src/HOL/Transcendental.thy
 author paulson Tue Apr 25 16:39:54 2017 +0100 (2017-04-25) changeset 65578 e4997c181cce parent 65552 f533820e7248 child 65583 8d53b3bebab4 permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
```     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 section \<open>Power Series, Transcendental Functions etc.\<close>
```
```     8
```
```     9 theory Transcendental
```
```    10 imports Series Deriv NthRoot
```
```    11 begin
```
```    12
```
```    13 text \<open>A fact theorem on reals.\<close>
```
```    14
```
```    15 lemma square_fact_le_2_fact: "fact n * fact n \<le> (fact (2 * n) :: real)"
```
```    16 proof (induct n)
```
```    17   case 0
```
```    18   then show ?case by simp
```
```    19 next
```
```    20   case (Suc n)
```
```    21   have "(fact (Suc n)) * (fact (Suc n)) = of_nat (Suc n) * of_nat (Suc n) * (fact n * fact n :: real)"
```
```    22     by (simp add: field_simps)
```
```    23   also have "\<dots> \<le> of_nat (Suc n) * of_nat (Suc n) * fact (2 * n)"
```
```    24     by (rule mult_left_mono [OF Suc]) simp
```
```    25   also have "\<dots> \<le> of_nat (Suc (Suc (2 * n))) * of_nat (Suc (2 * n)) * fact (2 * n)"
```
```    26     by (rule mult_right_mono)+ (auto simp: field_simps)
```
```    27   also have "\<dots> = fact (2 * Suc n)" by (simp add: field_simps)
```
```    28   finally show ?case .
```
```    29 qed
```
```    30
```
```    31 lemma fact_in_Reals: "fact n \<in> \<real>"
```
```    32   by (induction n) auto
```
```    33
```
```    34 lemma of_real_fact [simp]: "of_real (fact n) = fact n"
```
```    35   by (metis of_nat_fact of_real_of_nat_eq)
```
```    36
```
```    37 lemma pochhammer_of_real: "pochhammer (of_real x) n = of_real (pochhammer x n)"
```
```    38   by (simp add: pochhammer_prod)
```
```    39
```
```    40 lemma norm_fact [simp]: "norm (fact n :: 'a::real_normed_algebra_1) = fact n"
```
```    41 proof -
```
```    42   have "(fact n :: 'a) = of_real (fact n)"
```
```    43     by simp
```
```    44   also have "norm \<dots> = fact n"
```
```    45     by (subst norm_of_real) simp
```
```    46   finally show ?thesis .
```
```    47 qed
```
```    48
```
```    49 lemma root_test_convergence:
```
```    50   fixes f :: "nat \<Rightarrow> 'a::banach"
```
```    51   assumes f: "(\<lambda>n. root n (norm (f n))) \<longlonglongrightarrow> x" \<comment> "could be weakened to lim sup"
```
```    52     and "x < 1"
```
```    53   shows "summable f"
```
```    54 proof -
```
```    55   have "0 \<le> x"
```
```    56     by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1])
```
```    57   from \<open>x < 1\<close> obtain z where z: "x < z" "z < 1"
```
```    58     by (metis dense)
```
```    59   from f \<open>x < z\<close> have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
```
```    60     by (rule order_tendstoD)
```
```    61   then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially"
```
```    62     using eventually_ge_at_top
```
```    63   proof eventually_elim
```
```    64     fix n
```
```    65     assume less: "root n (norm (f n)) < z" and n: "1 \<le> n"
```
```    66     from power_strict_mono[OF less, of n] n show "norm (f n) \<le> z ^ n"
```
```    67       by simp
```
```    68   qed
```
```    69   then show "summable f"
```
```    70     unfolding eventually_sequentially
```
```    71     using z \<open>0 \<le> x\<close> by (auto intro!: summable_comparison_test[OF _  summable_geometric])
```
```    72 qed
```
```    73
```
```    74 subsection \<open>More facts about binomial coefficients\<close>
```
```    75
```
```    76 text \<open>
```
```    77   These facts could have been proven before, but having real numbers
```
```    78   makes the proofs a lot easier.
```
```    79 \<close>
```
```    80
```
```    81 lemma central_binomial_odd:
```
```    82   "odd n \<Longrightarrow> n choose (Suc (n div 2)) = n choose (n div 2)"
```
```    83 proof -
```
```    84   assume "odd n"
```
```    85   hence "Suc (n div 2) \<le> n" by presburger
```
```    86   hence "n choose (Suc (n div 2)) = n choose (n - Suc (n div 2))"
```
```    87     by (rule binomial_symmetric)
```
```    88   also from \<open>odd n\<close> have "n - Suc (n div 2) = n div 2" by presburger
```
```    89   finally show ?thesis .
```
```    90 qed
```
```    91
```
```    92 lemma binomial_less_binomial_Suc:
```
```    93   assumes k: "k < n div 2"
```
```    94   shows   "n choose k < n choose (Suc k)"
```
```    95 proof -
```
```    96   from k have k': "k \<le> n" "Suc k \<le> n" by simp_all
```
```    97   from k' have "real (n choose k) = fact n / (fact k * fact (n - k))"
```
```    98     by (simp add: binomial_fact)
```
```    99   also from k' have "n - k = Suc (n - Suc k)" by simp
```
```   100   also from k' have "fact \<dots> = (real n - real k) * fact (n - Suc k)"
```
```   101     by (subst fact_Suc) (simp_all add: of_nat_diff)
```
```   102   also from k have "fact k = fact (Suc k) / (real k + 1)" by (simp add: field_simps)
```
```   103   also have "fact n / (fact (Suc k) / (real k + 1) * ((real n - real k) * fact (n - Suc k))) =
```
```   104                (n choose (Suc k)) * ((real k + 1) / (real n - real k))"
```
```   105     using k by (simp add: divide_simps binomial_fact)
```
```   106   also from assms have "(real k + 1) / (real n - real k) < 1" by simp
```
```   107   finally show ?thesis using k by (simp add: mult_less_cancel_left)
```
```   108 qed
```
```   109
```
```   110 lemma binomial_strict_mono:
```
```   111   assumes "k < k'" "2*k' \<le> n"
```
```   112   shows   "n choose k < n choose k'"
```
```   113 proof -
```
```   114   from assms have "k \<le> k' - 1" by simp
```
```   115   thus ?thesis
```
```   116   proof (induction rule: inc_induct)
```
```   117     case base
```
```   118     with assms binomial_less_binomial_Suc[of "k' - 1" n]
```
```   119       show ?case by simp
```
```   120   next
```
```   121     case (step k)
```
```   122     from step.prems step.hyps assms have "n choose k < n choose (Suc k)"
```
```   123       by (intro binomial_less_binomial_Suc) simp_all
```
```   124     also have "\<dots> < n choose k'" by (rule step.IH)
```
```   125     finally show ?case .
```
```   126   qed
```
```   127 qed
```
```   128
```
```   129 lemma binomial_mono:
```
```   130   assumes "k \<le> k'" "2*k' \<le> n"
```
```   131   shows   "n choose k \<le> n choose k'"
```
```   132   using assms binomial_strict_mono[of k k' n] by (cases "k = k'") simp_all
```
```   133
```
```   134 lemma binomial_strict_antimono:
```
```   135   assumes "k < k'" "2 * k \<ge> n" "k' \<le> n"
```
```   136   shows   "n choose k > n choose k'"
```
```   137 proof -
```
```   138   from assms have "n choose (n - k) > n choose (n - k')"
```
```   139     by (intro binomial_strict_mono) (simp_all add: algebra_simps)
```
```   140   with assms show ?thesis by (simp add: binomial_symmetric [symmetric])
```
```   141 qed
```
```   142
```
```   143 lemma binomial_antimono:
```
```   144   assumes "k \<le> k'" "k \<ge> n div 2" "k' \<le> n"
```
```   145   shows   "n choose k \<ge> n choose k'"
```
```   146 proof (cases "k = k'")
```
```   147   case False
```
```   148   note not_eq = False
```
```   149   show ?thesis
```
```   150   proof (cases "k = n div 2 \<and> odd n")
```
```   151     case False
```
```   152     with assms(2) have "2*k \<ge> n" by presburger
```
```   153     with not_eq assms binomial_strict_antimono[of k k' n]
```
```   154       show ?thesis by simp
```
```   155   next
```
```   156     case True
```
```   157     have "n choose k' \<le> n choose (Suc (n div 2))"
```
```   158     proof (cases "k' = Suc (n div 2)")
```
```   159       case False
```
```   160       with assms True not_eq have "Suc (n div 2) < k'" by simp
```
```   161       with assms binomial_strict_antimono[of "Suc (n div 2)" k' n] True
```
```   162         show ?thesis by auto
```
```   163     qed simp_all
```
```   164     also from True have "\<dots> = n choose k" by (simp add: central_binomial_odd)
```
```   165     finally show ?thesis .
```
```   166   qed
```
```   167 qed simp_all
```
```   168
```
```   169 lemma binomial_maximum: "n choose k \<le> n choose (n div 2)"
```
```   170 proof -
```
```   171   have "k \<le> n div 2 \<longleftrightarrow> 2*k \<le> n" by linarith
```
```   172   consider "2*k \<le> n" | "2*k \<ge> n" "k \<le> n" | "k > n" by linarith
```
```   173   thus ?thesis
```
```   174   proof cases
```
```   175     case 1
```
```   176     thus ?thesis by (intro binomial_mono) linarith+
```
```   177   next
```
```   178     case 2
```
```   179     thus ?thesis by (intro binomial_antimono) simp_all
```
```   180   qed (simp_all add: binomial_eq_0)
```
```   181 qed
```
```   182
```
```   183 lemma binomial_maximum': "(2*n) choose k \<le> (2*n) choose n"
```
```   184   using binomial_maximum[of "2*n"] by simp
```
```   185
```
```   186 lemma central_binomial_lower_bound:
```
```   187   assumes "n > 0"
```
```   188   shows   "4^n / (2*real n) \<le> real ((2*n) choose n)"
```
```   189 proof -
```
```   190   from binomial[of 1 1 "2*n"]
```
```   191     have "4 ^ n = (\<Sum>k=0..2*n. (2*n) choose k)"
```
```   192     by (simp add: power_mult power2_eq_square One_nat_def [symmetric] del: One_nat_def)
```
```   193   also have "{0..2*n} = {0<..<2*n} \<union> {0,2*n}" by auto
```
```   194   also have "(\<Sum>k\<in>\<dots>. (2*n) choose k) =
```
```   195                (\<Sum>k\<in>{0<..<2*n}. (2*n) choose k) + (\<Sum>k\<in>{0,2*n}. (2*n) choose k)"
```
```   196     by (subst sum.union_disjoint) auto
```
```   197   also have "(\<Sum>k\<in>{0,2*n}. (2*n) choose k) \<le> (\<Sum>k\<le>1. (n choose k)\<^sup>2)"
```
```   198     by (cases n) simp_all
```
```   199   also from assms have "\<dots> \<le> (\<Sum>k\<le>n. (n choose k)\<^sup>2)"
```
```   200     by (intro sum_mono3) auto
```
```   201   also have "\<dots> = (2*n) choose n" by (rule choose_square_sum)
```
```   202   also have "(\<Sum>k\<in>{0<..<2*n}. (2*n) choose k) \<le> (\<Sum>k\<in>{0<..<2*n}. (2*n) choose n)"
```
```   203     by (intro sum_mono binomial_maximum')
```
```   204   also have "\<dots> = card {0<..<2*n} * ((2*n) choose n)" by simp
```
```   205   also have "card {0<..<2*n} \<le> 2*n - 1" by (cases n) simp_all
```
```   206   also have "(2 * n - 1) * (2 * n choose n) + (2 * n choose n) = ((2*n) choose n) * (2*n)"
```
```   207     using assms by (simp add: algebra_simps)
```
```   208   finally have "4 ^ n \<le> (2 * n choose n) * (2 * n)" by simp_all
```
```   209   hence "real (4 ^ n) \<le> real ((2 * n choose n) * (2 * n))"
```
```   210     by (subst of_nat_le_iff)
```
```   211   with assms show ?thesis by (simp add: field_simps)
```
```   212 qed
```
```   213
```
```   214
```
```   215 subsection \<open>Properties of Power Series\<close>
```
```   216
```
```   217 lemma powser_zero [simp]: "(\<Sum>n. f n * 0 ^ n) = f 0"
```
```   218   for f :: "nat \<Rightarrow> 'a::real_normed_algebra_1"
```
```   219 proof -
```
```   220   have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
```
```   221     by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
```
```   222   then show ?thesis by simp
```
```   223 qed
```
```   224
```
```   225 lemma powser_sums_zero: "(\<lambda>n. a n * 0^n) sums a 0"
```
```   226   for a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
```
```   227   using sums_finite [of "{0}" "\<lambda>n. a n * 0 ^ n"]
```
```   228   by simp
```
```   229
```
```   230 lemma powser_sums_zero_iff [simp]: "(\<lambda>n. a n * 0^n) sums x \<longleftrightarrow> a 0 = x"
```
```   231   for a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
```
```   232   using powser_sums_zero sums_unique2 by blast
```
```   233
```
```   234 text \<open>
```
```   235   Power series has a circle or radius of convergence: if it sums for \<open>x\<close>,
```
```   236   then it sums absolutely for \<open>z\<close> with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.\<close>
```
```   237
```
```   238 lemma powser_insidea:
```
```   239   fixes x z :: "'a::real_normed_div_algebra"
```
```   240   assumes 1: "summable (\<lambda>n. f n * x^n)"
```
```   241     and 2: "norm z < norm x"
```
```   242   shows "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   243 proof -
```
```   244   from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
```
```   245   from 1 have "(\<lambda>n. f n * x^n) \<longlonglongrightarrow> 0"
```
```   246     by (rule summable_LIMSEQ_zero)
```
```   247   then have "convergent (\<lambda>n. f n * x^n)"
```
```   248     by (rule convergentI)
```
```   249   then have "Cauchy (\<lambda>n. f n * x^n)"
```
```   250     by (rule convergent_Cauchy)
```
```   251   then have "Bseq (\<lambda>n. f n * x^n)"
```
```   252     by (rule Cauchy_Bseq)
```
```   253   then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x^n) \<le> K"
```
```   254     by (auto simp add: Bseq_def)
```
```   255   have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le> K * norm (z ^ n) * inverse (norm (x^n))"
```
```   256   proof (intro exI allI impI)
```
```   257     fix n :: nat
```
```   258     assume "0 \<le> n"
```
```   259     have "norm (norm (f n * z ^ n)) * norm (x^n) =
```
```   260           norm (f n * x^n) * norm (z ^ n)"
```
```   261       by (simp add: norm_mult abs_mult)
```
```   262     also have "\<dots> \<le> K * norm (z ^ n)"
```
```   263       by (simp only: mult_right_mono 4 norm_ge_zero)
```
```   264     also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x^n)) * norm (x^n))"
```
```   265       by (simp add: x_neq_0)
```
```   266     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x^n)) * norm (x^n)"
```
```   267       by (simp only: mult.assoc)
```
```   268     finally show "norm (norm (f n * z ^ n)) \<le> K * norm (z ^ n) * inverse (norm (x^n))"
```
```   269       by (simp add: mult_le_cancel_right x_neq_0)
```
```   270   qed
```
```   271   moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
```
```   272   proof -
```
```   273     from 2 have "norm (norm (z * inverse x)) < 1"
```
```   274       using x_neq_0
```
```   275       by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
```
```   276     then have "summable (\<lambda>n. norm (z * inverse x) ^ n)"
```
```   277       by (rule summable_geometric)
```
```   278     then have "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
```
```   279       by (rule summable_mult)
```
```   280     then show "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
```
```   281       using x_neq_0
```
```   282       by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
```
```   283           power_inverse norm_power mult.assoc)
```
```   284   qed
```
```   285   ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   286     by (rule summable_comparison_test)
```
```   287 qed
```
```   288
```
```   289 lemma powser_inside:
```
```   290   fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
```
```   291   shows
```
```   292     "summable (\<lambda>n. f n * (x^n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
```
```   293       summable (\<lambda>n. f n * (z ^ n))"
```
```   294   by (rule powser_insidea [THEN summable_norm_cancel])
```
```   295
```
```   296 lemma powser_times_n_limit_0:
```
```   297   fixes x :: "'a::{real_normed_div_algebra,banach}"
```
```   298   assumes "norm x < 1"
```
```   299     shows "(\<lambda>n. of_nat n * x ^ n) \<longlonglongrightarrow> 0"
```
```   300 proof -
```
```   301   have "norm x / (1 - norm x) \<ge> 0"
```
```   302     using assms by (auto simp: divide_simps)
```
```   303   moreover obtain N where N: "norm x / (1 - norm x) < of_int N"
```
```   304     using ex_le_of_int by (meson ex_less_of_int)
```
```   305   ultimately have N0: "N>0"
```
```   306     by auto
```
```   307   then have *: "real_of_int (N + 1) * norm x / real_of_int N < 1"
```
```   308     using N assms by (auto simp: field_simps)
```
```   309   have **: "real_of_int N * (norm x * (real_of_nat (Suc n) * norm (x ^ n))) \<le>
```
```   310       real_of_nat n * (norm x * ((1 + N) * norm (x ^ n)))" if "N \<le> int n" for n :: nat
```
```   311   proof -
```
```   312     from that have "real_of_int N * real_of_nat (Suc n) \<le> real_of_nat n * real_of_int (1 + N)"
```
```   313       by (simp add: algebra_simps)
```
```   314     then have "(real_of_int N * real_of_nat (Suc n)) * (norm x * norm (x ^ n)) \<le>
```
```   315         (real_of_nat n *  (1 + N)) * (norm x * norm (x ^ n))"
```
```   316       using N0 mult_mono by fastforce
```
```   317     then show ?thesis
```
```   318       by (simp add: algebra_simps)
```
```   319   qed
```
```   320   show ?thesis using *
```
```   321     by (rule summable_LIMSEQ_zero [OF summable_ratio_test, where N1="nat N"])
```
```   322       (simp add: N0 norm_mult field_simps ** del: of_nat_Suc of_int_add)
```
```   323 qed
```
```   324
```
```   325 corollary lim_n_over_pown:
```
```   326   fixes x :: "'a::{real_normed_field,banach}"
```
```   327   shows "1 < norm x \<Longrightarrow> ((\<lambda>n. of_nat n / x^n) \<longlongrightarrow> 0) sequentially"
```
```   328   using powser_times_n_limit_0 [of "inverse x"]
```
```   329   by (simp add: norm_divide divide_simps)
```
```   330
```
```   331 lemma sum_split_even_odd:
```
```   332   fixes f :: "nat \<Rightarrow> real"
```
```   333   shows "(\<Sum>i<2 * n. if even i then f i else g i) = (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
```
```   334 proof (induct n)
```
```   335   case 0
```
```   336   then show ?case by simp
```
```   337 next
```
```   338   case (Suc n)
```
```   339   have "(\<Sum>i<2 * Suc n. if even i then f i else g i) =
```
```   340     (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
```
```   341     using Suc.hyps unfolding One_nat_def by auto
```
```   342   also have "\<dots> = (\<Sum>i<Suc n. f (2 * i)) + (\<Sum>i<Suc n. g (2 * i + 1))"
```
```   343     by auto
```
```   344   finally show ?case .
```
```   345 qed
```
```   346
```
```   347 lemma sums_if':
```
```   348   fixes g :: "nat \<Rightarrow> real"
```
```   349   assumes "g sums x"
```
```   350   shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   351   unfolding sums_def
```
```   352 proof (rule LIMSEQ_I)
```
```   353   fix r :: real
```
```   354   assume "0 < r"
```
```   355   from \<open>g sums x\<close>[unfolded sums_def, THEN LIMSEQ_D, OF this]
```
```   356   obtain no where no_eq: "\<And>n. n \<ge> no \<Longrightarrow> (norm (sum g {..<n} - x) < r)"
```
```   357     by blast
```
```   358
```
```   359   let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)"
```
```   360   have "(norm (?SUM m - x) < r)" if "m \<ge> 2 * no" for m
```
```   361   proof -
```
```   362     from that have "m div 2 \<ge> no" by auto
```
```   363     have sum_eq: "?SUM (2 * (m div 2)) = sum g {..< m div 2}"
```
```   364       using sum_split_even_odd by auto
```
```   365     then have "(norm (?SUM (2 * (m div 2)) - x) < r)"
```
```   366       using no_eq unfolding sum_eq using \<open>m div 2 \<ge> no\<close> by auto
```
```   367     moreover
```
```   368     have "?SUM (2 * (m div 2)) = ?SUM m"
```
```   369     proof (cases "even m")
```
```   370       case True
```
```   371       then show ?thesis
```
```   372         by (auto simp add: even_two_times_div_two)
```
```   373     next
```
```   374       case False
```
```   375       then have eq: "Suc (2 * (m div 2)) = m" by simp
```
```   376       then have "even (2 * (m div 2))" using \<open>odd m\<close> by auto
```
```   377       have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
```
```   378       also have "\<dots> = ?SUM (2 * (m div 2))" using \<open>even (2 * (m div 2))\<close> by auto
```
```   379       finally show ?thesis by auto
```
```   380     qed
```
```   381     ultimately show ?thesis by auto
```
```   382   qed
```
```   383   then show "\<exists>no. \<forall> m \<ge> no. norm (?SUM m - x) < r"
```
```   384     by blast
```
```   385 qed
```
```   386
```
```   387 lemma sums_if:
```
```   388   fixes g :: "nat \<Rightarrow> real"
```
```   389   assumes "g sums x" and "f sums y"
```
```   390   shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
```
```   391 proof -
```
```   392   let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
```
```   393   have if_sum: "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
```
```   394     for B T E
```
```   395     by (cases B) auto
```
```   396   have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   397     using sums_if'[OF \<open>g sums x\<close>] .
```
```   398   have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)"
```
```   399     by auto
```
```   400   have "?s sums y" using sums_if'[OF \<open>f sums y\<close>] .
```
```   401   from this[unfolded sums_def, THEN LIMSEQ_Suc]
```
```   402   have "(\<lambda>n. if even n then f (n div 2) else 0) sums y"
```
```   403     by (simp add: lessThan_Suc_eq_insert_0 sum_atLeast1_atMost_eq image_Suc_lessThan
```
```   404         if_eq sums_def cong del: if_weak_cong)
```
```   405   from sums_add[OF g_sums this] show ?thesis
```
```   406     by (simp only: if_sum)
```
```   407 qed
```
```   408
```
```   409 subsection \<open>Alternating series test / Leibniz formula\<close>
```
```   410 (* FIXME: generalise these results from the reals via type classes? *)
```
```   411
```
```   412 lemma sums_alternating_upper_lower:
```
```   413   fixes a :: "nat \<Rightarrow> real"
```
```   414   assumes mono: "\<And>n. a (Suc n) \<le> a n"
```
```   415     and a_pos: "\<And>n. 0 \<le> a n"
```
```   416     and "a \<longlonglongrightarrow> 0"
```
```   417   shows "\<exists>l. ((\<forall>n. (\<Sum>i<2*n. (- 1)^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i<2*n. (- 1)^i*a i) \<longlonglongrightarrow> l) \<and>
```
```   418              ((\<forall>n. l \<le> (\<Sum>i<2*n + 1. (- 1)^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. (- 1)^i*a i) \<longlonglongrightarrow> l)"
```
```   419   (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
```
```   420 proof (rule nested_sequence_unique)
```
```   421   have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" by auto
```
```   422
```
```   423   show "\<forall>n. ?f n \<le> ?f (Suc n)"
```
```   424   proof
```
```   425     show "?f n \<le> ?f (Suc n)" for n
```
```   426       using mono[of "2*n"] by auto
```
```   427   qed
```
```   428   show "\<forall>n. ?g (Suc n) \<le> ?g n"
```
```   429   proof
```
```   430     show "?g (Suc n) \<le> ?g n" for n
```
```   431       using mono[of "Suc (2*n)"] by auto
```
```   432   qed
```
```   433   show "\<forall>n. ?f n \<le> ?g n"
```
```   434   proof
```
```   435     show "?f n \<le> ?g n" for n
```
```   436       using fg_diff a_pos by auto
```
```   437   qed
```
```   438   show "(\<lambda>n. ?f n - ?g n) \<longlonglongrightarrow> 0"
```
```   439     unfolding fg_diff
```
```   440   proof (rule LIMSEQ_I)
```
```   441     fix r :: real
```
```   442     assume "0 < r"
```
```   443     with \<open>a \<longlonglongrightarrow> 0\<close>[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
```
```   444       by auto
```
```   445     then have "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r"
```
```   446       by auto
```
```   447     then show "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r"
```
```   448       by auto
```
```   449   qed
```
```   450 qed
```
```   451
```
```   452 lemma summable_Leibniz':
```
```   453   fixes a :: "nat \<Rightarrow> real"
```
```   454   assumes a_zero: "a \<longlonglongrightarrow> 0"
```
```   455     and a_pos: "\<And>n. 0 \<le> a n"
```
```   456     and a_monotone: "\<And>n. a (Suc n) \<le> a n"
```
```   457   shows summable: "summable (\<lambda> n. (-1)^n * a n)"
```
```   458     and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
```
```   459     and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)"
```
```   460     and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i<2*n+1. (-1)^i*a i)"
```
```   461     and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)"
```
```   462 proof -
```
```   463   let ?S = "\<lambda>n. (-1)^n * a n"
```
```   464   let ?P = "\<lambda>n. \<Sum>i<n. ?S i"
```
```   465   let ?f = "\<lambda>n. ?P (2 * n)"
```
```   466   let ?g = "\<lambda>n. ?P (2 * n + 1)"
```
```   467   obtain l :: real
```
```   468     where below_l: "\<forall> n. ?f n \<le> l"
```
```   469       and "?f \<longlonglongrightarrow> l"
```
```   470       and above_l: "\<forall> n. l \<le> ?g n"
```
```   471       and "?g \<longlonglongrightarrow> l"
```
```   472     using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
```
```   473
```
```   474   let ?Sa = "\<lambda>m. \<Sum>n<m. ?S n"
```
```   475   have "?Sa \<longlonglongrightarrow> l"
```
```   476   proof (rule LIMSEQ_I)
```
```   477     fix r :: real
```
```   478     assume "0 < r"
```
```   479     with \<open>?f \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
```
```   480     obtain f_no where f: "\<And>n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r"
```
```   481       by auto
```
```   482     from \<open>0 < r\<close> \<open>?g \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
```
```   483     obtain g_no where g: "\<And>n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r"
```
```   484       by auto
```
```   485     have "norm (?Sa n - l) < r" if "n \<ge> (max (2 * f_no) (2 * g_no))" for n
```
```   486     proof -
```
```   487       from that have "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
```
```   488       show ?thesis
```
```   489       proof (cases "even n")
```
```   490         case True
```
```   491         then have n_eq: "2 * (n div 2) = n"
```
```   492           by (simp add: even_two_times_div_two)
```
```   493         with \<open>n \<ge> 2 * f_no\<close> have "n div 2 \<ge> f_no"
```
```   494           by auto
```
```   495         from f[OF this] show ?thesis
```
```   496           unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
```
```   497       next
```
```   498         case False
```
```   499         then have "even (n - 1)" by simp
```
```   500         then have n_eq: "2 * ((n - 1) div 2) = n - 1"
```
```   501           by (simp add: even_two_times_div_two)
```
```   502         then have range_eq: "n - 1 + 1 = n"
```
```   503           using odd_pos[OF False] by auto
```
```   504         from n_eq \<open>n \<ge> 2 * g_no\<close> have "(n - 1) div 2 \<ge> g_no"
```
```   505           by auto
```
```   506         from g[OF this] show ?thesis
```
```   507           by (simp only: n_eq range_eq)
```
```   508       qed
```
```   509     qed
```
```   510     then show "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
```
```   511   qed
```
```   512   then have sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
```
```   513     by (simp only: sums_def)
```
```   514   then show "summable ?S"
```
```   515     by (auto simp: summable_def)
```
```   516
```
```   517   have "l = suminf ?S" by (rule sums_unique[OF sums_l])
```
```   518
```
```   519   fix n
```
```   520   show "suminf ?S \<le> ?g n"
```
```   521     unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
```
```   522   show "?f n \<le> suminf ?S"
```
```   523     unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
```
```   524   show "?g \<longlonglongrightarrow> suminf ?S"
```
```   525     using \<open>?g \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto
```
```   526   show "?f \<longlonglongrightarrow> suminf ?S"
```
```   527     using \<open>?f \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto
```
```   528 qed
```
```   529
```
```   530 theorem summable_Leibniz:
```
```   531   fixes a :: "nat \<Rightarrow> real"
```
```   532   assumes a_zero: "a \<longlonglongrightarrow> 0"
```
```   533     and "monoseq a"
```
```   534   shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
```
```   535     and "0 < a 0 \<longrightarrow>
```
```   536       (\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n. (- 1)^i * a i .. \<Sum>i<2*n+1. (- 1)^i * a i})" (is "?pos")
```
```   537     and "a 0 < 0 \<longrightarrow>
```
```   538       (\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n+1. (- 1)^i * a i .. \<Sum>i<2*n. (- 1)^i * a i})" (is "?neg")
```
```   539     and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?f")
```
```   540     and "(\<lambda>n. \<Sum>i<2*n+1. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?g")
```
```   541 proof -
```
```   542   have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
```
```   543   proof (cases "(\<forall>n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
```
```   544     case True
```
```   545     then have ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m"
```
```   546       and ge0: "\<And>n. 0 \<le> a n"
```
```   547       by auto
```
```   548     have mono: "a (Suc n) \<le> a n" for n
```
```   549       using ord[where n="Suc n" and m=n] by auto
```
```   550     note leibniz = summable_Leibniz'[OF \<open>a \<longlonglongrightarrow> 0\<close> ge0]
```
```   551     from leibniz[OF mono]
```
```   552     show ?thesis using \<open>0 \<le> a 0\<close> by auto
```
```   553   next
```
```   554     let ?a = "\<lambda>n. - a n"
```
```   555     case False
```
```   556     with monoseq_le[OF \<open>monoseq a\<close> \<open>a \<longlonglongrightarrow> 0\<close>]
```
```   557     have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
```
```   558     then have ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
```
```   559       by auto
```
```   560     have monotone: "?a (Suc n) \<le> ?a n" for n
```
```   561       using ord[where n="Suc n" and m=n] by auto
```
```   562     note leibniz =
```
```   563       summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
```
```   564         OF tendsto_minus[OF \<open>a \<longlonglongrightarrow> 0\<close>, unfolded minus_zero] monotone]
```
```   565     have "summable (\<lambda> n. (-1)^n * ?a n)"
```
```   566       using leibniz(1) by auto
```
```   567     then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
```
```   568       unfolding summable_def by auto
```
```   569     from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
```
```   570       by auto
```
```   571     then have ?summable by (auto simp: summable_def)
```
```   572     moreover
```
```   573     have "\<bar>- a - - b\<bar> = \<bar>a - b\<bar>" for a b :: real
```
```   574       unfolding minus_diff_minus by auto
```
```   575
```
```   576     from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
```
```   577     have move_minus: "(\<Sum>n. - ((- 1) ^ n * a n)) = - (\<Sum>n. (- 1) ^ n * a n)"
```
```   578       by auto
```
```   579
```
```   580     have ?pos using \<open>0 \<le> ?a 0\<close> by auto
```
```   581     moreover have ?neg
```
```   582       using leibniz(2,4)
```
```   583       unfolding mult_minus_right sum_negf move_minus neg_le_iff_le
```
```   584       by auto
```
```   585     moreover have ?f and ?g
```
```   586       using leibniz(3,5)[unfolded mult_minus_right sum_negf move_minus, THEN tendsto_minus_cancel]
```
```   587       by auto
```
```   588     ultimately show ?thesis by auto
```
```   589   qed
```
```   590   then show ?summable and ?pos and ?neg and ?f and ?g
```
```   591     by safe
```
```   592 qed
```
```   593
```
```   594
```
```   595 subsection \<open>Term-by-Term Differentiability of Power Series\<close>
```
```   596
```
```   597 definition diffs :: "(nat \<Rightarrow> 'a::ring_1) \<Rightarrow> nat \<Rightarrow> 'a"
```
```   598   where "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))"
```
```   599
```
```   600 text \<open>Lemma about distributing negation over it.\<close>
```
```   601 lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
```
```   602   by (simp add: diffs_def)
```
```   603
```
```   604 lemma diffs_equiv:
```
```   605   fixes x :: "'a::{real_normed_vector,ring_1}"
```
```   606   shows "summable (\<lambda>n. diffs c n * x^n) \<Longrightarrow>
```
```   607     (\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)"
```
```   608   unfolding diffs_def
```
```   609   by (simp add: summable_sums sums_Suc_imp)
```
```   610
```
```   611 lemma lemma_termdiff1:
```
```   612   fixes z :: "'a :: {monoid_mult,comm_ring}"
```
```   613   shows "(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
```
```   614     (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
```
```   615   by (auto simp add: algebra_simps power_add [symmetric])
```
```   616
```
```   617 lemma sumr_diff_mult_const2: "sum f {..<n} - of_nat n * r = (\<Sum>i<n. f i - r)"
```
```   618   for r :: "'a::ring_1"
```
```   619   by (simp add: sum_subtractf)
```
```   620
```
```   621 lemma lemma_realpow_rev_sumr:
```
```   622   "(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) = (\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))"
```
```   623   by (subst nat_diff_sum_reindex[symmetric]) simp
```
```   624
```
```   625 lemma lemma_termdiff2:
```
```   626   fixes h :: "'a::field"
```
```   627   assumes h: "h \<noteq> 0"
```
```   628   shows "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
```
```   629     h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))"
```
```   630     (is "?lhs = ?rhs")
```
```   631   apply (subgoal_tac "h * ?lhs = h * ?rhs")
```
```   632    apply (simp add: h)
```
```   633   apply (simp add: right_diff_distrib diff_divide_distrib h)
```
```   634   apply (simp add: mult.assoc [symmetric])
```
```   635   apply (cases n)
```
```   636   apply simp
```
```   637   apply (simp add: diff_power_eq_sum h right_diff_distrib [symmetric] mult.assoc
```
```   638       del: power_Suc sum_lessThan_Suc of_nat_Suc)
```
```   639   apply (subst lemma_realpow_rev_sumr)
```
```   640   apply (subst sumr_diff_mult_const2)
```
```   641   apply simp
```
```   642   apply (simp only: lemma_termdiff1 sum_distrib_left)
```
```   643   apply (rule sum.cong [OF refl])
```
```   644   apply (simp add: less_iff_Suc_add)
```
```   645   apply clarify
```
```   646   apply (simp add: sum_distrib_left diff_power_eq_sum ac_simps
```
```   647       del: sum_lessThan_Suc power_Suc)
```
```   648   apply (subst mult.assoc [symmetric], subst power_add [symmetric])
```
```   649   apply (simp add: ac_simps)
```
```   650   done
```
```   651
```
```   652 lemma real_sum_nat_ivl_bounded2:
```
```   653   fixes K :: "'a::linordered_semidom"
```
```   654   assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
```
```   655     and K: "0 \<le> K"
```
```   656   shows "sum f {..<n-k} \<le> of_nat n * K"
```
```   657   apply (rule order_trans [OF sum_mono])
```
```   658    apply (rule f)
```
```   659    apply simp
```
```   660   apply (simp add: mult_right_mono K)
```
```   661   done
```
```   662
```
```   663 lemma lemma_termdiff3:
```
```   664   fixes h z :: "'a::real_normed_field"
```
```   665   assumes 1: "h \<noteq> 0"
```
```   666     and 2: "norm z \<le> K"
```
```   667     and 3: "norm (z + h) \<le> K"
```
```   668   shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) \<le>
```
```   669     of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   670 proof -
```
```   671   have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
```
```   672     norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
```
```   673     by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult)
```
```   674   also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
```
```   675   proof (rule mult_right_mono [OF _ norm_ge_zero])
```
```   676     from norm_ge_zero 2 have K: "0 \<le> K"
```
```   677       by (rule order_trans)
```
```   678     have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
```
```   679       apply (erule subst)
```
```   680       apply (simp only: norm_mult norm_power power_add)
```
```   681       apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
```
```   682       done
```
```   683     show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) \<le>
```
```   684         of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
```
```   685       apply (intro
```
```   686           order_trans [OF norm_sum]
```
```   687           real_sum_nat_ivl_bounded2
```
```   688           mult_nonneg_nonneg
```
```   689           of_nat_0_le_iff
```
```   690           zero_le_power K)
```
```   691       apply (rule le_Kn)
```
```   692       apply simp
```
```   693       done
```
```   694   qed
```
```   695   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   696     by (simp only: mult.assoc)
```
```   697   finally show ?thesis .
```
```   698 qed
```
```   699
```
```   700 lemma lemma_termdiff4:
```
```   701   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```   702     and k :: real
```
```   703   assumes k: "0 < k"
```
```   704     and le: "\<And>h. h \<noteq> 0 \<Longrightarrow> norm h < k \<Longrightarrow> norm (f h) \<le> K * norm h"
```
```   705   shows "f \<midarrow>0\<rightarrow> 0"
```
```   706 proof (rule tendsto_norm_zero_cancel)
```
```   707   show "(\<lambda>h. norm (f h)) \<midarrow>0\<rightarrow> 0"
```
```   708   proof (rule real_tendsto_sandwich)
```
```   709     show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)"
```
```   710       by simp
```
```   711     show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)"
```
```   712       using k by (auto simp add: eventually_at dist_norm le)
```
```   713     show "(\<lambda>h. 0) \<midarrow>(0::'a)\<rightarrow> (0::real)"
```
```   714       by (rule tendsto_const)
```
```   715     have "(\<lambda>h. K * norm h) \<midarrow>(0::'a)\<rightarrow> K * norm (0::'a)"
```
```   716       by (intro tendsto_intros)
```
```   717     then show "(\<lambda>h. K * norm h) \<midarrow>(0::'a)\<rightarrow> 0"
```
```   718       by simp
```
```   719   qed
```
```   720 qed
```
```   721
```
```   722 lemma lemma_termdiff5:
```
```   723   fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
```
```   724     and k :: real
```
```   725   assumes k: "0 < k"
```
```   726     and f: "summable f"
```
```   727     and le: "\<And>h n. h \<noteq> 0 \<Longrightarrow> norm h < k \<Longrightarrow> norm (g h n) \<le> f n * norm h"
```
```   728   shows "(\<lambda>h. suminf (g h)) \<midarrow>0\<rightarrow> 0"
```
```   729 proof (rule lemma_termdiff4 [OF k])
```
```   730   fix h :: 'a
```
```   731   assume "h \<noteq> 0" and "norm h < k"
```
```   732   then have 1: "\<forall>n. norm (g h n) \<le> f n * norm h"
```
```   733     by (simp add: le)
```
```   734   then have "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
```
```   735     by simp
```
```   736   moreover from f have 2: "summable (\<lambda>n. f n * norm h)"
```
```   737     by (rule summable_mult2)
```
```   738   ultimately have 3: "summable (\<lambda>n. norm (g h n))"
```
```   739     by (rule summable_comparison_test)
```
```   740   then have "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
```
```   741     by (rule summable_norm)
```
```   742   also from 1 3 2 have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
```
```   743     by (rule suminf_le)
```
```   744   also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
```
```   745     by (rule suminf_mult2 [symmetric])
```
```   746   finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
```
```   747 qed
```
```   748
```
```   749
```
```   750 (* FIXME: Long proofs *)
```
```   751
```
```   752 lemma termdiffs_aux:
```
```   753   fixes x :: "'a::{real_normed_field,banach}"
```
```   754   assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
```
```   755     and 2: "norm x < norm K"
```
```   756   shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0"
```
```   757 proof -
```
```   758   from dense [OF 2] obtain r where r1: "norm x < r" and r2: "r < norm K"
```
```   759     by fast
```
```   760   from norm_ge_zero r1 have r: "0 < r"
```
```   761     by (rule order_le_less_trans)
```
```   762   then have r_neq_0: "r \<noteq> 0" by simp
```
```   763   show ?thesis
```
```   764   proof (rule lemma_termdiff5)
```
```   765     show "0 < r - norm x"
```
```   766       using r1 by simp
```
```   767     from r r2 have "norm (of_real r::'a) < norm K"
```
```   768       by simp
```
```   769     with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
```
```   770       by (rule powser_insidea)
```
```   771     then have "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
```
```   772       using r by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
```
```   773     then have "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
```
```   774       by (rule diffs_equiv [THEN sums_summable])
```
```   775     also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
```
```   776       (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
```
```   777       apply (rule ext)
```
```   778       apply (simp add: diffs_def)
```
```   779       apply (case_tac n)
```
```   780        apply (simp_all add: r_neq_0)
```
```   781       done
```
```   782     finally have "summable
```
```   783       (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
```
```   784       by (rule diffs_equiv [THEN sums_summable])
```
```   785     also have
```
```   786       "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0)) =
```
```   787        (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
```
```   788       apply (rule ext)
```
```   789       apply (case_tac n)
```
```   790        apply simp
```
```   791       apply (rename_tac nat)
```
```   792       apply (case_tac nat)
```
```   793        apply simp
```
```   794       apply (simp add: r_neq_0)
```
```   795       done
```
```   796     finally show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
```
```   797   next
```
```   798     fix h :: 'a
```
```   799     fix n :: nat
```
```   800     assume h: "h \<noteq> 0"
```
```   801     assume "norm h < r - norm x"
```
```   802     then have "norm x + norm h < r" by simp
```
```   803     with norm_triangle_ineq have xh: "norm (x + h) < r"
```
```   804       by (rule order_le_less_trans)
```
```   805     show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<le>
```
```   806       norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
```
```   807       apply (simp only: norm_mult mult.assoc)
```
```   808       apply (rule mult_left_mono [OF _ norm_ge_zero])
```
```   809       apply (simp add: mult.assoc [symmetric])
```
```   810       apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
```
```   811       done
```
```   812   qed
```
```   813 qed
```
```   814
```
```   815 lemma termdiffs:
```
```   816   fixes K x :: "'a::{real_normed_field,banach}"
```
```   817   assumes 1: "summable (\<lambda>n. c n * K ^ n)"
```
```   818     and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
```
```   819     and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
```
```   820     and 4: "norm x < norm K"
```
```   821   shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. (diffs c) n * x^n)"
```
```   822   unfolding DERIV_def
```
```   823 proof (rule LIM_zero_cancel)
```
```   824   show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x^n)) / h
```
```   825             - suminf (\<lambda>n. diffs c n * x^n)) \<midarrow>0\<rightarrow> 0"
```
```   826   proof (rule LIM_equal2)
```
```   827     show "0 < norm K - norm x"
```
```   828       using 4 by (simp add: less_diff_eq)
```
```   829   next
```
```   830     fix h :: 'a
```
```   831     assume "norm (h - 0) < norm K - norm x"
```
```   832     then have "norm x + norm h < norm K" by simp
```
```   833     then have 5: "norm (x + h) < norm K"
```
```   834       by (rule norm_triangle_ineq [THEN order_le_less_trans])
```
```   835     have "summable (\<lambda>n. c n * x^n)"
```
```   836       and "summable (\<lambda>n. c n * (x + h) ^ n)"
```
```   837       and "summable (\<lambda>n. diffs c n * x^n)"
```
```   838       using 1 2 4 5 by (auto elim: powser_inside)
```
```   839     then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
```
```   840           (\<Sum>n. (c n * (x + h) ^ n - c n * x^n) / h - of_nat n * c n * x ^ (n - Suc 0))"
```
```   841       by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
```
```   842     then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
```
```   843           (\<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))"
```
```   844       by (simp add: algebra_simps)
```
```   845   next
```
```   846     show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0"
```
```   847       by (rule termdiffs_aux [OF 3 4])
```
```   848   qed
```
```   849 qed
```
```   850
```
```   851 subsection \<open>The Derivative of a Power Series Has the Same Radius of Convergence\<close>
```
```   852
```
```   853 lemma termdiff_converges:
```
```   854   fixes x :: "'a::{real_normed_field,banach}"
```
```   855   assumes K: "norm x < K"
```
```   856     and sm: "\<And>x. norm x < K \<Longrightarrow> summable(\<lambda>n. c n * x ^ n)"
```
```   857   shows "summable (\<lambda>n. diffs c n * x ^ n)"
```
```   858 proof (cases "x = 0")
```
```   859   case True
```
```   860   then show ?thesis
```
```   861     using powser_sums_zero sums_summable by auto
```
```   862 next
```
```   863   case False
```
```   864   then have "K > 0"
```
```   865     using K less_trans zero_less_norm_iff by blast
```
```   866   then obtain r :: real where r: "norm x < norm r" "norm r < K" "r > 0"
```
```   867     using K False
```
```   868     by (auto simp: field_simps abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"])
```
```   869   have "(\<lambda>n. of_nat n * (x / of_real r) ^ n) \<longlonglongrightarrow> 0"
```
```   870     using r by (simp add: norm_divide powser_times_n_limit_0 [of "x / of_real r"])
```
```   871   then obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> real_of_nat n * norm x ^ n < r ^ n"
```
```   872     using r unfolding LIMSEQ_iff
```
```   873     apply (drule_tac x=1 in spec)
```
```   874     apply (auto simp: norm_divide norm_mult norm_power field_simps)
```
```   875     done
```
```   876   have "summable (\<lambda>n. (of_nat n * c n) * x ^ n)"
```
```   877     apply (rule summable_comparison_test' [of "\<lambda>n. norm(c n * (of_real r) ^ n)" N])
```
```   878      apply (rule powser_insidea [OF sm [of "of_real ((r+K)/2)"]])
```
```   879     using N r norm_of_real [of "r + K", where 'a = 'a]
```
```   880       apply (auto simp add: norm_divide norm_mult norm_power field_simps)
```
```   881     apply (fastforce simp: less_eq_real_def)
```
```   882     done
```
```   883   then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ Suc n)"
```
```   884     using summable_iff_shift [of "\<lambda>n. of_nat n * c n * x ^ n" 1]
```
```   885     by simp
```
```   886   then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ n)"
```
```   887     using False summable_mult2 [of "\<lambda>n. (of_nat (Suc n) * c(Suc n) * x ^ n) * x" "inverse x"]
```
```   888     by (simp add: mult.assoc) (auto simp: ac_simps)
```
```   889   then show ?thesis
```
```   890     by (simp add: diffs_def)
```
```   891 qed
```
```   892
```
```   893 lemma termdiff_converges_all:
```
```   894   fixes x :: "'a::{real_normed_field,banach}"
```
```   895   assumes "\<And>x. summable (\<lambda>n. c n * x^n)"
```
```   896   shows "summable (\<lambda>n. diffs c n * x^n)"
```
```   897   apply (rule termdiff_converges [where K = "1 + norm x"])
```
```   898   using assms
```
```   899    apply auto
```
```   900   done
```
```   901
```
```   902 lemma termdiffs_strong:
```
```   903   fixes K x :: "'a::{real_normed_field,banach}"
```
```   904   assumes sm: "summable (\<lambda>n. c n * K ^ n)"
```
```   905     and K: "norm x < norm K"
```
```   906   shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. diffs c n * x^n)"
```
```   907 proof -
```
```   908   have K2: "norm ((of_real (norm K) + of_real (norm x)) / 2 :: 'a) < norm K"
```
```   909     using K
```
```   910     apply (auto simp: norm_divide field_simps)
```
```   911     apply (rule le_less_trans [of _ "of_real (norm K) + of_real (norm x)"])
```
```   912      apply (auto simp: mult_2_right norm_triangle_mono)
```
```   913     done
```
```   914   then have [simp]: "norm ((of_real (norm K) + of_real (norm x)) :: 'a) < norm K * 2"
```
```   915     by simp
```
```   916   have "summable (\<lambda>n. c n * (of_real (norm x + norm K) / 2) ^ n)"
```
```   917     by (metis K2 summable_norm_cancel [OF powser_insidea [OF sm]] add.commute of_real_add)
```
```   918   moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs c n * x ^ n)"
```
```   919     by (blast intro: sm termdiff_converges powser_inside)
```
```   920   moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs(diffs c) n * x ^ n)"
```
```   921     by (blast intro: sm termdiff_converges powser_inside)
```
```   922   ultimately show ?thesis
```
```   923     apply (rule termdiffs [where K = "of_real (norm x + norm K) / 2"])
```
```   924       apply (auto simp: field_simps)
```
```   925     using K
```
```   926     apply (simp_all add: of_real_add [symmetric] del: of_real_add)
```
```   927     done
```
```   928 qed
```
```   929
```
```   930 lemma termdiffs_strong_converges_everywhere:
```
```   931   fixes K x :: "'a::{real_normed_field,banach}"
```
```   932   assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)"
```
```   933   shows "((\<lambda>x. \<Sum>n. c n * x^n) has_field_derivative (\<Sum>n. diffs c n * x^n)) (at x)"
```
```   934   using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
```
```   935   by (force simp del: of_real_add)
```
```   936
```
```   937 lemma termdiffs_strong':
```
```   938   fixes z :: "'a :: {real_normed_field,banach}"
```
```   939   assumes "\<And>z. norm z < K \<Longrightarrow> summable (\<lambda>n. c n * z ^ n)"
```
```   940   assumes "norm z < K"
```
```   941   shows   "((\<lambda>z. \<Sum>n. c n * z^n) has_field_derivative (\<Sum>n. diffs c n * z^n)) (at z)"
```
```   942 proof (rule termdiffs_strong)
```
```   943   define L :: real where "L =  (norm z + K) / 2"
```
```   944   have "0 \<le> norm z" by simp
```
```   945   also note \<open>norm z < K\<close>
```
```   946   finally have K: "K \<ge> 0" by simp
```
```   947   from assms K have L: "L \<ge> 0" "norm z < L" "L < K" by (simp_all add: L_def)
```
```   948   from L show "norm z < norm (of_real L :: 'a)" by simp
```
```   949   from L show "summable (\<lambda>n. c n * of_real L ^ n)" by (intro assms(1)) simp_all
```
```   950 qed
```
```   951
```
```   952 lemma termdiffs_sums_strong:
```
```   953   fixes z :: "'a :: {banach,real_normed_field}"
```
```   954   assumes sums: "\<And>z. norm z < K \<Longrightarrow> (\<lambda>n. c n * z ^ n) sums f z"
```
```   955   assumes deriv: "(f has_field_derivative f') (at z)"
```
```   956   assumes norm: "norm z < K"
```
```   957   shows   "(\<lambda>n. diffs c n * z ^ n) sums f'"
```
```   958 proof -
```
```   959   have summable: "summable (\<lambda>n. diffs c n * z^n)"
```
```   960     by (intro termdiff_converges[OF norm] sums_summable[OF sums])
```
```   961   from norm have "eventually (\<lambda>z. z \<in> norm -` {..<K}) (nhds z)"
```
```   962     by (intro eventually_nhds_in_open open_vimage)
```
```   963        (simp_all add: continuous_on_norm continuous_on_id)
```
```   964   hence eq: "eventually (\<lambda>z. (\<Sum>n. c n * z^n) = f z) (nhds z)"
```
```   965     by eventually_elim (insert sums, simp add: sums_iff)
```
```   966
```
```   967   have "((\<lambda>z. \<Sum>n. c n * z^n) has_field_derivative (\<Sum>n. diffs c n * z^n)) (at z)"
```
```   968     by (intro termdiffs_strong'[OF _ norm] sums_summable[OF sums])
```
```   969   hence "(f has_field_derivative (\<Sum>n. diffs c n * z^n)) (at z)"
```
```   970     by (subst (asm) DERIV_cong_ev[OF refl eq refl])
```
```   971   from this and deriv have "(\<Sum>n. diffs c n * z^n) = f'" by (rule DERIV_unique)
```
```   972   with summable show ?thesis by (simp add: sums_iff)
```
```   973 qed
```
```   974
```
```   975 lemma isCont_powser:
```
```   976   fixes K x :: "'a::{real_normed_field,banach}"
```
```   977   assumes "summable (\<lambda>n. c n * K ^ n)"
```
```   978   assumes "norm x < norm K"
```
```   979   shows "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
```
```   980   using termdiffs_strong[OF assms] by (blast intro!: DERIV_isCont)
```
```   981
```
```   982 lemmas isCont_powser' = isCont_o2[OF _ isCont_powser]
```
```   983
```
```   984 lemma isCont_powser_converges_everywhere:
```
```   985   fixes K x :: "'a::{real_normed_field,banach}"
```
```   986   assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)"
```
```   987   shows "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
```
```   988   using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
```
```   989   by (force intro!: DERIV_isCont simp del: of_real_add)
```
```   990
```
```   991 lemma powser_limit_0:
```
```   992   fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
```
```   993   assumes s: "0 < s"
```
```   994     and sm: "\<And>x. norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
```
```   995   shows "(f \<longlongrightarrow> a 0) (at 0)"
```
```   996 proof -
```
```   997   have "summable (\<lambda>n. a n * (of_real s / 2) ^ n)"
```
```   998     apply (rule sums_summable [where l = "f (of_real s / 2)", OF sm])
```
```   999     using s
```
```  1000     apply (auto simp: norm_divide)
```
```  1001     done
```
```  1002   then have "((\<lambda>x. \<Sum>n. a n * x ^ n) has_field_derivative (\<Sum>n. diffs a n * 0 ^ n)) (at 0)"
```
```  1003     apply (rule termdiffs_strong)
```
```  1004     using s
```
```  1005     apply (auto simp: norm_divide)
```
```  1006     done
```
```  1007   then have "isCont (\<lambda>x. \<Sum>n. a n * x ^ n) 0"
```
```  1008     by (blast intro: DERIV_continuous)
```
```  1009   then have "((\<lambda>x. \<Sum>n. a n * x ^ n) \<longlongrightarrow> a 0) (at 0)"
```
```  1010     by (simp add: continuous_within)
```
```  1011   then show ?thesis
```
```  1012     apply (rule Lim_transform)
```
```  1013     apply (auto simp add: LIM_eq)
```
```  1014     apply (rule_tac x="s" in exI)
```
```  1015     using s
```
```  1016     apply (auto simp: sm [THEN sums_unique])
```
```  1017     done
```
```  1018 qed
```
```  1019
```
```  1020 lemma powser_limit_0_strong:
```
```  1021   fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  1022   assumes s: "0 < s"
```
```  1023     and sm: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
```
```  1024   shows "(f \<longlongrightarrow> a 0) (at 0)"
```
```  1025 proof -
```
```  1026   have *: "((\<lambda>x. if x = 0 then a 0 else f x) \<longlongrightarrow> a 0) (at 0)"
```
```  1027     apply (rule powser_limit_0 [OF s])
```
```  1028     apply (case_tac "x = 0")
```
```  1029      apply (auto simp add: powser_sums_zero sm)
```
```  1030     done
```
```  1031   show ?thesis
```
```  1032     apply (subst LIM_equal [where g = "(\<lambda>x. if x = 0 then a 0 else f x)"])
```
```  1033      apply (simp_all add: *)
```
```  1034     done
```
```  1035 qed
```
```  1036
```
```  1037
```
```  1038 subsection \<open>Derivability of power series\<close>
```
```  1039
```
```  1040 lemma DERIV_series':
```
```  1041   fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
```
```  1042   assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
```
```  1043     and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)"
```
```  1044     and x0_in_I: "x0 \<in> {a <..< b}"
```
```  1045     and "summable (f' x0)"
```
```  1046     and "summable L"
```
```  1047     and L_def: "\<And>n x y. x \<in> {a <..< b} \<Longrightarrow> y \<in> {a <..< b} \<Longrightarrow> \<bar>f x n - f y n\<bar> \<le> L n * \<bar>x - y\<bar>"
```
```  1048   shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
```
```  1049   unfolding DERIV_def
```
```  1050 proof (rule LIM_I)
```
```  1051   fix r :: real
```
```  1052   assume "0 < r" then have "0 < r/3" by auto
```
```  1053
```
```  1054   obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
```
```  1055     using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable L\<close>] by auto
```
```  1056
```
```  1057   obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
```
```  1058     using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable (f' x0)\<close>] by auto
```
```  1059
```
```  1060   let ?N = "Suc (max N_L N_f')"
```
```  1061   have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3")
```
```  1062     and L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3"
```
```  1063     using N_L[of "?N"] and N_f' [of "?N"] by auto
```
```  1064
```
```  1065   let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
```
```  1066
```
```  1067   let ?r = "r / (3 * real ?N)"
```
```  1068   from \<open>0 < r\<close> have "0 < ?r" by simp
```
```  1069
```
```  1070   let ?s = "\<lambda>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)"
```
```  1071   define S' where "S' = Min (?s ` {..< ?N })"
```
```  1072
```
```  1073   have "0 < S'"
```
```  1074     unfolding S'_def
```
```  1075   proof (rule iffD2[OF Min_gr_iff])
```
```  1076     show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
```
```  1077     proof
```
```  1078       fix x
```
```  1079       assume "x \<in> ?s ` {..<?N}"
```
```  1080       then obtain n where "x = ?s n" and "n \<in> {..<?N}"
```
```  1081         using image_iff[THEN iffD1] by blast
```
```  1082       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>, unfolded real_norm_def]
```
```  1083       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)"
```
```  1084         by auto
```
```  1085       have "0 < ?s n"
```
```  1086         by (rule someI2[where a=s]) (auto simp add: s_bound simp del: of_nat_Suc)
```
```  1087       then show "0 < x" by (simp only: \<open>x = ?s n\<close>)
```
```  1088     qed
```
```  1089   qed auto
```
```  1090
```
```  1091   define S where "S = min (min (x0 - a) (b - x0)) S'"
```
```  1092   then have "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
```
```  1093     and "S \<le> S'" using x0_in_I and \<open>0 < S'\<close>
```
```  1094     by auto
```
```  1095
```
```  1096   have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
```
```  1097     if "x \<noteq> 0" and "\<bar>x\<bar> < S" for x
```
```  1098   proof -
```
```  1099     from that have x_in_I: "x0 + x \<in> {a <..< b}"
```
```  1100       using S_a S_b by auto
```
```  1101
```
```  1102     note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```  1103     note div_smbl = summable_divide[OF diff_smbl]
```
```  1104     note all_smbl = summable_diff[OF div_smbl \<open>summable (f' x0)\<close>]
```
```  1105     note ign = summable_ignore_initial_segment[where k="?N"]
```
```  1106     note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
```
```  1107     note div_shft_smbl = summable_divide[OF diff_shft_smbl]
```
```  1108     note all_shft_smbl = summable_diff[OF div_smbl ign[OF \<open>summable (f' x0)\<close>]]
```
```  1109
```
```  1110     have 1: "\<bar>(\<bar>?diff (n + ?N) x\<bar>)\<bar> \<le> L (n + ?N)" for n
```
```  1111     proof -
```
```  1112       have "\<bar>?diff (n + ?N) x\<bar> \<le> L (n + ?N) * \<bar>(x0 + x) - x0\<bar> / \<bar>x\<bar>"
```
```  1113         using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
```
```  1114         by (simp only: abs_divide)
```
```  1115       with \<open>x \<noteq> 0\<close> show ?thesis by auto
```
```  1116     qed
```
```  1117     note 2 = summable_rabs_comparison_test[OF _ ign[OF \<open>summable L\<close>]]
```
```  1118     from 1 have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))"
```
```  1119       by (metis (lifting) abs_idempotent
```
```  1120           order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF \<open>summable L\<close>]]])
```
```  1121     then have "\<bar>\<Sum>i. ?diff (i + ?N) x\<bar> \<le> r / 3" (is "?L_part \<le> r/3")
```
```  1122       using L_estimate by auto
```
```  1123
```
```  1124     have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n\<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n\<bar>)" ..
```
```  1125     also have "\<dots> < (\<Sum>n<?N. ?r)"
```
```  1126     proof (rule sum_strict_mono)
```
```  1127       fix n
```
```  1128       assume "n \<in> {..< ?N}"
```
```  1129       have "\<bar>x\<bar> < S" using \<open>\<bar>x\<bar> < S\<close> .
```
```  1130       also have "S \<le> S'" using \<open>S \<le> S'\<close> .
```
```  1131       also have "S' \<le> ?s n"
```
```  1132         unfolding S'_def
```
```  1133       proof (rule Min_le_iff[THEN iffD2])
```
```  1134         have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n"
```
```  1135           using \<open>n \<in> {..< ?N}\<close> by auto
```
```  1136         then show "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n"
```
```  1137           by blast
```
```  1138       qed auto
```
```  1139       finally have "\<bar>x\<bar> < ?s n" .
```
```  1140
```
```  1141       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>,
```
```  1142           unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
```
```  1143       have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
```
```  1144       with \<open>x \<noteq> 0\<close> and \<open>\<bar>x\<bar> < ?s n\<close> show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
```
```  1145         by blast
```
```  1146     qed auto
```
```  1147     also have "\<dots> = of_nat (card {..<?N}) * ?r"
```
```  1148       by (rule sum_constant)
```
```  1149     also have "\<dots> = real ?N * ?r"
```
```  1150       by simp
```
```  1151     also have "\<dots> = r/3"
```
```  1152       by (auto simp del: of_nat_Suc)
```
```  1153     finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
```
```  1154
```
```  1155     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```  1156     have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
```
```  1157         \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
```
```  1158       unfolding suminf_diff[OF div_smbl \<open>summable (f' x0)\<close>, symmetric]
```
```  1159       using suminf_divide[OF diff_smbl, symmetric] by auto
```
```  1160     also have "\<dots> \<le> ?diff_part + \<bar>(\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N))\<bar>"
```
```  1161       unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
```
```  1162       unfolding suminf_diff[OF div_shft_smbl ign[OF \<open>summable (f' x0)\<close>]]
```
```  1163       apply (subst (5) add.commute)
```
```  1164       apply (rule abs_triangle_ineq)
```
```  1165       done
```
```  1166     also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
```
```  1167       using abs_triangle_ineq4 by auto
```
```  1168     also have "\<dots> < r /3 + r/3 + r/3"
```
```  1169       using \<open>?diff_part < r/3\<close> \<open>?L_part \<le> r/3\<close> and \<open>?f'_part < r/3\<close>
```
```  1170       by (rule add_strict_mono [OF add_less_le_mono])
```
```  1171     finally show ?thesis
```
```  1172       by auto
```
```  1173   qed
```
```  1174   then show "\<exists>s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
```
```  1175       norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
```
```  1176     using \<open>0 < S\<close> by auto
```
```  1177 qed
```
```  1178
```
```  1179 lemma DERIV_power_series':
```
```  1180   fixes f :: "nat \<Rightarrow> real"
```
```  1181   assumes converges: "\<And>x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda>n. f n * real (Suc n) * x^n)"
```
```  1182     and x0_in_I: "x0 \<in> {-R <..< R}"
```
```  1183     and "0 < R"
```
```  1184   shows "DERIV (\<lambda>x. (\<Sum>n. f n * x^(Suc n))) x0 :> (\<Sum>n. f n * real (Suc n) * x0^n)"
```
```  1185     (is "DERIV (\<lambda>x. suminf (?f x)) x0 :> suminf (?f' x0)")
```
```  1186 proof -
```
```  1187   have for_subinterval: "DERIV (\<lambda>x. suminf (?f x)) x0 :> suminf (?f' x0)"
```
```  1188     if "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'" for R'
```
```  1189   proof -
```
```  1190     from that have "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
```
```  1191       by auto
```
```  1192     show ?thesis
```
```  1193     proof (rule DERIV_series')
```
```  1194       show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
```
```  1195       proof -
```
```  1196         have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
```
```  1197           using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps)
```
```  1198         then have in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
```
```  1199           using \<open>R' < R\<close> by auto
```
```  1200         have "norm R' < norm ((R' + R) / 2)"
```
```  1201           using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps)
```
```  1202         from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
```
```  1203           by auto
```
```  1204       qed
```
```  1205     next
```
```  1206       fix n x y
```
```  1207       assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
```
```  1208       show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
```
```  1209       proof -
```
```  1210         have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
```
```  1211           (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>"
```
```  1212           unfolding right_diff_distrib[symmetric] diff_power_eq_sum abs_mult
```
```  1213           by auto
```
```  1214         also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
```
```  1215         proof (rule mult_left_mono)
```
```  1216           have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
```
```  1217             by (rule sum_abs)
```
```  1218           also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)"
```
```  1219           proof (rule sum_mono)
```
```  1220             fix p
```
```  1221             assume "p \<in> {..<Suc n}"
```
```  1222             then have "p \<le> n" by auto
```
```  1223             have "\<bar>x^n\<bar> \<le> R'^n" if  "x \<in> {-R'<..<R'}" for n and x :: real
```
```  1224             proof -
```
```  1225               from that have "\<bar>x\<bar> \<le> R'" by auto
```
```  1226               then show ?thesis
```
```  1227                 unfolding power_abs by (rule power_mono) auto
```
```  1228             qed
```
```  1229             from mult_mono[OF this[OF \<open>x \<in> {-R'<..<R'}\<close>, of p] this[OF \<open>y \<in> {-R'<..<R'}\<close>, of "n-p"]]
```
```  1230               and \<open>0 < R'\<close>
```
```  1231             have "\<bar>x^p * y^(n - p)\<bar> \<le> R'^p * R'^(n - p)"
```
```  1232               unfolding abs_mult by auto
```
```  1233             then show "\<bar>x^p * y^(n - p)\<bar> \<le> R'^n"
```
```  1234               unfolding power_add[symmetric] using \<open>p \<le> n\<close> by auto
```
```  1235           qed
```
```  1236           also have "\<dots> = real (Suc n) * R' ^ n"
```
```  1237             unfolding sum_constant card_atLeastLessThan by auto
```
```  1238           finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
```
```  1239             unfolding abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF \<open>0 < R'\<close>]]]
```
```  1240             by linarith
```
```  1241           show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
```
```  1242             unfolding abs_mult[symmetric] by auto
```
```  1243         qed
```
```  1244         also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
```
```  1245           unfolding abs_mult mult.assoc[symmetric] by algebra
```
```  1246         finally show ?thesis .
```
```  1247       qed
```
```  1248     next
```
```  1249       show "DERIV (\<lambda>x. ?f x n) x0 :> ?f' x0 n" for n
```
```  1250         by (auto intro!: derivative_eq_intros simp del: power_Suc)
```
```  1251     next
```
```  1252       fix x
```
```  1253       assume "x \<in> {-R' <..< R'}"
```
```  1254       then have "R' \<in> {-R <..< R}" and "norm x < norm R'"
```
```  1255         using assms \<open>R' < R\<close> by auto
```
```  1256       have "summable (\<lambda>n. f n * x^n)"
```
```  1257       proof (rule summable_comparison_test, intro exI allI impI)
```
```  1258         fix n
```
```  1259         have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
```
```  1260           by (rule mult_left_mono) auto
```
```  1261         show "norm (f n * x^n) \<le> norm (f n * real (Suc n) * x^n)"
```
```  1262           unfolding real_norm_def abs_mult
```
```  1263           using le mult_right_mono by fastforce
```
```  1264       qed (rule powser_insidea[OF converges[OF \<open>R' \<in> {-R <..< R}\<close>] \<open>norm x < norm R'\<close>])
```
```  1265       from this[THEN summable_mult2[where c=x], simplified mult.assoc, simplified mult.commute]
```
```  1266       show "summable (?f x)" by auto
```
```  1267     next
```
```  1268       show "summable (?f' x0)"
```
```  1269         using converges[OF \<open>x0 \<in> {-R <..< R}\<close>] .
```
```  1270       show "x0 \<in> {-R' <..< R'}"
```
```  1271         using \<open>x0 \<in> {-R' <..< R'}\<close> .
```
```  1272     qed
```
```  1273   qed
```
```  1274   let ?R = "(R + \<bar>x0\<bar>) / 2"
```
```  1275   have "\<bar>x0\<bar> < ?R"
```
```  1276     using assms by (auto simp: field_simps)
```
```  1277   then have "- ?R < x0"
```
```  1278   proof (cases "x0 < 0")
```
```  1279     case True
```
```  1280     then have "- x0 < ?R"
```
```  1281       using \<open>\<bar>x0\<bar> < ?R\<close> by auto
```
```  1282     then show ?thesis
```
```  1283       unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
```
```  1284   next
```
```  1285     case False
```
```  1286     have "- ?R < 0" using assms by auto
```
```  1287     also have "\<dots> \<le> x0" using False by auto
```
```  1288     finally show ?thesis .
```
```  1289   qed
```
```  1290   then have "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
```
```  1291     using assms by (auto simp: field_simps)
```
```  1292   from for_subinterval[OF this] show ?thesis .
```
```  1293 qed
```
```  1294
```
```  1295 lemma geometric_deriv_sums:
```
```  1296   fixes z :: "'a :: {real_normed_field,banach}"
```
```  1297   assumes "norm z < 1"
```
```  1298   shows   "(\<lambda>n. of_nat (Suc n) * z ^ n) sums (1 / (1 - z)^2)"
```
```  1299 proof -
```
```  1300   have "(\<lambda>n. diffs (\<lambda>n. 1) n * z^n) sums (1 / (1 - z)^2)"
```
```  1301   proof (rule termdiffs_sums_strong)
```
```  1302     fix z :: 'a assume "norm z < 1"
```
```  1303     thus "(\<lambda>n. 1 * z^n) sums (1 / (1 - z))" by (simp add: geometric_sums)
```
```  1304   qed (insert assms, auto intro!: derivative_eq_intros simp: power2_eq_square)
```
```  1305   thus ?thesis unfolding diffs_def by simp
```
```  1306 qed
```
```  1307
```
```  1308 lemma isCont_pochhammer [continuous_intros]: "isCont (\<lambda>z. pochhammer z n) z"
```
```  1309   for z :: "'a::real_normed_field"
```
```  1310   by (induct n) (auto simp: pochhammer_rec')
```
```  1311
```
```  1312 lemma continuous_on_pochhammer [continuous_intros]: "continuous_on A (\<lambda>z. pochhammer z n)"
```
```  1313   for A :: "'a::real_normed_field set"
```
```  1314   by (intro continuous_at_imp_continuous_on ballI isCont_pochhammer)
```
```  1315
```
```  1316
```
```  1317 subsection \<open>Exponential Function\<close>
```
```  1318
```
```  1319 definition exp :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```  1320   where "exp = (\<lambda>x. \<Sum>n. x^n /\<^sub>R fact n)"
```
```  1321
```
```  1322 lemma summable_exp_generic:
```
```  1323   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  1324   defines S_def: "S \<equiv> \<lambda>n. x^n /\<^sub>R fact n"
```
```  1325   shows "summable S"
```
```  1326 proof -
```
```  1327   have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R (Suc n)"
```
```  1328     unfolding S_def by (simp del: mult_Suc)
```
```  1329   obtain r :: real where r0: "0 < r" and r1: "r < 1"
```
```  1330     using dense [OF zero_less_one] by fast
```
```  1331   obtain N :: nat where N: "norm x < real N * r"
```
```  1332     using ex_less_of_nat_mult r0 by auto
```
```  1333   from r1 show ?thesis
```
```  1334   proof (rule summable_ratio_test [rule_format])
```
```  1335     fix n :: nat
```
```  1336     assume n: "N \<le> n"
```
```  1337     have "norm x \<le> real N * r"
```
```  1338       using N by (rule order_less_imp_le)
```
```  1339     also have "real N * r \<le> real (Suc n) * r"
```
```  1340       using r0 n by (simp add: mult_right_mono)
```
```  1341     finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
```
```  1342       using norm_ge_zero by (rule mult_right_mono)
```
```  1343     then have "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
```
```  1344       by (rule order_trans [OF norm_mult_ineq])
```
```  1345     then have "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
```
```  1346       by (simp add: pos_divide_le_eq ac_simps)
```
```  1347     then show "norm (S (Suc n)) \<le> r * norm (S n)"
```
```  1348       by (simp add: S_Suc inverse_eq_divide)
```
```  1349   qed
```
```  1350 qed
```
```  1351
```
```  1352 lemma summable_norm_exp: "summable (\<lambda>n. norm (x^n /\<^sub>R fact n))"
```
```  1353   for x :: "'a::{real_normed_algebra_1,banach}"
```
```  1354 proof (rule summable_norm_comparison_test [OF exI, rule_format])
```
```  1355   show "summable (\<lambda>n. norm x^n /\<^sub>R fact n)"
```
```  1356     by (rule summable_exp_generic)
```
```  1357   show "norm (x^n /\<^sub>R fact n) \<le> norm x^n /\<^sub>R fact n" for n
```
```  1358     by (simp add: norm_power_ineq)
```
```  1359 qed
```
```  1360
```
```  1361 lemma summable_exp: "summable (\<lambda>n. inverse (fact n) * x^n)"
```
```  1362   for x :: "'a::{real_normed_field,banach}"
```
```  1363   using summable_exp_generic [where x=x]
```
```  1364   by (simp add: scaleR_conv_of_real nonzero_of_real_inverse)
```
```  1365
```
```  1366 lemma exp_converges: "(\<lambda>n. x^n /\<^sub>R fact n) sums exp x"
```
```  1367   unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
```
```  1368
```
```  1369 lemma exp_fdiffs:
```
```  1370   "diffs (\<lambda>n. inverse (fact n)) = (\<lambda>n. inverse (fact n :: 'a::{real_normed_field,banach}))"
```
```  1371   by (simp add: diffs_def mult_ac nonzero_inverse_mult_distrib nonzero_of_real_inverse
```
```  1372       del: mult_Suc of_nat_Suc)
```
```  1373
```
```  1374 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
```
```  1375   by (simp add: diffs_def)
```
```  1376
```
```  1377 lemma DERIV_exp [simp]: "DERIV exp x :> exp x"
```
```  1378   unfolding exp_def scaleR_conv_of_real
```
```  1379   apply (rule DERIV_cong)
```
```  1380    apply (rule termdiffs [where K="of_real (1 + norm x)"])
```
```  1381       apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
```
```  1382      apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
```
```  1383   apply (simp del: of_real_add)
```
```  1384   done
```
```  1385
```
```  1386 declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
```
```  1387   and DERIV_exp[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  1388
```
```  1389 lemma norm_exp: "norm (exp x) \<le> exp (norm x)"
```
```  1390 proof -
```
```  1391   from summable_norm[OF summable_norm_exp, of x]
```
```  1392   have "norm (exp x) \<le> (\<Sum>n. inverse (fact n) * norm (x^n))"
```
```  1393     by (simp add: exp_def)
```
```  1394   also have "\<dots> \<le> exp (norm x)"
```
```  1395     using summable_exp_generic[of "norm x"] summable_norm_exp[of x]
```
```  1396     by (auto simp: exp_def intro!: suminf_le norm_power_ineq)
```
```  1397   finally show ?thesis .
```
```  1398 qed
```
```  1399
```
```  1400 lemma isCont_exp: "isCont exp x"
```
```  1401   for x :: "'a::{real_normed_field,banach}"
```
```  1402   by (rule DERIV_exp [THEN DERIV_isCont])
```
```  1403
```
```  1404 lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
```
```  1405   for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1406   by (rule isCont_o2 [OF _ isCont_exp])
```
```  1407
```
```  1408 lemma tendsto_exp [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) \<longlongrightarrow> exp a) F"
```
```  1409   for f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1410   by (rule isCont_tendsto_compose [OF isCont_exp])
```
```  1411
```
```  1412 lemma continuous_exp [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
```
```  1413   for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1414   unfolding continuous_def by (rule tendsto_exp)
```
```  1415
```
```  1416 lemma continuous_on_exp [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
```
```  1417   for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1418   unfolding continuous_on_def by (auto intro: tendsto_exp)
```
```  1419
```
```  1420
```
```  1421 subsubsection \<open>Properties of the Exponential Function\<close>
```
```  1422
```
```  1423 lemma exp_zero [simp]: "exp 0 = 1"
```
```  1424   unfolding exp_def by (simp add: scaleR_conv_of_real)
```
```  1425
```
```  1426 lemma exp_series_add_commuting:
```
```  1427   fixes x y :: "'a::{real_normed_algebra_1,banach}"
```
```  1428   defines S_def: "S \<equiv> \<lambda>x n. x^n /\<^sub>R fact n"
```
```  1429   assumes comm: "x * y = y * x"
```
```  1430   shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))"
```
```  1431 proof (induct n)
```
```  1432   case 0
```
```  1433   show ?case
```
```  1434     unfolding S_def by simp
```
```  1435 next
```
```  1436   case (Suc n)
```
```  1437   have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
```
```  1438     unfolding S_def by (simp del: mult_Suc)
```
```  1439   then have times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
```
```  1440     by simp
```
```  1441   have S_comm: "\<And>n. S x n * y = y * S x n"
```
```  1442     by (simp add: power_commuting_commutes comm S_def)
```
```  1443
```
```  1444   have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
```
```  1445     by (simp only: times_S)
```
```  1446   also have "\<dots> = (x + y) * (\<Sum>i\<le>n. S x i * S y (n - i))"
```
```  1447     by (simp only: Suc)
```
```  1448   also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n - i)) + y * (\<Sum>i\<le>n. S x i * S y (n - i))"
```
```  1449     by (rule distrib_right)
```
```  1450   also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n - i)) + (\<Sum>i\<le>n. S x i * y * S y (n - i))"
```
```  1451     by (simp add: sum_distrib_left ac_simps S_comm)
```
```  1452   also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n - i)) + (\<Sum>i\<le>n. S x i * (y * S y (n - i)))"
```
```  1453     by (simp add: ac_simps)
```
```  1454   also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i))) +
```
```  1455       (\<Sum>i\<le>n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))"
```
```  1456     by (simp add: times_S Suc_diff_le)
```
```  1457   also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i))) =
```
```  1458       (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n - i)))"
```
```  1459     by (subst sum_atMost_Suc_shift) simp
```
```  1460   also have "(\<Sum>i\<le>n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i))) =
```
```  1461       (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))"
```
```  1462     by simp
```
```  1463   also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n - i))) +
```
```  1464         (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i))) =
```
```  1465       (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n - i)))"
```
```  1466     by (simp only: sum.distrib [symmetric] scaleR_left_distrib [symmetric]
```
```  1467         of_nat_add [symmetric]) simp
```
```  1468   also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
```
```  1469     by (simp only: scaleR_right.sum)
```
```  1470   finally show "S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
```
```  1471     by (simp del: sum_cl_ivl_Suc)
```
```  1472 qed
```
```  1473
```
```  1474 lemma exp_add_commuting: "x * y = y * x \<Longrightarrow> exp (x + y) = exp x * exp y"
```
```  1475   by (simp only: exp_def Cauchy_product summable_norm_exp exp_series_add_commuting)
```
```  1476
```
```  1477 lemma exp_times_arg_commute: "exp A * A = A * exp A"
```
```  1478   by (simp add: exp_def suminf_mult[symmetric] summable_exp_generic power_commutes suminf_mult2)
```
```  1479
```
```  1480 lemma exp_add: "exp (x + y) = exp x * exp y"
```
```  1481   for x y :: "'a::{real_normed_field,banach}"
```
```  1482   by (rule exp_add_commuting) (simp add: ac_simps)
```
```  1483
```
```  1484 lemma exp_double: "exp(2 * z) = exp z ^ 2"
```
```  1485   by (simp add: exp_add_commuting mult_2 power2_eq_square)
```
```  1486
```
```  1487 lemmas mult_exp_exp = exp_add [symmetric]
```
```  1488
```
```  1489 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
```
```  1490   unfolding exp_def
```
```  1491   apply (subst suminf_of_real)
```
```  1492    apply (rule summable_exp_generic)
```
```  1493   apply (simp add: scaleR_conv_of_real)
```
```  1494   done
```
```  1495
```
```  1496 lemmas of_real_exp = exp_of_real[symmetric]
```
```  1497
```
```  1498 corollary exp_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> exp z \<in> \<real>"
```
```  1499   by (metis Reals_cases Reals_of_real exp_of_real)
```
```  1500
```
```  1501 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
```
```  1502 proof
```
```  1503   have "exp x * exp (- x) = 1"
```
```  1504     by (simp add: exp_add_commuting[symmetric])
```
```  1505   also assume "exp x = 0"
```
```  1506   finally show False by simp
```
```  1507 qed
```
```  1508
```
```  1509 lemma exp_minus_inverse [simp]: "exp x * exp (- x) = 1"
```
```  1510   by (simp add: exp_add_commuting[symmetric])
```
```  1511
```
```  1512 lemma exp_minus: "exp (- x) = inverse (exp x)"
```
```  1513   for x :: "'a::{real_normed_field,banach}"
```
```  1514   by (intro inverse_unique [symmetric] exp_minus_inverse)
```
```  1515
```
```  1516 lemma exp_diff: "exp (x - y) = exp x / exp y"
```
```  1517   for x :: "'a::{real_normed_field,banach}"
```
```  1518   using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
```
```  1519
```
```  1520 lemma exp_of_nat_mult [simp]: "exp (of_nat n * x) = exp x ^ n"
```
```  1521   for x :: "'a::{real_normed_field,banach}"
```
```  1522   by (induct n) (auto simp add: distrib_left exp_add mult.commute)
```
```  1523
```
```  1524 corollary exp_of_nat2_mult [simp]: "exp (x * of_nat n) = exp x ^ n"
```
```  1525   for x :: "'a::{real_normed_field,banach}"
```
```  1526   by (metis exp_of_nat_mult mult_of_nat_commute)
```
```  1527
```
```  1528 lemma exp_sum: "finite I \<Longrightarrow> exp (sum f I) = prod (\<lambda>x. exp (f x)) I"
```
```  1529   by (induct I rule: finite_induct) (auto simp: exp_add_commuting mult.commute)
```
```  1530
```
```  1531 lemma exp_divide_power_eq [simp]:
```
```  1532   fixes x :: "'a::{real_normed_field,banach}"
```
```  1533   assumes "n > 0"
```
```  1534   shows "exp (x / of_nat n) ^ n = exp x"
```
```  1535   using assms
```
```  1536 proof (induction n arbitrary: x)
```
```  1537   case 0
```
```  1538   then show ?case by simp
```
```  1539 next
```
```  1540   case (Suc n)
```
```  1541   show ?case
```
```  1542   proof (cases "n = 0")
```
```  1543     case True
```
```  1544     then show ?thesis by simp
```
```  1545   next
```
```  1546     case False
```
```  1547     then have [simp]: "x * of_nat n / (1 + of_nat n) / of_nat n = x / (1 + of_nat n)"
```
```  1548       by simp
```
```  1549     have [simp]: "x / (1 + of_nat n) + x * of_nat n / (1 + of_nat n) = x"
```
```  1550       apply (simp add: divide_simps)
```
```  1551       using of_nat_eq_0_iff apply (fastforce simp: distrib_left)
```
```  1552       done
```
```  1553     show ?thesis
```
```  1554       using Suc.IH [of "x * of_nat n / (1 + of_nat n)"] False
```
```  1555       by (simp add: exp_add [symmetric])
```
```  1556   qed
```
```  1557 qed
```
```  1558
```
```  1559
```
```  1560 subsubsection \<open>Properties of the Exponential Function on Reals\<close>
```
```  1561
```
```  1562 text \<open>Comparisons of @{term "exp x"} with zero.\<close>
```
```  1563
```
```  1564 text \<open>Proof: because every exponential can be seen as a square.\<close>
```
```  1565 lemma exp_ge_zero [simp]: "0 \<le> exp x"
```
```  1566   for x :: real
```
```  1567 proof -
```
```  1568   have "0 \<le> exp (x/2) * exp (x/2)"
```
```  1569     by simp
```
```  1570   then show ?thesis
```
```  1571     by (simp add: exp_add [symmetric])
```
```  1572 qed
```
```  1573
```
```  1574 lemma exp_gt_zero [simp]: "0 < exp x"
```
```  1575   for x :: real
```
```  1576   by (simp add: order_less_le)
```
```  1577
```
```  1578 lemma not_exp_less_zero [simp]: "\<not> exp x < 0"
```
```  1579   for x :: real
```
```  1580   by (simp add: not_less)
```
```  1581
```
```  1582 lemma not_exp_le_zero [simp]: "\<not> exp x \<le> 0"
```
```  1583   for x :: real
```
```  1584   by (simp add: not_le)
```
```  1585
```
```  1586 lemma abs_exp_cancel [simp]: "\<bar>exp x\<bar> = exp x"
```
```  1587   for x :: real
```
```  1588   by simp
```
```  1589
```
```  1590 text \<open>Strict monotonicity of exponential.\<close>
```
```  1591
```
```  1592 lemma exp_ge_add_one_self_aux:
```
```  1593   fixes x :: real
```
```  1594   assumes "0 \<le> x"
```
```  1595   shows "1 + x \<le> exp x"
```
```  1596   using order_le_imp_less_or_eq [OF assms]
```
```  1597 proof
```
```  1598   assume "0 < x"
```
```  1599   have "1 + x \<le> (\<Sum>n<2. inverse (fact n) * x^n)"
```
```  1600     by (auto simp add: numeral_2_eq_2)
```
```  1601   also have "\<dots> \<le> (\<Sum>n. inverse (fact n) * x^n)"
```
```  1602     apply (rule sum_le_suminf [OF summable_exp])
```
```  1603     using \<open>0 < x\<close>
```
```  1604     apply (auto  simp add:  zero_le_mult_iff)
```
```  1605     done
```
```  1606   finally show "1 + x \<le> exp x"
```
```  1607     by (simp add: exp_def)
```
```  1608 next
```
```  1609   assume "0 = x"
```
```  1610   then show "1 + x \<le> exp x"
```
```  1611     by auto
```
```  1612 qed
```
```  1613
```
```  1614 lemma exp_gt_one: "0 < x \<Longrightarrow> 1 < exp x"
```
```  1615   for x :: real
```
```  1616 proof -
```
```  1617   assume x: "0 < x"
```
```  1618   then have "1 < 1 + x" by simp
```
```  1619   also from x have "1 + x \<le> exp x"
```
```  1620     by (simp add: exp_ge_add_one_self_aux)
```
```  1621   finally show ?thesis .
```
```  1622 qed
```
```  1623
```
```  1624 lemma exp_less_mono:
```
```  1625   fixes x y :: real
```
```  1626   assumes "x < y"
```
```  1627   shows "exp x < exp y"
```
```  1628 proof -
```
```  1629   from \<open>x < y\<close> have "0 < y - x" by simp
```
```  1630   then have "1 < exp (y - x)" by (rule exp_gt_one)
```
```  1631   then have "1 < exp y / exp x" by (simp only: exp_diff)
```
```  1632   then show "exp x < exp y" by simp
```
```  1633 qed
```
```  1634
```
```  1635 lemma exp_less_cancel: "exp x < exp y \<Longrightarrow> x < y"
```
```  1636   for x y :: real
```
```  1637   unfolding linorder_not_le [symmetric]
```
```  1638   by (auto simp add: order_le_less exp_less_mono)
```
```  1639
```
```  1640 lemma exp_less_cancel_iff [iff]: "exp x < exp y \<longleftrightarrow> x < y"
```
```  1641   for x y :: real
```
```  1642   by (auto intro: exp_less_mono exp_less_cancel)
```
```  1643
```
```  1644 lemma exp_le_cancel_iff [iff]: "exp x \<le> exp y \<longleftrightarrow> x \<le> y"
```
```  1645   for x y :: real
```
```  1646   by (auto simp add: linorder_not_less [symmetric])
```
```  1647
```
```  1648 lemma exp_inj_iff [iff]: "exp x = exp y \<longleftrightarrow> x = y"
```
```  1649   for x y :: real
```
```  1650   by (simp add: order_eq_iff)
```
```  1651
```
```  1652 text \<open>Comparisons of @{term "exp x"} with one.\<close>
```
```  1653
```
```  1654 lemma one_less_exp_iff [simp]: "1 < exp x \<longleftrightarrow> 0 < x"
```
```  1655   for x :: real
```
```  1656   using exp_less_cancel_iff [where x = 0 and y = x] by simp
```
```  1657
```
```  1658 lemma exp_less_one_iff [simp]: "exp x < 1 \<longleftrightarrow> x < 0"
```
```  1659   for x :: real
```
```  1660   using exp_less_cancel_iff [where x = x and y = 0] by simp
```
```  1661
```
```  1662 lemma one_le_exp_iff [simp]: "1 \<le> exp x \<longleftrightarrow> 0 \<le> x"
```
```  1663   for x :: real
```
```  1664   using exp_le_cancel_iff [where x = 0 and y = x] by simp
```
```  1665
```
```  1666 lemma exp_le_one_iff [simp]: "exp x \<le> 1 \<longleftrightarrow> x \<le> 0"
```
```  1667   for x :: real
```
```  1668   using exp_le_cancel_iff [where x = x and y = 0] by simp
```
```  1669
```
```  1670 lemma exp_eq_one_iff [simp]: "exp x = 1 \<longleftrightarrow> x = 0"
```
```  1671   for x :: real
```
```  1672   using exp_inj_iff [where x = x and y = 0] by simp
```
```  1673
```
```  1674 lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x \<and> x \<le> y - 1 \<and> exp x = y"
```
```  1675   for y :: real
```
```  1676 proof (rule IVT)
```
```  1677   assume "1 \<le> y"
```
```  1678   then have "0 \<le> y - 1" by simp
```
```  1679   then have "1 + (y - 1) \<le> exp (y - 1)"
```
```  1680     by (rule exp_ge_add_one_self_aux)
```
```  1681   then show "y \<le> exp (y - 1)" by simp
```
```  1682 qed (simp_all add: le_diff_eq)
```
```  1683
```
```  1684 lemma exp_total: "0 < y \<Longrightarrow> \<exists>x. exp x = y"
```
```  1685   for y :: real
```
```  1686 proof (rule linorder_le_cases [of 1 y])
```
```  1687   assume "1 \<le> y"
```
```  1688   then show "\<exists>x. exp x = y"
```
```  1689     by (fast dest: lemma_exp_total)
```
```  1690 next
```
```  1691   assume "0 < y" and "y \<le> 1"
```
```  1692   then have "1 \<le> inverse y"
```
```  1693     by (simp add: one_le_inverse_iff)
```
```  1694   then obtain x where "exp x = inverse y"
```
```  1695     by (fast dest: lemma_exp_total)
```
```  1696   then have "exp (- x) = y"
```
```  1697     by (simp add: exp_minus)
```
```  1698   then show "\<exists>x. exp x = y" ..
```
```  1699 qed
```
```  1700
```
```  1701
```
```  1702 subsection \<open>Natural Logarithm\<close>
```
```  1703
```
```  1704 class ln = real_normed_algebra_1 + banach +
```
```  1705   fixes ln :: "'a \<Rightarrow> 'a"
```
```  1706   assumes ln_one [simp]: "ln 1 = 0"
```
```  1707
```
```  1708 definition powr :: "'a \<Rightarrow> 'a \<Rightarrow> 'a::ln"  (infixr "powr" 80)
```
```  1709   \<comment> \<open>exponentation via ln and exp\<close>
```
```  1710   where  [code del]: "x powr a \<equiv> if x = 0 then 0 else exp (a * ln x)"
```
```  1711
```
```  1712 lemma powr_0 [simp]: "0 powr z = 0"
```
```  1713   by (simp add: powr_def)
```
```  1714
```
```  1715
```
```  1716 instantiation real :: ln
```
```  1717 begin
```
```  1718
```
```  1719 definition ln_real :: "real \<Rightarrow> real"
```
```  1720   where "ln_real x = (THE u. exp u = x)"
```
```  1721
```
```  1722 instance
```
```  1723   by intro_classes (simp add: ln_real_def)
```
```  1724
```
```  1725 end
```
```  1726
```
```  1727 lemma powr_eq_0_iff [simp]: "w powr z = 0 \<longleftrightarrow> w = 0"
```
```  1728   by (simp add: powr_def)
```
```  1729
```
```  1730 lemma ln_exp [simp]: "ln (exp x) = x"
```
```  1731   for x :: real
```
```  1732   by (simp add: ln_real_def)
```
```  1733
```
```  1734 lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
```
```  1735   for x :: real
```
```  1736   by (auto dest: exp_total)
```
```  1737
```
```  1738 lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
```
```  1739   for x :: real
```
```  1740   by (metis exp_gt_zero exp_ln)
```
```  1741
```
```  1742 lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
```
```  1743   for x :: real
```
```  1744   by (erule subst) (rule ln_exp)
```
```  1745
```
```  1746 lemma ln_mult [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
```
```  1747   for x :: real
```
```  1748   by (rule ln_unique) (simp add: exp_add)
```
```  1749
```
```  1750 lemma ln_prod: "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i > 0) \<Longrightarrow> ln (prod f I) = sum (\<lambda>x. ln(f x)) I"
```
```  1751   for f :: "'a \<Rightarrow> real"
```
```  1752   by (induct I rule: finite_induct) (auto simp: ln_mult prod_pos)
```
```  1753
```
```  1754 lemma ln_inverse [simp]: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
```
```  1755   for x :: real
```
```  1756   by (rule ln_unique) (simp add: exp_minus)
```
```  1757
```
```  1758 lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
```
```  1759   for x :: real
```
```  1760   by (rule ln_unique) (simp add: exp_diff)
```
```  1761
```
```  1762 lemma ln_realpow [simp]: "0 < x \<Longrightarrow> ln (x^n) = real n * ln x"
```
```  1763   by (rule ln_unique) simp
```
```  1764
```
```  1765 lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
```
```  1766   for x :: real
```
```  1767   by (subst exp_less_cancel_iff [symmetric]) simp
```
```  1768
```
```  1769 lemma ln_le_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
```
```  1770   for x :: real
```
```  1771   by (simp add: linorder_not_less [symmetric])
```
```  1772
```
```  1773 lemma ln_inj_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
```
```  1774   for x :: real
```
```  1775   by (simp add: order_eq_iff)
```
```  1776
```
```  1777 lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
```
```  1778   for x :: real
```
```  1779   by (rule exp_le_cancel_iff [THEN iffD1]) (simp add: exp_ge_add_one_self_aux)
```
```  1780
```
```  1781 lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
```
```  1782   for x :: real
```
```  1783   by (rule order_less_le_trans [where y = "ln (1 + x)"]) simp_all
```
```  1784
```
```  1785 lemma ln_ge_iff: "\<And>x::real. 0 < x \<Longrightarrow> y \<le> ln x \<longleftrightarrow> exp y \<le> x"
```
```  1786   using exp_le_cancel_iff exp_total by force
```
```  1787
```
```  1788 lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
```
```  1789   for x :: real
```
```  1790   using ln_le_cancel_iff [of 1 x] by simp
```
```  1791
```
```  1792 lemma ln_ge_zero_imp_ge_one: "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
```
```  1793   for x :: real
```
```  1794   using ln_le_cancel_iff [of 1 x] by simp
```
```  1795
```
```  1796 lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
```
```  1797   for x :: real
```
```  1798   using ln_le_cancel_iff [of 1 x] by simp
```
```  1799
```
```  1800 lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
```
```  1801   for x :: real
```
```  1802   using ln_less_cancel_iff [of x 1] by simp
```
```  1803
```
```  1804 lemma ln_le_zero_iff [simp]: "0 < x \<Longrightarrow> ln x \<le> 0 \<longleftrightarrow> x \<le> 1"
```
```  1805   for x :: real
```
```  1806   by (metis less_numeral_extra(1) ln_le_cancel_iff ln_one)
```
```  1807
```
```  1808 lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
```
```  1809   for x :: real
```
```  1810   using ln_less_cancel_iff [of 1 x] by simp
```
```  1811
```
```  1812 lemma ln_gt_zero_imp_gt_one: "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
```
```  1813   for x :: real
```
```  1814   using ln_less_cancel_iff [of 1 x] by simp
```
```  1815
```
```  1816 lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
```
```  1817   for x :: real
```
```  1818   using ln_less_cancel_iff [of 1 x] by simp
```
```  1819
```
```  1820 lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
```
```  1821   for x :: real
```
```  1822   using ln_inj_iff [of x 1] by simp
```
```  1823
```
```  1824 lemma ln_less_zero: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
```
```  1825   for x :: real
```
```  1826   by simp
```
```  1827
```
```  1828 lemma ln_neg_is_const: "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)"
```
```  1829   for x :: real
```
```  1830   by (auto simp: ln_real_def intro!: arg_cong[where f = The])
```
```  1831
```
```  1832 lemma isCont_ln:
```
```  1833   fixes x :: real
```
```  1834   assumes "x \<noteq> 0"
```
```  1835   shows "isCont ln x"
```
```  1836 proof (cases "0 < x")
```
```  1837   case True
```
```  1838   then have "isCont ln (exp (ln x))"
```
```  1839     by (intro isCont_inv_fun[where d = "\<bar>x\<bar>" and f = exp]) auto
```
```  1840   with True show ?thesis
```
```  1841     by simp
```
```  1842 next
```
```  1843   case False
```
```  1844   with \<open>x \<noteq> 0\<close> show "isCont ln x"
```
```  1845     unfolding isCont_def
```
```  1846     by (subst filterlim_cong[OF _ refl, of _ "nhds (ln 0)" _ "\<lambda>_. ln 0"])
```
```  1847        (auto simp: ln_neg_is_const not_less eventually_at dist_real_def
```
```  1848          intro!: exI[of _ "\<bar>x\<bar>"])
```
```  1849 qed
```
```  1850
```
```  1851 lemma tendsto_ln [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) \<longlongrightarrow> ln a) F"
```
```  1852   for a :: real
```
```  1853   by (rule isCont_tendsto_compose [OF isCont_ln])
```
```  1854
```
```  1855 lemma continuous_ln:
```
```  1856   "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. ln (f x :: real))"
```
```  1857   unfolding continuous_def by (rule tendsto_ln)
```
```  1858
```
```  1859 lemma isCont_ln' [continuous_intros]:
```
```  1860   "continuous (at x) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x :: real))"
```
```  1861   unfolding continuous_at by (rule tendsto_ln)
```
```  1862
```
```  1863 lemma continuous_within_ln [continuous_intros]:
```
```  1864   "continuous (at x within s) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x :: real))"
```
```  1865   unfolding continuous_within by (rule tendsto_ln)
```
```  1866
```
```  1867 lemma continuous_on_ln [continuous_intros]:
```
```  1868   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. f x \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x :: real))"
```
```  1869   unfolding continuous_on_def by (auto intro: tendsto_ln)
```
```  1870
```
```  1871 lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
```
```  1872   for x :: real
```
```  1873   by (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
```
```  1874     (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln)
```
```  1875
```
```  1876 lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
```
```  1877   for x :: real
```
```  1878   by (rule DERIV_ln[THEN DERIV_cong]) (simp_all add: divide_inverse)
```
```  1879
```
```  1880 declare DERIV_ln_divide[THEN DERIV_chain2, derivative_intros]
```
```  1881   and DERIV_ln_divide[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  1882
```
```  1883 lemma ln_series:
```
```  1884   assumes "0 < x" and "x < 2"
```
```  1885   shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))"
```
```  1886     (is "ln x = suminf (?f (x - 1))")
```
```  1887 proof -
```
```  1888   let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n"
```
```  1889
```
```  1890   have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
```
```  1891   proof (rule DERIV_isconst3 [where x = x])
```
```  1892     fix x :: real
```
```  1893     assume "x \<in> {0 <..< 2}"
```
```  1894     then have "0 < x" and "x < 2" by auto
```
```  1895     have "norm (1 - x) < 1"
```
```  1896       using \<open>0 < x\<close> and \<open>x < 2\<close> by auto
```
```  1897     have "1 / x = 1 / (1 - (1 - x))" by auto
```
```  1898     also have "\<dots> = (\<Sum> n. (1 - x)^n)"
```
```  1899       using geometric_sums[OF \<open>norm (1 - x) < 1\<close>] by (rule sums_unique)
```
```  1900     also have "\<dots> = suminf (?f' x)"
```
```  1901       unfolding power_mult_distrib[symmetric]
```
```  1902       by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
```
```  1903     finally have "DERIV ln x :> suminf (?f' x)"
```
```  1904       using DERIV_ln[OF \<open>0 < x\<close>] unfolding divide_inverse by auto
```
```  1905     moreover
```
```  1906     have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
```
```  1907     have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :>
```
```  1908       (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
```
```  1909     proof (rule DERIV_power_series')
```
```  1910       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1"
```
```  1911         using \<open>0 < x\<close> \<open>x < 2\<close> by auto
```
```  1912     next
```
```  1913       fix x :: real
```
```  1914       assume "x \<in> {- 1<..<1}"
```
```  1915       then have "norm (-x) < 1" by auto
```
```  1916       show "summable (\<lambda>n. (- 1) ^ n * (1 / real (n + 1)) * real (Suc n) * x^n)"
```
```  1917         unfolding One_nat_def
```
```  1918         by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF \<open>norm (-x) < 1\<close>])
```
```  1919     qed
```
```  1920     then have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
```
```  1921       unfolding One_nat_def by auto
```
```  1922     then have "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
```
```  1923       unfolding DERIV_def repos .
```
```  1924     ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> suminf (?f' x) - suminf (?f' x)"
```
```  1925       by (rule DERIV_diff)
```
```  1926     then show "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
```
```  1927   qed (auto simp add: assms)
```
```  1928   then show ?thesis by auto
```
```  1929 qed
```
```  1930
```
```  1931 lemma exp_first_terms:
```
```  1932   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  1933   shows "exp x = (\<Sum>n<k. inverse(fact n) *\<^sub>R (x ^ n)) + (\<Sum>n. inverse(fact (n + k)) *\<^sub>R (x ^ (n + k)))"
```
```  1934 proof -
```
```  1935   have "exp x = suminf (\<lambda>n. inverse(fact n) *\<^sub>R (x^n))"
```
```  1936     by (simp add: exp_def)
```
```  1937   also from summable_exp_generic have "\<dots> = (\<Sum> n. inverse(fact(n+k)) *\<^sub>R (x ^ (n + k))) +
```
```  1938     (\<Sum> n::nat<k. inverse(fact n) *\<^sub>R (x^n))" (is "_ = _ + ?a")
```
```  1939     by (rule suminf_split_initial_segment)
```
```  1940   finally show ?thesis by simp
```
```  1941 qed
```
```  1942
```
```  1943 lemma exp_first_term: "exp x = 1 + (\<Sum>n. inverse (fact (Suc n)) *\<^sub>R (x ^ Suc n))"
```
```  1944   for x :: "'a::{real_normed_algebra_1,banach}"
```
```  1945   using exp_first_terms[of x 1] by simp
```
```  1946
```
```  1947 lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum>n. inverse (fact (n + 2)) *\<^sub>R (x ^ (n + 2)))"
```
```  1948   for x :: "'a::{real_normed_algebra_1,banach}"
```
```  1949   using exp_first_terms[of x 2] by (simp add: eval_nat_numeral)
```
```  1950
```
```  1951 lemma exp_bound:
```
```  1952   fixes x :: real
```
```  1953   assumes a: "0 \<le> x"
```
```  1954     and b: "x \<le> 1"
```
```  1955   shows "exp x \<le> 1 + x + x\<^sup>2"
```
```  1956 proof -
```
```  1957   have aux1: "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)" for n :: nat
```
```  1958   proof -
```
```  1959     have "(2::nat) * 2 ^ n \<le> fact (n + 2)"
```
```  1960       by (induct n) simp_all
```
```  1961     then have "real ((2::nat) * 2 ^ n) \<le> real_of_nat (fact (n + 2))"
```
```  1962       by (simp only: of_nat_le_iff)
```
```  1963     then have "((2::real) * 2 ^ n) \<le> fact (n + 2)"
```
```  1964       unfolding of_nat_fact by simp
```
```  1965     then have "inverse (fact (n + 2)) \<le> inverse ((2::real) * 2 ^ n)"
```
```  1966       by (rule le_imp_inverse_le) simp
```
```  1967     then have "inverse (fact (n + 2)) \<le> 1/(2::real) * (1/2)^n"
```
```  1968       by (simp add: power_inverse [symmetric])
```
```  1969     then have "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
```
```  1970       by (rule mult_mono) (rule mult_mono, simp_all add: power_le_one a b)
```
```  1971     then show ?thesis
```
```  1972       unfolding power_add by (simp add: ac_simps del: fact_Suc)
```
```  1973   qed
```
```  1974   have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
```
```  1975     by (intro sums_mult geometric_sums) simp
```
```  1976   then have aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
```
```  1977     by simp
```
```  1978   have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n + 2))) \<le> x\<^sup>2"
```
```  1979   proof -
```
```  1980     have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n + 2))) \<le> suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
```
```  1981       apply (rule suminf_le)
```
```  1982         apply (rule allI)
```
```  1983         apply (rule aux1)
```
```  1984        apply (rule summable_exp [THEN summable_ignore_initial_segment])
```
```  1985       apply (rule sums_summable)
```
```  1986       apply (rule aux2)
```
```  1987       done
```
```  1988     also have "\<dots> = x\<^sup>2"
```
```  1989       by (rule sums_unique [THEN sym]) (rule aux2)
```
```  1990     finally show ?thesis .
```
```  1991   qed
```
```  1992   then show ?thesis
```
```  1993     unfolding exp_first_two_terms by auto
```
```  1994 qed
```
```  1995
```
```  1996 corollary exp_half_le2: "exp(1/2) \<le> (2::real)"
```
```  1997   using exp_bound [of "1/2"]
```
```  1998   by (simp add: field_simps)
```
```  1999
```
```  2000 corollary exp_le: "exp 1 \<le> (3::real)"
```
```  2001   using exp_bound [of 1]
```
```  2002   by (simp add: field_simps)
```
```  2003
```
```  2004 lemma exp_bound_half: "norm z \<le> 1/2 \<Longrightarrow> norm (exp z) \<le> 2"
```
```  2005   by (blast intro: order_trans intro!: exp_half_le2 norm_exp)
```
```  2006
```
```  2007 lemma exp_bound_lemma:
```
```  2008   assumes "norm z \<le> 1/2"
```
```  2009   shows "norm (exp z) \<le> 1 + 2 * norm z"
```
```  2010 proof -
```
```  2011   have *: "(norm z)\<^sup>2 \<le> norm z * 1"
```
```  2012     unfolding power2_eq_square
```
```  2013     apply (rule mult_left_mono)
```
```  2014     using assms
```
```  2015      apply auto
```
```  2016     done
```
```  2017   show ?thesis
```
```  2018     apply (rule order_trans [OF norm_exp])
```
```  2019     apply (rule order_trans [OF exp_bound])
```
```  2020     using assms *
```
```  2021       apply auto
```
```  2022     done
```
```  2023 qed
```
```  2024
```
```  2025 lemma real_exp_bound_lemma: "0 \<le> x \<Longrightarrow> x \<le> 1/2 \<Longrightarrow> exp x \<le> 1 + 2 * x"
```
```  2026   for x :: real
```
```  2027   using exp_bound_lemma [of x] by simp
```
```  2028
```
```  2029 lemma ln_one_minus_pos_upper_bound:
```
```  2030   fixes x :: real
```
```  2031   assumes a: "0 \<le> x" and b: "x < 1"
```
```  2032   shows "ln (1 - x) \<le> - x"
```
```  2033 proof -
```
```  2034   have "(1 - x) * (1 + x + x\<^sup>2) = 1 - x^3"
```
```  2035     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
```
```  2036   also have "\<dots> \<le> 1"
```
```  2037     by (auto simp add: a)
```
```  2038   finally have "(1 - x) * (1 + x + x\<^sup>2) \<le> 1" .
```
```  2039   moreover have c: "0 < 1 + x + x\<^sup>2"
```
```  2040     by (simp add: add_pos_nonneg a)
```
```  2041   ultimately have "1 - x \<le> 1 / (1 + x + x\<^sup>2)"
```
```  2042     by (elim mult_imp_le_div_pos)
```
```  2043   also have "\<dots> \<le> 1 / exp x"
```
```  2044     by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs
```
```  2045         real_sqrt_pow2_iff real_sqrt_power)
```
```  2046   also have "\<dots> = exp (- x)"
```
```  2047     by (auto simp add: exp_minus divide_inverse)
```
```  2048   finally have "1 - x \<le> exp (- x)" .
```
```  2049   also have "1 - x = exp (ln (1 - x))"
```
```  2050     by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq)
```
```  2051   finally have "exp (ln (1 - x)) \<le> exp (- x)" .
```
```  2052   then show ?thesis
```
```  2053     by (auto simp only: exp_le_cancel_iff)
```
```  2054 qed
```
```  2055
```
```  2056 lemma exp_ge_add_one_self [simp]: "1 + x \<le> exp x"
```
```  2057   for x :: real
```
```  2058   apply (cases "0 \<le> x")
```
```  2059    apply (erule exp_ge_add_one_self_aux)
```
```  2060   apply (cases "x \<le> -1")
```
```  2061    apply (subgoal_tac "1 + x \<le> 0")
```
```  2062     apply (erule order_trans)
```
```  2063     apply simp
```
```  2064    apply simp
```
```  2065   apply (subgoal_tac "1 + x = exp (ln (1 + x))")
```
```  2066    apply (erule ssubst)
```
```  2067    apply (subst exp_le_cancel_iff)
```
```  2068    apply (subgoal_tac "ln (1 - (- x)) \<le> - (- x)")
```
```  2069     apply simp
```
```  2070    apply (rule ln_one_minus_pos_upper_bound)
```
```  2071     apply auto
```
```  2072   done
```
```  2073
```
```  2074 lemma ln_one_plus_pos_lower_bound:
```
```  2075   fixes x :: real
```
```  2076   assumes a: "0 \<le> x" and b: "x \<le> 1"
```
```  2077   shows "x - x\<^sup>2 \<le> ln (1 + x)"
```
```  2078 proof -
```
```  2079   have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
```
```  2080     by (rule exp_diff)
```
```  2081   also have "\<dots> \<le> (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
```
```  2082     by (metis a b divide_right_mono exp_bound exp_ge_zero)
```
```  2083   also have "\<dots> \<le> (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
```
```  2084     by (simp add: a divide_left_mono add_pos_nonneg)
```
```  2085   also from a have "\<dots> \<le> 1 + x"
```
```  2086     by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
```
```  2087   finally have "exp (x - x\<^sup>2) \<le> 1 + x" .
```
```  2088   also have "\<dots> = exp (ln (1 + x))"
```
```  2089   proof -
```
```  2090     from a have "0 < 1 + x" by auto
```
```  2091     then show ?thesis
```
```  2092       by (auto simp only: exp_ln_iff [THEN sym])
```
```  2093   qed
```
```  2094   finally have "exp (x - x\<^sup>2) \<le> exp (ln (1 + x))" .
```
```  2095   then show ?thesis
```
```  2096     by (metis exp_le_cancel_iff)
```
```  2097 qed
```
```  2098
```
```  2099 lemma ln_one_minus_pos_lower_bound:
```
```  2100   fixes x :: real
```
```  2101   assumes a: "0 \<le> x" and b: "x \<le> 1 / 2"
```
```  2102   shows "- x - 2 * x\<^sup>2 \<le> ln (1 - x)"
```
```  2103 proof -
```
```  2104   from b have c: "x < 1" by auto
```
```  2105   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
```
```  2106     apply (subst ln_inverse [symmetric])
```
```  2107      apply (simp add: field_simps)
```
```  2108     apply (rule arg_cong [where f=ln])
```
```  2109     apply (simp add: field_simps)
```
```  2110     done
```
```  2111   also have "- (x / (1 - x)) \<le> \<dots>"
```
```  2112   proof -
```
```  2113     have "ln (1 + x / (1 - x)) \<le> x / (1 - x)"
```
```  2114       using a c by (intro ln_add_one_self_le_self) auto
```
```  2115     then show ?thesis
```
```  2116       by auto
```
```  2117   qed
```
```  2118   also have "- (x / (1 - x)) = - x / (1 - x)"
```
```  2119     by auto
```
```  2120   finally have d: "- x / (1 - x) \<le> ln (1 - x)" .
```
```  2121   have "0 < 1 - x" using a b by simp
```
```  2122   then have e: "- x - 2 * x\<^sup>2 \<le> - x / (1 - x)"
```
```  2123     using mult_right_le_one_le[of "x * x" "2 * x"] a b
```
```  2124     by (simp add: field_simps power2_eq_square)
```
```  2125   from e d show "- x - 2 * x\<^sup>2 \<le> ln (1 - x)"
```
```  2126     by (rule order_trans)
```
```  2127 qed
```
```  2128
```
```  2129 lemma ln_add_one_self_le_self2:
```
```  2130   fixes x :: real
```
```  2131   shows "-1 < x \<Longrightarrow> ln (1 + x) \<le> x"
```
```  2132   apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)")
```
```  2133    apply simp
```
```  2134   apply (subst ln_le_cancel_iff)
```
```  2135     apply auto
```
```  2136   done
```
```  2137
```
```  2138 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
```
```  2139   fixes x :: real
```
```  2140   assumes x: "0 \<le> x" and x1: "x \<le> 1"
```
```  2141   shows "\<bar>ln (1 + x) - x\<bar> \<le> x\<^sup>2"
```
```  2142 proof -
```
```  2143   from x have "ln (1 + x) \<le> x"
```
```  2144     by (rule ln_add_one_self_le_self)
```
```  2145   then have "ln (1 + x) - x \<le> 0"
```
```  2146     by simp
```
```  2147   then have "\<bar>ln(1 + x) - x\<bar> = - (ln(1 + x) - x)"
```
```  2148     by (rule abs_of_nonpos)
```
```  2149   also have "\<dots> = x - ln (1 + x)"
```
```  2150     by simp
```
```  2151   also have "\<dots> \<le> x\<^sup>2"
```
```  2152   proof -
```
```  2153     from x x1 have "x - x\<^sup>2 \<le> ln (1 + x)"
```
```  2154       by (intro ln_one_plus_pos_lower_bound)
```
```  2155     then show ?thesis
```
```  2156       by simp
```
```  2157   qed
```
```  2158   finally show ?thesis .
```
```  2159 qed
```
```  2160
```
```  2161 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
```
```  2162   fixes x :: real
```
```  2163   assumes a: "-(1 / 2) \<le> x" and b: "x \<le> 0"
```
```  2164   shows "\<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2"
```
```  2165 proof -
```
```  2166   have "\<bar>ln (1 + x) - x\<bar> = x - ln (1 - (- x))"
```
```  2167     apply (subst abs_of_nonpos)
```
```  2168      apply simp
```
```  2169      apply (rule ln_add_one_self_le_self2)
```
```  2170     using a apply auto
```
```  2171     done
```
```  2172   also have "\<dots> \<le> 2 * x\<^sup>2"
```
```  2173     apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 \<le> ln (1 - (- x))")
```
```  2174      apply (simp add: algebra_simps)
```
```  2175     apply (rule ln_one_minus_pos_lower_bound)
```
```  2176     using a b apply auto
```
```  2177     done
```
```  2178   finally show ?thesis .
```
```  2179 qed
```
```  2180
```
```  2181 lemma abs_ln_one_plus_x_minus_x_bound:
```
```  2182   fixes x :: real
```
```  2183   shows "\<bar>x\<bar> \<le> 1 / 2 \<Longrightarrow> \<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2"
```
```  2184   apply (cases "0 \<le> x")
```
```  2185    apply (rule order_trans)
```
```  2186     apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
```
```  2187      apply auto
```
```  2188   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
```
```  2189    apply auto
```
```  2190   done
```
```  2191
```
```  2192 lemma ln_x_over_x_mono:
```
```  2193   fixes x :: real
```
```  2194   assumes x: "exp 1 \<le> x" "x \<le> y"
```
```  2195   shows "ln y / y \<le> ln x / x"
```
```  2196 proof -
```
```  2197   note x
```
```  2198   moreover have "0 < exp (1::real)" by simp
```
```  2199   ultimately have a: "0 < x" and b: "0 < y"
```
```  2200     by (fast intro: less_le_trans order_trans)+
```
```  2201   have "x * ln y - x * ln x = x * (ln y - ln x)"
```
```  2202     by (simp add: algebra_simps)
```
```  2203   also have "\<dots> = x * ln (y / x)"
```
```  2204     by (simp only: ln_div a b)
```
```  2205   also have "y / x = (x + (y - x)) / x"
```
```  2206     by simp
```
```  2207   also have "\<dots> = 1 + (y - x) / x"
```
```  2208     using x a by (simp add: field_simps)
```
```  2209   also have "x * ln (1 + (y - x) / x) \<le> x * ((y - x) / x)"
```
```  2210     using x a
```
```  2211     by (intro mult_left_mono ln_add_one_self_le_self) simp_all
```
```  2212   also have "\<dots> = y - x"
```
```  2213     using a by simp
```
```  2214   also have "\<dots> = (y - x) * ln (exp 1)" by simp
```
```  2215   also have "\<dots> \<le> (y - x) * ln x"
```
```  2216     apply (rule mult_left_mono)
```
```  2217      apply (subst ln_le_cancel_iff)
```
```  2218        apply fact
```
```  2219       apply (rule a)
```
```  2220      apply (rule x)
```
```  2221     using x apply simp
```
```  2222     done
```
```  2223   also have "\<dots> = y * ln x - x * ln x"
```
```  2224     by (rule left_diff_distrib)
```
```  2225   finally have "x * ln y \<le> y * ln x"
```
```  2226     by arith
```
```  2227   then have "ln y \<le> (y * ln x) / x"
```
```  2228     using a by (simp add: field_simps)
```
```  2229   also have "\<dots> = y * (ln x / x)" by simp
```
```  2230   finally show ?thesis
```
```  2231     using b by (simp add: field_simps)
```
```  2232 qed
```
```  2233
```
```  2234 lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1"
```
```  2235   for x :: real
```
```  2236   using exp_ge_add_one_self[of "ln x"] by simp
```
```  2237
```
```  2238 corollary ln_diff_le: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x - ln y \<le> (x - y) / y"
```
```  2239   for x :: real
```
```  2240   by (simp add: ln_div [symmetric] diff_divide_distrib ln_le_minus_one)
```
```  2241
```
```  2242 lemma ln_eq_minus_one:
```
```  2243   fixes x :: real
```
```  2244   assumes "0 < x" "ln x = x - 1"
```
```  2245   shows "x = 1"
```
```  2246 proof -
```
```  2247   let ?l = "\<lambda>y. ln y - y + 1"
```
```  2248   have D: "\<And>x::real. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
```
```  2249     by (auto intro!: derivative_eq_intros)
```
```  2250
```
```  2251   show ?thesis
```
```  2252   proof (cases rule: linorder_cases)
```
```  2253     assume "x < 1"
```
```  2254     from dense[OF \<open>x < 1\<close>] obtain a where "x < a" "a < 1" by blast
```
```  2255     from \<open>x < a\<close> have "?l x < ?l a"
```
```  2256     proof (rule DERIV_pos_imp_increasing, safe)
```
```  2257       fix y
```
```  2258       assume "x \<le> y" "y \<le> a"
```
```  2259       with \<open>0 < x\<close> \<open>a < 1\<close> have "0 < 1 / y - 1" "0 < y"
```
```  2260         by (auto simp: field_simps)
```
```  2261       with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z" by blast
```
```  2262     qed
```
```  2263     also have "\<dots> \<le> 0"
```
```  2264       using ln_le_minus_one \<open>0 < x\<close> \<open>x < a\<close> by (auto simp: field_simps)
```
```  2265     finally show "x = 1" using assms by auto
```
```  2266   next
```
```  2267     assume "1 < x"
```
```  2268     from dense[OF this] obtain a where "1 < a" "a < x" by blast
```
```  2269     from \<open>a < x\<close> have "?l x < ?l a"
```
```  2270     proof (rule DERIV_neg_imp_decreasing, safe)
```
```  2271       fix y
```
```  2272       assume "a \<le> y" "y \<le> x"
```
```  2273       with \<open>1 < a\<close> have "1 / y - 1 < 0" "0 < y"
```
```  2274         by (auto simp: field_simps)
```
```  2275       with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
```
```  2276         by blast
```
```  2277     qed
```
```  2278     also have "\<dots> \<le> 0"
```
```  2279       using ln_le_minus_one \<open>1 < a\<close> by (auto simp: field_simps)
```
```  2280     finally show "x = 1" using assms by auto
```
```  2281   next
```
```  2282     assume "x = 1"
```
```  2283     then show ?thesis by simp
```
```  2284   qed
```
```  2285 qed
```
```  2286
```
```  2287 lemma ln_x_over_x_tendsto_0: "((\<lambda>x::real. ln x / x) \<longlongrightarrow> 0) at_top"
```
```  2288 proof (rule lhospital_at_top_at_top[where f' = inverse and g' = "\<lambda>_. 1"])
```
```  2289   from eventually_gt_at_top[of "0::real"]
```
```  2290   show "\<forall>\<^sub>F x in at_top. (ln has_real_derivative inverse x) (at x)"
```
```  2291     by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
```
```  2292 qed (use tendsto_inverse_0 in
```
```  2293       \<open>auto simp: filterlim_ident dest!: tendsto_mono[OF at_top_le_at_infinity]\<close>)
```
```  2294
```
```  2295 lemma exp_ge_one_plus_x_over_n_power_n:
```
```  2296   assumes "x \<ge> - real n" "n > 0"
```
```  2297   shows "(1 + x / of_nat n) ^ n \<le> exp x"
```
```  2298 proof (cases "x = - of_nat n")
```
```  2299   case False
```
```  2300   from assms False have "(1 + x / of_nat n) ^ n = exp (of_nat n * ln (1 + x / of_nat n))"
```
```  2301     by (subst exp_of_nat_mult, subst exp_ln) (simp_all add: field_simps)
```
```  2302   also from assms False have "ln (1 + x / real n) \<le> x / real n"
```
```  2303     by (intro ln_add_one_self_le_self2) (simp_all add: field_simps)
```
```  2304   with assms have "exp (of_nat n * ln (1 + x / of_nat n)) \<le> exp x"
```
```  2305     by (simp add: field_simps del: exp_of_nat_mult)
```
```  2306   finally show ?thesis .
```
```  2307 next
```
```  2308   case True
```
```  2309   then show ?thesis by (simp add: zero_power)
```
```  2310 qed
```
```  2311
```
```  2312 lemma exp_ge_one_minus_x_over_n_power_n:
```
```  2313   assumes "x \<le> real n" "n > 0"
```
```  2314   shows "(1 - x / of_nat n) ^ n \<le> exp (-x)"
```
```  2315   using exp_ge_one_plus_x_over_n_power_n[of n "-x"] assms by simp
```
```  2316
```
```  2317 lemma exp_at_bot: "(exp \<longlongrightarrow> (0::real)) at_bot"
```
```  2318   unfolding tendsto_Zfun_iff
```
```  2319 proof (rule ZfunI, simp add: eventually_at_bot_dense)
```
```  2320   fix r :: real
```
```  2321   assume "0 < r"
```
```  2322   have "exp x < r" if "x < ln r" for x
```
```  2323   proof -
```
```  2324     from that have "exp x < exp (ln r)"
```
```  2325       by simp
```
```  2326     with \<open>0 < r\<close> show ?thesis
```
```  2327       by simp
```
```  2328   qed
```
```  2329   then show "\<exists>k. \<forall>n<k. exp n < r" by auto
```
```  2330 qed
```
```  2331
```
```  2332 lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
```
```  2333   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"])
```
```  2334     (auto intro: eventually_gt_at_top)
```
```  2335
```
```  2336 lemma lim_exp_minus_1: "((\<lambda>z::'a. (exp(z) - 1) / z) \<longlongrightarrow> 1) (at 0)"
```
```  2337   for x :: "'a::{real_normed_field,banach}"
```
```  2338 proof -
```
```  2339   have "((\<lambda>z::'a. exp(z) - 1) has_field_derivative 1) (at 0)"
```
```  2340     by (intro derivative_eq_intros | simp)+
```
```  2341   then show ?thesis
```
```  2342     by (simp add: Deriv.DERIV_iff2)
```
```  2343 qed
```
```  2344
```
```  2345 lemma ln_at_0: "LIM x at_right 0. ln (x::real) :> at_bot"
```
```  2346   by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  2347      (auto simp: eventually_at_filter)
```
```  2348
```
```  2349 lemma ln_at_top: "LIM x at_top. ln (x::real) :> at_top"
```
```  2350   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  2351      (auto intro: eventually_gt_at_top)
```
```  2352
```
```  2353 lemma filtermap_ln_at_top: "filtermap (ln::real \<Rightarrow> real) at_top = at_top"
```
```  2354   by (intro filtermap_fun_inverse[of exp] exp_at_top ln_at_top) auto
```
```  2355
```
```  2356 lemma filtermap_exp_at_top: "filtermap (exp::real \<Rightarrow> real) at_top = at_top"
```
```  2357   by (intro filtermap_fun_inverse[of ln] exp_at_top ln_at_top)
```
```  2358      (auto simp: eventually_at_top_dense)
```
```  2359
```
```  2360 lemma filtermap_ln_at_right: "filtermap ln (at_right (0::real)) = at_bot"
```
```  2361   by (auto intro!: filtermap_fun_inverse[where g="\<lambda>x. exp x"] ln_at_0
```
```  2362       simp: filterlim_at exp_at_bot)
```
```  2363
```
```  2364 lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) \<longlongrightarrow> (0::real)) at_top"
```
```  2365 proof (induct k)
```
```  2366   case 0
```
```  2367   show "((\<lambda>x. x ^ 0 / exp x) \<longlongrightarrow> (0::real)) at_top"
```
```  2368     by (simp add: inverse_eq_divide[symmetric])
```
```  2369        (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
```
```  2370          at_top_le_at_infinity order_refl)
```
```  2371 next
```
```  2372   case (Suc k)
```
```  2373   show ?case
```
```  2374   proof (rule lhospital_at_top_at_top)
```
```  2375     show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"
```
```  2376       by eventually_elim (intro derivative_eq_intros, auto)
```
```  2377     show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top"
```
```  2378       by eventually_elim auto
```
```  2379     show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
```
```  2380       by auto
```
```  2381     from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
```
```  2382     show "((\<lambda>x. real (Suc k) * x ^ k / exp x) \<longlongrightarrow> 0) at_top"
```
```  2383       by simp
```
```  2384   qed (rule exp_at_top)
```
```  2385 qed
```
```  2386
```
```  2387 subsubsection\<open> A couple of simple bounds\<close>
```
```  2388
```
```  2389 lemma exp_plus_inverse_exp:
```
```  2390   fixes x::real
```
```  2391   shows "2 \<le> exp x + inverse (exp x)"
```
```  2392 proof -
```
```  2393   have "2 \<le> exp x + exp (-x)"
```
```  2394     using exp_ge_add_one_self [of x] exp_ge_add_one_self [of "-x"]
```
```  2395     by linarith
```
```  2396   then show ?thesis
```
```  2397     by (simp add: exp_minus)
```
```  2398 qed
```
```  2399
```
```  2400 lemma real_le_x_sinh:
```
```  2401   fixes x::real
```
```  2402   assumes "0 \<le> x"
```
```  2403   shows "x \<le> (exp x - inverse(exp x)) / 2"
```
```  2404 proof -
```
```  2405   have *: "exp a - inverse(exp a) - 2*a \<le> exp b - inverse(exp b) - 2*b" if "a \<le> b" for a b::real
```
```  2406     apply (rule DERIV_nonneg_imp_nondecreasing [OF that])
```
```  2407     using exp_plus_inverse_exp
```
```  2408     apply (intro exI allI impI conjI derivative_eq_intros | force)+
```
```  2409     done
```
```  2410   show ?thesis
```
```  2411     using*[OF assms] by simp
```
```  2412 qed
```
```  2413
```
```  2414 lemma real_le_abs_sinh:
```
```  2415   fixes x::real
```
```  2416   shows "abs x \<le> abs((exp x - inverse(exp x)) / 2)"
```
```  2417 proof (cases "0 \<le> x")
```
```  2418   case True
```
```  2419   show ?thesis
```
```  2420     using real_le_x_sinh [OF True] True by (simp add: abs_if)
```
```  2421 next
```
```  2422   case False
```
```  2423   have "-x \<le> (exp(-x) - inverse(exp(-x))) / 2"
```
```  2424     by (meson False linear neg_le_0_iff_le real_le_x_sinh)
```
```  2425   also have "... \<le> \<bar>(exp x - inverse (exp x)) / 2\<bar>"
```
```  2426     by (metis (no_types, hide_lams) abs_divide abs_le_iff abs_minus_cancel
```
```  2427        add.inverse_inverse exp_minus minus_diff_eq order_refl)
```
```  2428   finally show ?thesis
```
```  2429     using False by linarith
```
```  2430 qed
```
```  2431
```
```  2432 subsection\<open>The general logarithm\<close>
```
```  2433
```
```  2434 definition log :: "real \<Rightarrow> real \<Rightarrow> real"
```
```  2435   \<comment> \<open>logarithm of @{term x} to base @{term a}\<close>
```
```  2436   where "log a x = ln x / ln a"
```
```  2437
```
```  2438 lemma tendsto_log [tendsto_intros]:
```
```  2439   "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> 0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow>
```
```  2440     ((\<lambda>x. log (f x) (g x)) \<longlongrightarrow> log a b) F"
```
```  2441   unfolding log_def by (intro tendsto_intros) auto
```
```  2442
```
```  2443 lemma continuous_log:
```
```  2444   assumes "continuous F f"
```
```  2445     and "continuous F g"
```
```  2446     and "0 < f (Lim F (\<lambda>x. x))"
```
```  2447     and "f (Lim F (\<lambda>x. x)) \<noteq> 1"
```
```  2448     and "0 < g (Lim F (\<lambda>x. x))"
```
```  2449   shows "continuous F (\<lambda>x. log (f x) (g x))"
```
```  2450   using assms unfolding continuous_def by (rule tendsto_log)
```
```  2451
```
```  2452 lemma continuous_at_within_log[continuous_intros]:
```
```  2453   assumes "continuous (at a within s) f"
```
```  2454     and "continuous (at a within s) g"
```
```  2455     and "0 < f a"
```
```  2456     and "f a \<noteq> 1"
```
```  2457     and "0 < g a"
```
```  2458   shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
```
```  2459   using assms unfolding continuous_within by (rule tendsto_log)
```
```  2460
```
```  2461 lemma isCont_log[continuous_intros, simp]:
```
```  2462   assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
```
```  2463   shows "isCont (\<lambda>x. log (f x) (g x)) a"
```
```  2464   using assms unfolding continuous_at by (rule tendsto_log)
```
```  2465
```
```  2466 lemma continuous_on_log[continuous_intros]:
```
```  2467   assumes "continuous_on s f" "continuous_on s g"
```
```  2468     and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
```
```  2469   shows "continuous_on s (\<lambda>x. log (f x) (g x))"
```
```  2470   using assms unfolding continuous_on_def by (fast intro: tendsto_log)
```
```  2471
```
```  2472 lemma powr_one_eq_one [simp]: "1 powr a = 1"
```
```  2473   by (simp add: powr_def)
```
```  2474
```
```  2475 lemma powr_zero_eq_one [simp]: "x powr 0 = (if x = 0 then 0 else 1)"
```
```  2476   by (simp add: powr_def)
```
```  2477
```
```  2478 lemma powr_one_gt_zero_iff [simp]: "x powr 1 = x \<longleftrightarrow> 0 \<le> x"
```
```  2479   for x :: real
```
```  2480   by (auto simp: powr_def)
```
```  2481 declare powr_one_gt_zero_iff [THEN iffD2, simp]
```
```  2482
```
```  2483 lemma powr_mult: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
```
```  2484   for a x y :: real
```
```  2485   by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
```
```  2486
```
```  2487 lemma powr_ge_pzero [simp]: "0 \<le> x powr y"
```
```  2488   for x y :: real
```
```  2489   by (simp add: powr_def)
```
```  2490
```
```  2491 lemma powr_divide: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
```
```  2492   for a b x :: real
```
```  2493   apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
```
```  2494   apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
```
```  2495   done
```
```  2496
```
```  2497 lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
```
```  2498   for a b x :: real
```
```  2499   apply (simp add: powr_def)
```
```  2500   apply (subst exp_diff [THEN sym])
```
```  2501   apply (simp add: left_diff_distrib)
```
```  2502   done
```
```  2503
```
```  2504 lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
```
```  2505   for a b x :: real
```
```  2506   by (simp add: powr_def exp_add [symmetric] distrib_right)
```
```  2507
```
```  2508 lemma powr_mult_base: "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
```
```  2509   for x :: real
```
```  2510   by (auto simp: powr_add)
```
```  2511
```
```  2512 lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
```
```  2513   for a b x :: real
```
```  2514   by (simp add: powr_def)
```
```  2515
```
```  2516 lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
```
```  2517   for a b x :: real
```
```  2518   by (simp add: powr_powr mult.commute)
```
```  2519
```
```  2520 lemma powr_minus: "x powr (- a) = inverse (x powr a)"
```
```  2521   for x a :: real
```
```  2522   by (simp add: powr_def exp_minus [symmetric])
```
```  2523
```
```  2524 lemma powr_minus_divide: "x powr (- a) = 1/(x powr a)"
```
```  2525   for x a :: real
```
```  2526   by (simp add: divide_inverse powr_minus)
```
```  2527
```
```  2528 lemma divide_powr_uminus: "a / b powr c = a * b powr (- c)"
```
```  2529   for a b c :: real
```
```  2530   by (simp add: powr_minus_divide)
```
```  2531
```
```  2532 lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
```
```  2533   for a b x :: real
```
```  2534   by (simp add: powr_def)
```
```  2535
```
```  2536 lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
```
```  2537   for a b x :: real
```
```  2538   by (simp add: powr_def)
```
```  2539
```
```  2540 lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> x powr a < x powr b \<longleftrightarrow> a < b"
```
```  2541   for a b x :: real
```
```  2542   by (blast intro: powr_less_cancel powr_less_mono)
```
```  2543
```
```  2544 lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> x powr a \<le> x powr b \<longleftrightarrow> a \<le> b"
```
```  2545   for a b x :: real
```
```  2546   by (simp add: linorder_not_less [symmetric])
```
```  2547
```
```  2548 lemma log_ln: "ln x = log (exp(1)) x"
```
```  2549   by (simp add: log_def)
```
```  2550
```
```  2551 lemma DERIV_log:
```
```  2552   assumes "x > 0"
```
```  2553   shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
```
```  2554 proof -
```
```  2555   define lb where "lb = 1 / ln b"
```
```  2556   moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
```
```  2557     using \<open>x > 0\<close> by (auto intro!: derivative_eq_intros)
```
```  2558   ultimately show ?thesis
```
```  2559     by (simp add: log_def)
```
```  2560 qed
```
```  2561
```
```  2562 lemmas DERIV_log[THEN DERIV_chain2, derivative_intros]
```
```  2563   and DERIV_log[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  2564
```
```  2565 lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
```
```  2566   by (simp add: powr_def log_def)
```
```  2567
```
```  2568 lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y"
```
```  2569   by (simp add: log_def powr_def)
```
```  2570
```
```  2571 lemma log_mult:
```
```  2572   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow>
```
```  2573     log a (x * y) = log a x + log a y"
```
```  2574   by (simp add: log_def ln_mult divide_inverse distrib_right)
```
```  2575
```
```  2576 lemma log_eq_div_ln_mult_log:
```
```  2577   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow>
```
```  2578     log a x = (ln b/ln a) * log b x"
```
```  2579   by (simp add: log_def divide_inverse)
```
```  2580
```
```  2581 text\<open>Base 10 logarithms\<close>
```
```  2582 lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x"
```
```  2583   by (simp add: log_def)
```
```  2584
```
```  2585 lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x"
```
```  2586   by (simp add: log_def)
```
```  2587
```
```  2588 lemma log_one [simp]: "log a 1 = 0"
```
```  2589   by (simp add: log_def)
```
```  2590
```
```  2591 lemma log_eq_one [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a a = 1"
```
```  2592   by (simp add: log_def)
```
```  2593
```
```  2594 lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x"
```
```  2595   apply (rule add_left_cancel [THEN iffD1, where a1 = "log a x"])
```
```  2596   apply (simp add: log_mult [symmetric])
```
```  2597   done
```
```  2598
```
```  2599 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"
```
```  2600   by (simp add: log_mult divide_inverse log_inverse)
```
```  2601
```
```  2602 lemma powr_gt_zero [simp]: "0 < x powr a \<longleftrightarrow> x \<noteq> 0"
```
```  2603   for a x :: real
```
```  2604   by (simp add: powr_def)
```
```  2605
```
```  2606 lemma log_add_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log b x + y = log b (x * b powr y)"
```
```  2607   and add_log_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> y + log b x = log b (b powr y * x)"
```
```  2608   and log_minus_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log b x - y = log b (x * b powr -y)"
```
```  2609   and minus_log_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> y - log b x = log b (b powr y / x)"
```
```  2610   by (simp_all add: log_mult log_divide)
```
```  2611
```
```  2612 lemma log_less_cancel_iff [simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
```
```  2613   apply safe
```
```  2614    apply (rule_tac [2] powr_less_cancel)
```
```  2615     apply (drule_tac a = "log a x" in powr_less_mono)
```
```  2616      apply auto
```
```  2617   done
```
```  2618
```
```  2619 lemma log_inj:
```
```  2620   assumes "1 < b"
```
```  2621   shows "inj_on (log b) {0 <..}"
```
```  2622 proof (rule inj_onI, simp)
```
```  2623   fix x y
```
```  2624   assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
```
```  2625   show "x = y"
```
```  2626   proof (cases rule: linorder_cases)
```
```  2627     assume "x = y"
```
```  2628     then show ?thesis by simp
```
```  2629   next
```
```  2630     assume "x < y"
```
```  2631     then have "log b x < log b y"
```
```  2632       using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp
```
```  2633     then show ?thesis using * by simp
```
```  2634   next
```
```  2635     assume "y < x"
```
```  2636     then have "log b y < log b x"
```
```  2637       using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp
```
```  2638     then show ?thesis using * by simp
```
```  2639   qed
```
```  2640 qed
```
```  2641
```
```  2642 lemma log_le_cancel_iff [simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x \<le> log a y \<longleftrightarrow> x \<le> y"
```
```  2643   by (simp add: linorder_not_less [symmetric])
```
```  2644
```
```  2645 lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
```
```  2646   using log_less_cancel_iff[of a 1 x] by simp
```
```  2647
```
```  2648 lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
```
```  2649   using log_le_cancel_iff[of a 1 x] by simp
```
```  2650
```
```  2651 lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
```
```  2652   using log_less_cancel_iff[of a x 1] by simp
```
```  2653
```
```  2654 lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
```
```  2655   using log_le_cancel_iff[of a x 1] by simp
```
```  2656
```
```  2657 lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
```
```  2658   using log_less_cancel_iff[of a a x] by simp
```
```  2659
```
```  2660 lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
```
```  2661   using log_le_cancel_iff[of a a x] by simp
```
```  2662
```
```  2663 lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
```
```  2664   using log_less_cancel_iff[of a x a] by simp
```
```  2665
```
```  2666 lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
```
```  2667   using log_le_cancel_iff[of a x a] by simp
```
```  2668
```
```  2669 lemma le_log_iff:
```
```  2670   fixes b x y :: real
```
```  2671   assumes "1 < b" "x > 0"
```
```  2672   shows "y \<le> log b x \<longleftrightarrow> b powr y \<le> x"
```
```  2673   using assms
```
```  2674   apply auto
```
```  2675    apply (metis (no_types, hide_lams) less_irrefl less_le_trans linear powr_le_cancel_iff
```
```  2676       powr_log_cancel zero_less_one)
```
```  2677   apply (metis not_less order.trans order_refl powr_le_cancel_iff powr_log_cancel zero_le_one)
```
```  2678   done
```
```  2679
```
```  2680 lemma less_log_iff:
```
```  2681   assumes "1 < b" "x > 0"
```
```  2682   shows "y < log b x \<longleftrightarrow> b powr y < x"
```
```  2683   by (metis assms dual_order.strict_trans less_irrefl powr_less_cancel_iff
```
```  2684     powr_log_cancel zero_less_one)
```
```  2685
```
```  2686 lemma
```
```  2687   assumes "1 < b" "x > 0"
```
```  2688   shows log_less_iff: "log b x < y \<longleftrightarrow> x < b powr y"
```
```  2689     and log_le_iff: "log b x \<le> y \<longleftrightarrow> x \<le> b powr y"
```
```  2690   using le_log_iff[OF assms, of y] less_log_iff[OF assms, of y]
```
```  2691   by auto
```
```  2692
```
```  2693 lemmas powr_le_iff = le_log_iff[symmetric]
```
```  2694   and powr_less_iff = le_log_iff[symmetric]
```
```  2695   and less_powr_iff = log_less_iff[symmetric]
```
```  2696   and le_powr_iff = log_le_iff[symmetric]
```
```  2697
```
```  2698 lemma gr_one_powr[simp]:
```
```  2699   fixes x y :: real shows "\<lbrakk> x > 1; y > 0 \<rbrakk> \<Longrightarrow> 1 < x powr y"
```
```  2700 by(simp add: less_powr_iff)
```
```  2701
```
```  2702 lemma floor_log_eq_powr_iff: "x > 0 \<Longrightarrow> b > 1 \<Longrightarrow> \<lfloor>log b x\<rfloor> = k \<longleftrightarrow> b powr k \<le> x \<and> x < b powr (k + 1)"
```
```  2703   by (auto simp add: floor_eq_iff powr_le_iff less_powr_iff)
```
```  2704
```
```  2705 lemma powr_realpow: "0 < x \<Longrightarrow> x powr (real n) = x^n"
```
```  2706   by (induct n) (simp_all add: ac_simps powr_add)
```
```  2707
```
```  2708 lemma powr_numeral [simp]: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
```
```  2709   by (metis of_nat_numeral powr_realpow)
```
```  2710
```
```  2711 lemma numeral_powr_numeral[simp]:
```
```  2712   "(numeral m :: real) powr (numeral n :: real) = numeral m ^ (numeral n)"
```
```  2713 by(simp add: powr_numeral)
```
```  2714
```
```  2715 lemma powr_real_of_int:
```
```  2716   "x > 0 \<Longrightarrow> x powr real_of_int n = (if n \<ge> 0 then x ^ nat n else inverse (x ^ nat (- n)))"
```
```  2717   using powr_realpow[of x "nat n"] powr_realpow[of x "nat (-n)"]
```
```  2718   by (auto simp: field_simps powr_minus)
```
```  2719
```
```  2720 lemma powr2_sqrt[simp]: "0 < x \<Longrightarrow> sqrt x powr 2 = x"
```
```  2721   by (simp add: powr_numeral)
```
```  2722
```
```  2723 lemma powr_realpow2: "0 \<le> x \<Longrightarrow> 0 < n \<Longrightarrow> x^n = (if (x = 0) then 0 else x powr (real n))"
```
```  2724   apply (cases "x = 0")
```
```  2725    apply simp_all
```
```  2726   apply (rule powr_realpow [THEN sym])
```
```  2727   apply simp
```
```  2728   done
```
```  2729
```
```  2730 lemma powr_int:
```
```  2731   assumes "x > 0"
```
```  2732   shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
```
```  2733 proof (cases "i < 0")
```
```  2734   case True
```
```  2735   have r: "x powr i = 1 / x powr (- i)"
```
```  2736     by (simp add: powr_minus field_simps)
```
```  2737   show ?thesis using \<open>i < 0\<close> \<open>x > 0\<close>
```
```  2738     by (simp add: r field_simps powr_realpow[symmetric])
```
```  2739 next
```
```  2740   case False
```
```  2741   then show ?thesis
```
```  2742     by (simp add: assms powr_realpow[symmetric])
```
```  2743 qed
```
```  2744
```
```  2745 lemma compute_powr[code]:
```
```  2746   fixes i :: real
```
```  2747   shows "b powr i =
```
```  2748     (if b \<le> 0 then Code.abort (STR ''op powr with nonpositive base'') (\<lambda>_. b powr i)
```
```  2749      else if \<lfloor>i\<rfloor> = i then (if 0 \<le> i then b ^ nat \<lfloor>i\<rfloor> else 1 / b ^ nat \<lfloor>- i\<rfloor>)
```
```  2750      else Code.abort (STR ''op powr with non-integer exponent'') (\<lambda>_. b powr i))"
```
```  2751   by (auto simp: powr_int)
```
```  2752
```
```  2753 lemma powr_one: "0 \<le> x \<Longrightarrow> x powr 1 = x"
```
```  2754   for x :: real
```
```  2755   using powr_realpow [of x 1] by simp
```
```  2756
```
```  2757 lemma powr_neg_one: "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
```
```  2758   for x :: real
```
```  2759   using powr_int [of x "- 1"] by simp
```
```  2760
```
```  2761 lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
```
```  2762   for x :: real
```
```  2763   using powr_int [of x "- numeral n"] by simp
```
```  2764
```
```  2765 lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
```
```  2766   by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
```
```  2767
```
```  2768 lemma ln_powr: "x \<noteq> 0 \<Longrightarrow> ln (x powr y) = y * ln x"
```
```  2769   for x :: real
```
```  2770   by (simp add: powr_def)
```
```  2771
```
```  2772 lemma ln_root: "n > 0 \<Longrightarrow> b > 0 \<Longrightarrow> ln (root n b) =  ln b / n"
```
```  2773   by (simp add: root_powr_inverse ln_powr)
```
```  2774
```
```  2775 lemma ln_sqrt: "0 < x \<Longrightarrow> ln (sqrt x) = ln x / 2"
```
```  2776   by (simp add: ln_powr ln_powr[symmetric] mult.commute)
```
```  2777
```
```  2778 lemma log_root: "n > 0 \<Longrightarrow> a > 0 \<Longrightarrow> log b (root n a) =  log b a / n"
```
```  2779   by (simp add: log_def ln_root)
```
```  2780
```
```  2781 lemma log_powr: "x \<noteq> 0 \<Longrightarrow> log b (x powr y) = y * log b x"
```
```  2782   by (simp add: log_def ln_powr)
```
```  2783
```
```  2784 (* [simp] is not worth it, interferes with some proofs *)
```
```  2785 lemma log_nat_power: "0 < x \<Longrightarrow> log b (x^n) = real n * log b x"
```
```  2786   by (simp add: log_powr powr_realpow [symmetric])
```
```  2787
```
```  2788 lemma le_log_of_power:
```
```  2789   assumes "1 < b" "b ^ n \<le> m"
```
```  2790   shows "n \<le> log b m"
```
```  2791 proof -
```
```  2792    from assms have "0 < m"
```
```  2793      by (metis less_trans zero_less_power less_le_trans zero_less_one)
```
```  2794    have "n = log b (b ^ n)"
```
```  2795      using assms(1) by (simp add: log_nat_power)
```
```  2796    also have "\<dots> \<le> log b m"
```
```  2797      using assms \<open>0 < m\<close> by simp
```
```  2798    finally show ?thesis .
```
```  2799 qed
```
```  2800
```
```  2801 lemma le_log2_of_power: "2 ^ n \<le> m \<Longrightarrow> n \<le> log 2 m"
```
```  2802   for m n :: nat
```
```  2803   using le_log_of_power[of 2] by simp
```
```  2804
```
```  2805 lemma log_base_change: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log b x = log a x / log a b"
```
```  2806   by (simp add: log_def)
```
```  2807
```
```  2808 lemma log_base_pow: "0 < a \<Longrightarrow> log (a ^ n) x = log a x / n"
```
```  2809   by (simp add: log_def ln_realpow)
```
```  2810
```
```  2811 lemma log_base_powr: "a \<noteq> 0 \<Longrightarrow> log (a powr b) x = log a x / b"
```
```  2812   by (simp add: log_def ln_powr)
```
```  2813
```
```  2814 lemma log_base_root: "n > 0 \<Longrightarrow> b > 0 \<Longrightarrow> log (root n b) x = n * (log b x)"
```
```  2815   by (simp add: log_def ln_root)
```
```  2816
```
```  2817 lemma ln_bound: "1 \<le> x \<Longrightarrow> ln x \<le> x"
```
```  2818   for x :: real
```
```  2819   apply (subgoal_tac "ln (1 + (x - 1)) \<le> x - 1")
```
```  2820    apply simp
```
```  2821   apply (rule ln_add_one_self_le_self)
```
```  2822   apply simp
```
```  2823   done
```
```  2824
```
```  2825 lemma powr_mono: "a \<le> b \<Longrightarrow> 1 \<le> x \<Longrightarrow> x powr a \<le> x powr b"
```
```  2826   for x :: real
```
```  2827   apply (cases "x = 1")
```
```  2828    apply simp
```
```  2829   apply (cases "a = b")
```
```  2830    apply simp
```
```  2831   apply (rule order_less_imp_le)
```
```  2832   apply (rule powr_less_mono)
```
```  2833    apply auto
```
```  2834   done
```
```  2835
```
```  2836 lemma ge_one_powr_ge_zero: "1 \<le> x \<Longrightarrow> 0 \<le> a \<Longrightarrow> 1 \<le> x powr a"
```
```  2837   for x :: real
```
```  2838   using powr_mono by fastforce
```
```  2839
```
```  2840 lemma powr_less_mono2: "0 < a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> x powr a < y powr a"
```
```  2841   for x :: real
```
```  2842   by (simp add: powr_def)
```
```  2843
```
```  2844 lemma powr_less_mono2_neg: "a < 0 \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> y powr a < x powr a"
```
```  2845   for x :: real
```
```  2846   by (simp add: powr_def)
```
```  2847
```
```  2848 lemma powr_mono2: "x powr a \<le> y powr a" if "0 \<le> a" "0 \<le> x" "x \<le> y"
```
```  2849   for x :: real
```
```  2850 proof (cases "a = 0")
```
```  2851   case True
```
```  2852   with that show ?thesis by simp
```
```  2853 next
```
```  2854   case False show ?thesis
```
```  2855   proof (cases "x = y")
```
```  2856     case True
```
```  2857     then show ?thesis by simp
```
```  2858   next
```
```  2859     case False
```
```  2860     then show ?thesis
```
```  2861       by (metis dual_order.strict_iff_order powr_less_mono2 that \<open>a \<noteq> 0\<close>)
```
```  2862   qed
```
```  2863 qed
```
```  2864
```
```  2865 lemma powr_le1: "0 \<le> a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> x powr a \<le> 1"
```
```  2866   for x :: real
```
```  2867   using powr_mono2 by fastforce
```
```  2868
```
```  2869 lemma powr_mono2':
```
```  2870   fixes a x y :: real
```
```  2871   assumes "a \<le> 0" "x > 0" "x \<le> y"
```
```  2872   shows "x powr a \<ge> y powr a"
```
```  2873 proof -
```
```  2874   from assms have "x powr - a \<le> y powr - a"
```
```  2875     by (intro powr_mono2) simp_all
```
```  2876   with assms show ?thesis
```
```  2877     by (auto simp add: powr_minus field_simps)
```
```  2878 qed
```
```  2879
```
```  2880 lemma powr_mono_both:
```
```  2881   fixes x :: real
```
```  2882   assumes "0 \<le> a" "a \<le> b" "1 \<le> x" "x \<le> y"
```
```  2883     shows "x powr a \<le> y powr b"
```
```  2884   by (meson assms order.trans powr_mono powr_mono2 zero_le_one)
```
```  2885
```
```  2886 lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
```
```  2887   for x :: real
```
```  2888   unfolding powr_def exp_inj_iff by simp
```
```  2889
```
```  2890 lemma powr_half_sqrt: "0 \<le> x \<Longrightarrow> x powr (1/2) = sqrt x"
```
```  2891   by (simp add: powr_def root_powr_inverse sqrt_def)
```
```  2892
```
```  2893 lemma ln_powr_bound: "1 \<le> x \<Longrightarrow> 0 < a \<Longrightarrow> ln x \<le> (x powr a) / a"
```
```  2894   for x :: real
```
```  2895   by (metis exp_gt_zero linear ln_eq_zero_iff ln_exp ln_less_self ln_powr mult.commute
```
```  2896       mult_imp_le_div_pos not_less powr_gt_zero)
```
```  2897
```
```  2898 lemma ln_powr_bound2:
```
```  2899   fixes x :: real
```
```  2900   assumes "1 < x" and "0 < a"
```
```  2901   shows "(ln x) powr a \<le> (a powr a) * x"
```
```  2902 proof -
```
```  2903   from assms have "ln x \<le> (x powr (1 / a)) / (1 / a)"
```
```  2904     by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff)
```
```  2905   also have "\<dots> = a * (x powr (1 / a))"
```
```  2906     by simp
```
```  2907   finally have "(ln x) powr a \<le> (a * (x powr (1 / a))) powr a"
```
```  2908     by (metis assms less_imp_le ln_gt_zero powr_mono2)
```
```  2909   also have "\<dots> = (a powr a) * ((x powr (1 / a)) powr a)"
```
```  2910     using assms powr_mult by auto
```
```  2911   also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
```
```  2912     by (rule powr_powr)
```
```  2913   also have "\<dots> = x" using assms
```
```  2914     by auto
```
```  2915   finally show ?thesis .
```
```  2916 qed
```
```  2917
```
```  2918 lemma tendsto_powr:
```
```  2919   fixes a b :: real
```
```  2920   assumes f: "(f \<longlongrightarrow> a) F"
```
```  2921     and g: "(g \<longlongrightarrow> b) F"
```
```  2922     and a: "a \<noteq> 0"
```
```  2923   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
```
```  2924   unfolding powr_def
```
```  2925 proof (rule filterlim_If)
```
```  2926   from f show "((\<lambda>x. 0) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))"
```
```  2927     by simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_mono dest: t1_space_nhds)
```
```  2928   from f g a show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a)))
```
```  2929       (inf F (principal {x. f x \<noteq> 0}))"
```
```  2930     by (auto intro!: tendsto_intros intro: tendsto_mono inf_le1)
```
```  2931 qed
```
```  2932
```
```  2933 lemma tendsto_powr'[tendsto_intros]:
```
```  2934   fixes a :: real
```
```  2935   assumes f: "(f \<longlongrightarrow> a) F"
```
```  2936     and g: "(g \<longlongrightarrow> b) F"
```
```  2937     and a: "a \<noteq> 0 \<or> (b > 0 \<and> eventually (\<lambda>x. f x \<ge> 0) F)"
```
```  2938   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
```
```  2939 proof -
```
```  2940   from a consider "a \<noteq> 0" | "a = 0" "b > 0" "eventually (\<lambda>x. f x \<ge> 0) F"
```
```  2941     by auto
```
```  2942   then show ?thesis
```
```  2943   proof cases
```
```  2944     case 1
```
```  2945     with f g show ?thesis by (rule tendsto_powr)
```
```  2946   next
```
```  2947     case 2
```
```  2948     have "((\<lambda>x. if f x = 0 then 0 else exp (g x * ln (f x))) \<longlongrightarrow> 0) F"
```
```  2949     proof (intro filterlim_If)
```
```  2950       have "filterlim f (principal {0<..}) (inf F (principal {z. f z \<noteq> 0}))"
```
```  2951         using \<open>eventually (\<lambda>x. f x \<ge> 0) F\<close>
```
```  2952         by (auto simp add: filterlim_iff eventually_inf_principal
```
```  2953             eventually_principal elim: eventually_mono)
```
```  2954       moreover have "filterlim f (nhds a) (inf F (principal {z. f z \<noteq> 0}))"
```
```  2955         by (rule tendsto_mono[OF _ f]) simp_all
```
```  2956       ultimately have f: "filterlim f (at_right 0) (inf F (principal {x. f x \<noteq> 0}))"
```
```  2957         by (simp add: at_within_def filterlim_inf \<open>a = 0\<close>)
```
```  2958       have g: "(g \<longlongrightarrow> b) (inf F (principal {z. f z \<noteq> 0}))"
```
```  2959         by (rule tendsto_mono[OF _ g]) simp_all
```
```  2960       show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> 0) (inf F (principal {x. f x \<noteq> 0}))"
```
```  2961         by (rule filterlim_compose[OF exp_at_bot] filterlim_tendsto_pos_mult_at_bot
```
```  2962                  filterlim_compose[OF ln_at_0] f g \<open>b > 0\<close>)+
```
```  2963     qed simp_all
```
```  2964     with \<open>a = 0\<close> show ?thesis
```
```  2965       by (simp add: powr_def)
```
```  2966   qed
```
```  2967 qed
```
```  2968
```
```  2969 lemma continuous_powr:
```
```  2970   assumes "continuous F f"
```
```  2971     and "continuous F g"
```
```  2972     and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```  2973   shows "continuous F (\<lambda>x. (f x) powr (g x :: real))"
```
```  2974   using assms unfolding continuous_def by (rule tendsto_powr)
```
```  2975
```
```  2976 lemma continuous_at_within_powr[continuous_intros]:
```
```  2977   fixes f g :: "_ \<Rightarrow> real"
```
```  2978   assumes "continuous (at a within s) f"
```
```  2979     and "continuous (at a within s) g"
```
```  2980     and "f a \<noteq> 0"
```
```  2981   shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
```
```  2982   using assms unfolding continuous_within by (rule tendsto_powr)
```
```  2983
```
```  2984 lemma isCont_powr[continuous_intros, simp]:
```
```  2985   fixes f g :: "_ \<Rightarrow> real"
```
```  2986   assumes "isCont f a" "isCont g a" "f a \<noteq> 0"
```
```  2987   shows "isCont (\<lambda>x. (f x) powr g x) a"
```
```  2988   using assms unfolding continuous_at by (rule tendsto_powr)
```
```  2989
```
```  2990 lemma continuous_on_powr[continuous_intros]:
```
```  2991   fixes f g :: "_ \<Rightarrow> real"
```
```  2992   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> 0"
```
```  2993   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
```
```  2994   using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
```
```  2995
```
```  2996 lemma tendsto_powr2:
```
```  2997   fixes a :: real
```
```  2998   assumes f: "(f \<longlongrightarrow> a) F"
```
```  2999     and g: "(g \<longlongrightarrow> b) F"
```
```  3000     and "\<forall>\<^sub>F x in F. 0 \<le> f x"
```
```  3001     and b: "0 < b"
```
```  3002   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
```
```  3003   using tendsto_powr'[of f a F g b] assms by auto
```
```  3004
```
```  3005 lemma DERIV_powr:
```
```  3006   fixes r :: real
```
```  3007   assumes g: "DERIV g x :> m"
```
```  3008     and pos: "g x > 0"
```
```  3009     and f: "DERIV f x :> r"
```
```  3010   shows "DERIV (\<lambda>x. g x powr f x) x :> (g x powr f x) * (r * ln (g x) + m * f x / g x)"
```
```  3011 proof -
```
```  3012   have "DERIV (\<lambda>x. exp (f x * ln (g x))) x :> (g x powr f x) * (r * ln (g x) + m * f x / g x)"
```
```  3013     using pos
```
```  3014     by (auto intro!: derivative_eq_intros g pos f simp: powr_def field_simps exp_diff)
```
```  3015   then show ?thesis
```
```  3016   proof (rule DERIV_cong_ev[OF refl _ refl, THEN iffD1, rotated])
```
```  3017     from DERIV_isCont[OF g] pos have "\<forall>\<^sub>F x in at x. 0 < g x"
```
```  3018       unfolding isCont_def by (rule order_tendstoD(1))
```
```  3019     with pos show "\<forall>\<^sub>F x in nhds x. exp (f x * ln (g x)) = g x powr f x"
```
```  3020       by (auto simp: eventually_at_filter powr_def elim: eventually_mono)
```
```  3021   qed
```
```  3022 qed
```
```  3023
```
```  3024 lemma DERIV_fun_powr:
```
```  3025   fixes r :: real
```
```  3026   assumes g: "DERIV g x :> m"
```
```  3027     and pos: "g x > 0"
```
```  3028   shows "DERIV (\<lambda>x. (g x) powr r) x :> r * (g x) powr (r - of_nat 1) * m"
```
```  3029   using DERIV_powr[OF g pos DERIV_const, of r] pos
```
```  3030   by (simp add: powr_divide2[symmetric] field_simps)
```
```  3031
```
```  3032 lemma has_real_derivative_powr:
```
```  3033   assumes "z > 0"
```
```  3034   shows "((\<lambda>z. z powr r) has_real_derivative r * z powr (r - 1)) (at z)"
```
```  3035 proof (subst DERIV_cong_ev[OF refl _ refl])
```
```  3036   from assms have "eventually (\<lambda>z. z \<noteq> 0) (nhds z)"
```
```  3037     by (intro t1_space_nhds) auto
```
```  3038   then show "eventually (\<lambda>z. z powr r = exp (r * ln z)) (nhds z)"
```
```  3039     unfolding powr_def by eventually_elim simp
```
```  3040   from assms show "((\<lambda>z. exp (r * ln z)) has_real_derivative r * z powr (r - 1)) (at z)"
```
```  3041     by (auto intro!: derivative_eq_intros simp: powr_def field_simps exp_diff)
```
```  3042 qed
```
```  3043
```
```  3044 declare has_real_derivative_powr[THEN DERIV_chain2, derivative_intros]
```
```  3045
```
```  3046 lemma tendsto_zero_powrI:
```
```  3047   assumes "(f \<longlongrightarrow> (0::real)) F" "(g \<longlongrightarrow> b) F" "\<forall>\<^sub>F x in F. 0 \<le> f x" "0 < b"
```
```  3048   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> 0) F"
```
```  3049   using tendsto_powr2[OF assms] by simp
```
```  3050
```
```  3051 lemma continuous_on_powr':
```
```  3052   fixes f g :: "_ \<Rightarrow> real"
```
```  3053   assumes "continuous_on s f" "continuous_on s g"
```
```  3054     and "\<forall>x\<in>s. f x \<ge> 0 \<and> (f x = 0 \<longrightarrow> g x > 0)"
```
```  3055   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
```
```  3056   unfolding continuous_on_def
```
```  3057 proof
```
```  3058   fix x
```
```  3059   assume x: "x \<in> s"
```
```  3060   from assms x show "((\<lambda>x. f x powr g x) \<longlongrightarrow> f x powr g x) (at x within s)"
```
```  3061   proof (cases "f x = 0")
```
```  3062     case True
```
```  3063     from assms(3) have "eventually (\<lambda>x. f x \<ge> 0) (at x within s)"
```
```  3064       by (auto simp: at_within_def eventually_inf_principal)
```
```  3065     with True x assms show ?thesis
```
```  3066       by (auto intro!: tendsto_zero_powrI[of f _ g "g x"] simp: continuous_on_def)
```
```  3067   next
```
```  3068     case False
```
```  3069     with assms x show ?thesis
```
```  3070       by (auto intro!: tendsto_powr' simp: continuous_on_def)
```
```  3071   qed
```
```  3072 qed
```
```  3073
```
```  3074 lemma tendsto_neg_powr:
```
```  3075   assumes "s < 0"
```
```  3076     and f: "LIM x F. f x :> at_top"
```
```  3077   shows "((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F"
```
```  3078 proof -
```
```  3079   have "((\<lambda>x. exp (s * ln (f x))) \<longlongrightarrow> (0::real)) F" (is "?X")
```
```  3080     by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_top]
```
```  3081         filterlim_tendsto_neg_mult_at_bot assms)
```
```  3082   also have "?X \<longleftrightarrow> ((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F"
```
```  3083     using f filterlim_at_top_dense[of f F]
```
```  3084     by (intro filterlim_cong[OF refl refl]) (auto simp: neq_iff powr_def elim: eventually_mono)
```
```  3085   finally show ?thesis .
```
```  3086 qed
```
```  3087
```
```  3088 lemma tendsto_exp_limit_at_right: "((\<lambda>y. (1 + x * y) powr (1 / y)) \<longlongrightarrow> exp x) (at_right 0)"
```
```  3089   for x :: real
```
```  3090 proof (cases "x = 0")
```
```  3091   case True
```
```  3092   then show ?thesis by simp
```
```  3093 next
```
```  3094   case False
```
```  3095   have "((\<lambda>y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)"
```
```  3096     by (auto intro!: derivative_eq_intros)
```
```  3097   then have "((\<lambda>y. ln (1 + x * y) / y) \<longlongrightarrow> x) (at 0)"
```
```  3098     by (auto simp add: has_field_derivative_def field_has_derivative_at)
```
```  3099   then have *: "((\<lambda>y. exp (ln (1 + x * y) / y)) \<longlongrightarrow> exp x) (at 0)"
```
```  3100     by (rule tendsto_intros)
```
```  3101   then show ?thesis
```
```  3102   proof (rule filterlim_mono_eventually)
```
```  3103     show "eventually (\<lambda>xa. exp (ln (1 + x * xa) / xa) = (1 + x * xa) powr (1 / xa)) (at_right 0)"
```
```  3104       unfolding eventually_at_right[OF zero_less_one]
```
```  3105       using False
```
```  3106       apply (intro exI[of _ "1 / \<bar>x\<bar>"])
```
```  3107       apply (auto simp: field_simps powr_def abs_if)
```
```  3108       apply (metis add_less_same_cancel1 mult_less_0_iff not_less_iff_gr_or_eq zero_less_one)
```
```  3109       done
```
```  3110   qed (simp_all add: at_eq_sup_left_right)
```
```  3111 qed
```
```  3112
```
```  3113 lemma tendsto_exp_limit_at_top: "((\<lambda>y. (1 + x / y) powr y) \<longlongrightarrow> exp x) at_top"
```
```  3114   for x :: real
```
```  3115   apply (subst filterlim_at_top_to_right)
```
```  3116   apply (simp add: inverse_eq_divide)
```
```  3117   apply (rule tendsto_exp_limit_at_right)
```
```  3118   done
```
```  3119
```
```  3120 lemma tendsto_exp_limit_sequentially: "(\<lambda>n. (1 + x / n) ^ n) \<longlonglongrightarrow> exp x"
```
```  3121   for x :: real
```
```  3122 proof (rule filterlim_mono_eventually)
```
```  3123   from reals_Archimedean2 [of "\<bar>x\<bar>"] obtain n :: nat where *: "real n > \<bar>x\<bar>" ..
```
```  3124   then have "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top"
```
```  3125     apply (intro eventually_sequentiallyI [of n])
```
```  3126     apply (cases "x \<ge> 0")
```
```  3127      apply (rule add_pos_nonneg)
```
```  3128       apply (auto intro: divide_nonneg_nonneg)
```
```  3129     apply (subgoal_tac "x / real xa > - 1")
```
```  3130      apply (auto simp add: field_simps)
```
```  3131     done
```
```  3132   then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top"
```
```  3133     by (rule eventually_mono) (erule powr_realpow)
```
```  3134   show "(\<lambda>n. (1 + x / real n) powr real n) \<longlonglongrightarrow> exp x"
```
```  3135     by (rule filterlim_compose [OF tendsto_exp_limit_at_top filterlim_real_sequentially])
```
```  3136 qed auto
```
```  3137
```
```  3138
```
```  3139 subsection \<open>Sine and Cosine\<close>
```
```  3140
```
```  3141 definition sin_coeff :: "nat \<Rightarrow> real"
```
```  3142   where "sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / (fact n))"
```
```  3143
```
```  3144 definition cos_coeff :: "nat \<Rightarrow> real"
```
```  3145   where "cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / (fact n) else 0)"
```
```  3146
```
```  3147 definition sin :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```  3148   where "sin = (\<lambda>x. \<Sum>n. sin_coeff n *\<^sub>R x^n)"
```
```  3149
```
```  3150 definition cos :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```  3151   where "cos = (\<lambda>x. \<Sum>n. cos_coeff n *\<^sub>R x^n)"
```
```  3152
```
```  3153 lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
```
```  3154   unfolding sin_coeff_def by simp
```
```  3155
```
```  3156 lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
```
```  3157   unfolding cos_coeff_def by simp
```
```  3158
```
```  3159 lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
```
```  3160   unfolding cos_coeff_def sin_coeff_def
```
```  3161   by (simp del: mult_Suc)
```
```  3162
```
```  3163 lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
```
```  3164   unfolding cos_coeff_def sin_coeff_def
```
```  3165   by (simp del: mult_Suc) (auto elim: oddE)
```
```  3166
```
```  3167 lemma summable_norm_sin: "summable (\<lambda>n. norm (sin_coeff n *\<^sub>R x^n))"
```
```  3168   for x :: "'a::{real_normed_algebra_1,banach}"
```
```  3169   unfolding sin_coeff_def
```
```  3170   apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]])
```
```  3171   apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  3172   done
```
```  3173
```
```  3174 lemma summable_norm_cos: "summable (\<lambda>n. norm (cos_coeff n *\<^sub>R x^n))"
```
```  3175   for x :: "'a::{real_normed_algebra_1,banach}"
```
```  3176   unfolding cos_coeff_def
```
```  3177   apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]])
```
```  3178   apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  3179   done
```
```  3180
```
```  3181 lemma sin_converges: "(\<lambda>n. sin_coeff n *\<^sub>R x^n) sums sin x"
```
```  3182   unfolding sin_def
```
```  3183   by (metis (full_types) summable_norm_cancel summable_norm_sin summable_sums)
```
```  3184
```
```  3185 lemma cos_converges: "(\<lambda>n. cos_coeff n *\<^sub>R x^n) sums cos x"
```
```  3186   unfolding cos_def
```
```  3187   by (metis (full_types) summable_norm_cancel summable_norm_cos summable_sums)
```
```  3188
```
```  3189 lemma sin_of_real: "sin (of_real x) = of_real (sin x)"
```
```  3190   for x :: real
```
```  3191 proof -
```
```  3192   have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R  x^n)) = (\<lambda>n. sin_coeff n *\<^sub>R  (of_real x)^n)"
```
```  3193   proof
```
```  3194     show "of_real (sin_coeff n *\<^sub>R  x^n) = sin_coeff n *\<^sub>R of_real x^n" for n
```
```  3195       by (simp add: scaleR_conv_of_real)
```
```  3196   qed
```
```  3197   also have "\<dots> sums (sin (of_real x))"
```
```  3198     by (rule sin_converges)
```
```  3199   finally have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) sums (sin (of_real x))" .
```
```  3200   then show ?thesis
```
```  3201     using sums_unique2 sums_of_real [OF sin_converges]
```
```  3202     by blast
```
```  3203 qed
```
```  3204
```
```  3205 corollary sin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> sin z \<in> \<real>"
```
```  3206   by (metis Reals_cases Reals_of_real sin_of_real)
```
```  3207
```
```  3208 lemma cos_of_real: "cos (of_real x) = of_real (cos x)"
```
```  3209   for x :: real
```
```  3210 proof -
```
```  3211   have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R  x^n)) = (\<lambda>n. cos_coeff n *\<^sub>R  (of_real x)^n)"
```
```  3212   proof
```
```  3213     show "of_real (cos_coeff n *\<^sub>R  x^n) = cos_coeff n *\<^sub>R of_real x^n" for n
```
```  3214       by (simp add: scaleR_conv_of_real)
```
```  3215   qed
```
```  3216   also have "\<dots> sums (cos (of_real x))"
```
```  3217     by (rule cos_converges)
```
```  3218   finally have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) sums (cos (of_real x))" .
```
```  3219   then show ?thesis
```
```  3220     using sums_unique2 sums_of_real [OF cos_converges]
```
```  3221     by blast
```
```  3222 qed
```
```  3223
```
```  3224 corollary cos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cos z \<in> \<real>"
```
```  3225   by (metis Reals_cases Reals_of_real cos_of_real)
```
```  3226
```
```  3227 lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
```
```  3228   by (simp add: diffs_def sin_coeff_Suc del: of_nat_Suc)
```
```  3229
```
```  3230 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
```
```  3231   by (simp add: diffs_def cos_coeff_Suc del: of_nat_Suc)
```
```  3232
```
```  3233 lemma sin_int_times_real: "sin (of_int m * of_real x) = of_real (sin (of_int m * x))"
```
```  3234   by (metis sin_of_real of_real_mult of_real_of_int_eq)
```
```  3235
```
```  3236 lemma cos_int_times_real: "cos (of_int m * of_real x) = of_real (cos (of_int m * x))"
```
```  3237   by (metis cos_of_real of_real_mult of_real_of_int_eq)
```
```  3238
```
```  3239 text \<open>Now at last we can get the derivatives of exp, sin and cos.\<close>
```
```  3240
```
```  3241 lemma DERIV_sin [simp]: "DERIV sin x :> cos x"
```
```  3242   for x :: "'a::{real_normed_field,banach}"
```
```  3243   unfolding sin_def cos_def scaleR_conv_of_real
```
```  3244   apply (rule DERIV_cong)
```
```  3245    apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
```
```  3246       apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff
```
```  3247               summable_minus_iff scaleR_conv_of_real [symmetric]
```
```  3248               summable_norm_sin [THEN summable_norm_cancel]
```
```  3249               summable_norm_cos [THEN summable_norm_cancel])
```
```  3250   done
```
```  3251
```
```  3252 declare DERIV_sin[THEN DERIV_chain2, derivative_intros]
```
```  3253   and DERIV_sin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  3254
```
```  3255 lemma DERIV_cos [simp]: "DERIV cos x :> - sin x"
```
```  3256   for x :: "'a::{real_normed_field,banach}"
```
```  3257   unfolding sin_def cos_def scaleR_conv_of_real
```
```  3258   apply (rule DERIV_cong)
```
```  3259    apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
```
```  3260       apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus
```
```  3261               diffs_sin_coeff diffs_cos_coeff
```
```  3262               summable_minus_iff scaleR_conv_of_real [symmetric]
```
```  3263               summable_norm_sin [THEN summable_norm_cancel]
```
```  3264               summable_norm_cos [THEN summable_norm_cancel])
```
```  3265   done
```
```  3266
```
```  3267 declare DERIV_cos[THEN DERIV_chain2, derivative_intros]
```
```  3268   and DERIV_cos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  3269
```
```  3270 lemma isCont_sin: "isCont sin x"
```
```  3271   for x :: "'a::{real_normed_field,banach}"
```
```  3272   by (rule DERIV_sin [THEN DERIV_isCont])
```
```  3273
```
```  3274 lemma isCont_cos: "isCont cos x"
```
```  3275   for x :: "'a::{real_normed_field,banach}"
```
```  3276   by (rule DERIV_cos [THEN DERIV_isCont])
```
```  3277
```
```  3278 lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
```
```  3279   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3280   by (rule isCont_o2 [OF _ isCont_sin])
```
```  3281
```
```  3282 (* FIXME a context for f would be better *)
```
```  3283
```
```  3284 lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
```
```  3285   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3286   by (rule isCont_o2 [OF _ isCont_cos])
```
```  3287
```
```  3288 lemma tendsto_sin [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) \<longlongrightarrow> sin a) F"
```
```  3289   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3290   by (rule isCont_tendsto_compose [OF isCont_sin])
```
```  3291
```
```  3292 lemma tendsto_cos [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) \<longlongrightarrow> cos a) F"
```
```  3293   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3294   by (rule isCont_tendsto_compose [OF isCont_cos])
```
```  3295
```
```  3296 lemma continuous_sin [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
```
```  3297   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3298   unfolding continuous_def by (rule tendsto_sin)
```
```  3299
```
```  3300 lemma continuous_on_sin [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
```
```  3301   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3302   unfolding continuous_on_def by (auto intro: tendsto_sin)
```
```  3303
```
```  3304 lemma continuous_within_sin: "continuous (at z within s) sin"
```
```  3305   for z :: "'a::{real_normed_field,banach}"
```
```  3306   by (simp add: continuous_within tendsto_sin)
```
```  3307
```
```  3308 lemma continuous_cos [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
```
```  3309   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3310   unfolding continuous_def by (rule tendsto_cos)
```
```  3311
```
```  3312 lemma continuous_on_cos [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
```
```  3313   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3314   unfolding continuous_on_def by (auto intro: tendsto_cos)
```
```  3315
```
```  3316 lemma continuous_within_cos: "continuous (at z within s) cos"
```
```  3317   for z :: "'a::{real_normed_field,banach}"
```
```  3318   by (simp add: continuous_within tendsto_cos)
```
```  3319
```
```  3320
```
```  3321 subsection \<open>Properties of Sine and Cosine\<close>
```
```  3322
```
```  3323 lemma sin_zero [simp]: "sin 0 = 0"
```
```  3324   by (simp add: sin_def sin_coeff_def scaleR_conv_of_real)
```
```  3325
```
```  3326 lemma cos_zero [simp]: "cos 0 = 1"
```
```  3327   by (simp add: cos_def cos_coeff_def scaleR_conv_of_real)
```
```  3328
```
```  3329 lemma DERIV_fun_sin: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. sin (g x)) x :> cos (g x) * m"
```
```  3330   by (auto intro!: derivative_intros)
```
```  3331
```
```  3332 lemma DERIV_fun_cos: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. cos(g x)) x :> - sin (g x) * m"
```
```  3333   by (auto intro!: derivative_eq_intros)
```
```  3334
```
```  3335
```
```  3336 subsection \<open>Deriving the Addition Formulas\<close>
```
```  3337
```
```  3338 text \<open>The product of two cosine series.\<close>
```
```  3339 lemma cos_x_cos_y:
```
```  3340   fixes x :: "'a::{real_normed_field,banach}"
```
```  3341   shows
```
```  3342     "(\<lambda>p. \<Sum>n\<le>p.
```
```  3343         if even p \<and> even n
```
```  3344         then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
```
```  3345       sums (cos x * cos y)"
```
```  3346 proof -
```
```  3347   have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p - n)) =
```
```  3348     (if even p \<and> even n then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p - n)
```
```  3349      else 0)"
```
```  3350     if "n \<le> p" for n p :: nat
```
```  3351   proof -
```
```  3352     from that have *: "even n \<Longrightarrow> even p \<Longrightarrow>
```
```  3353         (-1) ^ (n div 2) * (-1) ^ ((p - n) div 2) = (-1 :: real) ^ (p div 2)"
```
```  3354       by (metis div_add power_add le_add_diff_inverse odd_add)
```
```  3355     with that show ?thesis
```
```  3356       by (auto simp: algebra_simps cos_coeff_def binomial_fact)
```
```  3357   qed
```
```  3358   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> even n
```
```  3359                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
```
```  3360              (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
```
```  3361     by simp
```
```  3362   also have "\<dots> = (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n *\<^sub>R x^n) * (cos_coeff (p - n) *\<^sub>R y^(p-n)))"
```
```  3363     by (simp add: algebra_simps)
```
```  3364   also have "\<dots> sums (cos x * cos y)"
```
```  3365     using summable_norm_cos
```
```  3366     by (auto simp: cos_def scaleR_conv_of_real intro!: Cauchy_product_sums)
```
```  3367   finally show ?thesis .
```
```  3368 qed
```
```  3369
```
```  3370 text \<open>The product of two sine series.\<close>
```
```  3371 lemma sin_x_sin_y:
```
```  3372   fixes x :: "'a::{real_normed_field,banach}"
```
```  3373   shows
```
```  3374     "(\<lambda>p. \<Sum>n\<le>p.
```
```  3375         if even p \<and> odd n
```
```  3376         then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  3377         else 0)
```
```  3378       sums (sin x * sin y)"
```
```  3379 proof -
```
```  3380   have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
```
```  3381     (if even p \<and> odd n
```
```  3382      then -((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  3383      else 0)"
```
```  3384     if "n \<le> p" for n p :: nat
```
```  3385   proof -
```
```  3386     have "(-1) ^ ((n - Suc 0) div 2) * (-1) ^ ((p - Suc n) div 2) = - ((-1 :: real) ^ (p div 2))"
```
```  3387       if np: "odd n" "even p"
```
```  3388     proof -
```
```  3389       from \<open>n \<le> p\<close> np have *: "n - Suc 0 + (p - Suc n) = p - Suc (Suc 0)" "Suc (Suc 0) \<le> p"
```
```  3390         by arith+
```
```  3391       have "(p - Suc (Suc 0)) div 2 = p div 2 - Suc 0"
```
```  3392         by simp
```
```  3393       with \<open>n \<le> p\<close> np * show ?thesis
```
```  3394         apply (simp add: power_add [symmetric] div_add [symmetric] del: div_add)
```
```  3395         apply (metis (no_types) One_nat_def Suc_1 le_div_geq minus_minus
```
```  3396             mult.left_neutral mult_minus_left power.simps(2) zero_less_Suc)
```
```  3397         done
```
```  3398     qed
```
```  3399     then show ?thesis
```
```  3400       using \<open>n\<le>p\<close> by (auto simp: algebra_simps sin_coeff_def binomial_fact)
```
```  3401   qed
```
```  3402   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> odd n
```
```  3403                then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
```
```  3404              (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
```
```  3405     by simp
```
```  3406   also have "\<dots> = (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n *\<^sub>R x^n) * (sin_coeff (p - n) *\<^sub>R y^(p-n)))"
```
```  3407     by (simp add: algebra_simps)
```
```  3408   also have "\<dots> sums (sin x * sin y)"
```
```  3409     using summable_norm_sin
```
```  3410     by (auto simp: sin_def scaleR_conv_of_real intro!: Cauchy_product_sums)
```
```  3411   finally show ?thesis .
```
```  3412 qed
```
```  3413
```
```  3414 lemma sums_cos_x_plus_y:
```
```  3415   fixes x :: "'a::{real_normed_field,banach}"
```
```  3416   shows
```
```  3417     "(\<lambda>p. \<Sum>n\<le>p.
```
```  3418         if even p
```
```  3419         then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  3420         else 0)
```
```  3421       sums cos (x + y)"
```
```  3422 proof -
```
```  3423   have
```
```  3424     "(\<Sum>n\<le>p.
```
```  3425       if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  3426       else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)"
```
```  3427     for p :: nat
```
```  3428   proof -
```
```  3429     have
```
```  3430       "(\<Sum>n\<le>p. if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
```
```  3431        (if even p then \<Sum>n\<le>p. ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
```
```  3432       by simp
```
```  3433     also have "\<dots> =
```
```  3434        (if even p
```
```  3435         then of_real ((-1) ^ (p div 2) / (fact p)) * (\<Sum>n\<le>p. (p choose n) *\<^sub>R (x^n) * y^(p-n))
```
```  3436         else 0)"
```
```  3437       by (auto simp: sum_distrib_left field_simps scaleR_conv_of_real nonzero_of_real_divide)
```
```  3438     also have "\<dots> = cos_coeff p *\<^sub>R ((x + y) ^ p)"
```
```  3439       by (simp add: cos_coeff_def binomial_ring [of x y]  scaleR_conv_of_real atLeast0AtMost)
```
```  3440     finally show ?thesis .
```
```  3441   qed
```
```  3442   then have
```
```  3443     "(\<lambda>p. \<Sum>n\<le>p.
```
```  3444         if even p
```
```  3445         then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  3446         else 0) = (\<lambda>p. cos_coeff p *\<^sub>R ((x+y)^p))"
```
```  3447     by simp
```
```  3448    also have "\<dots> sums cos (x + y)"
```
```  3449     by (rule cos_converges)
```
```  3450    finally show ?thesis .
```
```  3451 qed
```
```  3452
```
```  3453 theorem cos_add:
```
```  3454   fixes x :: "'a::{real_normed_field,banach}"
```
```  3455   shows "cos (x + y) = cos x * cos y - sin x * sin y"
```
```  3456 proof -
```
```  3457   have
```
```  3458     "(if even p \<and> even n
```
```  3459       then ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) -
```
```  3460      (if even p \<and> odd n
```
```  3461       then - ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
```
```  3462      (if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
```
```  3463     if "n \<le> p" for n p :: nat
```
```  3464     by simp
```
```  3465   then have
```
```  3466     "(\<lambda>p. \<Sum>n\<le>p. (if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0))
```
```  3467       sums (cos x * cos y - sin x * sin y)"
```
```  3468     using sums_diff [OF cos_x_cos_y [of x y] sin_x_sin_y [of x y]]
```
```  3469     by (simp add: sum_subtractf [symmetric])
```
```  3470   then show ?thesis
```
```  3471     by (blast intro: sums_cos_x_plus_y sums_unique2)
```
```  3472 qed
```
```  3473
```
```  3474 lemma sin_minus_converges: "(\<lambda>n. - (sin_coeff n *\<^sub>R (-x)^n)) sums sin x"
```
```  3475 proof -
```
```  3476   have [simp]: "\<And>n. - (sin_coeff n *\<^sub>R (-x)^n) = (sin_coeff n *\<^sub>R x^n)"
```
```  3477     by (auto simp: sin_coeff_def elim!: oddE)
```
```  3478   show ?thesis
```
```  3479     by (simp add: sin_def summable_norm_sin [THEN summable_norm_cancel, THEN summable_sums])
```
```  3480 qed
```
```  3481
```
```  3482 lemma sin_minus [simp]: "sin (- x) = - sin x"
```
```  3483   for x :: "'a::{real_normed_algebra_1,banach}"
```
```  3484   using sin_minus_converges [of x]
```
```  3485   by (auto simp: sin_def summable_norm_sin [THEN summable_norm_cancel]
```
```  3486       suminf_minus sums_iff equation_minus_iff)
```
```  3487
```
```  3488 lemma cos_minus_converges: "(\<lambda>n. (cos_coeff n *\<^sub>R (-x)^n)) sums cos x"
```
```  3489 proof -
```
```  3490   have [simp]: "\<And>n. (cos_coeff n *\<^sub>R (-x)^n) = (cos_coeff n *\<^sub>R x^n)"
```
```  3491     by (auto simp: Transcendental.cos_coeff_def elim!: evenE)
```
```  3492   show ?thesis
```
```  3493     by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel, THEN summable_sums])
```
```  3494 qed
```
```  3495
```
```  3496 lemma cos_minus [simp]: "cos (-x) = cos x"
```
```  3497   for x :: "'a::{real_normed_algebra_1,banach}"
```
```  3498   using cos_minus_converges [of x]
```
```  3499   by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel]
```
```  3500       suminf_minus sums_iff equation_minus_iff)
```
```  3501
```
```  3502 lemma sin_cos_squared_add [simp]: "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
```
```  3503   for x :: "'a::{real_normed_field,banach}"
```
```  3504   using cos_add [of x "-x"]
```
```  3505   by (simp add: power2_eq_square algebra_simps)
```
```  3506
```
```  3507 lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
```
```  3508   for x :: "'a::{real_normed_field,banach}"
```
```  3509   by (subst add.commute, rule sin_cos_squared_add)
```
```  3510
```
```  3511 lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
```
```  3512   for x :: "'a::{real_normed_field,banach}"
```
```  3513   using sin_cos_squared_add2 [unfolded power2_eq_square] .
```
```  3514
```
```  3515 lemma sin_squared_eq: "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
```
```  3516   for x :: "'a::{real_normed_field,banach}"
```
```  3517   unfolding eq_diff_eq by (rule sin_cos_squared_add)
```
```  3518
```
```  3519 lemma cos_squared_eq: "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
```
```  3520   for x :: "'a::{real_normed_field,banach}"
```
```  3521   unfolding eq_diff_eq by (rule sin_cos_squared_add2)
```
```  3522
```
```  3523 lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
```
```  3524   for x :: real
```
```  3525   by (rule power2_le_imp_le) (simp_all add: sin_squared_eq)
```
```  3526
```
```  3527 lemma sin_ge_minus_one [simp]: "- 1 \<le> sin x"
```
```  3528   for x :: real
```
```  3529   using abs_sin_le_one [of x] by (simp add: abs_le_iff)
```
```  3530
```
```  3531 lemma sin_le_one [simp]: "sin x \<le> 1"
```
```  3532   for x :: real
```
```  3533   using abs_sin_le_one [of x] by (simp add: abs_le_iff)
```
```  3534
```
```  3535 lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
```
```  3536   for x :: real
```
```  3537   by (rule power2_le_imp_le) (simp_all add: cos_squared_eq)
```
```  3538
```
```  3539 lemma cos_ge_minus_one [simp]: "- 1 \<le> cos x"
```
```  3540   for x :: real
```
```  3541   using abs_cos_le_one [of x] by (simp add: abs_le_iff)
```
```  3542
```
```  3543 lemma cos_le_one [simp]: "cos x \<le> 1"
```
```  3544   for x :: real
```
```  3545   using abs_cos_le_one [of x] by (simp add: abs_le_iff)
```
```  3546
```
```  3547 lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
```
```  3548   for x :: "'a::{real_normed_field,banach}"
```
```  3549   using cos_add [of x "- y"] by simp
```
```  3550
```
```  3551 lemma cos_double: "cos(2*x) = (cos x)\<^sup>2 - (sin x)\<^sup>2"
```
```  3552   for x :: "'a::{real_normed_field,banach}"
```
```  3553   using cos_add [where x=x and y=x] by (simp add: power2_eq_square)
```
```  3554
```
```  3555 lemma sin_cos_le1: "\<bar>sin x * sin y + cos x * cos y\<bar> \<le> 1"
```
```  3556   for x :: real
```
```  3557   using cos_diff [of x y] by (metis abs_cos_le_one add.commute)
```
```  3558
```
```  3559 lemma DERIV_fun_pow: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
```
```  3560   by (auto intro!: derivative_eq_intros simp:)
```
```  3561
```
```  3562 lemma DERIV_fun_exp: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. exp (g x)) x :> exp (g x) * m"
```
```  3563   by (auto intro!: derivative_intros)
```
```  3564
```
```  3565
```
```  3566 subsection \<open>The Constant Pi\<close>
```
```  3567
```
```  3568 definition pi :: real
```
```  3569   where "pi = 2 * (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)"
```
```  3570
```
```  3571 text \<open>Show that there's a least positive @{term x} with @{term "cos x = 0"};
```
```  3572    hence define pi.\<close>
```
```  3573
```
```  3574 lemma sin_paired: "(\<lambda>n. (- 1) ^ n / (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums  sin x"
```
```  3575   for x :: real
```
```  3576 proof -
```
```  3577   have "(\<lambda>n. \<Sum>k = n*2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
```
```  3578     by (rule sums_group) (use sin_converges [of x, unfolded scaleR_conv_of_real] in auto)
```
```  3579   then show ?thesis
```
```  3580     by (simp add: sin_coeff_def ac_simps)
```
```  3581 qed
```
```  3582
```
```  3583 lemma sin_gt_zero_02:
```
```  3584   fixes x :: real
```
```  3585   assumes "0 < x" and "x < 2"
```
```  3586   shows "0 < sin x"
```
```  3587 proof -
```
```  3588   let ?f = "\<lambda>n::nat. \<Sum>k = n*2..<n*2+2. (- 1) ^ k / (fact (2*k+1)) * x^(2*k+1)"
```
```  3589   have pos: "\<forall>n. 0 < ?f n"
```
```  3590   proof
```
```  3591     fix n :: nat
```
```  3592     let ?k2 = "real (Suc (Suc (4 * n)))"
```
```  3593     let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
```
```  3594     have "x * x < ?k2 * ?k3"
```
```  3595       using assms by (intro mult_strict_mono', simp_all)
```
```  3596     then have "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
```
```  3597       by (intro mult_strict_right_mono zero_less_power \<open>0 < x\<close>)
```
```  3598     then show "0 < ?f n"
```
```  3599       by (simp add: divide_simps mult_ac del: mult_Suc)
```
```  3600 qed
```
```  3601   have sums: "?f sums sin x"
```
```  3602     by (rule sin_paired [THEN sums_group]) simp
```
```  3603   show "0 < sin x"
```
```  3604     unfolding sums_unique [OF sums]
```
```  3605     using sums_summable [OF sums] pos
```
```  3606     by (rule suminf_pos)
```
```  3607 qed
```
```  3608
```
```  3609 lemma cos_double_less_one: "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
```
```  3610   for x :: real
```
```  3611   using sin_gt_zero_02 [where x = x] by (auto simp: cos_squared_eq cos_double)
```
```  3612
```
```  3613 lemma cos_paired: "(\<lambda>n. (- 1) ^ n / (fact (2 * n)) * x ^ (2 * n)) sums cos x"
```
```  3614   for x :: real
```
```  3615 proof -
```
```  3616   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
```
```  3617     by (rule sums_group) (use cos_converges [of x, unfolded scaleR_conv_of_real] in auto)
```
```  3618   then show ?thesis
```
```  3619     by (simp add: cos_coeff_def ac_simps)
```
```  3620 qed
```
```  3621
```
```  3622 lemmas realpow_num_eq_if = power_eq_if
```
```  3623
```
```  3624 lemma sumr_pos_lt_pair:
```
```  3625   fixes f :: "nat \<Rightarrow> real"
```
```  3626   shows "summable f \<Longrightarrow>
```
```  3627     (\<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc (Suc 0) * d) + 1))) \<Longrightarrow>
```
```  3628     sum f {..<k} < suminf f"
```
```  3629   apply (simp only: One_nat_def)
```
```  3630   apply (subst suminf_split_initial_segment [where k=k])
```
```  3631    apply assumption
```
```  3632   apply simp
```
```  3633   apply (drule_tac k=k in summable_ignore_initial_segment)
```
```  3634   apply (drule_tac k="Suc (Suc 0)" in sums_group [OF summable_sums])
```
```  3635    apply simp
```
```  3636   apply simp
```
```  3637   apply (metis (no_types, lifting) add.commute suminf_pos summable_def sums_unique)
```
```  3638   done
```
```  3639
```
```  3640 lemma cos_two_less_zero [simp]: "cos 2 < (0::real)"
```
```  3641 proof -
```
```  3642   note fact_Suc [simp del]
```
```  3643   from sums_minus [OF cos_paired]
```
```  3644   have *: "(\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / fact (2 * n))) sums - cos (2::real)"
```
```  3645     by simp
```
```  3646   then have sm: "summable (\<lambda>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3647     by (rule sums_summable)
```
```  3648   have "0 < (\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3649     by (simp add: fact_num_eq_if realpow_num_eq_if)
```
```  3650   moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n  * 2 ^ (2 * n) / (fact (2 * n)))) <
```
```  3651     (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3652   proof -
```
```  3653     {
```
```  3654       fix d
```
```  3655       let ?six4d = "Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))"
```
```  3656       have "(4::real) * (fact (?six4d)) < (Suc (Suc (?six4d)) * fact (Suc (?six4d)))"
```
```  3657         unfolding of_nat_mult by (rule mult_strict_mono) (simp_all add: fact_less_mono)
```
```  3658       then have "(4::real) * (fact (?six4d)) < (fact (Suc (Suc (?six4d))))"
```
```  3659         by (simp only: fact_Suc [of "Suc (?six4d)"] of_nat_mult of_nat_fact)
```
```  3660       then have "(4::real) * inverse (fact (Suc (Suc (?six4d)))) < inverse (fact (?six4d))"
```
```  3661         by (simp add: inverse_eq_divide less_divide_eq)
```
```  3662     }
```
```  3663     then show ?thesis
```
```  3664       by (force intro!: sumr_pos_lt_pair [OF sm] simp add: divide_inverse algebra_simps)
```
```  3665   qed
```
```  3666   ultimately have "0 < (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3667     by (rule order_less_trans)
```
```  3668   moreover from * have "- cos 2 = (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3669     by (rule sums_unique)
```
```  3670   ultimately have "(0::real) < - cos 2" by simp
```
```  3671   then show ?thesis by simp
```
```  3672 qed
```
```  3673
```
```  3674 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
```
```  3675 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
```
```  3676
```
```  3677 lemma cos_is_zero: "\<exists>!x::real. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
```
```  3678 proof (rule ex_ex1I)
```
```  3679   show "\<exists>x::real. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
```
```  3680     by (rule IVT2) simp_all
```
```  3681 next
```
```  3682   fix x y :: real
```
```  3683   assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
```
```  3684   assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
```
```  3685   have [simp]: "\<forall>x::real. cos differentiable (at x)"
```
```  3686     unfolding real_differentiable_def by (auto intro: DERIV_cos)
```
```  3687   from x y less_linear [of x y] show "x = y"
```
```  3688     apply auto
```
```  3689      apply (drule_tac f = cos in Rolle)
```
```  3690         apply (drule_tac [5] f = cos in Rolle)
```
```  3691            apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  3692      apply (metis order_less_le_trans less_le sin_gt_zero_02)
```
```  3693     apply (metis order_less_le_trans less_le sin_gt_zero_02)
```
```  3694     done
```
```  3695 qed
```
```  3696
```
```  3697 lemma pi_half: "pi/2 = (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)"
```
```  3698   by (simp add: pi_def)
```
```  3699
```
```  3700 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
```
```  3701   by (simp add: pi_half cos_is_zero [THEN theI'])
```
```  3702
```
```  3703 lemma cos_of_real_pi_half [simp]: "cos ((of_real pi / 2) :: 'a) = 0"
```
```  3704   if "SORT_CONSTRAINT('a::{real_field,banach,real_normed_algebra_1})"
```
```  3705   by (metis cos_pi_half cos_of_real eq_numeral_simps(4)
```
```  3706       nonzero_of_real_divide of_real_0 of_real_numeral)
```
```  3707
```
```  3708 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
```
```  3709   apply (rule order_le_neq_trans)
```
```  3710    apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  3711   apply (metis cos_pi_half cos_zero zero_neq_one)
```
```  3712   done
```
```  3713
```
```  3714 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
```
```  3715 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
```
```  3716
```
```  3717 lemma pi_half_less_two [simp]: "pi / 2 < 2"
```
```  3718   apply (rule order_le_neq_trans)
```
```  3719    apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  3720   apply (metis cos_pi_half cos_two_neq_zero)
```
```  3721   done
```
```  3722
```
```  3723 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
```
```  3724 lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
```
```  3725
```
```  3726 lemma pi_gt_zero [simp]: "0 < pi"
```
```  3727   using pi_half_gt_zero by simp
```
```  3728
```
```  3729 lemma pi_ge_zero [simp]: "0 \<le> pi"
```
```  3730   by (rule pi_gt_zero [THEN order_less_imp_le])
```
```  3731
```
```  3732 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
```
```  3733   by (rule pi_gt_zero [THEN less_imp_neq, symmetric])
```
```  3734
```
```  3735 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
```
```  3736   by (simp add: linorder_not_less)
```
```  3737
```
```  3738 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
```
```  3739   by simp
```
```  3740
```
```  3741 lemma m2pi_less_pi: "- (2*pi) < pi"
```
```  3742   by simp
```
```  3743
```
```  3744 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
```
```  3745   using sin_cos_squared_add2 [where x = "pi/2"]
```
```  3746   using sin_gt_zero_02 [OF pi_half_gt_zero pi_half_less_two]
```
```  3747   by (simp add: power2_eq_1_iff)
```
```  3748
```
```  3749 lemma sin_of_real_pi_half [simp]: "sin ((of_real pi / 2) :: 'a) = 1"
```
```  3750   if "SORT_CONSTRAINT('a::{real_field,banach,real_normed_algebra_1})"
```
```  3751   using sin_pi_half
```
```  3752   by (metis sin_pi_half eq_numeral_simps(4) nonzero_of_real_divide of_real_1 of_real_numeral sin_of_real)
```
```  3753
```
```  3754 lemma sin_cos_eq: "sin x = cos (of_real pi / 2 - x)"
```
```  3755   for x :: "'a::{real_normed_field,banach}"
```
```  3756   by (simp add: cos_diff)
```
```  3757
```
```  3758 lemma minus_sin_cos_eq: "- sin x = cos (x + of_real pi / 2)"
```
```  3759   for x :: "'a::{real_normed_field,banach}"
```
```  3760   by (simp add: cos_add nonzero_of_real_divide)
```
```  3761
```
```  3762 lemma cos_sin_eq: "cos x = sin (of_real pi / 2 - x)"
```
```  3763   for x :: "'a::{real_normed_field,banach}"
```
```  3764   using sin_cos_eq [of "of_real pi / 2 - x"] by simp
```
```  3765
```
```  3766 lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
```
```  3767   for x :: "'a::{real_normed_field,banach}"
```
```  3768   using cos_add [of "of_real pi / 2 - x" "-y"]
```
```  3769   by (simp add: cos_sin_eq) (simp add: sin_cos_eq)
```
```  3770
```
```  3771 lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
```
```  3772   for x :: "'a::{real_normed_field,banach}"
```
```  3773   using sin_add [of x "- y"] by simp
```
```  3774
```
```  3775 lemma sin_double: "sin(2 * x) = 2 * sin x * cos x"
```
```  3776   for x :: "'a::{real_normed_field,banach}"
```
```  3777   using sin_add [where x=x and y=x] by simp
```
```  3778
```
```  3779 lemma cos_of_real_pi [simp]: "cos (of_real pi) = -1"
```
```  3780   using cos_add [where x = "pi/2" and y = "pi/2"]
```
```  3781   by (simp add: cos_of_real)
```
```  3782
```
```  3783 lemma sin_of_real_pi [simp]: "sin (of_real pi) = 0"
```
```  3784   using sin_add [where x = "pi/2" and y = "pi/2"]
```
```  3785   by (simp add: sin_of_real)
```
```  3786
```
```  3787 lemma cos_pi [simp]: "cos pi = -1"
```
```  3788   using cos_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  3789
```
```  3790 lemma sin_pi [simp]: "sin pi = 0"
```
```  3791   using sin_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  3792
```
```  3793 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
```
```  3794   by (simp add: sin_add)
```
```  3795
```
```  3796 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
```
```  3797   by (simp add: sin_add)
```
```  3798
```
```  3799 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
```
```  3800   by (simp add: cos_add)
```
```  3801
```
```  3802 lemma cos_periodic_pi2 [simp]: "cos (pi + x) = - cos x"
```
```  3803   by (simp add: cos_add)
```
```  3804
```
```  3805 lemma sin_periodic [simp]: "sin (x + 2 * pi) = sin x"
```
```  3806   by (simp add: sin_add sin_double cos_double)
```
```  3807
```
```  3808 lemma cos_periodic [simp]: "cos (x + 2 * pi) = cos x"
```
```  3809   by (simp add: cos_add sin_double cos_double)
```
```  3810
```
```  3811 lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n"
```
```  3812   by (induct n) (auto simp: distrib_right)
```
```  3813
```
```  3814 lemma cos_npi2 [simp]: "cos (pi * real n) = (- 1) ^ n"
```
```  3815   by (metis cos_npi mult.commute)
```
```  3816
```
```  3817 lemma sin_npi [simp]: "sin (real n * pi) = 0"
```
```  3818   for n :: nat
```
```  3819   by (induct n) (auto simp: distrib_right)
```
```  3820
```
```  3821 lemma sin_npi2 [simp]: "sin (pi * real n) = 0"
```
```  3822   for n :: nat
```
```  3823   by (simp add: mult.commute [of pi])
```
```  3824
```
```  3825 lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
```
```  3826   by (simp add: cos_double)
```
```  3827
```
```  3828 lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
```
```  3829   by (simp add: sin_double)
```
```  3830
```
```  3831 lemma sin_times_sin: "sin w * sin z = (cos (w - z) - cos (w + z)) / 2"
```
```  3832   for w :: "'a::{real_normed_field,banach}"
```
```  3833   by (simp add: cos_diff cos_add)
```
```  3834
```
```  3835 lemma sin_times_cos: "sin w * cos z = (sin (w + z) + sin (w - z)) / 2"
```
```  3836   for w :: "'a::{real_normed_field,banach}"
```
```  3837   by (simp add: sin_diff sin_add)
```
```  3838
```
```  3839 lemma cos_times_sin: "cos w * sin z = (sin (w + z) - sin (w - z)) / 2"
```
```  3840   for w :: "'a::{real_normed_field,banach}"
```
```  3841   by (simp add: sin_diff sin_add)
```
```  3842
```
```  3843 lemma cos_times_cos: "cos w * cos z = (cos (w - z) + cos (w + z)) / 2"
```
```  3844   for w :: "'a::{real_normed_field,banach}"
```
```  3845   by (simp add: cos_diff cos_add)
```
```  3846
```
```  3847 lemma sin_plus_sin: "sin w + sin z = 2 * sin ((w + z) / 2) * cos ((w - z) / 2)"
```
```  3848   for w :: "'a::{real_normed_field,banach,field}"  (* FIXME field should not be necessary *)
```
```  3849   apply (simp add: mult.assoc sin_times_cos)
```
```  3850   apply (simp add: field_simps)
```
```  3851   done
```
```  3852
```
```  3853 lemma sin_diff_sin: "sin w - sin z = 2 * sin ((w - z) / 2) * cos ((w + z) / 2)"
```
```  3854   for w :: "'a::{real_normed_field,banach,field}"
```
```  3855   apply (simp add: mult.assoc sin_times_cos)
```
```  3856   apply (simp add: field_simps)
```
```  3857   done
```
```  3858
```
```  3859 lemma cos_plus_cos: "cos w + cos z = 2 * cos ((w + z) / 2) * cos ((w - z) / 2)"
```
```  3860   for w :: "'a::{real_normed_field,banach,field}"
```
```  3861   apply (simp add: mult.assoc cos_times_cos)
```
```  3862   apply (simp add: field_simps)
```
```  3863   done
```
```  3864
```
```  3865 lemma cos_diff_cos: "cos w - cos z = 2 * sin ((w + z) / 2) * sin ((z - w) / 2)"
```
```  3866   for w :: "'a::{real_normed_field,banach,field}"
```
```  3867   apply (simp add: mult.assoc sin_times_sin)
```
```  3868   apply (simp add: field_simps)
```
```  3869   done
```
```  3870
```
```  3871 lemma cos_double_cos: "cos (2 * z) = 2 * cos z ^ 2 - 1"
```
```  3872   for z :: "'a::{real_normed_field,banach}"
```
```  3873   by (simp add: cos_double sin_squared_eq)
```
```  3874
```
```  3875 lemma cos_double_sin: "cos (2 * z) = 1 - 2 * sin z ^ 2"
```
```  3876   for z :: "'a::{real_normed_field,banach}"
```
```  3877   by (simp add: cos_double sin_squared_eq)
```
```  3878
```
```  3879 lemma sin_pi_minus [simp]: "sin (pi - x) = sin x"
```
```  3880   by (metis sin_minus sin_periodic_pi minus_minus uminus_add_conv_diff)
```
```  3881
```
```  3882 lemma cos_pi_minus [simp]: "cos (pi - x) = - (cos x)"
```
```  3883   by (metis cos_minus cos_periodic_pi uminus_add_conv_diff)
```
```  3884
```
```  3885 lemma sin_minus_pi [simp]: "sin (x - pi) = - (sin x)"
```
```  3886   by (simp add: sin_diff)
```
```  3887
```
```  3888 lemma cos_minus_pi [simp]: "cos (x - pi) = - (cos x)"
```
```  3889   by (simp add: cos_diff)
```
```  3890
```
```  3891 lemma sin_2pi_minus [simp]: "sin (2 * pi - x) = - (sin x)"
```
```  3892   by (metis sin_periodic_pi2 add_diff_eq mult_2 sin_pi_minus)
```
```  3893
```
```  3894 lemma cos_2pi_minus [simp]: "cos (2 * pi - x) = cos x"
```
```  3895   by (metis (no_types, hide_lams) cos_add cos_minus cos_two_pi sin_minus sin_two_pi
```
```  3896       diff_0_right minus_diff_eq mult_1 mult_zero_left uminus_add_conv_diff)
```
```  3897
```
```  3898 lemma sin_gt_zero2: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < sin x"
```
```  3899   by (metis sin_gt_zero_02 order_less_trans pi_half_less_two)
```
```  3900
```
```  3901 lemma sin_less_zero:
```
```  3902   assumes "- pi/2 < x" and "x < 0"
```
```  3903   shows "sin x < 0"
```
```  3904 proof -
```
```  3905   have "0 < sin (- x)"
```
```  3906     using assms by (simp only: sin_gt_zero2)
```
```  3907   then show ?thesis by simp
```
```  3908 qed
```
```  3909
```
```  3910 lemma pi_less_4: "pi < 4"
```
```  3911   using pi_half_less_two by auto
```
```  3912
```
```  3913 lemma cos_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cos x"
```
```  3914   by (simp add: cos_sin_eq sin_gt_zero2)
```
```  3915
```
```  3916 lemma cos_gt_zero_pi: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cos x"
```
```  3917   using cos_gt_zero [of x] cos_gt_zero [of "-x"]
```
```  3918   by (cases rule: linorder_cases [of x 0]) auto
```
```  3919
```
```  3920 lemma cos_ge_zero: "-(pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> 0 \<le> cos x"
```
```  3921   by (auto simp: order_le_less cos_gt_zero_pi)
```
```  3922     (metis cos_pi_half eq_divide_eq eq_numeral_simps(4))
```
```  3923
```
```  3924 lemma sin_gt_zero: "0 < x \<Longrightarrow> x < pi \<Longrightarrow> 0 < sin x"
```
```  3925   by (simp add: sin_cos_eq cos_gt_zero_pi)
```
```  3926
```
```  3927 lemma sin_lt_zero: "pi < x \<Longrightarrow> x < 2 * pi \<Longrightarrow> sin x < 0"
```
```  3928   using sin_gt_zero [of "x - pi"]
```
```  3929   by (simp add: sin_diff)
```
```  3930
```
```  3931 lemma pi_ge_two: "2 \<le> pi"
```
```  3932 proof (rule ccontr)
```
```  3933   assume "\<not> ?thesis"
```
```  3934   then have "pi < 2" by auto
```
```  3935   have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
```
```  3936   proof (cases "2 < 2 * pi")
```
```  3937     case True
```
```  3938     with dense[OF \<open>pi < 2\<close>] show ?thesis by auto
```
```  3939   next
```
```  3940     case False
```
```  3941     have "pi < 2 * pi" by auto
```
```  3942     from dense[OF this] and False show ?thesis by auto
```
```  3943   qed
```
```  3944   then obtain y where "pi < y" and "y < 2" and "y < 2 * pi"
```
```  3945     by blast
```
```  3946   then have "0 < sin y"
```
```  3947     using sin_gt_zero_02 by auto
```
```  3948   moreover have "sin y < 0"
```
```  3949     using sin_gt_zero[of "y - pi"] \<open>pi < y\<close> and \<open>y < 2 * pi\<close> sin_periodic_pi[of "y - pi"]
```
```  3950     by auto
```
```  3951   ultimately show False by auto
```
```  3952 qed
```
```  3953
```
```  3954 lemma sin_ge_zero: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> sin x"
```
```  3955   by (auto simp: order_le_less sin_gt_zero)
```
```  3956
```
```  3957 lemma sin_le_zero: "pi \<le> x \<Longrightarrow> x < 2 * pi \<Longrightarrow> sin x \<le> 0"
```
```  3958   using sin_ge_zero [of "x - pi"] by (simp add: sin_diff)
```
```  3959
```
```  3960 lemma sin_pi_divide_n_ge_0 [simp]:
```
```  3961   assumes "n \<noteq> 0"
```
```  3962   shows "0 \<le> sin (pi / real n)"
```
```  3963   by (rule sin_ge_zero) (use assms in \<open>simp_all add: divide_simps\<close>)
```
```  3964
```
```  3965 lemma sin_pi_divide_n_gt_0:
```
```  3966   assumes "2 \<le> n"
```
```  3967   shows "0 < sin (pi / real n)"
```
```  3968   by (rule sin_gt_zero) (use assms in \<open>simp_all add: divide_simps\<close>)
```
```  3969
```
```  3970 (* FIXME: This proof is almost identical to lemma \<open>cos_is_zero\<close>.
```
```  3971    It should be possible to factor out some of the common parts. *)
```
```  3972 lemma cos_total:
```
```  3973   assumes y: "- 1 \<le> y" "y \<le> 1"
```
```  3974   shows "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y"
```
```  3975 proof (rule ex_ex1I)
```
```  3976   show "\<exists>x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y"
```
```  3977     by (rule IVT2) (simp_all add: y)
```
```  3978 next
```
```  3979   fix a b
```
```  3980   assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
```
```  3981   assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
```
```  3982   have [simp]: "\<forall>x::real. cos differentiable (at x)"
```
```  3983     unfolding real_differentiable_def by (auto intro: DERIV_cos)
```
```  3984   from a b less_linear [of a b] show "a = b"
```
```  3985     apply auto
```
```  3986      apply (drule_tac f = cos in Rolle)
```
```  3987         apply (drule_tac [5] f = cos in Rolle)
```
```  3988            apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  3989      apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  3990     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  3991     done
```
```  3992 qed
```
```  3993
```
```  3994 lemma sin_total:
```
```  3995   assumes y: "-1 \<le> y" "y \<le> 1"
```
```  3996   shows "\<exists>!x. - (pi/2) \<le> x \<and> x \<le> pi/2 \<and> sin x = y"
```
```  3997 proof -
```
```  3998   from cos_total [OF y]
```
```  3999   obtain x where x: "0 \<le> x" "x \<le> pi" "cos x = y"
```
```  4000     and uniq: "\<And>x'. 0 \<le> x' \<Longrightarrow> x' \<le> pi \<Longrightarrow> cos x' = y \<Longrightarrow> x' = x "
```
```  4001     by blast
```
```  4002   show ?thesis
```
```  4003     apply (simp add: sin_cos_eq)
```
```  4004     apply (rule ex1I [where a="pi/2 - x"])
```
```  4005      apply (cut_tac [2] x'="pi/2 - xa" in uniq)
```
```  4006     using x
```
```  4007         apply auto
```
```  4008     done
```
```  4009 qed
```
```  4010
```
```  4011 lemma cos_zero_lemma:
```
```  4012   assumes "0 \<le> x" "cos x = 0"
```
```  4013   shows "\<exists>n. odd n \<and> x = of_nat n * (pi/2) \<and> n > 0"
```
```  4014 proof -
```
```  4015   have xle: "x < (1 + real_of_int \<lfloor>x/pi\<rfloor>) * pi"
```
```  4016     using floor_correct [of "x/pi"]
```
```  4017     by (simp add: add.commute divide_less_eq)
```
```  4018   obtain n where "real n * pi \<le> x" "x < real (Suc n) * pi"
```
```  4019     apply (rule that [of "nat \<lfloor>x/pi\<rfloor>"])
```
```  4020     using assms
```
```  4021      apply (simp_all add: xle)
```
```  4022     apply (metis floor_less_iff less_irrefl mult_imp_div_pos_less not_le pi_gt_zero)
```
```  4023     done
```
```  4024   then have x: "0 \<le> x - n * pi" "(x - n * pi) \<le> pi" "cos (x - n * pi) = 0"
```
```  4025     by (auto simp: algebra_simps cos_diff assms)
```
```  4026   then have "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = 0"
```
```  4027     by (auto simp: intro!: cos_total)
```
```  4028   then obtain \<theta> where \<theta>: "0 \<le> \<theta>" "\<theta> \<le> pi" "cos \<theta> = 0"
```
```  4029     and uniq: "\<And>\<phi>. 0 \<le> \<phi> \<Longrightarrow> \<phi> \<le> pi \<Longrightarrow> cos \<phi> = 0 \<Longrightarrow> \<phi> = \<theta>"
```
```  4030     by blast
```
```  4031   then have "x - real n * pi = \<theta>"
```
```  4032     using x by blast
```
```  4033   moreover have "pi/2 = \<theta>"
```
```  4034     using pi_half_ge_zero uniq by fastforce
```
```  4035   ultimately show ?thesis
```
```  4036     by (rule_tac x = "Suc (2 * n)" in exI) (simp add: algebra_simps)
```
```  4037 qed
```
```  4038
```
```  4039 lemma sin_zero_lemma: "0 \<le> x \<Longrightarrow> sin x = 0 \<Longrightarrow> \<exists>n::nat. even n \<and> x = real n * (pi/2)"
```
```  4040   using cos_zero_lemma [of "x + pi/2"]
```
```  4041   apply (clarsimp simp add: cos_add)
```
```  4042   apply (rule_tac x = "n - 1" in exI)
```
```  4043   apply (simp add: algebra_simps of_nat_diff)
```
```  4044   done
```
```  4045
```
```  4046 lemma cos_zero_iff:
```
```  4047   "cos x = 0 \<longleftrightarrow> ((\<exists>n. odd n \<and> x = real n * (pi/2)) \<or> (\<exists>n. odd n \<and> x = - (real n * (pi/2))))"
```
```  4048   (is "?lhs = ?rhs")
```
```  4049 proof -
```
```  4050   have *: "cos (real n * pi / 2) = 0" if "odd n" for n :: nat
```
```  4051   proof -
```
```  4052     from that obtain m where "n = 2 * m + 1" ..
```
```  4053     then show ?thesis
```
```  4054       by (simp add: field_simps) (simp add: cos_add add_divide_distrib)
```
```  4055   qed
```
```  4056   show ?thesis
```
```  4057   proof
```
```  4058     show ?rhs if ?lhs
```
```  4059       using that cos_zero_lemma [of x] cos_zero_lemma [of "-x"] by force
```
```  4060     show ?lhs if ?rhs
```
```  4061       using that by (auto dest: * simp del: eq_divide_eq_numeral1)
```
```  4062   qed
```
```  4063 qed
```
```  4064
```
```  4065 lemma sin_zero_iff:
```
```  4066   "sin x = 0 \<longleftrightarrow> ((\<exists>n. even n \<and> x = real n * (pi/2)) \<or> (\<exists>n. even n \<and> x = - (real n * (pi/2))))"
```
```  4067   (is "?lhs = ?rhs")
```
```  4068 proof
```
```  4069   show ?rhs if ?lhs
```
```  4070     using that sin_zero_lemma [of x] sin_zero_lemma [of "-x"] by force
```
```  4071   show ?lhs if ?rhs
```
```  4072     using that by (auto elim: evenE)
```
```  4073 qed
```
```  4074
```
```  4075 lemma cos_zero_iff_int: "cos x = 0 \<longleftrightarrow> (\<exists>n. odd n \<and> x = of_int n * (pi/2))"
```
```  4076 proof safe
```
```  4077   assume "cos x = 0"
```
```  4078   then show "\<exists>n. odd n \<and> x = of_int n * (pi/2)"
```
```  4079     apply (simp add: cos_zero_iff)
```
```  4080     apply safe
```
```  4081      apply (metis even_int_iff of_int_of_nat_eq)
```
```  4082     apply (rule_tac x="- (int n)" in exI)
```
```  4083     apply simp
```
```  4084     done
```
```  4085 next
```
```  4086   fix n :: int
```
```  4087   assume "odd n"
```
```  4088   then show "cos (of_int n * (pi / 2)) = 0"
```
```  4089     apply (simp add: cos_zero_iff)
```
```  4090     apply (cases n rule: int_cases2)
```
```  4091      apply simp_all
```
```  4092     done
```
```  4093 qed
```
```  4094
```
```  4095 lemma sin_zero_iff_int: "sin x = 0 \<longleftrightarrow> (\<exists>n. even n \<and> x = of_int n * (pi/2))"
```
```  4096 proof safe
```
```  4097   assume "sin x = 0"
```
```  4098   then show "\<exists>n. even n \<and> x = of_int n * (pi / 2)"
```
```  4099     apply (simp add: sin_zero_iff)
```
```  4100     apply safe
```
```  4101      apply (metis even_int_iff of_int_of_nat_eq)
```
```  4102     apply (rule_tac x="- (int n)" in exI)
```
```  4103     apply simp
```
```  4104     done
```
```  4105 next
```
```  4106   fix n :: int
```
```  4107   assume "even n"
```
```  4108   then show "sin (of_int n * (pi / 2)) = 0"
```
```  4109     apply (simp add: sin_zero_iff)
```
```  4110     apply (cases n rule: int_cases2)
```
```  4111      apply simp_all
```
```  4112     done
```
```  4113 qed
```
```  4114
```
```  4115 lemma sin_zero_iff_int2: "sin x = 0 \<longleftrightarrow> (\<exists>n::int. x = of_int n * pi)"
```
```  4116   apply (simp only: sin_zero_iff_int)
```
```  4117   apply (safe elim!: evenE)
```
```  4118    apply (simp_all add: field_simps)
```
```  4119   using dvd_triv_left apply fastforce
```
```  4120   done
```
```  4121
```
```  4122 lemma sin_npi_int [simp]: "sin (pi * of_int n) = 0"
```
```  4123   by (simp add: sin_zero_iff_int2)
```
```  4124
```
```  4125 lemma cos_monotone_0_pi:
```
```  4126   assumes "0 \<le> y" and "y < x" and "x \<le> pi"
```
```  4127   shows "cos x < cos y"
```
```  4128 proof -
```
```  4129   have "- (x - y) < 0" using assms by auto
```
```  4130   from MVT2[OF \<open>y < x\<close> DERIV_cos[THEN impI, THEN allI]]
```
```  4131   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z"
```
```  4132     by auto
```
```  4133   then have "0 < z" and "z < pi"
```
```  4134     using assms by auto
```
```  4135   then have "0 < sin z"
```
```  4136     using sin_gt_zero by auto
```
```  4137   then have "cos x - cos y < 0"
```
```  4138     unfolding cos_diff minus_mult_commute[symmetric]
```
```  4139     using \<open>- (x - y) < 0\<close> by (rule mult_pos_neg2)
```
```  4140   then show ?thesis by auto
```
```  4141 qed
```
```  4142
```
```  4143 lemma cos_monotone_0_pi_le:
```
```  4144   assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi"
```
```  4145   shows "cos x \<le> cos y"
```
```  4146 proof (cases "y < x")
```
```  4147   case True
```
```  4148   show ?thesis
```
```  4149     using cos_monotone_0_pi[OF \<open>0 \<le> y\<close> True \<open>x \<le> pi\<close>] by auto
```
```  4150 next
```
```  4151   case False
```
```  4152   then have "y = x" using \<open>y \<le> x\<close> by auto
```
```  4153   then show ?thesis by auto
```
```  4154 qed
```
```  4155
```
```  4156 lemma cos_monotone_minus_pi_0:
```
```  4157   assumes "- pi \<le> y" and "y < x" and "x \<le> 0"
```
```  4158   shows "cos y < cos x"
```
```  4159 proof -
```
```  4160   have "0 \<le> - x" and "- x < - y" and "- y \<le> pi"
```
```  4161     using assms by auto
```
```  4162   from cos_monotone_0_pi[OF this] show ?thesis
```
```  4163     unfolding cos_minus .
```
```  4164 qed
```
```  4165
```
```  4166 lemma cos_monotone_minus_pi_0':
```
```  4167   assumes "- pi \<le> y" and "y \<le> x" and "x \<le> 0"
```
```  4168   shows "cos y \<le> cos x"
```
```  4169 proof (cases "y < x")
```
```  4170   case True
```
```  4171   show ?thesis using cos_monotone_minus_pi_0[OF \<open>-pi \<le> y\<close> True \<open>x \<le> 0\<close>]
```
```  4172     by auto
```
```  4173 next
```
```  4174   case False
```
```  4175   then have "y = x" using \<open>y \<le> x\<close> by auto
```
```  4176   then show ?thesis by auto
```
```  4177 qed
```
```  4178
```
```  4179 lemma sin_monotone_2pi:
```
```  4180   assumes "- (pi/2) \<le> y" and "y < x" and "x \<le> pi/2"
```
```  4181   shows "sin y < sin x"
```
```  4182   apply (simp add: sin_cos_eq)
```
```  4183   apply (rule cos_monotone_0_pi)
```
```  4184   using assms
```
```  4185     apply auto
```
```  4186   done
```
```  4187
```
```  4188 lemma sin_monotone_2pi_le:
```
```  4189   assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2"
```
```  4190   shows "sin y \<le> sin x"
```
```  4191   by (metis assms le_less sin_monotone_2pi)
```
```  4192
```
```  4193 lemma sin_x_le_x:
```
```  4194   fixes x :: real
```
```  4195   assumes x: "x \<ge> 0"
```
```  4196   shows "sin x \<le> x"
```
```  4197 proof -
```
```  4198   let ?f = "\<lambda>x. x - sin x"
```
```  4199   from x have "?f x \<ge> ?f 0"
```
```  4200     apply (rule DERIV_nonneg_imp_nondecreasing)
```
```  4201     apply (intro allI impI exI[of _ "1 - cos x" for x])
```
```  4202     apply (auto intro!: derivative_eq_intros simp: field_simps)
```
```  4203     done
```
```  4204   then show "sin x \<le> x" by simp
```
```  4205 qed
```
```  4206
```
```  4207 lemma sin_x_ge_neg_x:
```
```  4208   fixes x :: real
```
```  4209   assumes x: "x \<ge> 0"
```
```  4210   shows "sin x \<ge> - x"
```
```  4211 proof -
```
```  4212   let ?f = "\<lambda>x. x + sin x"
```
```  4213   from x have "?f x \<ge> ?f 0"
```
```  4214     apply (rule DERIV_nonneg_imp_nondecreasing)
```
```  4215     apply (intro allI impI exI[of _ "1 + cos x" for x])
```
```  4216     apply (auto intro!: derivative_eq_intros simp: field_simps real_0_le_add_iff)
```
```  4217     done
```
```  4218   then show "sin x \<ge> -x" by simp
```
```  4219 qed
```
```  4220
```
```  4221 lemma abs_sin_x_le_abs_x: "\<bar>sin x\<bar> \<le> \<bar>x\<bar>"
```
```  4222   for x :: real
```
```  4223   using sin_x_ge_neg_x [of x] sin_x_le_x [of x] sin_x_ge_neg_x [of "-x"] sin_x_le_x [of "-x"]
```
```  4224   by (auto simp: abs_real_def)
```
```  4225
```
```  4226
```
```  4227 subsection \<open>More Corollaries about Sine and Cosine\<close>
```
```  4228
```
```  4229 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
```
```  4230 proof -
```
```  4231   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
```
```  4232     by (auto simp: algebra_simps sin_add)
```
```  4233   then show ?thesis
```
```  4234     by (simp add: distrib_right add_divide_distrib add.commute mult.commute [of pi])
```
```  4235 qed
```
```  4236
```
```  4237 lemma cos_2npi [simp]: "cos (2 * real n * pi) = 1"
```
```  4238   for n :: nat
```
```  4239   by (cases "even n") (simp_all add: cos_double mult.assoc)
```
```  4240
```
```  4241 lemma cos_3over2_pi [simp]: "cos (3/2*pi) = 0"
```
```  4242   apply (subgoal_tac "cos (pi + pi/2) = 0")
```
```  4243    apply simp
```
```  4244   apply (subst cos_add)
```
```  4245   apply simp
```
```  4246   done
```
```  4247
```
```  4248 lemma sin_2npi [simp]: "sin (2 * real n * pi) = 0"
```
```  4249   for n :: nat
```
```  4250   by (auto simp: mult.assoc sin_double)
```
```  4251
```
```  4252 lemma sin_3over2_pi [simp]: "sin (3/2*pi) = - 1"
```
```  4253   apply (subgoal_tac "sin (pi + pi/2) = - 1")
```
```  4254    apply simp
```
```  4255   apply (subst sin_add)
```
```  4256   apply simp
```
```  4257   done
```
```  4258
```
```  4259 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
```
```  4260   by (simp only: cos_add sin_add of_nat_Suc distrib_right distrib_left add_divide_distrib, auto)
```
```  4261
```
```  4262 lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)"
```
```  4263   by (auto intro!: derivative_eq_intros)
```
```  4264
```
```  4265 lemma sin_zero_norm_cos_one:
```
```  4266   fixes x :: "'a::{real_normed_field,banach}"
```
```  4267   assumes "sin x = 0"
```
```  4268   shows "norm (cos x) = 1"
```
```  4269   using sin_cos_squared_add [of x, unfolded assms]
```
```  4270   by (simp add: square_norm_one)
```
```  4271
```
```  4272 lemma sin_zero_abs_cos_one: "sin x = 0 \<Longrightarrow> \<bar>cos x\<bar> = (1::real)"
```
```  4273   using sin_zero_norm_cos_one by fastforce
```
```  4274
```
```  4275 lemma cos_one_sin_zero:
```
```  4276   fixes x :: "'a::{real_normed_field,banach}"
```
```  4277   assumes "cos x = 1"
```
```  4278   shows "sin x = 0"
```
```  4279   using sin_cos_squared_add [of x, unfolded assms]
```
```  4280   by simp
```
```  4281
```
```  4282 lemma sin_times_pi_eq_0: "sin (x * pi) = 0 \<longleftrightarrow> x \<in> \<int>"
```
```  4283   by (simp add: sin_zero_iff_int2) (metis Ints_cases Ints_of_int)
```
```  4284
```
```  4285 lemma cos_one_2pi: "cos x = 1 \<longleftrightarrow> (\<exists>n::nat. x = n * 2 * pi) | (\<exists>n::nat. x = - (n * 2 * pi))"
```
```  4286   (is "?lhs = ?rhs")
```
```  4287 proof
```
```  4288   assume ?lhs
```
```  4289   then have "sin x = 0"
```
```  4290     by (simp add: cos_one_sin_zero)
```
```  4291   then show ?rhs
```
```  4292   proof (simp only: sin_zero_iff, elim exE disjE conjE)
```
```  4293     fix n :: nat
```
```  4294     assume n: "even n" "x = real n * (pi/2)"
```
```  4295     then obtain m where m: "n = 2 * m"
```
```  4296       using dvdE by blast
```
```  4297     then have me: "even m" using \<open>?lhs\<close> n
```
```  4298       by (auto simp: field_simps) (metis one_neq_neg_one  power_minus_odd power_one)
```
```  4299     show ?rhs
```
```  4300       using m me n
```
```  4301       by (auto simp: field_simps elim!: evenE)
```
```  4302   next
```
```  4303     fix n :: nat
```
```  4304     assume n: "even n" "x = - (real n * (pi/2))"
```
```  4305     then obtain m where m: "n = 2 * m"
```
```  4306       using dvdE by blast
```
```  4307     then have me: "even m" using \<open>?lhs\<close> n
```
```  4308       by (auto simp: field_simps) (metis one_neq_neg_one  power_minus_odd power_one)
```
```  4309     show ?rhs
```
```  4310       using m me n
```
```  4311       by (auto simp: field_simps elim!: evenE)
```
```  4312   qed
```
```  4313 next
```
```  4314   assume ?rhs
```
```  4315   then show "cos x = 1"
```
```  4316     by (metis cos_2npi cos_minus mult.assoc mult.left_commute)
```
```  4317 qed
```
```  4318
```
```  4319 lemma cos_one_2pi_int: "cos x = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2 * pi)" (is "?lhs = ?rhs")
```
```  4320 proof
```
```  4321   assume "cos x = 1"
```
```  4322   then show ?rhs
```
```  4323     apply (auto simp: cos_one_2pi)
```
```  4324      apply (metis of_int_of_nat_eq)
```
```  4325     apply (metis mult_minus_right of_int_minus of_int_of_nat_eq)
```
```  4326     done
```
```  4327 next
```
```  4328   assume ?rhs
```
```  4329   then show "cos x = 1"
```
```  4330     by (clarsimp simp add: cos_one_2pi) (metis mult_minus_right of_int_of_nat)
```
```  4331 qed
```
```  4332
```
```  4333 lemma cos_npi_int [simp]:
```
```  4334   fixes n::int shows "cos (pi * of_int n) = (if even n then 1 else -1)"
```
```  4335     by (auto simp: algebra_simps cos_one_2pi_int elim!: oddE evenE)
```
```  4336
```
```  4337 lemma sin_cos_sqrt: "0 \<le> sin x \<Longrightarrow> sin x = sqrt (1 - (cos(x) ^ 2))"
```
```  4338   using sin_squared_eq real_sqrt_unique by fastforce
```
```  4339
```
```  4340 lemma sin_eq_0_pi: "- pi < x \<Longrightarrow> x < pi \<Longrightarrow> sin x = 0 \<Longrightarrow> x = 0"
```
```  4341   by (metis sin_gt_zero sin_minus minus_less_iff neg_0_less_iff_less not_less_iff_gr_or_eq)
```
```  4342
```
```  4343 lemma cos_treble_cos: "cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x"
```
```  4344   for x :: "'a::{real_normed_field,banach}"
```
```  4345 proof -
```
```  4346   have *: "(sin x * (sin x * 3)) = 3 - (cos x * (cos x * 3))"
```
```  4347     by (simp add: mult.assoc [symmetric] sin_squared_eq [unfolded power2_eq_square])
```
```  4348   have "cos(3 * x) = cos(2*x + x)"
```
```  4349     by simp
```
```  4350   also have "\<dots> = 4 * cos x ^ 3 - 3 * cos x"
```
```  4351     apply (simp only: cos_add cos_double sin_double)
```
```  4352     apply (simp add: * field_simps power2_eq_square power3_eq_cube)
```
```  4353     done
```
```  4354   finally show ?thesis .
```
```  4355 qed
```
```  4356
```
```  4357 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
```
```  4358 proof -
```
```  4359   let ?c = "cos (pi / 4)"
```
```  4360   let ?s = "sin (pi / 4)"
```
```  4361   have nonneg: "0 \<le> ?c"
```
```  4362     by (simp add: cos_ge_zero)
```
```  4363   have "0 = cos (pi / 4 + pi / 4)"
```
```  4364     by simp
```
```  4365   also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2"
```
```  4366     by (simp only: cos_add power2_eq_square)
```
```  4367   also have "\<dots> = 2 * ?c\<^sup>2 - 1"
```
```  4368     by (simp add: sin_squared_eq)
```
```  4369   finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2"
```
```  4370     by (simp add: power_divide)
```
```  4371   then show ?thesis
```
```  4372     using nonneg by (rule power2_eq_imp_eq) simp
```
```  4373 qed
```
```  4374
```
```  4375 lemma cos_30: "cos (pi / 6) = sqrt 3/2"
```
```  4376 proof -
```
```  4377   let ?c = "cos (pi / 6)"
```
```  4378   let ?s = "sin (pi / 6)"
```
```  4379   have pos_c: "0 < ?c"
```
```  4380     by (rule cos_gt_zero) simp_all
```
```  4381   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
```
```  4382     by simp
```
```  4383   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
```
```  4384     by (simp only: cos_add sin_add)
```
```  4385   also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)"
```
```  4386     by (simp add: algebra_simps power2_eq_square)
```
```  4387   finally have "?c\<^sup>2 = (sqrt 3/2)\<^sup>2"
```
```  4388     using pos_c by (simp add: sin_squared_eq power_divide)
```
```  4389   then show ?thesis
```
```  4390     using pos_c [THEN order_less_imp_le]
```
```  4391     by (rule power2_eq_imp_eq) simp
```
```  4392 qed
```
```  4393
```
```  4394 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
```
```  4395   by (simp add: sin_cos_eq cos_45)
```
```  4396
```
```  4397 lemma sin_60: "sin (pi / 3) = sqrt 3/2"
```
```  4398   by (simp add: sin_cos_eq cos_30)
```
```  4399
```
```  4400 lemma cos_60: "cos (pi / 3) = 1 / 2"
```
```  4401   apply (rule power2_eq_imp_eq)
```
```  4402     apply (simp add: cos_squared_eq sin_60 power_divide)
```
```  4403    apply (rule cos_ge_zero)
```
```  4404     apply (rule order_trans [where y=0])
```
```  4405      apply simp_all
```
```  4406   done
```
```  4407
```
```  4408 lemma sin_30: "sin (pi / 6) = 1 / 2"
```
```  4409   by (simp add: sin_cos_eq cos_60)
```
```  4410
```
```  4411 lemma cos_integer_2pi: "n \<in> \<int> \<Longrightarrow> cos(2 * pi * n) = 1"
```
```  4412   by (metis Ints_cases cos_one_2pi_int mult.assoc mult.commute)
```
```  4413
```
```  4414 lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2 * pi * n) = 0"
```
```  4415   by (metis sin_two_pi Ints_mult mult.assoc mult.commute sin_times_pi_eq_0)
```
```  4416
```
```  4417 lemma cos_int_2npi [simp]: "cos (2 * of_int n * pi) = 1"
```
```  4418   for n :: int
```
```  4419   by (simp add: cos_one_2pi_int)
```
```  4420
```
```  4421 lemma sin_int_2npi [simp]: "sin (2 * of_int n * pi) = 0"
```
```  4422   for n :: int
```
```  4423   by (metis Ints_of_int mult.assoc mult.commute sin_integer_2pi)
```
```  4424
```
```  4425 lemma sincos_principal_value: "\<exists>y. (- pi < y \<and> y \<le> pi) \<and> (sin y = sin x \<and> cos y = cos x)"
```
```  4426   apply (rule exI [where x="pi - (2 * pi) * frac ((pi - x) / (2 * pi))"])
```
```  4427   apply (auto simp: field_simps frac_lt_1)
```
```  4428    apply (simp_all add: frac_def divide_simps)
```
```  4429    apply (simp_all add: add_divide_distrib diff_divide_distrib)
```
```  4430    apply (simp_all add: sin_diff cos_diff mult.assoc [symmetric] cos_integer_2pi sin_integer_2pi)
```
```  4431   done
```
```  4432
```
```  4433
```
```  4434 subsection \<open>Tangent\<close>
```
```  4435
```
```  4436 definition tan :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4437   where "tan = (\<lambda>x. sin x / cos x)"
```
```  4438
```
```  4439 lemma tan_of_real: "of_real (tan x) = (tan (of_real x) :: 'a::{real_normed_field,banach})"
```
```  4440   by (simp add: tan_def sin_of_real cos_of_real)
```
```  4441
```
```  4442 lemma tan_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> tan z \<in> \<real>"
```
```  4443   for z :: "'a::{real_normed_field,banach}"
```
```  4444   by (simp add: tan_def)
```
```  4445
```
```  4446 lemma tan_zero [simp]: "tan 0 = 0"
```
```  4447   by (simp add: tan_def)
```
```  4448
```
```  4449 lemma tan_pi [simp]: "tan pi = 0"
```
```  4450   by (simp add: tan_def)
```
```  4451
```
```  4452 lemma tan_npi [simp]: "tan (real n * pi) = 0"
```
```  4453   for n :: nat
```
```  4454   by (simp add: tan_def)
```
```  4455
```
```  4456 lemma tan_minus [simp]: "tan (- x) = - tan x"
```
```  4457   by (simp add: tan_def)
```
```  4458
```
```  4459 lemma tan_periodic [simp]: "tan (x + 2 * pi) = tan x"
```
```  4460   by (simp add: tan_def)
```
```  4461
```
```  4462 lemma lemma_tan_add1: "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
```
```  4463   by (simp add: tan_def cos_add field_simps)
```
```  4464
```
```  4465 lemma add_tan_eq: "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
```
```  4466   for x :: "'a::{real_normed_field,banach}"
```
```  4467   by (simp add: tan_def sin_add field_simps)
```
```  4468
```
```  4469 lemma tan_add:
```
```  4470   "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> cos (x + y) \<noteq> 0 \<Longrightarrow> tan (x + y) = (tan x + tan y)/(1 - tan x * tan y)"
```
```  4471   for x :: "'a::{real_normed_field,banach}"
```
```  4472   by (simp add: add_tan_eq lemma_tan_add1 field_simps) (simp add: tan_def)
```
```  4473
```
```  4474 lemma tan_double: "cos x \<noteq> 0 \<Longrightarrow> cos (2 * x) \<noteq> 0 \<Longrightarrow> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)"
```
```  4475   for x :: "'a::{real_normed_field,banach}"
```
```  4476   using tan_add [of x x] by (simp add: power2_eq_square)
```
```  4477
```
```  4478 lemma tan_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < tan x"
```
```  4479   by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
```
```  4480
```
```  4481 lemma tan_less_zero:
```
```  4482   assumes "- pi/2 < x" and "x < 0"
```
```  4483   shows "tan x < 0"
```
```  4484 proof -
```
```  4485   have "0 < tan (- x)"
```
```  4486     using assms by (simp only: tan_gt_zero)
```
```  4487   then show ?thesis by simp
```
```  4488 qed
```
```  4489
```
```  4490 lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"
```
```  4491   for x :: "'a::{real_normed_field,banach,field}"
```
```  4492   unfolding tan_def sin_double cos_double sin_squared_eq
```
```  4493   by (simp add: power2_eq_square)
```
```  4494
```
```  4495 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
```
```  4496   unfolding tan_def by (simp add: sin_30 cos_30)
```
```  4497
```
```  4498 lemma tan_45: "tan (pi / 4) = 1"
```
```  4499   unfolding tan_def by (simp add: sin_45 cos_45)
```
```  4500
```
```  4501 lemma tan_60: "tan (pi / 3) = sqrt 3"
```
```  4502   unfolding tan_def by (simp add: sin_60 cos_60)
```
```  4503
```
```  4504 lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)"
```
```  4505   for x :: "'a::{real_normed_field,banach}"
```
```  4506   unfolding tan_def
```
```  4507   by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square)
```
```  4508
```
```  4509 lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
```
```  4510   for x :: "'a::{real_normed_field,banach}"
```
```  4511   by (rule DERIV_tan [THEN DERIV_isCont])
```
```  4512
```
```  4513 lemma isCont_tan' [simp,continuous_intros]:
```
```  4514   fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
```
```  4515   shows "isCont f a \<Longrightarrow> cos (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
```
```  4516   by (rule isCont_o2 [OF _ isCont_tan])
```
```  4517
```
```  4518 lemma tendsto_tan [tendsto_intros]:
```
```  4519   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4520   shows "(f \<longlongrightarrow> a) F \<Longrightarrow> cos a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. tan (f x)) \<longlongrightarrow> tan a) F"
```
```  4521   by (rule isCont_tendsto_compose [OF isCont_tan])
```
```  4522
```
```  4523 lemma continuous_tan:
```
```  4524   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4525   shows "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))"
```
```  4526   unfolding continuous_def by (rule tendsto_tan)
```
```  4527
```
```  4528 lemma continuous_on_tan [continuous_intros]:
```
```  4529   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4530   shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))"
```
```  4531   unfolding continuous_on_def by (auto intro: tendsto_tan)
```
```  4532
```
```  4533 lemma continuous_within_tan [continuous_intros]:
```
```  4534   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4535   shows "continuous (at x within s) f \<Longrightarrow>
```
```  4536     cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
```
```  4537   unfolding continuous_within by (rule tendsto_tan)
```
```  4538
```
```  4539 lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) \<midarrow>pi/2\<rightarrow> 0"
```
```  4540   by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
```
```  4541
```
```  4542 lemma lemma_tan_total: "0 < y \<Longrightarrow> \<exists>x. 0 < x \<and> x < pi/2 \<and> y < tan x"
```
```  4543   apply (insert LIM_cos_div_sin)
```
```  4544   apply (simp only: LIM_eq)
```
```  4545   apply (drule_tac x = "inverse y" in spec)
```
```  4546   apply safe
```
```  4547    apply force
```
```  4548   apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero])
```
```  4549   apply safe
```
```  4550   apply (rule_tac x = "(pi/2) - e" in exI)
```
```  4551   apply (simp (no_asm_simp))
```
```  4552   apply (drule_tac x = "(pi/2) - e" in spec)
```
```  4553   apply (auto simp add: tan_def sin_diff cos_diff)
```
```  4554   apply (rule inverse_less_iff_less [THEN iffD1])
```
```  4555     apply (auto simp add: divide_inverse)
```
```  4556    apply (rule mult_pos_pos)
```
```  4557     apply (subgoal_tac [3] "0 < sin e \<and> 0 < cos e")
```
```  4558      apply (auto intro: cos_gt_zero sin_gt_zero2 simp: mult.commute)
```
```  4559   done
```
```  4560
```
```  4561 lemma tan_total_pos: "0 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x \<and> x < pi/2 \<and> tan x = y"
```
```  4562   apply (frule order_le_imp_less_or_eq)
```
```  4563   apply safe
```
```  4564    prefer 2 apply force
```
```  4565   apply (drule lemma_tan_total)
```
```  4566   apply safe
```
```  4567   apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
```
```  4568   apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
```
```  4569   apply (drule_tac y = xa in order_le_imp_less_or_eq)
```
```  4570   apply (auto dest: cos_gt_zero)
```
```  4571   done
```
```  4572
```
```  4573 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y"
```
```  4574   apply (insert linorder_linear [of 0 y])
```
```  4575   apply safe
```
```  4576    apply (drule tan_total_pos)
```
```  4577    apply (cut_tac [2] y="-y" in tan_total_pos)
```
```  4578     apply safe
```
```  4579     apply (rule_tac [3] x = "-x" in exI)
```
```  4580     apply (auto del: exI intro!: exI)
```
```  4581   done
```
```  4582
```
```  4583 lemma tan_total: "\<exists>! x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y"
```
```  4584   apply (insert lemma_tan_total1 [where y = y])
```
```  4585   apply auto
```
```  4586   apply hypsubst_thin
```
```  4587   apply (cut_tac x = xa and y = y in linorder_less_linear)
```
```  4588   apply auto
```
```  4589    apply (subgoal_tac [2] "\<exists>z. y < z \<and> z < xa \<and> DERIV tan z :> 0")
```
```  4590     apply (subgoal_tac "\<exists>z. xa < z \<and> z < y \<and> DERIV tan z :> 0")
```
```  4591      apply (rule_tac [4] Rolle)
```
```  4592         apply (rule_tac [2] Rolle)
```
```  4593            apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
```
```  4594             simp add: real_differentiable_def)
```
```  4595        apply (rule_tac [!] DERIV_tan asm_rl)
```
```  4596        apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
```
```  4597         simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
```
```  4598   done
```
```  4599
```
```  4600 lemma tan_monotone:
```
```  4601   assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
```
```  4602   shows "tan y < tan x"
```
```  4603 proof -
```
```  4604   have "\<forall>x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse ((cos x')\<^sup>2)"
```
```  4605   proof (rule allI, rule impI)
```
```  4606     fix x' :: real
```
```  4607     assume "y \<le> x' \<and> x' \<le> x"
```
```  4608     then have "-(pi/2) < x'" and "x' < pi/2"
```
```  4609       using assms by auto
```
```  4610     from cos_gt_zero_pi[OF this]
```
```  4611     have "cos x' \<noteq> 0" by auto
```
```  4612     then show "DERIV tan x' :> inverse ((cos x')\<^sup>2)"
```
```  4613       by (rule DERIV_tan)
```
```  4614   qed
```
```  4615   from MVT2[OF \<open>y < x\<close> this]
```
```  4616   obtain z where "y < z" and "z < x"
```
```  4617     and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
```
```  4618   then have "- (pi / 2) < z" and "z < pi / 2"
```
```  4619     using assms by auto
```
```  4620   then have "0 < cos z"
```
```  4621     using cos_gt_zero_pi by auto
```
```  4622   then have inv_pos: "0 < inverse ((cos z)\<^sup>2)"
```
```  4623     by auto
```
```  4624   have "0 < x - y" using \<open>y < x\<close> by auto
```
```  4625   with inv_pos have "0 < tan x - tan y"
```
```  4626     unfolding tan_diff by auto
```
```  4627   then show ?thesis by auto
```
```  4628 qed
```
```  4629
```
```  4630 lemma tan_monotone':
```
```  4631   assumes "- (pi / 2) < y"
```
```  4632     and "y < pi / 2"
```
```  4633     and "- (pi / 2) < x"
```
```  4634     and "x < pi / 2"
```
```  4635   shows "y < x \<longleftrightarrow> tan y < tan x"
```
```  4636 proof
```
```  4637   assume "y < x"
```
```  4638   then show "tan y < tan x"
```
```  4639     using tan_monotone and \<open>- (pi / 2) < y\<close> and \<open>x < pi / 2\<close> by auto
```
```  4640 next
```
```  4641   assume "tan y < tan x"
```
```  4642   show "y < x"
```
```  4643   proof (rule ccontr)
```
```  4644     assume "\<not> ?thesis"
```
```  4645     then have "x \<le> y" by auto
```
```  4646     then have "tan x \<le> tan y"
```
```  4647     proof (cases "x = y")
```
```  4648       case True
```
```  4649       then show ?thesis by auto
```
```  4650     next
```
```  4651       case False
```
```  4652       then have "x < y" using \<open>x \<le> y\<close> by auto
```
```  4653       from tan_monotone[OF \<open>- (pi/2) < x\<close> this \<open>y < pi / 2\<close>] show ?thesis
```
```  4654         by auto
```
```  4655     qed
```
```  4656     then show False
```
```  4657       using \<open>tan y < tan x\<close> by auto
```
```  4658   qed
```
```  4659 qed
```
```  4660
```
```  4661 lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)"
```
```  4662   unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
```
```  4663
```
```  4664 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
```
```  4665   by (simp add: tan_def)
```
```  4666
```
```  4667 lemma tan_periodic_nat[simp]: "tan (x + real n * pi) = tan x"
```
```  4668   for n :: nat
```
```  4669 proof (induct n arbitrary: x)
```
```  4670   case 0
```
```  4671   then show ?case by simp
```
```  4672 next
```
```  4673   case (Suc n)
```
```  4674   have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi"
```
```  4675     unfolding Suc_eq_plus1 of_nat_add  distrib_right by auto
```
```  4676   show ?case
```
```  4677     unfolding split_pi_off using Suc by auto
```
```  4678 qed
```
```  4679
```
```  4680 lemma tan_periodic_int[simp]: "tan (x + of_int i * pi) = tan x"
```
```  4681 proof (cases "0 \<le> i")
```
```  4682   case True
```
```  4683   then have i_nat: "of_int i = of_int (nat i)" by auto
```
```  4684   show ?thesis unfolding i_nat
```
```  4685     by (metis of_int_of_nat_eq tan_periodic_nat)
```
```  4686 next
```
```  4687   case False
```
```  4688   then have i_nat: "of_int i = - of_int (nat (- i))" by auto
```
```  4689   have "tan x = tan (x + of_int i * pi - of_int i * pi)"
```
```  4690     by auto
```
```  4691   also have "\<dots> = tan (x + of_int i * pi)"
```
```  4692     unfolding i_nat mult_minus_left diff_minus_eq_add
```
```  4693     by (metis of_int_of_nat_eq tan_periodic_nat)
```
```  4694   finally show ?thesis by auto
```
```  4695 qed
```
```  4696
```
```  4697 lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
```
```  4698   using tan_periodic_int[of _ "numeral n" ] by simp
```
```  4699
```
```  4700 lemma tan_minus_45: "tan (-(pi/4)) = -1"
```
```  4701   unfolding tan_def by (simp add: sin_45 cos_45)
```
```  4702
```
```  4703 lemma tan_diff:
```
```  4704   "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> cos (x - y) \<noteq> 0 \<Longrightarrow> tan (x - y) = (tan x - tan y)/(1 + tan x * tan y)"
```
```  4705   for x :: "'a::{real_normed_field,banach}"
```
```  4706   using tan_add [of x "-y"] by simp
```
```  4707
```
```  4708 lemma tan_pos_pi2_le: "0 \<le> x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 \<le> tan x"
```
```  4709   using less_eq_real_def tan_gt_zero by auto
```
```  4710
```
```  4711 lemma cos_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> cos x = 1 / sqrt (1 + tan x ^ 2)"
```
```  4712   using cos_gt_zero_pi [of x]
```
```  4713   by (simp add: divide_simps tan_def real_sqrt_divide abs_if split: if_split_asm)
```
```  4714
```
```  4715 lemma sin_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> sin x = tan x / sqrt (1 + tan x ^ 2)"
```
```  4716   using cos_gt_zero [of "x"] cos_gt_zero [of "-x"]
```
```  4717   by (force simp add: divide_simps tan_def real_sqrt_divide abs_if split: if_split_asm)
```
```  4718
```
```  4719 lemma tan_mono_le: "-(pi/2) < x \<Longrightarrow> x \<le> y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x \<le> tan y"
```
```  4720   using less_eq_real_def tan_monotone by auto
```
```  4721
```
```  4722 lemma tan_mono_lt_eq:
```
```  4723   "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> -(pi/2) < y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x < tan y \<longleftrightarrow> x < y"
```
```  4724   using tan_monotone' by blast
```
```  4725
```
```  4726 lemma tan_mono_le_eq:
```
```  4727   "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> -(pi/2) < y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x \<le> tan y \<longleftrightarrow> x \<le> y"
```
```  4728   by (meson tan_mono_le not_le tan_monotone)
```
```  4729
```
```  4730 lemma tan_bound_pi2: "\<bar>x\<bar> < pi/4 \<Longrightarrow> \<bar>tan x\<bar> < 1"
```
```  4731   using tan_45 tan_monotone [of x "pi/4"] tan_monotone [of "-x" "pi/4"]
```
```  4732   by (auto simp: abs_if split: if_split_asm)
```
```  4733
```
```  4734 lemma tan_cot: "tan(pi/2 - x) = inverse(tan x)"
```
```  4735   by (simp add: tan_def sin_diff cos_diff)
```
```  4736
```
```  4737
```
```  4738 subsection \<open>Cotangent\<close>
```
```  4739
```
```  4740 definition cot :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4741   where "cot = (\<lambda>x. cos x / sin x)"
```
```  4742
```
```  4743 lemma cot_of_real: "of_real (cot x) = (cot (of_real x) :: 'a::{real_normed_field,banach})"
```
```  4744   by (simp add: cot_def sin_of_real cos_of_real)
```
```  4745
```
```  4746 lemma cot_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cot z \<in> \<real>"
```
```  4747   for z :: "'a::{real_normed_field,banach}"
```
```  4748   by (simp add: cot_def)
```
```  4749
```
```  4750 lemma cot_zero [simp]: "cot 0 = 0"
```
```  4751   by (simp add: cot_def)
```
```  4752
```
```  4753 lemma cot_pi [simp]: "cot pi = 0"
```
```  4754   by (simp add: cot_def)
```
```  4755
```
```  4756 lemma cot_npi [simp]: "cot (real n * pi) = 0"
```
```  4757   for n :: nat
```
```  4758   by (simp add: cot_def)
```
```  4759
```
```  4760 lemma cot_minus [simp]: "cot (- x) = - cot x"
```
```  4761   by (simp add: cot_def)
```
```  4762
```
```  4763 lemma cot_periodic [simp]: "cot (x + 2 * pi) = cot x"
```
```  4764   by (simp add: cot_def)
```
```  4765
```
```  4766 lemma cot_altdef: "cot x = inverse (tan x)"
```
```  4767   by (simp add: cot_def tan_def)
```
```  4768
```
```  4769 lemma tan_altdef: "tan x = inverse (cot x)"
```
```  4770   by (simp add: cot_def tan_def)
```
```  4771
```
```  4772 lemma tan_cot': "tan (pi/2 - x) = cot x"
```
```  4773   by (simp add: tan_cot cot_altdef)
```
```  4774
```
```  4775 lemma cot_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cot x"
```
```  4776   by (simp add: cot_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
```
```  4777
```
```  4778 lemma cot_less_zero:
```
```  4779   assumes lb: "- pi/2 < x" and "x < 0"
```
```  4780   shows "cot x < 0"
```
```  4781 proof -
```
```  4782   have "0 < cot (- x)"
```
```  4783     using assms by (simp only: cot_gt_zero)
```
```  4784   then show ?thesis by simp
```
```  4785 qed
```
```  4786
```
```  4787 lemma DERIV_cot [simp]: "sin x \<noteq> 0 \<Longrightarrow> DERIV cot x :> -inverse ((sin x)\<^sup>2)"
```
```  4788   for x :: "'a::{real_normed_field,banach}"
```
```  4789   unfolding cot_def using cos_squared_eq[of x]
```
```  4790   by (auto intro!: derivative_eq_intros) (simp add: divide_inverse power2_eq_square)
```
```  4791
```
```  4792 lemma isCont_cot: "sin x \<noteq> 0 \<Longrightarrow> isCont cot x"
```
```  4793   for x :: "'a::{real_normed_field,banach}"
```
```  4794   by (rule DERIV_cot [THEN DERIV_isCont])
```
```  4795
```
```  4796 lemma isCont_cot' [simp,continuous_intros]:
```
```  4797   "isCont f a \<Longrightarrow> sin (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. cot (f x)) a"
```
```  4798   for a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
```
```  4799   by (rule isCont_o2 [OF _ isCont_cot])
```
```  4800
```
```  4801 lemma tendsto_cot [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> sin a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. cot (f x)) \<longlongrightarrow> cot a) F"
```
```  4802   for f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4803   by (rule isCont_tendsto_compose [OF isCont_cot])
```
```  4804
```
```  4805 lemma continuous_cot:
```
```  4806   "continuous F f \<Longrightarrow> sin (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. cot (f x))"
```
```  4807   for f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4808   unfolding continuous_def by (rule tendsto_cot)
```
```  4809
```
```  4810 lemma continuous_on_cot [continuous_intros]:
```
```  4811   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4812   shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. sin (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. cot (f x))"
```
```  4813   unfolding continuous_on_def by (auto intro: tendsto_cot)
```
```  4814
```
```  4815 lemma continuous_within_cot [continuous_intros]:
```
```  4816   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4817   shows "continuous (at x within s) f \<Longrightarrow> sin (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. cot (f x))"
```
```  4818   unfolding continuous_within by (rule tendsto_cot)
```
```  4819
```
```  4820
```
```  4821 subsection \<open>Inverse Trigonometric Functions\<close>
```
```  4822
```
```  4823 definition arcsin :: "real \<Rightarrow> real"
```
```  4824   where "arcsin y = (THE x. -(pi/2) \<le> x \<and> x \<le> pi/2 \<and> sin x = y)"
```
```  4825
```
```  4826 definition arccos :: "real \<Rightarrow> real"
```
```  4827   where "arccos y = (THE x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y)"
```
```  4828
```
```  4829 definition arctan :: "real \<Rightarrow> real"
```
```  4830   where "arctan y = (THE x. -(pi/2) < x \<and> x < pi/2 \<and> tan x = y)"
```
```  4831
```
```  4832 lemma arcsin: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi/2 \<and> sin (arcsin y) = y"
```
```  4833   unfolding arcsin_def by (rule theI' [OF sin_total])
```
```  4834
```
```  4835 lemma arcsin_pi: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi \<and> sin (arcsin y) = y"
```
```  4836   by (drule (1) arcsin) (force intro: order_trans)
```
```  4837
```
```  4838 lemma sin_arcsin [simp]: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin (arcsin y) = y"
```
```  4839   by (blast dest: arcsin)
```
```  4840
```
```  4841 lemma arcsin_bounded: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi/2"
```
```  4842   by (blast dest: arcsin)
```
```  4843
```
```  4844 lemma arcsin_lbound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y"
```
```  4845   by (blast dest: arcsin)
```
```  4846
```
```  4847 lemma arcsin_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
```
```  4848   by (blast dest: arcsin)
```
```  4849
```
```  4850 lemma arcsin_lt_bounded: "- 1 < y \<Longrightarrow> y < 1 \<Longrightarrow> - (pi/2) < arcsin y \<and> arcsin y < pi/2"
```
```  4851   apply (frule order_less_imp_le)
```
```  4852   apply (frule_tac y = y in order_less_imp_le)
```
```  4853   apply (frule arcsin_bounded)
```
```  4854    apply safe
```
```  4855     apply simp
```
```  4856    apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
```
```  4857    apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq)
```
```  4858    apply safe
```
```  4859    apply (drule_tac [!] f = sin in arg_cong)
```
```  4860    apply auto
```
```  4861   done
```
```  4862
```
```  4863 lemma arcsin_sin: "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> arcsin (sin x) = x"
```
```  4864   apply (unfold arcsin_def)
```
```  4865   apply (rule the1_equality)
```
```  4866    apply (rule sin_total)
```
```  4867     apply auto
```
```  4868   done
```
```  4869
```
```  4870 lemma arcsin_0 [simp]: "arcsin 0 = 0"
```
```  4871   using arcsin_sin [of 0] by simp
```
```  4872
```
```  4873 lemma arcsin_1 [simp]: "arcsin 1 = pi/2"
```
```  4874   using arcsin_sin [of "pi/2"] by simp
```
```  4875
```
```  4876 lemma arcsin_minus_1 [simp]: "arcsin (- 1) = - (pi/2)"
```
```  4877   using arcsin_sin [of "- pi/2"] by simp
```
```  4878
```
```  4879 lemma arcsin_minus: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin (- x) = - arcsin x"
```
```  4880   by (metis (no_types, hide_lams) arcsin arcsin_sin minus_minus neg_le_iff_le sin_minus)
```
```  4881
```
```  4882 lemma arcsin_eq_iff: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x = arcsin y \<longleftrightarrow> x = y"
```
```  4883   by (metis abs_le_iff arcsin minus_le_iff)
```
```  4884
```
```  4885 lemma cos_arcsin_nonzero: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> cos (arcsin x) \<noteq> 0"
```
```  4886   using arcsin_lt_bounded cos_gt_zero_pi by force
```
```  4887
```
```  4888 lemma arccos: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi \<and> cos (arccos y) = y"
```
```  4889   unfolding arccos_def by (rule theI' [OF cos_total])
```
```  4890
```
```  4891 lemma cos_arccos [simp]: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> cos (arccos y) = y"
```
```  4892   by (blast dest: arccos)
```
```  4893
```
```  4894 lemma arccos_bounded: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi"
```
```  4895   by (blast dest: arccos)
```
```  4896
```
```  4897 lemma arccos_lbound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y"
```
```  4898   by (blast dest: arccos)
```
```  4899
```
```  4900 lemma arccos_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> pi"
```
```  4901   by (blast dest: arccos)
```
```  4902
```
```  4903 lemma arccos_lt_bounded: "- 1 < y \<Longrightarrow> y < 1 \<Longrightarrow> 0 < arccos y \<and> arccos y < pi"
```
```  4904   apply (frule order_less_imp_le)
```
```  4905   apply (frule_tac y = y in order_less_imp_le)
```
```  4906   apply (frule arccos_bounded)
```
```  4907    apply auto
```
```  4908    apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
```
```  4909    apply (drule_tac [2] y = pi in order_le_imp_less_or_eq)
```
```  4910    apply auto
```
```  4911    apply (drule_tac [!] f = cos in arg_cong)
```
```  4912    apply auto
```
```  4913   done
```
```  4914
```
```  4915 lemma arccos_cos: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> arccos (cos x) = x"
```
```  4916   by (auto simp: arccos_def intro!: the1_equality cos_total)
```
```  4917
```
```  4918 lemma arccos_cos2: "x \<le> 0 \<Longrightarrow> - pi \<le> x \<Longrightarrow> arccos (cos x) = -x"
```
```  4919   by (auto simp: arccos_def intro!: the1_equality cos_total)
```
```  4920
```
```  4921 lemma cos_arcsin: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
```
```  4922   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  4923    apply (rule power2_eq_imp_eq)
```
```  4924      apply (simp add: cos_squared_eq)
```
```  4925     apply (rule cos_ge_zero)
```
```  4926      apply (erule (1) arcsin_lbound)
```
```  4927     apply (erule (1) arcsin_ubound)
```
```  4928    apply simp
```
```  4929   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2")
```
```  4930    apply simp
```
```  4931   apply (rule power_mono)
```
```  4932    apply simp
```
```  4933   apply simp
```
```  4934   done
```
```  4935
```
```  4936 lemma sin_arccos: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
```
```  4937   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  4938    apply (rule power2_eq_imp_eq)
```
```  4939      apply (simp add: sin_squared_eq)
```
```  4940     apply (rule sin_ge_zero)
```
```  4941      apply (erule (1) arccos_lbound)
```
```  4942     apply (erule (1) arccos_ubound)
```
```  4943    apply simp
```
```  4944   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2")
```
```  4945    apply simp
```
```  4946   apply (rule power_mono)
```
```  4947    apply simp
```
```  4948   apply simp
```
```  4949   done
```
```  4950
```
```  4951 lemma arccos_0 [simp]: "arccos 0 = pi/2"
```
```  4952   by (metis arccos_cos cos_gt_zero cos_pi cos_pi_half pi_gt_zero
```
```  4953       pi_half_ge_zero not_le not_zero_less_neg_numeral numeral_One)
```
```  4954
```
```  4955 lemma arccos_1 [simp]: "arccos 1 = 0"
```
```  4956   using arccos_cos by force
```
```  4957
```
```  4958 lemma arccos_minus_1 [simp]: "arccos (- 1) = pi"
```
```  4959   by (metis arccos_cos cos_pi order_refl pi_ge_zero)
```
```  4960
```
```  4961 lemma arccos_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos (- x) = pi - arccos x"
```
```  4962   by (metis arccos_cos arccos_cos2 cos_minus_pi cos_total diff_le_0_iff_le le_add_same_cancel1
```
```  4963       minus_diff_eq uminus_add_conv_diff)
```
```  4964
```
```  4965 corollary arccos_minus_abs:
```
```  4966   assumes "\<bar>x\<bar> \<le> 1"
```
```  4967   shows "arccos (- x) = pi - arccos x"
```
```  4968 using assms by (simp add: arccos_minus)
```
```  4969
```
```  4970 lemma sin_arccos_nonzero: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> sin (arccos x) \<noteq> 0"
```
```  4971   using arccos_lt_bounded sin_gt_zero by force
```
```  4972
```
```  4973 lemma arctan: "- (pi/2) < arctan y \<and> arctan y < pi/2 \<and> tan (arctan y) = y"
```
```  4974   unfolding arctan_def by (rule theI' [OF tan_total])
```
```  4975
```
```  4976 lemma tan_arctan: "tan (arctan y) = y"
```
```  4977   by (simp add: arctan)
```
```  4978
```
```  4979 lemma arctan_bounded: "- (pi/2) < arctan y \<and> arctan y < pi/2"
```
```  4980   by (auto simp only: arctan)
```
```  4981
```
```  4982 lemma arctan_lbound: "- (pi/2) < arctan y"
```
```  4983   by (simp add: arctan)
```
```  4984
```
```  4985 lemma arctan_ubound: "arctan y < pi/2"
```
```  4986   by (auto simp only: arctan)
```
```  4987
```
```  4988 lemma arctan_unique:
```
```  4989   assumes "-(pi/2) < x"
```
```  4990     and "x < pi/2"
```
```  4991     and "tan x = y"
```
```  4992   shows "arctan y = x"
```
```  4993   using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
```
```  4994
```
```  4995 lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x"
```
```  4996   by (rule arctan_unique) simp_all
```
```  4997
```
```  4998 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
```
```  4999   by (rule arctan_unique) simp_all
```
```  5000
```
```  5001 lemma arctan_minus: "arctan (- x) = - arctan x"
```
```  5002   using arctan [of "x"] by (auto simp: arctan_unique)
```
```  5003
```
```  5004 lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
```
```  5005   by (intro less_imp_neq [symmetric] cos_gt_zero_pi arctan_lbound arctan_ubound)
```
```  5006
```
```  5007 lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)"
```
```  5008 proof (rule power2_eq_imp_eq)
```
```  5009   have "0 < 1 + x\<^sup>2" by (simp add: add_pos_nonneg)
```
```  5010   show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp
```
```  5011   show "0 \<le> cos (arctan x)"
```
```  5012     by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
```
```  5013   have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1"
```
```  5014     unfolding tan_def by (simp add: distrib_left power_divide)
```
```  5015   then show "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
```
```  5016     using \<open>0 < 1 + x\<^sup>2\<close> by (simp add: arctan power_divide eq_divide_eq)
```
```  5017 qed
```
```  5018
```
```  5019 lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)"
```
```  5020   using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
```
```  5021   using tan_arctan [of x] unfolding tan_def cos_arctan
```
```  5022   by (simp add: eq_divide_eq)
```
```  5023
```
```  5024 lemma tan_sec: "cos x \<noteq> 0 \<Longrightarrow> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
```
```  5025   for x :: "'a::{real_normed_field,banach,field}"
```
```  5026   apply (rule power_inverse [THEN subst])
```
```  5027   apply (rule_tac c1 = "(cos x)\<^sup>2" in mult_right_cancel [THEN iffD1])
```
```  5028    apply (auto simp add: tan_def field_simps)
```
```  5029   done
```
```  5030
```
```  5031 lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
```
```  5032   by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
```
```  5033
```
```  5034 lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
```
```  5035   by (simp only: not_less [symmetric] arctan_less_iff)
```
```  5036
```
```  5037 lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
```
```  5038   by (simp only: eq_iff [where 'a=real] arctan_le_iff)
```
```  5039
```
```  5040 lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
```
```  5041   using arctan_less_iff [of 0 x] by simp
```
```  5042
```
```  5043 lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
```
```  5044   using arctan_less_iff [of x 0] by simp
```
```  5045
```
```  5046 lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
```
```  5047   using arctan_le_iff [of 0 x] by simp
```
```  5048
```
```  5049 lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```  5050   using arctan_le_iff [of x 0] by simp
```
```  5051
```
```  5052 lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
```
```  5053   using arctan_eq_iff [of x 0] by simp
```
```  5054
```
```  5055 lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin"
```
```  5056 proof -
```
```  5057   have "continuous_on (sin ` {- pi / 2 .. pi / 2}) arcsin"
```
```  5058     by (rule continuous_on_inv) (auto intro: continuous_intros simp: arcsin_sin)
```
```  5059   also have "sin ` {- pi / 2 .. pi / 2} = {-1 .. 1}"
```
```  5060   proof safe
```
```  5061     fix x :: real
```
```  5062     assume "x \<in> {-1..1}"
```
```  5063     then show "x \<in> sin ` {- pi / 2..pi / 2}"
```
```  5064       using arcsin_lbound arcsin_ubound
```
```  5065       by (intro image_eqI[where x="arcsin x"]) auto
```
```  5066   qed simp
```
```  5067   finally show ?thesis .
```
```  5068 qed
```
```  5069
```
```  5070 lemma continuous_on_arcsin [continuous_intros]:
```
```  5071   "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))"
```
```  5072   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arcsin']]
```
```  5073   by (auto simp: comp_def subset_eq)
```
```  5074
```
```  5075 lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x"
```
```  5076   using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  5077   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  5078
```
```  5079 lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos"
```
```  5080 proof -
```
```  5081   have "continuous_on (cos ` {0 .. pi}) arccos"
```
```  5082     by (rule continuous_on_inv) (auto intro: continuous_intros simp: arccos_cos)
```
```  5083   also have "cos ` {0 .. pi} = {-1 .. 1}"
```
```  5084   proof safe
```
```  5085     fix x :: real
```
```  5086     assume "x \<in> {-1..1}"
```
```  5087     then show "x \<in> cos ` {0..pi}"
```
```  5088       using arccos_lbound arccos_ubound
```
```  5089       by (intro image_eqI[where x="arccos x"]) auto
```
```  5090   qed simp
```
```  5091   finally show ?thesis .
```
```  5092 qed
```
```  5093
```
```  5094 lemma continuous_on_arccos [continuous_intros]:
```
```  5095   "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))"
```
```  5096   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arccos']]
```
```  5097   by (auto simp: comp_def subset_eq)
```
```  5098
```
```  5099 lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x"
```
```  5100   using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  5101   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  5102
```
```  5103 lemma isCont_arctan: "isCont arctan x"
```
```  5104   apply (rule arctan_lbound [of x, THEN dense, THEN exE])
```
```  5105   apply clarify
```
```  5106   apply (rule arctan_ubound [of x, THEN dense, THEN exE])
```
```  5107   apply clarify
```
```  5108   apply (subgoal_tac "isCont arctan (tan (arctan x))")
```
```  5109    apply (simp add: arctan)
```
```  5110   apply (erule (1) isCont_inverse_function2 [where f=tan])
```
```  5111    apply (metis arctan_tan order_le_less_trans order_less_le_trans)
```
```  5112   apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
```
```  5113   done
```
```  5114
```
```  5115 lemma tendsto_arctan [tendsto_intros]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) \<longlongrightarrow> arctan x) F"
```
```  5116   by (rule isCont_tendsto_compose [OF isCont_arctan])
```
```  5117
```
```  5118 lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))"
```
```  5119   unfolding continuous_def by (rule tendsto_arctan)
```
```  5120
```
```  5121 lemma continuous_on_arctan [continuous_intros]:
```
```  5122   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
```
```  5123   unfolding continuous_on_def by (auto intro: tendsto_arctan)
```
```  5124
```
```  5125 lemma DERIV_arcsin: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
```
```  5126   apply (rule DERIV_inverse_function [where f=sin and a="-1" and b=1])
```
```  5127        apply (rule DERIV_cong [OF DERIV_sin])
```
```  5128        apply (simp add: cos_arcsin)
```
```  5129       apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2")
```
```  5130        apply simp
```
```  5131       apply (rule power_strict_mono)
```
```  5132         apply simp
```
```  5133        apply simp
```
```  5134       apply simp
```
```  5135      apply assumption
```
```  5136     apply assumption
```
```  5137    apply simp
```
```  5138   apply (erule (1) isCont_arcsin)
```
```  5139   done
```
```  5140
```
```  5141 lemma DERIV_arccos: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
```
```  5142   apply (rule DERIV_inverse_function [where f=cos and a="-1" and b=1])
```
```  5143        apply (rule DERIV_cong [OF DERIV_cos])
```
```  5144        apply (simp add: sin_arccos)
```
```  5145       apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2")
```
```  5146        apply simp
```
```  5147       apply (rule power_strict_mono)
```
```  5148         apply simp
```
```  5149        apply simp
```
```  5150       apply simp
```
```  5151      apply assumption
```
```  5152     apply assumption
```
```  5153    apply simp
```
```  5154   apply (erule (1) isCont_arccos)
```
```  5155   done
```
```  5156
```
```  5157 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
```
```  5158   apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
```
```  5159        apply (rule DERIV_cong [OF DERIV_tan])
```
```  5160         apply (rule cos_arctan_not_zero)
```
```  5161        apply (simp_all add: add_pos_nonneg arctan isCont_arctan)
```
```  5162    apply (simp add: arctan power_inverse [symmetric] tan_sec [symmetric])
```
```  5163   apply (subgoal_tac "0 < 1 + x\<^sup>2")
```
```  5164    apply simp
```
```  5165   apply (simp_all add: add_pos_nonneg arctan isCont_arctan)
```
```  5166   done
```
```  5167
```
```  5168 declare
```
```  5169   DERIV_arcsin[THEN DERIV_chain2, derivative_intros]
```
```  5170   DERIV_arcsin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  5171   DERIV_arccos[THEN DERIV_chain2, derivative_intros]
```
```  5172   DERIV_arccos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  5173   DERIV_arctan[THEN DERIV_chain2, derivative_intros]
```
```  5174   DERIV_arctan[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  5175
```
```  5176 lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- (pi/2)))"
```
```  5177   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])
```
```  5178      (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  5179            intro!: tan_monotone exI[of _ "pi/2"])
```
```  5180
```
```  5181 lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"
```
```  5182   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])
```
```  5183      (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  5184            intro!: tan_monotone exI[of _ "pi/2"])
```
```  5185
```
```  5186 lemma tendsto_arctan_at_top: "(arctan \<longlongrightarrow> (pi/2)) at_top"
```
```  5187 proof (rule tendstoI)
```
```  5188   fix e :: real
```
```  5189   assume "0 < e"
```
```  5190   define y where "y = pi/2 - min (pi/2) e"
```
```  5191   then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y"
```
```  5192     using \<open>0 < e\<close> by auto
```
```  5193   show "eventually (\<lambda>x. dist (arctan x) (pi / 2) < e) at_top"
```
```  5194   proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)
```
```  5195     fix x
```
```  5196     assume "tan y < x"
```
```  5197     then have "arctan (tan y) < arctan x"
```
```  5198       by (simp add: arctan_less_iff)
```
```  5199     with y have "y < arctan x"
```
```  5200       by (subst (asm) arctan_tan) simp_all
```
```  5201     with arctan_ubound[of x, arith] y \<open>0 < e\<close>
```
```  5202     show "dist (arctan x) (pi / 2) < e"
```
```  5203       by (simp add: dist_real_def)
```
```  5204   qed
```
```  5205 qed
```
```  5206
```
```  5207 lemma tendsto_arctan_at_bot: "(arctan \<longlongrightarrow> - (pi/2)) at_bot"
```
```  5208   unfolding filterlim_at_bot_mirror arctan_minus
```
```  5209   by (intro tendsto_minus tendsto_arctan_at_top)
```
```  5210
```
```  5211
```
```  5212 subsection \<open>Prove Totality of the Trigonometric Functions\<close>
```
```  5213
```
```  5214 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
```
```  5215   by (simp add: abs_le_iff)
```
```  5216
```
```  5217 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
```
```  5218   by (simp add: sin_arccos abs_le_iff)
```
```  5219
```
```  5220 lemma sin_mono_less_eq:
```
```  5221   "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x < sin y \<longleftrightarrow> x < y"
```
```  5222   by (metis not_less_iff_gr_or_eq sin_monotone_2pi)
```
```  5223
```
```  5224 lemma sin_mono_le_eq:
```
```  5225   "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x \<le> sin y \<longleftrightarrow> x \<le> y"
```
```  5226   by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le)
```
```  5227
```
```  5228 lemma sin_inj_pi:
```
```  5229   "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x = sin y \<Longrightarrow> x = y"
```
```  5230   by (metis arcsin_sin)
```
```  5231
```
```  5232 lemma cos_mono_less_eq: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x < cos y \<longleftrightarrow> y < x"
```
```  5233   by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear)
```
```  5234
```
```  5235 lemma cos_mono_le_eq: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x \<le> cos y \<longleftrightarrow> y \<le> x"
```
```  5236   by (metis arccos_cos cos_monotone_0_pi_le eq_iff linear)
```
```  5237
```
```  5238 lemma cos_inj_pi: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x = cos y \<Longrightarrow> x = y"
```
```  5239   by (metis arccos_cos)
```
```  5240
```
```  5241 lemma arccos_le_pi2: "\<lbrakk>0 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi/2"
```
```  5242   by (metis (mono_tags) arccos_0 arccos cos_le_one cos_monotone_0_pi_le
```
```  5243       cos_pi cos_pi_half pi_half_ge_zero antisym_conv less_eq_neg_nonpos linear minus_minus order.trans order_refl)
```
```  5244
```
```  5245 lemma sincos_total_pi_half:
```
```  5246   assumes "0 \<le> x" "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1"
```
```  5247   shows "\<exists>t. 0 \<le> t \<and> t \<le> pi/2 \<and> x = cos t \<and> y = sin t"
```
```  5248 proof -
```
```  5249   have x1: "x \<le> 1"
```
```  5250     using assms by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2)
```
```  5251   with assms have *: "0 \<le> arccos x" "cos (arccos x) = x"
```
```  5252     by (auto simp: arccos)
```
```  5253   from assms have "y = sqrt (1 - x\<^sup>2)"
```
```  5254     by (metis abs_of_nonneg add.commute add_diff_cancel real_sqrt_abs)
```
```  5255   with x1 * assms arccos_le_pi2 [of x] show ?thesis
```
```  5256     by (rule_tac x="arccos x" in exI) (auto simp: sin_arccos)
```
```  5257 qed
```
```  5258
```
```  5259 lemma sincos_total_pi:
```
```  5260   assumes "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1"
```
```  5261   shows "\<exists>t. 0 \<le> t \<and> t \<le> pi \<and> x = cos t \<and> y = sin t"
```
```  5262 proof (cases rule: le_cases [of 0 x])
```
```  5263   case le
```
```  5264   from sincos_total_pi_half [OF le] show ?thesis
```
```  5265     by (metis pi_ge_two pi_half_le_two add.commute add_le_cancel_left add_mono assms)
```
```  5266 next
```
```  5267   case ge
```
```  5268   then have "0 \<le> -x"
```
```  5269     by simp
```
```  5270   then obtain t where t: "t\<ge>0" "t \<le> pi/2" "-x = cos t" "y = sin t"
```
```  5271     using sincos_total_pi_half assms
```
```  5272     by auto (metis \<open>0 \<le> - x\<close> power2_minus)
```
```  5273   show ?thesis
```
```  5274     by (rule exI [where x = "pi -t"]) (use t in auto)
```
```  5275 qed
```
```  5276
```
```  5277 lemma sincos_total_2pi_le:
```
```  5278   assumes "x\<^sup>2 + y\<^sup>2 = 1"
```
```  5279   shows "\<exists>t. 0 \<le> t \<and> t \<le> 2 * pi \<and> x = cos t \<and> y = sin t"
```
```  5280 proof (cases rule: le_cases [of 0 y])
```
```  5281   case le
```
```  5282   from sincos_total_pi [OF le] show ?thesis
```
```  5283     by (metis assms le_add_same_cancel1 mult.commute mult_2_right order.trans)
```
```  5284 next
```
```  5285   case ge
```
```  5286   then have "0 \<le> -y"
```
```  5287     by simp
```
```  5288   then obtain t where t: "t\<ge>0" "t \<le> pi" "x = cos t" "-y = sin t"
```
```  5289     using sincos_total_pi assms
```
```  5290     by auto (metis \<open>0 \<le> - y\<close> power2_minus)
```
```  5291   show ?thesis
```
```  5292     by (rule exI [where x = "2 * pi - t"]) (use t in auto)
```
```  5293 qed
```
```  5294
```
```  5295 lemma sincos_total_2pi:
```
```  5296   assumes "x\<^sup>2 + y\<^sup>2 = 1"
```
```  5297   obtains t where "0 \<le> t" "t < 2*pi" "x = cos t" "y = sin t"
```
```  5298 proof -
```
```  5299   from sincos_total_2pi_le [OF assms]
```
```  5300   obtain t where t: "0 \<le> t" "t \<le> 2*pi" "x = cos t" "y = sin t"
```
```  5301     by blast
```
```  5302   show ?thesis
```
```  5303     by (cases "t = 2 * pi") (use t that in \<open>force+\<close>)
```
```  5304 qed
```
```  5305
```
```  5306 lemma arcsin_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x < arcsin y \<longleftrightarrow> x < y"
```
```  5307   by (rule trans [OF sin_mono_less_eq [symmetric]]) (use arcsin_ubound arcsin_lbound in auto)
```
```  5308
```
```  5309 lemma arcsin_le_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y \<longleftrightarrow> x \<le> y"
```
```  5310   using arcsin_less_mono not_le by blast
```
```  5311
```
```  5312 lemma arcsin_less_arcsin: "- 1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x < arcsin y"
```
```  5313   using arcsin_less_mono by auto
```
```  5314
```
```  5315 lemma arcsin_le_arcsin: "- 1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y"
```
```  5316   using arcsin_le_mono by auto
```
```  5317
```
```  5318 lemma arccos_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x < arccos y \<longleftrightarrow> y < x"
```
```  5319   by (rule trans [OF cos_mono_less_eq [symmetric]]) (use arccos_ubound arccos_lbound in auto)
```
```  5320
```
```  5321 lemma arccos_le_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x \<le> arccos y \<longleftrightarrow> y \<le> x"
```
```  5322   using arccos_less_mono [of y x] by (simp add: not_le [symmetric])
```
```  5323
```
```  5324 lemma arccos_less_arccos: "- 1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y < arccos x"
```
```  5325   using arccos_less_mono by auto
```
```  5326
```
```  5327 lemma arccos_le_arccos: "- 1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> arccos x"
```
```  5328   using arccos_le_mono by auto
```
```  5329
```
```  5330 lemma arccos_eq_iff: "\<bar>x\<bar> \<le> 1 \<and> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x = arccos y \<longleftrightarrow> x = y"
```
```  5331   using cos_arccos_abs by fastforce
```
```  5332
```
```  5333
```
```  5334 subsection \<open>Machin's formula\<close>
```
```  5335
```
```  5336 lemma arctan_one: "arctan 1 = pi / 4"
```
```  5337   by (rule arctan_unique) (simp_all add: tan_45 m2pi_less_pi)
```
```  5338
```
```  5339 lemma tan_total_pi4:
```
```  5340   assumes "\<bar>x\<bar> < 1"
```
```  5341   shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
```
```  5342 proof
```
```  5343   show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
```
```  5344     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  5345     unfolding arctan_less_iff
```
```  5346     using assms by (auto simp add: arctan)
```
```  5347 qed
```
```  5348
```
```  5349 lemma arctan_add:
```
```  5350   assumes "\<bar>x\<bar> \<le> 1" "\<bar>y\<bar> < 1"
```
```  5351   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
```
```  5352 proof (rule arctan_unique [symmetric])
```
```  5353   have "- (pi / 4) \<le> arctan x" "- (pi / 4) < arctan y"
```
```  5354     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  5355     unfolding arctan_le_iff arctan_less_iff
```
```  5356     using assms by auto
```
```  5357   from add_le_less_mono [OF this] show 1: "- (pi / 2) < arctan x + arctan y"
```
```  5358     by simp
```
```  5359   have "arctan x \<le> pi / 4" "arctan y < pi / 4"
```
```  5360     unfolding arctan_one [symmetric]
```
```  5361     unfolding arctan_le_iff arctan_less_iff
```
```  5362     using assms by auto
```
```  5363   from add_le_less_mono [OF this] show 2: "arctan x + arctan y < pi / 2"
```
```  5364     by simp
```
```  5365   show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
```
```  5366     using cos_gt_zero_pi [OF 1 2] by (simp add: arctan tan_add)
```
```  5367 qed
```
```  5368
```
```  5369 lemma arctan_double: "\<bar>x\<bar> < 1 \<Longrightarrow> 2 * arctan x = arctan ((2 * x) / (1 - x\<^sup>2))"
```
```  5370   by (metis arctan_add linear mult_2 not_less power2_eq_square)
```
```  5371
```
```  5372 theorem machin: "pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)"
```
```  5373 proof -
```
```  5374   have "\<bar>1 / 5\<bar> < (1 :: real)"
```
```  5375     by auto
```
```  5376   from arctan_add[OF less_imp_le[OF this] this] have "2 * arctan (1 / 5) = arctan (5 / 12)"
```
```  5377     by auto
```
```  5378   moreover
```
```  5379   have "\<bar>5 / 12\<bar> < (1 :: real)"
```
```  5380     by auto
```
```  5381   from arctan_add[OF less_imp_le[OF this] this] have "2 * arctan (5 / 12) = arctan (120 / 119)"
```
```  5382     by auto
```
```  5383   moreover
```
```  5384   have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)"
```
```  5385     by auto
```
```  5386   from arctan_add[OF this] have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)"
```
```  5387     by auto
```
```  5388   ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)"
```
```  5389     by auto
```
```  5390   then show ?thesis
```
```  5391     unfolding arctan_one by algebra
```
```  5392 qed
```
```  5393
```
```  5394 lemma machin_Euler: "5 * arctan (1 / 7) + 2 * arctan (3 / 79) = pi / 4"
```
```  5395 proof -
```
```  5396   have 17: "\<bar>1 / 7\<bar> < (1 :: real)" by auto
```
```  5397   with arctan_double have "2 * arctan (1 / 7) = arctan (7 / 24)"
```
```  5398     by simp (simp add: field_simps)
```
```  5399   moreover
```
```  5400   have "\<bar>7 / 24\<bar> < (1 :: real)" by auto
```
```  5401   with arctan_double have "2 * arctan (7 / 24) = arctan (336 / 527)"
```
```  5402     by simp (simp add: field_simps)
```
```  5403   moreover
```
```  5404   have "\<bar>336 / 527\<bar> < (1 :: real)" by auto
```
```  5405   from arctan_add[OF less_imp_le[OF 17] this]
```
```  5406   have "arctan(1/7) + arctan (336 / 527) = arctan (2879 / 3353)"
```
```  5407     by auto
```
```  5408   ultimately have I: "5 * arctan (1 / 7) = arctan (2879 / 3353)" by auto
```
```  5409   have 379: "\<bar>3 / 79\<bar> < (1 :: real)" by auto
```
```  5410   with arctan_double have II: "2 * arctan (3 / 79) = arctan (237 / 3116)"
```
```  5411     by simp (simp add: field_simps)
```
```  5412   have *: "\<bar>2879 / 3353\<bar> < (1 :: real)" by auto
```
```  5413   have "\<bar>237 / 3116\<bar> < (1 :: real)" by auto
```
```  5414   from arctan_add[OF less_imp_le[OF *] this] have "arctan (2879/3353) + arctan (237/3116) = pi/4"
```
```  5415     by (simp add: arctan_one)
```
```  5416   with I II show ?thesis by auto
```
```  5417 qed
```
```  5418
```
```  5419 (*But could also prove MACHIN_GAUSS:
```
```  5420   12 * arctan(1/18) + 8 * arctan(1/57) - 5 * arctan(1/239) = pi/4*)
```
```  5421
```
```  5422
```
```  5423 subsection \<open>Introducing the inverse tangent power series\<close>
```
```  5424
```
```  5425 lemma monoseq_arctan_series:
```
```  5426   fixes x :: real
```
```  5427   assumes "\<bar>x\<bar> \<le> 1"
```
```  5428   shows "monoseq (\<lambda>n. 1 / real (n * 2 + 1) * x^(n * 2 + 1))"
```
```  5429     (is "monoseq ?a")
```
```  5430 proof (cases "x = 0")
```
```  5431   case True
```
```  5432   then show ?thesis by (auto simp: monoseq_def)
```
```  5433 next
```
```  5434   case False
```
```  5435   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x"
```
```  5436     using assms by auto
```
```  5437   show "monoseq ?a"
```
```  5438   proof -
```
```  5439     have mono: "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le>
```
```  5440         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
```
```  5441       if "0 \<le> x" and "x \<le> 1" for n and x :: real
```
```  5442     proof (rule mult_mono)
```
```  5443       show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))"
```
```  5444         by (rule frac_le) simp_all
```
```  5445       show "0 \<le> 1 / real (Suc (n * 2))"
```
```  5446         by auto
```
```  5447       show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)"
```
```  5448         by (rule power_decreasing) (simp_all add: \<open>0 \<le> x\<close> \<open>x \<le> 1\<close>)
```
```  5449       show "0 \<le> x ^ Suc (Suc n * 2)"
```
```  5450         by (rule zero_le_power) (simp add: \<open>0 \<le> x\<close>)
```
```  5451     qed
```
```  5452     show ?thesis
```
```  5453     proof (cases "0 \<le> x")
```
```  5454       case True
```
```  5455       from mono[OF this \<open>x \<le> 1\<close>, THEN allI]
```
```  5456       show ?thesis
```
```  5457         unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI2)
```
```  5458     next
```
```  5459       case False
```
```  5460       then have "0 \<le> - x" and "- x \<le> 1"
```
```  5461         using \<open>-1 \<le> x\<close> by auto
```
```  5462       from mono[OF this]
```
```  5463       have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge>
```
```  5464           1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" for n
```
```  5465         using \<open>0 \<le> -x\<close> by auto
```
```  5466       then show ?thesis
```
```  5467         unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
```
```  5468     qed
```
```  5469   qed
```
```  5470 qed
```
```  5471
```
```  5472 lemma zeroseq_arctan_series:
```
```  5473   fixes x :: real
```
```  5474   assumes "\<bar>x\<bar> \<le> 1"
```
```  5475   shows "(\<lambda>n. 1 / real (n * 2 + 1) * x^(n * 2 + 1)) \<longlonglongrightarrow> 0"
```
```  5476     (is "?a \<longlonglongrightarrow> 0")
```
```  5477 proof (cases "x = 0")
```
```  5478   case True
```
```  5479   then show ?thesis by simp
```
```  5480 next
```
```  5481   case False
```
```  5482   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x"
```
```  5483     using assms by auto
```
```  5484   show "?a \<longlonglongrightarrow> 0"
```
```  5485   proof (cases "\<bar>x\<bar> < 1")
```
```  5486     case True
```
```  5487     then have "norm x < 1" by auto
```
```  5488     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF \<open>norm x < 1\<close>, THEN LIMSEQ_Suc]]
```
```  5489     have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) \<longlonglongrightarrow> 0"
```
```  5490       unfolding inverse_eq_divide Suc_eq_plus1 by simp
```
```  5491     then show ?thesis
```
```  5492       using pos2 by (rule LIMSEQ_linear)
```
```  5493   next
```
```  5494     case False
```
```  5495     then have "x = -1 \<or> x = 1"
```
```  5496       using \<open>\<bar>x\<bar> \<le> 1\<close> by auto
```
```  5497     then have n_eq: "\<And> n. x ^ (n * 2 + 1) = x"
```
```  5498       unfolding One_nat_def by auto
```
```  5499     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
```
```  5500     show ?thesis
```
```  5501       unfolding n_eq Suc_eq_plus1 by auto
```
```  5502   qed
```
```  5503 qed
```
```  5504
```
```  5505 lemma summable_arctan_series:
```
```  5506   fixes n :: nat
```
```  5507   assumes "\<bar>x\<bar> \<le> 1"
```
```  5508   shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  5509     (is "summable (?c x)")
```
```  5510   by (rule summable_Leibniz(1),
```
```  5511       rule zeroseq_arctan_series[OF assms],
```
```  5512       rule monoseq_arctan_series[OF assms])
```
```  5513
```
```  5514 lemma DERIV_arctan_series:
```
```  5515   assumes "\<bar>x\<bar> < 1"
```
```  5516   shows "DERIV (\<lambda>x'. \<Sum>k. (-1)^k * (1 / real (k * 2 + 1) * x' ^ (k * 2 + 1))) x :>
```
```  5517       (\<Sum>k. (-1)^k * x^(k * 2))"
```
```  5518     (is "DERIV ?arctan _ :> ?Int")
```
```  5519 proof -
```
```  5520   let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
```
```  5521
```
```  5522   have n_even: "even n \<Longrightarrow> 2 * (n div 2) = n" for n :: nat
```
```  5523     by presburger
```
```  5524   then have if_eq: "?f n * real (Suc n) * x'^n =
```
```  5525       (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)"
```
```  5526     for n x'
```
```  5527     by auto
```
```  5528
```
```  5529   have summable_Integral: "summable (\<lambda> n. (- 1) ^ n * x^(2 * n))" if "\<bar>x\<bar> < 1" for x :: real
```
```  5530   proof -
```
```  5531     from that have "x\<^sup>2 < 1"
```
```  5532       by (simp add: abs_square_less_1)
```
```  5533     have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)"
```
```  5534       by (rule summable_Leibniz(1))
```
```  5535         (auto intro!: LIMSEQ_realpow_zero monoseq_realpow \<open>x\<^sup>2 < 1\<close> order_less_imp_le[OF \<open>x\<^sup>2 < 1\<close>])
```
```  5536     then show ?thesis
```
```  5537       by (simp only: power_mult)
```
```  5538   qed
```
```  5539
```
```  5540   have sums_even: "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)"
```
```  5541     for f :: "nat \<Rightarrow> real"
```
```  5542   proof -
```
```  5543     have "f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x" for x :: real
```
```  5544     proof
```
```  5545       assume "f sums x"
```
```  5546       from sums_if[OF sums_zero this] show "(\<lambda>n. if even n then f (n div 2) else 0) sums x"
```
```  5547         by auto
```
```  5548     next
```
```  5549       assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  5550       from LIMSEQ_linear[OF this[simplified sums_def] pos2, simplified sum_split_even_odd[simplified mult.commute]]
```
```  5551       show "f sums x"
```
```  5552         unfolding sums_def by auto
```
```  5553     qed
```
```  5554     then show ?thesis ..
```
```  5555   qed
```
```  5556
```
```  5557   have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int"
```
```  5558     unfolding if_eq mult.commute[of _ 2]
```
```  5559       suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * x ^ (2 * n)", symmetric]
```
```  5560     by auto
```
```  5561
```
```  5562   have arctan_eq: "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x" for x
```
```  5563   proof -
```
```  5564     have if_eq': "\<And>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
```
```  5565       (if even n then (- 1) ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
```
```  5566       using n_even by auto
```
```  5567     have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)"
```
```  5568       by auto
```
```  5569     then show ?thesis
```
```  5570       unfolding if_eq' idx_eq suminf_def
```
```  5571         sums_even[of "\<lambda> n. (- 1) ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
```
```  5572       by auto
```
```  5573   qed
```
```  5574
```
```  5575   have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum>n. ?f n * real (Suc n) * x^n)"
```
```  5576   proof (rule DERIV_power_series')
```
```  5577     show "x \<in> {- 1 <..< 1}"
```
```  5578       using \<open>\<bar> x \<bar> < 1\<close> by auto
```
```  5579     show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)"
```
```  5580       if x'_bounds: "x' \<in> {- 1 <..< 1}" for x' :: real
```
```  5581     proof -
```
```  5582       from that have "\<bar>x'\<bar> < 1" by auto
```
```  5583       then have *: "summable (\<lambda>n. (- 1) ^ n * x' ^ (2 * n))"
```
```  5584         by (rule summable_Integral)
```
```  5585       show ?thesis
```
```  5586         unfolding if_eq
```
```  5587         apply (rule sums_summable [where l="0 + (\<Sum>n. (-1)^n * x'^(2 * n))"])
```
```  5588         apply (rule sums_if)
```
```  5589          apply (rule sums_zero)
```
```  5590         apply (rule summable_sums)
```
```  5591         apply (rule *)
```
```  5592         done
```
```  5593     qed
```
```  5594   qed auto
```
```  5595   then show ?thesis
```
```  5596     by (simp only: Int_eq arctan_eq)
```
```  5597 qed
```
```  5598
```
```  5599 lemma arctan_series:
```
```  5600   assumes "\<bar>x\<bar> \<le> 1"
```
```  5601   shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))"
```
```  5602     (is "_ = suminf (\<lambda> n. ?c x n)")
```
```  5603 proof -
```
```  5604   let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)"
```
```  5605
```
```  5606   have DERIV_arctan_suminf: "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))"
```
```  5607     if "0 < r" and "r < 1" and "\<bar>x\<bar> < r" for r x :: real
```
```  5608   proof (rule DERIV_arctan_series)
```
```  5609     from that show "\<bar>x\<bar> < 1"
```
```  5610       using \<open>r < 1\<close> and \<open>\<bar>x\<bar> < r\<close> by auto
```
```  5611   qed
```
```  5612
```
```  5613   {
```
```  5614     fix x :: real
```
```  5615     assume "\<bar>x\<bar> \<le> 1"
```
```  5616     note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]]
```
```  5617   } note arctan_series_borders = this
```
```  5618
```
```  5619   have when_less_one: "arctan x = (\<Sum>k. ?c x k)" if "\<bar>x\<bar> < 1" for x :: real
```
```  5620   proof -
```
```  5621     obtain r where "\<bar>x\<bar> < r" and "r < 1"
```
```  5622       using dense[OF \<open>\<bar>x\<bar> < 1\<close>] by blast
```
```  5623     then have "0 < r" and "- r < x" and "x < r" by auto
```
```  5624
```
```  5625     have suminf_eq_arctan_bounded: "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  5626       if "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b" for x a b
```
```  5627     proof -
```
```  5628       from that have "\<bar>x\<bar> < r" by auto
```
```  5629       show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  5630       proof (rule DERIV_isconst2[of "a" "b"])
```
```  5631         show "a < b" and "a \<le> x" and "x \<le> b"
```
```  5632           using \<open>a < b\<close> \<open>a \<le> x\<close> \<open>x \<le> b\<close> by auto
```
```  5633         have "\<forall>x. - r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  5634         proof (rule allI, rule impI)
```
```  5635           fix x
```
```  5636           assume "-r < x \<and> x < r"
```
```  5637           then have "\<bar>x\<bar> < r" by auto
```
```  5638           with \<open>r < 1\<close> have "\<bar>x\<bar> < 1" by auto
```
```  5639           have "\<bar>- (x\<^sup>2)\<bar> < 1" using abs_square_less_1 \<open>\<bar>x\<bar> < 1\<close> by auto
```
```  5640           then have "(\<lambda>n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  5641             unfolding real_norm_def[symmetric] by (rule geometric_sums)
```
```  5642           then have "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  5643             unfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2] by auto
```
```  5644           then have suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)"
```
```  5645             using sums_unique unfolding inverse_eq_divide by auto
```
```  5646           have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))"
```
```  5647             unfolding suminf_c'_eq_geom
```
```  5648             by (rule DERIV_arctan_suminf[OF \<open>0 < r\<close> \<open>r < 1\<close> \<open>\<bar>x\<bar> < r\<close>])
```
```  5649           from DERIV_diff [OF this DERIV_arctan] show "DERIV (\<lambda>x. suminf (?c x) - arctan x) x :> 0"
```
```  5650             by auto
```
```  5651         qed
```
```  5652         then have DERIV_in_rball: "\<forall>y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda>x. suminf (?c x) - arctan x) y :> 0"
```
```  5653           using \<open>-r < a\<close> \<open>b < r\<close> by auto
```
```  5654         then show "\<forall>y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda>x. suminf (?c x) - arctan x) y :> 0"
```
```  5655           using \<open>\<bar>x\<bar> < r\<close> by auto
```
```  5656         show "\<forall>y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda>x. suminf (?c x) - arctan x) y"
```
```  5657           using DERIV_in_rball DERIV_isCont by auto
```
```  5658       qed
```
```  5659     qed
```
```  5660
```
```  5661     have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
```
```  5662       unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero
```
```  5663       by auto
```
```  5664
```
```  5665     have "suminf (?c x) - arctan x = 0"
```
```  5666     proof (cases "x = 0")
```
```  5667       case True
```
```  5668       then show ?thesis
```
```  5669         using suminf_arctan_zero by auto
```
```  5670     next
```
```  5671       case False
```
```  5672       then have "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>"
```
```  5673         by auto
```
```  5674       have "suminf (?c (- \<bar>x\<bar>)) - arctan (- \<bar>x\<bar>) = suminf (?c 0) - arctan 0"
```
```  5675         by (rule suminf_eq_arctan_bounded[where x1="0" and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>", symmetric])
```
```  5676           (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>-\<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less)
```
```  5677       moreover
```
```  5678       have "suminf (?c x) - arctan x = suminf (?c (- \<bar>x\<bar>)) - arctan (- \<bar>x\<bar>)"
```
```  5679         by (rule suminf_eq_arctan_bounded[where x1="x" and a1="- \<bar>x\<bar>" and b1="\<bar>x\<bar>"])
```
```  5680            (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>- \<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less)
```
```  5681       ultimately show ?thesis
```
```  5682         using suminf_arctan_zero by auto
```
```  5683     qed
```
```  5684     then show ?thesis by auto
```
```  5685   qed
```
```  5686
```
```  5687   show "arctan x = suminf (\<lambda>n. ?c x n)"
```
```  5688   proof (cases "\<bar>x\<bar> < 1")
```
```  5689     case True
```
```  5690     then show ?thesis by (rule when_less_one)
```
```  5691   next
```
```  5692     case False
```
```  5693     then have "\<bar>x\<bar> = 1" using \<open>\<bar>x\<bar> \<le> 1\<close> by auto
```
```  5694     let ?a = "\<lambda>x n. \<bar>1 / real (n * 2 + 1) * x^(n * 2 + 1)\<bar>"
```
```  5695     let ?diff = "\<lambda>x n. \<bar>arctan x - (\<Sum>i<n. ?c x i)\<bar>"
```
```  5696     have "?diff 1 n \<le> ?a 1 n" for n :: nat
```
```  5697     proof -
```
```  5698       have "0 < (1 :: real)" by auto
```
```  5699       moreover
```
```  5700       have "?diff x n \<le> ?a x n" if "0 < x" and "x < 1" for x :: real
```
```  5701       proof -
```
```  5702         from that have "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1"
```
```  5703           by auto
```
```  5704         from \<open>0 < x\<close> have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)"
```
```  5705           by auto
```
```  5706         note bounds = mp[OF arctan_series_borders(2)[OF \<open>\<bar>x\<bar> \<le> 1\<close>] this, unfolded when_less_one[OF \<open>\<bar>x\<bar> < 1\<close>, symmetric], THEN spec]
```
```  5707         have "0 < 1 / real (n*2+1) * x^(n*2+1)"
```
```  5708           by (rule mult_pos_pos) (simp_all only: zero_less_power[OF \<open>0 < x\<close>], auto)
```
```  5709         then have a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)"
```
```  5710           by (rule abs_of_pos)
```
```  5711         show ?thesis
```
```  5712         proof (cases "even n")
```
```  5713           case True
```
```  5714           then have sgn_pos: "(-1)^n = (1::real)" by auto
```
```  5715           from \<open>even n\<close> obtain m where "n = 2 * m" ..
```
```  5716           then have "2 * m = n" ..
```
```  5717           from bounds[of m, unfolded this atLeastAtMost_iff]
```
```  5718           have "\<bar>arctan x - (\<Sum>i<n. (?c x i))\<bar> \<le> (\<Sum>i<n + 1. (?c x i)) - (\<Sum>i<n. (?c x i))"
```
```  5719             by auto
```
```  5720           also have "\<dots> = ?c x n" by auto
```
```  5721           also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
```
```  5722           finally show ?thesis .
```
```  5723         next
```
```  5724           case False
```
```  5725           then have sgn_neg: "(-1)^n = (-1::real)" by auto
```
```  5726           from \<open>odd n\<close> obtain m where "n = 2 * m + 1" ..
```
```  5727           then have m_def: "2 * m + 1 = n" ..
```
```  5728           then have m_plus: "2 * (m + 1) = n + 1" by auto
```
```  5729           from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
```
```  5730           have "\<bar>arctan x - (\<Sum>i<n. (?c x i))\<bar> \<le> (\<Sum>i<n. (?c x i)) - (\<Sum>i<n+1. (?c x i))" by auto
```
```  5731           also have "\<dots> = - ?c x n" by auto
```
```  5732           also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
```
```  5733           finally show ?thesis .
```
```  5734         qed
```
```  5735       qed
```
```  5736       hence "\<forall>x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
```
```  5737       moreover have "isCont (\<lambda> x. ?a x n - ?diff x n) x" for x
```
```  5738         unfolding diff_conv_add_uminus divide_inverse
```
```  5739         by (auto intro!: isCont_add isCont_rabs continuous_ident isCont_minus isCont_arctan
```
```  5740           isCont_inverse isCont_mult isCont_power continuous_const isCont_sum
```
```  5741           simp del: add_uminus_conv_diff)
```
```  5742       ultimately have "0 \<le> ?a 1 n - ?diff 1 n"
```
```  5743         by (rule LIM_less_bound)
```
```  5744       then show ?thesis by auto
```
```  5745     qed
```
```  5746     have "?a 1 \<longlonglongrightarrow> 0"
```
```  5747       unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
```
```  5748       by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat simp del: of_nat_Suc)
```
```  5749     have "?diff 1 \<longlonglongrightarrow> 0"
```
```  5750     proof (rule LIMSEQ_I)
```
```  5751       fix r :: real
```
```  5752       assume "0 < r"
```
```  5753       obtain N :: nat where N_I: "N \<le> n \<Longrightarrow> ?a 1 n < r" for n
```
```  5754         using LIMSEQ_D[OF \<open>?a 1 \<longlonglongrightarrow> 0\<close> \<open>0 < r\<close>] by auto
```
```  5755       have "norm (?diff 1 n - 0) < r" if "N \<le> n" for n
```
```  5756         using \<open>?diff 1 n \<le> ?a 1 n\<close> N_I[OF that] by auto
```
```  5757       then show "\<exists>N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
```
```  5758     qed
```
```  5759     from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
```
```  5760     have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
```
```  5761     then have "arctan 1 = (\<Sum>i. ?c 1 i)" by (rule sums_unique)
```
```  5762
```
```  5763     show ?thesis
```
```  5764     proof (cases "x = 1")
```
```  5765       case True
```
```  5766       then show ?thesis by (simp add: \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close>)
```
```  5767     next
```
```  5768       case False
```
```  5769       then have "x = -1" using \<open>\<bar>x\<bar> = 1\<close> by auto
```
```  5770
```
```  5771       have "- (pi / 2) < 0" using pi_gt_zero by auto
```
```  5772       have "- (2 * pi) < 0" using pi_gt_zero by auto
```
```  5773
```
```  5774       have c_minus_minus: "?c (- 1) i = - ?c 1 i" for i by auto
```
```  5775
```
```  5776       have "arctan (- 1) = arctan (tan (-(pi / 4)))"
```
```  5777         unfolding tan_45 tan_minus ..
```
```  5778       also have "\<dots> = - (pi / 4)"
```
```  5779         by (rule arctan_tan) (auto simp: order_less_trans[OF \<open>- (pi / 2) < 0\<close> pi_gt_zero])
```
```  5780       also have "\<dots> = - (arctan (tan (pi / 4)))"
```
```  5781         unfolding neg_equal_iff_equal
```
```  5782         by (rule arctan_tan[symmetric]) (auto simp: order_less_trans[OF \<open>- (2 * pi) < 0\<close> pi_gt_zero])
```
```  5783       also have "\<dots> = - (arctan 1)"
```
```  5784         unfolding tan_45 ..
```
```  5785       also have "\<dots> = - (\<Sum> i. ?c 1 i)"
```
```  5786         using \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close> by auto
```
```  5787       also have "\<dots> = (\<Sum> i. ?c (- 1) i)"
```
```  5788         using suminf_minus[OF sums_summable[OF \<open>(?c 1) sums (arctan 1)\<close>]]
```
```  5789         unfolding c_minus_minus by auto
```
```  5790       finally show ?thesis using \<open>x = -1\<close> by auto
```
```  5791     qed
```
```  5792   qed
```
```  5793 qed
```
```  5794
```
```  5795 lemma arctan_half: "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))"
```
```  5796   for x :: real
```
```  5797 proof -
```
```  5798   obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x"
```
```  5799     using tan_total by blast
```
```  5800   then have low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2"
```
```  5801     by auto
```
```  5802
```
```  5803   have "0 < cos y" by (rule cos_gt_zero_pi[OF low high])
```
```  5804   then have "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y"
```
```  5805     by auto
```
```  5806
```
```  5807   have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  5808     unfolding tan_def power_divide ..
```
```  5809   also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  5810     using \<open>cos y \<noteq> 0\<close> by auto
```
```  5811   also have "\<dots> = 1 / (cos y)\<^sup>2"
```
```  5812     unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
```
```  5813   finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" .
```
```  5814
```
```  5815   have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)"
```
```  5816     unfolding tan_def using \<open>cos y \<noteq> 0\<close> by (simp add: field_simps)
```
```  5817   also have "\<dots> = tan y / (1 + 1 / cos y)"
```
```  5818     using \<open>cos y \<noteq> 0\<close> unfolding add_divide_distrib by auto
```
```  5819   also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))"
```
```  5820     unfolding cos_sqrt ..
```
```  5821   also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))"
```
```  5822     unfolding real_sqrt_divide by auto
```
```  5823   finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))"
```
```  5824     unfolding \<open>1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2\<close> .
```
```  5825
```
```  5826   have "arctan x = y"
```
```  5827     using arctan_tan low high y_eq by auto
```
```  5828   also have "\<dots> = 2 * (arctan (tan (y/2)))"
```
```  5829     using arctan_tan[OF low2 high2] by auto
```
```  5830   also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))"
```
```  5831     unfolding tan_half by auto
```
```  5832   finally show ?thesis
```
```  5833     unfolding eq \<open>tan y = x\<close> .
```
```  5834 qed
```
```  5835
```
```  5836 lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y"
```
```  5837   by (simp only: arctan_less_iff)
```
```  5838
```
```  5839 lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y"
```
```  5840   by (simp only: arctan_le_iff)
```
```  5841
```
```  5842 lemma arctan_inverse:
```
```  5843   assumes "x \<noteq> 0"
```
```  5844   shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
```
```  5845 proof (rule arctan_unique)
```
```  5846   show "- (pi / 2) < sgn x * pi / 2 - arctan x"
```
```  5847     using arctan_bounded [of x] assms
```
```  5848     unfolding sgn_real_def
```
```  5849     apply (auto simp add: arctan algebra_simps)
```
```  5850     apply (drule zero_less_arctan_iff [THEN iffD2])
```
```  5851     apply arith
```
```  5852     done
```
```  5853   show "sgn x * pi / 2 - arctan x < pi / 2"
```
```  5854     using arctan_bounded [of "- x"] assms
```
```  5855     unfolding sgn_real_def arctan_minus
```
```  5856     by (auto simp add: algebra_simps)
```
```  5857   show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
```
```  5858     unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
```
```  5859     unfolding sgn_real_def
```
```  5860     by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
```
```  5861 qed
```
```  5862
```
```  5863 theorem pi_series: "pi / 4 = (\<Sum>k. (-1)^k * 1 / real (k * 2 + 1))"
```
```  5864   (is "_ = ?SUM")
```
```  5865 proof -
```
```  5866   have "pi / 4 = arctan 1"
```
```  5867     using arctan_one by auto
```
```  5868   also have "\<dots> = ?SUM"
```
```  5869     using arctan_series[of 1] by auto
```
```  5870   finally show ?thesis by auto
```
```  5871 qed
```
```  5872
```
```  5873
```
```  5874 subsection \<open>Existence of Polar Coordinates\<close>
```
```  5875
```
```  5876 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1"
```
```  5877   by (rule power2_le_imp_le [OF _ zero_le_one])
```
```  5878     (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
```
```  5879
```
```  5880 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
```
```  5881
```
```  5882 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
```
```  5883
```
```  5884 lemma polar_Ex: "\<exists>r::real. \<exists>a. x = r * cos a \<and> y = r * sin a"
```
```  5885 proof -
```
```  5886   have polar_ex1: "0 < y \<Longrightarrow> \<exists>r a. x = r * cos a \<and> y = r * sin a" for y
```
```  5887     apply (rule exI [where x = "sqrt (x\<^sup>2 + y\<^sup>2)"])
```
```  5888     apply (rule exI [where x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))"])
```
```  5889     apply (simp add: cos_arccos_lemma1 sin_arccos_lemma1 power_divide
```
```  5890         real_sqrt_mult [symmetric] right_diff_distrib)
```
```  5891     done
```
```  5892   show ?thesis
```
```  5893   proof (cases "0::real" y rule: linorder_cases)
```
```  5894     case less
```
```  5895     then show ?thesis
```
```  5896       by (rule polar_ex1)
```
```  5897   next
```
```  5898     case equal
```
```  5899     then show ?thesis
```
```  5900       by (force simp add: intro!: cos_zero sin_zero)
```
```  5901   next
```
```  5902     case greater
```
```  5903     with polar_ex1 [where y="-y"] show ?thesis
```
```  5904       by auto (metis cos_minus minus_minus minus_mult_right sin_minus)
```
```  5905   qed
```
```  5906 qed
```
```  5907
```
```  5908
```
```  5909 subsection \<open>Basics about polynomial functions: products, extremal behaviour and root counts\<close>
```
```  5910
```
```  5911 lemma pairs_le_eq_Sigma: "{(i, j). i + j \<le> m} = Sigma (atMost m) (\<lambda>r. atMost (m - r))"
```
```  5912   for m :: nat
```
```  5913   by auto
```
```  5914
```
```  5915 lemma sum_up_index_split: "(\<Sum>k\<le>m + n. f k) = (\<Sum>k\<le>m. f k) + (\<Sum>k = Suc m..m + n. f k)"
```
```  5916   by (metis atLeast0AtMost Suc_eq_plus1 le0 sum_ub_add_nat)
```
```  5917
```
```  5918 lemma Sigma_interval_disjoint: "(SIGMA i:A. {..v i}) \<inter> (SIGMA i:A.{v i<..w}) = {}"
```
```  5919   for w :: "'a::order"
```
```  5920   by auto
```
```  5921
```
```  5922 lemma product_atMost_eq_Un: "A \<times> {..m} = (SIGMA i:A.{..m - i}) \<union> (SIGMA i:A.{m - i<..m})"
```
```  5923   for m :: nat
```
```  5924   by auto
```
```  5925
```
```  5926 lemma polynomial_product: (*with thanks to Chaitanya Mangla*)
```
```  5927   fixes x :: "'a::idom"
```
```  5928   assumes m: "\<And>i. i > m \<Longrightarrow> a i = 0"
```
```  5929     and n: "\<And>j. j > n \<Longrightarrow> b j = 0"
```
```  5930   shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
```
```  5931     (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
```
```  5932 proof -
```
```  5933   have "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) = (\<Sum>i\<le>m. \<Sum>j\<le>n. (a i * x ^ i) * (b j * x ^ j))"
```
```  5934     by (rule sum_product)
```
```  5935   also have "\<dots> = (\<Sum>i\<le>m + n. \<Sum>j\<le>n + m. a i * x ^ i * (b j * x ^ j))"
```
```  5936     using assms by (auto simp: sum_up_index_split)
```
```  5937   also have "\<dots> = (\<Sum>r\<le>m + n. \<Sum>j\<le>m + n - r. a r * x ^ r * (b j * x ^ j))"
```
```  5938     apply (simp add: add_ac sum.Sigma product_atMost_eq_Un)
```
```  5939     apply (clarsimp simp add: sum_Un Sigma_interval_disjoint intro!: sum.neutral)
```
```  5940     apply (metis add_diff_assoc2 add.commute add_lessD1 leD m n nat_le_linear neqE)
```
```  5941     done
```
```  5942   also have "\<dots> = (\<Sum>(i,j)\<in>{(i,j). i+j \<le> m+n}. (a i * x ^ i) * (b j * x ^ j))"
```
```  5943     by (auto simp: pairs_le_eq_Sigma sum.Sigma)
```
```  5944   also have "\<dots> = (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
```
```  5945     apply (subst sum_triangle_reindex_eq)
```
```  5946     apply (auto simp: algebra_simps sum_distrib_left intro!: sum.cong)
```
```  5947     apply (metis le_add_diff_inverse power_add)
```
```  5948     done
```
```  5949   finally show ?thesis .
```
```  5950 qed
```
```  5951
```
```  5952 lemma polynomial_product_nat:
```
```  5953   fixes x :: nat
```
```  5954   assumes m: "\<And>i. i > m \<Longrightarrow> a i = 0"
```
```  5955     and n: "\<And>j. j > n \<Longrightarrow> b j = 0"
```
```  5956   shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
```
```  5957     (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
```
```  5958   using polynomial_product [of m a n b x] assms
```
```  5959   by (simp only: of_nat_mult [symmetric] of_nat_power [symmetric]
```
```  5960       of_nat_eq_iff Int.int_sum [symmetric])
```
```  5961
```
```  5962 lemma polyfun_diff: (*COMPLEX_SUB_POLYFUN in HOL Light*)
```
```  5963   fixes x :: "'a::idom"
```
```  5964   assumes "1 \<le> n"
```
```  5965   shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
```
```  5966     (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)"
```
```  5967 proof -
```
```  5968   have h: "bij_betw (\<lambda>(i,j). (j,i)) ((SIGMA i : atMost n. lessThan i)) (SIGMA j : lessThan n. {Suc j..n})"
```
```  5969     by (auto simp: bij_betw_def inj_on_def)
```
```  5970   have "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) = (\<Sum>i\<le>n. a i * (x^i - y^i))"
```
```  5971     by (simp add: right_diff_distrib sum_subtractf)
```
```  5972   also have "\<dots> = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))"
```
```  5973     by (simp add: power_diff_sumr2 mult.assoc)
```
```  5974   also have "\<dots> = (\<Sum>i\<le>n. \<Sum>j<i. a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5975     by (simp add: sum_distrib_left)
```
```  5976   also have "\<dots> = (\<Sum>(i,j) \<in> (SIGMA i : atMost n. lessThan i). a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5977     by (simp add: sum.Sigma)
```
```  5978   also have "\<dots> = (\<Sum>(j,i) \<in> (SIGMA j : lessThan n. {Suc j..n}). a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5979     by (auto simp add: sum.reindex_bij_betw [OF h, symmetric] intro: sum.strong_cong)
```
```  5980   also have "\<dots> = (\<Sum>j<n. \<Sum>i=Suc j..n. a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5981     by (simp add: sum.Sigma)
```
```  5982   also have "\<dots> = (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)"
```
```  5983     by (simp add: sum_distrib_left mult_ac)
```
```  5984   finally show ?thesis .
```
```  5985 qed
```
```  5986
```
```  5987 lemma polyfun_diff_alt: (*COMPLEX_SUB_POLYFUN_ALT in HOL Light*)
```
```  5988   fixes x :: "'a::idom"
```
```  5989   assumes "1 \<le> n"
```
```  5990   shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
```
```  5991     (x - y) * ((\<Sum>j<n. \<Sum>k<n-j. a(j + k + 1) * y^k * x^j))"
```
```  5992 proof -
```
```  5993   have "(\<Sum>i=Suc j..n. a i * y^(i - j - 1)) = (\<Sum>k<n-j. a(j+k+1) * y^k)"
```
```  5994     if "j < n" for j :: nat
```
```  5995   proof -
```
```  5996     have h: "bij_betw (\<lambda>i. i - (j + 1)) {Suc j..n} (lessThan (n-j))"
```
```  5997       apply (auto simp: bij_betw_def inj_on_def)
```
```  5998       apply (rule_tac x="x + Suc j" in image_eqI)
```
```  5999        apply (auto simp: )
```
```  6000       done
```
```  6001     then show ?thesis
```
```  6002       by (auto simp add: sum.reindex_bij_betw [OF h, symmetric] intro: sum.strong_cong)
```
```  6003   qed
```
```  6004   then show ?thesis
```
```  6005     by (simp add: polyfun_diff [OF assms] sum_distrib_right)
```
```  6006 qed
```
```  6007
```
```  6008 lemma polyfun_linear_factor:  (*COMPLEX_POLYFUN_LINEAR_FACTOR in HOL Light*)
```
```  6009   fixes a :: "'a::idom"
```
```  6010   shows "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c(i) * z^i) = (z - a) * (\<Sum>i<n. b(i) * z^i) + (\<Sum>i\<le>n. c(i) * a^i)"
```
```  6011 proof (cases "n = 0")
```
```  6012   case True then show ?thesis
```
```  6013     by simp
```
```  6014 next
```
```  6015   case False
```
```  6016   have "(\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) = (z - a) * (\<Sum>i<n. b i * z^i) + (\<Sum>i\<le>n. c i * a^i)) \<longleftrightarrow>
```
```  6017         (\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) - (\<Sum>i\<le>n. c i * a^i) = (z - a) * (\<Sum>i<n. b i * z^i))"
```
```  6018     by (simp add: algebra_simps)
```
```  6019   also have "\<dots> \<longleftrightarrow>
```
```  6020     (\<exists>b. \<forall>z. (z - a) * (\<Sum>j<n. (\<Sum>i = Suc j..n. c i * a^(i - Suc j)) * z^j) =
```
```  6021       (z - a) * (\<Sum>i<n. b i * z^i))"
```
```  6022     using False by (simp add: polyfun_diff)
```
```  6023   also have "\<dots> = True" by auto
```
```  6024   finally show ?thesis
```
```  6025     by simp
```
```  6026 qed
```
```  6027
```
```  6028 lemma polyfun_linear_factor_root:  (*COMPLEX_POLYFUN_LINEAR_FACTOR_ROOT in HOL Light*)
```
```  6029   fixes a :: "'a::idom"
```
```  6030   assumes "(\<Sum>i\<le>n. c(i) * a^i) = 0"
```
```  6031   obtains b where "\<And>z. (\<Sum>i\<le>n. c i * z^i) = (z - a) * (\<Sum>i<n. b i * z^i)"
```
```  6032   using polyfun_linear_factor [of c n a] assms by auto
```
```  6033
```
```  6034 (*The material of this section, up until this point, could go into a new theory of polynomials
```
```  6035   based on Main alone. The remaining material involves limits, continuity, series, etc.*)
```
```  6036
```
```  6037 lemma isCont_polynom: "isCont (\<lambda>w. \<Sum>i\<le>n. c i * w^i) a"
```
```  6038   for c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
```
```  6039   by simp
```
```  6040
```
```  6041 lemma zero_polynom_imp_zero_coeffs:
```
```  6042   fixes c :: "nat \<Rightarrow> 'a::{ab_semigroup_mult,real_normed_div_algebra}"
```
```  6043   assumes "\<And>w. (\<Sum>i\<le>n. c i * w^i) = 0"  "k \<le> n"
```
```  6044   shows "c k = 0"
```
```  6045   using assms
```
```  6046 proof (induction n arbitrary: c k)
```
```  6047   case 0
```