src/HOL/Transcendental.thy
 author wenzelm Tue Sep 01 22:32:58 2015 +0200 (2015-09-01) changeset 61076 bdc1e2f0a86a parent 61070 b72a990adfe2 child 61284 2314c2f62eb1 permissions -rw-r--r--
eliminated \<Colon>;
```     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 lemma reals_Archimedean4:
```
```    14   assumes "0 < y" "0 \<le> x"
```
```    15   obtains n where "real n * y \<le> x" "x < real (Suc n) * y"
```
```    16   using floor_correct [of "x/y"] assms
```
```    17   by (auto simp: Real.real_of_nat_Suc field_simps intro: that [of "nat (floor (x/y))"])
```
```    18
```
```    19 lemma of_real_fact [simp]: "of_real (fact n) = fact n"
```
```    20   by (metis of_nat_fact of_real_of_nat_eq)
```
```    21
```
```    22 lemma real_fact_nat [simp]: "real (fact n :: nat) = fact n"
```
```    23   by (simp add: real_of_nat_def)
```
```    24
```
```    25 lemma real_fact_int [simp]: "real (fact n :: int) = fact n"
```
```    26   by (metis of_int_of_nat_eq of_nat_fact real_of_int_def)
```
```    27
```
```    28 lemma root_test_convergence:
```
```    29   fixes f :: "nat \<Rightarrow> 'a::banach"
```
```    30   assumes f: "(\<lambda>n. root n (norm (f n))) ----> x" -- "could be weakened to lim sup"
```
```    31   assumes "x < 1"
```
```    32   shows "summable f"
```
```    33 proof -
```
```    34   have "0 \<le> x"
```
```    35     by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1])
```
```    36   from \<open>x < 1\<close> obtain z where z: "x < z" "z < 1"
```
```    37     by (metis dense)
```
```    38   from f \<open>x < z\<close>
```
```    39   have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
```
```    40     by (rule order_tendstoD)
```
```    41   then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially"
```
```    42     using eventually_ge_at_top
```
```    43   proof eventually_elim
```
```    44     fix n assume less: "root n (norm (f n)) < z" and n: "1 \<le> n"
```
```    45     from power_strict_mono[OF less, of n] n
```
```    46     show "norm (f n) \<le> z ^ n"
```
```    47       by simp
```
```    48   qed
```
```    49   then show "summable f"
```
```    50     unfolding eventually_sequentially
```
```    51     using z \<open>0 \<le> x\<close> by (auto intro!: summable_comparison_test[OF _  summable_geometric])
```
```    52 qed
```
```    53
```
```    54 subsection \<open>Properties of Power Series\<close>
```
```    55
```
```    56 lemma powser_zero:
```
```    57   fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra_1"
```
```    58   shows "(\<Sum>n. f n * 0 ^ n) = f 0"
```
```    59 proof -
```
```    60   have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
```
```    61     by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
```
```    62   thus ?thesis unfolding One_nat_def by simp
```
```    63 qed
```
```    64
```
```    65 lemma powser_sums_zero:
```
```    66   fixes a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
```
```    67   shows "(\<lambda>n. a n * 0^n) sums a 0"
```
```    68     using sums_finite [of "{0}" "\<lambda>n. a n * 0 ^ n"]
```
```    69     by simp
```
```    70
```
```    71 text\<open>Power series has a circle or radius of convergence: if it sums for @{term
```
```    72   x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.\<close>
```
```    73
```
```    74 lemma powser_insidea:
```
```    75   fixes x z :: "'a::real_normed_div_algebra"
```
```    76   assumes 1: "summable (\<lambda>n. f n * x^n)"
```
```    77     and 2: "norm z < norm x"
```
```    78   shows "summable (\<lambda>n. norm (f n * z ^ n))"
```
```    79 proof -
```
```    80   from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
```
```    81   from 1 have "(\<lambda>n. f n * x^n) ----> 0"
```
```    82     by (rule summable_LIMSEQ_zero)
```
```    83   hence "convergent (\<lambda>n. f n * x^n)"
```
```    84     by (rule convergentI)
```
```    85   hence "Cauchy (\<lambda>n. f n * x^n)"
```
```    86     by (rule convergent_Cauchy)
```
```    87   hence "Bseq (\<lambda>n. f n * x^n)"
```
```    88     by (rule Cauchy_Bseq)
```
```    89   then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x^n) \<le> K"
```
```    90     by (simp add: Bseq_def, safe)
```
```    91   have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
```
```    92                    K * norm (z ^ n) * inverse (norm (x^n))"
```
```    93   proof (intro exI allI impI)
```
```    94     fix n::nat
```
```    95     assume "0 \<le> n"
```
```    96     have "norm (norm (f n * z ^ n)) * norm (x^n) =
```
```    97           norm (f n * x^n) * norm (z ^ n)"
```
```    98       by (simp add: norm_mult abs_mult)
```
```    99     also have "\<dots> \<le> K * norm (z ^ n)"
```
```   100       by (simp only: mult_right_mono 4 norm_ge_zero)
```
```   101     also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x^n)) * norm (x^n))"
```
```   102       by (simp add: x_neq_0)
```
```   103     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x^n)) * norm (x^n)"
```
```   104       by (simp only: mult.assoc)
```
```   105     finally show "norm (norm (f n * z ^ n)) \<le>
```
```   106                   K * norm (z ^ n) * inverse (norm (x^n))"
```
```   107       by (simp add: mult_le_cancel_right x_neq_0)
```
```   108   qed
```
```   109   moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
```
```   110   proof -
```
```   111     from 2 have "norm (norm (z * inverse x)) < 1"
```
```   112       using x_neq_0
```
```   113       by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
```
```   114     hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
```
```   115       by (rule summable_geometric)
```
```   116     hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
```
```   117       by (rule summable_mult)
```
```   118     thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
```
```   119       using x_neq_0
```
```   120       by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
```
```   121                     power_inverse norm_power mult.assoc)
```
```   122   qed
```
```   123   ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   124     by (rule summable_comparison_test)
```
```   125 qed
```
```   126
```
```   127 lemma powser_inside:
```
```   128   fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
```
```   129   shows
```
```   130     "summable (\<lambda>n. f n * (x^n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
```
```   131       summable (\<lambda>n. f n * (z ^ n))"
```
```   132   by (rule powser_insidea [THEN summable_norm_cancel])
```
```   133
```
```   134 lemma powser_times_n_limit_0:
```
```   135   fixes x :: "'a::{real_normed_div_algebra,banach}"
```
```   136   assumes "norm x < 1"
```
```   137     shows "(\<lambda>n. of_nat n * x ^ n) ----> 0"
```
```   138 proof -
```
```   139   have "norm x / (1 - norm x) \<ge> 0"
```
```   140     using assms
```
```   141     by (auto simp: divide_simps)
```
```   142   moreover obtain N where N: "norm x / (1 - norm x) < of_int N"
```
```   143     using ex_le_of_int
```
```   144     by (meson ex_less_of_int)
```
```   145   ultimately have N0: "N>0"
```
```   146     by auto
```
```   147   then have *: "real (N + 1) * norm x / real N < 1"
```
```   148     using N assms
```
```   149     by (auto simp: field_simps)
```
```   150   { fix n::nat
```
```   151     assume "N \<le> int n"
```
```   152     then have "real N * real_of_nat (Suc n) \<le> real_of_nat n * real (1 + N)"
```
```   153       by (simp add: algebra_simps)
```
```   154     then have "(real N * real_of_nat (Suc n)) * (norm x * norm (x ^ n))
```
```   155                \<le> (real_of_nat n * real (1 + N)) * (norm x * norm (x ^ n))"
```
```   156       using N0 mult_mono by fastforce
```
```   157     then have "real N * (norm x * (real_of_nat (Suc n) * norm (x ^ n)))
```
```   158          \<le> real_of_nat n * (norm x * (real (1 + N) * norm (x ^ n)))"
```
```   159       by (simp add: algebra_simps)
```
```   160   } note ** = this
```
```   161   show ?thesis using *
```
```   162     apply (rule summable_LIMSEQ_zero [OF summable_ratio_test, where N1="nat N"])
```
```   163     apply (simp add: N0 norm_mult field_simps **
```
```   164                 del: of_nat_Suc real_of_int_add)
```
```   165     done
```
```   166 qed
```
```   167
```
```   168 corollary lim_n_over_pown:
```
```   169   fixes x :: "'a::{real_normed_field,banach}"
```
```   170   shows "1 < norm x \<Longrightarrow> ((\<lambda>n. of_nat n / x^n) ---> 0) sequentially"
```
```   171 using powser_times_n_limit_0 [of "inverse x"]
```
```   172 by (simp add: norm_divide divide_simps)
```
```   173
```
```   174 lemma lim_1_over_n: "((\<lambda>n. 1 / of_nat n) ---> (0::'a::real_normed_field)) sequentially"
```
```   175   apply (clarsimp simp: lim_sequentially norm_divide dist_norm divide_simps)
```
```   176   apply (auto simp: mult_ac dest!: ex_less_of_nat_mult [of _ 1])
```
```   177   by (metis le_eq_less_or_eq less_trans linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
```
```   178           of_nat_less_0_iff of_nat_less_iff zero_less_mult_iff zero_less_one)
```
```   179
```
```   180 lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) ---> (0::'a::real_normed_field)) sequentially"
```
```   181   using lim_1_over_n
```
```   182   by (simp add: inverse_eq_divide)
```
```   183
```
```   184 lemma sum_split_even_odd:
```
```   185   fixes f :: "nat \<Rightarrow> real"
```
```   186   shows
```
```   187     "(\<Sum>i<2 * n. if even i then f i else g i) =
```
```   188      (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
```
```   189 proof (induct n)
```
```   190   case 0
```
```   191   then show ?case by simp
```
```   192 next
```
```   193   case (Suc n)
```
```   194   have "(\<Sum>i<2 * Suc n. if even i then f i else g i) =
```
```   195     (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
```
```   196     using Suc.hyps unfolding One_nat_def by auto
```
```   197   also have "\<dots> = (\<Sum>i<Suc n. f (2 * i)) + (\<Sum>i<Suc n. g (2 * i + 1))"
```
```   198     by auto
```
```   199   finally show ?case .
```
```   200 qed
```
```   201
```
```   202 lemma sums_if':
```
```   203   fixes g :: "nat \<Rightarrow> real"
```
```   204   assumes "g sums x"
```
```   205   shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   206   unfolding sums_def
```
```   207 proof (rule LIMSEQ_I)
```
```   208   fix r :: real
```
```   209   assume "0 < r"
```
```   210   from \<open>g sums x\<close>[unfolded sums_def, THEN LIMSEQ_D, OF this]
```
```   211   obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g {..<n} - x) < r)" by blast
```
```   212
```
```   213   let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)"
```
```   214   {
```
```   215     fix m
```
```   216     assume "m \<ge> 2 * no"
```
```   217     hence "m div 2 \<ge> no" by auto
```
```   218     have sum_eq: "?SUM (2 * (m div 2)) = setsum g {..< m div 2}"
```
```   219       using sum_split_even_odd by auto
```
```   220     hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
```
```   221       using no_eq unfolding sum_eq using \<open>m div 2 \<ge> no\<close> by auto
```
```   222     moreover
```
```   223     have "?SUM (2 * (m div 2)) = ?SUM m"
```
```   224     proof (cases "even m")
```
```   225       case True
```
```   226       then show ?thesis by (auto simp add: even_two_times_div_two)
```
```   227     next
```
```   228       case False
```
```   229       then have eq: "Suc (2 * (m div 2)) = m" by simp
```
```   230       hence "even (2 * (m div 2))" using \<open>odd m\<close> by auto
```
```   231       have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
```
```   232       also have "\<dots> = ?SUM (2 * (m div 2))" using \<open>even (2 * (m div 2))\<close> by auto
```
```   233       finally show ?thesis by auto
```
```   234     qed
```
```   235     ultimately have "(norm (?SUM m - x) < r)" by auto
```
```   236   }
```
```   237   thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
```
```   238 qed
```
```   239
```
```   240 lemma sums_if:
```
```   241   fixes g :: "nat \<Rightarrow> real"
```
```   242   assumes "g sums x" and "f sums y"
```
```   243   shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
```
```   244 proof -
```
```   245   let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
```
```   246   {
```
```   247     fix B T E
```
```   248     have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
```
```   249       by (cases B) auto
```
```   250   } note if_sum = this
```
```   251   have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   252     using sums_if'[OF \<open>g sums x\<close>] .
```
```   253   {
```
```   254     have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
```
```   255
```
```   256     have "?s sums y" using sums_if'[OF \<open>f sums y\<close>] .
```
```   257     from this[unfolded sums_def, THEN LIMSEQ_Suc]
```
```   258     have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
```
```   259       by (simp add: lessThan_Suc_eq_insert_0 image_iff setsum.reindex if_eq sums_def cong del: if_cong)
```
```   260   }
```
```   261   from sums_add[OF g_sums this] show ?thesis unfolding if_sum .
```
```   262 qed
```
```   263
```
```   264 subsection \<open>Alternating series test / Leibniz formula\<close>
```
```   265
```
```   266 lemma sums_alternating_upper_lower:
```
```   267   fixes a :: "nat \<Rightarrow> real"
```
```   268   assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
```
```   269   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) ----> l) \<and>
```
```   270              ((\<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) ----> l)"
```
```   271   (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
```
```   272 proof (rule nested_sequence_unique)
```
```   273   have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
```
```   274
```
```   275   show "\<forall>n. ?f n \<le> ?f (Suc n)"
```
```   276   proof
```
```   277     fix n
```
```   278     show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
```
```   279   qed
```
```   280   show "\<forall>n. ?g (Suc n) \<le> ?g n"
```
```   281   proof
```
```   282     fix n
```
```   283     show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
```
```   284       unfolding One_nat_def by auto
```
```   285   qed
```
```   286   show "\<forall>n. ?f n \<le> ?g n"
```
```   287   proof
```
```   288     fix n
```
```   289     show "?f n \<le> ?g n" using fg_diff a_pos
```
```   290       unfolding One_nat_def by auto
```
```   291   qed
```
```   292   show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff
```
```   293   proof (rule LIMSEQ_I)
```
```   294     fix r :: real
```
```   295     assume "0 < r"
```
```   296     with \<open>a ----> 0\<close>[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
```
```   297       by auto
```
```   298     hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   299     thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   300   qed
```
```   301 qed
```
```   302
```
```   303 lemma summable_Leibniz':
```
```   304   fixes a :: "nat \<Rightarrow> real"
```
```   305   assumes a_zero: "a ----> 0"
```
```   306     and a_pos: "\<And> n. 0 \<le> a n"
```
```   307     and a_monotone: "\<And> n. a (Suc n) \<le> a n"
```
```   308   shows summable: "summable (\<lambda> n. (-1)^n * a n)"
```
```   309     and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
```
```   310     and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   311     and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i<2*n+1. (-1)^i*a i)"
```
```   312     and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   313 proof -
```
```   314   let ?S = "\<lambda>n. (-1)^n * a n"
```
```   315   let ?P = "\<lambda>n. \<Sum>i<n. ?S i"
```
```   316   let ?f = "\<lambda>n. ?P (2 * n)"
```
```   317   let ?g = "\<lambda>n. ?P (2 * n + 1)"
```
```   318   obtain l :: real
```
```   319     where below_l: "\<forall> n. ?f n \<le> l"
```
```   320       and "?f ----> l"
```
```   321       and above_l: "\<forall> n. l \<le> ?g n"
```
```   322       and "?g ----> l"
```
```   323     using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
```
```   324
```
```   325   let ?Sa = "\<lambda>m. \<Sum>n<m. ?S n"
```
```   326   have "?Sa ----> l"
```
```   327   proof (rule LIMSEQ_I)
```
```   328     fix r :: real
```
```   329     assume "0 < r"
```
```   330     with \<open>?f ----> l\<close>[THEN LIMSEQ_D]
```
```   331     obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
```
```   332
```
```   333     from \<open>0 < r\<close> \<open>?g ----> l\<close>[THEN LIMSEQ_D]
```
```   334     obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
```
```   335
```
```   336     {
```
```   337       fix n :: nat
```
```   338       assume "n \<ge> (max (2 * f_no) (2 * g_no))"
```
```   339       hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
```
```   340       have "norm (?Sa n - l) < r"
```
```   341       proof (cases "even n")
```
```   342         case True
```
```   343         then have n_eq: "2 * (n div 2) = n" by (simp add: even_two_times_div_two)
```
```   344         with \<open>n \<ge> 2 * f_no\<close> have "n div 2 \<ge> f_no"
```
```   345           by auto
```
```   346         from f[OF this] show ?thesis
```
```   347           unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
```
```   348       next
```
```   349         case False
```
```   350         hence "even (n - 1)" by simp
```
```   351         then have n_eq: "2 * ((n - 1) div 2) = n - 1"
```
```   352           by (simp add: even_two_times_div_two)
```
```   353         hence range_eq: "n - 1 + 1 = n"
```
```   354           using odd_pos[OF False] by auto
```
```   355
```
```   356         from n_eq \<open>n \<ge> 2 * g_no\<close> have "(n - 1) div 2 \<ge> g_no"
```
```   357           by auto
```
```   358         from g[OF this] show ?thesis
```
```   359           unfolding n_eq range_eq .
```
```   360       qed
```
```   361     }
```
```   362     thus "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
```
```   363   qed
```
```   364   hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
```
```   365     unfolding sums_def .
```
```   366   thus "summable ?S" using summable_def by auto
```
```   367
```
```   368   have "l = suminf ?S" using sums_unique[OF sums_l] .
```
```   369
```
```   370   fix n
```
```   371   show "suminf ?S \<le> ?g n"
```
```   372     unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
```
```   373   show "?f n \<le> suminf ?S"
```
```   374     unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
```
```   375   show "?g ----> suminf ?S"
```
```   376     using \<open>?g ----> l\<close> \<open>l = suminf ?S\<close> by auto
```
```   377   show "?f ----> suminf ?S"
```
```   378     using \<open>?f ----> l\<close> \<open>l = suminf ?S\<close> by auto
```
```   379 qed
```
```   380
```
```   381 theorem summable_Leibniz:
```
```   382   fixes a :: "nat \<Rightarrow> real"
```
```   383   assumes a_zero: "a ----> 0" and "monoseq a"
```
```   384   shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
```
```   385     and "0 < a 0 \<longrightarrow>
```
```   386       (\<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")
```
```   387     and "a 0 < 0 \<longrightarrow>
```
```   388       (\<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")
```
```   389     and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) ----> (\<Sum>i. (- 1)^i*a i)" (is "?f")
```
```   390     and "(\<lambda>n. \<Sum>i<2*n+1. (- 1)^i*a i) ----> (\<Sum>i. (- 1)^i*a i)" (is "?g")
```
```   391 proof -
```
```   392   have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
```
```   393   proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
```
```   394     case True
```
```   395     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n"
```
```   396       by auto
```
```   397     {
```
```   398       fix n
```
```   399       have "a (Suc n) \<le> a n"
```
```   400         using ord[where n="Suc n" and m=n] by auto
```
```   401     } note mono = this
```
```   402     note leibniz = summable_Leibniz'[OF \<open>a ----> 0\<close> ge0]
```
```   403     from leibniz[OF mono]
```
```   404     show ?thesis using \<open>0 \<le> a 0\<close> by auto
```
```   405   next
```
```   406     let ?a = "\<lambda> n. - a n"
```
```   407     case False
```
```   408     with monoseq_le[OF \<open>monoseq a\<close> \<open>a ----> 0\<close>]
```
```   409     have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
```
```   410     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
```
```   411       by auto
```
```   412     {
```
```   413       fix n
```
```   414       have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
```
```   415         by auto
```
```   416     } note monotone = this
```
```   417     note leibniz =
```
```   418       summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
```
```   419         OF tendsto_minus[OF \<open>a ----> 0\<close>, unfolded minus_zero] monotone]
```
```   420     have "summable (\<lambda> n. (-1)^n * ?a n)"
```
```   421       using leibniz(1) by auto
```
```   422     then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
```
```   423       unfolding summable_def by auto
```
```   424     from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
```
```   425       by auto
```
```   426     hence ?summable unfolding summable_def by auto
```
```   427     moreover
```
```   428     have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
```
```   429       unfolding minus_diff_minus by auto
```
```   430
```
```   431     from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
```
```   432     have move_minus: "(\<Sum>n. - ((- 1) ^ n * a n)) = - (\<Sum>n. (- 1) ^ n * a n)"
```
```   433       by auto
```
```   434
```
```   435     have ?pos using \<open>0 \<le> ?a 0\<close> by auto
```
```   436     moreover have ?neg
```
```   437       using leibniz(2,4)
```
```   438       unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
```
```   439       by auto
```
```   440     moreover have ?f and ?g
```
```   441       using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
```
```   442       by auto
```
```   443     ultimately show ?thesis by auto
```
```   444   qed
```
```   445   then show ?summable and ?pos and ?neg and ?f and ?g
```
```   446     by safe
```
```   447 qed
```
```   448
```
```   449 subsection \<open>Term-by-Term Differentiability of Power Series\<close>
```
```   450
```
```   451 definition diffs :: "(nat \<Rightarrow> 'a::ring_1) \<Rightarrow> nat \<Rightarrow> 'a"
```
```   452   where "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))"
```
```   453
```
```   454 text\<open>Lemma about distributing negation over it\<close>
```
```   455 lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
```
```   456   by (simp add: diffs_def)
```
```   457
```
```   458 lemma sums_Suc_imp:
```
```   459   "(f::nat \<Rightarrow> 'a::real_normed_vector) 0 = 0 \<Longrightarrow> (\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
```
```   460   using sums_Suc_iff[of f] by simp
```
```   461
```
```   462 lemma diffs_equiv:
```
```   463   fixes x :: "'a::{real_normed_vector, ring_1}"
```
```   464   shows "summable (\<lambda>n. diffs c n * x^n) \<Longrightarrow>
```
```   465       (\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)"
```
```   466   unfolding diffs_def
```
```   467   by (simp add: summable_sums sums_Suc_imp)
```
```   468
```
```   469 lemma lemma_termdiff1:
```
```   470   fixes z :: "'a :: {monoid_mult,comm_ring}" shows
```
```   471   "(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
```
```   472    (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
```
```   473   by (auto simp add: algebra_simps power_add [symmetric])
```
```   474
```
```   475 lemma sumr_diff_mult_const2:
```
```   476   "setsum f {..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i<n. f i - r)"
```
```   477   by (simp add: setsum_subtractf)
```
```   478
```
```   479 lemma lemma_realpow_rev_sumr:
```
```   480    "(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) =
```
```   481     (\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))"
```
```   482   by (subst nat_diff_setsum_reindex[symmetric]) simp
```
```   483
```
```   484 lemma lemma_termdiff2:
```
```   485   fixes h :: "'a :: {field}"
```
```   486   assumes h: "h \<noteq> 0"
```
```   487   shows
```
```   488     "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
```
```   489      h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p.
```
```   490           (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
```
```   491   apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
```
```   492   apply (simp add: right_diff_distrib diff_divide_distrib h)
```
```   493   apply (simp add: mult.assoc [symmetric])
```
```   494   apply (cases "n", simp)
```
```   495   apply (simp add: diff_power_eq_setsum h
```
```   496                    right_diff_distrib [symmetric] mult.assoc
```
```   497               del: power_Suc setsum_lessThan_Suc of_nat_Suc)
```
```   498   apply (subst lemma_realpow_rev_sumr)
```
```   499   apply (subst sumr_diff_mult_const2)
```
```   500   apply simp
```
```   501   apply (simp only: lemma_termdiff1 setsum_right_distrib)
```
```   502   apply (rule setsum.cong [OF refl])
```
```   503   apply (simp add: less_iff_Suc_add)
```
```   504   apply (clarify)
```
```   505   apply (simp add: setsum_right_distrib diff_power_eq_setsum ac_simps
```
```   506               del: setsum_lessThan_Suc power_Suc)
```
```   507   apply (subst mult.assoc [symmetric], subst power_add [symmetric])
```
```   508   apply (simp add: ac_simps)
```
```   509   done
```
```   510
```
```   511 lemma real_setsum_nat_ivl_bounded2:
```
```   512   fixes K :: "'a::linordered_semidom"
```
```   513   assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
```
```   514     and K: "0 \<le> K"
```
```   515   shows "setsum f {..<n-k} \<le> of_nat n * K"
```
```   516   apply (rule order_trans [OF setsum_mono])
```
```   517   apply (rule f, simp)
```
```   518   apply (simp add: mult_right_mono K)
```
```   519   done
```
```   520
```
```   521 lemma lemma_termdiff3:
```
```   522   fixes h z :: "'a::{real_normed_field}"
```
```   523   assumes 1: "h \<noteq> 0"
```
```   524     and 2: "norm z \<le> K"
```
```   525     and 3: "norm (z + h) \<le> K"
```
```   526   shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
```
```   527           \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   528 proof -
```
```   529   have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
```
```   530         norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p.
```
```   531           (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
```
```   532     by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult)
```
```   533   also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
```
```   534   proof (rule mult_right_mono [OF _ norm_ge_zero])
```
```   535     from norm_ge_zero 2 have K: "0 \<le> K"
```
```   536       by (rule order_trans)
```
```   537     have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
```
```   538       apply (erule subst)
```
```   539       apply (simp only: norm_mult norm_power power_add)
```
```   540       apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
```
```   541       done
```
```   542     show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))
```
```   543           \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
```
```   544       apply (intro
```
```   545          order_trans [OF norm_setsum]
```
```   546          real_setsum_nat_ivl_bounded2
```
```   547          mult_nonneg_nonneg
```
```   548          of_nat_0_le_iff
```
```   549          zero_le_power K)
```
```   550       apply (rule le_Kn, simp)
```
```   551       done
```
```   552   qed
```
```   553   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   554     by (simp only: mult.assoc)
```
```   555   finally show ?thesis .
```
```   556 qed
```
```   557
```
```   558 lemma lemma_termdiff4:
```
```   559   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```   560   assumes k: "0 < (k::real)"
```
```   561     and le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
```
```   562   shows "f -- 0 --> 0"
```
```   563 proof (rule tendsto_norm_zero_cancel)
```
```   564   show "(\<lambda>h. norm (f h)) -- 0 --> 0"
```
```   565   proof (rule real_tendsto_sandwich)
```
```   566     show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)"
```
```   567       by simp
```
```   568     show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)"
```
```   569       using k by (auto simp add: eventually_at dist_norm le)
```
```   570     show "(\<lambda>h. 0) -- (0::'a) --> (0::real)"
```
```   571       by (rule tendsto_const)
```
```   572     have "(\<lambda>h. K * norm h) -- (0::'a) --> K * norm (0::'a)"
```
```   573       by (intro tendsto_intros)
```
```   574     then show "(\<lambda>h. K * norm h) -- (0::'a) --> 0"
```
```   575       by simp
```
```   576   qed
```
```   577 qed
```
```   578
```
```   579 lemma lemma_termdiff5:
```
```   580   fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
```
```   581   assumes k: "0 < (k::real)"
```
```   582   assumes f: "summable f"
```
```   583   assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
```
```   584   shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
```
```   585 proof (rule lemma_termdiff4 [OF k])
```
```   586   fix h::'a
```
```   587   assume "h \<noteq> 0" and "norm h < k"
```
```   588   hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
```
```   589     by (simp add: le)
```
```   590   hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
```
```   591     by simp
```
```   592   moreover from f have B: "summable (\<lambda>n. f n * norm h)"
```
```   593     by (rule summable_mult2)
```
```   594   ultimately have C: "summable (\<lambda>n. norm (g h n))"
```
```   595     by (rule summable_comparison_test)
```
```   596   hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
```
```   597     by (rule summable_norm)
```
```   598   also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
```
```   599     by (rule suminf_le)
```
```   600   also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
```
```   601     by (rule suminf_mult2 [symmetric])
```
```   602   finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
```
```   603 qed
```
```   604
```
```   605
```
```   606 text\<open>FIXME: Long proofs\<close>
```
```   607
```
```   608 lemma termdiffs_aux:
```
```   609   fixes x :: "'a::{real_normed_field,banach}"
```
```   610   assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
```
```   611     and 2: "norm x < norm K"
```
```   612   shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h
```
```   613              - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   614 proof -
```
```   615   from dense [OF 2]
```
```   616   obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
```
```   617   from norm_ge_zero r1 have r: "0 < r"
```
```   618     by (rule order_le_less_trans)
```
```   619   hence r_neq_0: "r \<noteq> 0" by simp
```
```   620   show ?thesis
```
```   621   proof (rule lemma_termdiff5)
```
```   622     show "0 < r - norm x" using r1 by simp
```
```   623     from r r2 have "norm (of_real r::'a) < norm K"
```
```   624       by simp
```
```   625     with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
```
```   626       by (rule powser_insidea)
```
```   627     hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
```
```   628       using r
```
```   629       by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
```
```   630     hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
```
```   631       by (rule diffs_equiv [THEN sums_summable])
```
```   632     also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
```
```   633       (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
```
```   634       apply (rule ext)
```
```   635       apply (simp add: diffs_def)
```
```   636       apply (case_tac n, simp_all add: r_neq_0)
```
```   637       done
```
```   638     finally have "summable
```
```   639       (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
```
```   640       by (rule diffs_equiv [THEN sums_summable])
```
```   641     also have
```
```   642       "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
```
```   643            r ^ (n - Suc 0)) =
```
```   644        (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
```
```   645       apply (rule ext)
```
```   646       apply (case_tac "n", simp)
```
```   647       apply (rename_tac nat)
```
```   648       apply (case_tac "nat", simp)
```
```   649       apply (simp add: r_neq_0)
```
```   650       done
```
```   651     finally
```
```   652     show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
```
```   653   next
```
```   654     fix h::'a and n::nat
```
```   655     assume h: "h \<noteq> 0"
```
```   656     assume "norm h < r - norm x"
```
```   657     hence "norm x + norm h < r" by simp
```
```   658     with norm_triangle_ineq have xh: "norm (x + h) < r"
```
```   659       by (rule order_le_less_trans)
```
```   660     show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))
```
```   661           \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
```
```   662       apply (simp only: norm_mult mult.assoc)
```
```   663       apply (rule mult_left_mono [OF _ norm_ge_zero])
```
```   664       apply (simp add: mult.assoc [symmetric])
```
```   665       apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
```
```   666       done
```
```   667   qed
```
```   668 qed
```
```   669
```
```   670 lemma termdiffs:
```
```   671   fixes K x :: "'a::{real_normed_field,banach}"
```
```   672   assumes 1: "summable (\<lambda>n. c n * K ^ n)"
```
```   673       and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
```
```   674       and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
```
```   675       and 4: "norm x < norm K"
```
```   676   shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. (diffs c) n * x^n)"
```
```   677   unfolding DERIV_def
```
```   678 proof (rule LIM_zero_cancel)
```
```   679   show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x^n)) / h
```
```   680             - suminf (\<lambda>n. diffs c n * x^n)) -- 0 --> 0"
```
```   681   proof (rule LIM_equal2)
```
```   682     show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
```
```   683   next
```
```   684     fix h :: 'a
```
```   685     assume "norm (h - 0) < norm K - norm x"
```
```   686     hence "norm x + norm h < norm K" by simp
```
```   687     hence 5: "norm (x + h) < norm K"
```
```   688       by (rule norm_triangle_ineq [THEN order_le_less_trans])
```
```   689     have "summable (\<lambda>n. c n * x^n)"
```
```   690       and "summable (\<lambda>n. c n * (x + h) ^ n)"
```
```   691       and "summable (\<lambda>n. diffs c n * x^n)"
```
```   692       using 1 2 4 5 by (auto elim: powser_inside)
```
```   693     then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
```
```   694           (\<Sum>n. (c n * (x + h) ^ n - c n * x^n) / h - of_nat n * c n * x ^ (n - Suc 0))"
```
```   695       by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
```
```   696     then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
```
```   697           (\<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))"
```
```   698       by (simp add: algebra_simps)
```
```   699   next
```
```   700     show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   701       by (rule termdiffs_aux [OF 3 4])
```
```   702   qed
```
```   703 qed
```
```   704
```
```   705 subsection \<open>The Derivative of a Power Series Has the Same Radius of Convergence\<close>
```
```   706
```
```   707 lemma termdiff_converges:
```
```   708   fixes x :: "'a::{real_normed_field,banach}"
```
```   709   assumes K: "norm x < K"
```
```   710       and sm: "\<And>x. norm x < K \<Longrightarrow> summable(\<lambda>n. c n * x ^ n)"
```
```   711     shows "summable (\<lambda>n. diffs c n * x ^ n)"
```
```   712 proof (cases "x = 0")
```
```   713   case True then show ?thesis
```
```   714   using powser_sums_zero sums_summable by auto
```
```   715 next
```
```   716   case False
```
```   717   then have "K>0"
```
```   718     using K less_trans zero_less_norm_iff by blast
```
```   719   then obtain r::real where r: "norm x < norm r" "norm r < K" "r>0"
```
```   720     using K False
```
```   721     by (auto simp: abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"])
```
```   722   have "(\<lambda>n. of_nat n * (x / of_real r) ^ n) ----> 0"
```
```   723     using r by (simp add: norm_divide powser_times_n_limit_0 [of "x / of_real r"])
```
```   724   then obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> real_of_nat n * norm x ^ n < r ^ n"
```
```   725     using r unfolding LIMSEQ_iff
```
```   726     apply (drule_tac x=1 in spec)
```
```   727     apply (auto simp: norm_divide norm_mult norm_power field_simps)
```
```   728     done
```
```   729   have "summable (\<lambda>n. (of_nat n * c n) * x ^ n)"
```
```   730     apply (rule summable_comparison_test' [of "\<lambda>n. norm(c n * (of_real r) ^ n)" N])
```
```   731     apply (rule powser_insidea [OF sm [of "of_real ((r+K)/2)"]])
```
```   732     using N r norm_of_real [of "r+K", where 'a = 'a]
```
```   733     apply (auto simp add: norm_divide norm_mult norm_power )
```
```   734     using less_eq_real_def by fastforce
```
```   735   then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ Suc n)"
```
```   736     using summable_iff_shift [of "\<lambda>n. of_nat n * c n * x ^ n" 1]
```
```   737     by simp
```
```   738   then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ n)"
```
```   739     using False summable_mult2 [of "\<lambda>n. (of_nat (Suc n) * c(Suc n) * x ^ n) * x" "inverse x"]
```
```   740     by (simp add: mult.assoc) (auto simp: ac_simps)
```
```   741   then show ?thesis
```
```   742     by (simp add: diffs_def)
```
```   743 qed
```
```   744
```
```   745 lemma termdiff_converges_all:
```
```   746   fixes x :: "'a::{real_normed_field,banach}"
```
```   747   assumes "\<And>x. summable (\<lambda>n. c n * x^n)"
```
```   748     shows "summable (\<lambda>n. diffs c n * x^n)"
```
```   749   apply (rule termdiff_converges [where K = "1 + norm x"])
```
```   750   using assms
```
```   751   apply auto
```
```   752   done
```
```   753
```
```   754 lemma termdiffs_strong:
```
```   755   fixes K x :: "'a::{real_normed_field,banach}"
```
```   756   assumes sm: "summable (\<lambda>n. c n * K ^ n)"
```
```   757       and K: "norm x < norm K"
```
```   758   shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. diffs c n * x^n)"
```
```   759 proof -
```
```   760   have K2: "norm ((of_real (norm K) + of_real (norm x)) / 2 :: 'a) < norm K"
```
```   761     using K
```
```   762     apply (auto simp: norm_divide)
```
```   763     apply (rule le_less_trans [of _ "of_real (norm K) + of_real (norm x)"])
```
```   764     apply (auto simp: mult_2_right norm_triangle_mono)
```
```   765     done
```
```   766   then have [simp]: "norm ((of_real (norm K) + of_real (norm x)) :: 'a) < norm K * 2"
```
```   767     by simp
```
```   768   have "summable (\<lambda>n. c n * (of_real (norm x + norm K) / 2) ^ n)"
```
```   769     by (metis K2 summable_norm_cancel [OF powser_insidea [OF sm]] add.commute of_real_add)
```
```   770   moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs c n * x ^ n)"
```
```   771     by (blast intro: sm termdiff_converges powser_inside)
```
```   772   moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs(diffs c) n * x ^ n)"
```
```   773     by (blast intro: sm termdiff_converges powser_inside)
```
```   774   ultimately show ?thesis
```
```   775     apply (rule termdiffs [where K = "of_real (norm x + norm K) / 2"])
```
```   776     apply (auto simp: algebra_simps)
```
```   777     using K
```
```   778     apply (simp_all add: of_real_add [symmetric] del: of_real_add)
```
```   779     done
```
```   780 qed
```
```   781
```
```   782 lemma powser_limit_0:
```
```   783   fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
```
```   784   assumes s: "0 < s"
```
```   785       and sm: "\<And>x. norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
```
```   786     shows "(f ---> a 0) (at 0)"
```
```   787 proof -
```
```   788   have "summable (\<lambda>n. a n * (of_real s / 2) ^ n)"
```
```   789     apply (rule sums_summable [where l = "f (of_real s / 2)", OF sm])
```
```   790     using s
```
```   791     apply (auto simp: norm_divide)
```
```   792     done
```
```   793   then have "((\<lambda>x. \<Sum>n. a n * x ^ n) has_field_derivative (\<Sum>n. diffs a n * 0 ^ n)) (at 0)"
```
```   794     apply (rule termdiffs_strong)
```
```   795     using s
```
```   796     apply (auto simp: norm_divide)
```
```   797     done
```
```   798   then have "isCont (\<lambda>x. \<Sum>n. a n * x ^ n) 0"
```
```   799     by (blast intro: DERIV_continuous)
```
```   800   then have "((\<lambda>x. \<Sum>n. a n * x ^ n) ---> a 0) (at 0)"
```
```   801     by (simp add: continuous_within powser_zero)
```
```   802   then show ?thesis
```
```   803     apply (rule Lim_transform)
```
```   804     apply (auto simp add: LIM_eq)
```
```   805     apply (rule_tac x="s" in exI)
```
```   806     using s
```
```   807     apply (auto simp: sm [THEN sums_unique])
```
```   808     done
```
```   809 qed
```
```   810
```
```   811 lemma powser_limit_0_strong:
```
```   812   fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
```
```   813   assumes s: "0 < s"
```
```   814       and sm: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
```
```   815     shows "(f ---> a 0) (at 0)"
```
```   816 proof -
```
```   817   have *: "((\<lambda>x. if x = 0 then a 0 else f x) ---> a 0) (at 0)"
```
```   818     apply (rule powser_limit_0 [OF s])
```
```   819     apply (case_tac "x=0")
```
```   820     apply (auto simp add: powser_sums_zero sm)
```
```   821     done
```
```   822   show ?thesis
```
```   823     apply (subst LIM_equal [where g = "(\<lambda>x. if x = 0 then a 0 else f x)"])
```
```   824     apply (simp_all add: *)
```
```   825     done
```
```   826 qed
```
```   827
```
```   828
```
```   829 subsection \<open>Derivability of power series\<close>
```
```   830
```
```   831 lemma DERIV_series':
```
```   832   fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
```
```   833   assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
```
```   834     and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
```
```   835     and "summable (f' x0)"
```
```   836     and "summable L"
```
```   837     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>"
```
```   838   shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
```
```   839   unfolding DERIV_def
```
```   840 proof (rule LIM_I)
```
```   841   fix r :: real
```
```   842   assume "0 < r" hence "0 < r/3" by auto
```
```   843
```
```   844   obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
```
```   845     using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable L\<close>] by auto
```
```   846
```
```   847   obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
```
```   848     using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable (f' x0)\<close>] by auto
```
```   849
```
```   850   let ?N = "Suc (max N_L N_f')"
```
```   851   have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
```
```   852     L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
```
```   853
```
```   854   let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
```
```   855
```
```   856   let ?r = "r / (3 * real ?N)"
```
```   857   from \<open>0 < r\<close> have "0 < ?r" by simp
```
```   858
```
```   859   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)"
```
```   860   def S' \<equiv> "Min (?s ` {..< ?N })"
```
```   861
```
```   862   have "0 < S'" unfolding S'_def
```
```   863   proof (rule iffD2[OF Min_gr_iff])
```
```   864     show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
```
```   865     proof
```
```   866       fix x
```
```   867       assume "x \<in> ?s ` {..<?N}"
```
```   868       then obtain n where "x = ?s n" and "n \<in> {..<?N}"
```
```   869         using image_iff[THEN iffD1] by blast
```
```   870       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>, unfolded real_norm_def]
```
```   871       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)"
```
```   872         by auto
```
```   873       have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound)
```
```   874       thus "0 < x" unfolding \<open>x = ?s n\<close> .
```
```   875     qed
```
```   876   qed auto
```
```   877
```
```   878   def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
```
```   879   hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
```
```   880     and "S \<le> S'" using x0_in_I and \<open>0 < S'\<close>
```
```   881     by auto
```
```   882
```
```   883   {
```
```   884     fix x
```
```   885     assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
```
```   886     hence x_in_I: "x0 + x \<in> { a <..< b }"
```
```   887       using S_a S_b by auto
```
```   888
```
```   889     note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   890     note div_smbl = summable_divide[OF diff_smbl]
```
```   891     note all_smbl = summable_diff[OF div_smbl \<open>summable (f' x0)\<close>]
```
```   892     note ign = summable_ignore_initial_segment[where k="?N"]
```
```   893     note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
```
```   894     note div_shft_smbl = summable_divide[OF diff_shft_smbl]
```
```   895     note all_shft_smbl = summable_diff[OF div_smbl ign[OF \<open>summable (f' x0)\<close>]]
```
```   896
```
```   897     { fix n
```
```   898       have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
```
```   899         using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
```
```   900         unfolding abs_divide .
```
```   901       hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)"
```
```   902         using \<open>x \<noteq> 0\<close> by auto }
```
```   903     note 1 = this and 2 = summable_rabs_comparison_test[OF _ ign[OF \<open>summable L\<close>]]
```
```   904     then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))"
```
```   905       by (metis (lifting) abs_idempotent order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF \<open>summable L\<close>]]])
```
```   906     then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3")
```
```   907       using L_estimate by auto
```
```   908
```
```   909     have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n \<bar>)" ..
```
```   910     also have "\<dots> < (\<Sum>n<?N. ?r)"
```
```   911     proof (rule setsum_strict_mono)
```
```   912       fix n
```
```   913       assume "n \<in> {..< ?N}"
```
```   914       have "\<bar>x\<bar> < S" using \<open>\<bar>x\<bar> < S\<close> .
```
```   915       also have "S \<le> S'" using \<open>S \<le> S'\<close> .
```
```   916       also have "S' \<le> ?s n" unfolding S'_def
```
```   917       proof (rule Min_le_iff[THEN iffD2])
```
```   918         have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n"
```
```   919           using \<open>n \<in> {..< ?N}\<close> by auto
```
```   920         thus "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n" by blast
```
```   921       qed auto
```
```   922       finally have "\<bar>x\<bar> < ?s n" .
```
```   923
```
```   924       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]
```
```   925       have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
```
```   926       with \<open>x \<noteq> 0\<close> and \<open>\<bar>x\<bar> < ?s n\<close> show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
```
```   927         by blast
```
```   928     qed auto
```
```   929     also have "\<dots> = of_nat (card {..<?N}) * ?r"
```
```   930       by (rule setsum_constant)
```
```   931     also have "\<dots> = real ?N * ?r"
```
```   932       unfolding real_eq_of_nat by auto
```
```   933     also have "\<dots> = r/3" by auto
```
```   934     finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
```
```   935
```
```   936     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   937     have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
```
```   938         \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
```
```   939       unfolding suminf_diff[OF div_smbl \<open>summable (f' x0)\<close>, symmetric]
```
```   940       using suminf_divide[OF diff_smbl, symmetric] by auto
```
```   941     also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>"
```
```   942       unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
```
```   943       unfolding suminf_diff[OF div_shft_smbl ign[OF \<open>summable (f' x0)\<close>]]
```
```   944       apply (subst (5) add.commute)
```
```   945       by (rule abs_triangle_ineq)
```
```   946     also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
```
```   947       using abs_triangle_ineq4 by auto
```
```   948     also have "\<dots> < r /3 + r/3 + r/3"
```
```   949       using \<open>?diff_part < r/3\<close> \<open>?L_part \<le> r/3\<close> and \<open>?f'_part < r/3\<close>
```
```   950       by (rule add_strict_mono [OF add_less_le_mono])
```
```   951     finally have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
```
```   952       by auto
```
```   953   }
```
```   954   thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
```
```   955       norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
```
```   956     using \<open>0 < S\<close> unfolding real_norm_def diff_0_right by blast
```
```   957 qed
```
```   958
```
```   959 lemma DERIV_power_series':
```
```   960   fixes f :: "nat \<Rightarrow> real"
```
```   961   assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
```
```   962     and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
```
```   963   shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
```
```   964   (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
```
```   965 proof -
```
```   966   {
```
```   967     fix R'
```
```   968     assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
```
```   969     hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
```
```   970       by auto
```
```   971     have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
```
```   972     proof (rule DERIV_series')
```
```   973       show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
```
```   974       proof -
```
```   975         have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
```
```   976           using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by auto
```
```   977         hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
```
```   978           using \<open>R' < R\<close> by auto
```
```   979         have "norm R' < norm ((R' + R) / 2)"
```
```   980           using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by auto
```
```   981         from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
```
```   982           by auto
```
```   983       qed
```
```   984       {
```
```   985         fix n x y
```
```   986         assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
```
```   987         show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
```
```   988         proof -
```
```   989           have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
```
```   990             (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>"
```
```   991             unfolding right_diff_distrib[symmetric] diff_power_eq_setsum abs_mult
```
```   992             by auto
```
```   993           also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
```
```   994           proof (rule mult_left_mono)
```
```   995             have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
```
```   996               by (rule setsum_abs)
```
```   997             also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)"
```
```   998             proof (rule setsum_mono)
```
```   999               fix p
```
```  1000               assume "p \<in> {..<Suc n}"
```
```  1001               hence "p \<le> n" by auto
```
```  1002               {
```
```  1003                 fix n
```
```  1004                 fix x :: real
```
```  1005                 assume "x \<in> {-R'<..<R'}"
```
```  1006                 hence "\<bar>x\<bar> \<le> R'"  by auto
```
```  1007                 hence "\<bar>x^n\<bar> \<le> R'^n"
```
```  1008                   unfolding power_abs by (rule power_mono, auto)
```
```  1009               }
```
```  1010               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>
```
```  1011               have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
```
```  1012                 unfolding abs_mult by auto
```
```  1013               thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
```
```  1014                 unfolding power_add[symmetric] using \<open>p \<le> n\<close> by auto
```
```  1015             qed
```
```  1016             also have "\<dots> = real (Suc n) * R' ^ n"
```
```  1017               unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
```
```  1018             finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
```
```  1019               unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF \<open>0 < R'\<close>]]] .
```
```  1020             show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
```
```  1021               unfolding abs_mult[symmetric] by auto
```
```  1022           qed
```
```  1023           also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
```
```  1024             unfolding abs_mult mult.assoc[symmetric] by algebra
```
```  1025           finally show ?thesis .
```
```  1026         qed
```
```  1027       }
```
```  1028       {
```
```  1029         fix n
```
```  1030         show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
```
```  1031           by (auto intro!: derivative_eq_intros simp del: power_Suc simp: real_of_nat_def)
```
```  1032       }
```
```  1033       {
```
```  1034         fix x
```
```  1035         assume "x \<in> {-R' <..< R'}"
```
```  1036         hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
```
```  1037           using assms \<open>R' < R\<close> by auto
```
```  1038         have "summable (\<lambda> n. f n * x^n)"
```
```  1039         proof (rule summable_comparison_test, intro exI allI impI)
```
```  1040           fix n
```
```  1041           have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
```
```  1042             by (rule mult_left_mono) auto
```
```  1043           show "norm (f n * x^n) \<le> norm (f n * real (Suc n) * x^n)"
```
```  1044             unfolding real_norm_def abs_mult
```
```  1045             by (rule mult_right_mono) (auto simp add: le[unfolded mult_1_right])
```
```  1046         qed (rule powser_insidea[OF converges[OF \<open>R' \<in> {-R <..< R}\<close>] \<open>norm x < norm R'\<close>])
```
```  1047         from this[THEN summable_mult2[where c=x], unfolded mult.assoc, unfolded mult.commute]
```
```  1048         show "summable (?f x)" by auto
```
```  1049       }
```
```  1050       show "summable (?f' x0)"
```
```  1051         using converges[OF \<open>x0 \<in> {-R <..< R}\<close>] .
```
```  1052       show "x0 \<in> {-R' <..< R'}"
```
```  1053         using \<open>x0 \<in> {-R' <..< R'}\<close> .
```
```  1054     qed
```
```  1055   } note for_subinterval = this
```
```  1056   let ?R = "(R + \<bar>x0\<bar>) / 2"
```
```  1057   have "\<bar>x0\<bar> < ?R" using assms by auto
```
```  1058   hence "- ?R < x0"
```
```  1059   proof (cases "x0 < 0")
```
```  1060     case True
```
```  1061     hence "- x0 < ?R" using \<open>\<bar>x0\<bar> < ?R\<close> by auto
```
```  1062     thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
```
```  1063   next
```
```  1064     case False
```
```  1065     have "- ?R < 0" using assms by auto
```
```  1066     also have "\<dots> \<le> x0" using False by auto
```
```  1067     finally show ?thesis .
```
```  1068   qed
```
```  1069   hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
```
```  1070     using assms by auto
```
```  1071   from for_subinterval[OF this]
```
```  1072   show ?thesis .
```
```  1073 qed
```
```  1074
```
```  1075
```
```  1076 subsection \<open>Exponential Function\<close>
```
```  1077
```
```  1078 definition exp :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```  1079   where "exp = (\<lambda>x. \<Sum>n. x^n /\<^sub>R fact n)"
```
```  1080
```
```  1081 lemma summable_exp_generic:
```
```  1082   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  1083   defines S_def: "S \<equiv> \<lambda>n. x^n /\<^sub>R fact n"
```
```  1084   shows "summable S"
```
```  1085 proof -
```
```  1086   have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R (Suc n)"
```
```  1087     unfolding S_def by (simp del: mult_Suc)
```
```  1088   obtain r :: real where r0: "0 < r" and r1: "r < 1"
```
```  1089     using dense [OF zero_less_one] by fast
```
```  1090   obtain N :: nat where N: "norm x < real N * r"
```
```  1091     using ex_less_of_nat_mult r0 real_of_nat_def by auto
```
```  1092   from r1 show ?thesis
```
```  1093   proof (rule summable_ratio_test [rule_format])
```
```  1094     fix n :: nat
```
```  1095     assume n: "N \<le> n"
```
```  1096     have "norm x \<le> real N * r"
```
```  1097       using N by (rule order_less_imp_le)
```
```  1098     also have "real N * r \<le> real (Suc n) * r"
```
```  1099       using r0 n by (simp add: mult_right_mono)
```
```  1100     finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
```
```  1101       using norm_ge_zero by (rule mult_right_mono)
```
```  1102     hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
```
```  1103       by (rule order_trans [OF norm_mult_ineq])
```
```  1104     hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
```
```  1105       by (simp add: pos_divide_le_eq ac_simps)
```
```  1106     thus "norm (S (Suc n)) \<le> r * norm (S n)"
```
```  1107       by (simp add: S_Suc inverse_eq_divide)
```
```  1108   qed
```
```  1109 qed
```
```  1110
```
```  1111 lemma summable_norm_exp:
```
```  1112   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  1113   shows "summable (\<lambda>n. norm (x^n /\<^sub>R fact n))"
```
```  1114 proof (rule summable_norm_comparison_test [OF exI, rule_format])
```
```  1115   show "summable (\<lambda>n. norm x^n /\<^sub>R fact n)"
```
```  1116     by (rule summable_exp_generic)
```
```  1117   fix n
```
```  1118   show "norm (x^n /\<^sub>R fact n) \<le> norm x^n /\<^sub>R fact n"
```
```  1119     by (simp add: norm_power_ineq)
```
```  1120 qed
```
```  1121
```
```  1122 lemma summable_exp:
```
```  1123   fixes x :: "'a::{real_normed_field,banach}"
```
```  1124   shows "summable (\<lambda>n. inverse (fact n) * x^n)"
```
```  1125   using summable_exp_generic [where x=x]
```
```  1126   by (simp add: scaleR_conv_of_real nonzero_of_real_inverse)
```
```  1127
```
```  1128 lemma exp_converges: "(\<lambda>n. x^n /\<^sub>R fact n) sums exp x"
```
```  1129   unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
```
```  1130
```
```  1131 lemma exp_fdiffs:
```
```  1132   "diffs (\<lambda>n. inverse (fact n)) = (\<lambda>n. inverse (fact n :: 'a::{real_normed_field,banach}))"
```
```  1133   by (simp add: diffs_def mult_ac nonzero_inverse_mult_distrib nonzero_of_real_inverse
```
```  1134            del: mult_Suc of_nat_Suc)
```
```  1135
```
```  1136 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
```
```  1137   by (simp add: diffs_def)
```
```  1138
```
```  1139 lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
```
```  1140   unfolding exp_def scaleR_conv_of_real
```
```  1141   apply (rule DERIV_cong)
```
```  1142   apply (rule termdiffs [where K="of_real (1 + norm x)"])
```
```  1143   apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
```
```  1144   apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
```
```  1145   apply (simp del: of_real_add)
```
```  1146   done
```
```  1147
```
```  1148 declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
```
```  1149
```
```  1150 lemma norm_exp: "norm (exp x) \<le> exp (norm x)"
```
```  1151 proof -
```
```  1152   from summable_norm[OF summable_norm_exp, of x]
```
```  1153   have "norm (exp x) \<le> (\<Sum>n. inverse (fact n) * norm (x^n))"
```
```  1154     by (simp add: exp_def)
```
```  1155   also have "\<dots> \<le> exp (norm x)"
```
```  1156     using summable_exp_generic[of "norm x"] summable_norm_exp[of x]
```
```  1157     by (auto simp: exp_def intro!: suminf_le norm_power_ineq)
```
```  1158   finally show ?thesis .
```
```  1159 qed
```
```  1160
```
```  1161 lemma isCont_exp:
```
```  1162   fixes x::"'a::{real_normed_field,banach}"
```
```  1163   shows "isCont exp x"
```
```  1164   by (rule DERIV_exp [THEN DERIV_isCont])
```
```  1165
```
```  1166 lemma isCont_exp' [simp]:
```
```  1167   fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1168   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
```
```  1169   by (rule isCont_o2 [OF _ isCont_exp])
```
```  1170
```
```  1171 lemma tendsto_exp [tendsto_intros]:
```
```  1172   fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1173   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
```
```  1174   by (rule isCont_tendsto_compose [OF isCont_exp])
```
```  1175
```
```  1176 lemma continuous_exp [continuous_intros]:
```
```  1177   fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1178   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
```
```  1179   unfolding continuous_def by (rule tendsto_exp)
```
```  1180
```
```  1181 lemma continuous_on_exp [continuous_intros]:
```
```  1182   fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1183   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
```
```  1184   unfolding continuous_on_def by (auto intro: tendsto_exp)
```
```  1185
```
```  1186
```
```  1187 subsubsection \<open>Properties of the Exponential Function\<close>
```
```  1188
```
```  1189 lemma exp_zero [simp]: "exp 0 = 1"
```
```  1190   unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
```
```  1191
```
```  1192 lemma exp_series_add_commuting:
```
```  1193   fixes x y :: "'a::{real_normed_algebra_1, banach}"
```
```  1194   defines S_def: "S \<equiv> \<lambda>x n. x^n /\<^sub>R fact n"
```
```  1195   assumes comm: "x * y = y * x"
```
```  1196   shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))"
```
```  1197 proof (induct n)
```
```  1198   case 0
```
```  1199   show ?case
```
```  1200     unfolding S_def by simp
```
```  1201 next
```
```  1202   case (Suc n)
```
```  1203   have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
```
```  1204     unfolding S_def by (simp del: mult_Suc)
```
```  1205   hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
```
```  1206     by simp
```
```  1207   have S_comm: "\<And>n. S x n * y = y * S x n"
```
```  1208     by (simp add: power_commuting_commutes comm S_def)
```
```  1209
```
```  1210   have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
```
```  1211     by (simp only: times_S)
```
```  1212   also have "\<dots> = (x + y) * (\<Sum>i\<le>n. S x i * S y (n-i))"
```
```  1213     by (simp only: Suc)
```
```  1214   also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n-i))
```
```  1215                 + y * (\<Sum>i\<le>n. S x i * S y (n-i))"
```
```  1216     by (rule distrib_right)
```
```  1217   also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i))
```
```  1218                 + (\<Sum>i\<le>n. S x i * y * S y (n-i))"
```
```  1219     by (simp add: setsum_right_distrib ac_simps S_comm)
```
```  1220   also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i))
```
```  1221                 + (\<Sum>i\<le>n. S x i * (y * S y (n-i)))"
```
```  1222     by (simp add: ac_simps)
```
```  1223   also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
```
```  1224                 + (\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1225     by (simp add: times_S Suc_diff_le)
```
```  1226   also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
```
```  1227              (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1228     by (subst setsum_atMost_Suc_shift) simp
```
```  1229   also have "(\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```  1230              (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1231     by simp
```
```  1232   also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
```
```  1233              (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```  1234              (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1235     by (simp only: setsum.distrib [symmetric] scaleR_left_distrib [symmetric]
```
```  1236                    real_of_nat_add [symmetric]) simp
```
```  1237   also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n-i))"
```
```  1238     by (simp only: scaleR_right.setsum)
```
```  1239   finally show
```
```  1240     "S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
```
```  1241     by (simp del: setsum_cl_ivl_Suc)
```
```  1242 qed
```
```  1243
```
```  1244 lemma exp_add_commuting: "x * y = y * x \<Longrightarrow> exp (x + y) = exp x * exp y"
```
```  1245   unfolding exp_def
```
```  1246   by (simp only: Cauchy_product summable_norm_exp exp_series_add_commuting)
```
```  1247
```
```  1248 lemma exp_add:
```
```  1249   fixes x y::"'a::{real_normed_field,banach}"
```
```  1250   shows "exp (x + y) = exp x * exp y"
```
```  1251   by (rule exp_add_commuting) (simp add: ac_simps)
```
```  1252
```
```  1253 lemma exp_double: "exp(2 * z) = exp z ^ 2"
```
```  1254   by (simp add: exp_add_commuting mult_2 power2_eq_square)
```
```  1255
```
```  1256 lemmas mult_exp_exp = exp_add [symmetric]
```
```  1257
```
```  1258 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
```
```  1259   unfolding exp_def
```
```  1260   apply (subst suminf_of_real)
```
```  1261   apply (rule summable_exp_generic)
```
```  1262   apply (simp add: scaleR_conv_of_real)
```
```  1263   done
```
```  1264
```
```  1265 corollary exp_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> exp z \<in> \<real>"
```
```  1266   by (metis Reals_cases Reals_of_real exp_of_real)
```
```  1267
```
```  1268 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
```
```  1269 proof
```
```  1270   have "exp x * exp (- x) = 1" by (simp add: exp_add_commuting[symmetric])
```
```  1271   also assume "exp x = 0"
```
```  1272   finally show "False" by simp
```
```  1273 qed
```
```  1274
```
```  1275 lemma exp_minus_inverse:
```
```  1276   shows "exp x * exp (- x) = 1"
```
```  1277   by (simp add: exp_add_commuting[symmetric])
```
```  1278
```
```  1279 lemma exp_minus:
```
```  1280   fixes x :: "'a::{real_normed_field, banach}"
```
```  1281   shows "exp (- x) = inverse (exp x)"
```
```  1282   by (intro inverse_unique [symmetric] exp_minus_inverse)
```
```  1283
```
```  1284 lemma exp_diff:
```
```  1285   fixes x :: "'a::{real_normed_field, banach}"
```
```  1286   shows "exp (x - y) = exp x / exp y"
```
```  1287   using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
```
```  1288
```
```  1289 lemma exp_of_nat_mult:
```
```  1290   fixes x :: "'a::{real_normed_field,banach}"
```
```  1291   shows "exp(of_nat n * x) = exp(x) ^ n"
```
```  1292     by (induct n) (auto simp add: distrib_left exp_add mult.commute)
```
```  1293
```
```  1294 corollary exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
```
```  1295   by (simp add: exp_of_nat_mult real_of_nat_def)
```
```  1296
```
```  1297 lemma exp_setsum: "finite I \<Longrightarrow> exp(setsum f I) = setprod (\<lambda>x. exp(f x)) I"
```
```  1298   by (induction I rule: finite_induct) (auto simp: exp_add_commuting mult.commute)
```
```  1299
```
```  1300
```
```  1301 subsubsection \<open>Properties of the Exponential Function on Reals\<close>
```
```  1302
```
```  1303 text \<open>Comparisons of @{term "exp x"} with zero.\<close>
```
```  1304
```
```  1305 text\<open>Proof: because every exponential can be seen as a square.\<close>
```
```  1306 lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
```
```  1307 proof -
```
```  1308   have "0 \<le> exp (x/2) * exp (x/2)" by simp
```
```  1309   thus ?thesis by (simp add: exp_add [symmetric])
```
```  1310 qed
```
```  1311
```
```  1312 lemma exp_gt_zero [simp]: "0 < exp (x::real)"
```
```  1313   by (simp add: order_less_le)
```
```  1314
```
```  1315 lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
```
```  1316   by (simp add: not_less)
```
```  1317
```
```  1318 lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
```
```  1319   by (simp add: not_le)
```
```  1320
```
```  1321 lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
```
```  1322   by simp
```
```  1323
```
```  1324 text \<open>Strict monotonicity of exponential.\<close>
```
```  1325
```
```  1326 lemma exp_ge_add_one_self_aux:
```
```  1327   assumes "0 \<le> (x::real)" shows "1+x \<le> exp(x)"
```
```  1328 using order_le_imp_less_or_eq [OF assms]
```
```  1329 proof
```
```  1330   assume "0 < x"
```
```  1331   have "1+x \<le> (\<Sum>n<2. inverse (fact n) * x^n)"
```
```  1332     by (auto simp add: numeral_2_eq_2)
```
```  1333   also have "... \<le> (\<Sum>n. inverse (fact n) * x^n)"
```
```  1334     apply (rule setsum_le_suminf [OF summable_exp])
```
```  1335     using \<open>0 < x\<close>
```
```  1336     apply (auto  simp add:  zero_le_mult_iff)
```
```  1337     done
```
```  1338   finally show "1+x \<le> exp x"
```
```  1339     by (simp add: exp_def)
```
```  1340 next
```
```  1341   assume "0 = x"
```
```  1342   then show "1 + x \<le> exp x"
```
```  1343     by auto
```
```  1344 qed
```
```  1345
```
```  1346 lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
```
```  1347 proof -
```
```  1348   assume x: "0 < x"
```
```  1349   hence "1 < 1 + x" by simp
```
```  1350   also from x have "1 + x \<le> exp x"
```
```  1351     by (simp add: exp_ge_add_one_self_aux)
```
```  1352   finally show ?thesis .
```
```  1353 qed
```
```  1354
```
```  1355 lemma exp_less_mono:
```
```  1356   fixes x y :: real
```
```  1357   assumes "x < y"
```
```  1358   shows "exp x < exp y"
```
```  1359 proof -
```
```  1360   from \<open>x < y\<close> have "0 < y - x" by simp
```
```  1361   hence "1 < exp (y - x)" by (rule exp_gt_one)
```
```  1362   hence "1 < exp y / exp x" by (simp only: exp_diff)
```
```  1363   thus "exp x < exp y" by simp
```
```  1364 qed
```
```  1365
```
```  1366 lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
```
```  1367   unfolding linorder_not_le [symmetric]
```
```  1368   by (auto simp add: order_le_less exp_less_mono)
```
```  1369
```
```  1370 lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
```
```  1371   by (auto intro: exp_less_mono exp_less_cancel)
```
```  1372
```
```  1373 lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
```
```  1374   by (auto simp add: linorder_not_less [symmetric])
```
```  1375
```
```  1376 lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
```
```  1377   by (simp add: order_eq_iff)
```
```  1378
```
```  1379 text \<open>Comparisons of @{term "exp x"} with one.\<close>
```
```  1380
```
```  1381 lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
```
```  1382   using exp_less_cancel_iff [where x=0 and y=x] by simp
```
```  1383
```
```  1384 lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
```
```  1385   using exp_less_cancel_iff [where x=x and y=0] by simp
```
```  1386
```
```  1387 lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
```
```  1388   using exp_le_cancel_iff [where x=0 and y=x] by simp
```
```  1389
```
```  1390 lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
```
```  1391   using exp_le_cancel_iff [where x=x and y=0] by simp
```
```  1392
```
```  1393 lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
```
```  1394   using exp_inj_iff [where x=x and y=0] by simp
```
```  1395
```
```  1396 lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
```
```  1397 proof (rule IVT)
```
```  1398   assume "1 \<le> y"
```
```  1399   hence "0 \<le> y - 1" by simp
```
```  1400   hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
```
```  1401   thus "y \<le> exp (y - 1)" by simp
```
```  1402 qed (simp_all add: le_diff_eq)
```
```  1403
```
```  1404 lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y"
```
```  1405 proof (rule linorder_le_cases [of 1 y])
```
```  1406   assume "1 \<le> y"
```
```  1407   thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total)
```
```  1408 next
```
```  1409   assume "0 < y" and "y \<le> 1"
```
```  1410   hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
```
```  1411   then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
```
```  1412   hence "exp (- x) = y" by (simp add: exp_minus)
```
```  1413   thus "\<exists>x. exp x = y" ..
```
```  1414 qed
```
```  1415
```
```  1416
```
```  1417 subsection \<open>Natural Logarithm\<close>
```
```  1418
```
```  1419 class ln = real_normed_algebra_1 + banach +
```
```  1420   fixes ln :: "'a \<Rightarrow> 'a"
```
```  1421   assumes ln_one [simp]: "ln 1 = 0"
```
```  1422
```
```  1423 definition powr :: "['a,'a] => 'a::ln"     (infixr "powr" 80)
```
```  1424   -- \<open>exponentation via ln and exp\<close>
```
```  1425   where  [code del]: "x powr a \<equiv> if x = 0 then 0 else exp(a * ln x)"
```
```  1426
```
```  1427 lemma powr_0 [simp]: "0 powr z = 0"
```
```  1428   by (simp add: powr_def)
```
```  1429
```
```  1430
```
```  1431 instantiation real :: ln
```
```  1432 begin
```
```  1433
```
```  1434 definition ln_real :: "real \<Rightarrow> real"
```
```  1435   where "ln_real x = (THE u. exp u = x)"
```
```  1436
```
```  1437 instance
```
```  1438 by intro_classes (simp add: ln_real_def)
```
```  1439
```
```  1440 end
```
```  1441
```
```  1442 lemma powr_eq_0_iff [simp]: "w powr z = 0 \<longleftrightarrow> w = 0"
```
```  1443   by (simp add: powr_def)
```
```  1444
```
```  1445 lemma ln_exp [simp]:
```
```  1446   fixes x::real shows "ln (exp x) = x"
```
```  1447   by (simp add: ln_real_def)
```
```  1448
```
```  1449 lemma exp_ln [simp]:
```
```  1450   fixes x::real shows "0 < x \<Longrightarrow> exp (ln x) = x"
```
```  1451   by (auto dest: exp_total)
```
```  1452
```
```  1453 lemma exp_ln_iff [simp]:
```
```  1454   fixes x::real shows "exp (ln x) = x \<longleftrightarrow> 0 < x"
```
```  1455   by (metis exp_gt_zero exp_ln)
```
```  1456
```
```  1457 lemma ln_unique:
```
```  1458   fixes x::real shows "exp y = x \<Longrightarrow> ln x = y"
```
```  1459   by (erule subst, rule ln_exp)
```
```  1460
```
```  1461 lemma ln_mult:
```
```  1462   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
```
```  1463   by (rule ln_unique) (simp add: exp_add)
```
```  1464
```
```  1465 lemma ln_setprod:
```
```  1466   fixes f:: "'a => real"
```
```  1467   shows
```
```  1468     "\<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"
```
```  1469   by (induction I rule: finite_induct) (auto simp: ln_mult setprod_pos)
```
```  1470
```
```  1471 lemma ln_inverse:
```
```  1472   fixes x::real shows "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
```
```  1473   by (rule ln_unique) (simp add: exp_minus)
```
```  1474
```
```  1475 lemma ln_div:
```
```  1476   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
```
```  1477   by (rule ln_unique) (simp add: exp_diff)
```
```  1478
```
```  1479 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x^n) = real n * ln x"
```
```  1480   by (rule ln_unique) (simp add: exp_real_of_nat_mult)
```
```  1481
```
```  1482 lemma ln_less_cancel_iff [simp]:
```
```  1483   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
```
```  1484   by (subst exp_less_cancel_iff [symmetric]) simp
```
```  1485
```
```  1486 lemma ln_le_cancel_iff [simp]:
```
```  1487   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
```
```  1488   by (simp add: linorder_not_less [symmetric])
```
```  1489
```
```  1490 lemma ln_inj_iff [simp]:
```
```  1491   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
```
```  1492   by (simp add: order_eq_iff)
```
```  1493
```
```  1494 lemma ln_add_one_self_le_self [simp]:
```
```  1495   fixes x::real shows "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
```
```  1496   apply (rule exp_le_cancel_iff [THEN iffD1])
```
```  1497   apply (simp add: exp_ge_add_one_self_aux)
```
```  1498   done
```
```  1499
```
```  1500 lemma ln_less_self [simp]:
```
```  1501   fixes x::real shows "0 < x \<Longrightarrow> ln x < x"
```
```  1502   by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
```
```  1503
```
```  1504 lemma ln_ge_zero [simp]:
```
```  1505   fixes x::real shows "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
```
```  1506   using ln_le_cancel_iff [of 1 x] by simp
```
```  1507
```
```  1508 lemma ln_ge_zero_imp_ge_one:
```
```  1509   fixes x::real shows "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
```
```  1510   using ln_le_cancel_iff [of 1 x] by simp
```
```  1511
```
```  1512 lemma ln_ge_zero_iff [simp]:
```
```  1513   fixes x::real shows "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
```
```  1514   using ln_le_cancel_iff [of 1 x] by simp
```
```  1515
```
```  1516 lemma ln_less_zero_iff [simp]:
```
```  1517   fixes x::real shows "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
```
```  1518   using ln_less_cancel_iff [of x 1] by simp
```
```  1519
```
```  1520 lemma ln_gt_zero:
```
```  1521   fixes x::real shows "1 < x \<Longrightarrow> 0 < ln x"
```
```  1522   using ln_less_cancel_iff [of 1 x] by simp
```
```  1523
```
```  1524 lemma ln_gt_zero_imp_gt_one:
```
```  1525   fixes x::real shows "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
```
```  1526   using ln_less_cancel_iff [of 1 x] by simp
```
```  1527
```
```  1528 lemma ln_gt_zero_iff [simp]:
```
```  1529   fixes x::real shows "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
```
```  1530   using ln_less_cancel_iff [of 1 x] by simp
```
```  1531
```
```  1532 lemma ln_eq_zero_iff [simp]:
```
```  1533   fixes x::real shows "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
```
```  1534   using ln_inj_iff [of x 1] by simp
```
```  1535
```
```  1536 lemma ln_less_zero:
```
```  1537   fixes x::real shows "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
```
```  1538   by simp
```
```  1539
```
```  1540 lemma ln_neg_is_const:
```
```  1541   fixes x::real shows "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)"
```
```  1542   by (auto simp add: ln_real_def intro!: arg_cong[where f=The])
```
```  1543
```
```  1544 lemma isCont_ln:
```
```  1545   fixes x::real assumes "x \<noteq> 0" shows "isCont ln x"
```
```  1546 proof cases
```
```  1547   assume "0 < x"
```
```  1548   moreover then have "isCont ln (exp (ln x))"
```
```  1549     by (intro isCont_inv_fun[where d="\<bar>x\<bar>" and f=exp]) auto
```
```  1550   ultimately show ?thesis
```
```  1551     by simp
```
```  1552 next
```
```  1553   assume "\<not> 0 < x" with \<open>x \<noteq> 0\<close> show "isCont ln x"
```
```  1554     unfolding isCont_def
```
```  1555     by (subst filterlim_cong[OF _ refl, of _ "nhds (ln 0)" _ "\<lambda>_. ln 0"])
```
```  1556        (auto simp: ln_neg_is_const not_less eventually_at dist_real_def
```
```  1557                 intro!: exI[of _ "\<bar>x\<bar>"])
```
```  1558 qed
```
```  1559
```
```  1560 lemma tendsto_ln [tendsto_intros]:
```
```  1561   fixes a::real shows
```
```  1562   "(f ---> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
```
```  1563   by (rule isCont_tendsto_compose [OF isCont_ln])
```
```  1564
```
```  1565 lemma continuous_ln:
```
```  1566   "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. ln (f x :: real))"
```
```  1567   unfolding continuous_def by (rule tendsto_ln)
```
```  1568
```
```  1569 lemma isCont_ln' [continuous_intros]:
```
```  1570   "continuous (at x) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x :: real))"
```
```  1571   unfolding continuous_at by (rule tendsto_ln)
```
```  1572
```
```  1573 lemma continuous_within_ln [continuous_intros]:
```
```  1574   "continuous (at x within s) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x :: real))"
```
```  1575   unfolding continuous_within by (rule tendsto_ln)
```
```  1576
```
```  1577 lemma continuous_on_ln [continuous_intros]:
```
```  1578   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. f x \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x :: real))"
```
```  1579   unfolding continuous_on_def by (auto intro: tendsto_ln)
```
```  1580
```
```  1581 lemma DERIV_ln:
```
```  1582   fixes x::real shows "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
```
```  1583   apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
```
```  1584   apply (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln)
```
```  1585   done
```
```  1586
```
```  1587 lemma DERIV_ln_divide:
```
```  1588   fixes x::real shows "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
```
```  1589   by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
```
```  1590
```
```  1591 declare DERIV_ln_divide[THEN DERIV_chain2, derivative_intros]
```
```  1592
```
```  1593 lemma ln_series:
```
```  1594   assumes "0 < x" and "x < 2"
```
```  1595   shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))"
```
```  1596   (is "ln x = suminf (?f (x - 1))")
```
```  1597 proof -
```
```  1598   let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n"
```
```  1599
```
```  1600   have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
```
```  1601   proof (rule DERIV_isconst3[where x=x])
```
```  1602     fix x :: real
```
```  1603     assume "x \<in> {0 <..< 2}"
```
```  1604     hence "0 < x" and "x < 2" by auto
```
```  1605     have "norm (1 - x) < 1"
```
```  1606       using \<open>0 < x\<close> and \<open>x < 2\<close> by auto
```
```  1607     have "1 / x = 1 / (1 - (1 - x))" by auto
```
```  1608     also have "\<dots> = (\<Sum> n. (1 - x)^n)"
```
```  1609       using geometric_sums[OF \<open>norm (1 - x) < 1\<close>] by (rule sums_unique)
```
```  1610     also have "\<dots> = suminf (?f' x)"
```
```  1611       unfolding power_mult_distrib[symmetric]
```
```  1612       by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
```
```  1613     finally have "DERIV ln x :> suminf (?f' x)"
```
```  1614       using DERIV_ln[OF \<open>0 < x\<close>] unfolding divide_inverse by auto
```
```  1615     moreover
```
```  1616     have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
```
```  1617     have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :>
```
```  1618       (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
```
```  1619     proof (rule DERIV_power_series')
```
```  1620       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1"
```
```  1621         using \<open>0 < x\<close> \<open>x < 2\<close> by auto
```
```  1622       fix x :: real
```
```  1623       assume "x \<in> {- 1<..<1}"
```
```  1624       hence "norm (-x) < 1" by auto
```
```  1625       show "summable (\<lambda>n. (- 1) ^ n * (1 / real (n + 1)) * real (Suc n) * x^n)"
```
```  1626         unfolding One_nat_def
```
```  1627         by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF \<open>norm (-x) < 1\<close>])
```
```  1628     qed
```
```  1629     hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
```
```  1630       unfolding One_nat_def by auto
```
```  1631     hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
```
```  1632       unfolding DERIV_def repos .
```
```  1633     ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
```
```  1634       by (rule DERIV_diff)
```
```  1635     thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
```
```  1636   qed (auto simp add: assms)
```
```  1637   thus ?thesis by auto
```
```  1638 qed
```
```  1639
```
```  1640 lemma exp_first_two_terms:
```
```  1641   fixes x :: "'a::{real_normed_field,banach}"
```
```  1642   shows "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) * (x ^ (n+2)))"
```
```  1643 proof -
```
```  1644   have "exp x = suminf (\<lambda>n. inverse(fact n) * (x^n))"
```
```  1645     by (simp add: exp_def scaleR_conv_of_real nonzero_of_real_inverse)
```
```  1646   also from summable_exp have "... = (\<Sum> n. inverse(fact(n+2)) * (x ^ (n+2))) +
```
```  1647     (\<Sum> n::nat<2. inverse(fact n) * (x^n))" (is "_ = _ + ?a")
```
```  1648     by (rule suminf_split_initial_segment)
```
```  1649   also have "?a = 1 + x"
```
```  1650     by (simp add: numeral_2_eq_2)
```
```  1651   finally show ?thesis
```
```  1652     by simp
```
```  1653 qed
```
```  1654
```
```  1655 lemma exp_bound: "0 <= (x::real) \<Longrightarrow> x <= 1 \<Longrightarrow> exp x <= 1 + x + x\<^sup>2"
```
```  1656 proof -
```
```  1657   assume a: "0 <= x"
```
```  1658   assume b: "x <= 1"
```
```  1659   {
```
```  1660     fix n :: nat
```
```  1661     have "(2::nat) * 2 ^ n \<le> fact (n + 2)"
```
```  1662       by (induct n) simp_all
```
```  1663     hence "real ((2::nat) * 2 ^ n) \<le> real_of_nat (fact (n + 2))"
```
```  1664       by (simp only: real_of_nat_le_iff)
```
```  1665     hence "((2::real) * 2 ^ n) \<le> fact (n + 2)"
```
```  1666       unfolding of_nat_fact real_of_nat_def
```
```  1667       by (simp add: of_nat_mult of_nat_power)
```
```  1668     hence "inverse (fact (n + 2)) \<le> inverse ((2::real) * 2 ^ n)"
```
```  1669       by (rule le_imp_inverse_le) simp
```
```  1670     hence "inverse (fact (n + 2)) \<le> 1/(2::real) * (1/2)^n"
```
```  1671       by (simp add: power_inverse [symmetric])
```
```  1672     hence "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
```
```  1673       by (rule mult_mono)
```
```  1674         (rule mult_mono, simp_all add: power_le_one a b)
```
```  1675     hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)"
```
```  1676       unfolding power_add by (simp add: ac_simps del: fact.simps) }
```
```  1677   note aux1 = this
```
```  1678   have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
```
```  1679     by (intro sums_mult geometric_sums, simp)
```
```  1680   hence aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
```
```  1681     by simp
```
```  1682   have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2"
```
```  1683   proof -
```
```  1684     have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <=
```
```  1685         suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
```
```  1686       apply (rule suminf_le)
```
```  1687       apply (rule allI, rule aux1)
```
```  1688       apply (rule summable_exp [THEN summable_ignore_initial_segment])
```
```  1689       by (rule sums_summable, rule aux2)
```
```  1690     also have "... = x\<^sup>2"
```
```  1691       by (rule sums_unique [THEN sym], rule aux2)
```
```  1692     finally show ?thesis .
```
```  1693   qed
```
```  1694   thus ?thesis unfolding exp_first_two_terms by auto
```
```  1695 qed
```
```  1696
```
```  1697 corollary exp_half_le2: "exp(1/2) \<le> (2::real)"
```
```  1698   using exp_bound [of "1/2"]
```
```  1699   by (simp add: field_simps)
```
```  1700
```
```  1701 corollary exp_le: "exp 1 \<le> (3::real)"
```
```  1702   using exp_bound [of 1]
```
```  1703   by (simp add: field_simps)
```
```  1704
```
```  1705 lemma exp_bound_half: "norm(z) \<le> 1/2 \<Longrightarrow> norm(exp z) \<le> 2"
```
```  1706   by (blast intro: order_trans intro!: exp_half_le2 norm_exp)
```
```  1707
```
```  1708 lemma exp_bound_lemma:
```
```  1709   assumes "norm(z) \<le> 1/2" shows "norm(exp z) \<le> 1 + 2 * norm(z)"
```
```  1710 proof -
```
```  1711   have n: "(norm z)\<^sup>2 \<le> norm z * 1"
```
```  1712     unfolding power2_eq_square
```
```  1713     apply (rule mult_left_mono)
```
```  1714     using assms
```
```  1715     apply auto
```
```  1716     done
```
```  1717   show ?thesis
```
```  1718     apply (rule order_trans [OF norm_exp])
```
```  1719     apply (rule order_trans [OF exp_bound])
```
```  1720     using assms n
```
```  1721     apply auto
```
```  1722     done
```
```  1723 qed
```
```  1724
```
```  1725 lemma real_exp_bound_lemma:
```
```  1726   fixes x :: real
```
```  1727   shows "0 \<le> x \<Longrightarrow> x \<le> 1/2 \<Longrightarrow> exp(x) \<le> 1 + 2 * x"
```
```  1728 using exp_bound_lemma [of x]
```
```  1729 by simp
```
```  1730
```
```  1731 lemma ln_one_minus_pos_upper_bound:
```
```  1732   fixes x::real shows "0 <= x \<Longrightarrow> x < 1 \<Longrightarrow> ln (1 - x) <= - x"
```
```  1733 proof -
```
```  1734   assume a: "0 <= (x::real)" and b: "x < 1"
```
```  1735   have "(1 - x) * (1 + x + x\<^sup>2) = (1 - x^3)"
```
```  1736     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
```
```  1737   also have "... <= 1"
```
```  1738     by (auto simp add: a)
```
```  1739   finally have "(1 - x) * (1 + x + x\<^sup>2) <= 1" .
```
```  1740   moreover have c: "0 < 1 + x + x\<^sup>2"
```
```  1741     by (simp add: add_pos_nonneg a)
```
```  1742   ultimately have "1 - x <= 1 / (1 + x + x\<^sup>2)"
```
```  1743     by (elim mult_imp_le_div_pos)
```
```  1744   also have "... <= 1 / exp x"
```
```  1745     by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs
```
```  1746               real_sqrt_pow2_iff real_sqrt_power)
```
```  1747   also have "... = exp (-x)"
```
```  1748     by (auto simp add: exp_minus divide_inverse)
```
```  1749   finally have "1 - x <= exp (- x)" .
```
```  1750   also have "1 - x = exp (ln (1 - x))"
```
```  1751     by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq)
```
```  1752   finally have "exp (ln (1 - x)) <= exp (- x)" .
```
```  1753   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```  1754 qed
```
```  1755
```
```  1756 lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
```
```  1757   apply (case_tac "0 <= x")
```
```  1758   apply (erule exp_ge_add_one_self_aux)
```
```  1759   apply (case_tac "x <= -1")
```
```  1760   apply (subgoal_tac "1 + x <= 0")
```
```  1761   apply (erule order_trans)
```
```  1762   apply simp
```
```  1763   apply simp
```
```  1764   apply (subgoal_tac "1 + x = exp(ln (1 + x))")
```
```  1765   apply (erule ssubst)
```
```  1766   apply (subst exp_le_cancel_iff)
```
```  1767   apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
```
```  1768   apply simp
```
```  1769   apply (rule ln_one_minus_pos_upper_bound)
```
```  1770   apply auto
```
```  1771 done
```
```  1772
```
```  1773 lemma ln_one_plus_pos_lower_bound:
```
```  1774   fixes x::real shows "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> x - x\<^sup>2 <= ln (1 + x)"
```
```  1775 proof -
```
```  1776   assume a: "0 <= x" and b: "x <= 1"
```
```  1777   have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
```
```  1778     by (rule exp_diff)
```
```  1779   also have "... <= (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
```
```  1780     by (metis a b divide_right_mono exp_bound exp_ge_zero)
```
```  1781   also have "... <= (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
```
```  1782     by (simp add: a divide_left_mono add_pos_nonneg)
```
```  1783   also from a have "... <= 1 + x"
```
```  1784     by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
```
```  1785   finally have "exp (x - x\<^sup>2) <= 1 + x" .
```
```  1786   also have "... = exp (ln (1 + x))"
```
```  1787   proof -
```
```  1788     from a have "0 < 1 + x" by auto
```
```  1789     thus ?thesis
```
```  1790       by (auto simp only: exp_ln_iff [THEN sym])
```
```  1791   qed
```
```  1792   finally have "exp (x - x\<^sup>2) <= exp (ln (1 + x))" .
```
```  1793   thus ?thesis
```
```  1794     by (metis exp_le_cancel_iff)
```
```  1795 qed
```
```  1796
```
```  1797 lemma ln_one_minus_pos_lower_bound:
```
```  1798   fixes x::real
```
```  1799   shows "0 <= x \<Longrightarrow> x <= (1 / 2) \<Longrightarrow> - x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1800 proof -
```
```  1801   assume a: "0 <= x" and b: "x <= (1 / 2)"
```
```  1802   from b have c: "x < 1" by auto
```
```  1803   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
```
```  1804     apply (subst ln_inverse [symmetric])
```
```  1805     apply (simp add: field_simps)
```
```  1806     apply (rule arg_cong [where f=ln])
```
```  1807     apply (simp add: field_simps)
```
```  1808     done
```
```  1809   also have "- (x / (1 - x)) <= ..."
```
```  1810   proof -
```
```  1811     have "ln (1 + x / (1 - x)) <= x / (1 - x)"
```
```  1812       using a c by (intro ln_add_one_self_le_self) auto
```
```  1813     thus ?thesis
```
```  1814       by auto
```
```  1815   qed
```
```  1816   also have "- (x / (1 - x)) = -x / (1 - x)"
```
```  1817     by auto
```
```  1818   finally have d: "- x / (1 - x) <= ln (1 - x)" .
```
```  1819   have "0 < 1 - x" using a b by simp
```
```  1820   hence e: "-x - 2 * x\<^sup>2 <= - x / (1 - x)"
```
```  1821     using mult_right_le_one_le[of "x*x" "2*x"] a b
```
```  1822     by (simp add: field_simps power2_eq_square)
```
```  1823   from e d show "- x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1824     by (rule order_trans)
```
```  1825 qed
```
```  1826
```
```  1827 lemma ln_add_one_self_le_self2:
```
```  1828   fixes x::real shows "-1 < x \<Longrightarrow> ln(1 + x) <= x"
```
```  1829   apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
```
```  1830   apply (subst ln_le_cancel_iff)
```
```  1831   apply auto
```
```  1832   done
```
```  1833
```
```  1834 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
```
```  1835   fixes x::real shows "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> abs(ln (1 + x) - x) <= x\<^sup>2"
```
```  1836 proof -
```
```  1837   assume x: "0 <= x"
```
```  1838   assume x1: "x <= 1"
```
```  1839   from x have "ln (1 + x) <= x"
```
```  1840     by (rule ln_add_one_self_le_self)
```
```  1841   then have "ln (1 + x) - x <= 0"
```
```  1842     by simp
```
```  1843   then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
```
```  1844     by (rule abs_of_nonpos)
```
```  1845   also have "... = x - ln (1 + x)"
```
```  1846     by simp
```
```  1847   also have "... <= x\<^sup>2"
```
```  1848   proof -
```
```  1849     from x x1 have "x - x\<^sup>2 <= ln (1 + x)"
```
```  1850       by (intro ln_one_plus_pos_lower_bound)
```
```  1851     thus ?thesis
```
```  1852       by simp
```
```  1853   qed
```
```  1854   finally show ?thesis .
```
```  1855 qed
```
```  1856
```
```  1857 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
```
```  1858   fixes x::real shows "-(1 / 2) <= x \<Longrightarrow> x <= 0 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
```
```  1859 proof -
```
```  1860   assume a: "-(1 / 2) <= x"
```
```  1861   assume b: "x <= 0"
```
```  1862   have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
```
```  1863     apply (subst abs_of_nonpos)
```
```  1864     apply simp
```
```  1865     apply (rule ln_add_one_self_le_self2)
```
```  1866     using a apply auto
```
```  1867     done
```
```  1868   also have "... <= 2 * x\<^sup>2"
```
```  1869     apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 <= ln (1 - (-x))")
```
```  1870     apply (simp add: algebra_simps)
```
```  1871     apply (rule ln_one_minus_pos_lower_bound)
```
```  1872     using a b apply auto
```
```  1873     done
```
```  1874   finally show ?thesis .
```
```  1875 qed
```
```  1876
```
```  1877 lemma abs_ln_one_plus_x_minus_x_bound:
```
```  1878   fixes x::real shows "abs x <= 1 / 2 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
```
```  1879   apply (case_tac "0 <= x")
```
```  1880   apply (rule order_trans)
```
```  1881   apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
```
```  1882   apply auto
```
```  1883   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
```
```  1884   apply auto
```
```  1885   done
```
```  1886
```
```  1887 lemma ln_x_over_x_mono:
```
```  1888   fixes x::real shows "exp 1 <= x \<Longrightarrow> x <= y \<Longrightarrow> (ln y / y) <= (ln x / x)"
```
```  1889 proof -
```
```  1890   assume x: "exp 1 <= x" "x <= y"
```
```  1891   moreover have "0 < exp (1::real)" by simp
```
```  1892   ultimately have a: "0 < x" and b: "0 < y"
```
```  1893     by (fast intro: less_le_trans order_trans)+
```
```  1894   have "x * ln y - x * ln x = x * (ln y - ln x)"
```
```  1895     by (simp add: algebra_simps)
```
```  1896   also have "... = x * ln(y / x)"
```
```  1897     by (simp only: ln_div a b)
```
```  1898   also have "y / x = (x + (y - x)) / x"
```
```  1899     by simp
```
```  1900   also have "... = 1 + (y - x) / x"
```
```  1901     using x a by (simp add: field_simps)
```
```  1902   also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
```
```  1903     using x a
```
```  1904     by (intro mult_left_mono ln_add_one_self_le_self) simp_all
```
```  1905   also have "... = y - x" using a by simp
```
```  1906   also have "... = (y - x) * ln (exp 1)" by simp
```
```  1907   also have "... <= (y - x) * ln x"
```
```  1908     apply (rule mult_left_mono)
```
```  1909     apply (subst ln_le_cancel_iff)
```
```  1910     apply fact
```
```  1911     apply (rule a)
```
```  1912     apply (rule x)
```
```  1913     using x apply simp
```
```  1914     done
```
```  1915   also have "... = y * ln x - x * ln x"
```
```  1916     by (rule left_diff_distrib)
```
```  1917   finally have "x * ln y <= y * ln x"
```
```  1918     by arith
```
```  1919   then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
```
```  1920   also have "... = y * (ln x / x)" by simp
```
```  1921   finally show ?thesis using b by (simp add: field_simps)
```
```  1922 qed
```
```  1923
```
```  1924 lemma ln_le_minus_one:
```
```  1925   fixes x::real shows "0 < x \<Longrightarrow> ln x \<le> x - 1"
```
```  1926   using exp_ge_add_one_self[of "ln x"] by simp
```
```  1927
```
```  1928 corollary ln_diff_le:
```
```  1929   fixes x::real
```
```  1930   shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x - ln y \<le> (x - y) / y"
```
```  1931   by (simp add: ln_div [symmetric] diff_divide_distrib ln_le_minus_one)
```
```  1932
```
```  1933 lemma ln_eq_minus_one:
```
```  1934   fixes x::real
```
```  1935   assumes "0 < x" "ln x = x - 1"
```
```  1936   shows "x = 1"
```
```  1937 proof -
```
```  1938   let ?l = "\<lambda>y. ln y - y + 1"
```
```  1939   have D: "\<And>x::real. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
```
```  1940     by (auto intro!: derivative_eq_intros)
```
```  1941
```
```  1942   show ?thesis
```
```  1943   proof (cases rule: linorder_cases)
```
```  1944     assume "x < 1"
```
```  1945     from dense[OF \<open>x < 1\<close>] obtain a where "x < a" "a < 1" by blast
```
```  1946     from \<open>x < a\<close> have "?l x < ?l a"
```
```  1947     proof (rule DERIV_pos_imp_increasing, safe)
```
```  1948       fix y
```
```  1949       assume "x \<le> y" "y \<le> a"
```
```  1950       with \<open>0 < x\<close> \<open>a < 1\<close> have "0 < 1 / y - 1" "0 < y"
```
```  1951         by (auto simp: field_simps)
```
```  1952       with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
```
```  1953         by auto
```
```  1954     qed
```
```  1955     also have "\<dots> \<le> 0"
```
```  1956       using ln_le_minus_one \<open>0 < x\<close> \<open>x < a\<close> by (auto simp: field_simps)
```
```  1957     finally show "x = 1" using assms by auto
```
```  1958   next
```
```  1959     assume "1 < x"
```
```  1960     from dense[OF this] obtain a where "1 < a" "a < x" by blast
```
```  1961     from \<open>a < x\<close> have "?l x < ?l a"
```
```  1962     proof (rule DERIV_neg_imp_decreasing, safe)
```
```  1963       fix y
```
```  1964       assume "a \<le> y" "y \<le> x"
```
```  1965       with \<open>1 < a\<close> have "1 / y - 1 < 0" "0 < y"
```
```  1966         by (auto simp: field_simps)
```
```  1967       with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
```
```  1968         by blast
```
```  1969     qed
```
```  1970     also have "\<dots> \<le> 0"
```
```  1971       using ln_le_minus_one \<open>1 < a\<close> by (auto simp: field_simps)
```
```  1972     finally show "x = 1" using assms by auto
```
```  1973   next
```
```  1974     assume "x = 1"
```
```  1975     then show ?thesis by simp
```
```  1976   qed
```
```  1977 qed
```
```  1978
```
```  1979 lemma exp_at_bot: "(exp ---> (0::real)) at_bot"
```
```  1980   unfolding tendsto_Zfun_iff
```
```  1981 proof (rule ZfunI, simp add: eventually_at_bot_dense)
```
```  1982   fix r :: real assume "0 < r"
```
```  1983   {
```
```  1984     fix x
```
```  1985     assume "x < ln r"
```
```  1986     then have "exp x < exp (ln r)"
```
```  1987       by simp
```
```  1988     with \<open>0 < r\<close> have "exp x < r"
```
```  1989       by simp
```
```  1990   }
```
```  1991   then show "\<exists>k. \<forall>n<k. exp n < r" by auto
```
```  1992 qed
```
```  1993
```
```  1994 lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
```
```  1995   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"])
```
```  1996      (auto intro: eventually_gt_at_top)
```
```  1997
```
```  1998 lemma lim_exp_minus_1:
```
```  1999   fixes x :: "'a::{real_normed_field,banach}"
```
```  2000   shows "((\<lambda>z::'a. (exp(z) - 1) / z) ---> 1) (at 0)"
```
```  2001 proof -
```
```  2002   have "((\<lambda>z::'a. exp(z) - 1) has_field_derivative 1) (at 0)"
```
```  2003     by (intro derivative_eq_intros | simp)+
```
```  2004   then show ?thesis
```
```  2005     by (simp add: Deriv.DERIV_iff2)
```
```  2006 qed
```
```  2007
```
```  2008 lemma ln_at_0: "LIM x at_right 0. ln (x::real) :> at_bot"
```
```  2009   by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  2010      (auto simp: eventually_at_filter)
```
```  2011
```
```  2012 lemma ln_at_top: "LIM x at_top. ln (x::real) :> at_top"
```
```  2013   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  2014      (auto intro: eventually_gt_at_top)
```
```  2015
```
```  2016 lemma filtermap_ln_at_top: "filtermap (ln::real \<Rightarrow> real) at_top = at_top"
```
```  2017   by (intro filtermap_fun_inverse[of exp] exp_at_top ln_at_top) auto
```
```  2018
```
```  2019 lemma filtermap_exp_at_top: "filtermap (exp::real \<Rightarrow> real) at_top = at_top"
```
```  2020   by (intro filtermap_fun_inverse[of ln] exp_at_top ln_at_top)
```
```  2021      (auto simp: eventually_at_top_dense)
```
```  2022
```
```  2023 lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) ---> (0::real)) at_top"
```
```  2024 proof (induct k)
```
```  2025   case 0
```
```  2026   show "((\<lambda>x. x ^ 0 / exp x) ---> (0::real)) at_top"
```
```  2027     by (simp add: inverse_eq_divide[symmetric])
```
```  2028        (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
```
```  2029               at_top_le_at_infinity order_refl)
```
```  2030 next
```
```  2031   case (Suc k)
```
```  2032   show ?case
```
```  2033   proof (rule lhospital_at_top_at_top)
```
```  2034     show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"
```
```  2035       by eventually_elim (intro derivative_eq_intros, auto)
```
```  2036     show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top"
```
```  2037       by eventually_elim auto
```
```  2038     show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
```
```  2039       by auto
```
```  2040     from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
```
```  2041     show "((\<lambda>x. real (Suc k) * x ^ k / exp x) ---> 0) at_top"
```
```  2042       by simp
```
```  2043   qed (rule exp_at_top)
```
```  2044 qed
```
```  2045
```
```  2046
```
```  2047 definition log :: "[real,real] => real"
```
```  2048   -- \<open>logarithm of @{term x} to base @{term a}\<close>
```
```  2049   where "log a x = ln x / ln a"
```
```  2050
```
```  2051
```
```  2052 lemma tendsto_log [tendsto_intros]:
```
```  2053   "\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a; a \<noteq> 1; 0 < b\<rbrakk> \<Longrightarrow> ((\<lambda>x. log (f x) (g x)) ---> log a b) F"
```
```  2054   unfolding log_def by (intro tendsto_intros) auto
```
```  2055
```
```  2056 lemma continuous_log:
```
```  2057   assumes "continuous F f"
```
```  2058     and "continuous F g"
```
```  2059     and "0 < f (Lim F (\<lambda>x. x))"
```
```  2060     and "f (Lim F (\<lambda>x. x)) \<noteq> 1"
```
```  2061     and "0 < g (Lim F (\<lambda>x. x))"
```
```  2062   shows "continuous F (\<lambda>x. log (f x) (g x))"
```
```  2063   using assms unfolding continuous_def by (rule tendsto_log)
```
```  2064
```
```  2065 lemma continuous_at_within_log[continuous_intros]:
```
```  2066   assumes "continuous (at a within s) f"
```
```  2067     and "continuous (at a within s) g"
```
```  2068     and "0 < f a"
```
```  2069     and "f a \<noteq> 1"
```
```  2070     and "0 < g a"
```
```  2071   shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
```
```  2072   using assms unfolding continuous_within by (rule tendsto_log)
```
```  2073
```
```  2074 lemma isCont_log[continuous_intros, simp]:
```
```  2075   assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
```
```  2076   shows "isCont (\<lambda>x. log (f x) (g x)) a"
```
```  2077   using assms unfolding continuous_at by (rule tendsto_log)
```
```  2078
```
```  2079 lemma continuous_on_log[continuous_intros]:
```
```  2080   assumes "continuous_on s f" "continuous_on s g"
```
```  2081     and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
```
```  2082   shows "continuous_on s (\<lambda>x. log (f x) (g x))"
```
```  2083   using assms unfolding continuous_on_def by (fast intro: tendsto_log)
```
```  2084
```
```  2085 lemma powr_one_eq_one [simp]: "1 powr a = 1"
```
```  2086   by (simp add: powr_def)
```
```  2087
```
```  2088 lemma powr_zero_eq_one [simp]: "x powr 0 = (if x=0 then 0 else 1)"
```
```  2089   by (simp add: powr_def)
```
```  2090
```
```  2091 lemma powr_one_gt_zero_iff [simp]:
```
```  2092   fixes x::real shows "(x powr 1 = x) = (0 \<le> x)"
```
```  2093   by (auto simp: powr_def)
```
```  2094 declare powr_one_gt_zero_iff [THEN iffD2, simp]
```
```  2095
```
```  2096 lemma powr_mult:
```
```  2097   fixes x::real shows "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
```
```  2098   by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
```
```  2099
```
```  2100 lemma powr_ge_pzero [simp]:
```
```  2101   fixes x::real shows "0 <= x powr y"
```
```  2102   by (simp add: powr_def)
```
```  2103
```
```  2104 lemma powr_divide:
```
```  2105   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
```
```  2106   apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
```
```  2107   apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
```
```  2108   done
```
```  2109
```
```  2110 lemma powr_divide2:
```
```  2111   fixes x::real shows "x powr a / x powr b = x powr (a - b)"
```
```  2112   apply (simp add: powr_def)
```
```  2113   apply (subst exp_diff [THEN sym])
```
```  2114   apply (simp add: left_diff_distrib)
```
```  2115   done
```
```  2116
```
```  2117 lemma powr_add:
```
```  2118   fixes x::real shows "x powr (a + b) = (x powr a) * (x powr b)"
```
```  2119   by (simp add: powr_def exp_add [symmetric] distrib_right)
```
```  2120
```
```  2121 lemma powr_mult_base:
```
```  2122   fixes x::real shows "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
```
```  2123   using assms by (auto simp: powr_add)
```
```  2124
```
```  2125 lemma powr_powr:
```
```  2126   fixes x::real shows "(x powr a) powr b = x powr (a * b)"
```
```  2127   by (simp add: powr_def)
```
```  2128
```
```  2129 lemma powr_powr_swap:
```
```  2130   fixes x::real shows "(x powr a) powr b = (x powr b) powr a"
```
```  2131   by (simp add: powr_powr mult.commute)
```
```  2132
```
```  2133 lemma powr_minus:
```
```  2134   fixes x::real shows "x powr (-a) = inverse (x powr a)"
```
```  2135   by (simp add: powr_def exp_minus [symmetric])
```
```  2136
```
```  2137 lemma powr_minus_divide:
```
```  2138   fixes x::real shows "x powr (-a) = 1/(x powr a)"
```
```  2139   by (simp add: divide_inverse powr_minus)
```
```  2140
```
```  2141 lemma divide_powr_uminus:
```
```  2142   fixes a::real shows "a / b powr c = a * b powr (- c)"
```
```  2143   by (simp add: powr_minus_divide)
```
```  2144
```
```  2145 lemma powr_less_mono:
```
```  2146   fixes x::real shows "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
```
```  2147   by (simp add: powr_def)
```
```  2148
```
```  2149 lemma powr_less_cancel:
```
```  2150   fixes x::real shows "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
```
```  2151   by (simp add: powr_def)
```
```  2152
```
```  2153 lemma powr_less_cancel_iff [simp]:
```
```  2154   fixes x::real shows "1 < x \<Longrightarrow> (x powr a < x powr b) = (a < b)"
```
```  2155   by (blast intro: powr_less_cancel powr_less_mono)
```
```  2156
```
```  2157 lemma powr_le_cancel_iff [simp]:
```
```  2158   fixes x::real shows "1 < x \<Longrightarrow> (x powr a \<le> x powr b) = (a \<le> b)"
```
```  2159   by (simp add: linorder_not_less [symmetric])
```
```  2160
```
```  2161 lemma log_ln: "ln x = log (exp(1)) x"
```
```  2162   by (simp add: log_def)
```
```  2163
```
```  2164 lemma DERIV_log:
```
```  2165   assumes "x > 0"
```
```  2166   shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
```
```  2167 proof -
```
```  2168   def lb \<equiv> "1 / ln b"
```
```  2169   moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
```
```  2170     using \<open>x > 0\<close> by (auto intro!: derivative_eq_intros)
```
```  2171   ultimately show ?thesis
```
```  2172     by (simp add: log_def)
```
```  2173 qed
```
```  2174
```
```  2175 lemmas DERIV_log[THEN DERIV_chain2, derivative_intros]
```
```  2176
```
```  2177 lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
```
```  2178   by (simp add: powr_def log_def)
```
```  2179
```
```  2180 lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y"
```
```  2181   by (simp add: log_def powr_def)
```
```  2182
```
```  2183 lemma log_mult:
```
```  2184   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow>
```
```  2185     log a (x * y) = log a x + log a y"
```
```  2186   by (simp add: log_def ln_mult divide_inverse distrib_right)
```
```  2187
```
```  2188 lemma log_eq_div_ln_mult_log:
```
```  2189   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow>
```
```  2190     log a x = (ln b/ln a) * log b x"
```
```  2191   by (simp add: log_def divide_inverse)
```
```  2192
```
```  2193 text\<open>Base 10 logarithms\<close>
```
```  2194 lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x"
```
```  2195   by (simp add: log_def)
```
```  2196
```
```  2197 lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x"
```
```  2198   by (simp add: log_def)
```
```  2199
```
```  2200 lemma log_one [simp]: "log a 1 = 0"
```
```  2201   by (simp add: log_def)
```
```  2202
```
```  2203 lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
```
```  2204   by (simp add: log_def)
```
```  2205
```
```  2206 lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x"
```
```  2207   apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
```
```  2208   apply (simp add: log_mult [symmetric])
```
```  2209   done
```
```  2210
```
```  2211 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"
```
```  2212   by (simp add: log_mult divide_inverse log_inverse)
```
```  2213
```
```  2214 lemma powr_gt_zero [simp]: "0 < x powr a \<longleftrightarrow> (x::real) \<noteq> 0"
```
```  2215   by (simp add: powr_def)
```
```  2216
```
```  2217 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)"
```
```  2218   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)"
```
```  2219   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)"
```
```  2220   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)"
```
```  2221   by (simp_all add: log_mult log_divide)
```
```  2222
```
```  2223 lemma log_less_cancel_iff [simp]:
```
```  2224   "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
```
```  2225   apply safe
```
```  2226   apply (rule_tac [2] powr_less_cancel)
```
```  2227   apply (drule_tac a = "log a x" in powr_less_mono, auto)
```
```  2228   done
```
```  2229
```
```  2230 lemma log_inj:
```
```  2231   assumes "1 < b"
```
```  2232   shows "inj_on (log b) {0 <..}"
```
```  2233 proof (rule inj_onI, simp)
```
```  2234   fix x y
```
```  2235   assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
```
```  2236   show "x = y"
```
```  2237   proof (cases rule: linorder_cases)
```
```  2238     assume "x = y"
```
```  2239     then show ?thesis by simp
```
```  2240   next
```
```  2241     assume "x < y" hence "log b x < log b y"
```
```  2242       using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp
```
```  2243     then show ?thesis using * by simp
```
```  2244   next
```
```  2245     assume "y < x" hence "log b y < log b x"
```
```  2246       using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp
```
```  2247     then show ?thesis using * by simp
```
```  2248   qed
```
```  2249 qed
```
```  2250
```
```  2251 lemma log_le_cancel_iff [simp]:
```
```  2252   "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (log a x \<le> log a y) = (x \<le> y)"
```
```  2253   by (simp add: linorder_not_less [symmetric])
```
```  2254
```
```  2255 lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
```
```  2256   using log_less_cancel_iff[of a 1 x] by simp
```
```  2257
```
```  2258 lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
```
```  2259   using log_le_cancel_iff[of a 1 x] by simp
```
```  2260
```
```  2261 lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
```
```  2262   using log_less_cancel_iff[of a x 1] by simp
```
```  2263
```
```  2264 lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
```
```  2265   using log_le_cancel_iff[of a x 1] by simp
```
```  2266
```
```  2267 lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
```
```  2268   using log_less_cancel_iff[of a a x] by simp
```
```  2269
```
```  2270 lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
```
```  2271   using log_le_cancel_iff[of a a x] by simp
```
```  2272
```
```  2273 lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
```
```  2274   using log_less_cancel_iff[of a x a] by simp
```
```  2275
```
```  2276 lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
```
```  2277   using log_le_cancel_iff[of a x a] by simp
```
```  2278
```
```  2279 lemma le_log_iff:
```
```  2280   assumes "1 < b" "x > 0"
```
```  2281   shows "y \<le> log b x \<longleftrightarrow> b powr y \<le> (x::real)"
```
```  2282   using assms
```
```  2283   apply auto
```
```  2284   apply (metis (no_types, hide_lams) less_irrefl less_le_trans linear powr_le_cancel_iff
```
```  2285                powr_log_cancel zero_less_one)
```
```  2286   apply (metis not_less order.trans order_refl powr_le_cancel_iff powr_log_cancel zero_le_one)
```
```  2287   done
```
```  2288
```
```  2289 lemma less_log_iff:
```
```  2290   assumes "1 < b" "x > 0"
```
```  2291   shows "y < log b x \<longleftrightarrow> b powr y < x"
```
```  2292   by (metis assms dual_order.strict_trans less_irrefl powr_less_cancel_iff
```
```  2293     powr_log_cancel zero_less_one)
```
```  2294
```
```  2295 lemma
```
```  2296   assumes "1 < b" "x > 0"
```
```  2297   shows log_less_iff: "log b x < y \<longleftrightarrow> x < b powr y"
```
```  2298     and log_le_iff: "log b x \<le> y \<longleftrightarrow> x \<le> b powr y"
```
```  2299   using le_log_iff[OF assms, of y] less_log_iff[OF assms, of y]
```
```  2300   by auto
```
```  2301
```
```  2302 lemmas powr_le_iff = le_log_iff[symmetric]
```
```  2303   and powr_less_iff = le_log_iff[symmetric]
```
```  2304   and less_powr_iff = log_less_iff[symmetric]
```
```  2305   and le_powr_iff = log_le_iff[symmetric]
```
```  2306
```
```  2307 lemma
```
```  2308   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)"
```
```  2309   by (auto simp add: floor_eq_iff powr_le_iff less_powr_iff)
```
```  2310
```
```  2311 lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
```
```  2312   by (induct n) (simp_all add: ac_simps powr_add real_of_nat_Suc)
```
```  2313
```
```  2314 lemma powr_realpow_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
```
```  2315   unfolding real_of_nat_numeral [symmetric] by (rule powr_realpow)
```
```  2316
```
```  2317 lemma powr2_sqrt[simp]: "0 < x \<Longrightarrow> sqrt x powr 2 = x"
```
```  2318 by(simp add: powr_realpow_numeral)
```
```  2319
```
```  2320 lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
```
```  2321   apply (case_tac "x = 0", simp, simp)
```
```  2322   apply (rule powr_realpow [THEN sym], simp)
```
```  2323   done
```
```  2324
```
```  2325 lemma powr_int:
```
```  2326   assumes "x > 0"
```
```  2327   shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
```
```  2328 proof (cases "i < 0")
```
```  2329   case True
```
```  2330   have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
```
```  2331   show ?thesis using \<open>i < 0\<close> \<open>x > 0\<close> by (simp add: r field_simps powr_realpow[symmetric])
```
```  2332 next
```
```  2333   case False
```
```  2334   then show ?thesis by (simp add: assms powr_realpow[symmetric])
```
```  2335 qed
```
```  2336
```
```  2337 lemma compute_powr[code]:
```
```  2338   fixes i::real
```
```  2339   shows "b powr i =
```
```  2340     (if b \<le> 0 then Code.abort (STR ''op powr with nonpositive base'') (\<lambda>_. b powr i)
```
```  2341     else if floor i = i then (if 0 \<le> i then b ^ nat(floor i) else 1 / b ^ nat(floor (- i)))
```
```  2342     else Code.abort (STR ''op powr with non-integer exponent'') (\<lambda>_. b powr i))"
```
```  2343   by (auto simp: powr_int)
```
```  2344
```
```  2345 lemma powr_one:
```
```  2346   fixes x::real shows "0 \<le> x \<Longrightarrow> x powr 1 = x"
```
```  2347   using powr_realpow [of x 1]
```
```  2348   by simp
```
```  2349
```
```  2350 lemma powr_numeral:
```
```  2351   fixes x::real shows "0 < x \<Longrightarrow> x powr numeral n = x ^ numeral n"
```
```  2352   by (fact powr_realpow_numeral)
```
```  2353
```
```  2354 lemma powr_neg_one:
```
```  2355   fixes x::real shows "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
```
```  2356   using powr_int [of x "- 1"] by simp
```
```  2357
```
```  2358 lemma powr_neg_numeral:
```
```  2359   fixes x::real shows "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
```
```  2360   using powr_int [of x "- numeral n"] by simp
```
```  2361
```
```  2362 lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
```
```  2363   by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
```
```  2364
```
```  2365 lemma ln_powr:
```
```  2366   fixes x::real shows "x \<noteq> 0 \<Longrightarrow> ln (x powr y) = y * ln x"
```
```  2367   by (simp add: powr_def)
```
```  2368
```
```  2369 lemma ln_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> ln (root n b) =  ln b / n"
```
```  2370 by(simp add: root_powr_inverse ln_powr)
```
```  2371
```
```  2372 lemma ln_sqrt: "0 < x \<Longrightarrow> ln (sqrt x) = ln x / 2"
```
```  2373   by (simp add: ln_powr powr_numeral ln_powr[symmetric] mult.commute)
```
```  2374
```
```  2375 lemma log_root: "\<lbrakk> n > 0; a > 0 \<rbrakk> \<Longrightarrow> log b (root n a) =  log b a / n"
```
```  2376 by(simp add: log_def ln_root)
```
```  2377
```
```  2378 lemma log_powr: "x \<noteq> 0 \<Longrightarrow> log b (x powr y) = y * log b x"
```
```  2379   by (simp add: log_def ln_powr)
```
```  2380
```
```  2381 lemma log_nat_power: "0 < x \<Longrightarrow> log b (x^n) = real n * log b x"
```
```  2382   by (simp add: log_powr powr_realpow [symmetric])
```
```  2383
```
```  2384 lemma log_base_change: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log b x = log a x / log a b"
```
```  2385   by (simp add: log_def)
```
```  2386
```
```  2387 lemma log_base_pow: "0 < a \<Longrightarrow> log (a ^ n) x = log a x / n"
```
```  2388   by (simp add: log_def ln_realpow)
```
```  2389
```
```  2390 lemma log_base_powr: "a \<noteq> 0 \<Longrightarrow> log (a powr b) x = log a x / b"
```
```  2391   by (simp add: log_def ln_powr)
```
```  2392
```
```  2393 lemma log_base_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> log (root n b) x = n * (log b x)"
```
```  2394 by(simp add: log_def ln_root)
```
```  2395
```
```  2396 lemma ln_bound:
```
```  2397   fixes x::real shows "1 <= x ==> ln x <= x"
```
```  2398   apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
```
```  2399   apply simp
```
```  2400   apply (rule ln_add_one_self_le_self, simp)
```
```  2401   done
```
```  2402
```
```  2403 lemma powr_mono:
```
```  2404   fixes x::real shows "a <= b ==> 1 <= x ==> x powr a <= x powr b"
```
```  2405   apply (cases "x = 1", simp)
```
```  2406   apply (cases "a = b", simp)
```
```  2407   apply (rule order_less_imp_le)
```
```  2408   apply (rule powr_less_mono, auto)
```
```  2409   done
```
```  2410
```
```  2411 lemma ge_one_powr_ge_zero:
```
```  2412   fixes x::real shows "1 <= x ==> 0 <= a ==> 1 <= x powr a"
```
```  2413 using powr_mono by fastforce
```
```  2414
```
```  2415 lemma powr_less_mono2:
```
```  2416   fixes x::real shows "0 < a ==> 0 \<le> x ==> x < y ==> x powr a < y powr a"
```
```  2417   by (simp add: powr_def)
```
```  2418
```
```  2419 lemma powr_less_mono2_neg:
```
```  2420   fixes x::real shows "a < 0 ==> 0 < x ==> x < y ==> y powr a < x powr a"
```
```  2421   by (simp add: powr_def)
```
```  2422
```
```  2423 lemma powr_mono2:
```
```  2424   fixes x::real shows "0 <= a ==> 0 \<le> x ==> x <= y ==> x powr a <= y powr a"
```
```  2425   apply (case_tac "a = 0", simp)
```
```  2426   apply (case_tac "x = y", simp)
```
```  2427   apply (metis dual_order.strict_iff_order powr_less_mono2)
```
```  2428   done
```
```  2429
```
```  2430 lemma powr_inj:
```
```  2431   fixes x::real shows "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
```
```  2432   unfolding powr_def exp_inj_iff by simp
```
```  2433
```
```  2434 lemma powr_half_sqrt: "0 \<le> x \<Longrightarrow> x powr (1/2) = sqrt x"
```
```  2435   by (simp add: powr_def root_powr_inverse sqrt_def)
```
```  2436
```
```  2437 lemma ln_powr_bound:
```
```  2438   fixes x::real shows "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
```
```  2439 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)
```
```  2440
```
```  2441
```
```  2442 lemma ln_powr_bound2:
```
```  2443   fixes x::real
```
```  2444   assumes "1 < x" and "0 < a"
```
```  2445   shows "(ln x) powr a <= (a powr a) * x"
```
```  2446 proof -
```
```  2447   from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
```
```  2448     by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff)
```
```  2449   also have "... = a * (x powr (1 / a))"
```
```  2450     by simp
```
```  2451   finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
```
```  2452     by (metis assms less_imp_le ln_gt_zero powr_mono2)
```
```  2453   also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
```
```  2454     using assms powr_mult by auto
```
```  2455   also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
```
```  2456     by (rule powr_powr)
```
```  2457   also have "... = x" using assms
```
```  2458     by auto
```
```  2459   finally show ?thesis .
```
```  2460 qed
```
```  2461
```
```  2462 lemma tendsto_powr [tendsto_intros]:
```
```  2463   fixes a::real
```
```  2464   assumes f: "(f ---> a) F" and g: "(g ---> b) F" and a: "a \<noteq> 0"
```
```  2465   shows "((\<lambda>x. f x powr g x) ---> a powr b) F"
```
```  2466   unfolding powr_def
```
```  2467 proof (rule filterlim_If)
```
```  2468   from f show "((\<lambda>x. 0) ---> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))"
```
```  2469     by simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_elim1 dest: t1_space_nhds)
```
```  2470 qed (insert f g a, auto intro!: tendsto_intros intro: tendsto_mono inf_le1)
```
```  2471
```
```  2472 lemma continuous_powr:
```
```  2473   assumes "continuous F f"
```
```  2474     and "continuous F g"
```
```  2475     and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```  2476   shows "continuous F (\<lambda>x. (f x) powr (g x :: real))"
```
```  2477   using assms unfolding continuous_def by (rule tendsto_powr)
```
```  2478
```
```  2479 lemma continuous_at_within_powr[continuous_intros]:
```
```  2480   assumes "continuous (at a within s) f"
```
```  2481     and "continuous (at a within s) g"
```
```  2482     and "f a \<noteq> 0"
```
```  2483   shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x :: real))"
```
```  2484   using assms unfolding continuous_within by (rule tendsto_powr)
```
```  2485
```
```  2486 lemma isCont_powr[continuous_intros, simp]:
```
```  2487   assumes "isCont f a" "isCont g a" "f a \<noteq> (0::real)"
```
```  2488   shows "isCont (\<lambda>x. (f x) powr g x) a"
```
```  2489   using assms unfolding continuous_at by (rule tendsto_powr)
```
```  2490
```
```  2491 lemma continuous_on_powr[continuous_intros]:
```
```  2492   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> (0::real)"
```
```  2493   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
```
```  2494   using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
```
```  2495
```
```  2496 lemma tendsto_powr2:
```
```  2497   fixes a::real
```
```  2498   assumes f: "(f ---> a) F" and g: "(g ---> b) F" and f_nonneg: "\<forall>\<^sub>F x in F. 0 \<le> f x" and b: "0 < b"
```
```  2499   shows "((\<lambda>x. f x powr g x) ---> a powr b) F"
```
```  2500   unfolding powr_def
```
```  2501 proof (rule filterlim_If)
```
```  2502   from f show "((\<lambda>x. 0) ---> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))"
```
```  2503     by simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_elim1 dest: t1_space_nhds)
```
```  2504 next
```
```  2505   { assume "a = 0"
```
```  2506     with f f_nonneg have "LIM x inf F (principal {x. f x \<noteq> 0}). f x :> at_right 0"
```
```  2507       by (auto simp add: filterlim_at eventually_inf_principal le_less
```
```  2508                elim: eventually_elim1 intro: tendsto_mono inf_le1)
```
```  2509     then have "((\<lambda>x. exp (g x * ln (f x))) ---> 0) (inf F (principal {x. f x \<noteq> 0}))"
```
```  2510       by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_0]
```
```  2511                        filterlim_tendsto_pos_mult_at_bot[OF _ \<open>0 < b\<close>]
```
```  2512                intro: tendsto_mono inf_le1 g) }
```
```  2513   then show "((\<lambda>x. exp (g x * ln (f x))) ---> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x \<noteq> 0}))"
```
```  2514     using f g by (auto intro!: tendsto_intros intro: tendsto_mono inf_le1)
```
```  2515 qed
```
```  2516
```
```  2517 lemma DERIV_powr:
```
```  2518   fixes r::real
```
```  2519   assumes g: "DERIV g x :> m" and pos: "g x > 0" and f: "DERIV f x :> r"
```
```  2520   shows  "DERIV (\<lambda>x. g x powr f x) x :> (g x powr f x) * (r * ln (g x) + m * f x / g x)"
```
```  2521 proof -
```
```  2522   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)"
```
```  2523     using pos
```
```  2524     by (auto intro!: derivative_eq_intros g pos f simp: powr_def field_simps exp_diff)
```
```  2525   then show ?thesis
```
```  2526   proof (rule DERIV_cong_ev[OF refl _ refl, THEN iffD1, rotated])
```
```  2527     from DERIV_isCont[OF g] pos have "\<forall>\<^sub>F x in at x. 0 < g x"
```
```  2528       unfolding isCont_def by (rule order_tendstoD(1))
```
```  2529     with pos show "\<forall>\<^sub>F x in nhds x. exp (f x * ln (g x)) = g x powr f x"
```
```  2530       by (auto simp: eventually_at_filter powr_def elim: eventually_elim1)
```
```  2531   qed
```
```  2532 qed
```
```  2533
```
```  2534 lemma DERIV_fun_powr:
```
```  2535   fixes r::real
```
```  2536   assumes g: "DERIV g x :> m" and pos: "g x > 0"
```
```  2537   shows  "DERIV (\<lambda>x. (g x) powr r) x :> r * (g x) powr (r - of_nat 1) * m"
```
```  2538   using DERIV_powr[OF g pos DERIV_const, of r] pos
```
```  2539   by (simp add: powr_divide2[symmetric] field_simps)
```
```  2540
```
```  2541 lemma tendsto_zero_powrI:
```
```  2542   assumes "(f ---> (0::real)) F" "(g ---> b) F" "\<forall>\<^sub>F x in F. 0 \<le> f x" "0 < b"
```
```  2543   shows "((\<lambda>x. f x powr g x) ---> 0) F"
```
```  2544   using tendsto_powr2[OF assms] by simp
```
```  2545
```
```  2546 lemma tendsto_neg_powr:
```
```  2547   assumes "s < 0"
```
```  2548     and f: "LIM x F. f x :> at_top"
```
```  2549   shows "((\<lambda>x. f x powr s) ---> (0::real)) F"
```
```  2550 proof -
```
```  2551   have "((\<lambda>x. exp (s * ln (f x))) ---> (0::real)) F" (is "?X")
```
```  2552     by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_top]
```
```  2553                      filterlim_tendsto_neg_mult_at_bot assms)
```
```  2554   also have "?X \<longleftrightarrow> ((\<lambda>x. f x powr s) ---> (0::real)) F"
```
```  2555     using f filterlim_at_top_dense[of f F]
```
```  2556     by (intro filterlim_cong[OF refl refl]) (auto simp: neq_iff powr_def elim: eventually_elim1)
```
```  2557   finally show ?thesis .
```
```  2558 qed
```
```  2559
```
```  2560 lemma tendsto_exp_limit_at_right:
```
```  2561   fixes x :: real
```
```  2562   shows "((\<lambda>y. (1 + x * y) powr (1 / y)) ---> exp x) (at_right 0)"
```
```  2563 proof cases
```
```  2564   assume "x \<noteq> 0"
```
```  2565   have "((\<lambda>y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)"
```
```  2566     by (auto intro!: derivative_eq_intros)
```
```  2567   then have "((\<lambda>y. ln (1 + x * y) / y) ---> x) (at 0)"
```
```  2568     by (auto simp add: has_field_derivative_def field_has_derivative_at)
```
```  2569   then have *: "((\<lambda>y. exp (ln (1 + x * y) / y)) ---> exp x) (at 0)"
```
```  2570     by (rule tendsto_intros)
```
```  2571   then show ?thesis
```
```  2572   proof (rule filterlim_mono_eventually)
```
```  2573     show "eventually (\<lambda>xa. exp (ln (1 + x * xa) / xa) = (1 + x * xa) powr (1 / xa)) (at_right 0)"
```
```  2574       unfolding eventually_at_right[OF zero_less_one]
```
```  2575       using \<open>x \<noteq> 0\<close>
```
```  2576       apply  (intro exI[of _ "1 / \<bar>x\<bar>"])
```
```  2577       apply (auto simp: field_simps powr_def abs_if)
```
```  2578       by (metis add_less_same_cancel1 mult_less_0_iff not_less_iff_gr_or_eq zero_less_one)
```
```  2579   qed (simp_all add: at_eq_sup_left_right)
```
```  2580 qed simp
```
```  2581
```
```  2582 lemma tendsto_exp_limit_at_top:
```
```  2583   fixes x :: real
```
```  2584   shows "((\<lambda>y. (1 + x / y) powr y) ---> exp x) at_top"
```
```  2585   apply (subst filterlim_at_top_to_right)
```
```  2586   apply (simp add: inverse_eq_divide)
```
```  2587   apply (rule tendsto_exp_limit_at_right)
```
```  2588   done
```
```  2589
```
```  2590 lemma tendsto_exp_limit_sequentially:
```
```  2591   fixes x :: real
```
```  2592   shows "(\<lambda>n. (1 + x / n) ^ n) ----> exp x"
```
```  2593 proof (rule filterlim_mono_eventually)
```
```  2594   from reals_Archimedean2 [of "abs x"] obtain n :: nat where *: "real n > abs x" ..
```
```  2595   hence "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top"
```
```  2596     apply (intro eventually_sequentiallyI [of n])
```
```  2597     apply (case_tac "x \<ge> 0")
```
```  2598     apply (rule add_pos_nonneg, auto intro: divide_nonneg_nonneg)
```
```  2599     apply (subgoal_tac "x / real xa > -1")
```
```  2600     apply (auto simp add: field_simps)
```
```  2601     done
```
```  2602   then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top"
```
```  2603     by (rule eventually_elim1) (erule powr_realpow)
```
```  2604   show "(\<lambda>n. (1 + x / real n) powr real n) ----> exp x"
```
```  2605     by (rule filterlim_compose [OF tendsto_exp_limit_at_top filterlim_real_sequentially])
```
```  2606 qed auto
```
```  2607
```
```  2608 subsection \<open>Sine and Cosine\<close>
```
```  2609
```
```  2610 definition sin_coeff :: "nat \<Rightarrow> real" where
```
```  2611   "sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / (fact n))"
```
```  2612
```
```  2613 definition cos_coeff :: "nat \<Rightarrow> real" where
```
```  2614   "cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / (fact n) else 0)"
```
```  2615
```
```  2616 definition sin :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```  2617   where "sin = (\<lambda>x. \<Sum>n. sin_coeff n *\<^sub>R x^n)"
```
```  2618
```
```  2619 definition cos :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```  2620   where "cos = (\<lambda>x. \<Sum>n. cos_coeff n *\<^sub>R x^n)"
```
```  2621
```
```  2622 lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
```
```  2623   unfolding sin_coeff_def by simp
```
```  2624
```
```  2625 lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
```
```  2626   unfolding cos_coeff_def by simp
```
```  2627
```
```  2628 lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
```
```  2629   unfolding cos_coeff_def sin_coeff_def
```
```  2630   by (simp del: mult_Suc)
```
```  2631
```
```  2632 lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
```
```  2633   unfolding cos_coeff_def sin_coeff_def
```
```  2634   by (simp del: mult_Suc) (auto elim: oddE)
```
```  2635
```
```  2636 lemma summable_norm_sin:
```
```  2637   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  2638   shows "summable (\<lambda>n. norm (sin_coeff n *\<^sub>R x^n))"
```
```  2639   unfolding sin_coeff_def
```
```  2640   apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]])
```
```  2641   apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2642   done
```
```  2643
```
```  2644 lemma summable_norm_cos:
```
```  2645   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  2646   shows "summable (\<lambda>n. norm (cos_coeff n *\<^sub>R x^n))"
```
```  2647   unfolding cos_coeff_def
```
```  2648   apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]])
```
```  2649   apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2650   done
```
```  2651
```
```  2652 lemma sin_converges: "(\<lambda>n. sin_coeff n *\<^sub>R x^n) sums sin(x)"
```
```  2653 unfolding sin_def
```
```  2654   by (metis (full_types) summable_norm_cancel summable_norm_sin summable_sums)
```
```  2655
```
```  2656 lemma cos_converges: "(\<lambda>n. cos_coeff n *\<^sub>R x^n) sums cos(x)"
```
```  2657 unfolding cos_def
```
```  2658   by (metis (full_types) summable_norm_cancel summable_norm_cos summable_sums)
```
```  2659
```
```  2660 lemma sin_of_real:
```
```  2661   fixes x::real
```
```  2662   shows "sin (of_real x) = of_real (sin x)"
```
```  2663 proof -
```
```  2664   have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R  x^n)) = (\<lambda>n. sin_coeff n *\<^sub>R  (of_real x)^n)"
```
```  2665   proof
```
```  2666     fix n
```
```  2667     show "of_real (sin_coeff n *\<^sub>R  x^n) = sin_coeff n *\<^sub>R of_real x^n"
```
```  2668       by (simp add: scaleR_conv_of_real)
```
```  2669   qed
```
```  2670   also have "... sums (sin (of_real x))"
```
```  2671     by (rule sin_converges)
```
```  2672   finally have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) sums (sin (of_real x))" .
```
```  2673   then show ?thesis
```
```  2674     using sums_unique2 sums_of_real [OF sin_converges]
```
```  2675     by blast
```
```  2676 qed
```
```  2677
```
```  2678 corollary sin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> sin z \<in> \<real>"
```
```  2679   by (metis Reals_cases Reals_of_real sin_of_real)
```
```  2680
```
```  2681 lemma cos_of_real:
```
```  2682   fixes x::real
```
```  2683   shows "cos (of_real x) = of_real (cos x)"
```
```  2684 proof -
```
```  2685   have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R  x^n)) = (\<lambda>n. cos_coeff n *\<^sub>R  (of_real x)^n)"
```
```  2686   proof
```
```  2687     fix n
```
```  2688     show "of_real (cos_coeff n *\<^sub>R  x^n) = cos_coeff n *\<^sub>R of_real x^n"
```
```  2689       by (simp add: scaleR_conv_of_real)
```
```  2690   qed
```
```  2691   also have "... sums (cos (of_real x))"
```
```  2692     by (rule cos_converges)
```
```  2693   finally have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) sums (cos (of_real x))" .
```
```  2694   then show ?thesis
```
```  2695     using sums_unique2 sums_of_real [OF cos_converges]
```
```  2696     by blast
```
```  2697 qed
```
```  2698
```
```  2699 corollary cos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cos z \<in> \<real>"
```
```  2700   by (metis Reals_cases Reals_of_real cos_of_real)
```
```  2701
```
```  2702 lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
```
```  2703   by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  2704
```
```  2705 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
```
```  2706   by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  2707
```
```  2708 text\<open>Now at last we can get the derivatives of exp, sin and cos\<close>
```
```  2709
```
```  2710 lemma DERIV_sin [simp]:
```
```  2711   fixes x :: "'a::{real_normed_field,banach}"
```
```  2712   shows "DERIV sin x :> cos(x)"
```
```  2713   unfolding sin_def cos_def scaleR_conv_of_real
```
```  2714   apply (rule DERIV_cong)
```
```  2715   apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
```
```  2716   apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff
```
```  2717               summable_minus_iff scaleR_conv_of_real [symmetric]
```
```  2718               summable_norm_sin [THEN summable_norm_cancel]
```
```  2719               summable_norm_cos [THEN summable_norm_cancel])
```
```  2720   done
```
```  2721
```
```  2722 declare DERIV_sin[THEN DERIV_chain2, derivative_intros]
```
```  2723
```
```  2724 lemma DERIV_cos [simp]:
```
```  2725   fixes x :: "'a::{real_normed_field,banach}"
```
```  2726   shows "DERIV cos x :> -sin(x)"
```
```  2727   unfolding sin_def cos_def scaleR_conv_of_real
```
```  2728   apply (rule DERIV_cong)
```
```  2729   apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
```
```  2730   apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus
```
```  2731               diffs_sin_coeff diffs_cos_coeff
```
```  2732               summable_minus_iff scaleR_conv_of_real [symmetric]
```
```  2733               summable_norm_sin [THEN summable_norm_cancel]
```
```  2734               summable_norm_cos [THEN summable_norm_cancel])
```
```  2735   done
```
```  2736
```
```  2737 declare DERIV_cos[THEN DERIV_chain2, derivative_intros]
```
```  2738
```
```  2739 lemma isCont_sin:
```
```  2740   fixes x :: "'a::{real_normed_field,banach}"
```
```  2741   shows "isCont sin x"
```
```  2742   by (rule DERIV_sin [THEN DERIV_isCont])
```
```  2743
```
```  2744 lemma isCont_cos:
```
```  2745   fixes x :: "'a::{real_normed_field,banach}"
```
```  2746   shows "isCont cos x"
```
```  2747   by (rule DERIV_cos [THEN DERIV_isCont])
```
```  2748
```
```  2749 lemma isCont_sin' [simp]:
```
```  2750   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2751   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
```
```  2752   by (rule isCont_o2 [OF _ isCont_sin])
```
```  2753
```
```  2754 (*FIXME A CONTEXT FOR F WOULD BE BETTER*)
```
```  2755
```
```  2756 lemma isCont_cos' [simp]:
```
```  2757   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2758   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
```
```  2759   by (rule isCont_o2 [OF _ isCont_cos])
```
```  2760
```
```  2761 lemma tendsto_sin [tendsto_intros]:
```
```  2762   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2763   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
```
```  2764   by (rule isCont_tendsto_compose [OF isCont_sin])
```
```  2765
```
```  2766 lemma tendsto_cos [tendsto_intros]:
```
```  2767   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2768   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
```
```  2769   by (rule isCont_tendsto_compose [OF isCont_cos])
```
```  2770
```
```  2771 lemma continuous_sin [continuous_intros]:
```
```  2772   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2773   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
```
```  2774   unfolding continuous_def by (rule tendsto_sin)
```
```  2775
```
```  2776 lemma continuous_on_sin [continuous_intros]:
```
```  2777   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2778   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
```
```  2779   unfolding continuous_on_def by (auto intro: tendsto_sin)
```
```  2780
```
```  2781 lemma continuous_within_sin:
```
```  2782   fixes z :: "'a::{real_normed_field,banach}"
```
```  2783   shows "continuous (at z within s) sin"
```
```  2784   by (simp add: continuous_within tendsto_sin)
```
```  2785
```
```  2786 lemma continuous_cos [continuous_intros]:
```
```  2787   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2788   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
```
```  2789   unfolding continuous_def by (rule tendsto_cos)
```
```  2790
```
```  2791 lemma continuous_on_cos [continuous_intros]:
```
```  2792   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2793   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
```
```  2794   unfolding continuous_on_def by (auto intro: tendsto_cos)
```
```  2795
```
```  2796 lemma continuous_within_cos:
```
```  2797   fixes z :: "'a::{real_normed_field,banach}"
```
```  2798   shows "continuous (at z within s) cos"
```
```  2799   by (simp add: continuous_within tendsto_cos)
```
```  2800
```
```  2801 subsection \<open>Properties of Sine and Cosine\<close>
```
```  2802
```
```  2803 lemma sin_zero [simp]: "sin 0 = 0"
```
```  2804   unfolding sin_def sin_coeff_def by (simp add: scaleR_conv_of_real powser_zero)
```
```  2805
```
```  2806 lemma cos_zero [simp]: "cos 0 = 1"
```
```  2807   unfolding cos_def cos_coeff_def by (simp add: scaleR_conv_of_real powser_zero)
```
```  2808
```
```  2809 lemma DERIV_fun_sin:
```
```  2810      "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. sin(g x)) x :> cos(g x) * m"
```
```  2811   by (auto intro!: derivative_intros)
```
```  2812
```
```  2813 lemma DERIV_fun_cos:
```
```  2814      "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. cos(g x)) x :> -sin(g x) * m"
```
```  2815   by (auto intro!: derivative_eq_intros simp: real_of_nat_def)
```
```  2816
```
```  2817 subsection \<open>Deriving the Addition Formulas\<close>
```
```  2818
```
```  2819 text\<open>The the product of two cosine series\<close>
```
```  2820 lemma cos_x_cos_y:
```
```  2821   fixes x :: "'a::{real_normed_field,banach}"
```
```  2822   shows "(\<lambda>p. \<Sum>n\<le>p.
```
```  2823           if even p \<and> even n
```
```  2824           then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
```
```  2825          sums (cos x * cos y)"
```
```  2826 proof -
```
```  2827   { fix n p::nat
```
```  2828     assume "n\<le>p"
```
```  2829     then have *: "even n \<Longrightarrow> even p \<Longrightarrow> (-1) ^ (n div 2) * (-1) ^ ((p - n) div 2) = (-1 :: real) ^ (p div 2)"
```
```  2830       by (metis div_add power_add le_add_diff_inverse odd_add)
```
```  2831     have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
```
```  2832           (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)"
```
```  2833     using \<open>n\<le>p\<close>
```
```  2834       by (auto simp: * algebra_simps cos_coeff_def binomial_fact real_of_nat_def)
```
```  2835   }
```
```  2836   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> even n
```
```  2837                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
```
```  2838              (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
```
```  2839     by simp
```
```  2840   also have "... = (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n *\<^sub>R x^n) * (cos_coeff (p - n) *\<^sub>R y^(p-n)))"
```
```  2841     by (simp add: algebra_simps)
```
```  2842   also have "... sums (cos x * cos y)"
```
```  2843     using summable_norm_cos
```
```  2844     by (auto simp: cos_def scaleR_conv_of_real intro!: Cauchy_product_sums)
```
```  2845   finally show ?thesis .
```
```  2846 qed
```
```  2847
```
```  2848 text\<open>The product of two sine series\<close>
```
```  2849 lemma sin_x_sin_y:
```
```  2850   fixes x :: "'a::{real_normed_field,banach}"
```
```  2851   shows "(\<lambda>p. \<Sum>n\<le>p.
```
```  2852           if even p \<and> odd n
```
```  2853                then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
```
```  2854          sums (sin x * sin y)"
```
```  2855 proof -
```
```  2856   { fix n p::nat
```
```  2857     assume "n\<le>p"
```
```  2858     { assume np: "odd n" "even p"
```
```  2859         with \<open>n\<le>p\<close> have "n - Suc 0 + (p - Suc n) = p - Suc (Suc 0)" "Suc (Suc 0) \<le> p"
```
```  2860         by arith+
```
```  2861       moreover have "(p - Suc (Suc 0)) div 2 = p div 2 - Suc 0"
```
```  2862         by simp
```
```  2863       ultimately have *: "(-1) ^ ((n - Suc 0) div 2) * (-1) ^ ((p - Suc n) div 2) = - ((-1 :: real) ^ (p div 2))"
```
```  2864         using np \<open>n\<le>p\<close>
```
```  2865         apply (simp add: power_add [symmetric] div_add [symmetric] del: div_add)
```
```  2866         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)
```
```  2867         done
```
```  2868     } then
```
```  2869     have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
```
```  2870           (if even p \<and> odd n
```
```  2871           then -((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
```
```  2872     using \<open>n\<le>p\<close>
```
```  2873       by (auto simp:  algebra_simps sin_coeff_def binomial_fact real_of_nat_def)
```
```  2874   }
```
```  2875   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> odd n
```
```  2876                then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
```
```  2877              (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
```
```  2878     by simp
```
```  2879   also have "... = (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n *\<^sub>R x^n) * (sin_coeff (p - n) *\<^sub>R y^(p-n)))"
```
```  2880     by (simp add: algebra_simps)
```
```  2881   also have "... sums (sin x * sin y)"
```
```  2882     using summable_norm_sin
```
```  2883     by (auto simp: sin_def scaleR_conv_of_real intro!: Cauchy_product_sums)
```
```  2884   finally show ?thesis .
```
```  2885 qed
```
```  2886
```
```  2887 lemma sums_cos_x_plus_y:
```
```  2888   fixes x :: "'a::{real_normed_field,banach}"
```
```  2889   shows
```
```  2890   "(\<lambda>p. \<Sum>n\<le>p. if even p
```
```  2891                then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2892                else 0)
```
```  2893         sums cos (x + y)"
```
```  2894 proof -
```
```  2895   { fix p::nat
```
```  2896     have "(\<Sum>n\<le>p. if even p
```
```  2897                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2898                   else 0) =
```
```  2899           (if even p
```
```  2900                   then \<Sum>n\<le>p. ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2901                   else 0)"
```
```  2902       by simp
```
```  2903     also have "... = (if even p
```
```  2904                   then of_real ((-1) ^ (p div 2) / (fact p)) * (\<Sum>n\<le>p. (p choose n) *\<^sub>R (x^n) * y^(p-n))
```
```  2905                   else 0)"
```
```  2906       by (auto simp: setsum_right_distrib field_simps scaleR_conv_of_real nonzero_of_real_divide)
```
```  2907     also have "... = cos_coeff p *\<^sub>R ((x + y) ^ p)"
```
```  2908       by (simp add: cos_coeff_def binomial_ring [of x y]  scaleR_conv_of_real real_of_nat_def atLeast0AtMost)
```
```  2909     finally have "(\<Sum>n\<le>p. if even p
```
```  2910                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2911                   else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)" .
```
```  2912   }
```
```  2913   then have "(\<lambda>p. \<Sum>n\<le>p.
```
```  2914                if even p
```
```  2915                then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2916                else 0)
```
```  2917         = (\<lambda>p. cos_coeff p *\<^sub>R ((x+y)^p))"
```
```  2918         by simp
```
```  2919    also have "... sums cos (x + y)"
```
```  2920     by (rule cos_converges)
```
```  2921    finally show ?thesis .
```
```  2922 qed
```
```  2923
```
```  2924 theorem cos_add:
```
```  2925   fixes x :: "'a::{real_normed_field,banach}"
```
```  2926   shows "cos (x + y) = cos x * cos y - sin x * sin y"
```
```  2927 proof -
```
```  2928   { fix n p::nat
```
```  2929     assume "n\<le>p"
```
```  2930     then have "(if even p \<and> even n
```
```  2931                then ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) -
```
```  2932           (if even p \<and> odd n
```
```  2933                then - ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
```
```  2934           = (if even p
```
```  2935                then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
```
```  2936       by simp
```
```  2937   }
```
```  2938   then have "(\<lambda>p. \<Sum>n\<le>p. (if even p
```
```  2939                then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0))
```
```  2940         sums (cos x * cos y - sin x * sin y)"
```
```  2941     using sums_diff [OF cos_x_cos_y [of x y] sin_x_sin_y [of x y]]
```
```  2942     by (simp add: setsum_subtractf [symmetric])
```
```  2943   then show ?thesis
```
```  2944     by (blast intro: sums_cos_x_plus_y sums_unique2)
```
```  2945 qed
```
```  2946
```
```  2947 lemma sin_minus_converges: "(\<lambda>n. - (sin_coeff n *\<^sub>R (-x)^n)) sums sin(x)"
```
```  2948 proof -
```
```  2949   have [simp]: "\<And>n. - (sin_coeff n *\<^sub>R (-x)^n) = (sin_coeff n *\<^sub>R x^n)"
```
```  2950     by (auto simp: sin_coeff_def elim!: oddE)
```
```  2951   show ?thesis
```
```  2952     by (simp add: sin_def summable_norm_sin [THEN summable_norm_cancel, THEN summable_sums])
```
```  2953 qed
```
```  2954
```
```  2955 lemma sin_minus [simp]:
```
```  2956   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  2957   shows "sin (-x) = -sin(x)"
```
```  2958 using sin_minus_converges [of x]
```
```  2959 by (auto simp: sin_def summable_norm_sin [THEN summable_norm_cancel] suminf_minus sums_iff equation_minus_iff)
```
```  2960
```
```  2961 lemma cos_minus_converges: "(\<lambda>n. (cos_coeff n *\<^sub>R (-x)^n)) sums cos(x)"
```
```  2962 proof -
```
```  2963   have [simp]: "\<And>n. (cos_coeff n *\<^sub>R (-x)^n) = (cos_coeff n *\<^sub>R x^n)"
```
```  2964     by (auto simp: Transcendental.cos_coeff_def elim!: evenE)
```
```  2965   show ?thesis
```
```  2966     by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel, THEN summable_sums])
```
```  2967 qed
```
```  2968
```
```  2969 lemma cos_minus [simp]:
```
```  2970   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  2971   shows "cos (-x) = cos(x)"
```
```  2972 using cos_minus_converges [of x]
```
```  2973 by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel]
```
```  2974               suminf_minus sums_iff equation_minus_iff)
```
```  2975
```
```  2976 lemma sin_cos_squared_add [simp]:
```
```  2977   fixes x :: "'a::{real_normed_field,banach}"
```
```  2978   shows "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
```
```  2979 using cos_add [of x "-x"]
```
```  2980 by (simp add: power2_eq_square algebra_simps)
```
```  2981
```
```  2982 lemma sin_cos_squared_add2 [simp]:
```
```  2983   fixes x :: "'a::{real_normed_field,banach}"
```
```  2984   shows "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
```
```  2985   by (subst add.commute, rule sin_cos_squared_add)
```
```  2986
```
```  2987 lemma sin_cos_squared_add3 [simp]:
```
```  2988   fixes x :: "'a::{real_normed_field,banach}"
```
```  2989   shows "cos x * cos x + sin x * sin x = 1"
```
```  2990   using sin_cos_squared_add2 [unfolded power2_eq_square] .
```
```  2991
```
```  2992 lemma sin_squared_eq:
```
```  2993   fixes x :: "'a::{real_normed_field,banach}"
```
```  2994   shows "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
```
```  2995   unfolding eq_diff_eq by (rule sin_cos_squared_add)
```
```  2996
```
```  2997 lemma cos_squared_eq:
```
```  2998   fixes x :: "'a::{real_normed_field,banach}"
```
```  2999   shows "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
```
```  3000   unfolding eq_diff_eq by (rule sin_cos_squared_add2)
```
```  3001
```
```  3002 lemma abs_sin_le_one [simp]:
```
```  3003   fixes x :: real
```
```  3004   shows "\<bar>sin x\<bar> \<le> 1"
```
```  3005   by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
```
```  3006
```
```  3007 lemma sin_ge_minus_one [simp]:
```
```  3008   fixes x :: real
```
```  3009   shows "-1 \<le> sin x"
```
```  3010   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  3011
```
```  3012 lemma sin_le_one [simp]:
```
```  3013   fixes x :: real
```
```  3014   shows "sin x \<le> 1"
```
```  3015   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  3016
```
```  3017 lemma abs_cos_le_one [simp]:
```
```  3018   fixes x :: real
```
```  3019   shows "\<bar>cos x\<bar> \<le> 1"
```
```  3020   by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
```
```  3021
```
```  3022 lemma cos_ge_minus_one [simp]:
```
```  3023   fixes x :: real
```
```  3024   shows "-1 \<le> cos x"
```
```  3025   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  3026
```
```  3027 lemma cos_le_one [simp]:
```
```  3028   fixes x :: real
```
```  3029   shows "cos x \<le> 1"
```
```  3030   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  3031
```
```  3032 lemma cos_diff:
```
```  3033   fixes x :: "'a::{real_normed_field,banach}"
```
```  3034   shows "cos (x - y) = cos x * cos y + sin x * sin y"
```
```  3035   using cos_add [of x "- y"] by simp
```
```  3036
```
```  3037 lemma cos_double:
```
```  3038   fixes x :: "'a::{real_normed_field,banach}"
```
```  3039   shows "cos(2*x) = (cos x)\<^sup>2 - (sin x)\<^sup>2"
```
```  3040   using cos_add [where x=x and y=x]
```
```  3041   by (simp add: power2_eq_square)
```
```  3042
```
```  3043 lemma DERIV_fun_pow: "DERIV g x :> m ==>
```
```  3044       DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
```
```  3045   by (auto intro!: derivative_eq_intros simp: real_of_nat_def)
```
```  3046
```
```  3047 lemma DERIV_fun_exp:
```
```  3048      "DERIV g x :> m ==> DERIV (\<lambda>x. exp(g x)) x :> exp(g x) * m"
```
```  3049   by (auto intro!: derivative_intros)
```
```  3050
```
```  3051 subsection \<open>The Constant Pi\<close>
```
```  3052
```
```  3053 definition pi :: real
```
```  3054   where "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
```
```  3055
```
```  3056 text\<open>Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
```
```  3057    hence define pi.\<close>
```
```  3058
```
```  3059 lemma sin_paired:
```
```  3060   fixes x :: real
```
```  3061   shows "(\<lambda>n. (- 1) ^ n / (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums  sin x"
```
```  3062 proof -
```
```  3063   have "(\<lambda>n. \<Sum>k = n*2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
```
```  3064     apply (rule sums_group)
```
```  3065     using sin_converges [of x, unfolded scaleR_conv_of_real]
```
```  3066     by auto
```
```  3067   thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: ac_simps)
```
```  3068 qed
```
```  3069
```
```  3070 lemma sin_gt_zero_02:
```
```  3071   fixes x :: real
```
```  3072   assumes "0 < x" and "x < 2"
```
```  3073   shows "0 < sin x"
```
```  3074 proof -
```
```  3075   let ?f = "\<lambda>n::nat. \<Sum>k = n*2..<n*2+2. (- 1) ^ k / (fact (2*k+1)) * x^(2*k+1)"
```
```  3076   have pos: "\<forall>n. 0 < ?f n"
```
```  3077   proof
```
```  3078     fix n :: nat
```
```  3079     let ?k2 = "real (Suc (Suc (4 * n)))"
```
```  3080     let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
```
```  3081     have "x * x < ?k2 * ?k3"
```
```  3082       using assms by (intro mult_strict_mono', simp_all)
```
```  3083     hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
```
```  3084       by (intro mult_strict_right_mono zero_less_power \<open>0 < x\<close>)
```
```  3085     thus "0 < ?f n"
```
```  3086       by (simp add: real_of_nat_def divide_simps mult_ac del: mult_Suc)
```
```  3087 qed
```
```  3088   have sums: "?f sums sin x"
```
```  3089     by (rule sin_paired [THEN sums_group], simp)
```
```  3090   show "0 < sin x"
```
```  3091     unfolding sums_unique [OF sums]
```
```  3092     using sums_summable [OF sums] pos
```
```  3093     by (rule suminf_pos)
```
```  3094 qed
```
```  3095
```
```  3096 lemma cos_double_less_one:
```
```  3097   fixes x :: real
```
```  3098   shows "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
```
```  3099   using sin_gt_zero_02 [where x = x] by (auto simp: cos_squared_eq cos_double)
```
```  3100
```
```  3101 lemma cos_paired:
```
```  3102   fixes x :: real
```
```  3103   shows "(\<lambda>n. (- 1) ^ n / (fact (2 * n)) * x ^ (2 * n)) sums cos x"
```
```  3104 proof -
```
```  3105   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
```
```  3106     apply (rule sums_group)
```
```  3107     using cos_converges [of x, unfolded scaleR_conv_of_real]
```
```  3108     by auto
```
```  3109   thus ?thesis unfolding cos_coeff_def by (simp add: ac_simps)
```
```  3110 qed
```
```  3111
```
```  3112 lemmas realpow_num_eq_if = power_eq_if
```
```  3113
```
```  3114 lemma sumr_pos_lt_pair:
```
```  3115   fixes f :: "nat \<Rightarrow> real"
```
```  3116   shows "\<lbrakk>summable f;
```
```  3117         \<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk>
```
```  3118       \<Longrightarrow> setsum f {..<k} < suminf f"
```
```  3119 unfolding One_nat_def
```
```  3120 apply (subst suminf_split_initial_segment [where k=k], assumption, simp)
```
```  3121 apply (drule_tac k=k in summable_ignore_initial_segment)
```
```  3122 apply (drule_tac k="Suc (Suc 0)" in sums_group [OF summable_sums], simp)
```
```  3123 apply simp
```
```  3124 by (metis (no_types, lifting) add.commute suminf_pos summable_def sums_unique)
```
```  3125
```
```  3126 lemma cos_two_less_zero [simp]:
```
```  3127   "cos 2 < (0::real)"
```
```  3128 proof -
```
```  3129   note fact.simps(2) [simp del]
```
```  3130   from sums_minus [OF cos_paired]
```
```  3131   have *: "(\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / fact (2 * n))) sums - cos (2::real)"
```
```  3132     by simp
```
```  3133   then have sm: "summable (\<lambda>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3134     by (rule sums_summable)
```
```  3135   have "0 < (\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3136     by (simp add: fact_num_eq_if realpow_num_eq_if)
```
```  3137   moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n  * 2 ^ (2 * n) / (fact (2 * n))))
```
```  3138                  < (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3139   proof -
```
```  3140     { fix d
```
```  3141       let ?six4d = "Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))"
```
```  3142       have "(4::real) * (fact (?six4d)) < (Suc (Suc (?six4d)) * fact (Suc (?six4d)))"
```
```  3143         unfolding real_of_nat_mult
```
```  3144         by (rule mult_strict_mono) (simp_all add: fact_less_mono)
```
```  3145       then have "(4::real) * (fact (?six4d)) < (fact (Suc (Suc (?six4d))))"
```
```  3146         by (simp only: fact.simps(2) [of "Suc (?six4d)"] real_of_nat_def of_nat_mult of_nat_fact)
```
```  3147       then have "(4::real) * inverse (fact (Suc (Suc (?six4d)))) < inverse (fact (?six4d))"
```
```  3148         by (simp add: inverse_eq_divide less_divide_eq)
```
```  3149     }
```
```  3150     then show ?thesis
```
```  3151       by (force intro!: sumr_pos_lt_pair [OF sm] simp add: divide_inverse algebra_simps)
```
```  3152   qed
```
```  3153   ultimately have "0 < (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3154     by (rule order_less_trans)
```
```  3155   moreover from * have "- cos 2 = (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3156     by (rule sums_unique)
```
```  3157   ultimately have "(0::real) < - cos 2" by simp
```
```  3158   then show ?thesis by simp
```
```  3159 qed
```
```  3160
```
```  3161 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
```
```  3162 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
```
```  3163
```
```  3164 lemma cos_is_zero: "EX! x::real. 0 \<le> x & x \<le> 2 \<and> cos x = 0"
```
```  3165 proof (rule ex_ex1I)
```
```  3166   show "\<exists>x::real. 0 \<le> x & x \<le> 2 & cos x = 0"
```
```  3167     by (rule IVT2, simp_all)
```
```  3168 next
```
```  3169   fix x::real and y::real
```
```  3170   assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
```
```  3171   assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
```
```  3172   have [simp]: "\<forall>x::real. cos differentiable (at x)"
```
```  3173     unfolding real_differentiable_def by (auto intro: DERIV_cos)
```
```  3174   from x y show "x = y"
```
```  3175     apply (cut_tac less_linear [of x y], auto)
```
```  3176     apply (drule_tac f = cos in Rolle)
```
```  3177     apply (drule_tac [5] f = cos in Rolle)
```
```  3178     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  3179     apply (metis order_less_le_trans less_le sin_gt_zero_02)
```
```  3180     apply (metis order_less_le_trans less_le sin_gt_zero_02)
```
```  3181     done
```
```  3182 qed
```
```  3183
```
```  3184 lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
```
```  3185   by (simp add: pi_def)
```
```  3186
```
```  3187 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
```
```  3188   by (simp add: pi_half cos_is_zero [THEN theI'])
```
```  3189
```
```  3190 lemma cos_of_real_pi_half [simp]:
```
```  3191   fixes x :: "'a :: {real_field,banach,real_normed_algebra_1}"
```
```  3192   shows "cos ((of_real pi / 2) :: 'a) = 0"
```
```  3193 by (metis cos_pi_half cos_of_real eq_numeral_simps(4) nonzero_of_real_divide of_real_0 of_real_numeral)
```
```  3194
```
```  3195 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
```
```  3196   apply (rule order_le_neq_trans)
```
```  3197   apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  3198   apply (metis cos_pi_half cos_zero zero_neq_one)
```
```  3199   done
```
```  3200
```
```  3201 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
```
```  3202 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
```
```  3203
```
```  3204 lemma pi_half_less_two [simp]: "pi / 2 < 2"
```
```  3205   apply (rule order_le_neq_trans)
```
```  3206   apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  3207   apply (metis cos_pi_half cos_two_neq_zero)
```
```  3208   done
```
```  3209
```
```  3210 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
```
```  3211 lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
```
```  3212
```
```  3213 lemma pi_gt_zero [simp]: "0 < pi"
```
```  3214   using pi_half_gt_zero by simp
```
```  3215
```
```  3216 lemma pi_ge_zero [simp]: "0 \<le> pi"
```
```  3217   by (rule pi_gt_zero [THEN order_less_imp_le])
```
```  3218
```
```  3219 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
```
```  3220   by (rule pi_gt_zero [THEN less_imp_neq, symmetric])
```
```  3221
```
```  3222 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
```
```  3223   by (simp add: linorder_not_less)
```
```  3224
```
```  3225 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
```
```  3226   by simp
```
```  3227
```
```  3228 lemma m2pi_less_pi: "- (2*pi) < pi"
```
```  3229   by simp
```
```  3230
```
```  3231 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
```
```  3232   using sin_cos_squared_add2 [where x = "pi/2"]
```
```  3233   using sin_gt_zero_02 [OF pi_half_gt_zero pi_half_less_two]
```
```  3234   by (simp add: power2_eq_1_iff)
```
```  3235
```
```  3236 lemma sin_of_real_pi_half [simp]:
```
```  3237   fixes x :: "'a :: {real_field,banach,real_normed_algebra_1}"
```
```  3238   shows "sin ((of_real pi / 2) :: 'a) = 1"
```
```  3239   using sin_pi_half
```
```  3240 by (metis sin_pi_half eq_numeral_simps(4) nonzero_of_real_divide of_real_1 of_real_numeral sin_of_real)
```
```  3241
```
```  3242 lemma sin_cos_eq:
```
```  3243   fixes x :: "'a::{real_normed_field,banach}"
```
```  3244   shows "sin x = cos (of_real pi / 2 - x)"
```
```  3245   by (simp add: cos_diff)
```
```  3246
```
```  3247 lemma minus_sin_cos_eq:
```
```  3248   fixes x :: "'a::{real_normed_field,banach}"
```
```  3249   shows "-sin x = cos (x + of_real pi / 2)"
```
```  3250   by (simp add: cos_add nonzero_of_real_divide)
```
```  3251
```
```  3252 lemma cos_sin_eq:
```
```  3253   fixes x :: "'a::{real_normed_field,banach}"
```
```  3254   shows "cos x = sin (of_real pi / 2 - x)"
```
```  3255   using sin_cos_eq [of "of_real pi / 2 - x"]
```
```  3256   by simp
```
```  3257
```
```  3258 lemma sin_add:
```
```  3259   fixes x :: "'a::{real_normed_field,banach}"
```
```  3260   shows "sin (x + y) = sin x * cos y + cos x * sin y"
```
```  3261   using cos_add [of "of_real pi / 2 - x" "-y"]
```
```  3262   by (simp add: cos_sin_eq) (simp add: sin_cos_eq)
```
```  3263
```
```  3264 lemma sin_diff:
```
```  3265   fixes x :: "'a::{real_normed_field,banach}"
```
```  3266   shows "sin (x - y) = sin x * cos y - cos x * sin y"
```
```  3267   using sin_add [of x "- y"] by simp
```
```  3268
```
```  3269 lemma sin_double:
```
```  3270   fixes x :: "'a::{real_normed_field,banach}"
```
```  3271   shows "sin(2 * x) = 2 * sin x * cos x"
```
```  3272   using sin_add [where x=x and y=x] by simp
```
```  3273
```
```  3274
```
```  3275 lemma cos_of_real_pi [simp]: "cos (of_real pi) = -1"
```
```  3276   using cos_add [where x = "pi/2" and y = "pi/2"]
```
```  3277   by (simp add: cos_of_real)
```
```  3278
```
```  3279 lemma sin_of_real_pi [simp]: "sin (of_real pi) = 0"
```
```  3280   using sin_add [where x = "pi/2" and y = "pi/2"]
```
```  3281   by (simp add: sin_of_real)
```
```  3282
```
```  3283 lemma cos_pi [simp]: "cos pi = -1"
```
```  3284   using cos_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  3285
```
```  3286 lemma sin_pi [simp]: "sin pi = 0"
```
```  3287   using sin_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  3288
```
```  3289 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
```
```  3290   by (simp add: sin_add)
```
```  3291
```
```  3292 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
```
```  3293   by (simp add: sin_add)
```
```  3294
```
```  3295 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
```
```  3296   by (simp add: cos_add)
```
```  3297
```
```  3298 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
```
```  3299   by (simp add: sin_add sin_double cos_double)
```
```  3300
```
```  3301 lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
```
```  3302   by (simp add: cos_add sin_double cos_double)
```
```  3303
```
```  3304 lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n"
```
```  3305   by (induct n) (auto simp: real_of_nat_Suc distrib_right)
```
```  3306
```
```  3307 lemma cos_npi2 [simp]: "cos (pi * real n) = (- 1) ^ n"
```
```  3308   by (metis cos_npi mult.commute)
```
```  3309
```
```  3310 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
```
```  3311   by (induct n) (auto simp: real_of_nat_Suc distrib_right)
```
```  3312
```
```  3313 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
```
```  3314   by (simp add: mult.commute [of pi])
```
```  3315
```
```  3316 lemma cos_two_pi [simp]: "cos (2*pi) = 1"
```
```  3317   by (simp add: cos_double)
```
```  3318
```
```  3319 lemma sin_two_pi [simp]: "sin (2*pi) = 0"
```
```  3320   by (simp add: sin_double)
```
```  3321
```
```  3322
```
```  3323 lemma sin_times_sin:
```
```  3324   fixes w :: "'a::{real_normed_field,banach}"
```
```  3325   shows "sin(w) * sin(z) = (cos(w - z) - cos(w + z)) / 2"
```
```  3326   by (simp add: cos_diff cos_add)
```
```  3327
```
```  3328 lemma sin_times_cos:
```
```  3329   fixes w :: "'a::{real_normed_field,banach}"
```
```  3330   shows "sin(w) * cos(z) = (sin(w + z) + sin(w - z)) / 2"
```
```  3331   by (simp add: sin_diff sin_add)
```
```  3332
```
```  3333 lemma cos_times_sin:
```
```  3334   fixes w :: "'a::{real_normed_field,banach}"
```
```  3335   shows "cos(w) * sin(z) = (sin(w + z) - sin(w - z)) / 2"
```
```  3336   by (simp add: sin_diff sin_add)
```
```  3337
```
```  3338 lemma cos_times_cos:
```
```  3339   fixes w :: "'a::{real_normed_field,banach}"
```
```  3340   shows "cos(w) * cos(z) = (cos(w - z) + cos(w + z)) / 2"
```
```  3341   by (simp add: cos_diff cos_add)
```
```  3342
```
```  3343 lemma sin_plus_sin:  (*FIXME field should not be necessary*)
```
```  3344   fixes w :: "'a::{real_normed_field,banach,field}"
```
```  3345   shows "sin(w) + sin(z) = 2 * sin((w + z) / 2) * cos((w - z) / 2)"
```
```  3346   apply (simp add: mult.assoc sin_times_cos)
```
```  3347   apply (simp add: field_simps)
```
```  3348   done
```
```  3349
```
```  3350 lemma sin_diff_sin:
```
```  3351   fixes w :: "'a::{real_normed_field,banach,field}"
```
```  3352   shows "sin(w) - sin(z) = 2 * sin((w - z) / 2) * cos((w + z) / 2)"
```
```  3353   apply (simp add: mult.assoc sin_times_cos)
```
```  3354   apply (simp add: field_simps)
```
```  3355   done
```
```  3356
```
```  3357 lemma cos_plus_cos:
```
```  3358   fixes w :: "'a::{real_normed_field,banach,field}"
```
```  3359   shows "cos(w) + cos(z) = 2 * cos((w + z) / 2) * cos((w - z) / 2)"
```
```  3360   apply (simp add: mult.assoc cos_times_cos)
```
```  3361   apply (simp add: field_simps)
```
```  3362   done
```
```  3363
```
```  3364 lemma cos_diff_cos:
```
```  3365   fixes w :: "'a::{real_normed_field,banach,field}"
```
```  3366   shows "cos(w) - cos(z) = 2 * sin((w + z) / 2) * sin((z - w) / 2)"
```
```  3367   apply (simp add: mult.assoc sin_times_sin)
```
```  3368   apply (simp add: field_simps)
```
```  3369   done
```
```  3370
```
```  3371 lemma cos_double_cos:
```
```  3372   fixes z :: "'a::{real_normed_field,banach}"
```
```  3373   shows "cos(2 * z) = 2 * cos z ^ 2 - 1"
```
```  3374 by (simp add: cos_double sin_squared_eq)
```
```  3375
```
```  3376 lemma cos_double_sin:
```
```  3377   fixes z :: "'a::{real_normed_field,banach}"
```
```  3378   shows "cos(2 * z) = 1 - 2 * sin z ^ 2"
```
```  3379 by (simp add: cos_double sin_squared_eq)
```
```  3380
```
```  3381 lemma sin_pi_minus [simp]: "sin (pi - x) = sin x"
```
```  3382   by (metis sin_minus sin_periodic_pi minus_minus uminus_add_conv_diff)
```
```  3383
```
```  3384 lemma cos_pi_minus [simp]: "cos (pi - x) = -(cos x)"
```
```  3385   by (metis cos_minus cos_periodic_pi uminus_add_conv_diff)
```
```  3386
```
```  3387 lemma sin_minus_pi [simp]: "sin (x - pi) = - (sin x)"
```
```  3388   by (simp add: sin_diff)
```
```  3389
```
```  3390 lemma cos_minus_pi [simp]: "cos (x - pi) = -(cos x)"
```
```  3391   by (simp add: cos_diff)
```
```  3392
```
```  3393 lemma sin_2pi_minus [simp]: "sin (2*pi - x) = -(sin x)"
```
```  3394   by (metis sin_periodic_pi2 add_diff_eq mult_2 sin_pi_minus)
```
```  3395
```
```  3396 lemma cos_2pi_minus [simp]: "cos (2*pi - x) = cos x"
```
```  3397   by (metis (no_types, hide_lams) cos_add cos_minus cos_two_pi sin_minus sin_two_pi
```
```  3398            diff_0_right minus_diff_eq mult_1 mult_zero_left uminus_add_conv_diff)
```
```  3399
```
```  3400 lemma sin_gt_zero2: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < sin x"
```
```  3401   by (metis sin_gt_zero_02 order_less_trans pi_half_less_two)
```
```  3402
```
```  3403 lemma sin_less_zero:
```
```  3404   assumes "- pi/2 < x" and "x < 0"
```
```  3405   shows "sin x < 0"
```
```  3406 proof -
```
```  3407   have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
```
```  3408   thus ?thesis by simp
```
```  3409 qed
```
```  3410
```
```  3411 lemma pi_less_4: "pi < 4"
```
```  3412   using pi_half_less_two by auto
```
```  3413
```
```  3414 lemma cos_gt_zero: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < cos x"
```
```  3415   by (simp add: cos_sin_eq sin_gt_zero2)
```
```  3416
```
```  3417 lemma cos_gt_zero_pi: "\<lbrakk>-(pi/2) < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < cos x"
```
```  3418   using cos_gt_zero [of x] cos_gt_zero [of "-x"]
```
```  3419   by (cases rule: linorder_cases [of x 0]) auto
```
```  3420
```
```  3421 lemma cos_ge_zero: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2\<rbrakk> \<Longrightarrow> 0 \<le> cos x"
```
```  3422   apply (auto simp: order_le_less cos_gt_zero_pi)
```
```  3423   by (metis cos_pi_half eq_divide_eq eq_numeral_simps(4))
```
```  3424
```
```  3425 lemma sin_gt_zero: "\<lbrakk>0 < x; x < pi \<rbrakk> \<Longrightarrow> 0 < sin x"
```
```  3426   by (simp add: sin_cos_eq cos_gt_zero_pi)
```
```  3427
```
```  3428 lemma sin_lt_zero: "pi < x \<Longrightarrow> x < 2*pi \<Longrightarrow> sin x < 0"
```
```  3429   using sin_gt_zero [of "x-pi"]
```
```  3430   by (simp add: sin_diff)
```
```  3431
```
```  3432 lemma pi_ge_two: "2 \<le> pi"
```
```  3433 proof (rule ccontr)
```
```  3434   assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
```
```  3435   have "\<exists>y > pi. y < 2 \<and> y < 2*pi"
```
```  3436   proof (cases "2 < 2*pi")
```
```  3437     case True with dense[OF \<open>pi < 2\<close>] show ?thesis by auto
```
```  3438   next
```
```  3439     case False have "pi < 2*pi" by auto
```
```  3440     from dense[OF this] and False show ?thesis by auto
```
```  3441   qed
```
```  3442   then obtain y where "pi < y" and "y < 2" and "y < 2*pi" by blast
```
```  3443   hence "0 < sin y" using sin_gt_zero_02 by auto
```
```  3444   moreover
```
```  3445   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
```
```  3446   ultimately show False by auto
```
```  3447 qed
```
```  3448
```
```  3449 lemma sin_ge_zero: "\<lbrakk>0 \<le> x; x \<le> pi\<rbrakk> \<Longrightarrow> 0 \<le> sin x"
```
```  3450   by (auto simp: order_le_less sin_gt_zero)
```
```  3451
```
```  3452 lemma sin_le_zero: "pi \<le> x \<Longrightarrow> x < 2*pi \<Longrightarrow> sin x \<le> 0"
```
```  3453   using sin_ge_zero [of "x-pi"]
```
```  3454   by (simp add: sin_diff)
```
```  3455
```
```  3456 text \<open>FIXME: This proof is almost identical to lemma @{text cos_is_zero}.
```
```  3457   It should be possible to factor out some of the common parts.\<close>
```
```  3458
```
```  3459 lemma cos_total: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
```
```  3460 proof (rule ex_ex1I)
```
```  3461   assume y: "-1 \<le> y" "y \<le> 1"
```
```  3462   show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
```
```  3463     by (rule IVT2, simp_all add: y)
```
```  3464 next
```
```  3465   fix a b
```
```  3466   assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
```
```  3467   assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
```
```  3468   have [simp]: "\<forall>x::real. cos differentiable (at x)"
```
```  3469     unfolding real_differentiable_def by (auto intro: DERIV_cos)
```
```  3470   from a b show "a = b"
```
```  3471     apply (cut_tac less_linear [of a b], auto)
```
```  3472     apply (drule_tac f = cos in Rolle)
```
```  3473     apply (drule_tac [5] f = cos in Rolle)
```
```  3474     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  3475     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  3476     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  3477     done
```
```  3478 qed
```
```  3479
```
```  3480 lemma sin_total:
```
```  3481   assumes y: "-1 \<le> y" "y \<le> 1"
```
```  3482     shows "\<exists>! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
```
```  3483 proof -
```
```  3484   from cos_total [OF y]
```
```  3485   obtain x where x: "0 \<le> x" "x \<le> pi" "cos x = y"
```
```  3486            and uniq: "\<And>x'. 0 \<le> x' \<Longrightarrow> x' \<le> pi \<Longrightarrow> cos x' = y \<Longrightarrow> x' = x "
```
```  3487     by blast
```
```  3488   show ?thesis
```
```  3489     apply (simp add: sin_cos_eq)
```
```  3490     apply (rule ex1I [where a="pi/2 - x"])
```
```  3491     apply (cut_tac [2] x'="pi/2 - xa" in uniq)
```
```  3492     using x
```
```  3493     apply auto
```
```  3494     done
```
```  3495 qed
```
```  3496
```
```  3497 lemma reals_Archimedean4':
```
```  3498      "\<lbrakk>0 < y; 0 \<le> x\<rbrakk> \<Longrightarrow> \<exists>n. real n * y \<le> x \<and> x < real (Suc n) * y"
```
```  3499 apply (rule_tac x="nat (floor (x/y))" in exI)
```
```  3500 using floor_correct [of "x/y"]
```
```  3501 apply (auto simp: Real.real_of_nat_Suc field_simps)
```
```  3502 done
```
```  3503
```
```  3504 lemma cos_zero_lemma:
```
```  3505      "\<lbrakk>0 \<le> x; cos x = 0\<rbrakk> \<Longrightarrow>
```
```  3506       \<exists>n::nat. odd n & x = real n * (pi/2)"
```
```  3507 apply (erule reals_Archimedean4 [OF pi_gt_zero])
```
```  3508 apply (auto simp: )
```
```  3509 apply (subgoal_tac "0 \<le> x - real n * pi &
```
```  3510                     (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
```
```  3511 apply (auto simp: algebra_simps real_of_nat_Suc)
```
```  3512  prefer 2 apply (simp add: cos_diff)
```
```  3513 apply (simp add: cos_diff)
```
```  3514 apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
```
```  3515 apply (rule_tac [2] cos_total, safe)
```
```  3516 apply (drule_tac x = "x - real n * pi" in spec)
```
```  3517 apply (drule_tac x = "pi/2" in spec)
```
```  3518 apply (simp add: cos_diff)
```
```  3519 apply (rule_tac x = "Suc (2 * n)" in exI)
```
```  3520 apply (simp add: real_of_nat_Suc algebra_simps, auto)
```
```  3521 done
```
```  3522
```
```  3523 lemma sin_zero_lemma:
```
```  3524      "\<lbrakk>0 \<le> x; sin x = 0\<rbrakk> \<Longrightarrow>
```
```  3525       \<exists>n::nat. even n & x = real n * (pi/2)"
```
```  3526 apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
```
```  3527  apply (clarify, rule_tac x = "n - 1" in exI)
```
```  3528  apply (auto elim!: oddE simp add: real_of_nat_Suc field_simps)[1]
```
```  3529  apply (rule cos_zero_lemma)
```
```  3530  apply (auto simp: cos_add)
```
```  3531 done
```
```  3532
```
```  3533 lemma cos_zero_iff:
```
```  3534      "(cos x = 0) =
```
```  3535       ((\<exists>n::nat. odd n & (x = real n * (pi/2))) |
```
```  3536        (\<exists>n::nat. odd n & (x = -(real n * (pi/2)))))"
```
```  3537 proof -
```
```  3538   { fix n :: nat
```
```  3539     assume "odd n"
```
```  3540     then obtain m where "n = 2 * m + 1" ..
```
```  3541     then have "cos (real n * pi / 2) = 0"
```
```  3542       by (simp add: field_simps real_of_nat_Suc) (simp add: cos_add add_divide_distrib)
```
```  3543   } note * = this
```
```  3544   show ?thesis
```
```  3545   apply (rule iffI)
```
```  3546   apply (cut_tac linorder_linear [of 0 x], safe)
```
```  3547   apply (drule cos_zero_lemma, assumption+)
```
```  3548   apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
```
```  3549   apply (auto dest: *)
```
```  3550   done
```
```  3551 qed
```
```  3552
```
```  3553 (* ditto: but to a lesser extent *)
```
```  3554 lemma sin_zero_iff:
```
```  3555      "(sin x = 0) =
```
```  3556       ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
```
```  3557        (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
```
```  3558 apply (rule iffI)
```
```  3559 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  3560 apply (drule sin_zero_lemma, assumption+)
```
```  3561 apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
```
```  3562 apply (force simp add: minus_equation_iff [of x])
```
```  3563 apply (auto elim: evenE)
```
```  3564 done
```
```  3565
```
```  3566
```
```  3567 lemma cos_zero_iff_int:
```
```  3568      "cos x = 0 \<longleftrightarrow> (\<exists>n::int. odd n & x = real n * (pi/2))"
```
```  3569 proof safe
```
```  3570   assume "cos x = 0"
```
```  3571   then show "\<exists>n::int. odd n & x = real n * (pi/2)"
```
```  3572     apply (simp add: cos_zero_iff, safe)
```
```  3573     apply (metis even_int_iff real_of_int_of_nat_eq)
```
```  3574     apply (rule_tac x="- (int n)" in exI, simp)
```
```  3575     done
```
```  3576 next
```
```  3577   fix n::int
```
```  3578   assume "odd n"
```
```  3579   then show "cos (real n * (pi / 2)) = 0"
```
```  3580     apply (simp add: cos_zero_iff)
```
```  3581     apply (case_tac n rule: int_cases2, simp)
```
```  3582     apply (rule disjI2)
```
```  3583     apply (rule_tac x="nat (-n)" in exI, simp)
```
```  3584     done
```
```  3585 qed
```
```  3586
```
```  3587 lemma sin_zero_iff_int:
```
```  3588      "sin x = 0 \<longleftrightarrow> (\<exists>n::int. even n & (x = real n * (pi/2)))"
```
```  3589 proof safe
```
```  3590   assume "sin x = 0"
```
```  3591   then show "\<exists>n::int. even n \<and> x = real n * (pi / 2)"
```
```  3592     apply (simp add: sin_zero_iff, safe)
```
```  3593     apply (metis even_int_iff real_of_int_of_nat_eq)
```
```  3594     apply (rule_tac x="- (int n)" in exI, simp)
```
```  3595     done
```
```  3596 next
```
```  3597   fix n::int
```
```  3598   assume "even n"
```
```  3599   then show "sin (real n * (pi / 2)) = 0"
```
```  3600     apply (simp add: sin_zero_iff)
```
```  3601     apply (case_tac n rule: int_cases2, simp)
```
```  3602     apply (rule disjI2)
```
```  3603     apply (rule_tac x="nat (-n)" in exI, simp)
```
```  3604     done
```
```  3605 qed
```
```  3606
```
```  3607 lemma sin_zero_iff_int2:
```
```  3608   "sin x = 0 \<longleftrightarrow> (\<exists>n::int. x = real n * pi)"
```
```  3609   apply (simp only: sin_zero_iff_int)
```
```  3610   apply (safe elim!: evenE)
```
```  3611   apply (simp_all add: field_simps)
```
```  3612   apply (subst real_numeral(1) [symmetric])
```
```  3613   apply (simp only: real_of_int_mult [symmetric] real_of_int_inject)
```
```  3614   apply auto
```
```  3615   done
```
```  3616
```
```  3617 lemma cos_monotone_0_pi:
```
```  3618   assumes "0 \<le> y" and "y < x" and "x \<le> pi"
```
```  3619   shows "cos x < cos y"
```
```  3620 proof -
```
```  3621   have "- (x - y) < 0" using assms by auto
```
```  3622
```
```  3623   from MVT2[OF \<open>y < x\<close> DERIV_cos[THEN impI, THEN allI]]
```
```  3624   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z"
```
```  3625     by auto
```
```  3626   hence "0 < z" and "z < pi" using assms by auto
```
```  3627   hence "0 < sin z" using sin_gt_zero by auto
```
```  3628   hence "cos x - cos y < 0"
```
```  3629     unfolding cos_diff minus_mult_commute[symmetric]
```
```  3630     using \<open>- (x - y) < 0\<close> by (rule mult_pos_neg2)
```
```  3631   thus ?thesis by auto
```
```  3632 qed
```
```  3633
```
```  3634 lemma cos_monotone_0_pi_le:
```
```  3635   assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi"
```
```  3636   shows "cos x \<le> cos y"
```
```  3637 proof (cases "y < x")
```
```  3638   case True
```
```  3639   show ?thesis
```
```  3640     using cos_monotone_0_pi[OF \<open>0 \<le> y\<close> True \<open>x \<le> pi\<close>] by auto
```
```  3641 next
```
```  3642   case False
```
```  3643   hence "y = x" using \<open>y \<le> x\<close> by auto
```
```  3644   thus ?thesis by auto
```
```  3645 qed
```
```  3646
```
```  3647 lemma cos_monotone_minus_pi_0:
```
```  3648   assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
```
```  3649   shows "cos y < cos x"
```
```  3650 proof -
```
```  3651   have "0 \<le> -x" and "-x < -y" and "-y \<le> pi"
```
```  3652     using assms by auto
```
```  3653   from cos_monotone_0_pi[OF this] show ?thesis
```
```  3654     unfolding cos_minus .
```
```  3655 qed
```
```  3656
```
```  3657 lemma cos_monotone_minus_pi_0':
```
```  3658   assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0"
```
```  3659   shows "cos y \<le> cos x"
```
```  3660 proof (cases "y < x")
```
```  3661   case True
```
```  3662   show ?thesis using cos_monotone_minus_pi_0[OF \<open>-pi \<le> y\<close> True \<open>x \<le> 0\<close>]
```
```  3663     by auto
```
```  3664 next
```
```  3665   case False
```
```  3666   hence "y = x" using \<open>y \<le> x\<close> by auto
```
```  3667   thus ?thesis by auto
```
```  3668 qed
```
```  3669
```
```  3670 lemma sin_monotone_2pi:
```
```  3671   assumes "- (pi/2) \<le> y" and "y < x" and "x \<le> pi/2"
```
```  3672   shows "sin y < sin x"
```
```  3673     apply (simp add: sin_cos_eq)
```
```  3674     apply (rule cos_monotone_0_pi)
```
```  3675     using assms
```
```  3676     apply auto
```
```  3677     done
```
```  3678
```
```  3679 lemma sin_monotone_2pi_le:
```
```  3680   assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2"
```
```  3681   shows "sin y \<le> sin x"
```
```  3682   by (metis assms le_less sin_monotone_2pi)
```
```  3683
```
```  3684 lemma sin_x_le_x:
```
```  3685   fixes x::real assumes x: "x \<ge> 0" shows "sin x \<le> x"
```
```  3686 proof -
```
```  3687   let ?f = "\<lambda>x. x - sin x"
```
```  3688   from x have "?f x \<ge> ?f 0"
```
```  3689     apply (rule DERIV_nonneg_imp_nondecreasing)
```
```  3690     apply (intro allI impI exI[of _ "1 - cos x" for x])
```
```  3691     apply (auto intro!: derivative_eq_intros simp: field_simps)
```
```  3692     done
```
```  3693   thus "sin x \<le> x" by simp
```
```  3694 qed
```
```  3695
```
```  3696 lemma sin_x_ge_neg_x:
```
```  3697   fixes x::real assumes x: "x \<ge> 0" shows "sin x \<ge> - x"
```
```  3698 proof -
```
```  3699   let ?f = "\<lambda>x. x + sin x"
```
```  3700   from x have "?f x \<ge> ?f 0"
```
```  3701     apply (rule DERIV_nonneg_imp_nondecreasing)
```
```  3702     apply (intro allI impI exI[of _ "1 + cos x" for x])
```
```  3703     apply (auto intro!: derivative_eq_intros simp: field_simps real_0_le_add_iff)
```
```  3704     done
```
```  3705   thus "sin x \<ge> -x" by simp
```
```  3706 qed
```
```  3707
```
```  3708 lemma abs_sin_x_le_abs_x:
```
```  3709   fixes x::real shows "\<bar>sin x\<bar> \<le> \<bar>x\<bar>"
```
```  3710   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"]
```
```  3711   by (auto simp: abs_real_def)
```
```  3712
```
```  3713
```
```  3714 subsection \<open>More Corollaries about Sine and Cosine\<close>
```
```  3715
```
```  3716 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
```
```  3717 proof -
```
```  3718   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
```
```  3719     by (auto simp: algebra_simps sin_add)
```
```  3720   thus ?thesis
```
```  3721     by (simp add: real_of_nat_Suc distrib_right add_divide_distrib
```
```  3722                   mult.commute [of pi])
```
```  3723 qed
```
```  3724
```
```  3725 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
```
```  3726   by (cases "even n") (simp_all add: cos_double mult.assoc)
```
```  3727
```
```  3728 lemma cos_3over2_pi [simp]: "cos (3/2*pi) = 0"
```
```  3729   apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
```
```  3730   apply (subst cos_add, simp)
```
```  3731   done
```
```  3732
```
```  3733 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
```
```  3734   by (auto simp: mult.assoc sin_double)
```
```  3735
```
```  3736 lemma sin_3over2_pi [simp]: "sin (3/2*pi) = - 1"
```
```  3737   apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
```
```  3738   apply (subst sin_add, simp)
```
```  3739   done
```
```  3740
```
```  3741 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
```
```  3742 by (simp only: cos_add sin_add real_of_nat_Suc distrib_right distrib_left add_divide_distrib, auto)
```
```  3743
```
```  3744 lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)"
```
```  3745   by (auto intro!: derivative_eq_intros)
```
```  3746
```
```  3747 lemma sin_zero_norm_cos_one:
```
```  3748   fixes x :: "'a::{real_normed_field,banach}"
```
```  3749   assumes "sin x = 0" shows "norm (cos x) = 1"
```
```  3750   using sin_cos_squared_add [of x, unfolded assms]
```
```  3751   by (simp add: square_norm_one)
```
```  3752
```
```  3753 lemma sin_zero_abs_cos_one: "sin x = 0 \<Longrightarrow> \<bar>cos x\<bar> = (1::real)"
```
```  3754   using sin_zero_norm_cos_one by fastforce
```
```  3755
```
```  3756 lemma cos_one_sin_zero:
```
```  3757   fixes x :: "'a::{real_normed_field,banach}"
```
```  3758   assumes "cos x = 1" shows "sin x = 0"
```
```  3759   using sin_cos_squared_add [of x, unfolded assms]
```
```  3760   by simp
```
```  3761
```
```  3762 lemma sin_times_pi_eq_0: "sin(x * pi) = 0 \<longleftrightarrow> x \<in> \<int>"
```
```  3763   by (simp add: sin_zero_iff_int2) (metis Ints_cases Ints_real_of_int real_of_int_def)
```
```  3764
```
```  3765 lemma cos_one_2pi:
```
```  3766     "cos(x) = 1 \<longleftrightarrow> (\<exists>n::nat. x = n * 2*pi) | (\<exists>n::nat. x = -(n * 2*pi))"
```
```  3767     (is "?lhs = ?rhs")
```
```  3768 proof
```
```  3769   assume "cos(x) = 1"
```
```  3770   then have "sin x = 0"
```
```  3771     by (simp add: cos_one_sin_zero)
```
```  3772   then show ?rhs
```
```  3773   proof (simp only: sin_zero_iff, elim exE disjE conjE)
```
```  3774     fix n::nat
```
```  3775     assume n: "even n" "x = real n * (pi/2)"
```
```  3776     then obtain m where m: "n = 2 * m"
```
```  3777       using dvdE by blast
```
```  3778     then have me: "even m" using \<open>?lhs\<close> n
```
```  3779       by (auto simp: field_simps) (metis one_neq_neg_one  power_minus_odd power_one)
```
```  3780     show ?rhs
```
```  3781       using m me n
```
```  3782       by (auto simp: field_simps elim!: evenE)
```
```  3783   next
```
```  3784     fix n::nat
```
```  3785     assume n: "even n" "x = - (real n * (pi/2))"
```
```  3786     then obtain m where m: "n = 2 * m"
```
```  3787       using dvdE by blast
```
```  3788     then have me: "even m" using \<open>?lhs\<close> n
```
```  3789       by (auto simp: field_simps) (metis one_neq_neg_one  power_minus_odd power_one)
```
```  3790     show ?rhs
```
```  3791       using m me n
```
```  3792       by (auto simp: field_simps elim!: evenE)
```
```  3793   qed
```
```  3794 next
```
```  3795   assume "?rhs"
```
```  3796   then show "cos x = 1"
```
```  3797     by (metis cos_2npi cos_minus mult.assoc mult.left_commute)
```
```  3798 qed
```
```  3799
```
```  3800 lemma cos_one_2pi_int: "cos(x) = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2*pi)"
```
```  3801   apply auto  --\<open>FIXME simproc bug\<close>
```
```  3802   apply (auto simp: cos_one_2pi)
```
```  3803   apply (metis real_of_int_of_nat_eq)
```
```  3804   apply (metis mult_minus_right real_of_int_minus real_of_int_of_nat_eq)
```
```  3805   by (metis mult_minus_right of_int_of_nat real_of_int_def real_of_nat_def)
```
```  3806
```
```  3807 lemma sin_cos_sqrt: "0 \<le> sin(x) \<Longrightarrow> (sin(x) = sqrt(1 - (cos(x) ^ 2)))"
```
```  3808   using sin_squared_eq real_sqrt_unique by fastforce
```
```  3809
```
```  3810 lemma sin_eq_0_pi: "-pi < x \<Longrightarrow> x < pi \<Longrightarrow> sin(x) = 0 \<Longrightarrow> x = 0"
```
```  3811   by (metis sin_gt_zero sin_minus minus_less_iff neg_0_less_iff_less not_less_iff_gr_or_eq)
```
```  3812
```
```  3813 lemma cos_treble_cos:
```
```  3814   fixes x :: "'a::{real_normed_field,banach}"
```
```  3815   shows "cos(3 * x) = 4 * cos(x) ^ 3 - 3 * cos x"
```
```  3816 proof -
```
```  3817   have *: "(sin x * (sin x * 3)) = 3 - (cos x * (cos x * 3))"
```
```  3818     by (simp add: mult.assoc [symmetric] sin_squared_eq [unfolded power2_eq_square])
```
```  3819   have "cos(3 * x) = cos(2*x + x)"
```
```  3820     by simp
```
```  3821   also have "... = 4 * cos(x) ^ 3 - 3 * cos x"
```
```  3822     apply (simp only: cos_add cos_double sin_double)
```
```  3823     apply (simp add: * field_simps power2_eq_square power3_eq_cube)
```
```  3824     done
```
```  3825   finally show ?thesis .
```
```  3826 qed
```
```  3827
```
```  3828 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
```
```  3829 proof -
```
```  3830   let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
```
```  3831   have nonneg: "0 \<le> ?c"
```
```  3832     by (simp add: cos_ge_zero)
```
```  3833   have "0 = cos (pi / 4 + pi / 4)"
```
```  3834     by simp
```
```  3835   also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2"
```
```  3836     by (simp only: cos_add power2_eq_square)
```
```  3837   also have "\<dots> = 2 * ?c\<^sup>2 - 1"
```
```  3838     by (simp add: sin_squared_eq)
```
```  3839   finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2"
```
```  3840     by (simp add: power_divide)
```
```  3841   thus ?thesis
```
```  3842     using nonneg by (rule power2_eq_imp_eq) simp
```
```  3843 qed
```
```  3844
```
```  3845 lemma cos_30: "cos (pi / 6) = sqrt 3/2"
```
```  3846 proof -
```
```  3847   let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
```
```  3848   have pos_c: "0 < ?c"
```
```  3849     by (rule cos_gt_zero, simp, simp)
```
```  3850   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
```
```  3851     by simp
```
```  3852   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
```
```  3853     by (simp only: cos_add sin_add)
```
```  3854   also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)"
```
```  3855     by (simp add: algebra_simps power2_eq_square)
```
```  3856   finally have "?c\<^sup>2 = (sqrt 3/2)\<^sup>2"
```
```  3857     using pos_c by (simp add: sin_squared_eq power_divide)
```
```  3858   thus ?thesis
```
```  3859     using pos_c [THEN order_less_imp_le]
```
```  3860     by (rule power2_eq_imp_eq) simp
```
```  3861 qed
```
```  3862
```
```  3863 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
```
```  3864   by (simp add: sin_cos_eq cos_45)
```
```  3865
```
```  3866 lemma sin_60: "sin (pi / 3) = sqrt 3/2"
```
```  3867   by (simp add: sin_cos_eq cos_30)
```
```  3868
```
```  3869 lemma cos_60: "cos (pi / 3) = 1 / 2"
```
```  3870   apply (rule power2_eq_imp_eq)
```
```  3871   apply (simp add: cos_squared_eq sin_60 power_divide)
```
```  3872   apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
```
```  3873   done
```
```  3874
```
```  3875 lemma sin_30: "sin (pi / 6) = 1 / 2"
```
```  3876   by (simp add: sin_cos_eq cos_60)
```
```  3877
```
```  3878 lemma cos_integer_2pi: "n \<in> \<int> \<Longrightarrow> cos(2*pi * n) = 1"
```
```  3879   by (metis Ints_cases cos_one_2pi_int mult.assoc mult.commute real_of_int_def)
```
```  3880
```
```  3881 lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2*pi * n) = 0"
```
```  3882   by (metis sin_two_pi Ints_mult mult.assoc mult.commute sin_times_pi_eq_0)
```
```  3883
```
```  3884 lemma cos_int_2npi [simp]: "cos (2 * real (n::int) * pi) = 1"
```
```  3885   by (simp add: cos_one_2pi_int)
```
```  3886
```
```  3887 lemma sin_int_2npi [simp]: "sin (2 * real (n::int) * pi) = 0"
```
```  3888   by (metis Ints_real_of_int mult.assoc mult.commute sin_integer_2pi)
```
```  3889
```
```  3890 lemma sincos_principal_value: "\<exists>y. (-pi < y \<and> y \<le> pi) \<and> (sin(y) = sin(x) \<and> cos(y) = cos(x))"
```
```  3891   apply (rule exI [where x="pi - (2*pi) * frac((pi - x) / (2*pi))"])
```
```  3892   apply (auto simp: field_simps frac_lt_1)
```
```  3893   apply (simp_all add: frac_def divide_simps)
```
```  3894   apply (simp_all add: add_divide_distrib diff_divide_distrib)
```
```  3895   apply (simp_all add: sin_diff cos_diff mult.assoc [symmetric] cos_integer_2pi sin_integer_2pi)
```
```  3896   done
```
```  3897
```
```  3898
```
```  3899 subsection \<open>Tangent\<close>
```
```  3900
```
```  3901 definition tan :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3902   where "tan = (\<lambda>x. sin x / cos x)"
```
```  3903
```
```  3904 lemma tan_of_real:
```
```  3905   "of_real (tan x) = (tan (of_real x) :: 'a::{real_normed_field,banach})"
```
```  3906   by (simp add: tan_def sin_of_real cos_of_real)
```
```  3907
```
```  3908 lemma tan_in_Reals [simp]:
```
```  3909   fixes z :: "'a::{real_normed_field,banach}"
```
```  3910   shows "z \<in> \<real> \<Longrightarrow> tan z \<in> \<real>"
```
```  3911   by (simp add: tan_def)
```
```  3912
```
```  3913 lemma tan_zero [simp]: "tan 0 = 0"
```
```  3914   by (simp add: tan_def)
```
```  3915
```
```  3916 lemma tan_pi [simp]: "tan pi = 0"
```
```  3917   by (simp add: tan_def)
```
```  3918
```
```  3919 lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
```
```  3920   by (simp add: tan_def)
```
```  3921
```
```  3922 lemma tan_minus [simp]: "tan (-x) = - tan x"
```
```  3923   by (simp add: tan_def)
```
```  3924
```
```  3925 lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
```
```  3926   by (simp add: tan_def)
```
```  3927
```
```  3928 lemma lemma_tan_add1:
```
```  3929   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
```
```  3930   by (simp add: tan_def cos_add field_simps)
```
```  3931
```
```  3932 lemma add_tan_eq:
```
```  3933   fixes x :: "'a::{real_normed_field,banach}"
```
```  3934   shows "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
```
```  3935   by (simp add: tan_def sin_add field_simps)
```
```  3936
```
```  3937 lemma tan_add:
```
```  3938   fixes x :: "'a::{real_normed_field,banach}"
```
```  3939   shows
```
```  3940      "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0\<rbrakk>
```
```  3941       \<Longrightarrow> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
```
```  3942       by (simp add: add_tan_eq lemma_tan_add1 field_simps) (simp add: tan_def)
```
```  3943
```
```  3944 lemma tan_double:
```
```  3945   fixes x :: "'a::{real_normed_field,banach}"
```
```  3946   shows
```
```  3947      "\<lbrakk>cos x \<noteq> 0; cos (2 * x) \<noteq> 0\<rbrakk>
```
```  3948       \<Longrightarrow> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)"
```
```  3949   using tan_add [of x x] by (simp add: power2_eq_square)
```
```  3950
```
```  3951 lemma tan_gt_zero: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < tan x"
```
```  3952   by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
```
```  3953
```
```  3954 lemma tan_less_zero:
```
```  3955   assumes lb: "- pi/2 < x" and "x < 0"
```
```  3956   shows "tan x < 0"
```
```  3957 proof -
```
```  3958   have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
```
```  3959   thus ?thesis by simp
```
```  3960 qed
```
```  3961
```
```  3962 lemma tan_half:
```
```  3963   fixes x :: "'a::{real_normed_field,banach,field}"
```
```  3964   shows  "tan x = sin (2 * x) / (cos (2 * x) + 1)"
```
```  3965   unfolding tan_def sin_double cos_double sin_squared_eq
```
```  3966   by (simp add: power2_eq_square)
```
```  3967
```
```  3968 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
```
```  3969   unfolding tan_def by (simp add: sin_30 cos_30)
```
```  3970
```
```  3971 lemma tan_45: "tan (pi / 4) = 1"
```
```  3972   unfolding tan_def by (simp add: sin_45 cos_45)
```
```  3973
```
```  3974 lemma tan_60: "tan (pi / 3) = sqrt 3"
```
```  3975   unfolding tan_def by (simp add: sin_60 cos_60)
```
```  3976
```
```  3977 lemma DERIV_tan [simp]:
```
```  3978   fixes x :: "'a::{real_normed_field,banach}"
```
```  3979   shows "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)"
```
```  3980   unfolding tan_def
```
```  3981   by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square)
```
```  3982
```
```  3983 lemma isCont_tan:
```
```  3984   fixes x :: "'a::{real_normed_field,banach}"
```
```  3985   shows "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
```
```  3986   by (rule DERIV_tan [THEN DERIV_isCont])
```
```  3987
```
```  3988 lemma isCont_tan' [simp,continuous_intros]:
```
```  3989   fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
```
```  3990   shows "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
```
```  3991   by (rule isCont_o2 [OF _ isCont_tan])
```
```  3992
```
```  3993 lemma tendsto_tan [tendsto_intros]:
```
```  3994   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3995   shows "\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F"
```
```  3996   by (rule isCont_tendsto_compose [OF isCont_tan])
```
```  3997
```
```  3998 lemma continuous_tan:
```
```  3999   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4000   shows "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))"
```
```  4001   unfolding continuous_def by (rule tendsto_tan)
```
```  4002
```
```  4003 lemma continuous_on_tan [continuous_intros]:
```
```  4004   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4005   shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))"
```
```  4006   unfolding continuous_on_def by (auto intro: tendsto_tan)
```
```  4007
```
```  4008 lemma continuous_within_tan [continuous_intros]:
```
```  4009   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  4010   shows
```
```  4011   "continuous (at x within s) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
```
```  4012   unfolding continuous_within by (rule tendsto_tan)
```
```  4013
```
```  4014 lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) -- pi/2 --> 0"
```
```  4015   by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
```
```  4016
```
```  4017 lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
```
```  4018   apply (cut_tac LIM_cos_div_sin)
```
```  4019   apply (simp only: LIM_eq)
```
```  4020   apply (drule_tac x = "inverse y" in spec, safe, force)
```
```  4021   apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
```
```  4022   apply (rule_tac x = "(pi/2) - e" in exI)
```
```  4023   apply (simp (no_asm_simp))
```
```  4024   apply (drule_tac x = "(pi/2) - e" in spec)
```
```  4025   apply (auto simp add: tan_def sin_diff cos_diff)
```
```  4026   apply (rule inverse_less_iff_less [THEN iffD1])
```
```  4027   apply (auto simp add: divide_inverse)
```
```  4028   apply (rule mult_pos_pos)
```
```  4029   apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
```
```  4030   apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult.commute)
```
```  4031   done
```
```  4032
```
```  4033 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
```
```  4034   apply (frule order_le_imp_less_or_eq, safe)
```
```  4035    prefer 2 apply force
```
```  4036   apply (drule lemma_tan_total, safe)
```
```  4037   apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
```
```  4038   apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
```
```  4039   apply (drule_tac y = xa in order_le_imp_less_or_eq)
```
```  4040   apply (auto dest: cos_gt_zero)
```
```  4041   done
```
```  4042
```
```  4043 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  4044   apply (cut_tac linorder_linear [of 0 y], safe)
```
```  4045   apply (drule tan_total_pos)
```
```  4046   apply (cut_tac [2] y="-y" in tan_total_pos, safe)
```
```  4047   apply (rule_tac [3] x = "-x" in exI)
```
```  4048   apply (auto del: exI intro!: exI)
```
```  4049   done
```
```  4050
```
```  4051 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  4052   apply (cut_tac y = y in lemma_tan_total1, auto)
```
```  4053   apply hypsubst_thin
```
```  4054   apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
```
```  4055   apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
```
```  4056   apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
```
```  4057   apply (rule_tac [4] Rolle)
```
```  4058   apply (rule_tac [2] Rolle)
```
```  4059   apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
```
```  4060               simp add: real_differentiable_def)
```
```  4061   txt\<open>Now, simulate TRYALL\<close>
```
```  4062   apply (rule_tac [!] DERIV_tan asm_rl)
```
```  4063   apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
```
```  4064               simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
```
```  4065   done
```
```  4066
```
```  4067 lemma tan_monotone:
```
```  4068   assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
```
```  4069   shows "tan y < tan x"
```
```  4070 proof -
```
```  4071   have "\<forall>x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse ((cos x')\<^sup>2)"
```
```  4072   proof (rule allI, rule impI)
```
```  4073     fix x' :: real
```
```  4074     assume "y \<le> x' \<and> x' \<le> x"
```
```  4075     hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
```
```  4076     from cos_gt_zero_pi[OF this]
```
```  4077     have "cos x' \<noteq> 0" by auto
```
```  4078     thus "DERIV tan x' :> inverse ((cos x')\<^sup>2)" by (rule DERIV_tan)
```
```  4079   qed
```
```  4080   from MVT2[OF \<open>y < x\<close> this]
```
```  4081   obtain z where "y < z" and "z < x"
```
```  4082     and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
```
```  4083   hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
```
```  4084   hence "0 < cos z" using cos_gt_zero_pi by auto
```
```  4085   hence inv_pos: "0 < inverse ((cos z)\<^sup>2)" by auto
```
```  4086   have "0 < x - y" using \<open>y < x\<close> by auto
```
```  4087   with inv_pos have "0 < tan x - tan y" unfolding tan_diff by auto
```
```  4088   thus ?thesis by auto
```
```  4089 qed
```
```  4090
```
```  4091 lemma tan_monotone':
```
```  4092   assumes "- (pi / 2) < y"
```
```  4093     and "y < pi / 2"
```
```  4094     and "- (pi / 2) < x"
```
```  4095     and "x < pi / 2"
```
```  4096   shows "(y < x) = (tan y < tan x)"
```
```  4097 proof
```
```  4098   assume "y < x"
```
```  4099   thus "tan y < tan x"
```
```  4100     using tan_monotone and \<open>- (pi / 2) < y\<close> and \<open>x < pi / 2\<close> by auto
```
```  4101 next
```
```  4102   assume "tan y < tan x"
```
```  4103   show "y < x"
```
```  4104   proof (rule ccontr)
```
```  4105     assume "\<not> y < x" hence "x \<le> y" by auto
```
```  4106     hence "tan x \<le> tan y"
```
```  4107     proof (cases "x = y")
```
```  4108       case True thus ?thesis by auto
```
```  4109     next
```
```  4110       case False hence "x < y" using \<open>x \<le> y\<close> by auto
```
```  4111       from tan_monotone[OF \<open>- (pi/2) < x\<close> this \<open>y < pi / 2\<close>] show ?thesis by auto
```
```  4112     qed
```
```  4113     thus False using \<open>tan y < tan x\<close> by auto
```
```  4114   qed
```
```  4115 qed
```
```  4116
```
```  4117 lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)"
```
```  4118   unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
```
```  4119
```
```  4120 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
```
```  4121   by (simp add: tan_def)
```
```  4122
```
```  4123 lemma tan_periodic_nat[simp]:
```
```  4124   fixes n :: nat
```
```  4125   shows "tan (x + real n * pi) = tan x"
```
```  4126 proof (induct n arbitrary: x)
```
```  4127   case 0
```
```  4128   then show ?case by simp
```
```  4129 next
```
```  4130   case (Suc n)
```
```  4131   have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi"
```
```  4132     unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
```
```  4133   show ?case unfolding split_pi_off using Suc by auto
```
```  4134 qed
```
```  4135
```
```  4136 lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
```
```  4137 proof (cases "0 \<le> i")
```
```  4138   case True
```
```  4139   hence i_nat: "real i = real (nat i)" by auto
```
```  4140   show ?thesis unfolding i_nat by auto
```
```  4141 next
```
```  4142   case False
```
```  4143   hence i_nat: "real i = - real (nat (-i))" by auto
```
```  4144   have "tan x = tan (x + real i * pi - real i * pi)"
```
```  4145     by auto
```
```  4146   also have "\<dots> = tan (x + real i * pi)"
```
```  4147     unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
```
```  4148   finally show ?thesis by auto
```
```  4149 qed
```
```  4150
```
```  4151 lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
```
```  4152   using tan_periodic_int[of _ "numeral n" ] unfolding real_numeral .
```
```  4153
```
```  4154 lemma tan_minus_45: "tan (-(pi/4)) = -1"
```
```  4155   unfolding tan_def by (simp add: sin_45 cos_45)
```
```  4156
```
```  4157 lemma tan_diff:
```
```  4158   fixes x :: "'a::{real_normed_field,banach}"
```
```  4159   shows
```
```  4160      "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0; cos (x - y) \<noteq> 0\<rbrakk>
```
```  4161       \<Longrightarrow> tan(x - y) = (tan(x) - tan(y))/(1 + tan(x) * tan(y))"
```
```  4162   using tan_add [of x "-y"]
```
```  4163   by simp
```
```  4164
```
```  4165
```
```  4166 lemma tan_pos_pi2_le: "0 \<le> x ==> x < pi/2 \<Longrightarrow> 0 \<le> tan x"
```
```  4167   using less_eq_real_def tan_gt_zero by auto
```
```  4168
```
```  4169 lemma cos_tan: "abs(x) < pi/2 \<Longrightarrow> cos(x) = 1 / sqrt(1 + tan(x) ^ 2)"
```
```  4170   using cos_gt_zero_pi [of x]
```
```  4171   by (simp add: divide_simps tan_def real_sqrt_divide abs_if split: split_if_asm)
```
```  4172
```
```  4173 lemma sin_tan: "abs(x) < pi/2 \<Longrightarrow> sin(x) = tan(x) / sqrt(1 + tan(x) ^ 2)"
```
```  4174   using cos_gt_zero [of "x"] cos_gt_zero [of "-x"]
```
```  4175   by (force simp add: divide_simps tan_def real_sqrt_divide abs_if split: split_if_asm)
```
```  4176
```
```  4177 lemma tan_mono_le: "-(pi/2) < x ==> x \<le> y ==> y < pi/2 \<Longrightarrow> tan(x) \<le> tan(y)"
```
```  4178   using less_eq_real_def tan_monotone by auto
```
```  4179
```
```  4180 lemma tan_mono_lt_eq: "-(pi/2) < x ==> x < pi/2 ==> -(pi/2) < y ==> y < pi/2
```
```  4181          \<Longrightarrow> (tan(x) < tan(y) \<longleftrightarrow> x < y)"
```
```  4182   using tan_monotone' by blast
```
```  4183
```
```  4184 lemma tan_mono_le_eq: "-(pi/2) < x ==> x < pi/2 ==> -(pi/2) < y ==> y < pi/2
```
```  4185          \<Longrightarrow> (tan(x) \<le> tan(y) \<longleftrightarrow> x \<le> y)"
```
```  4186   by (meson tan_mono_le not_le tan_monotone)
```
```  4187
```
```  4188 lemma tan_bound_pi2: "abs(x) < pi/4 \<Longrightarrow> abs(tan x) < 1"
```
```  4189   using tan_45 tan_monotone [of x "pi/4"] tan_monotone [of "-x" "pi/4"]
```
```  4190   by (auto simp: abs_if split: split_if_asm)
```
```  4191
```
```  4192 lemma tan_cot: "tan(pi/2 - x) = inverse(tan x)"
```
```  4193   by (simp add: tan_def sin_diff cos_diff)
```
```  4194
```
```  4195 subsection \<open>Inverse Trigonometric Functions\<close>
```
```  4196
```
```  4197 definition arcsin :: "real => real"
```
```  4198   where "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
```
```  4199
```
```  4200 definition arccos :: "real => real"
```
```  4201   where "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
```
```  4202
```
```  4203 definition arctan :: "real => real"
```
```  4204   where "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
```
```  4205
```
```  4206 lemma arcsin:
```
```  4207   "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow>
```
```  4208     -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2 & sin(arcsin y) = y"
```
```  4209   unfolding arcsin_def by (rule theI' [OF sin_total])
```
```  4210
```
```  4211 lemma arcsin_pi:
```
```  4212   "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
```
```  4213   apply (drule (1) arcsin)
```
```  4214   apply (force intro: order_trans)
```
```  4215   done
```
```  4216
```
```  4217 lemma sin_arcsin [simp]: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin(arcsin y) = y"
```
```  4218   by (blast dest: arcsin)
```
```  4219
```
```  4220 lemma arcsin_bounded: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
```
```  4221   by (blast dest: arcsin)
```
```  4222
```
```  4223 lemma arcsin_lbound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y"
```
```  4224   by (blast dest: arcsin)
```
```  4225
```
```  4226 lemma arcsin_ubound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
```
```  4227   by (blast dest: arcsin)
```
```  4228
```
```  4229 lemma arcsin_lt_bounded:
```
```  4230      "\<lbrakk>-1 < y; y < 1\<rbrakk> \<Longrightarrow> -(pi/2) < arcsin y & arcsin y < pi/2"
```
```  4231   apply (frule order_less_imp_le)
```
```  4232   apply (frule_tac y = y in order_less_imp_le)
```
```  4233   apply (frule arcsin_bounded)
```
```  4234   apply (safe, simp)
```
```  4235   apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
```
```  4236   apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
```
```  4237   apply (drule_tac [!] f = sin in arg_cong, auto)
```
```  4238   done
```
```  4239
```
```  4240 lemma arcsin_sin: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2\<rbrakk> \<Longrightarrow> arcsin(sin x) = x"
```
```  4241   apply (unfold arcsin_def)
```
```  4242   apply (rule the1_equality)
```
```  4243   apply (rule sin_total, auto)
```
```  4244   done
```
```  4245
```
```  4246 lemma arcsin_0 [simp]: "arcsin 0 = 0"
```
```  4247   using arcsin_sin [of 0]
```
```  4248   by simp
```
```  4249
```
```  4250 lemma arcsin_1 [simp]: "arcsin 1 = pi/2"
```
```  4251   using arcsin_sin [of "pi/2"]
```
```  4252   by simp
```
```  4253
```
```  4254 lemma arcsin_minus_1 [simp]: "arcsin (-1) = - (pi/2)"
```
```  4255   using arcsin_sin [of "-pi/2"]
```
```  4256   by simp
```
```  4257
```
```  4258 lemma arcsin_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin(-x) = -arcsin x"
```
```  4259   by (metis (no_types, hide_lams) arcsin arcsin_sin minus_minus neg_le_iff_le sin_minus)
```
```  4260
```
```  4261 lemma arcsin_eq_iff: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> (arcsin x = arcsin y \<longleftrightarrow> x = y)"
```
```  4262   by (metis abs_le_interval_iff arcsin)
```
```  4263
```
```  4264 lemma cos_arcsin_nonzero: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> cos(arcsin x) \<noteq> 0"
```
```  4265   using arcsin_lt_bounded cos_gt_zero_pi by force
```
```  4266
```
```  4267 lemma arccos:
```
```  4268      "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk>
```
```  4269       \<Longrightarrow> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
```
```  4270   unfolding arccos_def by (rule theI' [OF cos_total])
```
```  4271
```
```  4272 lemma cos_arccos [simp]: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> cos(arccos y) = y"
```
```  4273   by (blast dest: arccos)
```
```  4274
```
```  4275 lemma arccos_bounded: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> arccos y & arccos y \<le> pi"
```
```  4276   by (blast dest: arccos)
```
```  4277
```
```  4278 lemma arccos_lbound: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> arccos y"
```
```  4279   by (blast dest: arccos)
```
```  4280
```
```  4281 lemma arccos_ubound: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi"
```
```  4282   by (blast dest: arccos)
```
```  4283
```
```  4284 lemma arccos_lt_bounded:
```
```  4285      "\<lbrakk>-1 < y; y < 1\<rbrakk> \<Longrightarrow> 0 < arccos y & arccos y < pi"
```
```  4286   apply (frule order_less_imp_le)
```
```  4287   apply (frule_tac y = y in order_less_imp_le)
```
```  4288   apply (frule arccos_bounded, auto)
```
```  4289   apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
```
```  4290   apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
```
```  4291   apply (drule_tac [!] f = cos in arg_cong, auto)
```
```  4292   done
```
```  4293
```
```  4294 lemma arccos_cos: "\<lbrakk>0 \<le> x; x \<le> pi\<rbrakk> \<Longrightarrow> arccos(cos x) = x"
```
```  4295   apply (simp add: arccos_def)
```
```  4296   apply (auto intro!: the1_equality cos_total)
```
```  4297   done
```
```  4298
```
```  4299 lemma arccos_cos2: "\<lbrakk>x \<le> 0; -pi \<le> x\<rbrakk> \<Longrightarrow> arccos(cos x) = -x"
```
```  4300   apply (simp add: arccos_def)
```
```  4301   apply (auto intro!: the1_equality cos_total)
```
```  4302   done
```
```  4303
```
```  4304 lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
```
```  4305   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  4306   apply (rule power2_eq_imp_eq)
```
```  4307   apply (simp add: cos_squared_eq)
```
```  4308   apply (rule cos_ge_zero)
```
```  4309   apply (erule (1) arcsin_lbound)
```
```  4310   apply (erule (1) arcsin_ubound)
```
```  4311   apply simp
```
```  4312   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  4313   apply (rule power_mono, simp, simp)
```
```  4314   done
```
```  4315
```
```  4316 lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
```
```  4317   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  4318   apply (rule power2_eq_imp_eq)
```
```  4319   apply (simp add: sin_squared_eq)
```
```  4320   apply (rule sin_ge_zero)
```
```  4321   apply (erule (1) arccos_lbound)
```
```  4322   apply (erule (1) arccos_ubound)
```
```  4323   apply simp
```
```  4324   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  4325   apply (rule power_mono, simp, simp)
```
```  4326   done
```
```  4327
```
```  4328 lemma arccos_0 [simp]: "arccos 0 = pi/2"
```
```  4329 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)
```
```  4330
```
```  4331 lemma arccos_1 [simp]: "arccos 1 = 0"
```
```  4332   using arccos_cos by force
```
```  4333
```
```  4334 lemma arccos_minus_1 [simp]: "arccos(-1) = pi"
```
```  4335   by (metis arccos_cos cos_pi order_refl pi_ge_zero)
```
```  4336
```
```  4337 lemma arccos_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos(-x) = pi - arccos x"
```
```  4338   by (metis arccos_cos arccos_cos2 cos_minus_pi cos_total diff_le_0_iff_le le_add_same_cancel1
```
```  4339     minus_diff_eq uminus_add_conv_diff)
```
```  4340
```
```  4341 lemma sin_arccos_nonzero: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> ~(sin(arccos x) = 0)"
```
```  4342   using arccos_lt_bounded sin_gt_zero by force
```
```  4343
```
```  4344 lemma arctan: "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
```
```  4345   unfolding arctan_def by (rule theI' [OF tan_total])
```
```  4346
```
```  4347 lemma tan_arctan: "tan (arctan y) = y"
```
```  4348   by (simp add: arctan)
```
```  4349
```
```  4350 lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
```
```  4351   by (auto simp only: arctan)
```
```  4352
```
```  4353 lemma arctan_lbound: "- (pi/2) < arctan y"
```
```  4354   by (simp add: arctan)
```
```  4355
```
```  4356 lemma arctan_ubound: "arctan y < pi/2"
```
```  4357   by (auto simp only: arctan)
```
```  4358
```
```  4359 lemma arctan_unique:
```
```  4360   assumes "-(pi/2) < x"
```
```  4361     and "x < pi/2"
```
```  4362     and "tan x = y"
```
```  4363   shows "arctan y = x"
```
```  4364   using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
```
```  4365
```
```  4366 lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x"
```
```  4367   by (rule arctan_unique) simp_all
```
```  4368
```
```  4369 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
```
```  4370   by (rule arctan_unique) simp_all
```
```  4371
```
```  4372 lemma arctan_minus: "arctan (- x) = - arctan x"
```
```  4373   apply (rule arctan_unique)
```
```  4374   apply (simp only: neg_less_iff_less arctan_ubound)
```
```  4375   apply (metis minus_less_iff arctan_lbound, simp add: arctan)
```
```  4376   done
```
```  4377
```
```  4378 lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
```
```  4379   by (intro less_imp_neq [symmetric] cos_gt_zero_pi
```
```  4380     arctan_lbound arctan_ubound)
```
```  4381
```
```  4382 lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)"
```
```  4383 proof (rule power2_eq_imp_eq)
```
```  4384   have "0 < 1 + x\<^sup>2" by (simp add: add_pos_nonneg)
```
```  4385   show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp
```
```  4386   show "0 \<le> cos (arctan x)"
```
```  4387     by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
```
```  4388   have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1"
```
```  4389     unfolding tan_def by (simp add: distrib_left power_divide)
```
```  4390   thus "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
```
```  4391     using \<open>0 < 1 + x\<^sup>2\<close> by (simp add: arctan power_divide eq_divide_eq)
```
```  4392 qed
```
```  4393
```
```  4394 lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)"
```
```  4395   using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
```
```  4396   using tan_arctan [of x] unfolding tan_def cos_arctan
```
```  4397   by (simp add: eq_divide_eq)
```
```  4398
```
```  4399 lemma tan_sec:
```
```  4400   fixes x :: "'a::{real_normed_field,banach,field}"
```
```  4401   shows "cos x \<noteq> 0 \<Longrightarrow> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
```
```  4402   apply (rule power_inverse [THEN subst])
```
```  4403   apply (rule_tac c1 = "(cos x)\<^sup>2" in mult_right_cancel [THEN iffD1])
```
```  4404   apply (auto simp add: tan_def field_simps)
```
```  4405   done
```
```  4406
```
```  4407 lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
```
```  4408   by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
```
```  4409
```
```  4410 lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
```
```  4411   by (simp only: not_less [symmetric] arctan_less_iff)
```
```  4412
```
```  4413 lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
```
```  4414   by (simp only: eq_iff [where 'a=real] arctan_le_iff)
```
```  4415
```
```  4416 lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
```
```  4417   using arctan_less_iff [of 0 x] by simp
```
```  4418
```
```  4419 lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
```
```  4420   using arctan_less_iff [of x 0] by simp
```
```  4421
```
```  4422 lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
```
```  4423   using arctan_le_iff [of 0 x] by simp
```
```  4424
```
```  4425 lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```  4426   using arctan_le_iff [of x 0] by simp
```
```  4427
```
```  4428 lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
```
```  4429   using arctan_eq_iff [of x 0] by simp
```
```  4430
```
```  4431 lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin"
```
```  4432 proof -
```
```  4433   have "continuous_on (sin ` {- pi / 2 .. pi / 2}) arcsin"
```
```  4434     by (rule continuous_on_inv) (auto intro: continuous_intros simp: arcsin_sin)
```
```  4435   also have "sin ` {- pi / 2 .. pi / 2} = {-1 .. 1}"
```
```  4436   proof safe
```
```  4437     fix x :: real
```
```  4438     assume "x \<in> {-1..1}"
```
```  4439     then show "x \<in> sin ` {- pi / 2..pi / 2}"
```
```  4440       using arcsin_lbound arcsin_ubound
```
```  4441       by (intro image_eqI[where x="arcsin x"]) auto
```
```  4442   qed simp
```
```  4443   finally show ?thesis .
```
```  4444 qed
```
```  4445
```
```  4446 lemma continuous_on_arcsin [continuous_intros]:
```
```  4447   "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))"
```
```  4448   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arcsin']]
```
```  4449   by (auto simp: comp_def subset_eq)
```
```  4450
```
```  4451 lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x"
```
```  4452   using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  4453   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  4454
```
```  4455 lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos"
```
```  4456 proof -
```
```  4457   have "continuous_on (cos ` {0 .. pi}) arccos"
```
```  4458     by (rule continuous_on_inv) (auto intro: continuous_intros simp: arccos_cos)
```
```  4459   also have "cos ` {0 .. pi} = {-1 .. 1}"
```
```  4460   proof safe
```
```  4461     fix x :: real
```
```  4462     assume "x \<in> {-1..1}"
```
```  4463     then show "x \<in> cos ` {0..pi}"
```
```  4464       using arccos_lbound arccos_ubound
```
```  4465       by (intro image_eqI[where x="arccos x"]) auto
```
```  4466   qed simp
```
```  4467   finally show ?thesis .
```
```  4468 qed
```
```  4469
```
```  4470 lemma continuous_on_arccos [continuous_intros]:
```
```  4471   "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))"
```
```  4472   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arccos']]
```
```  4473   by (auto simp: comp_def subset_eq)
```
```  4474
```
```  4475 lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x"
```
```  4476   using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  4477   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  4478
```
```  4479 lemma isCont_arctan: "isCont arctan x"
```
```  4480   apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
```
```  4481   apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
```
```  4482   apply (subgoal_tac "isCont arctan (tan (arctan x))", simp add: arctan)
```
```  4483   apply (erule (1) isCont_inverse_function2 [where f=tan])
```
```  4484   apply (metis arctan_tan order_le_less_trans order_less_le_trans)
```
```  4485   apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
```
```  4486   done
```
```  4487
```
```  4488 lemma tendsto_arctan [tendsto_intros]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) ---> arctan x) F"
```
```  4489   by (rule isCont_tendsto_compose [OF isCont_arctan])
```
```  4490
```
```  4491 lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))"
```
```  4492   unfolding continuous_def by (rule tendsto_arctan)
```
```  4493
```
```  4494 lemma continuous_on_arctan [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
```
```  4495   unfolding continuous_on_def by (auto intro: tendsto_arctan)
```
```  4496
```
```  4497 lemma DERIV_arcsin:
```
```  4498   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
```
```  4499   apply (rule DERIV_inverse_function [where f=sin and a="-1" and b=1])
```
```  4500   apply (rule DERIV_cong [OF DERIV_sin])
```
```  4501   apply (simp add: cos_arcsin)
```
```  4502   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  4503   apply (rule power_strict_mono, simp, simp, simp, assumption, assumption)
```
```  4504   apply simp
```
```  4505   apply (erule (1) isCont_arcsin)
```
```  4506   done
```
```  4507
```
```  4508 lemma DERIV_arccos:
```
```  4509   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
```
```  4510   apply (rule DERIV_inverse_function [where f=cos and a="-1" and b=1])
```
```  4511   apply (rule DERIV_cong [OF DERIV_cos])
```
```  4512   apply (simp add: sin_arccos)
```
```  4513   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  4514   apply (rule power_strict_mono, simp, simp, simp, assumption, assumption)
```
```  4515   apply simp
```
```  4516   apply (erule (1) isCont_arccos)
```
```  4517   done
```
```  4518
```
```  4519 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
```
```  4520   apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
```
```  4521   apply (rule DERIV_cong [OF DERIV_tan])
```
```  4522   apply (rule cos_arctan_not_zero)
```
```  4523   apply (simp_all add: add_pos_nonneg arctan isCont_arctan)
```
```  4524   apply (simp add: arctan power_inverse [symmetric] tan_sec [symmetric])
```
```  4525   apply (subgoal_tac "0 < 1 + x\<^sup>2", simp)
```
```  4526   apply (simp_all add: add_pos_nonneg arctan isCont_arctan)
```
```  4527   done
```
```  4528
```
```  4529 declare
```
```  4530   DERIV_arcsin[THEN DERIV_chain2, derivative_intros]
```
```  4531   DERIV_arccos[THEN DERIV_chain2, derivative_intros]
```
```  4532   DERIV_arctan[THEN DERIV_chain2, derivative_intros]
```
```  4533
```
```  4534 lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- pi/2))"
```
```  4535   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])
```
```  4536      (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  4537            intro!: tan_monotone exI[of _ "pi/2"])
```
```  4538
```
```  4539 lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"
```
```  4540   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])
```
```  4541      (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  4542            intro!: tan_monotone exI[of _ "pi/2"])
```
```  4543
```
```  4544 lemma tendsto_arctan_at_top: "(arctan ---> (pi/2)) at_top"
```
```  4545 proof (rule tendstoI)
```
```  4546   fix e :: real
```
```  4547   assume "0 < e"
```
```  4548   def y \<equiv> "pi/2 - min (pi/2) e"
```
```  4549   then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y"
```
```  4550     using \<open>0 < e\<close> by auto
```
```  4551
```
```  4552   show "eventually (\<lambda>x. dist (arctan x) (pi / 2) < e) at_top"
```
```  4553   proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)
```
```  4554     fix x
```
```  4555     assume "tan y < x"
```
```  4556     then have "arctan (tan y) < arctan x"
```
```  4557       by (simp add: arctan_less_iff)
```
```  4558     with y have "y < arctan x"
```
```  4559       by (subst (asm) arctan_tan) simp_all
```
```  4560     with arctan_ubound[of x, arith] y \<open>0 < e\<close>
```
```  4561     show "dist (arctan x) (pi / 2) < e"
```
```  4562       by (simp add: dist_real_def)
```
```  4563   qed
```
```  4564 qed
```
```  4565
```
```  4566 lemma tendsto_arctan_at_bot: "(arctan ---> - (pi/2)) at_bot"
```
```  4567   unfolding filterlim_at_bot_mirror arctan_minus
```
```  4568   by (intro tendsto_minus tendsto_arctan_at_top)
```
```  4569
```
```  4570
```
```  4571 subsection\<open>Prove Totality of the Trigonometric Functions\<close>
```
```  4572
```
```  4573 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
```
```  4574   by (simp add: abs_le_iff)
```
```  4575
```
```  4576 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
```
```  4577   by (simp add: sin_arccos abs_le_iff)
```
```  4578
```
```  4579 lemma sin_mono_less_eq: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2; -(pi/2) \<le> y; y \<le> pi/2\<rbrakk>
```
```  4580          \<Longrightarrow> (sin(x) < sin(y) \<longleftrightarrow> x < y)"
```
```  4581 by (metis not_less_iff_gr_or_eq sin_monotone_2pi)
```
```  4582
```
```  4583 lemma sin_mono_le_eq: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2; -(pi/2) \<le> y; y \<le> pi/2\<rbrakk>
```
```  4584          \<Longrightarrow> (sin(x) \<le> sin(y) \<longleftrightarrow> x \<le> y)"
```
```  4585 by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le)
```
```  4586
```
```  4587 lemma sin_inj_pi:
```
```  4588     "\<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"
```
```  4589 by (metis arcsin_sin)
```
```  4590
```
```  4591 lemma cos_mono_less_eq:
```
```  4592     "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi \<Longrightarrow> (cos(x) < cos(y) \<longleftrightarrow> y < x)"
```
```  4593 by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear)
```
```  4594
```
```  4595 lemma cos_mono_le_eq: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi
```
```  4596          \<Longrightarrow> (cos(x) \<le> cos(y) \<longleftrightarrow> y \<le> x)"
```
```  4597   by (metis arccos_cos cos_monotone_0_pi_le eq_iff linear)
```
```  4598
```
```  4599 lemma cos_inj_pi: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi ==> cos(x) = cos(y)
```
```  4600          \<Longrightarrow> x = y"
```
```  4601 by (metis arccos_cos)
```
```  4602
```
```  4603 lemma arccos_le_pi2: "\<lbrakk>0 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi/2"
```
```  4604   by (metis (mono_tags) arccos_0 arccos cos_le_one cos_monotone_0_pi_le
```
```  4605       cos_pi cos_pi_half pi_half_ge_zero antisym_conv less_eq_neg_nonpos linear minus_minus order.trans order_refl)
```
```  4606
```
```  4607 lemma sincos_total_pi_half:
```
```  4608   assumes "0 \<le> x" "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1"
```
```  4609     shows "\<exists>t. 0 \<le> t \<and> t \<le> pi/2 \<and> x = cos t \<and> y = sin t"
```
```  4610 proof -
```
```  4611   have x1: "x \<le> 1"
```
```  4612     using assms
```
```  4613     by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2)
```
```  4614   moreover with assms have ax: "0 \<le> arccos x" "cos(arccos x) = x"
```
```  4615     by (auto simp: arccos)
```
```  4616   moreover have "y = sqrt (1 - x\<^sup>2)" using assms
```
```  4617     by (metis abs_of_nonneg add.commute add_diff_cancel real_sqrt_abs)
```
```  4618   ultimately show ?thesis using assms arccos_le_pi2 [of x]
```
```  4619     by (rule_tac x="arccos x" in exI) (auto simp: sin_arccos)
```
```  4620 qed
```
```  4621
```
```  4622 lemma sincos_total_pi:
```
```  4623   assumes "0 \<le> y" and "x\<^sup>2 + y\<^sup>2 = 1"
```
```  4624     shows "\<exists>t. 0 \<le> t \<and> t \<le> pi \<and> x = cos t \<and> y = sin t"
```
```  4625 proof (cases rule: le_cases [of 0 x])
```
```  4626   case le from sincos_total_pi_half [OF le]
```
```  4627   show ?thesis
```
```  4628     by (metis pi_ge_two pi_half_le_two add.commute add_le_cancel_left add_mono assms)
```
```  4629 next
```
```  4630   case ge
```
```  4631   then have "0 \<le> -x"
```
```  4632     by simp
```
```  4633   then obtain t where "t\<ge>0" "t \<le> pi/2" "-x = cos t" "y = sin t"
```
```  4634     using sincos_total_pi_half assms
```
```  4635     apply auto
```
```  4636     by (metis \<open>0 \<le> - x\<close> power2_minus)
```
```  4637   then show ?thesis
```
```  4638     by (rule_tac x="pi-t" in exI, auto)
```
```  4639 qed
```
```  4640
```
```  4641 lemma sincos_total_2pi_le:
```
```  4642   assumes "x\<^sup>2 + y\<^sup>2 = 1"
```
```  4643     shows "\<exists>t. 0 \<le> t \<and> t \<le> 2*pi \<and> x = cos t \<and> y = sin t"
```
```  4644 proof (cases rule: le_cases [of 0 y])
```
```  4645   case le from sincos_total_pi [OF le]
```
```  4646   show ?thesis
```
```  4647     by (metis assms le_add_same_cancel1 mult.commute mult_2_right order.trans)
```
```  4648 next
```
```  4649   case ge
```
```  4650   then have "0 \<le> -y"
```
```  4651     by simp
```
```  4652   then obtain t where "t\<ge>0" "t \<le> pi" "x = cos t" "-y = sin t"
```
```  4653     using sincos_total_pi assms
```
```  4654     apply auto
```
```  4655     by (metis \<open>0 \<le> - y\<close> power2_minus)
```
```  4656   then show ?thesis
```
```  4657     by (rule_tac x="2*pi-t" in exI, auto)
```
```  4658 qed
```
```  4659
```
```  4660 lemma sincos_total_2pi:
```
```  4661   assumes "x\<^sup>2 + y\<^sup>2 = 1"
```
```  4662     obtains t where "0 \<le> t" "t < 2*pi" "x = cos t" "y = sin t"
```
```  4663 proof -
```
```  4664   from sincos_total_2pi_le [OF assms]
```
```  4665   obtain t where t: "0 \<le> t" "t \<le> 2*pi" "x = cos t" "y = sin t"
```
```  4666     by blast
```
```  4667   show ?thesis
```
```  4668     apply (cases "t = 2*pi")
```
```  4669     using t that
```
```  4670     apply force+
```
```  4671     done
```
```  4672 qed
```
```  4673
```
```  4674 lemma arcsin_less_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> arcsin x < arcsin y \<longleftrightarrow> x < y"
```
```  4675   apply (rule trans [OF sin_mono_less_eq [symmetric]])
```
```  4676   using arcsin_ubound arcsin_lbound
```
```  4677   apply auto
```
```  4678   done
```
```  4679
```
```  4680 lemma arcsin_le_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y \<longleftrightarrow> x \<le> y"
```
```  4681   using arcsin_less_mono not_le by blast
```
```  4682
```
```  4683 lemma arcsin_less_arcsin: "-1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x < arcsin y"
```
```  4684   using arcsin_less_mono by auto
```
```  4685
```
```  4686 lemma arcsin_le_arcsin: "-1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y"
```
```  4687   using arcsin_le_mono by auto
```
```  4688
```
```  4689 lemma arccos_less_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> (arccos x < arccos y \<longleftrightarrow> y < x)"
```
```  4690   apply (rule trans [OF cos_mono_less_eq [symmetric]])
```
```  4691   using arccos_ubound arccos_lbound
```
```  4692   apply auto
```
```  4693   done
```
```  4694
```
```  4695 lemma arccos_le_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> arccos x \<le> arccos y \<longleftrightarrow> y \<le> x"
```
```  4696   using arccos_less_mono [of y x]
```
```  4697   by (simp add: not_le [symmetric])
```
```  4698
```
```  4699 lemma arccos_less_arccos: "-1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y < arccos x"
```
```  4700   using arccos_less_mono by auto
```
```  4701
```
```  4702 lemma arccos_le_arccos: "-1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> arccos x"
```
```  4703   using arccos_le_mono by auto
```
```  4704
```
```  4705 lemma arccos_eq_iff: "abs x \<le> 1 & abs y \<le> 1 \<Longrightarrow> (arccos x = arccos y \<longleftrightarrow> x = y)"
```
```  4706   using cos_arccos_abs by fastforce
```
```  4707
```
```  4708 subsection \<open>Machins formula\<close>
```
```  4709
```
```  4710 lemma arctan_one: "arctan 1 = pi / 4"
```
```  4711   by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi)
```
```  4712
```
```  4713 lemma tan_total_pi4:
```
```  4714   assumes "\<bar>x\<bar> < 1"
```
```  4715   shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
```
```  4716 proof
```
```  4717   show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
```
```  4718     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  4719     unfolding arctan_less_iff using assms  by (auto simp add: arctan)
```
```  4720
```
```  4721 qed
```
```  4722
```
```  4723 lemma arctan_add:
```
```  4724   assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
```
```  4725   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
```
```  4726 proof (rule arctan_unique [symmetric])
```
```  4727   have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y"
```
```  4728     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  4729     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  4730   from add_le_less_mono [OF this]
```
```  4731   show 1: "- (pi / 2) < arctan x + arctan y" by simp
```
```  4732   have "arctan x \<le> pi / 4" and "arctan y < pi / 4"
```
```  4733     unfolding arctan_one [symmetric]
```
```  4734     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  4735   from add_le_less_mono [OF this]
```
```  4736   show 2: "arctan x + arctan y < pi / 2" by simp
```
```  4737   show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
```
```  4738     using cos_gt_zero_pi [OF 1 2] by (simp add: arctan tan_add)
```
```  4739 qed
```
```  4740
```
```  4741 lemma arctan_double:
```
```  4742   assumes "\<bar>x\<bar> < 1"
```
```  4743   shows "2 * arctan x = arctan ((2*x) / (1 - x\<^sup>2))"
```
```  4744   by (metis assms arctan_add linear mult_2 not_less power2_eq_square)
```
```  4745
```
```  4746 theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
```
```  4747 proof -
```
```  4748   have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
```
```  4749   from arctan_add[OF less_imp_le[OF this] this]
```
```  4750   have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
```
```  4751   moreover
```
```  4752   have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
```
```  4753   from arctan_add[OF less_imp_le[OF this] this]
```
```  4754   have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
```
```  4755   moreover
```
```  4756   have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
```
```  4757   from arctan_add[OF this]
```
```  4758   have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
```
```  4759   ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
```
```  4760   thus ?thesis unfolding arctan_one by algebra
```
```  4761 qed
```
```  4762
```
```  4763 lemma machin_Euler: "5 * arctan(1/7) + 2 * arctan(3/79) = pi/4"
```
```  4764 proof -
```
```  4765   have 17: "\<bar>1/7\<bar> < (1 :: real)" by auto
```
```  4766   with arctan_double
```
```  4767   have "2 * arctan (1/7) = arctan (7/24)"  by auto
```
```  4768   moreover
```
```  4769   have "\<bar>7/24\<bar> < (1 :: real)" by auto
```
```  4770   with arctan_double
```
```  4771   have "2 * arctan (7/24) = arctan (336/527)"  by auto
```
```  4772   moreover
```
```  4773   have "\<bar>336/527\<bar> < (1 :: real)" by auto
```
```  4774   from arctan_add[OF less_imp_le[OF 17] this]
```
```  4775   have "arctan(1/7) + arctan (336/527) = arctan (2879/3353)"  by auto
```
```  4776   ultimately have I: "5 * arctan(1/7) = arctan (2879/3353)"  by auto
```
```  4777   have 379: "\<bar>3/79\<bar> < (1 :: real)" by auto
```
```  4778   with arctan_double
```
```  4779   have II: "2 * arctan (3/79) = arctan (237/3116)"  by auto
```
```  4780   have *: "\<bar>2879/3353\<bar> < (1 :: real)" by auto
```
```  4781   have "\<bar>237/3116\<bar> < (1 :: real)" by auto
```
```  4782   from arctan_add[OF less_imp_le[OF *] this]
```
```  4783   have "arctan (2879/3353) + arctan (237/3116) = pi/4"
```
```  4784     by (simp add: arctan_one)
```
```  4785   then show ?thesis using I II
```
```  4786     by auto
```
```  4787 qed
```
```  4788
```
```  4789 (*But could also prove MACHIN_GAUSS:
```
```  4790   12 * arctan(1/18) + 8 * arctan(1/57) - 5 * arctan(1/239) = pi/4*)
```
```  4791
```
```  4792
```
```  4793 subsection \<open>Introducing the inverse tangent power series\<close>
```
```  4794
```
```  4795 lemma monoseq_arctan_series:
```
```  4796   fixes x :: real
```
```  4797   assumes "\<bar>x\<bar> \<le> 1"
```
```  4798   shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
```
```  4799 proof (cases "x = 0")
```
```  4800   case True
```
```  4801   thus ?thesis unfolding monoseq_def One_nat_def by auto
```
```  4802 next
```
```  4803   case False
```
```  4804   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  4805   show "monoseq ?a"
```
```  4806   proof -
```
```  4807     {
```
```  4808       fix n
```
```  4809       fix x :: real
```
```  4810       assume "0 \<le> x" and "x \<le> 1"
```
```  4811       have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le>
```
```  4812         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
```
```  4813       proof (rule mult_mono)
```
```  4814         show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))"
```
```  4815           by (rule frac_le) simp_all
```
```  4816         show "0 \<le> 1 / real (Suc (n * 2))"
```
```  4817           by auto
```
```  4818         show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)"
```
```  4819           by (rule power_decreasing) (simp_all add: \<open>0 \<le> x\<close> \<open>x \<le> 1\<close>)
```
```  4820         show "0 \<le> x ^ Suc (Suc n * 2)"
```
```  4821           by (rule zero_le_power) (simp add: \<open>0 \<le> x\<close>)
```
```  4822       qed
```
```  4823     } note mono = this
```
```  4824
```
```  4825     show ?thesis
```
```  4826     proof (cases "0 \<le> x")
```
```  4827       case True from mono[OF this \<open>x \<le> 1\<close>, THEN allI]
```
```  4828       show ?thesis unfolding Suc_eq_plus1[symmetric]
```
```  4829         by (rule mono_SucI2)
```
```  4830     next
```
```  4831       case False
```
```  4832       hence "0 \<le> -x" and "-x \<le> 1" using \<open>-1 \<le> x\<close> by auto
```
```  4833       from mono[OF this]
```
```  4834       have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge>
```
```  4835         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using \<open>0 \<le> -x\<close> by auto
```
```  4836       thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
```
```  4837     qed
```
```  4838   qed
```
```  4839 qed
```
```  4840
```
```  4841 lemma zeroseq_arctan_series:
```
```  4842   fixes x :: real
```
```  4843   assumes "\<bar>x\<bar> \<le> 1"
```
```  4844   shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
```
```  4845 proof (cases "x = 0")
```
```  4846   case True
```
```  4847   thus ?thesis
```
```  4848     unfolding One_nat_def by auto
```
```  4849 next
```
```  4850   case False
```
```  4851   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  4852   show "?a ----> 0"
```
```  4853   proof (cases "\<bar>x\<bar> < 1")
```
```  4854     case True
```
```  4855     hence "norm x < 1" by auto
```
```  4856     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF \<open>norm x < 1\<close>, THEN LIMSEQ_Suc]]
```
```  4857     have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"
```
```  4858       unfolding inverse_eq_divide Suc_eq_plus1 by simp
```
```  4859     then show ?thesis using pos2 by (rule LIMSEQ_linear)
```
```  4860   next
```
```  4861     case False
```
```  4862     hence "x = -1 \<or> x = 1" using \<open>\<bar>x\<bar> \<le> 1\<close> by auto
```
```  4863     hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x"
```
```  4864       unfolding One_nat_def by auto
```
```  4865     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
```
```  4866     show ?thesis unfolding n_eq Suc_eq_plus1 by auto
```
```  4867   qed
```
```  4868 qed
```
```  4869
```
```  4870 text\<open>FIXME: generalise from the reals via type classes?\<close>
```
```  4871 lemma summable_arctan_series:
```
```  4872   fixes x :: real and n :: nat
```
```  4873   assumes "\<bar>x\<bar> \<le> 1"
```
```  4874   shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  4875   (is "summable (?c x)")
```
```  4876   by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
```
```  4877
```
```  4878 lemma DERIV_arctan_series:
```
```  4879   assumes "\<bar> x \<bar> < 1"
```
```  4880   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))"
```
```  4881   (is "DERIV ?arctan _ :> ?Int")
```
```  4882 proof -
```
```  4883   let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
```
```  4884
```
```  4885   have n_even: "\<And>n :: nat. even n \<Longrightarrow> 2 * (n div 2) = n"
```
```  4886     by presburger
```
```  4887   then have if_eq: "\<And>n x'. ?f n * real (Suc n) * x'^n =
```
```  4888     (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)"
```
```  4889     by auto
```
```  4890
```
```  4891   {
```
```  4892     fix x :: real
```
```  4893     assume "\<bar>x\<bar> < 1"
```
```  4894     hence "x\<^sup>2 < 1" by (simp add: abs_square_less_1)
```
```  4895     have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)"
```
```  4896       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>])
```
```  4897     hence "summable (\<lambda> n. (- 1) ^ n * x^(2*n))" unfolding power_mult .
```
```  4898   } note summable_Integral = this
```
```  4899
```
```  4900   {
```
```  4901     fix f :: "nat \<Rightarrow> real"
```
```  4902     have "\<And>x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  4903     proof
```
```  4904       fix x :: real
```
```  4905       assume "f sums x"
```
```  4906       from sums_if[OF sums_zero this]
```
```  4907       show "(\<lambda>n. if even n then f (n div 2) else 0) sums x"
```
```  4908         by auto
```
```  4909     next
```
```  4910       fix x :: real
```
```  4911       assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  4912       from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult.commute]]
```
```  4913       show "f sums x" unfolding sums_def by auto
```
```  4914     qed
```
```  4915     hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
```
```  4916   } note sums_even = this
```
```  4917
```
```  4918   have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int"
```
```  4919     unfolding if_eq mult.commute[of _ 2] suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * x ^ (2 * n)", symmetric]
```
```  4920     by auto
```
```  4921
```
```  4922   {
```
```  4923     fix x :: real
```
```  4924     have if_eq': "\<And>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
```
```  4925       (if even n then (- 1) ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
```
```  4926       using n_even by auto
```
```  4927     have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)" by auto
```
```  4928     have "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x"
```
```  4929       unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
```
```  4930       by auto
```
```  4931   } note arctan_eq = this
```
```  4932
```
```  4933   have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
```
```  4934   proof (rule DERIV_power_series')
```
```  4935     show "x \<in> {- 1 <..< 1}" using \<open>\<bar> x \<bar> < 1\<close> by auto
```
```  4936     {
```
```  4937       fix x' :: real
```
```  4938       assume x'_bounds: "x' \<in> {- 1 <..< 1}"
```
```  4939       then have "\<bar>x'\<bar> < 1" by auto
```
```  4940       then
```
```  4941         have *: "summable (\<lambda>n. (- 1) ^ n * x' ^ (2 * n))"
```
```  4942         by (rule summable_Integral)
```
```  4943       let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
```
```  4944       show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
```
```  4945         apply (rule sums_summable [where l="0 + ?S"])
```
```  4946         apply (rule sums_if)
```
```  4947         apply (rule sums_zero)
```
```  4948         apply (rule summable_sums)
```
```  4949         apply (rule *)
```
```  4950         done
```
```  4951     }
```
```  4952   qed auto
```
```  4953   thus ?thesis unfolding Int_eq arctan_eq .
```
```  4954 qed
```
```  4955
```
```  4956 lemma arctan_series:
```
```  4957   assumes "\<bar> x \<bar> \<le> 1"
```
```  4958   shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  4959   (is "_ = suminf (\<lambda> n. ?c x n)")
```
```  4960 proof -
```
```  4961   let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)"
```
```  4962
```
```  4963   {
```
```  4964     fix r x :: real
```
```  4965     assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
```
```  4966     have "\<bar>x\<bar> < 1" using \<open>r < 1\<close> and \<open>\<bar>x\<bar> < r\<close> by auto
```
```  4967     from DERIV_arctan_series[OF this] have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
```
```  4968   } note DERIV_arctan_suminf = this
```
```  4969
```
```  4970   {
```
```  4971     fix x :: real
```
```  4972     assume "\<bar>x\<bar> \<le> 1"
```
```  4973     note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]]
```
```  4974   } note arctan_series_borders = this
```
```  4975
```
```  4976   {
```
```  4977     fix x :: real
```
```  4978     assume "\<bar>x\<bar> < 1"
```
```  4979     have "arctan x = (\<Sum>k. ?c x k)"
```
```  4980     proof -
```
```  4981       obtain r where "\<bar>x\<bar> < r" and "r < 1"
```
```  4982         using dense[OF \<open>\<bar>x\<bar> < 1\<close>] by blast
```
```  4983       hence "0 < r" and "-r < x" and "x < r" by auto
```
```  4984
```
```  4985       have suminf_eq_arctan_bounded: "\<And>x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow>
```
```  4986         suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  4987       proof -
```
```  4988         fix x a b
```
```  4989         assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
```
```  4990         hence "\<bar>x\<bar> < r" by auto
```
```  4991         show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  4992         proof (rule DERIV_isconst2[of "a" "b"])
```
```  4993           show "a < b" and "a \<le> x" and "x \<le> b"
```
```  4994             using \<open>a < b\<close> \<open>a \<le> x\<close> \<open>x \<le> b\<close> by auto
```
```  4995           have "\<forall>x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  4996           proof (rule allI, rule impI)
```
```  4997             fix x
```
```  4998             assume "-r < x \<and> x < r"
```
```  4999             hence "\<bar>x\<bar> < r" by auto
```
```  5000             hence "\<bar>x\<bar> < 1" using \<open>r < 1\<close> by auto
```
```  5001             have "\<bar> - (x\<^sup>2) \<bar> < 1"
```
```  5002               using abs_square_less_1 \<open>\<bar>x\<bar> < 1\<close> by auto
```
```  5003             hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  5004               unfolding real_norm_def[symmetric] by (rule geometric_sums)
```
```  5005             hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  5006               unfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2] by auto
```
```  5007             hence suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)"
```
```  5008               using sums_unique unfolding inverse_eq_divide by auto
```
```  5009             have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))"
```
```  5010               unfolding suminf_c'_eq_geom
```
```  5011               by (rule DERIV_arctan_suminf[OF \<open>0 < r\<close> \<open>r < 1\<close> \<open>\<bar>x\<bar> < r\<close>])
```
```  5012             from DERIV_diff [OF this DERIV_arctan]
```
```  5013             show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  5014               by auto
```
```  5015           qed
```
```  5016           hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
```
```  5017             using \<open>-r < a\<close> \<open>b < r\<close> by auto
```
```  5018           thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
```
```  5019             using \<open>\<bar>x\<bar> < r\<close> by auto
```
```  5020           show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y"
```
```  5021             using DERIV_in_rball DERIV_isCont by auto
```
```  5022         qed
```
```  5023       qed
```
```  5024
```
```  5025       have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
```
```  5026         unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero
```
```  5027         by auto
```
```  5028
```
```  5029       have "suminf (?c x) - arctan x = 0"
```
```  5030       proof (cases "x = 0")
```
```  5031         case True
```
```  5032         thus ?thesis using suminf_arctan_zero by auto
```
```  5033       next
```
```  5034         case False
```
```  5035         hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
```
```  5036         have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
```
```  5037           by (rule suminf_eq_arctan_bounded[where x1="0" and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>", symmetric])
```
```  5038             (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>-\<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less)
```
```  5039         moreover
```
```  5040         have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
```
```  5041           by (rule suminf_eq_arctan_bounded[where x1="x" and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>"])
```
```  5042              (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>-\<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less)
```
```  5043         ultimately
```
```  5044         show ?thesis using suminf_arctan_zero by auto
```
```  5045       qed
```
```  5046       thus ?thesis by auto
```
```  5047     qed
```
```  5048   } note when_less_one = this
```
```  5049
```
```  5050   show "arctan x = suminf (\<lambda> n. ?c x n)"
```
```  5051   proof (cases "\<bar>x\<bar> < 1")
```
```  5052     case True
```
```  5053     thus ?thesis by (rule when_less_one)
```
```  5054   next
```
```  5055     case False
```
```  5056     hence "\<bar>x\<bar> = 1" using \<open>\<bar>x\<bar> \<le> 1\<close> by auto
```
```  5057     let ?a = "\<lambda>x n. \<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
```
```  5058     let ?diff = "\<lambda> x n. \<bar> arctan x - (\<Sum> i<n. ?c x i)\<bar>"
```
```  5059     {
```
```  5060       fix n :: nat
```
```  5061       have "0 < (1 :: real)" by auto
```
```  5062       moreover
```
```  5063       {
```
```  5064         fix x :: real
```
```  5065         assume "0 < x" and "x < 1"
```
```  5066         hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
```
```  5067         from \<open>0 < x\<close> have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)"
```
```  5068           by auto
```
```  5069         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]
```
```  5070         have "0 < 1 / real (n*2+1) * x^(n*2+1)"
```
```  5071           by (rule mult_pos_pos, auto simp only: zero_less_power[OF \<open>0 < x\<close>], auto)
```
```  5072         hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)"
```
```  5073           by (rule abs_of_pos)
```
```  5074         have "?diff x n \<le> ?a x n"
```
```  5075         proof (cases "even n")
```
```  5076           case True
```
```  5077           hence sgn_pos: "(-1)^n = (1::real)" by auto
```
```  5078           from \<open>even n\<close> obtain m where "n = 2 * m" ..
```
```  5079           then have "2 * m = n" ..
```
```  5080           from bounds[of m, unfolded this atLeastAtMost_iff]
```
```  5081           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))"
```
```  5082             by auto
```
```  5083           also have "\<dots> = ?c x n" unfolding One_nat_def by auto
```
```  5084           also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
```
```  5085           finally show ?thesis .
```
```  5086         next
```
```  5087           case False
```
```  5088           hence sgn_neg: "(-1)^n = (-1::real)" by auto
```
```  5089           from \<open>odd n\<close> obtain m where "n = 2 * m + 1" ..
```
```  5090           then have m_def: "2 * m + 1 = n" ..
```
```  5091           hence m_plus: "2 * (m + 1) = n + 1" by auto
```
```  5092           from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
```
```  5093           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))"
```
```  5094             by auto
```
```  5095           also have "\<dots> = - ?c x n" unfolding One_nat_def by auto
```
```  5096           also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
```
```  5097           finally show ?thesis .
```
```  5098         qed
```
```  5099         hence "0 \<le> ?a x n - ?diff x n" by auto
```
```  5100       }
```
```  5101       hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
```
```  5102       moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
```
```  5103         unfolding diff_conv_add_uminus divide_inverse
```
```  5104         by (auto intro!: isCont_add isCont_rabs continuous_ident isCont_minus isCont_arctan
```
```  5105           isCont_inverse isCont_mult isCont_power continuous_const isCont_setsum
```
```  5106           simp del: add_uminus_conv_diff)
```
```  5107       ultimately have "0 \<le> ?a 1 n - ?diff 1 n"
```
```  5108         by (rule LIM_less_bound)
```
```  5109       hence "?diff 1 n \<le> ?a 1 n" by auto
```
```  5110     }
```
```  5111     have "?a 1 ----> 0"
```
```  5112       unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
```
```  5113       by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)
```
```  5114     have "?diff 1 ----> 0"
```
```  5115     proof (rule LIMSEQ_I)
```
```  5116       fix r :: real
```
```  5117       assume "0 < r"
```
```  5118       obtain N :: nat where N_I: "\<And>n. N \<le> n \<Longrightarrow> ?a 1 n < r"
```
```  5119         using LIMSEQ_D[OF \<open>?a 1 ----> 0\<close> \<open>0 < r\<close>] by auto
```
```  5120       {
```
```  5121         fix n
```
```  5122         assume "N \<le> n" from \<open>?diff 1 n \<le> ?a 1 n\<close> N_I[OF this]
```
```  5123         have "norm (?diff 1 n - 0) < r" by auto
```
```  5124       }
```
```  5125       thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
```
```  5126     qed
```
```  5127     from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
```
```  5128     have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
```
```  5129     hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
```
```  5130
```
```  5131     show ?thesis
```
```  5132     proof (cases "x = 1")
```
```  5133       case True
```
```  5134       then show ?thesis by (simp add: \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close>)
```
```  5135     next
```
```  5136       case False
```
```  5137       hence "x = -1" using \<open>\<bar>x\<bar> = 1\<close> by auto
```
```  5138
```
```  5139       have "- (pi / 2) < 0" using pi_gt_zero by auto
```
```  5140       have "- (2 * pi) < 0" using pi_gt_zero by auto
```
```  5141
```
```  5142       have c_minus_minus: "\<And>i. ?c (- 1) i = - ?c 1 i"
```
```  5143         unfolding One_nat_def by auto
```
```  5144
```
```  5145       have "arctan (- 1) = arctan (tan (-(pi / 4)))"
```
```  5146         unfolding tan_45 tan_minus ..
```
```  5147       also have "\<dots> = - (pi / 4)"
```
```  5148         by (rule arctan_tan, auto simp add: order_less_trans[OF \<open>- (pi / 2) < 0\<close> pi_gt_zero])
```
```  5149       also have "\<dots> = - (arctan (tan (pi / 4)))"
```
```  5150         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])
```
```  5151       also have "\<dots> = - (arctan 1)"
```
```  5152         unfolding tan_45 ..
```
```  5153       also have "\<dots> = - (\<Sum> i. ?c 1 i)"
```
```  5154         using \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close> by auto
```
```  5155       also have "\<dots> = (\<Sum> i. ?c (- 1) i)"
```
```  5156         using suminf_minus[OF sums_summable[OF \<open>(?c 1) sums (arctan 1)\<close>]]
```
```  5157         unfolding c_minus_minus by auto
```
```  5158       finally show ?thesis using \<open>x = -1\<close> by auto
```
```  5159     qed
```
```  5160   qed
```
```  5161 qed
```
```  5162
```
```  5163 lemma arctan_half:
```
```  5164   fixes x :: real
```
```  5165   shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))"
```
```  5166 proof -
```
```  5167   obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x"
```
```  5168     using tan_total by blast
```
```  5169   hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2"
```
```  5170     by auto
```
```  5171
```
```  5172   have "0 < cos y" using cos_gt_zero_pi[OF low high] .
```
```  5173   hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y"
```
```  5174     by auto
```
```  5175
```
```  5176   have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  5177     unfolding tan_def power_divide ..
```
```  5178   also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  5179     using \<open>cos y \<noteq> 0\<close> by auto
```
```  5180   also have "\<dots> = 1 / (cos y)\<^sup>2"
```
```  5181     unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
```
```  5182   finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" .
```
```  5183
```
```  5184   have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)"
```
```  5185     unfolding tan_def using \<open>cos y \<noteq> 0\<close> by (simp add: field_simps)
```
```  5186   also have "\<dots> = tan y / (1 + 1 / cos y)"
```
```  5187     using \<open>cos y \<noteq> 0\<close> unfolding add_divide_distrib by auto
```
```  5188   also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))"
```
```  5189     unfolding cos_sqrt ..
```
```  5190   also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))"
```
```  5191     unfolding real_sqrt_divide by auto
```
```  5192   finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))"
```
```  5193     unfolding \<open>1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2\<close> .
```
```  5194
```
```  5195   have "arctan x = y"
```
```  5196     using arctan_tan low high y_eq by auto
```
```  5197   also have "\<dots> = 2 * (arctan (tan (y/2)))"
```
```  5198     using arctan_tan[OF low2 high2] by auto
```
```  5199   also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))"
```
```  5200     unfolding tan_half by auto
```
```  5201   finally show ?thesis
```
```  5202     unfolding eq \<open>tan y = x\<close> .
```
```  5203 qed
```
```  5204
```
```  5205 lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y"
```
```  5206   by (simp only: arctan_less_iff)
```
```  5207
```
```  5208 lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y"
```
```  5209   by (simp only: arctan_le_iff)
```
```  5210
```
```  5211 lemma arctan_inverse:
```
```  5212   assumes "x \<noteq> 0"
```
```  5213   shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
```
```  5214 proof (rule arctan_unique)
```
```  5215   show "- (pi / 2) < sgn x * pi / 2 - arctan x"
```
```  5216     using arctan_bounded [of x] assms
```
```  5217     unfolding sgn_real_def
```
```  5218     apply (auto simp add: arctan algebra_simps)
```
```  5219     apply (drule zero_less_arctan_iff [THEN iffD2])
```
```  5220     apply arith
```
```  5221     done
```
```  5222   show "sgn x * pi / 2 - arctan x < pi / 2"
```
```  5223     using arctan_bounded [of "- x"] assms
```
```  5224     unfolding sgn_real_def arctan_minus
```
```  5225     by (auto simp add: algebra_simps)
```
```  5226   show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
```
```  5227     unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
```
```  5228     unfolding sgn_real_def
```
```  5229     by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
```
```  5230 qed
```
```  5231
```
```  5232 theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
```
```  5233 proof -
```
```  5234   have "pi / 4 = arctan 1" using arctan_one by auto
```
```  5235   also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
```
```  5236   finally show ?thesis by auto
```
```  5237 qed
```
```  5238
```
```  5239
```
```  5240 subsection \<open>Existence of Polar Coordinates\<close>
```
```  5241
```
```  5242 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1"
```
```  5243   apply (rule power2_le_imp_le [OF _ zero_le_one])
```
```  5244   apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
```
```  5245   done
```
```  5246
```
```  5247 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
```
```  5248
```
```  5249 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
```
```  5250
```
```  5251 lemma polar_Ex: "\<exists>r::real. \<exists>a. x = r * cos a & y = r * sin a"
```
```  5252 proof -
```
```  5253   have polar_ex1: "\<And>y. 0 < y \<Longrightarrow> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  5254     apply (rule_tac x = "sqrt (x\<^sup>2 + y\<^sup>2)" in exI)
```
```  5255     apply (rule_tac x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))" in exI)
```
```  5256     apply (simp add: cos_arccos_lemma1 sin_arccos_lemma1 power_divide
```
```  5257                      real_sqrt_mult [symmetric] right_diff_distrib)
```
```  5258     done
```
```  5259   show ?thesis
```
```  5260   proof (cases "0::real" y rule: linorder_cases)
```
```  5261     case less
```
```  5262       then show ?thesis by (rule polar_ex1)
```
```  5263   next
```
```  5264     case equal
```
```  5265       then show ?thesis
```
```  5266         by (force simp add: intro!: cos_zero sin_zero)
```
```  5267   next
```
```  5268     case greater
```
```  5269       then show ?thesis
```
```  5270      using polar_ex1 [where y="-y"]
```
```  5271     by auto (metis cos_minus minus_minus minus_mult_right sin_minus)
```
```  5272   qed
```
```  5273 qed
```
```  5274
```
```  5275
```
```  5276 subsection\<open>Basics about polynomial functions: products, extremal behaviour and root counts\<close>
```
```  5277
```
```  5278 lemma pairs_le_eq_Sigma:
```
```  5279   fixes m::nat
```
```  5280   shows "{(i,j). i+j \<le> m} = Sigma (atMost m) (\<lambda>r. atMost (m-r))"
```
```  5281 by auto
```
```  5282
```
```  5283 lemma setsum_up_index_split:
```
```  5284     "(\<Sum>k\<le>m + n. f k) = (\<Sum>k\<le>m. f k) + (\<Sum>k = Suc m..m + n. f k)"
```
```  5285   by (metis atLeast0AtMost Suc_eq_plus1 le0 setsum_ub_add_nat)
```
```  5286
```
```  5287 lemma Sigma_interval_disjoint:
```
```  5288   fixes w :: "'a::order"
```
```  5289   shows "(SIGMA i:A. {..v i}) \<inter> (SIGMA i:A.{v i<..w}) = {}"
```
```  5290     by auto
```
```  5291
```
```  5292 lemma product_atMost_eq_Un:
```
```  5293   fixes m :: nat
```
```  5294   shows "A \<times> {..m} = (SIGMA i:A.{..m - i}) \<union> (SIGMA i:A.{m - i<..m})"
```
```  5295     by auto
```
```  5296
```
```  5297 lemma polynomial_product: (*with thanks to Chaitanya Mangla*)
```
```  5298   fixes x:: "'a :: idom"
```
```  5299   assumes m: "\<And>i. i>m \<Longrightarrow> (a i) = 0" and n: "\<And>j. j>n \<Longrightarrow> (b j) = 0"
```
```  5300   shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
```
```  5301          (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
```
```  5302 proof -
```
```  5303   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))"
```
```  5304     by (rule setsum_product)
```
```  5305   also have "... = (\<Sum>i\<le>m + n. \<Sum>j\<le>n + m. a i * x ^ i * (b j * x ^ j))"
```
```  5306     using assms by (auto simp: setsum_up_index_split)
```
```  5307   also have "... = (\<Sum>r\<le>m + n. \<Sum>j\<le>m + n - r. a r * x ^ r * (b j * x ^ j))"
```
```  5308     apply (simp add: add_ac setsum.Sigma product_atMost_eq_Un)
```
```  5309     apply (clarsimp simp add: setsum_Un Sigma_interval_disjoint intro!: setsum.neutral)
```
```  5310     by (metis add_diff_assoc2 add.commute add_lessD1 leD m n nat_le_linear neqE)
```
```  5311   also have "... = (\<Sum>(i,j)\<in>{(i,j). i+j \<le> m+n}. (a i * x ^ i) * (b j * x ^ j))"
```
```  5312     by (auto simp: pairs_le_eq_Sigma setsum.Sigma)
```
```  5313   also have "... = (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
```
```  5314     apply (subst setsum_triangle_reindex_eq)
```
```  5315     apply (auto simp: algebra_simps setsum_right_distrib intro!: setsum.cong)
```
```  5316     by (metis le_add_diff_inverse power_add)
```
```  5317   finally show ?thesis .
```
```  5318 qed
```
```  5319
```
```  5320 lemma polynomial_product_nat:
```
```  5321   fixes x:: nat
```
```  5322   assumes m: "\<And>i. i>m \<Longrightarrow> (a i) = 0" and n: "\<And>j. j>n \<Longrightarrow> (b j) = 0"
```
```  5323   shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
```
```  5324          (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
```
```  5325   using polynomial_product [of m a n b x] assms
```
```  5326   by (simp add: Int.zpower_int Int.zmult_int Int.int_setsum [symmetric])
```
```  5327
```
```  5328 lemma polyfun_diff: (*COMPLEX_SUB_POLYFUN in HOL Light*)
```
```  5329     fixes x :: "'a::idom"
```
```  5330   assumes "1 \<le> n"
```
```  5331     shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
```
```  5332            (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)"
```
```  5333 proof -
```
```  5334   have h: "bij_betw (\<lambda>(i,j). (j,i)) ((SIGMA i : atMost n. lessThan i)) (SIGMA j : lessThan n. {Suc j..n})"
```
```  5335     by (auto simp: bij_betw_def inj_on_def)
```
```  5336   have "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
```
```  5337         (\<Sum>i\<le>n. a i * (x^i - y^i))"
```
```  5338     by (simp add: right_diff_distrib setsum_subtractf)
```
```  5339   also have "... = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))"
```
```  5340     by (simp add: power_diff_sumr2 mult.assoc)
```
```  5341   also have "... = (\<Sum>i\<le>n. \<Sum>j<i. a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5342     by (simp add: setsum_right_distrib)
```
```  5343   also have "... = (\<Sum>(i,j) \<in> (SIGMA i : atMost n. lessThan i). a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5344     by (simp add: setsum.Sigma)
```
```  5345   also have "... = (\<Sum>(j,i) \<in> (SIGMA j : lessThan n. {Suc j..n}). a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5346     by (auto simp add: setsum.reindex_bij_betw [OF h, symmetric] intro: setsum.strong_cong)
```
```  5347   also have "... = (\<Sum>j<n. \<Sum>i=Suc j..n. a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5348     by (simp add: setsum.Sigma)
```
```  5349   also have "... = (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)"
```
```  5350     by (simp add: setsum_right_distrib mult_ac)
```
```  5351   finally show ?thesis .
```
```  5352 qed
```
```  5353
```
```  5354 lemma polyfun_diff_alt: (*COMPLEX_SUB_POLYFUN_ALT in HOL Light*)
```
```  5355     fixes x :: "'a::idom"
```
```  5356   assumes "1 \<le> n"
```
```  5357     shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
```
```  5358            (x - y) * ((\<Sum>j<n. \<Sum>k<n-j. a(j+k+1) * y^k * x^j))"
```
```  5359 proof -
```
```  5360   { fix j::nat
```
```  5361     assume "j<n"
```
```  5362     have h: "bij_betw (\<lambda>i. i - (j + 1)) {Suc j..n} (lessThan (n-j))"
```
```  5363       apply (auto simp: bij_betw_def inj_on_def)
```
```  5364       apply (rule_tac x="x + Suc j" in image_eqI)
```
```  5365       apply (auto simp: )
```
```  5366       done
```
```  5367     have "(\<Sum>i=Suc j..n. a i * y^(i - j - 1)) = (\<Sum>k<n-j. a(j+k+1) * y^k)"
```
```  5368       by (auto simp add: setsum.reindex_bij_betw [OF h, symmetric] intro: setsum.strong_cong)
```
```  5369   }
```
```  5370   then show ?thesis
```
```  5371     by (simp add: polyfun_diff [OF assms] setsum_left_distrib)
```
```  5372 qed
```
```  5373
```
```  5374 lemma polyfun_linear_factor:  (*COMPLEX_POLYFUN_LINEAR_FACTOR in HOL Light*)
```
```  5375   fixes a :: "'a::idom"
```
```  5376   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)"
```
```  5377 proof (cases "n=0")
```
```  5378   case True then show ?thesis
```
```  5379     by simp
```
```  5380 next
```
```  5381   case False
```
```  5382   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)) =
```
```  5383         (\<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))"
```
```  5384     by (simp add: algebra_simps)
```
```  5385   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))"
```
```  5386     using False by (simp add: polyfun_diff)
```
```  5387   also have "... = True"
```
```  5388     by auto
```
```  5389   finally show ?thesis
```
```  5390     by simp
```
```  5391 qed
```
```  5392
```
```  5393 lemma polyfun_linear_factor_root:  (*COMPLEX_POLYFUN_LINEAR_FACTOR_ROOT in HOL Light*)
```
```  5394   fixes a :: "'a::idom"
```
```  5395   assumes "(\<Sum>i\<le>n. c(i) * a^i) = 0"
```
```  5396   obtains b where "\<And>z. (\<Sum>i\<le>n. c(i) * z^i) = (z - a) * (\<Sum>i<n. b(i) * z^i)"
```
```  5397   using polyfun_linear_factor [of c n a] assms
```
```  5398   by auto
```
```  5399
```
```  5400 (*The material of this section, up until this point, could go into a new theory of polynomials
```
```  5401   based on Main alone. The remaining material involves limits, continuity, series, etc.*)
```
```  5402
```
```  5403 lemma isCont_polynom:
```
```  5404   fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
```
```  5405   shows "isCont (\<lambda>w. \<Sum>i\<le>n. c i * w^i) a"
```
```  5406   by simp
```
```  5407
```
```  5408 lemma zero_polynom_imp_zero_coeffs:
```
```  5409     fixes c :: "nat \<Rightarrow> 'a::{ab_semigroup_mult,real_normed_div_algebra}"
```
```  5410   assumes "\<And>w. (\<Sum>i\<le>n. c i * w^i) = 0"  "k \<le> n"
```
```  5411     shows "c k = 0"
```
```  5412 using assms
```
```  5413 proof (induction n arbitrary: c k)
```
```  5414   case 0
```
```  5415   then show ?case
```
```  5416     by simp
```
```  5417 next
```
```  5418   case (Suc n c k)
```
```  5419   have [simp]: "c 0 = 0" using Suc.prems(1) [of 0]
```
```  5420     by simp
```
```  5421   { fix w
```
```  5422     have "(\<Sum>i\<le>Suc n. c i * w^i) = (\<Sum>i\<le>n. c (Suc i) * w ^ Suc i)"
```
```  5423       unfolding Set_Interval.setsum_atMost_Suc_shift
```
```  5424       by simp
```
```  5425     also have "... = w * (\<Sum>i\<le>n. c (Suc i) * w^i)"
```
```  5426       by (simp add: setsum_right_distrib ac_simps)
```
```  5427     finally have "(\<Sum>i\<le>Suc n. c i * w^i) = w * (\<Sum>i\<le>n. c (Suc i) * w^i)" .
```
```  5428   }
```
```  5429   then have wnz: "\<And>w. w \<noteq> 0 \<Longrightarrow> (\<Sum>i\<le>n. c (Suc i) * w^i) = 0"
```
```  5430     using Suc  by auto
```
```  5431   then have "(\<lambda>h. \<Sum>i\<le>n. c (Suc i) * h^i) -- 0 --> 0"
```
```  5432     by (simp cong: LIM_cong)                   --\<open>the case @{term"w=0"} by continuity\<close>
```
```  5433   then have "(\<Sum>i\<le>n. c (Suc i) * 0^i) = 0"
```
```  5434     using isCont_polynom [of 0 "\<lambda>i. c (Suc i)" n] LIM_unique
```
```  5435     by (force simp add: Limits.isCont_iff)
```
```  5436   then have "\<And>w. (\<Sum>i\<le>n. c (Suc i) * w^i) = 0" using wnz
```
```  5437     by metis
```
```  5438   then have "\<And>i. i\<le>n \<Longrightarrow> c (Suc i) = 0"
```
```  5439     using Suc.IH [of "\<lambda>i. c (Suc i)"]
```
```  5440     by blast
```
```  5441   then show ?case using \<open>k \<le> Suc n\<close>
```
```  5442     by (cases k) auto
```
```  5443 qed
```
```  5444
```
```  5445 lemma polyfun_rootbound: (*COMPLEX_POLYFUN_ROOTBOUND in HOL Light*)
```
```  5446     fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5447   assumes "c k \<noteq> 0" "k\<le>n"
```
```  5448     shows "finite {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<and>
```
```  5449              card {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<le> n"
```
```  5450 using assms
```
```  5451 proof (induction n arbitrary: c k)
```
```  5452   case 0
```
```  5453   then show ?case
```
```  5454     by simp
```
```  5455 next
```
```  5456   case (Suc m c k)
```
```  5457   let ?succase = ?case
```
```  5458   show ?case
```
```  5459   proof (cases "{z. (\<Sum>i\<le>Suc m. c(i) * z^i) = 0} = {}")
```
```  5460     case True
```
```  5461     then show ?succase
```
```  5462       by simp
```
```  5463   next
```
```  5464     case False
```
```  5465     then obtain z0 where z0: "(\<Sum>i\<le>Suc m. c(i) * z0^i) = 0"
```
```  5466       by blast
```
```  5467     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)"
```
```  5468       using polyfun_linear_factor_root [OF z0, unfolded lessThan_Suc_atMost]
```
```  5469       by blast
```
```  5470     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}"
```
```  5471       by auto
```
```  5472     have "~(\<forall>k\<le>m. b k = 0)"
```
```  5473     proof
```
```  5474       assume [simp]: "\<forall>k\<le>m. b k = 0"
```
```  5475       then have "\<And>w. (\<Sum>i\<le>m. b i * w^i) = 0"
```
```  5476         by simp
```
```  5477       then have "\<And>w. (\<Sum>i\<le>Suc m. c i * w^i) = 0"
```
```  5478         using b by simp
```
```  5479       then have "\<And>k. k \<le> Suc m \<Longrightarrow> c k = 0"
```
```  5480         using zero_polynom_imp_zero_coeffs
```
```  5481         by blast
```
```  5482       then show False using Suc.prems
```
```  5483         by blast
```
```  5484     qed
```
```  5485     then obtain k' where bk': "b k' \<noteq> 0" "k' \<le> m"
```
```  5486       by blast
```
```  5487     show ?succase
```
```  5488       using Suc.IH [of b k'] bk'
```
```  5489       by (simp add: eq card_insert_if del: setsum_atMost_Suc)
```
```  5490     qed
```
```  5491 qed
```
```  5492
```
```  5493 lemma
```
```  5494     fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5495   assumes "c k \<noteq> 0" "k\<le>n"
```
```  5496     shows polyfun_roots_finite: "finite {z. (\<Sum>i\<le>n. c(i) * z^i) = 0}"
```
```  5497       and polyfun_roots_card:   "card {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<le> n"
```
```  5498 using polyfun_rootbound assms
```
```  5499   by auto
```
```  5500
```
```  5501 lemma polyfun_finite_roots: (*COMPLEX_POLYFUN_FINITE_ROOTS in HOL Light*)
```
```  5502   fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5503   shows "finite {x. (\<Sum>i\<le>n. c i * x^i) = 0} \<longleftrightarrow> (\<exists>i\<le>n. c i \<noteq> 0)"
```
```  5504         (is "?lhs = ?rhs")
```
```  5505 proof
```
```  5506   assume ?lhs
```
```  5507   moreover
```
```  5508   { assume "\<forall>i\<le>n. c i = 0"
```
```  5509     then have "\<And>x. (\<Sum>i\<le>n. c i * x^i) = 0"
```
```  5510       by simp
```
```  5511     then have "\<not> finite {x. (\<Sum>i\<le>n. c i * x^i) = 0}"
```
```  5512       using ex_new_if_finite [OF infinite_UNIV_char_0 [where 'a='a]]
```
```  5513       by auto
```
```  5514   }
```
```  5515   ultimately show ?rhs
```
```  5516   by metis
```
```  5517 next
```
```  5518   assume ?rhs
```
```  5519   then show ?lhs
```
```  5520     using polyfun_rootbound
```
```  5521     by blast
```
```  5522 qed
```
```  5523
```
```  5524 lemma polyfun_eq_0: (*COMPLEX_POLYFUN_EQ_0 in HOL Light*)
```
```  5525   fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5526   shows "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = 0) \<longleftrightarrow> (\<forall>i\<le>n. c i = 0)"
```
```  5527   using zero_polynom_imp_zero_coeffs by auto
```
```  5528
```
```  5529 lemma polyfun_eq_coeffs:
```
```  5530   fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5531   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)"
```
```  5532 proof -
```
```  5533   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)"
```
```  5534     by (simp add: left_diff_distrib Groups_Big.setsum_subtractf)
```
```  5535   also have "... \<longleftrightarrow> (\<forall>i\<le>n. c i - d i = 0)"
```
```  5536     by (rule polyfun_eq_0)
```
```  5537   finally show ?thesis
```
```  5538     by simp
```
```  5539 qed
```
```  5540
```
```  5541 lemma polyfun_eq_const: (*COMPLEX_POLYFUN_EQ_CONST in HOL Light*)
```
```  5542   fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5543   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)"
```
```  5544         (is "?lhs = ?rhs")
```
```  5545 proof -
```
```  5546   have *: "\<forall>x. (\<Sum>i\<le>n. (if i=0 then k else 0) * x^i) = k"
```
```  5547     by (induct n) auto
```
```  5548   show ?thesis
```
```  5549   proof
```
```  5550     assume ?lhs
```
```  5551     with * have "(\<forall>i\<le>n. c i = (if i=0 then k else 0))"
```
```  5552       by (simp add: polyfun_eq_coeffs [symmetric])
```
```  5553     then show ?rhs
```
```  5554       by simp
```
```  5555   next
```
```  5556     assume ?rhs then show ?lhs
```
```  5557       by (induct n) auto
```
```  5558   qed
```
```  5559 qed
```
```  5560
```
```  5561 lemma root_polyfun:
```
```  5562   fixes z:: "'a::idom"
```
```  5563   assumes "1 \<le> n"
```
```  5564     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"
```
```  5565   using assms
```
```  5566   by (cases n; simp add: setsum_head_Suc atLeast0AtMost [symmetric])
```
```  5567
```
```  5568 lemma
```
```  5569     fixes zz :: "'a::{idom,real_normed_div_algebra}"
```
```  5570   assumes "1 \<le> n"
```
```  5571     shows finite_roots_unity: "finite {z::'a. z^n = 1}"
```
```  5572       and card_roots_unity:   "card {z::'a. z^n = 1} \<le> n"
```
```  5573   using polyfun_rootbound [of "\<lambda>i. if i = 0 then -1 else if i=n then 1 else 0" n n] assms
```
```  5574   by (auto simp add: root_polyfun [OF assms])
```
```  5575
```
```  5576 end
```