [where x=x] by simp
```
```  2416
```
```  2417 lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
```
```  2418   using sin_add [of x "- y"] by simp
```
```  2419
```
```  2420 lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
```
```  2421   by (simp add: sin_diff mult_commute)
```
```  2422
```
```  2423 lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
```
```  2424   using cos_add [of x "- y"] by simp
```
```  2425
```
```  2426 lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
```
```  2427   by (simp add: cos_diff mult_commute)
```
```  2428
```
```  2429 lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
```
```  2430   using sin_add [where x=x and y=x] by simp
```
```  2431
```
```  2432 lemma cos_double: "cos(2* x) = ((cos x)\<^sup>2) - ((sin x)\<^sup>2)"
```
```  2433   using cos_add [where x=x and y=x]
```
```  2434   by (simp add: power2_eq_square)
```
```  2435
```
```  2436
```
```  2437 subsection {* The Constant Pi *}
```
```  2438
```
```  2439 definition pi :: real
```
```  2440   where "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
```
```  2441
```
```  2442 text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
```
```  2443    hence define pi.*}
```
```  2444
```
```  2445 lemma sin_paired:
```
```  2446   "(\<lambda>n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) sums  sin x"
```
```  2447 proof -
```
```  2448   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
```
```  2449     by (rule sin_converges [THEN sums_group], simp)
```
```  2450   thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)
```
```  2451 qed
```
```  2452
```
```  2453 lemma sin_gt_zero:
```
```  2454   assumes "0 < x" and "x < 2"
```
```  2455   shows "0 < sin x"
```
```  2456 proof -
```
```  2457   let ?f = "\<lambda>n. \<Sum>k = n*2..<n*2+2. -1 ^ k / real (fact (2*k+1)) * x^(2*k+1)"
```
```  2458   have pos: "\<forall>n. 0 < ?f n"
```
```  2459   proof
```
```  2460     fix n :: nat
```
```  2461     let ?k2 = "real (Suc (Suc (4 * n)))"
```
```  2462     let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
```
```  2463     have "x * x < ?k2 * ?k3"
```
```  2464       using assms by (intro mult_strict_mono', simp_all)
```
```  2465     hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
```
```  2466       by (intro mult_strict_right_mono zero_less_power `0 < x`)
```
```  2467     thus "0 < ?f n"
```
```  2468       by (simp del: mult_Suc,
```
```  2469         simp add: less_divide_eq mult_pos_pos field_simps del: mult_Suc)
```
```  2470   qed
```
```  2471   have sums: "?f sums sin x"
```
```  2472     by (rule sin_paired [THEN sums_group], simp)
```
```  2473   show "0 < sin x"
```
```  2474     unfolding sums_unique [OF sums]
```
```  2475     using sums_summable [OF sums] pos
```
```  2476     by (rule suminf_gt_zero)
```
```  2477 qed
```
```  2478
```
```  2479 lemma cos_double_less_one: "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
```
```  2480   using sin_gt_zero [where x = x] by (auto simp add: cos_squared_eq cos_double)
```
```  2481
```
```  2482 lemma cos_paired: "(\<lambda>n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
```
```  2483 proof -
```
```  2484   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
```
```  2485     by (rule cos_converges [THEN sums_group], simp)
```
```  2486   thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)
```
```  2487 qed
```
```  2488
```
```  2489 lemma real_mult_inverse_cancel:
```
```  2490      "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
```
```  2491       ==> inverse x * y < inverse x1 * u"
```
```  2492   apply (rule_tac c=x in mult_less_imp_less_left)
```
```  2493   apply (auto simp add: mult_assoc [symmetric])
```
```  2494   apply (simp (no_asm) add: mult_ac)
```
```  2495   apply (rule_tac c=x1 in mult_less_imp_less_right)
```
```  2496   apply (auto simp add: mult_ac)
```
```  2497   done
```
```  2498
```
```  2499 lemma real_mult_inverse_cancel2:
```
```  2500      "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
```
```  2501   by (auto dest: real_mult_inverse_cancel simp add: mult_ac)
```
```  2502
```
```  2503 lemmas realpow_num_eq_if = power_eq_if
```
```  2504
```
```  2505 lemma cos_two_less_zero [simp]:
```
```  2506   "cos 2 < 0"
```
```  2507 proof -
```
```  2508   note fact_Suc [simp del]
```
```  2509   from cos_paired
```
```  2510   have "(\<lambda>n. - (-1 ^ n / real (fact (2 * n)) * 2 ^ (2 * n))) sums - cos 2"
```
```  2511     by (rule sums_minus)
```
```  2512   then have *: "(\<lambda>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n)))) sums - cos 2"
```
```  2513     by simp
```
```  2514   then have **: "summable (\<lambda>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2515     by (rule sums_summable)
```
```  2516   have "0 < (\<Sum>n = 0..<Suc (Suc (Suc 0)). - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2517     by (simp add: fact_num_eq_if_nat realpow_num_eq_if)
```
```  2518   moreover have "(\<Sum>n = 0..<Suc (Suc (Suc 0)). - (-1 ^ n  * 2 ^ (2 * n) / real (fact (2 * n))))
```
```  2519     < (\<Sum>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2520   proof -
```
```  2521     { fix d
```
```  2522       have "4 * real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))
```
```  2523        < real (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))) *
```
```  2524            fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))"
```
```  2525         by (simp only: real_of_nat_mult) (auto intro!: mult_strict_mono fact_less_mono_nat)
```
```  2526       then have "4 * real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))
```
```  2527         < real (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))))"
```
```  2528         by (simp only: fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
```
```  2529       then have "4 * inverse (real (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))))
```
```  2530         < inverse (real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))"
```
```  2531         by (simp add: inverse_eq_divide less_divide_eq)
```
```  2532     }
```
```  2533     note *** = this
```
```  2534     have [simp]: "\<And>x y::real. 0 < x - y \<longleftrightarrow> y < x" by arith
```
```  2535     from ** show ?thesis by (rule sumr_pos_lt_pair)
```
```  2536       (simp add: divide_inverse mult_assoc [symmetric] ***)
```
```  2537   qed
```
```  2538   ultimately have "0 < (\<Sum>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2539     by (rule order_less_trans)
```
```  2540   moreover from * have "- cos 2 = (\<Sum>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2541     by (rule sums_unique)
```
```  2542   ultimately have "0 < - cos 2" by simp
```
```  2543   then show ?thesis by simp
```
```  2544 qed
```
```  2545
```
```  2546 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
```
```  2547 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
```
```  2548
```
```  2549 lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 \<and> cos x = 0"
```
```  2550 proof (rule ex_ex1I)
```
```  2551   show "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0"
```
```  2552     by (rule IVT2, simp_all)
```
```  2553 next
```
```  2554   fix x y
```
```  2555   assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
```
```  2556   assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
```
```  2557   have [simp]: "\<forall>x. cos differentiable x"
```
```  2558     unfolding differentiable_def by (auto intro: DERIV_cos)
```
```  2559   from x y show "x = y"
```
```  2560     apply (cut_tac less_linear [of x y], auto)
```
```  2561     apply (drule_tac f = cos in Rolle)
```
```  2562     apply (drule_tac [5] f = cos in Rolle)
```
```  2563     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  2564     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  2565     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  2566     done
```
```  2567 qed
```
```  2568
```
```  2569 lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
```
```  2570   by (simp add: pi_def)
```
```  2571
```
```  2572 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
```
```  2573   by (simp add: pi_half cos_is_zero [THEN theI'])
```
```  2574
```
```  2575 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
```
```  2576   apply (rule order_le_neq_trans)
```
```  2577   apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  2578   apply (rule notI, drule arg_cong [where f=cos], simp)
```
```  2579   done
```
```  2580
```
```  2581 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
```
```  2582 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
```
```  2583
```
```  2584 lemma pi_half_less_two [simp]: "pi / 2 < 2"
```
```  2585   apply (rule order_le_neq_trans)
```
```  2586   apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  2587   apply (rule notI, drule arg_cong [where f=cos], simp)
```
```  2588   done
```
```  2589
```
```  2590 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
```
```  2591 lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
```
```  2592
```
```  2593 lemma pi_gt_zero [simp]: "0 < pi"
```
```  2594   using pi_half_gt_zero by simp
```
```  2595
```
```  2596 lemma pi_ge_zero [simp]: "0 \<le> pi"
```
```  2597   by (rule pi_gt_zero [THEN order_less_imp_le])
```
```  2598
```
```  2599 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
```
```  2600   by (rule pi_gt_zero [THEN less_imp_neq, symmetric])
```
```  2601
```
```  2602 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
```
```  2603   by (simp add: linorder_not_less)
```
```  2604
```
```  2605 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
```
```  2606   by simp
```
```  2607
```
```  2608 lemma m2pi_less_pi: "- (2 * pi) < pi"
```
```  2609   by simp
```
```  2610
```
```  2611 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
```
```  2612   using sin_cos_squared_add2 [where x = "pi/2"]
```
```  2613   using sin_gt_zero [OF pi_half_gt_zero pi_half_less_two]
```
```  2614   by (simp add: power2_eq_1_iff)
```
```  2615
```
```  2616 lemma cos_pi [simp]: "cos pi = -1"
```
```  2617   using cos_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  2618
```
```  2619 lemma sin_pi [simp]: "sin pi = 0"
```
```  2620   using sin_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  2621
```
```  2622 lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
```
```  2623   by (simp add: cos_diff)
```
```  2624
```
```  2625 lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
```
```  2626   by (simp add: cos_add)
```
```  2627
```
```  2628 lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
```
```  2629   by (simp add: sin_diff)
```
```  2630
```
```  2631 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
```
```  2632   by (simp add: sin_add)
```
```  2633
```
```  2634 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
```
```  2635   by (simp add: sin_add)
```
```  2636
```
```  2637 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
```
```  2638   by (simp add: cos_add)
```
```  2639
```
```  2640 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
```
```  2641   by (simp add: sin_add cos_double)
```
```  2642
```
```  2643 lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
```
```  2644   by (simp add: cos_add cos_double)
```
```  2645
```
```  2646 lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
```
```  2647   by (induct n) (auto simp add: real_of_nat_Suc distrib_right)
```
```  2648
```
```  2649 lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
```
```  2650 proof -
```
```  2651   have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
```
```  2652   also have "... = -1 ^ n" by (rule cos_npi)
```
```  2653   finally show ?thesis .
```
```  2654 qed
```
```  2655
```
```  2656 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
```
```  2657   by (induct n) (auto simp add: real_of_nat_Suc distrib_right)
```
```  2658
```
```  2659 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
```
```  2660   by (simp add: mult_commute [of pi])
```
```  2661
```
```  2662 lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
```
```  2663   by (simp add: cos_double)
```
```  2664
```
```  2665 lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
```
```  2666   by simp
```
```  2667
```
```  2668 lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
```
```  2669   apply (rule sin_gt_zero, assumption)
```
```  2670   apply (rule order_less_trans, assumption)
```
```  2671   apply (rule pi_half_less_two)
```
```  2672   done
```
```  2673
```
```  2674 lemma sin_less_zero:
```
```  2675   assumes "- pi/2 < x" and "x < 0"
```
```  2676   shows "sin x < 0"
```
```  2677 proof -
```
```  2678   have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
```
```  2679   thus ?thesis by simp
```
```  2680 qed
```
```  2681
```
```  2682 lemma pi_less_4: "pi < 4"
```
```  2683   using pi_half_less_two by auto
```
```  2684
```
```  2685 lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
```
```  2686   apply (cut_tac pi_less_4)
```
```  2687   apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
```
```  2688   apply (cut_tac cos_is_zero, safe)
```
```  2689   apply (rename_tac y z)
```
```  2690   apply (drule_tac x = y in spec)
```
```  2691   apply (drule_tac x = "pi/2" in spec, simp)
```
```  2692   done
```
```  2693
```
```  2694 lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
```
```  2695   apply (rule_tac x = x and y = 0 in linorder_cases)
```
```  2696   apply (rule cos_minus [THEN subst])
```
```  2697   apply (rule cos_gt_zero)
```
```  2698   apply (auto intro: cos_gt_zero)
```
```  2699   done
```
```  2700
```
```  2701 lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
```
```  2702   apply (auto simp add: order_le_less cos_gt_zero_pi)
```
```  2703   apply (subgoal_tac "x = pi/2", auto)
```
```  2704   done
```
```  2705
```
```  2706 lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
```
```  2707   by (simp add: sin_cos_eq cos_gt_zero_pi)
```
```  2708
```
```  2709 lemma pi_ge_two: "2 \<le> pi"
```
```  2710 proof (rule ccontr)
```
```  2711   assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
```
```  2712   have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
```
```  2713   proof (cases "2 < 2 * pi")
```
```  2714     case True with dense[OF `pi < 2`] show ?thesis by auto
```
```  2715   next
```
```  2716     case False have "pi < 2 * pi" by auto
```
```  2717     from dense[OF this] and False show ?thesis by auto
```
```  2718   qed
```
```  2719   then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
```
```  2720   hence "0 < sin y" using sin_gt_zero by auto
```
```  2721   moreover
```
```  2722   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
```
```  2723   ultimately show False by auto
```
```  2724 qed
```
```  2725
```
```  2726 lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
```
```  2727   by (auto simp add: order_le_less sin_gt_zero_pi)
```
```  2728
```
```  2729 text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}.
```
```  2730   It should be possible to factor out some of the common parts. *}
```
```  2731
```
```  2732 lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
```
```  2733 proof (rule ex_ex1I)
```
```  2734   assume y: "-1 \<le> y" "y \<le> 1"
```
```  2735   show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
```
```  2736     by (rule IVT2, simp_all add: y)
```
```  2737 next
```
```  2738   fix a b
```
```  2739   assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
```
```  2740   assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
```
```  2741   have [simp]: "\<forall>x. cos differentiable x"
```
```  2742     unfolding differentiable_def by (auto intro: DERIV_cos)
```
```  2743   from a b show "a = b"
```
```  2744     apply (cut_tac less_linear [of a b], auto)
```
```  2745     apply (drule_tac f = cos in Rolle)
```
```  2746     apply (drule_tac [5] f = cos in Rolle)
```
```  2747     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  2748     apply (metis order_less_le_trans less_le sin_gt_zero_pi)
```
```  2749     apply (metis order_less_le_trans less_le sin_gt_zero_pi)
```
```  2750     done
```
```  2751 qed
```
```  2752
```
```  2753 lemma sin_total:
```
```  2754      "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
```
```  2755 apply (rule ccontr)
```
```  2756 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))")
```
```  2757 apply (erule contrapos_np)
```
```  2758 apply simp
```
```  2759 apply (cut_tac y="-y" in cos_total, simp) apply simp
```
```  2760 apply (erule ex1E)
```
```  2761 apply (rule_tac a = "x - (pi/2)" in ex1I)
```
```  2762 apply (simp (no_asm) add: add_assoc)
```
```  2763 apply (rotate_tac 3)
```
```  2764 apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all add: cos_add)
```
```  2765 done
```
```  2766
```
```  2767 lemma reals_Archimedean4:
```
```  2768      "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
```
```  2769 apply (auto dest!: reals_Archimedean3)
```
```  2770 apply (drule_tac x = x in spec, clarify)
```
```  2771 apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
```
```  2772  prefer 2 apply (erule LeastI)
```
```  2773 apply (case_tac "LEAST m::nat. x < real m * y", simp)
```
```  2774 apply (subgoal_tac "~ x < real nat * y")
```
```  2775  prefer 2 apply (rule not_less_Least, simp, force)
```
```  2776 done
```
```  2777
```
```  2778 (* Pre Isabelle99-2 proof was simpler- numerals arithmetic
```
```  2779    now causes some unwanted re-arrangements of literals!   *)
```
```  2780 lemma cos_zero_lemma:
```
```  2781      "[| 0 \<le> x; cos x = 0 |] ==>
```
```  2782       \<exists>n::nat. ~even n & x = real n * (pi/2)"
```
```  2783 apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
```
```  2784 apply (subgoal_tac "0 \<le> x - real n * pi &
```
```  2785                     (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
```
```  2786 apply (auto simp add: algebra_simps real_of_nat_Suc)
```
```  2787  prefer 2 apply (simp add: cos_diff)
```
```  2788 apply (simp add: cos_diff)
```
```  2789 apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
```
```  2790 apply (rule_tac [2] cos_total, safe)
```
```  2791 apply (drule_tac x = "x - real n * pi" in spec)
```
```  2792 apply (drule_tac x = "pi/2" in spec)
```
```  2793 apply (simp add: cos_diff)
```
```  2794 apply (rule_tac x = "Suc (2 * n)" in exI)
```
```  2795 apply (simp add: real_of_nat_Suc algebra_simps, auto)
```
```  2796 done
```
```  2797
```
```  2798 lemma sin_zero_lemma:
```
```  2799      "[| 0 \<le> x; sin x = 0 |] ==>
```
```  2800       \<exists>n::nat. even n & x = real n * (pi/2)"
```
```  2801 apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
```
```  2802  apply (clarify, rule_tac x = "n - 1" in exI)
```
```  2803  apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc distrib_right)
```
```  2804 apply (rule cos_zero_lemma)
```
```  2805 apply (simp_all add: cos_add)
```
```  2806 done
```
```  2807
```
```  2808
```
```  2809 lemma cos_zero_iff:
```
```  2810      "(cos x = 0) =
```
```  2811       ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
```
```  2812        (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
```
```  2813 apply (rule iffI)
```
```  2814 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  2815 apply (drule cos_zero_lemma, assumption+)
```
```  2816 apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
```
```  2817 apply (force simp add: minus_equation_iff [of x])
```
```  2818 apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc distrib_right)
```
```  2819 apply (auto simp add: cos_diff cos_add)
```
```  2820 done
```
```  2821
```
```  2822 (* ditto: but to a lesser extent *)
```
```  2823 lemma sin_zero_iff:
```
```  2824      "(sin x = 0) =
```
```  2825       ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
```
```  2826        (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
```
```  2827 apply (rule iffI)
```
```  2828 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  2829 apply (drule sin_zero_lemma, assumption+)
```
```  2830 apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
```
```  2831 apply (force simp add: minus_equation_iff [of x])
```
```  2832 apply (auto simp add: even_mult_two_ex)
```
```  2833 done
```
```  2834
```
```  2835 lemma cos_monotone_0_pi:
```
```  2836   assumes "0 \<le> y" and "y < x" and "x \<le> pi"
```
```  2837   shows "cos x < cos y"
```
```  2838 proof -
```
```  2839   have "- (x - y) < 0" using assms by auto
```
```  2840
```
```  2841   from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
```
```  2842   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z"
```
```  2843     by auto
```
```  2844   hence "0 < z" and "z < pi" using assms by auto
```
```  2845   hence "0 < sin z" using sin_gt_zero_pi by auto
```
```  2846   hence "cos x - cos y < 0"
```
```  2847     unfolding cos_diff minus_mult_commute[symmetric]
```
```  2848     using `- (x - y) < 0` by (rule mult_pos_neg2)
```
```  2849   thus ?thesis by auto
```
```  2850 qed
```
```  2851
```
```  2852 lemma cos_monotone_0_pi':
```
```  2853   assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi"
```
```  2854   shows "cos x \<le> cos y"
```
```  2855 proof (cases "y < x")
```
```  2856   case True
```
```  2857   show ?thesis
```
```  2858     using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
```
```  2859 next
```
```  2860   case False
```
```  2861   hence "y = x" using `y \<le> x` by auto
```
```  2862   thus ?thesis by auto
```
```  2863 qed
```
```  2864
```
```  2865 lemma cos_monotone_minus_pi_0:
```
```  2866   assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
```
```  2867   shows "cos y < cos x"
```
```  2868 proof -
```
```  2869   have "0 \<le> -x" and "-x < -y" and "-y \<le> pi"
```
```  2870     using assms by auto
```
```  2871   from cos_monotone_0_pi[OF this] show ?thesis
```
```  2872     unfolding cos_minus .
```
```  2873 qed
```
```  2874
```
```  2875 lemma cos_monotone_minus_pi_0':
```
```  2876   assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0"
```
```  2877   shows "cos y \<le> cos x"
```
```  2878 proof (cases "y < x")
```
```  2879   case True
```
```  2880   show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`]
```
```  2881     by auto
```
```  2882 next
```
```  2883   case False
```
```  2884   hence "y = x" using `y \<le> x` by auto
```
```  2885   thus ?thesis by auto
```
```  2886 qed
```
```  2887
```
```  2888 lemma sin_monotone_2pi':
```
```  2889   assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2"
```
```  2890   shows "sin y \<le> sin x"
```
```  2891 proof -
```
```  2892   have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi"
```
```  2893     using pi_ge_two and assms by auto
```
```  2894   from cos_monotone_0_pi'[OF this] show ?thesis
```
```  2895     unfolding minus_sin_cos_eq[symmetric] by auto
```
```  2896 qed
```
```  2897
```
```  2898
```
```  2899 subsection {* Tangent *}
```
```  2900
```
```  2901 definition tan :: "real \<Rightarrow> real"
```
```  2902   where "tan = (\<lambda>x. sin x / cos x)"
```
```  2903
```
```  2904 lemma tan_zero [simp]: "tan 0 = 0"
```
```  2905   by (simp add: tan_def)
```
```  2906
```
```  2907 lemma tan_pi [simp]: "tan pi = 0"
```
```  2908   by (simp add: tan_def)
```
```  2909
```
```  2910 lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
```
```  2911   by (simp add: tan_def)
```
```  2912
```
```  2913 lemma tan_minus [simp]: "tan (-x) = - tan x"
```
```  2914   by (simp add: tan_def)
```
```  2915
```
```  2916 lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
```
```  2917   by (simp add: tan_def)
```
```  2918
```
```  2919 lemma lemma_tan_add1:
```
```  2920   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
```
```  2921   by (simp add: tan_def cos_add field_simps)
```
```  2922
```
```  2923 lemma add_tan_eq:
```
```  2924   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
```
```  2925   by (simp add: tan_def sin_add field_simps)
```
```  2926
```
```  2927 lemma tan_add:
```
```  2928      "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
```
```  2929       ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
```
```  2930   by (simp add: add_tan_eq lemma_tan_add1, simp add: tan_def)
```
```  2931
```
```  2932 lemma tan_double:
```
```  2933      "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
```
```  2934       ==> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)"
```
```  2935   using tan_add [of x x] by (simp add: power2_eq_square)
```
```  2936
```
```  2937 lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
```
```  2938   by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
```
```  2939
```
```  2940 lemma tan_less_zero:
```
```  2941   assumes lb: "- pi/2 < x" and "x < 0"
```
```  2942   shows "tan x < 0"
```
```  2943 proof -
```
```  2944   have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
```
```  2945   thus ?thesis by simp
```
```  2946 qed
```
```  2947
```
```  2948 lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"
```
```  2949   unfolding tan_def sin_double cos_double sin_squared_eq
```
```  2950   by (simp add: power2_eq_square)
```
```  2951
```
```  2952 lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)"
```
```  2953   unfolding tan_def
```
```  2954   by (auto intro!: DERIV_intros, simp add: divide_inverse power2_eq_square)
```
```  2955
```
```  2956 lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
```
```  2957   by (rule DERIV_tan [THEN DERIV_isCont])
```
```  2958
```
```  2959 lemma isCont_tan' [simp]:
```
```  2960   "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
```
```  2961   by (rule isCont_o2 [OF _ isCont_tan])
```
```  2962
```
```  2963 lemma tendsto_tan [tendsto_intros]:
```
```  2964   "\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F"
```
```  2965   by (rule isCont_tendsto_compose [OF isCont_tan])
```
```  2966
```
```  2967 lemma continuous_tan:
```
```  2968   "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))"
```
```  2969   unfolding continuous_def by (rule tendsto_tan)
```
```  2970
```
```  2971 lemma isCont_tan'' [continuous_intros]:
```
```  2972   "continuous (at x) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. tan (f x))"
```
```  2973   unfolding continuous_at by (rule tendsto_tan)
```
```  2974
```
```  2975 lemma continuous_within_tan [continuous_intros]:
```
```  2976   "continuous (at x within s) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
```
```  2977   unfolding continuous_within by (rule tendsto_tan)
```
```  2978
```
```  2979 lemma continuous_on_tan [continuous_on_intros]:
```
```  2980   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))"
```
```  2981   unfolding continuous_on_def by (auto intro: tendsto_tan)
```
```  2982
```
```  2983 lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) -- pi/2 --> 0"
```
```  2984   by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
```
```  2985
```
```  2986 lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
```
```  2987   apply (cut_tac LIM_cos_div_sin)
```
```  2988   apply (simp only: LIM_eq)
```
```  2989   apply (drule_tac x = "inverse y" in spec, safe, force)
```
```  2990   apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
```
```  2991   apply (rule_tac x = "(pi/2) - e" in exI)
```
```  2992   apply (simp (no_asm_simp))
```
```  2993   apply (drule_tac x = "(pi/2) - e" in spec)
```
```  2994   apply (auto simp add: tan_def sin_diff cos_diff)
```
```  2995   apply (rule inverse_less_iff_less [THEN iffD1])
```
```  2996   apply (auto simp add: divide_inverse)
```
```  2997   apply (rule mult_pos_pos)
```
```  2998   apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
```
```  2999   apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
```
```  3000   done
```
```  3001
```
```  3002 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
```
```  3003   apply (frule order_le_imp_less_or_eq, safe)
```
```  3004    prefer 2 apply force
```
```  3005   apply (drule lemma_tan_total, safe)
```
```  3006   apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
```
```  3007   apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
```
```  3008   apply (drule_tac y = xa in order_le_imp_less_or_eq)
```
```  3009   apply (auto dest: cos_gt_zero)
```
```  3010   done
```
```  3011
```
```  3012 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  3013   apply (cut_tac linorder_linear [of 0 y], safe)
```
```  3014   apply (drule tan_total_pos)
```
```  3015   apply (cut_tac [2] y="-y" in tan_total_pos, safe)
```
```  3016   apply (rule_tac [3] x = "-x" in exI)
```
```  3017   apply (auto del: exI intro!: exI)
```
```  3018   done
```
```  3019
```
```  3020 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  3021   apply (cut_tac y = y in lemma_tan_total1, auto)
```
```  3022   apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
```
```  3023   apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
```
```  3024   apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
```
```  3025   apply (rule_tac [4] Rolle)
```
```  3026   apply (rule_tac [2] Rolle)
```
```  3027   apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
```
```  3028               simp add: differentiable_def)
```
```  3029   txt{*Now, simulate TRYALL*}
```
```  3030   apply (rule_tac [!] DERIV_tan asm_rl)
```
```  3031   apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
```
```  3032               simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
```
```  3033   done
```
```  3034
```
```  3035 lemma tan_monotone:
```
```  3036   assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
```
```  3037   shows "tan y < tan x"
```
```  3038 proof -
```
```  3039   have "\<forall>x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse ((cos x')\<^sup>2)"
```
```  3040   proof (rule allI, rule impI)
```
```  3041     fix x' :: real
```
```  3042     assume "y \<le> x' \<and> x' \<le> x"
```
```  3043     hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
```
```  3044     from cos_gt_zero_pi[OF this]
```
```  3045     have "cos x' \<noteq> 0" by auto
```
```  3046     thus "DERIV tan x' :> inverse ((cos x')\<^sup>2)" by (rule DERIV_tan)
```
```  3047   qed
```
```  3048   from MVT2[OF `y < x` this]
```
```  3049   obtain z where "y < z" and "z < x"
```
```  3050     and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
```
```  3051   hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
```
```  3052   hence "0 < cos z" using cos_gt_zero_pi by auto
```
```  3053   hence inv_pos: "0 < inverse ((cos z)\<^sup>2)" by auto
```
```  3054   have "0 < x - y" using `y < x` by auto
```
```  3055   from mult_pos_pos [OF this inv_pos]
```
```  3056   have "0 < tan x - tan y" unfolding tan_diff by auto
```
```  3057   thus ?thesis by auto
```
```  3058 qed
```
```  3059
```
```  3060 lemma tan_monotone':
```
```  3061   assumes "- (pi / 2) < y"
```
```  3062     and "y < pi / 2"
```
```  3063     and "- (pi / 2) < x"
```
```  3064     and "x < pi / 2"
```
```  3065   shows "(y < x) = (tan y < tan x)"
```
```  3066 proof
```
```  3067   assume "y < x"
```
```  3068   thus "tan y < tan x"
```
```  3069     using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
```
```  3070 next
```
```  3071   assume "tan y < tan x"
```
```  3072   show "y < x"
```
```  3073   proof (rule ccontr)
```
```  3074     assume "\<not> y < x" hence "x \<le> y" by auto
```
```  3075     hence "tan x \<le> tan y"
```
```  3076     proof (cases "x = y")
```
```  3077       case True thus ?thesis by auto
```
```  3078     next
```
```  3079       case False hence "x < y" using `x \<le> y` by auto
```
```  3080       from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
```
```  3081     qed
```
```  3082     thus False using `tan y < tan x` by auto
```
```  3083   qed
```
```  3084 qed
```
```  3085
```
```  3086 lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)"
```
```  3087   unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
```
```  3088
```
```  3089 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
```
```  3090   by (simp add: tan_def)
```
```  3091
```
```  3092 lemma tan_periodic_nat[simp]:
```
```  3093   fixes n :: nat
```
```  3094   shows "tan (x + real n * pi) = tan x"
```
```  3095 proof (induct n arbitrary: x)
```
```  3096   case 0
```
```  3097   then show ?case by simp
```
```  3098 next
```
```  3099   case (Suc n)
```
```  3100   have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi"
```
```  3101     unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
```
```  3102   show ?case unfolding split_pi_off using Suc by auto
```
```  3103 qed
```
```  3104
```
```  3105 lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
```
```  3106 proof (cases "0 \<le> i")
```
```  3107   case True
```
```  3108   hence i_nat: "real i = real (nat i)" by auto
```
```  3109   show ?thesis unfolding i_nat by auto
```
```  3110 next
```
```  3111   case False
```
```  3112   hence i_nat: "real i = - real (nat (-i))" by auto
```
```  3113   have "tan x = tan (x + real i * pi - real i * pi)"
```
```  3114     by auto
```
```  3115   also have "\<dots> = tan (x + real i * pi)"
```
```  3116     unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
```
```  3117   finally show ?thesis by auto
```
```  3118 qed
```
```  3119
```
```  3120 lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
```
```  3121   using tan_periodic_int[of _ "numeral n" ] unfolding real_numeral .
```
```  3122
```
```  3123 subsection {* Inverse Trigonometric Functions *}
```
```  3124
```
```  3125 definition arcsin :: "real => real"
```
```  3126   where "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
```
```  3127
```
```  3128 definition arccos :: "real => real"
```
```  3129   where "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
```
```  3130
```
```  3131 definition arctan :: "real => real"
```
```  3132   where "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
```
```  3133
```
```  3134 lemma arcsin:
```
```  3135   "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow>
```
```  3136     -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2 & sin(arcsin y) = y"
```
```  3137   unfolding arcsin_def by (rule theI' [OF sin_total])
```
```  3138
```
```  3139 lemma arcsin_pi:
```
```  3140   "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
```
```  3141   apply (drule (1) arcsin)
```
```  3142   apply (force intro: order_trans)
```
```  3143   done
```
```  3144
```
```  3145 lemma sin_arcsin [simp]: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin(arcsin y) = y"
```
```  3146   by (blast dest: arcsin)
```
```  3147
```
```  3148 lemma arcsin_bounded: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
```
```  3149   by (blast dest: arcsin)
```
```  3150
```
```  3151 lemma arcsin_lbound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y"
```
```  3152   by (blast dest: arcsin)
```
```  3153
```
```  3154 lemma arcsin_ubound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
```
```  3155   by (blast dest: arcsin)
```
```  3156
```
```  3157 lemma arcsin_lt_bounded:
```
```  3158      "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
```
```  3159   apply (frule order_less_imp_le)
```
```  3160   apply (frule_tac y = y in order_less_imp_le)
```
```  3161   apply (frule arcsin_bounded)
```
```  3162   apply (safe, simp)
```
```  3163   apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
```
```  3164   apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
```
```  3165   apply (drule_tac [!] f = sin in arg_cong, auto)
```
```  3166   done
```
```  3167
```
```  3168 lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
```
```  3169   apply (unfold arcsin_def)
```
```  3170   apply (rule the1_equality)
```
```  3171   apply (rule sin_total, auto)
```
```  3172   done
```
```  3173
```
```  3174 lemma arccos:
```
```  3175      "[| -1 \<le> y; y \<le> 1 |]
```
```  3176       ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
```
```  3177   unfolding arccos_def by (rule theI' [OF cos_total])
```
```  3178
```
```  3179 lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
```
```  3180   by (blast dest: arccos)
```
```  3181
```
```  3182 lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
```
```  3183   by (blast dest: arccos)
```
```  3184
```
```  3185 lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
```
```  3186   by (blast dest: arccos)
```
```  3187
```
```  3188 lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
```
```  3189   by (blast dest: arccos)
```
```  3190
```
```  3191 lemma arccos_lt_bounded:
```
```  3192      "[| -1 < y; y < 1 |]
```
```  3193       ==> 0 < arccos y & arccos y < pi"
```
```  3194   apply (frule order_less_imp_le)
```
```  3195   apply (frule_tac y = y in order_less_imp_le)
```
```  3196   apply (frule arccos_bounded, auto)
```
```  3197   apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
```
```  3198   apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
```
```  3199   apply (drule_tac [!] f = cos in arg_cong, auto)
```
```  3200   done
```
```  3201
```
```  3202 lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
```
```  3203   apply (simp add: arccos_def)
```
```  3204   apply (auto intro!: the1_equality cos_total)
```
```  3205   done
```
```  3206
```
```  3207 lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
```
```  3208   apply (simp add: arccos_def)
```
```  3209   apply (auto intro!: the1_equality cos_total)
```
```  3210   done
```
```  3211
```
```  3212 lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
```
```  3213   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  3214   apply (rule power2_eq_imp_eq)
```
```  3215   apply (simp add: cos_squared_eq)
```
```  3216   apply (rule cos_ge_zero)
```
```  3217   apply (erule (1) arcsin_lbound)
```
```  3218   apply (erule (1) arcsin_ubound)
```
```  3219   apply simp
```
```  3220   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  3221   apply (rule power_mono, simp, simp)
```
```  3222   done
```
```  3223
```
```  3224 lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
```
```  3225   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  3226   apply (rule power2_eq_imp_eq)
```
```  3227   apply (simp add: sin_squared_eq)
```
```  3228   apply (rule sin_ge_zero)
```
```  3229   apply (erule (1) arccos_lbound)
```
```  3230   apply (erule (1) arccos_ubound)
```
```  3231   apply simp
```
```  3232   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  3233   apply (rule power_mono, simp, simp)
```
```  3234   done
```
```  3235
```
```  3236 lemma arctan [simp]: "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
```
```  3237   unfolding arctan_def by (rule theI' [OF tan_total])
```
```  3238
```
```  3239 lemma tan_arctan: "tan (arctan y) = y"
```
```  3240   by auto
```
```  3241
```
```  3242 lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
```
```  3243   by (auto simp only: arctan)
```
```  3244
```
```  3245 lemma arctan_lbound: "- (pi/2) < arctan y"
```
```  3246   by auto
```
```  3247
```
```  3248 lemma arctan_ubound: "arctan y < pi/2"
```
```  3249   by (auto simp only: arctan)
```
```  3250
```
```  3251 lemma arctan_unique:
```
```  3252   assumes "-(pi/2) < x"
```
```  3253     and "x < pi/2"
```
```  3254     and "tan x = y"
```
```  3255   shows "arctan y = x"
```
```  3256   using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
```
```  3257
```
```  3258 lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x"
```
```  3259   by (rule arctan_unique) simp_all
```
```  3260
```
```  3261 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
```
```  3262   by (rule arctan_unique) simp_all
```
```  3263
```
```  3264 lemma arctan_minus: "arctan (- x) = - arctan x"
```
```  3265   apply (rule arctan_unique)
```
```  3266   apply (simp only: neg_less_iff_less arctan_ubound)
```
```  3267   apply (metis minus_less_iff arctan_lbound)
```
```  3268   apply simp
```
```  3269   done
```
```  3270
```
```  3271 lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
```
```  3272   by (intro less_imp_neq [symmetric] cos_gt_zero_pi
```
```  3273     arctan_lbound arctan_ubound)
```
```  3274
```
```  3275 lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)"
```
```  3276 proof (rule power2_eq_imp_eq)
```
```  3277   have "0 < 1 + x\<^sup>2" by (simp add: add_pos_nonneg)
```
```  3278   show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp
```
```  3279   show "0 \<le> cos (arctan x)"
```
```  3280     by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
```
```  3281   have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1"
```
```  3282     unfolding tan_def by (simp add: distrib_left power_divide)
```
```  3283   thus "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
```
```  3284     using `0 < 1 + x\<^sup>2` by (simp add: power_divide eq_divide_eq)
```
```  3285 qed
```
```  3286
```
```  3287 lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)"
```
```  3288   using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
```
```  3289   using tan_arctan [of x] unfolding tan_def cos_arctan
```
```  3290   by (simp add: eq_divide_eq)
```
```  3291
```
```  3292 lemma tan_sec: "cos x \<noteq> 0 ==> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
```
```  3293   apply (rule power_inverse [THEN subst])
```
```  3294   apply (rule_tac c1 = "(cos x)\<^sup>2" in real_mult_right_cancel [THEN iffD1])
```
```  3295   apply (auto dest: field_power_not_zero
```
```  3296           simp add: power_mult_distrib distrib_right power_divide tan_def
```
```  3297                     mult_assoc power_inverse [symmetric])
```
```  3298   done
```
```  3299
```
```  3300 lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
```
```  3301   by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
```
```  3302
```
```  3303 lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
```
```  3304   by (simp only: not_less [symmetric] arctan_less_iff)
```
```  3305
```
```  3306 lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
```
```  3307   by (simp only: eq_iff [where 'a=real] arctan_le_iff)
```
```  3308
```
```  3309 lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
```
```  3310   using arctan_less_iff [of 0 x] by simp
```
```  3311
```
```  3312 lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
```
```  3313   using arctan_less_iff [of x 0] by simp
```
```  3314
```
```  3315 lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
```
```  3316   using arctan_le_iff [of 0 x] by simp
```
```  3317
```
```  3318 lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```  3319   using arctan_le_iff [of x 0] by simp
```
```  3320
```
```  3321 lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
```
```  3322   using arctan_eq_iff [of x 0] by simp
```
```  3323
```
```  3324 lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin"
```
```  3325 proof -
```
```  3326   have "continuous_on (sin ` {- pi / 2 .. pi / 2}) arcsin"
```
```  3327     by (rule continuous_on_inv) (auto intro: continuous_on_intros simp: arcsin_sin)
```
```  3328   also have "sin ` {- pi / 2 .. pi / 2} = {-1 .. 1}"
```
```  3329   proof safe
```
```  3330     fix x :: real
```
```  3331     assume "x \<in> {-1..1}"
```
```  3332     then show "x \<in> sin ` {- pi / 2..pi / 2}"
```
```  3333       using arcsin_lbound arcsin_ubound
```
```  3334       by (intro image_eqI[where x="arcsin x"]) auto
```
```  3335   qed simp
```
```  3336   finally show ?thesis .
```
```  3337 qed
```
```  3338
```
```  3339 lemma continuous_on_arcsin [continuous_on_intros]:
```
```  3340   "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))"
```
```  3341   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arcsin']]
```
```  3342   by (auto simp: comp_def subset_eq)
```
```  3343
```
```  3344 lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x"
```
```  3345   using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  3346   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  3347
```
```  3348 lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos"
```
```  3349 proof -
```
```  3350   have "continuous_on (cos ` {0 .. pi}) arccos"
```
```  3351     by (rule continuous_on_inv) (auto intro: continuous_on_intros simp: arccos_cos)
```
```  3352   also have "cos ` {0 .. pi} = {-1 .. 1}"
```
```  3353   proof safe
```
```  3354     fix x :: real
```
```  3355     assume "x \<in> {-1..1}"
```
```  3356     then show "x \<in> cos ` {0..pi}"
```
```  3357       using arccos_lbound arccos_ubound
```
```  3358       by (intro image_eqI[where x="arccos x"]) auto
```
```  3359   qed simp
```
```  3360   finally show ?thesis .
```
```  3361 qed
```
```  3362
```
```  3363 lemma continuous_on_arccos [continuous_on_intros]:
```
```  3364   "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))"
```
```  3365   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arccos']]
```
```  3366   by (auto simp: comp_def subset_eq)
```
```  3367
```
```  3368 lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x"
```
```  3369   using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  3370   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  3371
```
```  3372 lemma isCont_arctan: "isCont arctan x"
```
```  3373   apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
```
```  3374   apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
```
```  3375   apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
```
```  3376   apply (erule (1) isCont_inverse_function2 [where f=tan])
```
```  3377   apply (metis arctan_tan order_le_less_trans order_less_le_trans)
```
```  3378   apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
```
```  3379   done
```
```  3380
```
```  3381 lemma tendsto_arctan [tendsto_intros]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) ---> arctan x) F"
```
```  3382   by (rule isCont_tendsto_compose [OF isCont_arctan])
```
```  3383
```
```  3384 lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))"
```
```  3385   unfolding continuous_def by (rule tendsto_arctan)
```
```  3386
```
```  3387 lemma continuous_on_arctan [continuous_on_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
```
```  3388   unfolding continuous_on_def by (auto intro: tendsto_arctan)
```
```  3389
```
```  3390 lemma DERIV_arcsin:
```
```  3391   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
```
```  3392   apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
```
```  3393   apply (rule DERIV_cong [OF DERIV_sin])
```
```  3394   apply (simp add: cos_arcsin)
```
```  3395   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  3396   apply (rule power_strict_mono, simp, simp, simp)
```
```  3397   apply assumption
```
```  3398   apply assumption
```
```  3399   apply simp
```
```  3400   apply (erule (1) isCont_arcsin)
```
```  3401   done
```
```  3402
```
```  3403 lemma DERIV_arccos:
```
```  3404   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
```
```  3405   apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
```
```  3406   apply (rule DERIV_cong [OF DERIV_cos])
```
```  3407   apply (simp add: sin_arccos)
```
```  3408   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  3409   apply (rule power_strict_mono, simp, simp, simp)
```
```  3410   apply assumption
```
```  3411   apply assumption
```
```  3412   apply simp
```
```  3413   apply (erule (1) isCont_arccos)
```
```  3414   done
```
```  3415
```
```  3416 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
```
```  3417   apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
```
```  3418   apply (rule DERIV_cong [OF DERIV_tan])
```
```  3419   apply (rule cos_arctan_not_zero)
```
```  3420   apply (simp add: power_inverse tan_sec [symmetric])
```
```  3421   apply (subgoal_tac "0 < 1 + x\<^sup>2", simp)
```
```  3422   apply (simp add: add_pos_nonneg)
```
```  3423   apply (simp, simp, simp, rule isCont_arctan)
```
```  3424   done
```
```  3425
```
```  3426 declare
```
```  3427   DERIV_arcsin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  3428   DERIV_arccos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  3429   DERIV_arctan[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  3430
```
```  3431 lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- pi/2))"
```
```  3432   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])
```
```  3433      (auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  3434            intro!: tan_monotone exI[of _ "pi/2"])
```
```  3435
```
```  3436 lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"
```
```  3437   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])
```
```  3438      (auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  3439            intro!: tan_monotone exI[of _ "pi/2"])
```
```  3440
```
```  3441 lemma tendsto_arctan_at_top: "(arctan ---> (pi/2)) at_top"
```
```  3442 proof (rule tendstoI)
```
```  3443   fix e :: real
```
```  3444   assume "0 < e"
```
```  3445   def y \<equiv> "pi/2 - min (pi/2) e"
```
```  3446   then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y"
```
```  3447     using `0 < e` by auto
```
```  3448
```
```  3449   show "eventually (\<lambda>x. dist (arctan x) (pi / 2) < e) at_top"
```
```  3450   proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)
```
```  3451     fix x
```
```  3452     assume "tan y < x"
```
```  3453     then have "arctan (tan y) < arctan x"
```
```  3454       by (simp add: arctan_less_iff)
```
```  3455     with y have "y < arctan x"
```
```  3456       by (subst (asm) arctan_tan) simp_all
```
```  3457     with arctan_ubound[of x, arith] y `0 < e`
```
```  3458     show "dist (arctan x) (pi / 2) < e"
```
```  3459       by (simp add: dist_real_def)
```
```  3460   qed
```
```  3461 qed
```
```  3462
```
```  3463 lemma tendsto_arctan_at_bot: "(arctan ---> - (pi/2)) at_bot"
```
```  3464   unfolding filterlim_at_bot_mirror arctan_minus
```
```  3465   by (intro tendsto_minus tendsto_arctan_at_top)
```
```  3466
```
```  3467
```
```  3468 subsection {* More Theorems about Sin and Cos *}
```
```  3469
```
```  3470 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
```
```  3471 proof -
```
```  3472   let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
```
```  3473   have nonneg: "0 \<le> ?c"
```
```  3474     by (simp add: cos_ge_zero)
```
```  3475   have "0 = cos (pi / 4 + pi / 4)"
```
```  3476     by simp
```
```  3477   also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2"
```
```  3478     by (simp only: cos_add power2_eq_square)
```
```  3479   also have "\<dots> = 2 * ?c\<^sup>2 - 1"
```
```  3480     by (simp add: sin_squared_eq)
```
```  3481   finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2"
```
```  3482     by (simp add: power_divide)
```
```  3483   thus ?thesis
```
```  3484     using nonneg by (rule power2_eq_imp_eq) simp
```
```  3485 qed
```
```  3486
```
```  3487 lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"
```
```  3488 proof -
```
```  3489   let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
```
```  3490   have pos_c: "0 < ?c"
```
```  3491     by (rule cos_gt_zero, simp, simp)
```
```  3492   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
```
```  3493     by simp
```
```  3494   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
```
```  3495     by (simp only: cos_add sin_add)
```
```  3496   also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)"
```
```  3497     by (simp add: algebra_simps power2_eq_square)
```
```  3498   finally have "?c\<^sup>2 = (sqrt 3 / 2)\<^sup>2"
```
```  3499     using pos_c by (simp add: sin_squared_eq power_divide)
```
```  3500   thus ?thesis
```
```  3501     using pos_c [THEN order_less_imp_le]
```
```  3502     by (rule power2_eq_imp_eq) simp
```
```  3503 qed
```
```  3504
```
```  3505 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
```
```  3506   by (simp add: sin_cos_eq cos_45)
```
```  3507
```
```  3508 lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
```
```  3509   by (simp add: sin_cos_eq cos_30)
```
```  3510
```
```  3511 lemma cos_60: "cos (pi / 3) = 1 / 2"
```
```  3512   apply (rule power2_eq_imp_eq)
```
```  3513   apply (simp add: cos_squared_eq sin_60 power_divide)
```
```  3514   apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
```
```  3515   done
```
```  3516
```
```  3517 lemma sin_30: "sin (pi / 6) = 1 / 2"
```
```  3518   by (simp add: sin_cos_eq cos_60)
```
```  3519
```
```  3520 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
```
```  3521   unfolding tan_def by (simp add: sin_30 cos_30)
```
```  3522
```
```  3523 lemma tan_45: "tan (pi / 4) = 1"
```
```  3524   unfolding tan_def by (simp add: sin_45 cos_45)
```
```  3525
```
```  3526 lemma tan_60: "tan (pi / 3) = sqrt 3"
```
```  3527   unfolding tan_def by (simp add: sin_60 cos_60)
```
```  3528
```
```  3529 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
```
```  3530 proof -
```
```  3531   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
```
```  3532     by (auto simp add: algebra_simps sin_add)
```
```  3533   thus ?thesis
```
```  3534     by (simp add: real_of_nat_Suc distrib_right add_divide_distrib
```
```  3535                   mult_commute [of pi])
```
```  3536 qed
```
```  3537
```
```  3538 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
```
```  3539   by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
```
```  3540
```
```  3541 lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
```
```  3542   apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
```
```  3543   apply (subst cos_add, simp)
```
```  3544   done
```
```  3545
```
```  3546 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
```
```  3547   by (auto simp add: mult_assoc)
```
```  3548
```
```  3549 lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
```
```  3550   apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
```
```  3551   apply (subst sin_add, simp)
```
```  3552   done
```
```  3553
```
```  3554 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
```
```  3555   apply (simp only: cos_add sin_add real_of_nat_Suc distrib_right distrib_left add_divide_distrib)
```
```  3556   apply auto
```
```  3557   done
```
```  3558
```
```  3559 lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)"
```
```  3560   by (auto intro!: DERIV_intros)
```
```  3561
```
```  3562 lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
```
```  3563   by (auto simp add: sin_zero_iff even_mult_two_ex)
```
```  3564
```
```  3565 lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
```
```  3566   using sin_cos_squared_add3 [where x = x] by auto
```
```  3567
```
```  3568
```
```  3569 subsection {* Machins formula *}
```
```  3570
```
```  3571 lemma arctan_one: "arctan 1 = pi / 4"
```
```  3572   by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi)
```
```  3573
```
```  3574 lemma tan_total_pi4:
```
```  3575   assumes "\<bar>x\<bar> < 1"
```
```  3576   shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
```
```  3577 proof
```
```  3578   show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
```
```  3579     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  3580     unfolding arctan_less_iff using assms by auto
```
```  3581 qed
```
```  3582
```
```  3583 lemma arctan_add:
```
```  3584   assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
```
```  3585   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
```
```  3586 proof (rule arctan_unique [symmetric])
```
```  3587   have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y"
```
```  3588     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  3589     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  3590   from add_le_less_mono [OF this]
```
```  3591   show 1: "- (pi / 2) < arctan x + arctan y" by simp
```
```  3592   have "arctan x \<le> pi / 4" and "arctan y < pi / 4"
```
```  3593     unfolding arctan_one [symmetric]
```
```  3594     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  3595   from add_le_less_mono [OF this]
```
```  3596   show 2: "arctan x + arctan y < pi / 2" by simp
```
```  3597   show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
```
```  3598     using cos_gt_zero_pi [OF 1 2] by (simp add: tan_add)
```
```  3599 qed
```
```  3600
```
```  3601 theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
```
```  3602 proof -
```
```  3603   have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
```
```  3604   from arctan_add[OF less_imp_le[OF this] this]
```
```  3605   have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
```
```  3606   moreover
```
```  3607   have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
```
```  3608   from arctan_add[OF less_imp_le[OF this] this]
```
```  3609   have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
```
```  3610   moreover
```
```  3611   have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
```
```  3612   from arctan_add[OF this]
```
```  3613   have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
```
```  3614   ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
```
```  3615   thus ?thesis unfolding arctan_one by algebra
```
```  3616 qed
```
```  3617
```
```  3618
```
```  3619 subsection {* Introducing the arcus tangens power series *}
```
```  3620
```
```  3621 lemma monoseq_arctan_series:
```
```  3622   fixes x :: real
```
```  3623   assumes "\<bar>x\<bar> \<le> 1"
```
```  3624   shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
```
```  3625 proof (cases "x = 0")
```
```  3626   case True
```
```  3627   thus ?thesis unfolding monoseq_def One_nat_def by auto
```
```  3628 next
```
```  3629   case False
```
```  3630   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  3631   show "monoseq ?a"
```
```  3632   proof -
```
```  3633     {
```
```  3634       fix n
```
```  3635       fix x :: real
```
```  3636       assume "0 \<le> x" and "x \<le> 1"
```
```  3637       have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le>
```
```  3638         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
```
```  3639       proof (rule mult_mono)
```
```  3640         show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))"
```
```  3641           by (rule frac_le) simp_all
```
```  3642         show "0 \<le> 1 / real (Suc (n * 2))"
```
```  3643           by auto
```
```  3644         show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)"
```
```  3645           by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`)
```
```  3646         show "0 \<le> x ^ Suc (Suc n * 2)"
```
```  3647           by (rule zero_le_power) (simp add: `0 \<le> x`)
```
```  3648       qed
```
```  3649     } note mono = this
```
```  3650
```
```  3651     show ?thesis
```
```  3652     proof (cases "0 \<le> x")
```
```  3653       case True from mono[OF this `x \<le> 1`, THEN allI]
```
```  3654       show ?thesis unfolding Suc_eq_plus1[symmetric]
```
```  3655         by (rule mono_SucI2)
```
```  3656     next
```
```  3657       case False
```
```  3658       hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto
```
```  3659       from mono[OF this]
```
```  3660       have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge>
```
```  3661         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using `0 \<le> -x` by auto
```
```  3662       thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
```
```  3663     qed
```
```  3664   qed
```
```  3665 qed
```
```  3666
```
```  3667 lemma zeroseq_arctan_series:
```
```  3668   fixes x :: real
```
```  3669   assumes "\<bar>x\<bar> \<le> 1"
```
```  3670   shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
```
```  3671 proof (cases "x = 0")
```
```  3672   case True
```
```  3673   thus ?thesis
```
```  3674     unfolding One_nat_def by (auto simp add: tendsto_const)
```
```  3675 next
```
```  3676   case False
```
```  3677   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  3678   show "?a ----> 0"
```
```  3679   proof (cases "\<bar>x\<bar> < 1")
```
```  3680     case True
```
```  3681     hence "norm x < 1" by auto
```
```  3682     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]]
```
```  3683     have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"
```
```  3684       unfolding inverse_eq_divide Suc_eq_plus1 by simp
```
```  3685     then show ?thesis using pos2 by (rule LIMSEQ_linear)
```
```  3686   next
```
```  3687     case False
```
```  3688     hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  3689     hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x"
```
```  3690       unfolding One_nat_def by auto
```
```  3691     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
```
```  3692     show ?thesis unfolding n_eq Suc_eq_plus1 by auto
```
```  3693   qed
```
```  3694 qed
```
```  3695
```
```  3696 lemma summable_arctan_series:
```
```  3697   fixes x :: real and n :: nat
```
```  3698   assumes "\<bar>x\<bar> \<le> 1"
```
```  3699   shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  3700   (is "summable (?c x)")
```
```  3701   by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
```
```  3702
```
```  3703 lemma less_one_imp_sqr_less_one:
```
```  3704   fixes x :: real
```
```  3705   assumes "\<bar>x\<bar> < 1"
```
```  3706   shows "x\<^sup>2 < 1"
```
```  3707 proof -
```
```  3708   from mult_left_mono[OF less_imp_le[OF `\<bar>x\<bar> < 1`] abs_ge_zero[of x]]
```
```  3709   have "\<bar>x\<^sup>2\<bar> < 1" using `\<bar>x\<bar> < 1` unfolding numeral_2_eq_2 power_Suc2 by auto
```
```  3710   thus ?thesis using zero_le_power2 by auto
```
```  3711 qed
```
```  3712
```
```  3713 lemma DERIV_arctan_series:
```
```  3714   assumes "\<bar> x \<bar> < 1"
```
```  3715   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))"
```
```  3716   (is "DERIV ?arctan _ :> ?Int")
```
```  3717 proof -
```
```  3718   let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
```
```  3719
```
```  3720   have n_even: "\<And>n :: nat. even n \<Longrightarrow> 2 * (n div 2) = n"
```
```  3721     by presburger
```
```  3722   then have if_eq: "\<And>n x'. ?f n * real (Suc n) * x'^n =
```
```  3723     (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)"
```
```  3724     by auto
```
```  3725
```
```  3726   {
```
```  3727     fix x :: real
```
```  3728     assume "\<bar>x\<bar> < 1"
```
```  3729     hence "x\<^sup>2 < 1" by (rule less_one_imp_sqr_less_one)
```
```  3730     have "summable (\<lambda> n. -1 ^ n * (x\<^sup>2) ^n)"
```
```  3731       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`])
```
```  3732     hence "summable (\<lambda> n. -1 ^ n * x^(2*n))" unfolding power_mult .
```
```  3733   } note summable_Integral = this
```
```  3734
```
```  3735   {
```
```  3736     fix f :: "nat \<Rightarrow> real"
```
```  3737     have "\<And>x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  3738     proof
```
```  3739       fix x :: real
```
```  3740       assume "f sums x"
```
```  3741       from sums_if[OF sums_zero this]
```
```  3742       show "(\<lambda>n. if even n then f (n div 2) else 0) sums x"
```
```  3743         by auto
```
```  3744     next
```
```  3745       fix x :: real
```
```  3746       assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  3747       from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult_commute]]
```
```  3748       show "f sums x" unfolding sums_def by auto
```
```  3749     qed
```
```  3750     hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
```
```  3751   } note sums_even = this
```
```  3752
```
```  3753   have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int"
```
```  3754     unfolding if_eq mult_commute[of _ 2] suminf_def sums_even[of "\<lambda> n. -1 ^ n * x ^ (2 * n)", symmetric]
```
```  3755     by auto
```
```  3756
```
```  3757   {
```
```  3758     fix x :: real
```
```  3759     have if_eq': "\<And>n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
```
```  3760       (if even n then -1 ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
```
```  3761       using n_even by auto
```
```  3762     have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)" by auto
```
```  3763     have "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x"
```
```  3764       unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. -1 ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
```
```  3765       by auto
```
```  3766   } note arctan_eq = this
```
```  3767
```
```  3768   have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
```
```  3769   proof (rule DERIV_power_series')
```
```  3770     show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto
```
```  3771     {
```
```  3772       fix x' :: real
```
```  3773       assume x'_bounds: "x' \<in> {- 1 <..< 1}"
```
```  3774       hence "\<bar>x'\<bar> < 1" by auto
```
```  3775
```
```  3776       let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
```
```  3777       show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
```
```  3778         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`])
```
```  3779     }
```
```  3780   qed auto
```
```  3781   thus ?thesis unfolding Int_eq arctan_eq .
```
```  3782 qed
```
```  3783
```
```  3784 lemma arctan_series:
```
```  3785   assumes "\<bar> x \<bar> \<le> 1"
```
```  3786   shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  3787   (is "_ = suminf (\<lambda> n. ?c x n)")
```
```  3788 proof -
```
```  3789   let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)"
```
```  3790
```
```  3791   {
```
```  3792     fix r x :: real
```
```  3793     assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
```
```  3794     have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto
```
```  3795     from DERIV_arctan_series[OF this] have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
```
```  3796   } note DERIV_arctan_suminf = this
```
```  3797
```
```  3798   {
```
```  3799     fix x :: real
```
```  3800     assume "\<bar>x\<bar> \<le> 1"
```
```  3801     note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]]
```
```  3802   } note arctan_series_borders = this
```
```  3803
```
```  3804   {
```
```  3805     fix x :: real
```
```  3806     assume "\<bar>x\<bar> < 1"
```
```  3807     have "arctan x = (\<Sum>k. ?c x k)"
```
```  3808     proof -
```
```  3809       obtain r where "\<bar>x\<bar> < r" and "r < 1"
```
```  3810         using dense[OF `\<bar>x\<bar> < 1`] by blast
```
```  3811       hence "0 < r" and "-r < x" and "x < r" by auto
```
```  3812
```
```  3813       have suminf_eq_arctan_bounded: "\<And>x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow>
```
```  3814         suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  3815       proof -
```
```  3816         fix x a b
```
```  3817         assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
```
```  3818         hence "\<bar>x\<bar> < r" by auto
```
```  3819         show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  3820         proof (rule DERIV_isconst2[of "a" "b"])
```
```  3821           show "a < b" and "a \<le> x" and "x \<le> b"
```
```  3822             using `a < b` `a \<le> x` `x \<le> b` by auto
```
```  3823           have "\<forall>x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  3824           proof (rule allI, rule impI)
```
```  3825             fix x
```
```  3826             assume "-r < x \<and> x < r"
```
```  3827             hence "\<bar>x\<bar> < r" by auto
```
```  3828             hence "\<bar>x\<bar> < 1" using `r < 1` by auto
```
```  3829             have "\<bar> - (x\<^sup>2) \<bar> < 1"
```
```  3830               using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
```
```  3831             hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  3832               unfolding real_norm_def[symmetric] by (rule geometric_sums)
```
```  3833             hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  3834               unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto
```
```  3835             hence suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)"
```
```  3836               using sums_unique unfolding inverse_eq_divide by auto
```
```  3837             have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))"
```
```  3838               unfolding suminf_c'_eq_geom
```
```  3839               by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
```
```  3840             from DERIV_add_minus[OF this DERIV_arctan]
```
```  3841             show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  3842               by auto
```
```  3843           qed
```
```  3844           hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
```
```  3845             using `-r < a` `b < r` by auto
```
```  3846           thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
```
```  3847             using `\<bar>x\<bar> < r` by auto
```
```  3848           show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y"
```
```  3849             using DERIV_in_rball DERIV_isCont by auto
```
```  3850         qed
```
```  3851       qed
```
```  3852
```
```  3853       have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
```
```  3854         unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero
```
```  3855         by auto
```
```  3856
```
```  3857       have "suminf (?c x) - arctan x = 0"
```
```  3858       proof (cases "x = 0")
```
```  3859         case True
```
```  3860         thus ?thesis using suminf_arctan_zero by auto
```
```  3861       next
```
```  3862         case False
```
```  3863         hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
```
```  3864         have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
```
```  3865           by (rule suminf_eq_arctan_bounded[where x="0" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>", symmetric])
```
```  3866             (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  3867         moreover
```
```  3868         have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
```
```  3869           by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"])
```
```  3870             (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  3871         ultimately
```
```  3872         show ?thesis using suminf_arctan_zero by auto
```
```  3873       qed
```
```  3874       thus ?thesis by auto
```
```  3875     qed
```
```  3876   } note when_less_one = this
```
```  3877
```
```  3878   show "arctan x = suminf (\<lambda> n. ?c x n)"
```
```  3879   proof (cases "\<bar>x\<bar> < 1")
```
```  3880     case True
```
```  3881     thus ?thesis by (rule when_less_one)
```
```  3882   next
```
```  3883     case False
```
```  3884     hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  3885     let ?a = "\<lambda>x n. \<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
```
```  3886     let ?diff = "\<lambda> x n. \<bar> arctan x - (\<Sum> i = 0..<n. ?c x i)\<bar>"
```
```  3887     {
```
```  3888       fix n :: nat
```
```  3889       have "0 < (1 :: real)" by auto
```
```  3890       moreover
```
```  3891       {
```
```  3892         fix x :: real
```
```  3893         assume "0 < x" and "x < 1"
```
```  3894         hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
```
```  3895         from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)"
```
```  3896           by auto
```
```  3897         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]
```
```  3898         have "0 < 1 / real (n*2+1) * x^(n*2+1)"
```
```  3899           by (rule mult_pos_pos, auto simp only: zero_less_power[OF `0 < x`], auto)
```
```  3900         hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)"
```
```  3901           by (rule abs_of_pos)
```
```  3902         have "?diff x n \<le> ?a x n"
```
```  3903         proof (cases "even n")
```
```  3904           case True
```
```  3905           hence sgn_pos: "(-1)^n = (1::real)" by auto
```
```  3906           from `even n` obtain m where "2 * m = n"
```
```  3907             unfolding even_mult_two_ex by auto
```
```  3908           from bounds[of m, unfolded this atLeastAtMost_iff]
```
```  3909           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))"
```
```  3910             by auto
```
```  3911           also have "\<dots> = ?c x n" unfolding One_nat_def by auto
```
```  3912           also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
```
```  3913           finally show ?thesis .
```
```  3914         next
```
```  3915           case False
```
```  3916           hence sgn_neg: "(-1)^n = (-1::real)" by auto
```
```  3917           from `odd n` obtain m where m_def: "2 * m + 1 = n"
```
```  3918             unfolding odd_Suc_mult_two_ex by auto
```
```  3919           hence m_plus: "2 * (m + 1) = n + 1" by auto
```
```  3920           from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
```
```  3921           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))"
```
```  3922             by auto
```
```  3923           also have "\<dots> = - ?c x n" unfolding One_nat_def by auto
```
```  3924           also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
```
```  3925           finally show ?thesis .
```
```  3926         qed
```
```  3927         hence "0 \<le> ?a x n - ?diff x n" by auto
```
```  3928       }
```
```  3929       hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
```
```  3930       moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
```
```  3931         unfolding diff_conv_add_uminus divide_inverse
```
```  3932         by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan
```
```  3933           isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum
```
```  3934           simp del: add_uminus_conv_diff)
```
```  3935       ultimately have "0 \<le> ?a 1 n - ?diff 1 n"
```
```  3936         by (rule LIM_less_bound)
```
```  3937       hence "?diff 1 n \<le> ?a 1 n" by auto
```
```  3938     }
```
```  3939     have "?a 1 ----> 0"
```
```  3940       unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
```
```  3941       by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)
```
```  3942     have "?diff 1 ----> 0"
```
```  3943     proof (rule LIMSEQ_I)
```
```  3944       fix r :: real
```
```  3945       assume "0 < r"
```
```  3946       obtain N :: nat where N_I: "\<And>n. N \<le> n \<Longrightarrow> ?a 1 n < r"
```
```  3947         using LIMSEQ_D[OF `?a 1 ----> 0` `0 < r`] by auto
```
```  3948       {
```
```  3949         fix n
```
```  3950         assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this]
```
```  3951         have "norm (?diff 1 n - 0) < r" by auto
```
```  3952       }
```
```  3953       thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
```
```  3954     qed
```
```  3955     from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
```
```  3956     have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
```
```  3957     hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
```
```  3958
```
```  3959     show ?thesis
```
```  3960     proof (cases "x = 1")
```
```  3961       case True
```
```  3962       then show ?thesis by (simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
```
```  3963     next
```
```  3964       case False
```
```  3965       hence "x = -1" using `\<bar>x\<bar> = 1` by auto
```
```  3966
```
```  3967       have "- (pi / 2) < 0" using pi_gt_zero by auto
```
```  3968       have "- (2 * pi) < 0" using pi_gt_zero by auto
```
```  3969
```
```  3970       have c_minus_minus: "\<And>i. ?c (- 1) i = - ?c 1 i"
```
```  3971         unfolding One_nat_def by auto
```
```  3972
```
```  3973       have "arctan (- 1) = arctan (tan (-(pi / 4)))"
```
```  3974         unfolding tan_45 tan_minus ..
```
```  3975       also have "\<dots> = - (pi / 4)"
```
```  3976         by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
```
```  3977       also have "\<dots> = - (arctan (tan (pi / 4)))"
```
```  3978         unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero])
```
```  3979       also have "\<dots> = - (arctan 1)"
```
```  3980         unfolding tan_45 ..
```
```  3981       also have "\<dots> = - (\<Sum> i. ?c 1 i)"
```
```  3982         using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto
```
```  3983       also have "\<dots> = (\<Sum> i. ?c (- 1) i)"
```
```  3984         using suminf_minus[OF sums_summable[OF `(?c 1) sums (arctan 1)`]]
```
```  3985         unfolding c_minus_minus by auto
```
```  3986       finally show ?thesis using `x = -1` by auto
```
```  3987     qed
```
```  3988   qed
```
```  3989 qed
```
```  3990
```
```  3991 lemma arctan_half:
```
```  3992   fixes x :: real
```
```  3993   shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))"
```
```  3994 proof -
```
```  3995   obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x"
```
```  3996     using tan_total by blast
```
```  3997   hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2"
```
```  3998     by auto
```
```  3999
```
```  4000   have divide_nonzero_divide: "\<And>A B C :: real. C \<noteq> 0 \<Longrightarrow> A / B = (A / C) / (B / C)"
```
```  4001     by auto
```
```  4002
```
```  4003   have "0 < cos y" using cos_gt_zero_pi[OF low high] .
```
```  4004   hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y"
```
```  4005     by auto
```
```  4006
```
```  4007   have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  4008     unfolding tan_def power_divide ..
```
```  4009   also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  4010     using `cos y \<noteq> 0` by auto
```
```  4011   also have "\<dots> = 1 / (cos y)\<^sup>2"
```
```  4012     unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
```
```  4013   finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" .
```
```  4014
```
```  4015   have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)"
```
```  4016     unfolding tan_def divide_nonzero_divide[OF `cos y \<noteq> 0`, symmetric] ..
```
```  4017   also have "\<dots> = tan y / (1 + 1 / cos y)"
```
```  4018     using `cos y \<noteq> 0` unfolding add_divide_distrib by auto
```
```  4019   also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))"
```
```  4020     unfolding cos_sqrt ..
```
```  4021   also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))"
```
```  4022     unfolding real_sqrt_divide by auto
```
```  4023   finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))"
```
```  4024     unfolding `1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2` .
```
```  4025
```
```  4026   have "arctan x = y"
```
```  4027     using arctan_tan low high y_eq by auto
```
```  4028   also have "\<dots> = 2 * (arctan (tan (y/2)))"
```
```  4029     using arctan_tan[OF low2 high2] by auto
```
```  4030   also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))"
```
```  4031     unfolding tan_half by auto
```
```  4032   finally show ?thesis
```
```  4033     unfolding eq `tan y = x` .
```
```  4034 qed
```
```  4035
```
```  4036 lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y"
```
```  4037   by (simp only: arctan_less_iff)
```
```  4038
```
```  4039 lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y"
```
```  4040   by (simp only: arctan_le_iff)
```
```  4041
```
```  4042 lemma arctan_inverse:
```
```  4043   assumes "x \<noteq> 0"
```
```  4044   shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
```
```  4045 proof (rule arctan_unique)
```
```  4046   show "- (pi / 2) < sgn x * pi / 2 - arctan x"
```
```  4047     using arctan_bounded [of x] assms
```
```  4048     unfolding sgn_real_def
```
```  4049     apply (auto simp add: algebra_simps)
```
```  4050     apply (drule zero_less_arctan_iff [THEN iffD2])
```
```  4051     apply arith
```
```  4052     done
```
```  4053   show "sgn x * pi / 2 - arctan x < pi / 2"
```
```  4054     using arctan_bounded [of "- x"] assms
```
```  4055     unfolding sgn_real_def arctan_minus
```
```  4056     by (auto simp add: algebra_simps)
```
```  4057   show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
```
```  4058     unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
```
```  4059     unfolding sgn_real_def
```
```  4060     by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
```
```  4061 qed
```
```  4062
```
```  4063 theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
```
```  4064 proof -
```
```  4065   have "pi / 4 = arctan 1" using arctan_one by auto
```
```  4066   also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
```
```  4067   finally show ?thesis by auto
```
```  4068 qed
```
```  4069
```
```  4070
```
```  4071 subsection {* Existence of Polar Coordinates *}
```
```  4072
```
```  4073 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1"
```
```  4074   apply (rule power2_le_imp_le [OF _ zero_le_one])
```
```  4075   apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
```
```  4076   done
```
```  4077
```
```  4078 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
```
```  4079   by (simp add: abs_le_iff)
```
```  4080
```
```  4081 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
```
```  4082   by (simp add: sin_arccos abs_le_iff)
```
```  4083
```
```  4084 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
```
```  4085
```
```  4086 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
```
```  4087
```
```  4088 lemma polar_ex1: "0 < y \<Longrightarrow> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  4089   apply (rule_tac x = "sqrt (x\<^sup>2 + y\<^sup>2)" in exI)
```
```  4090   apply (rule_tac x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))" in exI)
```
```  4091   apply (simp add: cos_arccos_lemma1)
```
```  4092   apply (simp add: sin_arccos_lemma1)
```
```  4093   apply (simp add: power_divide)
```
```  4094   apply (simp add: real_sqrt_mult [symmetric])
```
```  4095   apply (simp add: right_diff_distrib)
```
```  4096   done
```
```  4097
```
```  4098 lemma polar_ex2: "y < 0 ==> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  4099   using polar_ex1 [where x=x and y="-y"]
```
```  4100   apply simp
```
```  4101   apply clarify
```
```  4102   apply (metis cos_minus minus_minus minus_mult_right sin_minus)
```
```  4103   done
```
```  4104
```
```  4105 lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
```
```  4106   apply (rule_tac x=0 and y=y in linorder_cases)
```
```  4107   apply (erule polar_ex1)
```
```  4108   apply (rule_tac x=x in exI, rule_tac x=0 in exI, simp)
```
```  4109   apply (erule polar_ex2)
```
```  4110   done
```
```  4111
```
```  4112 end
```