src/HOL/Transcendental.thy
 author paulson Sat Apr 11 11:56:40 2015 +0100 (2015-04-11) changeset 60017 b785d6d06430 parent 59869 3b5b53eb78ba child 60020 065ecea354d0 permissions -rw-r--r--
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
```     1 (*  Title:      HOL/Transcendental.thy
```
```     2     Author:     Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
```
```     3     Author:     Lawrence C Paulson
```
```     4     Author:     Jeremy Avigad
```
```     5 *)
```
```     6
```
```     7 section{*Power Series, Transcendental Functions etc.*}
```
```     8
```
```     9 theory Transcendental
```
```    10 imports Binomial Series Deriv NthRoot
```
```    11 begin
```
```    12
```
```    13 lemma of_real_fact [simp]: "of_real (fact n) = fact n"
```
```    14   by (metis of_nat_fact of_real_of_nat_eq)
```
```    15
```
```    16 lemma real_fact_nat [simp]: "real (fact n :: nat) = fact n"
```
```    17   by (simp add: real_of_nat_def)
```
```    18
```
```    19 lemma real_fact_int [simp]: "real (fact n :: int) = fact n"
```
```    20   by (metis of_int_of_nat_eq of_nat_fact real_of_int_def)
```
```    21
```
```    22 lemma root_test_convergence:
```
```    23   fixes f :: "nat \<Rightarrow> 'a::banach"
```
```    24   assumes f: "(\<lambda>n. root n (norm (f n))) ----> x" -- "could be weakened to lim sup"
```
```    25   assumes "x < 1"
```
```    26   shows "summable f"
```
```    27 proof -
```
```    28   have "0 \<le> x"
```
```    29     by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1])
```
```    30   from `x < 1` obtain z where z: "x < z" "z < 1"
```
```    31     by (metis dense)
```
```    32   from f `x < z`
```
```    33   have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
```
```    34     by (rule order_tendstoD)
```
```    35   then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially"
```
```    36     using eventually_ge_at_top
```
```    37   proof eventually_elim
```
```    38     fix n assume less: "root n (norm (f n)) < z" and n: "1 \<le> n"
```
```    39     from power_strict_mono[OF less, of n] n
```
```    40     show "norm (f n) \<le> z ^ n"
```
```    41       by simp
```
```    42   qed
```
```    43   then show "summable f"
```
```    44     unfolding eventually_sequentially
```
```    45     using z `0 \<le> x` by (auto intro!: summable_comparison_test[OF _  summable_geometric])
```
```    46 qed
```
```    47
```
```    48 subsection {* Properties of Power Series *}
```
```    49
```
```    50 lemma lemma_realpow_diff:
```
```    51   fixes y :: "'a::monoid_mult"
```
```    52   shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
```
```    53 proof -
```
```    54   assume "p \<le> n"
```
```    55   hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
```
```    56   thus ?thesis by (simp add: power_commutes)
```
```    57 qed
```
```    58
```
```    59 lemma lemma_realpow_diff_sumr2:
```
```    60   fixes y :: "'a::{comm_ring,monoid_mult}"
```
```    61   shows
```
```    62     "x ^ (Suc n) - y ^ (Suc n) =
```
```    63       (x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))"
```
```    64 proof (induct n)
```
```    65   case (Suc n)
```
```    66   have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x^n) - y * (y * y ^ n)"
```
```    67     by simp
```
```    68   also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x^n)"
```
```    69     by (simp add: algebra_simps)
```
```    70   also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
```
```    71     by (simp only: Suc)
```
```    72   also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
```
```    73     by (simp only: mult.left_commute)
```
```    74   also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
```
```    75     by (simp add: field_simps Suc_diff_le setsum_left_distrib setsum_right_distrib)
```
```    76   finally show ?case .
```
```    77 qed simp
```
```    78
```
```    79 corollary power_diff_sumr2: --{* @{text COMPLEX_POLYFUN} in HOL Light *}
```
```    80   fixes x :: "'a::{comm_ring,monoid_mult}"
```
```    81   shows   "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)"
```
```    82 using lemma_realpow_diff_sumr2[of x "n - 1" y]
```
```    83 by (cases "n = 0") (simp_all add: field_simps)
```
```    84
```
```    85 lemma lemma_realpow_rev_sumr:
```
```    86    "(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) =
```
```    87     (\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))"
```
```    88   by (subst nat_diff_setsum_reindex[symmetric]) simp
```
```    89
```
```    90 lemma power_diff_1_eq:
```
```    91   fixes x :: "'a::{comm_ring,monoid_mult}"
```
```    92   shows "n \<noteq> 0 \<Longrightarrow> x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"
```
```    93 using lemma_realpow_diff_sumr2 [of x _ 1]
```
```    94   by (cases n) auto
```
```    95
```
```    96 lemma one_diff_power_eq':
```
```    97   fixes x :: "'a::{comm_ring,monoid_mult}"
```
```    98   shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))"
```
```    99 using lemma_realpow_diff_sumr2 [of 1 _ x]
```
```   100   by (cases n) auto
```
```   101
```
```   102 lemma one_diff_power_eq:
```
```   103   fixes x :: "'a::{comm_ring,monoid_mult}"
```
```   104   shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^i)"
```
```   105 by (metis one_diff_power_eq' [of n x] nat_diff_setsum_reindex)
```
```   106
```
```   107 text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
```
```   108   x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
```
```   109
```
```   110 lemma powser_insidea:
```
```   111   fixes x z :: "'a::real_normed_div_algebra"
```
```   112   assumes 1: "summable (\<lambda>n. f n * x^n)"
```
```   113     and 2: "norm z < norm x"
```
```   114   shows "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   115 proof -
```
```   116   from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
```
```   117   from 1 have "(\<lambda>n. f n * x^n) ----> 0"
```
```   118     by (rule summable_LIMSEQ_zero)
```
```   119   hence "convergent (\<lambda>n. f n * x^n)"
```
```   120     by (rule convergentI)
```
```   121   hence "Cauchy (\<lambda>n. f n * x^n)"
```
```   122     by (rule convergent_Cauchy)
```
```   123   hence "Bseq (\<lambda>n. f n * x^n)"
```
```   124     by (rule Cauchy_Bseq)
```
```   125   then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x^n) \<le> K"
```
```   126     by (simp add: Bseq_def, safe)
```
```   127   have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
```
```   128                    K * norm (z ^ n) * inverse (norm (x^n))"
```
```   129   proof (intro exI allI impI)
```
```   130     fix n::nat
```
```   131     assume "0 \<le> n"
```
```   132     have "norm (norm (f n * z ^ n)) * norm (x^n) =
```
```   133           norm (f n * x^n) * norm (z ^ n)"
```
```   134       by (simp add: norm_mult abs_mult)
```
```   135     also have "\<dots> \<le> K * norm (z ^ n)"
```
```   136       by (simp only: mult_right_mono 4 norm_ge_zero)
```
```   137     also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x^n)) * norm (x^n))"
```
```   138       by (simp add: x_neq_0)
```
```   139     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x^n)) * norm (x^n)"
```
```   140       by (simp only: mult.assoc)
```
```   141     finally show "norm (norm (f n * z ^ n)) \<le>
```
```   142                   K * norm (z ^ n) * inverse (norm (x^n))"
```
```   143       by (simp add: mult_le_cancel_right x_neq_0)
```
```   144   qed
```
```   145   moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
```
```   146   proof -
```
```   147     from 2 have "norm (norm (z * inverse x)) < 1"
```
```   148       using x_neq_0
```
```   149       by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
```
```   150     hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
```
```   151       by (rule summable_geometric)
```
```   152     hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
```
```   153       by (rule summable_mult)
```
```   154     thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
```
```   155       using x_neq_0
```
```   156       by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
```
```   157                     power_inverse norm_power mult.assoc)
```
```   158   qed
```
```   159   ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   160     by (rule summable_comparison_test)
```
```   161 qed
```
```   162
```
```   163 lemma powser_inside:
```
```   164   fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
```
```   165   shows
```
```   166     "summable (\<lambda>n. f n * (x^n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
```
```   167       summable (\<lambda>n. f n * (z ^ n))"
```
```   168   by (rule powser_insidea [THEN summable_norm_cancel])
```
```   169
```
```   170 lemma sum_split_even_odd:
```
```   171   fixes f :: "nat \<Rightarrow> real"
```
```   172   shows
```
```   173     "(\<Sum>i<2 * n. if even i then f i else g i) =
```
```   174      (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
```
```   175 proof (induct n)
```
```   176   case 0
```
```   177   then show ?case by simp
```
```   178 next
```
```   179   case (Suc n)
```
```   180   have "(\<Sum>i<2 * Suc n. if even i then f i else g i) =
```
```   181     (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
```
```   182     using Suc.hyps unfolding One_nat_def by auto
```
```   183   also have "\<dots> = (\<Sum>i<Suc n. f (2 * i)) + (\<Sum>i<Suc n. g (2 * i + 1))"
```
```   184     by auto
```
```   185   finally show ?case .
```
```   186 qed
```
```   187
```
```   188 lemma sums_if':
```
```   189   fixes g :: "nat \<Rightarrow> real"
```
```   190   assumes "g sums x"
```
```   191   shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   192   unfolding sums_def
```
```   193 proof (rule LIMSEQ_I)
```
```   194   fix r :: real
```
```   195   assume "0 < r"
```
```   196   from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
```
```   197   obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g {..<n} - x) < r)" by blast
```
```   198
```
```   199   let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)"
```
```   200   {
```
```   201     fix m
```
```   202     assume "m \<ge> 2 * no"
```
```   203     hence "m div 2 \<ge> no" by auto
```
```   204     have sum_eq: "?SUM (2 * (m div 2)) = setsum g {..< m div 2}"
```
```   205       using sum_split_even_odd by auto
```
```   206     hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
```
```   207       using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
```
```   208     moreover
```
```   209     have "?SUM (2 * (m div 2)) = ?SUM m"
```
```   210     proof (cases "even m")
```
```   211       case True
```
```   212       then show ?thesis by (auto simp add: even_two_times_div_two)
```
```   213     next
```
```   214       case False
```
```   215       then have eq: "Suc (2 * (m div 2)) = m" by simp
```
```   216       hence "even (2 * (m div 2))" using `odd m` by auto
```
```   217       have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
```
```   218       also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
```
```   219       finally show ?thesis by auto
```
```   220     qed
```
```   221     ultimately have "(norm (?SUM m - x) < r)" by auto
```
```   222   }
```
```   223   thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
```
```   224 qed
```
```   225
```
```   226 lemma sums_if:
```
```   227   fixes g :: "nat \<Rightarrow> real"
```
```   228   assumes "g sums x" and "f sums y"
```
```   229   shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
```
```   230 proof -
```
```   231   let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
```
```   232   {
```
```   233     fix B T E
```
```   234     have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
```
```   235       by (cases B) auto
```
```   236   } note if_sum = this
```
```   237   have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
```
```   238     using sums_if'[OF `g sums x`] .
```
```   239   {
```
```   240     have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
```
```   241
```
```   242     have "?s sums y" using sums_if'[OF `f sums y`] .
```
```   243     from this[unfolded sums_def, THEN LIMSEQ_Suc]
```
```   244     have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
```
```   245       by (simp add: lessThan_Suc_eq_insert_0 image_iff setsum.reindex if_eq sums_def cong del: if_cong)
```
```   246   }
```
```   247   from sums_add[OF g_sums this] show ?thesis unfolding if_sum .
```
```   248 qed
```
```   249
```
```   250 subsection {* Alternating series test / Leibniz formula *}
```
```   251
```
```   252 lemma sums_alternating_upper_lower:
```
```   253   fixes a :: "nat \<Rightarrow> real"
```
```   254   assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
```
```   255   shows "\<exists>l. ((\<forall>n. (\<Sum>i<2*n. (- 1)^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i<2*n. (- 1)^i*a i) ----> l) \<and>
```
```   256              ((\<forall>n. l \<le> (\<Sum>i<2*n + 1. (- 1)^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. (- 1)^i*a i) ----> l)"
```
```   257   (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
```
```   258 proof (rule nested_sequence_unique)
```
```   259   have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
```
```   260
```
```   261   show "\<forall>n. ?f n \<le> ?f (Suc n)"
```
```   262   proof
```
```   263     fix n
```
```   264     show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
```
```   265   qed
```
```   266   show "\<forall>n. ?g (Suc n) \<le> ?g n"
```
```   267   proof
```
```   268     fix n
```
```   269     show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
```
```   270       unfolding One_nat_def by auto
```
```   271   qed
```
```   272   show "\<forall>n. ?f n \<le> ?g n"
```
```   273   proof
```
```   274     fix n
```
```   275     show "?f n \<le> ?g n" using fg_diff a_pos
```
```   276       unfolding One_nat_def by auto
```
```   277   qed
```
```   278   show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff
```
```   279   proof (rule LIMSEQ_I)
```
```   280     fix r :: real
```
```   281     assume "0 < r"
```
```   282     with `a ----> 0`[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
```
```   283       by auto
```
```   284     hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   285     thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
```
```   286   qed
```
```   287 qed
```
```   288
```
```   289 lemma summable_Leibniz':
```
```   290   fixes a :: "nat \<Rightarrow> real"
```
```   291   assumes a_zero: "a ----> 0"
```
```   292     and a_pos: "\<And> n. 0 \<le> a n"
```
```   293     and a_monotone: "\<And> n. a (Suc n) \<le> a n"
```
```   294   shows summable: "summable (\<lambda> n. (-1)^n * a n)"
```
```   295     and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
```
```   296     and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   297     and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i<2*n+1. (-1)^i*a i)"
```
```   298     and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
```
```   299 proof -
```
```   300   let ?S = "\<lambda>n. (-1)^n * a n"
```
```   301   let ?P = "\<lambda>n. \<Sum>i<n. ?S i"
```
```   302   let ?f = "\<lambda>n. ?P (2 * n)"
```
```   303   let ?g = "\<lambda>n. ?P (2 * n + 1)"
```
```   304   obtain l :: real
```
```   305     where below_l: "\<forall> n. ?f n \<le> l"
```
```   306       and "?f ----> l"
```
```   307       and above_l: "\<forall> n. l \<le> ?g n"
```
```   308       and "?g ----> l"
```
```   309     using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
```
```   310
```
```   311   let ?Sa = "\<lambda>m. \<Sum>n<m. ?S n"
```
```   312   have "?Sa ----> l"
```
```   313   proof (rule LIMSEQ_I)
```
```   314     fix r :: real
```
```   315     assume "0 < r"
```
```   316     with `?f ----> l`[THEN LIMSEQ_D]
```
```   317     obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
```
```   318
```
```   319     from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
```
```   320     obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
```
```   321
```
```   322     {
```
```   323       fix n :: nat
```
```   324       assume "n \<ge> (max (2 * f_no) (2 * g_no))"
```
```   325       hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
```
```   326       have "norm (?Sa n - l) < r"
```
```   327       proof (cases "even n")
```
```   328         case True
```
```   329         then have n_eq: "2 * (n div 2) = n" by (simp add: even_two_times_div_two)
```
```   330         with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no"
```
```   331           by auto
```
```   332         from f[OF this] show ?thesis
```
```   333           unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
```
```   334       next
```
```   335         case False
```
```   336         hence "even (n - 1)" by simp
```
```   337         then have n_eq: "2 * ((n - 1) div 2) = n - 1"
```
```   338           by (simp add: even_two_times_div_two)
```
```   339         hence range_eq: "n - 1 + 1 = n"
```
```   340           using odd_pos[OF False] by auto
```
```   341
```
```   342         from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no"
```
```   343           by auto
```
```   344         from g[OF this] show ?thesis
```
```   345           unfolding n_eq range_eq .
```
```   346       qed
```
```   347     }
```
```   348     thus "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
```
```   349   qed
```
```   350   hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
```
```   351     unfolding sums_def .
```
```   352   thus "summable ?S" using summable_def by auto
```
```   353
```
```   354   have "l = suminf ?S" using sums_unique[OF sums_l] .
```
```   355
```
```   356   fix n
```
```   357   show "suminf ?S \<le> ?g n"
```
```   358     unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
```
```   359   show "?f n \<le> suminf ?S"
```
```   360     unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
```
```   361   show "?g ----> suminf ?S"
```
```   362     using `?g ----> l` `l = suminf ?S` by auto
```
```   363   show "?f ----> suminf ?S"
```
```   364     using `?f ----> l` `l = suminf ?S` by auto
```
```   365 qed
```
```   366
```
```   367 theorem summable_Leibniz:
```
```   368   fixes a :: "nat \<Rightarrow> real"
```
```   369   assumes a_zero: "a ----> 0" and "monoseq a"
```
```   370   shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
```
```   371     and "0 < a 0 \<longrightarrow>
```
```   372       (\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n. (- 1)^i * a i .. \<Sum>i<2*n+1. (- 1)^i * a i})" (is "?pos")
```
```   373     and "a 0 < 0 \<longrightarrow>
```
```   374       (\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n+1. (- 1)^i * a i .. \<Sum>i<2*n. (- 1)^i * a i})" (is "?neg")
```
```   375     and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) ----> (\<Sum>i. (- 1)^i*a i)" (is "?f")
```
```   376     and "(\<lambda>n. \<Sum>i<2*n+1. (- 1)^i*a i) ----> (\<Sum>i. (- 1)^i*a i)" (is "?g")
```
```   377 proof -
```
```   378   have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
```
```   379   proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
```
```   380     case True
```
```   381     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n"
```
```   382       by auto
```
```   383     {
```
```   384       fix n
```
```   385       have "a (Suc n) \<le> a n"
```
```   386         using ord[where n="Suc n" and m=n] by auto
```
```   387     } note mono = this
```
```   388     note leibniz = summable_Leibniz'[OF `a ----> 0` ge0]
```
```   389     from leibniz[OF mono]
```
```   390     show ?thesis using `0 \<le> a 0` by auto
```
```   391   next
```
```   392     let ?a = "\<lambda> n. - a n"
```
```   393     case False
```
```   394     with monoseq_le[OF `monoseq a` `a ----> 0`]
```
```   395     have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
```
```   396     hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
```
```   397       by auto
```
```   398     {
```
```   399       fix n
```
```   400       have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
```
```   401         by auto
```
```   402     } note monotone = this
```
```   403     note leibniz =
```
```   404       summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
```
```   405         OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
```
```   406     have "summable (\<lambda> n. (-1)^n * ?a n)"
```
```   407       using leibniz(1) by auto
```
```   408     then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
```
```   409       unfolding summable_def by auto
```
```   410     from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
```
```   411       by auto
```
```   412     hence ?summable unfolding summable_def by auto
```
```   413     moreover
```
```   414     have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
```
```   415       unfolding minus_diff_minus by auto
```
```   416
```
```   417     from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
```
```   418     have move_minus: "(\<Sum>n. - ((- 1) ^ n * a n)) = - (\<Sum>n. (- 1) ^ n * a n)"
```
```   419       by auto
```
```   420
```
```   421     have ?pos using `0 \<le> ?a 0` by auto
```
```   422     moreover have ?neg
```
```   423       using leibniz(2,4)
```
```   424       unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
```
```   425       by auto
```
```   426     moreover have ?f and ?g
```
```   427       using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
```
```   428       by auto
```
```   429     ultimately show ?thesis by auto
```
```   430   qed
```
```   431   then show ?summable and ?pos and ?neg and ?f and ?g
```
```   432     by safe
```
```   433 qed
```
```   434
```
```   435 subsection {* Term-by-Term Differentiability of Power Series *}
```
```   436
```
```   437 definition diffs :: "(nat \<Rightarrow> 'a::ring_1) \<Rightarrow> nat \<Rightarrow> 'a"
```
```   438   where "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))"
```
```   439
```
```   440 text{*Lemma about distributing negation over it*}
```
```   441 lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
```
```   442   by (simp add: diffs_def)
```
```   443
```
```   444 lemma sums_Suc_imp:
```
```   445   "(f::nat \<Rightarrow> 'a::real_normed_vector) 0 = 0 \<Longrightarrow> (\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
```
```   446   using sums_Suc_iff[of f] by simp
```
```   447
```
```   448 lemma diffs_equiv:
```
```   449   fixes x :: "'a::{real_normed_vector, ring_1}"
```
```   450   shows "summable (\<lambda>n. diffs c n * x^n) \<Longrightarrow>
```
```   451       (\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)"
```
```   452   unfolding diffs_def
```
```   453   by (simp add: summable_sums sums_Suc_imp)
```
```   454
```
```   455 lemma lemma_termdiff1:
```
```   456   fixes z :: "'a :: {monoid_mult,comm_ring}" shows
```
```   457   "(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
```
```   458    (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
```
```   459   by (auto simp add: algebra_simps power_add [symmetric])
```
```   460
```
```   461 lemma sumr_diff_mult_const2:
```
```   462   "setsum f {..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i<n. f i - r)"
```
```   463   by (simp add: setsum_subtractf)
```
```   464
```
```   465 lemma lemma_termdiff2:
```
```   466   fixes h :: "'a :: {field}"
```
```   467   assumes h: "h \<noteq> 0"
```
```   468   shows
```
```   469     "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
```
```   470      h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p.
```
```   471           (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
```
```   472   apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
```
```   473   apply (simp add: right_diff_distrib diff_divide_distrib h)
```
```   474   apply (simp add: mult.assoc [symmetric])
```
```   475   apply (cases "n", simp)
```
```   476   apply (simp add: lemma_realpow_diff_sumr2 h
```
```   477                    right_diff_distrib [symmetric] mult.assoc
```
```   478               del: power_Suc setsum_lessThan_Suc of_nat_Suc)
```
```   479   apply (subst lemma_realpow_rev_sumr)
```
```   480   apply (subst sumr_diff_mult_const2)
```
```   481   apply simp
```
```   482   apply (simp only: lemma_termdiff1 setsum_right_distrib)
```
```   483   apply (rule setsum.cong [OF refl])
```
```   484   apply (simp add: less_iff_Suc_add)
```
```   485   apply (clarify)
```
```   486   apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 ac_simps
```
```   487               del: setsum_lessThan_Suc power_Suc)
```
```   488   apply (subst mult.assoc [symmetric], subst power_add [symmetric])
```
```   489   apply (simp add: ac_simps)
```
```   490   done
```
```   491
```
```   492 lemma real_setsum_nat_ivl_bounded2:
```
```   493   fixes K :: "'a::linordered_semidom"
```
```   494   assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
```
```   495     and K: "0 \<le> K"
```
```   496   shows "setsum f {..<n-k} \<le> of_nat n * K"
```
```   497   apply (rule order_trans [OF setsum_mono])
```
```   498   apply (rule f, simp)
```
```   499   apply (simp add: mult_right_mono K)
```
```   500   done
```
```   501
```
```   502 lemma lemma_termdiff3:
```
```   503   fixes h z :: "'a::{real_normed_field}"
```
```   504   assumes 1: "h \<noteq> 0"
```
```   505     and 2: "norm z \<le> K"
```
```   506     and 3: "norm (z + h) \<le> K"
```
```   507   shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
```
```   508           \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   509 proof -
```
```   510   have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
```
```   511         norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p.
```
```   512           (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
```
```   513     by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult)
```
```   514   also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
```
```   515   proof (rule mult_right_mono [OF _ norm_ge_zero])
```
```   516     from norm_ge_zero 2 have K: "0 \<le> K"
```
```   517       by (rule order_trans)
```
```   518     have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
```
```   519       apply (erule subst)
```
```   520       apply (simp only: norm_mult norm_power power_add)
```
```   521       apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
```
```   522       done
```
```   523     show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))
```
```   524           \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
```
```   525       apply (intro
```
```   526          order_trans [OF norm_setsum]
```
```   527          real_setsum_nat_ivl_bounded2
```
```   528          mult_nonneg_nonneg
```
```   529          of_nat_0_le_iff
```
```   530          zero_le_power K)
```
```   531       apply (rule le_Kn, simp)
```
```   532       done
```
```   533   qed
```
```   534   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
```
```   535     by (simp only: mult.assoc)
```
```   536   finally show ?thesis .
```
```   537 qed
```
```   538
```
```   539 lemma lemma_termdiff4:
```
```   540   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```   541   assumes k: "0 < (k::real)"
```
```   542     and le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
```
```   543   shows "f -- 0 --> 0"
```
```   544 proof (rule tendsto_norm_zero_cancel)
```
```   545   show "(\<lambda>h. norm (f h)) -- 0 --> 0"
```
```   546   proof (rule real_tendsto_sandwich)
```
```   547     show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)"
```
```   548       by simp
```
```   549     show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)"
```
```   550       using k by (auto simp add: eventually_at dist_norm le)
```
```   551     show "(\<lambda>h. 0) -- (0::'a) --> (0::real)"
```
```   552       by (rule tendsto_const)
```
```   553     have "(\<lambda>h. K * norm h) -- (0::'a) --> K * norm (0::'a)"
```
```   554       by (intro tendsto_intros)
```
```   555     then show "(\<lambda>h. K * norm h) -- (0::'a) --> 0"
```
```   556       by simp
```
```   557   qed
```
```   558 qed
```
```   559
```
```   560 lemma lemma_termdiff5:
```
```   561   fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
```
```   562   assumes k: "0 < (k::real)"
```
```   563   assumes f: "summable f"
```
```   564   assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
```
```   565   shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
```
```   566 proof (rule lemma_termdiff4 [OF k])
```
```   567   fix h::'a
```
```   568   assume "h \<noteq> 0" and "norm h < k"
```
```   569   hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
```
```   570     by (simp add: le)
```
```   571   hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
```
```   572     by simp
```
```   573   moreover from f have B: "summable (\<lambda>n. f n * norm h)"
```
```   574     by (rule summable_mult2)
```
```   575   ultimately have C: "summable (\<lambda>n. norm (g h n))"
```
```   576     by (rule summable_comparison_test)
```
```   577   hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
```
```   578     by (rule summable_norm)
```
```   579   also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
```
```   580     by (rule suminf_le)
```
```   581   also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
```
```   582     by (rule suminf_mult2 [symmetric])
```
```   583   finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
```
```   584 qed
```
```   585
```
```   586
```
```   587 text{* FIXME: Long proofs*}
```
```   588
```
```   589 lemma termdiffs_aux:
```
```   590   fixes x :: "'a::{real_normed_field,banach}"
```
```   591   assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
```
```   592     and 2: "norm x < norm K"
```
```   593   shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h
```
```   594              - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   595 proof -
```
```   596   from dense [OF 2]
```
```   597   obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
```
```   598   from norm_ge_zero r1 have r: "0 < r"
```
```   599     by (rule order_le_less_trans)
```
```   600   hence r_neq_0: "r \<noteq> 0" by simp
```
```   601   show ?thesis
```
```   602   proof (rule lemma_termdiff5)
```
```   603     show "0 < r - norm x" using r1 by simp
```
```   604     from r r2 have "norm (of_real r::'a) < norm K"
```
```   605       by simp
```
```   606     with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
```
```   607       by (rule powser_insidea)
```
```   608     hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
```
```   609       using r
```
```   610       by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
```
```   611     hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
```
```   612       by (rule diffs_equiv [THEN sums_summable])
```
```   613     also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
```
```   614       (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
```
```   615       apply (rule ext)
```
```   616       apply (simp add: diffs_def)
```
```   617       apply (case_tac n, simp_all add: r_neq_0)
```
```   618       done
```
```   619     finally have "summable
```
```   620       (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
```
```   621       by (rule diffs_equiv [THEN sums_summable])
```
```   622     also have
```
```   623       "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
```
```   624            r ^ (n - Suc 0)) =
```
```   625        (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
```
```   626       apply (rule ext)
```
```   627       apply (case_tac "n", simp)
```
```   628       apply (rename_tac nat)
```
```   629       apply (case_tac "nat", simp)
```
```   630       apply (simp add: r_neq_0)
```
```   631       done
```
```   632     finally
```
```   633     show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
```
```   634   next
```
```   635     fix h::'a and n::nat
```
```   636     assume h: "h \<noteq> 0"
```
```   637     assume "norm h < r - norm x"
```
```   638     hence "norm x + norm h < r" by simp
```
```   639     with norm_triangle_ineq have xh: "norm (x + h) < r"
```
```   640       by (rule order_le_less_trans)
```
```   641     show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))
```
```   642           \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
```
```   643       apply (simp only: norm_mult mult.assoc)
```
```   644       apply (rule mult_left_mono [OF _ norm_ge_zero])
```
```   645       apply (simp add: mult.assoc [symmetric])
```
```   646       apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
```
```   647       done
```
```   648   qed
```
```   649 qed
```
```   650
```
```   651 lemma termdiffs:
```
```   652   fixes K x :: "'a::{real_normed_field,banach}"
```
```   653   assumes 1: "summable (\<lambda>n. c n * K ^ n)"
```
```   654       and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
```
```   655       and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
```
```   656       and 4: "norm x < norm K"
```
```   657   shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. (diffs c) n * x^n)"
```
```   658   unfolding DERIV_def
```
```   659 proof (rule LIM_zero_cancel)
```
```   660   show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x^n)) / h
```
```   661             - suminf (\<lambda>n. diffs c n * x^n)) -- 0 --> 0"
```
```   662   proof (rule LIM_equal2)
```
```   663     show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
```
```   664   next
```
```   665     fix h :: 'a
```
```   666     assume "norm (h - 0) < norm K - norm x"
```
```   667     hence "norm x + norm h < norm K" by simp
```
```   668     hence 5: "norm (x + h) < norm K"
```
```   669       by (rule norm_triangle_ineq [THEN order_le_less_trans])
```
```   670     have "summable (\<lambda>n. c n * x^n)"
```
```   671       and "summable (\<lambda>n. c n * (x + h) ^ n)"
```
```   672       and "summable (\<lambda>n. diffs c n * x^n)"
```
```   673       using 1 2 4 5 by (auto elim: powser_inside)
```
```   674     then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
```
```   675           (\<Sum>n. (c n * (x + h) ^ n - c n * x^n) / h - of_nat n * c n * x ^ (n - Suc 0))"
```
```   676       by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
```
```   677     then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
```
```   678           (\<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))"
```
```   679       by (simp add: algebra_simps)
```
```   680   next
```
```   681     show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
```
```   682       by (rule termdiffs_aux [OF 3 4])
```
```   683   qed
```
```   684 qed
```
```   685
```
```   686
```
```   687 subsection {* Derivability of power series *}
```
```   688
```
```   689 lemma DERIV_series':
```
```   690   fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
```
```   691   assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
```
```   692     and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
```
```   693     and "summable (f' x0)"
```
```   694     and "summable L"
```
```   695     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>"
```
```   696   shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
```
```   697   unfolding DERIV_def
```
```   698 proof (rule LIM_I)
```
```   699   fix r :: real
```
```   700   assume "0 < r" hence "0 < r/3" by auto
```
```   701
```
```   702   obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
```
```   703     using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
```
```   704
```
```   705   obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
```
```   706     using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
```
```   707
```
```   708   let ?N = "Suc (max N_L N_f')"
```
```   709   have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
```
```   710     L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
```
```   711
```
```   712   let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
```
```   713
```
```   714   let ?r = "r / (3 * real ?N)"
```
```   715   from `0 < r` have "0 < ?r" by simp
```
```   716
```
```   717   let ?s = "\<lambda>n. SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"
```
```   718   def S' \<equiv> "Min (?s ` {..< ?N })"
```
```   719
```
```   720   have "0 < S'" unfolding S'_def
```
```   721   proof (rule iffD2[OF Min_gr_iff])
```
```   722     show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
```
```   723     proof
```
```   724       fix x
```
```   725       assume "x \<in> ?s ` {..<?N}"
```
```   726       then obtain n where "x = ?s n" and "n \<in> {..<?N}"
```
```   727         using image_iff[THEN iffD1] by blast
```
```   728       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
```
```   729       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)"
```
```   730         by auto
```
```   731       have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound)
```
```   732       thus "0 < x" unfolding `x = ?s n` .
```
```   733     qed
```
```   734   qed auto
```
```   735
```
```   736   def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
```
```   737   hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
```
```   738     and "S \<le> S'" using x0_in_I and `0 < S'`
```
```   739     by auto
```
```   740
```
```   741   {
```
```   742     fix x
```
```   743     assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
```
```   744     hence x_in_I: "x0 + x \<in> { a <..< b }"
```
```   745       using S_a S_b by auto
```
```   746
```
```   747     note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   748     note div_smbl = summable_divide[OF diff_smbl]
```
```   749     note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
```
```   750     note ign = summable_ignore_initial_segment[where k="?N"]
```
```   751     note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
```
```   752     note div_shft_smbl = summable_divide[OF diff_shft_smbl]
```
```   753     note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
```
```   754
```
```   755     { fix n
```
```   756       have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
```
```   757         using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
```
```   758         unfolding abs_divide .
```
```   759       hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)"
```
```   760         using `x \<noteq> 0` by auto }
```
```   761     note 1 = this and 2 = summable_rabs_comparison_test[OF _ ign[OF `summable L`]]
```
```   762     then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))"
```
```   763       by (metis (lifting) abs_idempotent order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF `summable L`]]])
```
```   764     then have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3")
```
```   765       using L_estimate by auto
```
```   766
```
```   767     have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n \<bar>)" ..
```
```   768     also have "\<dots> < (\<Sum>n<?N. ?r)"
```
```   769     proof (rule setsum_strict_mono)
```
```   770       fix n
```
```   771       assume "n \<in> {..< ?N}"
```
```   772       have "\<bar>x\<bar> < S" using `\<bar>x\<bar> < S` .
```
```   773       also have "S \<le> S'" using `S \<le> S'` .
```
```   774       also have "S' \<le> ?s n" unfolding S'_def
```
```   775       proof (rule Min_le_iff[THEN iffD2])
```
```   776         have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n"
```
```   777           using `n \<in> {..< ?N}` by auto
```
```   778         thus "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n" by blast
```
```   779       qed auto
```
```   780       finally have "\<bar>x\<bar> < ?s n" .
```
```   781
```
```   782       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]
```
```   783       have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
```
```   784       with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n` show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
```
```   785         by blast
```
```   786     qed auto
```
```   787     also have "\<dots> = of_nat (card {..<?N}) * ?r"
```
```   788       by (rule setsum_constant)
```
```   789     also have "\<dots> = real ?N * ?r"
```
```   790       unfolding real_eq_of_nat by auto
```
```   791     also have "\<dots> = r/3" by auto
```
```   792     finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
```
```   793
```
```   794     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
```
```   795     have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
```
```   796         \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
```
```   797       unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric]
```
```   798       using suminf_divide[OF diff_smbl, symmetric] by auto
```
```   799     also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>"
```
```   800       unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
```
```   801       unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]]
```
```   802       apply (subst (5) add.commute)
```
```   803       by (rule abs_triangle_ineq)
```
```   804     also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
```
```   805       using abs_triangle_ineq4 by auto
```
```   806     also have "\<dots> < r /3 + r/3 + r/3"
```
```   807       using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
```
```   808       by (rule add_strict_mono [OF add_less_le_mono])
```
```   809     finally have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
```
```   810       by auto
```
```   811   }
```
```   812   thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
```
```   813       norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
```
```   814     using `0 < S` unfolding real_norm_def diff_0_right by blast
```
```   815 qed
```
```   816
```
```   817 lemma DERIV_power_series':
```
```   818   fixes f :: "nat \<Rightarrow> real"
```
```   819   assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
```
```   820     and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
```
```   821   shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
```
```   822   (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
```
```   823 proof -
```
```   824   {
```
```   825     fix R'
```
```   826     assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
```
```   827     hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
```
```   828       by auto
```
```   829     have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
```
```   830     proof (rule DERIV_series')
```
```   831       show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
```
```   832       proof -
```
```   833         have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
```
```   834           using `0 < R'` `0 < R` `R' < R` by auto
```
```   835         hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
```
```   836           using `R' < R` by auto
```
```   837         have "norm R' < norm ((R' + R) / 2)"
```
```   838           using `0 < R'` `0 < R` `R' < R` by auto
```
```   839         from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
```
```   840           by auto
```
```   841       qed
```
```   842       {
```
```   843         fix n x y
```
```   844         assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
```
```   845         show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
```
```   846         proof -
```
```   847           have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
```
```   848             (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>"
```
```   849             unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult
```
```   850             by auto
```
```   851           also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
```
```   852           proof (rule mult_left_mono)
```
```   853             have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
```
```   854               by (rule setsum_abs)
```
```   855             also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)"
```
```   856             proof (rule setsum_mono)
```
```   857               fix p
```
```   858               assume "p \<in> {..<Suc n}"
```
```   859               hence "p \<le> n" by auto
```
```   860               {
```
```   861                 fix n
```
```   862                 fix x :: real
```
```   863                 assume "x \<in> {-R'<..<R'}"
```
```   864                 hence "\<bar>x\<bar> \<le> R'"  by auto
```
```   865                 hence "\<bar>x^n\<bar> \<le> R'^n"
```
```   866                   unfolding power_abs by (rule power_mono, auto)
```
```   867               }
```
```   868               from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
```
```   869               have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
```
```   870                 unfolding abs_mult by auto
```
```   871               thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
```
```   872                 unfolding power_add[symmetric] using `p \<le> n` by auto
```
```   873             qed
```
```   874             also have "\<dots> = real (Suc n) * R' ^ n"
```
```   875               unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
```
```   876             finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
```
```   877               unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
```
```   878             show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
```
```   879               unfolding abs_mult[symmetric] by auto
```
```   880           qed
```
```   881           also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
```
```   882             unfolding abs_mult mult.assoc[symmetric] by algebra
```
```   883           finally show ?thesis .
```
```   884         qed
```
```   885       }
```
```   886       {
```
```   887         fix n
```
```   888         show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
```
```   889           by (auto intro!: derivative_eq_intros simp del: power_Suc simp: real_of_nat_def)
```
```   890       }
```
```   891       {
```
```   892         fix x
```
```   893         assume "x \<in> {-R' <..< R'}"
```
```   894         hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
```
```   895           using assms `R' < R` by auto
```
```   896         have "summable (\<lambda> n. f n * x^n)"
```
```   897         proof (rule summable_comparison_test, intro exI allI impI)
```
```   898           fix n
```
```   899           have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
```
```   900             by (rule mult_left_mono) auto
```
```   901           show "norm (f n * x^n) \<le> norm (f n * real (Suc n) * x^n)"
```
```   902             unfolding real_norm_def abs_mult
```
```   903             by (rule mult_right_mono) (auto simp add: le[unfolded mult_1_right])
```
```   904         qed (rule powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`])
```
```   905         from this[THEN summable_mult2[where c=x], unfolded mult.assoc, unfolded mult.commute]
```
```   906         show "summable (?f x)" by auto
```
```   907       }
```
```   908       show "summable (?f' x0)"
```
```   909         using converges[OF `x0 \<in> {-R <..< R}`] .
```
```   910       show "x0 \<in> {-R' <..< R'}"
```
```   911         using `x0 \<in> {-R' <..< R'}` .
```
```   912     qed
```
```   913   } note for_subinterval = this
```
```   914   let ?R = "(R + \<bar>x0\<bar>) / 2"
```
```   915   have "\<bar>x0\<bar> < ?R" using assms by auto
```
```   916   hence "- ?R < x0"
```
```   917   proof (cases "x0 < 0")
```
```   918     case True
```
```   919     hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
```
```   920     thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
```
```   921   next
```
```   922     case False
```
```   923     have "- ?R < 0" using assms by auto
```
```   924     also have "\<dots> \<le> x0" using False by auto
```
```   925     finally show ?thesis .
```
```   926   qed
```
```   927   hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
```
```   928     using assms by auto
```
```   929   from for_subinterval[OF this]
```
```   930   show ?thesis .
```
```   931 qed
```
```   932
```
```   933
```
```   934 subsection {* Exponential Function *}
```
```   935
```
```   936 definition exp :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```   937   where "exp = (\<lambda>x. \<Sum>n. x^n /\<^sub>R fact n)"
```
```   938
```
```   939 lemma summable_exp_generic:
```
```   940   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   941   defines S_def: "S \<equiv> \<lambda>n. x^n /\<^sub>R fact n"
```
```   942   shows "summable S"
```
```   943 proof -
```
```   944   have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R (Suc n)"
```
```   945     unfolding S_def by (simp del: mult_Suc)
```
```   946   obtain r :: real where r0: "0 < r" and r1: "r < 1"
```
```   947     using dense [OF zero_less_one] by fast
```
```   948   obtain N :: nat where N: "norm x < real N * r"
```
```   949     using reals_Archimedean3 [OF r0] by fast
```
```   950   from r1 show ?thesis
```
```   951   proof (rule summable_ratio_test [rule_format])
```
```   952     fix n :: nat
```
```   953     assume n: "N \<le> n"
```
```   954     have "norm x \<le> real N * r"
```
```   955       using N by (rule order_less_imp_le)
```
```   956     also have "real N * r \<le> real (Suc n) * r"
```
```   957       using r0 n by (simp add: mult_right_mono)
```
```   958     finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
```
```   959       using norm_ge_zero by (rule mult_right_mono)
```
```   960     hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
```
```   961       by (rule order_trans [OF norm_mult_ineq])
```
```   962     hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
```
```   963       by (simp add: pos_divide_le_eq ac_simps)
```
```   964     thus "norm (S (Suc n)) \<le> r * norm (S n)"
```
```   965       by (simp add: S_Suc inverse_eq_divide)
```
```   966   qed
```
```   967 qed
```
```   968
```
```   969 lemma summable_norm_exp:
```
```   970   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```   971   shows "summable (\<lambda>n. norm (x^n /\<^sub>R fact n))"
```
```   972 proof (rule summable_norm_comparison_test [OF exI, rule_format])
```
```   973   show "summable (\<lambda>n. norm x^n /\<^sub>R fact n)"
```
```   974     by (rule summable_exp_generic)
```
```   975   fix n
```
```   976   show "norm (x^n /\<^sub>R fact n) \<le> norm x^n /\<^sub>R fact n"
```
```   977     by (simp add: norm_power_ineq)
```
```   978 qed
```
```   979
```
```   980 lemma summable_exp:
```
```   981   fixes x :: "'a::{real_normed_field,banach}"
```
```   982   shows "summable (\<lambda>n. inverse (fact n) * x^n)"
```
```   983   using summable_exp_generic [where x=x]
```
```   984   by (simp add: scaleR_conv_of_real nonzero_of_real_inverse)
```
```   985
```
```   986 lemma exp_converges: "(\<lambda>n. x^n /\<^sub>R fact n) sums exp x"
```
```   987   unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
```
```   988
```
```   989 lemma exp_fdiffs:
```
```   990   fixes XXX :: "'a::{real_normed_field,banach}"
```
```   991   shows "diffs (\<lambda>n. inverse (fact n)) = (\<lambda>n. inverse (fact n :: 'a))"
```
```   992   by (simp add: diffs_def mult_ac nonzero_inverse_mult_distrib nonzero_of_real_inverse
```
```   993            del: mult_Suc of_nat_Suc)
```
```   994
```
```   995 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
```
```   996   by (simp add: diffs_def)
```
```   997
```
```   998 lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
```
```   999   unfolding exp_def scaleR_conv_of_real
```
```  1000   apply (rule DERIV_cong)
```
```  1001   apply (rule termdiffs [where K="of_real (1 + norm x)"])
```
```  1002   apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
```
```  1003   apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
```
```  1004   apply (simp del: of_real_add)
```
```  1005   done
```
```  1006
```
```  1007 declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
```
```  1008
```
```  1009 lemma norm_exp: "norm (exp x) \<le> exp (norm x)"
```
```  1010 proof -
```
```  1011   from summable_norm[OF summable_norm_exp, of x]
```
```  1012   have "norm (exp x) \<le> (\<Sum>n. inverse (fact n) * norm (x^n))"
```
```  1013     by (simp add: exp_def)
```
```  1014   also have "\<dots> \<le> exp (norm x)"
```
```  1015     using summable_exp_generic[of "norm x"] summable_norm_exp[of x]
```
```  1016     by (auto simp: exp_def intro!: suminf_le norm_power_ineq)
```
```  1017   finally show ?thesis .
```
```  1018 qed
```
```  1019
```
```  1020 lemma isCont_exp:
```
```  1021   fixes x::"'a::{real_normed_field,banach}"
```
```  1022   shows "isCont exp x"
```
```  1023   by (rule DERIV_exp [THEN DERIV_isCont])
```
```  1024
```
```  1025 lemma isCont_exp' [simp]:
```
```  1026   fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1027   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
```
```  1028   by (rule isCont_o2 [OF _ isCont_exp])
```
```  1029
```
```  1030 lemma tendsto_exp [tendsto_intros]:
```
```  1031   fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1032   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
```
```  1033   by (rule isCont_tendsto_compose [OF isCont_exp])
```
```  1034
```
```  1035 lemma continuous_exp [continuous_intros]:
```
```  1036   fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1037   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
```
```  1038   unfolding continuous_def by (rule tendsto_exp)
```
```  1039
```
```  1040 lemma continuous_on_exp [continuous_intros]:
```
```  1041   fixes f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
```
```  1042   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
```
```  1043   unfolding continuous_on_def by (auto intro: tendsto_exp)
```
```  1044
```
```  1045
```
```  1046 subsubsection {* Properties of the Exponential Function *}
```
```  1047
```
```  1048 lemma powser_zero:
```
```  1049   fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
```
```  1050   shows "(\<Sum>n. f n * 0 ^ n) = f 0"
```
```  1051 proof -
```
```  1052   have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
```
```  1053     by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
```
```  1054   thus ?thesis unfolding One_nat_def by simp
```
```  1055 qed
```
```  1056
```
```  1057 lemma exp_zero [simp]: "exp 0 = 1"
```
```  1058   unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
```
```  1059
```
```  1060 lemma exp_series_add_commuting:
```
```  1061   fixes x y :: "'a::{real_normed_algebra_1, banach}"
```
```  1062   defines S_def: "S \<equiv> \<lambda>x n. x^n /\<^sub>R fact n"
```
```  1063   assumes comm: "x * y = y * x"
```
```  1064   shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))"
```
```  1065 proof (induct n)
```
```  1066   case 0
```
```  1067   show ?case
```
```  1068     unfolding S_def by simp
```
```  1069 next
```
```  1070   case (Suc n)
```
```  1071   have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
```
```  1072     unfolding S_def by (simp del: mult_Suc)
```
```  1073   hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
```
```  1074     by simp
```
```  1075   have S_comm: "\<And>n. S x n * y = y * S x n"
```
```  1076     by (simp add: power_commuting_commutes comm S_def)
```
```  1077
```
```  1078   have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
```
```  1079     by (simp only: times_S)
```
```  1080   also have "\<dots> = (x + y) * (\<Sum>i\<le>n. S x i * S y (n-i))"
```
```  1081     by (simp only: Suc)
```
```  1082   also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n-i))
```
```  1083                 + y * (\<Sum>i\<le>n. S x i * S y (n-i))"
```
```  1084     by (rule distrib_right)
```
```  1085   also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i))
```
```  1086                 + (\<Sum>i\<le>n. S x i * y * S y (n-i))"
```
```  1087     by (simp add: setsum_right_distrib ac_simps S_comm)
```
```  1088   also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n-i))
```
```  1089                 + (\<Sum>i\<le>n. S x i * (y * S y (n-i)))"
```
```  1090     by (simp add: ac_simps)
```
```  1091   also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
```
```  1092                 + (\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1093     by (simp add: times_S Suc_diff_le)
```
```  1094   also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
```
```  1095              (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1096     by (subst setsum_atMost_Suc_shift) simp
```
```  1097   also have "(\<Sum>i\<le>n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```  1098              (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1099     by simp
```
```  1100   also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
```
```  1101              (\<Sum>i\<le>Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
```
```  1102              (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
```
```  1103     by (simp only: setsum.distrib [symmetric] scaleR_left_distrib [symmetric]
```
```  1104                    real_of_nat_add [symmetric]) simp
```
```  1105   also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n-i))"
```
```  1106     by (simp only: scaleR_right.setsum)
```
```  1107   finally show
```
```  1108     "S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
```
```  1109     by (simp del: setsum_cl_ivl_Suc)
```
```  1110 qed
```
```  1111
```
```  1112 lemma exp_add_commuting: "x * y = y * x \<Longrightarrow> exp (x + y) = exp x * exp y"
```
```  1113   unfolding exp_def
```
```  1114   by (simp only: Cauchy_product summable_norm_exp exp_series_add_commuting)
```
```  1115
```
```  1116 lemma exp_add:
```
```  1117   fixes x y::"'a::{real_normed_field,banach}"
```
```  1118   shows "exp (x + y) = exp x * exp y"
```
```  1119   by (rule exp_add_commuting) (simp add: ac_simps)
```
```  1120
```
```  1121 lemma exp_double: "exp(2 * z) = exp z ^ 2"
```
```  1122   by (simp add: exp_add_commuting mult_2 power2_eq_square)
```
```  1123
```
```  1124 lemmas mult_exp_exp = exp_add [symmetric]
```
```  1125
```
```  1126 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
```
```  1127   unfolding exp_def
```
```  1128   apply (subst suminf_of_real)
```
```  1129   apply (rule summable_exp_generic)
```
```  1130   apply (simp add: scaleR_conv_of_real)
```
```  1131   done
```
```  1132
```
```  1133 corollary exp_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> exp z \<in> \<real>"
```
```  1134   by (metis Reals_cases Reals_of_real exp_of_real)
```
```  1135
```
```  1136 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
```
```  1137 proof
```
```  1138   have "exp x * exp (- x) = 1" by (simp add: exp_add_commuting[symmetric])
```
```  1139   also assume "exp x = 0"
```
```  1140   finally show "False" by simp
```
```  1141 qed
```
```  1142
```
```  1143 lemma exp_minus_inverse:
```
```  1144   shows "exp x * exp (- x) = 1"
```
```  1145   by (simp add: exp_add_commuting[symmetric])
```
```  1146
```
```  1147 lemma exp_minus:
```
```  1148   fixes x :: "'a::{real_normed_field, banach}"
```
```  1149   shows "exp (- x) = inverse (exp x)"
```
```  1150   by (intro inverse_unique [symmetric] exp_minus_inverse)
```
```  1151
```
```  1152 lemma exp_diff:
```
```  1153   fixes x :: "'a::{real_normed_field, banach}"
```
```  1154   shows "exp (x - y) = exp x / exp y"
```
```  1155   using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
```
```  1156
```
```  1157 lemma exp_of_nat_mult:
```
```  1158   fixes x :: "'a::{real_normed_field,banach}"
```
```  1159   shows "exp(of_nat n * x) = exp(x) ^ n"
```
```  1160     by (induct n) (auto simp add: distrib_left exp_add mult.commute)
```
```  1161
```
```  1162 lemma exp_setsum: "finite I \<Longrightarrow> exp(setsum f I) = setprod (\<lambda>x. exp(f x)) I"
```
```  1163   by (induction I rule: finite_induct) (auto simp: exp_add_commuting mult.commute)
```
```  1164
```
```  1165
```
```  1166 subsubsection {* Properties of the Exponential Function on Reals *}
```
```  1167
```
```  1168 text {* Comparisons of @{term "exp x"} with zero. *}
```
```  1169
```
```  1170 text{*Proof: because every exponential can be seen as a square.*}
```
```  1171 lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
```
```  1172 proof -
```
```  1173   have "0 \<le> exp (x/2) * exp (x/2)" by simp
```
```  1174   thus ?thesis by (simp add: exp_add [symmetric])
```
```  1175 qed
```
```  1176
```
```  1177 lemma exp_gt_zero [simp]: "0 < exp (x::real)"
```
```  1178   by (simp add: order_less_le)
```
```  1179
```
```  1180 lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
```
```  1181   by (simp add: not_less)
```
```  1182
```
```  1183 lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
```
```  1184   by (simp add: not_le)
```
```  1185
```
```  1186 lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
```
```  1187   by simp
```
```  1188
```
```  1189 (*FIXME: superseded by exp_of_nat_mult*)
```
```  1190 lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
```
```  1191   by (induct n) (auto simp add: real_of_nat_Suc distrib_left exp_add mult.commute)
```
```  1192
```
```  1193 text {* Strict monotonicity of exponential. *}
```
```  1194
```
```  1195 lemma exp_ge_add_one_self_aux:
```
```  1196   assumes "0 \<le> (x::real)" shows "1+x \<le> exp(x)"
```
```  1197 using order_le_imp_less_or_eq [OF assms]
```
```  1198 proof
```
```  1199   assume "0 < x"
```
```  1200   have "1+x \<le> (\<Sum>n<2. inverse (fact n) * x^n)"
```
```  1201     by (auto simp add: numeral_2_eq_2)
```
```  1202   also have "... \<le> (\<Sum>n. inverse (fact n) * x^n)"
```
```  1203     apply (rule setsum_le_suminf [OF summable_exp])
```
```  1204     using `0 < x`
```
```  1205     apply (auto  simp add:  zero_le_mult_iff)
```
```  1206     done
```
```  1207   finally show "1+x \<le> exp x"
```
```  1208     by (simp add: exp_def)
```
```  1209 next
```
```  1210   assume "0 = x"
```
```  1211   then show "1 + x \<le> exp x"
```
```  1212     by auto
```
```  1213 qed
```
```  1214
```
```  1215 lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
```
```  1216 proof -
```
```  1217   assume x: "0 < x"
```
```  1218   hence "1 < 1 + x" by simp
```
```  1219   also from x have "1 + x \<le> exp x"
```
```  1220     by (simp add: exp_ge_add_one_self_aux)
```
```  1221   finally show ?thesis .
```
```  1222 qed
```
```  1223
```
```  1224 lemma exp_less_mono:
```
```  1225   fixes x y :: real
```
```  1226   assumes "x < y"
```
```  1227   shows "exp x < exp y"
```
```  1228 proof -
```
```  1229   from `x < y` have "0 < y - x" by simp
```
```  1230   hence "1 < exp (y - x)" by (rule exp_gt_one)
```
```  1231   hence "1 < exp y / exp x" by (simp only: exp_diff)
```
```  1232   thus "exp x < exp y" by simp
```
```  1233 qed
```
```  1234
```
```  1235 lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
```
```  1236   unfolding linorder_not_le [symmetric]
```
```  1237   by (auto simp add: order_le_less exp_less_mono)
```
```  1238
```
```  1239 lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
```
```  1240   by (auto intro: exp_less_mono exp_less_cancel)
```
```  1241
```
```  1242 lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
```
```  1243   by (auto simp add: linorder_not_less [symmetric])
```
```  1244
```
```  1245 lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
```
```  1246   by (simp add: order_eq_iff)
```
```  1247
```
```  1248 text {* Comparisons of @{term "exp x"} with one. *}
```
```  1249
```
```  1250 lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
```
```  1251   using exp_less_cancel_iff [where x=0 and y=x] by simp
```
```  1252
```
```  1253 lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
```
```  1254   using exp_less_cancel_iff [where x=x and y=0] by simp
```
```  1255
```
```  1256 lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
```
```  1257   using exp_le_cancel_iff [where x=0 and y=x] by simp
```
```  1258
```
```  1259 lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
```
```  1260   using exp_le_cancel_iff [where x=x and y=0] by simp
```
```  1261
```
```  1262 lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
```
```  1263   using exp_inj_iff [where x=x and y=0] by simp
```
```  1264
```
```  1265 lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
```
```  1266 proof (rule IVT)
```
```  1267   assume "1 \<le> y"
```
```  1268   hence "0 \<le> y - 1" by simp
```
```  1269   hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
```
```  1270   thus "y \<le> exp (y - 1)" by simp
```
```  1271 qed (simp_all add: le_diff_eq)
```
```  1272
```
```  1273 lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y"
```
```  1274 proof (rule linorder_le_cases [of 1 y])
```
```  1275   assume "1 \<le> y"
```
```  1276   thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total)
```
```  1277 next
```
```  1278   assume "0 < y" and "y \<le> 1"
```
```  1279   hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
```
```  1280   then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
```
```  1281   hence "exp (- x) = y" by (simp add: exp_minus)
```
```  1282   thus "\<exists>x. exp x = y" ..
```
```  1283 qed
```
```  1284
```
```  1285
```
```  1286 subsection {* Natural Logarithm *}
```
```  1287
```
```  1288 class ln = real_normed_algebra_1 + banach +
```
```  1289   fixes ln :: "'a \<Rightarrow> 'a"
```
```  1290   assumes ln_one [simp]: "ln 1 = 0"
```
```  1291
```
```  1292 definition powr :: "['a,'a] => 'a::ln"     (infixr "powr" 80)
```
```  1293   -- {*exponentation via ln and exp*}
```
```  1294   where "x powr a \<equiv> if x = 0 then 0 else exp(a * ln x)"
```
```  1295
```
```  1296
```
```  1297 instantiation real :: ln
```
```  1298 begin
```
```  1299
```
```  1300 definition ln_real :: "real \<Rightarrow> real"
```
```  1301   where "ln_real x = (THE u. exp u = x)"
```
```  1302
```
```  1303 instance
```
```  1304 by intro_classes (simp add: ln_real_def)
```
```  1305
```
```  1306 end
```
```  1307
```
```  1308 lemma powr_eq_0_iff [simp]: "w powr z = 0 \<longleftrightarrow> w = 0"
```
```  1309   by (simp add: powr_def)
```
```  1310
```
```  1311 lemma ln_exp [simp]:
```
```  1312   fixes x::real shows "ln (exp x) = x"
```
```  1313   by (simp add: ln_real_def)
```
```  1314
```
```  1315 lemma exp_ln [simp]:
```
```  1316   fixes x::real shows "0 < x \<Longrightarrow> exp (ln x) = x"
```
```  1317   by (auto dest: exp_total)
```
```  1318
```
```  1319 lemma exp_ln_iff [simp]:
```
```  1320   fixes x::real shows "exp (ln x) = x \<longleftrightarrow> 0 < x"
```
```  1321   by (metis exp_gt_zero exp_ln)
```
```  1322
```
```  1323 lemma ln_unique:
```
```  1324   fixes x::real shows "exp y = x \<Longrightarrow> ln x = y"
```
```  1325   by (erule subst, rule ln_exp)
```
```  1326
```
```  1327 lemma ln_mult:
```
```  1328   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
```
```  1329   by (rule ln_unique) (simp add: exp_add)
```
```  1330
```
```  1331 lemma ln_setprod:
```
```  1332   fixes f:: "'a => real"
```
```  1333   shows
```
```  1334     "\<lbrakk>finite I; \<And>i. i \<in> I \<Longrightarrow> f i > 0\<rbrakk> \<Longrightarrow> ln(setprod f I) = setsum (\<lambda>x. ln(f x)) I"
```
```  1335   by (induction I rule: finite_induct) (auto simp: ln_mult setprod_pos)
```
```  1336
```
```  1337 lemma ln_inverse:
```
```  1338   fixes x::real shows "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
```
```  1339   by (rule ln_unique) (simp add: exp_minus)
```
```  1340
```
```  1341 lemma ln_div:
```
```  1342   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
```
```  1343   by (rule ln_unique) (simp add: exp_diff)
```
```  1344
```
```  1345 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x^n) = real n * ln x"
```
```  1346   by (rule ln_unique) (simp add: exp_real_of_nat_mult)
```
```  1347
```
```  1348 lemma ln_less_cancel_iff [simp]:
```
```  1349   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
```
```  1350   by (subst exp_less_cancel_iff [symmetric]) simp
```
```  1351
```
```  1352 lemma ln_le_cancel_iff [simp]:
```
```  1353   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
```
```  1354   by (simp add: linorder_not_less [symmetric])
```
```  1355
```
```  1356 lemma ln_inj_iff [simp]:
```
```  1357   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
```
```  1358   by (simp add: order_eq_iff)
```
```  1359
```
```  1360 lemma ln_add_one_self_le_self [simp]:
```
```  1361   fixes x::real shows "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
```
```  1362   apply (rule exp_le_cancel_iff [THEN iffD1])
```
```  1363   apply (simp add: exp_ge_add_one_self_aux)
```
```  1364   done
```
```  1365
```
```  1366 lemma ln_less_self [simp]:
```
```  1367   fixes x::real shows "0 < x \<Longrightarrow> ln x < x"
```
```  1368   by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
```
```  1369
```
```  1370 lemma ln_ge_zero [simp]:
```
```  1371   fixes x::real shows "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
```
```  1372   using ln_le_cancel_iff [of 1 x] by simp
```
```  1373
```
```  1374 lemma ln_ge_zero_imp_ge_one:
```
```  1375   fixes x::real shows "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
```
```  1376   using ln_le_cancel_iff [of 1 x] by simp
```
```  1377
```
```  1378 lemma ln_ge_zero_iff [simp]:
```
```  1379   fixes x::real shows "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
```
```  1380   using ln_le_cancel_iff [of 1 x] by simp
```
```  1381
```
```  1382 lemma ln_less_zero_iff [simp]:
```
```  1383   fixes x::real shows "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
```
```  1384   using ln_less_cancel_iff [of x 1] by simp
```
```  1385
```
```  1386 lemma ln_gt_zero:
```
```  1387   fixes x::real shows "1 < x \<Longrightarrow> 0 < ln x"
```
```  1388   using ln_less_cancel_iff [of 1 x] by simp
```
```  1389
```
```  1390 lemma ln_gt_zero_imp_gt_one:
```
```  1391   fixes x::real shows "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
```
```  1392   using ln_less_cancel_iff [of 1 x] by simp
```
```  1393
```
```  1394 lemma ln_gt_zero_iff [simp]:
```
```  1395   fixes x::real shows "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
```
```  1396   using ln_less_cancel_iff [of 1 x] by simp
```
```  1397
```
```  1398 lemma ln_eq_zero_iff [simp]:
```
```  1399   fixes x::real shows "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
```
```  1400   using ln_inj_iff [of x 1] by simp
```
```  1401
```
```  1402 lemma ln_less_zero:
```
```  1403   fixes x::real shows "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
```
```  1404   by simp
```
```  1405
```
```  1406 lemma ln_neg_is_const:
```
```  1407   fixes x::real shows "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)"
```
```  1408   by (auto simp add: ln_real_def intro!: arg_cong[where f=The])
```
```  1409
```
```  1410 lemma isCont_ln:
```
```  1411   fixes x::real assumes "x \<noteq> 0" shows "isCont ln x"
```
```  1412 proof cases
```
```  1413   assume "0 < x"
```
```  1414   moreover then have "isCont ln (exp (ln x))"
```
```  1415     by (intro isCont_inv_fun[where d="\<bar>x\<bar>" and f=exp]) auto
```
```  1416   ultimately show ?thesis
```
```  1417     by simp
```
```  1418 next
```
```  1419   assume "\<not> 0 < x" with `x \<noteq> 0` show "isCont ln x"
```
```  1420     unfolding isCont_def
```
```  1421     by (subst filterlim_cong[OF _ refl, of _ "nhds (ln 0)" _ "\<lambda>_. ln 0"])
```
```  1422        (auto simp: ln_neg_is_const not_less eventually_at dist_real_def
```
```  1423                 intro!: exI[of _ "\<bar>x\<bar>"])
```
```  1424 qed
```
```  1425
```
```  1426 lemma tendsto_ln [tendsto_intros]:
```
```  1427   fixes a::real shows
```
```  1428   "(f ---> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
```
```  1429   by (rule isCont_tendsto_compose [OF isCont_ln])
```
```  1430
```
```  1431 lemma continuous_ln:
```
```  1432   "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. ln (f x :: real))"
```
```  1433   unfolding continuous_def by (rule tendsto_ln)
```
```  1434
```
```  1435 lemma isCont_ln' [continuous_intros]:
```
```  1436   "continuous (at x) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x :: real))"
```
```  1437   unfolding continuous_at by (rule tendsto_ln)
```
```  1438
```
```  1439 lemma continuous_within_ln [continuous_intros]:
```
```  1440   "continuous (at x within s) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x :: real))"
```
```  1441   unfolding continuous_within by (rule tendsto_ln)
```
```  1442
```
```  1443 lemma continuous_on_ln [continuous_intros]:
```
```  1444   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. f x \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x :: real))"
```
```  1445   unfolding continuous_on_def by (auto intro: tendsto_ln)
```
```  1446
```
```  1447 lemma DERIV_ln:
```
```  1448   fixes x::real shows "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
```
```  1449   apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
```
```  1450   apply (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln)
```
```  1451   done
```
```  1452
```
```  1453 lemma DERIV_ln_divide:
```
```  1454   fixes x::real shows "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
```
```  1455   by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
```
```  1456
```
```  1457 declare DERIV_ln_divide[THEN DERIV_chain2, derivative_intros]
```
```  1458
```
```  1459 lemma ln_series:
```
```  1460   assumes "0 < x" and "x < 2"
```
```  1461   shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))"
```
```  1462   (is "ln x = suminf (?f (x - 1))")
```
```  1463 proof -
```
```  1464   let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n"
```
```  1465
```
```  1466   have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
```
```  1467   proof (rule DERIV_isconst3[where x=x])
```
```  1468     fix x :: real
```
```  1469     assume "x \<in> {0 <..< 2}"
```
```  1470     hence "0 < x" and "x < 2" by auto
```
```  1471     have "norm (1 - x) < 1"
```
```  1472       using `0 < x` and `x < 2` by auto
```
```  1473     have "1 / x = 1 / (1 - (1 - x))" by auto
```
```  1474     also have "\<dots> = (\<Sum> n. (1 - x)^n)"
```
```  1475       using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
```
```  1476     also have "\<dots> = suminf (?f' x)"
```
```  1477       unfolding power_mult_distrib[symmetric]
```
```  1478       by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
```
```  1479     finally have "DERIV ln x :> suminf (?f' x)"
```
```  1480       using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
```
```  1481     moreover
```
```  1482     have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
```
```  1483     have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :>
```
```  1484       (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
```
```  1485     proof (rule DERIV_power_series')
```
```  1486       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1"
```
```  1487         using `0 < x` `x < 2` by auto
```
```  1488       fix x :: real
```
```  1489       assume "x \<in> {- 1<..<1}"
```
```  1490       hence "norm (-x) < 1" by auto
```
```  1491       show "summable (\<lambda>n. (- 1) ^ n * (1 / real (n + 1)) * real (Suc n) * x^n)"
```
```  1492         unfolding One_nat_def
```
```  1493         by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
```
```  1494     qed
```
```  1495     hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
```
```  1496       unfolding One_nat_def by auto
```
```  1497     hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
```
```  1498       unfolding DERIV_def repos .
```
```  1499     ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
```
```  1500       by (rule DERIV_diff)
```
```  1501     thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
```
```  1502   qed (auto simp add: assms)
```
```  1503   thus ?thesis by auto
```
```  1504 qed
```
```  1505
```
```  1506 lemma exp_first_two_terms:
```
```  1507   fixes x :: "'a::{real_normed_field,banach}"
```
```  1508   shows "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) * (x ^ (n+2)))"
```
```  1509 proof -
```
```  1510   have "exp x = suminf (\<lambda>n. inverse(fact n) * (x^n))"
```
```  1511     by (simp add: exp_def scaleR_conv_of_real nonzero_of_real_inverse)
```
```  1512   also from summable_exp have "... = (\<Sum> n. inverse(fact(n+2)) * (x ^ (n+2))) +
```
```  1513     (\<Sum> n::nat<2. inverse(fact n) * (x^n))" (is "_ = _ + ?a")
```
```  1514     by (rule suminf_split_initial_segment)
```
```  1515   also have "?a = 1 + x"
```
```  1516     by (simp add: numeral_2_eq_2)
```
```  1517   finally show ?thesis
```
```  1518     by simp
```
```  1519 qed
```
```  1520
```
```  1521 lemma exp_bound: "0 <= (x::real) \<Longrightarrow> x <= 1 \<Longrightarrow> exp x <= 1 + x + x\<^sup>2"
```
```  1522 proof -
```
```  1523   assume a: "0 <= x"
```
```  1524   assume b: "x <= 1"
```
```  1525   {
```
```  1526     fix n :: nat
```
```  1527     have "(2::nat) * 2 ^ n \<le> fact (n + 2)"
```
```  1528       by (induct n) simp_all
```
```  1529     hence "real ((2::nat) * 2 ^ n) \<le> real_of_nat (fact (n + 2))"
```
```  1530       by (simp only: real_of_nat_le_iff)
```
```  1531     hence "((2::real) * 2 ^ n) \<le> fact (n + 2)"
```
```  1532       unfolding of_nat_fact real_of_nat_def
```
```  1533       by (simp add: of_nat_mult of_nat_power)
```
```  1534     hence "inverse (fact (n + 2)) \<le> inverse ((2::real) * 2 ^ n)"
```
```  1535       by (rule le_imp_inverse_le) simp
```
```  1536     hence "inverse (fact (n + 2)) \<le> 1/(2::real) * (1/2)^n"
```
```  1537       by (simp add: power_inverse)
```
```  1538     hence "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
```
```  1539       by (rule mult_mono)
```
```  1540         (rule mult_mono, simp_all add: power_le_one a b)
```
```  1541     hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)"
```
```  1542       unfolding power_add by (simp add: ac_simps del: fact.simps) }
```
```  1543   note aux1 = this
```
```  1544   have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
```
```  1545     by (intro sums_mult geometric_sums, simp)
```
```  1546   hence aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
```
```  1547     by simp
```
```  1548   have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2"
```
```  1549   proof -
```
```  1550     have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <=
```
```  1551         suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
```
```  1552       apply (rule suminf_le)
```
```  1553       apply (rule allI, rule aux1)
```
```  1554       apply (rule summable_exp [THEN summable_ignore_initial_segment])
```
```  1555       by (rule sums_summable, rule aux2)
```
```  1556     also have "... = x\<^sup>2"
```
```  1557       by (rule sums_unique [THEN sym], rule aux2)
```
```  1558     finally show ?thesis .
```
```  1559   qed
```
```  1560   thus ?thesis unfolding exp_first_two_terms by auto
```
```  1561 qed
```
```  1562
```
```  1563 corollary exp_half_le2: "exp(1/2) \<le> (2::real)"
```
```  1564   using exp_bound [of "1/2"]
```
```  1565   by (simp add: field_simps)
```
```  1566
```
```  1567 corollary exp_le: "exp 1 \<le> (3::real)"
```
```  1568   using exp_bound [of 1]
```
```  1569   by (simp add: field_simps)
```
```  1570
```
```  1571 lemma exp_bound_half: "norm(z) \<le> 1/2 \<Longrightarrow> norm(exp z) \<le> 2"
```
```  1572   by (blast intro: order_trans intro!: exp_half_le2 norm_exp)
```
```  1573
```
```  1574 lemma exp_bound_lemma:
```
```  1575   assumes "norm(z) \<le> 1/2" shows "norm(exp z) \<le> 1 + 2 * norm(z)"
```
```  1576 proof -
```
```  1577   have n: "(norm z)\<^sup>2 \<le> norm z * 1"
```
```  1578     unfolding power2_eq_square
```
```  1579     apply (rule mult_left_mono)
```
```  1580     using assms
```
```  1581     apply auto
```
```  1582     done
```
```  1583   show ?thesis
```
```  1584     apply (rule order_trans [OF norm_exp])
```
```  1585     apply (rule order_trans [OF exp_bound])
```
```  1586     using assms n
```
```  1587     apply auto
```
```  1588     done
```
```  1589 qed
```
```  1590
```
```  1591 lemma real_exp_bound_lemma:
```
```  1592   fixes x :: real
```
```  1593   shows "0 \<le> x \<Longrightarrow> x \<le> 1/2 \<Longrightarrow> exp(x) \<le> 1 + 2 * x"
```
```  1594 using exp_bound_lemma [of x]
```
```  1595 by simp
```
```  1596
```
```  1597 lemma ln_one_minus_pos_upper_bound:
```
```  1598   fixes x::real shows "0 <= x \<Longrightarrow> x < 1 \<Longrightarrow> ln (1 - x) <= - x"
```
```  1599 proof -
```
```  1600   assume a: "0 <= (x::real)" and b: "x < 1"
```
```  1601   have "(1 - x) * (1 + x + x\<^sup>2) = (1 - x^3)"
```
```  1602     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
```
```  1603   also have "... <= 1"
```
```  1604     by (auto simp add: a)
```
```  1605   finally have "(1 - x) * (1 + x + x\<^sup>2) <= 1" .
```
```  1606   moreover have c: "0 < 1 + x + x\<^sup>2"
```
```  1607     by (simp add: add_pos_nonneg a)
```
```  1608   ultimately have "1 - x <= 1 / (1 + x + x\<^sup>2)"
```
```  1609     by (elim mult_imp_le_div_pos)
```
```  1610   also have "... <= 1 / exp x"
```
```  1611     by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs
```
```  1612               real_sqrt_pow2_iff real_sqrt_power)
```
```  1613   also have "... = exp (-x)"
```
```  1614     by (auto simp add: exp_minus divide_inverse)
```
```  1615   finally have "1 - x <= exp (- x)" .
```
```  1616   also have "1 - x = exp (ln (1 - x))"
```
```  1617     by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq)
```
```  1618   finally have "exp (ln (1 - x)) <= exp (- x)" .
```
```  1619   thus ?thesis by (auto simp only: exp_le_cancel_iff)
```
```  1620 qed
```
```  1621
```
```  1622 lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
```
```  1623   apply (case_tac "0 <= x")
```
```  1624   apply (erule exp_ge_add_one_self_aux)
```
```  1625   apply (case_tac "x <= -1")
```
```  1626   apply (subgoal_tac "1 + x <= 0")
```
```  1627   apply (erule order_trans)
```
```  1628   apply simp
```
```  1629   apply simp
```
```  1630   apply (subgoal_tac "1 + x = exp(ln (1 + x))")
```
```  1631   apply (erule ssubst)
```
```  1632   apply (subst exp_le_cancel_iff)
```
```  1633   apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
```
```  1634   apply simp
```
```  1635   apply (rule ln_one_minus_pos_upper_bound)
```
```  1636   apply auto
```
```  1637 done
```
```  1638
```
```  1639 lemma ln_one_plus_pos_lower_bound:
```
```  1640   fixes x::real shows "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> x - x\<^sup>2 <= ln (1 + x)"
```
```  1641 proof -
```
```  1642   assume a: "0 <= x" and b: "x <= 1"
```
```  1643   have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
```
```  1644     by (rule exp_diff)
```
```  1645   also have "... <= (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
```
```  1646     by (metis a b divide_right_mono exp_bound exp_ge_zero)
```
```  1647   also have "... <= (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
```
```  1648     by (simp add: a divide_left_mono add_pos_nonneg)
```
```  1649   also from a have "... <= 1 + x"
```
```  1650     by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
```
```  1651   finally have "exp (x - x\<^sup>2) <= 1 + x" .
```
```  1652   also have "... = exp (ln (1 + x))"
```
```  1653   proof -
```
```  1654     from a have "0 < 1 + x" by auto
```
```  1655     thus ?thesis
```
```  1656       by (auto simp only: exp_ln_iff [THEN sym])
```
```  1657   qed
```
```  1658   finally have "exp (x - x\<^sup>2) <= exp (ln (1 + x))" .
```
```  1659   thus ?thesis
```
```  1660     by (metis exp_le_cancel_iff)
```
```  1661 qed
```
```  1662
```
```  1663 lemma ln_one_minus_pos_lower_bound:
```
```  1664   fixes x::real
```
```  1665   shows "0 <= x \<Longrightarrow> x <= (1 / 2) \<Longrightarrow> - x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1666 proof -
```
```  1667   assume a: "0 <= x" and b: "x <= (1 / 2)"
```
```  1668   from b have c: "x < 1" by auto
```
```  1669   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
```
```  1670     apply (subst ln_inverse [symmetric])
```
```  1671     apply (simp add: field_simps)
```
```  1672     apply (rule arg_cong [where f=ln])
```
```  1673     apply (simp add: field_simps)
```
```  1674     done
```
```  1675   also have "- (x / (1 - x)) <= ..."
```
```  1676   proof -
```
```  1677     have "ln (1 + x / (1 - x)) <= x / (1 - x)"
```
```  1678       using a c by (intro ln_add_one_self_le_self) auto
```
```  1679     thus ?thesis
```
```  1680       by auto
```
```  1681   qed
```
```  1682   also have "- (x / (1 - x)) = -x / (1 - x)"
```
```  1683     by auto
```
```  1684   finally have d: "- x / (1 - x) <= ln (1 - x)" .
```
```  1685   have "0 < 1 - x" using a b by simp
```
```  1686   hence e: "-x - 2 * x\<^sup>2 <= - x / (1 - x)"
```
```  1687     using mult_right_le_one_le[of "x*x" "2*x"] a b
```
```  1688     by (simp add: field_simps power2_eq_square)
```
```  1689   from e d show "- x - 2 * x\<^sup>2 <= ln (1 - x)"
```
```  1690     by (rule order_trans)
```
```  1691 qed
```
```  1692
```
```  1693 lemma ln_add_one_self_le_self2:
```
```  1694   fixes x::real shows "-1 < x \<Longrightarrow> ln(1 + x) <= x"
```
```  1695   apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
```
```  1696   apply (subst ln_le_cancel_iff)
```
```  1697   apply auto
```
```  1698   done
```
```  1699
```
```  1700 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
```
```  1701   fixes x::real shows "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> abs(ln (1 + x) - x) <= x\<^sup>2"
```
```  1702 proof -
```
```  1703   assume x: "0 <= x"
```
```  1704   assume x1: "x <= 1"
```
```  1705   from x have "ln (1 + x) <= x"
```
```  1706     by (rule ln_add_one_self_le_self)
```
```  1707   then have "ln (1 + x) - x <= 0"
```
```  1708     by simp
```
```  1709   then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
```
```  1710     by (rule abs_of_nonpos)
```
```  1711   also have "... = x - ln (1 + x)"
```
```  1712     by simp
```
```  1713   also have "... <= x\<^sup>2"
```
```  1714   proof -
```
```  1715     from x x1 have "x - x\<^sup>2 <= ln (1 + x)"
```
```  1716       by (intro ln_one_plus_pos_lower_bound)
```
```  1717     thus ?thesis
```
```  1718       by simp
```
```  1719   qed
```
```  1720   finally show ?thesis .
```
```  1721 qed
```
```  1722
```
```  1723 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
```
```  1724   fixes x::real shows "-(1 / 2) <= x \<Longrightarrow> x <= 0 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
```
```  1725 proof -
```
```  1726   assume a: "-(1 / 2) <= x"
```
```  1727   assume b: "x <= 0"
```
```  1728   have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
```
```  1729     apply (subst abs_of_nonpos)
```
```  1730     apply simp
```
```  1731     apply (rule ln_add_one_self_le_self2)
```
```  1732     using a apply auto
```
```  1733     done
```
```  1734   also have "... <= 2 * x\<^sup>2"
```
```  1735     apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 <= ln (1 - (-x))")
```
```  1736     apply (simp add: algebra_simps)
```
```  1737     apply (rule ln_one_minus_pos_lower_bound)
```
```  1738     using a b apply auto
```
```  1739     done
```
```  1740   finally show ?thesis .
```
```  1741 qed
```
```  1742
```
```  1743 lemma abs_ln_one_plus_x_minus_x_bound:
```
```  1744   fixes x::real shows "abs x <= 1 / 2 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
```
```  1745   apply (case_tac "0 <= x")
```
```  1746   apply (rule order_trans)
```
```  1747   apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
```
```  1748   apply auto
```
```  1749   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
```
```  1750   apply auto
```
```  1751   done
```
```  1752
```
```  1753 lemma ln_x_over_x_mono:
```
```  1754   fixes x::real shows "exp 1 <= x \<Longrightarrow> x <= y \<Longrightarrow> (ln y / y) <= (ln x / x)"
```
```  1755 proof -
```
```  1756   assume x: "exp 1 <= x" "x <= y"
```
```  1757   moreover have "0 < exp (1::real)" by simp
```
```  1758   ultimately have a: "0 < x" and b: "0 < y"
```
```  1759     by (fast intro: less_le_trans order_trans)+
```
```  1760   have "x * ln y - x * ln x = x * (ln y - ln x)"
```
```  1761     by (simp add: algebra_simps)
```
```  1762   also have "... = x * ln(y / x)"
```
```  1763     by (simp only: ln_div a b)
```
```  1764   also have "y / x = (x + (y - x)) / x"
```
```  1765     by simp
```
```  1766   also have "... = 1 + (y - x) / x"
```
```  1767     using x a by (simp add: field_simps)
```
```  1768   also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
```
```  1769     using x a
```
```  1770     by (intro mult_left_mono ln_add_one_self_le_self) simp_all
```
```  1771   also have "... = y - x" using a by simp
```
```  1772   also have "... = (y - x) * ln (exp 1)" by simp
```
```  1773   also have "... <= (y - x) * ln x"
```
```  1774     apply (rule mult_left_mono)
```
```  1775     apply (subst ln_le_cancel_iff)
```
```  1776     apply fact
```
```  1777     apply (rule a)
```
```  1778     apply (rule x)
```
```  1779     using x apply simp
```
```  1780     done
```
```  1781   also have "... = y * ln x - x * ln x"
```
```  1782     by (rule left_diff_distrib)
```
```  1783   finally have "x * ln y <= y * ln x"
```
```  1784     by arith
```
```  1785   then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
```
```  1786   also have "... = y * (ln x / x)" by simp
```
```  1787   finally show ?thesis using b by (simp add: field_simps)
```
```  1788 qed
```
```  1789
```
```  1790 lemma ln_le_minus_one:
```
```  1791   fixes x::real shows "0 < x \<Longrightarrow> ln x \<le> x - 1"
```
```  1792   using exp_ge_add_one_self[of "ln x"] by simp
```
```  1793
```
```  1794 lemma ln_eq_minus_one:
```
```  1795   fixes x::real
```
```  1796   assumes "0 < x" "ln x = x - 1"
```
```  1797   shows "x = 1"
```
```  1798 proof -
```
```  1799   let ?l = "\<lambda>y. ln y - y + 1"
```
```  1800   have D: "\<And>x::real. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
```
```  1801     by (auto intro!: derivative_eq_intros)
```
```  1802
```
```  1803   show ?thesis
```
```  1804   proof (cases rule: linorder_cases)
```
```  1805     assume "x < 1"
```
```  1806     from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast
```
```  1807     from `x < a` have "?l x < ?l a"
```
```  1808     proof (rule DERIV_pos_imp_increasing, safe)
```
```  1809       fix y
```
```  1810       assume "x \<le> y" "y \<le> a"
```
```  1811       with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y"
```
```  1812         by (auto simp: field_simps)
```
```  1813       with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
```
```  1814         by auto
```
```  1815     qed
```
```  1816     also have "\<dots> \<le> 0"
```
```  1817       using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps)
```
```  1818     finally show "x = 1" using assms by auto
```
```  1819   next
```
```  1820     assume "1 < x"
```
```  1821     from dense[OF this] obtain a where "1 < a" "a < x" by blast
```
```  1822     from `a < x` have "?l x < ?l a"
```
```  1823     proof (rule DERIV_neg_imp_decreasing, safe)
```
```  1824       fix y
```
```  1825       assume "a \<le> y" "y \<le> x"
```
```  1826       with `1 < a` have "1 / y - 1 < 0" "0 < y"
```
```  1827         by (auto simp: field_simps)
```
```  1828       with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
```
```  1829         by blast
```
```  1830     qed
```
```  1831     also have "\<dots> \<le> 0"
```
```  1832       using ln_le_minus_one `1 < a` by (auto simp: field_simps)
```
```  1833     finally show "x = 1" using assms by auto
```
```  1834   next
```
```  1835     assume "x = 1"
```
```  1836     then show ?thesis by simp
```
```  1837   qed
```
```  1838 qed
```
```  1839
```
```  1840 lemma exp_at_bot: "(exp ---> (0::real)) at_bot"
```
```  1841   unfolding tendsto_Zfun_iff
```
```  1842 proof (rule ZfunI, simp add: eventually_at_bot_dense)
```
```  1843   fix r :: real assume "0 < r"
```
```  1844   {
```
```  1845     fix x
```
```  1846     assume "x < ln r"
```
```  1847     then have "exp x < exp (ln r)"
```
```  1848       by simp
```
```  1849     with `0 < r` have "exp x < r"
```
```  1850       by simp
```
```  1851   }
```
```  1852   then show "\<exists>k. \<forall>n<k. exp n < r" by auto
```
```  1853 qed
```
```  1854
```
```  1855 lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
```
```  1856   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"])
```
```  1857      (auto intro: eventually_gt_at_top)
```
```  1858
```
```  1859 lemma lim_exp_minus_1:
```
```  1860   fixes x :: "'a::{real_normed_field,banach}"
```
```  1861   shows "((\<lambda>z::'a. (exp(z) - 1) / z) ---> 1) (at 0)"
```
```  1862 proof -
```
```  1863   have "((\<lambda>z::'a. exp(z) - 1) has_field_derivative 1) (at 0)"
```
```  1864     by (intro derivative_eq_intros | simp)+
```
```  1865   then show ?thesis
```
```  1866     by (simp add: Deriv.DERIV_iff2)
```
```  1867 qed
```
```  1868
```
```  1869 lemma ln_at_0: "LIM x at_right 0. ln (x::real) :> at_bot"
```
```  1870   by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  1871      (auto simp: eventually_at_filter)
```
```  1872
```
```  1873 lemma ln_at_top: "LIM x at_top. ln (x::real) :> at_top"
```
```  1874   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
```
```  1875      (auto intro: eventually_gt_at_top)
```
```  1876
```
```  1877 lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) ---> (0::real)) at_top"
```
```  1878 proof (induct k)
```
```  1879   case 0
```
```  1880   show "((\<lambda>x. x ^ 0 / exp x) ---> (0::real)) at_top"
```
```  1881     by (simp add: inverse_eq_divide[symmetric])
```
```  1882        (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
```
```  1883               at_top_le_at_infinity order_refl)
```
```  1884 next
```
```  1885   case (Suc k)
```
```  1886   show ?case
```
```  1887   proof (rule lhospital_at_top_at_top)
```
```  1888     show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"
```
```  1889       by eventually_elim (intro derivative_eq_intros, auto)
```
```  1890     show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top"
```
```  1891       by eventually_elim auto
```
```  1892     show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
```
```  1893       by auto
```
```  1894     from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
```
```  1895     show "((\<lambda>x. real (Suc k) * x ^ k / exp x) ---> 0) at_top"
```
```  1896       by simp
```
```  1897   qed (rule exp_at_top)
```
```  1898 qed
```
```  1899
```
```  1900
```
```  1901 definition log :: "[real,real] => real"
```
```  1902   -- {*logarithm of @{term x} to base @{term a}*}
```
```  1903   where "log a x = ln x / ln a"
```
```  1904
```
```  1905
```
```  1906 lemma tendsto_log [tendsto_intros]:
```
```  1907   "\<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"
```
```  1908   unfolding log_def by (intro tendsto_intros) auto
```
```  1909
```
```  1910 lemma continuous_log:
```
```  1911   assumes "continuous F f"
```
```  1912     and "continuous F g"
```
```  1913     and "0 < f (Lim F (\<lambda>x. x))"
```
```  1914     and "f (Lim F (\<lambda>x. x)) \<noteq> 1"
```
```  1915     and "0 < g (Lim F (\<lambda>x. x))"
```
```  1916   shows "continuous F (\<lambda>x. log (f x) (g x))"
```
```  1917   using assms unfolding continuous_def by (rule tendsto_log)
```
```  1918
```
```  1919 lemma continuous_at_within_log[continuous_intros]:
```
```  1920   assumes "continuous (at a within s) f"
```
```  1921     and "continuous (at a within s) g"
```
```  1922     and "0 < f a"
```
```  1923     and "f a \<noteq> 1"
```
```  1924     and "0 < g a"
```
```  1925   shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
```
```  1926   using assms unfolding continuous_within by (rule tendsto_log)
```
```  1927
```
```  1928 lemma isCont_log[continuous_intros, simp]:
```
```  1929   assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
```
```  1930   shows "isCont (\<lambda>x. log (f x) (g x)) a"
```
```  1931   using assms unfolding continuous_at by (rule tendsto_log)
```
```  1932
```
```  1933 lemma continuous_on_log[continuous_intros]:
```
```  1934   assumes "continuous_on s f" "continuous_on s g"
```
```  1935     and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
```
```  1936   shows "continuous_on s (\<lambda>x. log (f x) (g x))"
```
```  1937   using assms unfolding continuous_on_def by (fast intro: tendsto_log)
```
```  1938
```
```  1939 lemma powr_one_eq_one [simp]: "1 powr a = 1"
```
```  1940   by (simp add: powr_def)
```
```  1941
```
```  1942 lemma powr_zero_eq_one [simp]: "x powr 0 = (if x=0 then 0 else 1)"
```
```  1943   by (simp add: powr_def)
```
```  1944
```
```  1945 lemma powr_one_gt_zero_iff [simp]:
```
```  1946   fixes x::real shows "(x powr 1 = x) = (0 \<le> x)"
```
```  1947   by (auto simp: powr_def)
```
```  1948 declare powr_one_gt_zero_iff [THEN iffD2, simp]
```
```  1949
```
```  1950 lemma powr_mult:
```
```  1951   fixes x::real shows "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
```
```  1952   by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
```
```  1953
```
```  1954 lemma powr_ge_pzero [simp]:
```
```  1955   fixes x::real shows "0 <= x powr y"
```
```  1956   by (simp add: powr_def)
```
```  1957
```
```  1958 lemma powr_divide:
```
```  1959   fixes x::real shows "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
```
```  1960   apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
```
```  1961   apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
```
```  1962   done
```
```  1963
```
```  1964 lemma powr_divide2:
```
```  1965   fixes x::real shows "x powr a / x powr b = x powr (a - b)"
```
```  1966   apply (simp add: powr_def)
```
```  1967   apply (subst exp_diff [THEN sym])
```
```  1968   apply (simp add: left_diff_distrib)
```
```  1969   done
```
```  1970
```
```  1971 lemma powr_add:
```
```  1972   fixes x::real shows "x powr (a + b) = (x powr a) * (x powr b)"
```
```  1973   by (simp add: powr_def exp_add [symmetric] distrib_right)
```
```  1974
```
```  1975 lemma powr_mult_base:
```
```  1976   fixes x::real shows "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
```
```  1977   using assms by (auto simp: powr_add)
```
```  1978
```
```  1979 lemma powr_powr:
```
```  1980   fixes x::real shows "(x powr a) powr b = x powr (a * b)"
```
```  1981   by (simp add: powr_def)
```
```  1982
```
```  1983 lemma powr_powr_swap:
```
```  1984   fixes x::real shows "(x powr a) powr b = (x powr b) powr a"
```
```  1985   by (simp add: powr_powr mult.commute)
```
```  1986
```
```  1987 lemma powr_minus:
```
```  1988   fixes x::real shows "x powr (-a) = inverse (x powr a)"
```
```  1989   by (simp add: powr_def exp_minus [symmetric])
```
```  1990
```
```  1991 lemma powr_minus_divide:
```
```  1992   fixes x::real shows "x powr (-a) = 1/(x powr a)"
```
```  1993   by (simp add: divide_inverse powr_minus)
```
```  1994
```
```  1995 lemma divide_powr_uminus:
```
```  1996   fixes a::real shows "a / b powr c = a * b powr (- c)"
```
```  1997   by (simp add: powr_minus_divide)
```
```  1998
```
```  1999 lemma powr_less_mono:
```
```  2000   fixes x::real shows "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
```
```  2001   by (simp add: powr_def)
```
```  2002
```
```  2003 lemma powr_less_cancel:
```
```  2004   fixes x::real shows "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
```
```  2005   by (simp add: powr_def)
```
```  2006
```
```  2007 lemma powr_less_cancel_iff [simp]:
```
```  2008   fixes x::real shows "1 < x \<Longrightarrow> (x powr a < x powr b) = (a < b)"
```
```  2009   by (blast intro: powr_less_cancel powr_less_mono)
```
```  2010
```
```  2011 lemma powr_le_cancel_iff [simp]:
```
```  2012   fixes x::real shows "1 < x \<Longrightarrow> (x powr a \<le> x powr b) = (a \<le> b)"
```
```  2013   by (simp add: linorder_not_less [symmetric])
```
```  2014
```
```  2015 lemma log_ln: "ln x = log (exp(1)) x"
```
```  2016   by (simp add: log_def)
```
```  2017
```
```  2018 lemma DERIV_log:
```
```  2019   assumes "x > 0"
```
```  2020   shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
```
```  2021 proof -
```
```  2022   def lb \<equiv> "1 / ln b"
```
```  2023   moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
```
```  2024     using `x > 0` by (auto intro!: derivative_eq_intros)
```
```  2025   ultimately show ?thesis
```
```  2026     by (simp add: log_def)
```
```  2027 qed
```
```  2028
```
```  2029 lemmas DERIV_log[THEN DERIV_chain2, derivative_intros]
```
```  2030
```
```  2031 lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
```
```  2032   by (simp add: powr_def log_def)
```
```  2033
```
```  2034 lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y"
```
```  2035   by (simp add: log_def powr_def)
```
```  2036
```
```  2037 lemma log_mult:
```
```  2038   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow>
```
```  2039     log a (x * y) = log a x + log a y"
```
```  2040   by (simp add: log_def ln_mult divide_inverse distrib_right)
```
```  2041
```
```  2042 lemma log_eq_div_ln_mult_log:
```
```  2043   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow>
```
```  2044     log a x = (ln b/ln a) * log b x"
```
```  2045   by (simp add: log_def divide_inverse)
```
```  2046
```
```  2047 text{*Base 10 logarithms*}
```
```  2048 lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x"
```
```  2049   by (simp add: log_def)
```
```  2050
```
```  2051 lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x"
```
```  2052   by (simp add: log_def)
```
```  2053
```
```  2054 lemma log_one [simp]: "log a 1 = 0"
```
```  2055   by (simp add: log_def)
```
```  2056
```
```  2057 lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
```
```  2058   by (simp add: log_def)
```
```  2059
```
```  2060 lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x"
```
```  2061   apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
```
```  2062   apply (simp add: log_mult [symmetric])
```
```  2063   done
```
```  2064
```
```  2065 lemma log_divide: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a (x/y) = log a x - log a y"
```
```  2066   by (simp add: log_mult divide_inverse log_inverse)
```
```  2067
```
```  2068 lemma powr_gt_zero [simp]: "0 < x powr a \<longleftrightarrow> (x::real) \<noteq> 0"
```
```  2069   by (simp add: powr_def)
```
```  2070
```
```  2071 lemma log_add_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log b x + y = log b (x * b powr y)"
```
```  2072   and add_log_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> y + log b x = log b (b powr y * x)"
```
```  2073   and log_minus_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log b x - y = log b (x * b powr -y)"
```
```  2074   and minus_log_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> y - log b x = log b (b powr y / x)"
```
```  2075   by (simp_all add: log_mult log_divide)
```
```  2076
```
```  2077 lemma log_less_cancel_iff [simp]:
```
```  2078   "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
```
```  2079   apply safe
```
```  2080   apply (rule_tac [2] powr_less_cancel)
```
```  2081   apply (drule_tac a = "log a x" in powr_less_mono, auto)
```
```  2082   done
```
```  2083
```
```  2084 lemma log_inj:
```
```  2085   assumes "1 < b"
```
```  2086   shows "inj_on (log b) {0 <..}"
```
```  2087 proof (rule inj_onI, simp)
```
```  2088   fix x y
```
```  2089   assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
```
```  2090   show "x = y"
```
```  2091   proof (cases rule: linorder_cases)
```
```  2092     assume "x = y"
```
```  2093     then show ?thesis by simp
```
```  2094   next
```
```  2095     assume "x < y" hence "log b x < log b y"
```
```  2096       using log_less_cancel_iff[OF `1 < b`] pos by simp
```
```  2097     then show ?thesis using * by simp
```
```  2098   next
```
```  2099     assume "y < x" hence "log b y < log b x"
```
```  2100       using log_less_cancel_iff[OF `1 < b`] pos by simp
```
```  2101     then show ?thesis using * by simp
```
```  2102   qed
```
```  2103 qed
```
```  2104
```
```  2105 lemma log_le_cancel_iff [simp]:
```
```  2106   "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (log a x \<le> log a y) = (x \<le> y)"
```
```  2107   by (simp add: linorder_not_less [symmetric])
```
```  2108
```
```  2109 lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
```
```  2110   using log_less_cancel_iff[of a 1 x] by simp
```
```  2111
```
```  2112 lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
```
```  2113   using log_le_cancel_iff[of a 1 x] by simp
```
```  2114
```
```  2115 lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
```
```  2116   using log_less_cancel_iff[of a x 1] by simp
```
```  2117
```
```  2118 lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
```
```  2119   using log_le_cancel_iff[of a x 1] by simp
```
```  2120
```
```  2121 lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
```
```  2122   using log_less_cancel_iff[of a a x] by simp
```
```  2123
```
```  2124 lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
```
```  2125   using log_le_cancel_iff[of a a x] by simp
```
```  2126
```
```  2127 lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
```
```  2128   using log_less_cancel_iff[of a x a] by simp
```
```  2129
```
```  2130 lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
```
```  2131   using log_le_cancel_iff[of a x a] by simp
```
```  2132
```
```  2133 lemma le_log_iff:
```
```  2134   assumes "1 < b" "x > 0"
```
```  2135   shows "y \<le> log b x \<longleftrightarrow> b powr y \<le> (x::real)"
```
```  2136   using assms
```
```  2137   apply auto
```
```  2138   apply (metis (no_types, hide_lams) less_irrefl less_le_trans linear powr_le_cancel_iff
```
```  2139                powr_log_cancel zero_less_one)
```
```  2140   apply (metis not_less order.trans order_refl powr_le_cancel_iff powr_log_cancel zero_le_one)
```
```  2141   done
```
```  2142
```
```  2143 lemma less_log_iff:
```
```  2144   assumes "1 < b" "x > 0"
```
```  2145   shows "y < log b x \<longleftrightarrow> b powr y < x"
```
```  2146   by (metis assms dual_order.strict_trans less_irrefl powr_less_cancel_iff
```
```  2147     powr_log_cancel zero_less_one)
```
```  2148
```
```  2149 lemma
```
```  2150   assumes "1 < b" "x > 0"
```
```  2151   shows log_less_iff: "log b x < y \<longleftrightarrow> x < b powr y"
```
```  2152     and log_le_iff: "log b x \<le> y \<longleftrightarrow> x \<le> b powr y"
```
```  2153   using le_log_iff[OF assms, of y] less_log_iff[OF assms, of y]
```
```  2154   by auto
```
```  2155
```
```  2156 lemmas powr_le_iff = le_log_iff[symmetric]
```
```  2157   and powr_less_iff = le_log_iff[symmetric]
```
```  2158   and less_powr_iff = log_less_iff[symmetric]
```
```  2159   and le_powr_iff = log_le_iff[symmetric]
```
```  2160
```
```  2161 lemma
```
```  2162   floor_log_eq_powr_iff: "x > 0 \<Longrightarrow> b > 1 \<Longrightarrow> \<lfloor>log b x\<rfloor> = k \<longleftrightarrow> b powr k \<le> x \<and> x < b powr (k + 1)"
```
```  2163   by (auto simp add: floor_eq_iff powr_le_iff less_powr_iff)
```
```  2164
```
```  2165 lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
```
```  2166   apply (induct n)
```
```  2167   apply simp
```
```  2168   apply (subgoal_tac "real(Suc n) = real n + 1")
```
```  2169   apply (erule ssubst)
```
```  2170   apply (subst powr_add, simp, simp)
```
```  2171   done
```
```  2172
```
```  2173 lemma powr_realpow_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
```
```  2174   unfolding real_of_nat_numeral [symmetric] by (rule powr_realpow)
```
```  2175
```
```  2176 lemma powr2_sqrt[simp]: "0 < x \<Longrightarrow> sqrt x powr 2 = x"
```
```  2177 by(simp add: powr_realpow_numeral)
```
```  2178
```
```  2179 lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
```
```  2180   apply (case_tac "x = 0", simp, simp)
```
```  2181   apply (rule powr_realpow [THEN sym], simp)
```
```  2182   done
```
```  2183
```
```  2184 lemma powr_int:
```
```  2185   assumes "x > 0"
```
```  2186   shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
```
```  2187 proof (cases "i < 0")
```
```  2188   case True
```
```  2189   have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
```
```  2190   show ?thesis using `i < 0` `x > 0` by (simp add: r field_simps powr_realpow[symmetric])
```
```  2191 next
```
```  2192   case False
```
```  2193   then show ?thesis by (simp add: assms powr_realpow[symmetric])
```
```  2194 qed
```
```  2195
```
```  2196 lemma compute_powr[code]:
```
```  2197   fixes i::real
```
```  2198   shows "b powr i =
```
```  2199     (if b \<le> 0 then Code.abort (STR ''op powr with nonpositive base'') (\<lambda>_. b powr i)
```
```  2200     else if floor i = i then (if 0 \<le> i then b ^ nat(floor i) else 1 / b ^ nat(floor (- i)))
```
```  2201     else Code.abort (STR ''op powr with non-integer exponent'') (\<lambda>_. b powr i))"
```
```  2202   by (auto simp: powr_int)
```
```  2203
```
```  2204 lemma powr_one:
```
```  2205   fixes x::real shows "0 \<le> x \<Longrightarrow> x powr 1 = x"
```
```  2206   using powr_realpow [of x 1]
```
```  2207   by simp
```
```  2208
```
```  2209 lemma powr_numeral:
```
```  2210   fixes x::real shows "0 < x \<Longrightarrow> x powr numeral n = x ^ numeral n"
```
```  2211   by (fact powr_realpow_numeral)
```
```  2212
```
```  2213 lemma powr_neg_one:
```
```  2214   fixes x::real shows "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
```
```  2215   using powr_int [of x "- 1"] by simp
```
```  2216
```
```  2217 lemma powr_neg_numeral:
```
```  2218   fixes x::real shows "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
```
```  2219   using powr_int [of x "- numeral n"] by simp
```
```  2220
```
```  2221 lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
```
```  2222   by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
```
```  2223
```
```  2224 lemma ln_powr:
```
```  2225   fixes x::real shows "x \<noteq> 0 \<Longrightarrow> ln (x powr y) = y * ln x"
```
```  2226   by (simp add: powr_def)
```
```  2227
```
```  2228 lemma ln_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> ln (root n b) =  ln b / n"
```
```  2229 by(simp add: root_powr_inverse ln_powr)
```
```  2230
```
```  2231 lemma ln_sqrt: "0 < x \<Longrightarrow> ln (sqrt x) = ln x / 2"
```
```  2232   by (simp add: ln_powr powr_numeral ln_powr[symmetric] mult.commute)
```
```  2233
```
```  2234 lemma log_root: "\<lbrakk> n > 0; a > 0 \<rbrakk> \<Longrightarrow> log b (root n a) =  log b a / n"
```
```  2235 by(simp add: log_def ln_root)
```
```  2236
```
```  2237 lemma log_powr: "x \<noteq> 0 \<Longrightarrow> log b (x powr y) = y * log b x"
```
```  2238   by (simp add: log_def ln_powr)
```
```  2239
```
```  2240 lemma log_nat_power: "0 < x \<Longrightarrow> log b (x^n) = real n * log b x"
```
```  2241   by (simp add: log_powr powr_realpow [symmetric])
```
```  2242
```
```  2243 lemma log_base_change: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log b x = log a x / log a b"
```
```  2244   by (simp add: log_def)
```
```  2245
```
```  2246 lemma log_base_pow: "0 < a \<Longrightarrow> log (a ^ n) x = log a x / n"
```
```  2247   by (simp add: log_def ln_realpow)
```
```  2248
```
```  2249 lemma log_base_powr: "a \<noteq> 0 \<Longrightarrow> log (a powr b) x = log a x / b"
```
```  2250   by (simp add: log_def ln_powr)
```
```  2251
```
```  2252 lemma log_base_root: "\<lbrakk> n > 0; b > 0 \<rbrakk> \<Longrightarrow> log (root n b) x = n * (log b x)"
```
```  2253 by(simp add: log_def ln_root)
```
```  2254
```
```  2255 lemma ln_bound:
```
```  2256   fixes x::real shows "1 <= x ==> ln x <= x"
```
```  2257   apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
```
```  2258   apply simp
```
```  2259   apply (rule ln_add_one_self_le_self, simp)
```
```  2260   done
```
```  2261
```
```  2262 lemma powr_mono:
```
```  2263   fixes x::real shows "a <= b ==> 1 <= x ==> x powr a <= x powr b"
```
```  2264   apply (cases "x = 1", simp)
```
```  2265   apply (cases "a = b", simp)
```
```  2266   apply (rule order_less_imp_le)
```
```  2267   apply (rule powr_less_mono, auto)
```
```  2268   done
```
```  2269
```
```  2270 lemma ge_one_powr_ge_zero:
```
```  2271   fixes x::real shows "1 <= x ==> 0 <= a ==> 1 <= x powr a"
```
```  2272 using powr_mono by fastforce
```
```  2273
```
```  2274 lemma powr_less_mono2:
```
```  2275   fixes x::real shows "0 < a ==> 0 < x ==> x < y ==> x powr a < y powr a"
```
```  2276   by (simp add: powr_def)
```
```  2277
```
```  2278
```
```  2279 lemma powr_less_mono2_neg:
```
```  2280   fixes x::real shows "a < 0 ==> 0 < x ==> x < y ==> y powr a < x powr a"
```
```  2281   by (simp add: powr_def)
```
```  2282
```
```  2283 lemma powr_mono2:
```
```  2284   fixes x::real shows "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
```
```  2285   apply (case_tac "a = 0", simp)
```
```  2286   apply (case_tac "x = y", simp)
```
```  2287   apply (metis less_eq_real_def powr_less_mono2)
```
```  2288   done
```
```  2289
```
```  2290 lemma powr_inj:
```
```  2291   fixes x::real shows "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
```
```  2292   unfolding powr_def exp_inj_iff by simp
```
```  2293
```
```  2294 lemma ln_powr_bound:
```
```  2295   fixes x::real shows "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
```
```  2296 by (metis exp_gt_zero linear ln_eq_zero_iff ln_exp ln_less_self ln_powr mult.commute mult_imp_le_div_pos not_less powr_gt_zero)
```
```  2297
```
```  2298
```
```  2299 lemma ln_powr_bound2:
```
```  2300   fixes x::real
```
```  2301   assumes "1 < x" and "0 < a"
```
```  2302   shows "(ln x) powr a <= (a powr a) * x"
```
```  2303 proof -
```
```  2304   from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
```
```  2305     by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff)
```
```  2306   also have "... = a * (x powr (1 / a))"
```
```  2307     by simp
```
```  2308   finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
```
```  2309     by (metis assms less_imp_le ln_gt_zero powr_mono2)
```
```  2310   also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
```
```  2311     using assms powr_mult by auto
```
```  2312   also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
```
```  2313     by (rule powr_powr)
```
```  2314   also have "... = x" using assms
```
```  2315     by auto
```
```  2316   finally show ?thesis .
```
```  2317 qed
```
```  2318
```
```  2319 lemma tendsto_powr [tendsto_intros]:  (*FIXME a mess, suggests a general rule about exceptions*)
```
```  2320   fixes a::real
```
```  2321   shows "\<lbrakk>(f ---> a) F; (g ---> b) F; a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x powr g x) ---> a powr b) F"
```
```  2322   apply (simp add: powr_def)
```
```  2323   apply (simp add: tendsto_def)
```
```  2324   apply (simp add: Topological_Spaces.eventually_conj_iff )
```
```  2325   apply safe
```
```  2326   apply (case_tac "0 \<in> S")
```
```  2327   apply (auto simp: )
```
```  2328   apply (subgoal_tac "\<exists>T. open T & a : T & 0 \<notin> T")
```
```  2329   apply clarify
```
```  2330   apply (drule_tac x="T" in spec)
```
```  2331   apply (simp add: )
```
```  2332   apply (metis (mono_tags, lifting) eventually_mono)
```
```  2333   apply (simp add: separation_t1)
```
```  2334   apply (subgoal_tac "((\<lambda>x. exp (g x * ln (f x))) ---> exp (b * ln a)) F")
```
```  2335   apply (simp add: tendsto_def)
```
```  2336   apply (metis (mono_tags, lifting) eventually_mono)
```
```  2337   apply (simp add: tendsto_def [symmetric])
```
```  2338   apply (intro tendsto_intros)
```
```  2339   apply (auto simp: )
```
```  2340   done
```
```  2341
```
```  2342 lemma continuous_powr:
```
```  2343   assumes "continuous F f"
```
```  2344     and "continuous F g"
```
```  2345     and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```  2346   shows "continuous F (\<lambda>x. (f x) powr (g x :: real))"
```
```  2347   using assms unfolding continuous_def by (rule tendsto_powr)
```
```  2348
```
```  2349 lemma continuous_at_within_powr[continuous_intros]:
```
```  2350   assumes "continuous (at a within s) f"
```
```  2351     and "continuous (at a within s) g"
```
```  2352     and "f a \<noteq> 0"
```
```  2353   shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x :: real))"
```
```  2354   using assms unfolding continuous_within by (rule tendsto_powr)
```
```  2355
```
```  2356 lemma isCont_powr[continuous_intros, simp]:
```
```  2357   assumes "isCont f a" "isCont g a" "f a \<noteq> (0::real)"
```
```  2358   shows "isCont (\<lambda>x. (f x) powr g x) a"
```
```  2359   using assms unfolding continuous_at by (rule tendsto_powr)
```
```  2360
```
```  2361 lemma continuous_on_powr[continuous_intros]:
```
```  2362   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> (0::real)"
```
```  2363   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
```
```  2364   using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
```
```  2365
```
```  2366 (* FIXME: generalize by replacing d by with g x and g ---> d? *)
```
```  2367 lemma tendsto_zero_powrI:
```
```  2368   assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> (0::real)) F"
```
```  2369     and "0 < d"
```
```  2370   shows "((\<lambda>x. f x powr d) ---> 0) F"
```
```  2371 proof (rule tendstoI)
```
```  2372   fix e :: real assume "0 < e"
```
```  2373   def Z \<equiv> "e powr (1 / d)"
```
```  2374   with `0 < e` have "0 < Z" by simp
```
```  2375   with assms have "eventually (\<lambda>x. 0 < f x \<and> dist (f x) 0 < Z) F"
```
```  2376     by (intro eventually_conj tendstoD)
```
```  2377   moreover
```
```  2378   from assms have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> x powr d < Z powr d"
```
```  2379     by (intro powr_less_mono2) (auto simp: dist_real_def)
```
```  2380   with assms `0 < e` have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> dist (x powr d) 0 < e"
```
```  2381     unfolding dist_real_def Z_def by (auto simp: powr_powr)
```
```  2382   ultimately
```
```  2383   show "eventually (\<lambda>x. dist (f x powr d) 0 < e) F" by (rule eventually_elim1)
```
```  2384 qed
```
```  2385
```
```  2386 lemma tendsto_neg_powr:
```
```  2387   assumes "s < 0"
```
```  2388     and "LIM x F. f x :> at_top"
```
```  2389   shows "((\<lambda>x. f x powr s) ---> (0::real)) F"
```
```  2390 proof (rule tendstoI)
```
```  2391   fix e :: real assume "0 < e"
```
```  2392   def Z \<equiv> "e powr (1 / s)"
```
```  2393   have "Z > 0"
```
```  2394     using Z_def `0 < e` by auto
```
```  2395   from assms have "eventually (\<lambda>x. Z < f x) F"
```
```  2396     by (simp add: filterlim_at_top_dense)
```
```  2397   moreover
```
```  2398   from assms have "\<And>x::real. Z < x \<Longrightarrow> x powr s < Z powr s"
```
```  2399     using `Z > 0`
```
```  2400     by (auto simp: Z_def intro!: powr_less_mono2_neg)
```
```  2401   with assms `0 < e` have "\<And>x. Z < x \<Longrightarrow> dist (x powr s) 0 < e"
```
```  2402     by (simp add: powr_powr Z_def dist_real_def)
```
```  2403   ultimately
```
```  2404   show "eventually (\<lambda>x. dist (f x powr s) 0 < e) F" by (rule eventually_elim1)
```
```  2405 qed
```
```  2406
```
```  2407 lemma tendsto_exp_limit_at_right:
```
```  2408   fixes x :: real
```
```  2409   shows "((\<lambda>y. (1 + x * y) powr (1 / y)) ---> exp x) (at_right 0)"
```
```  2410 proof cases
```
```  2411   assume "x \<noteq> 0"
```
```  2412   have "((\<lambda>y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)"
```
```  2413     by (auto intro!: derivative_eq_intros)
```
```  2414   then have "((\<lambda>y. ln (1 + x * y) / y) ---> x) (at 0)"
```
```  2415     by (auto simp add: has_field_derivative_def field_has_derivative_at)
```
```  2416   then have *: "((\<lambda>y. exp (ln (1 + x * y) / y)) ---> exp x) (at 0)"
```
```  2417     by (rule tendsto_intros)
```
```  2418   then show ?thesis
```
```  2419   proof (rule filterlim_mono_eventually)
```
```  2420     show "eventually (\<lambda>xa. exp (ln (1 + x * xa) / xa) = (1 + x * xa) powr (1 / xa)) (at_right 0)"
```
```  2421       unfolding eventually_at_right[OF zero_less_one]
```
```  2422       using `x \<noteq> 0`
```
```  2423       apply  (intro exI[of _ "1 / \<bar>x\<bar>"])
```
```  2424       apply (auto simp: field_simps powr_def abs_if)
```
```  2425       by (metis add_less_same_cancel1 mult_less_0_iff not_less_iff_gr_or_eq zero_less_one)
```
```  2426   qed (simp_all add: at_eq_sup_left_right)
```
```  2427 qed simp
```
```  2428
```
```  2429 lemma tendsto_exp_limit_at_top:
```
```  2430   fixes x :: real
```
```  2431   shows "((\<lambda>y. (1 + x / y) powr y) ---> exp x) at_top"
```
```  2432   apply (subst filterlim_at_top_to_right)
```
```  2433   apply (simp add: inverse_eq_divide)
```
```  2434   apply (rule tendsto_exp_limit_at_right)
```
```  2435   done
```
```  2436
```
```  2437 lemma tendsto_exp_limit_sequentially:
```
```  2438   fixes x :: real
```
```  2439   shows "(\<lambda>n. (1 + x / n) ^ n) ----> exp x"
```
```  2440 proof (rule filterlim_mono_eventually)
```
```  2441   from reals_Archimedean2 [of "abs x"] obtain n :: nat where *: "real n > abs x" ..
```
```  2442   hence "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top"
```
```  2443     apply (intro eventually_sequentiallyI [of n])
```
```  2444     apply (case_tac "x \<ge> 0")
```
```  2445     apply (rule add_pos_nonneg, auto intro: divide_nonneg_nonneg)
```
```  2446     apply (subgoal_tac "x / real xa > -1")
```
```  2447     apply (auto simp add: field_simps)
```
```  2448     done
```
```  2449   then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top"
```
```  2450     by (rule eventually_elim1) (erule powr_realpow)
```
```  2451   show "(\<lambda>n. (1 + x / real n) powr real n) ----> exp x"
```
```  2452     by (rule filterlim_compose [OF tendsto_exp_limit_at_top filterlim_real_sequentially])
```
```  2453 qed auto
```
```  2454
```
```  2455 subsection {* Sine and Cosine *}
```
```  2456
```
```  2457 definition sin_coeff :: "nat \<Rightarrow> real" where
```
```  2458   "sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / (fact n))"
```
```  2459
```
```  2460 definition cos_coeff :: "nat \<Rightarrow> real" where
```
```  2461   "cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / (fact n) else 0)"
```
```  2462
```
```  2463 definition sin :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```  2464   where "sin = (\<lambda>x. \<Sum>n. sin_coeff n *\<^sub>R x^n)"
```
```  2465
```
```  2466 definition cos :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
```
```  2467   where "cos = (\<lambda>x. \<Sum>n. cos_coeff n *\<^sub>R x^n)"
```
```  2468
```
```  2469 lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
```
```  2470   unfolding sin_coeff_def by simp
```
```  2471
```
```  2472 lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
```
```  2473   unfolding cos_coeff_def by simp
```
```  2474
```
```  2475 lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
```
```  2476   unfolding cos_coeff_def sin_coeff_def
```
```  2477   by (simp del: mult_Suc)
```
```  2478
```
```  2479 lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
```
```  2480   unfolding cos_coeff_def sin_coeff_def
```
```  2481   by (simp del: mult_Suc) (auto elim: oddE)
```
```  2482
```
```  2483 lemma summable_norm_sin:
```
```  2484   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  2485   shows "summable (\<lambda>n. norm (sin_coeff n *\<^sub>R x^n))"
```
```  2486   unfolding sin_coeff_def
```
```  2487   apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]])
```
```  2488   apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2489   done
```
```  2490
```
```  2491 lemma summable_norm_cos:
```
```  2492   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  2493   shows "summable (\<lambda>n. norm (cos_coeff n *\<^sub>R x^n))"
```
```  2494   unfolding cos_coeff_def
```
```  2495   apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]])
```
```  2496   apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
```
```  2497   done
```
```  2498
```
```  2499 lemma sin_converges: "(\<lambda>n. sin_coeff n *\<^sub>R x^n) sums sin(x)"
```
```  2500 unfolding sin_def
```
```  2501   by (metis (full_types) summable_norm_cancel summable_norm_sin summable_sums)
```
```  2502
```
```  2503 lemma cos_converges: "(\<lambda>n. cos_coeff n *\<^sub>R x^n) sums cos(x)"
```
```  2504 unfolding cos_def
```
```  2505   by (metis (full_types) summable_norm_cancel summable_norm_cos summable_sums)
```
```  2506
```
```  2507 lemma sin_of_real:
```
```  2508   fixes x::real
```
```  2509   shows "sin (of_real x) = of_real (sin x)"
```
```  2510 proof -
```
```  2511   have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R  x^n)) = (\<lambda>n. sin_coeff n *\<^sub>R  (of_real x)^n)"
```
```  2512   proof
```
```  2513     fix n
```
```  2514     show "of_real (sin_coeff n *\<^sub>R  x^n) = sin_coeff n *\<^sub>R of_real x^n"
```
```  2515       by (simp add: scaleR_conv_of_real)
```
```  2516   qed
```
```  2517   also have "... sums (sin (of_real x))"
```
```  2518     by (rule sin_converges)
```
```  2519   finally have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) sums (sin (of_real x))" .
```
```  2520   then show ?thesis
```
```  2521     using sums_unique2 sums_of_real [OF sin_converges]
```
```  2522     by blast
```
```  2523 qed
```
```  2524
```
```  2525 corollary sin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> sin z \<in> \<real>"
```
```  2526   by (metis Reals_cases Reals_of_real sin_of_real)
```
```  2527
```
```  2528 lemma cos_of_real:
```
```  2529   fixes x::real
```
```  2530   shows "cos (of_real x) = of_real (cos x)"
```
```  2531 proof -
```
```  2532   have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R  x^n)) = (\<lambda>n. cos_coeff n *\<^sub>R  (of_real x)^n)"
```
```  2533   proof
```
```  2534     fix n
```
```  2535     show "of_real (cos_coeff n *\<^sub>R  x^n) = cos_coeff n *\<^sub>R of_real x^n"
```
```  2536       by (simp add: scaleR_conv_of_real)
```
```  2537   qed
```
```  2538   also have "... sums (cos (of_real x))"
```
```  2539     by (rule cos_converges)
```
```  2540   finally have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) sums (cos (of_real x))" .
```
```  2541   then show ?thesis
```
```  2542     using sums_unique2 sums_of_real [OF cos_converges]
```
```  2543     by blast
```
```  2544 qed
```
```  2545
```
```  2546 corollary cos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cos z \<in> \<real>"
```
```  2547   by (metis Reals_cases Reals_of_real cos_of_real)
```
```  2548
```
```  2549 lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
```
```  2550   by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  2551
```
```  2552 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
```
```  2553   by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
```
```  2554
```
```  2555 text{*Now at last we can get the derivatives of exp, sin and cos*}
```
```  2556
```
```  2557 lemma DERIV_sin [simp]:
```
```  2558   fixes x :: "'a::{real_normed_field,banach}"
```
```  2559   shows "DERIV sin x :> cos(x)"
```
```  2560   unfolding sin_def cos_def scaleR_conv_of_real
```
```  2561   apply (rule DERIV_cong)
```
```  2562   apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
```
```  2563   apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff
```
```  2564               summable_minus_iff scaleR_conv_of_real [symmetric]
```
```  2565               summable_norm_sin [THEN summable_norm_cancel]
```
```  2566               summable_norm_cos [THEN summable_norm_cancel])
```
```  2567   done
```
```  2568
```
```  2569 declare DERIV_sin[THEN DERIV_chain2, derivative_intros]
```
```  2570
```
```  2571 lemma DERIV_cos [simp]:
```
```  2572   fixes x :: "'a::{real_normed_field,banach}"
```
```  2573   shows "DERIV cos x :> -sin(x)"
```
```  2574   unfolding sin_def cos_def scaleR_conv_of_real
```
```  2575   apply (rule DERIV_cong)
```
```  2576   apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
```
```  2577   apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus
```
```  2578               diffs_sin_coeff diffs_cos_coeff
```
```  2579               summable_minus_iff scaleR_conv_of_real [symmetric]
```
```  2580               summable_norm_sin [THEN summable_norm_cancel]
```
```  2581               summable_norm_cos [THEN summable_norm_cancel])
```
```  2582   done
```
```  2583
```
```  2584 declare DERIV_cos[THEN DERIV_chain2, derivative_intros]
```
```  2585
```
```  2586 lemma isCont_sin:
```
```  2587   fixes x :: "'a::{real_normed_field,banach}"
```
```  2588   shows "isCont sin x"
```
```  2589   by (rule DERIV_sin [THEN DERIV_isCont])
```
```  2590
```
```  2591 lemma isCont_cos:
```
```  2592   fixes x :: "'a::{real_normed_field,banach}"
```
```  2593   shows "isCont cos x"
```
```  2594   by (rule DERIV_cos [THEN DERIV_isCont])
```
```  2595
```
```  2596 lemma isCont_sin' [simp]:
```
```  2597   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2598   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
```
```  2599   by (rule isCont_o2 [OF _ isCont_sin])
```
```  2600
```
```  2601 (*FIXME A CONTEXT FOR F WOULD BE BETTER*)
```
```  2602
```
```  2603 lemma isCont_cos' [simp]:
```
```  2604   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2605   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
```
```  2606   by (rule isCont_o2 [OF _ isCont_cos])
```
```  2607
```
```  2608 lemma tendsto_sin [tendsto_intros]:
```
```  2609   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2610   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
```
```  2611   by (rule isCont_tendsto_compose [OF isCont_sin])
```
```  2612
```
```  2613 lemma tendsto_cos [tendsto_intros]:
```
```  2614   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2615   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
```
```  2616   by (rule isCont_tendsto_compose [OF isCont_cos])
```
```  2617
```
```  2618 lemma continuous_sin [continuous_intros]:
```
```  2619   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2620   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
```
```  2621   unfolding continuous_def by (rule tendsto_sin)
```
```  2622
```
```  2623 lemma continuous_on_sin [continuous_intros]:
```
```  2624   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2625   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
```
```  2626   unfolding continuous_on_def by (auto intro: tendsto_sin)
```
```  2627
```
```  2628 lemma continuous_within_sin:
```
```  2629   fixes z :: "'a::{real_normed_field,banach}"
```
```  2630   shows "continuous (at z within s) sin"
```
```  2631   by (simp add: continuous_within tendsto_sin)
```
```  2632
```
```  2633 lemma continuous_cos [continuous_intros]:
```
```  2634   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2635   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
```
```  2636   unfolding continuous_def by (rule tendsto_cos)
```
```  2637
```
```  2638 lemma continuous_on_cos [continuous_intros]:
```
```  2639   fixes f:: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  2640   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
```
```  2641   unfolding continuous_on_def by (auto intro: tendsto_cos)
```
```  2642
```
```  2643 lemma continuous_within_cos:
```
```  2644   fixes z :: "'a::{real_normed_field,banach}"
```
```  2645   shows "continuous (at z within s) cos"
```
```  2646   by (simp add: continuous_within tendsto_cos)
```
```  2647
```
```  2648 subsection {* Properties of Sine and Cosine *}
```
```  2649
```
```  2650 lemma sin_zero [simp]: "sin 0 = 0"
```
```  2651   unfolding sin_def sin_coeff_def by (simp add: scaleR_conv_of_real powser_zero)
```
```  2652
```
```  2653 lemma cos_zero [simp]: "cos 0 = 1"
```
```  2654   unfolding cos_def cos_coeff_def by (simp add: scaleR_conv_of_real powser_zero)
```
```  2655
```
```  2656 lemma DERIV_fun_sin:
```
```  2657      "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. sin(g x)) x :> cos(g x) * m"
```
```  2658   by (auto intro!: derivative_intros)
```
```  2659
```
```  2660 lemma DERIV_fun_cos:
```
```  2661      "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. cos(g x)) x :> -sin(g x) * m"
```
```  2662   by (auto intro!: derivative_eq_intros simp: real_of_nat_def)
```
```  2663
```
```  2664 subsection {*Deriving the Addition Formulas*}
```
```  2665
```
```  2666 text{*The the product of two cosine series*}
```
```  2667 lemma cos_x_cos_y:
```
```  2668   fixes x :: "'a::{real_normed_field,banach}"
```
```  2669   shows "(\<lambda>p. \<Sum>n\<le>p.
```
```  2670           if even p \<and> even n
```
```  2671           then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
```
```  2672          sums (cos x * cos y)"
```
```  2673 proof -
```
```  2674   { fix n p::nat
```
```  2675     assume "n\<le>p"
```
```  2676     then have *: "even n \<Longrightarrow> even p \<Longrightarrow> (-1) ^ (n div 2) * (-1) ^ ((p - n) div 2) = (-1 :: real) ^ (p div 2)"
```
```  2677       by (metis div_add power_add le_add_diff_inverse odd_add)
```
```  2678     have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
```
```  2679           (if even p \<and> even n then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
```
```  2680     using `n\<le>p`
```
```  2681       by (auto simp: * algebra_simps cos_coeff_def binomial_fact real_of_nat_def)
```
```  2682   }
```
```  2683   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> even n
```
```  2684                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
```
```  2685              (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
```
```  2686     by simp
```
```  2687   also have "... = (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n *\<^sub>R x^n) * (cos_coeff (p - n) *\<^sub>R y^(p-n)))"
```
```  2688     by (simp add: algebra_simps)
```
```  2689   also have "... sums (cos x * cos y)"
```
```  2690     using summable_norm_cos
```
```  2691     by (auto simp: cos_def scaleR_conv_of_real intro!: Cauchy_product_sums)
```
```  2692   finally show ?thesis .
```
```  2693 qed
```
```  2694
```
```  2695 text{*The product of two sine series*}
```
```  2696 lemma sin_x_sin_y:
```
```  2697   fixes x :: "'a::{real_normed_field,banach}"
```
```  2698   shows "(\<lambda>p. \<Sum>n\<le>p.
```
```  2699           if even p \<and> odd n
```
```  2700                then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
```
```  2701          sums (sin x * sin y)"
```
```  2702 proof -
```
```  2703   { fix n p::nat
```
```  2704     assume "n\<le>p"
```
```  2705     { assume np: "odd n" "even p"
```
```  2706         with `n\<le>p` have "n - Suc 0 + (p - Suc n) = p - Suc (Suc 0)" "Suc (Suc 0) \<le> p"
```
```  2707         by arith+
```
```  2708       moreover have "(p - Suc (Suc 0)) div 2 = p div 2 - Suc 0"
```
```  2709         by simp
```
```  2710       ultimately have *: "(-1) ^ ((n - Suc 0) div 2) * (-1) ^ ((p - Suc n) div 2) = - ((-1 :: real) ^ (p div 2))"
```
```  2711         using np `n\<le>p`
```
```  2712         apply (simp add: power_add [symmetric] div_add [symmetric] del: div_add)
```
```  2713         apply (metis (no_types) One_nat_def Suc_1 le_div_geq minus_minus mult.left_neutral mult_minus_left power.simps(2) zero_less_Suc)
```
```  2714         done
```
```  2715     } then
```
```  2716     have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
```
```  2717           (if even p \<and> odd n
```
```  2718           then -((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
```
```  2719     using `n\<le>p`
```
```  2720       by (auto simp:  algebra_simps sin_coeff_def binomial_fact real_of_nat_def)
```
```  2721   }
```
```  2722   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> odd n
```
```  2723                then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
```
```  2724              (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
```
```  2725     by simp
```
```  2726   also have "... = (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n *\<^sub>R x^n) * (sin_coeff (p - n) *\<^sub>R y^(p-n)))"
```
```  2727     by (simp add: algebra_simps)
```
```  2728   also have "... sums (sin x * sin y)"
```
```  2729     using summable_norm_sin
```
```  2730     by (auto simp: sin_def scaleR_conv_of_real intro!: Cauchy_product_sums)
```
```  2731   finally show ?thesis .
```
```  2732 qed
```
```  2733
```
```  2734 lemma sums_cos_x_plus_y:
```
```  2735   fixes x :: "'a::{real_normed_field,banach}"
```
```  2736   shows
```
```  2737   "(\<lambda>p. \<Sum>n\<le>p. if even p
```
```  2738                then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2739                else 0)
```
```  2740         sums cos (x + y)"
```
```  2741 proof -
```
```  2742   { fix p::nat
```
```  2743     have "(\<Sum>n\<le>p. if even p
```
```  2744                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2745                   else 0) =
```
```  2746           (if even p
```
```  2747                   then \<Sum>n\<le>p. ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2748                   else 0)"
```
```  2749       by simp
```
```  2750     also have "... = (if even p
```
```  2751                   then of_real ((-1) ^ (p div 2) / (fact p)) * (\<Sum>n\<le>p. (p choose n) *\<^sub>R (x^n) * y^(p-n))
```
```  2752                   else 0)"
```
```  2753       by (auto simp: setsum_right_distrib field_simps scaleR_conv_of_real nonzero_of_real_divide)
```
```  2754     also have "... = cos_coeff p *\<^sub>R ((x + y) ^ p)"
```
```  2755       by (simp add: cos_coeff_def binomial_ring [of x y]  scaleR_conv_of_real real_of_nat_def atLeast0AtMost)
```
```  2756     finally have "(\<Sum>n\<le>p. if even p
```
```  2757                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2758                   else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)" .
```
```  2759   }
```
```  2760   then have "(\<lambda>p. \<Sum>n\<le>p.
```
```  2761                if even p
```
```  2762                then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
```
```  2763                else 0)
```
```  2764         = (\<lambda>p. cos_coeff p *\<^sub>R ((x+y)^p))"
```
```  2765         by simp
```
```  2766    also have "... sums cos (x + y)"
```
```  2767     by (rule cos_converges)
```
```  2768    finally show ?thesis .
```
```  2769 qed
```
```  2770
```
```  2771 theorem cos_add:
```
```  2772   fixes x :: "'a::{real_normed_field,banach}"
```
```  2773   shows "cos (x + y) = cos x * cos y - sin x * sin y"
```
```  2774 proof -
```
```  2775   { fix n p::nat
```
```  2776     assume "n\<le>p"
```
```  2777     then have "(if even p \<and> even n
```
```  2778                then ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) -
```
```  2779           (if even p \<and> odd n
```
```  2780                then - ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
```
```  2781           = (if even p
```
```  2782                then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
```
```  2783       by simp
```
```  2784   }
```
```  2785   then have "(\<lambda>p. \<Sum>n\<le>p. (if even p
```
```  2786                then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0))
```
```  2787         sums (cos x * cos y - sin x * sin y)"
```
```  2788     using sums_diff [OF cos_x_cos_y [of x y] sin_x_sin_y [of x y]]
```
```  2789     by (simp add: setsum_subtractf [symmetric])
```
```  2790   then show ?thesis
```
```  2791     by (blast intro: sums_cos_x_plus_y sums_unique2)
```
```  2792 qed
```
```  2793
```
```  2794 lemma sin_minus_converges: "(\<lambda>n. - (sin_coeff n *\<^sub>R (-x)^n)) sums sin(x)"
```
```  2795 proof -
```
```  2796   have [simp]: "\<And>n. - (sin_coeff n *\<^sub>R (-x)^n) = (sin_coeff n *\<^sub>R x^n)"
```
```  2797     by (auto simp: sin_coeff_def elim!: oddE)
```
```  2798   show ?thesis
```
```  2799     by (simp add: sin_def summable_norm_sin [THEN summable_norm_cancel, THEN summable_sums])
```
```  2800 qed
```
```  2801
```
```  2802 lemma sin_minus [simp]:
```
```  2803   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  2804   shows "sin (-x) = -sin(x)"
```
```  2805 using sin_minus_converges [of x]
```
```  2806 by (auto simp: sin_def summable_norm_sin [THEN summable_norm_cancel] suminf_minus sums_iff equation_minus_iff)
```
```  2807
```
```  2808 lemma cos_minus_converges: "(\<lambda>n. (cos_coeff n *\<^sub>R (-x)^n)) sums cos(x)"
```
```  2809 proof -
```
```  2810   have [simp]: "\<And>n. (cos_coeff n *\<^sub>R (-x)^n) = (cos_coeff n *\<^sub>R x^n)"
```
```  2811     by (auto simp: Transcendental.cos_coeff_def elim!: evenE)
```
```  2812   show ?thesis
```
```  2813     by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel, THEN summable_sums])
```
```  2814 qed
```
```  2815
```
```  2816 lemma cos_minus [simp]:
```
```  2817   fixes x :: "'a::{real_normed_algebra_1,banach}"
```
```  2818   shows "cos (-x) = cos(x)"
```
```  2819 using cos_minus_converges [of x]
```
```  2820 by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel]
```
```  2821               suminf_minus sums_iff equation_minus_iff)
```
```  2822
```
```  2823 lemma sin_cos_squared_add [simp]:
```
```  2824   fixes x :: "'a::{real_normed_field,banach}"
```
```  2825   shows "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
```
```  2826 using cos_add [of x "-x"]
```
```  2827 by (simp add: power2_eq_square algebra_simps)
```
```  2828
```
```  2829 lemma sin_cos_squared_add2 [simp]:
```
```  2830   fixes x :: "'a::{real_normed_field,banach}"
```
```  2831   shows "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
```
```  2832   by (subst add.commute, rule sin_cos_squared_add)
```
```  2833
```
```  2834 lemma sin_cos_squared_add3 [simp]:
```
```  2835   fixes x :: "'a::{real_normed_field,banach}"
```
```  2836   shows "cos x * cos x + sin x * sin x = 1"
```
```  2837   using sin_cos_squared_add2 [unfolded power2_eq_square] .
```
```  2838
```
```  2839 lemma sin_squared_eq:
```
```  2840   fixes x :: "'a::{real_normed_field,banach}"
```
```  2841   shows "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
```
```  2842   unfolding eq_diff_eq by (rule sin_cos_squared_add)
```
```  2843
```
```  2844 lemma cos_squared_eq:
```
```  2845   fixes x :: "'a::{real_normed_field,banach}"
```
```  2846   shows "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
```
```  2847   unfolding eq_diff_eq by (rule sin_cos_squared_add2)
```
```  2848
```
```  2849 lemma abs_sin_le_one [simp]:
```
```  2850   fixes x :: real
```
```  2851   shows "\<bar>sin x\<bar> \<le> 1"
```
```  2852   by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
```
```  2853
```
```  2854 lemma sin_ge_minus_one [simp]:
```
```  2855   fixes x :: real
```
```  2856   shows "-1 \<le> sin x"
```
```  2857   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  2858
```
```  2859 lemma sin_le_one [simp]:
```
```  2860   fixes x :: real
```
```  2861   shows "sin x \<le> 1"
```
```  2862   using abs_sin_le_one [of x] unfolding abs_le_iff by simp
```
```  2863
```
```  2864 lemma abs_cos_le_one [simp]:
```
```  2865   fixes x :: real
```
```  2866   shows "\<bar>cos x\<bar> \<le> 1"
```
```  2867   by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
```
```  2868
```
```  2869 lemma cos_ge_minus_one [simp]:
```
```  2870   fixes x :: real
```
```  2871   shows "-1 \<le> cos x"
```
```  2872   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  2873
```
```  2874 lemma cos_le_one [simp]:
```
```  2875   fixes x :: real
```
```  2876   shows "cos x \<le> 1"
```
```  2877   using abs_cos_le_one [of x] unfolding abs_le_iff by simp
```
```  2878
```
```  2879 lemma cos_diff:
```
```  2880   fixes x :: "'a::{real_normed_field,banach}"
```
```  2881   shows "cos (x - y) = cos x * cos y + sin x * sin y"
```
```  2882   using cos_add [of x "- y"] by simp
```
```  2883
```
```  2884 lemma cos_double:
```
```  2885   fixes x :: "'a::{real_normed_field,banach}"
```
```  2886   shows "cos(2*x) = (cos x)\<^sup>2 - (sin x)\<^sup>2"
```
```  2887   using cos_add [where x=x and y=x]
```
```  2888   by (simp add: power2_eq_square)
```
```  2889
```
```  2890 lemma DERIV_fun_pow: "DERIV g x :> m ==>
```
```  2891       DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
```
```  2892   by (auto intro!: derivative_eq_intros simp: real_of_nat_def)
```
```  2893
```
```  2894 lemma DERIV_fun_exp:
```
```  2895      "DERIV g x :> m ==> DERIV (\<lambda>x. exp(g x)) x :> exp(g x) * m"
```
```  2896   by (auto intro!: derivative_intros)
```
```  2897
```
```  2898 subsection {* The Constant Pi *}
```
```  2899
```
```  2900 definition pi :: real
```
```  2901   where "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
```
```  2902
```
```  2903 text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
```
```  2904    hence define pi.*}
```
```  2905
```
```  2906 lemma sin_paired:
```
```  2907   fixes x :: real
```
```  2908   shows "(\<lambda>n. (- 1) ^ n / (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums  sin x"
```
```  2909 proof -
```
```  2910   have "(\<lambda>n. \<Sum>k = n*2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
```
```  2911     apply (rule sums_group)
```
```  2912     using sin_converges [of x, unfolded scaleR_conv_of_real]
```
```  2913     by auto
```
```  2914   thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: ac_simps)
```
```  2915 qed
```
```  2916
```
```  2917 lemma sin_gt_zero_02:
```
```  2918   fixes x :: real
```
```  2919   assumes "0 < x" and "x < 2"
```
```  2920   shows "0 < sin x"
```
```  2921 proof -
```
```  2922   let ?f = "\<lambda>n::nat. \<Sum>k = n*2..<n*2+2. (- 1) ^ k / (fact (2*k+1)) * x^(2*k+1)"
```
```  2923   have pos: "\<forall>n. 0 < ?f n"
```
```  2924   proof
```
```  2925     fix n :: nat
```
```  2926     let ?k2 = "real (Suc (Suc (4 * n)))"
```
```  2927     let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
```
```  2928     have "x * x < ?k2 * ?k3"
```
```  2929       using assms by (intro mult_strict_mono', simp_all)
```
```  2930     hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
```
```  2931       by (intro mult_strict_right_mono zero_less_power `0 < x`)
```
```  2932     thus "0 < ?f n"
```
```  2933       by (simp add: real_of_nat_def divide_simps mult_ac del: mult_Suc)
```
```  2934 qed
```
```  2935   have sums: "?f sums sin x"
```
```  2936     by (rule sin_paired [THEN sums_group], simp)
```
```  2937   show "0 < sin x"
```
```  2938     unfolding sums_unique [OF sums]
```
```  2939     using sums_summable [OF sums] pos
```
```  2940     by (rule suminf_pos)
```
```  2941 qed
```
```  2942
```
```  2943 lemma cos_double_less_one:
```
```  2944   fixes x :: real
```
```  2945   shows "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
```
```  2946   using sin_gt_zero_02 [where x = x] by (auto simp: cos_squared_eq cos_double)
```
```  2947
```
```  2948 lemma cos_paired:
```
```  2949   fixes x :: real
```
```  2950   shows "(\<lambda>n. (- 1) ^ n / (fact (2 * n)) * x ^ (2 * n)) sums cos x"
```
```  2951 proof -
```
```  2952   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
```
```  2953     apply (rule sums_group)
```
```  2954     using cos_converges [of x, unfolded scaleR_conv_of_real]
```
```  2955     by auto
```
```  2956   thus ?thesis unfolding cos_coeff_def by (simp add: ac_simps)
```
```  2957 qed
```
```  2958
```
```  2959 lemmas realpow_num_eq_if = power_eq_if
```
```  2960
```
```  2961 lemma sumr_pos_lt_pair:  (*FIXME A MESS, BUT THE REAL MESS IS THE NEXT THEOREM*)
```
```  2962   fixes f :: "nat \<Rightarrow> real"
```
```  2963   shows "\<lbrakk>summable f;
```
```  2964         \<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk>
```
```  2965       \<Longrightarrow> setsum f {..<k} < suminf f"
```
```  2966 unfolding One_nat_def
```
```  2967 apply (subst suminf_split_initial_segment [where k=k], assumption, simp)
```
```  2968 apply (drule_tac k=k in summable_ignore_initial_segment)
```
```  2969 apply (drule_tac k="Suc (Suc 0)" in sums_group [OF summable_sums], simp)
```
```  2970 apply simp
```
```  2971 apply (frule sums_unique)
```
```  2972 apply (drule sums_summable, simp)
```
```  2973 apply (erule suminf_pos)
```
```  2974 apply (simp add: ac_simps)
```
```  2975 done
```
```  2976
```
```  2977 lemma cos_two_less_zero [simp]:
```
```  2978   "cos 2 < (0::real)"
```
```  2979 proof -
```
```  2980   note fact.simps(2) [simp del]
```
```  2981   from sums_minus [OF cos_paired]
```
```  2982   have *: "(\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / fact (2 * n))) sums - cos (2::real)"
```
```  2983     by simp
```
```  2984   then have **: "summable (\<lambda>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  2985     by (rule sums_summable)
```
```  2986   have "0 < (\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  2987     by (simp add: fact_num_eq_if realpow_num_eq_if)
```
```  2988   moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n  * 2 ^ (2 * n) / (fact (2 * n))))
```
```  2989     < (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  2990   proof -
```
```  2991     { fix d
```
```  2992       have "(4::real) * (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))
```
```  2993             < (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))) *
```
```  2994               fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))"
```
```  2995         unfolding real_of_nat_mult
```
```  2996         by (rule mult_strict_mono) (simp_all add: fact_less_mono)
```
```  2997       then have "(4::real) * (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))
```
```  2998         <  (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))))"
```
```  2999         by (simp only: fact.simps(2) [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"] real_of_nat_def of_nat_mult of_nat_fact)
```
```  3000       then have "(4::real) * inverse (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))))
```
```  3001         < inverse (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))"
```
```  3002         by (simp add: inverse_eq_divide less_divide_eq)
```
```  3003     }
```
```  3004     note *** = this
```
```  3005     have [simp]: "\<And>x y::real. 0 < x - y \<longleftrightarrow> y < x" by arith
```
```  3006     from ** show ?thesis by (rule sumr_pos_lt_pair)
```
```  3007       (simp add: divide_inverse mult.assoc [symmetric] ***)
```
```  3008   qed
```
```  3009   ultimately have "0 < (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3010     by (rule order_less_trans)
```
```  3011   moreover from * have "- cos 2 = (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
```
```  3012     by (rule sums_unique)
```
```  3013   ultimately have "(0::real) < - cos 2" by simp
```
```  3014   then show ?thesis by simp
```
```  3015 qed
```
```  3016
```
```  3017 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
```
```  3018 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
```
```  3019
```
```  3020 lemma cos_is_zero: "EX! x::real. 0 \<le> x & x \<le> 2 \<and> cos x = 0"
```
```  3021 proof (rule ex_ex1I)
```
```  3022   show "\<exists>x::real. 0 \<le> x & x \<le> 2 & cos x = 0"
```
```  3023     by (rule IVT2, simp_all)
```
```  3024 next
```
```  3025   fix x::real and y::real
```
```  3026   assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
```
```  3027   assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
```
```  3028   have [simp]: "\<forall>x::real. cos differentiable (at x)"
```
```  3029     unfolding real_differentiable_def by (auto intro: DERIV_cos)
```
```  3030   from x y show "x = y"
```
```  3031     apply (cut_tac less_linear [of x y], auto)
```
```  3032     apply (drule_tac f = cos in Rolle)
```
```  3033     apply (drule_tac [5] f = cos in Rolle)
```
```  3034     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  3035     apply (metis order_less_le_trans less_le sin_gt_zero_02)
```
```  3036     apply (metis order_less_le_trans less_le sin_gt_zero_02)
```
```  3037     done
```
```  3038 qed
```
```  3039
```
```  3040 lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
```
```  3041   by (simp add: pi_def)
```
```  3042
```
```  3043 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
```
```  3044   by (simp add: pi_half cos_is_zero [THEN theI'])
```
```  3045
```
```  3046 lemma cos_of_real_pi_half [simp]:
```
```  3047   fixes x :: "'a :: {real_field,banach,real_normed_algebra_1}"
```
```  3048   shows "cos ((of_real pi / 2) :: 'a) = 0"
```
```  3049 by (metis cos_pi_half cos_of_real eq_numeral_simps(4) nonzero_of_real_divide of_real_0 of_real_numeral)
```
```  3050
```
```  3051 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
```
```  3052   apply (rule order_le_neq_trans)
```
```  3053   apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  3054   apply (metis cos_pi_half cos_zero zero_neq_one)
```
```  3055   done
```
```  3056
```
```  3057 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
```
```  3058 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
```
```  3059
```
```  3060 lemma pi_half_less_two [simp]: "pi / 2 < 2"
```
```  3061   apply (rule order_le_neq_trans)
```
```  3062   apply (simp add: pi_half cos_is_zero [THEN theI'])
```
```  3063   apply (metis cos_pi_half cos_two_neq_zero)
```
```  3064   done
```
```  3065
```
```  3066 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
```
```  3067 lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
```
```  3068
```
```  3069 lemma pi_gt_zero [simp]: "0 < pi"
```
```  3070   using pi_half_gt_zero by simp
```
```  3071
```
```  3072 lemma pi_ge_zero [simp]: "0 \<le> pi"
```
```  3073   by (rule pi_gt_zero [THEN order_less_imp_le])
```
```  3074
```
```  3075 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
```
```  3076   by (rule pi_gt_zero [THEN less_imp_neq, symmetric])
```
```  3077
```
```  3078 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
```
```  3079   by (simp add: linorder_not_less)
```
```  3080
```
```  3081 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
```
```  3082   by simp
```
```  3083
```
```  3084 lemma m2pi_less_pi: "- (2*pi) < pi"
```
```  3085   by simp
```
```  3086
```
```  3087 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
```
```  3088   using sin_cos_squared_add2 [where x = "pi/2"]
```
```  3089   using sin_gt_zero_02 [OF pi_half_gt_zero pi_half_less_two]
```
```  3090   by (simp add: power2_eq_1_iff)
```
```  3091
```
```  3092 lemma sin_of_real_pi_half [simp]:
```
```  3093   fixes x :: "'a :: {real_field,banach,real_normed_algebra_1}"
```
```  3094   shows "sin ((of_real pi / 2) :: 'a) = 1"
```
```  3095   using sin_pi_half
```
```  3096 by (metis sin_pi_half eq_numeral_simps(4) nonzero_of_real_divide of_real_1 of_real_numeral sin_of_real)
```
```  3097
```
```  3098 lemma sin_cos_eq:
```
```  3099   fixes x :: "'a::{real_normed_field,banach}"
```
```  3100   shows "sin x = cos (of_real pi / 2 - x)"
```
```  3101   by (simp add: cos_diff)
```
```  3102
```
```  3103 lemma minus_sin_cos_eq:
```
```  3104   fixes x :: "'a::{real_normed_field,banach}"
```
```  3105   shows "-sin x = cos (x + of_real pi / 2)"
```
```  3106   by (simp add: cos_add nonzero_of_real_divide)
```
```  3107
```
```  3108 lemma cos_sin_eq:
```
```  3109   fixes x :: "'a::{real_normed_field,banach}"
```
```  3110   shows "cos x = sin (of_real pi / 2 - x)"
```
```  3111   using sin_cos_eq [of "of_real pi / 2 - x"]
```
```  3112   by simp
```
```  3113
```
```  3114 lemma sin_add:
```
```  3115   fixes x :: "'a::{real_normed_field,banach}"
```
```  3116   shows "sin (x + y) = sin x * cos y + cos x * sin y"
```
```  3117   using cos_add [of "of_real pi / 2 - x" "-y"]
```
```  3118   by (simp add: cos_sin_eq) (simp add: sin_cos_eq)
```
```  3119
```
```  3120 lemma sin_diff:
```
```  3121   fixes x :: "'a::{real_normed_field,banach}"
```
```  3122   shows "sin (x - y) = sin x * cos y - cos x * sin y"
```
```  3123   using sin_add [of x "- y"] by simp
```
```  3124
```
```  3125 lemma sin_double:
```
```  3126   fixes x :: "'a::{real_normed_field,banach}"
```
```  3127   shows "sin(2 * x) = 2 * sin x * cos x"
```
```  3128   using sin_add [where x=x and y=x] by simp
```
```  3129
```
```  3130
```
```  3131 lemma cos_of_real_pi [simp]: "cos (of_real pi) = -1"
```
```  3132   using cos_add [where x = "pi/2" and y = "pi/2"]
```
```  3133   by (simp add: cos_of_real)
```
```  3134
```
```  3135 lemma sin_of_real_pi [simp]: "sin (of_real pi) = 0"
```
```  3136   using sin_add [where x = "pi/2" and y = "pi/2"]
```
```  3137   by (simp add: sin_of_real)
```
```  3138
```
```  3139 lemma cos_pi [simp]: "cos pi = -1"
```
```  3140   using cos_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  3141
```
```  3142 lemma sin_pi [simp]: "sin pi = 0"
```
```  3143   using sin_add [where x = "pi/2" and y = "pi/2"] by simp
```
```  3144
```
```  3145 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
```
```  3146   by (simp add: sin_add)
```
```  3147
```
```  3148 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
```
```  3149   by (simp add: sin_add)
```
```  3150
```
```  3151 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
```
```  3152   by (simp add: cos_add)
```
```  3153
```
```  3154 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
```
```  3155   by (simp add: sin_add sin_double cos_double)
```
```  3156
```
```  3157 lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
```
```  3158   by (simp add: cos_add sin_double cos_double)
```
```  3159
```
```  3160 lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n"
```
```  3161   by (induct n) (auto simp: real_of_nat_Suc distrib_right)
```
```  3162
```
```  3163 lemma cos_npi2 [simp]: "cos (pi * real n) = (- 1) ^ n"
```
```  3164   by (metis cos_npi mult.commute)
```
```  3165
```
```  3166 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
```
```  3167   by (induct n) (auto simp: real_of_nat_Suc distrib_right)
```
```  3168
```
```  3169 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
```
```  3170   by (simp add: mult.commute [of pi])
```
```  3171
```
```  3172 lemma cos_two_pi [simp]: "cos (2*pi) = 1"
```
```  3173   by (simp add: cos_double)
```
```  3174
```
```  3175 lemma sin_two_pi [simp]: "sin (2*pi) = 0"
```
```  3176   by (simp add: sin_double)
```
```  3177
```
```  3178
```
```  3179 lemma sin_times_sin:
```
```  3180   fixes w :: "'a::{real_normed_field,banach}"
```
```  3181   shows "sin(w) * sin(z) = (cos(w - z) - cos(w + z)) / 2"
```
```  3182   by (simp add: cos_diff cos_add)
```
```  3183
```
```  3184 lemma sin_times_cos:
```
```  3185   fixes w :: "'a::{real_normed_field,banach}"
```
```  3186   shows "sin(w) * cos(z) = (sin(w + z) + sin(w - z)) / 2"
```
```  3187   by (simp add: sin_diff sin_add)
```
```  3188
```
```  3189 lemma cos_times_sin:
```
```  3190   fixes w :: "'a::{real_normed_field,banach}"
```
```  3191   shows "cos(w) * sin(z) = (sin(w + z) - sin(w - z)) / 2"
```
```  3192   by (simp add: sin_diff sin_add)
```
```  3193
```
```  3194 lemma cos_times_cos:
```
```  3195   fixes w :: "'a::{real_normed_field,banach}"
```
```  3196   shows "cos(w) * cos(z) = (cos(w - z) + cos(w + z)) / 2"
```
```  3197   by (simp add: cos_diff cos_add)
```
```  3198
```
```  3199 lemma sin_plus_sin:  (*FIXME field should not be necessary*)
```
```  3200   fixes w :: "'a::{real_normed_field,banach,field}"
```
```  3201   shows "sin(w) + sin(z) = 2 * sin((w + z) / 2) * cos((w - z) / 2)"
```
```  3202   apply (simp add: mult.assoc sin_times_cos)
```
```  3203   apply (simp add: field_simps)
```
```  3204   done
```
```  3205
```
```  3206 lemma sin_diff_sin:
```
```  3207   fixes w :: "'a::{real_normed_field,banach,field}"
```
```  3208   shows "sin(w) - sin(z) = 2 * sin((w - z) / 2) * cos((w + z) / 2)"
```
```  3209   apply (simp add: mult.assoc sin_times_cos)
```
```  3210   apply (simp add: field_simps)
```
```  3211   done
```
```  3212
```
```  3213 lemma cos_plus_cos:
```
```  3214   fixes w :: "'a::{real_normed_field,banach,field}"
```
```  3215   shows "cos(w) + cos(z) = 2 * cos((w + z) / 2) * cos((w - z) / 2)"
```
```  3216   apply (simp add: mult.assoc cos_times_cos)
```
```  3217   apply (simp add: field_simps)
```
```  3218   done
```
```  3219
```
```  3220 lemma cos_diff_cos:
```
```  3221   fixes w :: "'a::{real_normed_field,banach,field}"
```
```  3222   shows "cos(w) - cos(z) = 2 * sin((w + z) / 2) * sin((z - w) / 2)"
```
```  3223   apply (simp add: mult.assoc sin_times_sin)
```
```  3224   apply (simp add: field_simps)
```
```  3225   done
```
```  3226
```
```  3227 lemma cos_double_cos:
```
```  3228   fixes z :: "'a::{real_normed_field,banach}"
```
```  3229   shows "cos(2 * z) = 2 * cos z ^ 2 - 1"
```
```  3230 by (simp add: cos_double sin_squared_eq)
```
```  3231
```
```  3232 lemma cos_double_sin:
```
```  3233   fixes z :: "'a::{real_normed_field,banach}"
```
```  3234   shows "cos(2 * z) = 1 - 2 * sin z ^ 2"
```
```  3235 by (simp add: cos_double sin_squared_eq)
```
```  3236
```
```  3237 lemma sin_pi_minus [simp]: "sin (pi - x) = sin x"
```
```  3238   by (metis sin_minus sin_periodic_pi minus_minus uminus_add_conv_diff)
```
```  3239
```
```  3240 lemma cos_pi_minus [simp]: "cos (pi - x) = -(cos x)"
```
```  3241   by (metis cos_minus cos_periodic_pi uminus_add_conv_diff)
```
```  3242
```
```  3243 lemma sin_minus_pi [simp]: "sin (x - pi) = - (sin x)"
```
```  3244   by (simp add: sin_diff)
```
```  3245
```
```  3246 lemma cos_minus_pi [simp]: "cos (x - pi) = -(cos x)"
```
```  3247   by (simp add: cos_diff)
```
```  3248
```
```  3249 lemma sin_2pi_minus [simp]: "sin (2*pi - x) = -(sin x)"
```
```  3250   by (metis sin_periodic_pi2 add_diff_eq mult_2 sin_pi_minus)
```
```  3251
```
```  3252 lemma cos_2pi_minus [simp]: "cos (2*pi - x) = cos x"
```
```  3253   by (metis (no_types, hide_lams) cos_add cos_minus cos_two_pi sin_minus sin_two_pi
```
```  3254            diff_0_right minus_diff_eq mult_1 mult_zero_left uminus_add_conv_diff)
```
```  3255
```
```  3256 lemma sin_gt_zero2: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < sin x"
```
```  3257   by (metis sin_gt_zero_02 order_less_trans pi_half_less_two)
```
```  3258
```
```  3259 lemma sin_less_zero:
```
```  3260   assumes "- pi/2 < x" and "x < 0"
```
```  3261   shows "sin x < 0"
```
```  3262 proof -
```
```  3263   have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
```
```  3264   thus ?thesis by simp
```
```  3265 qed
```
```  3266
```
```  3267 lemma pi_less_4: "pi < 4"
```
```  3268   using pi_half_less_two by auto
```
```  3269
```
```  3270 lemma cos_gt_zero: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < cos x"
```
```  3271   by (simp add: cos_sin_eq sin_gt_zero2)
```
```  3272
```
```  3273 lemma cos_gt_zero_pi: "\<lbrakk>-(pi/2) < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < cos x"
```
```  3274   using cos_gt_zero [of x] cos_gt_zero [of "-x"]
```
```  3275   by (cases rule: linorder_cases [of x 0]) auto
```
```  3276
```
```  3277 lemma cos_ge_zero: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2\<rbrakk> \<Longrightarrow> 0 \<le> cos x"
```
```  3278   apply (auto simp: order_le_less cos_gt_zero_pi)
```
```  3279   by (metis cos_pi_half eq_divide_eq eq_numeral_simps(4))
```
```  3280
```
```  3281 lemma sin_gt_zero: "\<lbrakk>0 < x; x < pi \<rbrakk> \<Longrightarrow> 0 < sin x"
```
```  3282   by (simp add: sin_cos_eq cos_gt_zero_pi)
```
```  3283
```
```  3284 lemma sin_lt_zero: "pi < x \<Longrightarrow> x < 2*pi \<Longrightarrow> sin x < 0"
```
```  3285   using sin_gt_zero [of "x-pi"]
```
```  3286   by (simp add: sin_diff)
```
```  3287
```
```  3288 lemma pi_ge_two: "2 \<le> pi"
```
```  3289 proof (rule ccontr)
```
```  3290   assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
```
```  3291   have "\<exists>y > pi. y < 2 \<and> y < 2*pi"
```
```  3292   proof (cases "2 < 2*pi")
```
```  3293     case True with dense[OF `pi < 2`] show ?thesis by auto
```
```  3294   next
```
```  3295     case False have "pi < 2*pi" by auto
```
```  3296     from dense[OF this] and False show ?thesis by auto
```
```  3297   qed
```
```  3298   then obtain y where "pi < y" and "y < 2" and "y < 2*pi" by blast
```
```  3299   hence "0 < sin y" using sin_gt_zero_02 by auto
```
```  3300   moreover
```
```  3301   have "sin y < 0" using sin_gt_zero[of "y - pi"] `pi < y` and `y < 2*pi` sin_periodic_pi[of "y - pi"] by auto
```
```  3302   ultimately show False by auto
```
```  3303 qed
```
```  3304
```
```  3305 lemma sin_ge_zero: "\<lbrakk>0 \<le> x; x \<le> pi\<rbrakk> \<Longrightarrow> 0 \<le> sin x"
```
```  3306   by (auto simp: order_le_less sin_gt_zero)
```
```  3307
```
```  3308 lemma sin_le_zero: "pi \<le> x \<Longrightarrow> x < 2*pi \<Longrightarrow> sin x \<le> 0"
```
```  3309   using sin_ge_zero [of "x-pi"]
```
```  3310   by (simp add: sin_diff)
```
```  3311
```
```  3312 text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}.
```
```  3313   It should be possible to factor out some of the common parts. *}
```
```  3314
```
```  3315 lemma cos_total: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
```
```  3316 proof (rule ex_ex1I)
```
```  3317   assume y: "-1 \<le> y" "y \<le> 1"
```
```  3318   show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
```
```  3319     by (rule IVT2, simp_all add: y)
```
```  3320 next
```
```  3321   fix a b
```
```  3322   assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
```
```  3323   assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
```
```  3324   have [simp]: "\<forall>x::real. cos differentiable (at x)"
```
```  3325     unfolding real_differentiable_def by (auto intro: DERIV_cos)
```
```  3326   from a b show "a = b"
```
```  3327     apply (cut_tac less_linear [of a b], auto)
```
```  3328     apply (drule_tac f = cos in Rolle)
```
```  3329     apply (drule_tac [5] f = cos in Rolle)
```
```  3330     apply (auto dest!: DERIV_cos [THEN DERIV_unique])
```
```  3331     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  3332     apply (metis order_less_le_trans less_le sin_gt_zero)
```
```  3333     done
```
```  3334 qed
```
```  3335
```
```  3336 lemma sin_total:
```
```  3337   assumes y: "-1 \<le> y" "y \<le> 1"
```
```  3338     shows "\<exists>! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
```
```  3339 proof -
```
```  3340   from cos_total [OF y]
```
```  3341   obtain x where x: "0 \<le> x" "x \<le> pi" "cos x = y"
```
```  3342            and uniq: "\<And>x'. 0 \<le> x' \<Longrightarrow> x' \<le> pi \<Longrightarrow> cos x' = y \<Longrightarrow> x' = x "
```
```  3343     by blast
```
```  3344   show ?thesis
```
```  3345     apply (simp add: sin_cos_eq)
```
```  3346     apply (rule ex1I [where a="pi/2 - x"])
```
```  3347     apply (cut_tac [2] x'="pi/2 - xa" in uniq)
```
```  3348     using x
```
```  3349     apply auto
```
```  3350     done
```
```  3351 qed
```
```  3352
```
```  3353 lemma reals_Archimedean4:
```
```  3354      "\<lbrakk>0 < y; 0 \<le> x\<rbrakk> \<Longrightarrow> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
```
```  3355 apply (auto dest!: reals_Archimedean3)
```
```  3356 apply (drule_tac x = x in spec, clarify)
```
```  3357 apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
```
```  3358  prefer 2 apply (erule LeastI)
```
```  3359 apply (case_tac "LEAST m::nat. x < real m * y", simp)
```
```  3360 apply (rename_tac m)
```
```  3361 apply (subgoal_tac "~ x < real m * y")
```
```  3362  prefer 2 apply (rule not_less_Least, simp, force)
```
```  3363 done
```
```  3364
```
```  3365 lemma cos_zero_lemma:
```
```  3366      "\<lbrakk>0 \<le> x; cos x = 0\<rbrakk> \<Longrightarrow>
```
```  3367       \<exists>n::nat. odd n & x = real n * (pi/2)"
```
```  3368 apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
```
```  3369 apply (subgoal_tac "0 \<le> x - real n * pi &
```
```  3370                     (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
```
```  3371 apply (auto simp: algebra_simps real_of_nat_Suc)
```
```  3372  prefer 2 apply (simp add: cos_diff)
```
```  3373 apply (simp add: cos_diff)
```
```  3374 apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
```
```  3375 apply (rule_tac [2] cos_total, safe)
```
```  3376 apply (drule_tac x = "x - real n * pi" in spec)
```
```  3377 apply (drule_tac x = "pi/2" in spec)
```
```  3378 apply (simp add: cos_diff)
```
```  3379 apply (rule_tac x = "Suc (2 * n)" in exI)
```
```  3380 apply (simp add: real_of_nat_Suc algebra_simps, auto)
```
```  3381 done
```
```  3382
```
```  3383 lemma sin_zero_lemma:
```
```  3384      "\<lbrakk>0 \<le> x; sin x = 0\<rbrakk> \<Longrightarrow>
```
```  3385       \<exists>n::nat. even n & x = real n * (pi/2)"
```
```  3386 apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
```
```  3387  apply (clarify, rule_tac x = "n - 1" in exI)
```
```  3388  apply (auto elim!: oddE simp add: real_of_nat_Suc field_simps)[1]
```
```  3389  apply (rule cos_zero_lemma)
```
```  3390  apply (auto simp: cos_add)
```
```  3391 done
```
```  3392
```
```  3393 lemma cos_zero_iff:
```
```  3394      "(cos x = 0) =
```
```  3395       ((\<exists>n::nat. odd n & (x = real n * (pi/2))) |
```
```  3396        (\<exists>n::nat. odd n & (x = -(real n * (pi/2)))))"
```
```  3397 proof -
```
```  3398   { fix n :: nat
```
```  3399     assume "odd n"
```
```  3400     then obtain m where "n = 2 * m + 1" ..
```
```  3401     then have "cos (real n * pi / 2) = 0"
```
```  3402       by (simp add: field_simps real_of_nat_Suc) (simp add: cos_add add_divide_distrib)
```
```  3403   } note * = this
```
```  3404   show ?thesis
```
```  3405   apply (rule iffI)
```
```  3406   apply (cut_tac linorder_linear [of 0 x], safe)
```
```  3407   apply (drule cos_zero_lemma, assumption+)
```
```  3408   apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
```
```  3409   apply (auto dest: *)
```
```  3410   done
```
```  3411 qed
```
```  3412
```
```  3413 (* ditto: but to a lesser extent *)
```
```  3414 lemma sin_zero_iff:
```
```  3415      "(sin x = 0) =
```
```  3416       ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
```
```  3417        (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
```
```  3418 apply (rule iffI)
```
```  3419 apply (cut_tac linorder_linear [of 0 x], safe)
```
```  3420 apply (drule sin_zero_lemma, assumption+)
```
```  3421 apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
```
```  3422 apply (force simp add: minus_equation_iff [of x])
```
```  3423 apply (auto elim: evenE)
```
```  3424 done
```
```  3425
```
```  3426
```
```  3427 lemma cos_zero_iff_int:
```
```  3428      "cos x = 0 \<longleftrightarrow> (\<exists>n::int. odd n & x = real n * (pi/2))"
```
```  3429 proof safe
```
```  3430   assume "cos x = 0"
```
```  3431   then show "\<exists>n::int. odd n & x = real n * (pi/2)"
```
```  3432     apply (simp add: cos_zero_iff, safe)
```
```  3433     apply (metis even_int_iff real_of_int_of_nat_eq)
```
```  3434     apply (rule_tac x="- (int n)" in exI, simp)
```
```  3435     done
```
```  3436 next
```
```  3437   fix n::int
```
```  3438   assume "odd n"
```
```  3439   then show "cos (real n * (pi / 2)) = 0"
```
```  3440     apply (simp add: cos_zero_iff)
```
```  3441     apply (case_tac n rule: int_cases2, simp)
```
```  3442     apply (rule disjI2)
```
```  3443     apply (rule_tac x="nat (-n)" in exI, simp)
```
```  3444     done
```
```  3445 qed
```
```  3446
```
```  3447 lemma sin_zero_iff_int:
```
```  3448      "sin x = 0 \<longleftrightarrow> (\<exists>n::int. even n & (x = real n * (pi/2)))"
```
```  3449 proof safe
```
```  3450   assume "sin x = 0"
```
```  3451   then show "\<exists>n::int. even n \<and> x = real n * (pi / 2)"
```
```  3452     apply (simp add: sin_zero_iff, safe)
```
```  3453     apply (metis even_int_iff real_of_int_of_nat_eq)
```
```  3454     apply (rule_tac x="- (int n)" in exI, simp)
```
```  3455     done
```
```  3456 next
```
```  3457   fix n::int
```
```  3458   assume "even n"
```
```  3459   then show "sin (real n * (pi / 2)) = 0"
```
```  3460     apply (simp add: sin_zero_iff)
```
```  3461     apply (case_tac n rule: int_cases2, simp)
```
```  3462     apply (rule disjI2)
```
```  3463     apply (rule_tac x="nat (-n)" in exI, simp)
```
```  3464     done
```
```  3465 qed
```
```  3466
```
```  3467 lemma sin_zero_iff_int2: "sin x = 0 \<longleftrightarrow> (\<exists>n::int. x = real n * pi)"
```
```  3468   apply (simp only: sin_zero_iff_int)
```
```  3469   apply (safe elim!: evenE)
```
```  3470   apply (simp_all add: field_simps)
```
```  3471   using dvd_triv_left by fastforce
```
```  3472
```
```  3473 lemma cos_monotone_0_pi:
```
```  3474   assumes "0 \<le> y" and "y < x" and "x \<le> pi"
```
```  3475   shows "cos x < cos y"
```
```  3476 proof -
```
```  3477   have "- (x - y) < 0" using assms by auto
```
```  3478
```
```  3479   from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
```
```  3480   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z"
```
```  3481     by auto
```
```  3482   hence "0 < z" and "z < pi" using assms by auto
```
```  3483   hence "0 < sin z" using sin_gt_zero by auto
```
```  3484   hence "cos x - cos y < 0"
```
```  3485     unfolding cos_diff minus_mult_commute[symmetric]
```
```  3486     using `- (x - y) < 0` by (rule mult_pos_neg2)
```
```  3487   thus ?thesis by auto
```
```  3488 qed
```
```  3489
```
```  3490 lemma cos_monotone_0_pi_le:
```
```  3491   assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi"
```
```  3492   shows "cos x \<le> cos y"
```
```  3493 proof (cases "y < x")
```
```  3494   case True
```
```  3495   show ?thesis
```
```  3496     using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
```
```  3497 next
```
```  3498   case False
```
```  3499   hence "y = x" using `y \<le> x` by auto
```
```  3500   thus ?thesis by auto
```
```  3501 qed
```
```  3502
```
```  3503 lemma cos_monotone_minus_pi_0:
```
```  3504   assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
```
```  3505   shows "cos y < cos x"
```
```  3506 proof -
```
```  3507   have "0 \<le> -x" and "-x < -y" and "-y \<le> pi"
```
```  3508     using assms by auto
```
```  3509   from cos_monotone_0_pi[OF this] show ?thesis
```
```  3510     unfolding cos_minus .
```
```  3511 qed
```
```  3512
```
```  3513 lemma cos_monotone_minus_pi_0':
```
```  3514   assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0"
```
```  3515   shows "cos y \<le> cos x"
```
```  3516 proof (cases "y < x")
```
```  3517   case True
```
```  3518   show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`]
```
```  3519     by auto
```
```  3520 next
```
```  3521   case False
```
```  3522   hence "y = x" using `y \<le> x` by auto
```
```  3523   thus ?thesis by auto
```
```  3524 qed
```
```  3525
```
```  3526 lemma sin_monotone_2pi:
```
```  3527   assumes "- (pi/2) \<le> y" and "y < x" and "x \<le> pi/2"
```
```  3528   shows "sin y < sin x"
```
```  3529     apply (simp add: sin_cos_eq)
```
```  3530     apply (rule cos_monotone_0_pi)
```
```  3531     using assms
```
```  3532     apply auto
```
```  3533     done
```
```  3534
```
```  3535 lemma sin_monotone_2pi_le:
```
```  3536   assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2"
```
```  3537   shows "sin y \<le> sin x"
```
```  3538   by (metis assms le_less sin_monotone_2pi)
```
```  3539
```
```  3540 lemma sin_x_le_x:
```
```  3541   fixes x::real assumes x: "x \<ge> 0" shows "sin x \<le> x"
```
```  3542 proof -
```
```  3543   let ?f = "\<lambda>x. x - sin x"
```
```  3544   from x have "?f x \<ge> ?f 0"
```
```  3545     apply (rule DERIV_nonneg_imp_nondecreasing)
```
```  3546     apply (intro allI impI exI[of _ "1 - cos x" for x])
```
```  3547     apply (auto intro!: derivative_eq_intros simp: field_simps)
```
```  3548     done
```
```  3549   thus "sin x \<le> x" by simp
```
```  3550 qed
```
```  3551
```
```  3552 lemma sin_x_ge_neg_x:
```
```  3553   fixes x::real assumes x: "x \<ge> 0" shows "sin x \<ge> - x"
```
```  3554 proof -
```
```  3555   let ?f = "\<lambda>x. x + sin x"
```
```  3556   from x have "?f x \<ge> ?f 0"
```
```  3557     apply (rule DERIV_nonneg_imp_nondecreasing)
```
```  3558     apply (intro allI impI exI[of _ "1 + cos x" for x])
```
```  3559     apply (auto intro!: derivative_eq_intros simp: field_simps real_0_le_add_iff)
```
```  3560     done
```
```  3561   thus "sin x \<ge> -x" by simp
```
```  3562 qed
```
```  3563
```
```  3564 lemma abs_sin_x_le_abs_x:
```
```  3565   fixes x::real shows "\<bar>sin x\<bar> \<le> \<bar>x\<bar>"
```
```  3566   using sin_x_ge_neg_x [of x] sin_x_le_x [of x] sin_x_ge_neg_x [of "-x"] sin_x_le_x [of "-x"]
```
```  3567   by (auto simp: abs_real_def)
```
```  3568
```
```  3569
```
```  3570 subsection {* More Corollaries about Sine and Cosine *}
```
```  3571
```
```  3572 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
```
```  3573 proof -
```
```  3574   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
```
```  3575     by (auto simp: algebra_simps sin_add)
```
```  3576   thus ?thesis
```
```  3577     by (simp add: real_of_nat_Suc distrib_right add_divide_distrib
```
```  3578                   mult.commute [of pi])
```
```  3579 qed
```
```  3580
```
```  3581 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
```
```  3582   by (cases "even n") (simp_all add: cos_double mult.assoc)
```
```  3583
```
```  3584 lemma cos_3over2_pi [simp]: "cos (3/2*pi) = 0"
```
```  3585   apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
```
```  3586   apply (subst cos_add, simp)
```
```  3587   done
```
```  3588
```
```  3589 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
```
```  3590   by (auto simp: mult.assoc sin_double)
```
```  3591
```
```  3592 lemma sin_3over2_pi [simp]: "sin (3/2*pi) = - 1"
```
```  3593   apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
```
```  3594   apply (subst sin_add, simp)
```
```  3595   done
```
```  3596
```
```  3597 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
```
```  3598 by (simp only: cos_add sin_add real_of_nat_Suc distrib_right distrib_left add_divide_distrib, auto)
```
```  3599
```
```  3600 lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)"
```
```  3601   by (auto intro!: derivative_eq_intros)
```
```  3602
```
```  3603 lemma sin_zero_norm_cos_one:
```
```  3604   fixes x :: "'a::{real_normed_field,banach}"
```
```  3605   assumes "sin x = 0" shows "norm (cos x) = 1"
```
```  3606   using sin_cos_squared_add [of x, unfolded assms]
```
```  3607   by (simp add: square_norm_one)
```
```  3608
```
```  3609 lemma sin_zero_abs_cos_one: "sin x = 0 \<Longrightarrow> \<bar>cos x\<bar> = (1::real)"
```
```  3610   using sin_zero_norm_cos_one by fastforce
```
```  3611
```
```  3612 lemma cos_one_sin_zero:
```
```  3613   fixes x :: "'a::{real_normed_field,banach}"
```
```  3614   assumes "cos x = 1" shows "sin x = 0"
```
```  3615   using sin_cos_squared_add [of x, unfolded assms]
```
```  3616   by simp
```
```  3617
```
```  3618 lemma sin_times_pi_eq_0: "sin(x * pi) = 0 \<longleftrightarrow> x \<in> Ints"
```
```  3619   by (simp add: sin_zero_iff_int2) (metis Ints_cases Ints_real_of_int real_of_int_def)
```
```  3620
```
```  3621 lemma cos_one_2pi:
```
```  3622     "cos(x) = 1 \<longleftrightarrow> (\<exists>n::nat. x = n * 2*pi) | (\<exists>n::nat. x = -(n * 2*pi))"
```
```  3623     (is "?lhs = ?rhs")
```
```  3624 proof
```
```  3625   assume "cos(x) = 1"
```
```  3626   then have "sin x = 0"
```
```  3627     by (simp add: cos_one_sin_zero)
```
```  3628   then show ?rhs
```
```  3629   proof (simp only: sin_zero_iff, elim exE disjE conjE)
```
```  3630     fix n::nat
```
```  3631     assume n: "even n" "x = real n * (pi/2)"
```
```  3632     then obtain m where m: "n = 2 * m"
```
```  3633       using dvdE by blast
```
```  3634     then have me: "even m" using `?lhs` n
```
```  3635       by (auto simp: field_simps) (metis one_neq_neg_one  power_minus_odd power_one)
```
```  3636     show ?rhs
```
```  3637       using m me n
```
```  3638       by (auto simp: field_simps elim!: evenE)
```
```  3639   next
```
```  3640     fix n::nat
```
```  3641     assume n: "even n" "x = - (real n * (pi/2))"
```
```  3642     then obtain m where m: "n = 2 * m"
```
```  3643       using dvdE by blast
```
```  3644     then have me: "even m" using `?lhs` n
```
```  3645       by (auto simp: field_simps) (metis one_neq_neg_one  power_minus_odd power_one)
```
```  3646     show ?rhs
```
```  3647       using m me n
```
```  3648       by (auto simp: field_simps elim!: evenE)
```
```  3649   qed
```
```  3650 next
```
```  3651   assume "?rhs"
```
```  3652   then show "cos x = 1"
```
```  3653     by (metis cos_2npi cos_minus mult.assoc mult.left_commute)
```
```  3654 qed
```
```  3655
```
```  3656 lemma cos_one_2pi_int: "cos(x) = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2*pi)"
```
```  3657   apply auto  --{*FIXME simproc bug*}
```
```  3658   apply (auto simp: cos_one_2pi)
```
```  3659   apply (metis real_of_int_of_nat_eq)
```
```  3660   apply (metis mult_minus_right real_of_int_minus real_of_int_of_nat_eq)
```
```  3661   by (metis mult_minus_right of_int_of_nat real_of_int_def real_of_nat_def)
```
```  3662
```
```  3663 lemma sin_cos_sqrt: "0 \<le> sin(x) \<Longrightarrow> (sin(x) = sqrt(1 - (cos(x) ^ 2)))"
```
```  3664   using sin_squared_eq real_sqrt_unique by fastforce
```
```  3665
```
```  3666 lemma sin_eq_0_pi: "-pi < x \<Longrightarrow> x < pi \<Longrightarrow> sin(x) = 0 \<Longrightarrow> x = 0"
```
```  3667   by (metis sin_gt_zero sin_minus minus_less_iff neg_0_less_iff_less not_less_iff_gr_or_eq)
```
```  3668
```
```  3669 lemma cos_treble_cos:
```
```  3670   fixes x :: "'a::{real_normed_field,banach}"
```
```  3671   shows "cos(3 * x) = 4 * cos(x) ^ 3 - 3 * cos x"
```
```  3672 proof -
```
```  3673   have *: "(sin x * (sin x * 3)) = 3 - (cos x * (cos x * 3))"
```
```  3674     by (simp add: mult.assoc [symmetric] sin_squared_eq [unfolded power2_eq_square])
```
```  3675   have "cos(3 * x) = cos(2*x + x)"
```
```  3676     by simp
```
```  3677   also have "... = 4 * cos(x) ^ 3 - 3 * cos x"
```
```  3678     apply (simp only: cos_add cos_double sin_double)
```
```  3679     apply (simp add: * field_simps power2_eq_square power3_eq_cube)
```
```  3680     done
```
```  3681   finally show ?thesis .
```
```  3682 qed
```
```  3683
```
```  3684 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
```
```  3685 proof -
```
```  3686   let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
```
```  3687   have nonneg: "0 \<le> ?c"
```
```  3688     by (simp add: cos_ge_zero)
```
```  3689   have "0 = cos (pi / 4 + pi / 4)"
```
```  3690     by simp
```
```  3691   also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2"
```
```  3692     by (simp only: cos_add power2_eq_square)
```
```  3693   also have "\<dots> = 2 * ?c\<^sup>2 - 1"
```
```  3694     by (simp add: sin_squared_eq)
```
```  3695   finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2"
```
```  3696     by (simp add: power_divide)
```
```  3697   thus ?thesis
```
```  3698     using nonneg by (rule power2_eq_imp_eq) simp
```
```  3699 qed
```
```  3700
```
```  3701 lemma cos_30: "cos (pi / 6) = sqrt 3/2"
```
```  3702 proof -
```
```  3703   let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
```
```  3704   have pos_c: "0 < ?c"
```
```  3705     by (rule cos_gt_zero, simp, simp)
```
```  3706   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
```
```  3707     by simp
```
```  3708   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
```
```  3709     by (simp only: cos_add sin_add)
```
```  3710   also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)"
```
```  3711     by (simp add: algebra_simps power2_eq_square)
```
```  3712   finally have "?c\<^sup>2 = (sqrt 3/2)\<^sup>2"
```
```  3713     using pos_c by (simp add: sin_squared_eq power_divide)
```
```  3714   thus ?thesis
```
```  3715     using pos_c [THEN order_less_imp_le]
```
```  3716     by (rule power2_eq_imp_eq) simp
```
```  3717 qed
```
```  3718
```
```  3719 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
```
```  3720   by (simp add: sin_cos_eq cos_45)
```
```  3721
```
```  3722 lemma sin_60: "sin (pi / 3) = sqrt 3/2"
```
```  3723   by (simp add: sin_cos_eq cos_30)
```
```  3724
```
```  3725 lemma cos_60: "cos (pi / 3) = 1 / 2"
```
```  3726   apply (rule power2_eq_imp_eq)
```
```  3727   apply (simp add: cos_squared_eq sin_60 power_divide)
```
```  3728   apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
```
```  3729   done
```
```  3730
```
```  3731 lemma sin_30: "sin (pi / 6) = 1 / 2"
```
```  3732   by (simp add: sin_cos_eq cos_60)
```
```  3733
```
```  3734 lemma cos_integer_2pi: "n \<in> Ints \<Longrightarrow> cos(2*pi * n) = 1"
```
```  3735   by (metis Ints_cases cos_one_2pi_int mult.assoc mult.commute real_of_int_def)
```
```  3736
```
```  3737 lemma sin_integer_2pi: "n \<in> Ints \<Longrightarrow> sin(2*pi * n) = 0"
```
```  3738   by (metis sin_two_pi Ints_mult mult.assoc mult.commute sin_times_pi_eq_0)
```
```  3739
```
```  3740 lemma cos_int_2npi [simp]: "cos (2 * real (n::int) * pi) = 1"
```
```  3741   by (simp add: cos_one_2pi_int)
```
```  3742
```
```  3743 lemma sin_int_2npi [simp]: "sin (2 * real (n::int) * pi) = 0"
```
```  3744   by (metis Ints_real_of_int mult.assoc mult.commute sin_integer_2pi)
```
```  3745
```
```  3746 lemma sincos_principal_value: "\<exists>y. (-pi < y \<and> y \<le> pi) \<and> (sin(y) = sin(x) \<and> cos(y) = cos(x))"
```
```  3747   apply (rule exI [where x="pi - (2*pi) * frac((pi - x) / (2*pi))"])
```
```  3748   apply (auto simp: field_simps frac_lt_1)
```
```  3749   apply (simp_all add: frac_def divide_simps)
```
```  3750   apply (simp_all add: add_divide_distrib diff_divide_distrib)
```
```  3751   apply (simp_all add: sin_diff cos_diff mult.assoc [symmetric] cos_integer_2pi sin_integer_2pi)
```
```  3752   done
```
```  3753
```
```  3754
```
```  3755 subsection {* Tangent *}
```
```  3756
```
```  3757 definition tan :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3758   where "tan = (\<lambda>x. sin x / cos x)"
```
```  3759
```
```  3760 lemma tan_of_real:
```
```  3761   fixes XXX :: "'a::{real_normed_field,banach}"
```
```  3762   shows  "of_real(tan x) = (tan(of_real x) :: 'a)"
```
```  3763   by (simp add: tan_def sin_of_real cos_of_real)
```
```  3764
```
```  3765 lemma tan_in_Reals [simp]:
```
```  3766   fixes z :: "'a::{real_normed_field,banach}"
```
```  3767   shows "z \<in> \<real> \<Longrightarrow> tan z \<in> \<real>"
```
```  3768   by (simp add: tan_def)
```
```  3769
```
```  3770 lemma tan_zero [simp]: "tan 0 = 0"
```
```  3771   by (simp add: tan_def)
```
```  3772
```
```  3773 lemma tan_pi [simp]: "tan pi = 0"
```
```  3774   by (simp add: tan_def)
```
```  3775
```
```  3776 lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
```
```  3777   by (simp add: tan_def)
```
```  3778
```
```  3779 lemma tan_minus [simp]: "tan (-x) = - tan x"
```
```  3780   by (simp add: tan_def)
```
```  3781
```
```  3782 lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
```
```  3783   by (simp add: tan_def)
```
```  3784
```
```  3785 lemma lemma_tan_add1:
```
```  3786   "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
```
```  3787   by (simp add: tan_def cos_add field_simps)
```
```  3788
```
```  3789 lemma add_tan_eq:
```
```  3790   fixes x :: "'a::{real_normed_field,banach}"
```
```  3791   shows "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
```
```  3792   by (simp add: tan_def sin_add field_simps)
```
```  3793
```
```  3794 lemma tan_add:
```
```  3795   fixes x :: "'a::{real_normed_field,banach}"
```
```  3796   shows
```
```  3797      "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0\<rbrakk>
```
```  3798       \<Longrightarrow> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
```
```  3799       by (simp add: add_tan_eq lemma_tan_add1 field_simps) (simp add: tan_def)
```
```  3800
```
```  3801 lemma tan_double:
```
```  3802   fixes x :: "'a::{real_normed_field,banach}"
```
```  3803   shows
```
```  3804      "\<lbrakk>cos x \<noteq> 0; cos (2 * x) \<noteq> 0\<rbrakk>
```
```  3805       \<Longrightarrow> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)"
```
```  3806   using tan_add [of x x] by (simp add: power2_eq_square)
```
```  3807
```
```  3808 lemma tan_gt_zero: "\<lbrakk>0 < x; x < pi/2\<rbrakk> \<Longrightarrow> 0 < tan x"
```
```  3809   by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
```
```  3810
```
```  3811 lemma tan_less_zero:
```
```  3812   assumes lb: "- pi/2 < x" and "x < 0"
```
```  3813   shows "tan x < 0"
```
```  3814 proof -
```
```  3815   have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
```
```  3816   thus ?thesis by simp
```
```  3817 qed
```
```  3818
```
```  3819 lemma tan_half:
```
```  3820   fixes x :: "'a::{real_normed_field,banach,field}"
```
```  3821   shows  "tan x = sin (2 * x) / (cos (2 * x) + 1)"
```
```  3822   unfolding tan_def sin_double cos_double sin_squared_eq
```
```  3823   by (simp add: power2_eq_square)
```
```  3824
```
```  3825 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
```
```  3826   unfolding tan_def by (simp add: sin_30 cos_30)
```
```  3827
```
```  3828 lemma tan_45: "tan (pi / 4) = 1"
```
```  3829   unfolding tan_def by (simp add: sin_45 cos_45)
```
```  3830
```
```  3831 lemma tan_60: "tan (pi / 3) = sqrt 3"
```
```  3832   unfolding tan_def by (simp add: sin_60 cos_60)
```
```  3833
```
```  3834 lemma DERIV_tan [simp]:
```
```  3835   fixes x :: "'a::{real_normed_field,banach}"
```
```  3836   shows "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)"
```
```  3837   unfolding tan_def
```
```  3838   by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square)
```
```  3839
```
```  3840 lemma isCont_tan:
```
```  3841   fixes x :: "'a::{real_normed_field,banach}"
```
```  3842   shows "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
```
```  3843   by (rule DERIV_tan [THEN DERIV_isCont])
```
```  3844
```
```  3845 lemma isCont_tan' [simp,continuous_intros]:
```
```  3846   fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
```
```  3847   shows "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
```
```  3848   by (rule isCont_o2 [OF _ isCont_tan])
```
```  3849
```
```  3850 lemma tendsto_tan [tendsto_intros]:
```
```  3851   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3852   shows "\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F"
```
```  3853   by (rule isCont_tendsto_compose [OF isCont_tan])
```
```  3854
```
```  3855 lemma continuous_tan:
```
```  3856   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3857   shows "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))"
```
```  3858   unfolding continuous_def by (rule tendsto_tan)
```
```  3859
```
```  3860 lemma continuous_on_tan [continuous_intros]:
```
```  3861   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3862   shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))"
```
```  3863   unfolding continuous_on_def by (auto intro: tendsto_tan)
```
```  3864
```
```  3865 lemma continuous_within_tan [continuous_intros]:
```
```  3866   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
```
```  3867   shows
```
```  3868   "continuous (at x within s) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
```
```  3869   unfolding continuous_within by (rule tendsto_tan)
```
```  3870
```
```  3871 lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) -- pi/2 --> 0"
```
```  3872   by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
```
```  3873
```
```  3874 lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
```
```  3875   apply (cut_tac LIM_cos_div_sin)
```
```  3876   apply (simp only: LIM_eq)
```
```  3877   apply (drule_tac x = "inverse y" in spec, safe, force)
```
```  3878   apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
```
```  3879   apply (rule_tac x = "(pi/2) - e" in exI)
```
```  3880   apply (simp (no_asm_simp))
```
```  3881   apply (drule_tac x = "(pi/2) - e" in spec)
```
```  3882   apply (auto simp add: tan_def sin_diff cos_diff)
```
```  3883   apply (rule inverse_less_iff_less [THEN iffD1])
```
```  3884   apply (auto simp add: divide_inverse)
```
```  3885   apply (rule mult_pos_pos)
```
```  3886   apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
```
```  3887   apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult.commute)
```
```  3888   done
```
```  3889
```
```  3890 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
```
```  3891   apply (frule order_le_imp_less_or_eq, safe)
```
```  3892    prefer 2 apply force
```
```  3893   apply (drule lemma_tan_total, safe)
```
```  3894   apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
```
```  3895   apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
```
```  3896   apply (drule_tac y = xa in order_le_imp_less_or_eq)
```
```  3897   apply (auto dest: cos_gt_zero)
```
```  3898   done
```
```  3899
```
```  3900 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  3901   apply (cut_tac linorder_linear [of 0 y], safe)
```
```  3902   apply (drule tan_total_pos)
```
```  3903   apply (cut_tac [2] y="-y" in tan_total_pos, safe)
```
```  3904   apply (rule_tac [3] x = "-x" in exI)
```
```  3905   apply (auto del: exI intro!: exI)
```
```  3906   done
```
```  3907
```
```  3908 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
```
```  3909   apply (cut_tac y = y in lemma_tan_total1, auto)
```
```  3910   apply hypsubst_thin
```
```  3911   apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
```
```  3912   apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
```
```  3913   apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
```
```  3914   apply (rule_tac [4] Rolle)
```
```  3915   apply (rule_tac [2] Rolle)
```
```  3916   apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
```
```  3917               simp add: real_differentiable_def)
```
```  3918   txt{*Now, simulate TRYALL*}
```
```  3919   apply (rule_tac [!] DERIV_tan asm_rl)
```
```  3920   apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
```
```  3921               simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
```
```  3922   done
```
```  3923
```
```  3924 lemma tan_monotone:
```
```  3925   assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
```
```  3926   shows "tan y < tan x"
```
```  3927 proof -
```
```  3928   have "\<forall>x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse ((cos x')\<^sup>2)"
```
```  3929   proof (rule allI, rule impI)
```
```  3930     fix x' :: real
```
```  3931     assume "y \<le> x' \<and> x' \<le> x"
```
```  3932     hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
```
```  3933     from cos_gt_zero_pi[OF this]
```
```  3934     have "cos x' \<noteq> 0" by auto
```
```  3935     thus "DERIV tan x' :> inverse ((cos x')\<^sup>2)" by (rule DERIV_tan)
```
```  3936   qed
```
```  3937   from MVT2[OF `y < x` this]
```
```  3938   obtain z where "y < z" and "z < x"
```
```  3939     and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
```
```  3940   hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
```
```  3941   hence "0 < cos z" using cos_gt_zero_pi by auto
```
```  3942   hence inv_pos: "0 < inverse ((cos z)\<^sup>2)" by auto
```
```  3943   have "0 < x - y" using `y < x` by auto
```
```  3944   with inv_pos have "0 < tan x - tan y" unfolding tan_diff by auto
```
```  3945   thus ?thesis by auto
```
```  3946 qed
```
```  3947
```
```  3948 lemma tan_monotone':
```
```  3949   assumes "- (pi / 2) < y"
```
```  3950     and "y < pi / 2"
```
```  3951     and "- (pi / 2) < x"
```
```  3952     and "x < pi / 2"
```
```  3953   shows "(y < x) = (tan y < tan x)"
```
```  3954 proof
```
```  3955   assume "y < x"
```
```  3956   thus "tan y < tan x"
```
```  3957     using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
```
```  3958 next
```
```  3959   assume "tan y < tan x"
```
```  3960   show "y < x"
```
```  3961   proof (rule ccontr)
```
```  3962     assume "\<not> y < x" hence "x \<le> y" by auto
```
```  3963     hence "tan x \<le> tan y"
```
```  3964     proof (cases "x = y")
```
```  3965       case True thus ?thesis by auto
```
```  3966     next
```
```  3967       case False hence "x < y" using `x \<le> y` by auto
```
```  3968       from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
```
```  3969     qed
```
```  3970     thus False using `tan y < tan x` by auto
```
```  3971   qed
```
```  3972 qed
```
```  3973
```
```  3974 lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)"
```
```  3975   unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
```
```  3976
```
```  3977 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
```
```  3978   by (simp add: tan_def)
```
```  3979
```
```  3980 lemma tan_periodic_nat[simp]:
```
```  3981   fixes n :: nat
```
```  3982   shows "tan (x + real n * pi) = tan x"
```
```  3983 proof (induct n arbitrary: x)
```
```  3984   case 0
```
```  3985   then show ?case by simp
```
```  3986 next
```
```  3987   case (Suc n)
```
```  3988   have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi"
```
```  3989     unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
```
```  3990   show ?case unfolding split_pi_off using Suc by auto
```
```  3991 qed
```
```  3992
```
```  3993 lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
```
```  3994 proof (cases "0 \<le> i")
```
```  3995   case True
```
```  3996   hence i_nat: "real i = real (nat i)" by auto
```
```  3997   show ?thesis unfolding i_nat by auto
```
```  3998 next
```
```  3999   case False
```
```  4000   hence i_nat: "real i = - real (nat (-i))" by auto
```
```  4001   have "tan x = tan (x + real i * pi - real i * pi)"
```
```  4002     by auto
```
```  4003   also have "\<dots> = tan (x + real i * pi)"
```
```  4004     unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
```
```  4005   finally show ?thesis by auto
```
```  4006 qed
```
```  4007
```
```  4008 lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
```
```  4009   using tan_periodic_int[of _ "numeral n" ] unfolding real_numeral .
```
```  4010
```
```  4011 lemma tan_minus_45: "tan (-(pi/4)) = -1"
```
```  4012   unfolding tan_def by (simp add: sin_45 cos_45)
```
```  4013
```
```  4014 lemma tan_diff:
```
```  4015   fixes x :: "'a::{real_normed_field,banach}"
```
```  4016   shows
```
```  4017      "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0; cos (x - y) \<noteq> 0\<rbrakk>
```
```  4018       \<Longrightarrow> tan(x - y) = (tan(x) - tan(y))/(1 + tan(x) * tan(y))"
```
```  4019   using tan_add [of x "-y"]
```
```  4020   by simp
```
```  4021
```
```  4022
```
```  4023 lemma tan_pos_pi2_le: "0 \<le> x ==> x < pi/2 \<Longrightarrow> 0 \<le> tan x"
```
```  4024   using less_eq_real_def tan_gt_zero by auto
```
```  4025
```
```  4026 lemma cos_tan: "abs(x) < pi/2 \<Longrightarrow> cos(x) = 1 / sqrt(1 + tan(x) ^ 2)"
```
```  4027   using cos_gt_zero_pi [of x]
```
```  4028   by (simp add: divide_simps tan_def real_sqrt_divide abs_if split: split_if_asm)
```
```  4029
```
```  4030 lemma sin_tan: "abs(x) < pi/2 \<Longrightarrow> sin(x) = tan(x) / sqrt(1 + tan(x) ^ 2)"
```
```  4031   using cos_gt_zero [of "x"] cos_gt_zero [of "-x"]
```
```  4032   by (force simp add: divide_simps tan_def real_sqrt_divide abs_if split: split_if_asm)
```
```  4033
```
```  4034 lemma tan_mono_le: "-(pi/2) < x ==> x \<le> y ==> y < pi/2 \<Longrightarrow> tan(x) \<le> tan(y)"
```
```  4035   using less_eq_real_def tan_monotone by auto
```
```  4036
```
```  4037 lemma tan_mono_lt_eq: "-(pi/2) < x ==> x < pi/2 ==> -(pi/2) < y ==> y < pi/2
```
```  4038          \<Longrightarrow> (tan(x) < tan(y) \<longleftrightarrow> x < y)"
```
```  4039   using tan_monotone' by blast
```
```  4040
```
```  4041 lemma tan_mono_le_eq: "-(pi/2) < x ==> x < pi/2 ==> -(pi/2) < y ==> y < pi/2
```
```  4042          \<Longrightarrow> (tan(x) \<le> tan(y) \<longleftrightarrow> x \<le> y)"
```
```  4043   by (meson tan_mono_le not_le tan_monotone)
```
```  4044
```
```  4045 lemma tan_bound_pi2: "abs(x) < pi/4 \<Longrightarrow> abs(tan x) < 1"
```
```  4046   using tan_45 tan_monotone [of x "pi/4"] tan_monotone [of "-x" "pi/4"]
```
```  4047   by (auto simp: abs_if split: split_if_asm)
```
```  4048
```
```  4049 lemma tan_cot: "tan(pi/2 - x) = inverse(tan x)"
```
```  4050   by (simp add: tan_def sin_diff cos_diff)
```
```  4051
```
```  4052 subsection {* Inverse Trigonometric Functions *}
```
```  4053
```
```  4054 definition arcsin :: "real => real"
```
```  4055   where "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
```
```  4056
```
```  4057 definition arccos :: "real => real"
```
```  4058   where "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
```
```  4059
```
```  4060 definition arctan :: "real => real"
```
```  4061   where "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
```
```  4062
```
```  4063 lemma arcsin:
```
```  4064   "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow>
```
```  4065     -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2 & sin(arcsin y) = y"
```
```  4066   unfolding arcsin_def by (rule theI' [OF sin_total])
```
```  4067
```
```  4068 lemma arcsin_pi:
```
```  4069   "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
```
```  4070   apply (drule (1) arcsin)
```
```  4071   apply (force intro: order_trans)
```
```  4072   done
```
```  4073
```
```  4074 lemma sin_arcsin [simp]: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin(arcsin y) = y"
```
```  4075   by (blast dest: arcsin)
```
```  4076
```
```  4077 lemma arcsin_bounded: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
```
```  4078   by (blast dest: arcsin)
```
```  4079
```
```  4080 lemma arcsin_lbound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y"
```
```  4081   by (blast dest: arcsin)
```
```  4082
```
```  4083 lemma arcsin_ubound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
```
```  4084   by (blast dest: arcsin)
```
```  4085
```
```  4086 lemma arcsin_lt_bounded:
```
```  4087      "\<lbrakk>-1 < y; y < 1\<rbrakk> \<Longrightarrow> -(pi/2) < arcsin y & arcsin y < pi/2"
```
```  4088   apply (frule order_less_imp_le)
```
```  4089   apply (frule_tac y = y in order_less_imp_le)
```
```  4090   apply (frule arcsin_bounded)
```
```  4091   apply (safe, simp)
```
```  4092   apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
```
```  4093   apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
```
```  4094   apply (drule_tac [!] f = sin in arg_cong, auto)
```
```  4095   done
```
```  4096
```
```  4097 lemma arcsin_sin: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2\<rbrakk> \<Longrightarrow> arcsin(sin x) = x"
```
```  4098   apply (unfold arcsin_def)
```
```  4099   apply (rule the1_equality)
```
```  4100   apply (rule sin_total, auto)
```
```  4101   done
```
```  4102
```
```  4103 lemma arcsin_0 [simp]: "arcsin 0 = 0"
```
```  4104   using arcsin_sin [of 0]
```
```  4105   by simp
```
```  4106
```
```  4107 lemma arcsin_1 [simp]: "arcsin 1 = pi/2"
```
```  4108   using arcsin_sin [of "pi/2"]
```
```  4109   by simp
```
```  4110
```
```  4111 lemma arcsin_minus_1 [simp]: "arcsin (-1) = - (pi/2)"
```
```  4112   using arcsin_sin [of "-pi/2"]
```
```  4113   by simp
```
```  4114
```
```  4115 lemma arcsin_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin(-x) = -arcsin x"
```
```  4116   by (metis (no_types, hide_lams) arcsin arcsin_sin minus_minus neg_le_iff_le sin_minus)
```
```  4117
```
```  4118 lemma arcsin_eq_iff: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> (arcsin x = arcsin y \<longleftrightarrow> x = y)"
```
```  4119   by (metis abs_le_interval_iff arcsin)
```
```  4120
```
```  4121 lemma cos_arcsin_nonzero: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> cos(arcsin x) \<noteq> 0"
```
```  4122   using arcsin_lt_bounded cos_gt_zero_pi by force
```
```  4123
```
```  4124 lemma arccos:
```
```  4125      "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk>
```
```  4126       \<Longrightarrow> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
```
```  4127   unfolding arccos_def by (rule theI' [OF cos_total])
```
```  4128
```
```  4129 lemma cos_arccos [simp]: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> cos(arccos y) = y"
```
```  4130   by (blast dest: arccos)
```
```  4131
```
```  4132 lemma arccos_bounded: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> arccos y & arccos y \<le> pi"
```
```  4133   by (blast dest: arccos)
```
```  4134
```
```  4135 lemma arccos_lbound: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> arccos y"
```
```  4136   by (blast dest: arccos)
```
```  4137
```
```  4138 lemma arccos_ubound: "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi"
```
```  4139   by (blast dest: arccos)
```
```  4140
```
```  4141 lemma arccos_lt_bounded:
```
```  4142      "\<lbrakk>-1 < y; y < 1\<rbrakk> \<Longrightarrow> 0 < arccos y & arccos y < pi"
```
```  4143   apply (frule order_less_imp_le)
```
```  4144   apply (frule_tac y = y in order_less_imp_le)
```
```  4145   apply (frule arccos_bounded, auto)
```
```  4146   apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
```
```  4147   apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
```
```  4148   apply (drule_tac [!] f = cos in arg_cong, auto)
```
```  4149   done
```
```  4150
```
```  4151 lemma arccos_cos: "\<lbrakk>0 \<le> x; x \<le> pi\<rbrakk> \<Longrightarrow> arccos(cos x) = x"
```
```  4152   apply (simp add: arccos_def)
```
```  4153   apply (auto intro!: the1_equality cos_total)
```
```  4154   done
```
```  4155
```
```  4156 lemma arccos_cos2: "\<lbrakk>x \<le> 0; -pi \<le> x\<rbrakk> \<Longrightarrow> arccos(cos x) = -x"
```
```  4157   apply (simp add: arccos_def)
```
```  4158   apply (auto intro!: the1_equality cos_total)
```
```  4159   done
```
```  4160
```
```  4161 lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
```
```  4162   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  4163   apply (rule power2_eq_imp_eq)
```
```  4164   apply (simp add: cos_squared_eq)
```
```  4165   apply (rule cos_ge_zero)
```
```  4166   apply (erule (1) arcsin_lbound)
```
```  4167   apply (erule (1) arcsin_ubound)
```
```  4168   apply simp
```
```  4169   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  4170   apply (rule power_mono, simp, simp)
```
```  4171   done
```
```  4172
```
```  4173 lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
```
```  4174   apply (subgoal_tac "x\<^sup>2 \<le> 1")
```
```  4175   apply (rule power2_eq_imp_eq)
```
```  4176   apply (simp add: sin_squared_eq)
```
```  4177   apply (rule sin_ge_zero)
```
```  4178   apply (erule (1) arccos_lbound)
```
```  4179   apply (erule (1) arccos_ubound)
```
```  4180   apply simp
```
```  4181   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
```
```  4182   apply (rule power_mono, simp, simp)
```
```  4183   done
```
```  4184
```
```  4185 lemma arccos_0 [simp]: "arccos 0 = pi/2"
```
```  4186 by (metis arccos_cos cos_gt_zero cos_pi cos_pi_half pi_gt_zero pi_half_ge_zero not_le not_zero_less_neg_numeral numeral_One)
```
```  4187
```
```  4188 lemma arccos_1 [simp]: "arccos 1 = 0"
```
```  4189   using arccos_cos by force
```
```  4190
```
```  4191 lemma arccos_minus_1 [simp]: "arccos(-1) = pi"
```
```  4192   by (metis arccos_cos cos_pi order_refl pi_ge_zero)
```
```  4193
```
```  4194 lemma arccos_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos(-x) = pi - arccos x"
```
```  4195   by (metis arccos_cos arccos_cos2 cos_minus_pi cos_total diff_le_0_iff_le le_add_same_cancel1
```
```  4196     minus_diff_eq uminus_add_conv_diff)
```
```  4197
```
```  4198 lemma sin_arccos_nonzero: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> ~(sin(arccos x) = 0)"
```
```  4199   using arccos_lt_bounded sin_gt_zero by force
```
```  4200
```
```  4201 lemma arctan: "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
```
```  4202   unfolding arctan_def by (rule theI' [OF tan_total])
```
```  4203
```
```  4204 lemma tan_arctan: "tan (arctan y) = y"
```
```  4205   by (simp add: arctan)
```
```  4206
```
```  4207 lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
```
```  4208   by (auto simp only: arctan)
```
```  4209
```
```  4210 lemma arctan_lbound: "- (pi/2) < arctan y"
```
```  4211   by (simp add: arctan)
```
```  4212
```
```  4213 lemma arctan_ubound: "arctan y < pi/2"
```
```  4214   by (auto simp only: arctan)
```
```  4215
```
```  4216 lemma arctan_unique:
```
```  4217   assumes "-(pi/2) < x"
```
```  4218     and "x < pi/2"
```
```  4219     and "tan x = y"
```
```  4220   shows "arctan y = x"
```
```  4221   using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
```
```  4222
```
```  4223 lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x"
```
```  4224   by (rule arctan_unique) simp_all
```
```  4225
```
```  4226 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
```
```  4227   by (rule arctan_unique) simp_all
```
```  4228
```
```  4229 lemma arctan_minus: "arctan (- x) = - arctan x"
```
```  4230   apply (rule arctan_unique)
```
```  4231   apply (simp only: neg_less_iff_less arctan_ubound)
```
```  4232   apply (metis minus_less_iff arctan_lbound, simp add: arctan)
```
```  4233   done
```
```  4234
```
```  4235 lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
```
```  4236   by (intro less_imp_neq [symmetric] cos_gt_zero_pi
```
```  4237     arctan_lbound arctan_ubound)
```
```  4238
```
```  4239 lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)"
```
```  4240 proof (rule power2_eq_imp_eq)
```
```  4241   have "0 < 1 + x\<^sup>2" by (simp add: add_pos_nonneg)
```
```  4242   show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp
```
```  4243   show "0 \<le> cos (arctan x)"
```
```  4244     by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
```
```  4245   have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1"
```
```  4246     unfolding tan_def by (simp add: distrib_left power_divide)
```
```  4247   thus "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
```
```  4248     using `0 < 1 + x\<^sup>2` by (simp add: arctan power_divide eq_divide_eq)
```
```  4249 qed
```
```  4250
```
```  4251 lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)"
```
```  4252   using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
```
```  4253   using tan_arctan [of x] unfolding tan_def cos_arctan
```
```  4254   by (simp add: eq_divide_eq)
```
```  4255
```
```  4256 lemma tan_sec:
```
```  4257   fixes x :: "'a::{real_normed_field,banach,field}"
```
```  4258   shows "cos x \<noteq> 0 \<Longrightarrow> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
```
```  4259   apply (rule power_inverse [THEN subst])
```
```  4260   apply (rule_tac c1 = "(cos x)\<^sup>2" in mult_right_cancel [THEN iffD1])
```
```  4261   apply (auto dest: field_power_not_zero
```
```  4262           simp add: power_mult_distrib distrib_right power_divide tan_def
```
```  4263                     mult.assoc power_inverse [symmetric])
```
```  4264   done
```
```  4265
```
```  4266 lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
```
```  4267   by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
```
```  4268
```
```  4269 lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
```
```  4270   by (simp only: not_less [symmetric] arctan_less_iff)
```
```  4271
```
```  4272 lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
```
```  4273   by (simp only: eq_iff [where 'a=real] arctan_le_iff)
```
```  4274
```
```  4275 lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
```
```  4276   using arctan_less_iff [of 0 x] by simp
```
```  4277
```
```  4278 lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
```
```  4279   using arctan_less_iff [of x 0] by simp
```
```  4280
```
```  4281 lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
```
```  4282   using arctan_le_iff [of 0 x] by simp
```
```  4283
```
```  4284 lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
```
```  4285   using arctan_le_iff [of x 0] by simp
```
```  4286
```
```  4287 lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
```
```  4288   using arctan_eq_iff [of x 0] by simp
```
```  4289
```
```  4290 lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin"
```
```  4291 proof -
```
```  4292   have "continuous_on (sin ` {- pi / 2 .. pi / 2}) arcsin"
```
```  4293     by (rule continuous_on_inv) (auto intro: continuous_intros simp: arcsin_sin)
```
```  4294   also have "sin ` {- pi / 2 .. pi / 2} = {-1 .. 1}"
```
```  4295   proof safe
```
```  4296     fix x :: real
```
```  4297     assume "x \<in> {-1..1}"
```
```  4298     then show "x \<in> sin ` {- pi / 2..pi / 2}"
```
```  4299       using arcsin_lbound arcsin_ubound
```
```  4300       by (intro image_eqI[where x="arcsin x"]) auto
```
```  4301   qed simp
```
```  4302   finally show ?thesis .
```
```  4303 qed
```
```  4304
```
```  4305 lemma continuous_on_arcsin [continuous_intros]:
```
```  4306   "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))"
```
```  4307   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arcsin']]
```
```  4308   by (auto simp: comp_def subset_eq)
```
```  4309
```
```  4310 lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x"
```
```  4311   using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  4312   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  4313
```
```  4314 lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos"
```
```  4315 proof -
```
```  4316   have "continuous_on (cos ` {0 .. pi}) arccos"
```
```  4317     by (rule continuous_on_inv) (auto intro: continuous_intros simp: arccos_cos)
```
```  4318   also have "cos ` {0 .. pi} = {-1 .. 1}"
```
```  4319   proof safe
```
```  4320     fix x :: real
```
```  4321     assume "x \<in> {-1..1}"
```
```  4322     then show "x \<in> cos ` {0..pi}"
```
```  4323       using arccos_lbound arccos_ubound
```
```  4324       by (intro image_eqI[where x="arccos x"]) auto
```
```  4325   qed simp
```
```  4326   finally show ?thesis .
```
```  4327 qed
```
```  4328
```
```  4329 lemma continuous_on_arccos [continuous_intros]:
```
```  4330   "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))"
```
```  4331   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arccos']]
```
```  4332   by (auto simp: comp_def subset_eq)
```
```  4333
```
```  4334 lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x"
```
```  4335   using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
```
```  4336   by (auto simp: continuous_on_eq_continuous_at subset_eq)
```
```  4337
```
```  4338 lemma isCont_arctan: "isCont arctan x"
```
```  4339   apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
```
```  4340   apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
```
```  4341   apply (subgoal_tac "isCont arctan (tan (arctan x))", simp add: arctan)
```
```  4342   apply (erule (1) isCont_inverse_function2 [where f=tan])
```
```  4343   apply (metis arctan_tan order_le_less_trans order_less_le_trans)
```
```  4344   apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
```
```  4345   done
```
```  4346
```
```  4347 lemma tendsto_arctan [tendsto_intros]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) ---> arctan x) F"
```
```  4348   by (rule isCont_tendsto_compose [OF isCont_arctan])
```
```  4349
```
```  4350 lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))"
```
```  4351   unfolding continuous_def by (rule tendsto_arctan)
```
```  4352
```
```  4353 lemma continuous_on_arctan [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
```
```  4354   unfolding continuous_on_def by (auto intro: tendsto_arctan)
```
```  4355
```
```  4356 lemma DERIV_arcsin:
```
```  4357   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
```
```  4358   apply (rule DERIV_inverse_function [where f=sin and a="-1" and b=1])
```
```  4359   apply (rule DERIV_cong [OF DERIV_sin])
```
```  4360   apply (simp add: cos_arcsin)
```
```  4361   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  4362   apply (rule power_strict_mono, simp, simp, simp, assumption, assumption)
```
```  4363   apply simp
```
```  4364   apply (erule (1) isCont_arcsin)
```
```  4365   done
```
```  4366
```
```  4367 lemma DERIV_arccos:
```
```  4368   "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
```
```  4369   apply (rule DERIV_inverse_function [where f=cos and a="-1" and b=1])
```
```  4370   apply (rule DERIV_cong [OF DERIV_cos])
```
```  4371   apply (simp add: sin_arccos)
```
```  4372   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
```
```  4373   apply (rule power_strict_mono, simp, simp, simp, assumption, assumption)
```
```  4374   apply simp
```
```  4375   apply (erule (1) isCont_arccos)
```
```  4376   done
```
```  4377
```
```  4378 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
```
```  4379   apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
```
```  4380   apply (rule DERIV_cong [OF DERIV_tan])
```
```  4381   apply (rule cos_arctan_not_zero)
```
```  4382   apply (simp add: arctan power_inverse tan_sec [symmetric])
```
```  4383   apply (subgoal_tac "0 < 1 + x\<^sup>2", simp)
```
```  4384   apply (simp_all add: add_pos_nonneg arctan isCont_arctan)
```
```  4385   done
```
```  4386
```
```  4387 declare
```
```  4388   DERIV_arcsin[THEN DERIV_chain2, derivative_intros]
```
```  4389   DERIV_arccos[THEN DERIV_chain2, derivative_intros]
```
```  4390   DERIV_arctan[THEN DERIV_chain2, derivative_intros]
```
```  4391
```
```  4392 lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- pi/2))"
```
```  4393   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])
```
```  4394      (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  4395            intro!: tan_monotone exI[of _ "pi/2"])
```
```  4396
```
```  4397 lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"
```
```  4398   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])
```
```  4399      (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
```
```  4400            intro!: tan_monotone exI[of _ "pi/2"])
```
```  4401
```
```  4402 lemma tendsto_arctan_at_top: "(arctan ---> (pi/2)) at_top"
```
```  4403 proof (rule tendstoI)
```
```  4404   fix e :: real
```
```  4405   assume "0 < e"
```
```  4406   def y \<equiv> "pi/2 - min (pi/2) e"
```
```  4407   then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y"
```
```  4408     using `0 < e` by auto
```
```  4409
```
```  4410   show "eventually (\<lambda>x. dist (arctan x) (pi / 2) < e) at_top"
```
```  4411   proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)
```
```  4412     fix x
```
```  4413     assume "tan y < x"
```
```  4414     then have "arctan (tan y) < arctan x"
```
```  4415       by (simp add: arctan_less_iff)
```
```  4416     with y have "y < arctan x"
```
```  4417       by (subst (asm) arctan_tan) simp_all
```
```  4418     with arctan_ubound[of x, arith] y `0 < e`
```
```  4419     show "dist (arctan x) (pi / 2) < e"
```
```  4420       by (simp add: dist_real_def)
```
```  4421   qed
```
```  4422 qed
```
```  4423
```
```  4424 lemma tendsto_arctan_at_bot: "(arctan ---> - (pi/2)) at_bot"
```
```  4425   unfolding filterlim_at_bot_mirror arctan_minus
```
```  4426   by (intro tendsto_minus tendsto_arctan_at_top)
```
```  4427
```
```  4428
```
```  4429 subsection{* Prove Totality of the Trigonometric Functions *}
```
```  4430
```
```  4431 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
```
```  4432   by (simp add: abs_le_iff)
```
```  4433
```
```  4434 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
```
```  4435   by (simp add: sin_arccos abs_le_iff)
```
```  4436
```
```  4437 lemma sin_mono_less_eq: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2; -(pi/2) \<le> y; y \<le> pi/2\<rbrakk>
```
```  4438          \<Longrightarrow> (sin(x) < sin(y) \<longleftrightarrow> x < y)"
```
```  4439 by (metis not_less_iff_gr_or_eq sin_monotone_2pi)
```
```  4440
```
```  4441 lemma sin_mono_le_eq: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2; -(pi/2) \<le> y; y \<le> pi/2\<rbrakk>
```
```  4442          \<Longrightarrow> (sin(x) \<le> sin(y) \<longleftrightarrow> x \<le> y)"
```
```  4443 by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le)
```
```  4444
```
```  4445 lemma sin_inj_pi:
```
```  4446     "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2;-(pi/2) \<le> y; y \<le> pi/2; sin(x) = sin(y)\<rbrakk> \<Longrightarrow> x = y"
```
```  4447 by (metis arcsin_sin)
```
```  4448
```
```  4449 lemma cos_mono_less_eq:
```
```  4450     "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi \<Longrightarrow> (cos(x) < cos(y) \<longleftrightarrow> y < x)"
```
```  4451 by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear)
```
```  4452
```
```  4453 lemma cos_mono_le_eq: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi
```
```  4454          \<Longrightarrow> (cos(x) \<le> cos(y) \<longleftrightarrow> y \<le> x)"
```
```  4455   by (metis arccos_cos cos_monotone_0_pi_le eq_iff linear)
```
```  4456
```
```  4457 lemma cos_inj_pi: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi ==> cos(x) = cos(y)
```
```  4458          \<Longrightarrow> x = y"
```
```  4459 by (metis arccos_cos)
```
```  4460
```
```  4461 lemma arccos_le_pi2: "\<lbrakk>0 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi/2"
```
```  4462   by (metis (mono_tags) arccos_0 arccos cos_le_one cos_monotone_0_pi_le
```
```  4463       cos_pi cos_pi_half pi_half_ge_zero antisym_conv less_eq_neg_nonpos linear minus_minus order.trans order_refl)
```
```  4464
```
```  4465 lemma sincos_total_pi_half:
```
```  4466   assumes "0 \<le> x" "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1"
```
```  4467     shows "\<exists>t. 0 \<le> t \<and> t \<le> pi/2 \<and> x = cos t \<and> y = sin t"
```
```  4468 proof -
```
```  4469   have x1: "x \<le> 1"
```
```  4470     using assms
```
```  4471     by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2)
```
```  4472   moreover with assms have ax: "0 \<le> arccos x" "cos(arccos x) = x"
```
```  4473     by (auto simp: arccos)
```
```  4474   moreover have "y = sqrt (1 - x\<^sup>2)" using assms
```
```  4475     by (metis abs_of_nonneg add.commute add_diff_cancel real_sqrt_abs)
```
```  4476   ultimately show ?thesis using assms arccos_le_pi2 [of x]
```
```  4477     by (rule_tac x="arccos x" in exI) (auto simp: sin_arccos)
```
```  4478 qed
```
```  4479
```
```  4480 lemma sincos_total_pi:
```
```  4481   assumes "0 \<le> y" and "x\<^sup>2 + y\<^sup>2 = 1"
```
```  4482     shows "\<exists>t. 0 \<le> t \<and> t \<le> pi \<and> x = cos t \<and> y = sin t"
```
```  4483 proof (cases rule: le_cases [of 0 x])
```
```  4484   case le from sincos_total_pi_half [OF le]
```
```  4485   show ?thesis
```
```  4486     by (metis pi_ge_two pi_half_le_two add.commute add_le_cancel_left add_mono assms)
```
```  4487 next
```
```  4488   case ge
```
```  4489   then have "0 \<le> -x"
```
```  4490     by simp
```
```  4491   then obtain t where "t\<ge>0" "t \<le> pi/2" "-x = cos t" "y = sin t"
```
```  4492     using sincos_total_pi_half assms
```
```  4493     apply auto
```
```  4494     by (metis `0 \<le> - x` power2_minus)
```
```  4495   then show ?thesis
```
```  4496     by (rule_tac x="pi-t" in exI, auto)
```
```  4497 qed
```
```  4498
```
```  4499 lemma sincos_total_2pi_le:
```
```  4500   assumes "x\<^sup>2 + y\<^sup>2 = 1"
```
```  4501     shows "\<exists>t. 0 \<le> t \<and> t \<le> 2*pi \<and> x = cos t \<and> y = sin t"
```
```  4502 proof (cases rule: le_cases [of 0 y])
```
```  4503   case le from sincos_total_pi [OF le]
```
```  4504   show ?thesis
```
```  4505     by (metis assms le_add_same_cancel1 mult.commute mult_2_right order.trans)
```
```  4506 next
```
```  4507   case ge
```
```  4508   then have "0 \<le> -y"
```
```  4509     by simp
```
```  4510   then obtain t where "t\<ge>0" "t \<le> pi" "x = cos t" "-y = sin t"
```
```  4511     using sincos_total_pi assms
```
```  4512     apply auto
```
```  4513     by (metis `0 \<le> - y` power2_minus)
```
```  4514   then show ?thesis
```
```  4515     by (rule_tac x="2*pi-t" in exI, auto)
```
```  4516 qed
```
```  4517
```
```  4518 lemma sincos_total_2pi:
```
```  4519   assumes "x\<^sup>2 + y\<^sup>2 = 1"
```
```  4520     obtains t where "0 \<le> t" "t < 2*pi" "x = cos t" "y = sin t"
```
```  4521 proof -
```
```  4522   from sincos_total_2pi_le [OF assms]
```
```  4523   obtain t where t: "0 \<le> t" "t \<le> 2*pi" "x = cos t" "y = sin t"
```
```  4524     by blast
```
```  4525   show ?thesis
```
```  4526     apply (cases "t = 2*pi")
```
```  4527     using t that
```
```  4528     apply force+
```
```  4529     done
```
```  4530 qed
```
```  4531
```
```  4532 lemma arcsin_less_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> arcsin x < arcsin y \<longleftrightarrow> x < y"
```
```  4533   apply (rule trans [OF sin_mono_less_eq [symmetric]])
```
```  4534   using arcsin_ubound arcsin_lbound
```
```  4535   apply auto
```
```  4536   done
```
```  4537
```
```  4538 lemma arcsin_le_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y \<longleftrightarrow> x \<le> y"
```
```  4539   using arcsin_less_mono not_le by blast
```
```  4540
```
```  4541 lemma arcsin_less_arcsin: "-1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x < arcsin y"
```
```  4542   using arcsin_less_mono by auto
```
```  4543
```
```  4544 lemma arcsin_le_arcsin: "-1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y"
```
```  4545   using arcsin_le_mono by auto
```
```  4546
```
```  4547 lemma arccos_less_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> (arccos x < arccos y \<longleftrightarrow> y < x)"
```
```  4548   apply (rule trans [OF cos_mono_less_eq [symmetric]])
```
```  4549   using arccos_ubound arccos_lbound
```
```  4550   apply auto
```
```  4551   done
```
```  4552
```
```  4553 lemma arccos_le_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> arccos x \<le> arccos y \<longleftrightarrow> y \<le> x"
```
```  4554   using arccos_less_mono [of y x]
```
```  4555   by (simp add: not_le [symmetric])
```
```  4556
```
```  4557 lemma arccos_less_arccos: "-1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y < arccos x"
```
```  4558   using arccos_less_mono by auto
```
```  4559
```
```  4560 lemma arccos_le_arccos: "-1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> arccos x"
```
```  4561   using arccos_le_mono by auto
```
```  4562
```
```  4563 lemma arccos_eq_iff: "abs x \<le> 1 & abs y \<le> 1 \<Longrightarrow> (arccos x = arccos y \<longleftrightarrow> x = y)"
```
```  4564   using cos_arccos_abs by fastforce
```
```  4565
```
```  4566 subsection {* Machins formula *}
```
```  4567
```
```  4568 lemma arctan_one: "arctan 1 = pi / 4"
```
```  4569   by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi)
```
```  4570
```
```  4571 lemma tan_total_pi4:
```
```  4572   assumes "\<bar>x\<bar> < 1"
```
```  4573   shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
```
```  4574 proof
```
```  4575   show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
```
```  4576     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  4577     unfolding arctan_less_iff using assms  by (auto simp add: arctan)
```
```  4578
```
```  4579 qed
```
```  4580
```
```  4581 lemma arctan_add:
```
```  4582   assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
```
```  4583   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
```
```  4584 proof (rule arctan_unique [symmetric])
```
```  4585   have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y"
```
```  4586     unfolding arctan_one [symmetric] arctan_minus [symmetric]
```
```  4587     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  4588   from add_le_less_mono [OF this]
```
```  4589   show 1: "- (pi / 2) < arctan x + arctan y" by simp
```
```  4590   have "arctan x \<le> pi / 4" and "arctan y < pi / 4"
```
```  4591     unfolding arctan_one [symmetric]
```
```  4592     unfolding arctan_le_iff arctan_less_iff using assms by auto
```
```  4593   from add_le_less_mono [OF this]
```
```  4594   show 2: "arctan x + arctan y < pi / 2" by simp
```
```  4595   show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
```
```  4596     using cos_gt_zero_pi [OF 1 2] by (simp add: arctan tan_add)
```
```  4597 qed
```
```  4598
```
```  4599 theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
```
```  4600 proof -
```
```  4601   have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
```
```  4602   from arctan_add[OF less_imp_le[OF this] this]
```
```  4603   have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
```
```  4604   moreover
```
```  4605   have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
```
```  4606   from arctan_add[OF less_imp_le[OF this] this]
```
```  4607   have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
```
```  4608   moreover
```
```  4609   have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
```
```  4610   from arctan_add[OF this]
```
```  4611   have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
```
```  4612   ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
```
```  4613   thus ?thesis unfolding arctan_one by algebra
```
```  4614 qed
```
```  4615
```
```  4616
```
```  4617 subsection {* Introducing the inverse tangent power series *}
```
```  4618
```
```  4619 lemma monoseq_arctan_series:
```
```  4620   fixes x :: real
```
```  4621   assumes "\<bar>x\<bar> \<le> 1"
```
```  4622   shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
```
```  4623 proof (cases "x = 0")
```
```  4624   case True
```
```  4625   thus ?thesis unfolding monoseq_def One_nat_def by auto
```
```  4626 next
```
```  4627   case False
```
```  4628   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  4629   show "monoseq ?a"
```
```  4630   proof -
```
```  4631     {
```
```  4632       fix n
```
```  4633       fix x :: real
```
```  4634       assume "0 \<le> x" and "x \<le> 1"
```
```  4635       have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le>
```
```  4636         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
```
```  4637       proof (rule mult_mono)
```
```  4638         show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))"
```
```  4639           by (rule frac_le) simp_all
```
```  4640         show "0 \<le> 1 / real (Suc (n * 2))"
```
```  4641           by auto
```
```  4642         show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)"
```
```  4643           by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`)
```
```  4644         show "0 \<le> x ^ Suc (Suc n * 2)"
```
```  4645           by (rule zero_le_power) (simp add: `0 \<le> x`)
```
```  4646       qed
```
```  4647     } note mono = this
```
```  4648
```
```  4649     show ?thesis
```
```  4650     proof (cases "0 \<le> x")
```
```  4651       case True from mono[OF this `x \<le> 1`, THEN allI]
```
```  4652       show ?thesis unfolding Suc_eq_plus1[symmetric]
```
```  4653         by (rule mono_SucI2)
```
```  4654     next
```
```  4655       case False
```
```  4656       hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto
```
```  4657       from mono[OF this]
```
```  4658       have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge>
```
```  4659         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using `0 \<le> -x` by auto
```
```  4660       thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
```
```  4661     qed
```
```  4662   qed
```
```  4663 qed
```
```  4664
```
```  4665 lemma zeroseq_arctan_series:
```
```  4666   fixes x :: real
```
```  4667   assumes "\<bar>x\<bar> \<le> 1"
```
```  4668   shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
```
```  4669 proof (cases "x = 0")
```
```  4670   case True
```
```  4671   thus ?thesis
```
```  4672     unfolding One_nat_def by auto
```
```  4673 next
```
```  4674   case False
```
```  4675   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
```
```  4676   show "?a ----> 0"
```
```  4677   proof (cases "\<bar>x\<bar> < 1")
```
```  4678     case True
```
```  4679     hence "norm x < 1" by auto
```
```  4680     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]]
```
```  4681     have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"
```
```  4682       unfolding inverse_eq_divide Suc_eq_plus1 by simp
```
```  4683     then show ?thesis using pos2 by (rule LIMSEQ_linear)
```
```  4684   next
```
```  4685     case False
```
```  4686     hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  4687     hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x"
```
```  4688       unfolding One_nat_def by auto
```
```  4689     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
```
```  4690     show ?thesis unfolding n_eq Suc_eq_plus1 by auto
```
```  4691   qed
```
```  4692 qed
```
```  4693
```
```  4694 text{*FIXME: generalise from the reals via type classes?*}
```
```  4695 lemma summable_arctan_series:
```
```  4696   fixes x :: real and n :: nat
```
```  4697   assumes "\<bar>x\<bar> \<le> 1"
```
```  4698   shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  4699   (is "summable (?c x)")
```
```  4700   by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
```
```  4701
```
```  4702 lemma DERIV_arctan_series:
```
```  4703   assumes "\<bar> x \<bar> < 1"
```
```  4704   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))"
```
```  4705   (is "DERIV ?arctan _ :> ?Int")
```
```  4706 proof -
```
```  4707   let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
```
```  4708
```
```  4709   have n_even: "\<And>n :: nat. even n \<Longrightarrow> 2 * (n div 2) = n"
```
```  4710     by presburger
```
```  4711   then have if_eq: "\<And>n x'. ?f n * real (Suc n) * x'^n =
```
```  4712     (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)"
```
```  4713     by auto
```
```  4714
```
```  4715   {
```
```  4716     fix x :: real
```
```  4717     assume "\<bar>x\<bar> < 1"
```
```  4718     hence "x\<^sup>2 < 1" by (simp add: abs_square_less_1)
```
```  4719     have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)"
```
```  4720       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`])
```
```  4721     hence "summable (\<lambda> n. (- 1) ^ n * x^(2*n))" unfolding power_mult .
```
```  4722   } note summable_Integral = this
```
```  4723
```
```  4724   {
```
```  4725     fix f :: "nat \<Rightarrow> real"
```
```  4726     have "\<And>x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  4727     proof
```
```  4728       fix x :: real
```
```  4729       assume "f sums x"
```
```  4730       from sums_if[OF sums_zero this]
```
```  4731       show "(\<lambda>n. if even n then f (n div 2) else 0) sums x"
```
```  4732         by auto
```
```  4733     next
```
```  4734       fix x :: real
```
```  4735       assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
```
```  4736       from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult.commute]]
```
```  4737       show "f sums x" unfolding sums_def by auto
```
```  4738     qed
```
```  4739     hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
```
```  4740   } note sums_even = this
```
```  4741
```
```  4742   have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int"
```
```  4743     unfolding if_eq mult.commute[of _ 2] suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * x ^ (2 * n)", symmetric]
```
```  4744     by auto
```
```  4745
```
```  4746   {
```
```  4747     fix x :: real
```
```  4748     have if_eq': "\<And>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
```
```  4749       (if even n then (- 1) ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
```
```  4750       using n_even by auto
```
```  4751     have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)" by auto
```
```  4752     have "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x"
```
```  4753       unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
```
```  4754       by auto
```
```  4755   } note arctan_eq = this
```
```  4756
```
```  4757   have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
```
```  4758   proof (rule DERIV_power_series')
```
```  4759     show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto
```
```  4760     {
```
```  4761       fix x' :: real
```
```  4762       assume x'_bounds: "x' \<in> {- 1 <..< 1}"
```
```  4763       then have "\<bar>x'\<bar> < 1" by auto
```
```  4764       then
```
```  4765         have *: "summable (\<lambda>n. (- 1) ^ n * x' ^ (2 * n))"
```
```  4766         by (rule summable_Integral)
```
```  4767       let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
```
```  4768       show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
```
```  4769         apply (rule sums_summable [where l="0 + ?S"])
```
```  4770         apply (rule sums_if)
```
```  4771         apply (rule sums_zero)
```
```  4772         apply (rule summable_sums)
```
```  4773         apply (rule *)
```
```  4774         done
```
```  4775     }
```
```  4776   qed auto
```
```  4777   thus ?thesis unfolding Int_eq arctan_eq .
```
```  4778 qed
```
```  4779
```
```  4780 lemma arctan_series:
```
```  4781   assumes "\<bar> x \<bar> \<le> 1"
```
```  4782   shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
```
```  4783   (is "_ = suminf (\<lambda> n. ?c x n)")
```
```  4784 proof -
```
```  4785   let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)"
```
```  4786
```
```  4787   {
```
```  4788     fix r x :: real
```
```  4789     assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
```
```  4790     have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto
```
```  4791     from DERIV_arctan_series[OF this] have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
```
```  4792   } note DERIV_arctan_suminf = this
```
```  4793
```
```  4794   {
```
```  4795     fix x :: real
```
```  4796     assume "\<bar>x\<bar> \<le> 1"
```
```  4797     note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]]
```
```  4798   } note arctan_series_borders = this
```
```  4799
```
```  4800   {
```
```  4801     fix x :: real
```
```  4802     assume "\<bar>x\<bar> < 1"
```
```  4803     have "arctan x = (\<Sum>k. ?c x k)"
```
```  4804     proof -
```
```  4805       obtain r where "\<bar>x\<bar> < r" and "r < 1"
```
```  4806         using dense[OF `\<bar>x\<bar> < 1`] by blast
```
```  4807       hence "0 < r" and "-r < x" and "x < r" by auto
```
```  4808
```
```  4809       have suminf_eq_arctan_bounded: "\<And>x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow>
```
```  4810         suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  4811       proof -
```
```  4812         fix x a b
```
```  4813         assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
```
```  4814         hence "\<bar>x\<bar> < r" by auto
```
```  4815         show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
```
```  4816         proof (rule DERIV_isconst2[of "a" "b"])
```
```  4817           show "a < b" and "a \<le> x" and "x \<le> b"
```
```  4818             using `a < b` `a \<le> x` `x \<le> b` by auto
```
```  4819           have "\<forall>x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  4820           proof (rule allI, rule impI)
```
```  4821             fix x
```
```  4822             assume "-r < x \<and> x < r"
```
```  4823             hence "\<bar>x\<bar> < r" by auto
```
```  4824             hence "\<bar>x\<bar> < 1" using `r < 1` by auto
```
```  4825             have "\<bar> - (x\<^sup>2) \<bar> < 1"
```
```  4826               using abs_square_less_1 `\<bar>x\<bar> < 1` by auto
```
```  4827             hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  4828               unfolding real_norm_def[symmetric] by (rule geometric_sums)
```
```  4829             hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"
```
```  4830               unfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2] by auto
```
```  4831             hence suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)"
```
```  4832               using sums_unique unfolding inverse_eq_divide by auto
```
```  4833             have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))"
```
```  4834               unfolding suminf_c'_eq_geom
```
```  4835               by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
```
```  4836             from DERIV_diff [OF this DERIV_arctan]
```
```  4837             show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
```
```  4838               by auto
```
```  4839           qed
```
```  4840           hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
```
```  4841             using `-r < a` `b < r` by auto
```
```  4842           thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
```
```  4843             using `\<bar>x\<bar> < r` by auto
```
```  4844           show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y"
```
```  4845             using DERIV_in_rball DERIV_isCont by auto
```
```  4846         qed
```
```  4847       qed
```
```  4848
```
```  4849       have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
```
```  4850         unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero
```
```  4851         by auto
```
```  4852
```
```  4853       have "suminf (?c x) - arctan x = 0"
```
```  4854       proof (cases "x = 0")
```
```  4855         case True
```
```  4856         thus ?thesis using suminf_arctan_zero by auto
```
```  4857       next
```
```  4858         case False
```
```  4859         hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
```
```  4860         have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
```
```  4861           by (rule suminf_eq_arctan_bounded[where x1="0" and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>", symmetric])
```
```  4862             (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  4863         moreover
```
```  4864         have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
```
```  4865           by (rule suminf_eq_arctan_bounded[where x1="x" and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>"])
```
```  4866              (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
```
```  4867         ultimately
```
```  4868         show ?thesis using suminf_arctan_zero by auto
```
```  4869       qed
```
```  4870       thus ?thesis by auto
```
```  4871     qed
```
```  4872   } note when_less_one = this
```
```  4873
```
```  4874   show "arctan x = suminf (\<lambda> n. ?c x n)"
```
```  4875   proof (cases "\<bar>x\<bar> < 1")
```
```  4876     case True
```
```  4877     thus ?thesis by (rule when_less_one)
```
```  4878   next
```
```  4879     case False
```
```  4880     hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto
```
```  4881     let ?a = "\<lambda>x n. \<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
```
```  4882     let ?diff = "\<lambda> x n. \<bar> arctan x - (\<Sum> i<n. ?c x i)\<bar>"
```
```  4883     {
```
```  4884       fix n :: nat
```
```  4885       have "0 < (1 :: real)" by auto
```
```  4886       moreover
```
```  4887       {
```
```  4888         fix x :: real
```
```  4889         assume "0 < x" and "x < 1"
```
```  4890         hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
```
```  4891         from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)"
```
```  4892           by auto
```
```  4893         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]
```
```  4894         have "0 < 1 / real (n*2+1) * x^(n*2+1)"
```
```  4895           by (rule mult_pos_pos, auto simp only: zero_less_power[OF `0 < x`], auto)
```
```  4896         hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)"
```
```  4897           by (rule abs_of_pos)
```
```  4898         have "?diff x n \<le> ?a x n"
```
```  4899         proof (cases "even n")
```
```  4900           case True
```
```  4901           hence sgn_pos: "(-1)^n = (1::real)" by auto
```
```  4902           from `even n` obtain m where "n = 2 * m" ..
```
```  4903           then have "2 * m = n" ..
```
```  4904           from bounds[of m, unfolded this atLeastAtMost_iff]
```
```  4905           have "\<bar>arctan x - (\<Sum>i<n. (?c x i))\<bar> \<le> (\<Sum>i<n + 1. (?c x i)) - (\<Sum>i<n. (?c x i))"
```
```  4906             by auto
```
```  4907           also have "\<dots> = ?c x n" unfolding One_nat_def by auto
```
```  4908           also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
```
```  4909           finally show ?thesis .
```
```  4910         next
```
```  4911           case False
```
```  4912           hence sgn_neg: "(-1)^n = (-1::real)" by auto
```
```  4913           from `odd n` obtain m where "n = 2 * m + 1" ..
```
```  4914           then have m_def: "2 * m + 1 = n" ..
```
```  4915           hence m_plus: "2 * (m + 1) = n + 1" by auto
```
```  4916           from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
```
```  4917           have "\<bar>arctan x - (\<Sum>i<n. (?c x i))\<bar> \<le> (\<Sum>i<n. (?c x i)) - (\<Sum>i<n+1. (?c x i))"
```
```  4918             by auto
```
```  4919           also have "\<dots> = - ?c x n" unfolding One_nat_def by auto
```
```  4920           also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
```
```  4921           finally show ?thesis .
```
```  4922         qed
```
```  4923         hence "0 \<le> ?a x n - ?diff x n" by auto
```
```  4924       }
```
```  4925       hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
```
```  4926       moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
```
```  4927         unfolding diff_conv_add_uminus divide_inverse
```
```  4928         by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan
```
```  4929           isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum
```
```  4930           simp del: add_uminus_conv_diff)
```
```  4931       ultimately have "0 \<le> ?a 1 n - ?diff 1 n"
```
```  4932         by (rule LIM_less_bound)
```
```  4933       hence "?diff 1 n \<le> ?a 1 n" by auto
```
```  4934     }
```
```  4935     have "?a 1 ----> 0"
```
```  4936       unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
```
```  4937       by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)
```
```  4938     have "?diff 1 ----> 0"
```
```  4939     proof (rule LIMSEQ_I)
```
```  4940       fix r :: real
```
```  4941       assume "0 < r"
```
```  4942       obtain N :: nat where N_I: "\<And>n. N \<le> n \<Longrightarrow> ?a 1 n < r"
```
```  4943         using LIMSEQ_D[OF `?a 1 ----> 0` `0 < r`] by auto
```
```  4944       {
```
```  4945         fix n
```
```  4946         assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this]
```
```  4947         have "norm (?diff 1 n - 0) < r" by auto
```
```  4948       }
```
```  4949       thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
```
```  4950     qed
```
```  4951     from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
```
```  4952     have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
```
```  4953     hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
```
```  4954
```
```  4955     show ?thesis
```
```  4956     proof (cases "x = 1")
```
```  4957       case True
```
```  4958       then show ?thesis by (simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
```
```  4959     next
```
```  4960       case False
```
```  4961       hence "x = -1" using `\<bar>x\<bar> = 1` by auto
```
```  4962
```
```  4963       have "- (pi / 2) < 0" using pi_gt_zero by auto
```
```  4964       have "- (2 * pi) < 0" using pi_gt_zero by auto
```
```  4965
```
```  4966       have c_minus_minus: "\<And>i. ?c (- 1) i = - ?c 1 i"
```
```  4967         unfolding One_nat_def by auto
```
```  4968
```
```  4969       have "arctan (- 1) = arctan (tan (-(pi / 4)))"
```
```  4970         unfolding tan_45 tan_minus ..
```
```  4971       also have "\<dots> = - (pi / 4)"
```
```  4972         by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
```
```  4973       also have "\<dots> = - (arctan (tan (pi / 4)))"
```
```  4974         unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero])
```
```  4975       also have "\<dots> = - (arctan 1)"
```
```  4976         unfolding tan_45 ..
```
```  4977       also have "\<dots> = - (\<Sum> i. ?c 1 i)"
```
```  4978         using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto
```
```  4979       also have "\<dots> = (\<Sum> i. ?c (- 1) i)"
```
```  4980         using suminf_minus[OF sums_summable[OF `(?c 1) sums (arctan 1)`]]
```
```  4981         unfolding c_minus_minus by auto
```
```  4982       finally show ?thesis using `x = -1` by auto
```
```  4983     qed
```
```  4984   qed
```
```  4985 qed
```
```  4986
```
```  4987 lemma arctan_half:
```
```  4988   fixes x :: real
```
```  4989   shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))"
```
```  4990 proof -
```
```  4991   obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x"
```
```  4992     using tan_total by blast
```
```  4993   hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2"
```
```  4994     by auto
```
```  4995
```
```  4996   have "0 < cos y" using cos_gt_zero_pi[OF low high] .
```
```  4997   hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y"
```
```  4998     by auto
```
```  4999
```
```  5000   have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  5001     unfolding tan_def power_divide ..
```
```  5002   also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
```
```  5003     using `cos y \<noteq> 0` by auto
```
```  5004   also have "\<dots> = 1 / (cos y)\<^sup>2"
```
```  5005     unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
```
```  5006   finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" .
```
```  5007
```
```  5008   have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)"
```
```  5009     unfolding tan_def using `cos y \<noteq> 0` by (simp add: field_simps)
```
```  5010   also have "\<dots> = tan y / (1 + 1 / cos y)"
```
```  5011     using `cos y \<noteq> 0` unfolding add_divide_distrib by auto
```
```  5012   also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))"
```
```  5013     unfolding cos_sqrt ..
```
```  5014   also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))"
```
```  5015     unfolding real_sqrt_divide by auto
```
```  5016   finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))"
```
```  5017     unfolding `1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2` .
```
```  5018
```
```  5019   have "arctan x = y"
```
```  5020     using arctan_tan low high y_eq by auto
```
```  5021   also have "\<dots> = 2 * (arctan (tan (y/2)))"
```
```  5022     using arctan_tan[OF low2 high2] by auto
```
```  5023   also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))"
```
```  5024     unfolding tan_half by auto
```
```  5025   finally show ?thesis
```
```  5026     unfolding eq `tan y = x` .
```
```  5027 qed
```
```  5028
```
```  5029 lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y"
```
```  5030   by (simp only: arctan_less_iff)
```
```  5031
```
```  5032 lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y"
```
```  5033   by (simp only: arctan_le_iff)
```
```  5034
```
```  5035 lemma arctan_inverse:
```
```  5036   assumes "x \<noteq> 0"
```
```  5037   shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
```
```  5038 proof (rule arctan_unique)
```
```  5039   show "- (pi / 2) < sgn x * pi / 2 - arctan x"
```
```  5040     using arctan_bounded [of x] assms
```
```  5041     unfolding sgn_real_def
```
```  5042     apply (auto simp add: arctan algebra_simps)
```
```  5043     apply (drule zero_less_arctan_iff [THEN iffD2])
```
```  5044     apply arith
```
```  5045     done
```
```  5046   show "sgn x * pi / 2 - arctan x < pi / 2"
```
```  5047     using arctan_bounded [of "- x"] assms
```
```  5048     unfolding sgn_real_def arctan_minus
```
```  5049     by (auto simp add: algebra_simps)
```
```  5050   show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
```
```  5051     unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
```
```  5052     unfolding sgn_real_def
```
```  5053     by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
```
```  5054 qed
```
```  5055
```
```  5056 theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
```
```  5057 proof -
```
```  5058   have "pi / 4 = arctan 1" using arctan_one by auto
```
```  5059   also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
```
```  5060   finally show ?thesis by auto
```
```  5061 qed
```
```  5062
```
```  5063
```
```  5064 subsection {* Existence of Polar Coordinates *}
```
```  5065
```
```  5066 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1"
```
```  5067   apply (rule power2_le_imp_le [OF _ zero_le_one])
```
```  5068   apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
```
```  5069   done
```
```  5070
```
```  5071 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
```
```  5072
```
```  5073 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
```
```  5074
```
```  5075 lemma polar_Ex: "\<exists>r::real. \<exists>a. x = r * cos a & y = r * sin a"
```
```  5076 proof -
```
```  5077   have polar_ex1: "\<And>y. 0 < y \<Longrightarrow> \<exists>r a. x = r * cos a & y = r * sin a"
```
```  5078     apply (rule_tac x = "sqrt (x\<^sup>2 + y\<^sup>2)" in exI)
```
```  5079     apply (rule_tac x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))" in exI)
```
```  5080     apply (simp add: cos_arccos_lemma1 sin_arccos_lemma1 power_divide
```
```  5081                      real_sqrt_mult [symmetric] right_diff_distrib)
```
```  5082     done
```
```  5083   show ?thesis
```
```  5084   proof (cases "0::real" y rule: linorder_cases)
```
```  5085     case less
```
```  5086       then show ?thesis by (rule polar_ex1)
```
```  5087   next
```
```  5088     case equal
```
```  5089       then show ?thesis
```
```  5090         by (force simp add: intro!: cos_zero sin_zero)
```
```  5091   next
```
```  5092     case greater
```
```  5093       then show ?thesis
```
```  5094      using polar_ex1 [where y="-y"]
```
```  5095     by auto (metis cos_minus minus_minus minus_mult_right sin_minus)
```
```  5096   qed
```
```  5097 qed
```
```  5098
```
```  5099
```
```  5100 subsection{*Basics about polynomial functions: extremal behaviour and root counts*}
```
```  5101 (*ALL COULD GO TO COMPLEX_MAIN OR EARLIER*)
```
```  5102
```
```  5103 lemma polyfun_diff: (*COMPLEX_SUB_POLYFUN in HOL Light*)
```
```  5104     fixes x :: "'a::idom"
```
```  5105   assumes "1 \<le> n"
```
```  5106     shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
```
```  5107            (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)"
```
```  5108 proof -
```
```  5109   have h: "bij_betw (\<lambda>(i,j). (j,i)) ((SIGMA i : atMost n. lessThan i)) (SIGMA j : lessThan n. {Suc j..n})"
```
```  5110     by (auto simp: bij_betw_def inj_on_def)
```
```  5111   have "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
```
```  5112         (\<Sum>i\<le>n. a i * (x^i - y^i))"
```
```  5113     by (simp add: right_diff_distrib setsum_subtractf)
```
```  5114   also have "... = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))"
```
```  5115     by (simp add: power_diff_sumr2 mult.assoc)
```
```  5116   also have "... = (\<Sum>i\<le>n. \<Sum>j<i. a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5117     by (simp add: setsum_right_distrib)
```
```  5118   also have "... = (\<Sum>(i,j) \<in> (SIGMA i : atMost n. lessThan i). a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5119     by (simp add: setsum.Sigma)
```
```  5120   also have "... = (\<Sum>(j,i) \<in> (SIGMA j : lessThan n. {Suc j..n}). a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5121     by (auto simp add: setsum.reindex_bij_betw [OF h, symmetric] intro: setsum.strong_cong)
```
```  5122   also have "... = (\<Sum>j<n. \<Sum>i=Suc j..n. a i * (x - y) * (y^(i - Suc j) * x^j))"
```
```  5123     by (simp add: setsum.Sigma)
```
```  5124   also have "... = (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)"
```
```  5125     by (simp add: setsum_right_distrib mult_ac)
```
```  5126   finally show ?thesis .
```
```  5127 qed
```
```  5128
```
```  5129 lemma polyfun_diff_alt: (*COMPLEX_SUB_POLYFUN_ALT in HOL Light*)
```
```  5130     fixes x :: "'a::idom"
```
```  5131   assumes "1 \<le> n"
```
```  5132     shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
```
```  5133            (x - y) * ((\<Sum>j<n. \<Sum>k<n-j. a(j+k+1) * y^k * x^j))"
```
```  5134 proof -
```
```  5135   { fix j::nat
```
```  5136     assume "j<n"
```
```  5137     have h: "bij_betw (\<lambda>i. i - (j + 1)) {Suc j..n} (lessThan (n-j))"
```
```  5138       apply (auto simp: bij_betw_def inj_on_def)
```
```  5139       apply (rule_tac x="x + Suc j" in image_eqI)
```
```  5140       apply (auto simp: )
```
```  5141       done
```
```  5142     have "(\<Sum>i=Suc j..n. a i * y^(i - j - 1)) = (\<Sum>k<n-j. a(j+k+1) * y^k)"
```
```  5143       by (auto simp add: setsum.reindex_bij_betw [OF h, symmetric] intro: setsum.strong_cong)
```
```  5144   }
```
```  5145   then show ?thesis
```
```  5146     by (simp add: polyfun_diff [OF assms] setsum_left_distrib)
```
```  5147 qed
```
```  5148
```
```  5149 lemma polyfun_linear_factor:  (*COMPLEX_POLYFUN_LINEAR_FACTOR in HOL Light*)
```
```  5150   fixes a :: "'a::idom"
```
```  5151   shows "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c(i) * z^i) = (z - a) * (\<Sum>i<n. b(i) * z^i) + (\<Sum>i\<le>n. c(i) * a^i)"
```
```  5152 proof (cases "n=0")
```
```  5153   case True then show ?thesis
```
```  5154     by simp
```
```  5155 next
```
```  5156   case False
```
```  5157   have "(\<exists>b. \<forall>z. (\<Sum>i\<le>n. c(i) * z^i) = (z - a) * (\<Sum>i<n. b(i) * z^i) + (\<Sum>i\<le>n. c(i) * a^i)) =
```
```  5158         (\<exists>b. \<forall>z. (\<Sum>i\<le>n. c(i) * z^i) - (\<Sum>i\<le>n. c(i) * a^i) = (z - a) * (\<Sum>i<n. b(i) * z^i))"
```
```  5159     by (simp add: algebra_simps)
```
```  5160   also have "... = (\<exists>b. \<forall>z. (z - a) * (\<Sum>j<n. (\<Sum>i = Suc j..n. c i * a^(i - Suc j)) * z^j) = (z - a) * (\<Sum>i<n. b(i) * z^i))"
```
```  5161     using False by (simp add: polyfun_diff)
```
```  5162   also have "... = True"
```
```  5163     by auto
```
```  5164   finally show ?thesis
```
```  5165     by simp
```
```  5166 qed
```
```  5167
```
```  5168 lemma polyfun_linear_factor_root:  (*COMPLEX_POLYFUN_LINEAR_FACTOR_ROOT in HOL Light*)
```
```  5169   fixes a :: "'a::idom"
```
```  5170   assumes "(\<Sum>i\<le>n. c(i) * a^i) = 0"
```
```  5171   obtains b where "\<And>z. (\<Sum>i\<le>n. c(i) * z^i) = (z - a) * (\<Sum>i<n. b(i) * z^i)"
```
```  5172   using polyfun_linear_factor [of c n a] assms
```
```  5173   by auto
```
```  5174
```
```  5175 lemma isCont_polynom:
```
```  5176   fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
```
```  5177   shows "isCont (\<lambda>w. \<Sum>i\<le>n. c i * w^i) a"
```
```  5178   by simp
```
```  5179
```
```  5180 lemma zero_polynom_imp_zero_coeffs:
```
```  5181     fixes c :: "nat \<Rightarrow> 'a::{ab_semigroup_mult,real_normed_div_algebra}"
```
```  5182   assumes "\<And>w. (\<Sum>i\<le>n. c i * w^i) = 0"  "k \<le> n"
```
```  5183     shows "c k = 0"
```
```  5184 using assms
```
```  5185 proof (induction n arbitrary: c k)
```
```  5186   case 0
```
```  5187   then show ?case
```
```  5188     by simp
```
```  5189 next
```
```  5190   case (Suc n c k)
```
```  5191   have [simp]: "c 0 = 0" using Suc.prems(1) [of 0]
```
```  5192     by simp
```
```  5193   { fix w
```
```  5194     have "(\<Sum>i\<le>Suc n. c i * w^i) = (\<Sum>i\<le>n. c (Suc i) * w ^ Suc i)"
```
```  5195       unfolding Set_Interval.setsum_atMost_Suc_shift
```
```  5196       by simp
```
```  5197     also have "... = w * (\<Sum>i\<le>n. c (Suc i) * w^i)"
```
```  5198       by (simp add: power_Suc mult_ac setsum_right_distrib del: setsum_atMost_Suc)
```
```  5199     finally have "(\<Sum>i\<le>Suc n. c i * w^i) = w * (\<Sum>i\<le>n. c (Suc i) * w^i)" .
```
```  5200   }
```
```  5201   then have wnz: "\<And>w. w \<noteq> 0 \<Longrightarrow> (\<Sum>i\<le>n. c (Suc i) * w^i) = 0"
```
```  5202     using Suc  by auto
```
```  5203   then have "(\<lambda>h. \<Sum>i\<le>n. c (Suc i) * h^i) -- 0 --> 0"
```
```  5204     by (simp cong: LIM_cong)                   --{*the case @{term"w=0"} by continuity}*}
```
```  5205   then have "(\<Sum>i\<le>n. c (Suc i) * 0^i) = 0"
```
```  5206     using isCont_polynom [of 0 "\<lambda>i. c (Suc i)" n] LIM_unique
```
```  5207     by (force simp add: Limits.isCont_iff)
```
```  5208   then have "\<And>w. (\<Sum>i\<le>n. c (Suc i) * w^i) = 0" using wnz
```
```  5209     by metis
```
```  5210   then have "\<And>i. i\<le>n \<Longrightarrow> c (Suc i) = 0"
```
```  5211     using Suc.IH [of "\<lambda>i. c (Suc i)"]
```
```  5212     by blast
```
```  5213   then show ?case using `k \<le> Suc n`
```
```  5214     by (cases k) auto
```
```  5215 qed
```
```  5216
```
```  5217 lemma polyfun_rootbound: (*COMPLEX_POLYFUN_ROOTBOUND in HOL Light*)
```
```  5218     fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5219   assumes "c k \<noteq> 0" "k\<le>n"
```
```  5220     shows "finite {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<and>
```
```  5221              card {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<le> n"
```
```  5222 using assms
```
```  5223 proof (induction n arbitrary: c k)
```
```  5224   case 0
```
```  5225   then show ?case
```
```  5226     by simp
```
```  5227 next
```
```  5228   case (Suc m c k)
```
```  5229   let ?succase = ?case
```
```  5230   show ?case
```
```  5231   proof (cases "{z. (\<Sum>i\<le>Suc m. c(i) * z^i) = 0} = {}")
```
```  5232     case True
```
```  5233     then show ?succase
```
```  5234       by simp
```
```  5235   next
```
```  5236     case False
```
```  5237     then obtain z0 where z0: "(\<Sum>i\<le>Suc m. c(i) * z0^i) = 0"
```
```  5238       by blast
```
```  5239     then obtain b where b: "\<And>w. (\<Sum>i\<le>Suc m. c i * w^i) = (w - z0) * (\<Sum>i\<le>m. b i * w^i)"
```
```  5240       using polyfun_linear_factor_root [OF z0, unfolded lessThan_Suc_atMost]
```
```  5241       by blast
```
```  5242     then have eq: "{z. (\<Sum>i\<le>Suc m. c(i) * z^i) = 0} = insert z0 {z. (\<Sum>i\<le>m. b(i) * z^i) = 0}"
```
```  5243       by auto
```
```  5244     have "~(\<forall>k\<le>m. b k = 0)"
```
```  5245     proof
```
```  5246       assume [simp]: "\<forall>k\<le>m. b k = 0"
```
```  5247       then have "\<And>w. (\<Sum>i\<le>m. b i * w^i) = 0"
```
```  5248         by simp
```
```  5249       then have "\<And>w. (\<Sum>i\<le>Suc m. c i * w^i) = 0"
```
```  5250         using b by simp
```
```  5251       then have "\<And>k. k \<le> Suc m \<Longrightarrow> c k = 0"
```
```  5252         using zero_polynom_imp_zero_coeffs
```
```  5253         by blast
```
```  5254       then show False using Suc.prems
```
```  5255         by blast
```
```  5256     qed
```
```  5257     then obtain k' where bk': "b k' \<noteq> 0" "k' \<le> m"
```
```  5258       by blast
```
```  5259     show ?succase
```
```  5260       using Suc.IH [of b k'] bk'
```
```  5261       by (simp add: eq card_insert_if del: setsum_atMost_Suc)
```
```  5262     qed
```
```  5263 qed
```
```  5264
```
```  5265 lemma
```
```  5266     fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5267   assumes "c k \<noteq> 0" "k\<le>n"
```
```  5268     shows polyfun_roots_finite: "finite {z. (\<Sum>i\<le>n. c(i) * z^i) = 0}"
```
```  5269       and polyfun_roots_card:   "card {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<le> n"
```
```  5270 using polyfun_rootbound assms
```
```  5271   by auto
```
```  5272
```
```  5273 lemma polyfun_finite_roots: (*COMPLEX_POLYFUN_FINITE_ROOTS in HOL Light*)
```
```  5274   fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5275   shows "finite {x. (\<Sum>i\<le>n. c i * x^i) = 0} \<longleftrightarrow> (\<exists>i\<le>n. c i \<noteq> 0)"
```
```  5276         (is "?lhs = ?rhs")
```
```  5277 proof
```
```  5278   assume ?lhs
```
```  5279   moreover
```
```  5280   { assume "\<forall>i\<le>n. c i = 0"
```
```  5281     then have "\<And>x. (\<Sum>i\<le>n. c i * x^i) = 0"
```
```  5282       by simp
```
```  5283     then have "\<not> finite {x. (\<Sum>i\<le>n. c i * x^i) = 0}"
```
```  5284       using ex_new_if_finite [OF infinite_UNIV_char_0 [where 'a='a]]
```
```  5285       by auto
```
```  5286   }
```
```  5287   ultimately show ?rhs
```
```  5288   by metis
```
```  5289 next
```
```  5290   assume ?rhs
```
```  5291   then show ?lhs
```
```  5292     using polyfun_rootbound
```
```  5293     by blast
```
```  5294 qed
```
```  5295
```
```  5296 lemma polyfun_eq_0: (*COMPLEX_POLYFUN_EQ_0 in HOL Light*)
```
```  5297   fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5298   shows "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = 0) \<longleftrightarrow> (\<forall>i\<le>n. c i = 0)"
```
```  5299   using zero_polynom_imp_zero_coeffs by auto
```
```  5300
```
```  5301 lemma polyfun_eq_coeffs:
```
```  5302   fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5303   shows "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = (\<Sum>i\<le>n. d i * x^i)) \<longleftrightarrow> (\<forall>i\<le>n. c i = d i)"
```
```  5304 proof -
```
```  5305   have "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = (\<Sum>i\<le>n. d i * x^i)) \<longleftrightarrow> (\<forall>x. (\<Sum>i\<le>n. (c i - d i) * x^i) = 0)"
```
```  5306     by (simp add: left_diff_distrib Groups_Big.setsum_subtractf)
```
```  5307   also have "... \<longleftrightarrow> (\<forall>i\<le>n. c i - d i = 0)"
```
```  5308     by (rule polyfun_eq_0)
```
```  5309   finally show ?thesis
```
```  5310     by simp
```
```  5311 qed
```
```  5312
```
```  5313 lemma polyfun_eq_const: (*COMPLEX_POLYFUN_EQ_CONST in HOL Light*)
```
```  5314   fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}"
```
```  5315   shows "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = k) \<longleftrightarrow> c 0 = k \<and> (\<forall>i \<in> {1..n}. c i = 0)"
```
```  5316         (is "?lhs = ?rhs")
```
```  5317 proof -
```
```  5318   have *: "\<forall>x. (\<Sum>i\<le>n. (if i=0 then k else 0) * x^i) = k"
```
```  5319     by (induct n) auto
```
```  5320   show ?thesis
```
```  5321   proof
```
```  5322     assume ?lhs
```
```  5323     with * have "(\<forall>i\<le>n. c i = (if i=0 then k else 0))"
```
```  5324       by (simp add: polyfun_eq_coeffs [symmetric])
```
```  5325     then show ?rhs
```
```  5326       by simp
```
```  5327   next
```
```  5328     assume ?rhs then show ?lhs
```
```  5329       by (induct n) auto
```
```  5330   qed
```
```  5331 qed
```
```  5332
```
```  5333 lemma root_polyfun:
```
```  5334   fixes z:: "'a::idom"
```
```  5335   assumes "1 \<le> n"
```
```  5336     shows "z^n = a \<longleftrightarrow> (\<Sum>i\<le>n. (if i = 0 then -a else if i=n then 1 else 0) * z^i) = 0"
```
```  5337   using assms
```
```  5338   by (cases n; simp add: setsum_head_Suc atLeast0AtMost [symmetric])
```
```  5339
```
```  5340 lemma
```
```  5341     fixes zz :: "'a::{idom,real_normed_div_algebra}"
```
```  5342   assumes "1 \<le> n"
```
```  5343     shows finite_roots_unity: "finite {z::'a. z^n = 1}"
```
```  5344       and card_roots_unity:   "card {z::'a. z^n = 1} \<le> n"
```
```  5345   using polyfun_rootbound [of "\<lambda>i. if i = 0 then -1 else if i=n then 1 else 0" n n] assms
```
```  5346   by (auto simp add: root_polyfun [OF assms])
```
```  5347
```
```  5348 end
```