src/HOL/Transcendental.thy
 author wenzelm Tue Sep 26 20:54:40 2017 +0200 (21 months ago) changeset 66695 91500c024c7f parent 66521 b48077ae8b12 child 66827 c94531b5007d permissions -rw-r--r--
tuned;
1 (*  Title:      HOL/Transcendental.thy
2     Author:     Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
3     Author:     Lawrence C Paulson
5 *)
7 section \<open>Power Series, Transcendental Functions etc.\<close>
9 theory Transcendental
10 imports Series Deriv NthRoot
11 begin
13 text \<open>A fact theorem on reals.\<close>
15 lemma square_fact_le_2_fact: "fact n * fact n \<le> (fact (2 * n) :: real)"
16 proof (induct n)
17   case 0
18   then show ?case by simp
19 next
20   case (Suc n)
21   have "(fact (Suc n)) * (fact (Suc n)) = of_nat (Suc n) * of_nat (Suc n) * (fact n * fact n :: real)"
23   also have "\<dots> \<le> of_nat (Suc n) * of_nat (Suc n) * fact (2 * n)"
24     by (rule mult_left_mono [OF Suc]) simp
25   also have "\<dots> \<le> of_nat (Suc (Suc (2 * n))) * of_nat (Suc (2 * n)) * fact (2 * n)"
26     by (rule mult_right_mono)+ (auto simp: field_simps)
27   also have "\<dots> = fact (2 * Suc n)" by (simp add: field_simps)
28   finally show ?case .
29 qed
31 lemma fact_in_Reals: "fact n \<in> \<real>"
32   by (induction n) auto
34 lemma of_real_fact [simp]: "of_real (fact n) = fact n"
35   by (metis of_nat_fact of_real_of_nat_eq)
37 lemma pochhammer_of_real: "pochhammer (of_real x) n = of_real (pochhammer x n)"
40 lemma norm_fact [simp]: "norm (fact n :: 'a::real_normed_algebra_1) = fact n"
41 proof -
42   have "(fact n :: 'a) = of_real (fact n)"
43     by simp
44   also have "norm \<dots> = fact n"
45     by (subst norm_of_real) simp
46   finally show ?thesis .
47 qed
49 lemma root_test_convergence:
50   fixes f :: "nat \<Rightarrow> 'a::banach"
51   assumes f: "(\<lambda>n. root n (norm (f n))) \<longlonglongrightarrow> x" \<comment> "could be weakened to lim sup"
52     and "x < 1"
53   shows "summable f"
54 proof -
55   have "0 \<le> x"
56     by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1])
57   from \<open>x < 1\<close> obtain z where z: "x < z" "z < 1"
58     by (metis dense)
59   from f \<open>x < z\<close> have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
60     by (rule order_tendstoD)
61   then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially"
62     using eventually_ge_at_top
63   proof eventually_elim
64     fix n
65     assume less: "root n (norm (f n)) < z" and n: "1 \<le> n"
66     from power_strict_mono[OF less, of n] n show "norm (f n) \<le> z ^ n"
67       by simp
68   qed
69   then show "summable f"
70     unfolding eventually_sequentially
71     using z \<open>0 \<le> x\<close> by (auto intro!: summable_comparison_test[OF _  summable_geometric])
72 qed
74 subsection \<open>More facts about binomial coefficients\<close>
76 text \<open>
77   These facts could have been proven before, but having real numbers
78   makes the proofs a lot easier.
79 \<close>
81 lemma central_binomial_odd:
82   "odd n \<Longrightarrow> n choose (Suc (n div 2)) = n choose (n div 2)"
83 proof -
84   assume "odd n"
85   hence "Suc (n div 2) \<le> n" by presburger
86   hence "n choose (Suc (n div 2)) = n choose (n - Suc (n div 2))"
87     by (rule binomial_symmetric)
88   also from \<open>odd n\<close> have "n - Suc (n div 2) = n div 2" by presburger
89   finally show ?thesis .
90 qed
92 lemma binomial_less_binomial_Suc:
93   assumes k: "k < n div 2"
94   shows   "n choose k < n choose (Suc k)"
95 proof -
96   from k have k': "k \<le> n" "Suc k \<le> n" by simp_all
97   from k' have "real (n choose k) = fact n / (fact k * fact (n - k))"
99   also from k' have "n - k = Suc (n - Suc k)" by simp
100   also from k' have "fact \<dots> = (real n - real k) * fact (n - Suc k)"
101     by (subst fact_Suc) (simp_all add: of_nat_diff)
102   also from k have "fact k = fact (Suc k) / (real k + 1)" by (simp add: field_simps)
103   also have "fact n / (fact (Suc k) / (real k + 1) * ((real n - real k) * fact (n - Suc k))) =
104                (n choose (Suc k)) * ((real k + 1) / (real n - real k))"
105     using k by (simp add: divide_simps binomial_fact)
106   also from assms have "(real k + 1) / (real n - real k) < 1" by simp
107   finally show ?thesis using k by (simp add: mult_less_cancel_left)
108 qed
110 lemma binomial_strict_mono:
111   assumes "k < k'" "2*k' \<le> n"
112   shows   "n choose k < n choose k'"
113 proof -
114   from assms have "k \<le> k' - 1" by simp
115   thus ?thesis
116   proof (induction rule: inc_induct)
117     case base
118     with assms binomial_less_binomial_Suc[of "k' - 1" n]
119       show ?case by simp
120   next
121     case (step k)
122     from step.prems step.hyps assms have "n choose k < n choose (Suc k)"
123       by (intro binomial_less_binomial_Suc) simp_all
124     also have "\<dots> < n choose k'" by (rule step.IH)
125     finally show ?case .
126   qed
127 qed
129 lemma binomial_mono:
130   assumes "k \<le> k'" "2*k' \<le> n"
131   shows   "n choose k \<le> n choose k'"
132   using assms binomial_strict_mono[of k k' n] by (cases "k = k'") simp_all
134 lemma binomial_strict_antimono:
135   assumes "k < k'" "2 * k \<ge> n" "k' \<le> n"
136   shows   "n choose k > n choose k'"
137 proof -
138   from assms have "n choose (n - k) > n choose (n - k')"
139     by (intro binomial_strict_mono) (simp_all add: algebra_simps)
140   with assms show ?thesis by (simp add: binomial_symmetric [symmetric])
141 qed
143 lemma binomial_antimono:
144   assumes "k \<le> k'" "k \<ge> n div 2" "k' \<le> n"
145   shows   "n choose k \<ge> n choose k'"
146 proof (cases "k = k'")
147   case False
148   note not_eq = False
149   show ?thesis
150   proof (cases "k = n div 2 \<and> odd n")
151     case False
152     with assms(2) have "2*k \<ge> n" by presburger
153     with not_eq assms binomial_strict_antimono[of k k' n]
154       show ?thesis by simp
155   next
156     case True
157     have "n choose k' \<le> n choose (Suc (n div 2))"
158     proof (cases "k' = Suc (n div 2)")
159       case False
160       with assms True not_eq have "Suc (n div 2) < k'" by simp
161       with assms binomial_strict_antimono[of "Suc (n div 2)" k' n] True
162         show ?thesis by auto
163     qed simp_all
164     also from True have "\<dots> = n choose k" by (simp add: central_binomial_odd)
165     finally show ?thesis .
166   qed
167 qed simp_all
169 lemma binomial_maximum: "n choose k \<le> n choose (n div 2)"
170 proof -
171   have "k \<le> n div 2 \<longleftrightarrow> 2*k \<le> n" by linarith
172   consider "2*k \<le> n" | "2*k \<ge> n" "k \<le> n" | "k > n" by linarith
173   thus ?thesis
174   proof cases
175     case 1
176     thus ?thesis by (intro binomial_mono) linarith+
177   next
178     case 2
179     thus ?thesis by (intro binomial_antimono) simp_all
181 qed
183 lemma binomial_maximum': "(2*n) choose k \<le> (2*n) choose n"
184   using binomial_maximum[of "2*n"] by simp
186 lemma central_binomial_lower_bound:
187   assumes "n > 0"
188   shows   "4^n / (2*real n) \<le> real ((2*n) choose n)"
189 proof -
190   from binomial[of 1 1 "2*n"]
191     have "4 ^ n = (\<Sum>k=0..2*n. (2*n) choose k)"
192     by (simp add: power_mult power2_eq_square One_nat_def [symmetric] del: One_nat_def)
193   also have "{0..2*n} = {0<..<2*n} \<union> {0,2*n}" by auto
194   also have "(\<Sum>k\<in>\<dots>. (2*n) choose k) =
195                (\<Sum>k\<in>{0<..<2*n}. (2*n) choose k) + (\<Sum>k\<in>{0,2*n}. (2*n) choose k)"
196     by (subst sum.union_disjoint) auto
197   also have "(\<Sum>k\<in>{0,2*n}. (2*n) choose k) \<le> (\<Sum>k\<le>1. (n choose k)\<^sup>2)"
198     by (cases n) simp_all
199   also from assms have "\<dots> \<le> (\<Sum>k\<le>n. (n choose k)\<^sup>2)"
200     by (intro sum_mono2) auto
201   also have "\<dots> = (2*n) choose n" by (rule choose_square_sum)
202   also have "(\<Sum>k\<in>{0<..<2*n}. (2*n) choose k) \<le> (\<Sum>k\<in>{0<..<2*n}. (2*n) choose n)"
203     by (intro sum_mono binomial_maximum')
204   also have "\<dots> = card {0<..<2*n} * ((2*n) choose n)" by simp
205   also have "card {0<..<2*n} \<le> 2*n - 1" by (cases n) simp_all
206   also have "(2 * n - 1) * (2 * n choose n) + (2 * n choose n) = ((2*n) choose n) * (2*n)"
207     using assms by (simp add: algebra_simps)
208   finally have "4 ^ n \<le> (2 * n choose n) * (2 * n)" by simp_all
209   hence "real (4 ^ n) \<le> real ((2 * n choose n) * (2 * n))"
210     by (subst of_nat_le_iff)
211   with assms show ?thesis by (simp add: field_simps)
212 qed
215 subsection \<open>Properties of Power Series\<close>
217 lemma powser_zero [simp]: "(\<Sum>n. f n * 0 ^ n) = f 0"
218   for f :: "nat \<Rightarrow> 'a::real_normed_algebra_1"
219 proof -
220   have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
221     by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
222   then show ?thesis by simp
223 qed
225 lemma powser_sums_zero: "(\<lambda>n. a n * 0^n) sums a 0"
226   for a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
227   using sums_finite [of "{0}" "\<lambda>n. a n * 0 ^ n"]
228   by simp
230 lemma powser_sums_zero_iff [simp]: "(\<lambda>n. a n * 0^n) sums x \<longleftrightarrow> a 0 = x"
231   for a :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
232   using powser_sums_zero sums_unique2 by blast
234 text \<open>
235   Power series has a circle or radius of convergence: if it sums for \<open>x\<close>,
236   then it sums absolutely for \<open>z\<close> with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.\<close>
238 lemma powser_insidea:
239   fixes x z :: "'a::real_normed_div_algebra"
240   assumes 1: "summable (\<lambda>n. f n * x^n)"
241     and 2: "norm z < norm x"
242   shows "summable (\<lambda>n. norm (f n * z ^ n))"
243 proof -
244   from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
245   from 1 have "(\<lambda>n. f n * x^n) \<longlonglongrightarrow> 0"
246     by (rule summable_LIMSEQ_zero)
247   then have "convergent (\<lambda>n. f n * x^n)"
248     by (rule convergentI)
249   then have "Cauchy (\<lambda>n. f n * x^n)"
250     by (rule convergent_Cauchy)
251   then have "Bseq (\<lambda>n. f n * x^n)"
252     by (rule Cauchy_Bseq)
253   then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x^n) \<le> K"
254     by (auto simp add: Bseq_def)
255   have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le> K * norm (z ^ n) * inverse (norm (x^n))"
256   proof (intro exI allI impI)
257     fix n :: nat
258     assume "0 \<le> n"
259     have "norm (norm (f n * z ^ n)) * norm (x^n) =
260           norm (f n * x^n) * norm (z ^ n)"
261       by (simp add: norm_mult abs_mult)
262     also have "\<dots> \<le> K * norm (z ^ n)"
263       by (simp only: mult_right_mono 4 norm_ge_zero)
264     also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x^n)) * norm (x^n))"
266     also have "\<dots> = K * norm (z ^ n) * inverse (norm (x^n)) * norm (x^n)"
267       by (simp only: mult.assoc)
268     finally show "norm (norm (f n * z ^ n)) \<le> K * norm (z ^ n) * inverse (norm (x^n))"
269       by (simp add: mult_le_cancel_right x_neq_0)
270   qed
271   moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
272   proof -
273     from 2 have "norm (norm (z * inverse x)) < 1"
274       using x_neq_0
275       by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
276     then have "summable (\<lambda>n. norm (z * inverse x) ^ n)"
277       by (rule summable_geometric)
278     then have "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
279       by (rule summable_mult)
280     then show "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))"
281       using x_neq_0
282       by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
283           power_inverse norm_power mult.assoc)
284   qed
285   ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
286     by (rule summable_comparison_test)
287 qed
289 lemma powser_inside:
290   fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
291   shows
292     "summable (\<lambda>n. f n * (x^n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
293       summable (\<lambda>n. f n * (z ^ n))"
294   by (rule powser_insidea [THEN summable_norm_cancel])
296 lemma powser_times_n_limit_0:
297   fixes x :: "'a::{real_normed_div_algebra,banach}"
298   assumes "norm x < 1"
299     shows "(\<lambda>n. of_nat n * x ^ n) \<longlonglongrightarrow> 0"
300 proof -
301   have "norm x / (1 - norm x) \<ge> 0"
302     using assms by (auto simp: divide_simps)
303   moreover obtain N where N: "norm x / (1 - norm x) < of_int N"
304     using ex_le_of_int by (meson ex_less_of_int)
305   ultimately have N0: "N>0"
306     by auto
307   then have *: "real_of_int (N + 1) * norm x / real_of_int N < 1"
308     using N assms by (auto simp: field_simps)
309   have **: "real_of_int N * (norm x * (real_of_nat (Suc n) * norm (x ^ n))) \<le>
310       real_of_nat n * (norm x * ((1 + N) * norm (x ^ n)))" if "N \<le> int n" for n :: nat
311   proof -
312     from that have "real_of_int N * real_of_nat (Suc n) \<le> real_of_nat n * real_of_int (1 + N)"
314     then have "(real_of_int N * real_of_nat (Suc n)) * (norm x * norm (x ^ n)) \<le>
315         (real_of_nat n *  (1 + N)) * (norm x * norm (x ^ n))"
316       using N0 mult_mono by fastforce
317     then show ?thesis
319   qed
320   show ?thesis using *
321     by (rule summable_LIMSEQ_zero [OF summable_ratio_test, where N1="nat N"])
323 qed
325 corollary lim_n_over_pown:
326   fixes x :: "'a::{real_normed_field,banach}"
327   shows "1 < norm x \<Longrightarrow> ((\<lambda>n. of_nat n / x^n) \<longlongrightarrow> 0) sequentially"
328   using powser_times_n_limit_0 [of "inverse x"]
329   by (simp add: norm_divide divide_simps)
331 lemma sum_split_even_odd:
332   fixes f :: "nat \<Rightarrow> real"
333   shows "(\<Sum>i<2 * n. if even i then f i else g i) = (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
334 proof (induct n)
335   case 0
336   then show ?case by simp
337 next
338   case (Suc n)
339   have "(\<Sum>i<2 * Suc n. if even i then f i else g i) =
340     (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
341     using Suc.hyps unfolding One_nat_def by auto
342   also have "\<dots> = (\<Sum>i<Suc n. f (2 * i)) + (\<Sum>i<Suc n. g (2 * i + 1))"
343     by auto
344   finally show ?case .
345 qed
347 lemma sums_if':
348   fixes g :: "nat \<Rightarrow> real"
349   assumes "g sums x"
350   shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
351   unfolding sums_def
352 proof (rule LIMSEQ_I)
353   fix r :: real
354   assume "0 < r"
355   from \<open>g sums x\<close>[unfolded sums_def, THEN LIMSEQ_D, OF this]
356   obtain no where no_eq: "\<And>n. n \<ge> no \<Longrightarrow> (norm (sum g {..<n} - x) < r)"
357     by blast
359   let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)"
360   have "(norm (?SUM m - x) < r)" if "m \<ge> 2 * no" for m
361   proof -
362     from that have "m div 2 \<ge> no" by auto
363     have sum_eq: "?SUM (2 * (m div 2)) = sum g {..< m div 2}"
364       using sum_split_even_odd by auto
365     then have "(norm (?SUM (2 * (m div 2)) - x) < r)"
366       using no_eq unfolding sum_eq using \<open>m div 2 \<ge> no\<close> by auto
367     moreover
368     have "?SUM (2 * (m div 2)) = ?SUM m"
369     proof (cases "even m")
370       case True
371       then show ?thesis
372         by (auto simp add: even_two_times_div_two)
373     next
374       case False
375       then have eq: "Suc (2 * (m div 2)) = m" by simp
376       then have "even (2 * (m div 2))" using \<open>odd m\<close> by auto
377       have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
378       also have "\<dots> = ?SUM (2 * (m div 2))" using \<open>even (2 * (m div 2))\<close> by auto
379       finally show ?thesis by auto
380     qed
381     ultimately show ?thesis by auto
382   qed
383   then show "\<exists>no. \<forall> m \<ge> no. norm (?SUM m - x) < r"
384     by blast
385 qed
387 lemma sums_if:
388   fixes g :: "nat \<Rightarrow> real"
389   assumes "g sums x" and "f sums y"
390   shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
391 proof -
392   let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
393   have if_sum: "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
394     for B T E
395     by (cases B) auto
396   have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
397     using sums_if'[OF \<open>g sums x\<close>] .
398   have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)"
399     by auto
400   have "?s sums y" using sums_if'[OF \<open>f sums y\<close>] .
401   from this[unfolded sums_def, THEN LIMSEQ_Suc]
402   have "(\<lambda>n. if even n then f (n div 2) else 0) sums y"
403     by (simp add: lessThan_Suc_eq_insert_0 sum_atLeast1_atMost_eq image_Suc_lessThan
404         if_eq sums_def cong del: if_weak_cong)
405   from sums_add[OF g_sums this] show ?thesis
406     by (simp only: if_sum)
407 qed
409 subsection \<open>Alternating series test / Leibniz formula\<close>
410 (* FIXME: generalise these results from the reals via type classes? *)
412 lemma sums_alternating_upper_lower:
413   fixes a :: "nat \<Rightarrow> real"
414   assumes mono: "\<And>n. a (Suc n) \<le> a n"
415     and a_pos: "\<And>n. 0 \<le> a n"
416     and "a \<longlonglongrightarrow> 0"
417   shows "\<exists>l. ((\<forall>n. (\<Sum>i<2*n. (- 1)^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i<2*n. (- 1)^i*a i) \<longlonglongrightarrow> l) \<and>
418              ((\<forall>n. l \<le> (\<Sum>i<2*n + 1. (- 1)^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. (- 1)^i*a i) \<longlonglongrightarrow> l)"
419   (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
420 proof (rule nested_sequence_unique)
421   have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" by auto
423   show "\<forall>n. ?f n \<le> ?f (Suc n)"
424   proof
425     show "?f n \<le> ?f (Suc n)" for n
426       using mono[of "2*n"] by auto
427   qed
428   show "\<forall>n. ?g (Suc n) \<le> ?g n"
429   proof
430     show "?g (Suc n) \<le> ?g n" for n
431       using mono[of "Suc (2*n)"] by auto
432   qed
433   show "\<forall>n. ?f n \<le> ?g n"
434   proof
435     show "?f n \<le> ?g n" for n
436       using fg_diff a_pos by auto
437   qed
438   show "(\<lambda>n. ?f n - ?g n) \<longlonglongrightarrow> 0"
439     unfolding fg_diff
440   proof (rule LIMSEQ_I)
441     fix r :: real
442     assume "0 < r"
443     with \<open>a \<longlonglongrightarrow> 0\<close>[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
444       by auto
445     then have "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r"
446       by auto
447     then show "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r"
448       by auto
449   qed
450 qed
452 lemma summable_Leibniz':
453   fixes a :: "nat \<Rightarrow> real"
454   assumes a_zero: "a \<longlonglongrightarrow> 0"
455     and a_pos: "\<And>n. 0 \<le> a n"
456     and a_monotone: "\<And>n. a (Suc n) \<le> a n"
457   shows summable: "summable (\<lambda> n. (-1)^n * a n)"
458     and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
459     and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)"
460     and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i<2*n+1. (-1)^i*a i)"
461     and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)"
462 proof -
463   let ?S = "\<lambda>n. (-1)^n * a n"
464   let ?P = "\<lambda>n. \<Sum>i<n. ?S i"
465   let ?f = "\<lambda>n. ?P (2 * n)"
466   let ?g = "\<lambda>n. ?P (2 * n + 1)"
467   obtain l :: real
468     where below_l: "\<forall> n. ?f n \<le> l"
469       and "?f \<longlonglongrightarrow> l"
470       and above_l: "\<forall> n. l \<le> ?g n"
471       and "?g \<longlonglongrightarrow> l"
472     using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
474   let ?Sa = "\<lambda>m. \<Sum>n<m. ?S n"
475   have "?Sa \<longlonglongrightarrow> l"
476   proof (rule LIMSEQ_I)
477     fix r :: real
478     assume "0 < r"
479     with \<open>?f \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
480     obtain f_no where f: "\<And>n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r"
481       by auto
482     from \<open>0 < r\<close> \<open>?g \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
483     obtain g_no where g: "\<And>n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r"
484       by auto
485     have "norm (?Sa n - l) < r" if "n \<ge> (max (2 * f_no) (2 * g_no))" for n
486     proof -
487       from that have "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
488       show ?thesis
489       proof (cases "even n")
490         case True
491         then have n_eq: "2 * (n div 2) = n"
493         with \<open>n \<ge> 2 * f_no\<close> have "n div 2 \<ge> f_no"
494           by auto
495         from f[OF this] show ?thesis
496           unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
497       next
498         case False
499         then have "even (n - 1)" by simp
500         then have n_eq: "2 * ((n - 1) div 2) = n - 1"
502         then have range_eq: "n - 1 + 1 = n"
503           using odd_pos[OF False] by auto
504         from n_eq \<open>n \<ge> 2 * g_no\<close> have "(n - 1) div 2 \<ge> g_no"
505           by auto
506         from g[OF this] show ?thesis
507           by (simp only: n_eq range_eq)
508       qed
509     qed
510     then show "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
511   qed
512   then have sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
513     by (simp only: sums_def)
514   then show "summable ?S"
515     by (auto simp: summable_def)
517   have "l = suminf ?S" by (rule sums_unique[OF sums_l])
519   fix n
520   show "suminf ?S \<le> ?g n"
521     unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
522   show "?f n \<le> suminf ?S"
523     unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
524   show "?g \<longlonglongrightarrow> suminf ?S"
525     using \<open>?g \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto
526   show "?f \<longlonglongrightarrow> suminf ?S"
527     using \<open>?f \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto
528 qed
530 theorem summable_Leibniz:
531   fixes a :: "nat \<Rightarrow> real"
532   assumes a_zero: "a \<longlonglongrightarrow> 0"
533     and "monoseq a"
534   shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
535     and "0 < a 0 \<longrightarrow>
536       (\<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")
537     and "a 0 < 0 \<longrightarrow>
538       (\<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")
539     and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?f")
540     and "(\<lambda>n. \<Sum>i<2*n+1. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?g")
541 proof -
542   have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
543   proof (cases "(\<forall>n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
544     case True
545     then have ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m"
546       and ge0: "\<And>n. 0 \<le> a n"
547       by auto
548     have mono: "a (Suc n) \<le> a n" for n
549       using ord[where n="Suc n" and m=n] by auto
550     note leibniz = summable_Leibniz'[OF \<open>a \<longlonglongrightarrow> 0\<close> ge0]
551     from leibniz[OF mono]
552     show ?thesis using \<open>0 \<le> a 0\<close> by auto
553   next
554     let ?a = "\<lambda>n. - a n"
555     case False
556     with monoseq_le[OF \<open>monoseq a\<close> \<open>a \<longlonglongrightarrow> 0\<close>]
557     have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
558     then have ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
559       by auto
560     have monotone: "?a (Suc n) \<le> ?a n" for n
561       using ord[where n="Suc n" and m=n] by auto
562     note leibniz =
563       summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
564         OF tendsto_minus[OF \<open>a \<longlonglongrightarrow> 0\<close>, unfolded minus_zero] monotone]
565     have "summable (\<lambda> n. (-1)^n * ?a n)"
566       using leibniz(1) by auto
567     then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
568       unfolding summable_def by auto
569     from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
570       by auto
571     then have ?summable by (auto simp: summable_def)
572     moreover
573     have "\<bar>- a - - b\<bar> = \<bar>a - b\<bar>" for a b :: real
574       unfolding minus_diff_minus by auto
576     from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
577     have move_minus: "(\<Sum>n. - ((- 1) ^ n * a n)) = - (\<Sum>n. (- 1) ^ n * a n)"
578       by auto
580     have ?pos using \<open>0 \<le> ?a 0\<close> by auto
581     moreover have ?neg
582       using leibniz(2,4)
583       unfolding mult_minus_right sum_negf move_minus neg_le_iff_le
584       by auto
585     moreover have ?f and ?g
586       using leibniz(3,5)[unfolded mult_minus_right sum_negf move_minus, THEN tendsto_minus_cancel]
587       by auto
588     ultimately show ?thesis by auto
589   qed
590   then show ?summable and ?pos and ?neg and ?f and ?g
591     by safe
592 qed
595 subsection \<open>Term-by-Term Differentiability of Power Series\<close>
597 definition diffs :: "(nat \<Rightarrow> 'a::ring_1) \<Rightarrow> nat \<Rightarrow> 'a"
598   where "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))"
600 text \<open>Lemma about distributing negation over it.\<close>
601 lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
604 lemma diffs_equiv:
605   fixes x :: "'a::{real_normed_vector,ring_1}"
606   shows "summable (\<lambda>n. diffs c n * x^n) \<Longrightarrow>
607     (\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)"
608   unfolding diffs_def
609   by (simp add: summable_sums sums_Suc_imp)
611 lemma lemma_termdiff1:
612   fixes z :: "'a :: {monoid_mult,comm_ring}"
613   shows "(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
614     (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
617 lemma sumr_diff_mult_const2: "sum f {..<n} - of_nat n * r = (\<Sum>i<n. f i - r)"
618   for r :: "'a::ring_1"
621 lemma lemma_realpow_rev_sumr:
622   "(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) = (\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))"
623   by (subst nat_diff_sum_reindex[symmetric]) simp
625 lemma lemma_termdiff2:
626   fixes h :: "'a::field"
627   assumes h: "h \<noteq> 0"
628   shows "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
629     h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))"
630     (is "?lhs = ?rhs")
631   apply (subgoal_tac "h * ?lhs = h * ?rhs")
633   apply (simp add: right_diff_distrib diff_divide_distrib h)
634   apply (simp add: mult.assoc [symmetric])
635   apply (cases n)
636   apply simp
637   apply (simp add: diff_power_eq_sum h right_diff_distrib [symmetric] mult.assoc
638       del: power_Suc sum_lessThan_Suc of_nat_Suc)
639   apply (subst lemma_realpow_rev_sumr)
640   apply (subst sumr_diff_mult_const2)
641   apply simp
642   apply (simp only: lemma_termdiff1 sum_distrib_left)
643   apply (rule sum.cong [OF refl])
645   apply clarify
646   apply (simp add: sum_distrib_left diff_power_eq_sum ac_simps
647       del: sum_lessThan_Suc power_Suc)
648   apply (subst mult.assoc [symmetric], subst power_add [symmetric])
650   done
652 lemma real_sum_nat_ivl_bounded2:
653   fixes K :: "'a::linordered_semidom"
654   assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
655     and K: "0 \<le> K"
656   shows "sum f {..<n-k} \<le> of_nat n * K"
657   apply (rule order_trans [OF sum_mono])
658    apply (rule f)
659    apply simp
660   apply (simp add: mult_right_mono K)
661   done
663 lemma lemma_termdiff3:
664   fixes h z :: "'a::real_normed_field"
665   assumes 1: "h \<noteq> 0"
666     and 2: "norm z \<le> K"
667     and 3: "norm (z + h) \<le> K"
668   shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) \<le>
669     of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
670 proof -
671   have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
672     norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
673     by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult)
674   also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
675   proof (rule mult_right_mono [OF _ norm_ge_zero])
676     from norm_ge_zero 2 have K: "0 \<le> K"
677       by (rule order_trans)
678     have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
679       apply (erule subst)
680       apply (simp only: norm_mult norm_power power_add)
681       apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
682       done
683     show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) \<le>
684         of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
685       apply (intro
686           order_trans [OF norm_sum]
687           real_sum_nat_ivl_bounded2
688           mult_nonneg_nonneg
689           of_nat_0_le_iff
690           zero_le_power K)
691       apply (rule le_Kn)
692       apply simp
693       done
694   qed
695   also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
696     by (simp only: mult.assoc)
697   finally show ?thesis .
698 qed
700 lemma lemma_termdiff4:
701   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
702     and k :: real
703   assumes k: "0 < k"
704     and le: "\<And>h. h \<noteq> 0 \<Longrightarrow> norm h < k \<Longrightarrow> norm (f h) \<le> K * norm h"
705   shows "f \<midarrow>0\<rightarrow> 0"
706 proof (rule tendsto_norm_zero_cancel)
707   show "(\<lambda>h. norm (f h)) \<midarrow>0\<rightarrow> 0"
708   proof (rule real_tendsto_sandwich)
709     show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)"
710       by simp
711     show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)"
712       using k by (auto simp add: eventually_at dist_norm le)
713     show "(\<lambda>h. 0) \<midarrow>(0::'a)\<rightarrow> (0::real)"
714       by (rule tendsto_const)
715     have "(\<lambda>h. K * norm h) \<midarrow>(0::'a)\<rightarrow> K * norm (0::'a)"
716       by (intro tendsto_intros)
717     then show "(\<lambda>h. K * norm h) \<midarrow>(0::'a)\<rightarrow> 0"
718       by simp
719   qed
720 qed
722 lemma lemma_termdiff5:
723   fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach"
724     and k :: real
725   assumes k: "0 < k"
726     and f: "summable f"
727     and le: "\<And>h n. h \<noteq> 0 \<Longrightarrow> norm h < k \<Longrightarrow> norm (g h n) \<le> f n * norm h"
728   shows "(\<lambda>h. suminf (g h)) \<midarrow>0\<rightarrow> 0"
729 proof (rule lemma_termdiff4 [OF k])
730   fix h :: 'a
731   assume "h \<noteq> 0" and "norm h < k"
732   then have 1: "\<forall>n. norm (g h n) \<le> f n * norm h"
734   then have "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
735     by simp
736   moreover from f have 2: "summable (\<lambda>n. f n * norm h)"
737     by (rule summable_mult2)
738   ultimately have 3: "summable (\<lambda>n. norm (g h n))"
739     by (rule summable_comparison_test)
740   then have "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
741     by (rule summable_norm)
742   also from 1 3 2 have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
743     by (rule suminf_le)
744   also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
745     by (rule suminf_mult2 [symmetric])
746   finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
747 qed
750 (* FIXME: Long proofs *)
752 lemma termdiffs_aux:
753   fixes x :: "'a::{real_normed_field,banach}"
754   assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
755     and 2: "norm x < norm K"
756   shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0"
757 proof -
758   from dense [OF 2] obtain r where r1: "norm x < r" and r2: "r < norm K"
759     by fast
760   from norm_ge_zero r1 have r: "0 < r"
761     by (rule order_le_less_trans)
762   then have r_neq_0: "r \<noteq> 0" by simp
763   show ?thesis
764   proof (rule lemma_termdiff5)
765     show "0 < r - norm x"
766       using r1 by simp
767     from r r2 have "norm (of_real r::'a) < norm K"
768       by simp
769     with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
770       by (rule powser_insidea)
771     then have "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
772       using r by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
773     then have "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
774       by (rule diffs_equiv [THEN sums_summable])
775     also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
776       (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
777       apply (rule ext)
779       apply (case_tac n)
781       done
782     finally have "summable
783       (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
784       by (rule diffs_equiv [THEN sums_summable])
785     also have
786       "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0)) =
787        (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
788       apply (rule ext)
789       apply (case_tac n)
790        apply simp
791       apply (rename_tac nat)
792       apply (case_tac nat)
793        apply simp
795       done
796     finally show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
797   next
798     fix h :: 'a
799     fix n :: nat
800     assume h: "h \<noteq> 0"
801     assume "norm h < r - norm x"
802     then have "norm x + norm h < r" by simp
803     with norm_triangle_ineq have xh: "norm (x + h) < r"
804       by (rule order_le_less_trans)
805     show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<le>
806       norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
807       apply (simp only: norm_mult mult.assoc)
808       apply (rule mult_left_mono [OF _ norm_ge_zero])
809       apply (simp add: mult.assoc [symmetric])
810       apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
811       done
812   qed
813 qed
815 lemma termdiffs:
816   fixes K x :: "'a::{real_normed_field,banach}"
817   assumes 1: "summable (\<lambda>n. c n * K ^ n)"
818     and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
819     and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
820     and 4: "norm x < norm K"
821   shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. (diffs c) n * x^n)"
822   unfolding DERIV_def
823 proof (rule LIM_zero_cancel)
824   show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x^n)) / h
825             - suminf (\<lambda>n. diffs c n * x^n)) \<midarrow>0\<rightarrow> 0"
826   proof (rule LIM_equal2)
827     show "0 < norm K - norm x"
828       using 4 by (simp add: less_diff_eq)
829   next
830     fix h :: 'a
831     assume "norm (h - 0) < norm K - norm x"
832     then have "norm x + norm h < norm K" by simp
833     then have 5: "norm (x + h) < norm K"
834       by (rule norm_triangle_ineq [THEN order_le_less_trans])
835     have "summable (\<lambda>n. c n * x^n)"
836       and "summable (\<lambda>n. c n * (x + h) ^ n)"
837       and "summable (\<lambda>n. diffs c n * x^n)"
838       using 1 2 4 5 by (auto elim: powser_inside)
839     then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
840           (\<Sum>n. (c n * (x + h) ^ n - c n * x^n) / h - of_nat n * c n * x ^ (n - Suc 0))"
841       by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
842     then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) =
843           (\<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))"
845   next
846     show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0"
847       by (rule termdiffs_aux [OF 3 4])
848   qed
849 qed
851 subsection \<open>The Derivative of a Power Series Has the Same Radius of Convergence\<close>
853 lemma termdiff_converges:
854   fixes x :: "'a::{real_normed_field,banach}"
855   assumes K: "norm x < K"
856     and sm: "\<And>x. norm x < K \<Longrightarrow> summable(\<lambda>n. c n * x ^ n)"
857   shows "summable (\<lambda>n. diffs c n * x ^ n)"
858 proof (cases "x = 0")
859   case True
860   then show ?thesis
861     using powser_sums_zero sums_summable by auto
862 next
863   case False
864   then have "K > 0"
865     using K less_trans zero_less_norm_iff by blast
866   then obtain r :: real where r: "norm x < norm r" "norm r < K" "r > 0"
867     using K False
868     by (auto simp: field_simps abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"])
869   have "(\<lambda>n. of_nat n * (x / of_real r) ^ n) \<longlonglongrightarrow> 0"
870     using r by (simp add: norm_divide powser_times_n_limit_0 [of "x / of_real r"])
871   then obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> real_of_nat n * norm x ^ n < r ^ n"
872     using r unfolding LIMSEQ_iff
873     apply (drule_tac x=1 in spec)
874     apply (auto simp: norm_divide norm_mult norm_power field_simps)
875     done
876   have "summable (\<lambda>n. (of_nat n * c n) * x ^ n)"
877     apply (rule summable_comparison_test' [of "\<lambda>n. norm(c n * (of_real r) ^ n)" N])
878      apply (rule powser_insidea [OF sm [of "of_real ((r+K)/2)"]])
879     using N r norm_of_real [of "r + K", where 'a = 'a]
880       apply (auto simp add: norm_divide norm_mult norm_power field_simps)
881     apply (fastforce simp: less_eq_real_def)
882     done
883   then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ Suc n)"
884     using summable_iff_shift [of "\<lambda>n. of_nat n * c n * x ^ n" 1]
885     by simp
886   then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ n)"
887     using False summable_mult2 [of "\<lambda>n. (of_nat (Suc n) * c(Suc n) * x ^ n) * x" "inverse x"]
888     by (simp add: mult.assoc) (auto simp: ac_simps)
889   then show ?thesis
891 qed
893 lemma termdiff_converges_all:
894   fixes x :: "'a::{real_normed_field,banach}"
895   assumes "\<And>x. summable (\<lambda>n. c n * x^n)"
896   shows "summable (\<lambda>n. diffs c n * x^n)"
897   apply (rule termdiff_converges [where K = "1 + norm x"])
898   using assms
899    apply auto
900   done
902 lemma termdiffs_strong:
903   fixes K x :: "'a::{real_normed_field,banach}"
904   assumes sm: "summable (\<lambda>n. c n * K ^ n)"
905     and K: "norm x < norm K"
906   shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. diffs c n * x^n)"
907 proof -
908   have K2: "norm ((of_real (norm K) + of_real (norm x)) / 2 :: 'a) < norm K"
909     using K
910     apply (auto simp: norm_divide field_simps)
911     apply (rule le_less_trans [of _ "of_real (norm K) + of_real (norm x)"])
912      apply (auto simp: mult_2_right norm_triangle_mono)
913     done
914   then have [simp]: "norm ((of_real (norm K) + of_real (norm x)) :: 'a) < norm K * 2"
915     by simp
916   have "summable (\<lambda>n. c n * (of_real (norm x + norm K) / 2) ^ n)"
918   moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs c n * x ^ n)"
919     by (blast intro: sm termdiff_converges powser_inside)
920   moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs(diffs c) n * x ^ n)"
921     by (blast intro: sm termdiff_converges powser_inside)
922   ultimately show ?thesis
923     apply (rule termdiffs [where K = "of_real (norm x + norm K) / 2"])
924       apply (auto simp: field_simps)
925     using K
927     done
928 qed
930 lemma termdiffs_strong_converges_everywhere:
931   fixes K x :: "'a::{real_normed_field,banach}"
932   assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)"
933   shows "((\<lambda>x. \<Sum>n. c n * x^n) has_field_derivative (\<Sum>n. diffs c n * x^n)) (at x)"
934   using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
935   by (force simp del: of_real_add)
937 lemma termdiffs_strong':
938   fixes z :: "'a :: {real_normed_field,banach}"
939   assumes "\<And>z. norm z < K \<Longrightarrow> summable (\<lambda>n. c n * z ^ n)"
940   assumes "norm z < K"
941   shows   "((\<lambda>z. \<Sum>n. c n * z^n) has_field_derivative (\<Sum>n. diffs c n * z^n)) (at z)"
942 proof (rule termdiffs_strong)
943   define L :: real where "L =  (norm z + K) / 2"
944   have "0 \<le> norm z" by simp
945   also note \<open>norm z < K\<close>
946   finally have K: "K \<ge> 0" by simp
947   from assms K have L: "L \<ge> 0" "norm z < L" "L < K" by (simp_all add: L_def)
948   from L show "norm z < norm (of_real L :: 'a)" by simp
949   from L show "summable (\<lambda>n. c n * of_real L ^ n)" by (intro assms(1)) simp_all
950 qed
952 lemma termdiffs_sums_strong:
953   fixes z :: "'a :: {banach,real_normed_field}"
954   assumes sums: "\<And>z. norm z < K \<Longrightarrow> (\<lambda>n. c n * z ^ n) sums f z"
955   assumes deriv: "(f has_field_derivative f') (at z)"
956   assumes norm: "norm z < K"
957   shows   "(\<lambda>n. diffs c n * z ^ n) sums f'"
958 proof -
959   have summable: "summable (\<lambda>n. diffs c n * z^n)"
960     by (intro termdiff_converges[OF norm] sums_summable[OF sums])
961   from norm have "eventually (\<lambda>z. z \<in> norm -` {..<K}) (nhds z)"
962     by (intro eventually_nhds_in_open open_vimage)
964   hence eq: "eventually (\<lambda>z. (\<Sum>n. c n * z^n) = f z) (nhds z)"
965     by eventually_elim (insert sums, simp add: sums_iff)
967   have "((\<lambda>z. \<Sum>n. c n * z^n) has_field_derivative (\<Sum>n. diffs c n * z^n)) (at z)"
968     by (intro termdiffs_strong'[OF _ norm] sums_summable[OF sums])
969   hence "(f has_field_derivative (\<Sum>n. diffs c n * z^n)) (at z)"
970     by (subst (asm) DERIV_cong_ev[OF refl eq refl])
971   from this and deriv have "(\<Sum>n. diffs c n * z^n) = f'" by (rule DERIV_unique)
972   with summable show ?thesis by (simp add: sums_iff)
973 qed
975 lemma isCont_powser:
976   fixes K x :: "'a::{real_normed_field,banach}"
977   assumes "summable (\<lambda>n. c n * K ^ n)"
978   assumes "norm x < norm K"
979   shows "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
980   using termdiffs_strong[OF assms] by (blast intro!: DERIV_isCont)
982 lemmas isCont_powser' = isCont_o2[OF _ isCont_powser]
984 lemma isCont_powser_converges_everywhere:
985   fixes K x :: "'a::{real_normed_field,banach}"
986   assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)"
987   shows "isCont (\<lambda>x. \<Sum>n. c n * x^n) x"
988   using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
989   by (force intro!: DERIV_isCont simp del: of_real_add)
991 lemma powser_limit_0:
992   fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
993   assumes s: "0 < s"
994     and sm: "\<And>x. norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
995   shows "(f \<longlongrightarrow> a 0) (at 0)"
996 proof -
997   have "summable (\<lambda>n. a n * (of_real s / 2) ^ n)"
998     apply (rule sums_summable [where l = "f (of_real s / 2)", OF sm])
999     using s
1000     apply (auto simp: norm_divide)
1001     done
1002   then have "((\<lambda>x. \<Sum>n. a n * x ^ n) has_field_derivative (\<Sum>n. diffs a n * 0 ^ n)) (at 0)"
1003     apply (rule termdiffs_strong)
1004     using s
1005     apply (auto simp: norm_divide)
1006     done
1007   then have "isCont (\<lambda>x. \<Sum>n. a n * x ^ n) 0"
1008     by (blast intro: DERIV_continuous)
1009   then have "((\<lambda>x. \<Sum>n. a n * x ^ n) \<longlongrightarrow> a 0) (at 0)"
1011   then show ?thesis
1012     apply (rule Lim_transform)
1013     apply (auto simp add: LIM_eq)
1014     apply (rule_tac x="s" in exI)
1015     using s
1016     apply (auto simp: sm [THEN sums_unique])
1017     done
1018 qed
1020 lemma powser_limit_0_strong:
1021   fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
1022   assumes s: "0 < s"
1023     and sm: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)"
1024   shows "(f \<longlongrightarrow> a 0) (at 0)"
1025 proof -
1026   have *: "((\<lambda>x. if x = 0 then a 0 else f x) \<longlongrightarrow> a 0) (at 0)"
1027     apply (rule powser_limit_0 [OF s])
1028     apply (case_tac "x = 0")
1029      apply (auto simp add: powser_sums_zero sm)
1030     done
1031   show ?thesis
1032     apply (subst LIM_equal [where g = "(\<lambda>x. if x = 0 then a 0 else f x)"])
1034     done
1035 qed
1038 subsection \<open>Derivability of power series\<close>
1040 lemma DERIV_series':
1041   fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
1042   assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
1043     and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)"
1044     and x0_in_I: "x0 \<in> {a <..< b}"
1045     and "summable (f' x0)"
1046     and "summable L"
1047     and L_def: "\<And>n x y. x \<in> {a <..< b} \<Longrightarrow> y \<in> {a <..< b} \<Longrightarrow> \<bar>f x n - f y n\<bar> \<le> L n * \<bar>x - y\<bar>"
1048   shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
1049   unfolding DERIV_def
1050 proof (rule LIM_I)
1051   fix r :: real
1052   assume "0 < r" then have "0 < r/3" by auto
1054   obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
1055     using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable L\<close>] by auto
1057   obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
1058     using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable (f' x0)\<close>] by auto
1060   let ?N = "Suc (max N_L N_f')"
1061   have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3")
1062     and L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3"
1063     using N_L[of "?N"] and N_f' [of "?N"] by auto
1065   let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
1067   let ?r = "r / (3 * real ?N)"
1068   from \<open>0 < r\<close> have "0 < ?r" by simp
1070   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)"
1071   define S' where "S' = Min (?s ` {..< ?N })"
1073   have "0 < S'"
1074     unfolding S'_def
1075   proof (rule iffD2[OF Min_gr_iff])
1076     show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x"
1077     proof
1078       fix x
1079       assume "x \<in> ?s ` {..<?N}"
1080       then obtain n where "x = ?s n" and "n \<in> {..<?N}"
1081         using image_iff[THEN iffD1] by blast
1082       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>, unfolded real_norm_def]
1083       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)"
1084         by auto
1085       have "0 < ?s n"
1086         by (rule someI2[where a=s]) (auto simp add: s_bound simp del: of_nat_Suc)
1087       then show "0 < x" by (simp only: \<open>x = ?s n\<close>)
1088     qed
1089   qed auto
1091   define S where "S = min (min (x0 - a) (b - x0)) S'"
1092   then have "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
1093     and "S \<le> S'" using x0_in_I and \<open>0 < S'\<close>
1094     by auto
1096   have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
1097     if "x \<noteq> 0" and "\<bar>x\<bar> < S" for x
1098   proof -
1099     from that have x_in_I: "x0 + x \<in> {a <..< b}"
1100       using S_a S_b by auto
1102     note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
1103     note div_smbl = summable_divide[OF diff_smbl]
1104     note all_smbl = summable_diff[OF div_smbl \<open>summable (f' x0)\<close>]
1105     note ign = summable_ignore_initial_segment[where k="?N"]
1106     note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
1107     note div_shft_smbl = summable_divide[OF diff_shft_smbl]
1108     note all_shft_smbl = summable_diff[OF div_smbl ign[OF \<open>summable (f' x0)\<close>]]
1110     have 1: "\<bar>(\<bar>?diff (n + ?N) x\<bar>)\<bar> \<le> L (n + ?N)" for n
1111     proof -
1112       have "\<bar>?diff (n + ?N) x\<bar> \<le> L (n + ?N) * \<bar>(x0 + x) - x0\<bar> / \<bar>x\<bar>"
1113         using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
1114         by (simp only: abs_divide)
1115       with \<open>x \<noteq> 0\<close> show ?thesis by auto
1116     qed
1117     note 2 = summable_rabs_comparison_test[OF _ ign[OF \<open>summable L\<close>]]
1118     from 1 have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))"
1119       by (metis (lifting) abs_idempotent
1120           order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF \<open>summable L\<close>]]])
1121     then have "\<bar>\<Sum>i. ?diff (i + ?N) x\<bar> \<le> r / 3" (is "?L_part \<le> r/3")
1122       using L_estimate by auto
1124     have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n\<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n\<bar>)" ..
1125     also have "\<dots> < (\<Sum>n<?N. ?r)"
1126     proof (rule sum_strict_mono)
1127       fix n
1128       assume "n \<in> {..< ?N}"
1129       have "\<bar>x\<bar> < S" using \<open>\<bar>x\<bar> < S\<close> .
1130       also have "S \<le> S'" using \<open>S \<le> S'\<close> .
1131       also have "S' \<le> ?s n"
1132         unfolding S'_def
1133       proof (rule Min_le_iff[THEN iffD2])
1134         have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n"
1135           using \<open>n \<in> {..< ?N}\<close> by auto
1136         then show "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n"
1137           by blast
1138       qed auto
1139       finally have "\<bar>x\<bar> < ?s n" .
1141       from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>,
1142           unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
1143       have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
1144       with \<open>x \<noteq> 0\<close> and \<open>\<bar>x\<bar> < ?s n\<close> show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
1145         by blast
1146     qed auto
1147     also have "\<dots> = of_nat (card {..<?N}) * ?r"
1148       by (rule sum_constant)
1149     also have "\<dots> = real ?N * ?r"
1150       by simp
1151     also have "\<dots> = r/3"
1152       by (auto simp del: of_nat_Suc)
1153     finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
1155     from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
1156     have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
1157         \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
1158       unfolding suminf_diff[OF div_smbl \<open>summable (f' x0)\<close>, symmetric]
1159       using suminf_divide[OF diff_smbl, symmetric] by auto
1160     also have "\<dots> \<le> ?diff_part + \<bar>(\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N))\<bar>"
1161       unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
1162       unfolding suminf_diff[OF div_shft_smbl ign[OF \<open>summable (f' x0)\<close>]]
1164       apply (rule abs_triangle_ineq)
1165       done
1166     also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
1167       using abs_triangle_ineq4 by auto
1168     also have "\<dots> < r /3 + r/3 + r/3"
1169       using \<open>?diff_part < r/3\<close> \<open>?L_part \<le> r/3\<close> and \<open>?f'_part < r/3\<close>
1171     finally show ?thesis
1172       by auto
1173   qed
1174   then show "\<exists>s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
1175       norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
1176     using \<open>0 < S\<close> by auto
1177 qed
1179 lemma DERIV_power_series':
1180   fixes f :: "nat \<Rightarrow> real"
1181   assumes converges: "\<And>x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda>n. f n * real (Suc n) * x^n)"
1182     and x0_in_I: "x0 \<in> {-R <..< R}"
1183     and "0 < R"
1184   shows "DERIV (\<lambda>x. (\<Sum>n. f n * x^(Suc n))) x0 :> (\<Sum>n. f n * real (Suc n) * x0^n)"
1185     (is "DERIV (\<lambda>x. suminf (?f x)) x0 :> suminf (?f' x0)")
1186 proof -
1187   have for_subinterval: "DERIV (\<lambda>x. suminf (?f x)) x0 :> suminf (?f' x0)"
1188     if "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'" for R'
1189   proof -
1190     from that have "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
1191       by auto
1192     show ?thesis
1193     proof (rule DERIV_series')
1194       show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
1195       proof -
1196         have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
1197           using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps)
1198         then have in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
1199           using \<open>R' < R\<close> by auto
1200         have "norm R' < norm ((R' + R) / 2)"
1201           using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps)
1202         from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
1203           by auto
1204       qed
1205     next
1206       fix n x y
1207       assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
1208       show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
1209       proof -
1210         have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
1211           (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>"
1212           unfolding right_diff_distrib[symmetric] diff_power_eq_sum abs_mult
1213           by auto
1214         also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
1215         proof (rule mult_left_mono)
1216           have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
1217             by (rule sum_abs)
1218           also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)"
1219           proof (rule sum_mono)
1220             fix p
1221             assume "p \<in> {..<Suc n}"
1222             then have "p \<le> n" by auto
1223             have "\<bar>x^n\<bar> \<le> R'^n" if  "x \<in> {-R'<..<R'}" for n and x :: real
1224             proof -
1225               from that have "\<bar>x\<bar> \<le> R'" by auto
1226               then show ?thesis
1227                 unfolding power_abs by (rule power_mono) auto
1228             qed
1229             from mult_mono[OF this[OF \<open>x \<in> {-R'<..<R'}\<close>, of p] this[OF \<open>y \<in> {-R'<..<R'}\<close>, of "n-p"]]
1230               and \<open>0 < R'\<close>
1231             have "\<bar>x^p * y^(n - p)\<bar> \<le> R'^p * R'^(n - p)"
1232               unfolding abs_mult by auto
1233             then show "\<bar>x^p * y^(n - p)\<bar> \<le> R'^n"
1234               unfolding power_add[symmetric] using \<open>p \<le> n\<close> by auto
1235           qed
1236           also have "\<dots> = real (Suc n) * R' ^ n"
1237             unfolding sum_constant card_atLeastLessThan by auto
1238           finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
1239             unfolding abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF \<open>0 < R'\<close>]]]
1240             by linarith
1241           show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
1242             unfolding abs_mult[symmetric] by auto
1243         qed
1244         also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
1245           unfolding abs_mult mult.assoc[symmetric] by algebra
1246         finally show ?thesis .
1247       qed
1248     next
1249       show "DERIV (\<lambda>x. ?f x n) x0 :> ?f' x0 n" for n
1250         by (auto intro!: derivative_eq_intros simp del: power_Suc)
1251     next
1252       fix x
1253       assume "x \<in> {-R' <..< R'}"
1254       then have "R' \<in> {-R <..< R}" and "norm x < norm R'"
1255         using assms \<open>R' < R\<close> by auto
1256       have "summable (\<lambda>n. f n * x^n)"
1257       proof (rule summable_comparison_test, intro exI allI impI)
1258         fix n
1259         have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
1260           by (rule mult_left_mono) auto
1261         show "norm (f n * x^n) \<le> norm (f n * real (Suc n) * x^n)"
1262           unfolding real_norm_def abs_mult
1263           using le mult_right_mono by fastforce
1264       qed (rule powser_insidea[OF converges[OF \<open>R' \<in> {-R <..< R}\<close>] \<open>norm x < norm R'\<close>])
1265       from this[THEN summable_mult2[where c=x], simplified mult.assoc, simplified mult.commute]
1266       show "summable (?f x)" by auto
1267     next
1268       show "summable (?f' x0)"
1269         using converges[OF \<open>x0 \<in> {-R <..< R}\<close>] .
1270       show "x0 \<in> {-R' <..< R'}"
1271         using \<open>x0 \<in> {-R' <..< R'}\<close> .
1272     qed
1273   qed
1274   let ?R = "(R + \<bar>x0\<bar>) / 2"
1275   have "\<bar>x0\<bar> < ?R"
1276     using assms by (auto simp: field_simps)
1277   then have "- ?R < x0"
1278   proof (cases "x0 < 0")
1279     case True
1280     then have "- x0 < ?R"
1281       using \<open>\<bar>x0\<bar> < ?R\<close> by auto
1282     then show ?thesis
1283       unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
1284   next
1285     case False
1286     have "- ?R < 0" using assms by auto
1287     also have "\<dots> \<le> x0" using False by auto
1288     finally show ?thesis .
1289   qed
1290   then have "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
1291     using assms by (auto simp: field_simps)
1292   from for_subinterval[OF this] show ?thesis .
1293 qed
1295 lemma geometric_deriv_sums:
1296   fixes z :: "'a :: {real_normed_field,banach}"
1297   assumes "norm z < 1"
1298   shows   "(\<lambda>n. of_nat (Suc n) * z ^ n) sums (1 / (1 - z)^2)"
1299 proof -
1300   have "(\<lambda>n. diffs (\<lambda>n. 1) n * z^n) sums (1 / (1 - z)^2)"
1301   proof (rule termdiffs_sums_strong)
1302     fix z :: 'a assume "norm z < 1"
1303     thus "(\<lambda>n. 1 * z^n) sums (1 / (1 - z))" by (simp add: geometric_sums)
1304   qed (insert assms, auto intro!: derivative_eq_intros simp: power2_eq_square)
1305   thus ?thesis unfolding diffs_def by simp
1306 qed
1308 lemma isCont_pochhammer [continuous_intros]: "isCont (\<lambda>z. pochhammer z n) z"
1309   for z :: "'a::real_normed_field"
1310   by (induct n) (auto simp: pochhammer_rec')
1312 lemma continuous_on_pochhammer [continuous_intros]: "continuous_on A (\<lambda>z. pochhammer z n)"
1313   for A :: "'a::real_normed_field set"
1314   by (intro continuous_at_imp_continuous_on ballI isCont_pochhammer)
1316 lemmas continuous_on_pochhammer' [continuous_intros] =
1317   continuous_on_compose2[OF continuous_on_pochhammer _ subset_UNIV]
1320 subsection \<open>Exponential Function\<close>
1322 definition exp :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
1323   where "exp = (\<lambda>x. \<Sum>n. x^n /\<^sub>R fact n)"
1325 lemma summable_exp_generic:
1326   fixes x :: "'a::{real_normed_algebra_1,banach}"
1327   defines S_def: "S \<equiv> \<lambda>n. x^n /\<^sub>R fact n"
1328   shows "summable S"
1329 proof -
1330   have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R (Suc n)"
1331     unfolding S_def by (simp del: mult_Suc)
1332   obtain r :: real where r0: "0 < r" and r1: "r < 1"
1333     using dense [OF zero_less_one] by fast
1334   obtain N :: nat where N: "norm x < real N * r"
1335     using ex_less_of_nat_mult r0 by auto
1336   from r1 show ?thesis
1337   proof (rule summable_ratio_test [rule_format])
1338     fix n :: nat
1339     assume n: "N \<le> n"
1340     have "norm x \<le> real N * r"
1341       using N by (rule order_less_imp_le)
1342     also have "real N * r \<le> real (Suc n) * r"
1343       using r0 n by (simp add: mult_right_mono)
1344     finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
1345       using norm_ge_zero by (rule mult_right_mono)
1346     then have "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
1347       by (rule order_trans [OF norm_mult_ineq])
1348     then have "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
1349       by (simp add: pos_divide_le_eq ac_simps)
1350     then show "norm (S (Suc n)) \<le> r * norm (S n)"
1351       by (simp add: S_Suc inverse_eq_divide)
1352   qed
1353 qed
1355 lemma summable_norm_exp: "summable (\<lambda>n. norm (x^n /\<^sub>R fact n))"
1356   for x :: "'a::{real_normed_algebra_1,banach}"
1357 proof (rule summable_norm_comparison_test [OF exI, rule_format])
1358   show "summable (\<lambda>n. norm x^n /\<^sub>R fact n)"
1359     by (rule summable_exp_generic)
1360   show "norm (x^n /\<^sub>R fact n) \<le> norm x^n /\<^sub>R fact n" for n
1362 qed
1364 lemma summable_exp: "summable (\<lambda>n. inverse (fact n) * x^n)"
1365   for x :: "'a::{real_normed_field,banach}"
1366   using summable_exp_generic [where x=x]
1367   by (simp add: scaleR_conv_of_real nonzero_of_real_inverse)
1369 lemma exp_converges: "(\<lambda>n. x^n /\<^sub>R fact n) sums exp x"
1370   unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
1372 lemma exp_fdiffs:
1373   "diffs (\<lambda>n. inverse (fact n)) = (\<lambda>n. inverse (fact n :: 'a::{real_normed_field,banach}))"
1374   by (simp add: diffs_def mult_ac nonzero_inverse_mult_distrib nonzero_of_real_inverse
1375       del: mult_Suc of_nat_Suc)
1377 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
1380 lemma DERIV_exp [simp]: "DERIV exp x :> exp x"
1381   unfolding exp_def scaleR_conv_of_real
1382   apply (rule DERIV_cong)
1383    apply (rule termdiffs [where K="of_real (1 + norm x)"])
1384       apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
1385      apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
1387   done
1389 declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
1390   and DERIV_exp[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
1392 lemma norm_exp: "norm (exp x) \<le> exp (norm x)"
1393 proof -
1394   from summable_norm[OF summable_norm_exp, of x]
1395   have "norm (exp x) \<le> (\<Sum>n. inverse (fact n) * norm (x^n))"
1397   also have "\<dots> \<le> exp (norm x)"
1398     using summable_exp_generic[of "norm x"] summable_norm_exp[of x]
1399     by (auto simp: exp_def intro!: suminf_le norm_power_ineq)
1400   finally show ?thesis .
1401 qed
1403 lemma isCont_exp: "isCont exp x"
1404   for x :: "'a::{real_normed_field,banach}"
1405   by (rule DERIV_exp [THEN DERIV_isCont])
1407 lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
1408   for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
1409   by (rule isCont_o2 [OF _ isCont_exp])
1411 lemma tendsto_exp [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) \<longlongrightarrow> exp a) F"
1412   for f:: "_ \<Rightarrow>'a::{real_normed_field,banach}"
1413   by (rule isCont_tendsto_compose [OF isCont_exp])
1415 lemma continuous_exp [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
1416   for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
1417   unfolding continuous_def by (rule tendsto_exp)
1419 lemma continuous_on_exp [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
1420   for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}"
1421   unfolding continuous_on_def by (auto intro: tendsto_exp)
1424 subsubsection \<open>Properties of the Exponential Function\<close>
1426 lemma exp_zero [simp]: "exp 0 = 1"
1427   unfolding exp_def by (simp add: scaleR_conv_of_real)
1430   fixes x y :: "'a::{real_normed_algebra_1,banach}"
1431   defines S_def: "S \<equiv> \<lambda>x n. x^n /\<^sub>R fact n"
1432   assumes comm: "x * y = y * x"
1433   shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))"
1434 proof (induct n)
1435   case 0
1436   show ?case
1437     unfolding S_def by simp
1438 next
1439   case (Suc n)
1440   have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
1441     unfolding S_def by (simp del: mult_Suc)
1442   then have times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
1443     by simp
1444   have S_comm: "\<And>n. S x n * y = y * S x n"
1445     by (simp add: power_commuting_commutes comm S_def)
1447   have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
1448     by (simp only: times_S)
1449   also have "\<dots> = (x + y) * (\<Sum>i\<le>n. S x i * S y (n - i))"
1450     by (simp only: Suc)
1451   also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n - i)) + y * (\<Sum>i\<le>n. S x i * S y (n - i))"
1452     by (rule distrib_right)
1453   also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n - i)) + (\<Sum>i\<le>n. S x i * y * S y (n - i))"
1454     by (simp add: sum_distrib_left ac_simps S_comm)
1455   also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n - i)) + (\<Sum>i\<le>n. S x i * (y * S y (n - i)))"
1457   also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i))) +
1458       (\<Sum>i\<le>n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))"
1459     by (simp add: times_S Suc_diff_le)
1460   also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i))) =
1461       (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n - i)))"
1462     by (subst sum_atMost_Suc_shift) simp
1463   also have "(\<Sum>i\<le>n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i))) =
1464       (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))"
1465     by simp
1466   also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n - i))) +
1467         (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i))) =
1468       (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n - i)))"
1469     by (simp only: sum.distrib [symmetric] scaleR_left_distrib [symmetric]
1471   also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
1472     by (simp only: scaleR_right.sum)
1473   finally show "S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))"
1474     by (simp del: sum_cl_ivl_Suc)
1475 qed
1477 lemma exp_add_commuting: "x * y = y * x \<Longrightarrow> exp (x + y) = exp x * exp y"
1478   by (simp only: exp_def Cauchy_product summable_norm_exp exp_series_add_commuting)
1480 lemma exp_times_arg_commute: "exp A * A = A * exp A"
1481   by (simp add: exp_def suminf_mult[symmetric] summable_exp_generic power_commutes suminf_mult2)
1483 lemma exp_add: "exp (x + y) = exp x * exp y"
1484   for x y :: "'a::{real_normed_field,banach}"
1487 lemma exp_double: "exp(2 * z) = exp z ^ 2"
1490 lemmas mult_exp_exp = exp_add [symmetric]
1492 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
1493   unfolding exp_def
1494   apply (subst suminf_of_real)
1495    apply (rule summable_exp_generic)
1497   done
1499 lemmas of_real_exp = exp_of_real[symmetric]
1501 corollary exp_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> exp z \<in> \<real>"
1502   by (metis Reals_cases Reals_of_real exp_of_real)
1504 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
1505 proof
1506   have "exp x * exp (- x) = 1"
1508   also assume "exp x = 0"
1509   finally show False by simp
1510 qed
1512 lemma exp_minus_inverse: "exp x * exp (- x) = 1"
1515 lemma exp_minus: "exp (- x) = inverse (exp x)"
1516   for x :: "'a::{real_normed_field,banach}"
1517   by (intro inverse_unique [symmetric] exp_minus_inverse)
1519 lemma exp_diff: "exp (x - y) = exp x / exp y"
1520   for x :: "'a::{real_normed_field,banach}"
1521   using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
1523 lemma exp_of_nat_mult: "exp (of_nat n * x) = exp x ^ n"
1524   for x :: "'a::{real_normed_field,banach}"
1527 corollary exp_of_nat2_mult: "exp (x * of_nat n) = exp x ^ n"
1528   for x :: "'a::{real_normed_field,banach}"
1529   by (metis exp_of_nat_mult mult_of_nat_commute)
1531 lemma exp_sum: "finite I \<Longrightarrow> exp (sum f I) = prod (\<lambda>x. exp (f x)) I"
1532   by (induct I rule: finite_induct) (auto simp: exp_add_commuting mult.commute)
1534 lemma exp_divide_power_eq:
1535   fixes x :: "'a::{real_normed_field,banach}"
1536   assumes "n > 0"
1537   shows "exp (x / of_nat n) ^ n = exp x"
1538   using assms
1539 proof (induction n arbitrary: x)
1540   case 0
1541   then show ?case by simp
1542 next
1543   case (Suc n)
1544   show ?case
1545   proof (cases "n = 0")
1546     case True
1547     then show ?thesis by simp
1548   next
1549     case False
1550     then have [simp]: "x * of_nat n / (1 + of_nat n) / of_nat n = x / (1 + of_nat n)"
1551       by simp
1552     have [simp]: "x / (1 + of_nat n) + x * of_nat n / (1 + of_nat n) = x"
1554       using of_nat_eq_0_iff apply (fastforce simp: distrib_left)
1555       done
1556     show ?thesis
1557       using Suc.IH [of "x * of_nat n / (1 + of_nat n)"] False
1559   qed
1560 qed
1563 subsubsection \<open>Properties of the Exponential Function on Reals\<close>
1565 text \<open>Comparisons of @{term "exp x"} with zero.\<close>
1567 text \<open>Proof: because every exponential can be seen as a square.\<close>
1568 lemma exp_ge_zero [simp]: "0 \<le> exp x"
1569   for x :: real
1570 proof -
1571   have "0 \<le> exp (x/2) * exp (x/2)"
1572     by simp
1573   then show ?thesis
1575 qed
1577 lemma exp_gt_zero [simp]: "0 < exp x"
1578   for x :: real
1581 lemma not_exp_less_zero [simp]: "\<not> exp x < 0"
1582   for x :: real
1585 lemma not_exp_le_zero [simp]: "\<not> exp x \<le> 0"
1586   for x :: real
1589 lemma abs_exp_cancel [simp]: "\<bar>exp x\<bar> = exp x"
1590   for x :: real
1591   by simp
1593 text \<open>Strict monotonicity of exponential.\<close>
1596   fixes x :: real
1597   assumes "0 \<le> x"
1598   shows "1 + x \<le> exp x"
1599   using order_le_imp_less_or_eq [OF assms]
1600 proof
1601   assume "0 < x"
1602   have "1 + x \<le> (\<Sum>n<2. inverse (fact n) * x^n)"
1603     by (auto simp add: numeral_2_eq_2)
1604   also have "\<dots> \<le> (\<Sum>n. inverse (fact n) * x^n)"
1605     apply (rule sum_le_suminf [OF summable_exp])
1606     using \<open>0 < x\<close>
1607     apply (auto  simp add:  zero_le_mult_iff)
1608     done
1609   finally show "1 + x \<le> exp x"
1611 next
1612   assume "0 = x"
1613   then show "1 + x \<le> exp x"
1614     by auto
1615 qed
1617 lemma exp_gt_one: "0 < x \<Longrightarrow> 1 < exp x"
1618   for x :: real
1619 proof -
1620   assume x: "0 < x"
1621   then have "1 < 1 + x" by simp
1622   also from x have "1 + x \<le> exp x"
1624   finally show ?thesis .
1625 qed
1627 lemma exp_less_mono:
1628   fixes x y :: real
1629   assumes "x < y"
1630   shows "exp x < exp y"
1631 proof -
1632   from \<open>x < y\<close> have "0 < y - x" by simp
1633   then have "1 < exp (y - x)" by (rule exp_gt_one)
1634   then have "1 < exp y / exp x" by (simp only: exp_diff)
1635   then show "exp x < exp y" by simp
1636 qed
1638 lemma exp_less_cancel: "exp x < exp y \<Longrightarrow> x < y"
1639   for x y :: real
1640   unfolding linorder_not_le [symmetric]
1641   by (auto simp add: order_le_less exp_less_mono)
1643 lemma exp_less_cancel_iff [iff]: "exp x < exp y \<longleftrightarrow> x < y"
1644   for x y :: real
1645   by (auto intro: exp_less_mono exp_less_cancel)
1647 lemma exp_le_cancel_iff [iff]: "exp x \<le> exp y \<longleftrightarrow> x \<le> y"
1648   for x y :: real
1649   by (auto simp add: linorder_not_less [symmetric])
1651 lemma exp_inj_iff [iff]: "exp x = exp y \<longleftrightarrow> x = y"
1652   for x y :: real
1655 text \<open>Comparisons of @{term "exp x"} with one.\<close>
1657 lemma one_less_exp_iff [simp]: "1 < exp x \<longleftrightarrow> 0 < x"
1658   for x :: real
1659   using exp_less_cancel_iff [where x = 0 and y = x] by simp
1661 lemma exp_less_one_iff [simp]: "exp x < 1 \<longleftrightarrow> x < 0"
1662   for x :: real
1663   using exp_less_cancel_iff [where x = x and y = 0] by simp
1665 lemma one_le_exp_iff [simp]: "1 \<le> exp x \<longleftrightarrow> 0 \<le> x"
1666   for x :: real
1667   using exp_le_cancel_iff [where x = 0 and y = x] by simp
1669 lemma exp_le_one_iff [simp]: "exp x \<le> 1 \<longleftrightarrow> x \<le> 0"
1670   for x :: real
1671   using exp_le_cancel_iff [where x = x and y = 0] by simp
1673 lemma exp_eq_one_iff [simp]: "exp x = 1 \<longleftrightarrow> x = 0"
1674   for x :: real
1675   using exp_inj_iff [where x = x and y = 0] by simp
1677 lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x \<and> x \<le> y - 1 \<and> exp x = y"
1678   for y :: real
1679 proof (rule IVT)
1680   assume "1 \<le> y"
1681   then have "0 \<le> y - 1" by simp
1682   then have "1 + (y - 1) \<le> exp (y - 1)"
1684   then show "y \<le> exp (y - 1)" by simp
1687 lemma exp_total: "0 < y \<Longrightarrow> \<exists>x. exp x = y"
1688   for y :: real
1689 proof (rule linorder_le_cases [of 1 y])
1690   assume "1 \<le> y"
1691   then show "\<exists>x. exp x = y"
1692     by (fast dest: lemma_exp_total)
1693 next
1694   assume "0 < y" and "y \<le> 1"
1695   then have "1 \<le> inverse y"
1697   then obtain x where "exp x = inverse y"
1698     by (fast dest: lemma_exp_total)
1699   then have "exp (- x) = y"
1701   then show "\<exists>x. exp x = y" ..
1702 qed
1705 subsection \<open>Natural Logarithm\<close>
1707 class ln = real_normed_algebra_1 + banach +
1708   fixes ln :: "'a \<Rightarrow> 'a"
1709   assumes ln_one [simp]: "ln 1 = 0"
1711 definition powr :: "'a \<Rightarrow> 'a \<Rightarrow> 'a::ln"  (infixr "powr" 80)
1712   \<comment> \<open>exponentation via ln and exp\<close>
1713   where  [code del]: "x powr a \<equiv> if x = 0 then 0 else exp (a * ln x)"
1715 lemma powr_0 [simp]: "0 powr z = 0"
1719 instantiation real :: ln
1720 begin
1722 definition ln_real :: "real \<Rightarrow> real"
1723   where "ln_real x = (THE u. exp u = x)"
1725 instance
1726   by intro_classes (simp add: ln_real_def)
1728 end
1730 lemma powr_eq_0_iff [simp]: "w powr z = 0 \<longleftrightarrow> w = 0"
1733 lemma ln_exp [simp]: "ln (exp x) = x"
1734   for x :: real
1737 lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
1738   for x :: real
1739   by (auto dest: exp_total)
1741 lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
1742   for x :: real
1743   by (metis exp_gt_zero exp_ln)
1745 lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
1746   for x :: real
1747   by (erule subst) (rule ln_exp)
1749 lemma ln_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
1750   for x :: real
1753 lemma ln_prod: "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i > 0) \<Longrightarrow> ln (prod f I) = sum (\<lambda>x. ln(f x)) I"
1754   for f :: "'a \<Rightarrow> real"
1755   by (induct I rule: finite_induct) (auto simp: ln_mult prod_pos)
1757 lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
1758   for x :: real
1759   by (rule ln_unique) (simp add: exp_minus)
1761 lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
1762   for x :: real
1763   by (rule ln_unique) (simp add: exp_diff)
1765 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x^n) = real n * ln x"
1766   by (rule ln_unique) (simp add: exp_of_nat_mult)
1768 lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
1769   for x :: real
1770   by (subst exp_less_cancel_iff [symmetric]) simp
1772 lemma ln_le_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
1773   for x :: real
1774   by (simp add: linorder_not_less [symmetric])
1776 lemma ln_inj_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
1777   for x :: real
1780 lemma ln_add_one_self_le_self: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
1781   for x :: real
1784 lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
1785   for x :: real
1786   by (rule order_less_le_trans [where y = "ln (1 + x)"]) (simp_all add: ln_add_one_self_le_self)
1788 lemma ln_ge_iff: "\<And>x::real. 0 < x \<Longrightarrow> y \<le> ln x \<longleftrightarrow> exp y \<le> x"
1789   using exp_le_cancel_iff exp_total by force
1791 lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
1792   for x :: real
1793   using ln_le_cancel_iff [of 1 x] by simp
1795 lemma ln_ge_zero_imp_ge_one: "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
1796   for x :: real
1797   using ln_le_cancel_iff [of 1 x] by simp
1799 lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
1800   for x :: real
1801   using ln_le_cancel_iff [of 1 x] by simp
1803 lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
1804   for x :: real
1805   using ln_less_cancel_iff [of x 1] by simp
1807 lemma ln_le_zero_iff [simp]: "0 < x \<Longrightarrow> ln x \<le> 0 \<longleftrightarrow> x \<le> 1"
1808   for x :: real
1809   by (metis less_numeral_extra(1) ln_le_cancel_iff ln_one)
1811 lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
1812   for x :: real
1813   using ln_less_cancel_iff [of 1 x] by simp
1815 lemma ln_gt_zero_imp_gt_one: "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
1816   for x :: real
1817   using ln_less_cancel_iff [of 1 x] by simp
1819 lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
1820   for x :: real
1821   using ln_less_cancel_iff [of 1 x] by simp
1823 lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
1824   for x :: real
1825   using ln_inj_iff [of x 1] by simp
1827 lemma ln_less_zero: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
1828   for x :: real
1829   by simp
1831 lemma ln_neg_is_const: "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)"
1832   for x :: real
1833   by (auto simp: ln_real_def intro!: arg_cong[where f = The])
1835 lemma isCont_ln:
1836   fixes x :: real
1837   assumes "x \<noteq> 0"
1838   shows "isCont ln x"
1839 proof (cases "0 < x")
1840   case True
1841   then have "isCont ln (exp (ln x))"
1842     by (intro isCont_inv_fun[where d = "\<bar>x\<bar>" and f = exp]) auto
1843   with True show ?thesis
1844     by simp
1845 next
1846   case False
1847   with \<open>x \<noteq> 0\<close> show "isCont ln x"
1848     unfolding isCont_def
1849     by (subst filterlim_cong[OF _ refl, of _ "nhds (ln 0)" _ "\<lambda>_. ln 0"])
1850        (auto simp: ln_neg_is_const not_less eventually_at dist_real_def
1851          intro!: exI[of _ "\<bar>x\<bar>"])
1852 qed
1854 lemma tendsto_ln [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) \<longlongrightarrow> ln a) F"
1855   for a :: real
1856   by (rule isCont_tendsto_compose [OF isCont_ln])
1858 lemma continuous_ln:
1859   "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. ln (f x :: real))"
1860   unfolding continuous_def by (rule tendsto_ln)
1862 lemma isCont_ln' [continuous_intros]:
1863   "continuous (at x) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x :: real))"
1864   unfolding continuous_at by (rule tendsto_ln)
1866 lemma continuous_within_ln [continuous_intros]:
1867   "continuous (at x within s) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x :: real))"
1868   unfolding continuous_within by (rule tendsto_ln)
1870 lemma continuous_on_ln [continuous_intros]:
1871   "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. f x \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x :: real))"
1872   unfolding continuous_on_def by (auto intro: tendsto_ln)
1874 lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
1875   for x :: real
1876   by (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
1877     (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln)
1879 lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
1880   for x :: real
1881   by (rule DERIV_ln[THEN DERIV_cong]) (simp_all add: divide_inverse)
1883 declare DERIV_ln_divide[THEN DERIV_chain2, derivative_intros]
1884   and DERIV_ln_divide[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
1886 lemma ln_series:
1887   assumes "0 < x" and "x < 2"
1888   shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))"
1889     (is "ln x = suminf (?f (x - 1))")
1890 proof -
1891   let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n"
1893   have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
1894   proof (rule DERIV_isconst3 [where x = x])
1895     fix x :: real
1896     assume "x \<in> {0 <..< 2}"
1897     then have "0 < x" and "x < 2" by auto
1898     have "norm (1 - x) < 1"
1899       using \<open>0 < x\<close> and \<open>x < 2\<close> by auto
1900     have "1 / x = 1 / (1 - (1 - x))" by auto
1901     also have "\<dots> = (\<Sum> n. (1 - x)^n)"
1902       using geometric_sums[OF \<open>norm (1 - x) < 1\<close>] by (rule sums_unique)
1903     also have "\<dots> = suminf (?f' x)"
1904       unfolding power_mult_distrib[symmetric]
1905       by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
1906     finally have "DERIV ln x :> suminf (?f' x)"
1907       using DERIV_ln[OF \<open>0 < x\<close>] unfolding divide_inverse by auto
1908     moreover
1909     have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
1910     have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :>
1911       (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
1912     proof (rule DERIV_power_series')
1913       show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1"
1914         using \<open>0 < x\<close> \<open>x < 2\<close> by auto
1915     next
1916       fix x :: real
1917       assume "x \<in> {- 1<..<1}"
1918       then have "norm (-x) < 1" by auto
1919       show "summable (\<lambda>n. (- 1) ^ n * (1 / real (n + 1)) * real (Suc n) * x^n)"
1920         unfolding One_nat_def
1921         by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF \<open>norm (-x) < 1\<close>])
1922     qed
1923     then have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
1924       unfolding One_nat_def by auto
1925     then have "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
1926       unfolding DERIV_def repos .
1927     ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> suminf (?f' x) - suminf (?f' x)"
1928       by (rule DERIV_diff)
1929     then show "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
1930   qed (auto simp add: assms)
1931   then show ?thesis by auto
1932 qed
1934 lemma exp_first_terms:
1935   fixes x :: "'a::{real_normed_algebra_1,banach}"
1936   shows "exp x = (\<Sum>n<k. inverse(fact n) *\<^sub>R (x ^ n)) + (\<Sum>n. inverse(fact (n + k)) *\<^sub>R (x ^ (n + k)))"
1937 proof -
1938   have "exp x = suminf (\<lambda>n. inverse(fact n) *\<^sub>R (x^n))"
1940   also from summable_exp_generic have "\<dots> = (\<Sum> n. inverse(fact(n+k)) *\<^sub>R (x ^ (n + k))) +
1941     (\<Sum> n::nat<k. inverse(fact n) *\<^sub>R (x^n))" (is "_ = _ + ?a")
1942     by (rule suminf_split_initial_segment)
1943   finally show ?thesis by simp
1944 qed
1946 lemma exp_first_term: "exp x = 1 + (\<Sum>n. inverse (fact (Suc n)) *\<^sub>R (x ^ Suc n))"
1947   for x :: "'a::{real_normed_algebra_1,banach}"
1948   using exp_first_terms[of x 1] by simp
1950 lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum>n. inverse (fact (n + 2)) *\<^sub>R (x ^ (n + 2)))"
1951   for x :: "'a::{real_normed_algebra_1,banach}"
1952   using exp_first_terms[of x 2] by (simp add: eval_nat_numeral)
1954 lemma exp_bound:
1955   fixes x :: real
1956   assumes a: "0 \<le> x"
1957     and b: "x \<le> 1"
1958   shows "exp x \<le> 1 + x + x\<^sup>2"
1959 proof -
1960   have aux1: "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)" for n :: nat
1961   proof -
1962     have "(2::nat) * 2 ^ n \<le> fact (n + 2)"
1963       by (induct n) simp_all
1964     then have "real ((2::nat) * 2 ^ n) \<le> real_of_nat (fact (n + 2))"
1965       by (simp only: of_nat_le_iff)
1966     then have "((2::real) * 2 ^ n) \<le> fact (n + 2)"
1967       unfolding of_nat_fact by simp
1968     then have "inverse (fact (n + 2)) \<le> inverse ((2::real) * 2 ^ n)"
1969       by (rule le_imp_inverse_le) simp
1970     then have "inverse (fact (n + 2)) \<le> 1/(2::real) * (1/2)^n"
1971       by (simp add: power_inverse [symmetric])
1972     then have "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
1973       by (rule mult_mono) (rule mult_mono, simp_all add: power_le_one a b)
1974     then show ?thesis
1976   qed
1977   have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
1978     by (intro sums_mult geometric_sums) simp
1979   then have aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
1980     by simp
1981   have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n + 2))) \<le> x\<^sup>2"
1982   proof -
1983     have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n + 2))) \<le> suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
1984       apply (rule suminf_le)
1985         apply (rule allI)
1986         apply (rule aux1)
1987        apply (rule summable_exp [THEN summable_ignore_initial_segment])
1988       apply (rule sums_summable)
1989       apply (rule aux2)
1990       done
1991     also have "\<dots> = x\<^sup>2"
1992       by (rule sums_unique [THEN sym]) (rule aux2)
1993     finally show ?thesis .
1994   qed
1995   then show ?thesis
1996     unfolding exp_first_two_terms by auto
1997 qed
1999 corollary exp_half_le2: "exp(1/2) \<le> (2::real)"
2000   using exp_bound [of "1/2"]
2003 corollary exp_le: "exp 1 \<le> (3::real)"
2004   using exp_bound [of 1]
2007 lemma exp_bound_half: "norm z \<le> 1/2 \<Longrightarrow> norm (exp z) \<le> 2"
2008   by (blast intro: order_trans intro!: exp_half_le2 norm_exp)
2010 lemma exp_bound_lemma:
2011   assumes "norm z \<le> 1/2"
2012   shows "norm (exp z) \<le> 1 + 2 * norm z"
2013 proof -
2014   have *: "(norm z)\<^sup>2 \<le> norm z * 1"
2015     unfolding power2_eq_square
2016     apply (rule mult_left_mono)
2017     using assms
2018      apply auto
2019     done
2020   show ?thesis
2021     apply (rule order_trans [OF norm_exp])
2022     apply (rule order_trans [OF exp_bound])
2023     using assms *
2024       apply auto
2025     done
2026 qed
2028 lemma real_exp_bound_lemma: "0 \<le> x \<Longrightarrow> x \<le> 1/2 \<Longrightarrow> exp x \<le> 1 + 2 * x"
2029   for x :: real
2030   using exp_bound_lemma [of x] by simp
2032 lemma ln_one_minus_pos_upper_bound:
2033   fixes x :: real
2034   assumes a: "0 \<le> x" and b: "x < 1"
2035   shows "ln (1 - x) \<le> - x"
2036 proof -
2037   have "(1 - x) * (1 + x + x\<^sup>2) = 1 - x^3"
2038     by (simp add: algebra_simps power2_eq_square power3_eq_cube)
2039   also have "\<dots> \<le> 1"
2040     by (auto simp add: a)
2041   finally have "(1 - x) * (1 + x + x\<^sup>2) \<le> 1" .
2042   moreover have c: "0 < 1 + x + x\<^sup>2"
2044   ultimately have "1 - x \<le> 1 / (1 + x + x\<^sup>2)"
2045     by (elim mult_imp_le_div_pos)
2046   also have "\<dots> \<le> 1 / exp x"
2047     by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs
2048         real_sqrt_pow2_iff real_sqrt_power)
2049   also have "\<dots> = exp (- x)"
2050     by (auto simp add: exp_minus divide_inverse)
2051   finally have "1 - x \<le> exp (- x)" .
2052   also have "1 - x = exp (ln (1 - x))"
2053     by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq)
2054   finally have "exp (ln (1 - x)) \<le> exp (- x)" .
2055   then show ?thesis
2056     by (auto simp only: exp_le_cancel_iff)
2057 qed
2059 lemma exp_ge_add_one_self [simp]: "1 + x \<le> exp x"
2060   for x :: real
2061   apply (cases "0 \<le> x")
2063   apply (cases "x \<le> -1")
2064    apply (subgoal_tac "1 + x \<le> 0")
2065     apply (erule order_trans)
2066     apply simp
2067    apply simp
2068   apply (subgoal_tac "1 + x = exp (ln (1 + x))")
2069    apply (erule ssubst)
2070    apply (subst exp_le_cancel_iff)
2071    apply (subgoal_tac "ln (1 - (- x)) \<le> - (- x)")
2072     apply simp
2073    apply (rule ln_one_minus_pos_upper_bound)
2074     apply auto
2075   done
2077 lemma ln_one_plus_pos_lower_bound:
2078   fixes x :: real
2079   assumes a: "0 \<le> x" and b: "x \<le> 1"
2080   shows "x - x\<^sup>2 \<le> ln (1 + x)"
2081 proof -
2082   have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
2083     by (rule exp_diff)
2084   also have "\<dots> \<le> (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
2085     by (metis a b divide_right_mono exp_bound exp_ge_zero)
2086   also have "\<dots> \<le> (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
2088   also from a have "\<dots> \<le> 1 + x"
2090   finally have "exp (x - x\<^sup>2) \<le> 1 + x" .
2091   also have "\<dots> = exp (ln (1 + x))"
2092   proof -
2093     from a have "0 < 1 + x" by auto
2094     then show ?thesis
2095       by (auto simp only: exp_ln_iff [THEN sym])
2096   qed
2097   finally have "exp (x - x\<^sup>2) \<le> exp (ln (1 + x))" .
2098   then show ?thesis
2099     by (metis exp_le_cancel_iff)
2100 qed
2102 lemma ln_one_minus_pos_lower_bound:
2103   fixes x :: real
2104   assumes a: "0 \<le> x" and b: "x \<le> 1 / 2"
2105   shows "- x - 2 * x\<^sup>2 \<le> ln (1 - x)"
2106 proof -
2107   from b have c: "x < 1" by auto
2108   then have "ln (1 - x) = - ln (1 + x / (1 - x))"
2109     apply (subst ln_inverse [symmetric])
2111     apply (rule arg_cong [where f=ln])
2113     done
2114   also have "- (x / (1 - x)) \<le> \<dots>"
2115   proof -
2116     have "ln (1 + x / (1 - x)) \<le> x / (1 - x)"
2117       using a c by (intro ln_add_one_self_le_self) auto
2118     then show ?thesis
2119       by auto
2120   qed
2121   also have "- (x / (1 - x)) = - x / (1 - x)"
2122     by auto
2123   finally have d: "- x / (1 - x) \<le> ln (1 - x)" .
2124   have "0 < 1 - x" using a b by simp
2125   then have e: "- x - 2 * x\<^sup>2 \<le> - x / (1 - x)"
2126     using mult_right_le_one_le[of "x * x" "2 * x"] a b
2127     by (simp add: field_simps power2_eq_square)
2128   from e d show "- x - 2 * x\<^sup>2 \<le> ln (1 - x)"
2129     by (rule order_trans)
2130 qed
2133   fixes x :: real
2134   shows "-1 < x \<Longrightarrow> ln (1 + x) \<le> x"
2135   apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)")
2136    apply simp
2137   apply (subst ln_le_cancel_iff)
2138     apply auto
2139   done
2141 lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
2142   fixes x :: real
2143   assumes x: "0 \<le> x" and x1: "x \<le> 1"
2144   shows "\<bar>ln (1 + x) - x\<bar> \<le> x\<^sup>2"
2145 proof -
2146   from x have "ln (1 + x) \<le> x"
2148   then have "ln (1 + x) - x \<le> 0"
2149     by simp
2150   then have "\<bar>ln(1 + x) - x\<bar> = - (ln(1 + x) - x)"
2151     by (rule abs_of_nonpos)
2152   also have "\<dots> = x - ln (1 + x)"
2153     by simp
2154   also have "\<dots> \<le> x\<^sup>2"
2155   proof -
2156     from x x1 have "x - x\<^sup>2 \<le> ln (1 + x)"
2157       by (intro ln_one_plus_pos_lower_bound)
2158     then show ?thesis
2159       by simp
2160   qed
2161   finally show ?thesis .
2162 qed
2164 lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
2165   fixes x :: real
2166   assumes a: "-(1 / 2) \<le> x" and b: "x \<le> 0"
2167   shows "\<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2"
2168 proof -
2169   have "\<bar>ln (1 + x) - x\<bar> = x - ln (1 - (- x))"
2170     apply (subst abs_of_nonpos)
2171      apply simp
2173     using a apply auto
2174     done
2175   also have "\<dots> \<le> 2 * x\<^sup>2"
2176     apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 \<le> ln (1 - (- x))")
2178     apply (rule ln_one_minus_pos_lower_bound)
2179     using a b apply auto
2180     done
2181   finally show ?thesis .
2182 qed
2184 lemma abs_ln_one_plus_x_minus_x_bound:
2185   fixes x :: real
2186   shows "\<bar>x\<bar> \<le> 1 / 2 \<Longrightarrow> \<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2"
2187   apply (cases "0 \<le> x")
2188    apply (rule order_trans)
2189     apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
2190      apply auto
2191   apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
2192    apply auto
2193   done
2195 lemma ln_x_over_x_mono:
2196   fixes x :: real
2197   assumes x: "exp 1 \<le> x" "x \<le> y"
2198   shows "ln y / y \<le> ln x / x"
2199 proof -
2200   note x
2201   moreover have "0 < exp (1::real)" by simp
2202   ultimately have a: "0 < x" and b: "0 < y"
2203     by (fast intro: less_le_trans order_trans)+
2204   have "x * ln y - x * ln x = x * (ln y - ln x)"
2206   also have "\<dots> = x * ln (y / x)"
2207     by (simp only: ln_div a b)
2208   also have "y / x = (x + (y - x)) / x"
2209     by simp
2210   also have "\<dots> = 1 + (y - x) / x"
2211     using x a by (simp add: field_simps)
2212   also have "x * ln (1 + (y - x) / x) \<le> x * ((y - x) / x)"
2213     using x a
2214     by (intro mult_left_mono ln_add_one_self_le_self) simp_all
2215   also have "\<dots> = y - x"
2216     using a by simp
2217   also have "\<dots> = (y - x) * ln (exp 1)" by simp
2218   also have "\<dots> \<le> (y - x) * ln x"
2219     apply (rule mult_left_mono)
2220      apply (subst ln_le_cancel_iff)
2221        apply fact
2222       apply (rule a)
2223      apply (rule x)
2224     using x apply simp
2225     done
2226   also have "\<dots> = y * ln x - x * ln x"
2227     by (rule left_diff_distrib)
2228   finally have "x * ln y \<le> y * ln x"
2229     by arith
2230   then have "ln y \<le> (y * ln x) / x"
2231     using a by (simp add: field_simps)
2232   also have "\<dots> = y * (ln x / x)" by simp
2233   finally show ?thesis
2234     using b by (simp add: field_simps)
2235 qed
2237 lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1"
2238   for x :: real
2239   using exp_ge_add_one_self[of "ln x"] by simp
2241 corollary ln_diff_le: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x - ln y \<le> (x - y) / y"
2242   for x :: real
2243   by (simp add: ln_div [symmetric] diff_divide_distrib ln_le_minus_one)
2245 lemma ln_eq_minus_one:
2246   fixes x :: real
2247   assumes "0 < x" "ln x = x - 1"
2248   shows "x = 1"
2249 proof -
2250   let ?l = "\<lambda>y. ln y - y + 1"
2251   have D: "\<And>x::real. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
2252     by (auto intro!: derivative_eq_intros)
2254   show ?thesis
2255   proof (cases rule: linorder_cases)
2256     assume "x < 1"
2257     from dense[OF \<open>x < 1\<close>] obtain a where "x < a" "a < 1" by blast
2258     from \<open>x < a\<close> have "?l x < ?l a"
2259     proof (rule DERIV_pos_imp_increasing, safe)
2260       fix y
2261       assume "x \<le> y" "y \<le> a"
2262       with \<open>0 < x\<close> \<open>a < 1\<close> have "0 < 1 / y - 1" "0 < y"
2263         by (auto simp: field_simps)
2264       with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z" by blast
2265     qed
2266     also have "\<dots> \<le> 0"
2267       using ln_le_minus_one \<open>0 < x\<close> \<open>x < a\<close> by (auto simp: field_simps)
2268     finally show "x = 1" using assms by auto
2269   next
2270     assume "1 < x"
2271     from dense[OF this] obtain a where "1 < a" "a < x" by blast
2272     from \<open>a < x\<close> have "?l x < ?l a"
2273     proof (rule DERIV_neg_imp_decreasing, safe)
2274       fix y
2275       assume "a \<le> y" "y \<le> x"
2276       with \<open>1 < a\<close> have "1 / y - 1 < 0" "0 < y"
2277         by (auto simp: field_simps)
2278       with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
2279         by blast
2280     qed
2281     also have "\<dots> \<le> 0"
2282       using ln_le_minus_one \<open>1 < a\<close> by (auto simp: field_simps)
2283     finally show "x = 1" using assms by auto
2284   next
2285     assume "x = 1"
2286     then show ?thesis by simp
2287   qed
2288 qed
2290 lemma ln_x_over_x_tendsto_0: "((\<lambda>x::real. ln x / x) \<longlongrightarrow> 0) at_top"
2291 proof (rule lhospital_at_top_at_top[where f' = inverse and g' = "\<lambda>_. 1"])
2292   from eventually_gt_at_top[of "0::real"]
2293   show "\<forall>\<^sub>F x in at_top. (ln has_real_derivative inverse x) (at x)"
2294     by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps)
2295 qed (use tendsto_inverse_0 in
2296       \<open>auto simp: filterlim_ident dest!: tendsto_mono[OF at_top_le_at_infinity]\<close>)
2298 lemma exp_ge_one_plus_x_over_n_power_n:
2299   assumes "x \<ge> - real n" "n > 0"
2300   shows "(1 + x / of_nat n) ^ n \<le> exp x"
2301 proof (cases "x = - of_nat n")
2302   case False
2303   from assms False have "(1 + x / of_nat n) ^ n = exp (of_nat n * ln (1 + x / of_nat n))"
2304     by (subst exp_of_nat_mult, subst exp_ln) (simp_all add: field_simps)
2305   also from assms False have "ln (1 + x / real n) \<le> x / real n"
2307   with assms have "exp (of_nat n * ln (1 + x / of_nat n)) \<le> exp x"
2308     by (simp add: field_simps del: exp_of_nat_mult)
2309   finally show ?thesis .
2310 next
2311   case True
2312   then show ?thesis by (simp add: zero_power)
2313 qed
2315 lemma exp_ge_one_minus_x_over_n_power_n:
2316   assumes "x \<le> real n" "n > 0"
2317   shows "(1 - x / of_nat n) ^ n \<le> exp (-x)"
2318   using exp_ge_one_plus_x_over_n_power_n[of n "-x"] assms by simp
2320 lemma exp_at_bot: "(exp \<longlongrightarrow> (0::real)) at_bot"
2321   unfolding tendsto_Zfun_iff
2322 proof (rule ZfunI, simp add: eventually_at_bot_dense)
2323   fix r :: real
2324   assume "0 < r"
2325   have "exp x < r" if "x < ln r" for x
2326   proof -
2327     from that have "exp x < exp (ln r)"
2328       by simp
2329     with \<open>0 < r\<close> show ?thesis
2330       by simp
2331   qed
2332   then show "\<exists>k. \<forall>n<k. exp n < r" by auto
2333 qed
2335 lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
2336   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"])
2337     (auto intro: eventually_gt_at_top)
2339 lemma lim_exp_minus_1: "((\<lambda>z::'a. (exp(z) - 1) / z) \<longlongrightarrow> 1) (at 0)"
2340   for x :: "'a::{real_normed_field,banach}"
2341 proof -
2342   have "((\<lambda>z::'a. exp(z) - 1) has_field_derivative 1) (at 0)"
2343     by (intro derivative_eq_intros | simp)+
2344   then show ?thesis
2346 qed
2348 lemma ln_at_0: "LIM x at_right 0. ln (x::real) :> at_bot"
2349   by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
2350      (auto simp: eventually_at_filter)
2352 lemma ln_at_top: "LIM x at_top. ln (x::real) :> at_top"
2353   by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
2354      (auto intro: eventually_gt_at_top)
2356 lemma filtermap_ln_at_top: "filtermap (ln::real \<Rightarrow> real) at_top = at_top"
2357   by (intro filtermap_fun_inverse[of exp] exp_at_top ln_at_top) auto
2359 lemma filtermap_exp_at_top: "filtermap (exp::real \<Rightarrow> real) at_top = at_top"
2360   by (intro filtermap_fun_inverse[of ln] exp_at_top ln_at_top)
2361      (auto simp: eventually_at_top_dense)
2363 lemma filtermap_ln_at_right: "filtermap ln (at_right (0::real)) = at_bot"
2364   by (auto intro!: filtermap_fun_inverse[where g="\<lambda>x. exp x"] ln_at_0
2365       simp: filterlim_at exp_at_bot)
2367 lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) \<longlongrightarrow> (0::real)) at_top"
2368 proof (induct k)
2369   case 0
2370   show "((\<lambda>x. x ^ 0 / exp x) \<longlongrightarrow> (0::real)) at_top"
2372        (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
2373          at_top_le_at_infinity order_refl)
2374 next
2375   case (Suc k)
2376   show ?case
2377   proof (rule lhospital_at_top_at_top)
2378     show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"
2379       by eventually_elim (intro derivative_eq_intros, auto)
2380     show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top"
2381       by eventually_elim auto
2382     show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
2383       by auto
2384     from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
2385     show "((\<lambda>x. real (Suc k) * x ^ k / exp x) \<longlongrightarrow> 0) at_top"
2386       by simp
2387   qed (rule exp_at_top)
2388 qed
2390 subsubsection\<open> A couple of simple bounds\<close>
2392 lemma exp_plus_inverse_exp:
2393   fixes x::real
2394   shows "2 \<le> exp x + inverse (exp x)"
2395 proof -
2396   have "2 \<le> exp x + exp (-x)"
2398     by linarith
2399   then show ?thesis
2401 qed
2403 lemma real_le_x_sinh:
2404   fixes x::real
2405   assumes "0 \<le> x"
2406   shows "x \<le> (exp x - inverse(exp x)) / 2"
2407 proof -
2408   have *: "exp a - inverse(exp a) - 2*a \<le> exp b - inverse(exp b) - 2*b" if "a \<le> b" for a b::real
2409     apply (rule DERIV_nonneg_imp_nondecreasing [OF that])
2410     using exp_plus_inverse_exp
2411     apply (intro exI allI impI conjI derivative_eq_intros | force)+
2412     done
2413   show ?thesis
2414     using*[OF assms] by simp
2415 qed
2417 lemma real_le_abs_sinh:
2418   fixes x::real
2419   shows "abs x \<le> abs((exp x - inverse(exp x)) / 2)"
2420 proof (cases "0 \<le> x")
2421   case True
2422   show ?thesis
2423     using real_le_x_sinh [OF True] True by (simp add: abs_if)
2424 next
2425   case False
2426   have "-x \<le> (exp(-x) - inverse(exp(-x))) / 2"
2427     by (meson False linear neg_le_0_iff_le real_le_x_sinh)
2428   also have "... \<le> \<bar>(exp x - inverse (exp x)) / 2\<bar>"
2429     by (metis (no_types, hide_lams) abs_divide abs_le_iff abs_minus_cancel
2431   finally show ?thesis
2432     using False by linarith
2433 qed
2435 subsection\<open>The general logarithm\<close>
2437 definition log :: "real \<Rightarrow> real \<Rightarrow> real"
2438   \<comment> \<open>logarithm of @{term x} to base @{term a}\<close>
2439   where "log a x = ln x / ln a"
2441 lemma tendsto_log [tendsto_intros]:
2442   "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> 0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow>
2443     ((\<lambda>x. log (f x) (g x)) \<longlongrightarrow> log a b) F"
2444   unfolding log_def by (intro tendsto_intros) auto
2446 lemma continuous_log:
2447   assumes "continuous F f"
2448     and "continuous F g"
2449     and "0 < f (Lim F (\<lambda>x. x))"
2450     and "f (Lim F (\<lambda>x. x)) \<noteq> 1"
2451     and "0 < g (Lim F (\<lambda>x. x))"
2452   shows "continuous F (\<lambda>x. log (f x) (g x))"
2453   using assms unfolding continuous_def by (rule tendsto_log)
2455 lemma continuous_at_within_log[continuous_intros]:
2456   assumes "continuous (at a within s) f"
2457     and "continuous (at a within s) g"
2458     and "0 < f a"
2459     and "f a \<noteq> 1"
2460     and "0 < g a"
2461   shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
2462   using assms unfolding continuous_within by (rule tendsto_log)
2464 lemma isCont_log[continuous_intros, simp]:
2465   assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
2466   shows "isCont (\<lambda>x. log (f x) (g x)) a"
2467   using assms unfolding continuous_at by (rule tendsto_log)
2469 lemma continuous_on_log[continuous_intros]:
2470   assumes "continuous_on s f" "continuous_on s g"
2471     and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
2472   shows "continuous_on s (\<lambda>x. log (f x) (g x))"
2473   using assms unfolding continuous_on_def by (fast intro: tendsto_log)
2475 lemma powr_one_eq_one [simp]: "1 powr a = 1"
2478 lemma powr_zero_eq_one [simp]: "x powr 0 = (if x = 0 then 0 else 1)"
2481 lemma powr_one_gt_zero_iff [simp]: "x powr 1 = x \<longleftrightarrow> 0 \<le> x"
2482   for x :: real
2483   by (auto simp: powr_def)
2484 declare powr_one_gt_zero_iff [THEN iffD2, simp]
2486 lemma powr_diff:
2487   fixes w:: "'a::{ln,real_normed_field}" shows  "w powr (z1 - z2) = w powr z1 / w powr z2"
2488   by (simp add: powr_def algebra_simps exp_diff)
2490 lemma powr_mult: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
2491   for a x y :: real
2494 lemma powr_ge_pzero [simp]: "0 \<le> x powr y"
2495   for x y :: real
2498 lemma powr_divide: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
2499   for a b x :: real
2500   apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
2502   done
2504 lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
2505   for a b x :: "'a::{ln,real_normed_field}"
2508 lemma powr_mult_base: "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
2509   for x :: real
2512 lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
2513   for a b x :: real
2516 lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
2517   for a b x :: real
2518   by (simp add: powr_powr mult.commute)
2520 lemma powr_minus: "x powr (- a) = inverse (x powr a)"
2521       for a x :: "'a::{ln,real_normed_field}"
2522   by (simp add: powr_def exp_minus [symmetric])
2524 lemma powr_minus_divide: "x powr (- a) = 1/(x powr a)"
2525   for x a :: real
2526   by (simp add: divide_inverse powr_minus)
2528 lemma divide_powr_uminus: "a / b powr c = a * b powr (- c)"
2529   for a b c :: real
2532 lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
2533   for a b x :: real
2536 lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
2537   for a b x :: real
2540 lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> x powr a < x powr b \<longleftrightarrow> a < b"
2541   for a b x :: real
2542   by (blast intro: powr_less_cancel powr_less_mono)
2544 lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> x powr a \<le> x powr b \<longleftrightarrow> a \<le> b"
2545   for a b x :: real
2546   by (simp add: linorder_not_less [symmetric])
2548 lemma powr_realpow: "0 < x \<Longrightarrow> x powr (real n) = x^n"
2551 lemma log_ln: "ln x = log (exp(1)) x"
2554 lemma DERIV_log:
2555   assumes "x > 0"
2556   shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
2557 proof -
2558   define lb where "lb = 1 / ln b"
2559   moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
2560     using \<open>x > 0\<close> by (auto intro!: derivative_eq_intros)
2561   ultimately show ?thesis
2563 qed
2565 lemmas DERIV_log[THEN DERIV_chain2, derivative_intros]
2566   and DERIV_log[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
2568 lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
2569   by (simp add: powr_def log_def)
2571 lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y"
2572   by (simp add: log_def powr_def)
2574 lemma log_mult:
2575   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow>
2576     log a (x * y) = log a x + log a y"
2577   by (simp add: log_def ln_mult divide_inverse distrib_right)
2579 lemma log_eq_div_ln_mult_log:
2580   "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow>
2581     log a x = (ln b/ln a) * log b x"
2582   by (simp add: log_def divide_inverse)
2584 text\<open>Base 10 logarithms\<close>
2585 lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x"
2588 lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x"
2591 lemma log_one [simp]: "log a 1 = 0"
2594 lemma log_eq_one [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a a = 1"
2597 lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x"
2598   apply (rule add_left_cancel [THEN iffD1, where a1 = "log a x"])
2599   apply (simp add: log_mult [symmetric])
2600   done
2602 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"
2603   by (simp add: log_mult divide_inverse log_inverse)
2605 lemma powr_gt_zero [simp]: "0 < x powr a \<longleftrightarrow> x \<noteq> 0"
2606   for a x :: real
2609 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)"
2610   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)"
2611   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)"
2612   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)"
2613   by (simp_all add: log_mult log_divide)
2615 lemma log_less_cancel_iff [simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
2616   apply safe
2617    apply (rule_tac [2] powr_less_cancel)
2618     apply (drule_tac a = "log a x" in powr_less_mono)
2619      apply auto
2620   done
2622 lemma log_inj:
2623   assumes "1 < b"
2624   shows "inj_on (log b) {0 <..}"
2625 proof (rule inj_onI, simp)
2626   fix x y
2627   assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
2628   show "x = y"
2629   proof (cases rule: linorder_cases)
2630     assume "x = y"
2631     then show ?thesis by simp
2632   next
2633     assume "x < y"
2634     then have "log b x < log b y"
2635       using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp
2636     then show ?thesis using * by simp
2637   next
2638     assume "y < x"
2639     then have "log b y < log b x"
2640       using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp
2641     then show ?thesis using * by simp
2642   qed
2643 qed
2645 lemma log_le_cancel_iff [simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x \<le> log a y \<longleftrightarrow> x \<le> y"
2646   by (simp add: linorder_not_less [symmetric])
2648 lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
2649   using log_less_cancel_iff[of a 1 x] by simp
2651 lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
2652   using log_le_cancel_iff[of a 1 x] by simp
2654 lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
2655   using log_less_cancel_iff[of a x 1] by simp
2657 lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
2658   using log_le_cancel_iff[of a x 1] by simp
2660 lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
2661   using log_less_cancel_iff[of a a x] by simp
2663 lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
2664   using log_le_cancel_iff[of a a x] by simp
2666 lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
2667   using log_less_cancel_iff[of a x a] by simp
2669 lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
2670   using log_le_cancel_iff[of a x a] by simp
2672 lemma le_log_iff:
2673   fixes b x y :: real
2674   assumes "1 < b" "x > 0"
2675   shows "y \<le> log b x \<longleftrightarrow> b powr y \<le> x"
2676   using assms
2677   apply auto
2678    apply (metis (no_types, hide_lams) less_irrefl less_le_trans linear powr_le_cancel_iff
2679       powr_log_cancel zero_less_one)
2680   apply (metis not_less order.trans order_refl powr_le_cancel_iff powr_log_cancel zero_le_one)
2681   done
2683 lemma less_log_iff:
2684   assumes "1 < b" "x > 0"
2685   shows "y < log b x \<longleftrightarrow> b powr y < x"
2686   by (metis assms dual_order.strict_trans less_irrefl powr_less_cancel_iff
2687     powr_log_cancel zero_less_one)
2689 lemma
2690   assumes "1 < b" "x > 0"
2691   shows log_less_iff: "log b x < y \<longleftrightarrow> x < b powr y"
2692     and log_le_iff: "log b x \<le> y \<longleftrightarrow> x \<le> b powr y"
2693   using le_log_iff[OF assms, of y] less_log_iff[OF assms, of y]
2694   by auto
2696 lemmas powr_le_iff = le_log_iff[symmetric]
2697   and powr_less_iff = less_log_iff[symmetric]
2698   and less_powr_iff = log_less_iff[symmetric]
2699   and le_powr_iff = log_le_iff[symmetric]
2701 lemma le_log_of_power:
2702   assumes "b ^ n \<le> m" "1 < b"
2703   shows "n \<le> log b m"
2704 proof -
2705   from assms have "0 < m" by (metis less_trans zero_less_power less_le_trans zero_less_one)
2706   thus ?thesis using assms by (simp add: le_log_iff powr_realpow)
2707 qed
2709 lemma le_log2_of_power: "2 ^ n \<le> m \<Longrightarrow> n \<le> log 2 m" for m n :: nat
2710 using le_log_of_power[of 2] by simp
2712 lemma log_of_power_le: "\<lbrakk> m \<le> b ^ n; b > 1; m > 0 \<rbrakk> \<Longrightarrow> log b (real m) \<le> n"
2713 by (simp add: log_le_iff powr_realpow)
2715 lemma log2_of_power_le: "\<lbrakk> m \<le> 2 ^ n; m > 0 \<rbrakk> \<Longrightarrow> log 2 m \<le> n" for m n :: nat
2716 using log_of_power_le[of _ 2] by simp
2718 lemma log_of_power_less: "\<lbrakk> m < b ^ n; b > 1; m > 0 \<rbrakk> \<Longrightarrow> log b (real m) < n"
2719 by (simp add: log_less_iff powr_realpow)
2721 lemma log2_of_power_less: "\<lbrakk> m < 2 ^ n; m > 0 \<rbrakk> \<Longrightarrow> log 2 m < n" for m n :: nat
2722 using log_of_power_less[of _ 2] by simp
2724 lemma less_log_of_power:
2725   assumes "b ^ n < m" "1 < b"
2726   shows "n < log b m"
2727 proof -
2728   have "0 < m" by (metis assms less_trans zero_less_power zero_less_one)
2729   thus ?thesis using assms by (simp add: less_log_iff powr_realpow)
2730 qed
2732 lemma less_log2_of_power: "2 ^ n < m \<Longrightarrow> n < log 2 m" for m n :: nat
2733 using less_log_of_power[of 2] by simp
2735 lemma gr_one_powr[simp]:
2736   fixes x y :: real shows "\<lbrakk> x > 1; y > 0 \<rbrakk> \<Longrightarrow> 1 < x powr y"
2739 lemma 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)"
2740   by (auto simp add: floor_eq_iff powr_le_iff less_powr_iff)
2742 lemma floor_log_nat_eq_powr_iff: fixes b n k :: nat
2743   shows "\<lbrakk> b \<ge> 2; k > 0 \<rbrakk> \<Longrightarrow>
2744   floor (log b (real k)) = n \<longleftrightarrow> b^n \<le> k \<and> k < b^(n+1)"
2745 by (auto simp: floor_log_eq_powr_iff powr_add powr_realpow
2746                of_nat_power[symmetric] of_nat_mult[symmetric] ac_simps
2747          simp del: of_nat_power of_nat_mult)
2749 lemma floor_log_nat_eq_if: fixes b n k :: nat
2750   assumes "b^n \<le> k" "k < b^(n+1)" "b \<ge> 2"
2751   shows "floor (log b (real k)) = n"
2752 proof -
2753   have "k \<ge> 1" using assms(1,3) one_le_power[of b n] by linarith
2754   with assms show ?thesis by(simp add: floor_log_nat_eq_powr_iff)
2755 qed
2757 lemma ceiling_log_eq_powr_iff: "\<lbrakk> x > 0; b > 1 \<rbrakk>
2758   \<Longrightarrow> \<lceil>log b x\<rceil> = int k + 1 \<longleftrightarrow> b powr k < x \<and> x \<le> b powr (k + 1)"
2759 by (auto simp add: ceiling_eq_iff powr_less_iff le_powr_iff)
2761 lemma ceiling_log_nat_eq_powr_iff: fixes b n k :: nat
2762   shows "\<lbrakk> b \<ge> 2; k > 0 \<rbrakk> \<Longrightarrow>
2763   ceiling (log b (real k)) = int n + 1 \<longleftrightarrow> (b^n < k \<and> k \<le> b^(n+1))"
2764 using ceiling_log_eq_powr_iff
2765 by (auto simp: powr_add powr_realpow of_nat_power[symmetric] of_nat_mult[symmetric] ac_simps
2766          simp del: of_nat_power of_nat_mult)
2768 lemma ceiling_log_nat_eq_if: fixes b n k :: nat
2769   assumes "b^n < k" "k \<le> b^(n+1)" "b \<ge> 2"
2770   shows "ceiling (log b (real k)) = int n + 1"
2771 proof -
2772   have "k \<ge> 1" using assms(1,3) one_le_power[of b n] by linarith
2773   with assms show ?thesis by(simp add: ceiling_log_nat_eq_powr_iff)
2774 qed
2776 lemma floor_log2_div2: fixes n :: nat assumes "n \<ge> 2"
2777 shows "floor(log 2 n) = floor(log 2 (n div 2)) + 1"
2778 proof cases
2779   assume "n=2" thus ?thesis by simp
2780 next
2781   let ?m = "n div 2"
2782   assume "n\<noteq>2"
2783   hence "1 \<le> ?m" using assms by arith
2784   then obtain i where i: "2 ^ i \<le> ?m" "?m < 2 ^ (i + 1)"
2785     using ex_power_ivl1[of 2 ?m] by auto
2786   have "2^(i+1) \<le> 2*?m" using i(1) by simp
2787   also have "2*?m \<le> n" by arith
2788   finally have *: "2^(i+1) \<le> \<dots>" .
2789   have "n < 2^(i+1+1)" using i(2) by simp
2790   from floor_log_nat_eq_if[OF * this] floor_log_nat_eq_if[OF i]
2791   show ?thesis by simp
2792 qed
2794 lemma ceiling_log2_div2: assumes "n \<ge> 2"
2795 shows "ceiling(log 2 (real n)) = ceiling(log 2 ((n-1) div 2 + 1)) + 1"
2796 proof cases
2797   assume "n=2" thus ?thesis by simp
2798 next
2799   let ?m = "(n-1) div 2 + 1"
2800   assume "n\<noteq>2"
2801   hence "2 \<le> ?m" using assms by arith
2802   then obtain i where i: "2 ^ i < ?m" "?m \<le> 2 ^ (i + 1)"
2803     using ex_power_ivl2[of 2 ?m] by auto
2804   have "n \<le> 2*?m" by arith
2805   also have "2*?m \<le> 2 ^ ((i+1)+1)" using i(2) by simp
2806   finally have *: "n \<le> \<dots>" .
2807   have "2^(i+1) < n" using i(1) by (auto simp add: less_Suc_eq_0_disj)
2808   from ceiling_log_nat_eq_if[OF this *] ceiling_log_nat_eq_if[OF i]
2809   show ?thesis by simp
2810 qed
2812 lemma powr_real_of_int:
2813   "x > 0 \<Longrightarrow> x powr real_of_int n = (if n \<ge> 0 then x ^ nat n else inverse (x ^ nat (- n)))"
2814   using powr_realpow[of x "nat n"] powr_realpow[of x "nat (-n)"]
2815   by (auto simp: field_simps powr_minus)
2817 lemma powr_numeral [simp]: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
2818   by (metis of_nat_numeral powr_realpow)
2820 lemma powr_int:
2821   assumes "x > 0"
2822   shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
2823 proof (cases "i < 0")
2824   case True
2825   have r: "x powr i = 1 / x powr (- i)"
2826     by (simp add: powr_minus field_simps)
2827   show ?thesis using \<open>i < 0\<close> \<open>x > 0\<close>
2828     by (simp add: r field_simps powr_realpow[symmetric])
2829 next
2830   case False
2831   then show ?thesis
2832     by (simp add: assms powr_realpow[symmetric])
2833 qed
2835 lemma compute_powr[code]:
2836   fixes i :: real
2837   shows "b powr i =
2838     (if b \<le> 0 then Code.abort (STR ''op powr with nonpositive base'') (\<lambda>_. b powr i)
2839      else if \<lfloor>i\<rfloor> = i then (if 0 \<le> i then b ^ nat \<lfloor>i\<rfloor> else 1 / b ^ nat \<lfloor>- i\<rfloor>)
2840      else Code.abort (STR ''op powr with non-integer exponent'') (\<lambda>_. b powr i))"
2841   by (auto simp: powr_int)
2843 lemma powr_one: "0 \<le> x \<Longrightarrow> x powr 1 = x"
2844   for x :: real
2845   using powr_realpow [of x 1] by simp
2847 lemma powr_neg_one: "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
2848   for x :: real
2849   using powr_int [of x "- 1"] by simp
2851 lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
2852   for x :: real
2853   using powr_int [of x "- numeral n"] by simp
2855 lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
2856   by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
2858 lemma ln_powr: "x \<noteq> 0 \<Longrightarrow> ln (x powr y) = y * ln x"
2859   for x :: real
2862 lemma ln_root: "n > 0 \<Longrightarrow> b > 0 \<Longrightarrow> ln (root n b) =  ln b / n"
2863   by (simp add: root_powr_inverse ln_powr)
2865 lemma ln_sqrt: "0 < x \<Longrightarrow> ln (sqrt x) = ln x / 2"
2866   by (simp add: ln_powr ln_powr[symmetric] mult.commute)
2868 lemma log_root: "n > 0 \<Longrightarrow> a > 0 \<Longrightarrow> log b (root n a) =  log b a / n"
2869   by (simp add: log_def ln_root)
2871 lemma log_powr: "x \<noteq> 0 \<Longrightarrow> log b (x powr y) = y * log b x"
2872   by (simp add: log_def ln_powr)
2874 (* [simp] is not worth it, interferes with some proofs *)
2875 lemma log_nat_power: "0 < x \<Longrightarrow> log b (x^n) = real n * log b x"
2876   by (simp add: log_powr powr_realpow [symmetric])
2878 lemma log_of_power_eq:
2879   assumes "m = b ^ n" "b > 1"
2880   shows "n = log b (real m)"
2881 proof -
2882   have "n = log b (b ^ n)" using assms(2) by (simp add: log_nat_power)
2883   also have "\<dots> = log b m" using assms by (simp)
2884   finally show ?thesis .
2885 qed
2887 lemma log2_of_power_eq: "m = 2 ^ n \<Longrightarrow> n = log 2 m" for m n :: nat
2888 using log_of_power_eq[of _ 2] by simp
2890 lemma log_base_change: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log b x = log a x / log a b"
2893 lemma log_base_pow: "0 < a \<Longrightarrow> log (a ^ n) x = log a x / n"
2894   by (simp add: log_def ln_realpow)
2896 lemma log_base_powr: "a \<noteq> 0 \<Longrightarrow> log (a powr b) x = log a x / b"
2897   by (simp add: log_def ln_powr)
2899 lemma log_base_root: "n > 0 \<Longrightarrow> b > 0 \<Longrightarrow> log (root n b) x = n * (log b x)"
2900   by (simp add: log_def ln_root)
2902 lemma ln_bound: "1 \<le> x \<Longrightarrow> ln x \<le> x"
2903   for x :: real
2904   apply (subgoal_tac "ln (1 + (x - 1)) \<le> x - 1")
2905    apply simp
2907   apply simp
2908   done
2910 lemma powr_mono: "a \<le> b \<Longrightarrow> 1 \<le> x \<Longrightarrow> x powr a \<le> x powr b"
2911   for x :: real
2912   apply (cases "x = 1")
2913    apply simp
2914   apply (cases "a = b")
2915    apply simp
2916   apply (rule order_less_imp_le)
2917   apply (rule powr_less_mono)
2918    apply auto
2919   done
2921 lemma ge_one_powr_ge_zero: "1 \<le> x \<Longrightarrow> 0 \<le> a \<Longrightarrow> 1 \<le> x powr a"
2922   for x :: real
2923   using powr_mono by fastforce
2925 lemma powr_less_mono2: "0 < a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> x powr a < y powr a"
2926   for x :: real
2929 lemma powr_less_mono2_neg: "a < 0 \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> y powr a < x powr a"
2930   for x :: real
2933 lemma powr_mono2: "x powr a \<le> y powr a" if "0 \<le> a" "0 \<le> x" "x \<le> y"
2934   for x :: real
2935 proof (cases "a = 0")
2936   case True
2937   with that show ?thesis by simp
2938 next
2939   case False show ?thesis
2940   proof (cases "x = y")
2941     case True
2942     then show ?thesis by simp
2943   next
2944     case False
2945     then show ?thesis
2946       by (metis dual_order.strict_iff_order powr_less_mono2 that \<open>a \<noteq> 0\<close>)
2947   qed
2948 qed
2950 lemma powr_le1: "0 \<le> a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> x powr a \<le> 1"
2951   for x :: real
2952   using powr_mono2 by fastforce
2954 lemma powr_mono2':
2955   fixes a x y :: real
2956   assumes "a \<le> 0" "x > 0" "x \<le> y"
2957   shows "x powr a \<ge> y powr a"
2958 proof -
2959   from assms have "x powr - a \<le> y powr - a"
2960     by (intro powr_mono2) simp_all
2961   with assms show ?thesis
2962     by (auto simp add: powr_minus field_simps)
2963 qed
2965 lemma powr_mono_both:
2966   fixes x :: real
2967   assumes "0 \<le> a" "a \<le> b" "1 \<le> x" "x \<le> y"
2968     shows "x powr a \<le> y powr b"
2969   by (meson assms order.trans powr_mono powr_mono2 zero_le_one)
2971 lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
2972   for x :: real
2973   unfolding powr_def exp_inj_iff by simp
2975 lemma powr_half_sqrt: "0 \<le> x \<Longrightarrow> x powr (1/2) = sqrt x"
2976   by (simp add: powr_def root_powr_inverse sqrt_def)
2978 lemma ln_powr_bound: "1 \<le> x \<Longrightarrow> 0 < a \<Longrightarrow> ln x \<le> (x powr a) / a"
2979   for x :: real
2980   by (metis exp_gt_zero linear ln_eq_zero_iff ln_exp ln_less_self ln_powr mult.commute
2981       mult_imp_le_div_pos not_less powr_gt_zero)
2983 lemma ln_powr_bound2:
2984   fixes x :: real
2985   assumes "1 < x" and "0 < a"
2986   shows "(ln x) powr a \<le> (a powr a) * x"
2987 proof -
2988   from assms have "ln x \<le> (x powr (1 / a)) / (1 / a)"
2989     by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff)
2990   also have "\<dots> = a * (x powr (1 / a))"
2991     by simp
2992   finally have "(ln x) powr a \<le> (a * (x powr (1 / a))) powr a"
2993     by (metis assms less_imp_le ln_gt_zero powr_mono2)
2994   also have "\<dots> = (a powr a) * ((x powr (1 / a)) powr a)"
2995     using assms powr_mult by auto
2996   also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
2997     by (rule powr_powr)
2998   also have "\<dots> = x" using assms
2999     by auto
3000   finally show ?thesis .
3001 qed
3003 lemma tendsto_powr:
3004   fixes a b :: real
3005   assumes f: "(f \<longlongrightarrow> a) F"
3006     and g: "(g \<longlongrightarrow> b) F"
3007     and a: "a \<noteq> 0"
3008   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
3009   unfolding powr_def
3010 proof (rule filterlim_If)
3011   from f show "((\<lambda>x. 0) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))"
3012     by simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_mono dest: t1_space_nhds)
3013   from f g a show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a)))
3014       (inf F (principal {x. f x \<noteq> 0}))"
3015     by (auto intro!: tendsto_intros intro: tendsto_mono inf_le1)
3016 qed
3018 lemma tendsto_powr'[tendsto_intros]:
3019   fixes a :: real
3020   assumes f: "(f \<longlongrightarrow> a) F"
3021     and g: "(g \<longlongrightarrow> b) F"
3022     and a: "a \<noteq> 0 \<or> (b > 0 \<and> eventually (\<lambda>x. f x \<ge> 0) F)"
3023   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
3024 proof -
3025   from a consider "a \<noteq> 0" | "a = 0" "b > 0" "eventually (\<lambda>x. f x \<ge> 0) F"
3026     by auto
3027   then show ?thesis
3028   proof cases
3029     case 1
3030     with f g show ?thesis by (rule tendsto_powr)
3031   next
3032     case 2
3033     have "((\<lambda>x. if f x = 0 then 0 else exp (g x * ln (f x))) \<longlongrightarrow> 0) F"
3034     proof (intro filterlim_If)
3035       have "filterlim f (principal {0<..}) (inf F (principal {z. f z \<noteq> 0}))"
3036         using \<open>eventually (\<lambda>x. f x \<ge> 0) F\<close>
3037         by (auto simp add: filterlim_iff eventually_inf_principal
3038             eventually_principal elim: eventually_mono)
3039       moreover have "filterlim f (nhds a) (inf F (principal {z. f z \<noteq> 0}))"
3040         by (rule tendsto_mono[OF _ f]) simp_all
3041       ultimately have f: "filterlim f (at_right 0) (inf F (principal {x. f x \<noteq> 0}))"
3042         by (simp add: at_within_def filterlim_inf \<open>a = 0\<close>)
3043       have g: "(g \<longlongrightarrow> b) (inf F (principal {z. f z \<noteq> 0}))"
3044         by (rule tendsto_mono[OF _ g]) simp_all
3045       show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> 0) (inf F (principal {x. f x \<noteq> 0}))"
3046         by (rule filterlim_compose[OF exp_at_bot] filterlim_tendsto_pos_mult_at_bot
3047                  filterlim_compose[OF ln_at_0] f g \<open>b > 0\<close>)+
3048     qed simp_all
3049     with \<open>a = 0\<close> show ?thesis
3051   qed
3052 qed
3054 lemma continuous_powr:
3055   assumes "continuous F f"
3056     and "continuous F g"
3057     and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
3058   shows "continuous F (\<lambda>x. (f x) powr (g x :: real))"
3059   using assms unfolding continuous_def by (rule tendsto_powr)
3061 lemma continuous_at_within_powr[continuous_intros]:
3062   fixes f g :: "_ \<Rightarrow> real"
3063   assumes "continuous (at a within s) f"
3064     and "continuous (at a within s) g"
3065     and "f a \<noteq> 0"
3066   shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
3067   using assms unfolding continuous_within by (rule tendsto_powr)
3069 lemma isCont_powr[continuous_intros, simp]:
3070   fixes f g :: "_ \<Rightarrow> real"
3071   assumes "isCont f a" "isCont g a" "f a \<noteq> 0"
3072   shows "isCont (\<lambda>x. (f x) powr g x) a"
3073   using assms unfolding continuous_at by (rule tendsto_powr)
3075 lemma continuous_on_powr[continuous_intros]:
3076   fixes f g :: "_ \<Rightarrow> real"
3077   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> 0"
3078   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
3079   using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
3081 lemma tendsto_powr2:
3082   fixes a :: real
3083   assumes f: "(f \<longlongrightarrow> a) F"
3084     and g: "(g \<longlongrightarrow> b) F"
3085     and "\<forall>\<^sub>F x in F. 0 \<le> f x"
3086     and b: "0 < b"
3087   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
3088   using tendsto_powr'[of f a F g b] assms by auto
3090 lemma DERIV_powr:
3091   fixes r :: real
3092   assumes g: "DERIV g x :> m"
3093     and pos: "g x > 0"
3094     and f: "DERIV f x :> r"
3095   shows "DERIV (\<lambda>x. g x powr f x) x :> (g x powr f x) * (r * ln (g x) + m * f x / g x)"
3096 proof -
3097   have "DERIV (\<lambda>x. exp (f x * ln (g x))) x :> (g x powr f x) * (r * ln (g x) + m * f x / g x)"
3098     using pos
3099     by (auto intro!: derivative_eq_intros g pos f simp: powr_def field_simps exp_diff)
3100   then show ?thesis
3101   proof (rule DERIV_cong_ev[OF refl _ refl, THEN iffD1, rotated])
3102     from DERIV_isCont[OF g] pos have "\<forall>\<^sub>F x in at x. 0 < g x"
3103       unfolding isCont_def by (rule order_tendstoD(1))
3104     with pos show "\<forall>\<^sub>F x in nhds x. exp (f x * ln (g x)) = g x powr f x"
3105       by (auto simp: eventually_at_filter powr_def elim: eventually_mono)
3106   qed
3107 qed
3109 lemma DERIV_fun_powr:
3110   fixes r :: real
3111   assumes g: "DERIV g x :> m"
3112     and pos: "g x > 0"
3113   shows "DERIV (\<lambda>x. (g x) powr r) x :> r * (g x) powr (r - of_nat 1) * m"
3114   using DERIV_powr[OF g pos DERIV_const, of r] pos
3115   by (simp add: powr_diff field_simps)
3117 lemma has_real_derivative_powr:
3118   assumes "z > 0"
3119   shows "((\<lambda>z. z powr r) has_real_derivative r * z powr (r - 1)) (at z)"
3120 proof (subst DERIV_cong_ev[OF refl _ refl])
3121   from assms have "eventually (\<lambda>z. z \<noteq> 0) (nhds z)"
3122     by (intro t1_space_nhds) auto
3123   then show "eventually (\<lambda>z. z powr r = exp (r * ln z)) (nhds z)"
3124     unfolding powr_def by eventually_elim simp
3125   from assms show "((\<lambda>z. exp (r * ln z)) has_real_derivative r * z powr (r - 1)) (at z)"
3126     by (auto intro!: derivative_eq_intros simp: powr_def field_simps exp_diff)
3127 qed
3129 declare has_real_derivative_powr[THEN DERIV_chain2, derivative_intros]
3131 lemma tendsto_zero_powrI:
3132   assumes "(f \<longlongrightarrow> (0::real)) F" "(g \<longlongrightarrow> b) F" "\<forall>\<^sub>F x in F. 0 \<le> f x" "0 < b"
3133   shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> 0) F"
3134   using tendsto_powr2[OF assms] by simp
3136 lemma continuous_on_powr':
3137   fixes f g :: "_ \<Rightarrow> real"
3138   assumes "continuous_on s f" "continuous_on s g"
3139     and "\<forall>x\<in>s. f x \<ge> 0 \<and> (f x = 0 \<longrightarrow> g x > 0)"
3140   shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
3141   unfolding continuous_on_def
3142 proof
3143   fix x
3144   assume x: "x \<in> s"
3145   from assms x show "((\<lambda>x. f x powr g x) \<longlongrightarrow> f x powr g x) (at x within s)"
3146   proof (cases "f x = 0")
3147     case True
3148     from assms(3) have "eventually (\<lambda>x. f x \<ge> 0) (at x within s)"
3149       by (auto simp: at_within_def eventually_inf_principal)
3150     with True x assms show ?thesis
3151       by (auto intro!: tendsto_zero_powrI[of f _ g "g x"] simp: continuous_on_def)
3152   next
3153     case False
3154     with assms x show ?thesis
3155       by (auto intro!: tendsto_powr' simp: continuous_on_def)
3156   qed
3157 qed
3159 lemma tendsto_neg_powr:
3160   assumes "s < 0"
3161     and f: "LIM x F. f x :> at_top"
3162   shows "((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F"
3163 proof -
3164   have "((\<lambda>x. exp (s * ln (f x))) \<longlongrightarrow> (0::real)) F" (is "?X")
3165     by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_top]
3166         filterlim_tendsto_neg_mult_at_bot assms)
3167   also have "?X \<longleftrightarrow> ((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F"
3168     using f filterlim_at_top_dense[of f F]
3169     by (intro filterlim_cong[OF refl refl]) (auto simp: neq_iff powr_def elim: eventually_mono)
3170   finally show ?thesis .
3171 qed
3173 lemma tendsto_exp_limit_at_right: "((\<lambda>y. (1 + x * y) powr (1 / y)) \<longlongrightarrow> exp x) (at_right 0)"
3174   for x :: real
3175 proof (cases "x = 0")
3176   case True
3177   then show ?thesis by simp
3178 next
3179   case False
3180   have "((\<lambda>y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)"
3181     by (auto intro!: derivative_eq_intros)
3182   then have "((\<lambda>y. ln (1 + x * y) / y) \<longlongrightarrow> x) (at 0)"
3183     by (auto simp add: has_field_derivative_def field_has_derivative_at)
3184   then have *: "((\<lambda>y. exp (ln (1 + x * y) / y)) \<longlongrightarrow> exp x) (at 0)"
3185     by (rule tendsto_intros)
3186   then show ?thesis
3187   proof (rule filterlim_mono_eventually)
3188     show "eventually (\<lambda>xa. exp (ln (1 + x * xa) / xa) = (1 + x * xa) powr (1 / xa)) (at_right 0)"
3189       unfolding eventually_at_right[OF zero_less_one]
3190       using False
3191       apply (intro exI[of _ "1 / \<bar>x\<bar>"])
3192       apply (auto simp: field_simps powr_def abs_if)
3193       apply (metis add_less_same_cancel1 mult_less_0_iff not_less_iff_gr_or_eq zero_less_one)
3194       done
3196 qed
3198 lemma tendsto_exp_limit_at_top: "((\<lambda>y. (1 + x / y) powr y) \<longlongrightarrow> exp x) at_top"
3199   for x :: real
3200   apply (subst filterlim_at_top_to_right)
3202   apply (rule tendsto_exp_limit_at_right)
3203   done
3205 lemma tendsto_exp_limit_sequentially: "(\<lambda>n. (1 + x / n) ^ n) \<longlonglongrightarrow> exp x"
3206   for x :: real
3207 proof (rule filterlim_mono_eventually)
3208   from reals_Archimedean2 [of "\<bar>x\<bar>"] obtain n :: nat where *: "real n > \<bar>x\<bar>" ..
3209   then have "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top"
3210     apply (intro eventually_sequentiallyI [of n])
3211     apply (cases "x \<ge> 0")
3213       apply (auto intro: divide_nonneg_nonneg)
3214     apply (subgoal_tac "x / real xa > - 1")
3215      apply (auto simp add: field_simps)
3216     done
3217   then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top"
3218     by (rule eventually_mono) (erule powr_realpow)
3219   show "(\<lambda>n. (1 + x / real n) powr real n) \<longlonglongrightarrow> exp x"
3220     by (rule filterlim_compose [OF tendsto_exp_limit_at_top filterlim_real_sequentially])
3221 qed auto
3224 subsection \<open>Sine and Cosine\<close>
3226 definition sin_coeff :: "nat \<Rightarrow> real"
3227   where "sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / (fact n))"
3229 definition cos_coeff :: "nat \<Rightarrow> real"
3230   where "cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / (fact n) else 0)"
3232 definition sin :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
3233   where "sin = (\<lambda>x. \<Sum>n. sin_coeff n *\<^sub>R x^n)"
3235 definition cos :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}"
3236   where "cos = (\<lambda>x. \<Sum>n. cos_coeff n *\<^sub>R x^n)"
3238 lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
3239   unfolding sin_coeff_def by simp
3241 lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
3242   unfolding cos_coeff_def by simp
3244 lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
3245   unfolding cos_coeff_def sin_coeff_def
3246   by (simp del: mult_Suc)
3248 lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
3249   unfolding cos_coeff_def sin_coeff_def
3250   by (simp del: mult_Suc) (auto elim: oddE)
3252 lemma summable_norm_sin: "summable (\<lambda>n. norm (sin_coeff n *\<^sub>R x^n))"
3253   for x :: "'a::{real_normed_algebra_1,banach}"
3254   unfolding sin_coeff_def
3255   apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]])
3256   apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
3257   done
3259 lemma summable_norm_cos: "summable (\<lambda>n. norm (cos_coeff n *\<^sub>R x^n))"
3260   for x :: "'a::{real_normed_algebra_1,banach}"
3261   unfolding cos_coeff_def
3262   apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]])
3263   apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
3264   done
3266 lemma sin_converges: "(\<lambda>n. sin_coeff n *\<^sub>R x^n) sums sin x"
3267   unfolding sin_def
3268   by (metis (full_types) summable_norm_cancel summable_norm_sin summable_sums)
3270 lemma cos_converges: "(\<lambda>n. cos_coeff n *\<^sub>R x^n) sums cos x"
3271   unfolding cos_def
3272   by (metis (full_types) summable_norm_cancel summable_norm_cos summable_sums)
3274 lemma sin_of_real: "sin (of_real x) = of_real (sin x)"
3275   for x :: real
3276 proof -
3277   have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R  x^n)) = (\<lambda>n. sin_coeff n *\<^sub>R  (of_real x)^n)"
3278   proof
3279     show "of_real (sin_coeff n *\<^sub>R  x^n) = sin_coeff n *\<^sub>R of_real x^n" for n
3281   qed
3282   also have "\<dots> sums (sin (of_real x))"
3283     by (rule sin_converges)
3284   finally have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) sums (sin (of_real x))" .
3285   then show ?thesis
3286     using sums_unique2 sums_of_real [OF sin_converges]
3287     by blast
3288 qed
3290 corollary sin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> sin z \<in> \<real>"
3291   by (metis Reals_cases Reals_of_real sin_of_real)
3293 lemma cos_of_real: "cos (of_real x) = of_real (cos x)"
3294   for x :: real
3295 proof -
3296   have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R  x^n)) = (\<lambda>n. cos_coeff n *\<^sub>R  (of_real x)^n)"
3297   proof
3298     show "of_real (cos_coeff n *\<^sub>R  x^n) = cos_coeff n *\<^sub>R of_real x^n" for n
3300   qed
3301   also have "\<dots> sums (cos (of_real x))"
3302     by (rule cos_converges)
3303   finally have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) sums (cos (of_real x))" .
3304   then show ?thesis
3305     using sums_unique2 sums_of_real [OF cos_converges]
3306     by blast
3307 qed
3309 corollary cos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cos z \<in> \<real>"
3310   by (metis Reals_cases Reals_of_real cos_of_real)
3312 lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
3313   by (simp add: diffs_def sin_coeff_Suc del: of_nat_Suc)
3315 lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
3316   by (simp add: diffs_def cos_coeff_Suc del: of_nat_Suc)
3318 lemma sin_int_times_real: "sin (of_int m * of_real x) = of_real (sin (of_int m * x))"
3319   by (metis sin_of_real of_real_mult of_real_of_int_eq)
3321 lemma cos_int_times_real: "cos (of_int m * of_real x) = of_real (cos (of_int m * x))"
3322   by (metis cos_of_real of_real_mult of_real_of_int_eq)
3324 text \<open>Now at last we can get the derivatives of exp, sin and cos.\<close>
3326 lemma DERIV_sin [simp]: "DERIV sin x :> cos x"
3327   for x :: "'a::{real_normed_field,banach}"
3328   unfolding sin_def cos_def scaleR_conv_of_real
3329   apply (rule DERIV_cong)
3330    apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
3331       apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff
3332               summable_minus_iff scaleR_conv_of_real [symmetric]
3333               summable_norm_sin [THEN summable_norm_cancel]
3334               summable_norm_cos [THEN summable_norm_cancel])
3335   done
3337 declare DERIV_sin[THEN DERIV_chain2, derivative_intros]
3338   and DERIV_sin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
3340 lemma DERIV_cos [simp]: "DERIV cos x :> - sin x"
3341   for x :: "'a::{real_normed_field,banach}"
3342   unfolding sin_def cos_def scaleR_conv_of_real
3343   apply (rule DERIV_cong)
3344    apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"])
3345       apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus
3346               diffs_sin_coeff diffs_cos_coeff
3347               summable_minus_iff scaleR_conv_of_real [symmetric]
3348               summable_norm_sin [THEN summable_norm_cancel]
3349               summable_norm_cos [THEN summable_norm_cancel])
3350   done
3352 declare DERIV_cos[THEN DERIV_chain2, derivative_intros]
3353   and DERIV_cos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
3355 lemma isCont_sin: "isCont sin x"
3356   for x :: "'a::{real_normed_field,banach}"
3357   by (rule DERIV_sin [THEN DERIV_isCont])
3359 lemma isCont_cos: "isCont cos x"
3360   for x :: "'a::{real_normed_field,banach}"
3361   by (rule DERIV_cos [THEN DERIV_isCont])
3363 lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
3364   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
3365   by (rule isCont_o2 [OF _ isCont_sin])
3367 (* FIXME a context for f would be better *)
3369 lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
3370   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
3371   by (rule isCont_o2 [OF _ isCont_cos])
3373 lemma tendsto_sin [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) \<longlongrightarrow> sin a) F"
3374   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
3375   by (rule isCont_tendsto_compose [OF isCont_sin])
3377 lemma tendsto_cos [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) \<longlongrightarrow> cos a) F"
3378   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
3379   by (rule isCont_tendsto_compose [OF isCont_cos])
3381 lemma continuous_sin [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
3382   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
3383   unfolding continuous_def by (rule tendsto_sin)
3385 lemma continuous_on_sin [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
3386   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
3387   unfolding continuous_on_def by (auto intro: tendsto_sin)
3389 lemma continuous_within_sin: "continuous (at z within s) sin"
3390   for z :: "'a::{real_normed_field,banach}"
3391   by (simp add: continuous_within tendsto_sin)
3393 lemma continuous_cos [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
3394   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
3395   unfolding continuous_def by (rule tendsto_cos)
3397 lemma continuous_on_cos [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
3398   for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}"
3399   unfolding continuous_on_def by (auto intro: tendsto_cos)
3401 lemma continuous_within_cos: "continuous (at z within s) cos"
3402   for z :: "'a::{real_normed_field,banach}"
3403   by (simp add: continuous_within tendsto_cos)
3406 subsection \<open>Properties of Sine and Cosine\<close>
3408 lemma sin_zero [simp]: "sin 0 = 0"
3409   by (simp add: sin_def sin_coeff_def scaleR_conv_of_real)
3411 lemma cos_zero [simp]: "cos 0 = 1"
3412   by (simp add: cos_def cos_coeff_def scaleR_conv_of_real)
3414 lemma DERIV_fun_sin: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. sin (g x)) x :> cos (g x) * m"
3415   by (auto intro!: derivative_intros)
3417 lemma DERIV_fun_cos: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. cos(g x)) x :> - sin (g x) * m"
3418   by (auto intro!: derivative_eq_intros)
3421 subsection \<open>Deriving the Addition Formulas\<close>
3423 text \<open>The product of two cosine series.\<close>
3424 lemma cos_x_cos_y:
3425   fixes x :: "'a::{real_normed_field,banach}"
3426   shows
3427     "(\<lambda>p. \<Sum>n\<le>p.
3428         if even p \<and> even n
3429         then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)
3430       sums (cos x * cos y)"
3431 proof -
3432   have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p - n)) =
3433     (if even p \<and> even n then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p - n)
3434      else 0)"
3435     if "n \<le> p" for n p :: nat
3436   proof -
3437     from that have *: "even n \<Longrightarrow> even p \<Longrightarrow>
3438         (-1) ^ (n div 2) * (-1) ^ ((p - n) div 2) = (-1 :: real) ^ (p div 2)"
3440     with that show ?thesis
3441       by (auto simp: algebra_simps cos_coeff_def binomial_fact)
3442   qed
3443   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> even n
3444                   then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
3445              (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
3446     by simp
3447   also have "\<dots> = (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n *\<^sub>R x^n) * (cos_coeff (p - n) *\<^sub>R y^(p-n)))"
3449   also have "\<dots> sums (cos x * cos y)"
3450     using summable_norm_cos
3451     by (auto simp: cos_def scaleR_conv_of_real intro!: Cauchy_product_sums)
3452   finally show ?thesis .
3453 qed
3455 text \<open>The product of two sine series.\<close>
3456 lemma sin_x_sin_y:
3457   fixes x :: "'a::{real_normed_field,banach}"
3458   shows
3459     "(\<lambda>p. \<Sum>n\<le>p.
3460         if even p \<and> odd n
3461         then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
3462         else 0)
3463       sums (sin x * sin y)"
3464 proof -
3465   have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) =
3466     (if even p \<and> odd n
3467      then -((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
3468      else 0)"
3469     if "n \<le> p" for n p :: nat
3470   proof -
3471     have "(-1) ^ ((n - Suc 0) div 2) * (-1) ^ ((p - Suc n) div 2) = - ((-1 :: real) ^ (p div 2))"
3472       if np: "odd n" "even p"
3473     proof -
3474       from \<open>n \<le> p\<close> np have *: "n - Suc 0 + (p - Suc n) = p - Suc (Suc 0)" "Suc (Suc 0) \<le> p"
3475         by arith+
3476       have "(p - Suc (Suc 0)) div 2 = p div 2 - Suc 0"
3477         by simp
3478       with \<open>n \<le> p\<close> np * show ?thesis
3480         apply (metis (no_types) One_nat_def Suc_1 le_div_geq minus_minus
3481             mult.left_neutral mult_minus_left power.simps(2) zero_less_Suc)
3482         done
3483     qed
3484     then show ?thesis
3485       using \<open>n\<le>p\<close> by (auto simp: algebra_simps sin_coeff_def binomial_fact)
3486   qed
3487   then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> odd n
3488                then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
3489              (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))"
3490     by simp
3491   also have "\<dots> = (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n *\<^sub>R x^n) * (sin_coeff (p - n) *\<^sub>R y^(p-n)))"
3493   also have "\<dots> sums (sin x * sin y)"
3494     using summable_norm_sin
3495     by (auto simp: sin_def scaleR_conv_of_real intro!: Cauchy_product_sums)
3496   finally show ?thesis .
3497 qed
3499 lemma sums_cos_x_plus_y:
3500   fixes x :: "'a::{real_normed_field,banach}"
3501   shows
3502     "(\<lambda>p. \<Sum>n\<le>p.
3503         if even p
3504         then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
3505         else 0)
3506       sums cos (x + y)"
3507 proof -
3508   have
3509     "(\<Sum>n\<le>p.
3510       if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
3511       else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)"
3512     for p :: nat
3513   proof -
3514     have
3515       "(\<Sum>n\<le>p. if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
3516        (if even p then \<Sum>n\<le>p. ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
3517       by simp
3518     also have "\<dots> =
3519        (if even p
3520         then of_real ((-1) ^ (p div 2) / (fact p)) * (\<Sum>n\<le>p. (p choose n) *\<^sub>R (x^n) * y^(p-n))
3521         else 0)"
3522       by (auto simp: sum_distrib_left field_simps scaleR_conv_of_real nonzero_of_real_divide)
3523     also have "\<dots> = cos_coeff p *\<^sub>R ((x + y) ^ p)"
3524       by (simp add: cos_coeff_def binomial_ring [of x y]  scaleR_conv_of_real atLeast0AtMost)
3525     finally show ?thesis .
3526   qed
3527   then have
3528     "(\<lambda>p. \<Sum>n\<le>p.
3529         if even p
3530         then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n)
3531         else 0) = (\<lambda>p. cos_coeff p *\<^sub>R ((x+y)^p))"
3532     by simp
3533    also have "\<dots> sums cos (x + y)"
3534     by (rule cos_converges)
3535    finally show ?thesis .
3536 qed
3539   fixes x :: "'a::{real_normed_field,banach}"
3540   shows "cos (x + y) = cos x * cos y - sin x * sin y"
3541 proof -
3542   have
3543     "(if even p \<and> even n
3544       then ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) -
3545      (if even p \<and> odd n
3546       then - ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) =
3547      (if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)"
3548     if "n \<le> p" for n p :: nat
3549     by simp
3550   then have
3551     "(\<lambda>p. \<Sum>n\<le>p. (if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0))
3552       sums (cos x * cos y - sin x * sin y)"
3553     using sums_diff [OF cos_x_cos_y [of x y] sin_x_sin_y [of x y]]
3554     by (simp add: sum_subtractf [symmetric])
3555   then show ?thesis
3556     by (blast intro: sums_cos_x_plus_y sums_unique2)
3557 qed
3559 lemma sin_minus_converges: "(\<lambda>n. - (sin_coeff n *\<^sub>R (-x)^n)) sums sin x"
3560 proof -
3561   have [simp]: "\<And>n. - (sin_coeff n *\<^sub>R (-x)^n) = (sin_coeff n *\<^sub>R x^n)"
3562     by (auto simp: sin_coeff_def elim!: oddE)
3563   show ?thesis
3564     by (simp add: sin_def summable_norm_sin [THEN summable_norm_cancel, THEN summable_sums])
3565 qed
3567 lemma sin_minus [simp]: "sin (- x) = - sin x"
3568   for x :: "'a::{real_normed_algebra_1,banach}"
3569   using sin_minus_converges [of x]
3570   by (auto simp: sin_def summable_norm_sin [THEN summable_norm_cancel]
3571       suminf_minus sums_iff equation_minus_iff)
3573 lemma cos_minus_converges: "(\<lambda>n. (cos_coeff n *\<^sub>R (-x)^n)) sums cos x"
3574 proof -
3575   have [simp]: "\<And>n. (cos_coeff n *\<^sub>R (-x)^n) = (cos_coeff n *\<^sub>R x^n)"
3576     by (auto simp: Transcendental.cos_coeff_def elim!: evenE)
3577   show ?thesis
3578     by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel, THEN summable_sums])
3579 qed
3581 lemma cos_minus [simp]: "cos (-x) = cos x"
3582   for x :: "'a::{real_normed_algebra_1,banach}"
3583   using cos_minus_converges [of x]
3584   by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel]
3585       suminf_minus sums_iff equation_minus_iff)
3587 lemma sin_cos_squared_add [simp]: "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
3588   for x :: "'a::{real_normed_field,banach}"
3589   using cos_add [of x "-x"]
3590   by (simp add: power2_eq_square algebra_simps)
3592 lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
3593   for x :: "'a::{real_normed_field,banach}"
3596 lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
3597   for x :: "'a::{real_normed_field,banach}"
3598   using sin_cos_squared_add2 [unfolded power2_eq_square] .
3600 lemma sin_squared_eq: "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
3601   for x :: "'a::{real_normed_field,banach}"
3602   unfolding eq_diff_eq by (rule sin_cos_squared_add)
3604 lemma cos_squared_eq: "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
3605   for x :: "'a::{real_normed_field,banach}"
3606   unfolding eq_diff_eq by (rule sin_cos_squared_add2)
3608 lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
3609   for x :: real
3610   by (rule power2_le_imp_le) (simp_all add: sin_squared_eq)
3612 lemma sin_ge_minus_one [simp]: "- 1 \<le> sin x"
3613   for x :: real
3614   using abs_sin_le_one [of x] by (simp add: abs_le_iff)
3616 lemma sin_le_one [simp]: "sin x \<le> 1"
3617   for x :: real
3618   using abs_sin_le_one [of x] by (simp add: abs_le_iff)
3620 lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
3621   for x :: real
3622   by (rule power2_le_imp_le) (simp_all add: cos_squared_eq)
3624 lemma cos_ge_minus_one [simp]: "- 1 \<le> cos x"
3625   for x :: real
3626   using abs_cos_le_one [of x] by (simp add: abs_le_iff)
3628 lemma cos_le_one [simp]: "cos x \<le> 1"
3629   for x :: real
3630   using abs_cos_le_one [of x] by (simp add: abs_le_iff)
3632 lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
3633   for x :: "'a::{real_normed_field,banach}"
3634   using cos_add [of x "- y"] by simp
3636 lemma cos_double: "cos(2*x) = (cos x)\<^sup>2 - (sin x)\<^sup>2"
3637   for x :: "'a::{real_normed_field,banach}"
3640 lemma sin_cos_le1: "\<bar>sin x * sin y + cos x * cos y\<bar> \<le> 1"
3641   for x :: real
3642   using cos_diff [of x y] by (metis abs_cos_le_one add.commute)
3644 lemma DERIV_fun_pow: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
3645   by (auto intro!: derivative_eq_intros simp:)
3647 lemma DERIV_fun_exp: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. exp (g x)) x :> exp (g x) * m"
3648   by (auto intro!: derivative_intros)
3651 subsection \<open>The Constant Pi\<close>
3653 definition pi :: real
3654   where "pi = 2 * (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)"
3656 text \<open>Show that there's a least positive @{term x} with @{term "cos x = 0"};
3657    hence define pi.\<close>
3659 lemma sin_paired: "(\<lambda>n. (- 1) ^ n / (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums  sin x"
3660   for x :: real
3661 proof -
3662   have "(\<lambda>n. \<Sum>k = n*2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
3663     by (rule sums_group) (use sin_converges [of x, unfolded scaleR_conv_of_real] in auto)
3664   then show ?thesis
3665     by (simp add: sin_coeff_def ac_simps)
3666 qed
3668 lemma sin_gt_zero_02:
3669   fixes x :: real
3670   assumes "0 < x" and "x < 2"
3671   shows "0 < sin x"
3672 proof -
3673   let ?f = "\<lambda>n::nat. \<Sum>k = n*2..<n*2+2. (- 1) ^ k / (fact (2*k+1)) * x^(2*k+1)"
3674   have pos: "\<forall>n. 0 < ?f n"
3675   proof
3676     fix n :: nat
3677     let ?k2 = "real (Suc (Suc (4 * n)))"
3678     let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
3679     have "x * x < ?k2 * ?k3"
3680       using assms by (intro mult_strict_mono', simp_all)
3681     then have "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
3682       by (intro mult_strict_right_mono zero_less_power \<open>0 < x\<close>)
3683     then show "0 < ?f n"
3684       by (simp add: divide_simps mult_ac del: mult_Suc)
3685 qed
3686   have sums: "?f sums sin x"
3687     by (rule sin_paired [THEN sums_group]) simp
3688   show "0 < sin x"
3689     unfolding sums_unique [OF sums]
3690     using sums_summable [OF sums] pos
3691     by (rule suminf_pos)
3692 qed
3694 lemma cos_double_less_one: "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
3695   for x :: real
3696   using sin_gt_zero_02 [where x = x] by (auto simp: cos_squared_eq cos_double)
3698 lemma cos_paired: "(\<lambda>n. (- 1) ^ n / (fact (2 * n)) * x ^ (2 * n)) sums cos x"
3699   for x :: real
3700 proof -
3701   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
3702     by (rule sums_group) (use cos_converges [of x, unfolded scaleR_conv_of_real] in auto)
3703   then show ?thesis
3704     by (simp add: cos_coeff_def ac_simps)
3705 qed
3707 lemmas realpow_num_eq_if = power_eq_if
3709 lemma sumr_pos_lt_pair:
3710   fixes f :: "nat \<Rightarrow> real"
3711   shows "summable f \<Longrightarrow>
3712     (\<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc (Suc 0) * d) + 1))) \<Longrightarrow>
3713     sum f {..<k} < suminf f"
3714   apply (simp only: One_nat_def)
3715   apply (subst suminf_split_initial_segment [where k=k])
3716    apply assumption
3717   apply simp
3718   apply (drule_tac k=k in summable_ignore_initial_segment)
3719   apply (drule_tac k="Suc (Suc 0)" in sums_group [OF summable_sums])
3720    apply simp
3721   apply simp
3722   apply (metis (no_types, lifting) add.commute suminf_pos summable_def sums_unique)
3723   done
3725 lemma cos_two_less_zero [simp]: "cos 2 < (0::real)"
3726 proof -
3727   note fact_Suc [simp del]
3728   from sums_minus [OF cos_paired]
3729   have *: "(\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / fact (2 * n))) sums - cos (2::real)"
3730     by simp
3731   then have sm: "summable (\<lambda>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
3732     by (rule sums_summable)
3733   have "0 < (\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
3734     by (simp add: fact_num_eq_if realpow_num_eq_if)
3735   moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n  * 2 ^ (2 * n) / (fact (2 * n)))) <
3736     (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
3737   proof -
3738     {
3739       fix d
3740       let ?six4d = "Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))"
3741       have "(4::real) * (fact (?six4d)) < (Suc (Suc (?six4d)) * fact (Suc (?six4d)))"
3742         unfolding of_nat_mult by (rule mult_strict_mono) (simp_all add: fact_less_mono)
3743       then have "(4::real) * (fact (?six4d)) < (fact (Suc (Suc (?six4d))))"
3744         by (simp only: fact_Suc [of "Suc (?six4d)"] of_nat_mult of_nat_fact)
3745       then have "(4::real) * inverse (fact (Suc (Suc (?six4d)))) < inverse (fact (?six4d))"
3746         by (simp add: inverse_eq_divide less_divide_eq)
3747     }
3748     then show ?thesis
3749       by (force intro!: sumr_pos_lt_pair [OF sm] simp add: divide_inverse algebra_simps)
3750   qed
3751   ultimately have "0 < (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
3752     by (rule order_less_trans)
3753   moreover from * have "- cos 2 = (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))"
3754     by (rule sums_unique)
3755   ultimately have "(0::real) < - cos 2" by simp
3756   then show ?thesis by simp
3757 qed
3759 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
3760 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
3762 lemma cos_is_zero: "\<exists>!x::real. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
3763 proof (rule ex_ex1I)
3764   show "\<exists>x::real. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
3765     by (rule IVT2) simp_all
3766 next
3767   fix x y :: real
3768   assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
3769   assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
3770   have [simp]: "\<forall>x::real. cos differentiable (at x)"
3771     unfolding real_differentiable_def by (auto intro: DERIV_cos)
3772   from x y less_linear [of x y] show "x = y"
3773     apply auto
3774      apply (drule_tac f = cos in Rolle)
3775         apply (drule_tac [5] f = cos in Rolle)
3776            apply (auto dest!: DERIV_cos [THEN DERIV_unique])
3777      apply (metis order_less_le_trans less_le sin_gt_zero_02)
3778     apply (metis order_less_le_trans less_le sin_gt_zero_02)
3779     done
3780 qed
3782 lemma pi_half: "pi/2 = (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)"
3785 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
3786   by (simp add: pi_half cos_is_zero [THEN theI'])
3788 lemma cos_of_real_pi_half [simp]: "cos ((of_real pi / 2) :: 'a) = 0"
3789   if "SORT_CONSTRAINT('a::{real_field,banach,real_normed_algebra_1})"
3790   by (metis cos_pi_half cos_of_real eq_numeral_simps(4)
3791       nonzero_of_real_divide of_real_0 of_real_numeral)
3793 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
3794   apply (rule order_le_neq_trans)
3795    apply (simp add: pi_half cos_is_zero [THEN theI'])
3796   apply (metis cos_pi_half cos_zero zero_neq_one)
3797   done
3799 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
3800 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
3802 lemma pi_half_less_two [simp]: "pi / 2 < 2"
3803   apply (rule order_le_neq_trans)
3804    apply (simp add: pi_half cos_is_zero [THEN theI'])
3805   apply (metis cos_pi_half cos_two_neq_zero)
3806   done
3808 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
3809 lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
3811 lemma pi_gt_zero [simp]: "0 < pi"
3812   using pi_half_gt_zero by simp
3814 lemma pi_ge_zero [simp]: "0 \<le> pi"
3815   by (rule pi_gt_zero [THEN order_less_imp_le])
3817 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
3818   by (rule pi_gt_zero [THEN less_imp_neq, symmetric])
3820 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
3823 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
3824   by simp
3826 lemma m2pi_less_pi: "- (2*pi) < pi"
3827   by simp
3829 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
3830   using sin_cos_squared_add2 [where x = "pi/2"]
3831   using sin_gt_zero_02 [OF pi_half_gt_zero pi_half_less_two]
3834 lemma sin_of_real_pi_half [simp]: "sin ((of_real pi / 2) :: 'a) = 1"
3835   if "SORT_CONSTRAINT('a::{real_field,banach,real_normed_algebra_1})"
3836   using sin_pi_half
3837   by (metis sin_pi_half eq_numeral_simps(4) nonzero_of_real_divide of_real_1 of_real_numeral sin_of_real)
3839 lemma sin_cos_eq: "sin x = cos (of_real pi / 2 - x)"
3840   for x :: "'a::{real_normed_field,banach}"
3843 lemma minus_sin_cos_eq: "- sin x = cos (x + of_real pi / 2)"
3844   for x :: "'a::{real_normed_field,banach}"
3847 lemma cos_sin_eq: "cos x = sin (of_real pi / 2 - x)"
3848   for x :: "'a::{real_normed_field,banach}"
3849   using sin_cos_eq [of "of_real pi / 2 - x"] by simp
3851 lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
3852   for x :: "'a::{real_normed_field,banach}"
3853   using cos_add [of "of_real pi / 2 - x" "-y"]
3856 lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
3857   for x :: "'a::{real_normed_field,banach}"
3858   using sin_add [of x "- y"] by simp
3860 lemma sin_double: "sin(2 * x) = 2 * sin x * cos x"
3861   for x :: "'a::{real_normed_field,banach}"
3862   using sin_add [where x=x and y=x] by simp
3864 lemma cos_of_real_pi [simp]: "cos (of_real pi) = -1"
3865   using cos_add [where x = "pi/2" and y = "pi/2"]
3868 lemma sin_of_real_pi [simp]: "sin (of_real pi) = 0"
3869   using sin_add [where x = "pi/2" and y = "pi/2"]
3872 lemma cos_pi [simp]: "cos pi = -1"
3873   using cos_add [where x = "pi/2" and y = "pi/2"] by simp
3875 lemma sin_pi [simp]: "sin pi = 0"
3876   using sin_add [where x = "pi/2" and y = "pi/2"] by simp
3878 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
3881 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
3884 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
3887 lemma cos_periodic_pi2 [simp]: "cos (pi + x) = - cos x"
3890 lemma sin_periodic [simp]: "sin (x + 2 * pi) = sin x"
3893 lemma cos_periodic [simp]: "cos (x + 2 * pi) = cos x"
3896 lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n"
3897   by (induct n) (auto simp: distrib_right)
3899 lemma cos_npi2 [simp]: "cos (pi * real n) = (- 1) ^ n"
3900   by (metis cos_npi mult.commute)
3902 lemma sin_npi [simp]: "sin (real n * pi) = 0"
3903   for n :: nat
3904   by (induct n) (auto simp: distrib_right)
3906 lemma sin_npi2 [simp]: "sin (pi * real n) = 0"
3907   for n :: nat
3908   by (simp add: mult.commute [of pi])
3910 lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
3913 lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
3916 lemma sin_times_sin: "sin w * sin z = (cos (w - z) - cos (w + z)) / 2"
3917   for w :: "'a::{real_normed_field,banach}"
3920 lemma sin_times_cos: "sin w * cos z = (sin (w + z) + sin (w - z)) / 2"
3921   for w :: "'a::{real_normed_field,banach}"
3924 lemma cos_times_sin: "cos w * sin z = (sin (w + z) - sin (w - z)) / 2"
3925   for w :: "'a::{real_normed_field,banach}"
3928 lemma cos_times_cos: "cos w * cos z = (cos (w - z) + cos (w + z)) / 2"
3929   for w :: "'a::{real_normed_field,banach}"
3932 lemma sin_plus_sin: "sin w + sin z = 2 * sin ((w + z) / 2) * cos ((w - z) / 2)"
3933   for w :: "'a::{real_normed_field,banach,field}"  (* FIXME field should not be necessary *)
3934   apply (simp add: mult.assoc sin_times_cos)
3936   done
3938 lemma sin_diff_sin: "sin w - sin z = 2 * sin ((w - z) / 2) * cos ((w + z) / 2)"
3939   for w :: "'a::{real_normed_field,banach,field}"
3940   apply (simp add: mult.assoc sin_times_cos)
3942   done
3944 lemma cos_plus_cos: "cos w + cos z = 2 * cos ((w + z) / 2) * cos ((w - z) / 2)"
3945   for w :: "'a::{real_normed_field,banach,field}"
3946   apply (simp add: mult.assoc cos_times_cos)
3948   done
3950 lemma cos_diff_cos: "cos w - cos z = 2 * sin ((w + z) / 2) * sin ((z - w) / 2)"
3951   for w :: "'a::{real_normed_field,banach,field}"
3952   apply (simp add: mult.assoc sin_times_sin)
3954   done
3956 lemma cos_double_cos: "cos (2 * z) = 2 * cos z ^ 2 - 1"
3957   for z :: "'a::{real_normed_field,banach}"
3958   by (simp add: cos_double sin_squared_eq)
3960 lemma cos_double_sin: "cos (2 * z) = 1 - 2 * sin z ^ 2"
3961   for z :: "'a::{real_normed_field,banach}"
3962   by (simp add: cos_double sin_squared_eq)
3964 lemma sin_pi_minus [simp]: "sin (pi - x) = sin x"
3965   by (metis sin_minus sin_periodic_pi minus_minus uminus_add_conv_diff)
3967 lemma cos_pi_minus [simp]: "cos (pi - x) = - (cos x)"
3968   by (metis cos_minus cos_periodic_pi uminus_add_conv_diff)
3970 lemma sin_minus_pi [simp]: "sin (x - pi) = - (sin x)"
3973 lemma cos_minus_pi [simp]: "cos (x - pi) = - (cos x)"
3976 lemma sin_2pi_minus [simp]: "sin (2 * pi - x) = - (sin x)"
3977   by (metis sin_periodic_pi2 add_diff_eq mult_2 sin_pi_minus)
3979 lemma cos_2pi_minus [simp]: "cos (2 * pi - x) = cos x"
3980   by (metis (no_types, hide_lams) cos_add cos_minus cos_two_pi sin_minus sin_two_pi
3981       diff_0_right minus_diff_eq mult_1 mult_zero_left uminus_add_conv_diff)
3983 lemma sin_gt_zero2: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < sin x"
3984   by (metis sin_gt_zero_02 order_less_trans pi_half_less_two)
3986 lemma sin_less_zero:
3987   assumes "- pi/2 < x" and "x < 0"
3988   shows "sin x < 0"
3989 proof -
3990   have "0 < sin (- x)"
3991     using assms by (simp only: sin_gt_zero2)
3992   then show ?thesis by simp
3993 qed
3995 lemma pi_less_4: "pi < 4"
3996   using pi_half_less_two by auto
3998 lemma cos_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cos x"
3999   by (simp add: cos_sin_eq sin_gt_zero2)
4001 lemma cos_gt_zero_pi: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cos x"
4002   using cos_gt_zero [of x] cos_gt_zero [of "-x"]
4003   by (cases rule: linorder_cases [of x 0]) auto
4005 lemma cos_ge_zero: "-(pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> 0 \<le> cos x"
4006   by (auto simp: order_le_less cos_gt_zero_pi)
4007     (metis cos_pi_half eq_divide_eq eq_numeral_simps(4))
4009 lemma sin_gt_zero: "0 < x \<Longrightarrow> x < pi \<Longrightarrow> 0 < sin x"
4010   by (simp add: sin_cos_eq cos_gt_zero_pi)
4012 lemma sin_lt_zero: "pi < x \<Longrightarrow> x < 2 * pi \<Longrightarrow> sin x < 0"
4013   using sin_gt_zero [of "x - pi"]
4016 lemma pi_ge_two: "2 \<le> pi"
4017 proof (rule ccontr)
4018   assume "\<not> ?thesis"
4019   then have "pi < 2" by auto
4020   have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
4021   proof (cases "2 < 2 * pi")
4022     case True
4023     with dense[OF \<open>pi < 2\<close>] show ?thesis by auto
4024   next
4025     case False
4026     have "pi < 2 * pi" by auto
4027     from dense[OF this] and False show ?thesis by auto
4028   qed
4029   then obtain y where "pi < y" and "y < 2" and "y < 2 * pi"
4030     by blast
4031   then have "0 < sin y"
4032     using sin_gt_zero_02 by auto
4033   moreover have "sin y < 0"
4034     using sin_gt_zero[of "y - pi"] \<open>pi < y\<close> and \<open>y < 2 * pi\<close> sin_periodic_pi[of "y - pi"]
4035     by auto
4036   ultimately show False by auto
4037 qed
4039 lemma sin_ge_zero: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> sin x"
4040   by (auto simp: order_le_less sin_gt_zero)
4042 lemma sin_le_zero: "pi \<le> x \<Longrightarrow> x < 2 * pi \<Longrightarrow> sin x \<le> 0"
4043   using sin_ge_zero [of "x - pi"] by (simp add: sin_diff)
4045 lemma sin_pi_divide_n_ge_0 [simp]:
4046   assumes "n \<noteq> 0"
4047   shows "0 \<le> sin (pi / real n)"
4048   by (rule sin_ge_zero) (use assms in \<open>simp_all add: divide_simps\<close>)
4050 lemma sin_pi_divide_n_gt_0:
4051   assumes "2 \<le> n"
4052   shows "0 < sin (pi / real n)"
4053   by (rule sin_gt_zero) (use assms in \<open>simp_all add: divide_simps\<close>)
4055 (* FIXME: This proof is almost identical to lemma \<open>cos_is_zero\<close>.
4056    It should be possible to factor out some of the common parts. *)
4057 lemma cos_total:
4058   assumes y: "- 1 \<le> y" "y \<le> 1"
4059   shows "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y"
4060 proof (rule ex_ex1I)
4061   show "\<exists>x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y"
4062     by (rule IVT2) (simp_all add: y)
4063 next
4064   fix a b
4065   assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
4066   assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
4067   have [simp]: "\<forall>x::real. cos differentiable (at x)"
4068     unfolding real_differentiable_def by (auto intro: DERIV_cos)
4069   from a b less_linear [of a b] show "a = b"
4070     apply auto
4071      apply (drule_tac f = cos in Rolle)
4072         apply (drule_tac [5] f = cos in Rolle)
4073            apply (auto dest!: DERIV_cos [THEN DERIV_unique])
4074      apply (metis order_less_le_trans less_le sin_gt_zero)
4075     apply (metis order_less_le_trans less_le sin_gt_zero)
4076     done
4077 qed
4079 lemma sin_total:
4080   assumes y: "-1 \<le> y" "y \<le> 1"
4081   shows "\<exists>!x. - (pi/2) \<le> x \<and> x \<le> pi/2 \<and> sin x = y"
4082 proof -
4083   from cos_total [OF y]
4084   obtain x where x: "0 \<le> x" "x \<le> pi" "cos x = y"
4085     and uniq: "\<And>x'. 0 \<le> x' \<Longrightarrow> x' \<le> pi \<Longrightarrow> cos x' = y \<Longrightarrow> x' = x "
4086     by blast
4087   show ?thesis
4089     apply (rule ex1I [where a="pi/2 - x"])
4090      apply (cut_tac [2] x'="pi/2 - xa" in uniq)
4091     using x
4092         apply auto
4093     done
4094 qed
4096 lemma cos_zero_lemma:
4097   assumes "0 \<le> x" "cos x = 0"
4098   shows "\<exists>n. odd n \<and> x = of_nat n * (pi/2) \<and> n > 0"
4099 proof -
4100   have xle: "x < (1 + real_of_int \<lfloor>x/pi\<rfloor>) * pi"
4101     using floor_correct [of "x/pi"]
4103   obtain n where "real n * pi \<le> x" "x < real (Suc n) * pi"
4104     apply (rule that [of "nat \<lfloor>x/pi\<rfloor>"])
4105     using assms
4107     apply (metis floor_less_iff less_irrefl mult_imp_div_pos_less not_le pi_gt_zero)
4108     done
4109   then have x: "0 \<le> x - n * pi" "(x - n * pi) \<le> pi" "cos (x - n * pi) = 0"
4110     by (auto simp: algebra_simps cos_diff assms)
4111   then have "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = 0"
4112     by (auto simp: intro!: cos_total)
4113   then obtain \<theta> where \<theta>: "0 \<le> \<theta>" "\<theta> \<le> pi" "cos \<theta> = 0"
4114     and uniq: "\<And>\<phi>. 0 \<le> \<phi> \<Longrightarrow> \<phi> \<le> pi \<Longrightarrow> cos \<phi> = 0 \<Longrightarrow> \<phi> = \<theta>"
4115     by blast
4116   then have "x - real n * pi = \<theta>"
4117     using x by blast
4118   moreover have "pi/2 = \<theta>"
4119     using pi_half_ge_zero uniq by fastforce
4120   ultimately show ?thesis
4121     by (rule_tac x = "Suc (2 * n)" in exI) (simp add: algebra_simps)
4122 qed
4124 lemma sin_zero_lemma: "0 \<le> x \<Longrightarrow> sin x = 0 \<Longrightarrow> \<exists>n::nat. even n \<and> x = real n * (pi/2)"
4125   using cos_zero_lemma [of "x + pi/2"]
4127   apply (rule_tac x = "n - 1" in exI)
4128   apply (simp add: algebra_simps of_nat_diff)
4129   done
4131 lemma cos_zero_iff:
4132   "cos x = 0 \<longleftrightarrow> ((\<exists>n. odd n \<and> x = real n * (pi/2)) \<or> (\<exists>n. odd n \<and> x = - (real n * (pi/2))))"
4133   (is "?lhs = ?rhs")
4134 proof -
4135   have *: "cos (real n * pi / 2) = 0" if "odd n" for n :: nat
4136   proof -
4137     from that obtain m where "n = 2 * m + 1" ..
4138     then show ?thesis
4140   qed
4141   show ?thesis
4142   proof
4143     show ?rhs if ?lhs
4144       using that cos_zero_lemma [of x] cos_zero_lemma [of "-x"] by force
4145     show ?lhs if ?rhs
4146       using that by (auto dest: * simp del: eq_divide_eq_numeral1)
4147   qed
4148 qed
4150 lemma sin_zero_iff:
4151   "sin x = 0 \<longleftrightarrow> ((\<exists>n. even n \<and> x = real n * (pi/2)) \<or> (\<exists>n. even n \<and> x = - (real n * (pi/2))))"
4152   (is "?lhs = ?rhs")
4153 proof
4154   show ?rhs if ?lhs
4155     using that sin_zero_lemma [of x] sin_zero_lemma [of "-x"] by force
4156   show ?lhs if ?rhs
4157     using that by (auto elim: evenE)
4158 qed
4160 lemma cos_zero_iff_int: "cos x = 0 \<longleftrightarrow> (\<exists>n. odd n \<and> x = of_int n * (pi/2))"
4161 proof safe
4162   assume "cos x = 0"
4163   then show "\<exists>n. odd n \<and> x = of_int n * (pi/2)"
4165     apply safe
4166      apply (metis even_int_iff of_int_of_nat_eq)
4167     apply (rule_tac x="- (int n)" in exI)
4168     apply simp
4169     done
4170 next
4171   fix n :: int
4172   assume "odd n"
4173   then show "cos (of_int n * (pi / 2)) = 0"
4175     apply (cases n rule: int_cases2)
4176      apply simp_all
4177     done
4178 qed
4180 lemma sin_zero_iff_int: "sin x = 0 \<longleftrightarrow> (\<exists>n. even n \<and> x = of_int n * (pi/2))"
4181 proof safe
4182   assume "sin x = 0"
4183   then show "\<exists>n. even n \<and> x = of_int n * (pi / 2)"
4185     apply safe
4186      apply (metis even_int_iff of_int_of_nat_eq)
4187     apply (rule_tac x="- (int n)" in exI)
4188     apply simp
4189     done
4190 next
4191   fix n :: int
4192   assume "even n"
4193   then show "sin (of_int n * (pi / 2)) = 0"
4195     apply (cases n rule: int_cases2)
4196      apply simp_all
4197     done
4198 qed
4200 lemma sin_zero_iff_int2: "sin x = 0 \<longleftrightarrow> (\<exists>n::int. x = of_int n * pi)"
4201   apply (simp only: sin_zero_iff_int)
4202   apply (safe elim!: evenE)
4204   using dvd_triv_left apply fastforce
4205   done
4207 lemma sin_npi_int [simp]: "sin (pi * of_int n) = 0"
4210 lemma cos_monotone_0_pi:
4211   assumes "0 \<le> y" and "y < x" and "x \<le> pi"
4212   shows "cos x < cos y"
4213 proof -
4214   have "- (x - y) < 0" using assms by auto
4215   from MVT2[OF \<open>y < x\<close> DERIV_cos[THEN impI, THEN allI]]
4216   obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z"
4217     by auto
4218   then have "0 < z" and "z < pi"
4219     using assms by auto
4220   then have "0 < sin z"
4221     using sin_gt_zero by auto
4222   then have "cos x - cos y < 0"
4223     unfolding cos_diff minus_mult_commute[symmetric]
4224     using \<open>- (x - y) < 0\<close> by (rule mult_pos_neg2)
4225   then show ?thesis by auto
4226 qed
4228 lemma cos_monotone_0_pi_le:
4229   assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi"
4230   shows "cos x \<le> cos y"
4231 proof (cases "y < x")
4232   case True
4233   show ?thesis
4234     using cos_monotone_0_pi[OF \<open>0 \<le> y\<close> True \<open>x \<le> pi\<close>] by auto
4235 next
4236   case False
4237   then have "y = x" using \<open>y \<le> x\<close> by auto
4238   then show ?thesis by auto
4239 qed
4241 lemma cos_monotone_minus_pi_0:
4242   assumes "- pi \<le> y" and "y < x" and "x \<le> 0"
4243   shows "cos y < cos x"
4244 proof -
4245   have "0 \<le> - x" and "- x < - y" and "- y \<le> pi"
4246     using assms by auto
4247   from cos_monotone_0_pi[OF this] show ?thesis
4248     unfolding cos_minus .
4249 qed
4251 lemma cos_monotone_minus_pi_0':
4252   assumes "- pi \<le> y" and "y \<le> x" and "x \<le> 0"
4253   shows "cos y \<le> cos x"
4254 proof (cases "y < x")
4255   case True
4256   show ?thesis using cos_monotone_minus_pi_0[OF \<open>-pi \<le> y\<close> True \<open>x \<le> 0\<close>]
4257     by auto
4258 next
4259   case False
4260   then have "y = x" using \<open>y \<le> x\<close> by auto
4261   then show ?thesis by auto
4262 qed
4264 lemma sin_monotone_2pi:
4265   assumes "- (pi/2) \<le> y" and "y < x" and "x \<le> pi/2"
4266   shows "sin y < sin x"
4268   apply (rule cos_monotone_0_pi)
4269   using assms
4270     apply auto
4271   done
4273 lemma sin_monotone_2pi_le:
4274   assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2"
4275   shows "sin y \<le> sin x"
4276   by (metis assms le_less sin_monotone_2pi)
4278 lemma sin_x_le_x:
4279   fixes x :: real
4280   assumes x: "x \<ge> 0"
4281   shows "sin x \<le> x"
4282 proof -
4283   let ?f = "\<lambda>x. x - sin x"
4284   from x have "?f x \<ge> ?f 0"
4285     apply (rule DERIV_nonneg_imp_nondecreasing)
4286     apply (intro allI impI exI[of _ "1 - cos x" for x])
4287     apply (auto intro!: derivative_eq_intros simp: field_simps)
4288     done
4289   then show "sin x \<le> x" by simp
4290 qed
4292 lemma sin_x_ge_neg_x:
4293   fixes x :: real
4294   assumes x: "x \<ge> 0"
4295   shows "sin x \<ge> - x"
4296 proof -
4297   let ?f = "\<lambda>x. x + sin x"
4298   from x have "?f x \<ge> ?f 0"
4299     apply (rule DERIV_nonneg_imp_nondecreasing)
4300     apply (intro allI impI exI[of _ "1 + cos x" for x])
4301     apply (auto intro!: derivative_eq_intros simp: field_simps real_0_le_add_iff)
4302     done
4303   then show "sin x \<ge> -x" by simp
4304 qed
4306 lemma abs_sin_x_le_abs_x: "\<bar>sin x\<bar> \<le> \<bar>x\<bar>"
4307   for x :: real
4308   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"]
4309   by (auto simp: abs_real_def)
4312 subsection \<open>More Corollaries about Sine and Cosine\<close>
4314 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
4315 proof -
4316   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
4317     by (auto simp: algebra_simps sin_add)
4318   then show ?thesis
4320 qed
4322 lemma cos_2npi [simp]: "cos (2 * real n * pi) = 1"
4323   for n :: nat
4324   by (cases "even n") (simp_all add: cos_double mult.assoc)
4326 lemma cos_3over2_pi [simp]: "cos (3/2*pi) = 0"
4327   apply (subgoal_tac "cos (pi + pi/2) = 0")
4328    apply simp
4330   apply simp
4331   done
4333 lemma sin_2npi [simp]: "sin (2 * real n * pi) = 0"
4334   for n :: nat
4335   by (auto simp: mult.assoc sin_double)
4337 lemma sin_3over2_pi [simp]: "sin (3/2*pi) = - 1"
4338   apply (subgoal_tac "sin (pi + pi/2) = - 1")
4339    apply simp
4341   apply simp
4342   done
4344 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
4347 lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)"
4348   by (auto intro!: derivative_eq_intros)
4350 lemma sin_zero_norm_cos_one:
4351   fixes x :: "'a::{real_normed_field,banach}"
4352   assumes "sin x = 0"
4353   shows "norm (cos x) = 1"
4354   using sin_cos_squared_add [of x, unfolded assms]
4357 lemma sin_zero_abs_cos_one: "sin x = 0 \<Longrightarrow> \<bar>cos x\<bar> = (1::real)"
4358   using sin_zero_norm_cos_one by fastforce
4360 lemma cos_one_sin_zero:
4361   fixes x :: "'a::{real_normed_field,banach}"
4362   assumes "cos x = 1"
4363   shows "sin x = 0"
4364   using sin_cos_squared_add [of x, unfolded assms]
4365   by simp
4367 lemma sin_times_pi_eq_0: "sin (x * pi) = 0 \<longleftrightarrow> x \<in> \<int>"
4368   by (simp add: sin_zero_iff_int2) (metis Ints_cases Ints_of_int)
4370 lemma cos_one_2pi: "cos x = 1 \<longleftrightarrow> (\<exists>n::nat. x = n * 2 * pi) | (\<exists>n::nat. x = - (n * 2 * pi))"
4371   (is "?lhs = ?rhs")
4372 proof
4373   assume ?lhs
4374   then have "sin x = 0"
4376   then show ?rhs
4377   proof (simp only: sin_zero_iff, elim exE disjE conjE)
4378     fix n :: nat
4379     assume n: "even n" "x = real n * (pi/2)"
4380     then obtain m where m: "n = 2 * m"
4381       using dvdE by blast
4382     then have me: "even m" using \<open>?lhs\<close> n
4383       by (auto simp: field_simps) (metis one_neq_neg_one  power_minus_odd power_one)
4384     show ?rhs
4385       using m me n
4386       by (auto simp: field_simps elim!: evenE)
4387   next
4388     fix n :: nat
4389     assume n: "even n" "x = - (real n * (pi/2))"
4390     then obtain m where m: "n = 2 * m"
4391       using dvdE by blast
4392     then have me: "even m" using \<open>?lhs\<close> n
4393       by (auto simp: field_simps) (metis one_neq_neg_one  power_minus_odd power_one)
4394     show ?rhs
4395       using m me n
4396       by (auto simp: field_simps elim!: evenE)
4397   qed
4398 next
4399   assume ?rhs
4400   then show "cos x = 1"
4401     by (metis cos_2npi cos_minus mult.assoc mult.left_commute)
4402 qed
4404 lemma cos_one_2pi_int: "cos x = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2 * pi)" (is "?lhs = ?rhs")
4405 proof
4406   assume "cos x = 1"
4407   then show ?rhs
4408     apply (auto simp: cos_one_2pi)
4409      apply (metis of_int_of_nat_eq)
4410     apply (metis mult_minus_right of_int_minus of_int_of_nat_eq)
4411     done
4412 next
4413   assume ?rhs
4414   then show "cos x = 1"
4415     by (clarsimp simp add: cos_one_2pi) (metis mult_minus_right of_int_of_nat)
4416 qed
4418 lemma cos_npi_int [simp]:
4419   fixes n::int shows "cos (pi * of_int n) = (if even n then 1 else -1)"
4420     by (auto simp: algebra_simps cos_one_2pi_int elim!: oddE evenE)
4422 lemma sin_cos_sqrt: "0 \<le> sin x \<Longrightarrow> sin x = sqrt (1 - (cos(x) ^ 2))"
4423   using sin_squared_eq real_sqrt_unique by fastforce
4425 lemma sin_eq_0_pi: "- pi < x \<Longrightarrow> x < pi \<Longrightarrow> sin x = 0 \<Longrightarrow> x = 0"
4426   by (metis sin_gt_zero sin_minus minus_less_iff neg_0_less_iff_less not_less_iff_gr_or_eq)
4428 lemma cos_treble_cos: "cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x"
4429   for x :: "'a::{real_normed_field,banach}"
4430 proof -
4431   have *: "(sin x * (sin x * 3)) = 3 - (cos x * (cos x * 3))"
4432     by (simp add: mult.assoc [symmetric] sin_squared_eq [unfolded power2_eq_square])
4433   have "cos(3 * x) = cos(2*x + x)"
4434     by simp
4435   also have "\<dots> = 4 * cos x ^ 3 - 3 * cos x"
4436     apply (simp only: cos_add cos_double sin_double)
4437     apply (simp add: * field_simps power2_eq_square power3_eq_cube)
4438     done
4439   finally show ?thesis .
4440 qed
4442 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
4443 proof -
4444   let ?c = "cos (pi / 4)"
4445   let ?s = "sin (pi / 4)"
4446   have nonneg: "0 \<le> ?c"
4448   have "0 = cos (pi / 4 + pi / 4)"
4449     by simp
4450   also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2"
4451     by (simp only: cos_add power2_eq_square)
4452   also have "\<dots> = 2 * ?c\<^sup>2 - 1"
4454   finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2"
4456   then show ?thesis
4457     using nonneg by (rule power2_eq_imp_eq) simp
4458 qed
4460 lemma cos_30: "cos (pi / 6) = sqrt 3/2"
4461 proof -
4462   let ?c = "cos (pi / 6)"
4463   let ?s = "sin (pi / 6)"
4464   have pos_c: "0 < ?c"
4465     by (rule cos_gt_zero) simp_all
4466   have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
4467     by simp
4468   also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
4470   also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)"
4471     by (simp add: algebra_simps power2_eq_square)
4472   finally have "?c\<^sup>2 = (sqrt 3/2)\<^sup>2"
4473     using pos_c by (simp add: sin_squared_eq power_divide)
4474   then show ?thesis
4475     using pos_c [THEN order_less_imp_le]
4476     by (rule power2_eq_imp_eq) simp
4477 qed
4479 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
4480   by (simp add: sin_cos_eq cos_45)
4482 lemma sin_60: "sin (pi / 3) = sqrt 3/2"
4483   by (simp add: sin_cos_eq cos_30)
4485 lemma cos_60: "cos (pi / 3) = 1 / 2"
4486   apply (rule power2_eq_imp_eq)
4487     apply (simp add: cos_squared_eq sin_60 power_divide)
4488    apply (rule cos_ge_zero)
4489     apply (rule order_trans [where y=0])
4490      apply simp_all
4491   done
4493 lemma sin_30: "sin (pi / 6) = 1 / 2"
4494   by (simp add: sin_cos_eq cos_60)
4496 lemma cos_integer_2pi: "n \<in> \<int> \<Longrightarrow> cos(2 * pi * n) = 1"
4497   by (metis Ints_cases cos_one_2pi_int mult.assoc mult.commute)
4499 lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2 * pi * n) = 0"
4500   by (metis sin_two_pi Ints_mult mult.assoc mult.commute sin_times_pi_eq_0)
4502 lemma cos_int_2npi [simp]: "cos (2 * of_int n * pi) = 1"
4503   for n :: int
4506 lemma sin_int_2npi [simp]: "sin (2 * of_int n * pi) = 0"
4507   for n :: int
4508   by (metis Ints_of_int mult.assoc mult.commute sin_integer_2pi)
4510 lemma sincos_principal_value: "\<exists>y. (- pi < y \<and> y \<le> pi) \<and> (sin y = sin x \<and> cos y = cos x)"
4511   apply (rule exI [where x="pi - (2 * pi) * frac ((pi - x) / (2 * pi))"])
4512   apply (auto simp: field_simps frac_lt_1)
4513    apply (simp_all add: frac_def divide_simps)
4515    apply (simp_all add: sin_diff cos_diff mult.assoc [symmetric] cos_integer_2pi sin_integer_2pi)
4516   done
4519 subsection \<open>Tangent\<close>
4521 definition tan :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
4522   where "tan = (\<lambda>x. sin x / cos x)"
4524 lemma tan_of_real: "of_real (tan x) = (tan (of_real x) :: 'a::{real_normed_field,banach})"
4525   by (simp add: tan_def sin_of_real cos_of_real)
4527 lemma tan_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> tan z \<in> \<real>"
4528   for z :: "'a::{real_normed_field,banach}"
4531 lemma tan_zero [simp]: "tan 0 = 0"
4534 lemma tan_pi [simp]: "tan pi = 0"
4537 lemma tan_npi [simp]: "tan (real n * pi) = 0"
4538   for n :: nat
4541 lemma tan_minus [simp]: "tan (- x) = - tan x"
4544 lemma tan_periodic [simp]: "tan (x + 2 * pi) = tan x"
4547 lemma lemma_tan_add1: "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
4550 lemma add_tan_eq: "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
4551   for x :: "'a::{real_normed_field,banach}"
4555   "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> cos (x + y) \<noteq> 0 \<Longrightarrow> tan (x + y) = (tan x + tan y)/(1 - tan x * tan y)"
4556   for x :: "'a::{real_normed_field,banach}"
4559 lemma tan_double: "cos x \<noteq> 0 \<Longrightarrow> cos (2 * x) \<noteq> 0 \<Longrightarrow> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)"
4560   for x :: "'a::{real_normed_field,banach}"
4563 lemma tan_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < tan x"
4564   by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
4566 lemma tan_less_zero:
4567   assumes "- pi/2 < x" and "x < 0"
4568   shows "tan x < 0"
4569 proof -
4570   have "0 < tan (- x)"
4571     using assms by (simp only: tan_gt_zero)
4572   then show ?thesis by simp
4573 qed
4575 lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"
4576   for x :: "'a::{real_normed_field,banach,field}"
4577   unfolding tan_def sin_double cos_double sin_squared_eq
4580 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
4581   unfolding tan_def by (simp add: sin_30 cos_30)
4583 lemma tan_45: "tan (pi / 4) = 1"
4584   unfolding tan_def by (simp add: sin_45 cos_45)
4586 lemma tan_60: "tan (pi / 3) = sqrt 3"
4587   unfolding tan_def by (simp add: sin_60 cos_60)
4589 lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)"
4590   for x :: "'a::{real_normed_field,banach}"
4591   unfolding tan_def
4592   by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square)
4594 lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
4595   for x :: "'a::{real_normed_field,banach}"
4596   by (rule DERIV_tan [THEN DERIV_isCont])
4598 lemma isCont_tan' [simp,continuous_intros]:
4599   fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
4600   shows "isCont f a \<Longrightarrow> cos (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
4601   by (rule isCont_o2 [OF _ isCont_tan])
4603 lemma tendsto_tan [tendsto_intros]:
4604   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
4605   shows "(f \<longlongrightarrow> a) F \<Longrightarrow> cos a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. tan (f x)) \<longlongrightarrow> tan a) F"
4606   by (rule isCont_tendsto_compose [OF isCont_tan])
4608 lemma continuous_tan:
4609   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
4610   shows "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))"
4611   unfolding continuous_def by (rule tendsto_tan)
4613 lemma continuous_on_tan [continuous_intros]:
4614   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
4615   shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))"
4616   unfolding continuous_on_def by (auto intro: tendsto_tan)
4618 lemma continuous_within_tan [continuous_intros]:
4619   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
4620   shows "continuous (at x within s) f \<Longrightarrow>
4621     cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
4622   unfolding continuous_within by (rule tendsto_tan)
4624 lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) \<midarrow>pi/2\<rightarrow> 0"
4625   by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
4627 lemma lemma_tan_total: "0 < y \<Longrightarrow> \<exists>x. 0 < x \<and> x < pi/2 \<and> y < tan x"
4628   apply (insert LIM_cos_div_sin)
4629   apply (simp only: LIM_eq)
4630   apply (drule_tac x = "inverse y" in spec)
4631   apply safe
4632    apply force
4633   apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero])
4634   apply safe
4635   apply (rule_tac x = "(pi/2) - e" in exI)
4636   apply (simp (no_asm_simp))
4637   apply (drule_tac x = "(pi/2) - e" in spec)
4638   apply (auto simp add: tan_def sin_diff cos_diff)
4639   apply (rule inverse_less_iff_less [THEN iffD1])
4640     apply (auto simp add: divide_inverse)
4641    apply (rule mult_pos_pos)
4642     apply (subgoal_tac [3] "0 < sin e \<and> 0 < cos e")
4643      apply (auto intro: cos_gt_zero sin_gt_zero2 simp: mult.commute)
4644   done
4646 lemma tan_total_pos: "0 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x \<and> x < pi/2 \<and> tan x = y"
4647   apply (frule order_le_imp_less_or_eq)
4648   apply safe
4649    prefer 2 apply force
4650   apply (drule lemma_tan_total)
4651   apply safe
4652   apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
4653   apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
4654   apply (drule_tac y = xa in order_le_imp_less_or_eq)
4655   apply (auto dest: cos_gt_zero)
4656   done
4658 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y"
4659   apply (insert linorder_linear [of 0 y])
4660   apply safe
4661    apply (drule tan_total_pos)
4662    apply (cut_tac [2] y="-y" in tan_total_pos)
4663     apply safe
4664     apply (rule_tac [3] x = "-x" in exI)
4665     apply (auto del: exI intro!: exI)
4666   done
4668 lemma tan_total: "\<exists>! x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y"
4669   apply (insert lemma_tan_total1 [where y = y])
4670   apply auto
4671   apply hypsubst_thin
4672   apply (cut_tac x = xa and y = y in linorder_less_linear)
4673   apply auto
4674    apply (subgoal_tac [2] "\<exists>z. y < z \<and> z < xa \<and> DERIV tan z :> 0")
4675     apply (subgoal_tac "\<exists>z. xa < z \<and> z < y \<and> DERIV tan z :> 0")
4676      apply (rule_tac [4] Rolle)
4677         apply (rule_tac [2] Rolle)
4678            apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
4680        apply (rule_tac [!] DERIV_tan asm_rl)
4681        apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
4682         simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
4683   done
4685 lemma tan_monotone:
4686   assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
4687   shows "tan y < tan x"
4688 proof -
4689   have "\<forall>x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse ((cos x')\<^sup>2)"
4690   proof (rule allI, rule impI)
4691     fix x' :: real
4692     assume "y \<le> x' \<and> x' \<le> x"
4693     then have "-(pi/2) < x'" and "x' < pi/2"
4694       using assms by auto
4695     from cos_gt_zero_pi[OF this]
4696     have "cos x' \<noteq> 0" by auto
4697     then show "DERIV tan x' :> inverse ((cos x')\<^sup>2)"
4698       by (rule DERIV_tan)
4699   qed
4700   from MVT2[OF \<open>y < x\<close> this]
4701   obtain z where "y < z" and "z < x"
4702     and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
4703   then have "- (pi / 2) < z" and "z < pi / 2"
4704     using assms by auto
4705   then have "0 < cos z"
4706     using cos_gt_zero_pi by auto
4707   then have inv_pos: "0 < inverse ((cos z)\<^sup>2)"
4708     by auto
4709   have "0 < x - y" using \<open>y < x\<close> by auto
4710   with inv_pos have "0 < tan x - tan y"
4711     unfolding tan_diff by auto
4712   then show ?thesis by auto
4713 qed
4715 lemma tan_monotone':
4716   assumes "- (pi / 2) < y"
4717     and "y < pi / 2"
4718     and "- (pi / 2) < x"
4719     and "x < pi / 2"
4720   shows "y < x \<longleftrightarrow> tan y < tan x"
4721 proof
4722   assume "y < x"
4723   then show "tan y < tan x"
4724     using tan_monotone and \<open>- (pi / 2) < y\<close> and \<open>x < pi / 2\<close> by auto
4725 next
4726   assume "tan y < tan x"
4727   show "y < x"
4728   proof (rule ccontr)
4729     assume "\<not> ?thesis"
4730     then have "x \<le> y" by auto
4731     then have "tan x \<le> tan y"
4732     proof (cases "x = y")
4733       case True
4734       then show ?thesis by auto
4735     next
4736       case False
4737       then have "x < y" using \<open>x \<le> y\<close> by auto
4738       from tan_monotone[OF \<open>- (pi/2) < x\<close> this \<open>y < pi / 2\<close>] show ?thesis
4739         by auto
4740     qed
4741     then show False
4742       using \<open>tan y < tan x\<close> by auto
4743   qed
4744 qed
4746 lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)"
4747   unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
4749 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
4752 lemma tan_periodic_nat[simp]: "tan (x + real n * pi) = tan x"
4753   for n :: nat
4754 proof (induct n arbitrary: x)
4755   case 0
4756   then show ?case by simp
4757 next
4758   case (Suc n)
4759   have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi"
4760     unfolding Suc_eq_plus1 of_nat_add  distrib_right by auto
4761   show ?case
4762     unfolding split_pi_off using Suc by auto
4763 qed
4765 lemma tan_periodic_int[simp]: "tan (x + of_int i * pi) = tan x"
4766 proof (cases "0 \<le> i")
4767   case True
4768   then have i_nat: "of_int i = of_int (nat i)" by auto
4769   show ?thesis unfolding i_nat
4770     by (metis of_int_of_nat_eq tan_periodic_nat)
4771 next
4772   case False
4773   then have i_nat: "of_int i = - of_int (nat (- i))" by auto
4774   have "tan x = tan (x + of_int i * pi - of_int i * pi)"
4775     by auto
4776   also have "\<dots> = tan (x + of_int i * pi)"
4778     by (metis of_int_of_nat_eq tan_periodic_nat)
4779   finally show ?thesis by auto
4780 qed
4782 lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
4783   using tan_periodic_int[of _ "numeral n" ] by simp
4785 lemma tan_minus_45: "tan (-(pi/4)) = -1"
4786   unfolding tan_def by (simp add: sin_45 cos_45)
4788 lemma tan_diff:
4789   "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> cos (x - y) \<noteq> 0 \<Longrightarrow> tan (x - y) = (tan x - tan y)/(1 + tan x * tan y)"
4790   for x :: "'a::{real_normed_field,banach}"
4791   using tan_add [of x "-y"] by simp
4793 lemma tan_pos_pi2_le: "0 \<le> x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 \<le> tan x"
4794   using less_eq_real_def tan_gt_zero by auto
4796 lemma cos_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> cos x = 1 / sqrt (1 + tan x ^ 2)"
4797   using cos_gt_zero_pi [of x]
4798   by (simp add: divide_simps tan_def real_sqrt_divide abs_if split: if_split_asm)
4800 lemma sin_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> sin x = tan x / sqrt (1 + tan x ^ 2)"
4801   using cos_gt_zero [of "x"] cos_gt_zero [of "-x"]
4802   by (force simp add: divide_simps tan_def real_sqrt_divide abs_if split: if_split_asm)
4804 lemma tan_mono_le: "-(pi/2) < x \<Longrightarrow> x \<le> y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x \<le> tan y"
4805   using less_eq_real_def tan_monotone by auto
4807 lemma tan_mono_lt_eq:
4808   "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> -(pi/2) < y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x < tan y \<longleftrightarrow> x < y"
4809   using tan_monotone' by blast
4811 lemma tan_mono_le_eq:
4812   "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> -(pi/2) < y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x \<le> tan y \<longleftrightarrow> x \<le> y"
4813   by (meson tan_mono_le not_le tan_monotone)
4815 lemma tan_bound_pi2: "\<bar>x\<bar> < pi/4 \<Longrightarrow> \<bar>tan x\<bar> < 1"
4816   using tan_45 tan_monotone [of x "pi/4"] tan_monotone [of "-x" "pi/4"]
4817   by (auto simp: abs_if split: if_split_asm)
4819 lemma tan_cot: "tan(pi/2 - x) = inverse(tan x)"
4820   by (simp add: tan_def sin_diff cos_diff)
4823 subsection \<open>Cotangent\<close>
4825 definition cot :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
4826   where "cot = (\<lambda>x. cos x / sin x)"
4828 lemma cot_of_real: "of_real (cot x) = (cot (of_real x) :: 'a::{real_normed_field,banach})"
4829   by (simp add: cot_def sin_of_real cos_of_real)
4831 lemma cot_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cot z \<in> \<real>"
4832   for z :: "'a::{real_normed_field,banach}"
4835 lemma cot_zero [simp]: "cot 0 = 0"
4838 lemma cot_pi [simp]: "cot pi = 0"
4841 lemma cot_npi [simp]: "cot (real n * pi) = 0"
4842   for n :: nat
4845 lemma cot_minus [simp]: "cot (- x) = - cot x"
4848 lemma cot_periodic [simp]: "cot (x + 2 * pi) = cot x"
4851 lemma cot_altdef: "cot x = inverse (tan x)"
4852   by (simp add: cot_def tan_def)
4854 lemma tan_altdef: "tan x = inverse (cot x)"
4855   by (simp add: cot_def tan_def)
4857 lemma tan_cot': "tan (pi/2 - x) = cot x"
4858   by (simp add: tan_cot cot_altdef)
4860 lemma cot_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cot x"
4861   by (simp add: cot_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
4863 lemma cot_less_zero:
4864   assumes lb: "- pi/2 < x" and "x < 0"
4865   shows "cot x < 0"
4866 proof -
4867   have "0 < cot (- x)"
4868     using assms by (simp only: cot_gt_zero)
4869   then show ?thesis by simp
4870 qed
4872 lemma DERIV_cot [simp]: "sin x \<noteq> 0 \<Longrightarrow> DERIV cot x :> -inverse ((sin x)\<^sup>2)"
4873   for x :: "'a::{real_normed_field,banach}"
4874   unfolding cot_def using cos_squared_eq[of x]
4875   by (auto intro!: derivative_eq_intros) (simp add: divide_inverse power2_eq_square)
4877 lemma isCont_cot: "sin x \<noteq> 0 \<Longrightarrow> isCont cot x"
4878   for x :: "'a::{real_normed_field,banach}"
4879   by (rule DERIV_cot [THEN DERIV_isCont])
4881 lemma isCont_cot' [simp,continuous_intros]:
4882   "isCont f a \<Longrightarrow> sin (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. cot (f x)) a"
4883   for a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a"
4884   by (rule isCont_o2 [OF _ isCont_cot])
4886 lemma tendsto_cot [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> sin a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. cot (f x)) \<longlongrightarrow> cot a) F"
4887   for f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
4888   by (rule isCont_tendsto_compose [OF isCont_cot])
4890 lemma continuous_cot:
4891   "continuous F f \<Longrightarrow> sin (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. cot (f x))"
4892   for f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
4893   unfolding continuous_def by (rule tendsto_cot)
4895 lemma continuous_on_cot [continuous_intros]:
4896   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
4897   shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. sin (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. cot (f x))"
4898   unfolding continuous_on_def by (auto intro: tendsto_cot)
4900 lemma continuous_within_cot [continuous_intros]:
4901   fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
4902   shows "continuous (at x within s) f \<Longrightarrow> sin (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. cot (f x))"
4903   unfolding continuous_within by (rule tendsto_cot)
4906 subsection \<open>Inverse Trigonometric Functions\<close>
4908 definition arcsin :: "real \<Rightarrow> real"
4909   where "arcsin y = (THE x. -(pi/2) \<le> x \<and> x \<le> pi/2 \<and> sin x = y)"
4911 definition arccos :: "real \<Rightarrow> real"
4912   where "arccos y = (THE x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y)"
4914 definition arctan :: "real \<Rightarrow> real"
4915   where "arctan y = (THE x. -(pi/2) < x \<and> x < pi/2 \<and> tan x = y)"
4917 lemma arcsin: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi/2 \<and> sin (arcsin y) = y"
4918   unfolding arcsin_def by (rule theI' [OF sin_total])
4920 lemma arcsin_pi: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi \<and> sin (arcsin y) = y"
4921   by (drule (1) arcsin) (force intro: order_trans)
4923 lemma sin_arcsin [simp]: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin (arcsin y) = y"
4924   by (blast dest: arcsin)
4926 lemma arcsin_bounded: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi/2"
4927   by (blast dest: arcsin)
4929 lemma arcsin_lbound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y"
4930   by (blast dest: arcsin)
4932 lemma arcsin_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
4933   by (blast dest: arcsin)
4935 lemma arcsin_lt_bounded: "- 1 < y \<Longrightarrow> y < 1 \<Longrightarrow> - (pi/2) < arcsin y \<and> arcsin y < pi/2"
4936   apply (frule order_less_imp_le)
4937   apply (frule_tac y = y in order_less_imp_le)
4938   apply (frule arcsin_bounded)
4939    apply safe
4940     apply simp
4941    apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
4942    apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq)
4943    apply safe
4944    apply (drule_tac [!] f = sin in arg_cong)
4945    apply auto
4946   done
4948 lemma arcsin_sin: "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> arcsin (sin x) = x"
4949   apply (unfold arcsin_def)
4950   apply (rule the1_equality)
4951    apply (rule sin_total)
4952     apply auto
4953   done
4955 lemma arcsin_0 [simp]: "arcsin 0 = 0"
4956   using arcsin_sin [of 0] by simp
4958 lemma arcsin_1 [simp]: "arcsin 1 = pi/2"
4959   using arcsin_sin [of "pi/2"] by simp
4961 lemma arcsin_minus_1 [simp]: "arcsin (- 1) = - (pi/2)"
4962   using arcsin_sin [of "- pi/2"] by simp
4964 lemma arcsin_minus: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin (- x) = - arcsin x"
4965   by (metis (no_types, hide_lams) arcsin arcsin_sin minus_minus neg_le_iff_le sin_minus)
4967 lemma arcsin_eq_iff: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x = arcsin y \<longleftrightarrow> x = y"
4968   by (metis abs_le_iff arcsin minus_le_iff)
4970 lemma cos_arcsin_nonzero: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> cos (arcsin x) \<noteq> 0"
4971   using arcsin_lt_bounded cos_gt_zero_pi by force
4973 lemma arccos: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi \<and> cos (arccos y) = y"
4974   unfolding arccos_def by (rule theI' [OF cos_total])
4976 lemma cos_arccos [simp]: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> cos (arccos y) = y"
4977   by (blast dest: arccos)
4979 lemma arccos_bounded: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi"
4980   by (blast dest: arccos)
4982 lemma arccos_lbound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y"
4983   by (blast dest: arccos)
4985 lemma arccos_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> pi"
4986   by (blast dest: arccos)
4988 lemma arccos_lt_bounded: "- 1 < y \<Longrightarrow> y < 1 \<Longrightarrow> 0 < arccos y \<and> arccos y < pi"
4989   apply (frule order_less_imp_le)
4990   apply (frule_tac y = y in order_less_imp_le)
4991   apply (frule arccos_bounded)
4992    apply auto
4993    apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
4994    apply (drule_tac [2] y = pi in order_le_imp_less_or_eq)
4995    apply auto
4996    apply (drule_tac [!] f = cos in arg_cong)
4997    apply auto
4998   done
5000 lemma arccos_cos: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> arccos (cos x) = x"
5001   by (auto simp: arccos_def intro!: the1_equality cos_total)
5003 lemma arccos_cos2: "x \<le> 0 \<Longrightarrow> - pi \<le> x \<Longrightarrow> arccos (cos x) = -x"
5004   by (auto simp: arccos_def intro!: the1_equality cos_total)
5006 lemma cos_arcsin: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
5007   apply (subgoal_tac "x\<^sup>2 \<le> 1")
5008    apply (rule power2_eq_imp_eq)
5010     apply (rule cos_ge_zero)
5011      apply (erule (1) arcsin_lbound)
5012     apply (erule (1) arcsin_ubound)
5013    apply simp
5014   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2")
5015    apply simp
5016   apply (rule power_mono)
5017    apply simp
5018   apply simp
5019   done
5021 lemma sin_arccos: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
5022   apply (subgoal_tac "x\<^sup>2 \<le> 1")
5023    apply (rule power2_eq_imp_eq)
5025     apply (rule sin_ge_zero)
5026      apply (erule (1) arccos_lbound)
5027     apply (erule (1) arccos_ubound)
5028    apply simp
5029   apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2")
5030    apply simp
5031   apply (rule power_mono)
5032    apply simp
5033   apply simp
5034   done
5036 lemma arccos_0 [simp]: "arccos 0 = pi/2"
5037   by (metis arccos_cos cos_gt_zero cos_pi cos_pi_half pi_gt_zero
5038       pi_half_ge_zero not_le not_zero_less_neg_numeral numeral_One)
5040 lemma arccos_1 [simp]: "arccos 1 = 0"
5041   using arccos_cos by force
5043 lemma arccos_minus_1 [simp]: "arccos (- 1) = pi"
5044   by (metis arccos_cos cos_pi order_refl pi_ge_zero)
5046 lemma arccos_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos (- x) = pi - arccos x"
5047   by (metis arccos_cos arccos_cos2 cos_minus_pi cos_total diff_le_0_iff_le le_add_same_cancel1
5050 corollary arccos_minus_abs:
5051   assumes "\<bar>x\<bar> \<le> 1"
5052   shows "arccos (- x) = pi - arccos x"
5053 using assms by (simp add: arccos_minus)
5055 lemma sin_arccos_nonzero: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> sin (arccos x) \<noteq> 0"
5056   using arccos_lt_bounded sin_gt_zero by force
5058 lemma arctan: "- (pi/2) < arctan y \<and> arctan y < pi/2 \<and> tan (arctan y) = y"
5059   unfolding arctan_def by (rule theI' [OF tan_total])
5061 lemma tan_arctan: "tan (arctan y) = y"
5064 lemma arctan_bounded: "- (pi/2) < arctan y \<and> arctan y < pi/2"
5065   by (auto simp only: arctan)
5067 lemma arctan_lbound: "- (pi/2) < arctan y"
5070 lemma arctan_ubound: "arctan y < pi/2"
5071   by (auto simp only: arctan)
5073 lemma arctan_unique:
5074   assumes "-(pi/2) < x"
5075     and "x < pi/2"
5076     and "tan x = y"
5077   shows "arctan y = x"
5078   using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
5080 lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x"
5081   by (rule arctan_unique) simp_all
5083 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
5084   by (rule arctan_unique) simp_all
5086 lemma arctan_minus: "arctan (- x) = - arctan x"
5087   using arctan [of "x"] by (auto simp: arctan_unique)
5089 lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
5090   by (intro less_imp_neq [symmetric] cos_gt_zero_pi arctan_lbound arctan_ubound)
5092 lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)"
5093 proof (rule power2_eq_imp_eq)
5095   show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp
5096   show "0 \<le> cos (arctan x)"
5097     by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
5098   have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1"
5099     unfolding tan_def by (simp add: distrib_left power_divide)
5100   then show "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
5101     using \<open>0 < 1 + x\<^sup>2\<close> by (simp add: arctan power_divide eq_divide_eq)
5102 qed
5104 lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)"
5105   using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
5106   using tan_arctan [of x] unfolding tan_def cos_arctan
5109 lemma tan_sec: "cos x \<noteq> 0 \<Longrightarrow> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
5110   for x :: "'a::{real_normed_field,banach,field}"
5111   apply (rule power_inverse [THEN subst])
5112   apply (rule_tac c1 = "(cos x)\<^sup>2" in mult_right_cancel [THEN iffD1])
5113    apply (auto simp add: tan_def field_simps)
5114   done
5116 lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
5117   by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
5119 lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
5120   by (simp only: not_less [symmetric] arctan_less_iff)
5122 lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
5123   by (simp only: eq_iff [where 'a=real] arctan_le_iff)
5125 lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
5126   using arctan_less_iff [of 0 x] by simp
5128 lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
5129   using arctan_less_iff [of x 0] by simp
5131 lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
5132   using arctan_le_iff [of 0 x] by simp
5134 lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
5135   using arctan_le_iff [of x 0] by simp
5137 lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
5138   using arctan_eq_iff [of x 0] by simp
5140 lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin"
5141 proof -
5142   have "continuous_on (sin ` {- pi / 2 .. pi / 2}) arcsin"
5143     by (rule continuous_on_inv) (auto intro: continuous_intros simp: arcsin_sin)
5144   also have "sin ` {- pi / 2 .. pi / 2} = {-1 .. 1}"
5145   proof safe
5146     fix x :: real
5147     assume "x \<in> {-1..1}"
5148     then show "x \<in> sin ` {- pi / 2..pi / 2}"
5149       using arcsin_lbound arcsin_ubound
5150       by (intro image_eqI[where x="arcsin x"]) auto
5151   qed simp
5152   finally show ?thesis .
5153 qed
5155 lemma continuous_on_arcsin [continuous_intros]:
5156   "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))"
5157   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arcsin']]
5158   by (auto simp: comp_def subset_eq)
5160 lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x"
5161   using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
5162   by (auto simp: continuous_on_eq_continuous_at subset_eq)
5164 lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos"
5165 proof -
5166   have "continuous_on (cos ` {0 .. pi}) arccos"
5167     by (rule continuous_on_inv) (auto intro: continuous_intros simp: arccos_cos)
5168   also have "cos ` {0 .. pi} = {-1 .. 1}"
5169   proof safe
5170     fix x :: real
5171     assume "x \<in> {-1..1}"
5172     then show "x \<in> cos ` {0..pi}"
5173       using arccos_lbound arccos_ubound
5174       by (intro image_eqI[where x="arccos x"]) auto
5175   qed simp
5176   finally show ?thesis .
5177 qed
5179 lemma continuous_on_arccos [continuous_intros]:
5180   "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))"
5181   using continuous_on_compose[of s f, OF _ continuous_on_subset[OF  continuous_on_arccos']]
5182   by (auto simp: comp_def subset_eq)
5184 lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x"
5185   using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"]
5186   by (auto simp: continuous_on_eq_continuous_at subset_eq)
5188 lemma isCont_arctan: "isCont arctan x"
5189   apply (rule arctan_lbound [of x, THEN dense, THEN exE])
5190   apply clarify
5191   apply (rule arctan_ubound [of x, THEN dense, THEN exE])
5192   apply clarify
5193   apply (subgoal_tac "isCont arctan (tan (arctan x))")
5195   apply (erule (1) isCont_inverse_function2 [where f=tan])
5196    apply (metis arctan_tan order_le_less_trans order_less_le_trans)
5197   apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
5198   done
5200 lemma tendsto_arctan [tendsto_intros]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) \<longlongrightarrow> arctan x) F"
5201   by (rule isCont_tendsto_compose [OF isCont_arctan])
5203 lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))"
5204   unfolding continuous_def by (rule tendsto_arctan)
5206 lemma continuous_on_arctan [continuous_intros]:
5207   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
5208   unfolding continuous_on_def by (auto intro: tendsto_arctan)
5210 lemma DERIV_arcsin: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
5211   apply (rule DERIV_inverse_function [where f=sin and a="-1" and b=1])
5212        apply (rule DERIV_cong [OF DERIV_sin])
5214       apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2")
5215        apply simp
5216       apply (rule power_strict_mono)
5217         apply simp
5218        apply simp
5219       apply simp
5220      apply assumption
5221     apply assumption
5222    apply simp
5223   apply (erule (1) isCont_arcsin)
5224   done
5226 lemma DERIV_arccos: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
5227   apply (rule DERIV_inverse_function [where f=cos and a="-1" and b=1])
5228        apply (rule DERIV_cong [OF DERIV_cos])
5230       apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2")
5231        apply simp
5232       apply (rule power_strict_mono)
5233         apply simp
5234        apply simp
5235       apply simp
5236      apply assumption
5237     apply assumption
5238    apply simp
5239   apply (erule (1) isCont_arccos)
5240   done
5242 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
5243   apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
5244        apply (rule DERIV_cong [OF DERIV_tan])
5245         apply (rule cos_arctan_not_zero)
5247    apply (simp add: arctan power_inverse [symmetric] tan_sec [symmetric])
5248   apply (subgoal_tac "0 < 1 + x\<^sup>2")
5249    apply simp
5251   done
5253 declare
5254   DERIV_arcsin[THEN DERIV_chain2, derivative_intros]
5255   DERIV_arcsin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
5256   DERIV_arccos[THEN DERIV_chain2, derivative_intros]
5257   DERIV_arccos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
5258   DERIV_arctan[THEN DERIV_chain2, derivative_intros]
5259   DERIV_arctan[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
5261 lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- (pi/2)))"
5262   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])
5263      (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
5264            intro!: tan_monotone exI[of _ "pi/2"])
5266 lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"
5267   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])
5268      (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
5269            intro!: tan_monotone exI[of _ "pi/2"])
5271 lemma tendsto_arctan_at_top: "(arctan \<longlongrightarrow> (pi/2)) at_top"
5272 proof (rule tendstoI)
5273   fix e :: real
5274   assume "0 < e"
5275   define y where "y = pi/2 - min (pi/2) e"
5276   then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y"
5277     using \<open>0 < e\<close> by auto
5278   show "eventually (\<lambda>x. dist (arctan x) (pi / 2) < e) at_top"
5279   proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)
5280     fix x
5281     assume "tan y < x"
5282     then have "arctan (tan y) < arctan x"
5284     with y have "y < arctan x"
5285       by (subst (asm) arctan_tan) simp_all
5286     with arctan_ubound[of x, arith] y \<open>0 < e\<close>
5287     show "dist (arctan x) (pi / 2) < e"
5289   qed
5290 qed
5292 lemma tendsto_arctan_at_bot: "(arctan \<longlongrightarrow> - (pi/2)) at_bot"
5293   unfolding filterlim_at_bot_mirror arctan_minus
5294   by (intro tendsto_minus tendsto_arctan_at_top)
5297 subsection \<open>Prove Totality of the Trigonometric Functions\<close>
5299 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
5302 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
5303   by (simp add: sin_arccos abs_le_iff)
5305 lemma sin_mono_less_eq:
5306   "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x < sin y \<longleftrightarrow> x < y"
5307   by (metis not_less_iff_gr_or_eq sin_monotone_2pi)
5309 lemma sin_mono_le_eq:
5310   "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x \<le> sin y \<longleftrightarrow> x \<le> y"
5311   by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le)
5313 lemma sin_inj_pi:
5314   "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x = sin y \<Longrightarrow> x = y"
5315   by (metis arcsin_sin)
5317 lemma cos_mono_less_eq: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x < cos y \<longleftrightarrow> y < x"
5318   by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear)
5320 lemma cos_mono_le_eq: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x \<le> cos y \<longleftrightarrow> y \<le> x"
5321   by (metis arccos_cos cos_monotone_0_pi_le eq_iff linear)
5323 lemma cos_inj_pi: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x = cos y \<Longrightarrow> x = y"
5324   by (metis arccos_cos)
5326 lemma arccos_le_pi2: "\<lbrakk>0 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi/2"
5327   by (metis (mono_tags) arccos_0 arccos cos_le_one cos_monotone_0_pi_le
5328       cos_pi cos_pi_half pi_half_ge_zero antisym_conv less_eq_neg_nonpos linear minus_minus order.trans order_refl)
5330 lemma sincos_total_pi_half:
5331   assumes "0 \<le> x" "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1"
5332   shows "\<exists>t. 0 \<le> t \<and> t \<le> pi/2 \<and> x = cos t \<and> y = sin t"
5333 proof -
5334   have x1: "x \<le> 1"
5335     using assms by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2)
5336   with assms have *: "0 \<le> arccos x" "cos (arccos x) = x"
5337     by (auto simp: arccos)
5338   from assms have "y = sqrt (1 - x\<^sup>2)"
5340   with x1 * assms arccos_le_pi2 [of x] show ?thesis
5341     by (rule_tac x="arccos x" in exI) (auto simp: sin_arccos)
5342 qed
5344 lemma sincos_total_pi:
5345   assumes "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1"
5346   shows "\<exists>t. 0 \<le> t \<and> t \<le> pi \<and> x = cos t \<and> y = sin t"
5347 proof (cases rule: le_cases [of 0 x])
5348   case le
5349   from sincos_total_pi_half [OF le] show ?thesis
5351 next
5352   case ge
5353   then have "0 \<le> -x"
5354     by simp
5355   then obtain t where t: "t\<ge>0" "t \<le> pi/2" "-x = cos t" "y = sin t"
5356     using sincos_total_pi_half assms
5357     by auto (metis \<open>0 \<le> - x\<close> power2_minus)
5358   show ?thesis
5359     by (rule exI [where x = "pi -t"]) (use t in auto)
5360 qed
5362 lemma sincos_total_2pi_le:
5363   assumes "x\<^sup>2 + y\<^sup>2 = 1"
5364   shows "\<exists>t. 0 \<le> t \<and> t \<le> 2 * pi \<and> x = cos t \<and> y = sin t"
5365 proof (cases rule: le_cases [of 0 y])
5366   case le
5367   from sincos_total_pi [OF le] show ?thesis
5368     by (metis assms le_add_same_cancel1 mult.commute mult_2_right order.trans)
5369 next
5370   case ge
5371   then have "0 \<le> -y"
5372     by simp
5373   then obtain t where t: "t\<ge>0" "t \<le> pi" "x = cos t" "-y = sin t"
5374     using sincos_total_pi assms
5375     by auto (metis \<open>0 \<le> - y\<close> power2_minus)
5376   show ?thesis
5377     by (rule exI [where x = "2 * pi - t"]) (use t in auto)
5378 qed
5380 lemma sincos_total_2pi:
5381   assumes "x\<^sup>2 + y\<^sup>2 = 1"
5382   obtains t where "0 \<le> t" "t < 2*pi" "x = cos t" "y = sin t"
5383 proof -
5384   from sincos_total_2pi_le [OF assms]
5385   obtain t where t: "0 \<le> t" "t \<le> 2*pi" "x = cos t" "y = sin t"
5386     by blast
5387   show ?thesis
5388     by (cases "t = 2 * pi") (use t that in \<open>force+\<close>)
5389 qed
5391 lemma arcsin_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x < arcsin y \<longleftrightarrow> x < y"
5392   by (rule trans [OF sin_mono_less_eq [symmetric]]) (use arcsin_ubound arcsin_lbound in auto)
5394 lemma arcsin_le_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y \<longleftrightarrow> x \<le> y"
5395   using arcsin_less_mono not_le by blast
5397 lemma arcsin_less_arcsin: "- 1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x < arcsin y"
5398   using arcsin_less_mono by auto
5400 lemma arcsin_le_arcsin: "- 1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y"
5401   using arcsin_le_mono by auto
5403 lemma arccos_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x < arccos y \<longleftrightarrow> y < x"
5404   by (rule trans [OF cos_mono_less_eq [symmetric]]) (use arccos_ubound arccos_lbound in auto)
5406 lemma arccos_le_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x \<le> arccos y \<longleftrightarrow> y \<le> x"
5407   using arccos_less_mono [of y x] by (simp add: not_le [symmetric])
5409 lemma arccos_less_arccos: "- 1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y < arccos x"
5410   using arccos_less_mono by auto
5412 lemma arccos_le_arccos: "- 1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> arccos x"
5413   using arccos_le_mono by auto
5415 lemma arccos_eq_iff: "\<bar>x\<bar> \<le> 1 \<and> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x = arccos y \<longleftrightarrow> x = y"
5416   using cos_arccos_abs by fastforce
5419 subsection \<open>Machin's formula\<close>
5421 lemma arctan_one: "arctan 1 = pi / 4"
5422   by (rule arctan_unique) (simp_all add: tan_45 m2pi_less_pi)
5424 lemma tan_total_pi4:
5425   assumes "\<bar>x\<bar> < 1"
5426   shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
5427 proof
5428   show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
5429     unfolding arctan_one [symmetric] arctan_minus [symmetric]
5430     unfolding arctan_less_iff
5431     using assms by (auto simp add: arctan)
5432 qed
5435   assumes "\<bar>x\<bar> \<le> 1" "\<bar>y\<bar> < 1"
5436   shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
5437 proof (rule arctan_unique [symmetric])
5438   have "- (pi / 4) \<le> arctan x" "- (pi / 4) < arctan y"
5439     unfolding arctan_one [symmetric] arctan_minus [symmetric]
5440     unfolding arctan_le_iff arctan_less_iff
5441     using assms by auto
5442   from add_le_less_mono [OF this] show 1: "- (pi / 2) < arctan x + arctan y"
5443     by simp
5444   have "arctan x \<le> pi / 4" "arctan y < pi / 4"
5445     unfolding arctan_one [symmetric]
5446     unfolding arctan_le_iff arctan_less_iff
5447     using assms by auto
5448   from add_le_less_mono [OF this] show 2: "arctan x + arctan y < pi / 2"
5449     by simp
5450   show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
5452 qed
5454 lemma arctan_double: "\<bar>x\<bar> < 1 \<Longrightarrow> 2 * arctan x = arctan ((2 * x) / (1 - x\<^sup>2))"
5455   by (metis arctan_add linear mult_2 not_less power2_eq_square)
5457 theorem machin: "pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)"
5458 proof -
5459   have "\<bar>1 / 5\<bar> < (1 :: real)"
5460     by auto
5461   from arctan_add[OF less_imp_le[OF this] this] have "2 * arctan (1 / 5) = arctan (5 / 12)"
5462     by auto
5463   moreover
5464   have "\<bar>5 / 12\<bar> < (1 :: real)"
5465     by auto
5466   from arctan_add[OF less_imp_le[OF this] this] have "2 * arctan (5 / 12) = arctan (120 / 119)"
5467     by auto
5468   moreover
5469   have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)"
5470     by auto
5471   from arctan_add[OF this] have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)"
5472     by auto
5473   ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)"
5474     by auto
5475   then show ?thesis
5476     unfolding arctan_one by algebra
5477 qed
5479 lemma machin_Euler: "5 * arctan (1 / 7) + 2 * arctan (3 / 79) = pi / 4"
5480 proof -
5481   have 17: "\<bar>1 / 7\<bar> < (1 :: real)" by auto
5482   with arctan_double have "2 * arctan (1 / 7) = arctan (7 / 24)"
5483     by simp (simp add: field_simps)
5484   moreover
5485   have "\<bar>7 / 24\<bar> < (1 :: real)" by auto
5486   with arctan_double have "2 * arctan (7 / 24) = arctan (336 / 527)"
5487     by simp (simp add: field_simps)
5488   moreover
5489   have "\<bar>336 / 527\<bar> < (1 :: real)" by auto
5490   from arctan_add[OF less_imp_le[OF 17] this]
5491   have "arctan(1/7) + arctan (336 / 527) = arctan (2879 / 3353)"
5492     by auto
5493   ultimately have I: "5 * arctan (1 / 7) = arctan (2879 / 3353)" by auto
5494   have 379: "\<bar>3 / 79\<bar> < (1 :: real)" by auto
5495   with arctan_double have II: "2 * arctan (3 / 79) = arctan (237 / 3116)"
5496     by simp (simp add: field_simps)
5497   have *: "\<bar>2879 / 3353\<bar> < (1 :: real)" by auto
5498   have "\<bar>237 / 3116\<bar> < (1 :: real)" by auto
5499   from arctan_add[OF less_imp_le[OF *] this] have "arctan (2879/3353) + arctan (237/3116) = pi/4"
5501   with I II show ?thesis by auto
5502 qed
5504 (*But could also prove MACHIN_GAUSS:
5505   12 * arctan(1/18) + 8 * arctan(1/57) - 5 * arctan(1/239) = pi/4*)
5508 subsection \<open>Introducing the inverse tangent power series\<close>
5510 lemma monoseq_arctan_series:
5511   fixes x :: real
5512   assumes "\<bar>x\<bar> \<le> 1"
5513   shows "monoseq (\<lambda>n. 1 / real (n * 2 + 1) * x^(n * 2 + 1))"
5514     (is "monoseq ?a")
5515 proof (cases "x = 0")
5516   case True
5517   then show ?thesis by (auto simp: monoseq_def)
5518 next
5519   case False
5520   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x"
5521     using assms by auto
5522   show "monoseq ?a"
5523   proof -
5524     have mono: "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le>
5525         1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
5526       if "0 \<le> x" and "x \<le> 1" for n and x :: real
5527     proof (rule mult_mono)
5528       show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))"
5529         by (rule frac_le) simp_all
5530       show "0 \<le> 1 / real (Suc (n * 2))"
5531         by auto
5532       show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)"
5533         by (rule power_decreasing) (simp_all add: \<open>0 \<le> x\<close> \<open>x \<le> 1\<close>)
5534       show "0 \<le> x ^ Suc (Suc n * 2)"
5535         by (rule zero_le_power) (simp add: \<open>0 \<le> x\<close>)
5536     qed
5537     show ?thesis
5538     proof (cases "0 \<le> x")
5539       case True
5540       from mono[OF this \<open>x \<le> 1\<close>, THEN allI]
5541       show ?thesis
5542         unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI2)
5543     next
5544       case False
5545       then have "0 \<le> - x" and "- x \<le> 1"
5546         using \<open>-1 \<le> x\<close> by auto
5547       from mono[OF this]
5548       have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge>
5549           1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" for n
5550         using \<open>0 \<le> -x\<close> by auto
5551       then show ?thesis
5552         unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
5553     qed
5554   qed
5555 qed
5557 lemma zeroseq_arctan_series:
5558   fixes x :: real
5559   assumes "\<bar>x\<bar> \<le> 1"
5560   shows "(\<lambda>n. 1 / real (n * 2 + 1) * x^(n * 2 + 1)) \<longlonglongrightarrow> 0"
5561     (is "?a \<longlonglongrightarrow> 0")
5562 proof (cases "x = 0")
5563   case True
5564   then show ?thesis by simp
5565 next
5566   case False
5567   have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x"
5568     using assms by auto
5569   show "?a \<longlonglongrightarrow> 0"
5570   proof (cases "\<bar>x\<bar> < 1")
5571     case True
5572     then have "norm x < 1" by auto
5573     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF \<open>norm x < 1\<close>, THEN LIMSEQ_Suc]]
5574     have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) \<longlonglongrightarrow> 0"
5575       unfolding inverse_eq_divide Suc_eq_plus1 by simp
5576     then show ?thesis
5577       using pos2 by (rule LIMSEQ_linear)
5578   next
5579     case False
5580     then have "x = -1 \<or> x = 1"
5581       using \<open>\<bar>x\<bar> \<le> 1\<close> by auto
5582     then have n_eq: "\<And> n. x ^ (n * 2 + 1) = x"
5583       unfolding One_nat_def by auto
5584     from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
5585     show ?thesis
5586       unfolding n_eq Suc_eq_plus1 by auto
5587   qed
5588 qed
5590 lemma summable_arctan_series:
5591   fixes n :: nat
5592   assumes "\<bar>x\<bar> \<le> 1"
5593   shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
5594     (is "summable (?c x)")
5595   by (rule summable_Leibniz(1),
5596       rule zeroseq_arctan_series[OF assms],
5597       rule monoseq_arctan_series[OF assms])
5599 lemma DERIV_arctan_series:
5600   assumes "\<bar>x\<bar> < 1"
5601   shows "DERIV (\<lambda>x'. \<Sum>k. (-1)^k * (1 / real (k * 2 + 1) * x' ^ (k * 2 + 1))) x :>
5602       (\<Sum>k. (-1)^k * x^(k * 2))"
5603     (is "DERIV ?arctan _ :> ?Int")
5604 proof -
5605   let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
5607   have n_even: "even n \<Longrightarrow> 2 * (n div 2) = n" for n :: nat
5608     by presburger
5609   then have if_eq: "?f n * real (Suc n) * x'^n =
5610       (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)"
5611     for n x'
5612     by auto
5614   have summable_Integral: "summable (\<lambda> n. (- 1) ^ n * x^(2 * n))" if "\<bar>x\<bar> < 1" for x :: real
5615   proof -
5616     from that have "x\<^sup>2 < 1"
5618     have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)"
5619       by (rule summable_Leibniz(1))
5620         (auto intro!: LIMSEQ_realpow_zero monoseq_realpow \<open>x\<^sup>2 < 1\<close> order_less_imp_le[OF \<open>x\<^sup>2 < 1\<close>])
5621     then show ?thesis
5622       by (simp only: power_mult)
5623   qed
5625   have sums_even: "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)"
5626     for f :: "nat \<Rightarrow> real"
5627   proof -
5628     have "f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x" for x :: real
5629     proof
5630       assume "f sums x"
5631       from sums_if[OF sums_zero this] show "(\<lambda>n. if even n then f (n div 2) else 0) sums x"
5632         by auto
5633     next
5634       assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
5635       from LIMSEQ_linear[OF this[simplified sums_def] pos2, simplified sum_split_even_odd[simplified mult.commute]]
5636       show "f sums x"
5637         unfolding sums_def by auto
5638     qed
5639     then show ?thesis ..
5640   qed
5642   have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int"
5643     unfolding if_eq mult.commute[of _ 2]
5644       suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * x ^ (2 * n)", symmetric]
5645     by auto
5647   have arctan_eq: "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x" for x
5648   proof -
5649     have if_eq': "\<And>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
5650       (if even n then (- 1) ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
5651       using n_even by auto
5652     have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)"
5653       by auto
5654     then show ?thesis
5655       unfolding if_eq' idx_eq suminf_def
5656         sums_even[of "\<lambda> n. (- 1) ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
5657       by auto
5658   qed
5660   have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum>n. ?f n * real (Suc n) * x^n)"
5661   proof (rule DERIV_power_series')
5662     show "x \<in> {- 1 <..< 1}"
5663       using \<open>\<bar> x \<bar> < 1\<close> by auto
5664     show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)"
5665       if x'_bounds: "x' \<in> {- 1 <..< 1}" for x' :: real
5666     proof -
5667       from that have "\<bar>x'\<bar> < 1" by auto
5668       then have *: "summable (\<lambda>n. (- 1) ^ n * x' ^ (2 * n))"
5669         by (rule summable_Integral)
5670       show ?thesis
5671         unfolding if_eq
5672         apply (rule sums_summable [where l="0 + (\<Sum>n. (-1)^n * x'^(2 * n))"])
5673         apply (rule sums_if)
5674          apply (rule sums_zero)
5675         apply (rule summable_sums)
5676         apply (rule *)
5677         done
5678     qed
5679   qed auto
5680   then show ?thesis
5681     by (simp only: Int_eq arctan_eq)
5682 qed
5684 lemma arctan_series:
5685   assumes "\<bar>x\<bar> \<le> 1"
5686   shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))"
5687     (is "_ = suminf (\<lambda> n. ?c x n)")
5688 proof -
5689   let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)"
5691   have DERIV_arctan_suminf: "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))"
5692     if "0 < r" and "r < 1" and "\<bar>x\<bar> < r" for r x :: real
5693   proof (rule DERIV_arctan_series)
5694     from that show "\<bar>x\<bar> < 1"
5695       using \<open>r < 1\<close> and \<open>\<bar>x\<bar> < r\<close> by auto
5696   qed
5698   {
5699     fix x :: real
5700     assume "\<bar>x\<bar> \<le> 1"
5701     note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]]
5702   } note arctan_series_borders = this
5704   have when_less_one: "arctan x = (\<Sum>k. ?c x k)" if "\<bar>x\<bar> < 1" for x :: real
5705   proof -
5706     obtain r where "\<bar>x\<bar> < r" and "r < 1"
5707       using dense[OF \<open>\<bar>x\<bar> < 1\<close>] by blast
5708     then have "0 < r" and "- r < x" and "x < r" by auto
5710     have suminf_eq_arctan_bounded: "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
5711       if "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b" for x a b
5712     proof -
5713       from that have "\<bar>x\<bar> < r" by auto
5714       show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
5715       proof (rule DERIV_isconst2[of "a" "b"])
5716         show "a < b" and "a \<le> x" and "x \<le> b"
5717           using \<open>a < b\<close> \<open>a \<le> x\<close> \<open>x \<le> b\<close> by auto
5718         have "\<forall>x. - r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
5719         proof (rule allI, rule impI)
5720           fix x
5721           assume "-r < x \<and> x < r"
5722           then have "\<bar>x\<bar> < r" by auto
5723           with \<open>r < 1\<close> have "\<bar>x\<bar> < 1" by auto
5724           have "\<bar>- (x\<^sup>2)\<bar> < 1" using abs_square_less_1 \<open>\<bar>x\<bar> < 1\<close> by auto
5725           then have "(\<lambda>n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
5726             unfolding real_norm_def[symmetric] by (rule geometric_sums)
5727           then have "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"
5728             unfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2] by auto
5729           then have suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)"
5730             using sums_unique unfolding inverse_eq_divide by auto
5731           have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))"
5732             unfolding suminf_c'_eq_geom
5733             by (rule DERIV_arctan_suminf[OF \<open>0 < r\<close> \<open>r < 1\<close> \<open>\<bar>x\<bar> < r\<close>])
5734           from DERIV_diff [OF this DERIV_arctan] show "DERIV (\<lambda>x. suminf (?c x) - arctan x) x :> 0"
5735             by auto
5736         qed
5737         then have DERIV_in_rball: "\<forall>y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda>x. suminf (?c x) - arctan x) y :> 0"
5738           using \<open>-r < a\<close> \<open>b < r\<close> by auto
5739         then show "\<forall>y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda>x. suminf (?c x) - arctan x) y :> 0"
5740           using \<open>\<bar>x\<bar> < r\<close> by auto
5741         show "\<forall>y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda>x. suminf (?c x) - arctan x) y"
5742           using DERIV_in_rball DERIV_isCont by auto
5743       qed
5744     qed
5746     have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
5747       unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero
5748       by auto
5750     have "suminf (?c x) - arctan x = 0"
5751     proof (cases "x = 0")
5752       case True
5753       then show ?thesis
5754         using suminf_arctan_zero by auto
5755     next
5756       case False
5757       then have "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>"
5758         by auto
5759       have "suminf (?c (- \<bar>x\<bar>)) - arctan (- \<bar>x\<bar>) = suminf (?c 0) - arctan 0"
5760         by (rule suminf_eq_arctan_bounded[where x1="0" and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>", symmetric])
5761           (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>-\<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less)
5762       moreover
5763       have "suminf (?c x) - arctan x = suminf (?c (- \<bar>x\<bar>)) - arctan (- \<bar>x\<bar>)"
5764         by (rule suminf_eq_arctan_bounded[where x1="x" and a1="- \<bar>x\<bar>" and b1="\<bar>x\<bar>"])
5765            (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>- \<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less)
5766       ultimately show ?thesis
5767         using suminf_arctan_zero by auto
5768     qed
5769     then show ?thesis by auto
5770   qed
5772   show "arctan x = suminf (\<lambda>n. ?c x n)"
5773   proof (cases "\<bar>x\<bar> < 1")
5774     case True
5775     then show ?thesis by (rule when_less_one)
5776   next
5777     case False
5778     then have "\<bar>x\<bar> = 1" using \<open>\<bar>x\<bar> \<le> 1\<close> by auto
5779     let ?a = "\<lambda>x n. \<bar>1 / real (n * 2 + 1) * x^(n * 2 + 1)\<bar>"
5780     let ?diff = "\<lambda>x n. \<bar>arctan x - (\<Sum>i<n. ?c x i)\<bar>"
5781     have "?diff 1 n \<le> ?a 1 n" for n :: nat
5782     proof -
5783       have "0 < (1 :: real)" by auto
5784       moreover
5785       have "?diff x n \<le> ?a x n" if "0 < x" and "x < 1" for x :: real
5786       proof -
5787         from that have "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1"
5788           by auto
5789         from \<open>0 < x\<close> have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)"
5790           by auto
5791         note bounds = mp[OF arctan_series_borders(2)[OF \<open>\<bar>x\<bar> \<le> 1\<close>] this, unfolded when_less_one[OF \<open>\<bar>x\<bar> < 1\<close>, symmetric], THEN spec]
5792         have "0 < 1 / real (n*2+1) * x^(n*2+1)"
5793           by (rule mult_pos_pos) (simp_all only: zero_less_power[OF \<open>0 < x\<close>], auto)
5794         then have a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)"
5795           by (rule abs_of_pos)
5796         show ?thesis
5797         proof (cases "even n")
5798           case True
5799           then have sgn_pos: "(-1)^n = (1::real)" by auto
5800           from \<open>even n\<close> obtain m where "n = 2 * m" ..
5801           then have "2 * m = n" ..
5802           from bounds[of m, unfolded this atLeastAtMost_iff]
5803           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))"
5804             by auto
5805           also have "\<dots> = ?c x n" by auto
5806           also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
5807           finally show ?thesis .
5808         next
5809           case False
5810           then have sgn_neg: "(-1)^n = (-1::real)" by auto
5811           from \<open>odd n\<close> obtain m where "n = 2 * m + 1" ..
5812           then have m_def: "2 * m + 1 = n" ..
5813           then have m_plus: "2 * (m + 1) = n + 1" by auto
5814           from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
5815           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))" by auto
5816           also have "\<dots> = - ?c x n" by auto
5817           also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
5818           finally show ?thesis .
5819         qed
5820       qed
5821       hence "\<forall>x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
5822       moreover have "isCont (\<lambda> x. ?a x n - ?diff x n) x" for x
5824         by (auto intro!: isCont_add isCont_rabs continuous_ident isCont_minus isCont_arctan
5825           isCont_inverse isCont_mult isCont_power continuous_const isCont_sum
5827       ultimately have "0 \<le> ?a 1 n - ?diff 1 n"
5828         by (rule LIM_less_bound)
5829       then show ?thesis by auto
5830     qed
5831     have "?a 1 \<longlonglongrightarrow> 0"
5832       unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
5833       by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat simp del: of_nat_Suc)
5834     have "?diff 1 \<longlonglongrightarrow> 0"
5835     proof (rule LIMSEQ_I)
5836       fix r :: real
5837       assume "0 < r"
5838       obtain N :: nat where N_I: "N \<le> n \<Longrightarrow> ?a 1 n < r" for n
5839         using LIMSEQ_D[OF \<open>?a 1 \<longlonglongrightarrow> 0\<close> \<open>0 < r\<close>] by auto
5840       have "norm (?diff 1 n - 0) < r" if "N \<le> n" for n
5841         using \<open>?diff 1 n \<le> ?a 1 n\<close> N_I[OF that] by auto
5842       then show "\<exists>N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
5843     qed
5844     from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
5845     have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
5846     then have "arctan 1 = (\<Sum>i. ?c 1 i)" by (rule sums_unique)
5848     show ?thesis
5849     proof (cases "x = 1")
5850       case True
5851       then show ?thesis by (simp add: \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close>)
5852     next
5853       case False
5854       then have "x = -1" using \<open>\<bar>x\<bar> = 1\<close> by auto
5856       have "- (pi / 2) < 0" using pi_gt_zero by auto
5857       have "- (2 * pi) < 0" using pi_gt_zero by auto
5859       have c_minus_minus: "?c (- 1) i = - ?c 1 i" for i by auto
5861       have "arctan (- 1) = arctan (tan (-(pi / 4)))"
5862         unfolding tan_45 tan_minus ..
5863       also have "\<dots> = - (pi / 4)"
5864         by (rule arctan_tan) (auto simp: order_less_trans[OF \<open>- (pi / 2) < 0\<close> pi_gt_zero])
5865       also have "\<dots> = - (arctan (tan (pi / 4)))"
5866         unfolding neg_equal_iff_equal
5867         by (rule arctan_tan[symmetric]) (auto simp: order_less_trans[OF \<open>- (2 * pi) < 0\<close> pi_gt_zero])
5868       also have "\<dots> = - (arctan 1)"
5869         unfolding tan_45 ..
5870       also have "\<dots> = - (\<Sum> i. ?c 1 i)"
5871         using \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close> by auto
5872       also have "\<dots> = (\<Sum> i. ?c (- 1) i)"
5873         using suminf_minus[OF sums_summable[OF \<open>(?c 1) sums (arctan 1)\<close>]]
5874         unfolding c_minus_minus by auto
5875       finally show ?thesis using \<open>x = -1\<close> by auto
5876     qed
5877   qed
5878 qed
5880 lemma arctan_half: "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))"
5881   for x :: real
5882 proof -
5883   obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x"
5884     using tan_total by blast
5885   then have low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2"
5886     by auto
5888   have "0 < cos y" by (rule cos_gt_zero_pi[OF low high])
5889   then have "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y"
5890     by auto
5892   have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
5893     unfolding tan_def power_divide ..
5894   also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
5895     using \<open>cos y \<noteq> 0\<close> by auto
5896   also have "\<dots> = 1 / (cos y)\<^sup>2"
5898   finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" .
5900   have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)"
5901     unfolding tan_def using \<open>cos y \<noteq> 0\<close> by (simp add: field_simps)
5902   also have "\<dots> = tan y / (1 + 1 / cos y)"
5903     using \<open>cos y \<noteq> 0\<close> unfolding add_divide_distrib by auto
5904   also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))"
5905     unfolding cos_sqrt ..
5906   also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))"
5907     unfolding real_sqrt_divide by auto
5908   finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))"
5909     unfolding \<open>1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2\<close> .
5911   have "arctan x = y"
5912     using arctan_tan low high y_eq by auto
5913   also have "\<dots> = 2 * (arctan (tan (y/2)))"
5914     using arctan_tan[OF low2 high2] by auto
5915   also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))"
5916     unfolding tan_half by auto
5917   finally show ?thesis
5918     unfolding eq \<open>tan y = x\<close> .
5919 qed
5921 lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y"
5922   by (simp only: arctan_less_iff)
5924 lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y"
5925   by (simp only: arctan_le_iff)
5927 lemma arctan_inverse:
5928   assumes "x \<noteq> 0"
5929   shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
5930 proof (rule arctan_unique)
5931   show "- (pi / 2) < sgn x * pi / 2 - arctan x"
5932     using arctan_bounded [of x] assms
5933     unfolding sgn_real_def
5934     apply (auto simp add: arctan algebra_simps)
5935     apply (drule zero_less_arctan_iff [THEN iffD2])
5936     apply arith
5937     done
5938   show "sgn x * pi / 2 - arctan x < pi / 2"
5939     using arctan_bounded [of "- x"] assms
5940     unfolding sgn_real_def arctan_minus
5941     by (auto simp add: algebra_simps)
5942   show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
5943     unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
5944     unfolding sgn_real_def
5945     by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
5946 qed
5948 theorem pi_series: "pi / 4 = (\<Sum>k. (-1)^k * 1 / real (k * 2 + 1))"
5949   (is "_ = ?SUM")
5950 proof -
5951   have "pi / 4 = arctan 1"
5952     using arctan_one by auto
5953   also have "\<dots> = ?SUM"
5954     using arctan_series[of 1] by auto
5955   finally show ?thesis by auto
5956 qed
5959 subsection \<open>Existence of Polar Coordinates\<close>
5961 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1"
5962   by (rule power2_le_imp_le [OF _ zero_le_one])
5963     (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
5965 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
5967 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
5969 lemma polar_Ex: "\<exists>r::real. \<exists>a. x = r * cos a \<and> y = r * sin a"
5970 proof -
5971   have polar_ex1: "0 < y \<Longrightarrow> \<exists>r a. x = r * cos a \<and> y = r * sin a" for y
5972     apply (rule exI [where x = "sqrt (x\<^sup>2 + y\<^sup>2)"])
5973     apply (rule exI [where x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))"])
5974     apply (simp add: cos_arccos_lemma1 sin_arccos_lemma1 power_divide
5975         real_sqrt_mult [symmetric] right_diff_distrib)
5976     done
5977   show ?thesis
5978   proof (cases "0::real" y rule: linorder_cases)
5979     case less
5980     then show ?thesis
5981       by (rule polar_ex1)
5982   next
5983     case equal
5984     then show ?thesis
5985       by (force simp add: intro!: cos_zero sin_zero)
5986   next
5987     case greater
5988     with polar_ex1 [where y="-y"] show ?thesis
5989       by auto (metis cos_minus minus_minus minus_mult_right sin_minus)
5990   qed
5991 qed
5994 subsection \<open>Basics about polynomial functions: products, extremal behaviour and root counts\<close>
5996 lemma pairs_le_eq_Sigma: "{(i, j). i + j \<le> m} = Sigma (atMost m) (\<lambda>r. atMost (m - r))"
5997   for m :: nat
5998   by auto
6000 lemma sum_up_index_split: "(\<Sum>k\<le>m + n. f k) = (\<Sum>k\<le>m. f k) + (\<Sum>k = Suc m..m + n. f k)"
6001   by (metis atLeast0AtMost Suc_eq_plus1 le0 sum_ub_add_nat)
6003 lemma Sigma_interval_disjoint: "(SIGMA i:A. {..v i}) \<inter> (SIGMA i:A.{v i<..w}) = {}"
6004   for w :: "'a::order"
6005   by auto
6007 lemma product_atMost_eq_Un: "A \<times> {..m} = (SIGMA i:A.{..m - i}) \<union> (SIGMA i:A.{m - i<..m})"
6008   for m :: nat
6009   by auto
6011 lemma polynomial_product: (*with thanks to Chaitanya Mangla*)
6012   fixes x :: "'a::idom"
6013   assumes m: "\<And>i. i > m \<Longrightarrow> a i = 0"
6014     and n: "\<And>j. j > n \<Longrightarrow> b j = 0"
6015   shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
6016     (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
6017 proof -
6018   have "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) = (\<Sum>i\<le>m. \<Sum>j\<le>n. (a i * x ^ i) * (b j * x ^ j))"
6019     by (rule sum_product)
6020   also have "\<dots> = (\<Sum>i\<le>m + n. \<Sum>j\<le>n + m. a i * x ^ i * (b j * x ^ j))"
6021     using assms by (auto simp: sum_up_index_split)
6022   also have "\<dots> = (\<Sum>r\<le>m + n. \<Sum>j\<le>m + n - r. a r * x ^ r * (b j * x ^ j))"
6024     apply (clarsimp simp add: sum_Un Sigma_interval_disjoint intro!: sum.neutral)
6026     done
6027   also have "\<dots> = (\<Sum>(i,j)\<in>{(i,j). i+j \<le> m+n}. (a i * x ^ i) * (b j * x ^ j))"
6028     by (auto simp: pairs_le_eq_Sigma sum.Sigma)
6029   also have "\<dots> = (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
6030     apply (subst sum_triangle_reindex_eq)
6031     apply (auto simp: algebra_simps sum_distrib_left intro!: sum.cong)
6033     done
6034   finally show ?thesis .
6035 qed
6037 lemma polynomial_product_nat:
6038   fixes x :: nat
6039   assumes m: "\<And>i. i > m \<Longrightarrow> a i = 0"
6040     and n: "\<And>j. j > n \<Longrightarrow> b j = 0"
6041   shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
6042     (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
6043   using polynomial_product [of m a n b x] assms
6044   by (simp only: of_nat_mult [symmetric] of_nat_power [symmetric]
6045       of_nat_eq_iff Int.int_sum [symmetric])
6047 lemma polyfun_diff: (*COMPLEX_SUB_POLYFUN in HOL Light*)
6048   fixes x :: "'a::idom"
6049   assumes "1 \<le> n"
6050   shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =