src/HOL/Transcendental.thy
 author wenzelm Sun Aug 18 18:49:45 2013 +0200 (2013-08-18) changeset 53076 47c9aff07725 parent 53015 a1119cf551e8 child 53079 ade63ccd6f4e permissions -rw-r--r--
more symbols;
```     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
```
```     8 header{*Power Series, Transcendental Functions etc.*}
```
```     9
```
```    10 theory Transcendental
```
```    11 imports Fact Series Deriv NthRoot
```
```    12 begin
```
```    13
```
```    14 subsection {* Properties of Power Series *}
```
```    15
```
```    16 lemma lemma_realpow_diff:
```
```    17   fixes y :: "'a::monoid_mult"
```
```    18   shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
```
```    19 proof -
```
```    20   assume "p \<le> n"
```
```    21   hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
```
```    22   thus ?thesis by (simp add: power_commutes)
```
```    23 qed
```
```    24
```
```    25 lemma lemma_realpow_diff_sumr:
```
```    26   fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows
```
```    27      "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
```
```    28       y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
```
```    29 by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac
```
```    30          del: setsum_op_ivl_Suc)
```
```    31
```
```    32 lemma lemma_realpow_diff_sumr2:
```
```    33   fixes y :: "'a::{comm_ring,monoid_mult}" shows
```
```    34      "x ^ (Suc n) - y ^ (Suc n) =
```
```    35       (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
```
```    36 apply (induct n, simp)
```
```    37 apply (simp del: setsum_op_ivl_Suc)
```
```    38 apply (subst setsum_op_ivl_Suc)
```
```    39 apply (subst lemma_realpow_diff_sumr)
```
```    40 apply (simp add: distrib_left del: setsum_op_ivl_Suc)
```
```    41 apply (subst mult_left_commute [of "x - y"])
```
```    42 apply (erule subst)
```
```    43 apply (simp add: algebra_simps)
```
```    44 done
```
```    45
```
```    46 lemma lemma_realpow_rev_sumr:
```
```    47      "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
```
```    48       (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
```
```    49 apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
```
```    50 apply (rule inj_onI, simp)
```
```    51 apply auto
```
```    52 apply (rule_tac x="n - x" in image_eqI, simp, simp)
```
```    53 done
```
```    54
```
```    55 text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
```
```    56 x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
```
```    57
```
```    58 lemma powser_insidea:
```
```    59   fixes x z :: "'a::real_normed_field"
```
```    60   assumes 1: "summable (\<lambda>n. f n * x ^ n)"
```
```    61   assumes 2: "norm z < norm x"
```
```    62   shows "summable (\<lambda>n. norm (f n * z ^ n))"
```
```    63 proof -
```
```    64   from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
```
```    65   from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
```
```    66     by (rule summable_LIMSEQ_zero)
```
```    67   hence "convergent (\<lambda>n. f n * x ^ n)"
```
```    68     by (rule convergentI)
```
```    69   hence "Cauchy (\<lambda>n. f n * x ^ n)"
```
```    70     by (rule convergent_Cauchy)
```
```    71   hence "Bseq (\<lambda>n. f n * x ^ n)"
```
```    72     by (rule Cauchy_Bseq)
```
```    73   then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
```
```    74     by (simp add: Bseq_def, safe)
```
```    75   have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
```
```    76                    K * norm (z ^ n) * inverse (norm (x ^ n))"
```
```    77   proof (intro exI allI impI)
```
```    78     fix n::nat assume "0 \<le> n"
```
```    79     have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
```
```    80           norm (f n * x ^ n) * norm (z ^ n)"
```
```    81       by (simp add: norm_mult abs_mult)
```
```    82     also have "\<dots> \<le> K * norm (z ^ n)"
```
```    83       by (simp only: mult_right_mono 4 norm_ge_zero)
```
```    84     also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
```
```    85       by (simp add: x_neq_0)
```
```    86     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
```
```    87       by (simp only: mult_assoc)
```
```    88     finally show "norm (norm (f n * z ^ n)) \<le>
```
```    89                   K * norm (z ^ n) * inverse (norm (x ^ n))"
```
```    90       by (simp add: mult_le_cancel_right x_neq_0)
```
```    91   qed
```
```    92   moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
```
```    93   proof -
```
```    94     from 2 have "norm (norm (z * inverse x)) < 1"
```
```    95       using x_neq_0
```
```    96       by (simp add: nonzero_norm_divide divide_inverse [symmetric])
```
```    97     hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
```
```    98       by (rule summable_geometric)
```
```    99     hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
```
```   100       by (rule summable_mult)
```
```   101     thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
```
```   102       using x_neq_0
```
```   103       by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
```
```   104                     power_inverse norm_power mult_assoc)
```
```   105   qed
```
```   106   ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   107     by (rule summable_comparison_test)
```
```   108 qed
```
```   109
```
```   110 lemma powser_inside:
```
```   111   fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" shows
```
```   112      "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]
```
```   113       ==> summable (%n. f(n) * (z ^ n))"
```
```   114 by (rule powser_insidea [THEN summable_norm_cancel])
```
```   115
```
```   116 lemma sum_split_even_odd: fixes f :: "nat \<Rightarrow> real" shows
```
```   117   "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
```
```   118    (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
```
```   119 proof (induct n)
```
```   120   case (Suc n)
```
```   121   have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =
```
```   122         (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
```
```   123     using Suc.hyps unfolding One_nat_def by auto
```
```   124   also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))" by auto
```
```   125   finally show ?case .
```
```   126 qed auto
```
```   127
```
```   128 lemma sums_if': fixes g :: "nat \<Rightarrow> real" assumes "g sums x"
```
```   129   shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   130   unfolding sums_def
```
```   131 proof (rule LIMSEQ_I)
```
```   132   fix r :: real assume "0 < r"
```
```   133   from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
```
```   134   obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast
```
```   135
```
```   136   let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
```
```   137   { fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> no" by auto
```
```   138     have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
```
```   139       using sum_split_even_odd by auto
```
```   140     hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
```
```   141     moreover
```
```   142     have "?SUM (2 * (m div 2)) = ?SUM m"
```
```   143     proof (cases "even m")
```
```   144       case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
```
```   145     next
```
```   146       case False hence "even (Suc m)" by auto
```
```   147       from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]] odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]
```
```   148       have eq: "Suc (2 * (m div 2)) = m" by auto
```
```   149       hence "even (2 * (m div 2))" using `odd m` by auto
```
```   150       have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
```
```   151       also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
```
```   152       finally show ?thesis by auto
```
```   153     qed
```
```   154     ultimately have "(norm (?SUM m - x) < r)" by auto
```
```   155   }
```
```   156   thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
```
```   157 qed
```
```   158
```
```   159 lemma sums_if: fixes g :: "nat \<Rightarrow> real" assumes "g sums x" and "f sums y"
```
```   160   shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
```
```   161 proof -
```
```   162   let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
```
```   163   { fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
```
```   164       by (cases B) auto } note if_sum = this
```
```   165   have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" using sums_if'[OF `g sums x`] .
```
```   166   {
```
```   167     have "?s 0 = 0" by auto
```
```   168     have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
```
```   169     have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
```
```   170
```
```   171     have "?s sums y" using sums_if'[OF `f sums y`] .
```
```   172     from this[unfolded sums_def, THEN LIMSEQ_Suc]
```
```   173     have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
```
```   174       unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
```
```   175                 image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
```
```   176                 even_Suc Suc_m1 if_eq .
```
```   177   } from sums_add[OF g_sums this]
```
```   178   show ?thesis unfolding if_sum .
```
```   179 qed
```
```   180
```
```   181 subsection {* Alternating series test / Leibniz formula *}
```
```   182
```
```   183 lemma sums_alternating_upper_lower:
```
```   184   fixes a :: "nat \<Rightarrow> real"
```
```   185   assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
```
```   186   shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and>
```
```   187              ((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. -1^i*a i) ----> l)"
```
```   188   (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
```
```   189 proof -
```
```   190   have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
```
```   191
```
```   192   have "\<forall> n. ?f n \<le> ?f (Suc n)"
```
```   193   proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed
```
```   194   moreover
```
```   195   have "\<forall> n. ?g (Suc n) \<le> ?g n"
```
```   196   proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
```
```   197     unfolding One_nat_def by auto qed
```
```   198   moreover
```
```   199   have "\<forall> n. ?f n \<le> ?g n"
```
```   200   proof fix n show "?f n \<le> ?g n" using fg_diff a_pos
```
```   201     unfolding One_nat_def by auto qed
```
```   202   moreover
```
```   203   have "(\<lambda> n. ?f n - ?g n) ----> 0" unfolding fg_diff
```
```   204   proof (rule LIMSEQ_I)
```
```   205     fix r :: real assume "0 < r"
```
```   206     with `a ----> 0`[THEN LIMSEQ_D]
```
```   207     obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" by auto
```
```   208     hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   209     thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   210   qed
```
```   211   ultimately
```
```   212   show ?thesis by (rule nested_sequence_unique)
```
```   213 qed
```
```   214
```
```   215 lemma summable_Leibniz': fixes a :: "nat \<Rightarrow> real"
```
```   216   assumes a_zero: "a ----> 0" and a_pos: "\<And> n. 0 \<le> a n"
```
```   217   and a_monotone: "\<And> n. a (Suc n) \<le> a n"
```
```   218   shows summable: "summable (\<lambda> n. (-1)^n * a n)"
```
```   219   and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
```
```   220   and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   221   and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
```
```   222   and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   223 proof -
```
```   224   let "?S n" = "(-1)^n * a n"
```
```   225   let "?P n" = "\<Sum>i=0..<n. ?S i"
```
```   226   let "?f n" = "?P (2 * n)"
```
```   227   let "?g n" = "?P (2 * n + 1)"
```
```   228   obtain l :: real where below_l: "\<forall> n. ?f n \<le> l" and "?f ----> l" and above_l: "\<forall> n. l \<le> ?g n" and "?g ----> l"
```
```   229     using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
```
```   230
```
```   231   let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n"
```
```   232   have "?Sa ----> l"
```
```   233   proof (rule LIMSEQ_I)
```
```   234     fix r :: real assume "0 < r"
```
```   235
```
```   236     with `?f ----> l`[THEN LIMSEQ_D]
```
```   237     obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
```
```   238
```
```   239     from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
```
```   240     obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
```
```   241
```
```   242     { fix n :: nat
```
```   243       assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
```
```   244       have "norm (?Sa n - l) < r"
```
```   245       proof (cases "even n")
```
```   246         case True from even_nat_div_two_times_two[OF this]
```
```   247         have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto
```
```   248         with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" by auto
```
```   249         from f[OF this]
```
```   250         show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
```
```   251       next
```
```   252         case False hence "even (n - 1)" by simp
```
```   253         from even_nat_div_two_times_two[OF this]
```
```   254         have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto
```
```   255         hence range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto
```
```   256
```
```   257         from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" by auto
```
```   258         from g[OF this]
```
```   259         show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
```
```   260       qed
```
```   261     }
```
```   262     thus "\<exists> no. \<forall> n \<ge> no. norm (?Sa n - l) < r" by blast
```
```   263   qed
```
```   264   hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
```
```   265   thus "summable ?S" using summable_def by auto
```
```   266
```
```   267   have "l = suminf ?S" using sums_unique[OF sums_l] .
```
```   268
```
```   269   { fix n show "suminf ?S \<le> ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto }
```
```   270   { fix n show "?f n \<le> suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_l by auto }
```
```   271   show "?g ----> suminf ?S" using `?g ----> l` `l = suminf ?S` by auto
```
```   272   show "?f ----> suminf ?S" using `?f ----> l` `l = suminf ?S` by auto
```
```   273 qed
```
```   274
```
```   275 theorem summable_Leibniz: fixes a :: "nat \<Rightarrow> real"
```
```   276   assumes a_zero: "a ----> 0" and "monoseq a"
```
```   277   shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
```
```   278   and "0 < a 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n. -1^i * a i .. \<Sum>i=0..<2*n+1. -1^i * a i})" (is "?pos")
```
```   279   and "a 0 < 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n+1. -1^i * a i .. \<Sum>i=0..<2*n. -1^i * a i})" (is "?neg")
```
```   280   and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
```
```   281   and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
```
```   282 proof -
```
```   283   have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
```
```   284   proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
```
```   285     case True
```
```   286     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" by auto
```
```   287     { fix n have "a (Suc n) \<le> a n" using ord[where n="Suc n" and m=n] by auto }
```
```   288     note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] and mono = this
```
```   289     from leibniz[OF mono]
```
```   290     show ?thesis using `0 \<le> a 0` by auto
```
```   291   next
```
```   292     let ?a = "\<lambda> n. - a n"
```
```   293     case False
```
```   294     with monoseq_le[OF `monoseq a` `a ----> 0`]
```
```   295     have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
```
```   296     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" by auto
```
```   297     { fix n have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] by auto }
```
```   298     note monotone = this
```
```   299     note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
```
```   300     have "summable (\<lambda> n. (-1)^n * ?a n)" using leibniz(1) by auto
```
```   301     then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" unfolding summable_def by auto
```
```   302     from this[THEN sums_minus]
```
```   303     have "(\<lambda> n. (-1)^n * a n) sums -l" by auto
```
```   304     hence ?summable unfolding summable_def by auto
```
```   305     moreover
```
```   306     have "\<And> a b :: real. \<bar> - a - - b \<bar> = \<bar>a - b\<bar>" unfolding minus_diff_minus by auto
```
```   307
```
```   308     from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
```
```   309     have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)" by auto
```
```   310
```
```   311     have ?pos using `0 \<le> ?a 0` by auto
```
```   312     moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto
```
```   313     moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel] by auto
```
```   314     ultimately show ?thesis by auto
```
```   315   qed
```
```   316   from this[THEN conjunct1] this[THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1]
```
```   317        this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]
```
```   318   show ?summable and ?pos and ?neg and ?f and ?g .
```
```   319 qed
```
```   320
```
```   321 subsection {* Term-by-Term Differentiability of Power Series *}
```
```   322
```
```   323 definition
```
```   324   diffs :: "(nat => 'a::ring_1) => nat => 'a" where
```
```   325   "diffs c = (%n. of_nat (Suc n) * c(Suc n))"
```
```   326
```
```   327 text{*Lemma about distributing negation over it*}
```
```   328 lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
```
```   329 by (simp add: diffs_def)
```
```   330
```
```   331 lemma sums_Suc_imp:
```
```   332   assumes f: "f 0 = 0"
```
```   333   shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
```
```   334 unfolding sums_def
```
```   335 apply (rule LIMSEQ_imp_Suc)
```
```   336 apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
```
```   337 apply (simp only: setsum_shift_bounds_Suc_ivl)
```
```   338 done
```
```   339
```
```   340 lemma diffs_equiv:
```
```   341   fixes x :: "'a::{real_normed_vector, ring_1}"
```
```   342   shows "summable (%n. (diffs c)(n) * (x ^ n)) ==>
```
```   343       (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
```
```   344          (\<Sum>n. (diffs c)(n) * (x ^ n))"
```
```   345 unfolding diffs_def
```
```   346 apply (drule summable_sums)
```
```   347 apply (rule sums_Suc_imp, simp_all)
```
```   348 done
```
```   349
```
```   350 lemma lemma_termdiff1:
```
```   351   fixes z :: "'a :: {monoid_mult,comm_ring}" shows
```
```   352   "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
```
```   353    (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
```
```   354 by(auto simp add: algebra_simps power_add [symmetric])
```
```   355
```
```   356 lemma sumr_diff_mult_const2:
```
```   357   "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
```
```   358 by (simp add: setsum_subtractf)
```
```   359
```
```   360 lemma lemma_termdiff2:
```
```   361   fixes h :: "'a :: {field}"
```
```   362   assumes h: "h \<noteq> 0" shows
```
```   363   "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
```
```   364    h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
```
```   365         (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
```
```   366 apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
```
```   367 apply (simp add: right_diff_distrib diff_divide_distrib h)
```
```   368 apply (simp add: mult_assoc [symmetric])
```
```   369 apply (cases "n", simp)
```
```   370 apply (simp add: lemma_realpow_diff_sumr2 h
```
```   371                  right_diff_distrib [symmetric] mult_assoc
```
```   372             del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
```
```   373 apply (subst lemma_realpow_rev_sumr)
```
```   374 apply (subst sumr_diff_mult_const2)
```
```   375 apply simp
```
```   376 apply (simp only: lemma_termdiff1 setsum_right_distrib)
```
```   377 apply (rule setsum_cong [OF refl])
```
```   378 apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
```
```   379 apply (clarify)
```
```   380 apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
```
```   381             del: setsum_op_ivl_Suc power_Suc)
```
```   382 apply (subst mult_assoc [symmetric], subst power_add [symmetric])
```
```   383 apply (simp add: mult_ac)
```
```   384 done
```
```   385
```
```   386 lemma real_setsum_nat_ivl_bounded2:
```
```   387   fixes K :: "'a::linordered_semidom"
```
```   388   assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
```
```   389   assumes K: "0 \<le> K"
```
```   390   shows "setsum f {0..<n-k} \<le> of_nat n * K"
```
```   391 apply (rule order_trans [OF setsum_mono])
```
```   392 apply (rule f, simp)
```
```   393 apply (simp add: mult_right_mono K)
```
```   394 done
```
```   395
```
```   396 lemma lemma_termdiff3:
```
```   397   fixes h z :: "'a::{real_normed_field}"
```
```   398   assumes 1: "h \<noteq> 0"
```
```   399   assumes 2: "norm z \<le> K"
```
```   400   assumes 3: "norm (z + h) \<le> K"
```
```   401   shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
```
```   402           \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   403 proof -
```
```   404   have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
```
```   405         norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
```
```   406           (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
```
```   407     apply (subst lemma_termdiff2 [OF 1])
```
```   408     apply (subst norm_mult)
```
```   409     apply (rule mult_commute)
```
```   410     done
```
```   411   also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
```
```   412   proof (rule mult_right_mono [OF _ norm_ge_zero])
```
```   413     from norm_ge_zero 2 have K: "0 \<le> K" by (rule order_trans)
```
```   414     have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
```
```   415       apply (erule subst)
```
```   416       apply (simp only: norm_mult norm_power power_add)
```
```   417       apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
```
```   418       done
```
```   419     show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
```
```   420               (z + h) ^ q * z ^ (n - 2 - q))
```
```   421           \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
```
```   422       apply (intro
```
```   423          order_trans [OF norm_setsum]
```
```   424          real_setsum_nat_ivl_bounded2
```
```   425          mult_nonneg_nonneg
```
```   426          of_nat_0_le_iff
```
```   427          zero_le_power K)
```
```   428       apply (rule le_Kn, simp)
```
```   429       done
```
```   430   qed
```
```   431   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   432     by (simp only: mult_assoc)
```
```   433   finally show ?thesis .
```
```   434 qed
```
```   435
```
```   436 lemma lemma_termdiff4:
```
```   437   fixes f :: "'a::{real_normed_field} \<Rightarrow>
```
```   438               'b::real_normed_vector"
```
```   439   assumes k: "0 < (k::real)"
```
```   440   assumes le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
```
```   441   shows "f -- 0 --> 0"
```
```   442 unfolding LIM_eq diff_0_right
```
```   443 proof (safe)
```
```   444   let ?h = "of_real (k / 2)::'a"
```
```   445   have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
```
```   446   hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
```
```   447   hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
```
```   448   hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
```
```   449
```
```   450   fix r::real assume r: "0 < r"
```
```   451   show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
```
```   452   proof (cases)
```
```   453     assume "K = 0"
```
```   454     with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
```
```   455       by simp
```
```   456     thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
```
```   457   next
```
```   458     assume K_neq_zero: "K \<noteq> 0"
```
```   459     with zero_le_K have K: "0 < K" by simp
```
```   460     show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
```
```   461     proof (rule exI, safe)
```
```   462       from k r K show "0 < min k (r * inverse K / 2)"
```
```   463         by (simp add: mult_pos_pos positive_imp_inverse_positive)
```
```   464     next
```
```   465       fix x::'a
```
```   466       assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
```
```   467       from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
```
```   468         by simp_all
```
```   469       from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
```
```   470       also from x4 K have "K * norm x < K * (r * inverse K / 2)"
```
```   471         by (rule mult_strict_left_mono)
```
```   472       also have "\<dots> = r / 2"
```
```   473         using K_neq_zero by simp
```
```   474       also have "r / 2 < r"
```
```   475         using r by simp
```
```   476       finally show "norm (f x) < r" .
```
```   477     qed
```
```   478   qed
```
```   479 qed
```
```   480
```
```   481 lemma lemma_termdiff5:
```
```   482   fixes g :: "'a::{real_normed_field} \<Rightarrow>
```
```   483               nat \<Rightarrow> 'b::banach"
```
```   484   assumes k: "0 < (k::real)"
```
```   485   assumes f: "summable f"
```
```   486   assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
```
```   487   shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
```
```   488 proof (rule lemma_termdiff4 [OF k])
```
```   489   fix h::'a assume "h \<noteq> 0" and "norm h < k"
```
```   490   hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
```
```   491     by (simp add: le)
```
```   492   hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
```
```   493     by simp
```
```   494   moreover from f have B: "summable (\<lambda>n. f n * norm h)"
```
```   495     by (rule summable_mult2)
```
```   496   ultimately have C: "summable (\<lambda>n. norm (g h n))"
```
```   497     by (rule summable_comparison_test)
```
```   498   hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
```
```   499     by (rule summable_norm)
```
```   500   also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
```
```   501     by (rule summable_le)
```
```   502   also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
```
```   503     by (rule suminf_mult2 [symmetric])
```
```   504   finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
```
```   505 qed
```
```   506
```
```   507
```
```   508 text{* FIXME: Long proofs*}
```
```   509
```
```   510 lemma termdiffs_aux:
```
```   511   fixes x :: "'a::{real_normed_field,banach}"
```
```   512   assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
```
```   513   assumes 2: "norm x < norm K"
```
```   514   shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
```
```   515              - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   516 proof -
```
```   517   from dense [OF 2]
```
```   518   obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
```
```   519   from norm_ge_zero r1 have r: "0 < r"
```
```   520     by (rule order_le_less_trans)
```
```   521   hence r_neq_0: "r \<noteq> 0" by simp
```
```   522   show ?thesis
```
```   523   proof (rule lemma_termdiff5)
```
```   524     show "0 < r - norm x" using r1 by simp
```
```   525   next
```
```   526     from r r2 have "norm (of_real r::'a) < norm K"
```
```   527       by simp
```
```   528     with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
```
```   529       by (rule powser_insidea)
```
```   530     hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
```
```   531       using r
```
```   532       by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
```
```   533     hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
```
```   534       by (rule diffs_equiv [THEN sums_summable])
```
```   535     also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))
```
```   536       = (\<lambda>n. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
```
```   537       apply (rule ext)
```
```   538       apply (simp add: diffs_def)
```
```   539       apply (case_tac n, simp_all add: r_neq_0)
```
```   540       done
```
```   541     finally have "summable
```
```   542       (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
```
```   543       by (rule diffs_equiv [THEN sums_summable])
```
```   544     also have
```
```   545       "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
```
```   546            r ^ (n - Suc 0)) =
```
```   547        (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
```
```   548       apply (rule ext)
```
```   549       apply (case_tac "n", simp)
```
```   550       apply (case_tac "nat", simp)
```
```   551       apply (simp add: r_neq_0)
```
```   552       done
```
```   553     finally show
```
```   554       "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
```
```   555   next
```
```   556     fix h::'a and n::nat
```
```   557     assume h: "h \<noteq> 0"
```
```   558     assume "norm h < r - norm x"
```
```   559     hence "norm x + norm h < r" by simp
```
```   560     with norm_triangle_ineq have xh: "norm (x + h) < r"
```
```   561       by (rule order_le_less_trans)
```
```   562     show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
```
```   563           \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
```
```   564       apply (simp only: norm_mult mult_assoc)
```
```   565       apply (rule mult_left_mono [OF _ norm_ge_zero])
```
```   566       apply (simp (no_asm) add: mult_assoc [symmetric])
```
```   567       apply (rule lemma_termdiff3)
```
```   568       apply (rule h)
```
```   569       apply (rule r1 [THEN order_less_imp_le])
```
```   570       apply (rule xh [THEN order_less_imp_le])
```
```   571       done
```
```   572   qed
```
```   573 qed
```
```   574
```
```   575 lemma termdiffs:
```
```   576   fixes K x :: "'a::{real_normed_field,banach}"
```
```   577   assumes 1: "summable (\<lambda>n. c n * K ^ n)"
```
```   578   assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
```
```   579   assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
```
```   580   assumes 4: "norm x < norm K"
```
```   581   shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
```
```   582 unfolding deriv_def
```
```   583 proof (rule LIM_zero_cancel)
```
```   584   show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
```
```   585             - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
```
```   586   proof (rule LIM_equal2)
```
```   587     show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
```
```   588   next
```
```   589     fix h :: 'a
```
```   590     assume "h \<noteq> 0"
```
```   591     assume "norm (h - 0) < norm K - norm x"
```
```   592     hence "norm x + norm h < norm K" by simp
```
```   593     hence 5: "norm (x + h) < norm K"
```
```   594       by (rule norm_triangle_ineq [THEN order_le_less_trans])
```
```   595     have A: "summable (\<lambda>n. c n * x ^ n)"
```
```   596       by (rule powser_inside [OF 1 4])
```
```   597     have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
```
```   598       by (rule powser_inside [OF 1 5])
```
```   599     have C: "summable (\<lambda>n. diffs c n * x ^ n)"
```
```   600       by (rule powser_inside [OF 2 4])
```
```   601     show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
```
```   602              - (\<Sum>n. diffs c n * x ^ n) =
```
```   603           (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
```
```   604       apply (subst sums_unique [OF diffs_equiv [OF C]])
```
```   605       apply (subst suminf_diff [OF B A])
```
```   606       apply (subst suminf_divide [symmetric])
```
```   607       apply (rule summable_diff [OF B A])
```
```   608       apply (subst suminf_diff)
```
```   609       apply (rule summable_divide)
```
```   610       apply (rule summable_diff [OF B A])
```
```   611       apply (rule sums_summable [OF diffs_equiv [OF C]])
```
```   612       apply (rule arg_cong [where f="suminf"], rule ext)
```
```   613       apply (simp add: algebra_simps)
```
```   614       done
```
```   615   next
```
```   616     show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h -
```
```   617                of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   618         by (rule termdiffs_aux [OF 3 4])
```
```   619   qed
```
```   620 qed
```
```   621
```
```   622
```
```   623 subsection {* Derivability of power series *}
```
```   624
```
```   625 lemma DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
```
```   626   assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
```
```   627   and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
```
```   628   and "summable (f' x0)"
```
```   629   and "summable L" 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>"
```
```   630   shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
```
```   631   unfolding deriv_def
```
```   632 proof (rule LIM_I)
```
```   633   fix r :: real assume "0 < r" hence "0 < r/3" by auto
```
```   634
```
```   635   obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
```
```   636     using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
```
```   637
```
```   638   obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
```
```   639     using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
```
```   640
```
```   641   let ?N = "Suc (max N_L N_f')"
```
```   642   have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
```
```   643     L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
```
```   644
```
```   645   let "?diff i x" = "(f (x0 + x) i - f x0 i) / x"
```
```   646
```
```   647   let ?r = "r / (3 * real ?N)"
```
```   648   have "0 < 3 * real ?N" by auto
```
```   649   from divide_pos_pos[OF `0 < r` this]
```
```   650   have "0 < ?r" .
```
```   651
```
```   652   let "?s n" = "SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"
```
```   653   def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
```
```   654
```
```   655   have "0 < S'" unfolding S'_def
```
```   656   proof (rule iffD2[OF Min_gr_iff])
```
```   657     show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x"
```
```   658     proof (rule ballI)
```
```   659       fix x assume "x \<in> ?s ` {0..<?N}"
```
```   660       then obtain n where "x = ?s n" and "n \<in> {0..<?N}" using image_iff[THEN iffD1] by blast
```
```   661       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
```
```   662       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)" by auto
```
```   663       have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)
```
```   664       thus "0 < x" unfolding `x = ?s n` .
```
```   665     qed
```
```   666   qed auto
```
```   667
```
```   668   def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
```
```   669   hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0" and "S \<le> S'" using x0_in_I and `0 < S'`
```
```   670     by auto
```
```   671
```
```   672   { fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
```
```   673     hence x_in_I: "x0 + x \<in> { a <..< b }" using S_a S_b by auto
```
```   674
```
```   675     note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   676     note div_smbl = summable_divide[OF diff_smbl]
```
```   677     note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
```
```   678     note ign = summable_ignore_initial_segment[where k="?N"]
```
```   679     note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
```
```   680     note div_shft_smbl = summable_divide[OF diff_shft_smbl]
```
```   681     note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
```
```   682
```
```   683     { fix n
```
```   684       have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
```
```   685         using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .
```
```   686       hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 0` by auto
```
```   687     } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
```
```   688     from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
```
```   689     have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
```
```   690     hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") using L_estimate by auto
```
```   691
```
```   692     have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..
```
```   693     also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
```
```   694     proof (rule setsum_strict_mono)
```
```   695       fix n assume "n \<in> { 0 ..< ?N}"
```
```   696       have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .
```
```   697       also have "S \<le> S'" using `S \<le> S'` .
```
```   698       also have "S' \<le> ?s n" unfolding S'_def
```
```   699       proof (rule Min_le_iff[THEN iffD2])
```
```   700         have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto
```
```   701         thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
```
```   702       qed auto
```
```   703       finally have "\<bar> x \<bar> < ?s n" .
```
```   704
```
```   705       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]
```
```   706       have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
```
```   707       with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n`
```
```   708       show "\<bar>?diff n x - f' x0 n\<bar> < ?r" by blast
```
```   709     qed auto
```
```   710     also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)
```
```   711     also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto
```
```   712     also have "\<dots> = r/3" by auto
```
```   713     finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
```
```   714
```
```   715     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   716     have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> =
```
```   717                     \<bar> \<Sum>n. ?diff n x - f' x0 n \<bar>" unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] using suminf_divide[OF diff_smbl, symmetric] by auto
```
```   718     also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>" unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]] by (rule abs_triangle_ineq)
```
```   719     also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto
```
```   720     also have "\<dots> < r /3 + r/3 + r/3"
```
```   721       using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
```
```   722       by (rule add_strict_mono [OF add_less_le_mono])
```
```   723     finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"
```
```   724       by auto
```
```   725   } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
```
```   726       norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`
```
```   727     unfolding real_norm_def diff_0_right by blast
```
```   728 qed
```
```   729
```
```   730 lemma DERIV_power_series': fixes f :: "nat \<Rightarrow> real"
```
```   731   assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
```
```   732   and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
```
```   733   shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
```
```   734   (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
```
```   735 proof -
```
```   736   { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
```
```   737     hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" by auto
```
```   738     have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
```
```   739     proof (rule DERIV_series')
```
```   740       show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
```
```   741       proof -
```
```   742         have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto
```
```   743         hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" using `R' < R` by auto
```
```   744         have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto
```
```   745         from powser_insidea[OF converges[OF in_Rball] this] show ?thesis by auto
```
```   746       qed
```
```   747       { fix n x y assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
```
```   748         show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
```
```   749         proof -
```
```   750           have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> = (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
```
```   751             unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto
```
```   752           also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
```
```   753           proof (rule mult_left_mono)
```
```   754             have "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p = 0..<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)" by (rule setsum_abs)
```
```   755             also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
```
```   756             proof (rule setsum_mono)
```
```   757               fix p assume "p \<in> {0..<Suc n}" hence "p \<le> n" by auto
```
```   758               { fix n fix x :: real assume "x \<in> {-R'<..<R'}"
```
```   759                 hence "\<bar>x\<bar> \<le> R'"  by auto
```
```   760                 hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto)
```
```   761               } from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
```
```   762               have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" unfolding abs_mult by auto
```
```   763               thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> n` by auto
```
```   764             qed
```
```   765             also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
```
```   766             finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>" unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
```
```   767             show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult[symmetric] by auto
```
```   768           qed
```
```   769           also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult mult_assoc[symmetric] by algebra
```
```   770           finally show ?thesis .
```
```   771         qed }
```
```   772       { fix n show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
```
```   773           by (auto intro!: DERIV_intros simp del: power_Suc) }
```
```   774       { fix x assume "x \<in> {-R' <..< R'}" hence "R' \<in> {-R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto
```
```   775         have "summable (\<lambda> n. f n * x^n)"
```
```   776         proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
```
```   777           fix n
```
```   778           have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto)
```
```   779           show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult
```
```   780             by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right])
```
```   781         qed
```
```   782         from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
```
```   783         show "summable (?f x)" by auto }
```
```   784       show "summable (?f' x0)" using converges[OF `x0 \<in> {-R <..< R}`] .
```
```   785       show "x0 \<in> {-R' <..< R'}" using `x0 \<in> {-R' <..< R'}` .
```
```   786     qed
```
```   787   } note for_subinterval = this
```
```   788   let ?R = "(R + \<bar>x0\<bar>) / 2"
```
```   789   have "\<bar>x0\<bar> < ?R" using assms by auto
```
```   790   hence "- ?R < x0"
```
```   791   proof (cases "x0 < 0")
```
```   792     case True
```
```   793     hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
```
```   794     thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
```
```   795   next
```
```   796     case False
```
```   797     have "- ?R < 0" using assms by auto
```
```   798     also have "\<dots> \<le> x0" using False by auto
```
```   799     finally show ?thesis .
```
```   800   qed
```
```   801   hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto
```
```   802   from for_subinterval[OF this]
```
```   803   show ?thesis .
```
```   804 qed
```
```   805
```
```   806 subsection {* Exponential Function *}
```
```   807
```
```   808 definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" where
```
```   809   "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
```
```   810
```
```   811 lemma summable_exp_generic:
```
```   812   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   813   defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
```
```   814   shows "summable S"
```
```   815 proof -
```
```   816   have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
```
```   817     unfolding S_def by (simp del: mult_Suc)
```
```   818   obtain r :: real where r0: "0 < r" and r1: "r < 1"
```
```   819     using dense [OF zero_less_one] by fast
```
```   820   obtain N :: nat where N: "norm x < real N * r"
```
```   821     using reals_Archimedean3 [OF r0] by fast
```
```   822   from r1 show ?thesis
```
```   823   proof (rule ratio_test [rule_format])
```
```   824     fix n :: nat
```
```   825     assume n: "N \<le> n"
```
```   826     have "norm x \<le> real N * r"
```
```   827       using N by (rule order_less_imp_le)
```
```   828     also have "real N * r \<le> real (Suc n) * r"
```
```   829       using r0 n by (simp add: mult_right_mono)
```
```   830     finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
```
```   831       using norm_ge_zero by (rule mult_right_mono)
```
```   832     hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
```
```   833       by (rule order_trans [OF norm_mult_ineq])
```
```   834     hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
```
```   835       by (simp add: pos_divide_le_eq mult_ac)
```
```   836     thus "norm (S (Suc n)) \<le> r * norm (S n)"
```
```   837       by (simp add: S_Suc inverse_eq_divide)
```
```   838   qed
```
```   839 qed
```
```   840
```
```   841 lemma summable_norm_exp:
```
```   842   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   843   shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
```
```   844 proof (rule summable_norm_comparison_test [OF exI, rule_format])
```
```   845   show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
```
```   846     by (rule summable_exp_generic)
```
```   847 next
```
```   848   fix n show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
```
```   849     by (simp add: norm_power_ineq)
```
```   850 qed
```
```   851
```
```   852 lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
```
```   853 by (insert summable_exp_generic [where x=x], simp)
```
```   854
```
```   855 lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
```
```   856 unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
```
```   857
```
```   858
```
```   859 lemma exp_fdiffs:
```
```   860       "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
```
```   861 by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
```
```   862          del: mult_Suc of_nat_Suc)
```
```   863
```
```   864 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
```
```   865 by (simp add: diffs_def)
```
```   866
```
```   867 lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
```
```   868 unfolding exp_def scaleR_conv_of_real
```
```   869 apply (rule DERIV_cong)
```
```   870 apply (rule termdiffs [where K="of_real (1 + norm x)"])
```
```   871 apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
```
```   872 apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
```
```   873 apply (simp del: of_real_add)
```
```   874 done
```
```   875
```
```   876 declare DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```   877
```
```   878 lemma isCont_exp: "isCont exp x"
```
```   879   by (rule DERIV_exp [THEN DERIV_isCont])
```
```   880
```
```   881 lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
```
```   882   by (rule isCont_o2 [OF _ isCont_exp])
```
```   883
```
```   884 lemma tendsto_exp [tendsto_intros]:
```
```   885   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
```
```   886   by (rule isCont_tendsto_compose [OF isCont_exp])
```
```   887
```
```   888 lemma continuous_exp [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
```
```   889   unfolding continuous_def by (rule tendsto_exp)
```
```   890
```
```   891 lemma continuous_on_exp [continuous_on_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
```
```   892   unfolding continuous_on_def by (auto intro: tendsto_exp)
```
```   893
```
```   894 subsubsection {* Properties of the Exponential Function *}
```
```   895
```
```   896 lemma powser_zero:
```
```   897   fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
```
```   898   shows "(\<Sum>n. f n * 0 ^ n) = f 0"
```
```   899 proof -
```
```   900   have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
```
```   901     by (rule sums_unique [OF series_zero], simp add: power_0_left)
```
```   902   thus ?thesis unfolding One_nat_def by simp
```
```   903 qed
```
```   904
```
```   905 lemma exp_zero [simp]: "exp 0 = 1"
```
```   906 unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
```
```   907
```
```   908 lemma setsum_cl_ivl_Suc2:
```
```   909   "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
```
```   910 by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
```
```   911          del: setsum_cl_ivl_Suc)
```
```   912
```
```   913 lemma exp_series_add:
```
```   914   fixes x y :: "'a::{real_field}"
```
```   915   defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
```
```   916   shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
```
```   917 proof (induct n)
```
```   918   case 0
```
```   919   show ?case
```
```   920     unfolding S_def by simp
```
```   921 next
```
```   922   case (Suc n)
```
```   923   have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
```
```   924     unfolding S_def by (simp del: mult_Suc)
```
```   925   hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
```
```   926     by simp
```
```   927
```
```   928   have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
```
```   929     by (simp only: times_S)
```
```   930   also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
```
```   931     by (simp only: Suc)
```
```   932   also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
```
```   933                 + y * (\<Sum>i=0..n. S x i * S y (n-i))"
```
```   934     by (rule distrib_right)
```
```   935   also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
```
```   936                 + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
```
```   937     by (simp only: setsum_right_distrib mult_ac)
```
```   938   also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
```
```   939                 + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```   940     by (simp add: times_S Suc_diff_le)
```
```   941   also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
```
```   942              (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
```
```   943     by (subst setsum_cl_ivl_Suc2, simp)
```
```   944   also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```   945              (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```   946     by (subst setsum_cl_ivl_Suc, simp)
```
```   947   also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
```
```   948              (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```   949              (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```   950     by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
```
```   951               real_of_nat_add [symmetric], simp)
```
```   952   also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
```
```   953     by (simp only: scaleR_right.setsum)
```
```   954   finally show
```
```   955     "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
```
```   956     by (simp del: setsum_cl_ivl_Suc)
```
```   957 qed
```
```   958
```
```   959 lemma exp_add: "exp (x + y) = exp x * exp y"
```
```   960 unfolding exp_def
```
```   961 by (simp only: Cauchy_product summable_norm_exp exp_series_add)
```
```   962
```
```   963 lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
```
```   964 by (rule exp_add [symmetric])
```
```   965
```
```   966 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
```
```   967 unfolding exp_def
```
```   968 apply (subst suminf_of_real)
```
```   969 apply (rule summable_exp_generic)
```
```   970 apply (simp add: scaleR_conv_of_real)
```
```   971 done
```
```   972
```
```   973 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
```
```   974 proof
```
```   975   have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
```
```   976   also assume "exp x = 0"
```
```   977   finally show "False" by simp
```
```   978 qed
```
```   979
```
```   980 lemma exp_minus: "exp (- x) = inverse (exp x)"
```
```   981 by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
```
```   982
```
```   983 lemma exp_diff: "exp (x - y) = exp x / exp y"
```
```   984   unfolding diff_minus divide_inverse
```
```   985   by (simp add: exp_add exp_minus)
```
```   986
```
```   987
```
```   988 subsubsection {* Properties of the Exponential Function on Reals *}
```
```   989
```
```   990 text {* Comparisons of @{term "exp x"} with zero. *}
```
```   991
```
```   992 text{*Proof: because every exponential can be seen as a square.*}
```
```   993 lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
```
```   994 proof -
```
```   995   have "0 \<le> exp (x/2) * exp (x/2)" by simp
```
```   996   thus ?thesis by (simp add: exp_add [symmetric])
```
```   997 qed
```
```   998
```
```   999 lemma exp_gt_zero [simp]: "0 < exp (x::real)"
```
```  1000 by (simp add: order_less_le)
```
```  1001
```
```  1002 lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
```
```  1003 by (simp add: not_less)
```
```  1004
```
```  1005 lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
```
```  1006 by (simp add: not_le)
```
```  1007
```
```  1008 lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
```
```  1009 by simp
```
```  1010
```
```  1011 lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
```
```  1012 apply (induct "n")
```
```  1013 apply (auto simp add: real_of_nat_Suc distrib_left exp_add mult_commute)
```
```  1014 done
```
```  1015
```
```  1016 text {* Strict monotonicity of exponential. *}
```
```  1017
```
```  1018 lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
```
```  1019 apply (drule order_le_imp_less_or_eq, auto)
```
```  1020 apply (simp add: exp_def)
```
```  1021 apply (rule order_trans)
```
```  1022 apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
```
```  1023 apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
```
```  1024 done
```
```  1025
```
```  1026 lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
```
```  1027 proof -
```
```  1028   assume x: "0 < x"
```
```  1029   hence "1 < 1 + x" by simp
```
```  1030   also from x have "1 + x \<le> exp x"
```
```  1031     by (simp add: exp_ge_add_one_self_aux)
```
```  1032   finally show ?thesis .
```
```  1033 qed
```
```  1034
```
```  1035 lemma exp_less_mono:
```
```  1036   fixes x y :: real
```
```  1037   assumes "x < y" shows "exp x < exp y"
```
```  1038 proof -
```
```  1039   from `x < y` have "0 < y - x" by simp
```
```  1040   hence "1 < exp (y - x)" by (rule exp_gt_one)
```
```  1041   hence "1 < exp y / exp x" by (simp only: exp_diff)
```
```  1042   thus "exp x < exp y" by simp
```
```  1043 qed
```
```  1044
```
```  1045 lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
```
```  1046 apply (simp add: linorder_not_le [symmetric])
```
```  1047 apply (auto simp add: order_le_less exp_less_mono)
```
```  1048 done
```
```  1049
```
```  1050 lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
```
```  1051 by (auto intro: exp_less_mono exp_less_cancel)
```
```  1052
```
```  1053 lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
```
```  1054 by (auto simp add: linorder_not_less [symmetric])
```
```  1055
```
```  1056 lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
```
```  1057 by (simp add: order_eq_iff)
```
```  1058
```
```  1059 text {* Comparisons of @{term "exp x"} with one. *}
```
```  1060
```
```  1061 lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
```
```  1062   using exp_less_cancel_iff [where x=0 and y=x] by simp
```
```  1063
```
```  1064 lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
```
```  1065   using exp_less_cancel_iff [where x=x and y=0] by simp
```
```  1066
```
```  1067 lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
```
```  1068   using exp_le_cancel_iff [where x=0 and y=x] by simp
```
```  1069
```
```  1070 lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
```
```  1071   using exp_le_cancel_iff [where x=x and y=0] by simp
```
```  1072
```
```  1073 lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
```
```  1074   using exp_inj_iff [where x=x and y=0] by simp
```
```  1075
```
```  1076 lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
```
```  1077 proof (rule IVT)
```
```  1078   assume "1 \<le> y"
```
```  1079   hence "0 \<le> y - 1" by simp
```
```  1080   hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
```
```  1081   thus "y \<le> exp (y - 1)" by simp
```
```  1082 qed (simp_all add: le_diff_eq)
```
```  1083
```
```  1084 lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
```
```  1085 proof (rule linorder_le_cases [of 1 y])
```
```  1086   assume "1 \<le> y" thus "\<exists>x. exp x = y"
```
```  1087     by (fast dest: lemma_exp_total)
```
```  1088 next
```
```  1089   assume "0 < y" and "y \<le> 1"
```
```  1090   hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
```
```  1091   then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
```
```  1092   hence "exp (- x) = y" by (simp add: exp_minus)
```
```  1093   thus "\<exists>x. exp x = y" ..
```
```  1094 qed
```
```  1095
```
```  1096
```
```  1097 subsection {* Natural Logarithm *}
```
```  1098
```
```  1099 definition ln :: "real \<Rightarrow> real" where
```
```  1100   "ln x = (THE u. exp u = x)"
```
```  1101
```
```  1102 lemma ln_exp [simp]: "ln (exp x) = x"
```
```  1103   by (simp add: ln_def)
```
```  1104
```
```  1105 lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
```
```  1106   by (auto dest: exp_total)
```
```  1107
```
```  1108 lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
```
```  1109   by (metis exp_gt_zero exp_ln)
```
```  1110
```
```  1111 lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
```
```  1112   by (erule subst, rule ln_exp)
```
```  1113
```
```  1114 lemma ln_one [simp]: "ln 1 = 0"
```
```  1115   by (rule ln_unique, simp)
```
```  1116
```
```  1117 lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"
```
```  1118   by (rule ln_unique, simp add: exp_add)
```
```  1119
```
```  1120 lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
```
```  1121   by (rule ln_unique, simp add: exp_minus)
```
```  1122
```
```  1123 lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"
```
```  1124   by (rule ln_unique, simp add: exp_diff)
```
```  1125
```
```  1126 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
```
```  1127   by (rule ln_unique, simp add: exp_real_of_nat_mult)
```
```  1128
```
```  1129 lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
```
```  1130   by (subst exp_less_cancel_iff [symmetric], simp)
```
```  1131
```
```  1132 lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
```
```  1133   by (simp add: linorder_not_less [symmetric])
```
```  1134
```
```  1135 lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
```
```  1136   by (simp add: order_eq_iff)
```
```  1137
```
```  1138 lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
```
```  1139   apply (rule exp_le_cancel_iff [THEN iffD1])
```
```  1140   apply (simp add: exp_ge_add_one_self_aux)
```
```  1141   done
```
```  1142
```
```  1143 lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
```
```  1144   by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
```
```  1145
```
```  1146 lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
```
```  1147   using ln_le_cancel_iff [of 1 x] by simp
```
```  1148
```
```  1149 lemma ln_ge_zero_imp_ge_one: "\<lbrakk>0 \<le> ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 \<le> x"
```
```  1150   using ln_le_cancel_iff [of 1 x] by simp
```
```  1151
```
```  1152 lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> (0 \<le> ln x) = (1 \<le> x)"
```
```  1153   using ln_le_cancel_iff [of 1 x] by simp
```
```  1154
```
```  1155 lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x < 0) = (x < 1)"
```
```  1156   using ln_less_cancel_iff [of x 1] by simp
```
```  1157
```
```  1158 lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
```
```  1159   using ln_less_cancel_iff [of 1 x] by simp
```
```  1160
```
```  1161 lemma ln_gt_zero_imp_gt_one: "\<lbrakk>0 < ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 < x"
```
```  1162   using ln_less_cancel_iff [of 1 x] by simp
```
```  1163
```
```  1164 lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> (0 < ln x) = (1 < x)"
```
```  1165   using ln_less_cancel_iff [of 1 x] by simp
```
```  1166
```
```  1167 lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x = 0) = (x = 1)"
```
```  1168   using ln_inj_iff [of x 1] by simp
```
```  1169
```
```  1170 lemma ln_less_zero: "\<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> ln x < 0"
```
```  1171   by simp
```
```  1172
```
```  1173 lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
```
```  1174   apply (subgoal_tac "isCont ln (exp (ln x))", simp)
```
```  1175   apply (rule isCont_inverse_function [where f=exp], simp_all)
```
```  1176   done
```
```  1177
```
```  1178 lemma tendsto_ln [tendsto_intros]:
```
```  1179   "\<lbrakk>(f ---> a) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
```
```  1180   by (rule isCont_tendsto_compose [OF isCont_ln])
```
```  1181
```
```  1182 lemma continuous_ln:
```
```  1183   "continuous F f \<Longrightarrow> 0 < f (Lim F (\<lambda>x. x)) \<Longrightarrow> continuous F (\<lambda>x. ln (f x))"
```
```  1184   unfolding continuous_def by (rule tendsto_ln)
```
```  1185
```
```  1186 lemma isCont_ln' [continuous_intros]:
```
```  1187   "continuous (at x) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x))"
```
```  1188   unfolding continuous_at by (rule tendsto_ln)
```
```  1189
```
```  1190 lemma continuous_within_ln [continuous_intros]:
```
```  1191   "continuous (at x within s) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x))"
```
```  1192   unfolding continuous_within by (rule tendsto_ln)
```
```  1193
```
```  1194 lemma continuous_on_ln [continuous_on_intros]:
```
```  1195   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. 0 < f x) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x))"
```
```  1196   unfolding continuous_on_def by (auto intro: tendsto_ln)
```
```  1197
```
```  1198 lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
```
```  1199   apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
```
```  1200   apply (erule DERIV_cong [OF DERIV_exp exp_ln])
```
```  1201   apply (simp_all add: abs_if isCont_ln)
```
```  1202   done
```
```  1203
```
```  1204 lemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x"
```
```  1205   by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
```
```  1206
```
```  1207 declare DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1208
```
```  1209 lemma ln_series: assumes "0 < x" and "x < 2"
```
```  1210   shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))")
```
```  1211 proof -
```
```  1212   let "?f' x n" = "(-1)^n * (x - 1)^n"
```
```  1213
```
```  1214   have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
```
```  1215   proof (rule DERIV_isconst3[where x=x])
```
```  1216     fix x :: real assume "x \<in> {0 <..< 2}" hence "0 < x" and "x < 2" by auto
```
```  1217     have "norm (1 - x) < 1" using `0 < x` and `x < 2` by auto
```
```  1218     have "1 / x = 1 / (1 - (1 - x))" by auto
```
```  1219     also have "\<dots> = (\<Sum> n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
```
```  1220     also have "\<dots> = suminf (?f' x)" unfolding power_mult_distrib[symmetric] by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
```
```  1221     finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
```
```  1222     moreover
```
```  1223     have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
```
```  1224     have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
```
```  1225     proof (rule DERIV_power_series')
```
```  1226       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto
```
```  1227       { fix x :: real assume "x \<in> {- 1<..<1}" hence "norm (-x) < 1" by auto
```
```  1228         show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
```
```  1229           unfolding One_nat_def
```
```  1230           by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
```
```  1231       }
```
```  1232     qed
```
```  1233     hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" unfolding One_nat_def by auto
```
```  1234     hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos .
```
```  1235     ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
```
```  1236       by (rule DERIV_diff)
```
```  1237     thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
```
```  1238   qed (auto simp add: assms)
```
```  1239   thus ?thesis by auto
```
```  1240 qed
```
```  1241
```
```  1242 lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) * (x ^ (n+2)))"
```
```  1243 proof -
```
```  1244   have "exp x = suminf (%n. inverse(fact n) * (x ^ n))"
```
```  1245     by (simp add: exp_def)
```
```  1246   also from summable_exp have "... = (\<Sum> n::nat = 0 ..< 2. inverse(fact n) * (x ^ n)) +
```
```  1247       (\<Sum> n. inverse(fact(n+2)) * (x ^ (n+2)))" (is "_ = ?a + _")
```
```  1248     by (rule suminf_split_initial_segment)
```
```  1249   also have "?a = 1 + x"
```
```  1250     by (simp add: numeral_2_eq_2)
```
```  1251   finally show ?thesis .
```
```  1252 qed
```
```  1253
```
```  1254 lemma exp_bound: "0 <= (x::real) ==> x <= 1 ==> exp x <= 1 + x + x\<^sup>2"
```
```  1255 proof -
```
```  1256   assume a: "0 <= x"
```
```  1257   assume b: "x <= 1"
```
```  1258   { fix n :: nat
```
```  1259     have "2 * 2 ^ n \<le> fact (n + 2)"
```
```  1260       by (induct n, simp, simp)
```
```  1261     hence "real ((2::nat) * 2 ^ n) \<le> real (fact (n + 2))"
```
```  1262       by (simp only: real_of_nat_le_iff)
```
```  1263     hence "2 * 2 ^ n \<le> real (fact (n + 2))"
```
```  1264       by simp
```
```  1265     hence "inverse (fact (n + 2)) \<le> inverse (2 * 2 ^ n)"
```
```  1266       by (rule le_imp_inverse_le) simp
```
```  1267     hence "inverse (fact (n + 2)) \<le> 1/2 * (1/2)^n"
```
```  1268       by (simp add: inverse_mult_distrib power_inverse)
```
```  1269     hence "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
```
```  1270       by (rule mult_mono)
```
```  1271         (rule mult_mono, simp_all add: power_le_one a b mult_nonneg_nonneg)
```
```  1272     hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)"
```
```  1273       unfolding power_add by (simp add: mult_ac del: fact_Suc) }
```
```  1274   note aux1 = this
```
```  1275   have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
```
```  1276     by (intro sums_mult geometric_sums, simp)
```
```  1277   hence aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
```
```  1278     by simp
```
```  1279   have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2"
```
```  1280   proof -
```
```  1281     have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <=
```
```  1282         suminf (%n. (x\<^sup>2/2) * ((1/2)^n))"
```
```  1283       apply (rule summable_le)
```
```  1284       apply (rule allI, rule aux1)
```
```  1285       apply (rule summable_exp [THEN summable_ignore_initial_segment])
```
```  1286       by (rule sums_summable, rule aux2)
```
```  1287     also have "... = x\<^sup>2"
```
```  1288       by (rule sums_unique [THEN sym], rule aux2)
```
```  1289     finally show ?thesis .
```
```  1290   qed
```
```  1291   thus ?thesis unfolding exp_first_two_terms by auto
```
```  1292 qed
```
```  1293
```
```  1294 lemma ln_one_minus_pos_upper_bound: "0 <= x ==> x < 1 ==> ln (1 - x) <= - x"
```
```  1295 proof -
```
```  1296   assume a: "0 <= (x::real)" and b: "x < 1"
```
```  1297   have "(1 - x) * (1 + x + x\<^sup>2) = (1 - x^3)"
```
```  1298     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
```
```  1299   also have "... <= 1"
```
```  1300     by (auto simp add: a)
```
```  1301   finally have "(1 - x) * (1 + x + x\<^sup>2) <= 1" .
```
```  1302   moreover have c: "0 < 1 + x + x\<^sup>2"
```
```  1303     by (simp add: add_pos_nonneg a)
```
```  1304   ultimately have "1 - x <= 1 / (1 + x + x\<^sup>2)"
```
```  1305     by (elim mult_imp_le_div_pos)
```
```  1306   also have "... <= 1 / exp x"
```
```  1307     apply (rule divide_left_mono)
```
```  1308     apply (rule exp_bound, rule a)
```
```  1309     apply (rule b [THEN less_imp_le])
```
```  1310     apply simp
```
```  1311     apply (rule mult_pos_pos)
```
```  1312     apply (rule c)
```
```  1313     apply simp
```
```  1314     done
```
```  1315   also have "... = exp (-x)"
```
```  1316     by (auto simp add: exp_minus divide_inverse)
```
```  1317   finally have "1 - x <= exp (- x)" .
```
```  1318   also have "1 - x = exp (ln (1 - x))"
```
```  1319   proof -
```
```  1320     have "0 < 1 - x"
```
```  1321       by (insert b, auto)
```
```  1322     thus ?thesis
```
```  1323       by (auto simp only: exp_ln_iff [THEN sym])
```
```  1324   qed
```
```  1325   finally have "exp (ln (1 - x)) <= exp (- x)" .
```
```  1326   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```  1327 qed
```
```  1328
```
```  1329 lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
```
```  1330   apply (case_tac "0 <= x")
```
```  1331   apply (erule exp_ge_add_one_self_aux)
```
```  1332   apply (case_tac "x <= -1")
```
```  1333   apply (subgoal_tac "1 + x <= 0")
```
```  1334   apply (erule order_trans)
```
```  1335   apply simp
```
```  1336   apply simp
```
```  1337   apply (subgoal_tac "1 + x = exp(ln (1 + x))")
```
```  1338   apply (erule ssubst)
```
```  1339   apply (subst exp_le_cancel_iff)
```
```  1340   apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
```
```  1341   apply simp
```
```  1342   apply (rule ln_one_minus_pos_upper_bound)
```
```  1343   apply auto
```
```  1344 done
```
```  1345
```
```  1346 lemma ln_one_plus_pos_lower_bound: "0 <= x ==> x <= 1 ==> x - x\<^sup>2 <= ln (1 + x)"
```
```  1347 proof -
```
```  1348   assume a: "0 <= x" and b: "x <= 1"
```
```  1349   have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
```
```  1350     by (rule exp_diff)
```
```  1351   also have "... <= (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
```
```  1352     apply (rule divide_right_mono)
```
```  1353     apply (rule exp_bound)
```
```  1354     apply (rule a, rule b)
```
```  1355     apply simp
```
```  1356     done
```
```  1357   also have "... <= (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
```
```  1358     apply (rule divide_left_mono)
```
```  1359     apply (simp add: exp_ge_add_one_self_aux)
```
```  1360     apply (simp add: a)
```
```  1361     apply (simp add: mult_pos_pos add_pos_nonneg)
```
```  1362     done
```
```  1363   also from a have "... <= 1 + x"
```
```  1364     by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
```
```  1365   finally have "exp (x - x\<^sup>2) <= 1 + x" .
```
```  1366   also have "... = exp (ln (1 + x))"
```
```  1367   proof -
```
```  1368     from a have "0 < 1 + x" by auto
```
```  1369     thus ?thesis
```
```  1370       by (auto simp only: exp_ln_iff [THEN sym])
```
```  1371   qed
```
```  1372   finally have "exp (x - x\<^sup>2) <= exp (ln (1 + x))" .
```
```  1373   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```  1374 qed
```
```  1375
```
```  1376 lemma aux5: "x < 1 ==> ln(1 - x) = - ln(1 + x / (1 - x))"
```
```  1377 proof -
```
```  1378   assume a: "x < 1"
```
```  1379   have "ln(1 - x) = - ln(1 / (1 - x))"
```
```  1380   proof -
```
```  1381     have "ln(1 - x) = - (- ln (1 - x))"
```
```  1382       by auto
```
```  1383     also have "- ln(1 - x) = ln 1 - ln(1 - x)"
```
```  1384       by simp
```
```  1385     also have "... = ln(1 / (1 - x))"
```
```  1386       apply (rule ln_div [THEN sym])
```
```  1387       by (insert a, auto)
```
```  1388     finally show ?thesis .
```
```  1389   qed
```
```  1390   also have " 1 / (1 - x) = 1 + x / (1 - x)" using a by(simp add:field_simps)
```
```  1391   finally show ?thesis .
```
```  1392 qed
```
```  1393
```
```  1394 lemma ln_one_minus_pos_lower_bound: "0 <= x ==> x <= (1 / 2) ==>
```
```  1395     - x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1396 proof -
```
```  1397   assume a: "0 <= x" and b: "x <= (1 / 2)"
```
```  1398   from b have c: "x < 1"
```
```  1399     by auto
```
```  1400   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
```
```  1401     by (rule aux5)
```
```  1402   also have "- (x / (1 - x)) <= ..."
```
```  1403   proof -
```
```  1404     have "ln (1 + x / (1 - x)) <= x / (1 - x)"
```
```  1405       apply (rule ln_add_one_self_le_self)
```
```  1406       apply (rule divide_nonneg_pos)
```
```  1407       by (insert a c, auto)
```
```  1408     thus ?thesis
```
```  1409       by auto
```
```  1410   qed
```
```  1411   also have "- (x / (1 - x)) = -x / (1 - x)"
```
```  1412     by auto
```
```  1413   finally have d: "- x / (1 - x) <= ln (1 - x)" .
```
```  1414   have "0 < 1 - x" using a b by simp
```
```  1415   hence e: "-x - 2 * x\<^sup>2 <= - x / (1 - x)"
```
```  1416     using mult_right_le_one_le[of "x*x" "2*x"] a b
```
```  1417     by (simp add:field_simps power2_eq_square)
```
```  1418   from e d show "- x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1419     by (rule order_trans)
```
```  1420 qed
```
```  1421
```
```  1422 lemma ln_add_one_self_le_self2: "-1 < x ==> ln(1 + x) <= x"
```
```  1423   apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
```
```  1424   apply (subst ln_le_cancel_iff)
```
```  1425   apply auto
```
```  1426 done
```
```  1427
```
```  1428 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
```
```  1429     "0 <= x ==> x <= 1 ==> abs(ln (1 + x) - x) <= x\<^sup>2"
```
```  1430 proof -
```
```  1431   assume x: "0 <= x"
```
```  1432   assume x1: "x <= 1"
```
```  1433   from x have "ln (1 + x) <= x"
```
```  1434     by (rule ln_add_one_self_le_self)
```
```  1435   then have "ln (1 + x) - x <= 0"
```
```  1436     by simp
```
```  1437   then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
```
```  1438     by (rule abs_of_nonpos)
```
```  1439   also have "... = x - ln (1 + x)"
```
```  1440     by simp
```
```  1441   also have "... <= x\<^sup>2"
```
```  1442   proof -
```
```  1443     from x x1 have "x - x\<^sup>2 <= ln (1 + x)"
```
```  1444       by (intro ln_one_plus_pos_lower_bound)
```
```  1445     thus ?thesis
```
```  1446       by simp
```
```  1447   qed
```
```  1448   finally show ?thesis .
```
```  1449 qed
```
```  1450
```
```  1451 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
```
```  1452     "-(1 / 2) <= x ==> x <= 0 ==> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
```
```  1453 proof -
```
```  1454   assume a: "-(1 / 2) <= x"
```
```  1455   assume b: "x <= 0"
```
```  1456   have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
```
```  1457     apply (subst abs_of_nonpos)
```
```  1458     apply simp
```
```  1459     apply (rule ln_add_one_self_le_self2)
```
```  1460     using a apply auto
```
```  1461     done
```
```  1462   also have "... <= 2 * x\<^sup>2"
```
```  1463     apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 <= ln (1 - (-x))")
```
```  1464     apply (simp add: algebra_simps)
```
```  1465     apply (rule ln_one_minus_pos_lower_bound)
```
```  1466     using a b apply auto
```
```  1467     done
```
```  1468   finally show ?thesis .
```
```  1469 qed
```
```  1470
```
```  1471 lemma abs_ln_one_plus_x_minus_x_bound:
```
```  1472     "abs x <= 1 / 2 ==> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
```
```  1473   apply (case_tac "0 <= x")
```
```  1474   apply (rule order_trans)
```
```  1475   apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
```
```  1476   apply auto
```
```  1477   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
```
```  1478   apply auto
```
```  1479 done
```
```  1480
```
```  1481 lemma ln_x_over_x_mono: "exp 1 <= x ==> x <= y ==> (ln y / y) <= (ln x / x)"
```
```  1482 proof -
```
```  1483   assume x: "exp 1 <= x" "x <= y"
```
```  1484   moreover have "0 < exp (1::real)" by simp
```
```  1485   ultimately have a: "0 < x" and b: "0 < y"
```
```  1486     by (fast intro: less_le_trans order_trans)+
```
```  1487   have "x * ln y - x * ln x = x * (ln y - ln x)"
```
```  1488     by (simp add: algebra_simps)
```
```  1489   also have "... = x * ln(y / x)"
```
```  1490     by (simp only: ln_div a b)
```
```  1491   also have "y / x = (x + (y - x)) / x"
```
```  1492     by simp
```
```  1493   also have "... = 1 + (y - x) / x"
```
```  1494     using x a by (simp add: field_simps)
```
```  1495   also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
```
```  1496     apply (rule mult_left_mono)
```
```  1497     apply (rule ln_add_one_self_le_self)
```
```  1498     apply (rule divide_nonneg_pos)
```
```  1499     using x a apply simp_all
```
```  1500     done
```
```  1501   also have "... = y - x" using a by simp
```
```  1502   also have "... = (y - x) * ln (exp 1)" by simp
```
```  1503   also have "... <= (y - x) * ln x"
```
```  1504     apply (rule mult_left_mono)
```
```  1505     apply (subst ln_le_cancel_iff)
```
```  1506     apply fact
```
```  1507     apply (rule a)
```
```  1508     apply (rule x)
```
```  1509     using x apply simp
```
```  1510     done
```
```  1511   also have "... = y * ln x - x * ln x"
```
```  1512     by (rule left_diff_distrib)
```
```  1513   finally have "x * ln y <= y * ln x"
```
```  1514     by arith
```
```  1515   then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
```
```  1516   also have "... = y * (ln x / x)" by simp
```
```  1517   finally show ?thesis using b by (simp add: field_simps)
```
```  1518 qed
```
```  1519
```
```  1520 lemma ln_le_minus_one:
```
```  1521   "0 < x \<Longrightarrow> ln x \<le> x - 1"
```
```  1522   using exp_ge_add_one_self[of "ln x"] by simp
```
```  1523
```
```  1524 lemma ln_eq_minus_one:
```
```  1525   assumes "0 < x" "ln x = x - 1" shows "x = 1"
```
```  1526 proof -
```
```  1527   let "?l y" = "ln y - y + 1"
```
```  1528   have D: "\<And>x. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
```
```  1529     by (auto intro!: DERIV_intros)
```
```  1530
```
```  1531   show ?thesis
```
```  1532   proof (cases rule: linorder_cases)
```
```  1533     assume "x < 1"
```
```  1534     from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast
```
```  1535     from `x < a` have "?l x < ?l a"
```
```  1536     proof (rule DERIV_pos_imp_increasing, safe)
```
```  1537       fix y assume "x \<le> y" "y \<le> a"
```
```  1538       with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y"
```
```  1539         by (auto simp: field_simps)
```
```  1540       with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
```
```  1541         by auto
```
```  1542     qed
```
```  1543     also have "\<dots> \<le> 0"
```
```  1544       using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps)
```
```  1545     finally show "x = 1" using assms by auto
```
```  1546   next
```
```  1547     assume "1 < x"
```
```  1548     from dense[OF `1 < x`] obtain a where "1 < a" "a < x" by blast
```
```  1549     from `a < x` have "?l x < ?l a"
```
```  1550     proof (rule DERIV_neg_imp_decreasing, safe)
```
```  1551       fix y assume "a \<le> y" "y \<le> x"
```
```  1552       with `1 < a` have "1 / y - 1 < 0" "0 < y"
```
```  1553         by (auto simp: field_simps)
```
```  1554       with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
```
```  1555         by blast
```
```  1556     qed
```
```  1557     also have "\<dots> \<le> 0"
```
```  1558       using ln_le_minus_one `1 < a` by (auto simp: field_simps)
```
```  1559     finally show "x = 1" using assms by auto
```
```  1560   qed simp
```
```  1561 qed
```
```  1562
```
```  1563 lemma exp_at_bot: "(exp ---> (0::real)) at_bot"
```
```  1564   unfolding tendsto_Zfun_iff
```
```  1565 proof (rule ZfunI, simp add: eventually_at_bot_dense)
```
```  1566   fix r :: real assume "0 < r"
```
```  1567   { fix x assume "x < ln r"
```
```  1568     then have "exp x < exp (ln r)"
```
```  1569       by simp
```
```  1570     with `0 < r` have "exp x < r"
```
```  1571       by simp }
```
```  1572   then show "\<exists>k. \<forall>n<k. exp n < r" by auto
```
```  1573 qed
```
```  1574
```
```  1575 lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
```
```  1576   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"])
```
```  1577      (auto intro: eventually_gt_at_top)
```
```  1578
```
```  1579 lemma ln_at_0: "LIM x at_right 0. ln x :> at_bot"
```
```  1580   by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  1581      (auto simp: eventually_at_filter)
```
```  1582
```
```  1583 lemma ln_at_top: "LIM x at_top. ln x :> at_top"
```
```  1584   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  1585      (auto intro: eventually_gt_at_top)
```
```  1586
```
```  1587 lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) ---> (0::real)) at_top"
```
```  1588 proof (induct k)
```
```  1589   show "((\<lambda>x. x ^ 0 / exp x) ---> (0::real)) at_top"
```
```  1590     by (simp add: inverse_eq_divide[symmetric])
```
```  1591        (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
```
```  1592               at_top_le_at_infinity order_refl)
```
```  1593 next
```
```  1594   case (Suc k)
```
```  1595   show ?case
```
```  1596   proof (rule lhospital_at_top_at_top)
```
```  1597     show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"
```
```  1598       by eventually_elim (intro DERIV_intros, simp, simp)
```
```  1599     show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top"
```
```  1600       by eventually_elim (auto intro!: DERIV_intros)
```
```  1601     show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
```
```  1602       by auto
```
```  1603     from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
```
```  1604     show "((\<lambda>x. real (Suc k) * x ^ k / exp x) ---> 0) at_top"
```
```  1605       by simp
```
```  1606   qed (rule exp_at_top)
```
```  1607 qed
```
```  1608
```
```  1609
```
```  1610 definition
```
```  1611   powr  :: "[real,real] => real"     (infixr "powr" 80) where
```
```  1612     --{*exponentation with real exponent*}
```
```  1613   "x powr a = exp(a * ln x)"
```
```  1614
```
```  1615 definition
```
```  1616   log :: "[real,real] => real" where
```
```  1617     --{*logarithm of @{term x} to base @{term a}*}
```
```  1618   "log a x = ln x / ln a"
```
```  1619
```
```  1620
```
```  1621 lemma tendsto_log [tendsto_intros]:
```
```  1622   "\<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"
```
```  1623   unfolding log_def by (intro tendsto_intros) auto
```
```  1624
```
```  1625 lemma continuous_log:
```
```  1626   assumes "continuous F f" and "continuous F g" and "0 < f (Lim F (\<lambda>x. x))" and "f (Lim F (\<lambda>x. x)) \<noteq> 1" and "0 < g (Lim F (\<lambda>x. x))"
```
```  1627   shows "continuous F (\<lambda>x. log (f x) (g x))"
```
```  1628   using assms unfolding continuous_def by (rule tendsto_log)
```
```  1629
```
```  1630 lemma continuous_at_within_log[continuous_intros]:
```
```  1631   assumes "continuous (at a within s) f" "continuous (at a within s) g" and "0 < f a" and "f a \<noteq> 1" and "0 < g a"
```
```  1632   shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
```
```  1633   using assms unfolding continuous_within by (rule tendsto_log)
```
```  1634
```
```  1635 lemma isCont_log[continuous_intros, simp]:
```
```  1636   assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
```
```  1637   shows "isCont (\<lambda>x. log (f x) (g x)) a"
```
```  1638   using assms unfolding continuous_at by (rule tendsto_log)
```
```  1639
```
```  1640 lemma continuous_on_log[continuous_on_intros]:
```
```  1641   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
```
```  1642   shows "continuous_on s (\<lambda>x. log (f x) (g x))"
```
```  1643   using assms unfolding continuous_on_def by (fast intro: tendsto_log)
```
```  1644
```
```  1645 lemma powr_one_eq_one [simp]: "1 powr a = 1"
```
```  1646 by (simp add: powr_def)
```
```  1647
```
```  1648 lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
```
```  1649 by (simp add: powr_def)
```
```  1650
```
```  1651 lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
```
```  1652 by (simp add: powr_def)
```
```  1653 declare powr_one_gt_zero_iff [THEN iffD2, simp]
```
```  1654
```
```  1655 lemma powr_mult:
```
```  1656       "[| 0 < x; 0 < y |] ==> (x * y) powr a = (x powr a) * (y powr a)"
```
```  1657 by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
```
```  1658
```
```  1659 lemma powr_gt_zero [simp]: "0 < x powr a"
```
```  1660 by (simp add: powr_def)
```
```  1661
```
```  1662 lemma powr_ge_pzero [simp]: "0 <= x powr y"
```
```  1663 by (rule order_less_imp_le, rule powr_gt_zero)
```
```  1664
```
```  1665 lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
```
```  1666 by (simp add: powr_def)
```
```  1667
```
```  1668 lemma powr_divide:
```
```  1669      "[| 0 < x; 0 < y |] ==> (x / y) powr a = (x powr a)/(y powr a)"
```
```  1670 apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
```
```  1671 apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
```
```  1672 done
```
```  1673
```
```  1674 lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
```
```  1675   apply (simp add: powr_def)
```
```  1676   apply (subst exp_diff [THEN sym])
```
```  1677   apply (simp add: left_diff_distrib)
```
```  1678 done
```
```  1679
```
```  1680 lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
```
```  1681 by (simp add: powr_def exp_add [symmetric] distrib_right)
```
```  1682
```
```  1683 lemma powr_mult_base:
```
```  1684   "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
```
```  1685 using assms by (auto simp: powr_add)
```
```  1686
```
```  1687 lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
```
```  1688 by (simp add: powr_def)
```
```  1689
```
```  1690 lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
```
```  1691 by (simp add: powr_powr mult_commute)
```
```  1692
```
```  1693 lemma powr_minus: "x powr (-a) = inverse (x powr a)"
```
```  1694 by (simp add: powr_def exp_minus [symmetric])
```
```  1695
```
```  1696 lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
```
```  1697 by (simp add: divide_inverse powr_minus)
```
```  1698
```
```  1699 lemma powr_less_mono: "[| a < b; 1 < x |] ==> x powr a < x powr b"
```
```  1700 by (simp add: powr_def)
```
```  1701
```
```  1702 lemma powr_less_cancel: "[| x powr a < x powr b; 1 < x |] ==> a < b"
```
```  1703 by (simp add: powr_def)
```
```  1704
```
```  1705 lemma powr_less_cancel_iff [simp]: "1 < x ==> (x powr a < x powr b) = (a < b)"
```
```  1706 by (blast intro: powr_less_cancel powr_less_mono)
```
```  1707
```
```  1708 lemma powr_le_cancel_iff [simp]: "1 < x ==> (x powr a \<le> x powr b) = (a \<le> b)"
```
```  1709 by (simp add: linorder_not_less [symmetric])
```
```  1710
```
```  1711 lemma log_ln: "ln x = log (exp(1)) x"
```
```  1712 by (simp add: log_def)
```
```  1713
```
```  1714 lemma DERIV_log: assumes "x > 0" shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
```
```  1715 proof -
```
```  1716   def lb \<equiv> "1 / ln b"
```
```  1717   moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
```
```  1718     using `x > 0` by (auto intro!: DERIV_intros)
```
```  1719   ultimately show ?thesis
```
```  1720     by (simp add: log_def)
```
```  1721 qed
```
```  1722
```
```  1723 lemmas DERIV_log[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  1724
```
```  1725 lemma powr_log_cancel [simp]:
```
```  1726      "[| 0 < a; a \<noteq> 1; 0 < x |] ==> a powr (log a x) = x"
```
```  1727 by (simp add: powr_def log_def)
```
```  1728
```
```  1729 lemma log_powr_cancel [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a (a powr y) = y"
```
```  1730 by (simp add: log_def powr_def)
```
```  1731
```
```  1732 lemma log_mult:
```
```  1733      "[| 0 < a; a \<noteq> 1; 0 < x; 0 < y |]
```
```  1734       ==> log a (x * y) = log a x + log a y"
```
```  1735 by (simp add: log_def ln_mult divide_inverse distrib_right)
```
```  1736
```
```  1737 lemma log_eq_div_ln_mult_log:
```
```  1738      "[| 0 < a; a \<noteq> 1; 0 < b; b \<noteq> 1; 0 < x |]
```
```  1739       ==> log a x = (ln b/ln a) * log b x"
```
```  1740 by (simp add: log_def divide_inverse)
```
```  1741
```
```  1742 text{*Base 10 logarithms*}
```
```  1743 lemma log_base_10_eq1: "0 < x ==> log 10 x = (ln (exp 1) / ln 10) * ln x"
```
```  1744 by (simp add: log_def)
```
```  1745
```
```  1746 lemma log_base_10_eq2: "0 < x ==> log 10 x = (log 10 (exp 1)) * ln x"
```
```  1747 by (simp add: log_def)
```
```  1748
```
```  1749 lemma log_one [simp]: "log a 1 = 0"
```
```  1750 by (simp add: log_def)
```
```  1751
```
```  1752 lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
```
```  1753 by (simp add: log_def)
```
```  1754
```
```  1755 lemma log_inverse:
```
```  1756      "[| 0 < a; a \<noteq> 1; 0 < x |] ==> log a (inverse x) = - log a x"
```
```  1757 apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
```
```  1758 apply (simp add: log_mult [symmetric])
```
```  1759 done
```
```  1760
```
```  1761 lemma log_divide:
```
```  1762      "[|0 < a; a \<noteq> 1; 0 < x; 0 < y|] ==> log a (x/y) = log a x - log a y"
```
```  1763 by (simp add: log_mult divide_inverse log_inverse)
```
```  1764
```
```  1765 lemma log_less_cancel_iff [simp]:
```
```  1766      "[| 1 < a; 0 < x; 0 < y |] ==> (log a x < log a y) = (x < y)"
```
```  1767 apply safe
```
```  1768 apply (rule_tac [2] powr_less_cancel)
```
```  1769 apply (drule_tac a = "log a x" in powr_less_mono, auto)
```
```  1770 done
```
```  1771
```
```  1772 lemma log_inj: assumes "1 < b" shows "inj_on (log b) {0 <..}"
```
```  1773 proof (rule inj_onI, simp)
```
```  1774   fix x y assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
```
```  1775   show "x = y"
```
```  1776   proof (cases rule: linorder_cases)
```
```  1777     assume "x < y" hence "log b x < log b y"
```
```  1778       using log_less_cancel_iff[OF `1 < b`] pos by simp
```
```  1779     thus ?thesis using * by simp
```
```  1780   next
```
```  1781     assume "y < x" hence "log b y < log b x"
```
```  1782       using log_less_cancel_iff[OF `1 < b`] pos by simp
```
```  1783     thus ?thesis using * by simp
```
```  1784   qed simp
```
```  1785 qed
```
```  1786
```
```  1787 lemma log_le_cancel_iff [simp]:
```
```  1788      "[| 1 < a; 0 < x; 0 < y |] ==> (log a x \<le> log a y) = (x \<le> y)"
```
```  1789 by (simp add: linorder_not_less [symmetric])
```
```  1790
```
```  1791 lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
```
```  1792   using log_less_cancel_iff[of a 1 x] by simp
```
```  1793
```
```  1794 lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
```
```  1795   using log_le_cancel_iff[of a 1 x] by simp
```
```  1796
```
```  1797 lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
```
```  1798   using log_less_cancel_iff[of a x 1] by simp
```
```  1799
```
```  1800 lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
```
```  1801   using log_le_cancel_iff[of a x 1] by simp
```
```  1802
```
```  1803 lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
```
```  1804   using log_less_cancel_iff[of a a x] by simp
```
```  1805
```
```  1806 lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
```
```  1807   using log_le_cancel_iff[of a a x] by simp
```
```  1808
```
```  1809 lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
```
```  1810   using log_less_cancel_iff[of a x a] by simp
```
```  1811
```
```  1812 lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
```
```  1813   using log_le_cancel_iff[of a x a] by simp
```
```  1814
```
```  1815 lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
```
```  1816   apply (induct n, simp)
```
```  1817   apply (subgoal_tac "real(Suc n) = real n + 1")
```
```  1818   apply (erule ssubst)
```
```  1819   apply (subst powr_add, simp, simp)
```
```  1820 done
```
```  1821
```
```  1822 lemma powr_realpow_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x^(numeral n)"
```
```  1823   unfolding real_of_nat_numeral[symmetric] by (rule powr_realpow)
```
```  1824
```
```  1825 lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
```
```  1826   apply (case_tac "x = 0", simp, simp)
```
```  1827   apply (rule powr_realpow [THEN sym], simp)
```
```  1828 done
```
```  1829
```
```  1830 lemma powr_int:
```
```  1831   assumes "x > 0"
```
```  1832   shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
```
```  1833 proof cases
```
```  1834   assume "i < 0"
```
```  1835   have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
```
```  1836   show ?thesis using `i < 0` `x > 0` by (simp add: r field_simps powr_realpow[symmetric])
```
```  1837 qed (simp add: assms powr_realpow[symmetric])
```
```  1838
```
```  1839 lemma powr_numeral: "0 < x \<Longrightarrow> x powr numeral n = x^numeral n"
```
```  1840   using powr_realpow[of x "numeral n"] by simp
```
```  1841
```
```  1842 lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr neg_numeral n = 1 / x^numeral n"
```
```  1843   using powr_int[of x "neg_numeral n"] by simp
```
```  1844
```
```  1845 lemma root_powr_inverse:
```
```  1846   "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
```
```  1847   by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
```
```  1848
```
```  1849 lemma ln_powr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
```
```  1850 by (unfold powr_def, simp)
```
```  1851
```
```  1852 lemma log_powr: "0 < x ==> 0 \<le> y ==> log b (x powr y) = y * log b x"
```
```  1853   apply (case_tac "y = 0")
```
```  1854   apply force
```
```  1855   apply (auto simp add: log_def ln_powr field_simps)
```
```  1856 done
```
```  1857
```
```  1858 lemma log_nat_power: "0 < x ==> log b (x^n) = real n * log b x"
```
```  1859   apply (subst powr_realpow [symmetric])
```
```  1860   apply (auto simp add: log_powr)
```
```  1861 done
```
```  1862
```
```  1863 lemma ln_bound: "1 <= x ==> ln x <= x"
```
```  1864   apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
```
```  1865   apply simp
```
```  1866   apply (rule ln_add_one_self_le_self, simp)
```
```  1867 done
```
```  1868
```
```  1869 lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
```
```  1870   apply (case_tac "x = 1", simp)
```
```  1871   apply (case_tac "a = b", simp)
```
```  1872   apply (rule order_less_imp_le)
```
```  1873   apply (rule powr_less_mono, auto)
```
```  1874 done
```
```  1875
```
```  1876 lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
```
```  1877   apply (subst powr_zero_eq_one [THEN sym])
```
```  1878   apply (rule powr_mono, assumption+)
```
```  1879 done
```
```  1880
```
```  1881 lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a <
```
```  1882     y powr a"
```
```  1883   apply (unfold powr_def)
```
```  1884   apply (rule exp_less_mono)
```
```  1885   apply (rule mult_strict_left_mono)
```
```  1886   apply (subst ln_less_cancel_iff, assumption)
```
```  1887   apply (rule order_less_trans)
```
```  1888   prefer 2
```
```  1889   apply assumption+
```
```  1890 done
```
```  1891
```
```  1892 lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a <
```
```  1893     x powr a"
```
```  1894   apply (unfold powr_def)
```
```  1895   apply (rule exp_less_mono)
```
```  1896   apply (rule mult_strict_left_mono_neg)
```
```  1897   apply (subst ln_less_cancel_iff)
```
```  1898   apply assumption
```
```  1899   apply (rule order_less_trans)
```
```  1900   prefer 2
```
```  1901   apply assumption+
```
```  1902 done
```
```  1903
```
```  1904 lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
```
```  1905   apply (case_tac "a = 0", simp)
```
```  1906   apply (case_tac "x = y", simp)
```
```  1907   apply (rule order_less_imp_le)
```
```  1908   apply (rule powr_less_mono2, auto)
```
```  1909 done
```
```  1910
```
```  1911 lemma powr_inj:
```
```  1912   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
```
```  1913   unfolding powr_def exp_inj_iff by simp
```
```  1914
```
```  1915 lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
```
```  1916   apply (rule mult_imp_le_div_pos)
```
```  1917   apply (assumption)
```
```  1918   apply (subst mult_commute)
```
```  1919   apply (subst ln_powr [THEN sym])
```
```  1920   apply auto
```
```  1921   apply (rule ln_bound)
```
```  1922   apply (erule ge_one_powr_ge_zero)
```
```  1923   apply (erule order_less_imp_le)
```
```  1924 done
```
```  1925
```
```  1926 lemma ln_powr_bound2:
```
```  1927   assumes "1 < x" and "0 < a"
```
```  1928   shows "(ln x) powr a <= (a powr a) * x"
```
```  1929 proof -
```
```  1930   from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
```
```  1931     apply (intro ln_powr_bound)
```
```  1932     apply (erule order_less_imp_le)
```
```  1933     apply (rule divide_pos_pos)
```
```  1934     apply simp_all
```
```  1935     done
```
```  1936   also have "... = a * (x powr (1 / a))"
```
```  1937     by simp
```
```  1938   finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
```
```  1939     apply (intro powr_mono2)
```
```  1940     apply (rule order_less_imp_le, rule assms)
```
```  1941     apply (rule ln_gt_zero)
```
```  1942     apply (rule assms)
```
```  1943     apply assumption
```
```  1944     done
```
```  1945   also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
```
```  1946     apply (rule powr_mult)
```
```  1947     apply (rule assms)
```
```  1948     apply (rule powr_gt_zero)
```
```  1949     done
```
```  1950   also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
```
```  1951     by (rule powr_powr)
```
```  1952   also have "... = x"
```
```  1953     apply simp
```
```  1954     apply (subgoal_tac "a ~= 0")
```
```  1955     using assms apply auto
```
```  1956     done
```
```  1957   finally show ?thesis .
```
```  1958 qed
```
```  1959
```
```  1960 lemma tendsto_powr [tendsto_intros]:
```
```  1961   "\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x powr g x) ---> a powr b) F"
```
```  1962   unfolding powr_def by (intro tendsto_intros)
```
```  1963
```
```  1964 lemma continuous_powr:
```
```  1965   assumes "continuous F f" and "continuous F g" and "0 < f (Lim F (\<lambda>x. x))"
```
```  1966   shows "continuous F (\<lambda>x. (f x) powr (g x))"
```
```  1967   using assms unfolding continuous_def by (rule tendsto_powr)
```
```  1968
```
```  1969 lemma continuous_at_within_powr[continuous_intros]:
```
```  1970   assumes "continuous (at a within s) f" "continuous (at a within s) g" and "0 < f a"
```
```  1971   shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
```
```  1972   using assms unfolding continuous_within by (rule tendsto_powr)
```
```  1973
```
```  1974 lemma isCont_powr[continuous_intros, simp]:
```
```  1975   assumes "isCont f a" "isCont g a" "0 < f a"
```
```  1976   shows "isCont (\<lambda>x. (f x) powr g x) a"
```
```  1977   using assms unfolding continuous_at by (rule tendsto_powr)
```
```  1978
```
```  1979 lemma continuous_on_powr[continuous_on_intros]:
```
```  1980   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. 0 < f x"
```
```  1981   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
```
```  1982   using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
```
```  1983
```
```  1984 (* FIXME: generalize by replacing d by with g x and g ---> d? *)
```
```  1985 lemma tendsto_zero_powrI:
```
```  1986   assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> 0) F"
```
```  1987   assumes "0 < d"
```
```  1988   shows "((\<lambda>x. f x powr d) ---> 0) F"
```
```  1989 proof (rule tendstoI)
```
```  1990   fix e :: real assume "0 < e"
```
```  1991   def Z \<equiv> "e powr (1 / d)"
```
```  1992   with `0 < e` have "0 < Z" by simp
```
```  1993   with assms have "eventually (\<lambda>x. 0 < f x \<and> dist (f x) 0 < Z) F"
```
```  1994     by (intro eventually_conj tendstoD)
```
```  1995   moreover
```
```  1996   from assms have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> x powr d < Z powr d"
```
```  1997     by (intro powr_less_mono2) (auto simp: dist_real_def)
```
```  1998   with assms `0 < e` have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> dist (x powr d) 0 < e"
```
```  1999     unfolding dist_real_def Z_def by (auto simp: powr_powr)
```
```  2000   ultimately
```
```  2001   show "eventually (\<lambda>x. dist (f x powr d) 0 < e) F" by (rule eventually_elim1)
```
```  2002 qed
```
```  2003
```
```  2004 lemma tendsto_neg_powr:
```
```  2005   assumes "s < 0" and "LIM x F. f x :> at_top"
```
```  2006   shows "((\<lambda>x. f x powr s) ---> 0) F"
```
```  2007 proof (rule tendstoI)
```
```  2008   fix e :: real assume "0 < e"
```
```  2009   def Z \<equiv> "e powr (1 / s)"
```
```  2010   from assms have "eventually (\<lambda>x. Z < f x) F"
```
```  2011     by (simp add: filterlim_at_top_dense)
```
```  2012   moreover
```
```  2013   from assms have "\<And>x. Z < x \<Longrightarrow> x powr s < Z powr s"
```
```  2014     by (auto simp: Z_def intro!: powr_less_mono2_neg)
```
```  2015   with assms `0 < e` have "\<And>x. Z < x \<Longrightarrow> dist (x powr s) 0 < e"
```
```  2016     by (simp add: powr_powr Z_def dist_real_def)
```
```  2017   ultimately
```
```  2018   show "eventually (\<lambda>x. dist (f x powr s) 0 < e) F" by (rule eventually_elim1)
```
```  2019 qed
```
```  2020
```
```  2021 subsection {* Sine and Cosine *}
```
```  2022
```
```  2023 definition sin_coeff :: "nat \<Rightarrow> real" where
```
```  2024   "sin_coeff = (\<lambda>n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"
```
```  2025
```
```  2026 definition cos_coeff :: "nat \<Rightarrow> real" where
```
```  2027   "cos_coeff = (\<lambda>n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"
```
```  2028
```
```  2029 definition sin :: "real \<Rightarrow> real" where
```
```  2030   "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
```
```  2031
```
```  2032 definition cos :: "real \<Rightarrow> real" where
```
```  2033   "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
```
```  2034
```
```  2035 lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
```
```  2036   unfolding sin_coeff_def by simp
```
```  2037
```
```  2038 lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
```
```  2039   unfolding cos_coeff_def by simp
```
```  2040
```
```  2041 lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
```
```  2042   unfolding cos_coeff_def sin_coeff_def
```
```  2043   by (simp del: mult_Suc)
```
```  2044
```
```  2045 lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
```
```  2046   unfolding cos_coeff_def sin_coeff_def
```
```  2047   by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)
```
```  2048
```
```  2049 lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
```
```  2050 unfolding sin_coeff_def
```
```  2051 apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
```
```  2052 apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2053 done
```
```  2054
```
```  2055 lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"
```
```  2056 unfolding cos_coeff_def
```
```  2057 apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
```
```  2058 apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2059 done
```
```  2060
```
```  2061 lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
```
```  2062 unfolding sin_def by (rule summable_sin [THEN summable_sums])
```
```  2063
```
```  2064 lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
```
```  2065 unfolding cos_def by (rule summable_cos [THEN summable_sums])
```
```  2066
```
```  2067 lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
```
```  2068   by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  2069
```
```  2070 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
```
```  2071   by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  2072
```
```  2073 text{*Now at last we can get the derivatives of exp, sin and cos*}
```
```  2074
```
```  2075 lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
```
```  2076   unfolding sin_def cos_def
```
```  2077   apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
```
```  2078   apply (simp_all add: diffs_sin_coeff diffs_cos_coeff
```
```  2079     summable_minus summable_sin summable_cos)
```
```  2080   done
```
```  2081
```
```  2082 declare DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  2083
```
```  2084 lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
```
```  2085   unfolding cos_def sin_def
```
```  2086   apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
```
```  2087   apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus
```
```  2088     summable_minus summable_sin summable_cos suminf_minus)
```
```  2089   done
```
```  2090
```
```  2091 declare DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  2092
```
```  2093 lemma isCont_sin: "isCont sin x"
```
```  2094   by (rule DERIV_sin [THEN DERIV_isCont])
```
```  2095
```
```  2096 lemma isCont_cos: "isCont cos x"
```
```  2097   by (rule DERIV_cos [THEN DERIV_isCont])
```
```  2098
```
```  2099 lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
```
```  2100   by (rule isCont_o2 [OF _ isCont_sin])
```
```  2101
```
```  2102 lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
```
```  2103   by (rule isCont_o2 [OF _ isCont_cos])
```
```  2104
```
```  2105 lemma tendsto_sin [tendsto_intros]:
```
```  2106   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
```
```  2107   by (rule isCont_tendsto_compose [OF isCont_sin])
```
```  2108
```
```  2109 lemma tendsto_cos [tendsto_intros]:
```
```  2110   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
```
```  2111   by (rule isCont_tendsto_compose [OF isCont_cos])
```
```  2112
```
```  2113 lemma continuous_sin [continuous_intros]:
```
```  2114   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
```
```  2115   unfolding continuous_def by (rule tendsto_sin)
```
```  2116
```
```  2117 lemma continuous_on_sin [continuous_on_intros]:
```
```  2118   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
```
```  2119   unfolding continuous_on_def by (auto intro: tendsto_sin)
```
```  2120
```
```  2121 lemma continuous_cos [continuous_intros]:
```
```  2122   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
```
```  2123   unfolding continuous_def by (rule tendsto_cos)
```
```  2124
```
```  2125 lemma continuous_on_cos [continuous_on_intros]:
```
```  2126   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
```
```  2127   unfolding continuous_on_def by (auto intro: tendsto_cos)
```
```  2128
```
```  2129 subsection {* Properties of Sine and Cosine *}
```
```  2130
```
```  2131 lemma sin_zero [simp]: "sin 0 = 0"
```
```  2132   unfolding sin_def sin_coeff_def by (simp add: powser_zero)
```
```  2133
```
```  2134 lemma cos_zero [simp]: "cos 0 = 1"
```
```  2135   unfolding cos_def cos_coeff_def by (simp add: powser_zero)
```
```  2136
```
```  2137 lemma sin_cos_squared_add [simp]: "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
```
```  2138 proof -
```
```  2139   have "\<forall>x. DERIV (\<lambda>x. (sin x)\<^sup>2 + (cos x)\<^sup>2) x :> 0"
```
```  2140     by (auto intro!: DERIV_intros)
```
```  2141   hence "(sin x)\<^sup>2 + (cos x)\<^sup>2 = (sin 0)\<^sup>2 + (cos 0)\<^sup>2"
```
```  2142     by (rule DERIV_isconst_all)
```
```  2143   thus "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1" by simp
```
```  2144 qed
```
```  2145
```
```  2146 lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
```
```  2147   by (subst add_commute, rule sin_cos_squared_add)
```
```  2148
```
```  2149 lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
```
```  2150   using sin_cos_squared_add2 [unfolded power2_eq_square] .
```
```  2151
```
```  2152 lemma sin_squared_eq: "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
```
```  2153   unfolding eq_diff_eq by (rule sin_cos_squared_add)
```
```  2154
```
```  2155 lemma cos_squared_eq: "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
```
```  2156   unfolding eq_diff_eq by (rule sin_cos_squared_add2)
```
```  2157
```
```  2158 lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
```
```  2159   by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
```
```  2160
```
```  2161 lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
```
```  2162   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  2163
```
```  2164 lemma sin_le_one [simp]: "sin x \<le> 1"
```
```  2165   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  2166
```
```  2167 lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
```
```  2168   by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
```
```  2169
```
```  2170 lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
```
```  2171   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  2172
```
```  2173 lemma cos_le_one [simp]: "cos x \<le> 1"
```
```  2174   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  2175
```
```  2176 lemma DERIV_fun_pow: "DERIV g x :> m ==>
```
```  2177       DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
```
```  2178   by (auto intro!: DERIV_intros)
```
```  2179
```
```  2180 lemma DERIV_fun_exp:
```
```  2181      "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
```
```  2182   by (auto intro!: DERIV_intros)
```
```  2183
```
```  2184 lemma DERIV_fun_sin:
```
```  2185      "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
```
```  2186   by (auto intro!: DERIV_intros)
```
```  2187
```
```  2188 lemma DERIV_fun_cos:
```
```  2189      "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
```
```  2190   by (auto intro!: DERIV_intros)
```
```  2191
```
```  2192 lemma sin_cos_add_lemma:
```
```  2193      "(sin (x + y) - (sin x * cos y + cos x * sin y))\<^sup>2 +
```
```  2194       (cos (x + y) - (cos x * cos y - sin x * sin y))\<^sup>2 = 0"
```
```  2195   (is "?f x = 0")
```
```  2196 proof -
```
```  2197   have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
```
```  2198     by (auto intro!: DERIV_intros simp add: algebra_simps)
```
```  2199   hence "?f x = ?f 0"
```
```  2200     by (rule DERIV_isconst_all)
```
```  2201   thus ?thesis by simp
```
```  2202 qed
```
```  2203
```
```  2204 lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
```
```  2205   using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
```
```  2206
```
```  2207 lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
```
```  2208   using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
```
```  2209
```
```  2210 lemma sin_cos_minus_lemma:
```
```  2211   "(sin(-x) + sin(x))\<^sup>2 + (cos(-x) - cos(x))\<^sup>2 = 0" (is "?f x = 0")
```
```  2212 proof -
```
```  2213   have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
```
```  2214     by (auto intro!: DERIV_intros simp add: algebra_simps)
```
```  2215   hence "?f x = ?f 0"
```
```  2216     by (rule DERIV_isconst_all)
```
```  2217   thus ?thesis by simp
```
```  2218 qed
```
```  2219
```
```  2220 lemma sin_minus [simp]: "sin (-x) = -sin(x)"
```
```  2221   using sin_cos_minus_lemma [where x=x] by simp
```
```  2222
```
```  2223 lemma cos_minus [simp]: "cos (-x) = cos(x)"
```
```  2224   using sin_cos_minus_lemma [where x=x] by simp
```
```  2225
```
```  2226 lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
```
```  2227   by (simp add: diff_minus sin_add)
```
```  2228
```
```  2229 lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
```
```  2230   by (simp add: sin_diff mult_commute)
```
```  2231
```
```  2232 lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
```
```  2233   by (simp add: diff_minus cos_add)
```
```  2234
```
```  2235 lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
```
```  2236   by (simp add: cos_diff mult_commute)
```
```  2237
```
```  2238 lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
```
```  2239   using sin_add [where x=x and y=x] by simp
```
```  2240
```
```  2241 lemma cos_double: "cos(2* x) = ((cos x)\<^sup>2) - ((sin x)\<^sup>2)"
```
```  2242   using cos_add [where x=x and y=x]
```
```  2243   by (simp add: power2_eq_square)
```
```  2244
```
```  2245
```
```  2246 subsection {* The Constant Pi *}
```
```  2247
```
```  2248 definition pi :: "real" where
```
```  2249   "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
```
```  2250
```
```  2251 text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
```
```  2252    hence define pi.*}
```
```  2253
```
```  2254 lemma sin_paired:
```
```  2255      "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1))
```
```  2256       sums  sin x"
```
```  2257 proof -
```
```  2258   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
```
```  2259     by (rule sin_converges [THEN sums_group], simp)
```
```  2260   thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)
```
```  2261 qed
```
```  2262
```
```  2263 lemma sin_gt_zero:
```
```  2264   assumes "0 < x" and "x < 2" shows "0 < sin x"
```
```  2265 proof -
```
```  2266   let ?f = "\<lambda>n. \<Sum>k = n*2..<n*2+2. -1 ^ k / real (fact (2*k+1)) * x^(2*k+1)"
```
```  2267   have pos: "\<forall>n. 0 < ?f n"
```
```  2268   proof
```
```  2269     fix n :: nat
```
```  2270     let ?k2 = "real (Suc (Suc (4 * n)))"
```
```  2271     let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
```
```  2272     have "x * x < ?k2 * ?k3"
```
```  2273       using assms by (intro mult_strict_mono', simp_all)
```
```  2274     hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
```
```  2275       by (intro mult_strict_right_mono zero_less_power `0 < x`)
```
```  2276     thus "0 < ?f n"
```
```  2277       by (simp del: mult_Suc,
```
```  2278         simp add: less_divide_eq mult_pos_pos field_simps del: mult_Suc)
```
```  2279   qed
```
```  2280   have sums: "?f sums sin x"
```
```  2281     by (rule sin_paired [THEN sums_group], simp)
```
```  2282   show "0 < sin x"
```
```  2283     unfolding sums_unique [OF sums]
```
```  2284     using sums_summable [OF sums] pos
```
```  2285     by (rule suminf_gt_zero)
```
```  2286 qed
```
```  2287
```
```  2288 lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
```
```  2289 apply (cut_tac x = x in sin_gt_zero)
```
```  2290 apply (auto simp add: cos_squared_eq cos_double)
```
```  2291 done
```
```  2292
```
```  2293 lemma cos_paired:
```
```  2294      "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
```
```  2295 proof -
```
```  2296   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
```
```  2297     by (rule cos_converges [THEN sums_group], simp)
```
```  2298   thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)
```
```  2299 qed
```
```  2300
```
```  2301 lemma real_mult_inverse_cancel:
```
```  2302      "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
```
```  2303       ==> inverse x * y < inverse x1 * u"
```
```  2304 apply (rule_tac c=x in mult_less_imp_less_left)
```
```  2305 apply (auto simp add: mult_assoc [symmetric])
```
```  2306 apply (simp (no_asm) add: mult_ac)
```
```  2307 apply (rule_tac c=x1 in mult_less_imp_less_right)
```
```  2308 apply (auto simp add: mult_ac)
```
```  2309 done
```
```  2310
```
```  2311 lemma real_mult_inverse_cancel2:
```
```  2312      "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
```
```  2313 apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)
```
```  2314 done
```
```  2315
```
```  2316 lemma realpow_num_eq_if:
```
```  2317   fixes m :: "'a::power"
```
```  2318   shows "m ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
```
```  2319 by (cases n, auto)
```
```  2320
```
```  2321 lemma cos_two_less_zero [simp]: "cos (2) < 0"
```
```  2322 apply (cut_tac x = 2 in cos_paired)
```
```  2323 apply (drule sums_minus)
```
```  2324 apply (rule neg_less_iff_less [THEN iffD1])
```
```  2325 apply (frule sums_unique, auto)
```
```  2326 apply (rule_tac y =
```
```  2327  "\<Sum>n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
```
```  2328        in order_less_trans)
```
```  2329 apply (simp (no_asm) add: fact_num_eq_if_nat realpow_num_eq_if del: fact_Suc)
```
```  2330 apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
```
```  2331 apply (rule sumr_pos_lt_pair)
```
```  2332 apply (erule sums_summable, safe)
```
```  2333 unfolding One_nat_def
```
```  2334 apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
```
```  2335             del: fact_Suc)
```
```  2336 apply (simp add: inverse_eq_divide less_divide_eq del: fact_Suc)
```
```  2337 apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
```
```  2338 apply (simp only: real_of_nat_mult)
```
```  2339 apply (rule mult_strict_mono, force)
```
```  2340   apply (rule_tac [3] real_of_nat_ge_zero)
```
```  2341  prefer 2 apply force
```
```  2342 apply (rule real_of_nat_less_iff [THEN iffD2])
```
```  2343 apply (rule fact_less_mono_nat, auto)
```
```  2344 done
```
```  2345
```
```  2346 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
```
```  2347 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
```
```  2348
```
```  2349 lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
```
```  2350 proof (rule ex_ex1I)
```
```  2351   show "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0"
```
```  2352     by (rule IVT2, simp_all)
```
```  2353 next
```
```  2354   fix x y
```
```  2355   assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
```
```  2356   assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
```
```  2357   have [simp]: "\<forall>x. cos differentiable x"
```
```  2358     unfolding differentiable_def by (auto intro: DERIV_cos)
```
```  2359   from x y show "x = y"
```
```  2360     apply (cut_tac less_linear [of x y], auto)
```
```  2361     apply (drule_tac f = cos in Rolle)
```
```  2362     apply (drule_tac [5] f = cos in Rolle)
```
```  2363     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  2364     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  2365     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  2366     done
```
```  2367 qed
```
```  2368
```
```  2369 lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
```
```  2370 by (simp add: pi_def)
```
```  2371
```
```  2372 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
```
```  2373 by (simp add: pi_half cos_is_zero [THEN theI'])
```
```  2374
```
```  2375 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
```
```  2376 apply (rule order_le_neq_trans)
```
```  2377 apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  2378 apply (rule notI, drule arg_cong [where f=cos], simp)
```
```  2379 done
```
```  2380
```
```  2381 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
```
```  2382 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
```
```  2383
```
```  2384 lemma pi_half_less_two [simp]: "pi / 2 < 2"
```
```  2385 apply (rule order_le_neq_trans)
```
```  2386 apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  2387 apply (rule notI, drule arg_cong [where f=cos], simp)
```
```  2388 done
```
```  2389
```
```  2390 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
```
```  2391 lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
```
```  2392
```
```  2393 lemma pi_gt_zero [simp]: "0 < pi"
```
```  2394 by (insert pi_half_gt_zero, simp)
```
```  2395
```
```  2396 lemma pi_ge_zero [simp]: "0 \<le> pi"
```
```  2397 by (rule pi_gt_zero [THEN order_less_imp_le])
```
```  2398
```
```  2399 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
```
```  2400 by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym])
```
```  2401
```
```  2402 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
```
```  2403 by (simp add: linorder_not_less)
```
```  2404
```
```  2405 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
```
```  2406 by simp
```
```  2407
```
```  2408 lemma m2pi_less_pi: "- (2 * pi) < pi"
```
```  2409 by simp
```
```  2410
```
```  2411 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
```
```  2412 apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
```
```  2413 apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
```
```  2414 apply (simp add: power2_eq_1_iff)
```
```  2415 done
```
```  2416
```
```  2417 lemma cos_pi [simp]: "cos pi = -1"
```
```  2418 by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
```
```  2419
```
```  2420 lemma sin_pi [simp]: "sin pi = 0"
```
```  2421 by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
```
```  2422
```
```  2423 lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
```
```  2424 by (simp add: cos_diff)
```
```  2425
```
```  2426 lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
```
```  2427 by (simp add: cos_add)
```
```  2428
```
```  2429 lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
```
```  2430 by (simp add: sin_diff)
```
```  2431
```
```  2432 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
```
```  2433 by (simp add: sin_add)
```
```  2434
```
```  2435 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
```
```  2436 by (simp add: sin_add)
```
```  2437
```
```  2438 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
```
```  2439 by (simp add: cos_add)
```
```  2440
```
```  2441 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
```
```  2442 by (simp add: sin_add cos_double)
```
```  2443
```
```  2444 lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
```
```  2445 by (simp add: cos_add cos_double)
```
```  2446
```
```  2447 lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
```
```  2448 apply (induct "n")
```
```  2449 apply (auto simp add: real_of_nat_Suc distrib_right)
```
```  2450 done
```
```  2451
```
```  2452 lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
```
```  2453 proof -
```
```  2454   have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
```
```  2455   also have "... = -1 ^ n" by (rule cos_npi)
```
```  2456   finally show ?thesis .
```
```  2457 qed
```
```  2458
```
```  2459 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
```
```  2460 apply (induct "n")
```
```  2461 apply (auto simp add: real_of_nat_Suc distrib_right)
```
```  2462 done
```
```  2463
```
```  2464 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
```
```  2465 by (simp add: mult_commute [of pi])
```
```  2466
```
```  2467 lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
```
```  2468 by (simp add: cos_double)
```
```  2469
```
```  2470 lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
```
```  2471 by simp
```
```  2472
```
```  2473 lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
```
```  2474 apply (rule sin_gt_zero, assumption)
```
```  2475 apply (rule order_less_trans, assumption)
```
```  2476 apply (rule pi_half_less_two)
```
```  2477 done
```
```  2478
```
```  2479 lemma sin_less_zero:
```
```  2480   assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
```
```  2481 proof -
```
```  2482   have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
```
```  2483   thus ?thesis by simp
```
```  2484 qed
```
```  2485
```
```  2486 lemma pi_less_4: "pi < 4"
```
```  2487 by (cut_tac pi_half_less_two, auto)
```
```  2488
```
```  2489 lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
```
```  2490 apply (cut_tac pi_less_4)
```
```  2491 apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
```
```  2492 apply (cut_tac cos_is_zero, safe)
```
```  2493 apply (rename_tac y z)
```
```  2494 apply (drule_tac x = y in spec)
```
```  2495 apply (drule_tac x = "pi/2" in spec, simp)
```
```  2496 done
```
```  2497
```
```  2498 lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
```
```  2499 apply (rule_tac x = x and y = 0 in linorder_cases)
```
```  2500 apply (rule cos_minus [THEN subst])
```
```  2501 apply (rule cos_gt_zero)
```
```  2502 apply (auto intro: cos_gt_zero)
```
```  2503 done
```
```  2504
```
```  2505 lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
```
```  2506 apply (auto simp add: order_le_less cos_gt_zero_pi)
```
```  2507 apply (subgoal_tac "x = pi/2", auto)
```
```  2508 done
```
```  2509
```
```  2510 lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
```
```  2511 by (simp add: sin_cos_eq cos_gt_zero_pi)
```
```  2512
```
```  2513 lemma pi_ge_two: "2 \<le> pi"
```
```  2514 proof (rule ccontr)
```
```  2515   assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
```
```  2516   have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
```
```  2517   proof (cases "2 < 2 * pi")
```
```  2518     case True with dense[OF `pi < 2`] show ?thesis by auto
```
```  2519   next
```
```  2520     case False have "pi < 2 * pi" by auto
```
```  2521     from dense[OF this] and False show ?thesis by auto
```
```  2522   qed
```
```  2523   then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
```
```  2524   hence "0 < sin y" using sin_gt_zero by auto
```
```  2525   moreover
```
```  2526   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
```
```  2527   ultimately show False by auto
```
```  2528 qed
```
```  2529
```
```  2530 lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
```
```  2531 by (auto simp add: order_le_less sin_gt_zero_pi)
```
```  2532
```
```  2533 text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}.
```
```  2534   It should be possible to factor out some of the common parts. *}
```
```  2535
```
```  2536 lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
```
```  2537 proof (rule ex_ex1I)
```
```  2538   assume y: "-1 \<le> y" "y \<le> 1"
```
```  2539   show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
```
```  2540     by (rule IVT2, simp_all add: y)
```
```  2541 next
```
```  2542   fix a b
```
```  2543   assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
```
```  2544   assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
```
```  2545   have [simp]: "\<forall>x. cos differentiable x"
```
```  2546     unfolding differentiable_def by (auto intro: DERIV_cos)
```
```  2547   from a b show "a = b"
```
```  2548     apply (cut_tac less_linear [of a b], auto)
```
```  2549     apply (drule_tac f = cos in Rolle)
```
```  2550     apply (drule_tac [5] f = cos in Rolle)
```
```  2551     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  2552     apply (metis order_less_le_trans less_le sin_gt_zero_pi)
```
```  2553     apply (metis order_less_le_trans less_le sin_gt_zero_pi)
```
```  2554     done
```
```  2555 qed
```
```  2556
```
```  2557 lemma sin_total:
```
```  2558      "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
```
```  2559 apply (rule ccontr)
```
```  2560 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))")
```
```  2561 apply (erule contrapos_np)
```
```  2562 apply simp
```
```  2563 apply (cut_tac y="-y" in cos_total, simp) apply simp
```
```  2564 apply (erule ex1E)
```
```  2565 apply (rule_tac a = "x - (pi/2)" in ex1I)
```
```  2566 apply (simp (no_asm) add: add_assoc)
```
```  2567 apply (rotate_tac 3)
```
```  2568 apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all add: cos_add)
```
```  2569 done
```
```  2570
```
```  2571 lemma reals_Archimedean4:
```
```  2572      "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
```
```  2573 apply (auto dest!: reals_Archimedean3)
```
```  2574 apply (drule_tac x = x in spec, clarify)
```
```  2575 apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
```
```  2576  prefer 2 apply (erule LeastI)
```
```  2577 apply (case_tac "LEAST m::nat. x < real m * y", simp)
```
```  2578 apply (subgoal_tac "~ x < real nat * y")
```
```  2579  prefer 2 apply (rule not_less_Least, simp, force)
```
```  2580 done
```
```  2581
```
```  2582 (* Pre Isabelle99-2 proof was simpler- numerals arithmetic
```
```  2583    now causes some unwanted re-arrangements of literals!   *)
```
```  2584 lemma cos_zero_lemma:
```
```  2585      "[| 0 \<le> x; cos x = 0 |] ==>
```
```  2586       \<exists>n::nat. ~even n & x = real n * (pi/2)"
```
```  2587 apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
```
```  2588 apply (subgoal_tac "0 \<le> x - real n * pi &
```
```  2589                     (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
```
```  2590 apply (auto simp add: algebra_simps real_of_nat_Suc)
```
```  2591  prefer 2 apply (simp add: cos_diff)
```
```  2592 apply (simp add: cos_diff)
```
```  2593 apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
```
```  2594 apply (rule_tac [2] cos_total, safe)
```
```  2595 apply (drule_tac x = "x - real n * pi" in spec)
```
```  2596 apply (drule_tac x = "pi/2" in spec)
```
```  2597 apply (simp add: cos_diff)
```
```  2598 apply (rule_tac x = "Suc (2 * n)" in exI)
```
```  2599 apply (simp add: real_of_nat_Suc algebra_simps, auto)
```
```  2600 done
```
```  2601
```
```  2602 lemma sin_zero_lemma:
```
```  2603      "[| 0 \<le> x; sin x = 0 |] ==>
```
```  2604       \<exists>n::nat. even n & x = real n * (pi/2)"
```
```  2605 apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
```
```  2606  apply (clarify, rule_tac x = "n - 1" in exI)
```
```  2607  apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc distrib_right)
```
```  2608 apply (rule cos_zero_lemma)
```
```  2609 apply (simp_all add: cos_add)
```
```  2610 done
```
```  2611
```
```  2612
```
```  2613 lemma cos_zero_iff:
```
```  2614      "(cos x = 0) =
```
```  2615       ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
```
```  2616        (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
```
```  2617 apply (rule iffI)
```
```  2618 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  2619 apply (drule cos_zero_lemma, assumption+)
```
```  2620 apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
```
```  2621 apply (force simp add: minus_equation_iff [of x])
```
```  2622 apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc distrib_right)
```
```  2623 apply (auto simp add: cos_add)
```
```  2624 done
```
```  2625
```
```  2626 (* ditto: but to a lesser extent *)
```
```  2627 lemma sin_zero_iff:
```
```  2628      "(sin x = 0) =
```
```  2629       ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
```
```  2630        (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
```
```  2631 apply (rule iffI)
```
```  2632 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  2633 apply (drule sin_zero_lemma, assumption+)
```
```  2634 apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
```
```  2635 apply (force simp add: minus_equation_iff [of x])
```
```  2636 apply (auto simp add: even_mult_two_ex)
```
```  2637 done
```
```  2638
```
```  2639 lemma cos_monotone_0_pi: assumes "0 \<le> y" and "y < x" and "x \<le> pi"
```
```  2640   shows "cos x < cos y"
```
```  2641 proof -
```
```  2642   have "- (x - y) < 0" using assms by auto
```
```  2643
```
```  2644   from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
```
```  2645   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" by auto
```
```  2646   hence "0 < z" and "z < pi" using assms by auto
```
```  2647   hence "0 < sin z" using sin_gt_zero_pi by auto
```
```  2648   hence "cos x - cos y < 0" unfolding cos_diff minus_mult_commute[symmetric] using `- (x - y) < 0` by (rule mult_pos_neg2)
```
```  2649   thus ?thesis by auto
```
```  2650 qed
```
```  2651
```
```  2652 lemma cos_monotone_0_pi': assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" shows "cos x \<le> cos y"
```
```  2653 proof (cases "y < x")
```
```  2654   case True show ?thesis using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
```
```  2655 next
```
```  2656   case False hence "y = x" using `y \<le> x` by auto
```
```  2657   thus ?thesis by auto
```
```  2658 qed
```
```  2659
```
```  2660 lemma cos_monotone_minus_pi_0: assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
```
```  2661   shows "cos y < cos x"
```
```  2662 proof -
```
```  2663   have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" using assms by auto
```
```  2664   from cos_monotone_0_pi[OF this]
```
```  2665   show ?thesis unfolding cos_minus .
```
```  2666 qed
```
```  2667
```
```  2668 lemma cos_monotone_minus_pi_0': assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0" shows "cos y \<le> cos x"
```
```  2669 proof (cases "y < x")
```
```  2670   case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] by auto
```
```  2671 next
```
```  2672   case False hence "y = x" using `y \<le> x` by auto
```
```  2673   thus ?thesis by auto
```
```  2674 qed
```
```  2675
```
```  2676 lemma sin_monotone_2pi': assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2" shows "sin y \<le> sin x"
```
```  2677 proof -
```
```  2678   have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi"
```
```  2679     using pi_ge_two and assms by auto
```
```  2680   from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] by auto
```
```  2681 qed
```
```  2682
```
```  2683 subsection {* Tangent *}
```
```  2684
```
```  2685 definition tan :: "real \<Rightarrow> real" where
```
```  2686   "tan = (\<lambda>x. sin x / cos x)"
```
```  2687
```
```  2688 lemma tan_zero [simp]: "tan 0 = 0"
```
```  2689   by (simp add: tan_def)
```
```  2690
```
```  2691 lemma tan_pi [simp]: "tan pi = 0"
```
```  2692   by (simp add: tan_def)
```
```  2693
```
```  2694 lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
```
```  2695   by (simp add: tan_def)
```
```  2696
```
```  2697 lemma tan_minus [simp]: "tan (-x) = - tan x"
```
```  2698   by (simp add: tan_def)
```
```  2699
```
```  2700 lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
```
```  2701   by (simp add: tan_def)
```
```  2702
```
```  2703 lemma lemma_tan_add1:
```
```  2704   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
```
```  2705   by (simp add: tan_def cos_add field_simps)
```
```  2706
```
```  2707 lemma add_tan_eq:
```
```  2708   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
```
```  2709   by (simp add: tan_def sin_add field_simps)
```
```  2710
```
```  2711 lemma tan_add:
```
```  2712      "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
```
```  2713       ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
```
```  2714   by (simp add: add_tan_eq lemma_tan_add1, simp add: tan_def)
```
```  2715
```
```  2716 lemma tan_double:
```
```  2717      "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
```
```  2718       ==> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)"
```
```  2719   using tan_add [of x x] by (simp add: power2_eq_square)
```
```  2720
```
```  2721 lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
```
```  2722 by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
```
```  2723
```
```  2724 lemma tan_less_zero:
```
```  2725   assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
```
```  2726 proof -
```
```  2727   have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
```
```  2728   thus ?thesis by simp
```
```  2729 qed
```
```  2730
```
```  2731 lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"
```
```  2732   unfolding tan_def sin_double cos_double sin_squared_eq
```
```  2733   by (simp add: power2_eq_square)
```
```  2734
```
```  2735 lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)"
```
```  2736   unfolding tan_def
```
```  2737   by (auto intro!: DERIV_intros, simp add: divide_inverse power2_eq_square)
```
```  2738
```
```  2739 lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
```
```  2740   by (rule DERIV_tan [THEN DERIV_isCont])
```
```  2741
```
```  2742 lemma isCont_tan' [simp]:
```
```  2743   "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
```
```  2744   by (rule isCont_o2 [OF _ isCont_tan])
```
```  2745
```
```  2746 lemma tendsto_tan [tendsto_intros]:
```
```  2747   "\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F"
```
```  2748   by (rule isCont_tendsto_compose [OF isCont_tan])
```
```  2749
```
```  2750 lemma continuous_tan:
```
```  2751   "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))"
```
```  2752   unfolding continuous_def by (rule tendsto_tan)
```
```  2753
```
```  2754 lemma isCont_tan'' [continuous_intros]:
```
```  2755   "continuous (at x) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. tan (f x))"
```
```  2756   unfolding continuous_at by (rule tendsto_tan)
```
```  2757
```
```  2758 lemma continuous_within_tan [continuous_intros]:
```
```  2759   "continuous (at x within s) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
```
```  2760   unfolding continuous_within by (rule tendsto_tan)
```
```  2761
```
```  2762 lemma continuous_on_tan [continuous_on_intros]:
```
```  2763   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))"
```
```  2764   unfolding continuous_on_def by (auto intro: tendsto_tan)
```
```  2765
```
```  2766 lemma LIM_cos_div_sin: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
```
```  2767   by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
```
```  2768
```
```  2769 lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
```
```  2770 apply (cut_tac LIM_cos_div_sin)
```
```  2771 apply (simp only: LIM_eq)
```
```  2772 apply (drule_tac x = "inverse y" in spec, safe, force)
```
```  2773 apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
```
```  2774 apply (rule_tac x = "(pi/2) - e" in exI)
```
```  2775 apply (simp (no_asm_simp))
```
```  2776 apply (drule_tac x = "(pi/2) - e" in spec)
```
```  2777 apply (auto simp add: tan_def sin_diff cos_diff)
```
```  2778 apply (rule inverse_less_iff_less [THEN iffD1])
```
```  2779 apply (auto simp add: divide_inverse)
```
```  2780 apply (rule mult_pos_pos)
```
```  2781 apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
```
```  2782 apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
```
```  2783 done
```
```  2784
```
```  2785 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
```
```  2786 apply (frule order_le_imp_less_or_eq, safe)
```
```  2787  prefer 2 apply force
```
```  2788 apply (drule lemma_tan_total, safe)
```
```  2789 apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
```
```  2790 apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
```
```  2791 apply (drule_tac y = xa in order_le_imp_less_or_eq)
```
```  2792 apply (auto dest: cos_gt_zero)
```
```  2793 done
```
```  2794
```
```  2795 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  2796 apply (cut_tac linorder_linear [of 0 y], safe)
```
```  2797 apply (drule tan_total_pos)
```
```  2798 apply (cut_tac [2] y="-y" in tan_total_pos, safe)
```
```  2799 apply (rule_tac [3] x = "-x" in exI)
```
```  2800 apply (auto del: exI intro!: exI)
```
```  2801 done
```
```  2802
```
```  2803 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  2804 apply (cut_tac y = y in lemma_tan_total1, auto)
```
```  2805 apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
```
```  2806 apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
```
```  2807 apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
```
```  2808 apply (rule_tac [4] Rolle)
```
```  2809 apply (rule_tac [2] Rolle)
```
```  2810 apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
```
```  2811             simp add: differentiable_def)
```
```  2812 txt{*Now, simulate TRYALL*}
```
```  2813 apply (rule_tac [!] DERIV_tan asm_rl)
```
```  2814 apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
```
```  2815             simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
```
```  2816 done
```
```  2817
```
```  2818 lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
```
```  2819   shows "tan y < tan x"
```
```  2820 proof -
```
```  2821   have "\<forall> x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse ((cos x')\<^sup>2)"
```
```  2822   proof (rule allI, rule impI)
```
```  2823     fix x' :: real assume "y \<le> x' \<and> x' \<le> x"
```
```  2824     hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
```
```  2825     from cos_gt_zero_pi[OF this]
```
```  2826     have "cos x' \<noteq> 0" by auto
```
```  2827     thus "DERIV tan x' :> inverse ((cos x')\<^sup>2)" by (rule DERIV_tan)
```
```  2828   qed
```
```  2829   from MVT2[OF `y < x` this]
```
```  2830   obtain z where "y < z" and "z < x" and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
```
```  2831   hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
```
```  2832   hence "0 < cos z" using cos_gt_zero_pi by auto
```
```  2833   hence inv_pos: "0 < inverse ((cos z)\<^sup>2)" by auto
```
```  2834   have "0 < x - y" using `y < x` by auto
```
```  2835   from mult_pos_pos [OF this inv_pos]
```
```  2836   have "0 < tan x - tan y" unfolding tan_diff by auto
```
```  2837   thus ?thesis by auto
```
```  2838 qed
```
```  2839
```
```  2840 lemma tan_monotone': assumes "- (pi / 2) < y" and "y < pi / 2" and "- (pi / 2) < x" and "x < pi / 2"
```
```  2841   shows "(y < x) = (tan y < tan x)"
```
```  2842 proof
```
```  2843   assume "y < x" thus "tan y < tan x" using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
```
```  2844 next
```
```  2845   assume "tan y < tan x"
```
```  2846   show "y < x"
```
```  2847   proof (rule ccontr)
```
```  2848     assume "\<not> y < x" hence "x \<le> y" by auto
```
```  2849     hence "tan x \<le> tan y"
```
```  2850     proof (cases "x = y")
```
```  2851       case True thus ?thesis by auto
```
```  2852     next
```
```  2853       case False hence "x < y" using `x \<le> y` by auto
```
```  2854       from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
```
```  2855     qed
```
```  2856     thus False using `tan y < tan x` by auto
```
```  2857   qed
```
```  2858 qed
```
```  2859
```
```  2860 lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)" unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
```
```  2861
```
```  2862 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
```
```  2863   by (simp add: tan_def)
```
```  2864
```
```  2865 lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x"
```
```  2866 proof (induct n arbitrary: x)
```
```  2867   case (Suc n)
```
```  2868   have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
```
```  2869   show ?case unfolding split_pi_off using Suc by auto
```
```  2870 qed auto
```
```  2871
```
```  2872 lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
```
```  2873 proof (cases "0 \<le> i")
```
```  2874   case True hence i_nat: "real i = real (nat i)" by auto
```
```  2875   show ?thesis unfolding i_nat by auto
```
```  2876 next
```
```  2877   case False hence i_nat: "real i = - real (nat (-i))" by auto
```
```  2878   have "tan x = tan (x + real i * pi - real i * pi)" by auto
```
```  2879   also have "\<dots> = tan (x + real i * pi)" unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
```
```  2880   finally show ?thesis by auto
```
```  2881 qed
```
```  2882
```
```  2883 lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
```
```  2884   using tan_periodic_int[of _ "numeral n" ] unfolding real_numeral .
```
```  2885
```
```  2886 subsection {* Inverse Trigonometric Functions *}
```
```  2887
```
```  2888 definition
```
```  2889   arcsin :: "real => real" where
```
```  2890   "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
```
```  2891
```
```  2892 definition
```
```  2893   arccos :: "real => real" where
```
```  2894   "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
```
```  2895
```
```  2896 definition
```
```  2897   arctan :: "real => real" where
```
```  2898   "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
```
```  2899
```
```  2900 lemma arcsin:
```
```  2901      "[| -1 \<le> y; y \<le> 1 |]
```
```  2902       ==> -(pi/2) \<le> arcsin y &
```
```  2903            arcsin y \<le> pi/2 & sin(arcsin y) = y"
```
```  2904 unfolding arcsin_def by (rule theI' [OF sin_total])
```
```  2905
```
```  2906 lemma arcsin_pi:
```
```  2907      "[| -1 \<le> y; y \<le> 1 |]
```
```  2908       ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
```
```  2909 apply (drule (1) arcsin)
```
```  2910 apply (force intro: order_trans)
```
```  2911 done
```
```  2912
```
```  2913 lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
```
```  2914 by (blast dest: arcsin)
```
```  2915
```
```  2916 lemma arcsin_bounded:
```
```  2917      "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
```
```  2918 by (blast dest: arcsin)
```
```  2919
```
```  2920 lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
```
```  2921 by (blast dest: arcsin)
```
```  2922
```
```  2923 lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
```
```  2924 by (blast dest: arcsin)
```
```  2925
```
```  2926 lemma arcsin_lt_bounded:
```
```  2927      "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
```
```  2928 apply (frule order_less_imp_le)
```
```  2929 apply (frule_tac y = y in order_less_imp_le)
```
```  2930 apply (frule arcsin_bounded)
```
```  2931 apply (safe, simp)
```
```  2932 apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
```
```  2933 apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
```
```  2934 apply (drule_tac [!] f = sin in arg_cong, auto)
```
```  2935 done
```
```  2936
```
```  2937 lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
```
```  2938 apply (unfold arcsin_def)
```
```  2939 apply (rule the1_equality)
```
```  2940 apply (rule sin_total, auto)
```
```  2941 done
```
```  2942
```
```  2943 lemma arccos:
```
```  2944      "[| -1 \<le> y; y \<le> 1 |]
```
```  2945       ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
```
```  2946 unfolding arccos_def by (rule theI' [OF cos_total])
```
```  2947
```
```  2948 lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
```
```  2949 by (blast dest: arccos)
```
```  2950
```
```  2951 lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
```
```  2952 by (blast dest: arccos)
```
```  2953
```
```  2954 lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
```
```  2955 by (blast dest: arccos)
```
```  2956
```
```  2957 lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
```
```  2958 by (blast dest: arccos)
```
```  2959
```
```  2960 lemma arccos_lt_bounded:
```
```  2961      "[| -1 < y; y < 1 |]
```
```  2962       ==> 0 < arccos y & arccos y < pi"
```
```  2963 apply (frule order_less_imp_le)
```
```  2964 apply (frule_tac y = y in order_less_imp_le)
```
```  2965 apply (frule arccos_bounded, auto)
```
```  2966 apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
```
```  2967 apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
```
```  2968 apply (drule_tac [!] f = cos in arg_cong, auto)
```
```  2969 done
```
```  2970
```
```  2971 lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
```
```  2972 apply (simp add: arccos_def)
```
```  2973 apply (auto intro!: the1_equality cos_total)
```
```  2974 done
```
```  2975
```
```  2976 lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
```
```  2977 apply (simp add: arccos_def)
```
```  2978 apply (auto intro!: the1_equality cos_total)
```
```  2979 done
```
```  2980
```
```  2981 lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
```
```  2982 apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  2983 apply (rule power2_eq_imp_eq)
```
```  2984 apply (simp add: cos_squared_eq)
```
```  2985 apply (rule cos_ge_zero)
```
```  2986 apply (erule (1) arcsin_lbound)
```
```  2987 apply (erule (1) arcsin_ubound)
```
```  2988 apply simp
```
```  2989 apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  2990 apply (rule power_mono, simp, simp)
```
```  2991 done
```
```  2992
```
```  2993 lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
```
```  2994 apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  2995 apply (rule power2_eq_imp_eq)
```
```  2996 apply (simp add: sin_squared_eq)
```
```  2997 apply (rule sin_ge_zero)
```
```  2998 apply (erule (1) arccos_lbound)
```
```  2999 apply (erule (1) arccos_ubound)
```
```  3000 apply simp
```
```  3001 apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  3002 apply (rule power_mono, simp, simp)
```
```  3003 done
```
```  3004
```
```  3005 lemma arctan [simp]:
```
```  3006      "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
```
```  3007 unfolding arctan_def by (rule theI' [OF tan_total])
```
```  3008
```
```  3009 lemma tan_arctan: "tan(arctan y) = y"
```
```  3010 by auto
```
```  3011
```
```  3012 lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
```
```  3013 by (auto simp only: arctan)
```
```  3014
```
```  3015 lemma arctan_lbound: "- (pi/2) < arctan y"
```
```  3016 by auto
```
```  3017
```
```  3018 lemma arctan_ubound: "arctan y < pi/2"
```
```  3019 by (auto simp only: arctan)
```
```  3020
```
```  3021 lemma arctan_unique:
```
```  3022   assumes "-(pi/2) < x" and "x < pi/2" and "tan x = y"
```
```  3023   shows "arctan y = x"
```
```  3024   using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
```
```  3025
```
```  3026 lemma arctan_tan:
```
```  3027       "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
```
```  3028   by (rule arctan_unique, simp_all)
```
```  3029
```
```  3030 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
```
```  3031   by (rule arctan_unique, simp_all)
```
```  3032
```
```  3033 lemma arctan_minus: "arctan (- x) = - arctan x"
```
```  3034   apply (rule arctan_unique)
```
```  3035   apply (simp only: neg_less_iff_less arctan_ubound)
```
```  3036   apply (metis minus_less_iff arctan_lbound)
```
```  3037   apply simp
```
```  3038   done
```
```  3039
```
```  3040 lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
```
```  3041   by (intro less_imp_neq [symmetric] cos_gt_zero_pi
```
```  3042     arctan_lbound arctan_ubound)
```
```  3043
```
```  3044 lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)"
```
```  3045 proof (rule power2_eq_imp_eq)
```
```  3046   have "0 < 1 + x\<^sup>2" by (simp add: add_pos_nonneg)
```
```  3047   show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp
```
```  3048   show "0 \<le> cos (arctan x)"
```
```  3049     by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
```
```  3050   have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1"
```
```  3051     unfolding tan_def by (simp add: distrib_left power_divide)
```
```  3052   thus "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
```
```  3053     using `0 < 1 + x\<^sup>2` by (simp add: power_divide eq_divide_eq)
```
```  3054 qed
```
```  3055
```
```  3056 lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)"
```
```  3057   using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
```
```  3058   using tan_arctan [of x] unfolding tan_def cos_arctan
```
```  3059   by (simp add: eq_divide_eq)
```
```  3060
```
```  3061 lemma tan_sec: "cos x \<noteq> 0 ==> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
```
```  3062 apply (rule power_inverse [THEN subst])
```
```  3063 apply (rule_tac c1 = "(cos x)\<^sup>2" in real_mult_right_cancel [THEN iffD1])
```
```  3064 apply (auto dest: field_power_not_zero
```
```  3065         simp add: power_mult_distrib distrib_right power_divide tan_def
```
```  3066                   mult_assoc power_inverse [symmetric])
```
```  3067 done
```
```  3068
```
```  3069 lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
```
```  3070   by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
```
```  3071
```
```  3072 lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
```
```  3073   by (simp only: not_less [symmetric] arctan_less_iff)
```
```  3074
```
```  3075 lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
```
```  3076   by (simp only: eq_iff [where 'a=real] arctan_le_iff)
```
```  3077
```
```  3078 lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
```
```  3079   using arctan_less_iff [of 0 x] by simp
```
```  3080
```
```  3081 lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
```
```  3082   using arctan_less_iff [of x 0] by simp
```
```  3083
```
```  3084 lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
```
```  3085   using arctan_le_iff [of 0 x] by simp
```
```  3086
```
```  3087 lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```  3088   using arctan_le_iff [of x 0] by simp
```
```  3089
```
```  3090 lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
```
```  3091   using arctan_eq_iff [of x 0] by simp
```
```  3092
```
```  3093 lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin"
```
```  3094 proof -
```
```  3095   have "continuous_on (sin ` {- pi / 2 .. pi / 2}) arcsin"
```
```  3096     by (rule continuous_on_inv) (auto intro: continuous_on_intros simp: arcsin_sin)
```
```  3097   also have "sin ` {- pi / 2 .. pi / 2} = {-1 .. 1}"
```
```  3098   proof safe
```
```  3099     fix x :: real assume "x \<in> {-1..1}" then show "x \<in> sin ` {- pi / 2..pi / 2}"
```
```  3100       using arcsin_lbound arcsin_ubound by (intro image_eqI[where x="arcsin x"]) auto
```
```  3101   qed simp
```
```  3102   finally show ?thesis .
```
```  3103 qed
```
```  3104
```
```  3105 lemma continuous_on_arcsin [continuous_on_intros]:
```
```  3106   "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))"
```
```  3107   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arcsin']]
```
```  3108   by (auto simp: comp_def subset_eq)
```
```  3109
```
```  3110 lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x"
```
```  3111   using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  3112   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  3113
```
```  3114 lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos"
```
```  3115 proof -
```
```  3116   have "continuous_on (cos ` {0 .. pi}) arccos"
```
```  3117     by (rule continuous_on_inv) (auto intro: continuous_on_intros simp: arccos_cos)
```
```  3118   also have "cos ` {0 .. pi} = {-1 .. 1}"
```
```  3119   proof safe
```
```  3120     fix x :: real assume "x \<in> {-1..1}" then show "x \<in> cos ` {0..pi}"
```
```  3121       using arccos_lbound arccos_ubound by (intro image_eqI[where x="arccos x"]) auto
```
```  3122   qed simp
```
```  3123   finally show ?thesis .
```
```  3124 qed
```
```  3125
```
```  3126 lemma continuous_on_arccos [continuous_on_intros]:
```
```  3127   "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))"
```
```  3128   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arccos']]
```
```  3129   by (auto simp: comp_def subset_eq)
```
```  3130
```
```  3131 lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x"
```
```  3132   using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  3133   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  3134
```
```  3135 lemma isCont_arctan: "isCont arctan x"
```
```  3136 apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
```
```  3137 apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
```
```  3138 apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
```
```  3139 apply (erule (1) isCont_inverse_function2 [where f=tan])
```
```  3140 apply (metis arctan_tan order_le_less_trans order_less_le_trans)
```
```  3141 apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
```
```  3142 done
```
```  3143
```
```  3144 lemma tendsto_arctan [tendsto_intros]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) ---> arctan x) F"
```
```  3145   by (rule isCont_tendsto_compose [OF isCont_arctan])
```
```  3146
```
```  3147 lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))"
```
```  3148   unfolding continuous_def by (rule tendsto_arctan)
```
```  3149
```
```  3150 lemma continuous_on_arctan [continuous_on_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
```
```  3151   unfolding continuous_on_def by (auto intro: tendsto_arctan)
```
```  3152
```
```  3153 lemma DERIV_arcsin:
```
```  3154   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
```
```  3155 apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
```
```  3156 apply (rule DERIV_cong [OF DERIV_sin])
```
```  3157 apply (simp add: cos_arcsin)
```
```  3158 apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  3159 apply (rule power_strict_mono, simp, simp, simp)
```
```  3160 apply assumption
```
```  3161 apply assumption
```
```  3162 apply simp
```
```  3163 apply (erule (1) isCont_arcsin)
```
```  3164 done
```
```  3165
```
```  3166 lemma DERIV_arccos:
```
```  3167   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
```
```  3168 apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
```
```  3169 apply (rule DERIV_cong [OF DERIV_cos])
```
```  3170 apply (simp add: sin_arccos)
```
```  3171 apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  3172 apply (rule power_strict_mono, simp, simp, simp)
```
```  3173 apply assumption
```
```  3174 apply assumption
```
```  3175 apply simp
```
```  3176 apply (erule (1) isCont_arccos)
```
```  3177 done
```
```  3178
```
```  3179 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
```
```  3180 apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
```
```  3181 apply (rule DERIV_cong [OF DERIV_tan])
```
```  3182 apply (rule cos_arctan_not_zero)
```
```  3183 apply (simp add: power_inverse tan_sec [symmetric])
```
```  3184 apply (subgoal_tac "0 < 1 + x\<^sup>2", simp)
```
```  3185 apply (simp add: add_pos_nonneg)
```
```  3186 apply (simp, simp, simp, rule isCont_arctan)
```
```  3187 done
```
```  3188
```
```  3189 declare
```
```  3190   DERIV_arcsin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  3191   DERIV_arccos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  3192   DERIV_arctan[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
```
```  3193
```
```  3194 lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- pi/2))"
```
```  3195   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])
```
```  3196      (auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  3197            intro!: tan_monotone exI[of _ "pi/2"])
```
```  3198
```
```  3199 lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"
```
```  3200   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])
```
```  3201      (auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  3202            intro!: tan_monotone exI[of _ "pi/2"])
```
```  3203
```
```  3204 lemma tendsto_arctan_at_top: "(arctan ---> (pi/2)) at_top"
```
```  3205 proof (rule tendstoI)
```
```  3206   fix e :: real assume "0 < e"
```
```  3207   def y \<equiv> "pi/2 - min (pi/2) e"
```
```  3208   then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y"
```
```  3209     using `0 < e` by auto
```
```  3210
```
```  3211   show "eventually (\<lambda>x. dist (arctan x) (pi / 2) < e) at_top"
```
```  3212   proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)
```
```  3213     fix x assume "tan y < x"
```
```  3214     then have "arctan (tan y) < arctan x"
```
```  3215       by (simp add: arctan_less_iff)
```
```  3216     with y have "y < arctan x"
```
```  3217       by (subst (asm) arctan_tan) simp_all
```
```  3218     with arctan_ubound[of x, arith] y `0 < e`
```
```  3219     show "dist (arctan x) (pi / 2) < e"
```
```  3220       by (simp add: dist_real_def)
```
```  3221   qed
```
```  3222 qed
```
```  3223
```
```  3224 lemma tendsto_arctan_at_bot: "(arctan ---> - (pi/2)) at_bot"
```
```  3225   unfolding filterlim_at_bot_mirror arctan_minus by (intro tendsto_minus tendsto_arctan_at_top)
```
```  3226
```
```  3227 subsection {* More Theorems about Sin and Cos *}
```
```  3228
```
```  3229 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
```
```  3230 proof -
```
```  3231   let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
```
```  3232   have nonneg: "0 \<le> ?c"
```
```  3233     by (simp add: cos_ge_zero)
```
```  3234   have "0 = cos (pi / 4 + pi / 4)"
```
```  3235     by simp
```
```  3236   also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2"
```
```  3237     by (simp only: cos_add power2_eq_square)
```
```  3238   also have "\<dots> = 2 * ?c\<^sup>2 - 1"
```
```  3239     by (simp add: sin_squared_eq)
```
```  3240   finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2"
```
```  3241     by (simp add: power_divide)
```
```  3242   thus ?thesis
```
```  3243     using nonneg by (rule power2_eq_imp_eq) simp
```
```  3244 qed
```
```  3245
```
```  3246 lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"
```
```  3247 proof -
```
```  3248   let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
```
```  3249   have pos_c: "0 < ?c"
```
```  3250     by (rule cos_gt_zero, simp, simp)
```
```  3251   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
```
```  3252     by simp
```
```  3253   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
```
```  3254     by (simp only: cos_add sin_add)
```
```  3255   also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)"
```
```  3256     by (simp add: algebra_simps power2_eq_square)
```
```  3257   finally have "?c\<^sup>2 = (sqrt 3 / 2)\<^sup>2"
```
```  3258     using pos_c by (simp add: sin_squared_eq power_divide)
```
```  3259   thus ?thesis
```
```  3260     using pos_c [THEN order_less_imp_le]
```
```  3261     by (rule power2_eq_imp_eq) simp
```
```  3262 qed
```
```  3263
```
```  3264 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
```
```  3265 by (simp add: sin_cos_eq cos_45)
```
```  3266
```
```  3267 lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
```
```  3268 by (simp add: sin_cos_eq cos_30)
```
```  3269
```
```  3270 lemma cos_60: "cos (pi / 3) = 1 / 2"
```
```  3271 apply (rule power2_eq_imp_eq)
```
```  3272 apply (simp add: cos_squared_eq sin_60 power_divide)
```
```  3273 apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
```
```  3274 done
```
```  3275
```
```  3276 lemma sin_30: "sin (pi / 6) = 1 / 2"
```
```  3277 by (simp add: sin_cos_eq cos_60)
```
```  3278
```
```  3279 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
```
```  3280 unfolding tan_def by (simp add: sin_30 cos_30)
```
```  3281
```
```  3282 lemma tan_45: "tan (pi / 4) = 1"
```
```  3283 unfolding tan_def by (simp add: sin_45 cos_45)
```
```  3284
```
```  3285 lemma tan_60: "tan (pi / 3) = sqrt 3"
```
```  3286 unfolding tan_def by (simp add: sin_60 cos_60)
```
```  3287
```
```  3288 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
```
```  3289 proof -
```
```  3290   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
```
```  3291     by (auto simp add: algebra_simps sin_add)
```
```  3292   thus ?thesis
```
```  3293     by (simp add: real_of_nat_Suc distrib_right add_divide_distrib
```
```  3294                   mult_commute [of pi])
```
```  3295 qed
```
```  3296
```
```  3297 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
```
```  3298 by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
```
```  3299
```
```  3300 lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
```
```  3301 apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
```
```  3302 apply (subst cos_add, simp)
```
```  3303 done
```
```  3304
```
```  3305 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
```
```  3306 by (auto simp add: mult_assoc)
```
```  3307
```
```  3308 lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
```
```  3309 apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
```
```  3310 apply (subst sin_add, simp)
```
```  3311 done
```
```  3312
```
```  3313 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
```
```  3314 by (simp only: cos_add sin_add real_of_nat_Suc distrib_right distrib_left add_divide_distrib, auto)
```
```  3315
```
```  3316 lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
```
```  3317   by (auto intro!: DERIV_intros)
```
```  3318
```
```  3319 lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
```
```  3320 by (auto simp add: sin_zero_iff even_mult_two_ex)
```
```  3321
```
```  3322 lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
```
```  3323 by (cut_tac x = x in sin_cos_squared_add3, auto)
```
```  3324
```
```  3325 subsection {* Machins formula *}
```
```  3326
```
```  3327 lemma arctan_one: "arctan 1 = pi / 4"
```
```  3328   by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi)
```
```  3329
```
```  3330 lemma tan_total_pi4: assumes "\<bar>x\<bar> < 1"
```
```  3331   shows "\<exists> z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
```
```  3332 proof
```
```  3333   show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
```
```  3334     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  3335     unfolding arctan_less_iff using assms by auto
```
```  3336 qed
```
```  3337
```
```  3338 lemma arctan_add: assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
```
```  3339   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
```
```  3340 proof (rule arctan_unique [symmetric])
```
```  3341   have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y"
```
```  3342     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  3343     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  3344   from add_le_less_mono [OF this]
```
```  3345   show 1: "- (pi / 2) < arctan x + arctan y" by simp
```
```  3346   have "arctan x \<le> pi / 4" and "arctan y < pi / 4"
```
```  3347     unfolding arctan_one [symmetric]
```
```  3348     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  3349   from add_le_less_mono [OF this]
```
```  3350   show 2: "arctan x + arctan y < pi / 2" by simp
```
```  3351   show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
```
```  3352     using cos_gt_zero_pi [OF 1 2] by (simp add: tan_add)
```
```  3353 qed
```
```  3354
```
```  3355 theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
```
```  3356 proof -
```
```  3357   have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
```
```  3358   from arctan_add[OF less_imp_le[OF this] this]
```
```  3359   have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
```
```  3360   moreover
```
```  3361   have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
```
```  3362   from arctan_add[OF less_imp_le[OF this] this]
```
```  3363   have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
```
```  3364   moreover
```
```  3365   have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
```
```  3366   from arctan_add[OF this]
```
```  3367   have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
```
```  3368   ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
```
```  3369   thus ?thesis unfolding arctan_one by algebra
```
```  3370 qed
```
```  3371
```
```  3372 subsection {* Introducing the arcus tangens power series *}
```
```  3373
```
```  3374 lemma monoseq_arctan_series: fixes x :: real
```
```  3375   assumes "\<bar>x\<bar> \<le> 1" shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
```
```  3376 proof (cases "x = 0") case True thus ?thesis unfolding monoseq_def One_nat_def by auto
```
```  3377 next
```
```  3378   case False
```
```  3379   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  3380   show "monoseq ?a"
```
```  3381   proof -
```
```  3382     { fix n fix x :: real assume "0 \<le> x" and "x \<le> 1"
```
```  3383       have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le> 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
```
```  3384       proof (rule mult_mono)
```
```  3385         show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))" by (rule frac_le) simp_all
```
```  3386         show "0 \<le> 1 / real (Suc (n * 2))" by auto
```
```  3387         show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)" by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`)
```
```  3388         show "0 \<le> x ^ Suc (Suc n * 2)" by (rule zero_le_power) (simp add: `0 \<le> x`)
```
```  3389       qed
```
```  3390     } note mono = this
```
```  3391
```
```  3392     show ?thesis
```
```  3393     proof (cases "0 \<le> x")
```
```  3394       case True from mono[OF this `x \<le> 1`, THEN allI]
```
```  3395       show ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI2)
```
```  3396     next
```
```  3397       case False hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto
```
```  3398       from mono[OF this]
```
```  3399       have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge> 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using `0 \<le> -x` by auto
```
```  3400       thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
```
```  3401     qed
```
```  3402   qed
```
```  3403 qed
```
```  3404
```
```  3405 lemma zeroseq_arctan_series: fixes x :: real
```
```  3406   assumes "\<bar>x\<bar> \<le> 1" shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
```
```  3407 proof (cases "x = 0") case True thus ?thesis unfolding One_nat_def by (auto simp add: tendsto_const)
```
```  3408 next
```
```  3409   case False
```
```  3410   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  3411   show "?a ----> 0"
```
```  3412   proof (cases "\<bar>x\<bar> < 1")
```
```  3413     case True hence "norm x < 1" by auto
```
```  3414     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]]
```
```  3415     have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"
```
```  3416       unfolding inverse_eq_divide Suc_eq_plus1 by simp
```
```  3417     then show ?thesis using pos2 by (rule LIMSEQ_linear)
```
```  3418   next
```
```  3419     case False hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  3420     hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x" unfolding One_nat_def by auto
```
```  3421     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
```
```  3422     show ?thesis unfolding n_eq Suc_eq_plus1 by auto
```
```  3423   qed
```
```  3424 qed
```
```  3425
```
```  3426 lemma summable_arctan_series: fixes x :: real and n :: nat
```
```  3427   assumes "\<bar>x\<bar> \<le> 1" shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "summable (?c x)")
```
```  3428   by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
```
```  3429
```
```  3430 lemma less_one_imp_sqr_less_one: fixes x :: real assumes "\<bar>x\<bar> < 1" shows "x\<^sup>2 < 1"
```
```  3431 proof -
```
```  3432   from mult_left_mono[OF less_imp_le[OF `\<bar>x\<bar> < 1`] abs_ge_zero[of x]]
```
```  3433   have "\<bar>x\<^sup>2\<bar> < 1" using `\<bar>x\<bar> < 1` unfolding numeral_2_eq_2 power_Suc2 by auto
```
```  3434   thus ?thesis using zero_le_power2 by auto
```
```  3435 qed
```
```  3436
```
```  3437 lemma DERIV_arctan_series: assumes "\<bar> x \<bar> < 1"
```
```  3438   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))" (is "DERIV ?arctan _ :> ?Int")
```
```  3439 proof -
```
```  3440   let "?f n" = "if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
```
```  3441
```
```  3442   { fix n :: nat assume "even n" hence "2 * (n div 2) = n" by presburger } note n_even=this
```
```  3443   have if_eq: "\<And> n x'. ?f n * real (Suc n) * x'^n = (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)" using n_even by auto
```
```  3444
```
```  3445   { fix x :: real assume "\<bar>x\<bar> < 1" hence "x\<^sup>2 < 1" by (rule less_one_imp_sqr_less_one)
```
```  3446     have "summable (\<lambda> n. -1 ^ n * (x\<^sup>2) ^n)"
```
```  3447       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`])
```
```  3448     hence "summable (\<lambda> n. -1 ^ n * x^(2*n))" unfolding power_mult .
```
```  3449   } note summable_Integral = this
```
```  3450
```
```  3451   { fix f :: "nat \<Rightarrow> real"
```
```  3452     have "\<And> x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  3453     proof
```
```  3454       fix x :: real assume "f sums x"
```
```  3455       from sums_if[OF sums_zero this]
```
```  3456       show "(\<lambda> n. if even n then f (n div 2) else 0) sums x" by auto
```
```  3457     next
```
```  3458       fix x :: real assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  3459       from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult_commute]]
```
```  3460       show "f sums x" unfolding sums_def by auto
```
```  3461     qed
```
```  3462     hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
```
```  3463   } note sums_even = this
```
```  3464
```
```  3465   have Int_eq: "(\<Sum> n. ?f n * real (Suc n) * x^n) = ?Int" unfolding if_eq mult_commute[of _ 2] suminf_def sums_even[of "\<lambda> n. -1 ^ n * x ^ (2 * n)", symmetric]
```
```  3466     by auto
```
```  3467
```
```  3468   { fix x :: real
```
```  3469     have if_eq': "\<And> n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
```
```  3470       (if even n then -1 ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
```
```  3471       using n_even by auto
```
```  3472     have idx_eq: "\<And> n. n * 2 + 1 = Suc (2 * n)" by auto
```
```  3473     have "(\<Sum> n. ?f n * x^(Suc n)) = ?arctan x" unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. -1 ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
```
```  3474       by auto
```
```  3475   } note arctan_eq = this
```
```  3476
```
```  3477   have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
```
```  3478   proof (rule DERIV_power_series')
```
```  3479     show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto
```
```  3480     { fix x' :: real assume x'_bounds: "x' \<in> {- 1 <..< 1}"
```
```  3481       hence "\<bar>x'\<bar> < 1" by auto
```
```  3482
```
```  3483       let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
```
```  3484       show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
```
```  3485         by (rule sums_summable[where l="0 + ?S"], rule sums_if, rule sums_zero, rule summable_sums, rule summable_Integral[OF `\<bar>x'\<bar> < 1`])
```
```  3486     }
```
```  3487   qed auto
```
```  3488   thus ?thesis unfolding Int_eq arctan_eq .
```
```  3489 qed
```
```  3490
```
```  3491 lemma arctan_series: assumes "\<bar> x \<bar> \<le> 1"
```
```  3492   shows "arctan x = (\<Sum> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "_ = suminf (\<lambda> n. ?c x n)")
```
```  3493 proof -
```
```  3494   let "?c' x n" = "(-1)^n * x^(n*2)"
```
```  3495
```
```  3496   { fix r x :: real assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
```
```  3497     have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto
```
```  3498     from DERIV_arctan_series[OF this]
```
```  3499     have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
```
```  3500   } note DERIV_arctan_suminf = this
```
```  3501
```
```  3502   { fix x :: real assume "\<bar>x\<bar> \<le> 1" note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] }
```
```  3503   note arctan_series_borders = this
```
```  3504
```
```  3505   { fix x :: real assume "\<bar>x\<bar> < 1" have "arctan x = (\<Sum> k. ?c x k)"
```
```  3506   proof -
```
```  3507     obtain r where "\<bar>x\<bar> < r" and "r < 1" using dense[OF `\<bar>x\<bar> < 1`] by blast
```
```  3508     hence "0 < r" and "-r < x" and "x < r" by auto
```
```  3509
```
```  3510     have suminf_eq_arctan_bounded: "\<And> x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow> suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  3511     proof -
```
```  3512       fix x a b assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
```
```  3513       hence "\<bar>x\<bar> < r" by auto
```
```  3514       show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  3515       proof (rule DERIV_isconst2[of "a" "b"])
```
```  3516         show "a < b" and "a \<le> x" and "x \<le> b" using `a < b` `a \<le> x` `x \<le> b` by auto
```
```  3517         have "\<forall> x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  3518         proof (rule allI, rule impI)
```
```  3519           fix x assume "-r < x \<and> x < r" hence "\<bar>x\<bar> < r" by auto
```
```  3520           hence "\<bar>x\<bar> < 1" using `r < 1` by auto
```
```  3521           have "\<bar> - (x\<^sup>2) \<bar> < 1" using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
```
```  3522           hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))" unfolding real_norm_def[symmetric] by (rule geometric_sums)
```
```  3523           hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))" unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto
```
```  3524           hence suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)" using sums_unique unfolding inverse_eq_divide by auto
```
```  3525           have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))" unfolding suminf_c'_eq_geom
```
```  3526             by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
```
```  3527           from DERIV_add_minus[OF this DERIV_arctan]
```
```  3528           show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0" unfolding diff_minus by auto
```
```  3529         qed
```
```  3530         hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0" using `-r < a` `b < r` by auto
```
```  3531         thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0" using `\<bar>x\<bar> < r` by auto
```
```  3532         show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y" using DERIV_in_rball DERIV_isCont by auto
```
```  3533       qed
```
```  3534     qed
```
```  3535
```
```  3536     have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
```
```  3537       unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero by auto
```
```  3538
```
```  3539     have "suminf (?c x) - arctan x = 0"
```
```  3540     proof (cases "x = 0")
```
```  3541       case True thus ?thesis using suminf_arctan_zero by auto
```
```  3542     next
```
```  3543       case False hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
```
```  3544       have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
```
```  3545         by (rule suminf_eq_arctan_bounded[where x="0" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>", symmetric])
```
```  3546           (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  3547       moreover
```
```  3548       have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
```
```  3549         by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"])
```
```  3550           (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  3551       ultimately
```
```  3552       show ?thesis using suminf_arctan_zero by auto
```
```  3553     qed
```
```  3554     thus ?thesis by auto
```
```  3555   qed } note when_less_one = this
```
```  3556
```
```  3557   show "arctan x = suminf (\<lambda> n. ?c x n)"
```
```  3558   proof (cases "\<bar>x\<bar> < 1")
```
```  3559     case True thus ?thesis by (rule when_less_one)
```
```  3560   next case False hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  3561     let "?a x n" = "\<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
```
```  3562     let "?diff x n" = "\<bar> arctan x - (\<Sum> i = 0..<n. ?c x i)\<bar>"
```
```  3563     { fix n :: nat
```
```  3564       have "0 < (1 :: real)" by auto
```
```  3565       moreover
```
```  3566       { fix x :: real assume "0 < x" and "x < 1" hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
```
```  3567         from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)" by auto
```
```  3568         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]
```
```  3569         have "0 < 1 / real (n*2+1) * x^(n*2+1)" by (rule mult_pos_pos, auto simp only: zero_less_power[OF `0 < x`], auto)
```
```  3570         hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)" by (rule abs_of_pos)
```
```  3571         have "?diff x n \<le> ?a x n"
```
```  3572         proof (cases "even n")
```
```  3573           case True hence sgn_pos: "(-1)^n = (1::real)" by auto
```
```  3574           from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
```
```  3575           from bounds[of m, unfolded this atLeastAtMost_iff]
```
```  3576           have "\<bar>arctan x - (\<Sum>i = 0..<n. (?c x i))\<bar> \<le> (\<Sum>i = 0..<n + 1. (?c x i)) - (\<Sum>i = 0..<n. (?c x i))" by auto
```
```  3577           also have "\<dots> = ?c x n" unfolding One_nat_def by auto
```
```  3578           also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
```
```  3579           finally show ?thesis .
```
```  3580         next
```
```  3581           case False hence sgn_neg: "(-1)^n = (-1::real)" by auto
```
```  3582           from `odd n` obtain m where m_def: "2 * m + 1 = n" unfolding odd_Suc_mult_two_ex by auto
```
```  3583           hence m_plus: "2 * (m + 1) = n + 1" by auto
```
```  3584           from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
```
```  3585           have "\<bar>arctan x - (\<Sum>i = 0..<n. (?c x i))\<bar> \<le> (\<Sum>i = 0..<n. (?c x i)) - (\<Sum>i = 0..<n+1. (?c x i))" by auto
```
```  3586           also have "\<dots> = - ?c x n" unfolding One_nat_def by auto
```
```  3587           also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
```
```  3588           finally show ?thesis .
```
```  3589         qed
```
```  3590         hence "0 \<le> ?a x n - ?diff x n" by auto
```
```  3591       }
```
```  3592       hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
```
```  3593       moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
```
```  3594         unfolding diff_minus divide_inverse
```
```  3595         by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum)
```
```  3596       ultimately have "0 \<le> ?a 1 n - ?diff 1 n" by (rule LIM_less_bound)
```
```  3597       hence "?diff 1 n \<le> ?a 1 n" by auto
```
```  3598     }
```
```  3599     have "?a 1 ----> 0"
```
```  3600       unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
```
```  3601       by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)
```
```  3602     have "?diff 1 ----> 0"
```
```  3603     proof (rule LIMSEQ_I)
```
```  3604       fix r :: real assume "0 < r"
```
```  3605       obtain N :: nat where N_I: "\<And> n. N \<le> n \<Longrightarrow> ?a 1 n < r" using LIMSEQ_D[OF `?a 1 ----> 0` `0 < r`] by auto
```
```  3606       { fix n assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this]
```
```  3607         have "norm (?diff 1 n - 0) < r" by auto }
```
```  3608       thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
```
```  3609     qed
```
```  3610     from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
```
```  3611     have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
```
```  3612     hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
```
```  3613
```
```  3614     show ?thesis
```
```  3615     proof (cases "x = 1", simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
```
```  3616       assume "x \<noteq> 1" hence "x = -1" using `\<bar>x\<bar> = 1` by auto
```
```  3617
```
```  3618       have "- (pi / 2) < 0" using pi_gt_zero by auto
```
```  3619       have "- (2 * pi) < 0" using pi_gt_zero by auto
```
```  3620
```
```  3621       have c_minus_minus: "\<And> i. ?c (- 1) i = - ?c 1 i" unfolding One_nat_def by auto
```
```  3622
```
```  3623       have "arctan (- 1) = arctan (tan (-(pi / 4)))" unfolding tan_45 tan_minus ..
```
```  3624       also have "\<dots> = - (pi / 4)" by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
```
```  3625       also have "\<dots> = - (arctan (tan (pi / 4)))" unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero])
```
```  3626       also have "\<dots> = - (arctan 1)" unfolding tan_45 ..
```
```  3627       also have "\<dots> = - (\<Sum> i. ?c 1 i)" using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto
```
```  3628       also have "\<dots> = (\<Sum> i. ?c (- 1) i)" using suminf_minus[OF sums_summable[OF `(?c 1) sums (arctan 1)`]] unfolding c_minus_minus by auto
```
```  3629       finally show ?thesis using `x = -1` by auto
```
```  3630     qed
```
```  3631   qed
```
```  3632 qed
```
```  3633
```
```  3634 lemma arctan_half: fixes x :: real
```
```  3635   shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))"
```
```  3636 proof -
```
```  3637   obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x" using tan_total by blast
```
```  3638   hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2" by auto
```
```  3639
```
```  3640   have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> A / B = (A / C) / (B / C)" by auto
```
```  3641
```
```  3642   have "0 < cos y" using cos_gt_zero_pi[OF low high] .
```
```  3643   hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y" by auto
```
```  3644
```
```  3645   have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2" unfolding tan_def power_divide ..
```
```  3646   also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2" using `cos y \<noteq> 0` by auto
```
```  3647   also have "\<dots> = 1 / (cos y)\<^sup>2" unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
```
```  3648   finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" .
```
```  3649
```
```  3650   have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)" unfolding tan_def divide_nonzero_divide[OF `cos y \<noteq> 0`, symmetric] ..
```
```  3651   also have "\<dots> = tan y / (1 + 1 / cos y)" using `cos y \<noteq> 0` unfolding add_divide_distrib by auto
```
```  3652   also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))" unfolding cos_sqrt ..
```
```  3653   also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))" unfolding real_sqrt_divide by auto
```
```  3654   finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))" unfolding `1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2` .
```
```  3655
```
```  3656   have "arctan x = y" using arctan_tan low high y_eq by auto
```
```  3657   also have "\<dots> = 2 * (arctan (tan (y/2)))" using arctan_tan[OF low2 high2] by auto
```
```  3658   also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" unfolding tan_half by auto
```
```  3659   finally show ?thesis unfolding eq `tan y = x` .
```
```  3660 qed
```
```  3661
```
```  3662 lemma arctan_monotone: assumes "x < y"
```
```  3663   shows "arctan x < arctan y"
```
```  3664   using assms by (simp only: arctan_less_iff)
```
```  3665
```
```  3666 lemma arctan_monotone': assumes "x \<le> y" shows "arctan x \<le> arctan y"
```
```  3667   using assms by (simp only: arctan_le_iff)
```
```  3668
```
```  3669 lemma arctan_inverse:
```
```  3670   assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
```
```  3671 proof (rule arctan_unique)
```
```  3672   show "- (pi / 2) < sgn x * pi / 2 - arctan x"
```
```  3673     using arctan_bounded [of x] assms
```
```  3674     unfolding sgn_real_def
```
```  3675     apply (auto simp add: algebra_simps)
```
```  3676     apply (drule zero_less_arctan_iff [THEN iffD2])
```
```  3677     apply arith
```
```  3678     done
```
```  3679   show "sgn x * pi / 2 - arctan x < pi / 2"
```
```  3680     using arctan_bounded [of "- x"] assms
```
```  3681     unfolding sgn_real_def arctan_minus
```
```  3682     by auto
```
```  3683   show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
```
```  3684     unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
```
```  3685     unfolding sgn_real_def
```
```  3686     by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
```
```  3687 qed
```
```  3688
```
```  3689 theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
```
```  3690 proof -
```
```  3691   have "pi / 4 = arctan 1" using arctan_one by auto
```
```  3692   also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
```
```  3693   finally show ?thesis by auto
```
```  3694 qed
```
```  3695
```
```  3696 subsection {* Existence of Polar Coordinates *}
```
```  3697
```
```  3698 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1"
```
```  3699 apply (rule power2_le_imp_le [OF _ zero_le_one])
```
```  3700 apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
```
```  3701 done
```
```  3702
```
```  3703 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
```
```  3704 by (simp add: abs_le_iff)
```
```  3705
```
```  3706 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
```
```  3707 by (simp add: sin_arccos abs_le_iff)
```
```  3708
```
```  3709 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
```
```  3710
```
```  3711 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
```
```  3712
```
```  3713 lemma polar_ex1:
```
```  3714      "0 < y ==> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  3715 apply (rule_tac x = "sqrt (x\<^sup>2 + y\<^sup>2)" in exI)
```
```  3716 apply (rule_tac x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))" in exI)
```
```  3717 apply (simp add: cos_arccos_lemma1)
```
```  3718 apply (simp add: sin_arccos_lemma1)
```
```  3719 apply (simp add: power_divide)
```
```  3720 apply (simp add: real_sqrt_mult [symmetric])
```
```  3721 apply (simp add: right_diff_distrib)
```
```  3722 done
```
```  3723
```
```  3724 lemma polar_ex2:
```
```  3725      "y < 0 ==> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  3726 apply (insert polar_ex1 [where x=x and y="-y"], simp, clarify)
```
```  3727 apply (metis cos_minus minus_minus minus_mult_right sin_minus)
```
```  3728 done
```
```  3729
```
```  3730 lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
```
```  3731 apply (rule_tac x=0 and y=y in linorder_cases)
```
```  3732 apply (erule polar_ex1)
```
```  3733 apply (rule_tac x=x in exI, rule_tac x=0 in exI, simp)
```
```  3734 apply (erule polar_ex2)
```
```  3735 done
```
```  3736
```
```  3737 end
```