src/HOL/Transcendental.thy
 author hoelzl Fri Feb 19 13:40:50 2016 +0100 (2016-02-19) changeset 62378 85ed00c1fe7c parent 62347 2230b7047376 child 62379 340738057c8c permissions -rw-r--r--
generalize more theorems to support enat and ennreal
```     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 Binomial Series Deriv NthRoot
```
```    11 begin
```
```    12
```
```    13 text \<open>A fact theorem on reals.\<close>
```
```    14
```
```    15 lemma square_fact_le_2_fact:
```
```    16   shows "fact n * fact n \<le> (fact (2 * n) :: real)"
```
```    17 proof (induct n)
```
```    18   case 0 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
```
```    32 lemma fact_in_Reals: "fact n \<in> \<real>"
```
```    33   by (induction n) auto
```
```    34
```
```    35 lemma of_real_fact [simp]: "of_real (fact n) = fact n"
```
```    36   by (metis of_nat_fact of_real_of_nat_eq)
```
```    37
```
```    38 lemma pochhammer_of_real: "pochhammer (of_real x) n = of_real (pochhammer x n)"
```
```    39   by (simp add: pochhammer_def)
```
```    40
```
```    41 lemma norm_fact [simp]:
```
```    42   "norm (fact n :: 'a :: {real_normed_algebra_1}) = fact n"
```
```    43 proof -
```
```    44   have "(fact n :: 'a) = of_real (fact n)" by simp
```
```    45   also have "norm \<dots> = fact n" 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   assumes "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>
```
```    60   have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
```
```    61     by (rule order_tendstoD)
```
```    62   then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially"
```
```    63     using eventually_ge_at_top
```
```    64   proof eventually_elim
```
```    65     fix n assume less: "root n (norm (f n)) < z" and n: "1 \<le> n"
```
```    66     from power_strict_mono[OF less, of n] n
```
```    67     show "norm (f n) \<le> z ^ n"
```
```    68       by simp
```
```    69   qed
```
```    70   then show "summable f"
```
```    71     unfolding eventually_sequentially
```
```    72     using z \<open>0 \<le> x\<close> by (auto intro!: summable_comparison_test[OF _  summable_geometric])
```
```    73 qed
```
```    74
```
```    75 subsection \<open>Properties of Power Series\<close>
```
```    76
```
```    77 lemma powser_zero:
```
```    78   fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra_1"
```
```    79   shows "(\<Sum>n. f n * 0 ^ n) = f 0"
```
```    80 proof -
```
```    81   have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
```
```    82     by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
```
```    83   thus ?thesis unfolding One_nat_def by simp
```
```    84 qed
```
```    85
```
```    86 lemma powser_sums_zero:
```
```    87   fixes a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
```
```    88   shows "(\<lambda>n. a n * 0^n) sums a 0"
```
```    89     using sums_finite [of "{0}" "\<lambda>n. a n * 0 ^ n"]
```
```    90     by simp
```
```    91
```
```    92 text\<open>Power series has a circle or radius of convergence: if it sums for @{term
```
```    93   x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.\<close>
```
```    94
```
```    95 lemma powser_insidea:
```
```    96   fixes x z :: "'a::real_normed_div_algebra"
```
```    97   assumes 1: "summable (\<lambda>n. f n * x^n)"
```
```    98     and 2: "norm z < norm x"
```
```    99   shows "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   100 proof -
```
```   101   from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
```
```   102   from 1 have "(\<lambda>n. f n * x^n) \<longlonglongrightarrow> 0"
```
```   103     by (rule summable_LIMSEQ_zero)
```
```   104   hence "convergent (\<lambda>n. f n * x^n)"
```
```   105     by (rule convergentI)
```
```   106   hence "Cauchy (\<lambda>n. f n * x^n)"
```
```   107     by (rule convergent_Cauchy)
```
```   108   hence "Bseq (\<lambda>n. f n * x^n)"
```
```   109     by (rule Cauchy_Bseq)
```
```   110   then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x^n) \<le> K"
```
```   111     by (simp add: Bseq_def, safe)
```
```   112   have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
```
```   113                    K * norm (z ^ n) * inverse (norm (x^n))"
```
```   114   proof (intro exI allI impI)
```
```   115     fix n::nat
```
```   116     assume "0 \<le> n"
```
```   117     have "norm (norm (f n * z ^ n)) * norm (x^n) =
```
```   118           norm (f n * x^n) * norm (z ^ n)"
```
```   119       by (simp add: norm_mult abs_mult)
```
```   120     also have "\<dots> \<le> K * norm (z ^ n)"
```
```   121       by (simp only: mult_right_mono 4 norm_ge_zero)
```
```   122     also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x^n)) * norm (x^n))"
```
```   123       by (simp add: x_neq_0)
```
```   124     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x^n)) * norm (x^n)"
```
```   125       by (simp only: mult.assoc)
```
```   126     finally show "norm (norm (f n * z ^ n)) \<le>
```
```   127                   K * norm (z ^ n) * inverse (norm (x^n))"
```
```   128       by (simp add: mult_le_cancel_right x_neq_0)
```
```   129   qed
```
```   130   moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
```
```   131   proof -
```
```   132     from 2 have "norm (norm (z * inverse x)) < 1"
```
```   133       using x_neq_0
```
```   134       by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
```
```   135     hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
```
```   136       by (rule summable_geometric)
```
```   137     hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
```
```   138       by (rule summable_mult)
```
```   139     thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
```
```   140       using x_neq_0
```
```   141       by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
```
```   142                     power_inverse norm_power mult.assoc)
```
```   143   qed
```
```   144   ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   145     by (rule summable_comparison_test)
```
```   146 qed
```
```   147
```
```   148 lemma powser_inside:
```
```   149   fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
```
```   150   shows
```
```   151     "summable (\<lambda>n. f n * (x^n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
```
```   152       summable (\<lambda>n. f n * (z ^ n))"
```
```   153   by (rule powser_insidea [THEN summable_norm_cancel])
```
```   154
```
```   155 lemma powser_times_n_limit_0:
```
```   156   fixes x :: "'a::{real_normed_div_algebra,banach}"
```
```   157   assumes "norm x < 1"
```
```   158     shows "(\<lambda>n. of_nat n * x ^ n) \<longlonglongrightarrow> 0"
```
```   159 proof -
```
```   160   have "norm x / (1 - norm x) \<ge> 0"
```
```   161     using assms
```
```   162     by (auto simp: divide_simps)
```
```   163   moreover obtain N where N: "norm x / (1 - norm x) < of_int N"
```
```   164     using ex_le_of_int
```
```   165     by (meson ex_less_of_int)
```
```   166   ultimately have N0: "N>0"
```
```   167     by auto
```
```   168   then have *: "real_of_int (N + 1) * norm x / real_of_int N < 1"
```
```   169     using N assms
```
```   170     by (auto simp: field_simps)
```
```   171   { fix n::nat
```
```   172     assume "N \<le> int n"
```
```   173     then have "real_of_int N * real_of_nat (Suc n) \<le> real_of_nat n * real_of_int (1 + N)"
```
```   174       by (simp add: algebra_simps)
```
```   175     then have "(real_of_int N * real_of_nat (Suc n)) * (norm x * norm (x ^ n))
```
```   176                \<le> (real_of_nat n *  (1 + N)) * (norm x * norm (x ^ n))"
```
```   177       using N0 mult_mono by fastforce
```
```   178     then have "real_of_int N * (norm x * (real_of_nat (Suc n) * norm (x ^ n)))
```
```   179          \<le> real_of_nat n * (norm x * ((1 + N) * norm (x ^ n)))"
```
```   180       by (simp add: algebra_simps)
```
```   181   } note ** = this
```
```   182   show ?thesis using *
```
```   183     apply (rule summable_LIMSEQ_zero [OF summable_ratio_test, where N1="nat N"])
```
```   184     apply (simp add: N0 norm_mult field_simps **
```
```   185                 del: of_nat_Suc of_int_add)
```
```   186     done
```
```   187 qed
```
```   188
```
```   189 corollary lim_n_over_pown:
```
```   190   fixes x :: "'a::{real_normed_field,banach}"
```
```   191   shows "1 < norm x \<Longrightarrow> ((\<lambda>n. of_nat n / x^n) \<longlongrightarrow> 0) sequentially"
```
```   192 using powser_times_n_limit_0 [of "inverse x"]
```
```   193 by (simp add: norm_divide divide_simps)
```
```   194
```
```   195 lemma sum_split_even_odd:
```
```   196   fixes f :: "nat \<Rightarrow> real"
```
```   197   shows
```
```   198     "(\<Sum>i<2 * n. if even i then f i else g i) =
```
```   199      (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
```
```   200 proof (induct n)
```
```   201   case 0
```
```   202   then show ?case by simp
```
```   203 next
```
```   204   case (Suc n)
```
```   205   have "(\<Sum>i<2 * Suc n. if even i then f i else g i) =
```
```   206     (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
```
```   207     using Suc.hyps unfolding One_nat_def by auto
```
```   208   also have "\<dots> = (\<Sum>i<Suc n. f (2 * i)) + (\<Sum>i<Suc n. g (2 * i + 1))"
```
```   209     by auto
```
```   210   finally show ?case .
```
```   211 qed
```
```   212
```
```   213 lemma sums_if':
```
```   214   fixes g :: "nat \<Rightarrow> real"
```
```   215   assumes "g sums x"
```
```   216   shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   217   unfolding sums_def
```
```   218 proof (rule LIMSEQ_I)
```
```   219   fix r :: real
```
```   220   assume "0 < r"
```
```   221   from \<open>g sums x\<close>[unfolded sums_def, THEN LIMSEQ_D, OF this]
```
```   222   obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g {..<n} - x) < r)" by blast
```
```   223
```
```   224   let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)"
```
```   225   {
```
```   226     fix m
```
```   227     assume "m \<ge> 2 * no"
```
```   228     hence "m div 2 \<ge> no" by auto
```
```   229     have sum_eq: "?SUM (2 * (m div 2)) = setsum g {..< m div 2}"
```
```   230       using sum_split_even_odd by auto
```
```   231     hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
```
```   232       using no_eq unfolding sum_eq using \<open>m div 2 \<ge> no\<close> by auto
```
```   233     moreover
```
```   234     have "?SUM (2 * (m div 2)) = ?SUM m"
```
```   235     proof (cases "even m")
```
```   236       case True
```
```   237       then show ?thesis by (auto simp add: even_two_times_div_two)
```
```   238     next
```
```   239       case False
```
```   240       then have eq: "Suc (2 * (m div 2)) = m" by simp
```
```   241       hence "even (2 * (m div 2))" using \<open>odd m\<close> by auto
```
```   242       have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
```
```   243       also have "\<dots> = ?SUM (2 * (m div 2))" using \<open>even (2 * (m div 2))\<close> by auto
```
```   244       finally show ?thesis by auto
```
```   245     qed
```
```   246     ultimately have "(norm (?SUM m - x) < r)" by auto
```
```   247   }
```
```   248   thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
```
```   249 qed
```
```   250
```
```   251 lemma sums_if:
```
```   252   fixes g :: "nat \<Rightarrow> real"
```
```   253   assumes "g sums x" and "f sums y"
```
```   254   shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
```
```   255 proof -
```
```   256   let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
```
```   257   {
```
```   258     fix B T E
```
```   259     have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
```
```   260       by (cases B) auto
```
```   261   } note if_sum = this
```
```   262   have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   263     using sums_if'[OF \<open>g sums x\<close>] .
```
```   264   {
```
```   265     have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
```
```   266
```
```   267     have "?s sums y" using sums_if'[OF \<open>f sums y\<close>] .
```
```   268     from this[unfolded sums_def, THEN LIMSEQ_Suc]
```
```   269     have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
```
```   270       by (simp add: lessThan_Suc_eq_insert_0 image_iff setsum.reindex if_eq sums_def cong del: if_cong)
```
```   271   }
```
```   272   from sums_add[OF g_sums this] show ?thesis unfolding if_sum .
```
```   273 qed
```
```   274
```
```   275 subsection \<open>Alternating series test / Leibniz formula\<close>
```
```   276 text\<open>FIXME: generalise these results from the reals via type classes?\<close>
```
```   277
```
```   278 lemma sums_alternating_upper_lower:
```
```   279   fixes a :: "nat \<Rightarrow> real"
```
```   280   assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a \<longlonglongrightarrow> 0"
```
```   281   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>
```
```   282              ((\<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)"
```
```   283   (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
```
```   284 proof (rule nested_sequence_unique)
```
```   285   have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
```
```   286
```
```   287   show "\<forall>n. ?f n \<le> ?f (Suc n)"
```
```   288   proof
```
```   289     fix n
```
```   290     show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
```
```   291   qed
```
```   292   show "\<forall>n. ?g (Suc n) \<le> ?g n"
```
```   293   proof
```
```   294     fix n
```
```   295     show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
```
```   296       unfolding One_nat_def by auto
```
```   297   qed
```
```   298   show "\<forall>n. ?f n \<le> ?g n"
```
```   299   proof
```
```   300     fix n
```
```   301     show "?f n \<le> ?g n" using fg_diff a_pos
```
```   302       unfolding One_nat_def by auto
```
```   303   qed
```
```   304   show "(\<lambda>n. ?f n - ?g n) \<longlonglongrightarrow> 0" unfolding fg_diff
```
```   305   proof (rule LIMSEQ_I)
```
```   306     fix r :: real
```
```   307     assume "0 < r"
```
```   308     with \<open>a \<longlonglongrightarrow> 0\<close>[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
```
```   309       by auto
```
```   310     hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   311     thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   312   qed
```
```   313 qed
```
```   314
```
```   315 lemma summable_Leibniz':
```
```   316   fixes a :: "nat \<Rightarrow> real"
```
```   317   assumes a_zero: "a \<longlonglongrightarrow> 0"
```
```   318     and a_pos: "\<And> n. 0 \<le> a n"
```
```   319     and a_monotone: "\<And> n. a (Suc n) \<le> a n"
```
```   320   shows summable: "summable (\<lambda> n. (-1)^n * a n)"
```
```   321     and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
```
```   322     and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)"
```
```   323     and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i<2*n+1. (-1)^i*a i)"
```
```   324     and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)"
```
```   325 proof -
```
```   326   let ?S = "\<lambda>n. (-1)^n * a n"
```
```   327   let ?P = "\<lambda>n. \<Sum>i<n. ?S i"
```
```   328   let ?f = "\<lambda>n. ?P (2 * n)"
```
```   329   let ?g = "\<lambda>n. ?P (2 * n + 1)"
```
```   330   obtain l :: real
```
```   331     where below_l: "\<forall> n. ?f n \<le> l"
```
```   332       and "?f \<longlonglongrightarrow> l"
```
```   333       and above_l: "\<forall> n. l \<le> ?g n"
```
```   334       and "?g \<longlonglongrightarrow> l"
```
```   335     using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
```
```   336
```
```   337   let ?Sa = "\<lambda>m. \<Sum>n<m. ?S n"
```
```   338   have "?Sa \<longlonglongrightarrow> l"
```
```   339   proof (rule LIMSEQ_I)
```
```   340     fix r :: real
```
```   341     assume "0 < r"
```
```   342     with \<open>?f \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
```
```   343     obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
```
```   344
```
```   345     from \<open>0 < r\<close> \<open>?g \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
```
```   346     obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
```
```   347
```
```   348     {
```
```   349       fix n :: nat
```
```   350       assume "n \<ge> (max (2 * f_no) (2 * g_no))"
```
```   351       hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
```
```   352       have "norm (?Sa n - l) < r"
```
```   353       proof (cases "even n")
```
```   354         case True
```
```   355         then have n_eq: "2 * (n div 2) = n" by (simp add: even_two_times_div_two)
```
```   356         with \<open>n \<ge> 2 * f_no\<close> have "n div 2 \<ge> f_no"
```
```   357           by auto
```
```   358         from f[OF this] show ?thesis
```
```   359           unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
```
```   360       next
```
```   361         case False
```
```   362         hence "even (n - 1)" by simp
```
```   363         then have n_eq: "2 * ((n - 1) div 2) = n - 1"
```
```   364           by (simp add: even_two_times_div_two)
```
```   365         hence range_eq: "n - 1 + 1 = n"
```
```   366           using odd_pos[OF False] by auto
```
```   367
```
```   368         from n_eq \<open>n \<ge> 2 * g_no\<close> have "(n - 1) div 2 \<ge> g_no"
```
```   369           by auto
```
```   370         from g[OF this] show ?thesis
```
```   371           unfolding n_eq range_eq .
```
```   372       qed
```
```   373     }
```
```   374     thus "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
```
```   375   qed
```
```   376   hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
```
```   377     unfolding sums_def .
```
```   378   thus "summable ?S" using summable_def by auto
```
```   379
```
```   380   have "l = suminf ?S" using sums_unique[OF sums_l] .
```
```   381
```
```   382   fix n
```
```   383   show "suminf ?S \<le> ?g n"
```
```   384     unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
```
```   385   show "?f n \<le> suminf ?S"
```
```   386     unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
```
```   387   show "?g \<longlonglongrightarrow> suminf ?S"
```
```   388     using \<open>?g \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto
```
```   389   show "?f \<longlonglongrightarrow> suminf ?S"
```
```   390     using \<open>?f \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto
```
```   391 qed
```
```   392
```
```   393 theorem summable_Leibniz:
```
```   394   fixes a :: "nat \<Rightarrow> real"
```
```   395   assumes a_zero: "a \<longlonglongrightarrow> 0" and "monoseq a"
```
```   396   shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
```
```   397     and "0 < a 0 \<longrightarrow>
```
```   398       (\<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")
```
```   399     and "a 0 < 0 \<longrightarrow>
```
```   400       (\<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")
```
```   401     and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?f")
```
```   402     and "(\<lambda>n. \<Sum>i<2*n+1. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?g")
```
```   403 proof -
```
```   404   have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
```
```   405   proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
```
```   406     case True
```
```   407     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n"
```
```   408       by auto
```
```   409     {
```
```   410       fix n
```
```   411       have "a (Suc n) \<le> a n"
```
```   412         using ord[where n="Suc n" and m=n] by auto
```
```   413     } note mono = this
```
```   414     note leibniz = summable_Leibniz'[OF \<open>a \<longlonglongrightarrow> 0\<close> ge0]
```
```   415     from leibniz[OF mono]
```
```   416     show ?thesis using \<open>0 \<le> a 0\<close> by auto
```
```   417   next
```
```   418     let ?a = "\<lambda> n. - a n"
```
```   419     case False
```
```   420     with monoseq_le[OF \<open>monoseq a\<close> \<open>a \<longlonglongrightarrow> 0\<close>]
```
```   421     have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
```
```   422     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
```
```   423       by auto
```
```   424     {
```
```   425       fix n
```
```   426       have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
```
```   427         by auto
```
```   428     } note monotone = this
```
```   429     note leibniz =
```
```   430       summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
```
```   431         OF tendsto_minus[OF \<open>a \<longlonglongrightarrow> 0\<close>, unfolded minus_zero] monotone]
```
```   432     have "summable (\<lambda> n. (-1)^n * ?a n)"
```
```   433       using leibniz(1) by auto
```
```   434     then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
```
```   435       unfolding summable_def by auto
```
```   436     from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
```
```   437       by auto
```
```   438     hence ?summable unfolding summable_def by auto
```
```   439     moreover
```
```   440     have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
```
```   441       unfolding minus_diff_minus by auto
```
```   442
```
```   443     from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
```
```   444     have move_minus: "(\<Sum>n. - ((- 1) ^ n * a n)) = - (\<Sum>n. (- 1) ^ n * a n)"
```
```   445       by auto
```
```   446
```
```   447     have ?pos using \<open>0 \<le> ?a 0\<close> by auto
```
```   448     moreover have ?neg
```
```   449       using leibniz(2,4)
```
```   450       unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
```
```   451       by auto
```
```   452     moreover have ?f and ?g
```
```   453       using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
```
```   454       by auto
```
```   455     ultimately show ?thesis by auto
```
```   456   qed
```
```   457   then show ?summable and ?pos and ?neg and ?f and ?g
```
```   458     by safe
```
```   459 qed
```
```   460
```
```   461 subsection \<open>Term-by-Term Differentiability of Power Series\<close>
```
```   462
```
```   463 definition diffs :: "(nat \<Rightarrow> 'a::ring_1) \<Rightarrow> nat \<Rightarrow> 'a"
```
```   464   where "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))"
```
```   465
```
```   466 text\<open>Lemma about distributing negation over it\<close>
```
```   467 lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
```
```   468   by (simp add: diffs_def)
```
```   469
```
```   470 lemma sums_Suc_imp:
```
```   471   "(f::nat \<Rightarrow> 'a::real_normed_vector) 0 = 0 \<Longrightarrow> (\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
```
```   472   using sums_Suc_iff[of f] by simp
```
```   473
```
```   474 lemma diffs_equiv:
```
```   475   fixes x :: "'a::{real_normed_vector, ring_1}"
```
```   476   shows "summable (\<lambda>n. diffs c n * x^n) \<Longrightarrow>
```
```   477       (\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)"
```
```   478   unfolding diffs_def
```
```   479   by (simp add: summable_sums sums_Suc_imp)
```
```   480
```
```   481 lemma lemma_termdiff1:
```
```   482   fixes z :: "'a :: {monoid_mult,comm_ring}" shows
```
```   483   "(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
```
```   484    (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
```
```   485   by (auto simp add: algebra_simps power_add [symmetric])
```
```   486
```
```   487 lemma sumr_diff_mult_const2:
```
```   488   "setsum f {..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i<n. f i - r)"
```
```   489   by (simp add: setsum_subtractf)
```
```   490
```
```   491 lemma lemma_realpow_rev_sumr:
```
```   492    "(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) =
```
```   493     (\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))"
```
```   494   by (subst nat_diff_setsum_reindex[symmetric]) simp
```
```   495
```
```   496 lemma lemma_termdiff2:
```
```   497   fixes h :: "'a :: {field}"
```
```   498   assumes h: "h \<noteq> 0"
```
```   499   shows
```
```   500     "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
```
```   501      h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p.
```
```   502           (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
```
```   503   apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
```
```   504   apply (simp add: right_diff_distrib diff_divide_distrib h)
```
```   505   apply (simp add: mult.assoc [symmetric])
```
```   506   apply (cases "n", simp)
```
```   507   apply (simp add: diff_power_eq_setsum h
```
```   508                    right_diff_distrib [symmetric] mult.assoc
```
```   509               del: power_Suc setsum_lessThan_Suc of_nat_Suc)
```
```   510   apply (subst lemma_realpow_rev_sumr)
```
```   511   apply (subst sumr_diff_mult_const2)
```
```   512   apply simp
```
```   513   apply (simp only: lemma_termdiff1 setsum_right_distrib)
```
```   514   apply (rule setsum.cong [OF refl])
```
```   515   apply (simp add: less_iff_Suc_add)
```
```   516   apply (clarify)
```
```   517   apply (simp add: setsum_right_distrib diff_power_eq_setsum ac_simps
```
```   518               del: setsum_lessThan_Suc power_Suc)
```
```   519   apply (subst mult.assoc [symmetric], subst power_add [symmetric])
```
```   520   apply (simp add: ac_simps)
```
```   521   done
```
```   522
```
```   523 lemma real_setsum_nat_ivl_bounded2:
```
```   524   fixes K :: "'a::linordered_semidom"
```
```   525   assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
```
```   526     and K: "0 \<le> K"
```
```   527   shows "setsum f {..<n-k} \<le> of_nat n * K"
```
```   528   apply (rule order_trans [OF setsum_mono])
```
```   529   apply (rule f, simp)
```
```   530   apply (simp add: mult_right_mono K)
```
```   531   done
```
```   532
```
```   533 lemma lemma_termdiff3:
```
```   534   fixes h z :: "'a::{real_normed_field}"
```
```   535   assumes 1: "h \<noteq> 0"
```
```   536     and 2: "norm z \<le> K"
```
```   537     and 3: "norm (z + h) \<le> K"
```
```   538   shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
```
```   539           \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   540 proof -
```
```   541   have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
```
```   542         norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p.
```
```   543           (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
```
```   544     by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult)
```
```   545   also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
```
```   546   proof (rule mult_right_mono [OF _ norm_ge_zero])
```
```   547     from norm_ge_zero 2 have K: "0 \<le> K"
```
```   548       by (rule order_trans)
```
```   549     have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
```
```   550       apply (erule subst)
```
```   551       apply (simp only: norm_mult norm_power power_add)
```
```   552       apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
```
```   553       done
```
```   554     show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))
```
```   555           \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
```
```   556       apply (intro
```
```   557          order_trans [OF norm_setsum]
```
```   558          real_setsum_nat_ivl_bounded2
```
```   559          mult_nonneg_nonneg
```
```   560          of_nat_0_le_iff
```
```   561          zero_le_power K)
```
```   562       apply (rule le_Kn, simp)
```
```   563       done
```
```   564   qed
```
```   565   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   566     by (simp only: mult.assoc)
```
```   567   finally show ?thesis .
```
```   568 qed
```
```   569
```
```   570 lemma lemma_termdiff4:
```
```   571   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```   572   assumes k: "0 < (k::real)"
```
```   573     and le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
```
```   574   shows "f \<midarrow>0\<rightarrow> 0"
```
```   575 proof (rule tendsto_norm_zero_cancel)
```
```   576   show "(\<lambda>h. norm (f h)) \<midarrow>0\<rightarrow> 0"
```
```   577   proof (rule real_tendsto_sandwich)
```
```   578     show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)"
```
```   579       by simp
```
```   580     show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)"
```
```   581       using k by (auto simp add: eventually_at dist_norm le)
```
```   582     show "(\<lambda>h. 0) \<midarrow>(0::'a)\<rightarrow> (0::real)"
```
```   583       by (rule tendsto_const)
```
```   584     have "(\<lambda>h. K * norm h) \<midarrow>(0::'a)\<rightarrow> K * norm (0::'a)"
```
```   585       by (intro tendsto_intros)
```
```   586     then show "(\<lambda>h. K * norm h) \<midarrow>(0::'a)\<rightarrow> 0"
```
```   587       by simp
```
```   588   qed
```
```   589 qed
```
```   590
```
```   591 lemma lemma_termdiff5:
```
```   592   fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
```
```   593   assumes k: "0 < (k::real)"
```
```   594   assumes f: "summable f"
```
```   595   assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
```
```   596   shows "(\<lambda>h. suminf (g h)) \<midarrow>0\<rightarrow> 0"
```
```   597 proof (rule lemma_termdiff4 [OF k])
```
```   598   fix h::'a
```
```   599   assume "h \<noteq> 0" and "norm h < k"
```
```   600   hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
```
```   601     by (simp add: le)
```
```   602   hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
```
```   603     by simp
```
```   604   moreover from f have B: "summable (\<lambda>n. f n * norm h)"
```
```   605     by (rule summable_mult2)
```
```   606   ultimately have C: "summable (\<lambda>n. norm (g h n))"
```
```   607     by (rule summable_comparison_test)
```
```   608   hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
```
```   609     by (rule summable_norm)
```
```   610   also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
```
```   611     by (rule suminf_le)
```
```   612   also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
```
```   613     by (rule suminf_mult2 [symmetric])
```
```   614   finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
```
```   615 qed
```
```   616
```
```   617
```
```   618 text\<open>FIXME: Long proofs\<close>
```
```   619
```
```   620 lemma termdiffs_aux:
```
```   621   fixes x :: "'a::{real_normed_field,banach}"
```
```   622   assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
```
```   623     and 2: "norm x < norm K"
```
```   624   shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h
```
```   625              - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0"
```
```   626 proof -
```
```   627   from dense [OF 2]
```
```   628   obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
```
```   629   from norm_ge_zero r1 have r: "0 < r"
```
```   630     by (rule order_le_less_trans)
```
```   631   hence r_neq_0: "r \<noteq> 0" by simp
```
```   632   show ?thesis
```
```   633   proof (rule lemma_termdiff5)
```
```   634     show "0 < r - norm x" using r1 by simp
```
```   635     from r r2 have "norm (of_real r::'a) < norm K"
```
```   636       by simp
```
```   637     with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
```
```   638       by (rule powser_insidea)
```
```   639     hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
```
```   640       using r
```
```   641       by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
```
```   642     hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
```
```   643       by (rule diffs_equiv [THEN sums_summable])
```
```   644     also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
```
```   645       (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
```
```   646       apply (rule ext)
```
```   647       apply (simp add: diffs_def)
```
```   648       apply (case_tac n, simp_all add: r_neq_0)
```
```   649       done
```
```   650     finally have "summable
```
```   651       (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
```
```   652       by (rule diffs_equiv [THEN sums_summable])
```
```   653     also have
```
```   654       "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
```
```   655            r ^ (n - Suc 0)) =
```
```   656        (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
```
```   657       apply (rule ext)
```
```   658       apply (case_tac "n", simp)
```
```   659       apply (rename_tac nat)
```
```   660       apply (case_tac "nat", simp)
```
```   661       apply (simp add: r_neq_0)
```
```   662       done
```
```   663     finally
```
```   664     show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
```
```   665   next
```
```   666     fix h::'a and n::nat
```
```   667     assume h: "h \<noteq> 0"
```
```   668     assume "norm h < r - norm x"
```
```   669     hence "norm x + norm h < r" by simp
```
```   670     with norm_triangle_ineq have xh: "norm (x + h) < r"
```
```   671       by (rule order_le_less_trans)
```
```   672     show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))
```
```   673           \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
```
```   674       apply (simp only: norm_mult mult.assoc)
```
```   675       apply (rule mult_left_mono [OF _ norm_ge_zero])
```
```   676       apply (simp add: mult.assoc [symmetric])
```
```   677       apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
```
```   678       done
```
```   679   qed
```
```   680 qed
```
```   681
```
```   682 lemma termdiffs:
```
```   683   fixes K x :: "'a::{real_normed_field,banach}"
```
```   684   assumes 1: "summable (\<lambda>n. c n * K ^ n)"
```
```   685       and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
```
```   686       and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
```
```   687       and 4: "norm x < norm K"
```
```   688   shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. (diffs c) n * x^n)"
```
```   689   unfolding DERIV_def
```
```   690 proof (rule LIM_zero_cancel)
```
```   691   show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x^n)) / h
```
```   692             - suminf (\<lambda>n. diffs c n * x^n)) \<midarrow>0\<rightarrow> 0"
```
```   693   proof (rule LIM_equal2)
```
```   694     show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
```
```   695   next
```
```   696     fix h :: 'a
```
```   697     assume "norm (h - 0) < norm K - norm x"
```
```   698     hence "norm x + norm h < norm K" by simp
```
```   699     hence 5: "norm (x + h) < norm K"
```
```   700       by (rule norm_triangle_ineq [THEN order_le_less_trans])
```
```   701     have "summable (\<lambda>n. c n * x^n)"
```
```   702       and "summable (\<lambda>n. c n * (x + h) ^ n)"
```
```   703       and "summable (\<lambda>n. diffs c n * x^n)"
```
```   704       using 1 2 4 5 by (auto elim: powser_inside)
```
```   705     then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
```
```   706           (\<Sum>n. (c n * (x + h) ^ n - c n * x^n) / h - of_nat n * c n * x ^ (n - Suc 0))"
```
```   707       by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
```
```   708     then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
```
```   709           (\<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))"
```
```   710       by (simp add: algebra_simps)
```
```   711   next
```
```   712     show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0"
```
```   713       by (rule termdiffs_aux [OF 3 4])
```
```   714   qed
```
```   715 qed
```
```   716
```
```   717 subsection \<open>The Derivative of a Power Series Has the Same Radius of Convergence\<close>
```
```   718
```
```   719 lemma termdiff_converges:
```
```   720   fixes x :: "'a::{real_normed_field,banach}"
```
```   721   assumes K: "norm x < K"
```
```   722       and sm: "\<And>x. norm x < K \<Longrightarrow> summable(\<lambda>n. c n * x ^ n)"
```
```   723     shows "summable (\<lambda>n. diffs c n * x ^ n)"
```
```   724 proof (cases "x = 0")
```
```   725   case True then show ?thesis
```
```   726   using powser_sums_zero sums_summable by auto
```
```   727 next
```
```   728   case False
```
```   729   then have "K>0"
```
```   730     using K less_trans zero_less_norm_iff by blast
```
```   731   then obtain r::real where r: "norm x < norm r" "norm r < K" "r>0"
```
```   732     using K False
```
```   733     by (auto simp: field_simps abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"])
```
```   734   have "(\<lambda>n. of_nat n * (x / of_real r) ^ n) \<longlonglongrightarrow> 0"
```
```   735     using r by (simp add: norm_divide powser_times_n_limit_0 [of "x / of_real r"])
```
```   736   then obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> real_of_nat n * norm x ^ n < r ^ n"
```
```   737     using r unfolding LIMSEQ_iff
```
```   738     apply (drule_tac x=1 in spec)
```
```   739     apply (auto simp: norm_divide norm_mult norm_power field_simps)
```
```   740     done
```
```   741   have "summable (\<lambda>n. (of_nat n * c n) * x ^ n)"
```
```   742     apply (rule summable_comparison_test' [of "\<lambda>n. norm(c n * (of_real r) ^ n)" N])
```
```   743     apply (rule powser_insidea [OF sm [of "of_real ((r+K)/2)"]])
```
```   744     using N r norm_of_real [of "r+K", where 'a = 'a]
```
```   745     apply (auto simp add: norm_divide norm_mult norm_power field_simps)
```
```   746     using less_eq_real_def by fastforce
```
```   747   then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ Suc n)"
```
```   748     using summable_iff_shift [of "\<lambda>n. of_nat n * c n * x ^ n" 1]
```
```   749     by simp
```
```   750   then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ n)"
```
```   751     using False summable_mult2 [of "\<lambda>n. (of_nat (Suc n) * c(Suc n) * x ^ n) * x" "inverse x"]
```
```   752     by (simp add: mult.assoc) (auto simp: ac_simps)
```
```   753   then show ?thesis
```
```   754     by (simp add: diffs_def)
```
```   755 qed
```
```   756
```
```   757 lemma termdiff_converges_all:
```
```   758   fixes x :: "'a::{real_normed_field,banach}"
```
```   759   assumes "\<And>x. summable (\<lambda>n. c n * x^n)"
```
```   760     shows "summable (\<lambda>n. diffs c n * x^n)"
```
```   761   apply (rule termdiff_converges [where K = "1 + norm x"])
```
```   762   using assms
```
```   763   apply auto
```
```   764   done
```
```   765
```
```   766 lemma termdiffs_strong:
```
```   767   fixes K x :: "'a::{real_normed_field,banach}"
```
```   768   assumes sm: "summable (\<lambda>n. c n * K ^ n)"
```
```   769       and K: "norm x < norm K"
```
```   770   shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. diffs c n * x^n)"
```
```   771 proof -
```
```   772   have K2: "norm ((of_real (norm K) + of_real (norm x)) / 2 :: 'a) < norm K"
```
```   773     using K
```
```   774     apply (auto simp: norm_divide field_simps)
```
```   775     apply (rule le_less_trans [of _ "of_real (norm K) + of_real (norm x)"])
```
```   776     apply (auto simp: mult_2_right norm_triangle_mono)
```
```   777     done
```
```   778   then have [simp]: "norm ((of_real (norm K) + of_real (norm x)) :: 'a) < norm K * 2"
```
```   779     by simp
```
```   780   have "summable (\<lambda>n. c n * (of_real (norm x + norm K) / 2) ^ n)"
```
```   781     by (metis K2 summable_norm_cancel [OF powser_insidea [OF sm]] add.commute of_real_add)
```
```   782   moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs c n * x ^ n)"
```
```   783     by (blast intro: sm termdiff_converges powser_inside)
```
```   784   moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs(diffs c) n * x ^ n)"
```
```   785     by (blast intro: sm termdiff_converges powser_inside)
```
```   786   ultimately show ?thesis
```
```   787     apply (rule termdiffs [where K = "of_real (norm x + norm K) / 2"])
```
```   788     apply (auto simp: field_simps)
```
```   789     using K
```
```   790     apply (simp_all add: of_real_add [symmetric] del: of_real_add)
```
```   791     done
```
```   792 qed
```
```   793
```
```   794 lemma termdiffs_strong_converges_everywhere:
```
```   795     fixes K x :: "'a::{real_normed_field,banach}"
```
```   796   assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)"
```
```   797   shows   "((\<lambda>x. \<Sum>n. c n * x^n) has_field_derivative (\<Sum>n. diffs c n * x^n)) (at x)"
```
```   798   using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
```
```   799   by (force simp del: of_real_add)
```
```   800
```
```   801 lemma isCont_powser:
```
```   802   fixes K x :: "'a::{real_normed_field,banach}"
```
```   803   assumes "summable (\<lambda>n. c n * K ^ n)"
```
```   804   assumes "norm x < norm K"
```
```   805   shows   "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
```
```   806   using termdiffs_strong[OF assms] by (blast intro!: DERIV_isCont)
```
```   807
```
```   808 lemmas isCont_powser' = isCont_o2[OF _ isCont_powser]
```
```   809
```
```   810 lemma isCont_powser_converges_everywhere:
```
```   811   fixes K x :: "'a::{real_normed_field,banach}"
```
```   812   assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)"
```
```   813   shows   "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
```
```   814   using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
```
```   815   by (force intro!: DERIV_isCont simp del: of_real_add)
```
```   816
```
```   817 lemma powser_limit_0:
```
```   818   fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
```
```   819   assumes s: "0 < s"
```
```   820       and sm: "\<And>x. norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
```
```   821     shows "(f \<longlongrightarrow> a 0) (at 0)"
```
```   822 proof -
```
```   823   have "summable (\<lambda>n. a n * (of_real s / 2) ^ n)"
```
```   824     apply (rule sums_summable [where l = "f (of_real s / 2)", OF sm])
```
```   825     using s
```
```   826     apply (auto simp: norm_divide)
```
```   827     done
```
```   828   then have "((\<lambda>x. \<Sum>n. a n * x ^ n) has_field_derivative (\<Sum>n. diffs a n * 0 ^ n)) (at 0)"
```
```   829     apply (rule termdiffs_strong)
```
```   830     using s
```
```   831     apply (auto simp: norm_divide)
```
```   832     done
```
```   833   then have "isCont (\<lambda>x. \<Sum>n. a n * x ^ n) 0"
```
```   834     by (blast intro: DERIV_continuous)
```
```   835   then have "((\<lambda>x. \<Sum>n. a n * x ^ n) \<longlongrightarrow> a 0) (at 0)"
```
```   836     by (simp add: continuous_within powser_zero)
```
```   837   then show ?thesis
```
```   838     apply (rule Lim_transform)
```
```   839     apply (auto simp add: LIM_eq)
```
```   840     apply (rule_tac x="s" in exI)
```
```   841     using s
```
```   842     apply (auto simp: sm [THEN sums_unique])
```
```   843     done
```
```   844 qed
```
```   845
```
```   846 lemma powser_limit_0_strong:
```
```   847   fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
```
```   848   assumes s: "0 < s"
```
```   849       and sm: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
```
```   850     shows "(f \<longlongrightarrow> a 0) (at 0)"
```
```   851 proof -
```
```   852   have *: "((\<lambda>x. if x = 0 then a 0 else f x) \<longlongrightarrow> a 0) (at 0)"
```
```   853     apply (rule powser_limit_0 [OF s])
```
```   854     apply (case_tac "x=0")
```
```   855     apply (auto simp add: powser_sums_zero sm)
```
```   856     done
```
```   857   show ?thesis
```
```   858     apply (subst LIM_equal [where g = "(\<lambda>x. if x = 0 then a 0 else f x)"])
```
```   859     apply (simp_all add: *)
```
```   860     done
```
```   861 qed
```
```   862
```
```   863
```
```   864 subsection \<open>Derivability of power series\<close>
```
```   865
```
```   866 lemma DERIV_series':
```
```   867   fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
```
```   868   assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
```
```   869     and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
```
```   870     and "summable (f' x0)"
```
```   871     and "summable L"
```
```   872     and L_def: "\<And>n x y. \<lbrakk> x \<in> { a <..< b} ; y \<in> { a <..< b} \<rbrakk> \<Longrightarrow> \<bar>f x n - f y n\<bar> \<le> L n * \<bar>x - y\<bar>"
```
```   873   shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
```
```   874   unfolding DERIV_def
```
```   875 proof (rule LIM_I)
```
```   876   fix r :: real
```
```   877   assume "0 < r" hence "0 < r/3" by auto
```
```   878
```
```   879   obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
```
```   880     using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable L\<close>] by auto
```
```   881
```
```   882   obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
```
```   883     using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable (f' x0)\<close>] by auto
```
```   884
```
```   885   let ?N = "Suc (max N_L N_f')"
```
```   886   have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
```
```   887     L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
```
```   888
```
```   889   let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
```
```   890
```
```   891   let ?r = "r / (3 * real ?N)"
```
```   892   from \<open>0 < r\<close> have "0 < ?r" by simp
```
```   893
```
```   894   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)"
```
```   895   def S' \<equiv> "Min (?s ` {..< ?N })"
```
```   896
```
```   897   have "0 < S'" unfolding S'_def
```
```   898   proof (rule iffD2[OF Min_gr_iff])
```
```   899     show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
```
```   900     proof
```
```   901       fix x
```
```   902       assume "x \<in> ?s ` {..<?N}"
```
```   903       then obtain n where "x = ?s n" and "n \<in> {..<?N}"
```
```   904         using image_iff[THEN iffD1] by blast
```
```   905       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>, unfolded real_norm_def]
```
```   906       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)"
```
```   907         by auto
```
```   908       have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound simp del: of_nat_Suc)
```
```   909       thus "0 < x" unfolding \<open>x = ?s n\<close> .
```
```   910     qed
```
```   911   qed auto
```
```   912
```
```   913   def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
```
```   914   hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
```
```   915     and "S \<le> S'" using x0_in_I and \<open>0 < S'\<close>
```
```   916     by auto
```
```   917
```
```   918   {
```
```   919     fix x
```
```   920     assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
```
```   921     hence x_in_I: "x0 + x \<in> { a <..< b }"
```
```   922       using S_a S_b by auto
```
```   923
```
```   924     note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   925     note div_smbl = summable_divide[OF diff_smbl]
```
```   926     note all_smbl = summable_diff[OF div_smbl \<open>summable (f' x0)\<close>]
```
```   927     note ign = summable_ignore_initial_segment[where k="?N"]
```
```   928     note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
```
```   929     note div_shft_smbl = summable_divide[OF diff_shft_smbl]
```
```   930     note all_shft_smbl = summable_diff[OF div_smbl ign[OF \<open>summable (f' x0)\<close>]]
```
```   931
```
```   932     { fix n
```
```   933       have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
```
```   934         using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
```
```   935         unfolding abs_divide .
```
```   936       hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)"
```
```   937         using \<open>x \<noteq> 0\<close> by auto }
```
```   938     note 1 = this and 2 = summable_rabs_comparison_test[OF _ ign[OF \<open>summable L\<close>]]
```
```   939     then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))"
```
```   940       by (metis (lifting) abs_idempotent order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF \<open>summable L\<close>]]])
```
```   941     then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3")
```
```   942       using L_estimate by auto
```
```   943
```
```   944     have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n \<bar>)" ..
```
```   945     also have "\<dots> < (\<Sum>n<?N. ?r)"
```
```   946     proof (rule setsum_strict_mono)
```
```   947       fix n
```
```   948       assume "n \<in> {..< ?N}"
```
```   949       have "\<bar>x\<bar> < S" using \<open>\<bar>x\<bar> < S\<close> .
```
```   950       also have "S \<le> S'" using \<open>S \<le> S'\<close> .
```
```   951       also have "S' \<le> ?s n" unfolding S'_def
```
```   952       proof (rule Min_le_iff[THEN iffD2])
```
```   953         have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n"
```
```   954           using \<open>n \<in> {..< ?N}\<close> by auto
```
```   955         thus "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n" by blast
```
```   956       qed auto
```
```   957       finally have "\<bar>x\<bar> < ?s n" .
```
```   958
```
```   959       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>, unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
```
```   960       have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
```
```   961       with \<open>x \<noteq> 0\<close> and \<open>\<bar>x\<bar> < ?s n\<close> show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
```
```   962         by blast
```
```   963     qed auto
```
```   964     also have "\<dots> = of_nat (card {..<?N}) * ?r"
```
```   965       by (rule setsum_constant)
```
```   966     also have "\<dots> = real ?N * ?r" by simp
```
```   967     also have "\<dots> = r/3" by (auto simp del: of_nat_Suc)
```
```   968     finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
```
```   969
```
```   970     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   971     have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
```
```   972         \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
```
```   973       unfolding suminf_diff[OF div_smbl \<open>summable (f' x0)\<close>, symmetric]
```
```   974       using suminf_divide[OF diff_smbl, symmetric] by auto
```
```   975     also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>"
```
```   976       unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
```
```   977       unfolding suminf_diff[OF div_shft_smbl ign[OF \<open>summable (f' x0)\<close>]]
```
```   978       apply (subst (5) add.commute)
```
```   979       by (rule abs_triangle_ineq)
```
```   980     also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
```
```   981       using abs_triangle_ineq4 by auto
```
```   982     also have "\<dots> < r /3 + r/3 + r/3"
```
```   983       using \<open>?diff_part < r/3\<close> \<open>?L_part \<le> r/3\<close> and \<open>?f'_part < r/3\<close>
```
```   984       by (rule add_strict_mono [OF add_less_le_mono])
```
```   985     finally have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
```
```   986       by auto
```
```   987   }
```
```   988   thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
```
```   989       norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
```
```   990     using \<open>0 < S\<close> unfolding real_norm_def diff_0_right by blast
```
```   991 qed
```
```   992
```
```   993 lemma DERIV_power_series':
```
```   994   fixes f :: "nat \<Rightarrow> real"
```
```   995   assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
```
```   996     and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
```
```   997   shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
```
```   998   (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
```
```   999 proof -
```
```  1000   {
```
```  1001     fix R'
```
```  1002     assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
```
```  1003     hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
```
```  1004       by auto
```
```  1005     have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
```
```  1006     proof (rule DERIV_series')
```
```  1007       show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
```
```  1008       proof -
```
```  1009         have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
```
```  1010           using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps)
```
```  1011         hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
```
```  1012           using \<open>R' < R\<close> by auto
```
```  1013         have "norm R' < norm ((R' + R) / 2)"
```
```  1014           using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps)
```
```  1015         from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
```
```  1016           by auto
```
```  1017       qed
```
```  1018       {
```
```  1019         fix n x y
```
```  1020         assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
```
```  1021         show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
```
```  1022         proof -
```
```  1023           have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
```
```  1024             (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>"
```
```  1025             unfolding right_diff_distrib[symmetric] diff_power_eq_setsum abs_mult
```
```  1026             by auto
```
```  1027           also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
```
```  1028           proof (rule mult_left_mono)
```
```  1029             have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
```
```  1030               by (rule setsum_abs)
```
```  1031             also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)"
```
```  1032             proof (rule setsum_mono)
```
```  1033               fix p
```
```  1034               assume "p \<in> {..<Suc n}"
```
```  1035               hence "p \<le> n" by auto
```
```  1036               {
```
```  1037                 fix n
```
```  1038                 fix x :: real
```
```  1039                 assume "x \<in> {-R'<..<R'}"
```
```  1040                 hence "\<bar>x\<bar> \<le> R'"  by auto
```
```  1041                 hence "\<bar>x^n\<bar> \<le> R'^n"
```
```  1042                   unfolding power_abs by (rule power_mono, auto)
```
```  1043               }
```
```  1044               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"]] \<open>0 < R'\<close>
```
```  1045               have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
```
```  1046                 unfolding abs_mult by auto
```
```  1047               thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
```
```  1048                 unfolding power_add[symmetric] using \<open>p \<le> n\<close> by auto
```
```  1049             qed
```
```  1050             also have "\<dots> = real (Suc n) * R' ^ n"
```
```  1051               unfolding setsum_constant card_atLeastLessThan by auto
```
```  1052             finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
```
```  1053               unfolding  abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF \<open>0 < R'\<close>]]]
```
```  1054               by linarith
```
```  1055             show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
```
```  1056               unfolding abs_mult[symmetric] by auto
```
```  1057           qed
```
```  1058           also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
```
```  1059             unfolding abs_mult mult.assoc[symmetric] by algebra
```
```  1060           finally show ?thesis .
```
```  1061         qed
```
```  1062       }
```
```  1063       {
```
```  1064         fix n
```
```  1065         show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
```
```  1066           by (auto intro!: derivative_eq_intros simp del: power_Suc)
```
```  1067       }
```
```  1068       {
```
```  1069         fix x
```
```  1070         assume "x \<in> {-R' <..< R'}"
```
```  1071         hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
```
```  1072           using assms \<open>R' < R\<close> by auto
```
```  1073         have "summable (\<lambda> n. f n * x^n)"
```
```  1074         proof (rule summable_comparison_test, intro exI allI impI)
```
```  1075           fix n
```
```  1076           have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
```
```  1077             by (rule mult_left_mono) auto
```
```  1078           show "norm (f n * x^n) \<le> norm (f n * real (Suc n) * x^n)"
```
```  1079             unfolding real_norm_def abs_mult
```
```  1080             using le mult_right_mono by fastforce
```
```  1081         qed (rule powser_insidea[OF converges[OF \<open>R' \<in> {-R <..< R}\<close>] \<open>norm x < norm R'\<close>])
```
```  1082         from this[THEN summable_mult2[where c=x], unfolded mult.assoc, unfolded mult.commute]
```
```  1083         show "summable (?f x)" by auto
```
```  1084       }
```
```  1085       show "summable (?f' x0)"
```
```  1086         using converges[OF \<open>x0 \<in> {-R <..< R}\<close>] .
```
```  1087       show "x0 \<in> {-R' <..< R'}"
```
```  1088         using \<open>x0 \<in> {-R' <..< R'}\<close> .
```
```  1089     qed
```
```  1090   } note for_subinterval = this
```
```  1091   let ?R = "(R + \<bar>x0\<bar>) / 2"
```
```  1092   have "\<bar>x0\<bar> < ?R" using assms by (auto simp: field_simps)
```
```  1093   hence "- ?R < x0"
```
```  1094   proof (cases "x0 < 0")
```
```  1095     case True
```
```  1096     hence "- x0 < ?R" using \<open>\<bar>x0\<bar> < ?R\<close> by auto
```
```  1097     thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
```
```  1098   next
```
```  1099     case False
```
```  1100     have "- ?R < 0" using assms by auto
```
```  1101     also have "\<dots> \<le> x0" using False by auto
```
```  1102     finally show ?thesis .
```
```  1103   qed
```
```  1104   hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
```
```  1105     using assms by (auto simp: field_simps)
```
```  1106   from for_subinterval[OF this]
```
```  1107   show ?thesis .
```
```  1108 qed
```
```  1109
```
```  1110
```
```  1111 lemma isCont_pochhammer [continuous_intros]: "isCont (\<lambda>z::'a::real_normed_field. pochhammer z n) z"
```
```  1112   by (induction n) (auto intro!: continuous_intros simp: pochhammer_rec')
```
```  1113
```
```  1114 lemma continuous_on_pochhammer [continuous_intros]:
```
```  1115   fixes A :: "'a :: real_normed_field set"
```
```  1116   shows "continuous_on A (\<lambda>z. pochhammer z n)"
```
```  1117   by (intro continuous_at_imp_continuous_on ballI isCont_pochhammer)
```
```  1118
```
```  1119
```
```  1120 subsection \<open>Exponential Function\<close>
```
```  1121
```
```  1122 definition exp :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```  1123   where "exp = (\<lambda>x. \<Sum>n. x^n /\<^sub>R fact n)"
```
```  1124
```
```  1125 lemma summable_exp_generic:
```
```  1126   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  1127   defines S_def: "S \<equiv> \<lambda>n. x^n /\<^sub>R fact n"
```
```  1128   shows "summable S"
```
```  1129 proof -
```
```  1130   have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R (Suc n)"
```
```  1131     unfolding S_def by (simp del: mult_Suc)
```
```  1132   obtain r :: real where r0: "0 < r" and r1: "r < 1"
```
```  1133     using dense [OF zero_less_one] by fast
```
```  1134   obtain N :: nat where N: "norm x < real N * r"
```
```  1135     using ex_less_of_nat_mult r0 by auto
```
```  1136   from r1 show ?thesis
```
```  1137   proof (rule summable_ratio_test [rule_format])
```
```  1138     fix n :: nat
```
```  1139     assume n: "N \<le> n"
```
```  1140     have "norm x \<le> real N * r"
```
```  1141       using N by (rule order_less_imp_le)
```
```  1142     also have "real N * r \<le> real (Suc n) * r"
```
```  1143       using r0 n by (simp add: mult_right_mono)
```
```  1144     finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
```
```  1145       using norm_ge_zero by (rule mult_right_mono)
```
```  1146     hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
```
```  1147       by (rule order_trans [OF norm_mult_ineq])
```
```  1148     hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
```
```  1149       by (simp add: pos_divide_le_eq ac_simps)
```
```  1150     thus "norm (S (Suc n)) \<le> r * norm (S n)"
```
```  1151       by (simp add: S_Suc inverse_eq_divide)
```
```  1152   qed
```
```  1153 qed
```
```  1154
```
```  1155 lemma summable_norm_exp:
```
```  1156   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  1157   shows "summable (\<lambda>n. norm (x^n /\<^sub>R fact n))"
```
```  1158 proof (rule summable_norm_comparison_test [OF exI, rule_format])
```
```  1159   show "summable (\<lambda>n. norm x^n /\<^sub>R fact n)"
```
```  1160     by (rule summable_exp_generic)
```
```  1161   fix n
```
```  1162   show "norm (x^n /\<^sub>R fact n) \<le> norm x^n /\<^sub>R fact n"
```
```  1163     by (simp add: norm_power_ineq)
```
```  1164 qed
```
```  1165
```
```  1166 lemma summable_exp:
```
```  1167   fixes x :: "'a::{real_normed_field,banach}"
```
```  1168   shows "summable (\<lambda>n. inverse (fact n) * x^n)"
```
```  1169   using summable_exp_generic [where x=x]
```
```  1170   by (simp add: scaleR_conv_of_real nonzero_of_real_inverse)
```
```  1171
```
```  1172 lemma exp_converges: "(\<lambda>n. x^n /\<^sub>R fact n) sums exp x"
```
```  1173   unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
```
```  1174
```
```  1175 lemma exp_fdiffs:
```
```  1176   "diffs (\<lambda>n. inverse (fact n)) = (\<lambda>n. inverse (fact n :: 'a::{real_normed_field,banach}))"
```
```  1177   by (simp add: diffs_def mult_ac nonzero_inverse_mult_distrib nonzero_of_real_inverse
```
```  1178            del: mult_Suc of_nat_Suc)
```
```  1179
```
```  1180 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
```
```  1181   by (simp add: diffs_def)
```
```  1182
```
```  1183 lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
```
```  1184   unfolding exp_def scaleR_conv_of_real
```
```  1185   apply (rule DERIV_cong)
```
```  1186   apply (rule termdiffs [where K="of_real (1 + norm x)"])
```
```  1187   apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
```
```  1188   apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
```
```  1189   apply (simp del: of_real_add)
```
```  1190   done
```
```  1191
```
```  1192 declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
```
```  1193         DERIV_exp[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  1194
```
```  1195 lemma norm_exp: "norm (exp x) \<le> exp (norm x)"
```
```  1196 proof -
```
```  1197   from summable_norm[OF summable_norm_exp, of x]
```
```  1198   have "norm (exp x) \<le> (\<Sum>n. inverse (fact n) * norm (x^n))"
```
```  1199     by (simp add: exp_def)
```
```  1200   also have "\<dots> \<le> exp (norm x)"
```
```  1201     using summable_exp_generic[of "norm x"] summable_norm_exp[of x]
```
```  1202     by (auto simp: exp_def intro!: suminf_le norm_power_ineq)
```
```  1203   finally show ?thesis .
```
```  1204 qed
```
```  1205
```
```  1206 lemma isCont_exp:
```
```  1207   fixes x::"'a::{real_normed_field,banach}"
```
```  1208   shows "isCont exp x"
```
```  1209   by (rule DERIV_exp [THEN DERIV_isCont])
```
```  1210
```
```  1211 lemma isCont_exp' [simp]:
```
```  1212   fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1213   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
```
```  1214   by (rule isCont_o2 [OF _ isCont_exp])
```
```  1215
```
```  1216 lemma tendsto_exp [tendsto_intros]:
```
```  1217   fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1218   shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) \<longlongrightarrow> exp a) F"
```
```  1219   by (rule isCont_tendsto_compose [OF isCont_exp])
```
```  1220
```
```  1221 lemma continuous_exp [continuous_intros]:
```
```  1222   fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1223   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
```
```  1224   unfolding continuous_def by (rule tendsto_exp)
```
```  1225
```
```  1226 lemma continuous_on_exp [continuous_intros]:
```
```  1227   fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1228   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
```
```  1229   unfolding continuous_on_def by (auto intro: tendsto_exp)
```
```  1230
```
```  1231
```
```  1232 subsubsection \<open>Properties of the Exponential Function\<close>
```
```  1233
```
```  1234 lemma exp_zero [simp]: "exp 0 = 1"
```
```  1235   unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
```
```  1236
```
```  1237 lemma exp_series_add_commuting:
```
```  1238   fixes x y :: "'a::{real_normed_algebra_1, banach}"
```
```  1239   defines S_def: "S \<equiv> \<lambda>x n. x^n /\<^sub>R fact n"
```
```  1240   assumes comm: "x * y = y * x"
```
```  1241   shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))"
```
```  1242 proof (induct n)
```
```  1243   case 0
```
```  1244   show ?case
```
```  1245     unfolding S_def by simp
```
```  1246 next
```
```  1247   case (Suc n)
```
```  1248   have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
```
```  1249     unfolding S_def by (simp del: mult_Suc)
```
```  1250   hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
```
```  1251     by simp
```
```  1252   have S_comm: "\<And>n. S x n * y = y * S x n"
```
```  1253     by (simp add: power_commuting_commutes comm S_def)
```
```  1254
```
```  1255   have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
```
```  1256     by (simp only: times_S)
```
```  1257   also have "\<dots> = (x + y) * (\<Sum>i\<le>n. S x i * S y (n-i))"
```
```  1258     by (simp only: Suc)
```
```  1259   also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n-i))
```
```  1260                 + y * (\<Sum>i\<le>n. S x i * S y (n-i))"
```
```  1261     by (rule distrib_right)
```
```  1262   also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i))
```
```  1263                 + (\<Sum>i\<le>n. S x i * y * S y (n-i))"
```
```  1264     by (simp add: setsum_right_distrib ac_simps S_comm)
```
```  1265   also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i))
```
```  1266                 + (\<Sum>i\<le>n. S x i * (y * S y (n-i)))"
```
```  1267     by (simp add: ac_simps)
```
```  1268   also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
```
```  1269                 + (\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1270     by (simp add: times_S Suc_diff_le)
```
```  1271   also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
```
```  1272              (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1273     by (subst setsum_atMost_Suc_shift) simp
```
```  1274   also have "(\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```  1275              (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1276     by simp
```
```  1277   also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
```
```  1278              (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```  1279              (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1280     by (simp only: setsum.distrib [symmetric] scaleR_left_distrib [symmetric]
```
```  1281                    of_nat_add [symmetric]) simp
```
```  1282   also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n-i))"
```
```  1283     by (simp only: scaleR_right.setsum)
```
```  1284   finally show
```
```  1285     "S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
```
```  1286     by (simp del: setsum_cl_ivl_Suc)
```
```  1287 qed
```
```  1288
```
```  1289 lemma exp_add_commuting: "x * y = y * x \<Longrightarrow> exp (x + y) = exp x * exp y"
```
```  1290   unfolding exp_def
```
```  1291   by (simp only: Cauchy_product summable_norm_exp exp_series_add_commuting)
```
```  1292
```
```  1293 lemma exp_add:
```
```  1294   fixes x y::"'a::{real_normed_field,banach}"
```
```  1295   shows "exp (x + y) = exp x * exp y"
```
```  1296   by (rule exp_add_commuting) (simp add: ac_simps)
```
```  1297
```
```  1298 lemma exp_double: "exp(2 * z) = exp z ^ 2"
```
```  1299   by (simp add: exp_add_commuting mult_2 power2_eq_square)
```
```  1300
```
```  1301 lemmas mult_exp_exp = exp_add [symmetric]
```
```  1302
```
```  1303 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
```
```  1304   unfolding exp_def
```
```  1305   apply (subst suminf_of_real)
```
```  1306   apply (rule summable_exp_generic)
```
```  1307   apply (simp add: scaleR_conv_of_real)
```
```  1308   done
```
```  1309
```
```  1310 corollary exp_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> exp z \<in> \<real>"
```
```  1311   by (metis Reals_cases Reals_of_real exp_of_real)
```
```  1312
```
```  1313 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
```
```  1314 proof
```
```  1315   have "exp x * exp (- x) = 1" by (simp add: exp_add_commuting[symmetric])
```
```  1316   also assume "exp x = 0"
```
```  1317   finally show "False" by simp
```
```  1318 qed
```
```  1319
```
```  1320 lemma exp_minus_inverse:
```
```  1321   shows "exp x * exp (- x) = 1"
```
```  1322   by (simp add: exp_add_commuting[symmetric])
```
```  1323
```
```  1324 lemma exp_minus:
```
```  1325   fixes x :: "'a::{real_normed_field, banach}"
```
```  1326   shows "exp (- x) = inverse (exp x)"
```
```  1327   by (intro inverse_unique [symmetric] exp_minus_inverse)
```
```  1328
```
```  1329 lemma exp_diff:
```
```  1330   fixes x :: "'a::{real_normed_field, banach}"
```
```  1331   shows "exp (x - y) = exp x / exp y"
```
```  1332   using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
```
```  1333
```
```  1334 lemma exp_of_nat_mult:
```
```  1335   fixes x :: "'a::{real_normed_field,banach}"
```
```  1336   shows "exp(of_nat n * x) = exp(x) ^ n"
```
```  1337     by (induct n) (auto simp add: distrib_left exp_add mult.commute)
```
```  1338
```
```  1339 corollary exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
```
```  1340   by (simp add: exp_of_nat_mult)
```
```  1341
```
```  1342 lemma exp_setsum: "finite I \<Longrightarrow> exp(setsum f I) = setprod (\<lambda>x. exp(f x)) I"
```
```  1343   by (induction I rule: finite_induct) (auto simp: exp_add_commuting mult.commute)
```
```  1344
```
```  1345
```
```  1346 subsubsection \<open>Properties of the Exponential Function on Reals\<close>
```
```  1347
```
```  1348 text \<open>Comparisons of @{term "exp x"} with zero.\<close>
```
```  1349
```
```  1350 text\<open>Proof: because every exponential can be seen as a square.\<close>
```
```  1351 lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
```
```  1352 proof -
```
```  1353   have "0 \<le> exp (x/2) * exp (x/2)" by simp
```
```  1354   thus ?thesis by (simp add: exp_add [symmetric])
```
```  1355 qed
```
```  1356
```
```  1357 lemma exp_gt_zero [simp]: "0 < exp (x::real)"
```
```  1358   by (simp add: order_less_le)
```
```  1359
```
```  1360 lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
```
```  1361   by (simp add: not_less)
```
```  1362
```
```  1363 lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
```
```  1364   by (simp add: not_le)
```
```  1365
```
```  1366 lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
```
```  1367   by simp
```
```  1368
```
```  1369 text \<open>Strict monotonicity of exponential.\<close>
```
```  1370
```
```  1371 lemma exp_ge_add_one_self_aux:
```
```  1372   assumes "0 \<le> (x::real)" shows "1+x \<le> exp(x)"
```
```  1373 using order_le_imp_less_or_eq [OF assms]
```
```  1374 proof
```
```  1375   assume "0 < x"
```
```  1376   have "1+x \<le> (\<Sum>n<2. inverse (fact n) * x^n)"
```
```  1377     by (auto simp add: numeral_2_eq_2)
```
```  1378   also have "... \<le> (\<Sum>n. inverse (fact n) * x^n)"
```
```  1379     apply (rule setsum_le_suminf [OF summable_exp])
```
```  1380     using \<open>0 < x\<close>
```
```  1381     apply (auto  simp add:  zero_le_mult_iff)
```
```  1382     done
```
```  1383   finally show "1+x \<le> exp x"
```
```  1384     by (simp add: exp_def)
```
```  1385 next
```
```  1386   assume "0 = x"
```
```  1387   then show "1 + x \<le> exp x"
```
```  1388     by auto
```
```  1389 qed
```
```  1390
```
```  1391 lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
```
```  1392 proof -
```
```  1393   assume x: "0 < x"
```
```  1394   hence "1 < 1 + x" by simp
```
```  1395   also from x have "1 + x \<le> exp x"
```
```  1396     by (simp add: exp_ge_add_one_self_aux)
```
```  1397   finally show ?thesis .
```
```  1398 qed
```
```  1399
```
```  1400 lemma exp_less_mono:
```
```  1401   fixes x y :: real
```
```  1402   assumes "x < y"
```
```  1403   shows "exp x < exp y"
```
```  1404 proof -
```
```  1405   from \<open>x < y\<close> have "0 < y - x" by simp
```
```  1406   hence "1 < exp (y - x)" by (rule exp_gt_one)
```
```  1407   hence "1 < exp y / exp x" by (simp only: exp_diff)
```
```  1408   thus "exp x < exp y" by simp
```
```  1409 qed
```
```  1410
```
```  1411 lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
```
```  1412   unfolding linorder_not_le [symmetric]
```
```  1413   by (auto simp add: order_le_less exp_less_mono)
```
```  1414
```
```  1415 lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
```
```  1416   by (auto intro: exp_less_mono exp_less_cancel)
```
```  1417
```
```  1418 lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
```
```  1419   by (auto simp add: linorder_not_less [symmetric])
```
```  1420
```
```  1421 lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
```
```  1422   by (simp add: order_eq_iff)
```
```  1423
```
```  1424 text \<open>Comparisons of @{term "exp x"} with one.\<close>
```
```  1425
```
```  1426 lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
```
```  1427   using exp_less_cancel_iff [where x=0 and y=x] by simp
```
```  1428
```
```  1429 lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
```
```  1430   using exp_less_cancel_iff [where x=x and y=0] by simp
```
```  1431
```
```  1432 lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
```
```  1433   using exp_le_cancel_iff [where x=0 and y=x] by simp
```
```  1434
```
```  1435 lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
```
```  1436   using exp_le_cancel_iff [where x=x and y=0] by simp
```
```  1437
```
```  1438 lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
```
```  1439   using exp_inj_iff [where x=x and y=0] by simp
```
```  1440
```
```  1441 lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
```
```  1442 proof (rule IVT)
```
```  1443   assume "1 \<le> y"
```
```  1444   hence "0 \<le> y - 1" by simp
```
```  1445   hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
```
```  1446   thus "y \<le> exp (y - 1)" by simp
```
```  1447 qed (simp_all add: le_diff_eq)
```
```  1448
```
```  1449 lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y"
```
```  1450 proof (rule linorder_le_cases [of 1 y])
```
```  1451   assume "1 \<le> y"
```
```  1452   thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total)
```
```  1453 next
```
```  1454   assume "0 < y" and "y \<le> 1"
```
```  1455   hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
```
```  1456   then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
```
```  1457   hence "exp (- x) = y" by (simp add: exp_minus)
```
```  1458   thus "\<exists>x. exp x = y" ..
```
```  1459 qed
```
```  1460
```
```  1461
```
```  1462 subsection \<open>Natural Logarithm\<close>
```
```  1463
```
```  1464 class ln = real_normed_algebra_1 + banach +
```
```  1465   fixes ln :: "'a \<Rightarrow> 'a"
```
```  1466   assumes ln_one [simp]: "ln 1 = 0"
```
```  1467
```
```  1468 definition powr :: "['a,'a] => 'a::ln"     (infixr "powr" 80)
```
```  1469   \<comment> \<open>exponentation via ln and exp\<close>
```
```  1470   where  [code del]: "x powr a \<equiv> if x = 0 then 0 else exp(a * ln x)"
```
```  1471
```
```  1472 lemma powr_0 [simp]: "0 powr z = 0"
```
```  1473   by (simp add: powr_def)
```
```  1474
```
```  1475
```
```  1476 instantiation real :: ln
```
```  1477 begin
```
```  1478
```
```  1479 definition ln_real :: "real \<Rightarrow> real"
```
```  1480   where "ln_real x = (THE u. exp u = x)"
```
```  1481
```
```  1482 instance
```
```  1483 by intro_classes (simp add: ln_real_def)
```
```  1484
```
```  1485 end
```
```  1486
```
```  1487 lemma powr_eq_0_iff [simp]: "w powr z = 0 \<longleftrightarrow> w = 0"
```
```  1488   by (simp add: powr_def)
```
```  1489
```
```  1490 lemma ln_exp [simp]:
```
```  1491   fixes x::real shows "ln (exp x) = x"
```
```  1492   by (simp add: ln_real_def)
```
```  1493
```
```  1494 lemma exp_ln [simp]:
```
```  1495   fixes x::real shows "0 < x \<Longrightarrow> exp (ln x) = x"
```
```  1496   by (auto dest: exp_total)
```
```  1497
```
```  1498 lemma exp_ln_iff [simp]:
```
```  1499   fixes x::real shows "exp (ln x) = x \<longleftrightarrow> 0 < x"
```
```  1500   by (metis exp_gt_zero exp_ln)
```
```  1501
```
```  1502 lemma ln_unique:
```
```  1503   fixes x::real shows "exp y = x \<Longrightarrow> ln x = y"
```
```  1504   by (erule subst, rule ln_exp)
```
```  1505
```
```  1506 lemma ln_mult:
```
```  1507   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
```
```  1508   by (rule ln_unique) (simp add: exp_add)
```
```  1509
```
```  1510 lemma ln_setprod:
```
```  1511   fixes f:: "'a => real"
```
```  1512   shows
```
```  1513     "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> f i > 0\<rbrakk> \<Longrightarrow> ln(setprod f I) = setsum (\<lambda>x. ln(f x)) I"
```
```  1514   by (induction I rule: finite_induct) (auto simp: ln_mult setprod_pos)
```
```  1515
```
```  1516 lemma ln_inverse:
```
```  1517   fixes x::real shows "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
```
```  1518   by (rule ln_unique) (simp add: exp_minus)
```
```  1519
```
```  1520 lemma ln_div:
```
```  1521   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
```
```  1522   by (rule ln_unique) (simp add: exp_diff)
```
```  1523
```
```  1524 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x^n) = real n * ln x"
```
```  1525   by (rule ln_unique) (simp add: exp_real_of_nat_mult)
```
```  1526
```
```  1527 lemma ln_less_cancel_iff [simp]:
```
```  1528   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
```
```  1529   by (subst exp_less_cancel_iff [symmetric]) simp
```
```  1530
```
```  1531 lemma ln_le_cancel_iff [simp]:
```
```  1532   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
```
```  1533   by (simp add: linorder_not_less [symmetric])
```
```  1534
```
```  1535 lemma ln_inj_iff [simp]:
```
```  1536   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
```
```  1537   by (simp add: order_eq_iff)
```
```  1538
```
```  1539 lemma ln_add_one_self_le_self [simp]:
```
```  1540   fixes x::real shows "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
```
```  1541   apply (rule exp_le_cancel_iff [THEN iffD1])
```
```  1542   apply (simp add: exp_ge_add_one_self_aux)
```
```  1543   done
```
```  1544
```
```  1545 lemma ln_less_self [simp]:
```
```  1546   fixes x::real shows "0 < x \<Longrightarrow> ln x < x"
```
```  1547   by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
```
```  1548
```
```  1549 lemma ln_ge_zero [simp]:
```
```  1550   fixes x::real shows "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
```
```  1551   using ln_le_cancel_iff [of 1 x] by simp
```
```  1552
```
```  1553 lemma ln_ge_zero_imp_ge_one:
```
```  1554   fixes x::real shows "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
```
```  1555   using ln_le_cancel_iff [of 1 x] by simp
```
```  1556
```
```  1557 lemma ln_ge_zero_iff [simp]:
```
```  1558   fixes x::real shows "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
```
```  1559   using ln_le_cancel_iff [of 1 x] by simp
```
```  1560
```
```  1561 lemma ln_less_zero_iff [simp]:
```
```  1562   fixes x::real shows "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
```
```  1563   using ln_less_cancel_iff [of x 1] by simp
```
```  1564
```
```  1565 lemma ln_gt_zero:
```
```  1566   fixes x::real shows "1 < x \<Longrightarrow> 0 < ln x"
```
```  1567   using ln_less_cancel_iff [of 1 x] by simp
```
```  1568
```
```  1569 lemma ln_gt_zero_imp_gt_one:
```
```  1570   fixes x::real shows "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
```
```  1571   using ln_less_cancel_iff [of 1 x] by simp
```
```  1572
```
```  1573 lemma ln_gt_zero_iff [simp]:
```
```  1574   fixes x::real shows "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
```
```  1575   using ln_less_cancel_iff [of 1 x] by simp
```
```  1576
```
```  1577 lemma ln_eq_zero_iff [simp]:
```
```  1578   fixes x::real shows "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
```
```  1579   using ln_inj_iff [of x 1] by simp
```
```  1580
```
```  1581 lemma ln_less_zero:
```
```  1582   fixes x::real shows "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
```
```  1583   by simp
```
```  1584
```
```  1585 lemma ln_neg_is_const:
```
```  1586   fixes x::real shows "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)"
```
```  1587   by (auto simp add: ln_real_def intro!: arg_cong[where f=The])
```
```  1588
```
```  1589 lemma isCont_ln:
```
```  1590   fixes x::real assumes "x \<noteq> 0" shows "isCont ln x"
```
```  1591 proof cases
```
```  1592   assume "0 < x"
```
```  1593   moreover then have "isCont ln (exp (ln x))"
```
```  1594     by (intro isCont_inv_fun[where d="\<bar>x\<bar>" and f=exp]) auto
```
```  1595   ultimately show ?thesis
```
```  1596     by simp
```
```  1597 next
```
```  1598   assume "\<not> 0 < x" with \<open>x \<noteq> 0\<close> show "isCont ln x"
```
```  1599     unfolding isCont_def
```
```  1600     by (subst filterlim_cong[OF _ refl, of _ "nhds (ln 0)" _ "\<lambda>_. ln 0"])
```
```  1601        (auto simp: ln_neg_is_const not_less eventually_at dist_real_def
```
```  1602                 intro!: exI[of _ "\<bar>x\<bar>"])
```
```  1603 qed
```
```  1604
```
```  1605 lemma tendsto_ln [tendsto_intros]:
```
```  1606   fixes a::real shows
```
```  1607   "(f \<longlongrightarrow> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) \<longlongrightarrow> ln a) F"
```
```  1608   by (rule isCont_tendsto_compose [OF isCont_ln])
```
```  1609
```
```  1610 lemma continuous_ln:
```
```  1611   "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. ln (f x :: real))"
```
```  1612   unfolding continuous_def by (rule tendsto_ln)
```
```  1613
```
```  1614 lemma isCont_ln' [continuous_intros]:
```
```  1615   "continuous (at x) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x :: real))"
```
```  1616   unfolding continuous_at by (rule tendsto_ln)
```
```  1617
```
```  1618 lemma continuous_within_ln [continuous_intros]:
```
```  1619   "continuous (at x within s) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x :: real))"
```
```  1620   unfolding continuous_within by (rule tendsto_ln)
```
```  1621
```
```  1622 lemma continuous_on_ln [continuous_intros]:
```
```  1623   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. f x \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x :: real))"
```
```  1624   unfolding continuous_on_def by (auto intro: tendsto_ln)
```
```  1625
```
```  1626 lemma DERIV_ln:
```
```  1627   fixes x::real shows "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
```
```  1628   apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
```
```  1629   apply (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln)
```
```  1630   done
```
```  1631
```
```  1632 lemma DERIV_ln_divide:
```
```  1633   fixes x::real shows "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
```
```  1634   by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
```
```  1635
```
```  1636 declare DERIV_ln_divide[THEN DERIV_chain2, derivative_intros]
```
```  1637         DERIV_ln_divide[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  1638
```
```  1639
```
```  1640 lemma ln_series:
```
```  1641   assumes "0 < x" and "x < 2"
```
```  1642   shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))"
```
```  1643   (is "ln x = suminf (?f (x - 1))")
```
```  1644 proof -
```
```  1645   let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n"
```
```  1646
```
```  1647   have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
```
```  1648   proof (rule DERIV_isconst3[where x=x])
```
```  1649     fix x :: real
```
```  1650     assume "x \<in> {0 <..< 2}"
```
```  1651     hence "0 < x" and "x < 2" by auto
```
```  1652     have "norm (1 - x) < 1"
```
```  1653       using \<open>0 < x\<close> and \<open>x < 2\<close> by auto
```
```  1654     have "1 / x = 1 / (1 - (1 - x))" by auto
```
```  1655     also have "\<dots> = (\<Sum> n. (1 - x)^n)"
```
```  1656       using geometric_sums[OF \<open>norm (1 - x) < 1\<close>] by (rule sums_unique)
```
```  1657     also have "\<dots> = suminf (?f' x)"
```
```  1658       unfolding power_mult_distrib[symmetric]
```
```  1659       by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
```
```  1660     finally have "DERIV ln x :> suminf (?f' x)"
```
```  1661       using DERIV_ln[OF \<open>0 < x\<close>] unfolding divide_inverse by auto
```
```  1662     moreover
```
```  1663     have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
```
```  1664     have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :>
```
```  1665       (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
```
```  1666     proof (rule DERIV_power_series')
```
```  1667       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1"
```
```  1668         using \<open>0 < x\<close> \<open>x < 2\<close> by auto
```
```  1669       fix x :: real
```
```  1670       assume "x \<in> {- 1<..<1}"
```
```  1671       hence "norm (-x) < 1" by auto
```
```  1672       show "summable (\<lambda>n. (- 1) ^ n * (1 / real (n + 1)) * real (Suc n) * x^n)"
```
```  1673         unfolding One_nat_def
```
```  1674         by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF \<open>norm (-x) < 1\<close>])
```
```  1675     qed
```
```  1676     hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
```
```  1677       unfolding One_nat_def by auto
```
```  1678     hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
```
```  1679       unfolding DERIV_def repos .
```
```  1680     ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
```
```  1681       by (rule DERIV_diff)
```
```  1682     thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
```
```  1683   qed (auto simp add: assms)
```
```  1684   thus ?thesis by auto
```
```  1685 qed
```
```  1686
```
```  1687 lemma exp_first_two_terms:
```
```  1688   fixes x :: "'a::{real_normed_field,banach}"
```
```  1689   shows "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) * (x ^ (n+2)))"
```
```  1690 proof -
```
```  1691   have "exp x = suminf (\<lambda>n. inverse(fact n) * (x^n))"
```
```  1692     by (simp add: exp_def scaleR_conv_of_real nonzero_of_real_inverse)
```
```  1693   also from summable_exp have "... = (\<Sum> n. inverse(fact(n+2)) * (x ^ (n+2))) +
```
```  1694     (\<Sum> n::nat<2. inverse(fact n) * (x^n))" (is "_ = _ + ?a")
```
```  1695     by (rule suminf_split_initial_segment)
```
```  1696   also have "?a = 1 + x"
```
```  1697     by (simp add: numeral_2_eq_2)
```
```  1698   finally show ?thesis
```
```  1699     by simp
```
```  1700 qed
```
```  1701
```
```  1702 lemma exp_bound: "0 <= (x::real) \<Longrightarrow> x <= 1 \<Longrightarrow> exp x <= 1 + x + x\<^sup>2"
```
```  1703 proof -
```
```  1704   assume a: "0 <= x"
```
```  1705   assume b: "x <= 1"
```
```  1706   {
```
```  1707     fix n :: nat
```
```  1708     have "(2::nat) * 2 ^ n \<le> fact (n + 2)"
```
```  1709       by (induct n) simp_all
```
```  1710     hence "real ((2::nat) * 2 ^ n) \<le> real_of_nat (fact (n + 2))"
```
```  1711       by (simp only: of_nat_le_iff)
```
```  1712     hence "((2::real) * 2 ^ n) \<le> fact (n + 2)"
```
```  1713       unfolding of_nat_fact
```
```  1714       by (simp add: of_nat_mult of_nat_power)
```
```  1715     hence "inverse (fact (n + 2)) \<le> inverse ((2::real) * 2 ^ n)"
```
```  1716       by (rule le_imp_inverse_le) simp
```
```  1717     hence "inverse (fact (n + 2)) \<le> 1/(2::real) * (1/2)^n"
```
```  1718       by (simp add: power_inverse [symmetric])
```
```  1719     hence "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
```
```  1720       by (rule mult_mono)
```
```  1721         (rule mult_mono, simp_all add: power_le_one a b)
```
```  1722     hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)"
```
```  1723       unfolding power_add by (simp add: ac_simps del: fact.simps) }
```
```  1724   note aux1 = this
```
```  1725   have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
```
```  1726     by (intro sums_mult geometric_sums, simp)
```
```  1727   hence aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
```
```  1728     by simp
```
```  1729   have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2"
```
```  1730   proof -
```
```  1731     have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <=
```
```  1732         suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
```
```  1733       apply (rule suminf_le)
```
```  1734       apply (rule allI, rule aux1)
```
```  1735       apply (rule summable_exp [THEN summable_ignore_initial_segment])
```
```  1736       by (rule sums_summable, rule aux2)
```
```  1737     also have "... = x\<^sup>2"
```
```  1738       by (rule sums_unique [THEN sym], rule aux2)
```
```  1739     finally show ?thesis .
```
```  1740   qed
```
```  1741   thus ?thesis unfolding exp_first_two_terms by auto
```
```  1742 qed
```
```  1743
```
```  1744 corollary exp_half_le2: "exp(1/2) \<le> (2::real)"
```
```  1745   using exp_bound [of "1/2"]
```
```  1746   by (simp add: field_simps)
```
```  1747
```
```  1748 corollary exp_le: "exp 1 \<le> (3::real)"
```
```  1749   using exp_bound [of 1]
```
```  1750   by (simp add: field_simps)
```
```  1751
```
```  1752 lemma exp_bound_half: "norm(z) \<le> 1/2 \<Longrightarrow> norm(exp z) \<le> 2"
```
```  1753   by (blast intro: order_trans intro!: exp_half_le2 norm_exp)
```
```  1754
```
```  1755 lemma exp_bound_lemma:
```
```  1756   assumes "norm(z) \<le> 1/2" shows "norm(exp z) \<le> 1 + 2 * norm(z)"
```
```  1757 proof -
```
```  1758   have n: "(norm z)\<^sup>2 \<le> norm z * 1"
```
```  1759     unfolding power2_eq_square
```
```  1760     apply (rule mult_left_mono)
```
```  1761     using assms
```
```  1762     apply auto
```
```  1763     done
```
```  1764   show ?thesis
```
```  1765     apply (rule order_trans [OF norm_exp])
```
```  1766     apply (rule order_trans [OF exp_bound])
```
```  1767     using assms n
```
```  1768     apply auto
```
```  1769     done
```
```  1770 qed
```
```  1771
```
```  1772 lemma real_exp_bound_lemma:
```
```  1773   fixes x :: real
```
```  1774   shows "0 \<le> x \<Longrightarrow> x \<le> 1/2 \<Longrightarrow> exp(x) \<le> 1 + 2 * x"
```
```  1775 using exp_bound_lemma [of x]
```
```  1776 by simp
```
```  1777
```
```  1778 lemma ln_one_minus_pos_upper_bound:
```
```  1779   fixes x::real shows "0 <= x \<Longrightarrow> x < 1 \<Longrightarrow> ln (1 - x) <= - x"
```
```  1780 proof -
```
```  1781   assume a: "0 <= (x::real)" and b: "x < 1"
```
```  1782   have "(1 - x) * (1 + x + x\<^sup>2) = (1 - x^3)"
```
```  1783     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
```
```  1784   also have "... <= 1"
```
```  1785     by (auto simp add: a)
```
```  1786   finally have "(1 - x) * (1 + x + x\<^sup>2) <= 1" .
```
```  1787   moreover have c: "0 < 1 + x + x\<^sup>2"
```
```  1788     by (simp add: add_pos_nonneg a)
```
```  1789   ultimately have "1 - x <= 1 / (1 + x + x\<^sup>2)"
```
```  1790     by (elim mult_imp_le_div_pos)
```
```  1791   also have "... <= 1 / exp x"
```
```  1792     by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs
```
```  1793               real_sqrt_pow2_iff real_sqrt_power)
```
```  1794   also have "... = exp (-x)"
```
```  1795     by (auto simp add: exp_minus divide_inverse)
```
```  1796   finally have "1 - x <= exp (- x)" .
```
```  1797   also have "1 - x = exp (ln (1 - x))"
```
```  1798     by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq)
```
```  1799   finally have "exp (ln (1 - x)) <= exp (- x)" .
```
```  1800   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```  1801 qed
```
```  1802
```
```  1803 lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
```
```  1804   apply (case_tac "0 <= x")
```
```  1805   apply (erule exp_ge_add_one_self_aux)
```
```  1806   apply (case_tac "x <= -1")
```
```  1807   apply (subgoal_tac "1 + x <= 0")
```
```  1808   apply (erule order_trans)
```
```  1809   apply simp
```
```  1810   apply simp
```
```  1811   apply (subgoal_tac "1 + x = exp(ln (1 + x))")
```
```  1812   apply (erule ssubst)
```
```  1813   apply (subst exp_le_cancel_iff)
```
```  1814   apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
```
```  1815   apply simp
```
```  1816   apply (rule ln_one_minus_pos_upper_bound)
```
```  1817   apply auto
```
```  1818 done
```
```  1819
```
```  1820 lemma ln_one_plus_pos_lower_bound:
```
```  1821   fixes x::real shows "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> x - x\<^sup>2 <= ln (1 + x)"
```
```  1822 proof -
```
```  1823   assume a: "0 <= x" and b: "x <= 1"
```
```  1824   have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
```
```  1825     by (rule exp_diff)
```
```  1826   also have "... <= (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
```
```  1827     by (metis a b divide_right_mono exp_bound exp_ge_zero)
```
```  1828   also have "... <= (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
```
```  1829     by (simp add: a divide_left_mono add_pos_nonneg)
```
```  1830   also from a have "... <= 1 + x"
```
```  1831     by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
```
```  1832   finally have "exp (x - x\<^sup>2) <= 1 + x" .
```
```  1833   also have "... = exp (ln (1 + x))"
```
```  1834   proof -
```
```  1835     from a have "0 < 1 + x" by auto
```
```  1836     thus ?thesis
```
```  1837       by (auto simp only: exp_ln_iff [THEN sym])
```
```  1838   qed
```
```  1839   finally have "exp (x - x\<^sup>2) <= exp (ln (1 + x))" .
```
```  1840   thus ?thesis
```
```  1841     by (metis exp_le_cancel_iff)
```
```  1842 qed
```
```  1843
```
```  1844 lemma ln_one_minus_pos_lower_bound:
```
```  1845   fixes x::real
```
```  1846   shows "0 <= x \<Longrightarrow> x <= (1 / 2) \<Longrightarrow> - x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1847 proof -
```
```  1848   assume a: "0 <= x" and b: "x <= (1 / 2)"
```
```  1849   from b have c: "x < 1" by auto
```
```  1850   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
```
```  1851     apply (subst ln_inverse [symmetric])
```
```  1852     apply (simp add: field_simps)
```
```  1853     apply (rule arg_cong [where f=ln])
```
```  1854     apply (simp add: field_simps)
```
```  1855     done
```
```  1856   also have "- (x / (1 - x)) <= ..."
```
```  1857   proof -
```
```  1858     have "ln (1 + x / (1 - x)) <= x / (1 - x)"
```
```  1859       using a c by (intro ln_add_one_self_le_self) auto
```
```  1860     thus ?thesis
```
```  1861       by auto
```
```  1862   qed
```
```  1863   also have "- (x / (1 - x)) = -x / (1 - x)"
```
```  1864     by auto
```
```  1865   finally have d: "- x / (1 - x) <= ln (1 - x)" .
```
```  1866   have "0 < 1 - x" using a b by simp
```
```  1867   hence e: "-x - 2 * x\<^sup>2 <= - x / (1 - x)"
```
```  1868     using mult_right_le_one_le[of "x*x" "2*x"] a b
```
```  1869     by (simp add: field_simps power2_eq_square)
```
```  1870   from e d show "- x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1871     by (rule order_trans)
```
```  1872 qed
```
```  1873
```
```  1874 lemma ln_add_one_self_le_self2:
```
```  1875   fixes x::real shows "-1 < x \<Longrightarrow> ln(1 + x) <= x"
```
```  1876   apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
```
```  1877   apply (subst ln_le_cancel_iff)
```
```  1878   apply auto
```
```  1879   done
```
```  1880
```
```  1881 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
```
```  1882   fixes x::real shows "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> \<bar>ln (1 + x) - x\<bar> <= x\<^sup>2"
```
```  1883 proof -
```
```  1884   assume x: "0 <= x"
```
```  1885   assume x1: "x <= 1"
```
```  1886   from x have "ln (1 + x) <= x"
```
```  1887     by (rule ln_add_one_self_le_self)
```
```  1888   then have "ln (1 + x) - x <= 0"
```
```  1889     by simp
```
```  1890   then have "\<bar>ln(1 + x) - x\<bar> = - (ln(1 + x) - x)"
```
```  1891     by (rule abs_of_nonpos)
```
```  1892   also have "... = x - ln (1 + x)"
```
```  1893     by simp
```
```  1894   also have "... <= x\<^sup>2"
```
```  1895   proof -
```
```  1896     from x x1 have "x - x\<^sup>2 <= ln (1 + x)"
```
```  1897       by (intro ln_one_plus_pos_lower_bound)
```
```  1898     thus ?thesis
```
```  1899       by simp
```
```  1900   qed
```
```  1901   finally show ?thesis .
```
```  1902 qed
```
```  1903
```
```  1904 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
```
```  1905   fixes x::real shows "-(1 / 2) <= x \<Longrightarrow> x <= 0 \<Longrightarrow> \<bar>ln (1 + x) - x\<bar> <= 2 * x\<^sup>2"
```
```  1906 proof -
```
```  1907   assume a: "-(1 / 2) <= x"
```
```  1908   assume b: "x <= 0"
```
```  1909   have "\<bar>ln (1 + x) - x\<bar> = x - ln(1 - (-x))"
```
```  1910     apply (subst abs_of_nonpos)
```
```  1911     apply simp
```
```  1912     apply (rule ln_add_one_self_le_self2)
```
```  1913     using a apply auto
```
```  1914     done
```
```  1915   also have "... <= 2 * x\<^sup>2"
```
```  1916     apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 <= ln (1 - (-x))")
```
```  1917     apply (simp add: algebra_simps)
```
```  1918     apply (rule ln_one_minus_pos_lower_bound)
```
```  1919     using a b apply auto
```
```  1920     done
```
```  1921   finally show ?thesis .
```
```  1922 qed
```
```  1923
```
```  1924 lemma abs_ln_one_plus_x_minus_x_bound:
```
```  1925   fixes x::real shows "\<bar>x\<bar> <= 1 / 2 \<Longrightarrow> \<bar>ln (1 + x) - x\<bar> <= 2 * x\<^sup>2"
```
```  1926   apply (case_tac "0 <= x")
```
```  1927   apply (rule order_trans)
```
```  1928   apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
```
```  1929   apply auto
```
```  1930   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
```
```  1931   apply auto
```
```  1932   done
```
```  1933
```
```  1934 lemma ln_x_over_x_mono:
```
```  1935   fixes x::real shows "exp 1 <= x \<Longrightarrow> x <= y \<Longrightarrow> (ln y / y) <= (ln x / x)"
```
```  1936 proof -
```
```  1937   assume x: "exp 1 <= x" "x <= y"
```
```  1938   moreover have "0 < exp (1::real)" by simp
```
```  1939   ultimately have a: "0 < x" and b: "0 < y"
```
```  1940     by (fast intro: less_le_trans order_trans)+
```
```  1941   have "x * ln y - x * ln x = x * (ln y - ln x)"
```
```  1942     by (simp add: algebra_simps)
```
```  1943   also have "... = x * ln(y / x)"
```
```  1944     by (simp only: ln_div a b)
```
```  1945   also have "y / x = (x + (y - x)) / x"
```
```  1946     by simp
```
```  1947   also have "... = 1 + (y - x) / x"
```
```  1948     using x a by (simp add: field_simps)
```
```  1949   also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
```
```  1950     using x a
```
```  1951     by (intro mult_left_mono ln_add_one_self_le_self) simp_all
```
```  1952   also have "... = y - x" using a by simp
```
```  1953   also have "... = (y - x) * ln (exp 1)" by simp
```
```  1954   also have "... <= (y - x) * ln x"
```
```  1955     apply (rule mult_left_mono)
```
```  1956     apply (subst ln_le_cancel_iff)
```
```  1957     apply fact
```
```  1958     apply (rule a)
```
```  1959     apply (rule x)
```
```  1960     using x apply simp
```
```  1961     done
```
```  1962   also have "... = y * ln x - x * ln x"
```
```  1963     by (rule left_diff_distrib)
```
```  1964   finally have "x * ln y <= y * ln x"
```
```  1965     by arith
```
```  1966   then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
```
```  1967   also have "... = y * (ln x / x)" by simp
```
```  1968   finally show ?thesis using b by (simp add: field_simps)
```
```  1969 qed
```
```  1970
```
```  1971 lemma ln_le_minus_one:
```
```  1972   fixes x::real shows "0 < x \<Longrightarrow> ln x \<le> x - 1"
```
```  1973   using exp_ge_add_one_self[of "ln x"] by simp
```
```  1974
```
```  1975 corollary ln_diff_le:
```
```  1976   fixes x::real
```
```  1977   shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x - ln y \<le> (x - y) / y"
```
```  1978   by (simp add: ln_div [symmetric] diff_divide_distrib ln_le_minus_one)
```
```  1979
```
```  1980 lemma ln_eq_minus_one:
```
```  1981   fixes x::real
```
```  1982   assumes "0 < x" "ln x = x - 1"
```
```  1983   shows "x = 1"
```
```  1984 proof -
```
```  1985   let ?l = "\<lambda>y. ln y - y + 1"
```
```  1986   have D: "\<And>x::real. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
```
```  1987     by (auto intro!: derivative_eq_intros)
```
```  1988
```
```  1989   show ?thesis
```
```  1990   proof (cases rule: linorder_cases)
```
```  1991     assume "x < 1"
```
```  1992     from dense[OF \<open>x < 1\<close>] obtain a where "x < a" "a < 1" by blast
```
```  1993     from \<open>x < a\<close> have "?l x < ?l a"
```
```  1994     proof (rule DERIV_pos_imp_increasing, safe)
```
```  1995       fix y
```
```  1996       assume "x \<le> y" "y \<le> a"
```
```  1997       with \<open>0 < x\<close> \<open>a < 1\<close> have "0 < 1 / y - 1" "0 < y"
```
```  1998         by (auto simp: field_simps)
```
```  1999       with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z" by blast
```
```  2000     qed
```
```  2001     also have "\<dots> \<le> 0"
```
```  2002       using ln_le_minus_one \<open>0 < x\<close> \<open>x < a\<close> by (auto simp: field_simps)
```
```  2003     finally show "x = 1" using assms by auto
```
```  2004   next
```
```  2005     assume "1 < x"
```
```  2006     from dense[OF this] obtain a where "1 < a" "a < x" by blast
```
```  2007     from \<open>a < x\<close> have "?l x < ?l a"
```
```  2008     proof (rule DERIV_neg_imp_decreasing, safe)
```
```  2009       fix y
```
```  2010       assume "a \<le> y" "y \<le> x"
```
```  2011       with \<open>1 < a\<close> have "1 / y - 1 < 0" "0 < y"
```
```  2012         by (auto simp: field_simps)
```
```  2013       with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
```
```  2014         by blast
```
```  2015     qed
```
```  2016     also have "\<dots> \<le> 0"
```
```  2017       using ln_le_minus_one \<open>1 < a\<close> by (auto simp: field_simps)
```
```  2018     finally show "x = 1" using assms by auto
```
```  2019   next
```
```  2020     assume "x = 1"
```
```  2021     then show ?thesis by simp
```
```  2022   qed
```
```  2023 qed
```
```  2024
```
```  2025 lemma exp_at_bot: "(exp \<longlongrightarrow> (0::real)) at_bot"
```
```  2026   unfolding tendsto_Zfun_iff
```
```  2027 proof (rule ZfunI, simp add: eventually_at_bot_dense)
```
```  2028   fix r :: real assume "0 < r"
```
```  2029   {
```
```  2030     fix x
```
```  2031     assume "x < ln r"
```
```  2032     then have "exp x < exp (ln r)"
```
```  2033       by simp
```
```  2034     with \<open>0 < r\<close> have "exp x < r"
```
```  2035       by simp
```
```  2036   }
```
```  2037   then show "\<exists>k. \<forall>n<k. exp n < r" by auto
```
```  2038 qed
```
```  2039
```
```  2040 lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
```
```  2041   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"])
```
```  2042      (auto intro: eventually_gt_at_top)
```
```  2043
```
```  2044 lemma lim_exp_minus_1:
```
```  2045   fixes x :: "'a::{real_normed_field,banach}"
```
```  2046   shows "((\<lambda>z::'a. (exp(z) - 1) / z) \<longlongrightarrow> 1) (at 0)"
```
```  2047 proof -
```
```  2048   have "((\<lambda>z::'a. exp(z) - 1) has_field_derivative 1) (at 0)"
```
```  2049     by (intro derivative_eq_intros | simp)+
```
```  2050   then show ?thesis
```
```  2051     by (simp add: Deriv.DERIV_iff2)
```
```  2052 qed
```
```  2053
```
```  2054 lemma ln_at_0: "LIM x at_right 0. ln (x::real) :> at_bot"
```
```  2055   by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  2056      (auto simp: eventually_at_filter)
```
```  2057
```
```  2058 lemma ln_at_top: "LIM x at_top. ln (x::real) :> at_top"
```
```  2059   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  2060      (auto intro: eventually_gt_at_top)
```
```  2061
```
```  2062 lemma filtermap_ln_at_top: "filtermap (ln::real \<Rightarrow> real) at_top = at_top"
```
```  2063   by (intro filtermap_fun_inverse[of exp] exp_at_top ln_at_top) auto
```
```  2064
```
```  2065 lemma filtermap_exp_at_top: "filtermap (exp::real \<Rightarrow> real) at_top = at_top"
```
```  2066   by (intro filtermap_fun_inverse[of ln] exp_at_top ln_at_top)
```
```  2067      (auto simp: eventually_at_top_dense)
```
```  2068
```
```  2069 lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) \<longlongrightarrow> (0::real)) at_top"
```
```  2070 proof (induct k)
```
```  2071   case 0
```
```  2072   show "((\<lambda>x. x ^ 0 / exp x) \<longlongrightarrow> (0::real)) at_top"
```
```  2073     by (simp add: inverse_eq_divide[symmetric])
```
```  2074        (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
```
```  2075               at_top_le_at_infinity order_refl)
```
```  2076 next
```
```  2077   case (Suc k)
```
```  2078   show ?case
```
```  2079   proof (rule lhospital_at_top_at_top)
```
```  2080     show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"
```
```  2081       by eventually_elim (intro derivative_eq_intros, auto)
```
```  2082     show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top"
```
```  2083       by eventually_elim auto
```
```  2084     show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
```
```  2085       by auto
```
```  2086     from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
```
```  2087     show "((\<lambda>x. real (Suc k) * x ^ k / exp x) \<longlongrightarrow> 0) at_top"
```
```  2088       by simp
```
```  2089   qed (rule exp_at_top)
```
```  2090 qed
```
```  2091
```
```  2092
```
```  2093 definition log :: "[real,real] => real"
```
```  2094   \<comment> \<open>logarithm of @{term x} to base @{term a}\<close>
```
```  2095   where "log a x = ln x / ln a"
```
```  2096
```
```  2097
```
```  2098 lemma tendsto_log [tendsto_intros]:
```
```  2099   "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F; 0 < a; a \<noteq> 1; 0 < b\<rbrakk> \<Longrightarrow> ((\<lambda>x. log (f x) (g x)) \<longlongrightarrow> log a b) F"
```
```  2100   unfolding log_def by (intro tendsto_intros) auto
```
```  2101
```
```  2102 lemma continuous_log:
```
```  2103   assumes "continuous F f"
```
```  2104     and "continuous F g"
```
```  2105     and "0 < f (Lim F (\<lambda>x. x))"
```
```  2106     and "f (Lim F (\<lambda>x. x)) \<noteq> 1"
```
```  2107     and "0 < g (Lim F (\<lambda>x. x))"
```
```  2108   shows "continuous F (\<lambda>x. log (f x) (g x))"
```
```  2109   using assms unfolding continuous_def by (rule tendsto_log)
```
```  2110
```
```  2111 lemma continuous_at_within_log[continuous_intros]:
```
```  2112   assumes "continuous (at a within s) f"
```
```  2113     and "continuous (at a within s) g"
```
```  2114     and "0 < f a"
```
```  2115     and "f a \<noteq> 1"
```
```  2116     and "0 < g a"
```
```  2117   shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
```
```  2118   using assms unfolding continuous_within by (rule tendsto_log)
```
```  2119
```
```  2120 lemma isCont_log[continuous_intros, simp]:
```
```  2121   assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
```
```  2122   shows "isCont (\<lambda>x. log (f x) (g x)) a"
```
```  2123   using assms unfolding continuous_at by (rule tendsto_log)
```
```  2124
```
```  2125 lemma continuous_on_log[continuous_intros]:
```
```  2126   assumes "continuous_on s f" "continuous_on s g"
```
```  2127     and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
```
```  2128   shows "continuous_on s (\<lambda>x. log (f x) (g x))"
```
```  2129   using assms unfolding continuous_on_def by (fast intro: tendsto_log)
```
```  2130
```
```  2131 lemma powr_one_eq_one [simp]: "1 powr a = 1"
```
```  2132   by (simp add: powr_def)
```
```  2133
```
```  2134 lemma powr_zero_eq_one [simp]: "x powr 0 = (if x=0 then 0 else 1)"
```
```  2135   by (simp add: powr_def)
```
```  2136
```
```  2137 lemma powr_one_gt_zero_iff [simp]:
```
```  2138   fixes x::real shows "(x powr 1 = x) = (0 \<le> x)"
```
```  2139   by (auto simp: powr_def)
```
```  2140 declare powr_one_gt_zero_iff [THEN iffD2, simp]
```
```  2141
```
```  2142 lemma powr_mult:
```
```  2143   fixes x::real shows "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
```
```  2144   by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
```
```  2145
```
```  2146 lemma powr_ge_pzero [simp]:
```
```  2147   fixes x::real shows "0 <= x powr y"
```
```  2148   by (simp add: powr_def)
```
```  2149
```
```  2150 lemma powr_divide:
```
```  2151   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
```
```  2152   apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
```
```  2153   apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
```
```  2154   done
```
```  2155
```
```  2156 lemma powr_divide2:
```
```  2157   fixes x::real shows "x powr a / x powr b = x powr (a - b)"
```
```  2158   apply (simp add: powr_def)
```
```  2159   apply (subst exp_diff [THEN sym])
```
```  2160   apply (simp add: left_diff_distrib)
```
```  2161   done
```
```  2162
```
```  2163 lemma powr_add:
```
```  2164   fixes x::real shows "x powr (a + b) = (x powr a) * (x powr b)"
```
```  2165   by (simp add: powr_def exp_add [symmetric] distrib_right)
```
```  2166
```
```  2167 lemma powr_mult_base:
```
```  2168   fixes x::real shows "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
```
```  2169   using assms by (auto simp: powr_add)
```
```  2170
```
```  2171 lemma powr_powr:
```
```  2172   fixes x::real shows "(x powr a) powr b = x powr (a * b)"
```
```  2173   by (simp add: powr_def)
```
```  2174
```
```  2175 lemma powr_powr_swap:
```
```  2176   fixes x::real shows "(x powr a) powr b = (x powr b) powr a"
```
```  2177   by (simp add: powr_powr mult.commute)
```
```  2178
```
```  2179 lemma powr_minus:
```
```  2180   fixes x::real shows "x powr (-a) = inverse (x powr a)"
```
```  2181   by (simp add: powr_def exp_minus [symmetric])
```
```  2182
```
```  2183 lemma powr_minus_divide:
```
```  2184   fixes x::real shows "x powr (-a) = 1/(x powr a)"
```
```  2185   by (simp add: divide_inverse powr_minus)
```
```  2186
```
```  2187 lemma divide_powr_uminus:
```
```  2188   fixes a::real shows "a / b powr c = a * b powr (- c)"
```
```  2189   by (simp add: powr_minus_divide)
```
```  2190
```
```  2191 lemma powr_less_mono:
```
```  2192   fixes x::real shows "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
```
```  2193   by (simp add: powr_def)
```
```  2194
```
```  2195 lemma powr_less_cancel:
```
```  2196   fixes x::real shows "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
```
```  2197   by (simp add: powr_def)
```
```  2198
```
```  2199 lemma powr_less_cancel_iff [simp]:
```
```  2200   fixes x::real shows "1 < x \<Longrightarrow> (x powr a < x powr b) = (a < b)"
```
```  2201   by (blast intro: powr_less_cancel powr_less_mono)
```
```  2202
```
```  2203 lemma powr_le_cancel_iff [simp]:
```
```  2204   fixes x::real shows "1 < x \<Longrightarrow> (x powr a \<le> x powr b) = (a \<le> b)"
```
```  2205   by (simp add: linorder_not_less [symmetric])
```
```  2206
```
```  2207 lemma log_ln: "ln x = log (exp(1)) x"
```
```  2208   by (simp add: log_def)
```
```  2209
```
```  2210 lemma DERIV_log:
```
```  2211   assumes "x > 0"
```
```  2212   shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
```
```  2213 proof -
```
```  2214   def lb \<equiv> "1 / ln b"
```
```  2215   moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
```
```  2216     using \<open>x > 0\<close> by (auto intro!: derivative_eq_intros)
```
```  2217   ultimately show ?thesis
```
```  2218     by (simp add: log_def)
```
```  2219 qed
```
```  2220
```
```  2221 lemmas DERIV_log[THEN DERIV_chain2, derivative_intros]
```
```  2222        DERIV_log[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  2223
```
```  2224 lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
```
```  2225   by (simp add: powr_def log_def)
```
```  2226
```
```  2227 lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y"
```
```  2228   by (simp add: log_def powr_def)
```
```  2229
```
```  2230 lemma log_mult:
```
```  2231   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow>
```
```  2232     log a (x * y) = log a x + log a y"
```
```  2233   by (simp add: log_def ln_mult divide_inverse distrib_right)
```
```  2234
```
```  2235 lemma log_eq_div_ln_mult_log:
```
```  2236   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow>
```
```  2237     log a x = (ln b/ln a) * log b x"
```
```  2238   by (simp add: log_def divide_inverse)
```
```  2239
```
```  2240 text\<open>Base 10 logarithms\<close>
```
```  2241 lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x"
```
```  2242   by (simp add: log_def)
```
```  2243
```
```  2244 lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x"
```
```  2245   by (simp add: log_def)
```
```  2246
```
```  2247 lemma log_one [simp]: "log a 1 = 0"
```
```  2248   by (simp add: log_def)
```
```  2249
```
```  2250 lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
```
```  2251   by (simp add: log_def)
```
```  2252
```
```  2253 lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x"
```
```  2254   apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
```
```  2255   apply (simp add: log_mult [symmetric])
```
```  2256   done
```
```  2257
```
```  2258 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"
```
```  2259   by (simp add: log_mult divide_inverse log_inverse)
```
```  2260
```
```  2261 lemma powr_gt_zero [simp]: "0 < x powr a \<longleftrightarrow> (x::real) \<noteq> 0"
```
```  2262   by (simp add: powr_def)
```
```  2263
```
```  2264 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)"
```
```  2265   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)"
```
```  2266   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)"
```
```  2267   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)"
```
```  2268   by (simp_all add: log_mult log_divide)
```
```  2269
```
```  2270 lemma log_less_cancel_iff [simp]:
```
```  2271   "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
```
```  2272   apply safe
```
```  2273   apply (rule_tac [2] powr_less_cancel)
```
```  2274   apply (drule_tac a = "log a x" in powr_less_mono, auto)
```
```  2275   done
```
```  2276
```
```  2277 lemma log_inj:
```
```  2278   assumes "1 < b"
```
```  2279   shows "inj_on (log b) {0 <..}"
```
```  2280 proof (rule inj_onI, simp)
```
```  2281   fix x y
```
```  2282   assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
```
```  2283   show "x = y"
```
```  2284   proof (cases rule: linorder_cases)
```
```  2285     assume "x = y"
```
```  2286     then show ?thesis by simp
```
```  2287   next
```
```  2288     assume "x < y" hence "log b x < log b y"
```
```  2289       using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp
```
```  2290     then show ?thesis using * by simp
```
```  2291   next
```
```  2292     assume "y < x" hence "log b y < log b x"
```
```  2293       using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp
```
```  2294     then show ?thesis using * by simp
```
```  2295   qed
```
```  2296 qed
```
```  2297
```
```  2298 lemma log_le_cancel_iff [simp]:
```
```  2299   "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (log a x \<le> log a y) = (x \<le> y)"
```
```  2300   by (simp add: linorder_not_less [symmetric])
```
```  2301
```
```  2302 lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
```
```  2303   using log_less_cancel_iff[of a 1 x] by simp
```
```  2304
```
```  2305 lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
```
```  2306   using log_le_cancel_iff[of a 1 x] by simp
```
```  2307
```
```  2308 lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
```
```  2309   using log_less_cancel_iff[of a x 1] by simp
```
```  2310
```
```  2311 lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
```
```  2312   using log_le_cancel_iff[of a x 1] by simp
```
```  2313
```
```  2314 lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
```
```  2315   using log_less_cancel_iff[of a a x] by simp
```
```  2316
```
```  2317 lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
```
```  2318   using log_le_cancel_iff[of a a x] by simp
```
```  2319
```
```  2320 lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
```
```  2321   using log_less_cancel_iff[of a x a] by simp
```
```  2322
```
```  2323 lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
```
```  2324   using log_le_cancel_iff[of a x a] by simp
```
```  2325
```
```  2326 lemma le_log_iff:
```
```  2327   assumes "1 < b" "x > 0"
```
```  2328   shows "y \<le> log b x \<longleftrightarrow> b powr y \<le> (x::real)"
```
```  2329   using assms
```
```  2330   apply auto
```
```  2331   apply (metis (no_types, hide_lams) less_irrefl less_le_trans linear powr_le_cancel_iff
```
```  2332                powr_log_cancel zero_less_one)
```
```  2333   apply (metis not_less order.trans order_refl powr_le_cancel_iff powr_log_cancel zero_le_one)
```
```  2334   done
```
```  2335
```
```  2336 lemma less_log_iff:
```
```  2337   assumes "1 < b" "x > 0"
```
```  2338   shows "y < log b x \<longleftrightarrow> b powr y < x"
```
```  2339   by (metis assms dual_order.strict_trans less_irrefl powr_less_cancel_iff
```
```  2340     powr_log_cancel zero_less_one)
```
```  2341
```
```  2342 lemma
```
```  2343   assumes "1 < b" "x > 0"
```
```  2344   shows log_less_iff: "log b x < y \<longleftrightarrow> x < b powr y"
```
```  2345     and log_le_iff: "log b x \<le> y \<longleftrightarrow> x \<le> b powr y"
```
```  2346   using le_log_iff[OF assms, of y] less_log_iff[OF assms, of y]
```
```  2347   by auto
```
```  2348
```
```  2349 lemmas powr_le_iff = le_log_iff[symmetric]
```
```  2350   and powr_less_iff = le_log_iff[symmetric]
```
```  2351   and less_powr_iff = log_less_iff[symmetric]
```
```  2352   and le_powr_iff = log_le_iff[symmetric]
```
```  2353
```
```  2354 lemma
```
```  2355   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)"
```
```  2356   by (auto simp add: floor_eq_iff powr_le_iff less_powr_iff)
```
```  2357
```
```  2358 lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
```
```  2359   by (induct n) (simp_all add: ac_simps powr_add of_nat_Suc)
```
```  2360
```
```  2361 lemma powr_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
```
```  2362   by (metis of_nat_numeral powr_realpow)
```
```  2363
```
```  2364 lemma powr_real_of_int:
```
```  2365   "x > 0 \<Longrightarrow> x powr real_of_int n = (if n \<ge> 0 then x ^ nat n else inverse (x ^ nat (-n)))"
```
```  2366   using powr_realpow[of x "nat n"] powr_realpow[of x "nat (-n)"]
```
```  2367   by (auto simp: field_simps powr_minus)
```
```  2368
```
```  2369 lemma powr2_sqrt[simp]: "0 < x \<Longrightarrow> sqrt x powr 2 = x"
```
```  2370 by(simp add: powr_numeral)
```
```  2371
```
```  2372 lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
```
```  2373   apply (case_tac "x = 0", simp, simp)
```
```  2374   apply (rule powr_realpow [THEN sym], simp)
```
```  2375   done
```
```  2376
```
```  2377 lemma powr_int:
```
```  2378   assumes "x > 0"
```
```  2379   shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
```
```  2380 proof (cases "i < 0")
```
```  2381   case True
```
```  2382   have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
```
```  2383   show ?thesis using \<open>i < 0\<close> \<open>x > 0\<close> by (simp add: r field_simps powr_realpow[symmetric])
```
```  2384 next
```
```  2385   case False
```
```  2386   then show ?thesis by (simp add: assms powr_realpow[symmetric])
```
```  2387 qed
```
```  2388
```
```  2389 lemma compute_powr[code]:
```
```  2390   fixes i::real
```
```  2391   shows "b powr i =
```
```  2392     (if b \<le> 0 then Code.abort (STR ''op powr with nonpositive base'') (\<lambda>_. b powr i)
```
```  2393     else if \<lfloor>i\<rfloor> = i then (if 0 \<le> i then b ^ nat \<lfloor>i\<rfloor> else 1 / b ^ nat \<lfloor>- i\<rfloor>)
```
```  2394     else Code.abort (STR ''op powr with non-integer exponent'') (\<lambda>_. b powr i))"
```
```  2395   by (auto simp: powr_int)
```
```  2396
```
```  2397 lemma powr_one:
```
```  2398   fixes x::real shows "0 \<le> x \<Longrightarrow> x powr 1 = x"
```
```  2399   using powr_realpow [of x 1]
```
```  2400   by simp
```
```  2401
```
```  2402 lemma powr_neg_one:
```
```  2403   fixes x::real shows "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
```
```  2404   using powr_int [of x "- 1"] by simp
```
```  2405
```
```  2406 lemma powr_neg_numeral:
```
```  2407   fixes x::real shows "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
```
```  2408   using powr_int [of x "- numeral n"] by simp
```
```  2409
```
```  2410 lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
```
```  2411   by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
```
```  2412
```
```  2413 lemma ln_powr:
```
```  2414   fixes x::real shows "x \<noteq> 0 \<Longrightarrow> ln (x powr y) = y * ln x"
```
```  2415   by (simp add: powr_def)
```
```  2416
```
```  2417 lemma ln_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> ln (root n b) =  ln b / n"
```
```  2418 by(simp add: root_powr_inverse ln_powr)
```
```  2419
```
```  2420 lemma ln_sqrt: "0 < x \<Longrightarrow> ln (sqrt x) = ln x / 2"
```
```  2421   by (simp add: ln_powr powr_numeral ln_powr[symmetric] mult.commute)
```
```  2422
```
```  2423 lemma log_root: "\<lbrakk> n > 0; a > 0 \<rbrakk> \<Longrightarrow> log b (root n a) =  log b a / n"
```
```  2424 by(simp add: log_def ln_root)
```
```  2425
```
```  2426 lemma log_powr: "x \<noteq> 0 \<Longrightarrow> log b (x powr y) = y * log b x"
```
```  2427   by (simp add: log_def ln_powr)
```
```  2428
```
```  2429 lemma log_nat_power: "0 < x \<Longrightarrow> log b (x^n) = real n * log b x"
```
```  2430   by (simp add: log_powr powr_realpow [symmetric])
```
```  2431
```
```  2432 lemma log_base_change: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log b x = log a x / log a b"
```
```  2433   by (simp add: log_def)
```
```  2434
```
```  2435 lemma log_base_pow: "0 < a \<Longrightarrow> log (a ^ n) x = log a x / n"
```
```  2436   by (simp add: log_def ln_realpow)
```
```  2437
```
```  2438 lemma log_base_powr: "a \<noteq> 0 \<Longrightarrow> log (a powr b) x = log a x / b"
```
```  2439   by (simp add: log_def ln_powr)
```
```  2440
```
```  2441 lemma log_base_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> log (root n b) x = n * (log b x)"
```
```  2442 by(simp add: log_def ln_root)
```
```  2443
```
```  2444 lemma ln_bound:
```
```  2445   fixes x::real shows "1 <= x ==> ln x <= x"
```
```  2446   apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
```
```  2447   apply simp
```
```  2448   apply (rule ln_add_one_self_le_self, simp)
```
```  2449   done
```
```  2450
```
```  2451 lemma powr_mono:
```
```  2452   fixes x::real shows "a <= b ==> 1 <= x ==> x powr a <= x powr b"
```
```  2453   apply (cases "x = 1", simp)
```
```  2454   apply (cases "a = b", simp)
```
```  2455   apply (rule order_less_imp_le)
```
```  2456   apply (rule powr_less_mono, auto)
```
```  2457   done
```
```  2458
```
```  2459 lemma ge_one_powr_ge_zero:
```
```  2460   fixes x::real shows "1 <= x ==> 0 <= a ==> 1 <= x powr a"
```
```  2461 using powr_mono by fastforce
```
```  2462
```
```  2463 lemma powr_less_mono2:
```
```  2464   fixes x::real shows "0 < a ==> 0 \<le> x ==> x < y ==> x powr a < y powr a"
```
```  2465   by (simp add: powr_def)
```
```  2466
```
```  2467 lemma powr_less_mono2_neg:
```
```  2468   fixes x::real shows "a < 0 ==> 0 < x ==> x < y ==> y powr a < x powr a"
```
```  2469   by (simp add: powr_def)
```
```  2470
```
```  2471 lemma powr_mono2:
```
```  2472   fixes x::real shows "0 <= a ==> 0 \<le> x ==> x <= y ==> x powr a <= y powr a"
```
```  2473   apply (case_tac "a = 0", simp)
```
```  2474   apply (case_tac "x = y", simp)
```
```  2475   apply (metis dual_order.strict_iff_order powr_less_mono2)
```
```  2476   done
```
```  2477
```
```  2478 lemma powr_mono2':
```
```  2479   assumes "a \<le> 0" "x > 0" "x \<le> (y::real)"
```
```  2480   shows   "x powr a \<ge> y powr a"
```
```  2481 proof -
```
```  2482   from assms have "x powr -a \<le> y powr -a" by (intro powr_mono2) simp_all
```
```  2483   with assms show ?thesis by (auto simp add: powr_minus field_simps)
```
```  2484 qed
```
```  2485
```
```  2486 lemma powr_inj:
```
```  2487   fixes x::real shows "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
```
```  2488   unfolding powr_def exp_inj_iff by simp
```
```  2489
```
```  2490 lemma powr_half_sqrt: "0 \<le> x \<Longrightarrow> x powr (1/2) = sqrt x"
```
```  2491   by (simp add: powr_def root_powr_inverse sqrt_def)
```
```  2492
```
```  2493 lemma ln_powr_bound:
```
```  2494   fixes x::real shows "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
```
```  2495 by (metis exp_gt_zero linear ln_eq_zero_iff ln_exp ln_less_self ln_powr mult.commute mult_imp_le_div_pos not_less powr_gt_zero)
```
```  2496
```
```  2497
```
```  2498 lemma ln_powr_bound2:
```
```  2499   fixes x::real
```
```  2500   assumes "1 < x" and "0 < a"
```
```  2501   shows "(ln x) powr a <= (a powr a) * x"
```
```  2502 proof -
```
```  2503   from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
```
```  2504     by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff)
```
```  2505   also have "... = a * (x powr (1 / a))"
```
```  2506     by simp
```
```  2507   finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
```
```  2508     by (metis assms less_imp_le ln_gt_zero powr_mono2)
```
```  2509   also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
```
```  2510     using assms powr_mult by auto
```
```  2511   also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
```
```  2512     by (rule powr_powr)
```
```  2513   also have "... = x" using assms
```
```  2514     by auto
```
```  2515   finally show ?thesis .
```
```  2516 qed
```
```  2517
```
```  2518 lemma tendsto_powr [tendsto_intros]:
```
```  2519   fixes a::real
```
```  2520   assumes f: "(f \<longlongrightarrow> a) F" and g: "(g \<longlongrightarrow> b) F" and a: "a \<noteq> 0"
```
```  2521   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
```
```  2522   unfolding powr_def
```
```  2523 proof (rule filterlim_If)
```
```  2524   from f show "((\<lambda>x. 0) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))"
```
```  2525     by simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_mono dest: t1_space_nhds)
```
```  2526 qed (insert f g a, auto intro!: tendsto_intros intro: tendsto_mono inf_le1)
```
```  2527
```
```  2528 lemma continuous_powr:
```
```  2529   assumes "continuous F f"
```
```  2530     and "continuous F g"
```
```  2531     and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```  2532   shows "continuous F (\<lambda>x. (f x) powr (g x :: real))"
```
```  2533   using assms unfolding continuous_def by (rule tendsto_powr)
```
```  2534
```
```  2535 lemma continuous_at_within_powr[continuous_intros]:
```
```  2536   assumes "continuous (at a within s) f"
```
```  2537     and "continuous (at a within s) g"
```
```  2538     and "f a \<noteq> 0"
```
```  2539   shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x :: real))"
```
```  2540   using assms unfolding continuous_within by (rule tendsto_powr)
```
```  2541
```
```  2542 lemma isCont_powr[continuous_intros, simp]:
```
```  2543   assumes "isCont f a" "isCont g a" "f a \<noteq> (0::real)"
```
```  2544   shows "isCont (\<lambda>x. (f x) powr g x) a"
```
```  2545   using assms unfolding continuous_at by (rule tendsto_powr)
```
```  2546
```
```  2547 lemma continuous_on_powr[continuous_intros]:
```
```  2548   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> (0::real)"
```
```  2549   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
```
```  2550   using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
```
```  2551
```
```  2552 lemma tendsto_powr2:
```
```  2553   fixes a::real
```
```  2554   assumes f: "(f \<longlongrightarrow> a) F" and g: "(g \<longlongrightarrow> b) F" and f_nonneg: "\<forall>\<^sub>F x in F. 0 \<le> f x" and b: "0 < b"
```
```  2555   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
```
```  2556   unfolding powr_def
```
```  2557 proof (rule filterlim_If)
```
```  2558   from f show "((\<lambda>x. 0) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))"
```
```  2559     by simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_mono dest: t1_space_nhds)
```
```  2560 next
```
```  2561   { assume "a = 0"
```
```  2562     with f f_nonneg have "LIM x inf F (principal {x. f x \<noteq> 0}). f x :> at_right 0"
```
```  2563       by (auto simp add: filterlim_at eventually_inf_principal le_less
```
```  2564                elim: eventually_mono intro: tendsto_mono inf_le1)
```
```  2565     then have "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> 0) (inf F (principal {x. f x \<noteq> 0}))"
```
```  2566       by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_0]
```
```  2567                        filterlim_tendsto_pos_mult_at_bot[OF _ \<open>0 < b\<close>]
```
```  2568                intro: tendsto_mono inf_le1 g) }
```
```  2569   then show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x \<noteq> 0}))"
```
```  2570     using f g by (auto intro!: tendsto_intros intro: tendsto_mono inf_le1)
```
```  2571 qed
```
```  2572
```
```  2573 lemma DERIV_powr:
```
```  2574   fixes r::real
```
```  2575   assumes g: "DERIV g x :> m" and pos: "g x > 0" and f: "DERIV f x :> r"
```
```  2576   shows  "DERIV (\<lambda>x. g x powr f x) x :> (g x powr f x) * (r * ln (g x) + m * f x / g x)"
```
```  2577 proof -
```
```  2578   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)"
```
```  2579     using pos
```
```  2580     by (auto intro!: derivative_eq_intros g pos f simp: powr_def field_simps exp_diff)
```
```  2581   then show ?thesis
```
```  2582   proof (rule DERIV_cong_ev[OF refl _ refl, THEN iffD1, rotated])
```
```  2583     from DERIV_isCont[OF g] pos have "\<forall>\<^sub>F x in at x. 0 < g x"
```
```  2584       unfolding isCont_def by (rule order_tendstoD(1))
```
```  2585     with pos show "\<forall>\<^sub>F x in nhds x. exp (f x * ln (g x)) = g x powr f x"
```
```  2586       by (auto simp: eventually_at_filter powr_def elim: eventually_mono)
```
```  2587   qed
```
```  2588 qed
```
```  2589
```
```  2590 lemma DERIV_fun_powr:
```
```  2591   fixes r::real
```
```  2592   assumes g: "DERIV g x :> m" and pos: "g x > 0"
```
```  2593   shows  "DERIV (\<lambda>x. (g x) powr r) x :> r * (g x) powr (r - of_nat 1) * m"
```
```  2594   using DERIV_powr[OF g pos DERIV_const, of r] pos
```
```  2595   by (simp add: powr_divide2[symmetric] field_simps)
```
```  2596
```
```  2597 lemma has_real_derivative_powr:
```
```  2598   assumes "z > 0"
```
```  2599   shows "((\<lambda>z. z powr r) has_real_derivative r * z powr (r - 1)) (at z)"
```
```  2600 proof (subst DERIV_cong_ev[OF refl _ refl])
```
```  2601   from assms have "eventually (\<lambda>z. z \<noteq> 0) (nhds z)" by (intro t1_space_nhds) auto
```
```  2602   thus "eventually (\<lambda>z. z powr r = exp (r * ln z)) (nhds z)"
```
```  2603     unfolding powr_def by eventually_elim simp
```
```  2604   from assms show "((\<lambda>z. exp (r * ln z)) has_real_derivative r * z powr (r - 1)) (at z)"
```
```  2605     by (auto intro!: derivative_eq_intros simp: powr_def field_simps exp_diff)
```
```  2606 qed
```
```  2607
```
```  2608 declare has_real_derivative_powr[THEN DERIV_chain2, derivative_intros]
```
```  2609
```
```  2610 lemma tendsto_zero_powrI:
```
```  2611   assumes "(f \<longlongrightarrow> (0::real)) F" "(g \<longlongrightarrow> b) F" "\<forall>\<^sub>F x in F. 0 \<le> f x" "0 < b"
```
```  2612   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> 0) F"
```
```  2613   using tendsto_powr2[OF assms] by simp
```
```  2614
```
```  2615 lemma tendsto_neg_powr:
```
```  2616   assumes "s < 0"
```
```  2617     and f: "LIM x F. f x :> at_top"
```
```  2618   shows "((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F"
```
```  2619 proof -
```
```  2620   have "((\<lambda>x. exp (s * ln (f x))) \<longlongrightarrow> (0::real)) F" (is "?X")
```
```  2621     by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_top]
```
```  2622                      filterlim_tendsto_neg_mult_at_bot assms)
```
```  2623   also have "?X \<longleftrightarrow> ((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F"
```
```  2624     using f filterlim_at_top_dense[of f F]
```
```  2625     by (intro filterlim_cong[OF refl refl]) (auto simp: neq_iff powr_def elim: eventually_mono)
```
```  2626   finally show ?thesis .
```
```  2627 qed
```
```  2628
```
```  2629 lemma tendsto_exp_limit_at_right:
```
```  2630   fixes x :: real
```
```  2631   shows "((\<lambda>y. (1 + x * y) powr (1 / y)) \<longlongrightarrow> exp x) (at_right 0)"
```
```  2632 proof cases
```
```  2633   assume "x \<noteq> 0"
```
```  2634   have "((\<lambda>y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)"
```
```  2635     by (auto intro!: derivative_eq_intros)
```
```  2636   then have "((\<lambda>y. ln (1 + x * y) / y) \<longlongrightarrow> x) (at 0)"
```
```  2637     by (auto simp add: has_field_derivative_def field_has_derivative_at)
```
```  2638   then have *: "((\<lambda>y. exp (ln (1 + x * y) / y)) \<longlongrightarrow> exp x) (at 0)"
```
```  2639     by (rule tendsto_intros)
```
```  2640   then show ?thesis
```
```  2641   proof (rule filterlim_mono_eventually)
```
```  2642     show "eventually (\<lambda>xa. exp (ln (1 + x * xa) / xa) = (1 + x * xa) powr (1 / xa)) (at_right 0)"
```
```  2643       unfolding eventually_at_right[OF zero_less_one]
```
```  2644       using \<open>x \<noteq> 0\<close>
```
```  2645       apply  (intro exI[of _ "1 / \<bar>x\<bar>"])
```
```  2646       apply (auto simp: field_simps powr_def abs_if)
```
```  2647       by (metis add_less_same_cancel1 mult_less_0_iff not_less_iff_gr_or_eq zero_less_one)
```
```  2648   qed (simp_all add: at_eq_sup_left_right)
```
```  2649 qed simp
```
```  2650
```
```  2651 lemma tendsto_exp_limit_at_top:
```
```  2652   fixes x :: real
```
```  2653   shows "((\<lambda>y. (1 + x / y) powr y) \<longlongrightarrow> exp x) at_top"
```
```  2654   apply (subst filterlim_at_top_to_right)
```
```  2655   apply (simp add: inverse_eq_divide)
```
```  2656   apply (rule tendsto_exp_limit_at_right)
```
```  2657   done
```
```  2658
```
```  2659 lemma tendsto_exp_limit_sequentially:
```
```  2660   fixes x :: real
```
```  2661   shows "(\<lambda>n. (1 + x / n) ^ n) \<longlonglongrightarrow> exp x"
```
```  2662 proof (rule filterlim_mono_eventually)
```
```  2663   from reals_Archimedean2 [of "\<bar>x\<bar>"] obtain n :: nat where *: "real n > \<bar>x\<bar>" ..
```
```  2664   hence "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top"
```
```  2665     apply (intro eventually_sequentiallyI [of n])
```
```  2666     apply (case_tac "x \<ge> 0")
```
```  2667     apply (rule add_pos_nonneg, auto intro: divide_nonneg_nonneg)
```
```  2668     apply (subgoal_tac "x / real xa > -1")
```
```  2669     apply (auto simp add: field_simps)
```
```  2670     done
```
```  2671   then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top"
```
```  2672     by (rule eventually_mono) (erule powr_realpow)
```
```  2673   show "(\<lambda>n. (1 + x / real n) powr real n) \<longlonglongrightarrow> exp x"
```
```  2674     by (rule filterlim_compose [OF tendsto_exp_limit_at_top filterlim_real_sequentially])
```
```  2675 qed auto
```
```  2676
```
```  2677 subsection \<open>Sine and Cosine\<close>
```
```  2678
```
```  2679 definition sin_coeff :: "nat \<Rightarrow> real" where
```
```  2680   "sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / (fact n))"
```
```  2681
```
```  2682 definition cos_coeff :: "nat \<Rightarrow> real" where
```
```  2683   "cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / (fact n) else 0)"
```
```  2684
```
```  2685 definition sin :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```  2686   where "sin = (\<lambda>x. \<Sum>n. sin_coeff n *\<^sub>R x^n)"
```
```  2687
```
```  2688 definition cos :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```  2689   where "cos = (\<lambda>x. \<Sum>n. cos_coeff n *\<^sub>R x^n)"
```
```  2690
```
```  2691 lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
```
```  2692   unfolding sin_coeff_def by simp
```
```  2693
```
```  2694 lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
```
```  2695   unfolding cos_coeff_def by simp
```
```  2696
```
```  2697 lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
```
```  2698   unfolding cos_coeff_def sin_coeff_def
```
```  2699   by (simp del: mult_Suc)
```
```  2700
```
```  2701 lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
```
```  2702   unfolding cos_coeff_def sin_coeff_def
```
```  2703   by (simp del: mult_Suc) (auto elim: oddE)
```
```  2704
```
```  2705 lemma summable_norm_sin:
```
```  2706   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  2707   shows "summable (\<lambda>n. norm (sin_coeff n *\<^sub>R x^n))"
```
```  2708   unfolding sin_coeff_def
```
```  2709   apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]])
```
```  2710   apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2711   done
```
```  2712
```
```  2713 lemma summable_norm_cos:
```
```  2714   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  2715   shows "summable (\<lambda>n. norm (cos_coeff n *\<^sub>R x^n))"
```
```  2716   unfolding cos_coeff_def
```
```  2717   apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]])
```
```  2718   apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2719   done
```
```  2720
```
```  2721 lemma sin_converges: "(\<lambda>n. sin_coeff n *\<^sub>R x^n) sums sin(x)"
```
```  2722 unfolding sin_def
```
```  2723   by (metis (full_types) summable_norm_cancel summable_norm_sin summable_sums)
```
```  2724
```
```  2725 lemma cos_converges: "(\<lambda>n. cos_coeff n *\<^sub>R x^n) sums cos(x)"
```
```  2726 unfolding cos_def
```
```  2727   by (metis (full_types) summable_norm_cancel summable_norm_cos summable_sums)
```
```  2728
```
```  2729 lemma sin_of_real:
```
```  2730   fixes x::real
```
```  2731   shows "sin (of_real x) = of_real (sin x)"
```
```  2732 proof -
```
```  2733   have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R  x^n)) = (\<lambda>n. sin_coeff n *\<^sub>R  (of_real x)^n)"
```
```  2734   proof
```
```  2735     fix n
```
```  2736     show "of_real (sin_coeff n *\<^sub>R  x^n) = sin_coeff n *\<^sub>R of_real x^n"
```
```  2737       by (simp add: scaleR_conv_of_real)
```
```  2738   qed
```
```  2739   also have "... sums (sin (of_real x))"
```
```  2740     by (rule sin_converges)
```
```  2741   finally have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) sums (sin (of_real x))" .
```
```  2742   then show ?thesis
```
```  2743     using sums_unique2 sums_of_real [OF sin_converges]
```
```  2744     by blast
```
```  2745 qed
```
```  2746
```
```  2747 corollary sin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> sin z \<in> \<real>"
```
```  2748   by (metis Reals_cases Reals_of_real sin_of_real)
```
```  2749
```
```  2750 lemma cos_of_real:
```
```  2751   fixes x::real
```
```  2752   shows "cos (of_real x) = of_real (cos x)"
```
```  2753 proof -
```
```  2754   have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R  x^n)) = (\<lambda>n. cos_coeff n *\<^sub>R  (of_real x)^n)"
```
```  2755   proof
```
```  2756     fix n
```
```  2757     show "of_real (cos_coeff n *\<^sub>R  x^n) = cos_coeff n *\<^sub>R of_real x^n"
```
```  2758       by (simp add: scaleR_conv_of_real)
```
```  2759   qed
```
```  2760   also have "... sums (cos (of_real x))"
```
```  2761     by (rule cos_converges)
```
```  2762   finally have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) sums (cos (of_real x))" .
```
```  2763   then show ?thesis
```
```  2764     using sums_unique2 sums_of_real [OF cos_converges]
```
```  2765     by blast
```
```  2766 qed
```
```  2767
```
```  2768 corollary cos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cos z \<in> \<real>"
```
```  2769   by (metis Reals_cases Reals_of_real cos_of_real)
```
```  2770
```
```  2771 lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
```
```  2772   by (simp add: diffs_def sin_coeff_Suc del: of_nat_Suc)
```
```  2773
```
```  2774 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
```
```  2775   by (simp add: diffs_def cos_coeff_Suc del: of_nat_Suc)
```
```  2776
```
```  2777 text\<open>Now at last we can get the derivatives of exp, sin and cos\<close>
```
```  2778
```
```  2779 lemma DERIV_sin [simp]:
```
```  2780   fixes x :: "'a::{real_normed_field,banach}"
```
```  2781   shows "DERIV sin x :> cos(x)"
```
```  2782   unfolding sin_def cos_def scaleR_conv_of_real
```
```  2783   apply (rule DERIV_cong)
```
```  2784   apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
```
```  2785   apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff
```
```  2786               summable_minus_iff scaleR_conv_of_real [symmetric]
```
```  2787               summable_norm_sin [THEN summable_norm_cancel]
```
```  2788               summable_norm_cos [THEN summable_norm_cancel])
```
```  2789   done
```
```  2790
```
```  2791 declare DERIV_sin[THEN DERIV_chain2, derivative_intros]
```
```  2792         DERIV_sin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  2793
```
```  2794 lemma DERIV_cos [simp]:
```
```  2795   fixes x :: "'a::{real_normed_field,banach}"
```
```  2796   shows "DERIV cos x :> -sin(x)"
```
```  2797   unfolding sin_def cos_def scaleR_conv_of_real
```
```  2798   apply (rule DERIV_cong)
```
```  2799   apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
```
```  2800   apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus
```
```  2801               diffs_sin_coeff diffs_cos_coeff
```
```  2802               summable_minus_iff scaleR_conv_of_real [symmetric]
```
```  2803               summable_norm_sin [THEN summable_norm_cancel]
```
```  2804               summable_norm_cos [THEN summable_norm_cancel])
```
```  2805   done
```
```  2806
```
```  2807 declare DERIV_cos[THEN DERIV_chain2, derivative_intros]
```
```  2808         DERIV_cos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  2809
```
```  2810 lemma isCont_sin:
```
```  2811   fixes x :: "'a::{real_normed_field,banach}"
```
```  2812   shows "isCont sin x"
```
```  2813   by (rule DERIV_sin [THEN DERIV_isCont])
```
```  2814
```
```  2815 lemma isCont_cos:
```
```  2816   fixes x :: "'a::{real_normed_field,banach}"
```
```  2817   shows "isCont cos x"
```
```  2818   by (rule DERIV_cos [THEN DERIV_isCont])
```
```  2819
```
```  2820 lemma isCont_sin' [simp]:
```
```  2821   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2822   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
```
```  2823   by (rule isCont_o2 [OF _ isCont_sin])
```
```  2824
```
```  2825 (*FIXME A CONTEXT FOR F WOULD BE BETTER*)
```
```  2826
```
```  2827 lemma isCont_cos' [simp]:
```
```  2828   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2829   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
```
```  2830   by (rule isCont_o2 [OF _ isCont_cos])
```
```  2831
```
```  2832 lemma tendsto_sin [tendsto_intros]:
```
```  2833   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2834   shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) \<longlongrightarrow> sin a) F"
```
```  2835   by (rule isCont_tendsto_compose [OF isCont_sin])
```
```  2836
```
```  2837 lemma tendsto_cos [tendsto_intros]:
```
```  2838   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2839   shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) \<longlongrightarrow> cos a) F"
```
```  2840   by (rule isCont_tendsto_compose [OF isCont_cos])
```
```  2841
```
```  2842 lemma continuous_sin [continuous_intros]:
```
```  2843   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2844   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
```
```  2845   unfolding continuous_def by (rule tendsto_sin)
```
```  2846
```
```  2847 lemma continuous_on_sin [continuous_intros]:
```
```  2848   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2849   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
```
```  2850   unfolding continuous_on_def by (auto intro: tendsto_sin)
```
```  2851
```
```  2852 lemma continuous_within_sin:
```
```  2853   fixes z :: "'a::{real_normed_field,banach}"
```
```  2854   shows "continuous (at z within s) sin"
```
```  2855   by (simp add: continuous_within tendsto_sin)
```
```  2856
```
```  2857 lemma continuous_cos [continuous_intros]:
```
```  2858   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2859   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
```
```  2860   unfolding continuous_def by (rule tendsto_cos)
```
```  2861
```
```  2862 lemma continuous_on_cos [continuous_intros]:
```
```  2863   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2864   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
```
```  2865   unfolding continuous_on_def by (auto intro: tendsto_cos)
```
```  2866
```
```  2867 lemma continuous_within_cos:
```
```  2868   fixes z :: "'a::{real_normed_field,banach}"
```
```  2869   shows "continuous (at z within s) cos"
```
```  2870   by (simp add: continuous_within tendsto_cos)
```
```  2871
```
```  2872 subsection \<open>Properties of Sine and Cosine\<close>
```
```  2873
```
```  2874 lemma sin_zero [simp]: "sin 0 = 0"
```
```  2875   unfolding sin_def sin_coeff_def by (simp add: scaleR_conv_of_real powser_zero)
```
```  2876
```
```  2877 lemma cos_zero [simp]: "cos 0 = 1"
```
```  2878   unfolding cos_def cos_coeff_def by (simp add: scaleR_conv_of_real powser_zero)
```
```  2879
```
```  2880 lemma DERIV_fun_sin:
```
```  2881      "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. sin(g x)) x :> cos(g x) * m"
```
```  2882   by (auto intro!: derivative_intros)
```
```  2883
```
```  2884 lemma DERIV_fun_cos:
```
```  2885      "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. cos(g x)) x :> -sin(g x) * m"
```
```  2886   by (auto intro!: derivative_eq_intros)
```
```  2887
```
```  2888 subsection \<open>Deriving the Addition Formulas\<close>
```
```  2889
```
```  2890 text\<open>The the product of two cosine series\<close>
```
```  2891 lemma cos_x_cos_y:
```
```  2892   fixes x :: "'a::{real_normed_field,banach}"
```
```  2893   shows "(\<lambda>p. \<Sum>n\<le>p.
```
```  2894           if even p \<and> even n
```
```  2895           then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
```
```  2896          sums (cos x * cos y)"
```
```  2897 proof -
```
```  2898   { fix n p::nat
```
```  2899     assume "n\<le>p"
```
```  2900     then have *: "even n \<Longrightarrow> even p \<Longrightarrow> (-1) ^ (n div 2) * (-1) ^ ((p - n) div 2) = (-1 :: real) ^ (p div 2)"
```
```  2901       by (metis div_add power_add le_add_diff_inverse odd_add)
```
```  2902     have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
```
```  2903           (if even p \<and> even n then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
```
```  2904     using \<open>n\<le>p\<close>
```
```  2905       by (auto simp: * algebra_simps cos_coeff_def binomial_fact)
```
```  2906   }
```
```  2907   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> even n
```
```  2908                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
```
```  2909              (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
```
```  2910     by simp
```
```  2911   also have "... = (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n *\<^sub>R x^n) * (cos_coeff (p - n) *\<^sub>R y^(p-n)))"
```
```  2912     by (simp add: algebra_simps)
```
```  2913   also have "... sums (cos x * cos y)"
```
```  2914     using summable_norm_cos
```
```  2915     by (auto simp: cos_def scaleR_conv_of_real intro!: Cauchy_product_sums)
```
```  2916   finally show ?thesis .
```
```  2917 qed
```
```  2918
```
```  2919 text\<open>The product of two sine series\<close>
```
```  2920 lemma sin_x_sin_y:
```
```  2921   fixes x :: "'a::{real_normed_field,banach}"
```
```  2922   shows "(\<lambda>p. \<Sum>n\<le>p.
```
```  2923           if even p \<and> odd n
```
```  2924                then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
```
```  2925          sums (sin x * sin y)"
```
```  2926 proof -
```
```  2927   { fix n p::nat
```
```  2928     assume "n\<le>p"
```
```  2929     { assume np: "odd n" "even p"
```
```  2930         with \<open>n\<le>p\<close> have "n - Suc 0 + (p - Suc n) = p - Suc (Suc 0)" "Suc (Suc 0) \<le> p"
```
```  2931         by arith+
```
```  2932       moreover have "(p - Suc (Suc 0)) div 2 = p div 2 - Suc 0"
```
```  2933         by simp
```
```  2934       ultimately have *: "(-1) ^ ((n - Suc 0) div 2) * (-1) ^ ((p - Suc n) div 2) = - ((-1 :: real) ^ (p div 2))"
```
```  2935         using np \<open>n\<le>p\<close>
```
```  2936         apply (simp add: power_add [symmetric] div_add [symmetric] del: div_add)
```
```  2937         apply (metis (no_types) One_nat_def Suc_1 le_div_geq minus_minus mult.left_neutral mult_minus_left power.simps(2) zero_less_Suc)
```
```  2938         done
```
```  2939     } then
```
```  2940     have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
```
```  2941           (if even p \<and> odd n
```
```  2942           then -((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
```
```  2943     using \<open>n\<le>p\<close>
```
```  2944       by (auto simp:  algebra_simps sin_coeff_def binomial_fact)
```
```  2945   }
```
```  2946   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> odd n
```
```  2947                then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
```
```  2948              (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
```
```  2949     by simp
```
```  2950   also have "... = (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n *\<^sub>R x^n) * (sin_coeff (p - n) *\<^sub>R y^(p-n)))"
```
```  2951     by (simp add: algebra_simps)
```
```  2952   also have "... sums (sin x * sin y)"
```
```  2953     using summable_norm_sin
```
```  2954     by (auto simp: sin_def scaleR_conv_of_real intro!: Cauchy_product_sums)
```
```  2955   finally show ?thesis .
```
```  2956 qed
```
```  2957
```
```  2958 lemma sums_cos_x_plus_y:
```
```  2959   fixes x :: "'a::{real_normed_field,banach}"
```
```  2960   shows
```
```  2961   "(\<lambda>p. \<Sum>n\<le>p. if even p
```
```  2962                then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2963                else 0)
```
```  2964         sums cos (x + y)"
```
```  2965 proof -
```
```  2966   { fix p::nat
```
```  2967     have "(\<Sum>n\<le>p. if even p
```
```  2968                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2969                   else 0) =
```
```  2970           (if even p
```
```  2971                   then \<Sum>n\<le>p. ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2972                   else 0)"
```
```  2973       by simp
```
```  2974     also have "... = (if even p
```
```  2975                   then of_real ((-1) ^ (p div 2) / (fact p)) * (\<Sum>n\<le>p. (p choose n) *\<^sub>R (x^n) * y^(p-n))
```
```  2976                   else 0)"
```
```  2977       by (auto simp: setsum_right_distrib field_simps scaleR_conv_of_real nonzero_of_real_divide)
```
```  2978     also have "... = cos_coeff p *\<^sub>R ((x + y) ^ p)"
```
```  2979       by (simp add: cos_coeff_def binomial_ring [of x y]  scaleR_conv_of_real atLeast0AtMost)
```
```  2980     finally have "(\<Sum>n\<le>p. if even p
```
```  2981                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2982                   else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)" .
```
```  2983   }
```
```  2984   then have "(\<lambda>p. \<Sum>n\<le>p.
```
```  2985                if even p
```
```  2986                then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2987                else 0)
```
```  2988         = (\<lambda>p. cos_coeff p *\<^sub>R ((x+y)^p))"
```
```  2989         by simp
```
```  2990    also have "... sums cos (x + y)"
```
```  2991     by (rule cos_converges)
```
```  2992    finally show ?thesis .
```
```  2993 qed
```
```  2994
```
```  2995 theorem cos_add:
```
```  2996   fixes x :: "'a::{real_normed_field,banach}"
```
```  2997   shows "cos (x + y) = cos x * cos y - sin x * sin y"
```
```  2998 proof -
```
```  2999   { fix n p::nat
```
```  3000     assume "n\<le>p"
```
```  3001     then have "(if even p \<and> even n
```
```  3002                then ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) -
```
```  3003           (if even p \<and> odd n
```
```  3004                then - ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
```
```  3005           = (if even p
```
```  3006                then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
```
```  3007       by simp
```
```  3008   }
```
```  3009   then have "(\<lambda>p. \<Sum>n\<le>p. (if even p
```
```  3010                then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0))
```
```  3011         sums (cos x * cos y - sin x * sin y)"
```
```  3012     using sums_diff [OF cos_x_cos_y [of x y] sin_x_sin_y [of x y]]
```
```  3013     by (simp add: setsum_subtractf [symmetric])
```
```  3014   then show ?thesis
```
```  3015     by (blast intro: sums_cos_x_plus_y sums_unique2)
```
```  3016 qed
```
```  3017
```
```  3018 lemma sin_minus_converges: "(\<lambda>n. - (sin_coeff n *\<^sub>R (-x)^n)) sums sin(x)"
```
```  3019 proof -
```
```  3020   have [simp]: "\<And>n. - (sin_coeff n *\<^sub>R (-x)^n) = (sin_coeff n *\<^sub>R x^n)"
```
```  3021     by (auto simp: sin_coeff_def elim!: oddE)
```
```  3022   show ?thesis
```
```  3023     by (simp add: sin_def summable_norm_sin [THEN summable_norm_cancel, THEN summable_sums])
```
```  3024 qed
```
```  3025
```
```  3026 lemma sin_minus [simp]:
```
```  3027   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  3028   shows "sin (-x) = -sin(x)"
```
```  3029 using sin_minus_converges [of x]
```
```  3030 by (auto simp: sin_def summable_norm_sin [THEN summable_norm_cancel] suminf_minus sums_iff equation_minus_iff)
```
```  3031
```
```  3032 lemma cos_minus_converges: "(\<lambda>n. (cos_coeff n *\<^sub>R (-x)^n)) sums cos(x)"
```
```  3033 proof -
```
```  3034   have [simp]: "\<And>n. (cos_coeff n *\<^sub>R (-x)^n) = (cos_coeff n *\<^sub>R x^n)"
```
```  3035     by (auto simp: Transcendental.cos_coeff_def elim!: evenE)
```
```  3036   show ?thesis
```
```  3037     by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel, THEN summable_sums])
```
```  3038 qed
```
```  3039
```
```  3040 lemma cos_minus [simp]:
```
```  3041   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  3042   shows "cos (-x) = cos(x)"
```
```  3043 using cos_minus_converges [of x]
```
```  3044 by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel]
```
```  3045               suminf_minus sums_iff equation_minus_iff)
```
```  3046
```
```  3047 lemma sin_cos_squared_add [simp]:
```
```  3048   fixes x :: "'a::{real_normed_field,banach}"
```
```  3049   shows "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
```
```  3050 using cos_add [of x "-x"]
```
```  3051 by (simp add: power2_eq_square algebra_simps)
```
```  3052
```
```  3053 lemma sin_cos_squared_add2 [simp]:
```
```  3054   fixes x :: "'a::{real_normed_field,banach}"
```
```  3055   shows "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
```
```  3056   by (subst add.commute, rule sin_cos_squared_add)
```
```  3057
```
```  3058 lemma sin_cos_squared_add3 [simp]:
```
```  3059   fixes x :: "'a::{real_normed_field,banach}"
```
```  3060   shows "cos x * cos x + sin x * sin x = 1"
```
```  3061   using sin_cos_squared_add2 [unfolded power2_eq_square] .
```
```  3062
```
```  3063 lemma sin_squared_eq:
```
```  3064   fixes x :: "'a::{real_normed_field,banach}"
```
```  3065   shows "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
```
```  3066   unfolding eq_diff_eq by (rule sin_cos_squared_add)
```
```  3067
```
```  3068 lemma cos_squared_eq:
```
```  3069   fixes x :: "'a::{real_normed_field,banach}"
```
```  3070   shows "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
```
```  3071   unfolding eq_diff_eq by (rule sin_cos_squared_add2)
```
```  3072
```
```  3073 lemma abs_sin_le_one [simp]:
```
```  3074   fixes x :: real
```
```  3075   shows "\<bar>sin x\<bar> \<le> 1"
```
```  3076   by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
```
```  3077
```
```  3078 lemma sin_ge_minus_one [simp]:
```
```  3079   fixes x :: real
```
```  3080   shows "-1 \<le> sin x"
```
```  3081   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  3082
```
```  3083 lemma sin_le_one [simp]:
```
```  3084   fixes x :: real
```
```  3085   shows "sin x \<le> 1"
```
```  3086   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  3087
```
```  3088 lemma abs_cos_le_one [simp]:
```
```  3089   fixes x :: real
```
```  3090   shows "\<bar>cos x\<bar> \<le> 1"
```
```  3091   by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
```
```  3092
```
```  3093 lemma cos_ge_minus_one [simp]:
```
```  3094   fixes x :: real
```
```  3095   shows "-1 \<le> cos x"
```
```  3096   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  3097
```
```  3098 lemma cos_le_one [simp]:
```
```  3099   fixes x :: real
```
```  3100   shows "cos x \<le> 1"
```
```  3101   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  3102
```
```  3103 lemma cos_diff:
```
```  3104   fixes x :: "'a::{real_normed_field,banach}"
```
```  3105   shows "cos (x - y) = cos x * cos y + sin x * sin y"
```
```  3106   using cos_add [of x "- y"] by simp
```
```  3107
```
```  3108 lemma cos_double:
```
```  3109   fixes x :: "'a::{real_normed_field,banach}"
```
```  3110   shows "cos(2*x) = (cos x)\<^sup>2 - (sin x)\<^sup>2"
```
```  3111   using cos_add [where x=x and y=x]
```
```  3112   by (simp add: power2_eq_square)
```
```  3113
```
```  3114 lemma sin_cos_le1:
```
```  3115   fixes x::real shows "\<bar>sin x * sin y + cos x * cos y\<bar> \<le> 1"
```
```  3116   using cos_diff [of x y]
```
```  3117   by (metis abs_cos_le_one add.commute)
```
```  3118
```
```  3119 lemma DERIV_fun_pow: "DERIV g x :> m ==>
```
```  3120       DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
```
```  3121   by (auto intro!: derivative_eq_intros simp:)
```
```  3122
```
```  3123 lemma DERIV_fun_exp:
```
```  3124      "DERIV g x :> m ==> DERIV (\<lambda>x. exp(g x)) x :> exp(g x) * m"
```
```  3125   by (auto intro!: derivative_intros)
```
```  3126
```
```  3127 subsection \<open>The Constant Pi\<close>
```
```  3128
```
```  3129 definition pi :: real
```
```  3130   where "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
```
```  3131
```
```  3132 text\<open>Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
```
```  3133    hence define pi.\<close>
```
```  3134
```
```  3135 lemma sin_paired:
```
```  3136   fixes x :: real
```
```  3137   shows "(\<lambda>n. (- 1) ^ n / (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums  sin x"
```
```  3138 proof -
```
```  3139   have "(\<lambda>n. \<Sum>k = n*2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
```
```  3140     apply (rule sums_group)
```
```  3141     using sin_converges [of x, unfolded scaleR_conv_of_real]
```
```  3142     by auto
```
```  3143   thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: ac_simps)
```
```  3144 qed
```
```  3145
```
```  3146 lemma sin_gt_zero_02:
```
```  3147   fixes x :: real
```
```  3148   assumes "0 < x" and "x < 2"
```
```  3149   shows "0 < sin x"
```
```  3150 proof -
```
```  3151   let ?f = "\<lambda>n::nat. \<Sum>k = n*2..<n*2+2. (- 1) ^ k / (fact (2*k+1)) * x^(2*k+1)"
```
```  3152   have pos: "\<forall>n. 0 < ?f n"
```
```  3153   proof
```
```  3154     fix n :: nat
```
```  3155     let ?k2 = "real (Suc (Suc (4 * n)))"
```
```  3156     let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
```
```  3157     have "x * x < ?k2 * ?k3"
```
```  3158       using assms by (intro mult_strict_mono', simp_all)
```
```  3159     hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
```
```  3160       by (intro mult_strict_right_mono zero_less_power \<open>0 < x\<close>)
```
```  3161     thus "0 < ?f n"
```
```  3162       by (simp add: divide_simps mult_ac del: mult_Suc)
```
```  3163 qed
```
```  3164   have sums: "?f sums sin x"
```
```  3165     by (rule sin_paired [THEN sums_group], simp)
```
```  3166   show "0 < sin x"
```
```  3167     unfolding sums_unique [OF sums]
```
```  3168     using sums_summable [OF sums] pos
```
```  3169     by (rule suminf_pos)
```
```  3170 qed
```
```  3171
```
```  3172 lemma cos_double_less_one:
```
```  3173   fixes x :: real
```
```  3174   shows "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
```
```  3175   using sin_gt_zero_02 [where x = x] by (auto simp: cos_squared_eq cos_double)
```
```  3176
```
```  3177 lemma cos_paired:
```
```  3178   fixes x :: real
```
```  3179   shows "(\<lambda>n. (- 1) ^ n / (fact (2 * n)) * x ^ (2 * n)) sums cos x"
```
```  3180 proof -
```
```  3181   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
```
```  3182     apply (rule sums_group)
```
```  3183     using cos_converges [of x, unfolded scaleR_conv_of_real]
```
```  3184     by auto
```
```  3185   thus ?thesis unfolding cos_coeff_def by (simp add: ac_simps)
```
```  3186 qed
```
```  3187
```
```  3188 lemmas realpow_num_eq_if = power_eq_if
```
```  3189
```
```  3190 lemma sumr_pos_lt_pair:
```
```  3191   fixes f :: "nat \<Rightarrow> real"
```
```  3192   shows "\<lbrakk>summable f;
```
```  3193         \<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk>
```
```  3194       \<Longrightarrow> setsum f {..<k} < suminf f"
```
```  3195 unfolding One_nat_def
```
```  3196 apply (subst suminf_split_initial_segment [where k=k], assumption, simp)
```
```  3197 apply (drule_tac k=k in summable_ignore_initial_segment)
```
```  3198 apply (drule_tac k="Suc (Suc 0)" in sums_group [OF summable_sums], simp)
```
```  3199 apply simp
```
```  3200 by (metis (no_types, lifting) add.commute suminf_pos summable_def sums_unique)
```
```  3201
```
```  3202 lemma cos_two_less_zero [simp]:
```
```  3203   "cos 2 < (0::real)"
```
```  3204 proof -
```
```  3205   note fact.simps(2) [simp del]
```
```  3206   from sums_minus [OF cos_paired]
```
```  3207   have *: "(\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / fact (2 * n))) sums - cos (2::real)"
```
```  3208     by simp
```
```  3209   then have sm: "summable (\<lambda>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3210     by (rule sums_summable)
```
```  3211   have "0 < (\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3212     by (simp add: fact_num_eq_if realpow_num_eq_if)
```
```  3213   moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n  * 2 ^ (2 * n) / (fact (2 * n))))
```
```  3214                  < (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3215   proof -
```
```  3216     { fix d
```
```  3217       let ?six4d = "Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))"
```
```  3218       have "(4::real) * (fact (?six4d)) < (Suc (Suc (?six4d)) * fact (Suc (?six4d)))"
```
```  3219         unfolding of_nat_mult   by (rule mult_strict_mono) (simp_all add: fact_less_mono)
```
```  3220       then have "(4::real) * (fact (?six4d)) < (fact (Suc (Suc (?six4d))))"
```
```  3221         by (simp only: fact.simps(2) [of "Suc (?six4d)"] of_nat_mult of_nat_fact)
```
```  3222       then have "(4::real) * inverse (fact (Suc (Suc (?six4d)))) < inverse (fact (?six4d))"
```
```  3223         by (simp add: inverse_eq_divide less_divide_eq)
```
```  3224     }
```
```  3225     then show ?thesis
```
```  3226       by (force intro!: sumr_pos_lt_pair [OF sm] simp add: divide_inverse algebra_simps)
```
```  3227   qed
```
```  3228   ultimately have "0 < (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3229     by (rule order_less_trans)
```
```  3230   moreover from * have "- cos 2 = (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3231     by (rule sums_unique)
```
```  3232   ultimately have "(0::real) < - cos 2" by simp
```
```  3233   then show ?thesis by simp
```
```  3234 qed
```
```  3235
```
```  3236 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
```
```  3237 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
```
```  3238
```
```  3239 lemma cos_is_zero: "EX! x::real. 0 \<le> x & x \<le> 2 \<and> cos x = 0"
```
```  3240 proof (rule ex_ex1I)
```
```  3241   show "\<exists>x::real. 0 \<le> x & x \<le> 2 & cos x = 0"
```
```  3242     by (rule IVT2, simp_all)
```
```  3243 next
```
```  3244   fix x::real and y::real
```
```  3245   assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
```
```  3246   assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
```
```  3247   have [simp]: "\<forall>x::real. cos differentiable (at x)"
```
```  3248     unfolding real_differentiable_def by (auto intro: DERIV_cos)
```
```  3249   from x y show "x = y"
```
```  3250     apply (cut_tac less_linear [of x y], auto)
```
```  3251     apply (drule_tac f = cos in Rolle)
```
```  3252     apply (drule_tac [5] f = cos in Rolle)
```
```  3253     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  3254     apply (metis order_less_le_trans less_le sin_gt_zero_02)
```
```  3255     apply (metis order_less_le_trans less_le sin_gt_zero_02)
```
```  3256     done
```
```  3257 qed
```
```  3258
```
```  3259 lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
```
```  3260   by (simp add: pi_def)
```
```  3261
```
```  3262 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
```
```  3263   by (simp add: pi_half cos_is_zero [THEN theI'])
```
```  3264
```
```  3265 lemma cos_of_real_pi_half [simp]:
```
```  3266   fixes x :: "'a :: {real_field,banach,real_normed_algebra_1}"
```
```  3267   shows "cos ((of_real pi / 2) :: 'a) = 0"
```
```  3268 by (metis cos_pi_half cos_of_real eq_numeral_simps(4) nonzero_of_real_divide of_real_0 of_real_numeral)
```
```  3269
```
```  3270 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
```
```  3271   apply (rule order_le_neq_trans)
```
```  3272   apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  3273   apply (metis cos_pi_half cos_zero zero_neq_one)
```
```  3274   done
```
```  3275
```
```  3276 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
```
```  3277 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
```
```  3278
```
```  3279 lemma pi_half_less_two [simp]: "pi / 2 < 2"
```
```  3280   apply (rule order_le_neq_trans)
```
```  3281   apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  3282   apply (metis cos_pi_half cos_two_neq_zero)
```
```  3283   done
```
```  3284
```
```  3285 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
```
```  3286 lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
```
```  3287
```
```  3288 lemma pi_gt_zero [simp]: "0 < pi"
```
```  3289   using pi_half_gt_zero by simp
```
```  3290
```
```  3291 lemma pi_ge_zero [simp]: "0 \<le> pi"
```
```  3292   by (rule pi_gt_zero [THEN order_less_imp_le])
```
```  3293
```
```  3294 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
```
```  3295   by (rule pi_gt_zero [THEN less_imp_neq, symmetric])
```
```  3296
```
```  3297 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
```
```  3298   by (simp add: linorder_not_less)
```
```  3299
```
```  3300 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
```
```  3301   by simp
```
```  3302
```
```  3303 lemma m2pi_less_pi: "- (2*pi) < pi"
```
```  3304   by simp
```
```  3305
```
```  3306 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
```
```  3307   using sin_cos_squared_add2 [where x = "pi/2"]
```
```  3308   using sin_gt_zero_02 [OF pi_half_gt_zero pi_half_less_two]
```
```  3309   by (simp add: power2_eq_1_iff)
```
```  3310
```
```  3311 lemma sin_of_real_pi_half [simp]:
```
```  3312   fixes x :: "'a :: {real_field,banach,real_normed_algebra_1}"
```
```  3313   shows "sin ((of_real pi / 2) :: 'a) = 1"
```
```  3314   using sin_pi_half
```
```  3315 by (metis sin_pi_half eq_numeral_simps(4) nonzero_of_real_divide of_real_1 of_real_numeral sin_of_real)
```
```  3316
```
```  3317 lemma sin_cos_eq:
```
```  3318   fixes x :: "'a::{real_normed_field,banach}"
```
```  3319   shows "sin x = cos (of_real pi / 2 - x)"
```
```  3320   by (simp add: cos_diff)
```
```  3321
```
```  3322 lemma minus_sin_cos_eq:
```
```  3323   fixes x :: "'a::{real_normed_field,banach}"
```
```  3324   shows "-sin x = cos (x + of_real pi / 2)"
```
```  3325   by (simp add: cos_add nonzero_of_real_divide)
```
```  3326
```
```  3327 lemma cos_sin_eq:
```
```  3328   fixes x :: "'a::{real_normed_field,banach}"
```
```  3329   shows "cos x = sin (of_real pi / 2 - x)"
```
```  3330   using sin_cos_eq [of "of_real pi / 2 - x"]
```
```  3331   by simp
```
```  3332
```
```  3333 lemma sin_add:
```
```  3334   fixes x :: "'a::{real_normed_field,banach}"
```
```  3335   shows "sin (x + y) = sin x * cos y + cos x * sin y"
```
```  3336   using cos_add [of "of_real pi / 2 - x" "-y"]
```
```  3337   by (simp add: cos_sin_eq) (simp add: sin_cos_eq)
```
```  3338
```
```  3339 lemma sin_diff:
```
```  3340   fixes x :: "'a::{real_normed_field,banach}"
```
```  3341   shows "sin (x - y) = sin x * cos y - cos x * sin y"
```
```  3342   using sin_add [of x "- y"] by simp
```
```  3343
```
```  3344 lemma sin_double:
```
```  3345   fixes x :: "'a::{real_normed_field,banach}"
```
```  3346   shows "sin(2 * x) = 2 * sin x * cos x"
```
```  3347   using sin_add [where x=x and y=x] by simp
```
```  3348
```
```  3349
```
```  3350 lemma cos_of_real_pi [simp]: "cos (of_real pi) = -1"
```
```  3351   using cos_add [where x = "pi/2" and y = "pi/2"]
```
```  3352   by (simp add: cos_of_real)
```
```  3353
```
```  3354 lemma sin_of_real_pi [simp]: "sin (of_real pi) = 0"
```
```  3355   using sin_add [where x = "pi/2" and y = "pi/2"]
```
```  3356   by (simp add: sin_of_real)
```
```  3357
```
```  3358 lemma cos_pi [simp]: "cos pi = -1"
```
```  3359   using cos_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  3360
```
```  3361 lemma sin_pi [simp]: "sin pi = 0"
```
```  3362   using sin_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  3363
```
```  3364 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
```
```  3365   by (simp add: sin_add)
```
```  3366
```
```  3367 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
```
```  3368   by (simp add: sin_add)
```
```  3369
```
```  3370 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
```
```  3371   by (simp add: cos_add)
```
```  3372
```
```  3373 lemma cos_periodic_pi2 [simp]: "cos (pi + x) = - cos x"
```
```  3374   by (simp add: cos_add)
```
```  3375
```
```  3376 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
```
```  3377   by (simp add: sin_add sin_double cos_double)
```
```  3378
```
```  3379 lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
```
```  3380   by (simp add: cos_add sin_double cos_double)
```
```  3381
```
```  3382 lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n"
```
```  3383   by (induct n) (auto simp: distrib_right)
```
```  3384
```
```  3385 lemma cos_npi2 [simp]: "cos (pi * real n) = (- 1) ^ n"
```
```  3386   by (metis cos_npi mult.commute)
```
```  3387
```
```  3388 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
```
```  3389   by (induct n) (auto simp: of_nat_Suc distrib_right)
```
```  3390
```
```  3391 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
```
```  3392   by (simp add: mult.commute [of pi])
```
```  3393
```
```  3394 lemma cos_two_pi [simp]: "cos (2*pi) = 1"
```
```  3395   by (simp add: cos_double)
```
```  3396
```
```  3397 lemma sin_two_pi [simp]: "sin (2*pi) = 0"
```
```  3398   by (simp add: sin_double)
```
```  3399
```
```  3400
```
```  3401 lemma sin_times_sin:
```
```  3402   fixes w :: "'a::{real_normed_field,banach}"
```
```  3403   shows "sin(w) * sin(z) = (cos(w - z) - cos(w + z)) / 2"
```
```  3404   by (simp add: cos_diff cos_add)
```
```  3405
```
```  3406 lemma sin_times_cos:
```
```  3407   fixes w :: "'a::{real_normed_field,banach}"
```
```  3408   shows "sin(w) * cos(z) = (sin(w + z) + sin(w - z)) / 2"
```
```  3409   by (simp add: sin_diff sin_add)
```
```  3410
```
```  3411 lemma cos_times_sin:
```
```  3412   fixes w :: "'a::{real_normed_field,banach}"
```
```  3413   shows "cos(w) * sin(z) = (sin(w + z) - sin(w - z)) / 2"
```
```  3414   by (simp add: sin_diff sin_add)
```
```  3415
```
```  3416 lemma cos_times_cos:
```
```  3417   fixes w :: "'a::{real_normed_field,banach}"
```
```  3418   shows "cos(w) * cos(z) = (cos(w - z) + cos(w + z)) / 2"
```
```  3419   by (simp add: cos_diff cos_add)
```
```  3420
```
```  3421 lemma sin_plus_sin:  (*FIXME field should not be necessary*)
```
```  3422   fixes w :: "'a::{real_normed_field,banach,field}"
```
```  3423   shows "sin(w) + sin(z) = 2 * sin((w + z) / 2) * cos((w - z) / 2)"
```
```  3424   apply (simp add: mult.assoc sin_times_cos)
```
```  3425   apply (simp add: field_simps)
```
```  3426   done
```
```  3427
```
```  3428 lemma sin_diff_sin:
```
```  3429   fixes w :: "'a::{real_normed_field,banach,field}"
```
```  3430   shows "sin(w) - sin(z) = 2 * sin((w - z) / 2) * cos((w + z) / 2)"
```
```  3431   apply (simp add: mult.assoc sin_times_cos)
```
```  3432   apply (simp add: field_simps)
```
```  3433   done
```
```  3434
```
```  3435 lemma cos_plus_cos:
```
```  3436   fixes w :: "'a::{real_normed_field,banach,field}"
```
```  3437   shows "cos(w) + cos(z) = 2 * cos((w + z) / 2) * cos((w - z) / 2)"
```
```  3438   apply (simp add: mult.assoc cos_times_cos)
```
```  3439   apply (simp add: field_simps)
```
```  3440   done
```
```  3441
```
```  3442 lemma cos_diff_cos:
```
```  3443   fixes w :: "'a::{real_normed_field,banach,field}"
```
```  3444   shows "cos(w) - cos(z) = 2 * sin((w + z) / 2) * sin((z - w) / 2)"
```
```  3445   apply (simp add: mult.assoc sin_times_sin)
```
```  3446   apply (simp add: field_simps)
```
```  3447   done
```
```  3448
```
```  3449 lemma cos_double_cos:
```
```  3450   fixes z :: "'a::{real_normed_field,banach}"
```
```  3451   shows "cos(2 * z) = 2 * cos z ^ 2 - 1"
```
```  3452 by (simp add: cos_double sin_squared_eq)
```
```  3453
```
```  3454 lemma cos_double_sin:
```
```  3455   fixes z :: "'a::{real_normed_field,banach}"
```
```  3456   shows "cos(2 * z) = 1 - 2 * sin z ^ 2"
```
```  3457 by (simp add: cos_double sin_squared_eq)
```
```  3458
```
```  3459 lemma sin_pi_minus [simp]: "sin (pi - x) = sin x"
```
```  3460   by (metis sin_minus sin_periodic_pi minus_minus uminus_add_conv_diff)
```
```  3461
```
```  3462 lemma cos_pi_minus [simp]: "cos (pi - x) = -(cos x)"
```
```  3463   by (metis cos_minus cos_periodic_pi uminus_add_conv_diff)
```
```  3464
```
```  3465 lemma sin_minus_pi [simp]: "sin (x - pi) = - (sin x)"
```
```  3466   by (simp add: sin_diff)
```
```  3467
```
```  3468 lemma cos_minus_pi [simp]: "cos (x - pi) = -(cos x)"
```
```  3469   by (simp add: cos_diff)
```
```  3470
```
```  3471 lemma sin_2pi_minus [simp]: "sin (2*pi - x) = -(sin x)"
```
```  3472   by (metis sin_periodic_pi2 add_diff_eq mult_2 sin_pi_minus)
```
```  3473
```
```  3474 lemma cos_2pi_minus [simp]: "cos (2*pi - x) = cos x"
```
```  3475   by (metis (no_types, hide_lams) cos_add cos_minus cos_two_pi sin_minus sin_two_pi
```
```  3476            diff_0_right minus_diff_eq mult_1 mult_zero_left uminus_add_conv_diff)
```
```  3477
```
```  3478 lemma sin_gt_zero2: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < sin x"
```
```  3479   by (metis sin_gt_zero_02 order_less_trans pi_half_less_two)
```
```  3480
```
```  3481 lemma sin_less_zero:
```
```  3482   assumes "- pi/2 < x" and "x < 0"
```
```  3483   shows "sin x < 0"
```
```  3484 proof -
```
```  3485   have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
```
```  3486   thus ?thesis by simp
```
```  3487 qed
```
```  3488
```
```  3489 lemma pi_less_4: "pi < 4"
```
```  3490   using pi_half_less_two by auto
```
```  3491
```
```  3492 lemma cos_gt_zero: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < cos x"
```
```  3493   by (simp add: cos_sin_eq sin_gt_zero2)
```
```  3494
```
```  3495 lemma cos_gt_zero_pi: "\<lbrakk>-(pi/2) < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < cos x"
```
```  3496   using cos_gt_zero [of x] cos_gt_zero [of "-x"]
```
```  3497   by (cases rule: linorder_cases [of x 0]) auto
```
```  3498
```
```  3499 lemma cos_ge_zero: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2\<rbrakk> \<Longrightarrow> 0 \<le> cos x"
```
```  3500   apply (auto simp: order_le_less cos_gt_zero_pi)
```
```  3501   by (metis cos_pi_half eq_divide_eq eq_numeral_simps(4))
```
```  3502
```
```  3503 lemma sin_gt_zero: "\<lbrakk>0 < x; x < pi \<rbrakk> \<Longrightarrow> 0 < sin x"
```
```  3504   by (simp add: sin_cos_eq cos_gt_zero_pi)
```
```  3505
```
```  3506 lemma sin_lt_zero: "pi < x \<Longrightarrow> x < 2*pi \<Longrightarrow> sin x < 0"
```
```  3507   using sin_gt_zero [of "x-pi"]
```
```  3508   by (simp add: sin_diff)
```
```  3509
```
```  3510 lemma pi_ge_two: "2 \<le> pi"
```
```  3511 proof (rule ccontr)
```
```  3512   assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
```
```  3513   have "\<exists>y > pi. y < 2 \<and> y < 2*pi"
```
```  3514   proof (cases "2 < 2*pi")
```
```  3515     case True with dense[OF \<open>pi < 2\<close>] show ?thesis by auto
```
```  3516   next
```
```  3517     case False have "pi < 2*pi" by auto
```
```  3518     from dense[OF this] and False show ?thesis by auto
```
```  3519   qed
```
```  3520   then obtain y where "pi < y" and "y < 2" and "y < 2*pi" by blast
```
```  3521   hence "0 < sin y" using sin_gt_zero_02 by auto
```
```  3522   moreover
```
```  3523   have "sin y < 0" using sin_gt_zero[of "y - pi"] \<open>pi < y\<close> and \<open>y < 2*pi\<close> sin_periodic_pi[of "y - pi"] by auto
```
```  3524   ultimately show False by auto
```
```  3525 qed
```
```  3526
```
```  3527 lemma sin_ge_zero: "\<lbrakk>0 \<le> x; x \<le> pi\<rbrakk> \<Longrightarrow> 0 \<le> sin x"
```
```  3528   by (auto simp: order_le_less sin_gt_zero)
```
```  3529
```
```  3530 lemma sin_le_zero: "pi \<le> x \<Longrightarrow> x < 2*pi \<Longrightarrow> sin x \<le> 0"
```
```  3531   using sin_ge_zero [of "x-pi"]
```
```  3532   by (simp add: sin_diff)
```
```  3533
```
```  3534 text \<open>FIXME: This proof is almost identical to lemma \<open>cos_is_zero\<close>.
```
```  3535   It should be possible to factor out some of the common parts.\<close>
```
```  3536
```
```  3537 lemma cos_total: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
```
```  3538 proof (rule ex_ex1I)
```
```  3539   assume y: "-1 \<le> y" "y \<le> 1"
```
```  3540   show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
```
```  3541     by (rule IVT2, simp_all add: y)
```
```  3542 next
```
```  3543   fix a b
```
```  3544   assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
```
```  3545   assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
```
```  3546   have [simp]: "\<forall>x::real. cos differentiable (at x)"
```
```  3547     unfolding real_differentiable_def by (auto intro: DERIV_cos)
```
```  3548   from a b show "a = b"
```
```  3549     apply (cut_tac less_linear [of a b], auto)
```
```  3550     apply (drule_tac f = cos in Rolle)
```
```  3551     apply (drule_tac [5] f = cos in Rolle)
```
```  3552     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  3553     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  3554     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  3555     done
```
```  3556 qed
```
```  3557
```
```  3558 lemma sin_total:
```
```  3559   assumes y: "-1 \<le> y" "y \<le> 1"
```
```  3560     shows "\<exists>! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
```
```  3561 proof -
```
```  3562   from cos_total [OF y]
```
```  3563   obtain x where x: "0 \<le> x" "x \<le> pi" "cos x = y"
```
```  3564            and uniq: "\<And>x'. 0 \<le> x' \<Longrightarrow> x' \<le> pi \<Longrightarrow> cos x' = y \<Longrightarrow> x' = x "
```
```  3565     by blast
```
```  3566   show ?thesis
```
```  3567     apply (simp add: sin_cos_eq)
```
```  3568     apply (rule ex1I [where a="pi/2 - x"])
```
```  3569     apply (cut_tac [2] x'="pi/2 - xa" in uniq)
```
```  3570     using x
```
```  3571     apply auto
```
```  3572     done
```
```  3573 qed
```
```  3574
```
```  3575 lemma cos_zero_lemma:
```
```  3576   assumes "0 \<le> x" "cos x = 0"
```
```  3577   shows "\<exists>n. odd n \<and> x = of_nat n * (pi/2) \<and> n > 0"
```
```  3578 proof -
```
```  3579   have xle: "x < (1 + real_of_int \<lfloor>x/pi\<rfloor>) * pi"
```
```  3580     using floor_correct [of "x/pi"]
```
```  3581     by (simp add: add.commute divide_less_eq)
```
```  3582   obtain n where "real n * pi \<le> x" "x < real (Suc n) * pi"
```
```  3583     apply (rule that [of "nat \<lfloor>x/pi\<rfloor>"])
```
```  3584     using assms
```
```  3585     apply (simp_all add: xle)
```
```  3586     apply (metis floor_less_iff less_irrefl mult_imp_div_pos_less not_le pi_gt_zero)
```
```  3587     done
```
```  3588   then have x: "0 \<le> x - n * pi" "(x - n * pi) \<le> pi" "cos (x - n * pi) = 0"
```
```  3589     by (auto simp: algebra_simps cos_diff assms)
```
```  3590   then have "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = 0"
```
```  3591     by (auto simp: intro!: cos_total)
```
```  3592   then obtain \<theta> where \<theta>: "0 \<le> \<theta>" "\<theta> \<le> pi" "cos \<theta> = 0"
```
```  3593                   and uniq: "\<And>\<phi>. \<lbrakk>0 \<le> \<phi>; \<phi> \<le> pi; cos \<phi> = 0\<rbrakk> \<Longrightarrow> \<phi> = \<theta>"
```
```  3594     by blast
```
```  3595   then have "x - real n * pi = \<theta>"
```
```  3596     using x by blast
```
```  3597   moreover have "pi/2 = \<theta>"
```
```  3598     using pi_half_ge_zero uniq by fastforce
```
```  3599   ultimately show ?thesis
```
```  3600     by (rule_tac x = "Suc (2 * n)" in exI) (simp add: algebra_simps)
```
```  3601 qed
```
```  3602
```
```  3603 lemma sin_zero_lemma:
```
```  3604      "\<lbrakk>0 \<le> x; sin x = 0\<rbrakk> \<Longrightarrow> \<exists>n::nat. even n & x = real n * (pi/2)"
```
```  3605   using cos_zero_lemma [of "x + pi/2"]
```
```  3606   apply (clarsimp simp add: cos_add)
```
```  3607   apply (rule_tac x = "n - 1" in exI)
```
```  3608   apply (simp add: algebra_simps of_nat_diff)
```
```  3609   done
```
```  3610
```
```  3611 lemma cos_zero_iff:
```
```  3612      "(cos x = 0) \<longleftrightarrow>
```
```  3613       ((\<exists>n. odd n & (x = real n * (pi/2))) \<or> (\<exists>n. odd n & (x = -(real n * (pi/2)))))"
```
```  3614       (is "?lhs = ?rhs")
```
```  3615 proof -
```
```  3616   { fix n :: nat
```
```  3617     assume "odd n"
```
```  3618     then obtain m where "n = 2 * m + 1" ..
```
```  3619     then have "cos (real n * pi / 2) = 0"
```
```  3620       by (simp add: field_simps) (simp add: cos_add add_divide_distrib)
```
```  3621   } note * = this
```
```  3622   show ?thesis
```
```  3623   proof
```
```  3624     assume "cos x = 0" then show ?rhs
```
```  3625       using cos_zero_lemma [of x] cos_zero_lemma [of "-x"] by force
```
```  3626   next
```
```  3627     assume ?rhs then show "cos x = 0"
```
```  3628     by (auto dest: * simp del: eq_divide_eq_numeral1)
```
```  3629   qed
```
```  3630 qed
```
```  3631
```
```  3632 lemma sin_zero_iff:
```
```  3633      "(sin x = 0) \<longleftrightarrow>
```
```  3634       ((\<exists>n. even n & (x = real n * (pi/2))) \<or> (\<exists>n. even n & (x = -(real n * (pi/2)))))"
```
```  3635       (is "?lhs = ?rhs")
```
```  3636 proof
```
```  3637   assume "sin x = 0" then show ?rhs
```
```  3638     using sin_zero_lemma [of x] sin_zero_lemma [of "-x"] by force
```
```  3639 next
```
```  3640   assume ?rhs then show "sin x = 0"
```
```  3641     by (auto elim: evenE)
```
```  3642 qed
```
```  3643
```
```  3644 lemma cos_zero_iff_int:
```
```  3645      "cos x = 0 \<longleftrightarrow> (\<exists>n. odd n \<and> x = of_int n * (pi/2))"
```
```  3646 proof safe
```
```  3647   assume "cos x = 0"
```
```  3648   then show "\<exists>n. odd n & x = of_int n * (pi/2)"
```
```  3649     apply (simp add: cos_zero_iff, safe)
```
```  3650     apply (metis even_int_iff of_int_of_nat_eq)
```
```  3651     apply (rule_tac x="- (int n)" in exI, simp)
```
```  3652     done
```
```  3653 next
```
```  3654   fix n::int
```
```  3655   assume "odd n"
```
```  3656   then show "cos (of_int n * (pi / 2)) = 0"
```
```  3657     apply (simp add: cos_zero_iff)
```
```  3658     apply (case_tac n rule: int_cases2, simp_all)
```
```  3659     done
```
```  3660 qed
```
```  3661
```
```  3662 lemma sin_zero_iff_int:
```
```  3663      "sin x = 0 \<longleftrightarrow> (\<exists>n. even n & (x = of_int n * (pi/2)))"
```
```  3664 proof safe
```
```  3665   assume "sin x = 0"
```
```  3666   then show "\<exists>n. even n \<and> x = of_int n * (pi / 2)"
```
```  3667     apply (simp add: sin_zero_iff, safe)
```
```  3668     apply (metis even_int_iff of_int_of_nat_eq)
```
```  3669     apply (rule_tac x="- (int n)" in exI, simp)
```
```  3670     done
```
```  3671 next
```
```  3672   fix n::int
```
```  3673   assume "even n"
```
```  3674   then show "sin (of_int n * (pi / 2)) = 0"
```
```  3675     apply (simp add: sin_zero_iff)
```
```  3676     apply (case_tac n rule: int_cases2, simp_all)
```
```  3677     done
```
```  3678 qed
```
```  3679
```
```  3680 lemma sin_zero_iff_int2:
```
```  3681   "sin x = 0 \<longleftrightarrow> (\<exists>n::int. x = of_int n * pi)"
```
```  3682   apply (simp only: sin_zero_iff_int)
```
```  3683   apply (safe elim!: evenE)
```
```  3684   apply (simp_all add: field_simps)
```
```  3685   using dvd_triv_left apply fastforce
```
```  3686   done
```
```  3687
```
```  3688 lemma cos_monotone_0_pi:
```
```  3689   assumes "0 \<le> y" and "y < x" and "x \<le> pi"
```
```  3690   shows "cos x < cos y"
```
```  3691 proof -
```
```  3692   have "- (x - y) < 0" using assms by auto
```
```  3693   from MVT2[OF \<open>y < x\<close> DERIV_cos[THEN impI, THEN allI]]
```
```  3694   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z"
```
```  3695     by auto
```
```  3696   hence "0 < z" and "z < pi" using assms by auto
```
```  3697   hence "0 < sin z" using sin_gt_zero by auto
```
```  3698   hence "cos x - cos y < 0"
```
```  3699     unfolding cos_diff minus_mult_commute[symmetric]
```
```  3700     using \<open>- (x - y) < 0\<close> by (rule mult_pos_neg2)
```
```  3701   thus ?thesis by auto
```
```  3702 qed
```
```  3703
```
```  3704 lemma cos_monotone_0_pi_le:
```
```  3705   assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi"
```
```  3706   shows "cos x \<le> cos y"
```
```  3707 proof (cases "y < x")
```
```  3708   case True
```
```  3709   show ?thesis
```
```  3710     using cos_monotone_0_pi[OF \<open>0 \<le> y\<close> True \<open>x \<le> pi\<close>] by auto
```
```  3711 next
```
```  3712   case False
```
```  3713   hence "y = x" using \<open>y \<le> x\<close> by auto
```
```  3714   thus ?thesis by auto
```
```  3715 qed
```
```  3716
```
```  3717 lemma cos_monotone_minus_pi_0:
```
```  3718   assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
```
```  3719   shows "cos y < cos x"
```
```  3720 proof -
```
```  3721   have "0 \<le> -x" and "-x < -y" and "-y \<le> pi"
```
```  3722     using assms by auto
```
```  3723   from cos_monotone_0_pi[OF this] show ?thesis
```
```  3724     unfolding cos_minus .
```
```  3725 qed
```
```  3726
```
```  3727 lemma cos_monotone_minus_pi_0':
```
```  3728   assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0"
```
```  3729   shows "cos y \<le> cos x"
```
```  3730 proof (cases "y < x")
```
```  3731   case True
```
```  3732   show ?thesis using cos_monotone_minus_pi_0[OF \<open>-pi \<le> y\<close> True \<open>x \<le> 0\<close>]
```
```  3733     by auto
```
```  3734 next
```
```  3735   case False
```
```  3736   hence "y = x" using \<open>y \<le> x\<close> by auto
```
```  3737   thus ?thesis by auto
```
```  3738 qed
```
```  3739
```
```  3740 lemma sin_monotone_2pi:
```
```  3741   assumes "- (pi/2) \<le> y" and "y < x" and "x \<le> pi/2"
```
```  3742   shows "sin y < sin x"
```
```  3743     apply (simp add: sin_cos_eq)
```
```  3744     apply (rule cos_monotone_0_pi)
```
```  3745     using assms
```
```  3746     apply auto
```
```  3747     done
```
```  3748
```
```  3749 lemma sin_monotone_2pi_le:
```
```  3750   assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2"
```
```  3751   shows "sin y \<le> sin x"
```
```  3752   by (metis assms le_less sin_monotone_2pi)
```
```  3753
```
```  3754 lemma sin_x_le_x:
```
```  3755   fixes x::real assumes x: "x \<ge> 0" shows "sin x \<le> x"
```
```  3756 proof -
```
```  3757   let ?f = "\<lambda>x. x - sin x"
```
```  3758   from x have "?f x \<ge> ?f 0"
```
```  3759     apply (rule DERIV_nonneg_imp_nondecreasing)
```
```  3760     apply (intro allI impI exI[of _ "1 - cos x" for x])
```
```  3761     apply (auto intro!: derivative_eq_intros simp: field_simps)
```
```  3762     done
```
```  3763   thus "sin x \<le> x" by simp
```
```  3764 qed
```
```  3765
```
```  3766 lemma sin_x_ge_neg_x:
```
```  3767   fixes x::real assumes x: "x \<ge> 0" shows "sin x \<ge> - x"
```
```  3768 proof -
```
```  3769   let ?f = "\<lambda>x. x + sin x"
```
```  3770   from x have "?f x \<ge> ?f 0"
```
```  3771     apply (rule DERIV_nonneg_imp_nondecreasing)
```
```  3772     apply (intro allI impI exI[of _ "1 + cos x" for x])
```
```  3773     apply (auto intro!: derivative_eq_intros simp: field_simps real_0_le_add_iff)
```
```  3774     done
```
```  3775   thus "sin x \<ge> -x" by simp
```
```  3776 qed
```
```  3777
```
```  3778 lemma abs_sin_x_le_abs_x:
```
```  3779   fixes x::real shows "\<bar>sin x\<bar> \<le> \<bar>x\<bar>"
```
```  3780   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"]
```
```  3781   by (auto simp: abs_real_def)
```
```  3782
```
```  3783
```
```  3784 subsection \<open>More Corollaries about Sine and Cosine\<close>
```
```  3785
```
```  3786 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
```
```  3787 proof -
```
```  3788   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
```
```  3789     by (auto simp: algebra_simps sin_add)
```
```  3790   thus ?thesis
```
```  3791     by (simp add: distrib_right add_divide_distrib add.commute mult.commute [of pi])
```
```  3792 qed
```
```  3793
```
```  3794 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
```
```  3795   by (cases "even n") (simp_all add: cos_double mult.assoc)
```
```  3796
```
```  3797 lemma cos_3over2_pi [simp]: "cos (3/2*pi) = 0"
```
```  3798   apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
```
```  3799   apply (subst cos_add, simp)
```
```  3800   done
```
```  3801
```
```  3802 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
```
```  3803   by (auto simp: mult.assoc sin_double)
```
```  3804
```
```  3805 lemma sin_3over2_pi [simp]: "sin (3/2*pi) = - 1"
```
```  3806   apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
```
```  3807   apply (subst sin_add, simp)
```
```  3808   done
```
```  3809
```
```  3810 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
```
```  3811 by (simp only: cos_add sin_add of_nat_Suc distrib_right distrib_left add_divide_distrib, auto)
```
```  3812
```
```  3813 lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)"
```
```  3814   by (auto intro!: derivative_eq_intros)
```
```  3815
```
```  3816 lemma sin_zero_norm_cos_one:
```
```  3817   fixes x :: "'a::{real_normed_field,banach}"
```
```  3818   assumes "sin x = 0" shows "norm (cos x) = 1"
```
```  3819   using sin_cos_squared_add [of x, unfolded assms]
```
```  3820   by (simp add: square_norm_one)
```
```  3821
```
```  3822 lemma sin_zero_abs_cos_one: "sin x = 0 \<Longrightarrow> \<bar>cos x\<bar> = (1::real)"
```
```  3823   using sin_zero_norm_cos_one by fastforce
```
```  3824
```
```  3825 lemma cos_one_sin_zero:
```
```  3826   fixes x :: "'a::{real_normed_field,banach}"
```
```  3827   assumes "cos x = 1" shows "sin x = 0"
```
```  3828   using sin_cos_squared_add [of x, unfolded assms]
```
```  3829   by simp
```
```  3830
```
```  3831 lemma sin_times_pi_eq_0: "sin(x * pi) = 0 \<longleftrightarrow> x \<in> \<int>"
```
```  3832   by (simp add: sin_zero_iff_int2) (metis Ints_cases Ints_of_int)
```
```  3833
```
```  3834 lemma cos_one_2pi:
```
```  3835     "cos(x) = 1 \<longleftrightarrow> (\<exists>n::nat. x = n * 2*pi) | (\<exists>n::nat. x = -(n * 2*pi))"
```
```  3836     (is "?lhs = ?rhs")
```
```  3837 proof
```
```  3838   assume "cos(x) = 1"
```
```  3839   then have "sin x = 0"
```
```  3840     by (simp add: cos_one_sin_zero)
```
```  3841   then show ?rhs
```
```  3842   proof (simp only: sin_zero_iff, elim exE disjE conjE)
```
```  3843     fix n::nat
```
```  3844     assume n: "even n" "x = real n * (pi/2)"
```
```  3845     then obtain m where m: "n = 2 * m"
```
```  3846       using dvdE by blast
```
```  3847     then have me: "even m" using \<open>?lhs\<close> n
```
```  3848       by (auto simp: field_simps) (metis one_neq_neg_one  power_minus_odd power_one)
```
```  3849     show ?rhs
```
```  3850       using m me n
```
```  3851       by (auto simp: field_simps elim!: evenE)
```
```  3852   next
```
```  3853     fix n::nat
```
```  3854     assume n: "even n" "x = - (real n * (pi/2))"
```
```  3855     then obtain m where m: "n = 2 * m"
```
```  3856       using dvdE by blast
```
```  3857     then have me: "even m" using \<open>?lhs\<close> n
```
```  3858       by (auto simp: field_simps) (metis one_neq_neg_one  power_minus_odd power_one)
```
```  3859     show ?rhs
```
```  3860       using m me n
```
```  3861       by (auto simp: field_simps elim!: evenE)
```
```  3862   qed
```
```  3863 next
```
```  3864   assume "?rhs"
```
```  3865   then show "cos x = 1"
```
```  3866     by (metis cos_2npi cos_minus mult.assoc mult.left_commute)
```
```  3867 qed
```
```  3868
```
```  3869 lemma cos_one_2pi_int: "cos(x) = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2*pi)"
```
```  3870   apply auto  \<comment>\<open>FIXME simproc bug\<close>
```
```  3871   apply (auto simp: cos_one_2pi)
```
```  3872   apply (metis of_int_of_nat_eq)
```
```  3873   apply (metis mult_minus_right of_int_minus of_int_of_nat_eq)
```
```  3874   by (metis mult_minus_right of_int_of_nat )
```
```  3875
```
```  3876 lemma sin_cos_sqrt: "0 \<le> sin(x) \<Longrightarrow> (sin(x) = sqrt(1 - (cos(x) ^ 2)))"
```
```  3877   using sin_squared_eq real_sqrt_unique by fastforce
```
```  3878
```
```  3879 lemma sin_eq_0_pi: "-pi < x \<Longrightarrow> x < pi \<Longrightarrow> sin(x) = 0 \<Longrightarrow> x = 0"
```
```  3880   by (metis sin_gt_zero sin_minus minus_less_iff neg_0_less_iff_less not_less_iff_gr_or_eq)
```
```  3881
```
```  3882 lemma cos_treble_cos:
```
```  3883   fixes x :: "'a::{real_normed_field,banach}"
```
```  3884   shows "cos(3 * x) = 4 * cos(x) ^ 3 - 3 * cos x"
```
```  3885 proof -
```
```  3886   have *: "(sin x * (sin x * 3)) = 3 - (cos x * (cos x * 3))"
```
```  3887     by (simp add: mult.assoc [symmetric] sin_squared_eq [unfolded power2_eq_square])
```
```  3888   have "cos(3 * x) = cos(2*x + x)"
```
```  3889     by simp
```
```  3890   also have "... = 4 * cos(x) ^ 3 - 3 * cos x"
```
```  3891     apply (simp only: cos_add cos_double sin_double)
```
```  3892     apply (simp add: * field_simps power2_eq_square power3_eq_cube)
```
```  3893     done
```
```  3894   finally show ?thesis .
```
```  3895 qed
```
```  3896
```
```  3897 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
```
```  3898 proof -
```
```  3899   let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
```
```  3900   have nonneg: "0 \<le> ?c"
```
```  3901     by (simp add: cos_ge_zero)
```
```  3902   have "0 = cos (pi / 4 + pi / 4)"
```
```  3903     by simp
```
```  3904   also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2"
```
```  3905     by (simp only: cos_add power2_eq_square)
```
```  3906   also have "\<dots> = 2 * ?c\<^sup>2 - 1"
```
```  3907     by (simp add: sin_squared_eq)
```
```  3908   finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2"
```
```  3909     by (simp add: power_divide)
```
```  3910   thus ?thesis
```
```  3911     using nonneg by (rule power2_eq_imp_eq) simp
```
```  3912 qed
```
```  3913
```
```  3914 lemma cos_30: "cos (pi / 6) = sqrt 3/2"
```
```  3915 proof -
```
```  3916   let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
```
```  3917   have pos_c: "0 < ?c"
```
```  3918     by (rule cos_gt_zero, simp, simp)
```
```  3919   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
```
```  3920     by simp
```
```  3921   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
```
```  3922     by (simp only: cos_add sin_add)
```
```  3923   also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)"
```
```  3924     by (simp add: algebra_simps power2_eq_square)
```
```  3925   finally have "?c\<^sup>2 = (sqrt 3/2)\<^sup>2"
```
```  3926     using pos_c by (simp add: sin_squared_eq power_divide)
```
```  3927   thus ?thesis
```
```  3928     using pos_c [THEN order_less_imp_le]
```
```  3929     by (rule power2_eq_imp_eq) simp
```
```  3930 qed
```
```  3931
```
```  3932 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
```
```  3933   by (simp add: sin_cos_eq cos_45)
```
```  3934
```
```  3935 lemma sin_60: "sin (pi / 3) = sqrt 3/2"
```
```  3936   by (simp add: sin_cos_eq cos_30)
```
```  3937
```
```  3938 lemma cos_60: "cos (pi / 3) = 1 / 2"
```
```  3939   apply (rule power2_eq_imp_eq)
```
```  3940   apply (simp add: cos_squared_eq sin_60 power_divide)
```
```  3941   apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
```
```  3942   done
```
```  3943
```
```  3944 lemma sin_30: "sin (pi / 6) = 1 / 2"
```
```  3945   by (simp add: sin_cos_eq cos_60)
```
```  3946
```
```  3947 lemma cos_integer_2pi: "n \<in> \<int> \<Longrightarrow> cos(2*pi * n) = 1"
```
```  3948   by (metis Ints_cases cos_one_2pi_int mult.assoc mult.commute)
```
```  3949
```
```  3950 lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2*pi * n) = 0"
```
```  3951   by (metis sin_two_pi Ints_mult mult.assoc mult.commute sin_times_pi_eq_0)
```
```  3952
```
```  3953 lemma cos_int_2npi [simp]: "cos (2 * of_int (n::int) * pi) = 1"
```
```  3954   by (simp add: cos_one_2pi_int)
```
```  3955
```
```  3956 lemma sin_int_2npi [simp]: "sin (2 * of_int (n::int) * pi) = 0"
```
```  3957   by (metis Ints_of_int mult.assoc mult.commute sin_integer_2pi)
```
```  3958
```
```  3959 lemma sincos_principal_value: "\<exists>y. (-pi < y \<and> y \<le> pi) \<and> (sin(y) = sin(x) \<and> cos(y) = cos(x))"
```
```  3960   apply (rule exI [where x="pi - (2*pi) * frac((pi - x) / (2*pi))"])
```
```  3961   apply (auto simp: field_simps frac_lt_1)
```
```  3962   apply (simp_all add: frac_def divide_simps)
```
```  3963   apply (simp_all add: add_divide_distrib diff_divide_distrib)
```
```  3964   apply (simp_all add: sin_diff cos_diff mult.assoc [symmetric] cos_integer_2pi sin_integer_2pi)
```
```  3965   done
```
```  3966
```
```  3967
```
```  3968 subsection \<open>Tangent\<close>
```
```  3969
```
```  3970 definition tan :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3971   where "tan = (\<lambda>x. sin x / cos x)"
```
```  3972
```
```  3973 lemma tan_of_real:
```
```  3974   "of_real (tan x) = (tan (of_real x) :: 'a::{real_normed_field,banach})"
```
```  3975   by (simp add: tan_def sin_of_real cos_of_real)
```
```  3976
```
```  3977 lemma tan_in_Reals [simp]:
```
```  3978   fixes z :: "'a::{real_normed_field,banach}"
```
```  3979   shows "z \<in> \<real> \<Longrightarrow> tan z \<in> \<real>"
```
```  3980   by (simp add: tan_def)
```
```  3981
```
```  3982 lemma tan_zero [simp]: "tan 0 = 0"
```
```  3983   by (simp add: tan_def)
```
```  3984
```
```  3985 lemma tan_pi [simp]: "tan pi = 0"
```
```  3986   by (simp add: tan_def)
```
```  3987
```
```  3988 lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
```
```  3989   by (simp add: tan_def)
```
```  3990
```
```  3991 lemma tan_minus [simp]: "tan (-x) = - tan x"
```
```  3992   by (simp add: tan_def)
```
```  3993
```
```  3994 lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
```
```  3995   by (simp add: tan_def)
```
```  3996
```
```  3997 lemma lemma_tan_add1:
```
```  3998   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
```
```  3999   by (simp add: tan_def cos_add field_simps)
```
```  4000
```
```  4001 lemma add_tan_eq:
```
```  4002   fixes x :: "'a::{real_normed_field,banach}"
```
```  4003   shows "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
```
```  4004   by (simp add: tan_def sin_add field_simps)
```
```  4005
```
```  4006 lemma tan_add:
```
```  4007   fixes x :: "'a::{real_normed_field,banach}"
```
```  4008   shows
```
```  4009      "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0\<rbrakk>
```
```  4010       \<Longrightarrow> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
```
```  4011       by (simp add: add_tan_eq lemma_tan_add1 field_simps) (simp add: tan_def)
```
```  4012
```
```  4013 lemma tan_double:
```
```  4014   fixes x :: "'a::{real_normed_field,banach}"
```
```  4015   shows
```
```  4016      "\<lbrakk>cos x \<noteq> 0; cos (2 * x) \<noteq> 0\<rbrakk>
```
```  4017       \<Longrightarrow> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)"
```
```  4018   using tan_add [of x x] by (simp add: power2_eq_square)
```
```  4019
```
```  4020 lemma tan_gt_zero: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < tan x"
```
```  4021   by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
```
```  4022
```
```  4023 lemma tan_less_zero:
```
```  4024   assumes lb: "- pi/2 < x" and "x < 0"
```
```  4025   shows "tan x < 0"
```
```  4026 proof -
```
```  4027   have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
```
```  4028   thus ?thesis by simp
```
```  4029 qed
```
```  4030
```
```  4031 lemma tan_half:
```
```  4032   fixes x :: "'a::{real_normed_field,banach,field}"
```
```  4033   shows  "tan x = sin (2 * x) / (cos (2 * x) + 1)"
```
```  4034   unfolding tan_def sin_double cos_double sin_squared_eq
```
```  4035   by (simp add: power2_eq_square)
```
```  4036
```
```  4037 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
```
```  4038   unfolding tan_def by (simp add: sin_30 cos_30)
```
```  4039
```
```  4040 lemma tan_45: "tan (pi / 4) = 1"
```
```  4041   unfolding tan_def by (simp add: sin_45 cos_45)
```
```  4042
```
```  4043 lemma tan_60: "tan (pi / 3) = sqrt 3"
```
```  4044   unfolding tan_def by (simp add: sin_60 cos_60)
```
```  4045
```
```  4046 lemma DERIV_tan [simp]:
```
```  4047   fixes x :: "'a::{real_normed_field,banach}"
```
```  4048   shows "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)"
```
```  4049   unfolding tan_def
```
```  4050   by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square)
```
```  4051
```
```  4052 lemma isCont_tan:
```
```  4053   fixes x :: "'a::{real_normed_field,banach}"
```
```  4054   shows "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
```
```  4055   by (rule DERIV_tan [THEN DERIV_isCont])
```
```  4056
```
```  4057 lemma isCont_tan' [simp,continuous_intros]:
```
```  4058   fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
```
```  4059   shows "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
```
```  4060   by (rule isCont_o2 [OF _ isCont_tan])
```
```  4061
```
```  4062 lemma tendsto_tan [tendsto_intros]:
```
```  4063   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4064   shows "\<lbrakk>(f \<longlongrightarrow> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) \<longlongrightarrow> tan a) F"
```
```  4065   by (rule isCont_tendsto_compose [OF isCont_tan])
```
```  4066
```
```  4067 lemma continuous_tan:
```
```  4068   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4069   shows "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))"
```
```  4070   unfolding continuous_def by (rule tendsto_tan)
```
```  4071
```
```  4072 lemma continuous_on_tan [continuous_intros]:
```
```  4073   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4074   shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))"
```
```  4075   unfolding continuous_on_def by (auto intro: tendsto_tan)
```
```  4076
```
```  4077 lemma continuous_within_tan [continuous_intros]:
```
```  4078   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4079   shows
```
```  4080   "continuous (at x within s) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
```
```  4081   unfolding continuous_within by (rule tendsto_tan)
```
```  4082
```
```  4083 lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) \<midarrow>pi/2\<rightarrow> 0"
```
```  4084   by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
```
```  4085
```
```  4086 lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
```
```  4087   apply (cut_tac LIM_cos_div_sin)
```
```  4088   apply (simp only: LIM_eq)
```
```  4089   apply (drule_tac x = "inverse y" in spec, safe, force)
```
```  4090   apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
```
```  4091   apply (rule_tac x = "(pi/2) - e" in exI)
```
```  4092   apply (simp (no_asm_simp))
```
```  4093   apply (drule_tac x = "(pi/2) - e" in spec)
```
```  4094   apply (auto simp add: tan_def sin_diff cos_diff)
```
```  4095   apply (rule inverse_less_iff_less [THEN iffD1])
```
```  4096   apply (auto simp add: divide_inverse)
```
```  4097   apply (rule mult_pos_pos)
```
```  4098   apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
```
```  4099   apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult.commute)
```
```  4100   done
```
```  4101
```
```  4102 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
```
```  4103   apply (frule order_le_imp_less_or_eq, safe)
```
```  4104    prefer 2 apply force
```
```  4105   apply (drule lemma_tan_total, safe)
```
```  4106   apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
```
```  4107   apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
```
```  4108   apply (drule_tac y = xa in order_le_imp_less_or_eq)
```
```  4109   apply (auto dest: cos_gt_zero)
```
```  4110   done
```
```  4111
```
```  4112 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  4113   apply (cut_tac linorder_linear [of 0 y], safe)
```
```  4114   apply (drule tan_total_pos)
```
```  4115   apply (cut_tac [2] y="-y" in tan_total_pos, safe)
```
```  4116   apply (rule_tac [3] x = "-x" in exI)
```
```  4117   apply (auto del: exI intro!: exI)
```
```  4118   done
```
```  4119
```
```  4120 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  4121   apply (cut_tac y = y in lemma_tan_total1, auto)
```
```  4122   apply hypsubst_thin
```
```  4123   apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
```
```  4124   apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
```
```  4125   apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
```
```  4126   apply (rule_tac [4] Rolle)
```
```  4127   apply (rule_tac [2] Rolle)
```
```  4128   apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
```
```  4129               simp add: real_differentiable_def)
```
```  4130   txt\<open>Now, simulate TRYALL\<close>
```
```  4131   apply (rule_tac [!] DERIV_tan asm_rl)
```
```  4132   apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
```
```  4133               simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
```
```  4134   done
```
```  4135
```
```  4136 lemma tan_monotone:
```
```  4137   assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
```
```  4138   shows "tan y < tan x"
```
```  4139 proof -
```
```  4140   have "\<forall>x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse ((cos x')\<^sup>2)"
```
```  4141   proof (rule allI, rule impI)
```
```  4142     fix x' :: real
```
```  4143     assume "y \<le> x' \<and> x' \<le> x"
```
```  4144     hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
```
```  4145     from cos_gt_zero_pi[OF this]
```
```  4146     have "cos x' \<noteq> 0" by auto
```
```  4147     thus "DERIV tan x' :> inverse ((cos x')\<^sup>2)" by (rule DERIV_tan)
```
```  4148   qed
```
```  4149   from MVT2[OF \<open>y < x\<close> this]
```
```  4150   obtain z where "y < z" and "z < x"
```
```  4151     and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
```
```  4152   hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
```
```  4153   hence "0 < cos z" using cos_gt_zero_pi by auto
```
```  4154   hence inv_pos: "0 < inverse ((cos z)\<^sup>2)" by auto
```
```  4155   have "0 < x - y" using \<open>y < x\<close> by auto
```
```  4156   with inv_pos have "0 < tan x - tan y" unfolding tan_diff by auto
```
```  4157   thus ?thesis by auto
```
```  4158 qed
```
```  4159
```
```  4160 lemma tan_monotone':
```
```  4161   assumes "- (pi / 2) < y"
```
```  4162     and "y < pi / 2"
```
```  4163     and "- (pi / 2) < x"
```
```  4164     and "x < pi / 2"
```
```  4165   shows "(y < x) = (tan y < tan x)"
```
```  4166 proof
```
```  4167   assume "y < x"
```
```  4168   thus "tan y < tan x"
```
```  4169     using tan_monotone and \<open>- (pi / 2) < y\<close> and \<open>x < pi / 2\<close> by auto
```
```  4170 next
```
```  4171   assume "tan y < tan x"
```
```  4172   show "y < x"
```
```  4173   proof (rule ccontr)
```
```  4174     assume "\<not> y < x" hence "x \<le> y" by auto
```
```  4175     hence "tan x \<le> tan y"
```
```  4176     proof (cases "x = y")
```
```  4177       case True thus ?thesis by auto
```
```  4178     next
```
```  4179       case False hence "x < y" using \<open>x \<le> y\<close> by auto
```
```  4180       from tan_monotone[OF \<open>- (pi/2) < x\<close> this \<open>y < pi / 2\<close>] show ?thesis by auto
```
```  4181     qed
```
```  4182     thus False using \<open>tan y < tan x\<close> by auto
```
```  4183   qed
```
```  4184 qed
```
```  4185
```
```  4186 lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)"
```
```  4187   unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
```
```  4188
```
```  4189 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
```
```  4190   by (simp add: tan_def)
```
```  4191
```
```  4192 lemma tan_periodic_nat[simp]:
```
```  4193   fixes n :: nat
```
```  4194   shows "tan (x + real n * pi) = tan x"
```
```  4195 proof (induct n arbitrary: x)
```
```  4196   case 0
```
```  4197   then show ?case by simp
```
```  4198 next
```
```  4199   case (Suc n)
```
```  4200   have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi"
```
```  4201     unfolding Suc_eq_plus1 of_nat_add  distrib_right by auto
```
```  4202   show ?case unfolding split_pi_off using Suc by auto
```
```  4203 qed
```
```  4204
```
```  4205 lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + of_int i * pi) = tan x"
```
```  4206 proof (cases "0 \<le> i")
```
```  4207   case True
```
```  4208   hence i_nat: "of_int i = of_int (nat i)" by auto
```
```  4209   show ?thesis unfolding i_nat
```
```  4210     by (metis of_int_of_nat_eq tan_periodic_nat)
```
```  4211 next
```
```  4212   case False
```
```  4213   hence i_nat: "of_int i = - of_int (nat (-i))" by auto
```
```  4214   have "tan x = tan (x + of_int i * pi - of_int i * pi)"
```
```  4215     by auto
```
```  4216   also have "\<dots> = tan (x + of_int i * pi)"
```
```  4217     unfolding i_nat mult_minus_left diff_minus_eq_add
```
```  4218     by (metis of_int_of_nat_eq tan_periodic_nat)
```
```  4219   finally show ?thesis by auto
```
```  4220 qed
```
```  4221
```
```  4222 lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
```
```  4223   using tan_periodic_int[of _ "numeral n" ] by simp
```
```  4224
```
```  4225 lemma tan_minus_45: "tan (-(pi/4)) = -1"
```
```  4226   unfolding tan_def by (simp add: sin_45 cos_45)
```
```  4227
```
```  4228 lemma tan_diff:
```
```  4229   fixes x :: "'a::{real_normed_field,banach}"
```
```  4230   shows
```
```  4231      "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0; cos (x - y) \<noteq> 0\<rbrakk>
```
```  4232       \<Longrightarrow> tan(x - y) = (tan(x) - tan(y))/(1 + tan(x) * tan(y))"
```
```  4233   using tan_add [of x "-y"]
```
```  4234   by simp
```
```  4235
```
```  4236
```
```  4237 lemma tan_pos_pi2_le: "0 \<le> x ==> x < pi/2 \<Longrightarrow> 0 \<le> tan x"
```
```  4238   using less_eq_real_def tan_gt_zero by auto
```
```  4239
```
```  4240 lemma cos_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> cos(x) = 1 / sqrt(1 + tan(x) ^ 2)"
```
```  4241   using cos_gt_zero_pi [of x]
```
```  4242   by (simp add: divide_simps tan_def real_sqrt_divide abs_if split: split_if_asm)
```
```  4243
```
```  4244 lemma sin_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> sin(x) = tan(x) / sqrt(1 + tan(x) ^ 2)"
```
```  4245   using cos_gt_zero [of "x"] cos_gt_zero [of "-x"]
```
```  4246   by (force simp add: divide_simps tan_def real_sqrt_divide abs_if split: split_if_asm)
```
```  4247
```
```  4248 lemma tan_mono_le: "-(pi/2) < x ==> x \<le> y ==> y < pi/2 \<Longrightarrow> tan(x) \<le> tan(y)"
```
```  4249   using less_eq_real_def tan_monotone by auto
```
```  4250
```
```  4251 lemma tan_mono_lt_eq: "-(pi/2) < x ==> x < pi/2 ==> -(pi/2) < y ==> y < pi/2
```
```  4252          \<Longrightarrow> (tan(x) < tan(y) \<longleftrightarrow> x < y)"
```
```  4253   using tan_monotone' by blast
```
```  4254
```
```  4255 lemma tan_mono_le_eq: "-(pi/2) < x ==> x < pi/2 ==> -(pi/2) < y ==> y < pi/2
```
```  4256          \<Longrightarrow> (tan(x) \<le> tan(y) \<longleftrightarrow> x \<le> y)"
```
```  4257   by (meson tan_mono_le not_le tan_monotone)
```
```  4258
```
```  4259 lemma tan_bound_pi2: "\<bar>x\<bar> < pi/4 \<Longrightarrow> \<bar>tan x\<bar> < 1"
```
```  4260   using tan_45 tan_monotone [of x "pi/4"] tan_monotone [of "-x" "pi/4"]
```
```  4261   by (auto simp: abs_if split: split_if_asm)
```
```  4262
```
```  4263 lemma tan_cot: "tan(pi/2 - x) = inverse(tan x)"
```
```  4264   by (simp add: tan_def sin_diff cos_diff)
```
```  4265
```
```  4266 subsection \<open>Cotangent\<close>
```
```  4267
```
```  4268 definition cot :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4269   where "cot = (\<lambda>x. cos x / sin x)"
```
```  4270
```
```  4271 lemma cot_of_real:
```
```  4272   "of_real (cot x) = (cot (of_real x) :: 'a::{real_normed_field,banach})"
```
```  4273   by (simp add: cot_def sin_of_real cos_of_real)
```
```  4274
```
```  4275 lemma cot_in_Reals [simp]:
```
```  4276   fixes z :: "'a::{real_normed_field,banach}"
```
```  4277   shows "z \<in> \<real> \<Longrightarrow> cot z \<in> \<real>"
```
```  4278   by (simp add: cot_def)
```
```  4279
```
```  4280 lemma cot_zero [simp]: "cot 0 = 0"
```
```  4281   by (simp add: cot_def)
```
```  4282
```
```  4283 lemma cot_pi [simp]: "cot pi = 0"
```
```  4284   by (simp add: cot_def)
```
```  4285
```
```  4286 lemma cot_npi [simp]: "cot (real (n::nat) * pi) = 0"
```
```  4287   by (simp add: cot_def)
```
```  4288
```
```  4289 lemma cot_minus [simp]: "cot (-x) = - cot x"
```
```  4290   by (simp add: cot_def)
```
```  4291
```
```  4292 lemma cot_periodic [simp]: "cot (x + 2*pi) = cot x"
```
```  4293   by (simp add: cot_def)
```
```  4294
```
```  4295 lemma cot_altdef: "cot x = inverse (tan x)"
```
```  4296   by (simp add: cot_def tan_def)
```
```  4297
```
```  4298 lemma tan_altdef: "tan x = inverse (cot x)"
```
```  4299   by (simp add: cot_def tan_def)
```
```  4300
```
```  4301 lemma tan_cot': "tan(pi/2 - x) = cot x"
```
```  4302   by (simp add: tan_cot cot_altdef)
```
```  4303
```
```  4304 lemma cot_gt_zero: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < cot x"
```
```  4305   by (simp add: cot_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
```
```  4306
```
```  4307 lemma cot_less_zero:
```
```  4308   assumes lb: "- pi/2 < x" and "x < 0"
```
```  4309   shows "cot x < 0"
```
```  4310 proof -
```
```  4311   have "0 < cot (- x)" using assms by (simp only: cot_gt_zero)
```
```  4312   thus ?thesis by simp
```
```  4313 qed
```
```  4314
```
```  4315 lemma DERIV_cot [simp]:
```
```  4316   fixes x :: "'a::{real_normed_field,banach}"
```
```  4317   shows "sin x \<noteq> 0 \<Longrightarrow> DERIV cot x :> -inverse ((sin x)\<^sup>2)"
```
```  4318   unfolding cot_def using cos_squared_eq[of x]
```
```  4319   by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square)
```
```  4320
```
```  4321 lemma isCont_cot:
```
```  4322   fixes x :: "'a::{real_normed_field,banach}"
```
```  4323   shows "sin x \<noteq> 0 \<Longrightarrow> isCont cot x"
```
```  4324   by (rule DERIV_cot [THEN DERIV_isCont])
```
```  4325
```
```  4326 lemma isCont_cot' [simp,continuous_intros]:
```
```  4327   fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
```
```  4328   shows "\<lbrakk>isCont f a; sin (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. cot (f x)) a"
```
```  4329   by (rule isCont_o2 [OF _ isCont_cot])
```
```  4330
```
```  4331 lemma tendsto_cot [tendsto_intros]:
```
```  4332   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4333   shows "\<lbrakk>(f \<longlongrightarrow> a) F; sin a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. cot (f x)) \<longlongrightarrow> cot a) F"
```
```  4334   by (rule isCont_tendsto_compose [OF isCont_cot])
```
```  4335
```
```  4336 lemma continuous_cot:
```
```  4337   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4338   shows "continuous F f \<Longrightarrow> sin (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. cot (f x))"
```
```  4339   unfolding continuous_def by (rule tendsto_cot)
```
```  4340
```
```  4341 lemma continuous_on_cot [continuous_intros]:
```
```  4342   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4343   shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. sin (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. cot (f x))"
```
```  4344   unfolding continuous_on_def by (auto intro: tendsto_cot)
```
```  4345
```
```  4346 lemma continuous_within_cot [continuous_intros]:
```
```  4347   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4348   shows
```
```  4349   "continuous (at x within s) f \<Longrightarrow> sin (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. cot (f x))"
```
```  4350   unfolding continuous_within by (rule tendsto_cot)
```
```  4351
```
```  4352
```
```  4353 subsection \<open>Inverse Trigonometric Functions\<close>
```
```  4354
```
```  4355 definition arcsin :: "real => real"
```
```  4356   where "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
```
```  4357
```
```  4358 definition arccos :: "real => real"
```
```  4359   where "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
```
```  4360
```
```  4361 definition arctan :: "real => real"
```
```  4362   where "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
```
```  4363
```
```  4364 lemma arcsin:
```
```  4365   "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow>
```
```  4366     -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2 & sin(arcsin y) = y"
```
```  4367   unfolding arcsin_def by (rule theI' [OF sin_total])
```
```  4368
```
```  4369 lemma arcsin_pi:
```
```  4370   "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
```
```  4371   apply (drule (1) arcsin)
```
```  4372   apply (force intro: order_trans)
```
```  4373   done
```
```  4374
```
```  4375 lemma sin_arcsin [simp]: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin(arcsin y) = y"
```
```  4376   by (blast dest: arcsin)
```
```  4377
```
```  4378 lemma arcsin_bounded: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
```
```  4379   by (blast dest: arcsin)
```
```  4380
```
```  4381 lemma arcsin_lbound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y"
```
```  4382   by (blast dest: arcsin)
```
```  4383
```
```  4384 lemma arcsin_ubound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
```
```  4385   by (blast dest: arcsin)
```
```  4386
```
```  4387 lemma arcsin_lt_bounded:
```
```  4388      "\<lbrakk>-1 < y; y < 1\<rbrakk> \<Longrightarrow> -(pi/2) < arcsin y & arcsin y < pi/2"
```
```  4389   apply (frule order_less_imp_le)
```
```  4390   apply (frule_tac y = y in order_less_imp_le)
```
```  4391   apply (frule arcsin_bounded)
```
```  4392   apply (safe, simp)
```
```  4393   apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
```
```  4394   apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
```
```  4395   apply (drule_tac [!] f = sin in arg_cong, auto)
```
```  4396   done
```
```  4397
```
```  4398 lemma arcsin_sin: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2\<rbrakk> \<Longrightarrow> arcsin(sin x) = x"
```
```  4399   apply (unfold arcsin_def)
```
```  4400   apply (rule the1_equality)
```
```  4401   apply (rule sin_total, auto)
```
```  4402   done
```
```  4403
```
```  4404 lemma arcsin_0 [simp]: "arcsin 0 = 0"
```
```  4405   using arcsin_sin [of 0]
```
```  4406   by simp
```
```  4407
```
```  4408 lemma arcsin_1 [simp]: "arcsin 1 = pi/2"
```
```  4409   using arcsin_sin [of "pi/2"]
```
```  4410   by simp
```
```  4411
```
```  4412 lemma arcsin_minus_1 [simp]: "arcsin (-1) = - (pi/2)"
```
```  4413   using arcsin_sin [of "-pi/2"]
```
```  4414   by simp
```
```  4415
```
```  4416 lemma arcsin_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin(-x) = -arcsin x"
```
```  4417   by (metis (no_types, hide_lams) arcsin arcsin_sin minus_minus neg_le_iff_le sin_minus)
```
```  4418
```
```  4419 lemma arcsin_eq_iff: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> (arcsin x = arcsin y \<longleftrightarrow> x = y)"
```
```  4420   by (metis abs_le_iff arcsin minus_le_iff)
```
```  4421
```
```  4422 lemma cos_arcsin_nonzero: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> cos(arcsin x) \<noteq> 0"
```
```  4423   using arcsin_lt_bounded cos_gt_zero_pi by force
```
```  4424
```
```  4425 lemma arccos:
```
```  4426      "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk>
```
```  4427       \<Longrightarrow> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
```
```  4428   unfolding arccos_def by (rule theI' [OF cos_total])
```
```  4429
```
```  4430 lemma cos_arccos [simp]: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> cos(arccos y) = y"
```
```  4431   by (blast dest: arccos)
```
```  4432
```
```  4433 lemma arccos_bounded: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> arccos y & arccos y \<le> pi"
```
```  4434   by (blast dest: arccos)
```
```  4435
```
```  4436 lemma arccos_lbound: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> arccos y"
```
```  4437   by (blast dest: arccos)
```
```  4438
```
```  4439 lemma arccos_ubound: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi"
```
```  4440   by (blast dest: arccos)
```
```  4441
```
```  4442 lemma arccos_lt_bounded:
```
```  4443      "\<lbrakk>-1 < y; y < 1\<rbrakk> \<Longrightarrow> 0 < arccos y & arccos y < pi"
```
```  4444   apply (frule order_less_imp_le)
```
```  4445   apply (frule_tac y = y in order_less_imp_le)
```
```  4446   apply (frule arccos_bounded, auto)
```
```  4447   apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
```
```  4448   apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
```
```  4449   apply (drule_tac [!] f = cos in arg_cong, auto)
```
```  4450   done
```
```  4451
```
```  4452 lemma arccos_cos: "\<lbrakk>0 \<le> x; x \<le> pi\<rbrakk> \<Longrightarrow> arccos(cos x) = x"
```
```  4453   apply (simp add: arccos_def)
```
```  4454   apply (auto intro!: the1_equality cos_total)
```
```  4455   done
```
```  4456
```
```  4457 lemma arccos_cos2: "\<lbrakk>x \<le> 0; -pi \<le> x\<rbrakk> \<Longrightarrow> arccos(cos x) = -x"
```
```  4458   apply (simp add: arccos_def)
```
```  4459   apply (auto intro!: the1_equality cos_total)
```
```  4460   done
```
```  4461
```
```  4462 lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
```
```  4463   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  4464   apply (rule power2_eq_imp_eq)
```
```  4465   apply (simp add: cos_squared_eq)
```
```  4466   apply (rule cos_ge_zero)
```
```  4467   apply (erule (1) arcsin_lbound)
```
```  4468   apply (erule (1) arcsin_ubound)
```
```  4469   apply simp
```
```  4470   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  4471   apply (rule power_mono, simp, simp)
```
```  4472   done
```
```  4473
```
```  4474 lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
```
```  4475   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  4476   apply (rule power2_eq_imp_eq)
```
```  4477   apply (simp add: sin_squared_eq)
```
```  4478   apply (rule sin_ge_zero)
```
```  4479   apply (erule (1) arccos_lbound)
```
```  4480   apply (erule (1) arccos_ubound)
```
```  4481   apply simp
```
```  4482   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  4483   apply (rule power_mono, simp, simp)
```
```  4484   done
```
```  4485
```
```  4486 lemma arccos_0 [simp]: "arccos 0 = pi/2"
```
```  4487 by (metis arccos_cos cos_gt_zero cos_pi cos_pi_half pi_gt_zero pi_half_ge_zero not_le not_zero_less_neg_numeral numeral_One)
```
```  4488
```
```  4489 lemma arccos_1 [simp]: "arccos 1 = 0"
```
```  4490   using arccos_cos by force
```
```  4491
```
```  4492 lemma arccos_minus_1 [simp]: "arccos(-1) = pi"
```
```  4493   by (metis arccos_cos cos_pi order_refl pi_ge_zero)
```
```  4494
```
```  4495 lemma arccos_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos(-x) = pi - arccos x"
```
```  4496   by (metis arccos_cos arccos_cos2 cos_minus_pi cos_total diff_le_0_iff_le le_add_same_cancel1
```
```  4497     minus_diff_eq uminus_add_conv_diff)
```
```  4498
```
```  4499 lemma sin_arccos_nonzero: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> ~(sin(arccos x) = 0)"
```
```  4500   using arccos_lt_bounded sin_gt_zero by force
```
```  4501
```
```  4502 lemma arctan: "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
```
```  4503   unfolding arctan_def by (rule theI' [OF tan_total])
```
```  4504
```
```  4505 lemma tan_arctan: "tan (arctan y) = y"
```
```  4506   by (simp add: arctan)
```
```  4507
```
```  4508 lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
```
```  4509   by (auto simp only: arctan)
```
```  4510
```
```  4511 lemma arctan_lbound: "- (pi/2) < arctan y"
```
```  4512   by (simp add: arctan)
```
```  4513
```
```  4514 lemma arctan_ubound: "arctan y < pi/2"
```
```  4515   by (auto simp only: arctan)
```
```  4516
```
```  4517 lemma arctan_unique:
```
```  4518   assumes "-(pi/2) < x"
```
```  4519     and "x < pi/2"
```
```  4520     and "tan x = y"
```
```  4521   shows "arctan y = x"
```
```  4522   using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
```
```  4523
```
```  4524 lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x"
```
```  4525   by (rule arctan_unique) simp_all
```
```  4526
```
```  4527 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
```
```  4528   by (rule arctan_unique) simp_all
```
```  4529
```
```  4530 lemma arctan_minus: "arctan (- x) = - arctan x"
```
```  4531   apply (rule arctan_unique)
```
```  4532   apply (simp only: neg_less_iff_less arctan_ubound)
```
```  4533   apply (metis minus_less_iff arctan_lbound, simp add: arctan)
```
```  4534   done
```
```  4535
```
```  4536 lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
```
```  4537   by (intro less_imp_neq [symmetric] cos_gt_zero_pi
```
```  4538     arctan_lbound arctan_ubound)
```
```  4539
```
```  4540 lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)"
```
```  4541 proof (rule power2_eq_imp_eq)
```
```  4542   have "0 < 1 + x\<^sup>2" by (simp add: add_pos_nonneg)
```
```  4543   show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp
```
```  4544   show "0 \<le> cos (arctan x)"
```
```  4545     by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
```
```  4546   have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1"
```
```  4547     unfolding tan_def by (simp add: distrib_left power_divide)
```
```  4548   thus "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
```
```  4549     using \<open>0 < 1 + x\<^sup>2\<close> by (simp add: arctan power_divide eq_divide_eq)
```
```  4550 qed
```
```  4551
```
```  4552 lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)"
```
```  4553   using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
```
```  4554   using tan_arctan [of x] unfolding tan_def cos_arctan
```
```  4555   by (simp add: eq_divide_eq)
```
```  4556
```
```  4557 lemma tan_sec:
```
```  4558   fixes x :: "'a::{real_normed_field,banach,field}"
```
```  4559   shows "cos x \<noteq> 0 \<Longrightarrow> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
```
```  4560   apply (rule power_inverse [THEN subst])
```
```  4561   apply (rule_tac c1 = "(cos x)\<^sup>2" in mult_right_cancel [THEN iffD1])
```
```  4562   apply (auto simp add: tan_def field_simps)
```
```  4563   done
```
```  4564
```
```  4565 lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
```
```  4566   by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
```
```  4567
```
```  4568 lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
```
```  4569   by (simp only: not_less [symmetric] arctan_less_iff)
```
```  4570
```
```  4571 lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
```
```  4572   by (simp only: eq_iff [where 'a=real] arctan_le_iff)
```
```  4573
```
```  4574 lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
```
```  4575   using arctan_less_iff [of 0 x] by simp
```
```  4576
```
```  4577 lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
```
```  4578   using arctan_less_iff [of x 0] by simp
```
```  4579
```
```  4580 lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
```
```  4581   using arctan_le_iff [of 0 x] by simp
```
```  4582
```
```  4583 lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```  4584   using arctan_le_iff [of x 0] by simp
```
```  4585
```
```  4586 lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
```
```  4587   using arctan_eq_iff [of x 0] by simp
```
```  4588
```
```  4589 lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin"
```
```  4590 proof -
```
```  4591   have "continuous_on (sin ` {- pi / 2 .. pi / 2}) arcsin"
```
```  4592     by (rule continuous_on_inv) (auto intro: continuous_intros simp: arcsin_sin)
```
```  4593   also have "sin ` {- pi / 2 .. pi / 2} = {-1 .. 1}"
```
```  4594   proof safe
```
```  4595     fix x :: real
```
```  4596     assume "x \<in> {-1..1}"
```
```  4597     then show "x \<in> sin ` {- pi / 2..pi / 2}"
```
```  4598       using arcsin_lbound arcsin_ubound
```
```  4599       by (intro image_eqI[where x="arcsin x"]) auto
```
```  4600   qed simp
```
```  4601   finally show ?thesis .
```
```  4602 qed
```
```  4603
```
```  4604 lemma continuous_on_arcsin [continuous_intros]:
```
```  4605   "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))"
```
```  4606   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arcsin']]
```
```  4607   by (auto simp: comp_def subset_eq)
```
```  4608
```
```  4609 lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x"
```
```  4610   using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  4611   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  4612
```
```  4613 lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos"
```
```  4614 proof -
```
```  4615   have "continuous_on (cos ` {0 .. pi}) arccos"
```
```  4616     by (rule continuous_on_inv) (auto intro: continuous_intros simp: arccos_cos)
```
```  4617   also have "cos ` {0 .. pi} = {-1 .. 1}"
```
```  4618   proof safe
```
```  4619     fix x :: real
```
```  4620     assume "x \<in> {-1..1}"
```
```  4621     then show "x \<in> cos ` {0..pi}"
```
```  4622       using arccos_lbound arccos_ubound
```
```  4623       by (intro image_eqI[where x="arccos x"]) auto
```
```  4624   qed simp
```
```  4625   finally show ?thesis .
```
```  4626 qed
```
```  4627
```
```  4628 lemma continuous_on_arccos [continuous_intros]:
```
```  4629   "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))"
```
```  4630   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arccos']]
```
```  4631   by (auto simp: comp_def subset_eq)
```
```  4632
```
```  4633 lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x"
```
```  4634   using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  4635   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  4636
```
```  4637 lemma isCont_arctan: "isCont arctan x"
```
```  4638   apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
```
```  4639   apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
```
```  4640   apply (subgoal_tac "isCont arctan (tan (arctan x))", simp add: arctan)
```
```  4641   apply (erule (1) isCont_inverse_function2 [where f=tan])
```
```  4642   apply (metis arctan_tan order_le_less_trans order_less_le_trans)
```
```  4643   apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
```
```  4644   done
```
```  4645
```
```  4646 lemma tendsto_arctan [tendsto_intros]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) \<longlongrightarrow> arctan x) F"
```
```  4647   by (rule isCont_tendsto_compose [OF isCont_arctan])
```
```  4648
```
```  4649 lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))"
```
```  4650   unfolding continuous_def by (rule tendsto_arctan)
```
```  4651
```
```  4652 lemma continuous_on_arctan [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
```
```  4653   unfolding continuous_on_def by (auto intro: tendsto_arctan)
```
```  4654
```
```  4655 lemma DERIV_arcsin:
```
```  4656   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
```
```  4657   apply (rule DERIV_inverse_function [where f=sin and a="-1" and b=1])
```
```  4658   apply (rule DERIV_cong [OF DERIV_sin])
```
```  4659   apply (simp add: cos_arcsin)
```
```  4660   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  4661   apply (rule power_strict_mono, simp, simp, simp, assumption, assumption)
```
```  4662   apply simp
```
```  4663   apply (erule (1) isCont_arcsin)
```
```  4664   done
```
```  4665
```
```  4666 lemma DERIV_arccos:
```
```  4667   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
```
```  4668   apply (rule DERIV_inverse_function [where f=cos and a="-1" and b=1])
```
```  4669   apply (rule DERIV_cong [OF DERIV_cos])
```
```  4670   apply (simp add: sin_arccos)
```
```  4671   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  4672   apply (rule power_strict_mono, simp, simp, simp, assumption, assumption)
```
```  4673   apply simp
```
```  4674   apply (erule (1) isCont_arccos)
```
```  4675   done
```
```  4676
```
```  4677 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
```
```  4678   apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
```
```  4679   apply (rule DERIV_cong [OF DERIV_tan])
```
```  4680   apply (rule cos_arctan_not_zero)
```
```  4681   apply (simp_all add: add_pos_nonneg arctan isCont_arctan)
```
```  4682   apply (simp add: arctan power_inverse [symmetric] tan_sec [symmetric])
```
```  4683   apply (subgoal_tac "0 < 1 + x\<^sup>2", simp)
```
```  4684   apply (simp_all add: add_pos_nonneg arctan isCont_arctan)
```
```  4685   done
```
```  4686
```
```  4687 declare
```
```  4688   DERIV_arcsin[THEN DERIV_chain2, derivative_intros]
```
```  4689   DERIV_arcsin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  4690   DERIV_arccos[THEN DERIV_chain2, derivative_intros]
```
```  4691   DERIV_arccos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  4692   DERIV_arctan[THEN DERIV_chain2, derivative_intros]
```
```  4693   DERIV_arctan[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
```
```  4694
```
```  4695 lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- (pi/2)))"
```
```  4696   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])
```
```  4697      (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  4698            intro!: tan_monotone exI[of _ "pi/2"])
```
```  4699
```
```  4700 lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"
```
```  4701   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])
```
```  4702      (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  4703            intro!: tan_monotone exI[of _ "pi/2"])
```
```  4704
```
```  4705 lemma tendsto_arctan_at_top: "(arctan \<longlongrightarrow> (pi/2)) at_top"
```
```  4706 proof (rule tendstoI)
```
```  4707   fix e :: real
```
```  4708   assume "0 < e"
```
```  4709   def y \<equiv> "pi/2 - min (pi/2) e"
```
```  4710   then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y"
```
```  4711     using \<open>0 < e\<close> by auto
```
```  4712
```
```  4713   show "eventually (\<lambda>x. dist (arctan x) (pi / 2) < e) at_top"
```
```  4714   proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)
```
```  4715     fix x
```
```  4716     assume "tan y < x"
```
```  4717     then have "arctan (tan y) < arctan x"
```
```  4718       by (simp add: arctan_less_iff)
```
```  4719     with y have "y < arctan x"
```
```  4720       by (subst (asm) arctan_tan) simp_all
```
```  4721     with arctan_ubound[of x, arith] y \<open>0 < e\<close>
```
```  4722     show "dist (arctan x) (pi / 2) < e"
```
```  4723       by (simp add: dist_real_def)
```
```  4724   qed
```
```  4725 qed
```
```  4726
```
```  4727 lemma tendsto_arctan_at_bot: "(arctan \<longlongrightarrow> - (pi/2)) at_bot"
```
```  4728   unfolding filterlim_at_bot_mirror arctan_minus
```
```  4729   by (intro tendsto_minus tendsto_arctan_at_top)
```
```  4730
```
```  4731
```
```  4732 subsection\<open>Prove Totality of the Trigonometric Functions\<close>
```
```  4733
```
```  4734 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
```
```  4735   by (simp add: abs_le_iff)
```
```  4736
```
```  4737 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
```
```  4738   by (simp add: sin_arccos abs_le_iff)
```
```  4739
```
```  4740 lemma sin_mono_less_eq: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2; -(pi/2) \<le> y; y \<le> pi/2\<rbrakk>
```
```  4741          \<Longrightarrow> (sin(x) < sin(y) \<longleftrightarrow> x < y)"
```
```  4742 by (metis not_less_iff_gr_or_eq sin_monotone_2pi)
```
```  4743
```
```  4744 lemma sin_mono_le_eq: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2; -(pi/2) \<le> y; y \<le> pi/2\<rbrakk>
```
```  4745          \<Longrightarrow> (sin(x) \<le> sin(y) \<longleftrightarrow> x \<le> y)"
```
```  4746 by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le)
```
```  4747
```
```  4748 lemma sin_inj_pi:
```
```  4749     "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2;-(pi/2) \<le> y; y \<le> pi/2; sin(x) = sin(y)\<rbrakk> \<Longrightarrow> x = y"
```
```  4750 by (metis arcsin_sin)
```
```  4751
```
```  4752 lemma cos_mono_less_eq:
```
```  4753     "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi \<Longrightarrow> (cos(x) < cos(y) \<longleftrightarrow> y < x)"
```
```  4754 by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear)
```
```  4755
```
```  4756 lemma cos_mono_le_eq: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi
```
```  4757          \<Longrightarrow> (cos(x) \<le> cos(y) \<longleftrightarrow> y \<le> x)"
```
```  4758   by (metis arccos_cos cos_monotone_0_pi_le eq_iff linear)
```
```  4759
```
```  4760 lemma cos_inj_pi: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi ==> cos(x) = cos(y)
```
```  4761          \<Longrightarrow> x = y"
```
```  4762 by (metis arccos_cos)
```
```  4763
```
```  4764 lemma arccos_le_pi2: "\<lbrakk>0 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi/2"
```
```  4765   by (metis (mono_tags) arccos_0 arccos cos_le_one cos_monotone_0_pi_le
```
```  4766       cos_pi cos_pi_half pi_half_ge_zero antisym_conv less_eq_neg_nonpos linear minus_minus order.trans order_refl)
```
```  4767
```
```  4768 lemma sincos_total_pi_half:
```
```  4769   assumes "0 \<le> x" "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1"
```
```  4770     shows "\<exists>t. 0 \<le> t \<and> t \<le> pi/2 \<and> x = cos t \<and> y = sin t"
```
```  4771 proof -
```
```  4772   have x1: "x \<le> 1"
```
```  4773     using assms
```
```  4774     by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2)
```
```  4775   moreover with assms have ax: "0 \<le> arccos x" "cos(arccos x) = x"
```
```  4776     by (auto simp: arccos)
```
```  4777   moreover have "y = sqrt (1 - x\<^sup>2)" using assms
```
```  4778     by (metis abs_of_nonneg add.commute add_diff_cancel real_sqrt_abs)
```
```  4779   ultimately show ?thesis using assms arccos_le_pi2 [of x]
```
```  4780     by (rule_tac x="arccos x" in exI) (auto simp: sin_arccos)
```
```  4781 qed
```
```  4782
```
```  4783 lemma sincos_total_pi:
```
```  4784   assumes "0 \<le> y" and "x\<^sup>2 + y\<^sup>2 = 1"
```
```  4785     shows "\<exists>t. 0 \<le> t \<and> t \<le> pi \<and> x = cos t \<and> y = sin t"
```
```  4786 proof (cases rule: le_cases [of 0 x])
```
```  4787   case le from sincos_total_pi_half [OF le]
```
```  4788   show ?thesis
```
```  4789     by (metis pi_ge_two pi_half_le_two add.commute add_le_cancel_left add_mono assms)
```
```  4790 next
```
```  4791   case ge
```
```  4792   then have "0 \<le> -x"
```
```  4793     by simp
```
```  4794   then obtain t where "t\<ge>0" "t \<le> pi/2" "-x = cos t" "y = sin t"
```
```  4795     using sincos_total_pi_half assms
```
```  4796     apply auto
```
```  4797     by (metis \<open>0 \<le> - x\<close> power2_minus)
```
```  4798   then show ?thesis
```
```  4799     by (rule_tac x="pi-t" in exI, auto)
```
```  4800 qed
```
```  4801
```
```  4802 lemma sincos_total_2pi_le:
```
```  4803   assumes "x\<^sup>2 + y\<^sup>2 = 1"
```
```  4804     shows "\<exists>t. 0 \<le> t \<and> t \<le> 2*pi \<and> x = cos t \<and> y = sin t"
```
```  4805 proof (cases rule: le_cases [of 0 y])
```
```  4806   case le from sincos_total_pi [OF le]
```
```  4807   show ?thesis
```
```  4808     by (metis assms le_add_same_cancel1 mult.commute mult_2_right order.trans)
```
```  4809 next
```
```  4810   case ge
```
```  4811   then have "0 \<le> -y"
```
```  4812     by simp
```
```  4813   then obtain t where "t\<ge>0" "t \<le> pi" "x = cos t" "-y = sin t"
```
```  4814     using sincos_total_pi assms
```
```  4815     apply auto
```
```  4816     by (metis \<open>0 \<le> - y\<close> power2_minus)
```
```  4817   then show ?thesis
```
```  4818     by (rule_tac x="2*pi-t" in exI, auto)
```
```  4819 qed
```
```  4820
```
```  4821 lemma sincos_total_2pi:
```
```  4822   assumes "x\<^sup>2 + y\<^sup>2 = 1"
```
```  4823     obtains t where "0 \<le> t" "t < 2*pi" "x = cos t" "y = sin t"
```
```  4824 proof -
```
```  4825   from sincos_total_2pi_le [OF assms]
```
```  4826   obtain t where t: "0 \<le> t" "t \<le> 2*pi" "x = cos t" "y = sin t"
```
```  4827     by blast
```
```  4828   show ?thesis
```
```  4829     apply (cases "t = 2*pi")
```
```  4830     using t that
```
```  4831     apply force+
```
```  4832     done
```
```  4833 qed
```
```  4834
```
```  4835 lemma arcsin_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x < arcsin y \<longleftrightarrow> x < y"
```
```  4836   apply (rule trans [OF sin_mono_less_eq [symmetric]])
```
```  4837   using arcsin_ubound arcsin_lbound
```
```  4838   apply auto
```
```  4839   done
```
```  4840
```
```  4841 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"
```
```  4842   using arcsin_less_mono not_le by blast
```
```  4843
```
```  4844 lemma arcsin_less_arcsin: "-1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x < arcsin y"
```
```  4845   using arcsin_less_mono by auto
```
```  4846
```
```  4847 lemma arcsin_le_arcsin: "-1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y"
```
```  4848   using arcsin_le_mono by auto
```
```  4849
```
```  4850 lemma arccos_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> (arccos x < arccos y \<longleftrightarrow> y < x)"
```
```  4851   apply (rule trans [OF cos_mono_less_eq [symmetric]])
```
```  4852   using arccos_ubound arccos_lbound
```
```  4853   apply auto
```
```  4854   done
```
```  4855
```
```  4856 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"
```
```  4857   using arccos_less_mono [of y x]
```
```  4858   by (simp add: not_le [symmetric])
```
```  4859
```
```  4860 lemma arccos_less_arccos: "-1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y < arccos x"
```
```  4861   using arccos_less_mono by auto
```
```  4862
```
```  4863 lemma arccos_le_arccos: "-1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> arccos x"
```
```  4864   using arccos_le_mono by auto
```
```  4865
```
```  4866 lemma arccos_eq_iff: "\<bar>x\<bar> \<le> 1 & \<bar>y\<bar> \<le> 1 \<Longrightarrow> (arccos x = arccos y \<longleftrightarrow> x = y)"
```
```  4867   using cos_arccos_abs by fastforce
```
```  4868
```
```  4869 subsection \<open>Machins formula\<close>
```
```  4870
```
```  4871 lemma arctan_one: "arctan 1 = pi / 4"
```
```  4872   by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi)
```
```  4873
```
```  4874 lemma tan_total_pi4:
```
```  4875   assumes "\<bar>x\<bar> < 1"
```
```  4876   shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
```
```  4877 proof
```
```  4878   show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
```
```  4879     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  4880     unfolding arctan_less_iff using assms  by (auto simp add: arctan)
```
```  4881
```
```  4882 qed
```
```  4883
```
```  4884 lemma arctan_add:
```
```  4885   assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
```
```  4886   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
```
```  4887 proof (rule arctan_unique [symmetric])
```
```  4888   have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y"
```
```  4889     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  4890     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  4891   from add_le_less_mono [OF this]
```
```  4892   show 1: "- (pi / 2) < arctan x + arctan y" by simp
```
```  4893   have "arctan x \<le> pi / 4" and "arctan y < pi / 4"
```
```  4894     unfolding arctan_one [symmetric]
```
```  4895     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  4896   from add_le_less_mono [OF this]
```
```  4897   show 2: "arctan x + arctan y < pi / 2" by simp
```
```  4898   show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
```
```  4899     using cos_gt_zero_pi [OF 1 2] by (simp add: arctan tan_add)
```
```  4900 qed
```
```  4901
```
```  4902 lemma arctan_double:
```
```  4903   assumes "\<bar>x\<bar> < 1"
```
```  4904   shows "2 * arctan x = arctan ((2*x) / (1 - x\<^sup>2))"
```
```  4905   by (metis assms arctan_add linear mult_2 not_less power2_eq_square)
```
```  4906
```
```  4907 theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
```
```  4908 proof -
```
```  4909   have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
```
```  4910   from arctan_add[OF less_imp_le[OF this] this]
```
```  4911   have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
```
```  4912   moreover
```
```  4913   have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
```
```  4914   from arctan_add[OF less_imp_le[OF this] this]
```
```  4915   have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
```
```  4916   moreover
```
```  4917   have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
```
```  4918   from arctan_add[OF this]
```
```  4919   have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
```
```  4920   ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
```
```  4921   thus ?thesis unfolding arctan_one by algebra
```
```  4922 qed
```
```  4923
```
```  4924 lemma machin_Euler: "5 * arctan(1/7) + 2 * arctan(3/79) = pi/4"
```
```  4925 proof -
```
```  4926   have 17: "\<bar>1/7\<bar> < (1 :: real)" by auto
```
```  4927   with arctan_double have "2 * arctan (1/7) = arctan (7/24)"
```
```  4928     by simp (simp add: field_simps)
```
```  4929   moreover have "\<bar>7/24\<bar> < (1 :: real)" by auto
```
```  4930   with arctan_double have "2 * arctan (7/24) = arctan (336/527)"  by simp (simp add: field_simps)
```
```  4931   moreover have "\<bar>336/527\<bar> < (1 :: real)" by auto
```
```  4932   from arctan_add[OF less_imp_le[OF 17] this]
```
```  4933   have "arctan(1/7) + arctan (336/527) = arctan (2879/3353)"  by auto
```
```  4934   ultimately have I: "5 * arctan(1/7) = arctan (2879/3353)"  by auto
```
```  4935   have 379: "\<bar>3/79\<bar> < (1 :: real)" by auto
```
```  4936   with arctan_double have II: "2 * arctan (3/79) = arctan (237/3116)"  by simp (simp add: field_simps)
```
```  4937   have *: "\<bar>2879/3353\<bar> < (1 :: real)" by auto
```
```  4938   have "\<bar>237/3116\<bar> < (1 :: real)" by auto
```
```  4939   from arctan_add[OF less_imp_le[OF *] this]
```
```  4940   have "arctan (2879/3353) + arctan (237/3116) = pi/4"
```
```  4941     by (simp add: arctan_one)
```
```  4942   then show ?thesis using I II
```
```  4943     by auto
```
```  4944 qed
```
```  4945
```
```  4946 (*But could also prove MACHIN_GAUSS:
```
```  4947   12 * arctan(1/18) + 8 * arctan(1/57) - 5 * arctan(1/239) = pi/4*)
```
```  4948
```
```  4949
```
```  4950 subsection \<open>Introducing the inverse tangent power series\<close>
```
```  4951
```
```  4952 lemma monoseq_arctan_series:
```
```  4953   fixes x :: real
```
```  4954   assumes "\<bar>x\<bar> \<le> 1"
```
```  4955   shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
```
```  4956 proof (cases "x = 0")
```
```  4957   case True
```
```  4958   thus ?thesis unfolding monoseq_def One_nat_def by auto
```
```  4959 next
```
```  4960   case False
```
```  4961   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  4962   show "monoseq ?a"
```
```  4963   proof -
```
```  4964     {
```
```  4965       fix n
```
```  4966       fix x :: real
```
```  4967       assume "0 \<le> x" and "x \<le> 1"
```
```  4968       have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le>
```
```  4969         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
```
```  4970       proof (rule mult_mono)
```
```  4971         show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))"
```
```  4972           by (rule frac_le) simp_all
```
```  4973         show "0 \<le> 1 / real (Suc (n * 2))"
```
```  4974           by auto
```
```  4975         show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)"
```
```  4976           by (rule power_decreasing) (simp_all add: \<open>0 \<le> x\<close> \<open>x \<le> 1\<close>)
```
```  4977         show "0 \<le> x ^ Suc (Suc n * 2)"
```
```  4978           by (rule zero_le_power) (simp add: \<open>0 \<le> x\<close>)
```
```  4979       qed
```
```  4980     } note mono = this
```
```  4981
```
```  4982     show ?thesis
```
```  4983     proof (cases "0 \<le> x")
```
```  4984       case True from mono[OF this \<open>x \<le> 1\<close>, THEN allI]
```
```  4985       show ?thesis unfolding Suc_eq_plus1[symmetric]
```
```  4986         by (rule mono_SucI2)
```
```  4987     next
```
```  4988       case False
```
```  4989       hence "0 \<le> -x" and "-x \<le> 1" using \<open>-1 \<le> x\<close> by auto
```
```  4990       from mono[OF this]
```
```  4991       have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge>
```
```  4992         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using \<open>0 \<le> -x\<close> by auto
```
```  4993       thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
```
```  4994     qed
```
```  4995   qed
```
```  4996 qed
```
```  4997
```
```  4998 lemma zeroseq_arctan_series:
```
```  4999   fixes x :: real
```
```  5000   assumes "\<bar>x\<bar> \<le> 1"
```
```  5001   shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) \<longlonglongrightarrow> 0" (is "?a \<longlonglongrightarrow> 0")
```
```  5002 proof (cases "x = 0")
```
```  5003   case True
```
```  5004   thus ?thesis
```
```  5005     unfolding One_nat_def by auto
```
```  5006 next
```
```  5007   case False
```
```  5008   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  5009   show "?a \<longlonglongrightarrow> 0"
```
```  5010   proof (cases "\<bar>x\<bar> < 1")
```
```  5011     case True
```
```  5012     hence "norm x < 1" by auto
```
```  5013     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF \<open>norm x < 1\<close>, THEN LIMSEQ_Suc]]
```
```  5014     have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) \<longlonglongrightarrow> 0"
```
```  5015       unfolding inverse_eq_divide Suc_eq_plus1 by simp
```
```  5016     then show ?thesis using pos2 by (rule LIMSEQ_linear)
```
```  5017   next
```
```  5018     case False
```
```  5019     hence "x = -1 \<or> x = 1" using \<open>\<bar>x\<bar> \<le> 1\<close> by auto
```
```  5020     hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x"
```
```  5021       unfolding One_nat_def by auto
```
```  5022     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
```
```  5023     show ?thesis unfolding n_eq Suc_eq_plus1 by auto
```
```  5024   qed
```
```  5025 qed
```
```  5026
```
```  5027 lemma summable_arctan_series:
```
```  5028   fixes n :: nat
```
```  5029   assumes "\<bar>x\<bar> \<le> 1"
```
```  5030   shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  5031   (is "summable (?c x)")
```
```  5032   by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
```
```  5033
```
```  5034 lemma DERIV_arctan_series:
```
```  5035   assumes "\<bar> x \<bar> < 1"
```
```  5036   shows "DERIV (\<lambda> x'. \<Sum> k. (-1)^k * (1 / real (k*2+1) * x' ^ (k*2+1))) x :> (\<Sum> k. (-1)^k * x^(k*2))"
```
```  5037   (is "DERIV ?arctan _ :> ?Int")
```
```  5038 proof -
```
```  5039   let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
```
```  5040
```
```  5041   have n_even: "\<And>n :: nat. even n \<Longrightarrow> 2 * (n div 2) = n"
```
```  5042     by presburger
```
```  5043   then have if_eq: "\<And>n x'. ?f n * real (Suc n) * x'^n =
```
```  5044     (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)"
```
```  5045     by auto
```
```  5046
```
```  5047   {
```
```  5048     fix x :: real
```
```  5049     assume "\<bar>x\<bar> < 1"
```
```  5050     hence "x\<^sup>2 < 1" by (simp add: abs_square_less_1)
```
```  5051     have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)"
```
```  5052       by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow \<open>x\<^sup>2 < 1\<close> order_less_imp_le[OF \<open>x\<^sup>2 < 1\<close>])
```
```  5053     hence "summable (\<lambda> n. (- 1) ^ n * x^(2*n))" unfolding power_mult .
```
```  5054   } note summable_Integral = this
```
```  5055
```
```  5056   {
```
```  5057     fix f :: "nat \<Rightarrow> real"
```
```  5058     have "\<And>x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  5059     proof
```
```  5060       fix x :: real
```
```  5061       assume "f sums x"
```
```  5062       from sums_if[OF sums_zero this]
```
```  5063       show "(\<lambda>n. if even n then f (n div 2) else 0) sums x"
```
```  5064         by auto
```
```  5065     next
```
```  5066       fix x :: real
```
```  5067       assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  5068       from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult.commute]]
```
```  5069       show "f sums x" unfolding sums_def by auto
```
```  5070     qed
```
```  5071     hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
```
```  5072   } note sums_even = this
```
```  5073
```
```  5074   have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int"
```
```  5075     unfolding if_eq mult.commute[of _ 2] suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * x ^ (2 * n)", symmetric]
```
```  5076     by auto
```
```  5077
```
```  5078   {
```
```  5079     fix x :: real
```
```  5080     have if_eq': "\<And>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
```
```  5081       (if even n then (- 1) ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
```
```  5082       using n_even by auto
```
```  5083     have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)" by auto
```
```  5084     have "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x"
```
```  5085       unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
```
```  5086       by auto
```
```  5087   } note arctan_eq = this
```
```  5088
```
```  5089   have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
```
```  5090   proof (rule DERIV_power_series')
```
```  5091     show "x \<in> {- 1 <..< 1}" using \<open>\<bar> x \<bar> < 1\<close> by auto
```
```  5092     {
```
```  5093       fix x' :: real
```
```  5094       assume x'_bounds: "x' \<in> {- 1 <..< 1}"
```
```  5095       then have "\<bar>x'\<bar> < 1" by auto
```
```  5096       then
```
```  5097         have *: "summable (\<lambda>n. (- 1) ^ n * x' ^ (2 * n))"
```
```  5098         by (rule summable_Integral)
```
```  5099       let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
```
```  5100       show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
```
```  5101         apply (rule sums_summable [where l="0 + ?S"])
```
```  5102         apply (rule sums_if)
```
```  5103         apply (rule sums_zero)
```
```  5104         apply (rule summable_sums)
```
```  5105         apply (rule *)
```
```  5106         done
```
```  5107     }
```
```  5108   qed auto
```
```  5109   thus ?thesis unfolding Int_eq arctan_eq .
```
```  5110 qed
```
```  5111
```
```  5112 lemma arctan_series:
```
```  5113   assumes "\<bar> x \<bar> \<le> 1"
```
```  5114   shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  5115   (is "_ = suminf (\<lambda> n. ?c x n)")
```
```  5116 proof -
```
```  5117   let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)"
```
```  5118
```
```  5119   {
```
```  5120     fix r x :: real
```
```  5121     assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
```
```  5122     have "\<bar>x\<bar> < 1" using \<open>r < 1\<close> and \<open>\<bar>x\<bar> < r\<close> by auto
```
```  5123     from DERIV_arctan_series[OF this] have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
```
```  5124   } note DERIV_arctan_suminf = this
```
```  5125
```
```  5126   {
```
```  5127     fix x :: real
```
```  5128     assume "\<bar>x\<bar> \<le> 1"
```
```  5129     note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]]
```
```  5130   } note arctan_series_borders = this
```
```  5131
```
```  5132   {
```
```  5133     fix x :: real
```
```  5134     assume "\<bar>x\<bar> < 1"
```
```  5135     have "arctan x = (\<Sum>k. ?c x k)"
```
```  5136     proof -
```
```  5137       obtain r where "\<bar>x\<bar> < r" and "r < 1"
```
```  5138         using dense[OF \<open>\<bar>x\<bar> < 1\<close>] by blast
```
```  5139       hence "0 < r" and "-r < x" and "x < r" by auto
```
```  5140
```
```  5141       have suminf_eq_arctan_bounded: "\<And>x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow>
```
```  5142         suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  5143       proof -
```
```  5144         fix x a b
```
```  5145         assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
```
```  5146         hence "\<bar>x\<bar> < r" by auto
```
```  5147         show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  5148         proof (rule DERIV_isconst2[of "a" "b"])
```
```  5149           show "a < b" and "a \<le> x" and "x \<le> b"
```
```  5150             using \<open>a < b\<close> \<open>a \<le> x\<close> \<open>x \<le> b\<close> by auto
```
```  5151           have "\<forall>x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  5152           proof (rule allI, rule impI)
```
```  5153             fix x
```
```  5154             assume "-r < x \<and> x < r"
```
```  5155             hence "\<bar>x\<bar> < r" by auto
```
```  5156             hence "\<bar>x\<bar> < 1" using \<open>r < 1\<close> by auto
```
```  5157             have "\<bar> - (x\<^sup>2) \<bar> < 1"
```
```  5158               using abs_square_less_1 \<open>\<bar>x\<bar> < 1\<close> by auto
```
```  5159             hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  5160               unfolding real_norm_def[symmetric] by (rule geometric_sums)
```
```  5161             hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  5162               unfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2] by auto
```
```  5163             hence suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)"
```
```  5164               using sums_unique unfolding inverse_eq_divide by auto
```
```  5165             have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))"
```
```  5166               unfolding suminf_c'_eq_geom
```
```  5167               by (rule DERIV_arctan_suminf[OF \<open>0 < r\<close> \<open>r < 1\<close> \<open>\<bar>x\<bar> < r\<close>])
```
```  5168             from DERIV_diff [OF this DERIV_arctan]
```
```  5169             show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  5170               by auto
```
```  5171           qed
```
```  5172           hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
```
```  5173             using \<open>-r < a\<close> \<open>b < r\<close> by auto
```
```  5174           thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
```
```  5175             using \<open>\<bar>x\<bar> < r\<close> by auto
```
```  5176           show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y"
```
```  5177             using DERIV_in_rball DERIV_isCont by auto
```
```  5178         qed
```
```  5179       qed
```
```  5180
```
```  5181       have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
```
```  5182         unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero
```
```  5183         by auto
```
```  5184
```
```  5185       have "suminf (?c x) - arctan x = 0"
```
```  5186       proof (cases "x = 0")
```
```  5187         case True
```
```  5188         thus ?thesis using suminf_arctan_zero by auto
```
```  5189       next
```
```  5190         case False
```
```  5191         hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
```
```  5192         have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
```
```  5193           by (rule suminf_eq_arctan_bounded[where x1="0" and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>", symmetric])
```
```  5194             (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>-\<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less)
```
```  5195         moreover
```
```  5196         have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
```
```  5197           by (rule suminf_eq_arctan_bounded[where x1="x" and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>"])
```
```  5198              (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>-\<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less)
```
```  5199         ultimately
```
```  5200         show ?thesis using suminf_arctan_zero by auto
```
```  5201       qed
```
```  5202       thus ?thesis by auto
```
```  5203     qed
```
```  5204   } note when_less_one = this
```
```  5205
```
```  5206   show "arctan x = suminf (\<lambda> n. ?c x n)"
```
```  5207   proof (cases "\<bar>x\<bar> < 1")
```
```  5208     case True
```
```  5209     thus ?thesis by (rule when_less_one)
```
```  5210   next
```
```  5211     case False
```
```  5212     hence "\<bar>x\<bar> = 1" using \<open>\<bar>x\<bar> \<le> 1\<close> by auto
```
```  5213     let ?a = "\<lambda>x n. \<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
```
```  5214     let ?diff = "\<lambda> x n. \<bar> arctan x - (\<Sum> i<n. ?c x i)\<bar>"
```
```  5215     {
```
```  5216       fix n :: nat
```
```  5217       have "0 < (1 :: real)" by auto
```
```  5218       moreover
```
```  5219       {
```
```  5220         fix x :: real
```
```  5221         assume "0 < x" and "x < 1"
```
```  5222         hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
```
```  5223         from \<open>0 < x\<close> have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)"
```
```  5224           by auto
```
```  5225         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]
```
```  5226         have "0 < 1 / real (n*2+1) * x^(n*2+1)"
```
```  5227           by (rule mult_pos_pos, auto simp only: zero_less_power[OF \<open>0 < x\<close>], auto)
```
```  5228         hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)"
```
```  5229           by (rule abs_of_pos)
```
```  5230         have "?diff x n \<le> ?a x n"
```
```  5231         proof (cases "even n")
```
```  5232           case True
```
```  5233           hence sgn_pos: "(-1)^n = (1::real)" by auto
```
```  5234           from \<open>even n\<close> obtain m where "n = 2 * m" ..
```
```  5235           then have "2 * m = n" ..
```
```  5236           from bounds[of m, unfolded this atLeastAtMost_iff]
```
```  5237           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))"
```
```  5238             by auto
```
```  5239           also have "\<dots> = ?c x n" unfolding One_nat_def by auto
```
```  5240           also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
```
```  5241           finally show ?thesis .
```
```  5242         next
```
```  5243           case False
```
```  5244           hence sgn_neg: "(-1)^n = (-1::real)" by auto
```
```  5245           from \<open>odd n\<close> obtain m where "n = 2 * m + 1" ..
```
```  5246           then have m_def: "2 * m + 1 = n" ..
```
```  5247           hence m_plus: "2 * (m + 1) = n + 1" by auto
```
```  5248           from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
```
```  5249           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))"
```
```  5250             by auto
```
```  5251           also have "\<dots> = - ?c x n" unfolding One_nat_def by auto
```
```  5252           also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
```
```  5253           finally show ?thesis .
```
```  5254         qed
```
```  5255         hence "0 \<le> ?a x n - ?diff x n" by auto
```
```  5256       }
```
```  5257       hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
```
```  5258       moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
```
```  5259         unfolding diff_conv_add_uminus divide_inverse
```
```  5260         by (auto intro!: isCont_add isCont_rabs continuous_ident isCont_minus isCont_arctan
```
```  5261           isCont_inverse isCont_mult isCont_power continuous_const isCont_setsum
```
```  5262           simp del: add_uminus_conv_diff)
```
```  5263       ultimately have "0 \<le> ?a 1 n - ?diff 1 n"
```
```  5264         by (rule LIM_less_bound)
```
```  5265       hence "?diff 1 n \<le> ?a 1 n" by auto
```
```  5266     }
```
```  5267     have "?a 1 \<longlonglongrightarrow> 0"
```
```  5268       unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
```
```  5269       by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat simp del: of_nat_Suc)
```
```  5270     have "?diff 1 \<longlonglongrightarrow> 0"
```
```  5271     proof (rule LIMSEQ_I)
```
```  5272       fix r :: real
```
```  5273       assume "0 < r"
```
```  5274       obtain N :: nat where N_I: "\<And>n. N \<le> n \<Longrightarrow> ?a 1 n < r"
```
```  5275         using LIMSEQ_D[OF \<open>?a 1 \<longlonglongrightarrow> 0\<close> \<open>0 < r\<close>] by auto
```
```  5276       {
```
```  5277         fix n
```
```  5278         assume "N \<le> n" from \<open>?diff 1 n \<le> ?a 1 n\<close> N_I[OF this]
```
```  5279         have "norm (?diff 1 n - 0) < r" by auto
```
```  5280       }
```
```  5281       thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
```
```  5282     qed
```
```  5283     from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
```
```  5284     have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
```
```  5285     hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
```
```  5286
```
```  5287     show ?thesis
```
```  5288     proof (cases "x = 1")
```
```  5289       case True
```
```  5290       then show ?thesis by (simp add: \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close>)
```
```  5291     next
```
```  5292       case False
```
```  5293       hence "x = -1" using \<open>\<bar>x\<bar> = 1\<close> by auto
```
```  5294
```
```  5295       have "- (pi / 2) < 0" using pi_gt_zero by auto
```
```  5296       have "- (2 * pi) < 0" using pi_gt_zero by auto
```
```  5297
```
```  5298       have c_minus_minus: "\<And>i. ?c (- 1) i = - ?c 1 i"
```
```  5299         unfolding One_nat_def by auto
```
```  5300
```
```  5301       have "arctan (- 1) = arctan (tan (-(pi / 4)))"
```
```  5302         unfolding tan_45 tan_minus ..
```
```  5303       also have "\<dots> = - (pi / 4)"
```
```  5304         by (rule arctan_tan, auto simp add: order_less_trans[OF \<open>- (pi / 2) < 0\<close> pi_gt_zero])
```
```  5305       also have "\<dots> = - (arctan (tan (pi / 4)))"
```
```  5306         unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF \<open>- (2 * pi) < 0\<close> pi_gt_zero])
```
```  5307       also have "\<dots> = - (arctan 1)"
```
```  5308         unfolding tan_45 ..
```
```  5309       also have "\<dots> = - (\<Sum> i. ?c 1 i)"
```
```  5310         using \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close> by auto
```
```  5311       also have "\<dots> = (\<Sum> i. ?c (- 1) i)"
```
```  5312         using suminf_minus[OF sums_summable[OF \<open>(?c 1) sums (arctan 1)\<close>]]
```
```  5313         unfolding c_minus_minus by auto
```
```  5314       finally show ?thesis using \<open>x = -1\<close> by auto
```
```  5315     qed
```
```  5316   qed
```
```  5317 qed
```
```  5318
```
```  5319 lemma arctan_half:
```
```  5320   fixes x :: real
```
```  5321   shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))"
```
```  5322 proof -
```
```  5323   obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x"
```
```  5324     using tan_total by blast
```
```  5325   hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2"
```
```  5326     by auto
```
```  5327
```
```  5328   have "0 < cos y" using cos_gt_zero_pi[OF low high] .
```
```  5329   hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y"
```
```  5330     by auto
```
```  5331
```
```  5332   have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  5333     unfolding tan_def power_divide ..
```
```  5334   also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  5335     using \<open>cos y \<noteq> 0\<close> by auto
```
```  5336   also have "\<dots> = 1 / (cos y)\<^sup>2"
```
```  5337     unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
```
```  5338   finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" .
```
```  5339
```
```  5340   have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)"
```
```  5341     unfolding tan_def using \<open>cos y \<noteq> 0\<close> by (simp add: field_simps)
```
```  5342   also have "\<dots> = tan y / (1 + 1 / cos y)"
```
```  5343     using \<open>cos y \<noteq> 0\<close> unfolding add_divide_distrib by auto
```
```  5344   also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))"
```
```  5345     unfolding cos_sqrt ..
```
```  5346   also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))"
```
```  5347     unfolding real_sqrt_divide by auto
```
```  5348   finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))"
```
```  5349     unfolding \<open>1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2\<close> .
```
```  5350
```
```  5351   have "arctan x = y"
```
```  5352     using arctan_tan low high y_eq by auto
```
```  5353   also have "\<dots> = 2 * (arctan (tan (y/2)))"
```
```  5354     using arctan_tan[OF low2 high2] by auto
```
```  5355   also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))"
```
```  5356     unfolding tan_half by auto
```
```  5357   finally show ?thesis
```
```  5358     unfolding eq \<open>tan y = x\<close> .
```
```  5359 qed
```
```  5360
```
```  5361 lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y"
```
```  5362   by (simp only: arctan_less_iff)
```
```  5363
```
```  5364 lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y"
```
```  5365   by (simp only: arctan_le_iff)
```
```  5366
```
```  5367 lemma arctan_inverse:
```
```  5368   assumes "x \<noteq> 0"
```
```  5369   shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
```
```  5370 proof (rule arctan_unique)
```
```  5371   show "- (pi / 2) < sgn x * pi / 2 - arctan x"
```
```  5372     using arctan_bounded [of x] assms
```
```  5373     unfolding sgn_real_def
```
```  5374     apply (auto simp add: arctan algebra_simps)
```
```  5375     apply (drule zero_less_arctan_iff [THEN iffD2])
```
```  5376     apply arith
```
```  5377     done
```
```  5378   show "sgn x * pi / 2 - arctan x < pi / 2"
```
```  5379     using arctan_bounded [of "- x"] assms
```
```  5380     unfolding sgn_real_def arctan_minus
```
```  5381     by (auto simp add: algebra_simps)
```
```  5382   show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
```
```  5383     unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
```
```  5384     unfolding sgn_real_def
```
```  5385     by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
```
```  5386 qed
```
```  5387
```
```  5388 theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
```
```  5389 proof -
```
```  5390   have "pi / 4 = arctan 1" using arctan_one by auto
```
```  5391   also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
```
```  5392   finally show ?thesis by auto
```
```  5393 qed
```
```  5394
```
```  5395
```
```  5396 subsection \<open>Existence of Polar Coordinates\<close>
```
```  5397
```
```  5398 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1"
```
```  5399   apply (rule power2_le_imp_le [OF _ zero_le_one])
```
```  5400   apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
```
```  5401   done
```
```  5402
```
```  5403 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
```
```  5404
```
```  5405 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
```
```  5406
```
```  5407 lemma polar_Ex: "\<exists>r::real. \<exists>a. x = r * cos a & y = r * sin a"
```
```  5408 proof -
```
```  5409   have polar_ex1: "\<And>y. 0 < y \<Longrightarrow> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  5410     apply (rule_tac x = "sqrt (x\<^sup>2 + y\<^sup>2)" in exI)
```
```  5411     apply (rule_tac x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))" in exI)
```
```  5412     apply (simp add: cos_arccos_lemma1 sin_arccos_lemma1 power_divide
```
```  5413                      real_sqrt_mult [symmetric] right_diff_distrib)
```
```  5414     done
```
```  5415   show ?thesis
```
```  5416   proof (cases "0::real" y rule: linorder_cases)
```
```  5417     case less
```
```  5418       then show ?thesis by (rule polar_ex1)
```
```  5419   next
```
```  5420     case equal
```
```  5421       then show ?thesis
```
```  5422         by (force simp add: intro!: cos_zero sin_zero)
```
```  5423   next
```
```  5424     case greater
```
```  5425       then show ?thesis
```
```  5426      using polar_ex1 [where y="-y"]
```
```  5427     by auto (metis cos_minus minus_minus minus_mult_right sin_minus)
```
```  5428   qed
```
```  5429 qed
```
```  5430
```
```  5431
```
```  5432 subsection\<open>Basics about polynomial functions: products, extremal behaviour and root counts\<close>
```
```  5433
```
```  5434 lemma pairs_le_eq_Sigma:
```
```  5435   fixes m::nat
```
```  5436   shows "{(i,j). i+j \<le> m} = Sigma (atMost m) (\<lambda>r. atMost (m-r))"
```
```  5437 by auto
```
```  5438
```
```  5439 lemma setsum_up_index_split:
```
```  5440     "(\<Sum>k\<le>m + n. f k) = (\<Sum>k\<le>m. f k) + (\<Sum>k = Suc m..m + n. f k)"
```
```  5441   by (metis atLeast0AtMost Suc_eq_plus1 le0 setsum_ub_add_nat)
```
```  5442
```
```  5443 lemma Sigma_interval_disjoint:
```
```  5444   fixes w :: "'a::order"
```
```  5445   shows "(SIGMA i:A. {..v i}) \<inter> (SIGMA i:A.{v i<..w}) = {}"
```
```  5446     by auto
```
```  5447
```
```  5448 lemma product_atMost_eq_Un:
```
```  5449   fixes m :: nat
```
```  5450   shows "A \<times> {..m} = (SIGMA i:A.{..m - i}) \<union> (SIGMA i:A.{m - i<..m})"
```
```  5451     by auto
```
```  5452
```
```  5453 lemma polynomial_product: (*with thanks to Chaitanya Mangla*)
```
```  5454   fixes x:: "'a :: idom"
```
```  5455   assumes m: "\<And>i. i>m \<Longrightarrow> (a i) = 0" and n: "\<And>j. j>n \<Longrightarrow> (b j) = 0"
```
```  5456   shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
```
```  5457          (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
```
```  5458 proof -
```
```  5459   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))"
```
```  5460     by (rule setsum_product)
```
```  5461   also have "... = (\<Sum>i\<le>m + n. \<Sum>j\<le>n + m. a i * x ^ i * (b j * x ^ j))"
```
```  5462     using assms by (auto simp: setsum_up_index_split)
```
```  5463   also have "... = (\<Sum>r\<le>m + n. \<Sum>j\<le>m + n - r. a r * x ^ r * (b j * x ^ j))"
```
```  5464     apply (simp add: add_ac setsum.Sigma product_atMost_eq_Un)
```
```  5465     apply (clarsimp simp add: setsum_Un Sigma_interval_disjoint intro!: setsum.neutral)
```
```  5466     by (metis add_diff_assoc2 add.commute add_lessD1 leD m n nat_le_linear neqE)
```
```  5467   also have "... = (\<Sum>(i,j)\<in>{(i,j). i+j \<le> m+n}. (a i * x ^ i) * (b j * x ^ j))"
```
```  5468     by (auto simp: pairs_le_eq_Sigma setsum.Sigma)
```
```  5469   also have "... = (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
```
```  5470     apply (subst setsum_triangle_reindex_eq)
```
```  5471     apply (auto simp: algebra_simps setsum_right_distrib intro!: setsum.cong)
```
```  5472     by (metis le_add_diff_inverse power_add)
```
```  5473   finally show ?thesis .
```
```  5474 qed
```
```  5475
```
```  5476 lemma polynomial_product_nat:
```
```  5477   fixes x:: nat
```
```  5478   assumes m: "\<And>i. i>m \<Longrightarrow> (a i) = 0" and n: "\<And>j. j>n \<Longrightarrow> (b j) = 0"
```
```  5479   shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
```
```  5480          (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
```
```  5481   using polynomial_product [of m a n b x] assms
```
```  5482   by (simp only: of_nat_mult [symmetric] of_nat_power [symmetric] of_nat_eq_iff Int.int_setsum [symmetric])
```
```  5483
```
```  5484 lemma polyfun_diff: (*COMPLEX_SUB_POLYFUN in HOL Light*)
```
```  5485     fixes x :: "'a::idom"
```
```  5486   assumes "1 \<le> n"
```
```  5487     shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
```
```  5488            (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)"
```
```  5489 proof -
```
```  5490   have h: "bij_betw (\<lambda>(i,j). (j,i)) ((SIGMA i : atMost n. lessThan i)) (SIGMA j : lessThan n. {Suc j..n})"
```
```  5491     by (auto simp: bij_betw_def inj_on_def)
```
```  5492   have "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
```
```  5493         (\<Sum>i\<le>n. a i * (x^i - y^i))"
```
```  5494     by (simp add: right_diff_distrib setsum_subtractf)
```
```  5495   also have "... = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))"
```
```  5496     by (simp add: power_diff_sumr2 mult.assoc)
```
```  5497   also have "... = (\<Sum>i\<le>n. \<Sum>j<i. a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5498     by (simp add: setsum_right_distrib)
```
```  5499   also have "... = (\<Sum>(i,j) \<in> (SIGMA i : atMost n. lessThan i). a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5500     by (simp add: setsum.Sigma)
```
```  5501   also have "... = (\<Sum>(j,i) \<in> (SIGMA j : lessThan n. {Suc j..n}). a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5502     by (auto simp add: setsum.reindex_bij_betw [OF h, symmetric] intro: setsum.strong_cong)
```
```  5503   also have "... = (\<Sum>j<n. \<Sum>i=Suc j..n. a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5504     by (simp add: setsum.Sigma)
```
```  5505   also have "... = (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)"
```
```  5506     by (simp add: setsum_right_distrib mult_ac)
```
```  5507   finally show ?thesis .
```
```  5508 qed
```
```  5509
```
```  5510 lemma polyfun_diff_alt: (*COMPLEX_SUB_POLYFUN_ALT in HOL Light*)
```
```  5511     fixes x :: "'a::idom"
```
```  5512   assumes "1 \<le> n"
```
```  5513     shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
```
```  5514            (x - y) * ((\<Sum>j<n. \<Sum>k<n-j. a(j+k+1) * y^k * x^j))"
```
```  5515 proof -
```
```  5516   { fix j::nat
```
```  5517     assume "j<n"
```
```  5518     have h: "bij_betw (\<lambda>i. i - (j + 1)) {Suc j..n} (lessThan (n-j))"
```
```  5519       apply (auto simp: bij_betw_def inj_on_def)
```
```  5520       apply (rule_tac x="x + Suc j" in image_eqI)
```
```  5521       apply (auto simp: )
```
```  5522       done
```
```  5523     have "(\<Sum>i=Suc j..n. a i * y^(i - j - 1)) = (\<Sum>k<n-j. a(j+k+1) * y^k)"
```
```  5524       by (auto simp add: setsum.reindex_bij_betw [OF h, symmetric] intro: setsum.strong_cong)
```
```  5525   }
```
```  5526   then show ?thesis
```
```  5527     by (simp add: polyfun_diff [OF assms] setsum_left_distrib)
```
```  5528 qed
```
```  5529
```
```  5530 lemma polyfun_linear_factor:  (*COMPLEX_POLYFUN_LINEAR_FACTOR in HOL Light*)
```
```  5531   fixes a :: "'a::idom"
```
```  5532   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)"
```
```  5533 proof (cases "n=0")
```
```  5534   case True then show ?thesis
```
```  5535     by simp
```
```  5536 next
```
```  5537   case False
```
```  5538   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)) =
```
```  5539         (\<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))"
```
```  5540     by (simp add: algebra_simps)
```
```  5541   also have "... = (\<exists>b. \<forall>z. (z - a) * (\<Sum>j<n. (\<Sum>i = Suc j..n. c i * a^(i - Suc j)) * z^j) = (z - a) * (\<Sum>i<n. b(i) * z^i))"
```
```  5542     using False by (simp add: polyfun_diff)
```
```  5543   also have "... = True"
```
```  5544     by auto
```
```  5545   finally show ?thesis
```
```  5546     by simp
```
```  5547 qed
```
```  5548
```
```  5549 lemma polyfun_linear_factor_root:  (*COMPLEX_POLYFUN_LINEAR_FACTOR_ROOT in HOL Light*)
```
```  5550   fixes a :: "'a::idom"
```
```  5551   assumes "(\<Sum>i\<le>n. c(i) * a^i) = 0"
```
```  5552   obtains b where "\<And>z. (\<Sum>i\<le>n. c(i) * z^i) = (z - a) * (\<Sum>i<n. b(i) * z^i)"
```
```  5553   using polyfun_linear_factor [of c n a] assms
```
```  5554   by auto
```
```  5555
```
```  5556 (*The material of this section, up until this point, could go into a new theory of polynomials
```
```  5557   based on Main alone. The remaining material involves limits, continuity, series, etc.*)
```
```  5558
```
```  5559 lemma isCont_polynom:
```
```  5560   fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
```
```  5561   shows "isCont (\<lambda>w. \<Sum>i\<le>n. c i * w^i) a"
```
```  5562   by simp
```
```  5563
```
```  5564 lemma zero_polynom_imp_zero_coeffs:
```
```  5565     fixes c :: "nat \<Rightarrow> 'a::{ab_semigroup_mult,real_normed_div_algebra}"
```
```  5566   assumes "\<And>w. (\<Sum>i\<le>n. c i * w^i) = 0"  "k \<le> n"
```
```  5567     shows "c k = 0"
```
```  5568 using assms
```
```  5569 proof (induction n arbitrary: c k)
```
```  5570   case 0
```
```  5571   then show ?case
```
```  5572     by simp
```
```  5573 next
```
```  5574   case (Suc n c k)
```
```  5575   have [simp]: "c 0 = 0" using Suc.prems(1) [of 0]
```
```  5576     by simp
```
```  5577   { fix w
```
```  5578     have "(\<Sum>i\<le>Suc n. c i * w^i) = (\<Sum>i\<le>n. c (Suc i) * w ^ Suc i)"
```
```  5579       unfolding Set_Interval.setsum_atMost_Suc_shift
```
```  5580       by simp
```
```  5581     also have "... = w * (\<Sum>i\<le>n. c (Suc i) * w^i)"
```
```  5582       by (simp add: setsum_right_distrib ac_simps)
```
```  5583     finally have "(\<Sum>i\<le>Suc n. c i * w^i) = w * (\<Sum>i\<le>n. c (Suc i) * w^i)" .
```
```  5584   }
```
```  5585   then have wnz: "\<And>w. w \<noteq> 0 \<Longrightarrow> (\<Sum>i\<le>n. c (Suc i) * w^i) = 0"
```
```  5586     using Suc  by auto
```
```  5587   then have "(\<lambda>h. \<Sum>i\<le>n. c (Suc i) * h^i) \<midarrow>0\<rightarrow> 0"
```
```  5588     by (simp cong: LIM_cong)                   \<comment>\<open>the case @{term"w=0"} by continuity\<close>
```
```  5589   then have "(\<Sum>i\<le>n. c (Suc i) * 0^i) = 0"
```
```  5590     using isCont_polynom [of 0 "\<lambda>i. c (Suc i)" n] LIM_unique
```
```  5591     by (force simp add: Limits.isCont_iff)
```
```  5592   then have "\<And>w. (\<Sum>i\<le>n. c (Suc i) * w^i) = 0" using wnz
```
```  5593     by metis
```
```  5594   then have "\<And>i. i\<le>n \<Longrightarrow> c (Suc i) = 0"
```
```  5595     using Suc.IH [of "\<lambda>i. c (Suc i)"]
```
```  5596     by blast
```
```  5597   then show ?case using \<open>k \<le> Suc n\<close>
```
```  5598     by (cases k) auto
```
```  5599 qed
```
```  5600
```
```  5601 lemma polyfun_rootbound: (*COMPLEX_POLYFUN_ROOTBOUND in HOL Light*)
```
```  5602     fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5603   assumes "c k \<noteq> 0" "k\<le>n"
```
```  5604     shows "finite {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<and>
```
```  5605              card {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<le> n"
```
```  5606 using assms
```
```  5607 proof (induction n arbitrary: c k)
```
```  5608   case 0
```
```  5609   then show ?case
```
```  5610     by simp
```
```  5611 next
```
```  5612   case (Suc m c k)
```
```  5613   let ?succase = ?case
```
```  5614   show ?case
```
```  5615   proof (cases "{z. (\<Sum>i\<le>Suc m. c(i) * z^i) = 0} = {}")
```
```  5616     case True
```
```  5617     then show ?succase
```
```  5618       by simp
```
```  5619   next
```
```  5620     case False
```
```  5621     then obtain z0 where z0: "(\<Sum>i\<le>Suc m. c(i) * z0^i) = 0"
```
```  5622       by blast
```
```  5623     then obtain b where b: "\<And>w. (\<Sum>i\<le>Suc m. c i * w^i) = (w - z0) * (\<Sum>i\<le>m. b i * w^i)"
```
```  5624       using polyfun_linear_factor_root [OF z0, unfolded lessThan_Suc_atMost]
```
```  5625       by blast
```
```  5626     then have eq: "{z. (\<Sum>i\<le>Suc m. c(i) * z^i) = 0} = insert z0 {z. (\<Sum>i\<le>m. b(i) * z^i) = 0}"
```
```  5627       by auto
```
```  5628     have "~(\<forall>k\<le>m. b k = 0)"
```
```  5629     proof
```
```  5630       assume [simp]: "\<forall>k\<le>m. b k = 0"
```
```  5631       then have "\<And>w. (\<Sum>i\<le>m. b i * w^i) = 0"
```
```  5632         by simp
```
```  5633       then have "\<And>w. (\<Sum>i\<le>Suc m. c i * w^i) = 0"
```
```  5634         using b by simp
```
```  5635       then have "\<And>k. k \<le> Suc m \<Longrightarrow> c k = 0"
```
```  5636         using zero_polynom_imp_zero_coeffs
```
```  5637         by blast
```
```  5638       then show False using Suc.prems
```
```  5639         by blast
```
```  5640     qed
```
```  5641     then obtain k' where bk': "b k' \<noteq> 0" "k' \<le> m"
```
```  5642       by blast
```
```  5643     show ?succase
```
```  5644       using Suc.IH [of b k'] bk'
```
```  5645       by (simp add: eq card_insert_if del: setsum_atMost_Suc)
```
```  5646     qed
```
```  5647 qed
```
```  5648
```
```  5649 lemma
```
```  5650     fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5651   assumes "c k \<noteq> 0" "k\<le>n"
```
```  5652     shows polyfun_roots_finite: "finite {z. (\<Sum>i\<le>n. c(i) * z^i) = 0}"
```
```  5653       and polyfun_roots_card:   "card {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<le> n"
```
```  5654 using polyfun_rootbound assms
```
```  5655   by auto
```
```  5656
```
```  5657 lemma polyfun_finite_roots: (*COMPLEX_POLYFUN_FINITE_ROOTS in HOL Light*)
```
```  5658   fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5659   shows "finite {x. (\<Sum>i\<le>n. c i * x^i) = 0} \<longleftrightarrow> (\<exists>i\<le>n. c i \<noteq> 0)"
```
```  5660         (is "?lhs = ?rhs")
```
```  5661 proof
```
```  5662   assume ?lhs
```
```  5663   moreover
```
```  5664   { assume "\<forall>i\<le>n. c i = 0"
```
```  5665     then have "\<And>x. (\<Sum>i\<le>n. c i * x^i) = 0"
```
```  5666       by simp
```
```  5667     then have "\<not> finite {x. (\<Sum>i\<le>n. c i * x^i) = 0}"
```
```  5668       using ex_new_if_finite [OF infinite_UNIV_char_0 [where 'a='a]]
```
```  5669       by auto
```
```  5670   }
```
```  5671   ultimately show ?rhs
```
```  5672   by metis
```
```  5673 next
```
```  5674   assume ?rhs
```
```  5675   then show ?lhs
```
```  5676     using polyfun_rootbound
```
```  5677     by blast
```
```  5678 qed
```
```  5679
```
```  5680 lemma polyfun_eq_0: (*COMPLEX_POLYFUN_EQ_0 in HOL Light*)
```
```  5681   fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5682   shows "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = 0) \<longleftrightarrow> (\<forall>i\<le>n. c i = 0)"
```
```  5683   using zero_polynom_imp_zero_coeffs by auto
```
```  5684
```
```  5685 lemma polyfun_eq_coeffs:
```
```  5686   fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5687   shows "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = (\<Sum>i\<le>n. d i * x^i)) \<longleftrightarrow> (\<forall>i\<le>n. c i = d i)"
```
```  5688 proof -
```
```  5689   have "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = (\<Sum>i\<le>n. d i * x^i)) \<longleftrightarrow> (\<forall>x. (\<Sum>i\<le>n. (c i - d i) * x^i) = 0)"
```
```  5690     by (simp add: left_diff_distrib Groups_Big.setsum_subtractf)
```
```  5691   also have "... \<longleftrightarrow> (\<forall>i\<le>n. c i - d i = 0)"
```
```  5692     by (rule polyfun_eq_0)
```
```  5693   finally show ?thesis
```
```  5694     by simp
```
```  5695 qed
```
```  5696
```
```  5697 lemma polyfun_eq_const: (*COMPLEX_POLYFUN_EQ_CONST in HOL Light*)
```
```  5698   fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5699   shows "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = k) \<longleftrightarrow> c 0 = k \<and> (\<forall>i \<in> {1..n}. c i = 0)"
```
```  5700         (is "?lhs = ?rhs")
```
```  5701 proof -
```
```  5702   have *: "\<forall>x. (\<Sum>i\<le>n. (if i=0 then k else 0) * x^i) = k"
```
```  5703     by (induct n) auto
```
```  5704   show ?thesis
```
```  5705   proof
```
```  5706     assume ?lhs
```
```  5707     with * have "(\<forall>i\<le>n. c i = (if i=0 then k else 0))"
```
```  5708       by (simp add: polyfun_eq_coeffs [symmetric])
```
```  5709     then show ?rhs
```
```  5710       by simp
```
```  5711   next
```
```  5712     assume ?rhs then show ?lhs
```
```  5713       by (induct n) auto
```
```  5714   qed
```
```  5715 qed
```
```  5716
```
```  5717 lemma root_polyfun:
```
```  5718   fixes z:: "'a::idom"
```
```  5719   assumes "1 \<le> n"
```
```  5720     shows "z^n = a \<longleftrightarrow> (\<Sum>i\<le>n. (if i = 0 then -a else if i=n then 1 else 0) * z^i) = 0"
```
```  5721   using assms
```
```  5722   by (cases n; simp add: setsum_head_Suc atLeast0AtMost [symmetric])
```
```  5723
```
```  5724 lemma
```
```  5725     fixes zz :: "'a::{idom,real_normed_div_algebra}"
```
```  5726   assumes "1 \<le> n"
```
```  5727     shows finite_roots_unity: "finite {z::'a. z^n = 1}"
```
```  5728       and card_roots_unity:   "card {z::'a. z^n = 1} \<le> n"
```
```  5729   using polyfun_rootbound [of "\<lambda>i. if i = 0 then -1 else if i=n then 1 else 0" n n] assms
```
```  5730   by (auto simp add: root_polyfun [OF assms])
```
```  5731
```
```  5732 end
```