1 (* Title: HOL/Transcendental.thy
2 Author: Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
3 Author: Lawrence C Paulson
6 header{*Power Series, Transcendental Functions etc.*}
9 imports Fact Series Deriv NthRoot
12 subsection {* Properties of Power Series *}
14 lemma lemma_realpow_diff:
15 fixes y :: "'a::monoid_mult"
16 shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
19 hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
20 thus ?thesis by (simp add: power_commutes)
23 lemma lemma_realpow_diff_sumr:
24 fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows
25 "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
26 y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
27 by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac
28 del: setsum_op_ivl_Suc)
30 lemma lemma_realpow_diff_sumr2:
31 fixes y :: "'a::{comm_ring,monoid_mult}" shows
32 "x ^ (Suc n) - y ^ (Suc n) =
33 (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
34 apply (induct n, simp)
35 apply (simp del: setsum_op_ivl_Suc)
36 apply (subst setsum_op_ivl_Suc)
37 apply (subst lemma_realpow_diff_sumr)
38 apply (simp add: right_distrib del: setsum_op_ivl_Suc)
39 apply (subst mult_left_commute [of "x - y"])
41 apply (simp add: algebra_simps)
44 lemma lemma_realpow_rev_sumr:
45 "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
46 (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
47 apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
48 apply (rule inj_onI, simp)
50 apply (rule_tac x="n - x" in image_eqI, simp, simp)
53 text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
54 x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
57 fixes x z :: "'a::{real_normed_field,banach}"
58 assumes 1: "summable (\<lambda>n. f n * x ^ n)"
59 assumes 2: "norm z < norm x"
60 shows "summable (\<lambda>n. norm (f n * z ^ n))"
62 from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
63 from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
64 by (rule summable_LIMSEQ_zero)
65 hence "convergent (\<lambda>n. f n * x ^ n)"
67 hence "Cauchy (\<lambda>n. f n * x ^ n)"
68 by (simp add: Cauchy_convergent_iff)
69 hence "Bseq (\<lambda>n. f n * x ^ n)"
71 then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
72 by (simp add: Bseq_def, safe)
73 have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
74 K * norm (z ^ n) * inverse (norm (x ^ n))"
75 proof (intro exI allI impI)
76 fix n::nat assume "0 \<le> n"
77 have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
78 norm (f n * x ^ n) * norm (z ^ n)"
79 by (simp add: norm_mult abs_mult)
80 also have "\<dots> \<le> K * norm (z ^ n)"
81 by (simp only: mult_right_mono 4 norm_ge_zero)
82 also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
83 by (simp add: x_neq_0)
84 also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
85 by (simp only: mult_assoc)
86 finally show "norm (norm (f n * z ^ n)) \<le>
87 K * norm (z ^ n) * inverse (norm (x ^ n))"
88 by (simp add: mult_le_cancel_right x_neq_0)
90 moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
92 from 2 have "norm (norm (z * inverse x)) < 1"
94 by (simp add: nonzero_norm_divide divide_inverse [symmetric])
95 hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
96 by (rule summable_geometric)
97 hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
98 by (rule summable_mult)
99 thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
101 by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
102 power_inverse norm_power mult_assoc)
104 ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
105 by (rule summable_comparison_test)
109 fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" shows
110 "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]
111 ==> summable (%n. f(n) * (z ^ n))"
112 by (rule powser_insidea [THEN summable_norm_cancel])
114 lemma sum_split_even_odd: fixes f :: "nat \<Rightarrow> real" shows
115 "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
116 (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
119 have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =
120 (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
121 using Suc.hyps unfolding One_nat_def by auto
122 also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))" by auto
126 lemma sums_if': fixes g :: "nat \<Rightarrow> real" assumes "g sums x"
127 shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
129 proof (rule LIMSEQ_I)
130 fix r :: real assume "0 < r"
131 from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
132 obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast
134 let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
135 { fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> no" by auto
136 have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
137 using sum_split_even_odd by auto
138 hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
140 have "?SUM (2 * (m div 2)) = ?SUM m"
141 proof (cases "even m")
142 case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
144 case False hence "even (Suc m)" by auto
145 from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]] odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]
146 have eq: "Suc (2 * (m div 2)) = m" by auto
147 hence "even (2 * (m div 2))" using `odd m` by auto
148 have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
149 also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
150 finally show ?thesis by auto
152 ultimately have "(norm (?SUM m - x) < r)" by auto
154 thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
157 lemma sums_if: fixes g :: "nat \<Rightarrow> real" assumes "g sums x" and "f sums y"
158 shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
160 let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
161 { fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
162 by (cases B) auto } note if_sum = this
163 have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" using sums_if'[OF `g sums x`] .
165 have "?s 0 = 0" by auto
166 have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
167 have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
169 have "?s sums y" using sums_if'[OF `f sums y`] .
170 from this[unfolded sums_def, THEN LIMSEQ_Suc]
171 have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
172 unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
173 image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
174 even_Suc Suc_m1 if_eq .
175 } from sums_add[OF g_sums this]
176 show ?thesis unfolding if_sum .
179 subsection {* Alternating series test / Leibniz formula *}
181 lemma sums_alternating_upper_lower:
182 fixes a :: "nat \<Rightarrow> real"
183 assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
184 shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and>
185 ((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. -1^i*a i) ----> l)"
186 (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
188 have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
190 have "\<forall> n. ?f n \<le> ?f (Suc n)"
191 proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed
193 have "\<forall> n. ?g (Suc n) \<le> ?g n"
194 proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
195 unfolding One_nat_def by auto qed
197 have "\<forall> n. ?f n \<le> ?g n"
198 proof fix n show "?f n \<le> ?g n" using fg_diff a_pos
199 unfolding One_nat_def by auto qed
201 have "(\<lambda> n. ?f n - ?g n) ----> 0" unfolding fg_diff
202 proof (rule LIMSEQ_I)
203 fix r :: real assume "0 < r"
204 with `a ----> 0`[THEN LIMSEQ_D]
205 obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" by auto
206 hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
207 thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
210 show ?thesis by (rule lemma_nest_unique)
213 lemma summable_Leibniz': fixes a :: "nat \<Rightarrow> real"
214 assumes a_zero: "a ----> 0" and a_pos: "\<And> n. 0 \<le> a n"
215 and a_monotone: "\<And> n. a (Suc n) \<le> a n"
216 shows summable: "summable (\<lambda> n. (-1)^n * a n)"
217 and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
218 and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
219 and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
220 and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
222 let "?S n" = "(-1)^n * a n"
223 let "?P n" = "\<Sum>i=0..<n. ?S i"
224 let "?f n" = "?P (2 * n)"
225 let "?g n" = "?P (2 * n + 1)"
226 obtain l :: real where below_l: "\<forall> n. ?f n \<le> l" and "?f ----> l" and above_l: "\<forall> n. l \<le> ?g n" and "?g ----> l"
227 using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
229 let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n"
231 proof (rule LIMSEQ_I)
232 fix r :: real assume "0 < r"
234 with `?f ----> l`[THEN LIMSEQ_D]
235 obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
237 from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
238 obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
241 assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
242 have "norm (?Sa n - l) < r"
243 proof (cases "even n")
244 case True from even_nat_div_two_times_two[OF this]
245 have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto
246 with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" by auto
248 show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
250 case False hence "even (n - 1)" by simp
251 from even_nat_div_two_times_two[OF this]
252 have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto
253 hence range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto
255 from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" by auto
257 show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
260 thus "\<exists> no. \<forall> n \<ge> no. norm (?Sa n - l) < r" by blast
262 hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
263 thus "summable ?S" using summable_def by auto
265 have "l = suminf ?S" using sums_unique[OF sums_l] .
267 { fix n show "suminf ?S \<le> ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto }
268 { fix n show "?f n \<le> suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_l by auto }
269 show "?g ----> suminf ?S" using `?g ----> l` `l = suminf ?S` by auto
270 show "?f ----> suminf ?S" using `?f ----> l` `l = suminf ?S` by auto
273 theorem summable_Leibniz: fixes a :: "nat \<Rightarrow> real"
274 assumes a_zero: "a ----> 0" and "monoseq a"
275 shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
276 and "0 < a 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n. -1^i * a i .. \<Sum>i=0..<2*n+1. -1^i * a i})" (is "?pos")
277 and "a 0 < 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n+1. -1^i * a i .. \<Sum>i=0..<2*n. -1^i * a i})" (is "?neg")
278 and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
279 and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
281 have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
282 proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
284 hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" by auto
285 { fix n have "a (Suc n) \<le> a n" using ord[where n="Suc n" and m=n] by auto }
286 note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] and mono = this
287 from leibniz[OF mono]
288 show ?thesis using `0 \<le> a 0` by auto
290 let ?a = "\<lambda> n. - a n"
292 with monoseq_le[OF `monoseq a` `a ----> 0`]
293 have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
294 hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" by auto
295 { fix n have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] by auto }
297 note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF LIMSEQ_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
298 have "summable (\<lambda> n. (-1)^n * ?a n)" using leibniz(1) by auto
299 then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" unfolding summable_def by auto
300 from this[THEN sums_minus]
301 have "(\<lambda> n. (-1)^n * a n) sums -l" by auto
302 hence ?summable unfolding summable_def by auto
304 have "\<And> a b :: real. \<bar> - a - - b \<bar> = \<bar>a - b\<bar>" unfolding minus_diff_minus by auto
306 from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
307 have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)" by auto
309 have ?pos using `0 \<le> ?a 0` by auto
310 moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto
311 moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN LIMSEQ_minus_cancel] by auto
312 ultimately show ?thesis by auto
314 from this[THEN conjunct1] this[THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1]
315 this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]
316 show ?summable and ?pos and ?neg and ?f and ?g .
319 subsection {* Term-by-Term Differentiability of Power Series *}
322 diffs :: "(nat => 'a::ring_1) => nat => 'a" where
323 "diffs c = (%n. of_nat (Suc n) * c(Suc n))"
325 text{*Lemma about distributing negation over it*}
326 lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
327 by (simp add: diffs_def)
331 shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
333 apply (rule LIMSEQ_imp_Suc)
334 apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
335 apply (simp only: setsum_shift_bounds_Suc_ivl)
339 fixes x :: "'a::{real_normed_vector, ring_1}"
340 shows "summable (%n. (diffs c)(n) * (x ^ n)) ==>
341 (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
342 (\<Sum>n. (diffs c)(n) * (x ^ n))"
344 apply (drule summable_sums)
345 apply (rule sums_Suc_imp, simp_all)
348 lemma lemma_termdiff1:
349 fixes z :: "'a :: {monoid_mult,comm_ring}" shows
350 "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
351 (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
352 by(auto simp add: algebra_simps power_add [symmetric])
354 lemma sumr_diff_mult_const2:
355 "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
356 by (simp add: setsum_subtractf)
358 lemma lemma_termdiff2:
359 fixes h :: "'a :: {field}"
360 assumes h: "h \<noteq> 0" shows
361 "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
362 h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
363 (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
364 apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
365 apply (simp add: right_diff_distrib diff_divide_distrib h)
366 apply (simp add: mult_assoc [symmetric])
367 apply (cases "n", simp)
368 apply (simp add: lemma_realpow_diff_sumr2 h
369 right_diff_distrib [symmetric] mult_assoc
370 del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
371 apply (subst lemma_realpow_rev_sumr)
372 apply (subst sumr_diff_mult_const2)
374 apply (simp only: lemma_termdiff1 setsum_right_distrib)
375 apply (rule setsum_cong [OF refl])
376 apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
378 apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
379 del: setsum_op_ivl_Suc power_Suc)
380 apply (subst mult_assoc [symmetric], subst power_add [symmetric])
381 apply (simp add: mult_ac)
384 lemma real_setsum_nat_ivl_bounded2:
385 fixes K :: "'a::linordered_semidom"
386 assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
387 assumes K: "0 \<le> K"
388 shows "setsum f {0..<n-k} \<le> of_nat n * K"
389 apply (rule order_trans [OF setsum_mono])
391 apply (simp add: mult_right_mono K)
394 lemma lemma_termdiff3:
395 fixes h z :: "'a::{real_normed_field}"
396 assumes 1: "h \<noteq> 0"
397 assumes 2: "norm z \<le> K"
398 assumes 3: "norm (z + h) \<le> K"
399 shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
400 \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
402 have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
403 norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
404 (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
405 apply (subst lemma_termdiff2 [OF 1])
406 apply (subst norm_mult)
407 apply (rule mult_commute)
409 also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
410 proof (rule mult_right_mono [OF _ norm_ge_zero])
411 from norm_ge_zero 2 have K: "0 \<le> K" by (rule order_trans)
412 have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
414 apply (simp only: norm_mult norm_power power_add)
415 apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
417 show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
418 (z + h) ^ q * z ^ (n - 2 - q))
419 \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
421 order_trans [OF norm_setsum]
422 real_setsum_nat_ivl_bounded2
426 apply (rule le_Kn, simp)
429 also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
430 by (simp only: mult_assoc)
431 finally show ?thesis .
434 lemma lemma_termdiff4:
435 fixes f :: "'a::{real_normed_field} \<Rightarrow>
436 'b::real_normed_vector"
437 assumes k: "0 < (k::real)"
438 assumes le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
440 unfolding LIM_eq diff_0_right
442 let ?h = "of_real (k / 2)::'a"
443 have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
444 hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
445 hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
446 hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
448 fix r::real assume r: "0 < r"
449 show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
452 with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
454 thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
456 assume K_neq_zero: "K \<noteq> 0"
457 with zero_le_K have K: "0 < K" by simp
458 show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
459 proof (rule exI, safe)
460 from k r K show "0 < min k (r * inverse K / 2)"
461 by (simp add: mult_pos_pos positive_imp_inverse_positive)
464 assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
465 from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
467 from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
468 also from x4 K have "K * norm x < K * (r * inverse K / 2)"
469 by (rule mult_strict_left_mono)
470 also have "\<dots> = r / 2"
471 using K_neq_zero by simp
472 also have "r / 2 < r"
474 finally show "norm (f x) < r" .
479 lemma lemma_termdiff5:
480 fixes g :: "'a::{real_normed_field} \<Rightarrow>
481 nat \<Rightarrow> 'b::banach"
482 assumes k: "0 < (k::real)"
483 assumes f: "summable f"
484 assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
485 shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
486 proof (rule lemma_termdiff4 [OF k])
487 fix h::'a assume "h \<noteq> 0" and "norm h < k"
488 hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
490 hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
492 moreover from f have B: "summable (\<lambda>n. f n * norm h)"
493 by (rule summable_mult2)
494 ultimately have C: "summable (\<lambda>n. norm (g h n))"
495 by (rule summable_comparison_test)
496 hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
497 by (rule summable_norm)
498 also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
499 by (rule summable_le)
500 also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
501 by (rule suminf_mult2 [symmetric])
502 finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
506 text{* FIXME: Long proofs*}
509 fixes x :: "'a::{real_normed_field,banach}"
510 assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
511 assumes 2: "norm x < norm K"
512 shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
513 - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
516 obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
517 from norm_ge_zero r1 have r: "0 < r"
518 by (rule order_le_less_trans)
519 hence r_neq_0: "r \<noteq> 0" by simp
521 proof (rule lemma_termdiff5)
522 show "0 < r - norm x" using r1 by simp
524 from r r2 have "norm (of_real r::'a) < norm K"
526 with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
527 by (rule powser_insidea)
528 hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
530 by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
531 hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
532 by (rule diffs_equiv [THEN sums_summable])
533 also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))
534 = (\<lambda>n. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
536 apply (simp add: diffs_def)
537 apply (case_tac n, simp_all add: r_neq_0)
539 finally have "summable
540 (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
541 by (rule diffs_equiv [THEN sums_summable])
543 "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
545 (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
547 apply (case_tac "n", simp)
548 apply (case_tac "nat", simp)
549 apply (simp add: r_neq_0)
552 "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
555 assume h: "h \<noteq> 0"
556 assume "norm h < r - norm x"
557 hence "norm x + norm h < r" by simp
558 with norm_triangle_ineq have xh: "norm (x + h) < r"
559 by (rule order_le_less_trans)
560 show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
561 \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
562 apply (simp only: norm_mult mult_assoc)
563 apply (rule mult_left_mono [OF _ norm_ge_zero])
564 apply (simp (no_asm) add: mult_assoc [symmetric])
565 apply (rule lemma_termdiff3)
567 apply (rule r1 [THEN order_less_imp_le])
568 apply (rule xh [THEN order_less_imp_le])
574 fixes K x :: "'a::{real_normed_field,banach}"
575 assumes 1: "summable (\<lambda>n. c n * K ^ n)"
576 assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
577 assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
578 assumes 4: "norm x < norm K"
579 shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
581 proof (rule LIM_zero_cancel)
582 show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
583 - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
584 proof (rule LIM_equal2)
585 show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
588 assume "h \<noteq> 0"
589 assume "norm (h - 0) < norm K - norm x"
590 hence "norm x + norm h < norm K" by simp
591 hence 5: "norm (x + h) < norm K"
592 by (rule norm_triangle_ineq [THEN order_le_less_trans])
593 have A: "summable (\<lambda>n. c n * x ^ n)"
594 by (rule powser_inside [OF 1 4])
595 have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
596 by (rule powser_inside [OF 1 5])
597 have C: "summable (\<lambda>n. diffs c n * x ^ n)"
598 by (rule powser_inside [OF 2 4])
599 show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
600 - (\<Sum>n. diffs c n * x ^ n) =
601 (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
602 apply (subst sums_unique [OF diffs_equiv [OF C]])
603 apply (subst suminf_diff [OF B A])
604 apply (subst suminf_divide [symmetric])
605 apply (rule summable_diff [OF B A])
606 apply (subst suminf_diff)
607 apply (rule summable_divide)
608 apply (rule summable_diff [OF B A])
609 apply (rule sums_summable [OF diffs_equiv [OF C]])
610 apply (rule arg_cong [where f="suminf"], rule ext)
611 apply (simp add: algebra_simps)
614 show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h -
615 of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
616 by (rule termdiffs_aux [OF 3 4])
621 subsection {* Derivability of power series *}
623 lemma DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
624 assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
625 and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
626 and "summable (f' x0)"
627 and "summable L" and L_def: "\<And> n x y. \<lbrakk> x \<in> { a <..< b} ; y \<in> { a <..< b} \<rbrakk> \<Longrightarrow> \<bar> f x n - f y n \<bar> \<le> L n * \<bar> x - y \<bar>"
628 shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
631 fix r :: real assume "0 < r" hence "0 < r/3" by auto
633 obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
634 using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
636 obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
637 using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
639 let ?N = "Suc (max N_L N_f')"
640 have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
641 L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
643 let "?diff i x" = "(f (x0 + x) i - f x0 i) / x"
645 let ?r = "r / (3 * real ?N)"
646 have "0 < 3 * real ?N" by auto
647 from divide_pos_pos[OF `0 < r` this]
650 let "?s n" = "SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"
651 def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
653 have "0 < S'" unfolding S'_def
654 proof (rule iffD2[OF Min_gr_iff])
655 show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x"
657 fix x assume "x \<in> ?s ` {0..<?N}"
658 then obtain n where "x = ?s n" and "n \<in> {0..<?N}" using image_iff[THEN iffD1] by blast
659 from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
660 obtain s where s_bound: "0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r)" by auto
661 have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)
662 thus "0 < x" unfolding `x = ?s n` .
666 def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
667 hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0" and "S \<le> S'" using x0_in_I and `0 < S'`
670 { fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
671 hence x_in_I: "x0 + x \<in> { a <..< b }" using S_a S_b by auto
673 note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
674 note div_smbl = summable_divide[OF diff_smbl]
675 note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
676 note ign = summable_ignore_initial_segment[where k="?N"]
677 note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
678 note div_shft_smbl = summable_divide[OF diff_shft_smbl]
679 note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
682 have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
683 using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .
684 hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 0` by auto
685 } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
686 from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
687 have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
688 hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") using L_estimate by auto
690 have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..
691 also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
692 proof (rule setsum_strict_mono)
693 fix n assume "n \<in> { 0 ..< ?N}"
694 have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .
695 also have "S \<le> S'" using `S \<le> S'` .
696 also have "S' \<le> ?s n" unfolding S'_def
697 proof (rule Min_le_iff[THEN iffD2])
698 have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto
699 thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
701 finally have "\<bar> x \<bar> < ?s n" .
703 from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
704 have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
705 with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n`
706 show "\<bar>?diff n x - f' x0 n\<bar> < ?r" by blast
708 also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)
709 also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto
710 also have "\<dots> = r/3" by auto
711 finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
713 from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
714 have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> =
715 \<bar> \<Sum>n. ?diff n x - f' x0 n \<bar>" unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] using suminf_divide[OF diff_smbl, symmetric] by auto
716 also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>" unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]] by (rule abs_triangle_ineq)
717 also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto
718 also have "\<dots> < r /3 + r/3 + r/3"
719 using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
720 by (rule add_strict_mono [OF add_less_le_mono])
721 finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"
723 } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
724 norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`
725 unfolding real_norm_def diff_0_right by blast
728 lemma DERIV_power_series': fixes f :: "nat \<Rightarrow> real"
729 assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
730 and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
731 shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
732 (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
734 { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
735 hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" by auto
736 have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
737 proof (rule DERIV_series')
738 show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
740 have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto
741 hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" using `R' < R` by auto
742 have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto
743 from powser_insidea[OF converges[OF in_Rball] this] show ?thesis by auto
745 { fix n x y assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
746 show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
748 have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> = (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
749 unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto
750 also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
751 proof (rule mult_left_mono)
752 have "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p = 0..<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)" by (rule setsum_abs)
753 also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
754 proof (rule setsum_mono)
755 fix p assume "p \<in> {0..<Suc n}" hence "p \<le> n" by auto
756 { fix n fix x :: real assume "x \<in> {-R'<..<R'}"
757 hence "\<bar>x\<bar> \<le> R'" by auto
758 hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto)
759 } from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
760 have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" unfolding abs_mult by auto
761 thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> n` by auto
763 also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
764 finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>" unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
765 show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult[symmetric] by auto
767 also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult mult_assoc[symmetric] by algebra
768 finally show ?thesis .
770 { fix n show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
771 by (auto intro!: DERIV_intros simp del: power_Suc) }
772 { fix x assume "x \<in> {-R' <..< R'}" hence "R' \<in> {-R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto
773 have "summable (\<lambda> n. f n * x^n)"
774 proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
776 have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto)
777 show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult
778 by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right])
780 from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
781 show "summable (?f x)" by auto }
782 show "summable (?f' x0)" using converges[OF `x0 \<in> {-R <..< R}`] .
783 show "x0 \<in> {-R' <..< R'}" using `x0 \<in> {-R' <..< R'}` .
785 } note for_subinterval = this
786 let ?R = "(R + \<bar>x0\<bar>) / 2"
787 have "\<bar>x0\<bar> < ?R" using assms by auto
789 proof (cases "x0 < 0")
791 hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
792 thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
795 have "- ?R < 0" using assms by auto
796 also have "\<dots> \<le> x0" using False by auto
797 finally show ?thesis .
799 hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto
800 from for_subinterval[OF this]
804 subsection {* Exponential Function *}
806 definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" where
807 "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
809 lemma summable_exp_generic:
810 fixes x :: "'a::{real_normed_algebra_1,banach}"
811 defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
814 have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
815 unfolding S_def by (simp del: mult_Suc)
816 obtain r :: real where r0: "0 < r" and r1: "r < 1"
817 using dense [OF zero_less_one] by fast
818 obtain N :: nat where N: "norm x < real N * r"
819 using reals_Archimedean3 [OF r0] by fast
821 proof (rule ratio_test [rule_format])
823 assume n: "N \<le> n"
824 have "norm x \<le> real N * r"
825 using N by (rule order_less_imp_le)
826 also have "real N * r \<le> real (Suc n) * r"
827 using r0 n by (simp add: mult_right_mono)
828 finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
829 using norm_ge_zero by (rule mult_right_mono)
830 hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
831 by (rule order_trans [OF norm_mult_ineq])
832 hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
833 by (simp add: pos_divide_le_eq mult_ac)
834 thus "norm (S (Suc n)) \<le> r * norm (S n)"
835 by (simp add: S_Suc inverse_eq_divide)
839 lemma summable_norm_exp:
840 fixes x :: "'a::{real_normed_algebra_1,banach}"
841 shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
842 proof (rule summable_norm_comparison_test [OF exI, rule_format])
843 show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
844 by (rule summable_exp_generic)
846 fix n show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
847 by (simp add: norm_power_ineq)
850 lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
851 by (insert summable_exp_generic [where x=x], simp)
853 lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
854 unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
858 "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
859 by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
860 del: mult_Suc of_nat_Suc)
862 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
863 by (simp add: diffs_def)
865 lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
866 unfolding exp_def scaleR_conv_of_real
867 apply (rule DERIV_cong)
868 apply (rule termdiffs [where K="of_real (1 + norm x)"])
869 apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
870 apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
871 apply (simp del: of_real_add)
874 lemma isCont_exp [simp]: "isCont exp x"
875 by (rule DERIV_exp [THEN DERIV_isCont])
878 subsubsection {* Properties of the Exponential Function *}
881 fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
882 shows "(\<Sum>n. f n * 0 ^ n) = f 0"
884 have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
885 by (rule sums_unique [OF series_zero], simp add: power_0_left)
886 thus ?thesis unfolding One_nat_def by simp
889 lemma exp_zero [simp]: "exp 0 = 1"
890 unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
892 lemma setsum_cl_ivl_Suc2:
893 "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
894 by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
895 del: setsum_cl_ivl_Suc)
897 lemma exp_series_add:
898 fixes x y :: "'a::{real_field}"
899 defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
900 shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
904 unfolding S_def by simp
907 have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
908 unfolding S_def by (simp del: mult_Suc)
909 hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
912 have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
913 by (simp only: times_S)
914 also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
916 also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
917 + y * (\<Sum>i=0..n. S x i * S y (n-i))"
918 by (rule left_distrib)
919 also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
920 + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
921 by (simp only: setsum_right_distrib mult_ac)
922 also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
923 + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
924 by (simp add: times_S Suc_diff_le)
925 also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
926 (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
927 by (subst setsum_cl_ivl_Suc2, simp)
928 also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
929 (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
930 by (subst setsum_cl_ivl_Suc, simp)
931 also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
932 (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
933 (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
934 by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
935 real_of_nat_add [symmetric], simp)
936 also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
937 by (simp only: scaleR_right.setsum)
939 "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
940 by (simp del: setsum_cl_ivl_Suc)
943 lemma exp_add: "exp (x + y) = exp x * exp y"
945 by (simp only: Cauchy_product summable_norm_exp exp_series_add)
947 lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
948 by (rule exp_add [symmetric])
950 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
952 apply (subst suminf_of_real)
953 apply (rule summable_exp_generic)
954 apply (simp add: scaleR_conv_of_real)
957 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
959 have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
960 also assume "exp x = 0"
961 finally show "False" by simp
964 lemma exp_minus: "exp (- x) = inverse (exp x)"
965 by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
967 lemma exp_diff: "exp (x - y) = exp x / exp y"
968 unfolding diff_minus divide_inverse
969 by (simp add: exp_add exp_minus)
972 subsubsection {* Properties of the Exponential Function on Reals *}
974 text {* Comparisons of @{term "exp x"} with zero. *}
976 text{*Proof: because every exponential can be seen as a square.*}
977 lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
979 have "0 \<le> exp (x/2) * exp (x/2)" by simp
980 thus ?thesis by (simp add: exp_add [symmetric])
983 lemma exp_gt_zero [simp]: "0 < exp (x::real)"
984 by (simp add: order_less_le)
986 lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
987 by (simp add: not_less)
989 lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
990 by (simp add: not_le)
992 lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
995 lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
997 apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
1000 text {* Strict monotonicity of exponential. *}
1002 lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
1003 apply (drule order_le_imp_less_or_eq, auto)
1004 apply (simp add: exp_def)
1005 apply (rule order_trans)
1006 apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
1007 apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
1010 lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
1013 hence "1 < 1 + x" by simp
1014 also from x have "1 + x \<le> exp x"
1015 by (simp add: exp_ge_add_one_self_aux)
1016 finally show ?thesis .
1019 lemma exp_less_mono:
1021 assumes "x < y" shows "exp x < exp y"
1023 from `x < y` have "0 < y - x" by simp
1024 hence "1 < exp (y - x)" by (rule exp_gt_one)
1025 hence "1 < exp y / exp x" by (simp only: exp_diff)
1026 thus "exp x < exp y" by simp
1029 lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
1030 apply (simp add: linorder_not_le [symmetric])
1031 apply (auto simp add: order_le_less exp_less_mono)
1034 lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
1035 by (auto intro: exp_less_mono exp_less_cancel)
1037 lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
1038 by (auto simp add: linorder_not_less [symmetric])
1040 lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
1041 by (simp add: order_eq_iff)
1043 text {* Comparisons of @{term "exp x"} with one. *}
1045 lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
1046 using exp_less_cancel_iff [where x=0 and y=x] by simp
1048 lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
1049 using exp_less_cancel_iff [where x=x and y=0] by simp
1051 lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
1052 using exp_le_cancel_iff [where x=0 and y=x] by simp
1054 lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
1055 using exp_le_cancel_iff [where x=x and y=0] by simp
1057 lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
1058 using exp_inj_iff [where x=x and y=0] by simp
1060 lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
1062 apply (auto intro: isCont_exp simp add: le_diff_eq)
1063 apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)")
1065 apply (rule exp_ge_add_one_self_aux, simp)
1068 lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
1069 apply (rule_tac x = 1 and y = y in linorder_cases)
1070 apply (drule order_less_imp_le [THEN lemma_exp_total])
1071 apply (rule_tac [2] x = 0 in exI)
1072 apply (frule_tac [3] one_less_inverse)
1073 apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
1074 apply (rule_tac x = "-x" in exI)
1075 apply (simp add: exp_minus)
1079 subsection {* Natural Logarithm *}
1081 definition ln :: "real \<Rightarrow> real" where
1082 "ln x = (THE u. exp u = x)"
1084 lemma ln_exp [simp]: "ln (exp x) = x"
1085 by (simp add: ln_def)
1087 lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
1088 by (auto dest: exp_total)
1090 lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
1091 by (metis exp_gt_zero exp_ln)
1093 lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
1094 by (erule subst, rule ln_exp)
1096 lemma ln_one [simp]: "ln 1 = 0"
1097 by (rule ln_unique, simp)
1099 lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"
1100 by (rule ln_unique, simp add: exp_add)
1102 lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
1103 by (rule ln_unique, simp add: exp_minus)
1105 lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"
1106 by (rule ln_unique, simp add: exp_diff)
1108 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
1109 by (rule ln_unique, simp add: exp_real_of_nat_mult)
1111 lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
1112 by (subst exp_less_cancel_iff [symmetric], simp)
1114 lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
1115 by (simp add: linorder_not_less [symmetric])
1117 lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
1118 by (simp add: order_eq_iff)
1120 lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
1121 apply (rule exp_le_cancel_iff [THEN iffD1])
1122 apply (simp add: exp_ge_add_one_self_aux)
1125 lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
1126 by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
1128 lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
1129 using ln_le_cancel_iff [of 1 x] by simp
1131 lemma ln_ge_zero_imp_ge_one: "\<lbrakk>0 \<le> ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 \<le> x"
1132 using ln_le_cancel_iff [of 1 x] by simp
1134 lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> (0 \<le> ln x) = (1 \<le> x)"
1135 using ln_le_cancel_iff [of 1 x] by simp
1137 lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x < 0) = (x < 1)"
1138 using ln_less_cancel_iff [of x 1] by simp
1140 lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
1141 using ln_less_cancel_iff [of 1 x] by simp
1143 lemma ln_gt_zero_imp_gt_one: "\<lbrakk>0 < ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 < x"
1144 using ln_less_cancel_iff [of 1 x] by simp
1146 lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> (0 < ln x) = (1 < x)"
1147 using ln_less_cancel_iff [of 1 x] by simp
1149 lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x = 0) = (x = 1)"
1150 using ln_inj_iff [of x 1] by simp
1152 lemma ln_less_zero: "\<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> ln x < 0"
1155 lemma exp_ln_eq: "exp u = x ==> ln x = u"
1156 by (rule ln_unique) (* TODO: delete *)
1158 lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
1159 apply (subgoal_tac "isCont ln (exp (ln x))", simp)
1160 apply (rule isCont_inverse_function [where f=exp], simp_all)
1163 lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
1164 apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
1165 apply (erule lemma_DERIV_subst [OF DERIV_exp exp_ln])
1166 apply (simp_all add: abs_if isCont_ln)
1169 lemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x"
1170 by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
1172 lemma ln_series: assumes "0 < x" and "x < 2"
1173 shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))")
1175 let "?f' x n" = "(-1)^n * (x - 1)^n"
1177 have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
1178 proof (rule DERIV_isconst3[where x=x])
1179 fix x :: real assume "x \<in> {0 <..< 2}" hence "0 < x" and "x < 2" by auto
1180 have "norm (1 - x) < 1" using `0 < x` and `x < 2` by auto
1181 have "1 / x = 1 / (1 - (1 - x))" by auto
1182 also have "\<dots> = (\<Sum> n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
1183 also have "\<dots> = suminf (?f' x)" unfolding power_mult_distrib[symmetric] by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
1184 finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
1186 have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
1187 have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
1188 proof (rule DERIV_power_series')
1189 show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto
1190 { fix x :: real assume "x \<in> {- 1<..<1}" hence "norm (-x) < 1" by auto
1191 show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
1192 unfolding One_nat_def
1193 by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
1196 hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" unfolding One_nat_def by auto
1197 hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos .
1198 ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
1199 by (rule DERIV_diff)
1200 thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
1201 qed (auto simp add: assms)
1202 thus ?thesis by auto
1205 subsection {* Sine and Cosine *}
1207 definition sin_coeff :: "nat \<Rightarrow> real" where
1208 "sin_coeff = (\<lambda>n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"
1210 definition cos_coeff :: "nat \<Rightarrow> real" where
1211 "cos_coeff = (\<lambda>n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"
1213 definition sin :: "real \<Rightarrow> real" where
1214 "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
1216 definition cos :: "real \<Rightarrow> real" where
1217 "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
1219 lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
1220 unfolding sin_coeff_def
1221 apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
1222 apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
1225 lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"
1226 unfolding cos_coeff_def
1227 apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
1228 apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
1231 lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
1232 unfolding sin_def by (rule summable_sin [THEN summable_sums])
1234 lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
1235 unfolding cos_def by (rule summable_cos [THEN summable_sums])
1237 lemma sin_fdiffs: "diffs sin_coeff = cos_coeff"
1238 unfolding sin_coeff_def cos_coeff_def
1239 by (auto intro!: ext
1240 simp add: diffs_def divide_inverse real_of_nat_def of_nat_mult
1241 simp del: mult_Suc of_nat_Suc)
1243 lemma sin_fdiffs2: "diffs sin_coeff n = cos_coeff n"
1244 by (simp only: sin_fdiffs)
1246 lemma cos_fdiffs: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
1247 unfolding sin_coeff_def cos_coeff_def
1248 by (auto intro!: ext
1249 simp add: diffs_def divide_inverse odd_Suc_mult_two_ex real_of_nat_def of_nat_mult
1250 simp del: mult_Suc of_nat_Suc)
1252 lemma cos_fdiffs2: "diffs cos_coeff n = - sin_coeff n"
1253 by (simp only: cos_fdiffs)
1255 text{*Now at last we can get the derivatives of exp, sin and cos*}
1257 lemma lemma_sin_minus: "- sin x = (\<Sum>n. - (sin_coeff n * x ^ n))"
1258 by (auto intro!: sums_unique sums_minus sin_converges)
1260 lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
1261 unfolding sin_def cos_def
1262 apply (auto simp add: sin_fdiffs2 [symmetric])
1263 apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
1264 apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs)
1267 lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
1269 apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
1270 apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
1271 apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus)
1274 lemma isCont_sin [simp]: "isCont sin x"
1275 by (rule DERIV_sin [THEN DERIV_isCont])
1277 lemma isCont_cos [simp]: "isCont cos x"
1278 by (rule DERIV_cos [THEN DERIV_isCont])
1282 DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
1283 DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
1284 DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
1285 DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
1287 subsection {* Properties of Sine and Cosine *}
1289 lemma sin_zero [simp]: "sin 0 = 0"
1290 unfolding sin_def sin_coeff_def by (simp add: powser_zero)
1292 lemma cos_zero [simp]: "cos 0 = 1"
1293 unfolding cos_def cos_coeff_def by (simp add: powser_zero)
1295 lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
1296 by (rule DERIV_cong) (* TODO: delete *)
1298 lemma sin_cos_squared_add [simp]: "(sin x)\<twosuperior> + (cos x)\<twosuperior> = 1"
1300 have "\<forall>x. DERIV (\<lambda>x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
1301 by (auto intro!: DERIV_intros)
1302 hence "(sin x)\<twosuperior> + (cos x)\<twosuperior> = (sin 0)\<twosuperior> + (cos 0)\<twosuperior>"
1303 by (rule DERIV_isconst_all)
1304 thus "(sin x)\<twosuperior> + (cos x)\<twosuperior> = 1" by simp
1307 lemma sin_cos_squared_add2 [simp]: "(cos x)\<twosuperior> + (sin x)\<twosuperior> = 1"
1308 by (subst add_commute, rule sin_cos_squared_add)
1310 lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
1311 using sin_cos_squared_add2 [unfolded power2_eq_square] .
1313 lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
1314 unfolding eq_diff_eq by (rule sin_cos_squared_add)
1316 lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
1317 unfolding eq_diff_eq by (rule sin_cos_squared_add2)
1319 lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
1320 by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
1322 lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
1323 using abs_sin_le_one [of x] unfolding abs_le_iff by simp
1325 lemma sin_le_one [simp]: "sin x \<le> 1"
1326 using abs_sin_le_one [of x] unfolding abs_le_iff by simp
1328 lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
1329 by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
1331 lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
1332 using abs_cos_le_one [of x] unfolding abs_le_iff by simp
1334 lemma cos_le_one [simp]: "cos x \<le> 1"
1335 using abs_cos_le_one [of x] unfolding abs_le_iff by simp
1337 lemma DERIV_fun_pow: "DERIV g x :> m ==>
1338 DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
1339 unfolding One_nat_def
1340 apply (rule DERIV_cong)
1341 apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2)
1342 apply (rule DERIV_pow, auto)
1345 lemma DERIV_fun_exp:
1346 "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
1347 apply (rule DERIV_cong)
1348 apply (rule_tac f = exp in DERIV_chain2)
1349 apply (rule DERIV_exp, auto)
1352 lemma DERIV_fun_sin:
1353 "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
1354 apply (rule DERIV_cong)
1355 apply (rule_tac f = sin in DERIV_chain2)
1356 apply (rule DERIV_sin, auto)
1359 lemma DERIV_fun_cos:
1360 "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
1361 apply (rule DERIV_cong)
1362 apply (rule_tac f = cos in DERIV_chain2)
1363 apply (rule DERIV_cos, auto)
1366 lemma sin_cos_add_lemma:
1367 "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
1368 (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
1371 have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
1372 by (auto intro!: DERIV_intros simp add: algebra_simps)
1374 by (rule DERIV_isconst_all)
1375 thus ?thesis by simp
1378 lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
1379 using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
1381 lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
1382 using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
1384 lemma sin_cos_minus_lemma:
1385 "(sin(-x) + sin(x))\<twosuperior> + (cos(-x) - cos(x))\<twosuperior> = 0" (is "?f x = 0")
1387 have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
1388 by (auto intro!: DERIV_intros simp add: algebra_simps)
1390 by (rule DERIV_isconst_all)
1391 thus ?thesis by simp
1394 lemma sin_minus [simp]: "sin (-x) = -sin(x)"
1395 using sin_cos_minus_lemma [where x=x] by simp
1397 lemma cos_minus [simp]: "cos (-x) = cos(x)"
1398 using sin_cos_minus_lemma [where x=x] by simp
1400 lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
1401 by (simp add: diff_minus sin_add)
1403 lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
1404 by (simp add: sin_diff mult_commute)
1406 lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
1407 by (simp add: diff_minus cos_add)
1409 lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
1410 by (simp add: cos_diff mult_commute)
1412 lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
1413 using sin_add [where x=x and y=x] by simp
1415 lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
1416 using cos_add [where x=x and y=x]
1417 by (simp add: power2_eq_square)
1420 subsection {* The Constant Pi *}
1422 definition pi :: "real" where
1423 "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
1425 text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
1429 "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1))
1432 have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
1434 by (rule sin_converges [THEN sums_summable, THEN sums_group], simp)
1435 thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)
1438 text {* FIXME: This is a long, ugly proof! *}
1439 lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
1441 "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
1442 -1 ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1))
1443 sums (\<Sum>n. -1 ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
1445 apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp)
1446 apply (rotate_tac 2)
1447 apply (drule sin_paired [THEN sums_unique, THEN ssubst])
1448 unfolding One_nat_def
1449 apply (auto simp del: fact_Suc)
1450 apply (frule sums_unique)
1451 apply (auto simp del: fact_Suc)
1452 apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])
1453 apply (auto simp del: fact_Suc)
1454 apply (erule sums_summable)
1455 apply (case_tac "m=0")
1456 apply (simp (no_asm_simp))
1457 apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x")
1458 apply (simp only: mult_less_cancel_left, simp)
1459 apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
1460 apply (subgoal_tac "x*x < 2*3", simp)
1461 apply (rule mult_strict_mono)
1462 apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc)
1463 apply (subst fact_Suc)
1464 apply (subst fact_Suc)
1465 apply (subst fact_Suc)
1466 apply (subst fact_Suc)
1467 apply (subst real_of_nat_mult)
1468 apply (subst real_of_nat_mult)
1469 apply (subst real_of_nat_mult)
1470 apply (subst real_of_nat_mult)
1471 apply (simp (no_asm) add: divide_inverse del: fact_Suc)
1472 apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
1473 apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right)
1474 apply (auto simp add: mult_assoc simp del: fact_Suc)
1475 apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right)
1476 apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
1477 apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)")
1478 apply (erule ssubst)+
1479 apply (auto simp del: fact_Suc)
1480 apply (subgoal_tac "0 < x ^ (4 * m) ")
1481 prefer 2 apply (simp only: zero_less_power)
1482 apply (simp (no_asm_simp) add: mult_less_cancel_left)
1483 apply (rule mult_strict_mono)
1484 apply (simp_all (no_asm_simp))
1487 lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"
1488 by (auto intro: sin_gt_zero)
1490 lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
1491 apply (cut_tac x = x in sin_gt_zero1)
1492 apply (auto simp add: cos_squared_eq cos_double)
1496 "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
1498 have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
1500 by (rule cos_converges [THEN sums_summable, THEN sums_group], simp)
1501 thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)
1504 lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
1507 lemma real_mult_inverse_cancel:
1508 "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
1509 ==> inverse x * y < inverse x1 * u"
1510 apply (rule_tac c=x in mult_less_imp_less_left)
1511 apply (auto simp add: mult_assoc [symmetric])
1512 apply (simp (no_asm) add: mult_ac)
1513 apply (rule_tac c=x1 in mult_less_imp_less_right)
1514 apply (auto simp add: mult_ac)
1517 lemma real_mult_inverse_cancel2:
1518 "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
1519 apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)
1522 lemma realpow_num_eq_if:
1523 fixes m :: "'a::power"
1524 shows "m ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
1527 lemma cos_two_less_zero [simp]: "cos (2) < 0"
1528 apply (cut_tac x = 2 in cos_paired)
1529 apply (drule sums_minus)
1530 apply (rule neg_less_iff_less [THEN iffD1])
1531 apply (frule sums_unique, auto)
1533 "\<Sum>n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
1534 in order_less_trans)
1535 apply (simp (no_asm) add: fact_num_eq_if_nat realpow_num_eq_if del: fact_Suc)
1536 apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
1537 apply (rule sumr_pos_lt_pair)
1538 apply (erule sums_summable, safe)
1539 unfolding One_nat_def
1540 apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
1542 apply (rule real_mult_inverse_cancel2)
1543 apply (simp del: fact_Suc)
1544 apply (simp del: fact_Suc)
1545 apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
1546 apply (subst fact_lemma)
1547 apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
1548 apply (simp only: real_of_nat_mult)
1549 apply (rule mult_strict_mono, force)
1550 apply (rule_tac [3] real_of_nat_ge_zero)
1551 prefer 2 apply force
1552 apply (rule real_of_nat_less_iff [THEN iffD2])
1553 apply (rule fact_less_mono_nat, auto)
1556 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
1557 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
1559 lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
1560 apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")
1561 apply (rule_tac [2] IVT2)
1562 apply (auto intro: DERIV_isCont DERIV_cos)
1563 apply (cut_tac x = xa and y = y in linorder_less_linear)
1565 apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")
1566 apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
1567 apply (drule_tac f = cos in Rolle)
1568 apply (drule_tac [5] f = cos in Rolle)
1569 apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
1570 apply (metis order_less_le_trans less_le sin_gt_zero)
1571 apply (metis order_less_le_trans less_le sin_gt_zero)
1574 lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
1575 by (simp add: pi_def)
1577 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
1578 by (simp add: pi_half cos_is_zero [THEN theI'])
1580 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
1581 apply (rule order_le_neq_trans)
1582 apply (simp add: pi_half cos_is_zero [THEN theI'])
1583 apply (rule notI, drule arg_cong [where f=cos], simp)
1586 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
1587 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
1589 lemma pi_half_less_two [simp]: "pi / 2 < 2"
1590 apply (rule order_le_neq_trans)
1591 apply (simp add: pi_half cos_is_zero [THEN theI'])
1592 apply (rule notI, drule arg_cong [where f=cos], simp)
1595 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
1596 lemmas pi_half_le_two [simp] = pi_half_less_two [THEN order_less_imp_le]
1598 lemma pi_gt_zero [simp]: "0 < pi"
1599 by (insert pi_half_gt_zero, simp)
1601 lemma pi_ge_zero [simp]: "0 \<le> pi"
1602 by (rule pi_gt_zero [THEN order_less_imp_le])
1604 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
1605 by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym])
1607 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
1608 by (simp add: linorder_not_less)
1610 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
1613 lemma m2pi_less_pi: "- (2 * pi) < pi"
1615 have "- (2 * pi) < 0" and "0 < pi" by auto
1616 from order_less_trans[OF this] show ?thesis .
1619 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
1620 apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
1621 apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
1622 apply (simp add: power2_eq_1_iff)
1625 lemma cos_pi [simp]: "cos pi = -1"
1626 by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
1628 lemma sin_pi [simp]: "sin pi = 0"
1629 by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
1631 lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
1632 by (simp add: diff_minus cos_add)
1633 declare sin_cos_eq [symmetric, simp]
1635 lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
1636 by (simp add: cos_add)
1637 declare minus_sin_cos_eq [symmetric, simp]
1639 lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
1640 by (simp add: diff_minus sin_add)
1641 declare cos_sin_eq [symmetric, simp]
1643 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
1644 by (simp add: sin_add)
1646 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
1647 by (simp add: sin_add)
1649 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
1650 by (simp add: cos_add)
1652 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
1653 by (simp add: sin_add cos_double)
1655 lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
1656 by (simp add: cos_add cos_double)
1658 lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
1660 apply (auto simp add: real_of_nat_Suc left_distrib)
1663 lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
1665 have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
1666 also have "... = -1 ^ n" by (rule cos_npi)
1667 finally show ?thesis .
1670 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
1672 apply (auto simp add: real_of_nat_Suc left_distrib)
1675 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
1676 by (simp add: mult_commute [of pi])
1678 lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
1679 by (simp add: cos_double)
1681 lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
1684 lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
1685 apply (rule sin_gt_zero, assumption)
1686 apply (rule order_less_trans, assumption)
1687 apply (rule pi_half_less_two)
1690 lemma sin_less_zero:
1691 assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
1693 have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
1694 thus ?thesis by simp
1697 lemma pi_less_4: "pi < 4"
1698 by (cut_tac pi_half_less_two, auto)
1700 lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
1701 apply (cut_tac pi_less_4)
1702 apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
1703 apply (cut_tac cos_is_zero, safe)
1704 apply (rename_tac y z)
1705 apply (drule_tac x = y in spec)
1706 apply (drule_tac x = "pi/2" in spec, simp)
1709 lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
1710 apply (rule_tac x = x and y = 0 in linorder_cases)
1711 apply (rule cos_minus [THEN subst])
1712 apply (rule cos_gt_zero)
1713 apply (auto intro: cos_gt_zero)
1716 lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
1717 apply (auto simp add: order_le_less cos_gt_zero_pi)
1718 apply (subgoal_tac "x = pi/2", auto)
1721 lemma sin_gt_zero_pi: "[| 0 < x; x < pi |] ==> 0 < sin x"
1722 apply (subst sin_cos_eq)
1723 apply (rotate_tac 1)
1724 apply (drule real_sum_of_halves [THEN ssubst])
1725 apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
1729 lemma pi_ge_two: "2 \<le> pi"
1731 assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
1732 have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
1733 proof (cases "2 < 2 * pi")
1734 case True with dense[OF `pi < 2`] show ?thesis by auto
1736 case False have "pi < 2 * pi" by auto
1737 from dense[OF this] and False show ?thesis by auto
1739 then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
1740 hence "0 < sin y" using sin_gt_zero by auto
1742 have "sin y < 0" using sin_gt_zero_pi[of "y - pi"] `pi < y` and `y < 2 * pi` sin_periodic_pi[of "y - pi"] by auto
1743 ultimately show False by auto
1746 lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
1747 by (auto simp add: order_le_less sin_gt_zero_pi)
1749 lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
1750 apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
1751 apply (rule_tac [2] IVT2)
1752 apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
1753 apply (cut_tac x = xa and y = y in linorder_less_linear)
1754 apply (rule ccontr, auto)
1755 apply (drule_tac f = cos in Rolle)
1756 apply (drule_tac [5] f = cos in Rolle)
1757 apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
1758 dest!: DERIV_cos [THEN DERIV_unique]
1759 simp add: differentiable_def)
1760 apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
1764 "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
1766 apply (subgoal_tac "\<forall>x. (- (pi/2) \<le> x & x \<le> pi/2 & (sin x = y)) = (0 \<le> (x + pi/2) & (x + pi/2) \<le> pi & (cos (x + pi/2) = -y))")
1767 apply (erule contrapos_np)
1768 apply (simp del: minus_sin_cos_eq [symmetric])
1769 apply (cut_tac y="-y" in cos_total, simp) apply simp
1771 apply (rule_tac a = "x - (pi/2)" in ex1I)
1772 apply (simp (no_asm) add: add_assoc)
1773 apply (rotate_tac 3)
1774 apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all)
1777 lemma reals_Archimedean4:
1778 "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
1779 apply (auto dest!: reals_Archimedean3)
1780 apply (drule_tac x = x in spec, clarify)
1781 apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
1782 prefer 2 apply (erule LeastI)
1783 apply (case_tac "LEAST m::nat. x < real m * y", simp)
1784 apply (subgoal_tac "~ x < real nat * y")
1785 prefer 2 apply (rule not_less_Least, simp, force)
1788 (* Pre Isabelle99-2 proof was simpler- numerals arithmetic
1789 now causes some unwanted re-arrangements of literals! *)
1790 lemma cos_zero_lemma:
1791 "[| 0 \<le> x; cos x = 0 |] ==>
1792 \<exists>n::nat. ~even n & x = real n * (pi/2)"
1793 apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
1794 apply (subgoal_tac "0 \<le> x - real n * pi &
1795 (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
1796 apply (auto simp add: algebra_simps real_of_nat_Suc)
1797 prefer 2 apply (simp add: cos_diff)
1798 apply (simp add: cos_diff)
1799 apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
1800 apply (rule_tac [2] cos_total, safe)
1801 apply (drule_tac x = "x - real n * pi" in spec)
1802 apply (drule_tac x = "pi/2" in spec)
1803 apply (simp add: cos_diff)
1804 apply (rule_tac x = "Suc (2 * n)" in exI)
1805 apply (simp add: real_of_nat_Suc algebra_simps, auto)
1808 lemma sin_zero_lemma:
1809 "[| 0 \<le> x; sin x = 0 |] ==>
1810 \<exists>n::nat. even n & x = real n * (pi/2)"
1811 apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
1812 apply (clarify, rule_tac x = "n - 1" in exI)
1813 apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
1814 apply (rule cos_zero_lemma)
1815 apply (simp_all add: add_increasing)
1821 ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
1822 (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
1824 apply (cut_tac linorder_linear [of 0 x], safe)
1825 apply (drule cos_zero_lemma, assumption+)
1826 apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
1827 apply (force simp add: minus_equation_iff [of x])
1828 apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
1829 apply (auto simp add: cos_add)
1832 (* ditto: but to a lesser extent *)
1835 ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
1836 (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
1838 apply (cut_tac linorder_linear [of 0 x], safe)
1839 apply (drule sin_zero_lemma, assumption+)
1840 apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
1841 apply (force simp add: minus_equation_iff [of x])
1842 apply (auto simp add: even_mult_two_ex)
1845 lemma cos_monotone_0_pi: assumes "0 \<le> y" and "y < x" and "x \<le> pi"
1846 shows "cos x < cos y"
1848 have "- (x - y) < 0" using assms by auto
1850 from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
1851 obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" by auto
1852 hence "0 < z" and "z < pi" using assms by auto
1853 hence "0 < sin z" using sin_gt_zero_pi by auto
1854 hence "cos x - cos y < 0" unfolding cos_diff minus_mult_commute[symmetric] using `- (x - y) < 0` by (rule mult_pos_neg2)
1855 thus ?thesis by auto
1858 lemma cos_monotone_0_pi': assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" shows "cos x \<le> cos y"
1859 proof (cases "y < x")
1860 case True show ?thesis using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
1862 case False hence "y = x" using `y \<le> x` by auto
1863 thus ?thesis by auto
1866 lemma cos_monotone_minus_pi_0: assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
1867 shows "cos y < cos x"
1869 have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" using assms by auto
1870 from cos_monotone_0_pi[OF this]
1871 show ?thesis unfolding cos_minus .
1874 lemma cos_monotone_minus_pi_0': assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0" shows "cos y \<le> cos x"
1875 proof (cases "y < x")
1876 case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] by auto
1878 case False hence "y = x" using `y \<le> x` by auto
1879 thus ?thesis by auto
1882 lemma sin_monotone_2pi': assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2" shows "sin y \<le> sin x"
1884 have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi"
1885 using pi_ge_two and assms by auto
1886 from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] by auto
1889 subsection {* Tangent *}
1892 tan :: "real => real" where
1893 "tan x = (sin x)/(cos x)"
1895 lemma tan_zero [simp]: "tan 0 = 0"
1896 by (simp add: tan_def)
1898 lemma tan_pi [simp]: "tan pi = 0"
1899 by (simp add: tan_def)
1901 lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
1902 by (simp add: tan_def)
1904 lemma tan_minus [simp]: "tan (-x) = - tan x"
1905 by (simp add: tan_def minus_mult_left)
1907 lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
1908 by (simp add: tan_def)
1910 lemma lemma_tan_add1:
1911 "[| cos x \<noteq> 0; cos y \<noteq> 0 |]
1912 ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
1913 apply (simp add: tan_def divide_inverse)
1914 apply (auto simp del: inverse_mult_distrib
1915 simp add: inverse_mult_distrib [symmetric] mult_ac)
1916 apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
1917 apply (auto simp del: inverse_mult_distrib
1918 simp add: mult_assoc left_diff_distrib cos_add)
1922 "[| cos x \<noteq> 0; cos y \<noteq> 0 |]
1923 ==> tan x + tan y = sin(x + y)/(cos x * cos y)"
1924 apply (simp add: tan_def)
1925 apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
1926 apply (auto simp add: mult_assoc left_distrib)
1927 apply (simp add: sin_add)
1931 "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
1932 ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
1933 apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)
1934 apply (simp add: tan_def)
1938 "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
1939 ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
1940 apply (insert tan_add [of x x])
1941 apply (simp add: mult_2 [symmetric])
1942 apply (auto simp add: numeral_2_eq_2)
1945 lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
1946 by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
1948 lemma tan_less_zero:
1949 assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
1951 have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
1952 thus ?thesis by simp
1955 lemma tan_half: fixes x :: real assumes "- (pi / 2) < x" and "x < pi / 2"
1956 shows "tan x = sin (2 * x) / (cos (2 * x) + 1)"
1958 from cos_gt_zero_pi[OF `- (pi / 2) < x` `x < pi / 2`]
1959 have "cos x \<noteq> 0" by auto
1961 have minus_cos_2x: "\<And>X. X - cos (2*x) = X - (cos x) ^ 2 + (sin x) ^ 2" unfolding cos_double by algebra
1963 have "tan x = (tan x + tan x) / 2" by auto
1964 also have "\<dots> = sin (x + x) / (cos x * cos x) / 2" unfolding add_tan_eq[OF `cos x \<noteq> 0` `cos x \<noteq> 0`] ..
1965 also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + (cos x) ^ 2 + cos (2*x) - cos (2*x))" unfolding divide_divide_eq_left numeral_2_eq_2 by auto
1966 also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + cos (2*x) + (sin x)^2)" unfolding minus_cos_2x by auto
1967 also have "\<dots> = sin (2 * x) / (cos (2*x) + 1)" by auto
1968 finally show ?thesis .
1971 lemma lemma_DERIV_tan:
1972 "cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"
1973 by (auto intro!: DERIV_intros simp add: field_simps numeral_2_eq_2)
1975 lemma DERIV_tan [simp]: "cos x \<noteq> 0 ==> DERIV tan x :> inverse((cos x)\<twosuperior>)"
1976 by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])
1978 lemma isCont_tan [simp]: "cos x \<noteq> 0 ==> isCont tan x"
1979 by (rule DERIV_tan [THEN DERIV_isCont])
1981 lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
1982 apply (subgoal_tac "(\<lambda>x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")
1983 apply (simp add: divide_inverse [symmetric])
1984 apply (rule LIM_mult)
1985 apply (rule_tac [2] inverse_1 [THEN subst])
1986 apply (rule_tac [2] LIM_inverse)
1987 apply (simp_all add: divide_inverse [symmetric])
1988 apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric])
1989 apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
1992 lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
1993 apply (cut_tac LIM_cos_div_sin)
1994 apply (simp only: LIM_eq)
1995 apply (drule_tac x = "inverse y" in spec, safe, force)
1996 apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
1997 apply (rule_tac x = "(pi/2) - e" in exI)
1998 apply (simp (no_asm_simp))
1999 apply (drule_tac x = "(pi/2) - e" in spec)
2000 apply (auto simp add: tan_def)
2001 apply (rule inverse_less_iff_less [THEN iffD1])
2002 apply (auto simp add: divide_inverse)
2003 apply (rule mult_pos_pos)
2004 apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
2005 apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
2008 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
2009 apply (frule order_le_imp_less_or_eq, safe)
2010 prefer 2 apply force
2011 apply (drule lemma_tan_total, safe)
2012 apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
2013 apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
2014 apply (drule_tac y = xa in order_le_imp_less_or_eq)
2015 apply (auto dest: cos_gt_zero)
2018 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
2019 apply (cut_tac linorder_linear [of 0 y], safe)
2020 apply (drule tan_total_pos)
2021 apply (cut_tac [2] y="-y" in tan_total_pos, safe)
2022 apply (rule_tac [3] x = "-x" in exI)
2023 apply (auto intro!: exI)
2026 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
2027 apply (cut_tac y = y in lemma_tan_total1, auto)
2028 apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
2029 apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
2030 apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
2031 apply (rule_tac [4] Rolle)
2032 apply (rule_tac [2] Rolle)
2033 apply (auto intro!: DERIV_tan DERIV_isCont exI
2034 simp add: differentiable_def)
2035 txt{*Now, simulate TRYALL*}
2036 apply (rule_tac [!] DERIV_tan asm_rl)
2037 apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
2038 simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
2041 lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
2042 shows "tan y < tan x"
2044 have "\<forall> x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse (cos x'^2)"
2045 proof (rule allI, rule impI)
2046 fix x' :: real assume "y \<le> x' \<and> x' \<le> x"
2047 hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
2048 from cos_gt_zero_pi[OF this]
2049 have "cos x' \<noteq> 0" by auto
2050 thus "DERIV tan x' :> inverse (cos x'^2)" by (rule DERIV_tan)
2052 from MVT2[OF `y < x` this]
2053 obtain z where "y < z" and "z < x" and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<twosuperior>)" by auto
2054 hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
2055 hence "0 < cos z" using cos_gt_zero_pi by auto
2056 hence inv_pos: "0 < inverse ((cos z)\<twosuperior>)" by auto
2057 have "0 < x - y" using `y < x` by auto
2058 from mult_pos_pos [OF this inv_pos]
2059 have "0 < tan x - tan y" unfolding tan_diff by auto
2060 thus ?thesis by auto
2063 lemma tan_monotone': assumes "- (pi / 2) < y" and "y < pi / 2" and "- (pi / 2) < x" and "x < pi / 2"
2064 shows "(y < x) = (tan y < tan x)"
2066 assume "y < x" thus "tan y < tan x" using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
2068 assume "tan y < tan x"
2071 assume "\<not> y < x" hence "x \<le> y" by auto
2072 hence "tan x \<le> tan y"
2073 proof (cases "x = y")
2074 case True thus ?thesis by auto
2076 case False hence "x < y" using `x \<le> y` by auto
2077 from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
2079 thus False using `tan y < tan x` by auto
2083 lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)" unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
2085 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
2086 by (simp add: tan_def)
2088 lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x"
2089 proof (induct n arbitrary: x)
2091 have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" unfolding Suc_eq_plus1 real_of_nat_add real_of_one left_distrib by auto
2092 show ?case unfolding split_pi_off using Suc by auto
2095 lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
2096 proof (cases "0 \<le> i")
2097 case True hence i_nat: "real i = real (nat i)" by auto
2098 show ?thesis unfolding i_nat by auto
2100 case False hence i_nat: "real i = - real (nat (-i))" by auto
2101 have "tan x = tan (x + real i * pi - real i * pi)" by auto
2102 also have "\<dots> = tan (x + real i * pi)" unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
2103 finally show ?thesis by auto
2106 lemma tan_periodic_n[simp]: "tan (x + number_of n * pi) = tan x"
2107 using tan_periodic_int[of _ "number_of n" ] unfolding real_number_of .
2109 subsection {* Inverse Trigonometric Functions *}
2112 arcsin :: "real => real" where
2113 "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
2116 arccos :: "real => real" where
2117 "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
2120 arctan :: "real => real" where
2121 "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
2124 "[| -1 \<le> y; y \<le> 1 |]
2125 ==> -(pi/2) \<le> arcsin y &
2126 arcsin y \<le> pi/2 & sin(arcsin y) = y"
2127 unfolding arcsin_def by (rule theI' [OF sin_total])
2130 "[| -1 \<le> y; y \<le> 1 |]
2131 ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
2132 apply (drule (1) arcsin)
2133 apply (force intro: order_trans)
2136 lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
2137 by (blast dest: arcsin)
2139 lemma arcsin_bounded:
2140 "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
2141 by (blast dest: arcsin)
2143 lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
2144 by (blast dest: arcsin)
2146 lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
2147 by (blast dest: arcsin)
2149 lemma arcsin_lt_bounded:
2150 "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
2151 apply (frule order_less_imp_le)
2152 apply (frule_tac y = y in order_less_imp_le)
2153 apply (frule arcsin_bounded)
2155 apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
2156 apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
2157 apply (drule_tac [!] f = sin in arg_cong, auto)
2160 lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
2161 apply (unfold arcsin_def)
2162 apply (rule the1_equality)
2163 apply (rule sin_total, auto)
2167 "[| -1 \<le> y; y \<le> 1 |]
2168 ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
2169 unfolding arccos_def by (rule theI' [OF cos_total])
2171 lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
2172 by (blast dest: arccos)
2174 lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
2175 by (blast dest: arccos)
2177 lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
2178 by (blast dest: arccos)
2180 lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
2181 by (blast dest: arccos)
2183 lemma arccos_lt_bounded:
2184 "[| -1 < y; y < 1 |]
2185 ==> 0 < arccos y & arccos y < pi"
2186 apply (frule order_less_imp_le)
2187 apply (frule_tac y = y in order_less_imp_le)
2188 apply (frule arccos_bounded, auto)
2189 apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
2190 apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
2191 apply (drule_tac [!] f = cos in arg_cong, auto)
2194 lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
2195 apply (simp add: arccos_def)
2196 apply (auto intro!: the1_equality cos_total)
2199 lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
2200 apply (simp add: arccos_def)
2201 apply (auto intro!: the1_equality cos_total)
2204 lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<twosuperior>)"
2205 apply (subgoal_tac "x\<twosuperior> \<le> 1")
2206 apply (rule power2_eq_imp_eq)
2207 apply (simp add: cos_squared_eq)
2208 apply (rule cos_ge_zero)
2209 apply (erule (1) arcsin_lbound)
2210 apply (erule (1) arcsin_ubound)
2212 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
2213 apply (rule power_mono, simp, simp)
2216 lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<twosuperior>)"
2217 apply (subgoal_tac "x\<twosuperior> \<le> 1")
2218 apply (rule power2_eq_imp_eq)
2219 apply (simp add: sin_squared_eq)
2220 apply (rule sin_ge_zero)
2221 apply (erule (1) arccos_lbound)
2222 apply (erule (1) arccos_ubound)
2224 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
2225 apply (rule power_mono, simp, simp)
2228 lemma arctan [simp]:
2229 "- (pi/2) < arctan y & arctan y < pi/2 & tan (arctan y) = y"
2230 unfolding arctan_def by (rule theI' [OF tan_total])
2232 lemma tan_arctan: "tan(arctan y) = y"
2235 lemma arctan_bounded: "- (pi/2) < arctan y & arctan y < pi/2"
2236 by (auto simp only: arctan)
2238 lemma arctan_lbound: "- (pi/2) < arctan y"
2241 lemma arctan_ubound: "arctan y < pi/2"
2242 by (auto simp only: arctan)
2245 "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
2246 apply (unfold arctan_def)
2247 apply (rule the1_equality)
2248 apply (rule tan_total, auto)
2251 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
2252 by (insert arctan_tan [of 0], simp)
2254 lemma cos_arctan_not_zero [simp]: "cos(arctan x) \<noteq> 0"
2255 apply (auto simp add: cos_zero_iff)
2256 apply (case_tac "n")
2257 apply (case_tac [3] "n")
2258 apply (cut_tac [2] y = x in arctan_ubound)
2259 apply (cut_tac [4] y = x in arctan_lbound)
2260 apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)
2263 lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
2264 apply (rule power_inverse [THEN subst])
2265 apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
2266 apply (auto dest: field_power_not_zero
2267 simp add: power_mult_distrib left_distrib power_divide tan_def
2268 mult_assoc power_inverse [symmetric])
2271 lemma isCont_inverse_function2:
2272 fixes f g :: "real \<Rightarrow> real" shows
2273 "\<lbrakk>a < x; x < b;
2274 \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
2275 \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
2276 \<Longrightarrow> isCont g (f x)"
2277 apply (rule isCont_inverse_function
2278 [where f=f and d="min (x - a) (b - x)"])
2279 apply (simp_all add: abs_le_iff)
2282 lemma isCont_arcsin: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arcsin x"
2283 apply (subgoal_tac "isCont arcsin (sin (arcsin x))", simp)
2284 apply (rule isCont_inverse_function2 [where f=sin])
2285 apply (erule (1) arcsin_lt_bounded [THEN conjunct1])
2286 apply (erule (1) arcsin_lt_bounded [THEN conjunct2])
2287 apply (fast intro: arcsin_sin, simp)
2290 lemma isCont_arccos: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arccos x"
2291 apply (subgoal_tac "isCont arccos (cos (arccos x))", simp)
2292 apply (rule isCont_inverse_function2 [where f=cos])
2293 apply (erule (1) arccos_lt_bounded [THEN conjunct1])
2294 apply (erule (1) arccos_lt_bounded [THEN conjunct2])
2295 apply (fast intro: arccos_cos, simp)
2298 lemma isCont_arctan: "isCont arctan x"
2299 apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
2300 apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
2301 apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
2302 apply (erule (1) isCont_inverse_function2 [where f=tan])
2303 apply (metis arctan_tan order_le_less_trans order_less_le_trans)
2304 apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
2308 "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<twosuperior>))"
2309 apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
2310 apply (rule DERIV_cong [OF DERIV_sin])
2311 apply (simp add: cos_arcsin)
2312 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
2313 apply (rule power_strict_mono, simp, simp, simp)
2317 apply (erule (1) isCont_arcsin)
2321 "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<twosuperior>))"
2322 apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
2323 apply (rule DERIV_cong [OF DERIV_cos])
2324 apply (simp add: sin_arccos)
2325 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
2326 apply (rule power_strict_mono, simp, simp, simp)
2330 apply (erule (1) isCont_arccos)
2333 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<twosuperior>)"
2334 apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
2335 apply (rule DERIV_cong [OF DERIV_tan])
2336 apply (rule cos_arctan_not_zero)
2337 apply (simp add: power_inverse tan_sec [symmetric])
2338 apply (subgoal_tac "0 < 1 + x\<twosuperior>", simp)
2339 apply (simp add: add_pos_nonneg)
2340 apply (simp, simp, simp, rule isCont_arctan)
2344 DERIV_arcsin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
2345 DERIV_arccos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
2346 DERIV_arctan[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
2348 subsection {* More Theorems about Sin and Cos *}
2350 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
2352 let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
2353 have nonneg: "0 \<le> ?c"
2354 by (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
2355 have "0 = cos (pi / 4 + pi / 4)"
2357 also have "cos (pi / 4 + pi / 4) = ?c\<twosuperior> - ?s\<twosuperior>"
2358 by (simp only: cos_add power2_eq_square)
2359 also have "\<dots> = 2 * ?c\<twosuperior> - 1"
2360 by (simp add: sin_squared_eq)
2361 finally have "?c\<twosuperior> = (sqrt 2 / 2)\<twosuperior>"
2362 by (simp add: power_divide)
2364 using nonneg by (rule power2_eq_imp_eq) simp
2367 lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"
2369 let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
2370 have pos_c: "0 < ?c"
2371 by (rule cos_gt_zero, simp, simp)
2372 have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
2374 also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
2375 by (simp only: cos_add sin_add)
2376 also have "\<dots> = ?c * (?c\<twosuperior> - 3 * ?s\<twosuperior>)"
2377 by (simp add: algebra_simps power2_eq_square)
2378 finally have "?c\<twosuperior> = (sqrt 3 / 2)\<twosuperior>"
2379 using pos_c by (simp add: sin_squared_eq power_divide)
2381 using pos_c [THEN order_less_imp_le]
2382 by (rule power2_eq_imp_eq) simp
2385 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
2387 have "sin (pi / 4) = cos (pi / 2 - pi / 4)" by (rule sin_cos_eq)
2388 also have "pi / 2 - pi / 4 = pi / 4" by simp
2389 also have "cos (pi / 4) = sqrt 2 / 2" by (rule cos_45)
2390 finally show ?thesis .
2393 lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
2395 have "sin (pi / 3) = cos (pi / 2 - pi / 3)" by (rule sin_cos_eq)
2396 also have "pi / 2 - pi / 3 = pi / 6" by simp
2397 also have "cos (pi / 6) = sqrt 3 / 2" by (rule cos_30)
2398 finally show ?thesis .
2401 lemma cos_60: "cos (pi / 3) = 1 / 2"
2402 apply (rule power2_eq_imp_eq)
2403 apply (simp add: cos_squared_eq sin_60 power_divide)
2404 apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
2407 lemma sin_30: "sin (pi / 6) = 1 / 2"
2409 have "sin (pi / 6) = cos (pi / 2 - pi / 6)" by (rule sin_cos_eq)
2410 also have "pi / 2 - pi / 6 = pi / 3" by simp
2411 also have "cos (pi / 3) = 1 / 2" by (rule cos_60)
2412 finally show ?thesis .
2415 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
2416 unfolding tan_def by (simp add: sin_30 cos_30)
2418 lemma tan_45: "tan (pi / 4) = 1"
2419 unfolding tan_def by (simp add: sin_45 cos_45)
2421 lemma tan_60: "tan (pi / 3) = sqrt 3"
2422 unfolding tan_def by (simp add: sin_60 cos_60)
2424 lemma DERIV_sin_add: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)"
2425 by (auto intro!: DERIV_intros) (* TODO: delete *)
2427 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
2429 have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
2430 by (auto simp add: algebra_simps sin_add)
2432 by (simp add: real_of_nat_Suc left_distrib add_divide_distrib
2433 mult_commute [of pi])
2436 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
2437 by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
2439 lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
2440 apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
2441 apply (subst cos_add, simp)
2444 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
2445 by (auto simp add: mult_assoc)
2447 lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
2448 apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
2449 apply (subst sin_add, simp)
2452 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
2453 by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto)
2455 lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
2456 by (auto intro!: DERIV_intros)
2458 lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
2459 by (auto simp add: sin_zero_iff even_mult_two_ex)
2461 lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
2462 by (cut_tac x = x in sin_cos_squared_add3, auto)
2464 subsection {* Machins formula *}
2466 lemma tan_total_pi4: assumes "\<bar>x\<bar> < 1"
2467 shows "\<exists> z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
2469 obtain z where "- (pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast
2470 have "tan (pi / 4) = 1" and "tan (- (pi / 4)) = - 1" using tan_45 tan_minus by auto
2471 have "z \<noteq> pi / 4"
2473 assume "\<not> (z \<noteq> pi / 4)" hence "z = pi / 4" by auto
2474 have "tan z = 1" unfolding `z = pi / 4` `tan (pi / 4) = 1` ..
2475 thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto
2477 have "z \<noteq> - (pi / 4)"
2479 assume "\<not> (z \<noteq> - (pi / 4))" hence "z = - (pi / 4)" by auto
2480 have "tan z = - 1" unfolding `z = - (pi / 4)` `tan (- (pi / 4)) = - 1` ..
2481 thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto
2486 assume "\<not> (z < pi / 4)" hence "pi / 4 < z" using `z \<noteq> pi / 4` by auto
2487 have "- (pi / 2) < pi / 4" using m2pi_less_pi by auto
2488 from tan_monotone[OF this `pi / 4 < z` `z < pi / 2`]
2489 have "1 < x" unfolding `tan z = x` `tan (pi / 4) = 1` .
2490 thus False using `\<bar>x\<bar> < 1` by auto
2493 have "-(pi / 4) < z"
2495 assume "\<not> (-(pi / 4) < z)" hence "z < - (pi / 4)" using `z \<noteq> - (pi / 4)` by auto
2496 have "-(pi / 4) < pi / 2" using m2pi_less_pi by auto
2497 from tan_monotone[OF `-(pi / 2) < z` `z < -(pi / 4)` this]
2498 have "x < - 1" unfolding `tan z = x` `tan (-(pi / 4)) = - 1` .
2499 thus False using `\<bar>x\<bar> < 1` by auto
2501 ultimately show ?thesis using `tan z = x` by auto
2504 lemma arctan_add: assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
2505 shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
2507 obtain y' where "-(pi/4) < y'" and "y' < pi/4" and "tan y' = y" using tan_total_pi4[OF `\<bar>y\<bar> < 1`] by blast
2509 have "pi / 4 < pi / 2" by auto
2511 have "\<exists> x'. -(pi/4) \<le> x' \<and> x' \<le> pi/4 \<and> tan x' = x"
2512 proof (cases "\<bar>x\<bar> < 1")
2513 case True from tan_total_pi4[OF this] obtain x' where "-(pi/4) < x'" and "x' < pi/4" and "tan x' = x" by blast
2514 hence "-(pi/4) \<le> x'" and "x' \<le> pi/4" and "tan x' = x" by auto
2515 thus ?thesis by auto
2519 proof (cases "x = 1")
2520 case True hence "tan (pi/4) = x" using tan_45 by auto
2522 have "- pi \<le> pi" unfolding minus_le_self_iff by auto
2523 hence "-(pi/4) \<le> pi/4" and "pi/4 \<le> pi/4" by auto
2524 ultimately show ?thesis by blast
2526 case False hence "x = -1" using `\<not> \<bar>x\<bar> < 1` and `\<bar>x\<bar> \<le> 1` by auto
2527 hence "tan (-(pi/4)) = x" using tan_45 tan_minus by auto
2529 have "- pi \<le> pi" unfolding minus_le_self_iff by auto
2530 hence "-(pi/4) \<le> pi/4" and "-(pi/4) \<le> -(pi/4)" by auto
2531 ultimately show ?thesis by blast
2534 then obtain x' where "-(pi/4) \<le> x'" and "x' \<le> pi/4" and "tan x' = x" by blast
2535 hence "-(pi/2) < x'" and "x' < pi/2" using order_le_less_trans[OF `x' \<le> pi/4` `pi / 4 < pi / 2`] by auto
2537 have "cos x' \<noteq> 0" using cos_gt_zero_pi[THEN less_imp_neq] and `-(pi/2) < x'` and `x' < pi/2` by auto
2538 moreover have "cos y' \<noteq> 0" using cos_gt_zero_pi[THEN less_imp_neq] and `-(pi/4) < y'` and `y' < pi/4` by auto
2539 ultimately have "cos x' * cos y' \<noteq> 0" by auto
2541 have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> (A / C) / (B / C) = A / B" by auto
2542 have divide_mult_commute: "\<And> A B C D :: real. A * B / (C * D) = (A / C) * (B / D)" by auto
2544 have "tan (x' + y') = sin (x' + y') / (cos x' * cos y' - sin x' * sin y')" unfolding tan_def cos_add ..
2545 also have "\<dots> = (tan x' + tan y') / ((cos x' * cos y' - sin x' * sin y') / (cos x' * cos y'))" unfolding add_tan_eq[OF `cos x' \<noteq> 0` `cos y' \<noteq> 0`] divide_nonzero_divide[OF `cos x' * cos y' \<noteq> 0`] ..
2546 also have "\<dots> = (tan x' + tan y') / (1 - tan x' * tan y')" unfolding tan_def diff_divide_distrib divide_self[OF `cos x' * cos y' \<noteq> 0`] unfolding divide_mult_commute ..
2547 finally have tan_eq: "tan (x' + y') = (x + y) / (1 - x * y)" unfolding `tan y' = y` `tan x' = x` .
2549 have "arctan (tan (x' + y')) = x' + y'" using `-(pi/4) < y'` `-(pi/4) \<le> x'` `y' < pi/4` and `x' \<le> pi/4` by (auto intro!: arctan_tan)
2550 moreover have "arctan (tan (x')) = x'" using `-(pi/2) < x'` and `x' < pi/2` by (auto intro!: arctan_tan)
2551 moreover have "arctan (tan (y')) = y'" using `-(pi/4) < y'` and `y' < pi/4` by (auto intro!: arctan_tan)
2552 ultimately have "arctan x + arctan y = arctan (tan (x' + y'))" unfolding `tan y' = y` [symmetric] `tan x' = x`[symmetric] by auto
2553 thus "arctan x + arctan y = arctan ((x + y) / (1 - x * y))" unfolding tan_eq .
2556 lemma arctan1_eq_pi4: "arctan 1 = pi / 4" unfolding tan_45[symmetric] by (rule arctan_tan, auto simp add: m2pi_less_pi)
2558 theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
2560 have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
2561 from arctan_add[OF less_imp_le[OF this] this]
2562 have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
2564 have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
2565 from arctan_add[OF less_imp_le[OF this] this]
2566 have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
2568 have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
2569 from arctan_add[OF this]
2570 have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
2571 ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
2572 thus ?thesis unfolding arctan1_eq_pi4 by algebra
2574 subsection {* Introducing the arcus tangens power series *}
2576 lemma monoseq_arctan_series: fixes x :: real
2577 assumes "\<bar>x\<bar> \<le> 1" shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
2578 proof (cases "x = 0") case True thus ?thesis unfolding monoseq_def One_nat_def by auto
2581 have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
2584 { fix n fix x :: real assume "0 \<le> x" and "x \<le> 1"
2585 have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le> 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
2586 proof (rule mult_mono)
2587 show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))" by (rule frac_le) simp_all
2588 show "0 \<le> 1 / real (Suc (n * 2))" by auto
2589 show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)" by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`)
2590 show "0 \<le> x ^ Suc (Suc n * 2)" by (rule zero_le_power) (simp add: `0 \<le> x`)
2595 proof (cases "0 \<le> x")
2596 case True from mono[OF this `x \<le> 1`, THEN allI]
2597 show ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI2)
2599 case False hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto
2601 have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge> 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using `0 \<le> -x` by auto
2602 thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
2607 lemma zeroseq_arctan_series: fixes x :: real
2608 assumes "\<bar>x\<bar> \<le> 1" shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
2609 proof (cases "x = 0") case True thus ?thesis unfolding One_nat_def by (auto simp add: LIMSEQ_const)
2612 have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
2614 proof (cases "\<bar>x\<bar> < 1")
2615 case True hence "norm x < 1" by auto
2616 from LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]]
2617 have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"
2618 unfolding inverse_eq_divide Suc_eq_plus1 by simp
2619 then show ?thesis using pos2 by (rule LIMSEQ_linear)
2621 case False hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto
2622 hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x" unfolding One_nat_def by auto
2623 from LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] LIMSEQ_const[of x]]
2624 show ?thesis unfolding n_eq Suc_eq_plus1 by auto
2628 lemma summable_arctan_series: fixes x :: real and n :: nat
2629 assumes "\<bar>x\<bar> \<le> 1" shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "summable (?c x)")
2630 by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
2632 lemma less_one_imp_sqr_less_one: fixes x :: real assumes "\<bar>x\<bar> < 1" shows "x^2 < 1"
2634 from mult_left_mono[OF less_imp_le[OF `\<bar>x\<bar> < 1`] abs_ge_zero[of x]]
2635 have "\<bar> x^2 \<bar> < 1" using `\<bar> x \<bar> < 1` unfolding numeral_2_eq_2 power_Suc2 by auto
2636 thus ?thesis using zero_le_power2 by auto
2639 lemma DERIV_arctan_series: assumes "\<bar> x \<bar> < 1"
2640 shows "DERIV (\<lambda> x'. \<Sum> k. (-1)^k * (1 / real (k*2+1) * x' ^ (k*2+1))) x :> (\<Sum> k. (-1)^k * x^(k*2))" (is "DERIV ?arctan _ :> ?Int")
2642 let "?f n" = "if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
2644 { fix n :: nat assume "even n" hence "2 * (n div 2) = n" by presburger } note n_even=this
2645 have if_eq: "\<And> n x'. ?f n * real (Suc n) * x'^n = (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)" using n_even by auto
2647 { fix x :: real assume "\<bar>x\<bar> < 1" hence "x^2 < 1" by (rule less_one_imp_sqr_less_one)
2648 have "summable (\<lambda> n. -1 ^ n * (x^2) ^n)"
2649 by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x^2 < 1` order_less_imp_le[OF `x^2 < 1`])
2650 hence "summable (\<lambda> n. -1 ^ n * x^(2*n))" unfolding power_mult .
2651 } note summable_Integral = this
2653 { fix f :: "nat \<Rightarrow> real"
2654 have "\<And> x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
2656 fix x :: real assume "f sums x"
2657 from sums_if[OF sums_zero this]
2658 show "(\<lambda> n. if even n then f (n div 2) else 0) sums x" by auto
2660 fix x :: real assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
2661 from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult_commute]]
2662 show "f sums x" unfolding sums_def by auto
2664 hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
2665 } note sums_even = this
2667 have Int_eq: "(\<Sum> n. ?f n * real (Suc n) * x^n) = ?Int" unfolding if_eq mult_commute[of _ 2] suminf_def sums_even[of "\<lambda> n. -1 ^ n * x ^ (2 * n)", symmetric]
2671 have if_eq': "\<And> n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
2672 (if even n then -1 ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
2673 using n_even by auto
2674 have idx_eq: "\<And> n. n * 2 + 1 = Suc (2 * n)" by auto
2675 have "(\<Sum> n. ?f n * x^(Suc n)) = ?arctan x" unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. -1 ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
2677 } note arctan_eq = this
2679 have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
2680 proof (rule DERIV_power_series')
2681 show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto
2682 { fix x' :: real assume x'_bounds: "x' \<in> {- 1 <..< 1}"
2683 hence "\<bar>x'\<bar> < 1" by auto
2685 let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
2686 show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
2687 by (rule sums_summable[where l="0 + ?S"], rule sums_if, rule sums_zero, rule summable_sums, rule summable_Integral[OF `\<bar>x'\<bar> < 1`])
2690 thus ?thesis unfolding Int_eq arctan_eq .
2693 lemma arctan_series: assumes "\<bar> x \<bar> \<le> 1"
2694 shows "arctan x = (\<Sum> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "_ = suminf (\<lambda> n. ?c x n)")
2696 let "?c' x n" = "(-1)^n * x^(n*2)"
2698 { fix r x :: real assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
2699 have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto
2700 from DERIV_arctan_series[OF this]
2701 have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
2702 } note DERIV_arctan_suminf = this
2704 { fix x :: real assume "\<bar>x\<bar> \<le> 1" note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] }
2705 note arctan_series_borders = this
2707 { fix x :: real assume "\<bar>x\<bar> < 1" have "arctan x = (\<Sum> k. ?c x k)"
2709 obtain r where "\<bar>x\<bar> < r" and "r < 1" using dense[OF `\<bar>x\<bar> < 1`] by blast
2710 hence "0 < r" and "-r < x" and "x < r" by auto
2712 have suminf_eq_arctan_bounded: "\<And> x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow> suminf (?c x) - arctan x = suminf (?c a) - arctan a"
2714 fix x a b assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
2715 hence "\<bar>x\<bar> < r" by auto
2716 show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
2717 proof (rule DERIV_isconst2[of "a" "b"])
2718 show "a < b" and "a \<le> x" and "x \<le> b" using `a < b` `a \<le> x` `x \<le> b` by auto
2719 have "\<forall> x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
2720 proof (rule allI, rule impI)
2721 fix x assume "-r < x \<and> x < r" hence "\<bar>x\<bar> < r" by auto
2722 hence "\<bar>x\<bar> < 1" using `r < 1` by auto
2723 have "\<bar> - (x^2) \<bar> < 1" using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
2724 hence "(\<lambda> n. (- (x^2)) ^ n) sums (1 / (1 - (- (x^2))))" unfolding real_norm_def[symmetric] by (rule geometric_sums)
2725 hence "(?c' x) sums (1 / (1 - (- (x^2))))" unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto
2726 hence suminf_c'_eq_geom: "inverse (1 + x^2) = suminf (?c' x)" using sums_unique unfolding inverse_eq_divide by auto
2727 have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x^2))" unfolding suminf_c'_eq_geom
2728 by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
2729 from DERIV_add_minus[OF this DERIV_arctan]
2730 show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0" unfolding diff_minus by auto
2732 hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0" using `-r < a` `b < r` by auto
2733 thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0" using `\<bar>x\<bar> < r` by auto
2734 show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y" using DERIV_in_rball DERIV_isCont by auto
2738 have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
2739 unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero by auto
2741 have "suminf (?c x) - arctan x = 0"
2742 proof (cases "x = 0")
2743 case True thus ?thesis using suminf_arctan_zero by auto
2745 case False hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
2746 have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
2747 by (rule suminf_eq_arctan_bounded[where x="0" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>", symmetric])
2748 (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
2750 have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
2751 by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"])
2752 (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
2754 show ?thesis using suminf_arctan_zero by auto
2756 thus ?thesis by auto
2757 qed } note when_less_one = this
2759 show "arctan x = suminf (\<lambda> n. ?c x n)"
2760 proof (cases "\<bar>x\<bar> < 1")
2761 case True thus ?thesis by (rule when_less_one)
2762 next case False hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto
2763 let "?a x n" = "\<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
2764 let "?diff x n" = "\<bar> arctan x - (\<Sum> i = 0..<n. ?c x i)\<bar>"
2766 have "0 < (1 :: real)" by auto
2768 { fix x :: real assume "0 < x" and "x < 1" hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
2769 from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)" by auto
2770 note bounds = mp[OF arctan_series_borders(2)[OF `\<bar>x\<bar> \<le> 1`] this, unfolded when_less_one[OF `\<bar>x\<bar> < 1`, symmetric], THEN spec]
2771 have "0 < 1 / real (n*2+1) * x^(n*2+1)" by (rule mult_pos_pos, auto simp only: zero_less_power[OF `0 < x`], auto)
2772 hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)" by (rule abs_of_pos)
2773 have "?diff x n \<le> ?a x n"
2774 proof (cases "even n")
2775 case True hence sgn_pos: "(-1)^n = (1::real)" by auto
2776 from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
2777 from bounds[of m, unfolded this atLeastAtMost_iff]
2778 have "\<bar>arctan x - (\<Sum>i = 0..<n. (?c x i))\<bar> \<le> (\<Sum>i = 0..<n + 1. (?c x i)) - (\<Sum>i = 0..<n. (?c x i))" by auto
2779 also have "\<dots> = ?c x n" unfolding One_nat_def by auto
2780 also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
2781 finally show ?thesis .
2783 case False hence sgn_neg: "(-1)^n = (-1::real)" by auto
2784 from `odd n` obtain m where m_def: "2 * m + 1 = n" unfolding odd_Suc_mult_two_ex by auto
2785 hence m_plus: "2 * (m + 1) = n + 1" by auto
2786 from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
2787 have "\<bar>arctan x - (\<Sum>i = 0..<n. (?c x i))\<bar> \<le> (\<Sum>i = 0..<n. (?c x i)) - (\<Sum>i = 0..<n+1. (?c x i))" by auto
2788 also have "\<dots> = - ?c x n" unfolding One_nat_def by auto
2789 also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
2790 finally show ?thesis .
2792 hence "0 \<le> ?a x n - ?diff x n" by auto
2794 hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
2795 moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
2796 unfolding diff_minus divide_inverse
2797 by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum)
2798 ultimately have "0 \<le> ?a 1 n - ?diff 1 n" by (rule LIM_less_bound)
2799 hence "?diff 1 n \<le> ?a 1 n" by auto
2802 unfolding LIMSEQ_rabs_zero power_one divide_inverse One_nat_def
2803 by (auto intro!: LIMSEQ_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)
2804 have "?diff 1 ----> 0"
2805 proof (rule LIMSEQ_I)
2806 fix r :: real assume "0 < r"
2807 obtain N :: nat where N_I: "\<And> n. N \<le> n \<Longrightarrow> ?a 1 n < r" using LIMSEQ_D[OF `?a 1 ----> 0` `0 < r`] by auto
2808 { fix n assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this]
2809 have "norm (?diff 1 n - 0) < r" by auto }
2810 thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
2812 from this[unfolded LIMSEQ_rabs_zero diff_minus add_commute[of "arctan 1"], THEN LIMSEQ_add_const, of "- arctan 1", THEN LIMSEQ_minus]
2813 have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
2814 hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
2817 proof (cases "x = 1", simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
2818 assume "x \<noteq> 1" hence "x = -1" using `\<bar>x\<bar> = 1` by auto
2820 have "- (pi / 2) < 0" using pi_gt_zero by auto
2821 have "- (2 * pi) < 0" using pi_gt_zero by auto
2823 have c_minus_minus: "\<And> i. ?c (- 1) i = - ?c 1 i" unfolding One_nat_def by auto
2825 have "arctan (- 1) = arctan (tan (-(pi / 4)))" unfolding tan_45 tan_minus ..
2826 also have "\<dots> = - (pi / 4)" by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
2827 also have "\<dots> = - (arctan (tan (pi / 4)))" unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero])
2828 also have "\<dots> = - (arctan 1)" unfolding tan_45 ..
2829 also have "\<dots> = - (\<Sum> i. ?c 1 i)" using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto
2830 also have "\<dots> = (\<Sum> i. ?c (- 1) i)" using suminf_minus[OF sums_summable[OF `(?c 1) sums (arctan 1)`]] unfolding c_minus_minus by auto
2831 finally show ?thesis using `x = -1` by auto
2836 lemma arctan_half: fixes x :: real
2837 shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x^2)))"
2839 obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x" using tan_total by blast
2840 hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2" by auto
2842 have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> A / B = (A / C) / (B / C)" by auto
2844 have "0 < cos y" using cos_gt_zero_pi[OF low high] .
2845 hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y) ^ 2) = cos y" by auto
2847 have "1 + (tan y)^2 = 1 + sin y^2 / cos y^2" unfolding tan_def power_divide ..
2848 also have "\<dots> = cos y^2 / cos y^2 + sin y^2 / cos y^2" using `cos y \<noteq> 0` by auto
2849 also have "\<dots> = 1 / cos y^2" unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
2850 finally have "1 + (tan y)^2 = 1 / cos y^2" .
2852 have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)" unfolding tan_def divide_nonzero_divide[OF `cos y \<noteq> 0`, symmetric] ..
2853 also have "\<dots> = tan y / (1 + 1 / cos y)" using `cos y \<noteq> 0` unfolding add_divide_distrib by auto
2854 also have "\<dots> = tan y / (1 + 1 / sqrt(cos y^2))" unfolding cos_sqrt ..
2855 also have "\<dots> = tan y / (1 + sqrt(1 / cos y^2))" unfolding real_sqrt_divide by auto
2856 finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)^2))" unfolding `1 + (tan y)^2 = 1 / cos y^2` .
2858 have "arctan x = y" using arctan_tan low high y_eq by auto
2859 also have "\<dots> = 2 * (arctan (tan (y/2)))" using arctan_tan[OF low2 high2] by auto
2860 also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" unfolding tan_half[OF low2 high2] by auto
2861 finally show ?thesis unfolding eq `tan y = x` .
2864 lemma arctan_monotone: assumes "x < y"
2865 shows "arctan x < arctan y"
2867 obtain z where "-(pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast
2868 obtain w where "-(pi / 2) < w" and "w < pi / 2" and "tan w = y" using tan_total by blast
2869 have "z < w" unfolding tan_monotone'[OF `-(pi / 2) < z` `z < pi / 2` `-(pi / 2) < w` `w < pi / 2`] `tan z = x` `tan w = y` using `x < y` .
2871 unfolding `tan z = x`[symmetric] arctan_tan[OF `-(pi / 2) < z` `z < pi / 2`]
2872 unfolding `tan w = y`[symmetric] arctan_tan[OF `-(pi / 2) < w` `w < pi / 2`] .
2875 lemma arctan_monotone': assumes "x \<le> y" shows "arctan x \<le> arctan y"
2876 proof (cases "x = y")
2877 case False hence "x < y" using `x \<le> y` by auto from arctan_monotone[OF this] show ?thesis by auto
2880 lemma arctan_minus: "arctan (- x) = - arctan x"
2882 obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast
2883 thus ?thesis unfolding `tan y = x`[symmetric] tan_minus[symmetric] using arctan_tan[of y] arctan_tan[of "-y"] by auto
2886 lemma arctan_inverse: assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
2888 obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast
2889 hence "y = arctan x" unfolding `tan y = x`[symmetric] using arctan_tan by auto
2891 { fix y x :: real assume "0 < y" and "y < pi /2" and "y = arctan x" and "tan y = x" hence "- (pi / 2) < y" by auto
2892 have "tan y > 0" using tan_monotone'[OF _ _ `- (pi / 2) < y` `y < pi / 2`, of 0] tan_zero `0 < y` by auto
2893 hence "x > 0" using `tan y = x` by auto
2895 have "- (pi / 2) < pi / 2 - y" using `y > 0` `y < pi / 2` by auto
2896 moreover have "pi / 2 - y < pi / 2" using `y > 0` `y < pi / 2` by auto
2897 ultimately have "arctan (1 / x) = pi / 2 - y" unfolding `tan y = x`[symmetric] tan_inverse using arctan_tan by auto
2898 hence "arctan (1 / x) = sgn x * pi / 2 - arctan x" unfolding `y = arctan x` real_sgn_pos[OF `x > 0`] by auto
2902 proof (cases "y > 0")
2903 case True from pos_y[OF this `y < pi / 2` `y = arctan x` `tan y = x`] show ?thesis .
2905 case False hence "y \<le> 0" by auto
2906 moreover have "y \<noteq> 0"
2908 assume "\<not> y \<noteq> 0" hence "y = 0" by auto
2909 have "x = 0" unfolding `tan y = x`[symmetric] `y = 0` tan_zero ..
2910 thus False using `x \<noteq> 0` by auto
2912 ultimately have "y < 0" by auto
2913 hence "0 < - y" and "-y < pi / 2" using `- (pi / 2) < y` by auto
2914 moreover have "-y = arctan (-x)" unfolding arctan_minus `y = arctan x` ..
2915 moreover have "tan (-y) = -x" unfolding tan_minus `tan y = x` ..
2916 ultimately have "arctan (1 / -x) = sgn (-x) * pi / 2 - arctan (-x)" using pos_y by blast
2917 hence "arctan (- (1 / x)) = - (sgn x * pi / 2 - arctan x)" unfolding arctan_minus[of x] divide_minus_right sgn_minus by auto
2918 thus ?thesis unfolding arctan_minus neg_equal_iff_equal .
2922 theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
2924 have "pi / 4 = arctan 1" using arctan1_eq_pi4 by auto
2925 also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
2926 finally show ?thesis by auto
2929 subsection {* Existence of Polar Coordinates *}
2931 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<twosuperior> + y\<twosuperior>)\<bar> \<le> 1"
2932 apply (rule power2_le_imp_le [OF _ zero_le_one])
2933 apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
2936 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
2937 by (simp add: abs_le_iff)
2939 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<twosuperior>)"
2940 by (simp add: sin_arccos abs_le_iff)
2942 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
2944 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
2947 "0 < y ==> \<exists>r a. x = r * cos a & y = r * sin a"
2948 apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>)" in exI)
2949 apply (rule_tac x = "arccos (x / sqrt (x\<twosuperior> + y\<twosuperior>))" in exI)
2950 apply (simp add: cos_arccos_lemma1)
2951 apply (simp add: sin_arccos_lemma1)
2952 apply (simp add: power_divide)
2953 apply (simp add: real_sqrt_mult [symmetric])
2954 apply (simp add: right_diff_distrib)
2958 "y < 0 ==> \<exists>r a. x = r * cos a & y = r * sin a"
2959 apply (insert polar_ex1 [where x=x and y="-y"], simp, clarify)
2960 apply (metis cos_minus minus_minus minus_mult_right sin_minus)
2963 lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
2964 apply (rule_tac x=0 and y=y in linorder_cases)
2965 apply (erule polar_ex1)
2966 apply (rule_tac x=x in exI, rule_tac x=0 in exI, simp)
2967 apply (erule polar_ex2)