src/HOL/Transcendental.thy
 author haftmann Tue Oct 21 21:10:44 2014 +0200 (2014-10-21) changeset 58740 cb9d84d3e7f2 parent 58729 e8ecc79aee43 child 58834 773b378d9313 permissions -rw-r--r--
turn even into an abbreviation
```     1 (*  Title:      HOL/Transcendental.thy
```
```     2     Author:     Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
```
```     3     Author:     Lawrence C Paulson
```
```     4     Author:     Jeremy Avigad
```
```     5 *)
```
```     6
```
```     7 header{*Power Series, Transcendental Functions etc.*}
```
```     8
```
```     9 theory Transcendental
```
```    10 imports Fact Series Deriv NthRoot
```
```    11 begin
```
```    12
```
```    13 lemma root_test_convergence:
```
```    14   fixes f :: "nat \<Rightarrow> 'a::banach"
```
```    15   assumes f: "(\<lambda>n. root n (norm (f n))) ----> x" -- "could be weakened to lim sup"
```
```    16   assumes "x < 1"
```
```    17   shows "summable f"
```
```    18 proof -
```
```    19   have "0 \<le> x"
```
```    20     by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1])
```
```    21   from `x < 1` obtain z where z: "x < z" "z < 1"
```
```    22     by (metis dense)
```
```    23   from f `x < z`
```
```    24   have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
```
```    25     by (rule order_tendstoD)
```
```    26   then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially"
```
```    27     using eventually_ge_at_top
```
```    28   proof eventually_elim
```
```    29     fix n assume less: "root n (norm (f n)) < z" and n: "1 \<le> n"
```
```    30     from power_strict_mono[OF less, of n] n
```
```    31     show "norm (f n) \<le> z ^ n"
```
```    32       by simp
```
```    33   qed
```
```    34   then show "summable f"
```
```    35     unfolding eventually_sequentially
```
```    36     using z `0 \<le> x` by (auto intro!: summable_comparison_test[OF _  summable_geometric])
```
```    37 qed
```
```    38
```
```    39 subsection {* Properties of Power Series *}
```
```    40
```
```    41 lemma lemma_realpow_diff:
```
```    42   fixes y :: "'a::monoid_mult"
```
```    43   shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
```
```    44 proof -
```
```    45   assume "p \<le> n"
```
```    46   hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
```
```    47   thus ?thesis by (simp add: power_commutes)
```
```    48 qed
```
```    49
```
```    50 lemma lemma_realpow_diff_sumr2:
```
```    51   fixes y :: "'a::{comm_ring,monoid_mult}"
```
```    52   shows
```
```    53     "x ^ (Suc n) - y ^ (Suc n) =
```
```    54       (x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))"
```
```    55 proof (induct n)
```
```    56   case (Suc n)
```
```    57   have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x ^ n) - y * (y * y ^ n)"
```
```    58     by simp
```
```    59   also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x ^ n)"
```
```    60     by (simp add: algebra_simps)
```
```    61   also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
```
```    62     by (simp only: Suc)
```
```    63   also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
```
```    64     by (simp only: mult.left_commute)
```
```    65   also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
```
```    66     by (simp add: field_simps Suc_diff_le setsum_left_distrib setsum_right_distrib)
```
```    67   finally show ?case .
```
```    68 qed simp
```
```    69
```
```    70 corollary power_diff_sumr2: --{* @{text COMPLEX_POLYFUN} in HOL Light *}
```
```    71   fixes x :: "'a::{comm_ring,monoid_mult}"
```
```    72   shows   "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)"
```
```    73 using lemma_realpow_diff_sumr2[of x "n - 1" y]
```
```    74 by (cases "n = 0") (simp_all add: field_simps)
```
```    75
```
```    76 lemma lemma_realpow_rev_sumr:
```
```    77    "(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) =
```
```    78     (\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))"
```
```    79   by (subst nat_diff_setsum_reindex[symmetric]) simp
```
```    80
```
```    81 lemma power_diff_1_eq:
```
```    82   fixes x :: "'a::{comm_ring,monoid_mult}"
```
```    83   shows "n \<noteq> 0 \<Longrightarrow> x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"
```
```    84 using lemma_realpow_diff_sumr2 [of x _ 1]
```
```    85   by (cases n) auto
```
```    86
```
```    87 lemma one_diff_power_eq':
```
```    88   fixes x :: "'a::{comm_ring,monoid_mult}"
```
```    89   shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))"
```
```    90 using lemma_realpow_diff_sumr2 [of 1 _ x]
```
```    91   by (cases n) auto
```
```    92
```
```    93 lemma one_diff_power_eq:
```
```    94   fixes x :: "'a::{comm_ring,monoid_mult}"
```
```    95   shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^i)"
```
```    96 by (metis one_diff_power_eq' [of n x] nat_diff_setsum_reindex)
```
```    97
```
```    98 text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
```
```    99   x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
```
```   100
```
```   101 lemma powser_insidea:
```
```   102   fixes x z :: "'a::real_normed_div_algebra"
```
```   103   assumes 1: "summable (\<lambda>n. f n * x ^ n)"
```
```   104     and 2: "norm z < norm x"
```
```   105   shows "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   106 proof -
```
```   107   from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
```
```   108   from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
```
```   109     by (rule summable_LIMSEQ_zero)
```
```   110   hence "convergent (\<lambda>n. f n * x ^ n)"
```
```   111     by (rule convergentI)
```
```   112   hence "Cauchy (\<lambda>n. f n * x ^ n)"
```
```   113     by (rule convergent_Cauchy)
```
```   114   hence "Bseq (\<lambda>n. f n * x ^ n)"
```
```   115     by (rule Cauchy_Bseq)
```
```   116   then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
```
```   117     by (simp add: Bseq_def, safe)
```
```   118   have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
```
```   119                    K * norm (z ^ n) * inverse (norm (x ^ n))"
```
```   120   proof (intro exI allI impI)
```
```   121     fix n::nat
```
```   122     assume "0 \<le> n"
```
```   123     have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
```
```   124           norm (f n * x ^ n) * norm (z ^ n)"
```
```   125       by (simp add: norm_mult abs_mult)
```
```   126     also have "\<dots> \<le> K * norm (z ^ n)"
```
```   127       by (simp only: mult_right_mono 4 norm_ge_zero)
```
```   128     also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
```
```   129       by (simp add: x_neq_0)
```
```   130     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
```
```   131       by (simp only: mult.assoc)
```
```   132     finally show "norm (norm (f n * z ^ n)) \<le>
```
```   133                   K * norm (z ^ n) * inverse (norm (x ^ n))"
```
```   134       by (simp add: mult_le_cancel_right x_neq_0)
```
```   135   qed
```
```   136   moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
```
```   137   proof -
```
```   138     from 2 have "norm (norm (z * inverse x)) < 1"
```
```   139       using x_neq_0
```
```   140       by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
```
```   141     hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
```
```   142       by (rule summable_geometric)
```
```   143     hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
```
```   144       by (rule summable_mult)
```
```   145     thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
```
```   146       using x_neq_0
```
```   147       by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
```
```   148                     power_inverse norm_power mult.assoc)
```
```   149   qed
```
```   150   ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   151     by (rule summable_comparison_test)
```
```   152 qed
```
```   153
```
```   154 lemma powser_inside:
```
```   155   fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
```
```   156   shows
```
```   157     "summable (\<lambda>n. f n * (x ^ n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
```
```   158       summable (\<lambda>n. f n * (z ^ n))"
```
```   159   by (rule powser_insidea [THEN summable_norm_cancel])
```
```   160
```
```   161 lemma sum_split_even_odd:
```
```   162   fixes f :: "nat \<Rightarrow> real"
```
```   163   shows
```
```   164     "(\<Sum>i<2 * n. if even i then f i else g i) =
```
```   165      (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
```
```   166 proof (induct n)
```
```   167   case 0
```
```   168   then show ?case by simp
```
```   169 next
```
```   170   case (Suc n)
```
```   171   have "(\<Sum>i<2 * Suc n. if even i then f i else g i) =
```
```   172     (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
```
```   173     using Suc.hyps unfolding One_nat_def by auto
```
```   174   also have "\<dots> = (\<Sum>i<Suc n. f (2 * i)) + (\<Sum>i<Suc n. g (2 * i + 1))"
```
```   175     by auto
```
```   176   finally show ?case .
```
```   177 qed
```
```   178
```
```   179 lemma sums_if':
```
```   180   fixes g :: "nat \<Rightarrow> real"
```
```   181   assumes "g sums x"
```
```   182   shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   183   unfolding sums_def
```
```   184 proof (rule LIMSEQ_I)
```
```   185   fix r :: real
```
```   186   assume "0 < r"
```
```   187   from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
```
```   188   obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g {..<n} - x) < r)" by blast
```
```   189
```
```   190   let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)"
```
```   191   {
```
```   192     fix m
```
```   193     assume "m \<ge> 2 * no"
```
```   194     hence "m div 2 \<ge> no" by auto
```
```   195     have sum_eq: "?SUM (2 * (m div 2)) = setsum g {..< m div 2}"
```
```   196       using sum_split_even_odd by auto
```
```   197     hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
```
```   198       using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
```
```   199     moreover
```
```   200     have "?SUM (2 * (m div 2)) = ?SUM m"
```
```   201     proof (cases "even m")
```
```   202       case True
```
```   203       then show ?thesis by (auto simp add: even_two_times_div_two)
```
```   204     next
```
```   205       case False
```
```   206       then have eq: "Suc (2 * (m div 2)) = m" by (simp add: odd_two_times_div_two_Suc)
```
```   207       hence "even (2 * (m div 2))" using `odd m` by auto
```
```   208       have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
```
```   209       also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
```
```   210       finally show ?thesis by auto
```
```   211     qed
```
```   212     ultimately have "(norm (?SUM m - x) < r)" by auto
```
```   213   }
```
```   214   thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
```
```   215 qed
```
```   216
```
```   217 lemma sums_if:
```
```   218   fixes g :: "nat \<Rightarrow> real"
```
```   219   assumes "g sums x" and "f sums y"
```
```   220   shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
```
```   221 proof -
```
```   222   let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
```
```   223   {
```
```   224     fix B T E
```
```   225     have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
```
```   226       by (cases B) auto
```
```   227   } note if_sum = this
```
```   228   have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   229     using sums_if'[OF `g sums x`] .
```
```   230   {
```
```   231     have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
```
```   232
```
```   233     have "?s sums y" using sums_if'[OF `f sums y`] .
```
```   234     from this[unfolded sums_def, THEN LIMSEQ_Suc]
```
```   235     have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
```
```   236       by (simp add: lessThan_Suc_eq_insert_0 image_iff setsum.reindex if_eq sums_def cong del: if_cong)
```
```   237   }
```
```   238   from sums_add[OF g_sums this] show ?thesis unfolding if_sum .
```
```   239 qed
```
```   240
```
```   241 subsection {* Alternating series test / Leibniz formula *}
```
```   242
```
```   243 lemma sums_alternating_upper_lower:
```
```   244   fixes a :: "nat \<Rightarrow> real"
```
```   245   assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
```
```   246   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>
```
```   247              ((\<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)"
```
```   248   (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
```
```   249 proof (rule nested_sequence_unique)
```
```   250   have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
```
```   251
```
```   252   show "\<forall>n. ?f n \<le> ?f (Suc n)"
```
```   253   proof
```
```   254     fix n
```
```   255     show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
```
```   256   qed
```
```   257   show "\<forall>n. ?g (Suc n) \<le> ?g n"
```
```   258   proof
```
```   259     fix n
```
```   260     show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
```
```   261       unfolding One_nat_def by auto
```
```   262   qed
```
```   263   show "\<forall>n. ?f n \<le> ?g n"
```
```   264   proof
```
```   265     fix n
```
```   266     show "?f n \<le> ?g n" using fg_diff a_pos
```
```   267       unfolding One_nat_def by auto
```
```   268   qed
```
```   269   show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff
```
```   270   proof (rule LIMSEQ_I)
```
```   271     fix r :: real
```
```   272     assume "0 < r"
```
```   273     with `a ----> 0`[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
```
```   274       by auto
```
```   275     hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   276     thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   277   qed
```
```   278 qed
```
```   279
```
```   280 lemma summable_Leibniz':
```
```   281   fixes a :: "nat \<Rightarrow> real"
```
```   282   assumes a_zero: "a ----> 0"
```
```   283     and a_pos: "\<And> n. 0 \<le> a n"
```
```   284     and a_monotone: "\<And> n. a (Suc n) \<le> a n"
```
```   285   shows summable: "summable (\<lambda> n. (-1)^n * a n)"
```
```   286     and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
```
```   287     and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   288     and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i<2*n+1. (-1)^i*a i)"
```
```   289     and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   290 proof -
```
```   291   let ?S = "\<lambda>n. (-1)^n * a n"
```
```   292   let ?P = "\<lambda>n. \<Sum>i<n. ?S i"
```
```   293   let ?f = "\<lambda>n. ?P (2 * n)"
```
```   294   let ?g = "\<lambda>n. ?P (2 * n + 1)"
```
```   295   obtain l :: real
```
```   296     where below_l: "\<forall> n. ?f n \<le> l"
```
```   297       and "?f ----> l"
```
```   298       and above_l: "\<forall> n. l \<le> ?g n"
```
```   299       and "?g ----> l"
```
```   300     using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
```
```   301
```
```   302   let ?Sa = "\<lambda>m. \<Sum>n<m. ?S n"
```
```   303   have "?Sa ----> l"
```
```   304   proof (rule LIMSEQ_I)
```
```   305     fix r :: real
```
```   306     assume "0 < r"
```
```   307     with `?f ----> l`[THEN LIMSEQ_D]
```
```   308     obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
```
```   309
```
```   310     from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
```
```   311     obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
```
```   312
```
```   313     {
```
```   314       fix n :: nat
```
```   315       assume "n \<ge> (max (2 * f_no) (2 * g_no))"
```
```   316       hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
```
```   317       have "norm (?Sa n - l) < r"
```
```   318       proof (cases "even n")
```
```   319         case True
```
```   320         then have n_eq: "2 * (n div 2) = n" by (simp add: even_two_times_div_two)
```
```   321         with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no"
```
```   322           by auto
```
```   323         from f[OF this] show ?thesis
```
```   324           unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
```
```   325       next
```
```   326         case False
```
```   327         hence "even (n - 1)" by simp
```
```   328         then have n_eq: "2 * ((n - 1) div 2) = n - 1"
```
```   329           by (simp add: even_two_times_div_two)
```
```   330         hence range_eq: "n - 1 + 1 = n"
```
```   331           using odd_pos[OF False] by auto
```
```   332
```
```   333         from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no"
```
```   334           by auto
```
```   335         from g[OF this] show ?thesis
```
```   336           unfolding n_eq range_eq .
```
```   337       qed
```
```   338     }
```
```   339     thus "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
```
```   340   qed
```
```   341   hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
```
```   342     unfolding sums_def .
```
```   343   thus "summable ?S" using summable_def by auto
```
```   344
```
```   345   have "l = suminf ?S" using sums_unique[OF sums_l] .
```
```   346
```
```   347   fix n
```
```   348   show "suminf ?S \<le> ?g n"
```
```   349     unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
```
```   350   show "?f n \<le> suminf ?S"
```
```   351     unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
```
```   352   show "?g ----> suminf ?S"
```
```   353     using `?g ----> l` `l = suminf ?S` by auto
```
```   354   show "?f ----> suminf ?S"
```
```   355     using `?f ----> l` `l = suminf ?S` by auto
```
```   356 qed
```
```   357
```
```   358 theorem summable_Leibniz:
```
```   359   fixes a :: "nat \<Rightarrow> real"
```
```   360   assumes a_zero: "a ----> 0" and "monoseq a"
```
```   361   shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
```
```   362     and "0 < a 0 \<longrightarrow>
```
```   363       (\<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")
```
```   364     and "a 0 < 0 \<longrightarrow>
```
```   365       (\<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")
```
```   366     and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) ----> (\<Sum>i. (- 1)^i*a i)" (is "?f")
```
```   367     and "(\<lambda>n. \<Sum>i<2*n+1. (- 1)^i*a i) ----> (\<Sum>i. (- 1)^i*a i)" (is "?g")
```
```   368 proof -
```
```   369   have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
```
```   370   proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
```
```   371     case True
```
```   372     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n"
```
```   373       by auto
```
```   374     {
```
```   375       fix n
```
```   376       have "a (Suc n) \<le> a n"
```
```   377         using ord[where n="Suc n" and m=n] by auto
```
```   378     } note mono = this
```
```   379     note leibniz = summable_Leibniz'[OF `a ----> 0` ge0]
```
```   380     from leibniz[OF mono]
```
```   381     show ?thesis using `0 \<le> a 0` by auto
```
```   382   next
```
```   383     let ?a = "\<lambda> n. - a n"
```
```   384     case False
```
```   385     with monoseq_le[OF `monoseq a` `a ----> 0`]
```
```   386     have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
```
```   387     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
```
```   388       by auto
```
```   389     {
```
```   390       fix n
```
```   391       have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
```
```   392         by auto
```
```   393     } note monotone = this
```
```   394     note leibniz =
```
```   395       summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
```
```   396         OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
```
```   397     have "summable (\<lambda> n. (-1)^n * ?a n)"
```
```   398       using leibniz(1) by auto
```
```   399     then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
```
```   400       unfolding summable_def by auto
```
```   401     from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
```
```   402       by auto
```
```   403     hence ?summable unfolding summable_def by auto
```
```   404     moreover
```
```   405     have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
```
```   406       unfolding minus_diff_minus by auto
```
```   407
```
```   408     from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
```
```   409     have move_minus: "(\<Sum>n. - ((- 1) ^ n * a n)) = - (\<Sum>n. (- 1) ^ n * a n)"
```
```   410       by auto
```
```   411
```
```   412     have ?pos using `0 \<le> ?a 0` by auto
```
```   413     moreover have ?neg
```
```   414       using leibniz(2,4)
```
```   415       unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
```
```   416       by auto
```
```   417     moreover have ?f and ?g
```
```   418       using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
```
```   419       by auto
```
```   420     ultimately show ?thesis by auto
```
```   421   qed
```
```   422   then show ?summable and ?pos and ?neg and ?f and ?g
```
```   423     by safe
```
```   424 qed
```
```   425
```
```   426 subsection {* Term-by-Term Differentiability of Power Series *}
```
```   427
```
```   428 definition diffs :: "(nat \<Rightarrow> 'a::ring_1) \<Rightarrow> nat \<Rightarrow> 'a"
```
```   429   where "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))"
```
```   430
```
```   431 text{*Lemma about distributing negation over it*}
```
```   432 lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
```
```   433   by (simp add: diffs_def)
```
```   434
```
```   435 lemma sums_Suc_imp:
```
```   436   "(f::nat \<Rightarrow> 'a::real_normed_vector) 0 = 0 \<Longrightarrow> (\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
```
```   437   using sums_Suc_iff[of f] by simp
```
```   438
```
```   439 lemma diffs_equiv:
```
```   440   fixes x :: "'a::{real_normed_vector, ring_1}"
```
```   441   shows "summable (\<lambda>n. diffs c n * x^n) \<Longrightarrow>
```
```   442       (\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)"
```
```   443   unfolding diffs_def
```
```   444   by (simp add: summable_sums sums_Suc_imp)
```
```   445
```
```   446 lemma lemma_termdiff1:
```
```   447   fixes z :: "'a :: {monoid_mult,comm_ring}" shows
```
```   448   "(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
```
```   449    (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
```
```   450   by (auto simp add: algebra_simps power_add [symmetric])
```
```   451
```
```   452 lemma sumr_diff_mult_const2:
```
```   453   "setsum f {..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i<n. f i - r)"
```
```   454   by (simp add: setsum_subtractf)
```
```   455
```
```   456 lemma lemma_termdiff2:
```
```   457   fixes h :: "'a :: {field}"
```
```   458   assumes h: "h \<noteq> 0"
```
```   459   shows
```
```   460     "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
```
```   461      h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p.
```
```   462           (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
```
```   463   apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
```
```   464   apply (simp add: right_diff_distrib diff_divide_distrib h)
```
```   465   apply (simp add: mult.assoc [symmetric])
```
```   466   apply (cases "n", simp)
```
```   467   apply (simp add: lemma_realpow_diff_sumr2 h
```
```   468                    right_diff_distrib [symmetric] mult.assoc
```
```   469               del: power_Suc setsum_lessThan_Suc of_nat_Suc)
```
```   470   apply (subst lemma_realpow_rev_sumr)
```
```   471   apply (subst sumr_diff_mult_const2)
```
```   472   apply simp
```
```   473   apply (simp only: lemma_termdiff1 setsum_right_distrib)
```
```   474   apply (rule setsum.cong [OF refl])
```
```   475   apply (simp add: less_iff_Suc_add)
```
```   476   apply (clarify)
```
```   477   apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 ac_simps
```
```   478               del: setsum_lessThan_Suc power_Suc)
```
```   479   apply (subst mult.assoc [symmetric], subst power_add [symmetric])
```
```   480   apply (simp add: ac_simps)
```
```   481   done
```
```   482
```
```   483 lemma real_setsum_nat_ivl_bounded2:
```
```   484   fixes K :: "'a::linordered_semidom"
```
```   485   assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
```
```   486     and K: "0 \<le> K"
```
```   487   shows "setsum f {..<n-k} \<le> of_nat n * K"
```
```   488   apply (rule order_trans [OF setsum_mono])
```
```   489   apply (rule f, simp)
```
```   490   apply (simp add: mult_right_mono K)
```
```   491   done
```
```   492
```
```   493 lemma lemma_termdiff3:
```
```   494   fixes h z :: "'a::{real_normed_field}"
```
```   495   assumes 1: "h \<noteq> 0"
```
```   496     and 2: "norm z \<le> K"
```
```   497     and 3: "norm (z + h) \<le> K"
```
```   498   shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
```
```   499           \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   500 proof -
```
```   501   have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
```
```   502         norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p.
```
```   503           (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
```
```   504     by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult)
```
```   505   also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
```
```   506   proof (rule mult_right_mono [OF _ norm_ge_zero])
```
```   507     from norm_ge_zero 2 have K: "0 \<le> K"
```
```   508       by (rule order_trans)
```
```   509     have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
```
```   510       apply (erule subst)
```
```   511       apply (simp only: norm_mult norm_power power_add)
```
```   512       apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
```
```   513       done
```
```   514     show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))
```
```   515           \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
```
```   516       apply (intro
```
```   517          order_trans [OF norm_setsum]
```
```   518          real_setsum_nat_ivl_bounded2
```
```   519          mult_nonneg_nonneg
```
```   520          of_nat_0_le_iff
```
```   521          zero_le_power K)
```
```   522       apply (rule le_Kn, simp)
```
```   523       done
```
```   524   qed
```
```   525   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   526     by (simp only: mult.assoc)
```
```   527   finally show ?thesis .
```
```   528 qed
```
```   529
```
```   530 lemma lemma_termdiff4:
```
```   531   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```   532   assumes k: "0 < (k::real)"
```
```   533     and le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
```
```   534   shows "f -- 0 --> 0"
```
```   535 proof (rule tendsto_norm_zero_cancel)
```
```   536   show "(\<lambda>h. norm (f h)) -- 0 --> 0"
```
```   537   proof (rule real_tendsto_sandwich)
```
```   538     show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)"
```
```   539       by simp
```
```   540     show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)"
```
```   541       using k by (auto simp add: eventually_at dist_norm le)
```
```   542     show "(\<lambda>h. 0) -- (0::'a) --> (0::real)"
```
```   543       by (rule tendsto_const)
```
```   544     have "(\<lambda>h. K * norm h) -- (0::'a) --> K * norm (0::'a)"
```
```   545       by (intro tendsto_intros)
```
```   546     then show "(\<lambda>h. K * norm h) -- (0::'a) --> 0"
```
```   547       by simp
```
```   548   qed
```
```   549 qed
```
```   550
```
```   551 lemma lemma_termdiff5:
```
```   552   fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
```
```   553   assumes k: "0 < (k::real)"
```
```   554   assumes f: "summable f"
```
```   555   assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
```
```   556   shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
```
```   557 proof (rule lemma_termdiff4 [OF k])
```
```   558   fix h::'a
```
```   559   assume "h \<noteq> 0" and "norm h < k"
```
```   560   hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
```
```   561     by (simp add: le)
```
```   562   hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
```
```   563     by simp
```
```   564   moreover from f have B: "summable (\<lambda>n. f n * norm h)"
```
```   565     by (rule summable_mult2)
```
```   566   ultimately have C: "summable (\<lambda>n. norm (g h n))"
```
```   567     by (rule summable_comparison_test)
```
```   568   hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
```
```   569     by (rule summable_norm)
```
```   570   also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
```
```   571     by (rule suminf_le)
```
```   572   also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
```
```   573     by (rule suminf_mult2 [symmetric])
```
```   574   finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
```
```   575 qed
```
```   576
```
```   577
```
```   578 text{* FIXME: Long proofs*}
```
```   579
```
```   580 lemma termdiffs_aux:
```
```   581   fixes x :: "'a::{real_normed_field,banach}"
```
```   582   assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
```
```   583     and 2: "norm x < norm K"
```
```   584   shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
```
```   585              - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   586 proof -
```
```   587   from dense [OF 2]
```
```   588   obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
```
```   589   from norm_ge_zero r1 have r: "0 < r"
```
```   590     by (rule order_le_less_trans)
```
```   591   hence r_neq_0: "r \<noteq> 0" by simp
```
```   592   show ?thesis
```
```   593   proof (rule lemma_termdiff5)
```
```   594     show "0 < r - norm x" using r1 by simp
```
```   595     from r r2 have "norm (of_real r::'a) < norm K"
```
```   596       by simp
```
```   597     with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
```
```   598       by (rule powser_insidea)
```
```   599     hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
```
```   600       using r
```
```   601       by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
```
```   602     hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
```
```   603       by (rule diffs_equiv [THEN sums_summable])
```
```   604     also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
```
```   605       (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
```
```   606       apply (rule ext)
```
```   607       apply (simp add: diffs_def)
```
```   608       apply (case_tac n, simp_all add: r_neq_0)
```
```   609       done
```
```   610     finally have "summable
```
```   611       (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
```
```   612       by (rule diffs_equiv [THEN sums_summable])
```
```   613     also have
```
```   614       "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
```
```   615            r ^ (n - Suc 0)) =
```
```   616        (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
```
```   617       apply (rule ext)
```
```   618       apply (case_tac "n", simp)
```
```   619       apply (rename_tac nat)
```
```   620       apply (case_tac "nat", simp)
```
```   621       apply (simp add: r_neq_0)
```
```   622       done
```
```   623     finally
```
```   624     show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
```
```   625   next
```
```   626     fix h::'a and n::nat
```
```   627     assume h: "h \<noteq> 0"
```
```   628     assume "norm h < r - norm x"
```
```   629     hence "norm x + norm h < r" by simp
```
```   630     with norm_triangle_ineq have xh: "norm (x + h) < r"
```
```   631       by (rule order_le_less_trans)
```
```   632     show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
```
```   633           \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
```
```   634       apply (simp only: norm_mult mult.assoc)
```
```   635       apply (rule mult_left_mono [OF _ norm_ge_zero])
```
```   636       apply (simp add: mult.assoc [symmetric])
```
```   637       apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
```
```   638       done
```
```   639   qed
```
```   640 qed
```
```   641
```
```   642 lemma termdiffs:
```
```   643   fixes K x :: "'a::{real_normed_field,banach}"
```
```   644   assumes 1: "summable (\<lambda>n. c n * K ^ n)"
```
```   645       and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
```
```   646       and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
```
```   647       and 4: "norm x < norm K"
```
```   648   shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
```
```   649   unfolding DERIV_def
```
```   650 proof (rule LIM_zero_cancel)
```
```   651   show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
```
```   652             - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
```
```   653   proof (rule LIM_equal2)
```
```   654     show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
```
```   655   next
```
```   656     fix h :: 'a
```
```   657     assume "norm (h - 0) < norm K - norm x"
```
```   658     hence "norm x + norm h < norm K" by simp
```
```   659     hence 5: "norm (x + h) < norm K"
```
```   660       by (rule norm_triangle_ineq [THEN order_le_less_trans])
```
```   661     have "summable (\<lambda>n. c n * x ^ n)"
```
```   662       and "summable (\<lambda>n. c n * (x + h) ^ n)"
```
```   663       and "summable (\<lambda>n. diffs c n * x ^ n)"
```
```   664       using 1 2 4 5 by (auto elim: powser_inside)
```
```   665     then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h - (\<Sum>n. diffs c n * x ^ n) =
```
```   666           (\<Sum>n. (c n * (x + h) ^ n - c n * x ^ n) / h - of_nat n * c n * x ^ (n - Suc 0))"
```
```   667       by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
```
```   668     then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h - (\<Sum>n. diffs c n * x ^ n) =
```
```   669           (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
```
```   670       by (simp add: algebra_simps)
```
```   671   next
```
```   672     show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   673       by (rule termdiffs_aux [OF 3 4])
```
```   674   qed
```
```   675 qed
```
```   676
```
```   677
```
```   678 subsection {* Derivability of power series *}
```
```   679
```
```   680 lemma DERIV_series':
```
```   681   fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
```
```   682   assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
```
```   683     and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
```
```   684     and "summable (f' x0)"
```
```   685     and "summable L"
```
```   686     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>"
```
```   687   shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
```
```   688   unfolding DERIV_def
```
```   689 proof (rule LIM_I)
```
```   690   fix r :: real
```
```   691   assume "0 < r" hence "0 < r/3" by auto
```
```   692
```
```   693   obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
```
```   694     using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
```
```   695
```
```   696   obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
```
```   697     using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
```
```   698
```
```   699   let ?N = "Suc (max N_L N_f')"
```
```   700   have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
```
```   701     L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
```
```   702
```
```   703   let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
```
```   704
```
```   705   let ?r = "r / (3 * real ?N)"
```
```   706   from `0 < r` have "0 < ?r" by simp
```
```   707
```
```   708   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)"
```
```   709   def S' \<equiv> "Min (?s ` {..< ?N })"
```
```   710
```
```   711   have "0 < S'" unfolding S'_def
```
```   712   proof (rule iffD2[OF Min_gr_iff])
```
```   713     show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
```
```   714     proof
```
```   715       fix x
```
```   716       assume "x \<in> ?s ` {..<?N}"
```
```   717       then obtain n where "x = ?s n" and "n \<in> {..<?N}"
```
```   718         using image_iff[THEN iffD1] by blast
```
```   719       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
```
```   720       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)"
```
```   721         by auto
```
```   722       have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound)
```
```   723       thus "0 < x" unfolding `x = ?s n` .
```
```   724     qed
```
```   725   qed auto
```
```   726
```
```   727   def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
```
```   728   hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
```
```   729     and "S \<le> S'" using x0_in_I and `0 < S'`
```
```   730     by auto
```
```   731
```
```   732   {
```
```   733     fix x
```
```   734     assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
```
```   735     hence x_in_I: "x0 + x \<in> { a <..< b }"
```
```   736       using S_a S_b by auto
```
```   737
```
```   738     note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   739     note div_smbl = summable_divide[OF diff_smbl]
```
```   740     note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
```
```   741     note ign = summable_ignore_initial_segment[where k="?N"]
```
```   742     note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
```
```   743     note div_shft_smbl = summable_divide[OF diff_shft_smbl]
```
```   744     note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
```
```   745
```
```   746     { fix n
```
```   747       have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
```
```   748         using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
```
```   749         unfolding abs_divide .
```
```   750       hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)"
```
```   751         using `x \<noteq> 0` by auto }
```
```   752     note 1 = this and 2 = summable_rabs_comparison_test[OF _ ign[OF `summable L`]]
```
```   753     then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))"
```
```   754       by (metis (lifting) abs_idempotent order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF `summable L`]]])
```
```   755     then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3")
```
```   756       using L_estimate by auto
```
```   757
```
```   758     have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n \<bar>)" ..
```
```   759     also have "\<dots> < (\<Sum>n<?N. ?r)"
```
```   760     proof (rule setsum_strict_mono)
```
```   761       fix n
```
```   762       assume "n \<in> {..< ?N}"
```
```   763       have "\<bar>x\<bar> < S" using `\<bar>x\<bar> < S` .
```
```   764       also have "S \<le> S'" using `S \<le> S'` .
```
```   765       also have "S' \<le> ?s n" unfolding S'_def
```
```   766       proof (rule Min_le_iff[THEN iffD2])
```
```   767         have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n"
```
```   768           using `n \<in> {..< ?N}` by auto
```
```   769         thus "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n" by blast
```
```   770       qed auto
```
```   771       finally have "\<bar>x\<bar> < ?s n" .
```
```   772
```
```   773       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
```
```   774       have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
```
```   775       with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n` show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
```
```   776         by blast
```
```   777     qed auto
```
```   778     also have "\<dots> = of_nat (card {..<?N}) * ?r"
```
```   779       by (rule setsum_constant)
```
```   780     also have "\<dots> = real ?N * ?r"
```
```   781       unfolding real_eq_of_nat by auto
```
```   782     also have "\<dots> = r/3" by auto
```
```   783     finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
```
```   784
```
```   785     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   786     have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
```
```   787         \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
```
```   788       unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric]
```
```   789       using suminf_divide[OF diff_smbl, symmetric] by auto
```
```   790     also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>"
```
```   791       unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
```
```   792       unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]]
```
```   793       apply (subst (5) add.commute)
```
```   794       by (rule abs_triangle_ineq)
```
```   795     also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
```
```   796       using abs_triangle_ineq4 by auto
```
```   797     also have "\<dots> < r /3 + r/3 + r/3"
```
```   798       using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
```
```   799       by (rule add_strict_mono [OF add_less_le_mono])
```
```   800     finally have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
```
```   801       by auto
```
```   802   }
```
```   803   thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
```
```   804       norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
```
```   805     using `0 < S` unfolding real_norm_def diff_0_right by blast
```
```   806 qed
```
```   807
```
```   808 lemma DERIV_power_series':
```
```   809   fixes f :: "nat \<Rightarrow> real"
```
```   810   assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
```
```   811     and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
```
```   812   shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
```
```   813   (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
```
```   814 proof -
```
```   815   {
```
```   816     fix R'
```
```   817     assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
```
```   818     hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
```
```   819       by auto
```
```   820     have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
```
```   821     proof (rule DERIV_series')
```
```   822       show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
```
```   823       proof -
```
```   824         have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
```
```   825           using `0 < R'` `0 < R` `R' < R` by auto
```
```   826         hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
```
```   827           using `R' < R` by auto
```
```   828         have "norm R' < norm ((R' + R) / 2)"
```
```   829           using `0 < R'` `0 < R` `R' < R` by auto
```
```   830         from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
```
```   831           by auto
```
```   832       qed
```
```   833       {
```
```   834         fix n x y
```
```   835         assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
```
```   836         show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
```
```   837         proof -
```
```   838           have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
```
```   839             (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>"
```
```   840             unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult
```
```   841             by auto
```
```   842           also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
```
```   843           proof (rule mult_left_mono)
```
```   844             have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
```
```   845               by (rule setsum_abs)
```
```   846             also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)"
```
```   847             proof (rule setsum_mono)
```
```   848               fix p
```
```   849               assume "p \<in> {..<Suc n}"
```
```   850               hence "p \<le> n" by auto
```
```   851               {
```
```   852                 fix n
```
```   853                 fix x :: real
```
```   854                 assume "x \<in> {-R'<..<R'}"
```
```   855                 hence "\<bar>x\<bar> \<le> R'"  by auto
```
```   856                 hence "\<bar>x^n\<bar> \<le> R'^n"
```
```   857                   unfolding power_abs by (rule power_mono, auto)
```
```   858               }
```
```   859               from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
```
```   860               have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
```
```   861                 unfolding abs_mult by auto
```
```   862               thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
```
```   863                 unfolding power_add[symmetric] using `p \<le> n` by auto
```
```   864             qed
```
```   865             also have "\<dots> = real (Suc n) * R' ^ n"
```
```   866               unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
```
```   867             finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
```
```   868               unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
```
```   869             show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
```
```   870               unfolding abs_mult[symmetric] by auto
```
```   871           qed
```
```   872           also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
```
```   873             unfolding abs_mult mult.assoc[symmetric] by algebra
```
```   874           finally show ?thesis .
```
```   875         qed
```
```   876       }
```
```   877       {
```
```   878         fix n
```
```   879         show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
```
```   880           by (auto intro!: derivative_eq_intros simp del: power_Suc simp: real_of_nat_def)
```
```   881       }
```
```   882       {
```
```   883         fix x
```
```   884         assume "x \<in> {-R' <..< R'}"
```
```   885         hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
```
```   886           using assms `R' < R` by auto
```
```   887         have "summable (\<lambda> n. f n * x^n)"
```
```   888         proof (rule summable_comparison_test, intro exI allI impI)
```
```   889           fix n
```
```   890           have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
```
```   891             by (rule mult_left_mono) auto
```
```   892           show "norm (f n * x ^ n) \<le> norm (f n * real (Suc n) * x ^ n)"
```
```   893             unfolding real_norm_def abs_mult
```
```   894             by (rule mult_right_mono) (auto simp add: le[unfolded mult_1_right])
```
```   895         qed (rule powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`])
```
```   896         from this[THEN summable_mult2[where c=x], unfolded mult.assoc, unfolded mult.commute]
```
```   897         show "summable (?f x)" by auto
```
```   898       }
```
```   899       show "summable (?f' x0)"
```
```   900         using converges[OF `x0 \<in> {-R <..< R}`] .
```
```   901       show "x0 \<in> {-R' <..< R'}"
```
```   902         using `x0 \<in> {-R' <..< R'}` .
```
```   903     qed
```
```   904   } note for_subinterval = this
```
```   905   let ?R = "(R + \<bar>x0\<bar>) / 2"
```
```   906   have "\<bar>x0\<bar> < ?R" using assms by auto
```
```   907   hence "- ?R < x0"
```
```   908   proof (cases "x0 < 0")
```
```   909     case True
```
```   910     hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
```
```   911     thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
```
```   912   next
```
```   913     case False
```
```   914     have "- ?R < 0" using assms by auto
```
```   915     also have "\<dots> \<le> x0" using False by auto
```
```   916     finally show ?thesis .
```
```   917   qed
```
```   918   hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
```
```   919     using assms by auto
```
```   920   from for_subinterval[OF this]
```
```   921   show ?thesis .
```
```   922 qed
```
```   923
```
```   924
```
```   925 subsection {* Exponential Function *}
```
```   926
```
```   927 definition exp :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```   928   where "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
```
```   929
```
```   930 lemma summable_exp_generic:
```
```   931   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   932   defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
```
```   933   shows "summable S"
```
```   934 proof -
```
```   935   have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
```
```   936     unfolding S_def by (simp del: mult_Suc)
```
```   937   obtain r :: real where r0: "0 < r" and r1: "r < 1"
```
```   938     using dense [OF zero_less_one] by fast
```
```   939   obtain N :: nat where N: "norm x < real N * r"
```
```   940     using reals_Archimedean3 [OF r0] by fast
```
```   941   from r1 show ?thesis
```
```   942   proof (rule summable_ratio_test [rule_format])
```
```   943     fix n :: nat
```
```   944     assume n: "N \<le> n"
```
```   945     have "norm x \<le> real N * r"
```
```   946       using N by (rule order_less_imp_le)
```
```   947     also have "real N * r \<le> real (Suc n) * r"
```
```   948       using r0 n by (simp add: mult_right_mono)
```
```   949     finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
```
```   950       using norm_ge_zero by (rule mult_right_mono)
```
```   951     hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
```
```   952       by (rule order_trans [OF norm_mult_ineq])
```
```   953     hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
```
```   954       by (simp add: pos_divide_le_eq ac_simps)
```
```   955     thus "norm (S (Suc n)) \<le> r * norm (S n)"
```
```   956       by (simp add: S_Suc inverse_eq_divide)
```
```   957   qed
```
```   958 qed
```
```   959
```
```   960 lemma summable_norm_exp:
```
```   961   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   962   shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
```
```   963 proof (rule summable_norm_comparison_test [OF exI, rule_format])
```
```   964   show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
```
```   965     by (rule summable_exp_generic)
```
```   966   fix n
```
```   967   show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
```
```   968     by (simp add: norm_power_ineq)
```
```   969 qed
```
```   970
```
```   971 lemma summable_exp: "summable (\<lambda>n. inverse (real (fact n)) * x ^ n)"
```
```   972   using summable_exp_generic [where x=x] by simp
```
```   973
```
```   974 lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
```
```   975   unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
```
```   976
```
```   977
```
```   978 lemma exp_fdiffs:
```
```   979       "diffs (\<lambda>n. inverse(real (fact n))) = (\<lambda>n. inverse(real (fact n)))"
```
```   980   by (simp add: diffs_def mult.assoc [symmetric] real_of_nat_def of_nat_mult
```
```   981         del: mult_Suc of_nat_Suc)
```
```   982
```
```   983 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
```
```   984   by (simp add: diffs_def)
```
```   985
```
```   986 lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
```
```   987   unfolding exp_def scaleR_conv_of_real
```
```   988   apply (rule DERIV_cong)
```
```   989   apply (rule termdiffs [where K="of_real (1 + norm x)"])
```
```   990   apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
```
```   991   apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
```
```   992   apply (simp del: of_real_add)
```
```   993   done
```
```   994
```
```   995 declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
```
```   996
```
```   997 lemma norm_exp: "norm (exp x) \<le> exp (norm x)"
```
```   998 proof -
```
```   999   from summable_norm[OF summable_norm_exp, of x]
```
```  1000   have "norm (exp x) \<le> (\<Sum>n. inverse (real (fact n)) * norm (x ^ n))"
```
```  1001     by (simp add: exp_def)
```
```  1002   also have "\<dots> \<le> exp (norm x)"
```
```  1003     using summable_exp_generic[of "norm x"] summable_norm_exp[of x]
```
```  1004     by (auto simp: exp_def intro!: suminf_le norm_power_ineq)
```
```  1005   finally show ?thesis .
```
```  1006 qed
```
```  1007
```
```  1008 lemma isCont_exp:
```
```  1009   fixes x::"'a::{real_normed_field,banach}"
```
```  1010   shows "isCont exp x"
```
```  1011   by (rule DERIV_exp [THEN DERIV_isCont])
```
```  1012
```
```  1013 lemma isCont_exp' [simp]:
```
```  1014   fixes f::"_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1015   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
```
```  1016   by (rule isCont_o2 [OF _ isCont_exp])
```
```  1017
```
```  1018 lemma tendsto_exp [tendsto_intros]:
```
```  1019   fixes f::"_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1020   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
```
```  1021   by (rule isCont_tendsto_compose [OF isCont_exp])
```
```  1022
```
```  1023 lemma continuous_exp [continuous_intros]:
```
```  1024   fixes f::"_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1025   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
```
```  1026   unfolding continuous_def by (rule tendsto_exp)
```
```  1027
```
```  1028 lemma continuous_on_exp [continuous_intros]:
```
```  1029   fixes f::"_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1030   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
```
```  1031   unfolding continuous_on_def by (auto intro: tendsto_exp)
```
```  1032
```
```  1033
```
```  1034 subsubsection {* Properties of the Exponential Function *}
```
```  1035
```
```  1036 lemma powser_zero:
```
```  1037   fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
```
```  1038   shows "(\<Sum>n. f n * 0 ^ n) = f 0"
```
```  1039 proof -
```
```  1040   have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
```
```  1041     by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
```
```  1042   thus ?thesis unfolding One_nat_def by simp
```
```  1043 qed
```
```  1044
```
```  1045 lemma exp_zero [simp]: "exp 0 = 1"
```
```  1046   unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
```
```  1047
```
```  1048 lemma exp_series_add_commuting:
```
```  1049   fixes x y :: "'a::{real_normed_algebra_1, banach}"
```
```  1050   defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
```
```  1051   assumes comm: "x * y = y * x"
```
```  1052   shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))"
```
```  1053 proof (induct n)
```
```  1054   case 0
```
```  1055   show ?case
```
```  1056     unfolding S_def by simp
```
```  1057 next
```
```  1058   case (Suc n)
```
```  1059   have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
```
```  1060     unfolding S_def by (simp del: mult_Suc)
```
```  1061   hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
```
```  1062     by simp
```
```  1063   have S_comm: "\<And>n. S x n * y = y * S x n"
```
```  1064     by (simp add: power_commuting_commutes comm S_def)
```
```  1065
```
```  1066   have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
```
```  1067     by (simp only: times_S)
```
```  1068   also have "\<dots> = (x + y) * (\<Sum>i\<le>n. S x i * S y (n-i))"
```
```  1069     by (simp only: Suc)
```
```  1070   also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n-i))
```
```  1071                 + y * (\<Sum>i\<le>n. S x i * S y (n-i))"
```
```  1072     by (rule distrib_right)
```
```  1073   also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i))
```
```  1074                 + (\<Sum>i\<le>n. S x i * y * S y (n-i))"
```
```  1075     by (simp add: setsum_right_distrib ac_simps S_comm)
```
```  1076   also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i))
```
```  1077                 + (\<Sum>i\<le>n. S x i * (y * S y (n-i)))"
```
```  1078     by (simp add: ac_simps)
```
```  1079   also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
```
```  1080                 + (\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1081     by (simp add: times_S Suc_diff_le)
```
```  1082   also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
```
```  1083              (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1084     by (subst setsum_atMost_Suc_shift) simp
```
```  1085   also have "(\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```  1086              (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1087     by simp
```
```  1088   also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
```
```  1089              (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```  1090              (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1091     by (simp only: setsum.distrib [symmetric] scaleR_left_distrib [symmetric]
```
```  1092                    real_of_nat_add [symmetric]) simp
```
```  1093   also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n-i))"
```
```  1094     by (simp only: scaleR_right.setsum)
```
```  1095   finally show
```
```  1096     "S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
```
```  1097     by (simp del: setsum_cl_ivl_Suc)
```
```  1098 qed
```
```  1099
```
```  1100 lemma exp_add_commuting: "x * y = y * x \<Longrightarrow> exp (x + y) = exp x * exp y"
```
```  1101   unfolding exp_def
```
```  1102   by (simp only: Cauchy_product summable_norm_exp exp_series_add_commuting)
```
```  1103
```
```  1104 lemma exp_add:
```
```  1105   fixes x y::"'a::{real_normed_field,banach}"
```
```  1106   shows "exp (x + y) = exp x * exp y"
```
```  1107   by (rule exp_add_commuting) (simp add: ac_simps)
```
```  1108
```
```  1109 lemmas mult_exp_exp = exp_add [symmetric]
```
```  1110
```
```  1111 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
```
```  1112   unfolding exp_def
```
```  1113   apply (subst suminf_of_real)
```
```  1114   apply (rule summable_exp_generic)
```
```  1115   apply (simp add: scaleR_conv_of_real)
```
```  1116   done
```
```  1117
```
```  1118 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
```
```  1119 proof
```
```  1120   have "exp x * exp (- x) = 1" by (simp add: exp_add_commuting[symmetric])
```
```  1121   also assume "exp x = 0"
```
```  1122   finally show "False" by simp
```
```  1123 qed
```
```  1124
```
```  1125 lemma exp_minus_inverse:
```
```  1126   shows "exp x * exp (- x) = 1"
```
```  1127   by (simp add: exp_add_commuting[symmetric])
```
```  1128
```
```  1129 lemma exp_minus:
```
```  1130   fixes x :: "'a::{real_normed_field, banach}"
```
```  1131   shows "exp (- x) = inverse (exp x)"
```
```  1132   by (intro inverse_unique [symmetric] exp_minus_inverse)
```
```  1133
```
```  1134 lemma exp_diff:
```
```  1135   fixes x :: "'a::{real_normed_field, banach}"
```
```  1136   shows "exp (x - y) = exp x / exp y"
```
```  1137   using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
```
```  1138
```
```  1139
```
```  1140 subsubsection {* Properties of the Exponential Function on Reals *}
```
```  1141
```
```  1142 text {* Comparisons of @{term "exp x"} with zero. *}
```
```  1143
```
```  1144 text{*Proof: because every exponential can be seen as a square.*}
```
```  1145 lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
```
```  1146 proof -
```
```  1147   have "0 \<le> exp (x/2) * exp (x/2)" by simp
```
```  1148   thus ?thesis by (simp add: exp_add [symmetric])
```
```  1149 qed
```
```  1150
```
```  1151 lemma exp_gt_zero [simp]: "0 < exp (x::real)"
```
```  1152   by (simp add: order_less_le)
```
```  1153
```
```  1154 lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
```
```  1155   by (simp add: not_less)
```
```  1156
```
```  1157 lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
```
```  1158   by (simp add: not_le)
```
```  1159
```
```  1160 lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
```
```  1161   by simp
```
```  1162
```
```  1163 lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
```
```  1164   by (induct n) (auto simp add: real_of_nat_Suc distrib_left exp_add mult.commute)
```
```  1165
```
```  1166 text {* Strict monotonicity of exponential. *}
```
```  1167
```
```  1168 lemma exp_ge_add_one_self_aux:
```
```  1169   assumes "0 \<le> (x::real)" shows "1+x \<le> exp(x)"
```
```  1170 using order_le_imp_less_or_eq [OF assms]
```
```  1171 proof
```
```  1172   assume "0 < x"
```
```  1173   have "1+x \<le> (\<Sum>n<2. inverse (real (fact n)) * x ^ n)"
```
```  1174     by (auto simp add: numeral_2_eq_2)
```
```  1175   also have "... \<le> (\<Sum>n. inverse (real (fact n)) * x ^ n)"
```
```  1176     apply (rule setsum_le_suminf [OF summable_exp])
```
```  1177     using `0 < x`
```
```  1178     apply (auto  simp add:  zero_le_mult_iff)
```
```  1179     done
```
```  1180   finally show "1+x \<le> exp x"
```
```  1181     by (simp add: exp_def)
```
```  1182 next
```
```  1183   assume "0 = x"
```
```  1184   then show "1 + x \<le> exp x"
```
```  1185     by auto
```
```  1186 qed
```
```  1187
```
```  1188 lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
```
```  1189 proof -
```
```  1190   assume x: "0 < x"
```
```  1191   hence "1 < 1 + x" by simp
```
```  1192   also from x have "1 + x \<le> exp x"
```
```  1193     by (simp add: exp_ge_add_one_self_aux)
```
```  1194   finally show ?thesis .
```
```  1195 qed
```
```  1196
```
```  1197 lemma exp_less_mono:
```
```  1198   fixes x y :: real
```
```  1199   assumes "x < y"
```
```  1200   shows "exp x < exp y"
```
```  1201 proof -
```
```  1202   from `x < y` have "0 < y - x" by simp
```
```  1203   hence "1 < exp (y - x)" by (rule exp_gt_one)
```
```  1204   hence "1 < exp y / exp x" by (simp only: exp_diff)
```
```  1205   thus "exp x < exp y" by simp
```
```  1206 qed
```
```  1207
```
```  1208 lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
```
```  1209   unfolding linorder_not_le [symmetric]
```
```  1210   by (auto simp add: order_le_less exp_less_mono)
```
```  1211
```
```  1212 lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
```
```  1213   by (auto intro: exp_less_mono exp_less_cancel)
```
```  1214
```
```  1215 lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
```
```  1216   by (auto simp add: linorder_not_less [symmetric])
```
```  1217
```
```  1218 lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
```
```  1219   by (simp add: order_eq_iff)
```
```  1220
```
```  1221 text {* Comparisons of @{term "exp x"} with one. *}
```
```  1222
```
```  1223 lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
```
```  1224   using exp_less_cancel_iff [where x=0 and y=x] by simp
```
```  1225
```
```  1226 lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
```
```  1227   using exp_less_cancel_iff [where x=x and y=0] by simp
```
```  1228
```
```  1229 lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
```
```  1230   using exp_le_cancel_iff [where x=0 and y=x] by simp
```
```  1231
```
```  1232 lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
```
```  1233   using exp_le_cancel_iff [where x=x and y=0] by simp
```
```  1234
```
```  1235 lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
```
```  1236   using exp_inj_iff [where x=x and y=0] by simp
```
```  1237
```
```  1238 lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
```
```  1239 proof (rule IVT)
```
```  1240   assume "1 \<le> y"
```
```  1241   hence "0 \<le> y - 1" by simp
```
```  1242   hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
```
```  1243   thus "y \<le> exp (y - 1)" by simp
```
```  1244 qed (simp_all add: le_diff_eq)
```
```  1245
```
```  1246 lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y"
```
```  1247 proof (rule linorder_le_cases [of 1 y])
```
```  1248   assume "1 \<le> y"
```
```  1249   thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total)
```
```  1250 next
```
```  1251   assume "0 < y" and "y \<le> 1"
```
```  1252   hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
```
```  1253   then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
```
```  1254   hence "exp (- x) = y" by (simp add: exp_minus)
```
```  1255   thus "\<exists>x. exp x = y" ..
```
```  1256 qed
```
```  1257
```
```  1258
```
```  1259 subsection {* Natural Logarithm *}
```
```  1260
```
```  1261 definition ln :: "real \<Rightarrow> real"
```
```  1262   where "ln x = (THE u. exp u = x)"
```
```  1263
```
```  1264 lemma ln_exp [simp]: "ln (exp x) = x"
```
```  1265   by (simp add: ln_def)
```
```  1266
```
```  1267 lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
```
```  1268   by (auto dest: exp_total)
```
```  1269
```
```  1270 lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
```
```  1271   by (metis exp_gt_zero exp_ln)
```
```  1272
```
```  1273 lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
```
```  1274   by (erule subst, rule ln_exp)
```
```  1275
```
```  1276 lemma ln_one [simp]: "ln 1 = 0"
```
```  1277   by (rule ln_unique) simp
```
```  1278
```
```  1279 lemma ln_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
```
```  1280   by (rule ln_unique) (simp add: exp_add)
```
```  1281
```
```  1282 lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
```
```  1283   by (rule ln_unique) (simp add: exp_minus)
```
```  1284
```
```  1285 lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
```
```  1286   by (rule ln_unique) (simp add: exp_diff)
```
```  1287
```
```  1288 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
```
```  1289   by (rule ln_unique) (simp add: exp_real_of_nat_mult)
```
```  1290
```
```  1291 lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
```
```  1292   by (subst exp_less_cancel_iff [symmetric]) simp
```
```  1293
```
```  1294 lemma ln_le_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
```
```  1295   by (simp add: linorder_not_less [symmetric])
```
```  1296
```
```  1297 lemma ln_inj_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
```
```  1298   by (simp add: order_eq_iff)
```
```  1299
```
```  1300 lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
```
```  1301   apply (rule exp_le_cancel_iff [THEN iffD1])
```
```  1302   apply (simp add: exp_ge_add_one_self_aux)
```
```  1303   done
```
```  1304
```
```  1305 lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
```
```  1306   by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
```
```  1307
```
```  1308 lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
```
```  1309   using ln_le_cancel_iff [of 1 x] by simp
```
```  1310
```
```  1311 lemma ln_ge_zero_imp_ge_one: "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
```
```  1312   using ln_le_cancel_iff [of 1 x] by simp
```
```  1313
```
```  1314 lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
```
```  1315   using ln_le_cancel_iff [of 1 x] by simp
```
```  1316
```
```  1317 lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
```
```  1318   using ln_less_cancel_iff [of x 1] by simp
```
```  1319
```
```  1320 lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
```
```  1321   using ln_less_cancel_iff [of 1 x] by simp
```
```  1322
```
```  1323 lemma ln_gt_zero_imp_gt_one: "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
```
```  1324   using ln_less_cancel_iff [of 1 x] by simp
```
```  1325
```
```  1326 lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
```
```  1327   using ln_less_cancel_iff [of 1 x] by simp
```
```  1328
```
```  1329 lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
```
```  1330   using ln_inj_iff [of x 1] by simp
```
```  1331
```
```  1332 lemma ln_less_zero: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
```
```  1333   by simp
```
```  1334
```
```  1335 lemma ln_neg_is_const: "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)"
```
```  1336   by (auto simp add: ln_def intro!: arg_cong[where f=The])
```
```  1337
```
```  1338 lemma isCont_ln: assumes "x \<noteq> 0" shows "isCont ln x"
```
```  1339 proof cases
```
```  1340   assume "0 < x"
```
```  1341   moreover then have "isCont ln (exp (ln x))"
```
```  1342     by (intro isCont_inv_fun[where d="\<bar>x\<bar>" and f=exp]) auto
```
```  1343   ultimately show ?thesis
```
```  1344     by simp
```
```  1345 next
```
```  1346   assume "\<not> 0 < x" with `x \<noteq> 0` show "isCont ln x"
```
```  1347     unfolding isCont_def
```
```  1348     by (subst filterlim_cong[OF _ refl, of _ "nhds (ln 0)" _ "\<lambda>_. ln 0"])
```
```  1349        (auto simp: ln_neg_is_const not_less eventually_at dist_real_def
```
```  1350                 intro!: exI[of _ "\<bar>x\<bar>"])
```
```  1351 qed
```
```  1352
```
```  1353 lemma tendsto_ln [tendsto_intros]:
```
```  1354   "(f ---> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
```
```  1355   by (rule isCont_tendsto_compose [OF isCont_ln])
```
```  1356
```
```  1357 lemma continuous_ln:
```
```  1358   "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. ln (f x))"
```
```  1359   unfolding continuous_def by (rule tendsto_ln)
```
```  1360
```
```  1361 lemma isCont_ln' [continuous_intros]:
```
```  1362   "continuous (at x) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x))"
```
```  1363   unfolding continuous_at by (rule tendsto_ln)
```
```  1364
```
```  1365 lemma continuous_within_ln [continuous_intros]:
```
```  1366   "continuous (at x within s) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x))"
```
```  1367   unfolding continuous_within by (rule tendsto_ln)
```
```  1368
```
```  1369 lemma continuous_on_ln [continuous_intros]:
```
```  1370   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. f x \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x))"
```
```  1371   unfolding continuous_on_def by (auto intro: tendsto_ln)
```
```  1372
```
```  1373 lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
```
```  1374   apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
```
```  1375   apply (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln)
```
```  1376   done
```
```  1377
```
```  1378 lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
```
```  1379   by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
```
```  1380
```
```  1381 declare DERIV_ln_divide[THEN DERIV_chain2, derivative_intros]
```
```  1382
```
```  1383 lemma ln_series:
```
```  1384   assumes "0 < x" and "x < 2"
```
```  1385   shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))"
```
```  1386   (is "ln x = suminf (?f (x - 1))")
```
```  1387 proof -
```
```  1388   let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n"
```
```  1389
```
```  1390   have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
```
```  1391   proof (rule DERIV_isconst3[where x=x])
```
```  1392     fix x :: real
```
```  1393     assume "x \<in> {0 <..< 2}"
```
```  1394     hence "0 < x" and "x < 2" by auto
```
```  1395     have "norm (1 - x) < 1"
```
```  1396       using `0 < x` and `x < 2` by auto
```
```  1397     have "1 / x = 1 / (1 - (1 - x))" by auto
```
```  1398     also have "\<dots> = (\<Sum> n. (1 - x)^n)"
```
```  1399       using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
```
```  1400     also have "\<dots> = suminf (?f' x)"
```
```  1401       unfolding power_mult_distrib[symmetric]
```
```  1402       by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
```
```  1403     finally have "DERIV ln x :> suminf (?f' x)"
```
```  1404       using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
```
```  1405     moreover
```
```  1406     have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
```
```  1407     have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :>
```
```  1408       (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
```
```  1409     proof (rule DERIV_power_series')
```
```  1410       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1"
```
```  1411         using `0 < x` `x < 2` by auto
```
```  1412       fix x :: real
```
```  1413       assume "x \<in> {- 1<..<1}"
```
```  1414       hence "norm (-x) < 1" by auto
```
```  1415       show "summable (\<lambda>n. (- 1) ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
```
```  1416         unfolding One_nat_def
```
```  1417         by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
```
```  1418     qed
```
```  1419     hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
```
```  1420       unfolding One_nat_def by auto
```
```  1421     hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
```
```  1422       unfolding DERIV_def repos .
```
```  1423     ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
```
```  1424       by (rule DERIV_diff)
```
```  1425     thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
```
```  1426   qed (auto simp add: assms)
```
```  1427   thus ?thesis by auto
```
```  1428 qed
```
```  1429
```
```  1430 lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) * (x ^ (n+2)))"
```
```  1431 proof -
```
```  1432   have "exp x = suminf (\<lambda>n. inverse(fact n) * (x ^ n))"
```
```  1433     by (simp add: exp_def)
```
```  1434   also from summable_exp have "... = (\<Sum> n. inverse(fact(n+2)) * (x ^ (n+2))) +
```
```  1435     (\<Sum> n::nat<2. inverse(fact n) * (x ^ n))" (is "_ = _ + ?a")
```
```  1436     by (rule suminf_split_initial_segment)
```
```  1437   also have "?a = 1 + x"
```
```  1438     by (simp add: numeral_2_eq_2)
```
```  1439   finally show ?thesis
```
```  1440     by simp
```
```  1441 qed
```
```  1442
```
```  1443 lemma exp_bound: "0 <= (x::real) \<Longrightarrow> x <= 1 \<Longrightarrow> exp x <= 1 + x + x\<^sup>2"
```
```  1444 proof -
```
```  1445   assume a: "0 <= x"
```
```  1446   assume b: "x <= 1"
```
```  1447   {
```
```  1448     fix n :: nat
```
```  1449     have "2 * 2 ^ n \<le> fact (n + 2)"
```
```  1450       by (induct n) simp_all
```
```  1451     hence "real ((2::nat) * 2 ^ n) \<le> real (fact (n + 2))"
```
```  1452       by (simp only: real_of_nat_le_iff)
```
```  1453     hence "2 * 2 ^ n \<le> real (fact (n + 2))"
```
```  1454       by simp
```
```  1455     hence "inverse (fact (n + 2)) \<le> inverse (2 * 2 ^ n)"
```
```  1456       by (rule le_imp_inverse_le) simp
```
```  1457     hence "inverse (fact (n + 2)) \<le> 1/2 * (1/2)^n"
```
```  1458       by (simp add: power_inverse)
```
```  1459     hence "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
```
```  1460       by (rule mult_mono)
```
```  1461         (rule mult_mono, simp_all add: power_le_one a b)
```
```  1462     hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)"
```
```  1463       unfolding power_add by (simp add: ac_simps del: fact_Suc) }
```
```  1464   note aux1 = this
```
```  1465   have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
```
```  1466     by (intro sums_mult geometric_sums, simp)
```
```  1467   hence aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
```
```  1468     by simp
```
```  1469   have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2"
```
```  1470   proof -
```
```  1471     have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <=
```
```  1472         suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
```
```  1473       apply (rule suminf_le)
```
```  1474       apply (rule allI, rule aux1)
```
```  1475       apply (rule summable_exp [THEN summable_ignore_initial_segment])
```
```  1476       by (rule sums_summable, rule aux2)
```
```  1477     also have "... = x\<^sup>2"
```
```  1478       by (rule sums_unique [THEN sym], rule aux2)
```
```  1479     finally show ?thesis .
```
```  1480   qed
```
```  1481   thus ?thesis unfolding exp_first_two_terms by auto
```
```  1482 qed
```
```  1483
```
```  1484 lemma ln_one_minus_pos_upper_bound: "0 <= x \<Longrightarrow> x < 1 \<Longrightarrow> ln (1 - x) <= - x"
```
```  1485 proof -
```
```  1486   assume a: "0 <= (x::real)" and b: "x < 1"
```
```  1487   have "(1 - x) * (1 + x + x\<^sup>2) = (1 - x^3)"
```
```  1488     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
```
```  1489   also have "... <= 1"
```
```  1490     by (auto simp add: a)
```
```  1491   finally have "(1 - x) * (1 + x + x\<^sup>2) <= 1" .
```
```  1492   moreover have c: "0 < 1 + x + x\<^sup>2"
```
```  1493     by (simp add: add_pos_nonneg a)
```
```  1494   ultimately have "1 - x <= 1 / (1 + x + x\<^sup>2)"
```
```  1495     by (elim mult_imp_le_div_pos)
```
```  1496   also have "... <= 1 / exp x"
```
```  1497     by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs
```
```  1498               real_sqrt_pow2_iff real_sqrt_power)
```
```  1499   also have "... = exp (-x)"
```
```  1500     by (auto simp add: exp_minus divide_inverse)
```
```  1501   finally have "1 - x <= exp (- x)" .
```
```  1502   also have "1 - x = exp (ln (1 - x))"
```
```  1503     by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq)
```
```  1504   finally have "exp (ln (1 - x)) <= exp (- x)" .
```
```  1505   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```  1506 qed
```
```  1507
```
```  1508 lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
```
```  1509   apply (case_tac "0 <= x")
```
```  1510   apply (erule exp_ge_add_one_self_aux)
```
```  1511   apply (case_tac "x <= -1")
```
```  1512   apply (subgoal_tac "1 + x <= 0")
```
```  1513   apply (erule order_trans)
```
```  1514   apply simp
```
```  1515   apply simp
```
```  1516   apply (subgoal_tac "1 + x = exp(ln (1 + x))")
```
```  1517   apply (erule ssubst)
```
```  1518   apply (subst exp_le_cancel_iff)
```
```  1519   apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
```
```  1520   apply simp
```
```  1521   apply (rule ln_one_minus_pos_upper_bound)
```
```  1522   apply auto
```
```  1523 done
```
```  1524
```
```  1525 lemma ln_one_plus_pos_lower_bound: "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> x - x\<^sup>2 <= ln (1 + x)"
```
```  1526 proof -
```
```  1527   assume a: "0 <= x" and b: "x <= 1"
```
```  1528   have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
```
```  1529     by (rule exp_diff)
```
```  1530   also have "... <= (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
```
```  1531     by (metis a b divide_right_mono exp_bound exp_ge_zero)
```
```  1532   also have "... <= (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
```
```  1533     by (simp add: a divide_left_mono add_pos_nonneg)
```
```  1534   also from a have "... <= 1 + x"
```
```  1535     by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
```
```  1536   finally have "exp (x - x\<^sup>2) <= 1 + x" .
```
```  1537   also have "... = exp (ln (1 + x))"
```
```  1538   proof -
```
```  1539     from a have "0 < 1 + x" by auto
```
```  1540     thus ?thesis
```
```  1541       by (auto simp only: exp_ln_iff [THEN sym])
```
```  1542   qed
```
```  1543   finally have "exp (x - x\<^sup>2) <= exp (ln (1 + x))" .
```
```  1544   thus ?thesis
```
```  1545     by (metis exp_le_cancel_iff)
```
```  1546 qed
```
```  1547
```
```  1548 lemma ln_one_minus_pos_lower_bound:
```
```  1549   "0 <= x \<Longrightarrow> x <= (1 / 2) \<Longrightarrow> - x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1550 proof -
```
```  1551   assume a: "0 <= x" and b: "x <= (1 / 2)"
```
```  1552   from b have c: "x < 1" by auto
```
```  1553   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
```
```  1554     apply (subst ln_inverse [symmetric])
```
```  1555     apply (simp add: field_simps)
```
```  1556     apply (rule arg_cong [where f=ln])
```
```  1557     apply (simp add: field_simps)
```
```  1558     done
```
```  1559   also have "- (x / (1 - x)) <= ..."
```
```  1560   proof -
```
```  1561     have "ln (1 + x / (1 - x)) <= x / (1 - x)"
```
```  1562       using a c by (intro ln_add_one_self_le_self) auto
```
```  1563     thus ?thesis
```
```  1564       by auto
```
```  1565   qed
```
```  1566   also have "- (x / (1 - x)) = -x / (1 - x)"
```
```  1567     by auto
```
```  1568   finally have d: "- x / (1 - x) <= ln (1 - x)" .
```
```  1569   have "0 < 1 - x" using a b by simp
```
```  1570   hence e: "-x - 2 * x\<^sup>2 <= - x / (1 - x)"
```
```  1571     using mult_right_le_one_le[of "x*x" "2*x"] a b
```
```  1572     by (simp add: field_simps power2_eq_square)
```
```  1573   from e d show "- x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1574     by (rule order_trans)
```
```  1575 qed
```
```  1576
```
```  1577 lemma ln_add_one_self_le_self2: "-1 < x \<Longrightarrow> ln(1 + x) <= x"
```
```  1578   apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
```
```  1579   apply (subst ln_le_cancel_iff)
```
```  1580   apply auto
```
```  1581   done
```
```  1582
```
```  1583 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
```
```  1584   "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> abs(ln (1 + x) - x) <= x\<^sup>2"
```
```  1585 proof -
```
```  1586   assume x: "0 <= x"
```
```  1587   assume x1: "x <= 1"
```
```  1588   from x have "ln (1 + x) <= x"
```
```  1589     by (rule ln_add_one_self_le_self)
```
```  1590   then have "ln (1 + x) - x <= 0"
```
```  1591     by simp
```
```  1592   then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
```
```  1593     by (rule abs_of_nonpos)
```
```  1594   also have "... = x - ln (1 + x)"
```
```  1595     by simp
```
```  1596   also have "... <= x\<^sup>2"
```
```  1597   proof -
```
```  1598     from x x1 have "x - x\<^sup>2 <= ln (1 + x)"
```
```  1599       by (intro ln_one_plus_pos_lower_bound)
```
```  1600     thus ?thesis
```
```  1601       by simp
```
```  1602   qed
```
```  1603   finally show ?thesis .
```
```  1604 qed
```
```  1605
```
```  1606 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
```
```  1607   "-(1 / 2) <= x \<Longrightarrow> x <= 0 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
```
```  1608 proof -
```
```  1609   assume a: "-(1 / 2) <= x"
```
```  1610   assume b: "x <= 0"
```
```  1611   have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
```
```  1612     apply (subst abs_of_nonpos)
```
```  1613     apply simp
```
```  1614     apply (rule ln_add_one_self_le_self2)
```
```  1615     using a apply auto
```
```  1616     done
```
```  1617   also have "... <= 2 * x\<^sup>2"
```
```  1618     apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 <= ln (1 - (-x))")
```
```  1619     apply (simp add: algebra_simps)
```
```  1620     apply (rule ln_one_minus_pos_lower_bound)
```
```  1621     using a b apply auto
```
```  1622     done
```
```  1623   finally show ?thesis .
```
```  1624 qed
```
```  1625
```
```  1626 lemma abs_ln_one_plus_x_minus_x_bound:
```
```  1627     "abs x <= 1 / 2 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
```
```  1628   apply (case_tac "0 <= x")
```
```  1629   apply (rule order_trans)
```
```  1630   apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
```
```  1631   apply auto
```
```  1632   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
```
```  1633   apply auto
```
```  1634   done
```
```  1635
```
```  1636 lemma ln_x_over_x_mono: "exp 1 <= x \<Longrightarrow> x <= y \<Longrightarrow> (ln y / y) <= (ln x / x)"
```
```  1637 proof -
```
```  1638   assume x: "exp 1 <= x" "x <= y"
```
```  1639   moreover have "0 < exp (1::real)" by simp
```
```  1640   ultimately have a: "0 < x" and b: "0 < y"
```
```  1641     by (fast intro: less_le_trans order_trans)+
```
```  1642   have "x * ln y - x * ln x = x * (ln y - ln x)"
```
```  1643     by (simp add: algebra_simps)
```
```  1644   also have "... = x * ln(y / x)"
```
```  1645     by (simp only: ln_div a b)
```
```  1646   also have "y / x = (x + (y - x)) / x"
```
```  1647     by simp
```
```  1648   also have "... = 1 + (y - x) / x"
```
```  1649     using x a by (simp add: field_simps)
```
```  1650   also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
```
```  1651     using x a
```
```  1652     by (intro mult_left_mono ln_add_one_self_le_self) simp_all
```
```  1653   also have "... = y - x" using a by simp
```
```  1654   also have "... = (y - x) * ln (exp 1)" by simp
```
```  1655   also have "... <= (y - x) * ln x"
```
```  1656     apply (rule mult_left_mono)
```
```  1657     apply (subst ln_le_cancel_iff)
```
```  1658     apply fact
```
```  1659     apply (rule a)
```
```  1660     apply (rule x)
```
```  1661     using x apply simp
```
```  1662     done
```
```  1663   also have "... = y * ln x - x * ln x"
```
```  1664     by (rule left_diff_distrib)
```
```  1665   finally have "x * ln y <= y * ln x"
```
```  1666     by arith
```
```  1667   then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
```
```  1668   also have "... = y * (ln x / x)" by simp
```
```  1669   finally show ?thesis using b by (simp add: field_simps)
```
```  1670 qed
```
```  1671
```
```  1672 lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1"
```
```  1673   using exp_ge_add_one_self[of "ln x"] by simp
```
```  1674
```
```  1675 lemma ln_eq_minus_one:
```
```  1676   assumes "0 < x" "ln x = x - 1"
```
```  1677   shows "x = 1"
```
```  1678 proof -
```
```  1679   let ?l = "\<lambda>y. ln y - y + 1"
```
```  1680   have D: "\<And>x. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
```
```  1681     by (auto intro!: derivative_eq_intros)
```
```  1682
```
```  1683   show ?thesis
```
```  1684   proof (cases rule: linorder_cases)
```
```  1685     assume "x < 1"
```
```  1686     from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast
```
```  1687     from `x < a` have "?l x < ?l a"
```
```  1688     proof (rule DERIV_pos_imp_increasing, safe)
```
```  1689       fix y
```
```  1690       assume "x \<le> y" "y \<le> a"
```
```  1691       with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y"
```
```  1692         by (auto simp: field_simps)
```
```  1693       with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
```
```  1694         by auto
```
```  1695     qed
```
```  1696     also have "\<dots> \<le> 0"
```
```  1697       using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps)
```
```  1698     finally show "x = 1" using assms by auto
```
```  1699   next
```
```  1700     assume "1 < x"
```
```  1701     from dense[OF this] obtain a where "1 < a" "a < x" by blast
```
```  1702     from `a < x` have "?l x < ?l a"
```
```  1703     proof (rule DERIV_neg_imp_decreasing, safe)
```
```  1704       fix y
```
```  1705       assume "a \<le> y" "y \<le> x"
```
```  1706       with `1 < a` have "1 / y - 1 < 0" "0 < y"
```
```  1707         by (auto simp: field_simps)
```
```  1708       with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
```
```  1709         by blast
```
```  1710     qed
```
```  1711     also have "\<dots> \<le> 0"
```
```  1712       using ln_le_minus_one `1 < a` by (auto simp: field_simps)
```
```  1713     finally show "x = 1" using assms by auto
```
```  1714   next
```
```  1715     assume "x = 1"
```
```  1716     then show ?thesis by simp
```
```  1717   qed
```
```  1718 qed
```
```  1719
```
```  1720 lemma exp_at_bot: "(exp ---> (0::real)) at_bot"
```
```  1721   unfolding tendsto_Zfun_iff
```
```  1722 proof (rule ZfunI, simp add: eventually_at_bot_dense)
```
```  1723   fix r :: real assume "0 < r"
```
```  1724   {
```
```  1725     fix x
```
```  1726     assume "x < ln r"
```
```  1727     then have "exp x < exp (ln r)"
```
```  1728       by simp
```
```  1729     with `0 < r` have "exp x < r"
```
```  1730       by simp
```
```  1731   }
```
```  1732   then show "\<exists>k. \<forall>n<k. exp n < r" by auto
```
```  1733 qed
```
```  1734
```
```  1735 lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
```
```  1736   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"])
```
```  1737      (auto intro: eventually_gt_at_top)
```
```  1738
```
```  1739 lemma ln_at_0: "LIM x at_right 0. ln x :> at_bot"
```
```  1740   by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  1741      (auto simp: eventually_at_filter)
```
```  1742
```
```  1743 lemma ln_at_top: "LIM x at_top. ln x :> at_top"
```
```  1744   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  1745      (auto intro: eventually_gt_at_top)
```
```  1746
```
```  1747 lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) ---> (0::real)) at_top"
```
```  1748 proof (induct k)
```
```  1749   case 0
```
```  1750   show "((\<lambda>x. x ^ 0 / exp x) ---> (0::real)) at_top"
```
```  1751     by (simp add: inverse_eq_divide[symmetric])
```
```  1752        (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
```
```  1753               at_top_le_at_infinity order_refl)
```
```  1754 next
```
```  1755   case (Suc k)
```
```  1756   show ?case
```
```  1757   proof (rule lhospital_at_top_at_top)
```
```  1758     show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"
```
```  1759       by eventually_elim (intro derivative_eq_intros, auto)
```
```  1760     show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top"
```
```  1761       by eventually_elim auto
```
```  1762     show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
```
```  1763       by auto
```
```  1764     from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
```
```  1765     show "((\<lambda>x. real (Suc k) * x ^ k / exp x) ---> 0) at_top"
```
```  1766       by simp
```
```  1767   qed (rule exp_at_top)
```
```  1768 qed
```
```  1769
```
```  1770
```
```  1771 definition powr :: "[real,real] => real"  (infixr "powr" 80)
```
```  1772   -- {*exponentation with real exponent*}
```
```  1773   where "x powr a = exp(a * ln x)"
```
```  1774
```
```  1775 definition log :: "[real,real] => real"
```
```  1776   -- {*logarithm of @{term x} to base @{term a}*}
```
```  1777   where "log a x = ln x / ln a"
```
```  1778
```
```  1779
```
```  1780 lemma tendsto_log [tendsto_intros]:
```
```  1781   "\<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"
```
```  1782   unfolding log_def by (intro tendsto_intros) auto
```
```  1783
```
```  1784 lemma continuous_log:
```
```  1785   assumes "continuous F f"
```
```  1786     and "continuous F g"
```
```  1787     and "0 < f (Lim F (\<lambda>x. x))"
```
```  1788     and "f (Lim F (\<lambda>x. x)) \<noteq> 1"
```
```  1789     and "0 < g (Lim F (\<lambda>x. x))"
```
```  1790   shows "continuous F (\<lambda>x. log (f x) (g x))"
```
```  1791   using assms unfolding continuous_def by (rule tendsto_log)
```
```  1792
```
```  1793 lemma continuous_at_within_log[continuous_intros]:
```
```  1794   assumes "continuous (at a within s) f"
```
```  1795     and "continuous (at a within s) g"
```
```  1796     and "0 < f a"
```
```  1797     and "f a \<noteq> 1"
```
```  1798     and "0 < g a"
```
```  1799   shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
```
```  1800   using assms unfolding continuous_within by (rule tendsto_log)
```
```  1801
```
```  1802 lemma isCont_log[continuous_intros, simp]:
```
```  1803   assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
```
```  1804   shows "isCont (\<lambda>x. log (f x) (g x)) a"
```
```  1805   using assms unfolding continuous_at by (rule tendsto_log)
```
```  1806
```
```  1807 lemma continuous_on_log[continuous_intros]:
```
```  1808   assumes "continuous_on s f" "continuous_on s g"
```
```  1809     and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
```
```  1810   shows "continuous_on s (\<lambda>x. log (f x) (g x))"
```
```  1811   using assms unfolding continuous_on_def by (fast intro: tendsto_log)
```
```  1812
```
```  1813 lemma powr_one_eq_one [simp]: "1 powr a = 1"
```
```  1814   by (simp add: powr_def)
```
```  1815
```
```  1816 lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
```
```  1817   by (simp add: powr_def)
```
```  1818
```
```  1819 lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
```
```  1820   by (simp add: powr_def)
```
```  1821 declare powr_one_gt_zero_iff [THEN iffD2, simp]
```
```  1822
```
```  1823 lemma powr_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
```
```  1824   by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
```
```  1825
```
```  1826 lemma powr_gt_zero [simp]: "0 < x powr a"
```
```  1827   by (simp add: powr_def)
```
```  1828
```
```  1829 lemma powr_ge_pzero [simp]: "0 <= x powr y"
```
```  1830   by (rule order_less_imp_le, rule powr_gt_zero)
```
```  1831
```
```  1832 lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
```
```  1833   by (simp add: powr_def)
```
```  1834
```
```  1835 lemma powr_divide: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
```
```  1836   apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
```
```  1837   apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
```
```  1838   done
```
```  1839
```
```  1840 lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
```
```  1841   apply (simp add: powr_def)
```
```  1842   apply (subst exp_diff [THEN sym])
```
```  1843   apply (simp add: left_diff_distrib)
```
```  1844   done
```
```  1845
```
```  1846 lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
```
```  1847   by (simp add: powr_def exp_add [symmetric] distrib_right)
```
```  1848
```
```  1849 lemma powr_mult_base: "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
```
```  1850   using assms by (auto simp: powr_add)
```
```  1851
```
```  1852 lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
```
```  1853   by (simp add: powr_def)
```
```  1854
```
```  1855 lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
```
```  1856   by (simp add: powr_powr mult.commute)
```
```  1857
```
```  1858 lemma powr_minus: "x powr (-a) = inverse (x powr a)"
```
```  1859   by (simp add: powr_def exp_minus [symmetric])
```
```  1860
```
```  1861 lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
```
```  1862   by (simp add: divide_inverse powr_minus)
```
```  1863
```
```  1864 lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
```
```  1865   by (simp add: powr_def)
```
```  1866
```
```  1867 lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
```
```  1868   by (simp add: powr_def)
```
```  1869
```
```  1870 lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a < x powr b) = (a < b)"
```
```  1871   by (blast intro: powr_less_cancel powr_less_mono)
```
```  1872
```
```  1873 lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a \<le> x powr b) = (a \<le> b)"
```
```  1874   by (simp add: linorder_not_less [symmetric])
```
```  1875
```
```  1876 lemma log_ln: "ln x = log (exp(1)) x"
```
```  1877   by (simp add: log_def)
```
```  1878
```
```  1879 lemma DERIV_log:
```
```  1880   assumes "x > 0"
```
```  1881   shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
```
```  1882 proof -
```
```  1883   def lb \<equiv> "1 / ln b"
```
```  1884   moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
```
```  1885     using `x > 0` by (auto intro!: derivative_eq_intros)
```
```  1886   ultimately show ?thesis
```
```  1887     by (simp add: log_def)
```
```  1888 qed
```
```  1889
```
```  1890 lemmas DERIV_log[THEN DERIV_chain2, derivative_intros]
```
```  1891
```
```  1892 lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
```
```  1893   by (simp add: powr_def log_def)
```
```  1894
```
```  1895 lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y"
```
```  1896   by (simp add: log_def powr_def)
```
```  1897
```
```  1898 lemma log_mult:
```
```  1899   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow>
```
```  1900     log a (x * y) = log a x + log a y"
```
```  1901   by (simp add: log_def ln_mult divide_inverse distrib_right)
```
```  1902
```
```  1903 lemma log_eq_div_ln_mult_log:
```
```  1904   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow>
```
```  1905     log a x = (ln b/ln a) * log b x"
```
```  1906   by (simp add: log_def divide_inverse)
```
```  1907
```
```  1908 text{*Base 10 logarithms*}
```
```  1909 lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x"
```
```  1910   by (simp add: log_def)
```
```  1911
```
```  1912 lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x"
```
```  1913   by (simp add: log_def)
```
```  1914
```
```  1915 lemma log_one [simp]: "log a 1 = 0"
```
```  1916   by (simp add: log_def)
```
```  1917
```
```  1918 lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
```
```  1919   by (simp add: log_def)
```
```  1920
```
```  1921 lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x"
```
```  1922   apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
```
```  1923   apply (simp add: log_mult [symmetric])
```
```  1924   done
```
```  1925
```
```  1926 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"
```
```  1927   by (simp add: log_mult divide_inverse log_inverse)
```
```  1928
```
```  1929 lemma log_less_cancel_iff [simp]:
```
```  1930   "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
```
```  1931   apply safe
```
```  1932   apply (rule_tac [2] powr_less_cancel)
```
```  1933   apply (drule_tac a = "log a x" in powr_less_mono, auto)
```
```  1934   done
```
```  1935
```
```  1936 lemma log_inj:
```
```  1937   assumes "1 < b"
```
```  1938   shows "inj_on (log b) {0 <..}"
```
```  1939 proof (rule inj_onI, simp)
```
```  1940   fix x y
```
```  1941   assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
```
```  1942   show "x = y"
```
```  1943   proof (cases rule: linorder_cases)
```
```  1944     assume "x = y"
```
```  1945     then show ?thesis by simp
```
```  1946   next
```
```  1947     assume "x < y" hence "log b x < log b y"
```
```  1948       using log_less_cancel_iff[OF `1 < b`] pos by simp
```
```  1949     then show ?thesis using * by simp
```
```  1950   next
```
```  1951     assume "y < x" hence "log b y < log b x"
```
```  1952       using log_less_cancel_iff[OF `1 < b`] pos by simp
```
```  1953     then show ?thesis using * by simp
```
```  1954   qed
```
```  1955 qed
```
```  1956
```
```  1957 lemma log_le_cancel_iff [simp]:
```
```  1958   "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (log a x \<le> log a y) = (x \<le> y)"
```
```  1959   by (simp add: linorder_not_less [symmetric])
```
```  1960
```
```  1961 lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
```
```  1962   using log_less_cancel_iff[of a 1 x] by simp
```
```  1963
```
```  1964 lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
```
```  1965   using log_le_cancel_iff[of a 1 x] by simp
```
```  1966
```
```  1967 lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
```
```  1968   using log_less_cancel_iff[of a x 1] by simp
```
```  1969
```
```  1970 lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
```
```  1971   using log_le_cancel_iff[of a x 1] by simp
```
```  1972
```
```  1973 lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
```
```  1974   using log_less_cancel_iff[of a a x] by simp
```
```  1975
```
```  1976 lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
```
```  1977   using log_le_cancel_iff[of a a x] by simp
```
```  1978
```
```  1979 lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
```
```  1980   using log_less_cancel_iff[of a x a] by simp
```
```  1981
```
```  1982 lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
```
```  1983   using log_le_cancel_iff[of a x a] by simp
```
```  1984
```
```  1985 lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
```
```  1986   apply (induct n)
```
```  1987   apply simp
```
```  1988   apply (subgoal_tac "real(Suc n) = real n + 1")
```
```  1989   apply (erule ssubst)
```
```  1990   apply (subst powr_add, simp, simp)
```
```  1991   done
```
```  1992
```
```  1993 lemma powr_realpow_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
```
```  1994   unfolding real_of_nat_numeral [symmetric] by (rule powr_realpow)
```
```  1995
```
```  1996 lemma powr2_sqrt[simp]: "0 < x \<Longrightarrow> sqrt x powr 2 = x"
```
```  1997 by(simp add: powr_realpow_numeral)
```
```  1998
```
```  1999 lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
```
```  2000   apply (case_tac "x = 0", simp, simp)
```
```  2001   apply (rule powr_realpow [THEN sym], simp)
```
```  2002   done
```
```  2003
```
```  2004 lemma powr_int:
```
```  2005   assumes "x > 0"
```
```  2006   shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
```
```  2007 proof (cases "i < 0")
```
```  2008   case True
```
```  2009   have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
```
```  2010   show ?thesis using `i < 0` `x > 0` by (simp add: r field_simps powr_realpow[symmetric])
```
```  2011 next
```
```  2012   case False
```
```  2013   then show ?thesis by (simp add: assms powr_realpow[symmetric])
```
```  2014 qed
```
```  2015
```
```  2016 lemma powr_one: "0 < x \<Longrightarrow> x powr 1 = x"
```
```  2017   using powr_realpow [of x 1] by simp
```
```  2018
```
```  2019 lemma powr_numeral: "0 < x \<Longrightarrow> x powr numeral n = x ^ numeral n"
```
```  2020   by (fact powr_realpow_numeral)
```
```  2021
```
```  2022 lemma powr_neg_one: "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
```
```  2023   using powr_int [of x "- 1"] by simp
```
```  2024
```
```  2025 lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
```
```  2026   using powr_int [of x "- numeral n"] by simp
```
```  2027
```
```  2028 lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
```
```  2029   by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
```
```  2030
```
```  2031 lemma ln_powr: "ln (x powr y) = y * ln x"
```
```  2032   by (simp add: powr_def)
```
```  2033
```
```  2034 lemma ln_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> ln (root n b) =  ln b / n"
```
```  2035 by(simp add: root_powr_inverse ln_powr)
```
```  2036
```
```  2037 lemma ln_sqrt: "0 < x \<Longrightarrow> ln (sqrt x) = ln x / 2"
```
```  2038   by (simp add: ln_powr powr_numeral ln_powr[symmetric] mult.commute)
```
```  2039
```
```  2040 lemma log_root: "\<lbrakk> n > 0; a > 0 \<rbrakk> \<Longrightarrow> log b (root n a) =  log b a / n"
```
```  2041 by(simp add: log_def ln_root)
```
```  2042
```
```  2043 lemma log_powr: "log b (x powr y) = y * log b x"
```
```  2044   by (simp add: log_def ln_powr)
```
```  2045
```
```  2046 lemma log_nat_power: "0 < x \<Longrightarrow> log b (x ^ n) = real n * log b x"
```
```  2047   by (simp add: log_powr powr_realpow [symmetric])
```
```  2048
```
```  2049 lemma log_base_change: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log b x = log a x / log a b"
```
```  2050   by (simp add: log_def)
```
```  2051
```
```  2052 lemma log_base_pow: "0 < a \<Longrightarrow> log (a ^ n) x = log a x / n"
```
```  2053   by (simp add: log_def ln_realpow)
```
```  2054
```
```  2055 lemma log_base_powr: "log (a powr b) x = log a x / b"
```
```  2056   by (simp add: log_def ln_powr)
```
```  2057
```
```  2058 lemma log_base_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> log (root n b) x = n * (log b x)"
```
```  2059 by(simp add: log_def ln_root)
```
```  2060
```
```  2061 lemma ln_bound: "1 <= x ==> ln x <= x"
```
```  2062   apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
```
```  2063   apply simp
```
```  2064   apply (rule ln_add_one_self_le_self, simp)
```
```  2065   done
```
```  2066
```
```  2067 lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
```
```  2068   apply (cases "x = 1", simp)
```
```  2069   apply (cases "a = b", simp)
```
```  2070   apply (rule order_less_imp_le)
```
```  2071   apply (rule powr_less_mono, auto)
```
```  2072   done
```
```  2073
```
```  2074 lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
```
```  2075   apply (subst powr_zero_eq_one [THEN sym])
```
```  2076   apply (rule powr_mono, assumption+)
```
```  2077   done
```
```  2078
```
```  2079 lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a < y powr a"
```
```  2080   apply (unfold powr_def)
```
```  2081   apply (rule exp_less_mono)
```
```  2082   apply (rule mult_strict_left_mono)
```
```  2083   apply (subst ln_less_cancel_iff, assumption)
```
```  2084   apply (rule order_less_trans)
```
```  2085   prefer 2
```
```  2086   apply assumption+
```
```  2087   done
```
```  2088
```
```  2089 lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a < x powr a"
```
```  2090   apply (unfold powr_def)
```
```  2091   apply (rule exp_less_mono)
```
```  2092   apply (rule mult_strict_left_mono_neg)
```
```  2093   apply (subst ln_less_cancel_iff)
```
```  2094   apply assumption
```
```  2095   apply (rule order_less_trans)
```
```  2096   prefer 2
```
```  2097   apply assumption+
```
```  2098   done
```
```  2099
```
```  2100 lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
```
```  2101   apply (case_tac "a = 0", simp)
```
```  2102   apply (case_tac "x = y", simp)
```
```  2103   apply (metis less_eq_real_def powr_less_mono2)
```
```  2104   done
```
```  2105
```
```  2106 lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
```
```  2107   unfolding powr_def exp_inj_iff by simp
```
```  2108
```
```  2109 lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
```
```  2110   by (metis less_eq_real_def ln_less_self mult_imp_le_div_pos ln_powr mult.commute
```
```  2111             order.strict_trans2 powr_gt_zero zero_less_one)
```
```  2112
```
```  2113 lemma ln_powr_bound2:
```
```  2114   assumes "1 < x" and "0 < a"
```
```  2115   shows "(ln x) powr a <= (a powr a) * x"
```
```  2116 proof -
```
```  2117   from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
```
```  2118     by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff)
```
```  2119   also have "... = a * (x powr (1 / a))"
```
```  2120     by simp
```
```  2121   finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
```
```  2122     by (metis assms less_imp_le ln_gt_zero powr_mono2)
```
```  2123   also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
```
```  2124     by (metis assms(2) powr_mult powr_gt_zero)
```
```  2125   also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
```
```  2126     by (rule powr_powr)
```
```  2127   also have "... = x" using assms
```
```  2128     by auto
```
```  2129   finally show ?thesis .
```
```  2130 qed
```
```  2131
```
```  2132 lemma tendsto_powr [tendsto_intros]:
```
```  2133   "\<lbrakk>(f ---> a) F; (g ---> b) F; a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x powr g x) ---> a powr b) F"
```
```  2134   unfolding powr_def by (intro tendsto_intros)
```
```  2135
```
```  2136 lemma continuous_powr:
```
```  2137   assumes "continuous F f"
```
```  2138     and "continuous F g"
```
```  2139     and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```  2140   shows "continuous F (\<lambda>x. (f x) powr (g x))"
```
```  2141   using assms unfolding continuous_def by (rule tendsto_powr)
```
```  2142
```
```  2143 lemma continuous_at_within_powr[continuous_intros]:
```
```  2144   assumes "continuous (at a within s) f"
```
```  2145     and "continuous (at a within s) g"
```
```  2146     and "f a \<noteq> 0"
```
```  2147   shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
```
```  2148   using assms unfolding continuous_within by (rule tendsto_powr)
```
```  2149
```
```  2150 lemma isCont_powr[continuous_intros, simp]:
```
```  2151   assumes "isCont f a" "isCont g a" "f a \<noteq> 0"
```
```  2152   shows "isCont (\<lambda>x. (f x) powr g x) a"
```
```  2153   using assms unfolding continuous_at by (rule tendsto_powr)
```
```  2154
```
```  2155 lemma continuous_on_powr[continuous_intros]:
```
```  2156   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> 0"
```
```  2157   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
```
```  2158   using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
```
```  2159
```
```  2160 (* FIXME: generalize by replacing d by with g x and g ---> d? *)
```
```  2161 lemma tendsto_zero_powrI:
```
```  2162   assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> 0) F"
```
```  2163     and "0 < d"
```
```  2164   shows "((\<lambda>x. f x powr d) ---> 0) F"
```
```  2165 proof (rule tendstoI)
```
```  2166   fix e :: real assume "0 < e"
```
```  2167   def Z \<equiv> "e powr (1 / d)"
```
```  2168   with `0 < e` have "0 < Z" by simp
```
```  2169   with assms have "eventually (\<lambda>x. 0 < f x \<and> dist (f x) 0 < Z) F"
```
```  2170     by (intro eventually_conj tendstoD)
```
```  2171   moreover
```
```  2172   from assms have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> x powr d < Z powr d"
```
```  2173     by (intro powr_less_mono2) (auto simp: dist_real_def)
```
```  2174   with assms `0 < e` have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> dist (x powr d) 0 < e"
```
```  2175     unfolding dist_real_def Z_def by (auto simp: powr_powr)
```
```  2176   ultimately
```
```  2177   show "eventually (\<lambda>x. dist (f x powr d) 0 < e) F" by (rule eventually_elim1)
```
```  2178 qed
```
```  2179
```
```  2180 lemma tendsto_neg_powr:
```
```  2181   assumes "s < 0"
```
```  2182     and "LIM x F. f x :> at_top"
```
```  2183   shows "((\<lambda>x. f x powr s) ---> 0) F"
```
```  2184 proof (rule tendstoI)
```
```  2185   fix e :: real assume "0 < e"
```
```  2186   def Z \<equiv> "e powr (1 / s)"
```
```  2187   from assms have "eventually (\<lambda>x. Z < f x) F"
```
```  2188     by (simp add: filterlim_at_top_dense)
```
```  2189   moreover
```
```  2190   from assms have "\<And>x. Z < x \<Longrightarrow> x powr s < Z powr s"
```
```  2191     by (auto simp: Z_def intro!: powr_less_mono2_neg)
```
```  2192   with assms `0 < e` have "\<And>x. Z < x \<Longrightarrow> dist (x powr s) 0 < e"
```
```  2193     by (simp add: powr_powr Z_def dist_real_def)
```
```  2194   ultimately
```
```  2195   show "eventually (\<lambda>x. dist (f x powr s) 0 < e) F" by (rule eventually_elim1)
```
```  2196 qed
```
```  2197
```
```  2198 (* it is funny that this isn't in the library! It could go in Transcendental *)
```
```  2199 lemma tendsto_exp_limit_at_right:
```
```  2200   fixes x :: real
```
```  2201   shows "((\<lambda>y. (1 + x * y) powr (1 / y)) ---> exp x) (at_right 0)"
```
```  2202 proof cases
```
```  2203   assume "x \<noteq> 0"
```
```  2204
```
```  2205   have "((\<lambda>y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)"
```
```  2206     by (auto intro!: derivative_eq_intros)
```
```  2207   then have "((\<lambda>y. ln (1 + x * y) / y) ---> x) (at 0)"
```
```  2208     by (auto simp add: has_field_derivative_def field_has_derivative_at)
```
```  2209   then have *: "((\<lambda>y. exp (ln (1 + x * y) / y)) ---> exp x) (at 0)"
```
```  2210     by (rule tendsto_intros)
```
```  2211   then show ?thesis
```
```  2212   proof (rule filterlim_mono_eventually)
```
```  2213     show "eventually (\<lambda>xa. exp (ln (1 + x * xa) / xa) = (1 + x * xa) powr (1 / xa)) (at_right 0)"
```
```  2214       unfolding eventually_at_right[OF zero_less_one]
```
```  2215       using `x \<noteq> 0` by (intro exI[of _ "1 / \<bar>x\<bar>"]) (auto simp: field_simps powr_def)
```
```  2216   qed (simp_all add: at_eq_sup_left_right)
```
```  2217 qed simp
```
```  2218
```
```  2219 lemma tendsto_exp_limit_at_top:
```
```  2220   fixes x :: real
```
```  2221   shows "((\<lambda>y. (1 + x / y) powr y) ---> exp x) at_top"
```
```  2222   apply (subst filterlim_at_top_to_right)
```
```  2223   apply (simp add: inverse_eq_divide)
```
```  2224   apply (rule tendsto_exp_limit_at_right)
```
```  2225   done
```
```  2226
```
```  2227 lemma tendsto_exp_limit_sequentially:
```
```  2228   fixes x :: real
```
```  2229   shows "(\<lambda>n. (1 + x / n) ^ n) ----> exp x"
```
```  2230 proof (rule filterlim_mono_eventually)
```
```  2231   from reals_Archimedean2 [of "abs x"] obtain n :: nat where *: "real n > abs x" ..
```
```  2232   hence "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top"
```
```  2233     apply (intro eventually_sequentiallyI [of n])
```
```  2234     apply (case_tac "x \<ge> 0")
```
```  2235     apply (rule add_pos_nonneg, auto intro: divide_nonneg_nonneg)
```
```  2236     apply (subgoal_tac "x / real xa > -1")
```
```  2237     apply (auto simp add: field_simps)
```
```  2238     done
```
```  2239   then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top"
```
```  2240     by (rule eventually_elim1) (erule powr_realpow)
```
```  2241   show "(\<lambda>n. (1 + x / real n) powr real n) ----> exp x"
```
```  2242     by (rule filterlim_compose [OF tendsto_exp_limit_at_top filterlim_real_sequentially])
```
```  2243 qed auto
```
```  2244
```
```  2245 subsection {* Sine and Cosine *}
```
```  2246
```
```  2247 definition sin_coeff :: "nat \<Rightarrow> real" where
```
```  2248   "sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / real (fact n))"
```
```  2249
```
```  2250 definition cos_coeff :: "nat \<Rightarrow> real" where
```
```  2251   "cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / real (fact n) else 0)"
```
```  2252
```
```  2253 definition sin :: "real \<Rightarrow> real"
```
```  2254   where "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
```
```  2255
```
```  2256 definition cos :: "real \<Rightarrow> real"
```
```  2257   where "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
```
```  2258
```
```  2259 lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
```
```  2260   unfolding sin_coeff_def by simp
```
```  2261
```
```  2262 lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
```
```  2263   unfolding cos_coeff_def by simp
```
```  2264
```
```  2265 lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
```
```  2266   unfolding cos_coeff_def sin_coeff_def
```
```  2267   by (simp del: mult_Suc)
```
```  2268
```
```  2269 lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
```
```  2270   unfolding cos_coeff_def sin_coeff_def
```
```  2271   by (simp del: mult_Suc) (auto elim: oddE)
```
```  2272
```
```  2273 lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
```
```  2274   unfolding sin_coeff_def
```
```  2275   apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
```
```  2276   apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2277   done
```
```  2278
```
```  2279 lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"
```
```  2280   unfolding cos_coeff_def
```
```  2281   apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
```
```  2282   apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2283   done
```
```  2284
```
```  2285 lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
```
```  2286   unfolding sin_def by (rule summable_sin [THEN summable_sums])
```
```  2287
```
```  2288 lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
```
```  2289   unfolding cos_def by (rule summable_cos [THEN summable_sums])
```
```  2290
```
```  2291 lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
```
```  2292   by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  2293
```
```  2294 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
```
```  2295   by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  2296
```
```  2297 text{*Now at last we can get the derivatives of exp, sin and cos*}
```
```  2298
```
```  2299 lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
```
```  2300   unfolding sin_def cos_def
```
```  2301   apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
```
```  2302   apply (simp_all add: diffs_sin_coeff diffs_cos_coeff
```
```  2303     summable_minus summable_sin summable_cos)
```
```  2304   done
```
```  2305
```
```  2306 declare DERIV_sin[THEN DERIV_chain2, derivative_intros]
```
```  2307
```
```  2308 lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
```
```  2309   unfolding cos_def sin_def
```
```  2310   apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
```
```  2311   apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus
```
```  2312     summable_minus summable_sin summable_cos suminf_minus)
```
```  2313   done
```
```  2314
```
```  2315 declare DERIV_cos[THEN DERIV_chain2, derivative_intros]
```
```  2316
```
```  2317 lemma isCont_sin: "isCont sin x"
```
```  2318   by (rule DERIV_sin [THEN DERIV_isCont])
```
```  2319
```
```  2320 lemma isCont_cos: "isCont cos x"
```
```  2321   by (rule DERIV_cos [THEN DERIV_isCont])
```
```  2322
```
```  2323 lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
```
```  2324   by (rule isCont_o2 [OF _ isCont_sin])
```
```  2325
```
```  2326 lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
```
```  2327   by (rule isCont_o2 [OF _ isCont_cos])
```
```  2328
```
```  2329 lemma tendsto_sin [tendsto_intros]:
```
```  2330   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
```
```  2331   by (rule isCont_tendsto_compose [OF isCont_sin])
```
```  2332
```
```  2333 lemma tendsto_cos [tendsto_intros]:
```
```  2334   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
```
```  2335   by (rule isCont_tendsto_compose [OF isCont_cos])
```
```  2336
```
```  2337 lemma continuous_sin [continuous_intros]:
```
```  2338   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
```
```  2339   unfolding continuous_def by (rule tendsto_sin)
```
```  2340
```
```  2341 lemma continuous_on_sin [continuous_intros]:
```
```  2342   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
```
```  2343   unfolding continuous_on_def by (auto intro: tendsto_sin)
```
```  2344
```
```  2345 lemma continuous_cos [continuous_intros]:
```
```  2346   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
```
```  2347   unfolding continuous_def by (rule tendsto_cos)
```
```  2348
```
```  2349 lemma continuous_on_cos [continuous_intros]:
```
```  2350   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
```
```  2351   unfolding continuous_on_def by (auto intro: tendsto_cos)
```
```  2352
```
```  2353 subsection {* Properties of Sine and Cosine *}
```
```  2354
```
```  2355 lemma sin_zero [simp]: "sin 0 = 0"
```
```  2356   unfolding sin_def sin_coeff_def by (simp add: powser_zero)
```
```  2357
```
```  2358 lemma cos_zero [simp]: "cos 0 = 1"
```
```  2359   unfolding cos_def cos_coeff_def by (simp add: powser_zero)
```
```  2360
```
```  2361 lemma sin_cos_squared_add [simp]: "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
```
```  2362 proof -
```
```  2363   have "\<forall>x. DERIV (\<lambda>x. (sin x)\<^sup>2 + (cos x)\<^sup>2) x :> 0"
```
```  2364     by (auto intro!: derivative_eq_intros)
```
```  2365   hence "(sin x)\<^sup>2 + (cos x)\<^sup>2 = (sin 0)\<^sup>2 + (cos 0)\<^sup>2"
```
```  2366     by (rule DERIV_isconst_all)
```
```  2367   thus "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1" by simp
```
```  2368 qed
```
```  2369
```
```  2370 lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
```
```  2371   by (subst add.commute, rule sin_cos_squared_add)
```
```  2372
```
```  2373 lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
```
```  2374   using sin_cos_squared_add2 [unfolded power2_eq_square] .
```
```  2375
```
```  2376 lemma sin_squared_eq: "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
```
```  2377   unfolding eq_diff_eq by (rule sin_cos_squared_add)
```
```  2378
```
```  2379 lemma cos_squared_eq: "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
```
```  2380   unfolding eq_diff_eq by (rule sin_cos_squared_add2)
```
```  2381
```
```  2382 lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
```
```  2383   by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
```
```  2384
```
```  2385 lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
```
```  2386   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  2387
```
```  2388 lemma sin_le_one [simp]: "sin x \<le> 1"
```
```  2389   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  2390
```
```  2391 lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
```
```  2392   by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
```
```  2393
```
```  2394 lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
```
```  2395   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  2396
```
```  2397 lemma cos_le_one [simp]: "cos x \<le> 1"
```
```  2398   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  2399
```
```  2400 lemma DERIV_fun_pow: "DERIV g x :> m ==>
```
```  2401       DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
```
```  2402   by (auto intro!: derivative_eq_intros simp: real_of_nat_def)
```
```  2403
```
```  2404 lemma DERIV_fun_exp:
```
```  2405      "DERIV g x :> m ==> DERIV (\<lambda>x. exp(g x)) x :> exp(g x) * m"
```
```  2406   by (auto intro!: derivative_intros)
```
```  2407
```
```  2408 lemma DERIV_fun_sin:
```
```  2409      "DERIV g x :> m ==> DERIV (\<lambda>x. sin(g x)) x :> cos(g x) * m"
```
```  2410   by (auto intro!: derivative_intros)
```
```  2411
```
```  2412 lemma DERIV_fun_cos:
```
```  2413      "DERIV g x :> m ==> DERIV (\<lambda>x. cos(g x)) x :> -sin(g x) * m"
```
```  2414   by (auto intro!: derivative_eq_intros simp: real_of_nat_def)
```
```  2415
```
```  2416 lemma sin_cos_add_lemma:
```
```  2417   "(sin (x + y) - (sin x * cos y + cos x * sin y))\<^sup>2 +
```
```  2418     (cos (x + y) - (cos x * cos y - sin x * sin y))\<^sup>2 = 0"
```
```  2419   (is "?f x = 0")
```
```  2420 proof -
```
```  2421   have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
```
```  2422     by (auto intro!: derivative_eq_intros simp add: algebra_simps)
```
```  2423   hence "?f x = ?f 0"
```
```  2424     by (rule DERIV_isconst_all)
```
```  2425   thus ?thesis by simp
```
```  2426 qed
```
```  2427
```
```  2428 lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
```
```  2429   using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
```
```  2430
```
```  2431 lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
```
```  2432   using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
```
```  2433
```
```  2434 lemma sin_cos_minus_lemma:
```
```  2435   "(sin(-x) + sin(x))\<^sup>2 + (cos(-x) - cos(x))\<^sup>2 = 0" (is "?f x = 0")
```
```  2436 proof -
```
```  2437   have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
```
```  2438     by (auto intro!: derivative_eq_intros simp add: algebra_simps)
```
```  2439   hence "?f x = ?f 0"
```
```  2440     by (rule DERIV_isconst_all)
```
```  2441   thus ?thesis by simp
```
```  2442 qed
```
```  2443
```
```  2444 lemma sin_minus [simp]: "sin (-x) = -sin(x)"
```
```  2445   using sin_cos_minus_lemma [where x=x] by simp
```
```  2446
```
```  2447 lemma cos_minus [simp]: "cos (-x) = cos(x)"
```
```  2448   using sin_cos_minus_lemma [where x=x] by simp
```
```  2449
```
```  2450 lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
```
```  2451   using sin_add [of x "- y"] by simp
```
```  2452
```
```  2453 lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
```
```  2454   by (simp add: sin_diff mult.commute)
```
```  2455
```
```  2456 lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
```
```  2457   using cos_add [of x "- y"] by simp
```
```  2458
```
```  2459 lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
```
```  2460   by (simp add: cos_diff mult.commute)
```
```  2461
```
```  2462 lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
```
```  2463   using sin_add [where x=x and y=x] by simp
```
```  2464
```
```  2465 lemma cos_double: "cos(2* x) = ((cos x)\<^sup>2) - ((sin x)\<^sup>2)"
```
```  2466   using cos_add [where x=x and y=x]
```
```  2467   by (simp add: power2_eq_square)
```
```  2468
```
```  2469 lemma sin_x_le_x: assumes x: "x \<ge> 0" shows "sin x \<le> x"
```
```  2470 proof -
```
```  2471   let ?f = "\<lambda>x. x - sin x"
```
```  2472   from x have "?f x \<ge> ?f 0"
```
```  2473     apply (rule DERIV_nonneg_imp_nondecreasing)
```
```  2474     apply (intro allI impI exI[of _ "1 - cos x" for x])
```
```  2475     apply (auto intro!: derivative_eq_intros simp: field_simps)
```
```  2476     done
```
```  2477   thus "sin x \<le> x" by simp
```
```  2478 qed
```
```  2479
```
```  2480 lemma sin_x_ge_neg_x: assumes x: "x \<ge> 0" shows "sin x \<ge> - x"
```
```  2481 proof -
```
```  2482   let ?f = "\<lambda>x. x + sin x"
```
```  2483   from x have "?f x \<ge> ?f 0"
```
```  2484     apply (rule DERIV_nonneg_imp_nondecreasing)
```
```  2485     apply (intro allI impI exI[of _ "1 + cos x" for x])
```
```  2486     apply (auto intro!: derivative_eq_intros simp: field_simps real_0_le_add_iff)
```
```  2487     done
```
```  2488   thus "sin x \<ge> -x" by simp
```
```  2489 qed
```
```  2490
```
```  2491 lemma abs_sin_x_le_abs_x: "\<bar>sin x\<bar> \<le> \<bar>x\<bar>"
```
```  2492   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"]
```
```  2493   by (auto simp: abs_real_def)
```
```  2494
```
```  2495 subsection {* The Constant Pi *}
```
```  2496
```
```  2497 definition pi :: real
```
```  2498   where "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
```
```  2499
```
```  2500 text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
```
```  2501    hence define pi.*}
```
```  2502
```
```  2503 lemma sin_paired:
```
```  2504   "(\<lambda>n. (- 1) ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) sums  sin x"
```
```  2505 proof -
```
```  2506   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
```
```  2507     by (rule sin_converges [THEN sums_group], simp)
```
```  2508   thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: ac_simps)
```
```  2509 qed
```
```  2510
```
```  2511 lemma sin_gt_zero:
```
```  2512   assumes "0 < x" and "x < 2"
```
```  2513   shows "0 < sin x"
```
```  2514 proof -
```
```  2515   let ?f = "\<lambda>n. \<Sum>k = n*2..<n*2+2. (- 1) ^ k / real (fact (2*k+1)) * x^(2*k+1)"
```
```  2516   have pos: "\<forall>n. 0 < ?f n"
```
```  2517   proof
```
```  2518     fix n :: nat
```
```  2519     let ?k2 = "real (Suc (Suc (4 * n)))"
```
```  2520     let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
```
```  2521     have "x * x < ?k2 * ?k3"
```
```  2522       using assms by (intro mult_strict_mono', simp_all)
```
```  2523     hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
```
```  2524       by (intro mult_strict_right_mono zero_less_power `0 < x`)
```
```  2525     thus "0 < ?f n"
```
```  2526       by (simp del: mult_Suc,
```
```  2527         simp add: less_divide_eq field_simps del: mult_Suc)
```
```  2528   qed
```
```  2529   have sums: "?f sums sin x"
```
```  2530     by (rule sin_paired [THEN sums_group], simp)
```
```  2531   show "0 < sin x"
```
```  2532     unfolding sums_unique [OF sums]
```
```  2533     using sums_summable [OF sums] pos
```
```  2534     by (rule suminf_pos)
```
```  2535 qed
```
```  2536
```
```  2537 lemma cos_double_less_one: "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
```
```  2538   using sin_gt_zero [where x = x] by (auto simp add: cos_squared_eq cos_double)
```
```  2539
```
```  2540 lemma cos_paired: "(\<lambda>n. (- 1) ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
```
```  2541 proof -
```
```  2542   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
```
```  2543     by (rule cos_converges [THEN sums_group], simp)
```
```  2544   thus ?thesis unfolding cos_coeff_def by (simp add: ac_simps)
```
```  2545 qed
```
```  2546
```
```  2547 lemmas realpow_num_eq_if = power_eq_if
```
```  2548
```
```  2549 lemma sumr_pos_lt_pair:
```
```  2550   fixes f :: "nat \<Rightarrow> real"
```
```  2551   shows "\<lbrakk>summable f;
```
```  2552         \<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk>
```
```  2553       \<Longrightarrow> setsum f {..<k} < suminf f"
```
```  2554 unfolding One_nat_def
```
```  2555 apply (subst suminf_split_initial_segment [where k="k"])
```
```  2556 apply assumption
```
```  2557 apply simp
```
```  2558 apply (drule_tac k="k" in summable_ignore_initial_segment)
```
```  2559 apply (drule_tac k="Suc (Suc 0)" in sums_group [OF summable_sums], simp)
```
```  2560 apply simp
```
```  2561 apply (frule sums_unique)
```
```  2562 apply (drule sums_summable)
```
```  2563 apply simp
```
```  2564 apply (erule suminf_pos)
```
```  2565 apply (simp add: ac_simps)
```
```  2566 done
```
```  2567
```
```  2568 lemma cos_two_less_zero [simp]:
```
```  2569   "cos 2 < 0"
```
```  2570 proof -
```
```  2571   note fact_Suc [simp del]
```
```  2572   from cos_paired
```
```  2573   have "(\<lambda>n. - ((- 1) ^ n / real (fact (2 * n)) * 2 ^ (2 * n))) sums - cos 2"
```
```  2574     by (rule sums_minus)
```
```  2575   then have *: "(\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n)))) sums - cos 2"
```
```  2576     by simp
```
```  2577   then have **: "summable (\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2578     by (rule sums_summable)
```
```  2579   have "0 < (\<Sum>n<Suc (Suc (Suc 0)). - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2580     by (simp add: fact_num_eq_if_nat realpow_num_eq_if)
```
```  2581   moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1) ^ n  * 2 ^ (2 * n) / real (fact (2 * n))))
```
```  2582     < (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2583   proof -
```
```  2584     { fix d
```
```  2585       have "4 * real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))
```
```  2586        < real (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))) *
```
```  2587            fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))"
```
```  2588         by (simp only: real_of_nat_mult) (auto intro!: mult_strict_mono fact_less_mono_nat)
```
```  2589       then have "4 * real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))
```
```  2590         < real (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))))"
```
```  2591         by (simp only: fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
```
```  2592       then have "4 * inverse (real (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))))
```
```  2593         < inverse (real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))"
```
```  2594         by (simp add: inverse_eq_divide less_divide_eq)
```
```  2595     }
```
```  2596     note *** = this
```
```  2597     have [simp]: "\<And>x y::real. 0 < x - y \<longleftrightarrow> y < x" by arith
```
```  2598     from ** show ?thesis by (rule sumr_pos_lt_pair)
```
```  2599       (simp add: divide_inverse mult.assoc [symmetric] ***)
```
```  2600   qed
```
```  2601   ultimately have "0 < (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2602     by (rule order_less_trans)
```
```  2603   moreover from * have "- cos 2 = (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
```
```  2604     by (rule sums_unique)
```
```  2605   ultimately have "0 < - cos 2" by simp
```
```  2606   then show ?thesis by simp
```
```  2607 qed
```
```  2608
```
```  2609 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
```
```  2610 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
```
```  2611
```
```  2612 lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 \<and> cos x = 0"
```
```  2613 proof (rule ex_ex1I)
```
```  2614   show "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0"
```
```  2615     by (rule IVT2, simp_all)
```
```  2616 next
```
```  2617   fix x y
```
```  2618   assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
```
```  2619   assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
```
```  2620   have [simp]: "\<forall>x. cos differentiable (at x)"
```
```  2621     unfolding real_differentiable_def by (auto intro: DERIV_cos)
```
```  2622   from x y show "x = y"
```
```  2623     apply (cut_tac less_linear [of x y], auto)
```
```  2624     apply (drule_tac f = cos in Rolle)
```
```  2625     apply (drule_tac [5] f = cos in Rolle)
```
```  2626     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  2627     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  2628     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  2629     done
```
```  2630 qed
```
```  2631
```
```  2632 lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
```
```  2633   by (simp add: pi_def)
```
```  2634
```
```  2635 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
```
```  2636   by (simp add: pi_half cos_is_zero [THEN theI'])
```
```  2637
```
```  2638 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
```
```  2639   apply (rule order_le_neq_trans)
```
```  2640   apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  2641   apply (metis cos_pi_half cos_zero zero_neq_one)
```
```  2642   done
```
```  2643
```
```  2644 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
```
```  2645 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
```
```  2646
```
```  2647 lemma pi_half_less_two [simp]: "pi / 2 < 2"
```
```  2648   apply (rule order_le_neq_trans)
```
```  2649   apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  2650   apply (metis cos_pi_half cos_two_neq_zero)
```
```  2651   done
```
```  2652
```
```  2653 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
```
```  2654 lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
```
```  2655
```
```  2656 lemma pi_gt_zero [simp]: "0 < pi"
```
```  2657   using pi_half_gt_zero by simp
```
```  2658
```
```  2659 lemma pi_ge_zero [simp]: "0 \<le> pi"
```
```  2660   by (rule pi_gt_zero [THEN order_less_imp_le])
```
```  2661
```
```  2662 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
```
```  2663   by (rule pi_gt_zero [THEN less_imp_neq, symmetric])
```
```  2664
```
```  2665 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
```
```  2666   by (simp add: linorder_not_less)
```
```  2667
```
```  2668 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
```
```  2669   by simp
```
```  2670
```
```  2671 lemma m2pi_less_pi: "- (2 * pi) < pi"
```
```  2672   by simp
```
```  2673
```
```  2674 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
```
```  2675   using sin_cos_squared_add2 [where x = "pi/2"]
```
```  2676   using sin_gt_zero [OF pi_half_gt_zero pi_half_less_two]
```
```  2677   by (simp add: power2_eq_1_iff)
```
```  2678
```
```  2679 lemma cos_pi [simp]: "cos pi = -1"
```
```  2680   using cos_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  2681
```
```  2682 lemma sin_pi [simp]: "sin pi = 0"
```
```  2683   using sin_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  2684
```
```  2685 lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
```
```  2686   by (simp add: cos_diff)
```
```  2687
```
```  2688 lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
```
```  2689   by (simp add: cos_add)
```
```  2690
```
```  2691 lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
```
```  2692   by (simp add: sin_diff)
```
```  2693
```
```  2694 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
```
```  2695   by (simp add: sin_add)
```
```  2696
```
```  2697 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
```
```  2698   by (simp add: sin_add)
```
```  2699
```
```  2700 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
```
```  2701   by (simp add: cos_add)
```
```  2702
```
```  2703 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
```
```  2704   by (simp add: sin_add cos_double)
```
```  2705
```
```  2706 lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
```
```  2707   by (simp add: cos_add cos_double)
```
```  2708
```
```  2709 lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n"
```
```  2710   by (induct n) (auto simp add: real_of_nat_Suc distrib_right)
```
```  2711
```
```  2712 lemma cos_npi2 [simp]: "cos (pi * real n) = (- 1) ^ n"
```
```  2713   by (metis cos_npi mult.commute)
```
```  2714
```
```  2715 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
```
```  2716   by (induct n) (auto simp add: real_of_nat_Suc distrib_right)
```
```  2717
```
```  2718 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
```
```  2719   by (simp add: mult.commute [of pi])
```
```  2720
```
```  2721 lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
```
```  2722   by (simp add: cos_double)
```
```  2723
```
```  2724 lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
```
```  2725   by simp
```
```  2726
```
```  2727 lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
```
```  2728   by (metis sin_gt_zero order_less_trans pi_half_less_two)
```
```  2729
```
```  2730 lemma sin_less_zero:
```
```  2731   assumes "- pi/2 < x" and "x < 0"
```
```  2732   shows "sin x < 0"
```
```  2733 proof -
```
```  2734   have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
```
```  2735   thus ?thesis by simp
```
```  2736 qed
```
```  2737
```
```  2738 lemma pi_less_4: "pi < 4"
```
```  2739   using pi_half_less_two by auto
```
```  2740
```
```  2741 lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
```
```  2742   apply (cut_tac pi_less_4)
```
```  2743   apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
```
```  2744   apply (cut_tac cos_is_zero, safe)
```
```  2745   apply (rename_tac y z)
```
```  2746   apply (drule_tac x = y in spec)
```
```  2747   apply (drule_tac x = "pi/2" in spec, simp)
```
```  2748   done
```
```  2749
```
```  2750 lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
```
```  2751   apply (rule_tac x = x and y = 0 in linorder_cases)
```
```  2752   apply (metis cos_gt_zero cos_minus minus_less_iff neg_0_less_iff_less)
```
```  2753   apply (auto intro: cos_gt_zero)
```
```  2754   done
```
```  2755
```
```  2756 lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
```
```  2757   apply (auto simp add: order_le_less cos_gt_zero_pi)
```
```  2758   apply (subgoal_tac "x = pi/2", auto)
```
```  2759   done
```
```  2760
```
```  2761 lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
```
```  2762   by (simp add: sin_cos_eq cos_gt_zero_pi)
```
```  2763
```
```  2764 lemma pi_ge_two: "2 \<le> pi"
```
```  2765 proof (rule ccontr)
```
```  2766   assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
```
```  2767   have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
```
```  2768   proof (cases "2 < 2 * pi")
```
```  2769     case True with dense[OF `pi < 2`] show ?thesis by auto
```
```  2770   next
```
```  2771     case False have "pi < 2 * pi" by auto
```
```  2772     from dense[OF this] and False show ?thesis by auto
```
```  2773   qed
```
```  2774   then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
```
```  2775   hence "0 < sin y" using sin_gt_zero by auto
```
```  2776   moreover
```
```  2777   have "sin y < 0" using sin_gt_zero_pi[of "y - pi"] `pi < y` and `y < 2 * pi` sin_periodic_pi[of "y - pi"] by auto
```
```  2778   ultimately show False by auto
```
```  2779 qed
```
```  2780
```
```  2781 lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
```
```  2782   by (auto simp add: order_le_less sin_gt_zero_pi)
```
```  2783
```
```  2784 text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}.
```
```  2785   It should be possible to factor out some of the common parts. *}
```
```  2786
```
```  2787 lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
```
```  2788 proof (rule ex_ex1I)
```
```  2789   assume y: "-1 \<le> y" "y \<le> 1"
```
```  2790   show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
```
```  2791     by (rule IVT2, simp_all add: y)
```
```  2792 next
```
```  2793   fix a b
```
```  2794   assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
```
```  2795   assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
```
```  2796   have [simp]: "\<forall>x. cos differentiable (at x)"
```
```  2797     unfolding real_differentiable_def by (auto intro: DERIV_cos)
```
```  2798   from a b show "a = b"
```
```  2799     apply (cut_tac less_linear [of a b], auto)
```
```  2800     apply (drule_tac f = cos in Rolle)
```
```  2801     apply (drule_tac [5] f = cos in Rolle)
```
```  2802     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  2803     apply (metis order_less_le_trans less_le sin_gt_zero_pi)
```
```  2804     apply (metis order_less_le_trans less_le sin_gt_zero_pi)
```
```  2805     done
```
```  2806 qed
```
```  2807
```
```  2808 lemma sin_total:
```
```  2809      "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
```
```  2810 apply (rule ccontr)
```
```  2811 apply (subgoal_tac "\<forall>x. (- (pi/2) \<le> x & x \<le> pi/2 & (sin x = y)) = (0 \<le> (x + pi/2) & (x + pi/2) \<le> pi & (cos (x + pi/2) = -y))")
```
```  2812 apply (erule contrapos_np)
```
```  2813 apply simp
```
```  2814 apply (cut_tac y="-y" in cos_total, simp) apply simp
```
```  2815 apply (erule ex1E)
```
```  2816 apply (rule_tac a = "x - (pi/2)" in ex1I)
```
```  2817 apply (simp (no_asm) add: add.assoc)
```
```  2818 apply (rotate_tac 3)
```
```  2819 apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all add: cos_add)
```
```  2820 done
```
```  2821
```
```  2822 lemma reals_Archimedean4:
```
```  2823      "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
```
```  2824 apply (auto dest!: reals_Archimedean3)
```
```  2825 apply (drule_tac x = x in spec, clarify)
```
```  2826 apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
```
```  2827  prefer 2 apply (erule LeastI)
```
```  2828 apply (case_tac "LEAST m::nat. x < real m * y", simp)
```
```  2829 apply (rename_tac m)
```
```  2830 apply (subgoal_tac "~ x < real m * y")
```
```  2831  prefer 2 apply (rule not_less_Least, simp, force)
```
```  2832 done
```
```  2833
```
```  2834 (* Pre Isabelle99-2 proof was simpler- numerals arithmetic
```
```  2835    now causes some unwanted re-arrangements of literals!   *)
```
```  2836 lemma cos_zero_lemma:
```
```  2837      "[| 0 \<le> x; cos x = 0 |] ==>
```
```  2838       \<exists>n::nat. ~even n & x = real n * (pi/2)"
```
```  2839 apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
```
```  2840 apply (subgoal_tac "0 \<le> x - real n * pi &
```
```  2841                     (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
```
```  2842 apply (auto simp add: algebra_simps real_of_nat_Suc)
```
```  2843  prefer 2 apply (simp add: cos_diff)
```
```  2844 apply (simp add: cos_diff)
```
```  2845 apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
```
```  2846 apply (rule_tac [2] cos_total, safe)
```
```  2847 apply (drule_tac x = "x - real n * pi" in spec)
```
```  2848 apply (drule_tac x = "pi/2" in spec)
```
```  2849 apply (simp add: cos_diff)
```
```  2850 apply (rule_tac x = "Suc (2 * n)" in exI)
```
```  2851 apply (simp add: real_of_nat_Suc algebra_simps, auto)
```
```  2852 done
```
```  2853
```
```  2854 lemma sin_zero_lemma:
```
```  2855      "[| 0 \<le> x; sin x = 0 |] ==>
```
```  2856       \<exists>n::nat. even n & x = real n * (pi/2)"
```
```  2857 apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
```
```  2858  apply (clarify, rule_tac x = "n - 1" in exI)
```
```  2859  apply (auto elim!: oddE simp add: real_of_nat_Suc field_simps)[1]
```
```  2860  apply (rule cos_zero_lemma)
```
```  2861  apply (auto simp add: cos_add)
```
```  2862 done
```
```  2863
```
```  2864 lemma cos_zero_iff:
```
```  2865      "(cos x = 0) =
```
```  2866       ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
```
```  2867        (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
```
```  2868 proof -
```
```  2869   { fix n :: nat
```
```  2870     assume "odd n"
```
```  2871     then obtain m where "n = 2 * m + 1" ..
```
```  2872     then have "cos (real n * pi / 2) = 0"
```
```  2873       by (simp add: field_simps real_of_nat_Suc) (simp add: cos_add add_divide_distrib)
```
```  2874   } note * = this
```
```  2875   show ?thesis
```
```  2876   apply (rule iffI)
```
```  2877   apply (cut_tac linorder_linear [of 0 x], safe)
```
```  2878   apply (drule cos_zero_lemma, assumption+)
```
```  2879   apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
```
```  2880   apply (auto dest: *)
```
```  2881   done
```
```  2882 qed
```
```  2883
```
```  2884 (* ditto: but to a lesser extent *)
```
```  2885 lemma sin_zero_iff:
```
```  2886      "(sin x = 0) =
```
```  2887       ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
```
```  2888        (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
```
```  2889 apply (rule iffI)
```
```  2890 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  2891 apply (drule sin_zero_lemma, assumption+)
```
```  2892 apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
```
```  2893 apply (force simp add: minus_equation_iff [of x])
```
```  2894 apply (auto elim: evenE)
```
```  2895 done
```
```  2896
```
```  2897 lemma cos_monotone_0_pi:
```
```  2898   assumes "0 \<le> y" and "y < x" and "x \<le> pi"
```
```  2899   shows "cos x < cos y"
```
```  2900 proof -
```
```  2901   have "- (x - y) < 0" using assms by auto
```
```  2902
```
```  2903   from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
```
```  2904   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z"
```
```  2905     by auto
```
```  2906   hence "0 < z" and "z < pi" using assms by auto
```
```  2907   hence "0 < sin z" using sin_gt_zero_pi by auto
```
```  2908   hence "cos x - cos y < 0"
```
```  2909     unfolding cos_diff minus_mult_commute[symmetric]
```
```  2910     using `- (x - y) < 0` by (rule mult_pos_neg2)
```
```  2911   thus ?thesis by auto
```
```  2912 qed
```
```  2913
```
```  2914 lemma cos_monotone_0_pi':
```
```  2915   assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi"
```
```  2916   shows "cos x \<le> cos y"
```
```  2917 proof (cases "y < x")
```
```  2918   case True
```
```  2919   show ?thesis
```
```  2920     using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
```
```  2921 next
```
```  2922   case False
```
```  2923   hence "y = x" using `y \<le> x` by auto
```
```  2924   thus ?thesis by auto
```
```  2925 qed
```
```  2926
```
```  2927 lemma cos_monotone_minus_pi_0:
```
```  2928   assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
```
```  2929   shows "cos y < cos x"
```
```  2930 proof -
```
```  2931   have "0 \<le> -x" and "-x < -y" and "-y \<le> pi"
```
```  2932     using assms by auto
```
```  2933   from cos_monotone_0_pi[OF this] show ?thesis
```
```  2934     unfolding cos_minus .
```
```  2935 qed
```
```  2936
```
```  2937 lemma cos_monotone_minus_pi_0':
```
```  2938   assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0"
```
```  2939   shows "cos y \<le> cos x"
```
```  2940 proof (cases "y < x")
```
```  2941   case True
```
```  2942   show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`]
```
```  2943     by auto
```
```  2944 next
```
```  2945   case False
```
```  2946   hence "y = x" using `y \<le> x` by auto
```
```  2947   thus ?thesis by auto
```
```  2948 qed
```
```  2949
```
```  2950 lemma sin_monotone_2pi':
```
```  2951   assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2"
```
```  2952   shows "sin y \<le> sin x"
```
```  2953 proof -
```
```  2954   have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi"
```
```  2955     using pi_ge_two and assms by auto
```
```  2956   from cos_monotone_0_pi'[OF this] show ?thesis
```
```  2957     unfolding minus_sin_cos_eq[symmetric] by auto
```
```  2958 qed
```
```  2959
```
```  2960
```
```  2961 subsection {* Tangent *}
```
```  2962
```
```  2963 definition tan :: "real \<Rightarrow> real"
```
```  2964   where "tan = (\<lambda>x. sin x / cos x)"
```
```  2965
```
```  2966 lemma tan_zero [simp]: "tan 0 = 0"
```
```  2967   by (simp add: tan_def)
```
```  2968
```
```  2969 lemma tan_pi [simp]: "tan pi = 0"
```
```  2970   by (simp add: tan_def)
```
```  2971
```
```  2972 lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
```
```  2973   by (simp add: tan_def)
```
```  2974
```
```  2975 lemma tan_minus [simp]: "tan (-x) = - tan x"
```
```  2976   by (simp add: tan_def)
```
```  2977
```
```  2978 lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
```
```  2979   by (simp add: tan_def)
```
```  2980
```
```  2981 lemma lemma_tan_add1:
```
```  2982   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
```
```  2983   by (simp add: tan_def cos_add field_simps)
```
```  2984
```
```  2985 lemma add_tan_eq:
```
```  2986   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
```
```  2987   by (simp add: tan_def sin_add field_simps)
```
```  2988
```
```  2989 lemma tan_add:
```
```  2990      "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
```
```  2991       ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
```
```  2992   by (simp add: add_tan_eq lemma_tan_add1, simp add: tan_def)
```
```  2993
```
```  2994 lemma tan_double:
```
```  2995      "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
```
```  2996       ==> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)"
```
```  2997   using tan_add [of x x] by (simp add: power2_eq_square)
```
```  2998
```
```  2999 lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
```
```  3000   by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
```
```  3001
```
```  3002 lemma tan_less_zero:
```
```  3003   assumes lb: "- pi/2 < x" and "x < 0"
```
```  3004   shows "tan x < 0"
```
```  3005 proof -
```
```  3006   have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
```
```  3007   thus ?thesis by simp
```
```  3008 qed
```
```  3009
```
```  3010 lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"
```
```  3011   unfolding tan_def sin_double cos_double sin_squared_eq
```
```  3012   by (simp add: power2_eq_square)
```
```  3013
```
```  3014 lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)"
```
```  3015   unfolding tan_def
```
```  3016   by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square)
```
```  3017
```
```  3018 lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
```
```  3019   by (rule DERIV_tan [THEN DERIV_isCont])
```
```  3020
```
```  3021 lemma isCont_tan' [simp]:
```
```  3022   "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
```
```  3023   by (rule isCont_o2 [OF _ isCont_tan])
```
```  3024
```
```  3025 lemma tendsto_tan [tendsto_intros]:
```
```  3026   "\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F"
```
```  3027   by (rule isCont_tendsto_compose [OF isCont_tan])
```
```  3028
```
```  3029 lemma continuous_tan:
```
```  3030   "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))"
```
```  3031   unfolding continuous_def by (rule tendsto_tan)
```
```  3032
```
```  3033 lemma isCont_tan'' [continuous_intros]:
```
```  3034   "continuous (at x) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. tan (f x))"
```
```  3035   unfolding continuous_at by (rule tendsto_tan)
```
```  3036
```
```  3037 lemma continuous_within_tan [continuous_intros]:
```
```  3038   "continuous (at x within s) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
```
```  3039   unfolding continuous_within by (rule tendsto_tan)
```
```  3040
```
```  3041 lemma continuous_on_tan [continuous_intros]:
```
```  3042   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))"
```
```  3043   unfolding continuous_on_def by (auto intro: tendsto_tan)
```
```  3044
```
```  3045 lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) -- pi/2 --> 0"
```
```  3046   by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
```
```  3047
```
```  3048 lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
```
```  3049   apply (cut_tac LIM_cos_div_sin)
```
```  3050   apply (simp only: LIM_eq)
```
```  3051   apply (drule_tac x = "inverse y" in spec, safe, force)
```
```  3052   apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
```
```  3053   apply (rule_tac x = "(pi/2) - e" in exI)
```
```  3054   apply (simp (no_asm_simp))
```
```  3055   apply (drule_tac x = "(pi/2) - e" in spec)
```
```  3056   apply (auto simp add: tan_def sin_diff cos_diff)
```
```  3057   apply (rule inverse_less_iff_less [THEN iffD1])
```
```  3058   apply (auto simp add: divide_inverse)
```
```  3059   apply (rule mult_pos_pos)
```
```  3060   apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
```
```  3061   apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult.commute)
```
```  3062   done
```
```  3063
```
```  3064 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
```
```  3065   apply (frule order_le_imp_less_or_eq, safe)
```
```  3066    prefer 2 apply force
```
```  3067   apply (drule lemma_tan_total, safe)
```
```  3068   apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
```
```  3069   apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
```
```  3070   apply (drule_tac y = xa in order_le_imp_less_or_eq)
```
```  3071   apply (auto dest: cos_gt_zero)
```
```  3072   done
```
```  3073
```
```  3074 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  3075   apply (cut_tac linorder_linear [of 0 y], safe)
```
```  3076   apply (drule tan_total_pos)
```
```  3077   apply (cut_tac [2] y="-y" in tan_total_pos, safe)
```
```  3078   apply (rule_tac [3] x = "-x" in exI)
```
```  3079   apply (auto del: exI intro!: exI)
```
```  3080   done
```
```  3081
```
```  3082 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  3083   apply (cut_tac y = y in lemma_tan_total1, auto)
```
```  3084   apply hypsubst_thin
```
```  3085   apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
```
```  3086   apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
```
```  3087   apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
```
```  3088   apply (rule_tac [4] Rolle)
```
```  3089   apply (rule_tac [2] Rolle)
```
```  3090   apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
```
```  3091               simp add: real_differentiable_def)
```
```  3092   txt{*Now, simulate TRYALL*}
```
```  3093   apply (rule_tac [!] DERIV_tan asm_rl)
```
```  3094   apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
```
```  3095               simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
```
```  3096   done
```
```  3097
```
```  3098 lemma tan_monotone:
```
```  3099   assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
```
```  3100   shows "tan y < tan x"
```
```  3101 proof -
```
```  3102   have "\<forall>x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse ((cos x')\<^sup>2)"
```
```  3103   proof (rule allI, rule impI)
```
```  3104     fix x' :: real
```
```  3105     assume "y \<le> x' \<and> x' \<le> x"
```
```  3106     hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
```
```  3107     from cos_gt_zero_pi[OF this]
```
```  3108     have "cos x' \<noteq> 0" by auto
```
```  3109     thus "DERIV tan x' :> inverse ((cos x')\<^sup>2)" by (rule DERIV_tan)
```
```  3110   qed
```
```  3111   from MVT2[OF `y < x` this]
```
```  3112   obtain z where "y < z" and "z < x"
```
```  3113     and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
```
```  3114   hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
```
```  3115   hence "0 < cos z" using cos_gt_zero_pi by auto
```
```  3116   hence inv_pos: "0 < inverse ((cos z)\<^sup>2)" by auto
```
```  3117   have "0 < x - y" using `y < x` by auto
```
```  3118   with inv_pos have "0 < tan x - tan y" unfolding tan_diff by auto
```
```  3119   thus ?thesis by auto
```
```  3120 qed
```
```  3121
```
```  3122 lemma tan_monotone':
```
```  3123   assumes "- (pi / 2) < y"
```
```  3124     and "y < pi / 2"
```
```  3125     and "- (pi / 2) < x"
```
```  3126     and "x < pi / 2"
```
```  3127   shows "(y < x) = (tan y < tan x)"
```
```  3128 proof
```
```  3129   assume "y < x"
```
```  3130   thus "tan y < tan x"
```
```  3131     using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
```
```  3132 next
```
```  3133   assume "tan y < tan x"
```
```  3134   show "y < x"
```
```  3135   proof (rule ccontr)
```
```  3136     assume "\<not> y < x" hence "x \<le> y" by auto
```
```  3137     hence "tan x \<le> tan y"
```
```  3138     proof (cases "x = y")
```
```  3139       case True thus ?thesis by auto
```
```  3140     next
```
```  3141       case False hence "x < y" using `x \<le> y` by auto
```
```  3142       from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
```
```  3143     qed
```
```  3144     thus False using `tan y < tan x` by auto
```
```  3145   qed
```
```  3146 qed
```
```  3147
```
```  3148 lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)"
```
```  3149   unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
```
```  3150
```
```  3151 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
```
```  3152   by (simp add: tan_def)
```
```  3153
```
```  3154 lemma tan_periodic_nat[simp]:
```
```  3155   fixes n :: nat
```
```  3156   shows "tan (x + real n * pi) = tan x"
```
```  3157 proof (induct n arbitrary: x)
```
```  3158   case 0
```
```  3159   then show ?case by simp
```
```  3160 next
```
```  3161   case (Suc n)
```
```  3162   have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi"
```
```  3163     unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
```
```  3164   show ?case unfolding split_pi_off using Suc by auto
```
```  3165 qed
```
```  3166
```
```  3167 lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
```
```  3168 proof (cases "0 \<le> i")
```
```  3169   case True
```
```  3170   hence i_nat: "real i = real (nat i)" by auto
```
```  3171   show ?thesis unfolding i_nat by auto
```
```  3172 next
```
```  3173   case False
```
```  3174   hence i_nat: "real i = - real (nat (-i))" by auto
```
```  3175   have "tan x = tan (x + real i * pi - real i * pi)"
```
```  3176     by auto
```
```  3177   also have "\<dots> = tan (x + real i * pi)"
```
```  3178     unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
```
```  3179   finally show ?thesis by auto
```
```  3180 qed
```
```  3181
```
```  3182 lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
```
```  3183   using tan_periodic_int[of _ "numeral n" ] unfolding real_numeral .
```
```  3184
```
```  3185 subsection {* Inverse Trigonometric Functions *}
```
```  3186
```
```  3187 definition arcsin :: "real => real"
```
```  3188   where "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
```
```  3189
```
```  3190 definition arccos :: "real => real"
```
```  3191   where "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
```
```  3192
```
```  3193 definition arctan :: "real => real"
```
```  3194   where "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
```
```  3195
```
```  3196 lemma arcsin:
```
```  3197   "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow>
```
```  3198     -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2 & sin(arcsin y) = y"
```
```  3199   unfolding arcsin_def by (rule theI' [OF sin_total])
```
```  3200
```
```  3201 lemma arcsin_pi:
```
```  3202   "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
```
```  3203   apply (drule (1) arcsin)
```
```  3204   apply (force intro: order_trans)
```
```  3205   done
```
```  3206
```
```  3207 lemma sin_arcsin [simp]: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin(arcsin y) = y"
```
```  3208   by (blast dest: arcsin)
```
```  3209
```
```  3210 lemma arcsin_bounded: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
```
```  3211   by (blast dest: arcsin)
```
```  3212
```
```  3213 lemma arcsin_lbound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y"
```
```  3214   by (blast dest: arcsin)
```
```  3215
```
```  3216 lemma arcsin_ubound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
```
```  3217   by (blast dest: arcsin)
```
```  3218
```
```  3219 lemma arcsin_lt_bounded:
```
```  3220      "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
```
```  3221   apply (frule order_less_imp_le)
```
```  3222   apply (frule_tac y = y in order_less_imp_le)
```
```  3223   apply (frule arcsin_bounded)
```
```  3224   apply (safe, simp)
```
```  3225   apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
```
```  3226   apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
```
```  3227   apply (drule_tac [!] f = sin in arg_cong, auto)
```
```  3228   done
```
```  3229
```
```  3230 lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
```
```  3231   apply (unfold arcsin_def)
```
```  3232   apply (rule the1_equality)
```
```  3233   apply (rule sin_total, auto)
```
```  3234   done
```
```  3235
```
```  3236 lemma arccos:
```
```  3237      "[| -1 \<le> y; y \<le> 1 |]
```
```  3238       ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
```
```  3239   unfolding arccos_def by (rule theI' [OF cos_total])
```
```  3240
```
```  3241 lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
```
```  3242   by (blast dest: arccos)
```
```  3243
```
```  3244 lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
```
```  3245   by (blast dest: arccos)
```
```  3246
```
```  3247 lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
```
```  3248   by (blast dest: arccos)
```
```  3249
```
```  3250 lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
```
```  3251   by (blast dest: arccos)
```
```  3252
```
```  3253 lemma arccos_lt_bounded:
```
```  3254      "[| -1 < y; y < 1 |]
```
```  3255       ==> 0 < arccos y & arccos y < pi"
```
```  3256   apply (frule order_less_imp_le)
```
```  3257   apply (frule_tac y = y in order_less_imp_le)
```
```  3258   apply (frule arccos_bounded, auto)
```
```  3259   apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
```
```  3260   apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
```
```  3261   apply (drule_tac [!] f = cos in arg_cong, auto)
```
```  3262   done
```
```  3263
```
```  3264 lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
```
```  3265   apply (simp add: arccos_def)
```
```  3266   apply (auto intro!: the1_equality cos_total)
```
```  3267   done
```
```  3268
```
```  3269 lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
```
```  3270   apply (simp add: arccos_def)
```
```  3271   apply (auto intro!: the1_equality cos_total)
```
```  3272   done
```
```  3273
```
```  3274 lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
```
```  3275   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  3276   apply (rule power2_eq_imp_eq)
```
```  3277   apply (simp add: cos_squared_eq)
```
```  3278   apply (rule cos_ge_zero)
```
```  3279   apply (erule (1) arcsin_lbound)
```
```  3280   apply (erule (1) arcsin_ubound)
```
```  3281   apply simp
```
```  3282   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  3283   apply (rule power_mono, simp, simp)
```
```  3284   done
```
```  3285
```
```  3286 lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
```
```  3287   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  3288   apply (rule power2_eq_imp_eq)
```
```  3289   apply (simp add: sin_squared_eq)
```
```  3290   apply (rule sin_ge_zero)
```
```  3291   apply (erule (1) arccos_lbound)
```
```  3292   apply (erule (1) arccos_ubound)
```
```  3293   apply simp
```
```  3294   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  3295   apply (rule power_mono, simp, simp)
```
```  3296   done
```
```  3297
```
```  3298 lemma arctan [simp]: "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
```
```  3299   unfolding arctan_def by (rule theI' [OF tan_total])
```
```  3300
```
```  3301 lemma tan_arctan: "tan (arctan y) = y"
```
```  3302   by auto
```
```  3303
```
```  3304 lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
```
```  3305   by (auto simp only: arctan)
```
```  3306
```
```  3307 lemma arctan_lbound: "- (pi/2) < arctan y"
```
```  3308   by auto
```
```  3309
```
```  3310 lemma arctan_ubound: "arctan y < pi/2"
```
```  3311   by (auto simp only: arctan)
```
```  3312
```
```  3313 lemma arctan_unique:
```
```  3314   assumes "-(pi/2) < x"
```
```  3315     and "x < pi/2"
```
```  3316     and "tan x = y"
```
```  3317   shows "arctan y = x"
```
```  3318   using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
```
```  3319
```
```  3320 lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x"
```
```  3321   by (rule arctan_unique) simp_all
```
```  3322
```
```  3323 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
```
```  3324   by (rule arctan_unique) simp_all
```
```  3325
```
```  3326 lemma arctan_minus: "arctan (- x) = - arctan x"
```
```  3327   apply (rule arctan_unique)
```
```  3328   apply (simp only: neg_less_iff_less arctan_ubound)
```
```  3329   apply (metis minus_less_iff arctan_lbound)
```
```  3330   apply simp
```
```  3331   done
```
```  3332
```
```  3333 lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
```
```  3334   by (intro less_imp_neq [symmetric] cos_gt_zero_pi
```
```  3335     arctan_lbound arctan_ubound)
```
```  3336
```
```  3337 lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)"
```
```  3338 proof (rule power2_eq_imp_eq)
```
```  3339   have "0 < 1 + x\<^sup>2" by (simp add: add_pos_nonneg)
```
```  3340   show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp
```
```  3341   show "0 \<le> cos (arctan x)"
```
```  3342     by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
```
```  3343   have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1"
```
```  3344     unfolding tan_def by (simp add: distrib_left power_divide)
```
```  3345   thus "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
```
```  3346     using `0 < 1 + x\<^sup>2` by (simp add: power_divide eq_divide_eq)
```
```  3347 qed
```
```  3348
```
```  3349 lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)"
```
```  3350   using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
```
```  3351   using tan_arctan [of x] unfolding tan_def cos_arctan
```
```  3352   by (simp add: eq_divide_eq)
```
```  3353
```
```  3354 lemma tan_sec: "cos x \<noteq> 0 ==> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
```
```  3355   apply (rule power_inverse [THEN subst])
```
```  3356   apply (rule_tac c1 = "(cos x)\<^sup>2" in mult_right_cancel [THEN iffD1])
```
```  3357   apply (auto dest: field_power_not_zero
```
```  3358           simp add: power_mult_distrib distrib_right power_divide tan_def
```
```  3359                     mult.assoc power_inverse [symmetric])
```
```  3360   done
```
```  3361
```
```  3362 lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
```
```  3363   by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
```
```  3364
```
```  3365 lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
```
```  3366   by (simp only: not_less [symmetric] arctan_less_iff)
```
```  3367
```
```  3368 lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
```
```  3369   by (simp only: eq_iff [where 'a=real] arctan_le_iff)
```
```  3370
```
```  3371 lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
```
```  3372   using arctan_less_iff [of 0 x] by simp
```
```  3373
```
```  3374 lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
```
```  3375   using arctan_less_iff [of x 0] by simp
```
```  3376
```
```  3377 lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
```
```  3378   using arctan_le_iff [of 0 x] by simp
```
```  3379
```
```  3380 lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```  3381   using arctan_le_iff [of x 0] by simp
```
```  3382
```
```  3383 lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
```
```  3384   using arctan_eq_iff [of x 0] by simp
```
```  3385
```
```  3386 lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin"
```
```  3387 proof -
```
```  3388   have "continuous_on (sin ` {- pi / 2 .. pi / 2}) arcsin"
```
```  3389     by (rule continuous_on_inv) (auto intro: continuous_intros simp: arcsin_sin)
```
```  3390   also have "sin ` {- pi / 2 .. pi / 2} = {-1 .. 1}"
```
```  3391   proof safe
```
```  3392     fix x :: real
```
```  3393     assume "x \<in> {-1..1}"
```
```  3394     then show "x \<in> sin ` {- pi / 2..pi / 2}"
```
```  3395       using arcsin_lbound arcsin_ubound
```
```  3396       by (intro image_eqI[where x="arcsin x"]) auto
```
```  3397   qed simp
```
```  3398   finally show ?thesis .
```
```  3399 qed
```
```  3400
```
```  3401 lemma continuous_on_arcsin [continuous_intros]:
```
```  3402   "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))"
```
```  3403   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arcsin']]
```
```  3404   by (auto simp: comp_def subset_eq)
```
```  3405
```
```  3406 lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x"
```
```  3407   using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  3408   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  3409
```
```  3410 lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos"
```
```  3411 proof -
```
```  3412   have "continuous_on (cos ` {0 .. pi}) arccos"
```
```  3413     by (rule continuous_on_inv) (auto intro: continuous_intros simp: arccos_cos)
```
```  3414   also have "cos ` {0 .. pi} = {-1 .. 1}"
```
```  3415   proof safe
```
```  3416     fix x :: real
```
```  3417     assume "x \<in> {-1..1}"
```
```  3418     then show "x \<in> cos ` {0..pi}"
```
```  3419       using arccos_lbound arccos_ubound
```
```  3420       by (intro image_eqI[where x="arccos x"]) auto
```
```  3421   qed simp
```
```  3422   finally show ?thesis .
```
```  3423 qed
```
```  3424
```
```  3425 lemma continuous_on_arccos [continuous_intros]:
```
```  3426   "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))"
```
```  3427   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arccos']]
```
```  3428   by (auto simp: comp_def subset_eq)
```
```  3429
```
```  3430 lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x"
```
```  3431   using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  3432   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  3433
```
```  3434 lemma isCont_arctan: "isCont arctan x"
```
```  3435   apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
```
```  3436   apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
```
```  3437   apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
```
```  3438   apply (erule (1) isCont_inverse_function2 [where f=tan])
```
```  3439   apply (metis arctan_tan order_le_less_trans order_less_le_trans)
```
```  3440   apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
```
```  3441   done
```
```  3442
```
```  3443 lemma tendsto_arctan [tendsto_intros]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) ---> arctan x) F"
```
```  3444   by (rule isCont_tendsto_compose [OF isCont_arctan])
```
```  3445
```
```  3446 lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))"
```
```  3447   unfolding continuous_def by (rule tendsto_arctan)
```
```  3448
```
```  3449 lemma continuous_on_arctan [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
```
```  3450   unfolding continuous_on_def by (auto intro: tendsto_arctan)
```
```  3451
```
```  3452 lemma DERIV_arcsin:
```
```  3453   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
```
```  3454   apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
```
```  3455   apply (rule DERIV_cong [OF DERIV_sin])
```
```  3456   apply (simp add: cos_arcsin)
```
```  3457   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  3458   apply (rule power_strict_mono, simp, simp, simp)
```
```  3459   apply assumption
```
```  3460   apply assumption
```
```  3461   apply simp
```
```  3462   apply (erule (1) isCont_arcsin)
```
```  3463   done
```
```  3464
```
```  3465 lemma DERIV_arccos:
```
```  3466   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
```
```  3467   apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
```
```  3468   apply (rule DERIV_cong [OF DERIV_cos])
```
```  3469   apply (simp add: sin_arccos)
```
```  3470   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  3471   apply (rule power_strict_mono, simp, simp, simp)
```
```  3472   apply assumption
```
```  3473   apply assumption
```
```  3474   apply simp
```
```  3475   apply (erule (1) isCont_arccos)
```
```  3476   done
```
```  3477
```
```  3478 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
```
```  3479   apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
```
```  3480   apply (rule DERIV_cong [OF DERIV_tan])
```
```  3481   apply (rule cos_arctan_not_zero)
```
```  3482   apply (simp add: power_inverse tan_sec [symmetric])
```
```  3483   apply (subgoal_tac "0 < 1 + x\<^sup>2", simp)
```
```  3484   apply (simp add: add_pos_nonneg)
```
```  3485   apply (simp, simp, simp, rule isCont_arctan)
```
```  3486   done
```
```  3487
```
```  3488 declare
```
```  3489   DERIV_arcsin[THEN DERIV_chain2, derivative_intros]
```
```  3490   DERIV_arccos[THEN DERIV_chain2, derivative_intros]
```
```  3491   DERIV_arctan[THEN DERIV_chain2, derivative_intros]
```
```  3492
```
```  3493 lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- pi/2))"
```
```  3494   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])
```
```  3495      (auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  3496            intro!: tan_monotone exI[of _ "pi/2"])
```
```  3497
```
```  3498 lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"
```
```  3499   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])
```
```  3500      (auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  3501            intro!: tan_monotone exI[of _ "pi/2"])
```
```  3502
```
```  3503 lemma tendsto_arctan_at_top: "(arctan ---> (pi/2)) at_top"
```
```  3504 proof (rule tendstoI)
```
```  3505   fix e :: real
```
```  3506   assume "0 < e"
```
```  3507   def y \<equiv> "pi/2 - min (pi/2) e"
```
```  3508   then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y"
```
```  3509     using `0 < e` by auto
```
```  3510
```
```  3511   show "eventually (\<lambda>x. dist (arctan x) (pi / 2) < e) at_top"
```
```  3512   proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)
```
```  3513     fix x
```
```  3514     assume "tan y < x"
```
```  3515     then have "arctan (tan y) < arctan x"
```
```  3516       by (simp add: arctan_less_iff)
```
```  3517     with y have "y < arctan x"
```
```  3518       by (subst (asm) arctan_tan) simp_all
```
```  3519     with arctan_ubound[of x, arith] y `0 < e`
```
```  3520     show "dist (arctan x) (pi / 2) < e"
```
```  3521       by (simp add: dist_real_def)
```
```  3522   qed
```
```  3523 qed
```
```  3524
```
```  3525 lemma tendsto_arctan_at_bot: "(arctan ---> - (pi/2)) at_bot"
```
```  3526   unfolding filterlim_at_bot_mirror arctan_minus
```
```  3527   by (intro tendsto_minus tendsto_arctan_at_top)
```
```  3528
```
```  3529
```
```  3530 subsection {* More Theorems about Sin and Cos *}
```
```  3531
```
```  3532 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
```
```  3533 proof -
```
```  3534   let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
```
```  3535   have nonneg: "0 \<le> ?c"
```
```  3536     by (simp add: cos_ge_zero)
```
```  3537   have "0 = cos (pi / 4 + pi / 4)"
```
```  3538     by simp
```
```  3539   also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2"
```
```  3540     by (simp only: cos_add power2_eq_square)
```
```  3541   also have "\<dots> = 2 * ?c\<^sup>2 - 1"
```
```  3542     by (simp add: sin_squared_eq)
```
```  3543   finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2"
```
```  3544     by (simp add: power_divide)
```
```  3545   thus ?thesis
```
```  3546     using nonneg by (rule power2_eq_imp_eq) simp
```
```  3547 qed
```
```  3548
```
```  3549 lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"
```
```  3550 proof -
```
```  3551   let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
```
```  3552   have pos_c: "0 < ?c"
```
```  3553     by (rule cos_gt_zero, simp, simp)
```
```  3554   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
```
```  3555     by simp
```
```  3556   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
```
```  3557     by (simp only: cos_add sin_add)
```
```  3558   also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)"
```
```  3559     by (simp add: algebra_simps power2_eq_square)
```
```  3560   finally have "?c\<^sup>2 = (sqrt 3 / 2)\<^sup>2"
```
```  3561     using pos_c by (simp add: sin_squared_eq power_divide)
```
```  3562   thus ?thesis
```
```  3563     using pos_c [THEN order_less_imp_le]
```
```  3564     by (rule power2_eq_imp_eq) simp
```
```  3565 qed
```
```  3566
```
```  3567 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
```
```  3568   by (simp add: sin_cos_eq cos_45)
```
```  3569
```
```  3570 lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
```
```  3571   by (simp add: sin_cos_eq cos_30)
```
```  3572
```
```  3573 lemma cos_60: "cos (pi / 3) = 1 / 2"
```
```  3574   apply (rule power2_eq_imp_eq)
```
```  3575   apply (simp add: cos_squared_eq sin_60 power_divide)
```
```  3576   apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
```
```  3577   done
```
```  3578
```
```  3579 lemma sin_30: "sin (pi / 6) = 1 / 2"
```
```  3580   by (simp add: sin_cos_eq cos_60)
```
```  3581
```
```  3582 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
```
```  3583   unfolding tan_def by (simp add: sin_30 cos_30)
```
```  3584
```
```  3585 lemma tan_45: "tan (pi / 4) = 1"
```
```  3586   unfolding tan_def by (simp add: sin_45 cos_45)
```
```  3587
```
```  3588 lemma tan_60: "tan (pi / 3) = sqrt 3"
```
```  3589   unfolding tan_def by (simp add: sin_60 cos_60)
```
```  3590
```
```  3591 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
```
```  3592 proof -
```
```  3593   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
```
```  3594     by (auto simp add: algebra_simps sin_add)
```
```  3595   thus ?thesis
```
```  3596     by (simp add: real_of_nat_Suc distrib_right add_divide_distrib
```
```  3597                   mult.commute [of pi])
```
```  3598 qed
```
```  3599
```
```  3600 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
```
```  3601   by (cases "even n") (simp_all add: cos_double mult.assoc)
```
```  3602
```
```  3603 lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
```
```  3604   apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
```
```  3605   apply (subst cos_add, simp)
```
```  3606   done
```
```  3607
```
```  3608 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
```
```  3609   by (auto simp add: mult.assoc)
```
```  3610
```
```  3611 lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
```
```  3612   apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
```
```  3613   apply (subst sin_add, simp)
```
```  3614   done
```
```  3615
```
```  3616 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
```
```  3617   apply (simp only: cos_add sin_add real_of_nat_Suc distrib_right distrib_left add_divide_distrib)
```
```  3618   apply auto
```
```  3619   done
```
```  3620
```
```  3621 lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)"
```
```  3622   by (auto intro!: derivative_eq_intros)
```
```  3623
```
```  3624 lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
```
```  3625   by (auto simp add: sin_zero_iff elim: evenE)
```
```  3626
```
```  3627 lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
```
```  3628   using sin_cos_squared_add3 [where x = x] by auto
```
```  3629
```
```  3630
```
```  3631 subsection {* Machins formula *}
```
```  3632
```
```  3633 lemma arctan_one: "arctan 1 = pi / 4"
```
```  3634   by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi)
```
```  3635
```
```  3636 lemma tan_total_pi4:
```
```  3637   assumes "\<bar>x\<bar> < 1"
```
```  3638   shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
```
```  3639 proof
```
```  3640   show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
```
```  3641     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  3642     unfolding arctan_less_iff using assms by auto
```
```  3643 qed
```
```  3644
```
```  3645 lemma arctan_add:
```
```  3646   assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
```
```  3647   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
```
```  3648 proof (rule arctan_unique [symmetric])
```
```  3649   have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y"
```
```  3650     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  3651     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  3652   from add_le_less_mono [OF this]
```
```  3653   show 1: "- (pi / 2) < arctan x + arctan y" by simp
```
```  3654   have "arctan x \<le> pi / 4" and "arctan y < pi / 4"
```
```  3655     unfolding arctan_one [symmetric]
```
```  3656     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  3657   from add_le_less_mono [OF this]
```
```  3658   show 2: "arctan x + arctan y < pi / 2" by simp
```
```  3659   show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
```
```  3660     using cos_gt_zero_pi [OF 1 2] by (simp add: tan_add)
```
```  3661 qed
```
```  3662
```
```  3663 theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
```
```  3664 proof -
```
```  3665   have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
```
```  3666   from arctan_add[OF less_imp_le[OF this] this]
```
```  3667   have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
```
```  3668   moreover
```
```  3669   have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
```
```  3670   from arctan_add[OF less_imp_le[OF this] this]
```
```  3671   have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
```
```  3672   moreover
```
```  3673   have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
```
```  3674   from arctan_add[OF this]
```
```  3675   have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
```
```  3676   ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
```
```  3677   thus ?thesis unfolding arctan_one by algebra
```
```  3678 qed
```
```  3679
```
```  3680
```
```  3681 subsection {* Introducing the arcus tangens power series *}
```
```  3682
```
```  3683 lemma monoseq_arctan_series:
```
```  3684   fixes x :: real
```
```  3685   assumes "\<bar>x\<bar> \<le> 1"
```
```  3686   shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
```
```  3687 proof (cases "x = 0")
```
```  3688   case True
```
```  3689   thus ?thesis unfolding monoseq_def One_nat_def by auto
```
```  3690 next
```
```  3691   case False
```
```  3692   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  3693   show "monoseq ?a"
```
```  3694   proof -
```
```  3695     {
```
```  3696       fix n
```
```  3697       fix x :: real
```
```  3698       assume "0 \<le> x" and "x \<le> 1"
```
```  3699       have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le>
```
```  3700         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
```
```  3701       proof (rule mult_mono)
```
```  3702         show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))"
```
```  3703           by (rule frac_le) simp_all
```
```  3704         show "0 \<le> 1 / real (Suc (n * 2))"
```
```  3705           by auto
```
```  3706         show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)"
```
```  3707           by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`)
```
```  3708         show "0 \<le> x ^ Suc (Suc n * 2)"
```
```  3709           by (rule zero_le_power) (simp add: `0 \<le> x`)
```
```  3710       qed
```
```  3711     } note mono = this
```
```  3712
```
```  3713     show ?thesis
```
```  3714     proof (cases "0 \<le> x")
```
```  3715       case True from mono[OF this `x \<le> 1`, THEN allI]
```
```  3716       show ?thesis unfolding Suc_eq_plus1[symmetric]
```
```  3717         by (rule mono_SucI2)
```
```  3718     next
```
```  3719       case False
```
```  3720       hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto
```
```  3721       from mono[OF this]
```
```  3722       have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge>
```
```  3723         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using `0 \<le> -x` by auto
```
```  3724       thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
```
```  3725     qed
```
```  3726   qed
```
```  3727 qed
```
```  3728
```
```  3729 lemma zeroseq_arctan_series:
```
```  3730   fixes x :: real
```
```  3731   assumes "\<bar>x\<bar> \<le> 1"
```
```  3732   shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
```
```  3733 proof (cases "x = 0")
```
```  3734   case True
```
```  3735   thus ?thesis
```
```  3736     unfolding One_nat_def by auto
```
```  3737 next
```
```  3738   case False
```
```  3739   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  3740   show "?a ----> 0"
```
```  3741   proof (cases "\<bar>x\<bar> < 1")
```
```  3742     case True
```
```  3743     hence "norm x < 1" by auto
```
```  3744     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]]
```
```  3745     have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"
```
```  3746       unfolding inverse_eq_divide Suc_eq_plus1 by simp
```
```  3747     then show ?thesis using pos2 by (rule LIMSEQ_linear)
```
```  3748   next
```
```  3749     case False
```
```  3750     hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  3751     hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x"
```
```  3752       unfolding One_nat_def by auto
```
```  3753     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
```
```  3754     show ?thesis unfolding n_eq Suc_eq_plus1 by auto
```
```  3755   qed
```
```  3756 qed
```
```  3757
```
```  3758 lemma summable_arctan_series:
```
```  3759   fixes x :: real and n :: nat
```
```  3760   assumes "\<bar>x\<bar> \<le> 1"
```
```  3761   shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  3762   (is "summable (?c x)")
```
```  3763   by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
```
```  3764
```
```  3765 lemma less_one_imp_sqr_less_one:
```
```  3766   fixes x :: real
```
```  3767   assumes "\<bar>x\<bar> < 1"
```
```  3768   shows "x\<^sup>2 < 1"
```
```  3769 proof -
```
```  3770   have "\<bar>x\<^sup>2\<bar> < 1"
```
```  3771     by (metis abs_power2 assms pos2 power2_abs power_0 power_strict_decreasing zero_eq_power2 zero_less_abs_iff)
```
```  3772   thus ?thesis using zero_le_power2 by auto
```
```  3773 qed
```
```  3774
```
```  3775 lemma DERIV_arctan_series:
```
```  3776   assumes "\<bar> x \<bar> < 1"
```
```  3777   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))"
```
```  3778   (is "DERIV ?arctan _ :> ?Int")
```
```  3779 proof -
```
```  3780   let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
```
```  3781
```
```  3782   have n_even: "\<And>n :: nat. even n \<Longrightarrow> 2 * (n div 2) = n"
```
```  3783     by presburger
```
```  3784   then have if_eq: "\<And>n x'. ?f n * real (Suc n) * x'^n =
```
```  3785     (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)"
```
```  3786     by auto
```
```  3787
```
```  3788   {
```
```  3789     fix x :: real
```
```  3790     assume "\<bar>x\<bar> < 1"
```
```  3791     hence "x\<^sup>2 < 1" by (rule less_one_imp_sqr_less_one)
```
```  3792     have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)"
```
```  3793       by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x\<^sup>2 < 1` order_less_imp_le[OF `x\<^sup>2 < 1`])
```
```  3794     hence "summable (\<lambda> n. (- 1) ^ n * x^(2*n))" unfolding power_mult .
```
```  3795   } note summable_Integral = this
```
```  3796
```
```  3797   {
```
```  3798     fix f :: "nat \<Rightarrow> real"
```
```  3799     have "\<And>x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  3800     proof
```
```  3801       fix x :: real
```
```  3802       assume "f sums x"
```
```  3803       from sums_if[OF sums_zero this]
```
```  3804       show "(\<lambda>n. if even n then f (n div 2) else 0) sums x"
```
```  3805         by auto
```
```  3806     next
```
```  3807       fix x :: real
```
```  3808       assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  3809       from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult.commute]]
```
```  3810       show "f sums x" unfolding sums_def by auto
```
```  3811     qed
```
```  3812     hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
```
```  3813   } note sums_even = this
```
```  3814
```
```  3815   have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int"
```
```  3816     unfolding if_eq mult.commute[of _ 2] suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * x ^ (2 * n)", symmetric]
```
```  3817     by auto
```
```  3818
```
```  3819   {
```
```  3820     fix x :: real
```
```  3821     have if_eq': "\<And>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
```
```  3822       (if even n then (- 1) ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
```
```  3823       using n_even by auto
```
```  3824     have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)" by auto
```
```  3825     have "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x"
```
```  3826       unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
```
```  3827       by auto
```
```  3828   } note arctan_eq = this
```
```  3829
```
```  3830   have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
```
```  3831   proof (rule DERIV_power_series')
```
```  3832     show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto
```
```  3833     {
```
```  3834       fix x' :: real
```
```  3835       assume x'_bounds: "x' \<in> {- 1 <..< 1}"
```
```  3836       then have "\<bar>x'\<bar> < 1" by auto
```
```  3837       then
```
```  3838         have *: "summable (\<lambda>n. (- 1) ^ n * x' ^ (2 * n))"
```
```  3839         by (rule summable_Integral)
```
```  3840       let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
```
```  3841       show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
```
```  3842         apply (rule sums_summable [where l="0 + ?S"])
```
```  3843         apply (rule sums_if)
```
```  3844         apply (rule sums_zero)
```
```  3845         apply (rule summable_sums)
```
```  3846         apply (rule *)
```
```  3847         done
```
```  3848     }
```
```  3849   qed auto
```
```  3850   thus ?thesis unfolding Int_eq arctan_eq .
```
```  3851 qed
```
```  3852
```
```  3853 lemma arctan_series:
```
```  3854   assumes "\<bar> x \<bar> \<le> 1"
```
```  3855   shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  3856   (is "_ = suminf (\<lambda> n. ?c x n)")
```
```  3857 proof -
```
```  3858   let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)"
```
```  3859
```
```  3860   {
```
```  3861     fix r x :: real
```
```  3862     assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
```
```  3863     have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto
```
```  3864     from DERIV_arctan_series[OF this] have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
```
```  3865   } note DERIV_arctan_suminf = this
```
```  3866
```
```  3867   {
```
```  3868     fix x :: real
```
```  3869     assume "\<bar>x\<bar> \<le> 1"
```
```  3870     note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]]
```
```  3871   } note arctan_series_borders = this
```
```  3872
```
```  3873   {
```
```  3874     fix x :: real
```
```  3875     assume "\<bar>x\<bar> < 1"
```
```  3876     have "arctan x = (\<Sum>k. ?c x k)"
```
```  3877     proof -
```
```  3878       obtain r where "\<bar>x\<bar> < r" and "r < 1"
```
```  3879         using dense[OF `\<bar>x\<bar> < 1`] by blast
```
```  3880       hence "0 < r" and "-r < x" and "x < r" by auto
```
```  3881
```
```  3882       have suminf_eq_arctan_bounded: "\<And>x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow>
```
```  3883         suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  3884       proof -
```
```  3885         fix x a b
```
```  3886         assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
```
```  3887         hence "\<bar>x\<bar> < r" by auto
```
```  3888         show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  3889         proof (rule DERIV_isconst2[of "a" "b"])
```
```  3890           show "a < b" and "a \<le> x" and "x \<le> b"
```
```  3891             using `a < b` `a \<le> x` `x \<le> b` by auto
```
```  3892           have "\<forall>x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  3893           proof (rule allI, rule impI)
```
```  3894             fix x
```
```  3895             assume "-r < x \<and> x < r"
```
```  3896             hence "\<bar>x\<bar> < r" by auto
```
```  3897             hence "\<bar>x\<bar> < 1" using `r < 1` by auto
```
```  3898             have "\<bar> - (x\<^sup>2) \<bar> < 1"
```
```  3899               using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
```
```  3900             hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  3901               unfolding real_norm_def[symmetric] by (rule geometric_sums)
```
```  3902             hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  3903               unfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2] by auto
```
```  3904             hence suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)"
```
```  3905               using sums_unique unfolding inverse_eq_divide by auto
```
```  3906             have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))"
```
```  3907               unfolding suminf_c'_eq_geom
```
```  3908               by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
```
```  3909             from DERIV_diff [OF this DERIV_arctan]
```
```  3910             show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  3911               by auto
```
```  3912           qed
```
```  3913           hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
```
```  3914             using `-r < a` `b < r` by auto
```
```  3915           thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
```
```  3916             using `\<bar>x\<bar> < r` by auto
```
```  3917           show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y"
```
```  3918             using DERIV_in_rball DERIV_isCont by auto
```
```  3919         qed
```
```  3920       qed
```
```  3921
```
```  3922       have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
```
```  3923         unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero
```
```  3924         by auto
```
```  3925
```
```  3926       have "suminf (?c x) - arctan x = 0"
```
```  3927       proof (cases "x = 0")
```
```  3928         case True
```
```  3929         thus ?thesis using suminf_arctan_zero by auto
```
```  3930       next
```
```  3931         case False
```
```  3932         hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
```
```  3933         have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
```
```  3934           by (rule suminf_eq_arctan_bounded[where x="0" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>", symmetric])
```
```  3935             (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  3936         moreover
```
```  3937         have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
```
```  3938           by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"])
```
```  3939              (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  3940         ultimately
```
```  3941         show ?thesis using suminf_arctan_zero by auto
```
```  3942       qed
```
```  3943       thus ?thesis by auto
```
```  3944     qed
```
```  3945   } note when_less_one = this
```
```  3946
```
```  3947   show "arctan x = suminf (\<lambda> n. ?c x n)"
```
```  3948   proof (cases "\<bar>x\<bar> < 1")
```
```  3949     case True
```
```  3950     thus ?thesis by (rule when_less_one)
```
```  3951   next
```
```  3952     case False
```
```  3953     hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  3954     let ?a = "\<lambda>x n. \<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
```
```  3955     let ?diff = "\<lambda> x n. \<bar> arctan x - (\<Sum> i<n. ?c x i)\<bar>"
```
```  3956     {
```
```  3957       fix n :: nat
```
```  3958       have "0 < (1 :: real)" by auto
```
```  3959       moreover
```
```  3960       {
```
```  3961         fix x :: real
```
```  3962         assume "0 < x" and "x < 1"
```
```  3963         hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
```
```  3964         from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)"
```
```  3965           by auto
```
```  3966         note bounds = mp[OF arctan_series_borders(2)[OF `\<bar>x\<bar> \<le> 1`] this, unfolded when_less_one[OF `\<bar>x\<bar> < 1`, symmetric], THEN spec]
```
```  3967         have "0 < 1 / real (n*2+1) * x^(n*2+1)"
```
```  3968           by (rule mult_pos_pos, auto simp only: zero_less_power[OF `0 < x`], auto)
```
```  3969         hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)"
```
```  3970           by (rule abs_of_pos)
```
```  3971         have "?diff x n \<le> ?a x n"
```
```  3972         proof (cases "even n")
```
```  3973           case True
```
```  3974           hence sgn_pos: "(-1)^n = (1::real)" by auto
```
```  3975           from `even n` obtain m where "n = 2 * m" ..
```
```  3976           then have "2 * m = n" ..
```
```  3977           from bounds[of m, unfolded this atLeastAtMost_iff]
```
```  3978           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))"
```
```  3979             by auto
```
```  3980           also have "\<dots> = ?c x n" unfolding One_nat_def by auto
```
```  3981           also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
```
```  3982           finally show ?thesis .
```
```  3983         next
```
```  3984           case False
```
```  3985           hence sgn_neg: "(-1)^n = (-1::real)" by auto
```
```  3986           from `odd n` obtain m where "n = 2 * m + 1" ..
```
```  3987           then have m_def: "2 * m + 1 = n" ..
```
```  3988           hence m_plus: "2 * (m + 1) = n + 1" by auto
```
```  3989           from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
```
```  3990           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))"
```
```  3991             by auto
```
```  3992           also have "\<dots> = - ?c x n" unfolding One_nat_def by auto
```
```  3993           also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
```
```  3994           finally show ?thesis .
```
```  3995         qed
```
```  3996         hence "0 \<le> ?a x n - ?diff x n" by auto
```
```  3997       }
```
```  3998       hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
```
```  3999       moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
```
```  4000         unfolding diff_conv_add_uminus divide_inverse
```
```  4001         by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan
```
```  4002           isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum
```
```  4003           simp del: add_uminus_conv_diff)
```
```  4004       ultimately have "0 \<le> ?a 1 n - ?diff 1 n"
```
```  4005         by (rule LIM_less_bound)
```
```  4006       hence "?diff 1 n \<le> ?a 1 n" by auto
```
```  4007     }
```
```  4008     have "?a 1 ----> 0"
```
```  4009       unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
```
```  4010       by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)
```
```  4011     have "?diff 1 ----> 0"
```
```  4012     proof (rule LIMSEQ_I)
```
```  4013       fix r :: real
```
```  4014       assume "0 < r"
```
```  4015       obtain N :: nat where N_I: "\<And>n. N \<le> n \<Longrightarrow> ?a 1 n < r"
```
```  4016         using LIMSEQ_D[OF `?a 1 ----> 0` `0 < r`] by auto
```
```  4017       {
```
```  4018         fix n
```
```  4019         assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this]
```
```  4020         have "norm (?diff 1 n - 0) < r" by auto
```
```  4021       }
```
```  4022       thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
```
```  4023     qed
```
```  4024     from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
```
```  4025     have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
```
```  4026     hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
```
```  4027
```
```  4028     show ?thesis
```
```  4029     proof (cases "x = 1")
```
```  4030       case True
```
```  4031       then show ?thesis by (simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
```
```  4032     next
```
```  4033       case False
```
```  4034       hence "x = -1" using `\<bar>x\<bar> = 1` by auto
```
```  4035
```
```  4036       have "- (pi / 2) < 0" using pi_gt_zero by auto
```
```  4037       have "- (2 * pi) < 0" using pi_gt_zero by auto
```
```  4038
```
```  4039       have c_minus_minus: "\<And>i. ?c (- 1) i = - ?c 1 i"
```
```  4040         unfolding One_nat_def by auto
```
```  4041
```
```  4042       have "arctan (- 1) = arctan (tan (-(pi / 4)))"
```
```  4043         unfolding tan_45 tan_minus ..
```
```  4044       also have "\<dots> = - (pi / 4)"
```
```  4045         by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
```
```  4046       also have "\<dots> = - (arctan (tan (pi / 4)))"
```
```  4047         unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero])
```
```  4048       also have "\<dots> = - (arctan 1)"
```
```  4049         unfolding tan_45 ..
```
```  4050       also have "\<dots> = - (\<Sum> i. ?c 1 i)"
```
```  4051         using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto
```
```  4052       also have "\<dots> = (\<Sum> i. ?c (- 1) i)"
```
```  4053         using suminf_minus[OF sums_summable[OF `(?c 1) sums (arctan 1)`]]
```
```  4054         unfolding c_minus_minus by auto
```
```  4055       finally show ?thesis using `x = -1` by auto
```
```  4056     qed
```
```  4057   qed
```
```  4058 qed
```
```  4059
```
```  4060 lemma arctan_half:
```
```  4061   fixes x :: real
```
```  4062   shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))"
```
```  4063 proof -
```
```  4064   obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x"
```
```  4065     using tan_total by blast
```
```  4066   hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2"
```
```  4067     by auto
```
```  4068
```
```  4069   have "0 < cos y" using cos_gt_zero_pi[OF low high] .
```
```  4070   hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y"
```
```  4071     by auto
```
```  4072
```
```  4073   have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  4074     unfolding tan_def power_divide ..
```
```  4075   also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  4076     using `cos y \<noteq> 0` by auto
```
```  4077   also have "\<dots> = 1 / (cos y)\<^sup>2"
```
```  4078     unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
```
```  4079   finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" .
```
```  4080
```
```  4081   have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)"
```
```  4082     unfolding tan_def using `cos y \<noteq> 0` by (simp add: field_simps)
```
```  4083   also have "\<dots> = tan y / (1 + 1 / cos y)"
```
```  4084     using `cos y \<noteq> 0` unfolding add_divide_distrib by auto
```
```  4085   also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))"
```
```  4086     unfolding cos_sqrt ..
```
```  4087   also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))"
```
```  4088     unfolding real_sqrt_divide by auto
```
```  4089   finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))"
```
```  4090     unfolding `1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2` .
```
```  4091
```
```  4092   have "arctan x = y"
```
```  4093     using arctan_tan low high y_eq by auto
```
```  4094   also have "\<dots> = 2 * (arctan (tan (y/2)))"
```
```  4095     using arctan_tan[OF low2 high2] by auto
```
```  4096   also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))"
```
```  4097     unfolding tan_half by auto
```
```  4098   finally show ?thesis
```
```  4099     unfolding eq `tan y = x` .
```
```  4100 qed
```
```  4101
```
```  4102 lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y"
```
```  4103   by (simp only: arctan_less_iff)
```
```  4104
```
```  4105 lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y"
```
```  4106   by (simp only: arctan_le_iff)
```
```  4107
```
```  4108 lemma arctan_inverse:
```
```  4109   assumes "x \<noteq> 0"
```
```  4110   shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
```
```  4111 proof (rule arctan_unique)
```
```  4112   show "- (pi / 2) < sgn x * pi / 2 - arctan x"
```
```  4113     using arctan_bounded [of x] assms
```
```  4114     unfolding sgn_real_def
```
```  4115     apply (auto simp add: algebra_simps)
```
```  4116     apply (drule zero_less_arctan_iff [THEN iffD2])
```
```  4117     apply arith
```
```  4118     done
```
```  4119   show "sgn x * pi / 2 - arctan x < pi / 2"
```
```  4120     using arctan_bounded [of "- x"] assms
```
```  4121     unfolding sgn_real_def arctan_minus
```
```  4122     by (auto simp add: algebra_simps)
```
```  4123   show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
```
```  4124     unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
```
```  4125     unfolding sgn_real_def
```
```  4126     by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
```
```  4127 qed
```
```  4128
```
```  4129 theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
```
```  4130 proof -
```
```  4131   have "pi / 4 = arctan 1" using arctan_one by auto
```
```  4132   also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
```
```  4133   finally show ?thesis by auto
```
```  4134 qed
```
```  4135
```
```  4136
```
```  4137 subsection {* Existence of Polar Coordinates *}
```
```  4138
```
```  4139 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1"
```
```  4140   apply (rule power2_le_imp_le [OF _ zero_le_one])
```
```  4141   apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
```
```  4142   done
```
```  4143
```
```  4144 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
```
```  4145   by (simp add: abs_le_iff)
```
```  4146
```
```  4147 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
```
```  4148   by (simp add: sin_arccos abs_le_iff)
```
```  4149
```
```  4150 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
```
```  4151
```
```  4152 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
```
```  4153
```
```  4154 lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
```
```  4155 proof -
```
```  4156   have polar_ex1: "\<And>y. 0 < y \<Longrightarrow> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  4157     apply (rule_tac x = "sqrt (x\<^sup>2 + y\<^sup>2)" in exI)
```
```  4158     apply (rule_tac x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))" in exI)
```
```  4159     apply (simp add: cos_arccos_lemma1 sin_arccos_lemma1 power_divide
```
```  4160                      real_sqrt_mult [symmetric] right_diff_distrib)
```
```  4161     done
```
```  4162   show ?thesis
```
```  4163   proof (cases "0::real" y rule: linorder_cases)
```
```  4164     case less
```
```  4165       then show ?thesis by (rule polar_ex1)
```
```  4166   next
```
```  4167     case equal
```
```  4168       then show ?thesis
```
```  4169         by (force simp add: intro!: cos_zero sin_zero)
```
```  4170   next
```
```  4171     case greater
```
```  4172       then show ?thesis
```
```  4173      using polar_ex1 [where y="-y"]
```
```  4174     by auto (metis cos_minus minus_minus minus_mult_right sin_minus)
```
```  4175   qed
```
```  4176 qed
```
```  4177
```
```  4178 end
```