1 (* Title : Transcendental.thy
2 Author : Jacques D. Fleuriot
3 Copyright : 1998,1999 University of Cambridge
4 1999,2001 University of Edinburgh
5 Conversion to Isar and new proofs by Lawrence C Paulson, 2004
8 header{*Power Series, Transcendental Functions etc.*}
11 imports Fact Series Deriv NthRoot
14 subsection {* Properties of Power Series *}
16 lemma lemma_realpow_diff:
17 fixes y :: "'a::monoid_mult"
18 shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
21 hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
22 thus ?thesis by (simp add: power_commutes)
25 lemma lemma_realpow_diff_sumr:
26 fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows
27 "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
28 y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
29 by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac
30 del: setsum_op_ivl_Suc cong: strong_setsum_cong)
32 lemma lemma_realpow_diff_sumr2:
33 fixes y :: "'a::{comm_ring,monoid_mult}" shows
34 "x ^ (Suc n) - y ^ (Suc n) =
35 (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
36 apply (induct n, simp)
37 apply (simp del: setsum_op_ivl_Suc)
38 apply (subst setsum_op_ivl_Suc)
39 apply (subst lemma_realpow_diff_sumr)
40 apply (simp add: right_distrib del: setsum_op_ivl_Suc)
41 apply (subst mult_left_commute [where a="x - y"])
43 apply (simp add: algebra_simps)
46 lemma lemma_realpow_rev_sumr:
47 "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
48 (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
49 apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
50 apply (rule inj_onI, simp)
52 apply (rule_tac x="n - x" in image_eqI, simp, simp)
55 text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
56 x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
59 fixes x z :: "'a::{real_normed_field,banach}"
60 assumes 1: "summable (\<lambda>n. f n * x ^ n)"
61 assumes 2: "norm z < norm x"
62 shows "summable (\<lambda>n. norm (f n * z ^ n))"
64 from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
65 from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
66 by (rule summable_LIMSEQ_zero)
67 hence "convergent (\<lambda>n. f n * x ^ n)"
69 hence "Cauchy (\<lambda>n. f n * x ^ n)"
70 by (simp add: Cauchy_convergent_iff)
71 hence "Bseq (\<lambda>n. f n * x ^ n)"
73 then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
74 by (simp add: Bseq_def, safe)
75 have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
76 K * norm (z ^ n) * inverse (norm (x ^ n))"
77 proof (intro exI allI impI)
78 fix n::nat assume "0 \<le> n"
79 have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
80 norm (f n * x ^ n) * norm (z ^ n)"
81 by (simp add: norm_mult abs_mult)
82 also have "\<dots> \<le> K * norm (z ^ n)"
83 by (simp only: mult_right_mono 4 norm_ge_zero)
84 also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
85 by (simp add: x_neq_0)
86 also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
87 by (simp only: mult_assoc)
88 finally show "norm (norm (f n * z ^ n)) \<le>
89 K * norm (z ^ n) * inverse (norm (x ^ n))"
90 by (simp add: mult_le_cancel_right x_neq_0)
92 moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
94 from 2 have "norm (norm (z * inverse x)) < 1"
96 by (simp add: nonzero_norm_divide divide_inverse [symmetric])
97 hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
98 by (rule summable_geometric)
99 hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
100 by (rule summable_mult)
101 thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
103 by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
104 power_inverse norm_power mult_assoc)
106 ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
107 by (rule summable_comparison_test)
111 fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" shows
112 "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]
113 ==> summable (%n. f(n) * (z ^ n))"
114 by (rule powser_insidea [THEN summable_norm_cancel])
116 lemma sum_split_even_odd: fixes f :: "nat \<Rightarrow> real" shows
117 "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
118 (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
121 have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =
122 (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
123 using Suc.hyps unfolding One_nat_def by auto
124 also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))" by auto
128 lemma sums_if': fixes g :: "nat \<Rightarrow> real" assumes "g sums x"
129 shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
131 proof (rule LIMSEQ_I)
132 fix r :: real assume "0 < r"
133 from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
134 obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast
136 let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
137 { fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> no" by auto
138 have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
139 using sum_split_even_odd by auto
140 hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
142 have "?SUM (2 * (m div 2)) = ?SUM m"
143 proof (cases "even m")
144 case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
146 case False hence "even (Suc m)" by auto
147 from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]] odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]
148 have eq: "Suc (2 * (m div 2)) = m" by auto
149 hence "even (2 * (m div 2))" using `odd m` by auto
150 have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
151 also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
152 finally show ?thesis by auto
154 ultimately have "(norm (?SUM m - x) < r)" by auto
156 thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
159 lemma sums_if: fixes g :: "nat \<Rightarrow> real" assumes "g sums x" and "f sums y"
160 shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
162 let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
163 { fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
164 by (cases B) auto } note if_sum = this
165 have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" using sums_if'[OF `g sums x`] .
167 have "?s 0 = 0" by auto
168 have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
169 { fix B T E have "(if \<not> B then T else E) = (if B then E else T)" by auto } note if_eq = this
171 have "?s sums y" using sums_if'[OF `f sums y`] .
172 from this[unfolded sums_def, THEN LIMSEQ_Suc]
173 have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
174 unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
175 image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
176 even_Suc Suc_m1 if_eq .
177 } from sums_add[OF g_sums this]
178 show ?thesis unfolding if_sum .
181 subsection {* Alternating series test / Leibniz formula *}
183 lemma sums_alternating_upper_lower:
184 fixes a :: "nat \<Rightarrow> real"
185 assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
186 shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and>
187 ((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. -1^i*a i) ----> l)"
188 (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
190 have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
192 have "\<forall> n. ?f n \<le> ?f (Suc n)"
193 proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed
195 have "\<forall> n. ?g (Suc n) \<le> ?g n"
196 proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
197 unfolding One_nat_def by auto qed
199 have "\<forall> n. ?f n \<le> ?g n"
200 proof fix n show "?f n \<le> ?g n" using fg_diff a_pos
201 unfolding One_nat_def by auto qed
203 have "(\<lambda> n. ?f n - ?g n) ----> 0" unfolding fg_diff
204 proof (rule LIMSEQ_I)
205 fix r :: real assume "0 < r"
206 with `a ----> 0`[THEN LIMSEQ_D]
207 obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" by auto
208 hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
209 thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
212 show ?thesis by (rule lemma_nest_unique)
215 lemma summable_Leibniz': fixes a :: "nat \<Rightarrow> real"
216 assumes a_zero: "a ----> 0" and a_pos: "\<And> n. 0 \<le> a n"
217 and a_monotone: "\<And> n. a (Suc n) \<le> a n"
218 shows summable: "summable (\<lambda> n. (-1)^n * a n)"
219 and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
220 and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
221 and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
222 and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
224 let "?S n" = "(-1)^n * a n"
225 let "?P n" = "\<Sum>i=0..<n. ?S i"
226 let "?f n" = "?P (2 * n)"
227 let "?g n" = "?P (2 * n + 1)"
228 obtain l :: real where below_l: "\<forall> n. ?f n \<le> l" and "?f ----> l" and above_l: "\<forall> n. l \<le> ?g n" and "?g ----> l"
229 using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
231 let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n"
233 proof (rule LIMSEQ_I)
234 fix r :: real assume "0 < r"
236 with `?f ----> l`[THEN LIMSEQ_D]
237 obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
239 from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
240 obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
243 assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
244 have "norm (?Sa n - l) < r"
245 proof (cases "even n")
246 case True from even_nat_div_two_times_two[OF this]
247 have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto
248 with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" by auto
250 show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
252 case False hence "even (n - 1)" using even_num_iff odd_pos by auto
253 from even_nat_div_two_times_two[OF this]
254 have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto
255 hence range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto
257 from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" by auto
259 show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
262 thus "\<exists> no. \<forall> n \<ge> no. norm (?Sa n - l) < r" by blast
264 hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
265 thus "summable ?S" using summable_def by auto
267 have "l = suminf ?S" using sums_unique[OF sums_l] .
269 { fix n show "suminf ?S \<le> ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto }
270 { fix n show "?f n \<le> suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_l by auto }
271 show "?g ----> suminf ?S" using `?g ----> l` `l = suminf ?S` by auto
272 show "?f ----> suminf ?S" using `?f ----> l` `l = suminf ?S` by auto
275 theorem summable_Leibniz: fixes a :: "nat \<Rightarrow> real"
276 assumes a_zero: "a ----> 0" and "monoseq a"
277 shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
278 and "0 < a 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n. -1^i * a i .. \<Sum>i=0..<2*n+1. -1^i * a i})" (is "?pos")
279 and "a 0 < 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n+1. -1^i * a i .. \<Sum>i=0..<2*n. -1^i * a i})" (is "?neg")
280 and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
281 and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
283 have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
284 proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
286 hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" by auto
287 { fix n have "a (Suc n) \<le> a n" using ord[where n="Suc n" and m=n] by auto }
288 note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] and mono = this
289 from leibniz[OF mono]
290 show ?thesis using `0 \<le> a 0` by auto
292 let ?a = "\<lambda> n. - a n"
294 with monoseq_le[OF `monoseq a` `a ----> 0`]
295 have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
296 hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" by auto
297 { fix n have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] by auto }
299 note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF LIMSEQ_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
300 have "summable (\<lambda> n. (-1)^n * ?a n)" using leibniz(1) by auto
301 then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" unfolding summable_def by auto
302 from this[THEN sums_minus]
303 have "(\<lambda> n. (-1)^n * a n) sums -l" by auto
304 hence ?summable unfolding summable_def by auto
306 have "\<And> a b :: real. \<bar> - a - - b \<bar> = \<bar>a - b\<bar>" unfolding minus_diff_minus by auto
308 from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
309 have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)" by auto
311 have ?pos using `0 \<le> ?a 0` by auto
312 moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto
313 moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN LIMSEQ_minus_cancel] by auto
314 ultimately show ?thesis by auto
316 from this[THEN conjunct1] this[THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1]
317 this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]
318 show ?summable and ?pos and ?neg and ?f and ?g .
321 subsection {* Term-by-Term Differentiability of Power Series *}
324 diffs :: "(nat => 'a::ring_1) => nat => 'a" where
325 "diffs c = (%n. of_nat (Suc n) * c(Suc n))"
327 text{*Lemma about distributing negation over it*}
328 lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
329 by (simp add: diffs_def)
333 shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
335 apply (rule LIMSEQ_imp_Suc)
336 apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
337 apply (simp only: setsum_shift_bounds_Suc_ivl)
341 "summable (%n. (diffs c)(n) * (x ^ n)) ==>
342 (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
343 (\<Sum>n. (diffs c)(n) * (x ^ n))"
345 apply (drule summable_sums)
346 apply (rule sums_Suc_imp, simp_all)
349 lemma lemma_termdiff1:
350 fixes z :: "'a :: {monoid_mult,comm_ring}" shows
351 "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
352 (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
353 by(auto simp add: algebra_simps power_add [symmetric] cong: strong_setsum_cong)
355 lemma sumr_diff_mult_const2:
356 "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
357 by (simp add: setsum_subtractf)
359 lemma lemma_termdiff2:
360 fixes h :: "'a :: {field}"
361 assumes h: "h \<noteq> 0" shows
362 "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
363 h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
364 (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
365 apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
366 apply (simp add: right_diff_distrib diff_divide_distrib h)
367 apply (simp add: mult_assoc [symmetric])
368 apply (cases "n", simp)
369 apply (simp add: lemma_realpow_diff_sumr2 h
370 right_diff_distrib [symmetric] mult_assoc
371 del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
372 apply (subst lemma_realpow_rev_sumr)
373 apply (subst sumr_diff_mult_const2)
375 apply (simp only: lemma_termdiff1 setsum_right_distrib)
376 apply (rule setsum_cong [OF refl])
377 apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
379 apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
380 del: setsum_op_ivl_Suc power_Suc)
381 apply (subst mult_assoc [symmetric], subst power_add [symmetric])
382 apply (simp add: mult_ac)
385 lemma real_setsum_nat_ivl_bounded2:
386 fixes K :: "'a::ordered_semidom"
387 assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
388 assumes K: "0 \<le> K"
389 shows "setsum f {0..<n-k} \<le> of_nat n * K"
390 apply (rule order_trans [OF setsum_mono])
392 apply (simp add: mult_right_mono K)
395 lemma lemma_termdiff3:
396 fixes h z :: "'a::{real_normed_field}"
397 assumes 1: "h \<noteq> 0"
398 assumes 2: "norm z \<le> K"
399 assumes 3: "norm (z + h) \<le> K"
400 shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
401 \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
403 have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
404 norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
405 (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
406 apply (subst lemma_termdiff2 [OF 1])
407 apply (subst norm_mult)
408 apply (rule mult_commute)
410 also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
411 proof (rule mult_right_mono [OF _ norm_ge_zero])
412 from norm_ge_zero 2 have K: "0 \<le> K" by (rule order_trans)
413 have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
415 apply (simp only: norm_mult norm_power power_add)
416 apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
418 show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
419 (z + h) ^ q * z ^ (n - 2 - q))
420 \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
422 order_trans [OF norm_setsum]
423 real_setsum_nat_ivl_bounded2
427 apply (rule le_Kn, simp)
430 also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
431 by (simp only: mult_assoc)
432 finally show ?thesis .
435 lemma lemma_termdiff4:
436 fixes f :: "'a::{real_normed_field} \<Rightarrow>
437 'b::real_normed_vector"
438 assumes k: "0 < (k::real)"
439 assumes le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
441 unfolding LIM_def diff_0_right
443 let ?h = "of_real (k / 2)::'a"
444 have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
445 hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
446 hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
447 hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
449 fix r::real assume r: "0 < r"
450 show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
453 with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
455 thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
457 assume K_neq_zero: "K \<noteq> 0"
458 with zero_le_K have K: "0 < K" by simp
459 show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
460 proof (rule exI, safe)
461 from k r K show "0 < min k (r * inverse K / 2)"
462 by (simp add: mult_pos_pos positive_imp_inverse_positive)
465 assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
466 from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
468 from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
469 also from x4 K have "K * norm x < K * (r * inverse K / 2)"
470 by (rule mult_strict_left_mono)
471 also have "\<dots> = r / 2"
472 using K_neq_zero by simp
473 also have "r / 2 < r"
475 finally show "norm (f x) < r" .
480 lemma lemma_termdiff5:
481 fixes g :: "'a::{real_normed_field} \<Rightarrow>
482 nat \<Rightarrow> 'b::banach"
483 assumes k: "0 < (k::real)"
484 assumes f: "summable f"
485 assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
486 shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
487 proof (rule lemma_termdiff4 [OF k])
488 fix h::'a assume "h \<noteq> 0" and "norm h < k"
489 hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
491 hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
493 moreover from f have B: "summable (\<lambda>n. f n * norm h)"
494 by (rule summable_mult2)
495 ultimately have C: "summable (\<lambda>n. norm (g h n))"
496 by (rule summable_comparison_test)
497 hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
498 by (rule summable_norm)
499 also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
500 by (rule summable_le)
501 also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
502 by (rule suminf_mult2 [symmetric])
503 finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
507 text{* FIXME: Long proofs*}
510 fixes x :: "'a::{real_normed_field,banach}"
511 assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
512 assumes 2: "norm x < norm K"
513 shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
514 - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
517 obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
518 from norm_ge_zero r1 have r: "0 < r"
519 by (rule order_le_less_trans)
520 hence r_neq_0: "r \<noteq> 0" by simp
522 proof (rule lemma_termdiff5)
523 show "0 < r - norm x" using r1 by simp
525 from r r2 have "norm (of_real r::'a) < norm K"
527 with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
528 by (rule powser_insidea)
529 hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
531 by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
532 hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
533 by (rule diffs_equiv [THEN sums_summable])
534 also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))
535 = (\<lambda>n. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
537 apply (simp add: diffs_def)
538 apply (case_tac n, simp_all add: r_neq_0)
540 finally have "summable
541 (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
542 by (rule diffs_equiv [THEN sums_summable])
544 "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
546 (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
548 apply (case_tac "n", simp)
549 apply (case_tac "nat", simp)
550 apply (simp add: r_neq_0)
553 "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
556 assume h: "h \<noteq> 0"
557 assume "norm h < r - norm x"
558 hence "norm x + norm h < r" by simp
559 with norm_triangle_ineq have xh: "norm (x + h) < r"
560 by (rule order_le_less_trans)
561 show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
562 \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
563 apply (simp only: norm_mult mult_assoc)
564 apply (rule mult_left_mono [OF _ norm_ge_zero])
565 apply (simp (no_asm) add: mult_assoc [symmetric])
566 apply (rule lemma_termdiff3)
568 apply (rule r1 [THEN order_less_imp_le])
569 apply (rule xh [THEN order_less_imp_le])
575 fixes K x :: "'a::{real_normed_field,banach}"
576 assumes 1: "summable (\<lambda>n. c n * K ^ n)"
577 assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
578 assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
579 assumes 4: "norm x < norm K"
580 shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
582 proof (rule LIM_zero_cancel)
583 show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
584 - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
585 proof (rule LIM_equal2)
586 show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
589 assume "h \<noteq> 0"
590 assume "norm (h - 0) < norm K - norm x"
591 hence "norm x + norm h < norm K" by simp
592 hence 5: "norm (x + h) < norm K"
593 by (rule norm_triangle_ineq [THEN order_le_less_trans])
594 have A: "summable (\<lambda>n. c n * x ^ n)"
595 by (rule powser_inside [OF 1 4])
596 have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
597 by (rule powser_inside [OF 1 5])
598 have C: "summable (\<lambda>n. diffs c n * x ^ n)"
599 by (rule powser_inside [OF 2 4])
600 show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
601 - (\<Sum>n. diffs c n * x ^ n) =
602 (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
603 apply (subst sums_unique [OF diffs_equiv [OF C]])
604 apply (subst suminf_diff [OF B A])
605 apply (subst suminf_divide [symmetric])
606 apply (rule summable_diff [OF B A])
607 apply (subst suminf_diff)
608 apply (rule summable_divide)
609 apply (rule summable_diff [OF B A])
610 apply (rule sums_summable [OF diffs_equiv [OF C]])
611 apply (rule arg_cong [where f="suminf"], rule ext)
612 apply (simp add: algebra_simps)
615 show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h -
616 of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
617 by (rule termdiffs_aux [OF 3 4])
622 subsection{* Some properties of factorials *}
624 lemma real_of_nat_fact_not_zero [simp]: "real (fact n) \<noteq> 0"
627 lemma real_of_nat_fact_gt_zero [simp]: "0 < real(fact n)"
630 lemma real_of_nat_fact_ge_zero [simp]: "0 \<le> real(fact n)"
633 lemma inv_real_of_nat_fact_gt_zero [simp]: "0 < inverse (real (fact n))"
634 by (auto simp add: positive_imp_inverse_positive)
636 lemma inv_real_of_nat_fact_ge_zero [simp]: "0 \<le> inverse (real (fact n))"
637 by (auto intro: order_less_imp_le)
639 subsection {* Derivability of power series *}
641 lemma DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
642 assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
643 and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
644 and "summable (f' x0)"
645 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>"
646 shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
649 fix r :: real assume "0 < r" hence "0 < r/3" by auto
651 obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
652 using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
654 obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
655 using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
657 let ?N = "Suc (max N_L N_f')"
658 have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
659 L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
661 let "?diff i x" = "(f (x0 + x) i - f x0 i) / x"
663 let ?r = "r / (3 * real ?N)"
664 have "0 < 3 * real ?N" by auto
665 from divide_pos_pos[OF `0 < r` this]
668 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)"
669 def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
671 have "0 < S'" unfolding S'_def
672 proof (rule iffD2[OF Min_gr_iff])
673 show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x"
675 fix x assume "x \<in> ?s ` {0..<?N}"
676 then obtain n where "x = ?s n" and "n \<in> {0..<?N}" using image_iff[THEN iffD1] by blast
677 from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
678 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
679 have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)
680 thus "0 < x" unfolding `x = ?s n` .
684 def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
685 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'`
688 { fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
689 hence x_in_I: "x0 + x \<in> { a <..< b }" using S_a S_b by auto
691 note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
692 note div_smbl = summable_divide[OF diff_smbl]
693 note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
694 note ign = summable_ignore_initial_segment[where k="?N"]
695 note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
696 note div_shft_smbl = summable_divide[OF diff_shft_smbl]
697 note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
700 have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
701 using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .
702 hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 0` by auto
703 } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
704 from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
705 have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
706 hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") using L_estimate by auto
708 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>)" ..
709 also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
710 proof (rule setsum_strict_mono)
711 fix n assume "n \<in> { 0 ..< ?N}"
712 have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .
713 also have "S \<le> S'" using `S \<le> S'` .
714 also have "S' \<le> ?s n" unfolding S'_def
715 proof (rule Min_le_iff[THEN iffD2])
716 have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto
717 thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
719 finally have "\<bar> x \<bar> < ?s n" .
721 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]
722 have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
723 with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n`
724 show "\<bar>?diff n x - f' x0 n\<bar> < ?r" by blast
726 also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)
727 also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto
728 also have "\<dots> = r/3" by auto
729 finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
731 from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
732 have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> =
733 \<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
734 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)
735 also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto
736 also have "\<dots> < r /3 + r/3 + r/3"
737 using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3` by auto
738 finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"
740 } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
741 norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`
742 unfolding real_norm_def diff_0_right by blast
745 lemma DERIV_power_series': fixes f :: "nat \<Rightarrow> real"
746 assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
747 and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
748 shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
749 (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
751 { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
752 hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" by auto
753 have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
754 proof (rule DERIV_series')
755 show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
757 have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto
758 hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" using `R' < R` by auto
759 have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto
760 from powser_insidea[OF converges[OF in_Rball] this] show ?thesis by auto
762 { fix n x y assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
763 show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
765 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>"
766 unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto
767 also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
768 proof (rule mult_left_mono)
769 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)
770 also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
771 proof (rule setsum_mono)
772 fix p assume "p \<in> {0..<Suc n}" hence "p \<le> n" by auto
773 { fix n fix x :: real assume "x \<in> {-R'<..<R'}"
774 hence "\<bar>x\<bar> \<le> R'" by auto
775 hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto)
776 } from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
777 have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" unfolding abs_mult by auto
778 thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> n` by auto
780 also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
781 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'`]]] .
782 show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult[symmetric] by auto
784 also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult real_mult_assoc[symmetric] by algebra
785 finally show ?thesis .
788 from DERIV_pow[of "Suc n" x0, THEN DERIV_cmult[where c="f n"]]
789 show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)" unfolding real_mult_assoc by auto }
790 { fix x assume "x \<in> {-R' <..< R'}" hence "R' \<in> {-R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto
791 have "summable (\<lambda> n. f n * x^n)"
792 proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
794 have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto)
795 show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult
796 by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right])
798 from this[THEN summable_mult2[where c=x], unfolded real_mult_assoc, unfolded real_mult_commute]
799 show "summable (?f x)" by auto }
800 show "summable (?f' x0)" using converges[OF `x0 \<in> {-R <..< R}`] .
801 show "x0 \<in> {-R' <..< R'}" using `x0 \<in> {-R' <..< R'}` .
803 } note for_subinterval = this
804 let ?R = "(R + \<bar>x0\<bar>) / 2"
805 have "\<bar>x0\<bar> < ?R" using assms by auto
807 proof (cases "x0 < 0")
809 hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
810 thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
813 have "- ?R < 0" using assms by auto
814 also have "\<dots> \<le> x0" using False by auto
815 finally show ?thesis .
817 hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto
818 from for_subinterval[OF this]
822 subsection {* Exponential Function *}
825 exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" where
826 "exp x = (\<Sum>n. x ^ n /\<^sub>R real (fact n))"
828 lemma summable_exp_generic:
829 fixes x :: "'a::{real_normed_algebra_1,banach}"
830 defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
833 have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
834 unfolding S_def by (simp del: mult_Suc)
835 obtain r :: real where r0: "0 < r" and r1: "r < 1"
836 using dense [OF zero_less_one] by fast
837 obtain N :: nat where N: "norm x < real N * r"
838 using reals_Archimedean3 [OF r0] by fast
840 proof (rule ratio_test [rule_format])
842 assume n: "N \<le> n"
843 have "norm x \<le> real N * r"
844 using N by (rule order_less_imp_le)
845 also have "real N * r \<le> real (Suc n) * r"
846 using r0 n by (simp add: mult_right_mono)
847 finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
848 using norm_ge_zero by (rule mult_right_mono)
849 hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
850 by (rule order_trans [OF norm_mult_ineq])
851 hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
852 by (simp add: pos_divide_le_eq mult_ac)
853 thus "norm (S (Suc n)) \<le> r * norm (S n)"
854 by (simp add: S_Suc norm_scaleR inverse_eq_divide)
858 lemma summable_norm_exp:
859 fixes x :: "'a::{real_normed_algebra_1,banach}"
860 shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
861 proof (rule summable_norm_comparison_test [OF exI, rule_format])
862 show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
863 by (rule summable_exp_generic)
865 fix n show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
866 by (simp add: norm_scaleR norm_power_ineq)
869 lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
870 by (insert summable_exp_generic [where x=x], simp)
872 lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
873 unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
877 "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
878 by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
879 del: mult_Suc of_nat_Suc)
881 lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
882 by (simp add: diffs_def)
884 lemma lemma_exp_ext: "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
885 by (auto intro!: ext simp add: exp_def)
887 lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
888 apply (simp add: exp_def)
889 apply (subst lemma_exp_ext)
890 apply (subgoal_tac "DERIV (\<lambda>u. \<Sum>n. of_real (inverse (real (fact n))) * u ^ n) x :> (\<Sum>n. diffs (\<lambda>n. of_real (inverse (real (fact n)))) n * x ^ n)")
891 apply (rule_tac [2] K = "of_real (1 + norm x)" in termdiffs)
892 apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
893 apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
894 apply (simp del: of_real_add)
897 lemma isCont_exp [simp]: "isCont exp x"
898 by (rule DERIV_exp [THEN DERIV_isCont])
901 subsubsection {* Properties of the Exponential Function *}
904 fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
905 shows "(\<Sum>n. f n * 0 ^ n) = f 0"
907 have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
908 by (rule sums_unique [OF series_zero], simp add: power_0_left)
909 thus ?thesis unfolding One_nat_def by simp
912 lemma exp_zero [simp]: "exp 0 = 1"
913 unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
915 lemma setsum_cl_ivl_Suc2:
916 "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
917 by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
918 del: setsum_cl_ivl_Suc)
920 lemma exp_series_add:
921 fixes x y :: "'a::{real_field}"
922 defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
923 shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
927 unfolding S_def by simp
930 have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
931 unfolding S_def by (simp del: mult_Suc)
932 hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
935 have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
936 by (simp only: times_S)
937 also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
939 also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
940 + y * (\<Sum>i=0..n. S x i * S y (n-i))"
941 by (rule left_distrib)
942 also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
943 + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
944 by (simp only: setsum_right_distrib mult_ac)
945 also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
946 + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
947 by (simp add: times_S Suc_diff_le)
948 also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
949 (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
950 by (subst setsum_cl_ivl_Suc2, simp)
951 also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
952 (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
953 by (subst setsum_cl_ivl_Suc, simp)
954 also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
955 (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
956 (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
957 by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
958 real_of_nat_add [symmetric], simp)
959 also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
960 by (simp only: scaleR_right.setsum)
962 "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
963 by (simp add: scaleR_cancel_left del: setsum_cl_ivl_Suc)
966 lemma exp_add: "exp (x + y) = exp x * exp y"
968 by (simp only: Cauchy_product summable_norm_exp exp_series_add)
970 lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
971 by (rule exp_add [symmetric])
973 lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
975 apply (subst of_real.suminf)
976 apply (rule summable_exp_generic)
977 apply (simp add: scaleR_conv_of_real)
980 lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
982 have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
983 also assume "exp x = 0"
984 finally show "False" by simp
987 lemma exp_minus: "exp (- x) = inverse (exp x)"
988 by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
990 lemma exp_diff: "exp (x - y) = exp x / exp y"
991 unfolding diff_minus divide_inverse
992 by (simp add: exp_add exp_minus)
995 subsubsection {* Properties of the Exponential Function on Reals *}
997 text {* Comparisons of @{term "exp x"} with zero. *}
999 text{*Proof: because every exponential can be seen as a square.*}
1000 lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
1002 have "0 \<le> exp (x/2) * exp (x/2)" by simp
1003 thus ?thesis by (simp add: exp_add [symmetric])
1006 lemma exp_gt_zero [simp]: "0 < exp (x::real)"
1007 by (simp add: order_less_le)
1009 lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
1010 by (simp add: not_less)
1012 lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
1013 by (simp add: not_le)
1015 lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
1018 lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
1020 apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
1023 text {* Strict monotonicity of exponential. *}
1025 lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
1026 apply (drule order_le_imp_less_or_eq, auto)
1027 apply (simp add: exp_def)
1028 apply (rule real_le_trans)
1029 apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
1030 apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
1033 lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
1036 hence "1 < 1 + x" by simp
1037 also from x have "1 + x \<le> exp x"
1038 by (simp add: exp_ge_add_one_self_aux)
1039 finally show ?thesis .
1042 lemma exp_less_mono:
1044 assumes "x < y" shows "exp x < exp y"
1046 from `x < y` have "0 < y - x" by simp
1047 hence "1 < exp (y - x)" by (rule exp_gt_one)
1048 hence "1 < exp y / exp x" by (simp only: exp_diff)
1049 thus "exp x < exp y" by simp
1052 lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
1053 apply (simp add: linorder_not_le [symmetric])
1054 apply (auto simp add: order_le_less exp_less_mono)
1057 lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
1058 by (auto intro: exp_less_mono exp_less_cancel)
1060 lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
1061 by (auto simp add: linorder_not_less [symmetric])
1063 lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
1064 by (simp add: order_eq_iff)
1066 text {* Comparisons of @{term "exp x"} with one. *}
1068 lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
1069 using exp_less_cancel_iff [where x=0 and y=x] by simp
1071 lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
1072 using exp_less_cancel_iff [where x=x and y=0] by simp
1074 lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
1075 using exp_le_cancel_iff [where x=0 and y=x] by simp
1077 lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
1078 using exp_le_cancel_iff [where x=x and y=0] by simp
1080 lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
1081 using exp_inj_iff [where x=x and y=0] by simp
1083 lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
1085 apply (auto intro: isCont_exp simp add: le_diff_eq)
1086 apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)")
1088 apply (rule exp_ge_add_one_self_aux, simp)
1091 lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
1092 apply (rule_tac x = 1 and y = y in linorder_cases)
1093 apply (drule order_less_imp_le [THEN lemma_exp_total])
1094 apply (rule_tac [2] x = 0 in exI)
1095 apply (frule_tac [3] real_inverse_gt_one)
1096 apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
1097 apply (rule_tac x = "-x" in exI)
1098 apply (simp add: exp_minus)
1102 subsection {* Natural Logarithm *}
1105 ln :: "real => real" where
1106 "ln x = (THE u. exp u = x)"
1108 lemma ln_exp [simp]: "ln (exp x) = x"
1109 by (simp add: ln_def)
1111 lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
1112 by (auto dest: exp_total)
1114 lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
1116 apply (erule subst, rule exp_gt_zero)
1117 apply (erule exp_ln)
1120 lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
1121 by (erule subst, rule ln_exp)
1123 lemma ln_one [simp]: "ln 1 = 0"
1124 by (rule ln_unique, simp)
1126 lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"
1127 by (rule ln_unique, simp add: exp_add)
1129 lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
1130 by (rule ln_unique, simp add: exp_minus)
1132 lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"
1133 by (rule ln_unique, simp add: exp_diff)
1135 lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
1136 by (rule ln_unique, simp add: exp_real_of_nat_mult)
1138 lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
1139 by (subst exp_less_cancel_iff [symmetric], simp)
1141 lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
1142 by (simp add: linorder_not_less [symmetric])
1144 lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
1145 by (simp add: order_eq_iff)
1147 lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
1148 apply (rule exp_le_cancel_iff [THEN iffD1])
1149 apply (simp add: exp_ge_add_one_self_aux)
1152 lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
1153 by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
1155 lemma ln_ge_zero [simp]:
1156 assumes x: "1 \<le> x" shows "0 \<le> ln x"
1158 have "0 < x" using x by arith
1159 hence "exp 0 \<le> exp (ln x)"
1161 thus ?thesis by (simp only: exp_le_cancel_iff)
1164 lemma ln_ge_zero_imp_ge_one:
1165 assumes ln: "0 \<le> ln x"
1169 from ln have "ln 1 \<le> ln x" by simp
1170 thus ?thesis by (simp add: x del: ln_one)
1173 lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
1174 by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
1176 lemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)"
1177 by (insert ln_ge_zero_iff [of x], arith)
1180 assumes x: "1 < x" shows "0 < ln x"
1182 have "0 < x" using x by arith
1183 hence "exp 0 < exp (ln x)" by (simp add: x)
1184 thus ?thesis by (simp only: exp_less_cancel_iff)
1187 lemma ln_gt_zero_imp_gt_one:
1188 assumes ln: "0 < ln x"
1192 from ln have "ln 1 < ln x" by simp
1193 thus ?thesis by (simp add: x del: ln_one)
1196 lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
1197 by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
1199 lemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)"
1200 by (insert ln_less_zero_iff [of x] ln_gt_zero_iff [of x], arith)
1202 lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
1205 lemma exp_ln_eq: "exp u = x ==> ln x = u"
1208 lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
1209 apply (subgoal_tac "isCont ln (exp (ln x))", simp)
1210 apply (rule isCont_inverse_function [where f=exp], simp_all)
1213 lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
1214 apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
1215 apply (erule lemma_DERIV_subst [OF DERIV_exp exp_ln])
1216 apply (simp_all add: abs_if isCont_ln)
1219 lemma ln_series: assumes "0 < x" and "x < 2"
1220 shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))")
1222 let "?f' x n" = "(-1)^n * (x - 1)^n"
1224 have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
1225 proof (rule DERIV_isconst3[where x=x])
1226 fix x :: real assume "x \<in> {0 <..< 2}" hence "0 < x" and "x < 2" by auto
1227 have "norm (1 - x) < 1" using `0 < x` and `x < 2` by auto
1228 have "1 / x = 1 / (1 - (1 - x))" by auto
1229 also have "\<dots> = (\<Sum> n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
1230 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)
1231 finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding real_divide_def by auto
1233 have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
1234 have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
1235 proof (rule DERIV_power_series')
1236 show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto
1237 { fix x :: real assume "x \<in> {- 1<..<1}" hence "norm (-x) < 1" by auto
1238 show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
1239 unfolding One_nat_def
1240 by (auto simp del: power_mult_distrib simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
1243 hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" unfolding One_nat_def by auto
1244 hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos .
1245 ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
1246 by (rule DERIV_diff)
1247 thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
1248 qed (auto simp add: assms)
1249 thus ?thesis by (auto simp add: suminf_zero)
1252 subsection {* Sine and Cosine *}
1255 sin :: "real => real" where
1256 "sin x = (\<Sum>n. (if even(n) then 0 else
1257 (-1 ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n)"
1260 cos :: "real => real" where
1261 "cos x = (\<Sum>n. (if even(n) then (-1 ^ (n div 2))/(real (fact n))
1267 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) *
1269 apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
1270 apply (rule_tac [2] summable_exp)
1271 apply (rule_tac x = 0 in exI)
1272 apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
1278 -1 ^ (n div 2)/(real (fact n)) else 0) * x ^ n)"
1279 apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
1280 apply (rule_tac [2] summable_exp)
1281 apply (rule_tac x = 0 in exI)
1282 apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
1285 lemma lemma_STAR_sin:
1287 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"
1288 by (induct "n", auto)
1290 lemma lemma_STAR_cos:
1292 -1 ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
1293 by (induct "n", auto)
1295 lemma lemma_STAR_cos1:
1297 (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
1298 by (induct "n", auto)
1300 lemma lemma_STAR_cos2:
1301 "(\<Sum>n=1..<n. if even n then -1 ^ (n div 2)/(real (fact n)) * 0 ^ n
1304 apply (case_tac [2] "n", auto)
1307 lemma sin_converges:
1308 "(%n. (if even n then 0
1309 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) *
1311 unfolding sin_def by (rule summable_sin [THEN summable_sums])
1313 lemma cos_converges:
1314 "(%n. (if even n then
1315 -1 ^ (n div 2)/(real (fact n))
1316 else 0) * x ^ n) sums cos(x)"
1317 unfolding cos_def by (rule summable_cos [THEN summable_sums])
1320 "diffs(%n. if even n then 0
1321 else -1 ^ ((n - Suc 0) div 2)/(real (fact n)))
1322 = (%n. if even n then
1323 -1 ^ (n div 2)/(real (fact n))
1325 by (auto intro!: ext
1326 simp add: diffs_def divide_inverse real_of_nat_def of_nat_mult
1327 simp del: mult_Suc of_nat_Suc)
1330 "diffs(%n. if even n then 0
1331 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) n
1333 -1 ^ (n div 2)/(real (fact n))
1335 by (simp only: sin_fdiffs)
1338 "diffs(%n. if even n then
1339 -1 ^ (n div 2)/(real (fact n)) else 0)
1340 = (%n. - (if even n then 0
1341 else -1 ^ ((n - Suc 0)div 2)/(real (fact n))))"
1342 by (auto intro!: ext
1343 simp add: diffs_def divide_inverse odd_Suc_mult_two_ex real_of_nat_def of_nat_mult
1344 simp del: mult_Suc of_nat_Suc)
1348 "diffs(%n. if even n then
1349 -1 ^ (n div 2)/(real (fact n)) else 0) n
1350 = - (if even n then 0
1351 else -1 ^ ((n - Suc 0)div 2)/(real (fact n)))"
1352 by (simp only: cos_fdiffs)
1354 text{*Now at last we can get the derivatives of exp, sin and cos*}
1356 lemma lemma_sin_minus:
1357 "- sin x = (\<Sum>n. - ((if even n then 0
1358 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))"
1359 by (auto intro!: sums_unique sums_minus sin_converges)
1361 lemma lemma_sin_ext:
1362 "sin = (%x. \<Sum>n.
1364 else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) *
1366 by (auto intro!: ext simp add: sin_def)
1368 lemma lemma_cos_ext:
1369 "cos = (%x. \<Sum>n.
1370 (if even n then -1 ^ (n div 2)/(real (fact n)) else 0) *
1372 by (auto intro!: ext simp add: cos_def)
1374 lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
1375 apply (simp add: cos_def)
1376 apply (subst lemma_sin_ext)
1377 apply (auto simp add: sin_fdiffs2 [symmetric])
1378 apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
1379 apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs)
1382 lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
1383 apply (subst lemma_cos_ext)
1384 apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
1385 apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
1386 apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus)
1389 lemma isCont_sin [simp]: "isCont sin x"
1390 by (rule DERIV_sin [THEN DERIV_isCont])
1392 lemma isCont_cos [simp]: "isCont cos x"
1393 by (rule DERIV_cos [THEN DERIV_isCont])
1396 subsection {* Properties of Sine and Cosine *}
1398 lemma sin_zero [simp]: "sin 0 = 0"
1399 unfolding sin_def by (simp add: powser_zero)
1401 lemma cos_zero [simp]: "cos 0 = 1"
1402 unfolding cos_def by (simp add: powser_zero)
1404 lemma DERIV_sin_sin_mult [simp]:
1405 "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)"
1406 by (rule DERIV_mult, auto)
1408 lemma DERIV_sin_sin_mult2 [simp]:
1409 "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"
1410 apply (cut_tac x = x in DERIV_sin_sin_mult)
1411 apply (auto simp add: mult_assoc)
1414 lemma DERIV_sin_realpow2 [simp]:
1415 "DERIV (%x. (sin x)\<twosuperior>) x :> cos(x) * sin(x) + cos(x) * sin(x)"
1416 by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
1418 lemma DERIV_sin_realpow2a [simp]:
1419 "DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"
1420 by (auto simp add: numeral_2_eq_2)
1422 lemma DERIV_cos_cos_mult [simp]:
1423 "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
1424 by (rule DERIV_mult, auto)
1426 lemma DERIV_cos_cos_mult2 [simp]:
1427 "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"
1428 apply (cut_tac x = x in DERIV_cos_cos_mult)
1429 apply (auto simp add: mult_ac)
1432 lemma DERIV_cos_realpow2 [simp]:
1433 "DERIV (%x. (cos x)\<twosuperior>) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
1434 by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
1436 lemma DERIV_cos_realpow2a [simp]:
1437 "DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"
1438 by (auto simp add: numeral_2_eq_2)
1440 lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
1443 lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\<twosuperior>) x :> -(2 * cos(x) * sin(x))"
1444 apply (rule lemma_DERIV_subst)
1445 apply (rule DERIV_cos_realpow2a, auto)
1449 lemma DERIV_cos_cos_mult3 [simp]:
1450 "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
1451 apply (rule lemma_DERIV_subst)
1452 apply (rule DERIV_cos_cos_mult2, auto)
1455 lemma DERIV_sin_circle_all:
1456 "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>
1457 (2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
1458 apply (simp only: diff_minus, safe)
1459 apply (rule DERIV_add)
1460 apply (auto simp add: numeral_2_eq_2)
1463 lemma DERIV_sin_circle_all_zero [simp]:
1464 "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
1465 by (cut_tac DERIV_sin_circle_all, auto)
1467 lemma sin_cos_squared_add [simp]: "((sin x)\<twosuperior>) + ((cos x)\<twosuperior>) = 1"
1468 apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all])
1469 apply (auto simp add: numeral_2_eq_2)
1472 lemma sin_cos_squared_add2 [simp]: "((cos x)\<twosuperior>) + ((sin x)\<twosuperior>) = 1"
1473 apply (subst add_commute)
1474 apply (rule sin_cos_squared_add)
1477 lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
1478 apply (cut_tac x = x in sin_cos_squared_add2)
1479 apply (simp add: power2_eq_square)
1482 lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
1483 apply (rule_tac a1 = "(cos x)\<twosuperior>" in add_right_cancel [THEN iffD1])
1487 lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
1488 apply (rule_tac a1 = "(sin x)\<twosuperior>" in add_right_cancel [THEN iffD1])
1492 lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
1493 by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
1495 lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
1496 apply (insert abs_sin_le_one [of x])
1497 apply (simp add: abs_le_iff del: abs_sin_le_one)
1500 lemma sin_le_one [simp]: "sin x \<le> 1"
1501 apply (insert abs_sin_le_one [of x])
1502 apply (simp add: abs_le_iff del: abs_sin_le_one)
1505 lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
1506 by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
1508 lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
1509 apply (insert abs_cos_le_one [of x])
1510 apply (simp add: abs_le_iff del: abs_cos_le_one)
1513 lemma cos_le_one [simp]: "cos x \<le> 1"
1514 apply (insert abs_cos_le_one [of x])
1515 apply (simp add: abs_le_iff del: abs_cos_le_one)
1518 lemma DERIV_fun_pow: "DERIV g x :> m ==>
1519 DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
1520 unfolding One_nat_def
1521 apply (rule lemma_DERIV_subst)
1522 apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2)
1523 apply (rule DERIV_pow, auto)
1526 lemma DERIV_fun_exp:
1527 "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
1528 apply (rule lemma_DERIV_subst)
1529 apply (rule_tac f = exp in DERIV_chain2)
1530 apply (rule DERIV_exp, auto)
1533 lemma DERIV_fun_sin:
1534 "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
1535 apply (rule lemma_DERIV_subst)
1536 apply (rule_tac f = sin in DERIV_chain2)
1537 apply (rule DERIV_sin, auto)
1540 lemma DERIV_fun_cos:
1541 "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
1542 apply (rule lemma_DERIV_subst)
1543 apply (rule_tac f = cos in DERIV_chain2)
1544 apply (rule DERIV_cos, auto)
1547 lemmas DERIV_intros = DERIV_ident DERIV_const DERIV_cos DERIV_cmult
1548 DERIV_sin DERIV_exp DERIV_inverse DERIV_pow
1549 DERIV_add DERIV_diff DERIV_mult DERIV_minus
1550 DERIV_inverse_fun DERIV_quotient DERIV_fun_pow
1551 DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos
1554 lemma lemma_DERIV_sin_cos_add:
1556 DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
1557 (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"
1558 apply (safe, rule lemma_DERIV_subst)
1559 apply (best intro!: DERIV_intros intro: DERIV_chain2)
1560 --{*replaces the old @{text DERIV_tac}*}
1561 apply (auto simp add: algebra_simps)
1564 lemma sin_cos_add [simp]:
1565 "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
1566 (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
1567 apply (cut_tac y = 0 and x = x and y7 = y
1568 in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])
1569 apply (auto simp add: numeral_2_eq_2)
1572 lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
1573 apply (cut_tac x = x and y = y in sin_cos_add)
1574 apply (simp del: sin_cos_add)
1577 lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
1578 apply (cut_tac x = x and y = y in sin_cos_add)
1579 apply (simp del: sin_cos_add)
1582 lemma lemma_DERIV_sin_cos_minus:
1583 "\<forall>x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"
1584 apply (safe, rule lemma_DERIV_subst)
1585 apply (best intro!: DERIV_intros intro: DERIV_chain2)
1586 apply (simp add: algebra_simps)
1589 lemma sin_cos_minus:
1590 "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"
1591 apply (cut_tac y = 0 and x = x
1592 in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])
1596 lemma sin_minus [simp]: "sin (-x) = -sin(x)"
1597 using sin_cos_minus [where x=x] by simp
1599 lemma cos_minus [simp]: "cos (-x) = cos(x)"
1600 using sin_cos_minus [where x=x] by simp
1602 lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
1603 by (simp add: diff_minus sin_add)
1605 lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
1606 by (simp add: sin_diff mult_commute)
1608 lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
1609 by (simp add: diff_minus cos_add)
1611 lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
1612 by (simp add: cos_diff mult_commute)
1614 lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
1615 using sin_add [where x=x and y=x] by simp
1617 lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
1618 using cos_add [where x=x and y=x]
1619 by (simp add: power2_eq_square)
1622 subsection {* The Constant Pi *}
1626 "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
1628 text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
1632 "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1))
1635 have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
1637 else -1 ^ ((k - Suc 0) div 2) / real (fact k)) *
1641 by (rule sin_converges [THEN sums_summable, THEN sums_group], simp)
1642 thus ?thesis unfolding One_nat_def by (simp add: mult_ac)
1645 text {* FIXME: This is a long, ugly proof! *}
1646 lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
1648 "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
1649 -1 ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1))
1650 sums (\<Sum>n. -1 ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
1652 apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp)
1653 apply (rotate_tac 2)
1654 apply (drule sin_paired [THEN sums_unique, THEN ssubst])
1655 unfolding One_nat_def
1656 apply (auto simp del: fact_Suc)
1657 apply (frule sums_unique)
1658 apply (auto simp del: fact_Suc)
1659 apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])
1660 apply (auto simp del: fact_Suc)
1661 apply (erule sums_summable)
1662 apply (case_tac "m=0")
1663 apply (simp (no_asm_simp))
1664 apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x")
1665 apply (simp only: mult_less_cancel_left, simp)
1666 apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
1667 apply (subgoal_tac "x*x < 2*3", simp)
1668 apply (rule mult_strict_mono)
1669 apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc)
1670 apply (subst fact_Suc)
1671 apply (subst fact_Suc)
1672 apply (subst fact_Suc)
1673 apply (subst fact_Suc)
1674 apply (subst real_of_nat_mult)
1675 apply (subst real_of_nat_mult)
1676 apply (subst real_of_nat_mult)
1677 apply (subst real_of_nat_mult)
1678 apply (simp (no_asm) add: divide_inverse del: fact_Suc)
1679 apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
1680 apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right)
1681 apply (auto simp add: mult_assoc simp del: fact_Suc)
1682 apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right)
1683 apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
1684 apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)")
1685 apply (erule ssubst)+
1686 apply (auto simp del: fact_Suc)
1687 apply (subgoal_tac "0 < x ^ (4 * m) ")
1688 prefer 2 apply (simp only: zero_less_power)
1689 apply (simp (no_asm_simp) add: mult_less_cancel_left)
1690 apply (rule mult_strict_mono)
1691 apply (simp_all (no_asm_simp))
1694 lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"
1695 by (auto intro: sin_gt_zero)
1697 lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
1698 apply (cut_tac x = x in sin_gt_zero1)
1699 apply (auto simp add: cos_squared_eq cos_double)
1703 "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
1705 have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
1706 (if even k then -1 ^ (k div 2) / real (fact k) else 0) *
1710 by (rule cos_converges [THEN sums_summable, THEN sums_group], simp)
1711 thus ?thesis by (simp add: mult_ac)
1714 lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
1717 lemma cos_two_less_zero [simp]: "cos (2) < 0"
1718 apply (cut_tac x = 2 in cos_paired)
1719 apply (drule sums_minus)
1720 apply (rule neg_less_iff_less [THEN iffD1])
1721 apply (frule sums_unique, auto)
1723 "\<Sum>n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
1724 in order_less_trans)
1725 apply (simp (no_asm) add: fact_num_eq_if realpow_num_eq_if del: fact_Suc)
1726 apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
1727 apply (rule sumr_pos_lt_pair)
1728 apply (erule sums_summable, safe)
1729 unfolding One_nat_def
1730 apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
1732 apply (rule real_mult_inverse_cancel2)
1733 apply (rule real_of_nat_fact_gt_zero)+
1734 apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
1735 apply (subst fact_lemma)
1736 apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
1737 apply (simp only: real_of_nat_mult)
1738 apply (rule mult_strict_mono, force)
1739 apply (rule_tac [3] real_of_nat_ge_zero)
1740 prefer 2 apply force
1741 apply (rule real_of_nat_less_iff [THEN iffD2])
1742 apply (rule fact_less_mono, auto)
1745 lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
1746 lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
1748 lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
1749 apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")
1750 apply (rule_tac [2] IVT2)
1751 apply (auto intro: DERIV_isCont DERIV_cos)
1752 apply (cut_tac x = xa and y = y in linorder_less_linear)
1754 apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")
1755 apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
1756 apply (drule_tac f = cos in Rolle)
1757 apply (drule_tac [5] f = cos in Rolle)
1758 apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
1759 apply (drule_tac y1 = xa in order_le_less_trans [THEN sin_gt_zero])
1760 apply (assumption, rule_tac y=y in order_less_le_trans, simp_all)
1761 apply (drule_tac y1 = y in order_le_less_trans [THEN sin_gt_zero], assumption, simp_all)
1764 lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
1765 by (simp add: pi_def)
1767 lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
1768 by (simp add: pi_half cos_is_zero [THEN theI'])
1770 lemma pi_half_gt_zero [simp]: "0 < pi / 2"
1771 apply (rule order_le_neq_trans)
1772 apply (simp add: pi_half cos_is_zero [THEN theI'])
1773 apply (rule notI, drule arg_cong [where f=cos], simp)
1776 lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
1777 lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
1779 lemma pi_half_less_two [simp]: "pi / 2 < 2"
1780 apply (rule order_le_neq_trans)
1781 apply (simp add: pi_half cos_is_zero [THEN theI'])
1782 apply (rule notI, drule arg_cong [where f=cos], simp)
1785 lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
1786 lemmas pi_half_le_two [simp] = pi_half_less_two [THEN order_less_imp_le]
1788 lemma pi_gt_zero [simp]: "0 < pi"
1789 by (insert pi_half_gt_zero, simp)
1791 lemma pi_ge_zero [simp]: "0 \<le> pi"
1792 by (rule pi_gt_zero [THEN order_less_imp_le])
1794 lemma pi_neq_zero [simp]: "pi \<noteq> 0"
1795 by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym])
1797 lemma pi_not_less_zero [simp]: "\<not> pi < 0"
1798 by (simp add: linorder_not_less)
1800 lemma minus_pi_half_less_zero: "-(pi/2) < 0"
1803 lemma m2pi_less_pi: "- (2 * pi) < pi"
1805 have "- (2 * pi) < 0" and "0 < pi" by auto
1806 from order_less_trans[OF this] show ?thesis .
1809 lemma sin_pi_half [simp]: "sin(pi/2) = 1"
1810 apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
1811 apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
1812 apply (simp add: power2_eq_square)
1815 lemma cos_pi [simp]: "cos pi = -1"
1816 by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
1818 lemma sin_pi [simp]: "sin pi = 0"
1819 by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
1821 lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
1822 by (simp add: diff_minus cos_add)
1823 declare sin_cos_eq [symmetric, simp]
1825 lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
1826 by (simp add: cos_add)
1827 declare minus_sin_cos_eq [symmetric, simp]
1829 lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
1830 by (simp add: diff_minus sin_add)
1831 declare cos_sin_eq [symmetric, simp]
1833 lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
1834 by (simp add: sin_add)
1836 lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
1837 by (simp add: sin_add)
1839 lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
1840 by (simp add: cos_add)
1842 lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
1843 by (simp add: sin_add cos_double)
1845 lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
1846 by (simp add: cos_add cos_double)
1848 lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
1850 apply (auto simp add: real_of_nat_Suc left_distrib)
1853 lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
1855 have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
1856 also have "... = -1 ^ n" by (rule cos_npi)
1857 finally show ?thesis .
1860 lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
1862 apply (auto simp add: real_of_nat_Suc left_distrib)
1865 lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
1866 by (simp add: mult_commute [of pi])
1868 lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
1869 by (simp add: cos_double)
1871 lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
1874 lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
1875 apply (rule sin_gt_zero, assumption)
1876 apply (rule order_less_trans, assumption)
1877 apply (rule pi_half_less_two)
1880 lemma sin_less_zero:
1881 assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
1883 have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2)
1884 thus ?thesis by simp
1887 lemma pi_less_4: "pi < 4"
1888 by (cut_tac pi_half_less_two, auto)
1890 lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
1891 apply (cut_tac pi_less_4)
1892 apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
1893 apply (cut_tac cos_is_zero, safe)
1894 apply (rename_tac y z)
1895 apply (drule_tac x = y in spec)
1896 apply (drule_tac x = "pi/2" in spec, simp)
1899 lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
1900 apply (rule_tac x = x and y = 0 in linorder_cases)
1901 apply (rule cos_minus [THEN subst])
1902 apply (rule cos_gt_zero)
1903 apply (auto intro: cos_gt_zero)
1906 lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
1907 apply (auto simp add: order_le_less cos_gt_zero_pi)
1908 apply (subgoal_tac "x = pi/2", auto)
1911 lemma sin_gt_zero_pi: "[| 0 < x; x < pi |] ==> 0 < sin x"
1912 apply (subst sin_cos_eq)
1913 apply (rotate_tac 1)
1914 apply (drule real_sum_of_halves [THEN ssubst])
1915 apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
1919 lemma pi_ge_two: "2 \<le> pi"
1921 assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
1922 have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
1923 proof (cases "2 < 2 * pi")
1924 case True with dense[OF `pi < 2`] show ?thesis by auto
1926 case False have "pi < 2 * pi" by auto
1927 from dense[OF this] and False show ?thesis by auto
1929 then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
1930 hence "0 < sin y" using sin_gt_zero by auto
1932 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
1933 ultimately show False by auto
1936 lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
1937 by (auto simp add: order_le_less sin_gt_zero_pi)
1939 lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
1940 apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
1941 apply (rule_tac [2] IVT2)
1942 apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
1943 apply (cut_tac x = xa and y = y in linorder_less_linear)
1944 apply (rule ccontr, auto)
1945 apply (drule_tac f = cos in Rolle)
1946 apply (drule_tac [5] f = cos in Rolle)
1947 apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
1948 dest!: DERIV_cos [THEN DERIV_unique]
1949 simp add: differentiable_def)
1950 apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
1954 "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
1956 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))")
1957 apply (erule contrapos_np)
1958 apply (simp del: minus_sin_cos_eq [symmetric])
1959 apply (cut_tac y="-y" in cos_total, simp) apply simp
1961 apply (rule_tac a = "x - (pi/2)" in ex1I)
1962 apply (simp (no_asm) add: add_assoc)
1963 apply (rotate_tac 3)
1964 apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all)
1967 lemma reals_Archimedean4:
1968 "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
1969 apply (auto dest!: reals_Archimedean3)
1970 apply (drule_tac x = x in spec, clarify)
1971 apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
1972 prefer 2 apply (erule LeastI)
1973 apply (case_tac "LEAST m::nat. x < real m * y", simp)
1974 apply (subgoal_tac "~ x < real nat * y")
1975 prefer 2 apply (rule not_less_Least, simp, force)
1978 (* Pre Isabelle99-2 proof was simpler- numerals arithmetic
1979 now causes some unwanted re-arrangements of literals! *)
1980 lemma cos_zero_lemma:
1981 "[| 0 \<le> x; cos x = 0 |] ==>
1982 \<exists>n::nat. ~even n & x = real n * (pi/2)"
1983 apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
1984 apply (subgoal_tac "0 \<le> x - real n * pi &
1985 (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
1986 apply (auto simp add: algebra_simps real_of_nat_Suc)
1987 prefer 2 apply (simp add: cos_diff)
1988 apply (simp add: cos_diff)
1989 apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
1990 apply (rule_tac [2] cos_total, safe)
1991 apply (drule_tac x = "x - real n * pi" in spec)
1992 apply (drule_tac x = "pi/2" in spec)
1993 apply (simp add: cos_diff)
1994 apply (rule_tac x = "Suc (2 * n)" in exI)
1995 apply (simp add: real_of_nat_Suc algebra_simps, auto)
1998 lemma sin_zero_lemma:
1999 "[| 0 \<le> x; sin x = 0 |] ==>
2000 \<exists>n::nat. even n & x = real n * (pi/2)"
2001 apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
2002 apply (clarify, rule_tac x = "n - 1" in exI)
2003 apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
2004 apply (rule cos_zero_lemma)
2005 apply (simp_all add: add_increasing)
2011 ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
2012 (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
2014 apply (cut_tac linorder_linear [of 0 x], safe)
2015 apply (drule cos_zero_lemma, assumption+)
2016 apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
2017 apply (force simp add: minus_equation_iff [of x])
2018 apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
2019 apply (auto simp add: cos_add)
2022 (* ditto: but to a lesser extent *)
2025 ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
2026 (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
2028 apply (cut_tac linorder_linear [of 0 x], safe)
2029 apply (drule sin_zero_lemma, assumption+)
2030 apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
2031 apply (force simp add: minus_equation_iff [of x])
2032 apply (auto simp add: even_mult_two_ex)
2035 lemma cos_monotone_0_pi: assumes "0 \<le> y" and "y < x" and "x \<le> pi"
2036 shows "cos x < cos y"
2038 have "- (x - y) < 0" by (auto!)
2040 from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
2041 obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" by auto
2042 hence "0 < z" and "z < pi" by (auto!)
2043 hence "0 < sin z" using sin_gt_zero_pi by auto
2044 hence "cos x - cos y < 0" unfolding cos_diff minus_mult_commute[symmetric] using `- (x - y) < 0` by (rule mult_pos_neg2)
2045 thus ?thesis by auto
2048 lemma cos_monotone_0_pi': assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" shows "cos x \<le> cos y"
2049 proof (cases "y < x")
2050 case True show ?thesis using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
2052 case False hence "y = x" using `y \<le> x` by auto
2053 thus ?thesis by auto
2056 lemma cos_monotone_minus_pi_0: assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
2057 shows "cos y < cos x"
2059 have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" by (auto!)
2060 from cos_monotone_0_pi[OF this]
2061 show ?thesis unfolding cos_minus .
2064 lemma cos_monotone_minus_pi_0': assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0" shows "cos y \<le> cos x"
2065 proof (cases "y < x")
2066 case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] by auto
2068 case False hence "y = x" using `y \<le> x` by auto
2069 thus ?thesis by auto
2072 lemma sin_monotone_2pi': assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2" shows "sin y \<le> sin x"
2074 have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi" using pi_ge_two by (auto!)
2075 from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] by auto
2078 subsection {* Tangent *}
2081 tan :: "real => real" where
2082 "tan x = (sin x)/(cos x)"
2084 lemma tan_zero [simp]: "tan 0 = 0"
2085 by (simp add: tan_def)
2087 lemma tan_pi [simp]: "tan pi = 0"
2088 by (simp add: tan_def)
2090 lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
2091 by (simp add: tan_def)
2093 lemma tan_minus [simp]: "tan (-x) = - tan x"
2094 by (simp add: tan_def minus_mult_left)
2096 lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
2097 by (simp add: tan_def)
2099 lemma lemma_tan_add1:
2100 "[| cos x \<noteq> 0; cos y \<noteq> 0 |]
2101 ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
2102 apply (simp add: tan_def divide_inverse)
2103 apply (auto simp del: inverse_mult_distrib
2104 simp add: inverse_mult_distrib [symmetric] mult_ac)
2105 apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
2106 apply (auto simp del: inverse_mult_distrib
2107 simp add: mult_assoc left_diff_distrib cos_add)
2111 "[| cos x \<noteq> 0; cos y \<noteq> 0 |]
2112 ==> tan x + tan y = sin(x + y)/(cos x * cos y)"
2113 apply (simp add: tan_def)
2114 apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
2115 apply (auto simp add: mult_assoc left_distrib)
2116 apply (simp add: sin_add)
2120 "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
2121 ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
2122 apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)
2123 apply (simp add: tan_def)
2127 "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
2128 ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
2129 apply (insert tan_add [of x x])
2130 apply (simp add: mult_2 [symmetric])
2131 apply (auto simp add: numeral_2_eq_2)
2134 lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
2135 by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
2137 lemma tan_less_zero:
2138 assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
2140 have "0 < tan (- x)" using prems by (simp only: tan_gt_zero)
2141 thus ?thesis by simp
2144 lemma tan_half: fixes x :: real assumes "- (pi / 2) < x" and "x < pi / 2"
2145 shows "tan x = sin (2 * x) / (cos (2 * x) + 1)"
2147 from cos_gt_zero_pi[OF `- (pi / 2) < x` `x < pi / 2`]
2148 have "cos x \<noteq> 0" by auto
2150 have minus_cos_2x: "\<And>X. X - cos (2*x) = X - (cos x) ^ 2 + (sin x) ^ 2" unfolding cos_double by algebra
2152 have "tan x = (tan x + tan x) / 2" by auto
2153 also have "\<dots> = sin (x + x) / (cos x * cos x) / 2" unfolding add_tan_eq[OF `cos x \<noteq> 0` `cos x \<noteq> 0`] ..
2154 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
2155 also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + cos (2*x) + (sin x)^2)" unfolding minus_cos_2x by auto
2156 also have "\<dots> = sin (2 * x) / (cos (2*x) + 1)" by auto
2157 finally show ?thesis .
2160 lemma lemma_DERIV_tan:
2161 "cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"
2162 apply (rule lemma_DERIV_subst)
2163 apply (best intro!: DERIV_intros intro: DERIV_chain2)
2164 apply (auto simp add: divide_inverse numeral_2_eq_2)
2167 lemma DERIV_tan [simp]: "cos x \<noteq> 0 ==> DERIV tan x :> inverse((cos x)\<twosuperior>)"
2168 by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])
2170 lemma isCont_tan [simp]: "cos x \<noteq> 0 ==> isCont tan x"
2171 by (rule DERIV_tan [THEN DERIV_isCont])
2173 lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
2174 apply (subgoal_tac "(\<lambda>x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")
2175 apply (simp add: divide_inverse [symmetric])
2176 apply (rule LIM_mult)
2177 apply (rule_tac [2] inverse_1 [THEN subst])
2178 apply (rule_tac [2] LIM_inverse)
2179 apply (simp_all add: divide_inverse [symmetric])
2180 apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric])
2181 apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
2184 lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
2185 apply (cut_tac LIM_cos_div_sin)
2186 apply (simp only: LIM_def)
2187 apply (drule_tac x = "inverse y" in spec, safe, force)
2188 apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
2189 apply (rule_tac x = "(pi/2) - e" in exI)
2190 apply (simp (no_asm_simp))
2191 apply (drule_tac x = "(pi/2) - e" in spec)
2192 apply (auto simp add: tan_def)
2193 apply (rule inverse_less_iff_less [THEN iffD1])
2194 apply (auto simp add: divide_inverse)
2195 apply (rule real_mult_order)
2196 apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
2197 apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
2200 lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
2201 apply (frule order_le_imp_less_or_eq, safe)
2202 prefer 2 apply force
2203 apply (drule lemma_tan_total, safe)
2204 apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
2205 apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
2206 apply (drule_tac y = xa in order_le_imp_less_or_eq)
2207 apply (auto dest: cos_gt_zero)
2210 lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
2211 apply (cut_tac linorder_linear [of 0 y], safe)
2212 apply (drule tan_total_pos)
2213 apply (cut_tac [2] y="-y" in tan_total_pos, safe)
2214 apply (rule_tac [3] x = "-x" in exI)
2215 apply (auto intro!: exI)
2218 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
2219 apply (cut_tac y = y in lemma_tan_total1, auto)
2220 apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
2221 apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
2222 apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
2223 apply (rule_tac [4] Rolle)
2224 apply (rule_tac [2] Rolle)
2225 apply (auto intro!: DERIV_tan DERIV_isCont exI
2226 simp add: differentiable_def)
2227 txt{*Now, simulate TRYALL*}
2228 apply (rule_tac [!] DERIV_tan asm_rl)
2229 apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
2230 simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
2233 lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
2234 shows "tan y < tan x"
2236 have "\<forall> x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse (cos x'^2)"
2237 proof (rule allI, rule impI)
2238 fix x' :: real assume "y \<le> x' \<and> x' \<le> x"
2239 hence "-(pi/2) < x'" and "x' < pi/2" by (auto!)
2240 from cos_gt_zero_pi[OF this]
2241 have "cos x' \<noteq> 0" by auto
2242 thus "DERIV tan x' :> inverse (cos x'^2)" by (rule DERIV_tan)
2244 from MVT2[OF `y < x` this]
2245 obtain z where "y < z" and "z < x" and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<twosuperior>)" by auto
2246 hence "- (pi / 2) < z" and "z < pi / 2" by (auto!)
2247 hence "0 < cos z" using cos_gt_zero_pi by auto
2248 hence inv_pos: "0 < inverse ((cos z)\<twosuperior>)" by auto
2249 have "0 < x - y" using `y < x` by auto
2250 from real_mult_order[OF this inv_pos]
2251 have "0 < tan x - tan y" unfolding tan_diff by auto
2252 thus ?thesis by auto
2255 lemma tan_monotone': assumes "- (pi / 2) < y" and "y < pi / 2" and "- (pi / 2) < x" and "x < pi / 2"
2256 shows "(y < x) = (tan y < tan x)"
2258 assume "y < x" thus "tan y < tan x" using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
2260 assume "tan y < tan x"
2263 assume "\<not> y < x" hence "x \<le> y" by auto
2264 hence "tan x \<le> tan y"
2265 proof (cases "x = y")
2266 case True thus ?thesis by auto
2268 case False hence "x < y" using `x \<le> y` by auto
2269 from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
2271 thus False using `tan y < tan x` by auto
2275 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
2277 lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
2278 by (simp add: tan_def)
2280 lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x"
2281 proof (induct n arbitrary: x)
2283 have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" unfolding Suc_plus1 real_of_nat_add real_of_one real_add_mult_distrib by auto
2284 show ?case unfolding split_pi_off using Suc by auto
2287 lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
2288 proof (cases "0 \<le> i")
2289 case True hence i_nat: "real i = real (nat i)" by auto
2290 show ?thesis unfolding i_nat by auto
2292 case False hence i_nat: "real i = - real (nat (-i))" by auto
2293 have "tan x = tan (x + real i * pi - real i * pi)" by auto
2294 also have "\<dots> = tan (x + real i * pi)" unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
2295 finally show ?thesis by auto
2298 lemma tan_periodic_n[simp]: "tan (x + number_of n * pi) = tan x"
2299 using tan_periodic_int[of _ "number_of n" ] unfolding real_number_of .
2301 subsection {* Inverse Trigonometric Functions *}
2304 arcsin :: "real => real" where
2305 "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
2308 arccos :: "real => real" where
2309 "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
2312 arctan :: "real => real" where
2313 "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
2316 "[| -1 \<le> y; y \<le> 1 |]
2317 ==> -(pi/2) \<le> arcsin y &
2318 arcsin y \<le> pi/2 & sin(arcsin y) = y"
2319 unfolding arcsin_def by (rule theI' [OF sin_total])
2322 "[| -1 \<le> y; y \<le> 1 |]
2323 ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
2324 apply (drule (1) arcsin)
2325 apply (force intro: order_trans)
2328 lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
2329 by (blast dest: arcsin)
2331 lemma arcsin_bounded:
2332 "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
2333 by (blast dest: arcsin)
2335 lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
2336 by (blast dest: arcsin)
2338 lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
2339 by (blast dest: arcsin)
2341 lemma arcsin_lt_bounded:
2342 "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
2343 apply (frule order_less_imp_le)
2344 apply (frule_tac y = y in order_less_imp_le)
2345 apply (frule arcsin_bounded)
2347 apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
2348 apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
2349 apply (drule_tac [!] f = sin in arg_cong, auto)
2352 lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
2353 apply (unfold arcsin_def)
2354 apply (rule the1_equality)
2355 apply (rule sin_total, auto)
2359 "[| -1 \<le> y; y \<le> 1 |]
2360 ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
2361 unfolding arccos_def by (rule theI' [OF cos_total])
2363 lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
2364 by (blast dest: arccos)
2366 lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
2367 by (blast dest: arccos)
2369 lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
2370 by (blast dest: arccos)
2372 lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
2373 by (blast dest: arccos)
2375 lemma arccos_lt_bounded:
2376 "[| -1 < y; y < 1 |]
2377 ==> 0 < arccos y & arccos y < pi"
2378 apply (frule order_less_imp_le)
2379 apply (frule_tac y = y in order_less_imp_le)
2380 apply (frule arccos_bounded, auto)
2381 apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
2382 apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
2383 apply (drule_tac [!] f = cos in arg_cong, auto)
2386 lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
2387 apply (simp add: arccos_def)
2388 apply (auto intro!: the1_equality cos_total)
2391 lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
2392 apply (simp add: arccos_def)
2393 apply (auto intro!: the1_equality cos_total)
2396 lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<twosuperior>)"
2397 apply (subgoal_tac "x\<twosuperior> \<le> 1")
2398 apply (rule power2_eq_imp_eq)
2399 apply (simp add: cos_squared_eq)
2400 apply (rule cos_ge_zero)
2401 apply (erule (1) arcsin_lbound)
2402 apply (erule (1) arcsin_ubound)
2404 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
2405 apply (rule power_mono, simp, simp)
2408 lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<twosuperior>)"
2409 apply (subgoal_tac "x\<twosuperior> \<le> 1")
2410 apply (rule power2_eq_imp_eq)
2411 apply (simp add: sin_squared_eq)
2412 apply (rule sin_ge_zero)
2413 apply (erule (1) arccos_lbound)
2414 apply (erule (1) arccos_ubound)
2416 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
2417 apply (rule power_mono, simp, simp)
2420 lemma arctan [simp]:
2421 "- (pi/2) < arctan y & arctan y < pi/2 & tan (arctan y) = y"
2422 unfolding arctan_def by (rule theI' [OF tan_total])
2424 lemma tan_arctan: "tan(arctan y) = y"
2427 lemma arctan_bounded: "- (pi/2) < arctan y & arctan y < pi/2"
2428 by (auto simp only: arctan)
2430 lemma arctan_lbound: "- (pi/2) < arctan y"
2433 lemma arctan_ubound: "arctan y < pi/2"
2434 by (auto simp only: arctan)
2437 "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
2438 apply (unfold arctan_def)
2439 apply (rule the1_equality)
2440 apply (rule tan_total, auto)
2443 lemma arctan_zero_zero [simp]: "arctan 0 = 0"
2444 by (insert arctan_tan [of 0], simp)
2446 lemma cos_arctan_not_zero [simp]: "cos(arctan x) \<noteq> 0"
2447 apply (auto simp add: cos_zero_iff)
2448 apply (case_tac "n")
2449 apply (case_tac [3] "n")
2450 apply (cut_tac [2] y = x in arctan_ubound)
2451 apply (cut_tac [4] y = x in arctan_lbound)
2452 apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)
2455 lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
2456 apply (rule power_inverse [THEN subst])
2457 apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
2458 apply (auto dest: field_power_not_zero
2459 simp add: power_mult_distrib left_distrib power_divide tan_def
2460 mult_assoc power_inverse [symmetric])
2463 lemma isCont_inverse_function2:
2464 fixes f g :: "real \<Rightarrow> real" shows
2465 "\<lbrakk>a < x; x < b;
2466 \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
2467 \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
2468 \<Longrightarrow> isCont g (f x)"
2469 apply (rule isCont_inverse_function
2470 [where f=f and d="min (x - a) (b - x)"])
2471 apply (simp_all add: abs_le_iff)
2474 lemma isCont_arcsin: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arcsin x"
2475 apply (subgoal_tac "isCont arcsin (sin (arcsin x))", simp)
2476 apply (rule isCont_inverse_function2 [where f=sin])
2477 apply (erule (1) arcsin_lt_bounded [THEN conjunct1])
2478 apply (erule (1) arcsin_lt_bounded [THEN conjunct2])
2479 apply (fast intro: arcsin_sin, simp)
2482 lemma isCont_arccos: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arccos x"
2483 apply (subgoal_tac "isCont arccos (cos (arccos x))", simp)
2484 apply (rule isCont_inverse_function2 [where f=cos])
2485 apply (erule (1) arccos_lt_bounded [THEN conjunct1])
2486 apply (erule (1) arccos_lt_bounded [THEN conjunct2])
2487 apply (fast intro: arccos_cos, simp)
2490 lemma isCont_arctan: "isCont arctan x"
2491 apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
2492 apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
2493 apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
2494 apply (erule (1) isCont_inverse_function2 [where f=tan])
2495 apply (clarify, rule arctan_tan)
2496 apply (erule (1) order_less_le_trans)
2497 apply (erule (1) order_le_less_trans)
2498 apply (clarify, rule isCont_tan)
2499 apply (rule less_imp_neq [symmetric])
2500 apply (rule cos_gt_zero_pi)
2501 apply (erule (1) order_less_le_trans)
2502 apply (erule (1) order_le_less_trans)
2506 "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<twosuperior>))"
2507 apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
2508 apply (rule lemma_DERIV_subst [OF DERIV_sin])
2509 apply (simp add: cos_arcsin)
2510 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
2511 apply (rule power_strict_mono, simp, simp, simp)
2515 apply (erule (1) isCont_arcsin)
2519 "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<twosuperior>))"
2520 apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
2521 apply (rule lemma_DERIV_subst [OF DERIV_cos])
2522 apply (simp add: sin_arccos)
2523 apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
2524 apply (rule power_strict_mono, simp, simp, simp)
2528 apply (erule (1) isCont_arccos)
2531 lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<twosuperior>)"
2532 apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
2533 apply (rule lemma_DERIV_subst [OF DERIV_tan])
2534 apply (rule cos_arctan_not_zero)
2535 apply (simp add: power_inverse tan_sec [symmetric])
2536 apply (subgoal_tac "0 < 1 + x\<twosuperior>", simp)
2537 apply (simp add: add_pos_nonneg)
2538 apply (simp, simp, simp, rule isCont_arctan)
2541 subsection {* More Theorems about Sin and Cos *}
2543 lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
2545 let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
2546 have nonneg: "0 \<le> ?c"
2547 by (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
2548 have "0 = cos (pi / 4 + pi / 4)"
2550 also have "cos (pi / 4 + pi / 4) = ?c\<twosuperior> - ?s\<twosuperior>"
2551 by (simp only: cos_add power2_eq_square)
2552 also have "\<dots> = 2 * ?c\<twosuperior> - 1"
2553 by (simp add: sin_squared_eq)
2554 finally have "?c\<twosuperior> = (sqrt 2 / 2)\<twosuperior>"
2555 by (simp add: power_divide)
2557 using nonneg by (rule power2_eq_imp_eq) simp
2560 lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"
2562 let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
2563 have pos_c: "0 < ?c"
2564 by (rule cos_gt_zero, simp, simp)
2565 have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
2567 also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
2568 by (simp only: cos_add sin_add)
2569 also have "\<dots> = ?c * (?c\<twosuperior> - 3 * ?s\<twosuperior>)"
2570 by (simp add: algebra_simps power2_eq_square)
2571 finally have "?c\<twosuperior> = (sqrt 3 / 2)\<twosuperior>"
2572 using pos_c by (simp add: sin_squared_eq power_divide)
2574 using pos_c [THEN order_less_imp_le]
2575 by (rule power2_eq_imp_eq) simp
2578 lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
2580 have "sin (pi / 4) = cos (pi / 2 - pi / 4)" by (rule sin_cos_eq)
2581 also have "pi / 2 - pi / 4 = pi / 4" by simp
2582 also have "cos (pi / 4) = sqrt 2 / 2" by (rule cos_45)
2583 finally show ?thesis .
2586 lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
2588 have "sin (pi / 3) = cos (pi / 2 - pi / 3)" by (rule sin_cos_eq)
2589 also have "pi / 2 - pi / 3 = pi / 6" by simp
2590 also have "cos (pi / 6) = sqrt 3 / 2" by (rule cos_30)
2591 finally show ?thesis .
2594 lemma cos_60: "cos (pi / 3) = 1 / 2"
2595 apply (rule power2_eq_imp_eq)
2596 apply (simp add: cos_squared_eq sin_60 power_divide)
2597 apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
2600 lemma sin_30: "sin (pi / 6) = 1 / 2"
2602 have "sin (pi / 6) = cos (pi / 2 - pi / 6)" by (rule sin_cos_eq)
2603 also have "pi / 2 - pi / 6 = pi / 3" by simp
2604 also have "cos (pi / 3) = 1 / 2" by (rule cos_60)
2605 finally show ?thesis .
2608 lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
2609 unfolding tan_def by (simp add: sin_30 cos_30)
2611 lemma tan_45: "tan (pi / 4) = 1"
2612 unfolding tan_def by (simp add: sin_45 cos_45)
2614 lemma tan_60: "tan (pi / 3) = sqrt 3"
2615 unfolding tan_def by (simp add: sin_60 cos_60)
2619 "sin (x + 1 / 2 * real (Suc m) * pi) =
2620 cos (x + 1 / 2 * real (m) * pi)"
2621 by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib, auto)
2625 "sin (x + real (Suc m) * pi / 2) =
2626 cos (x + real (m) * pi / 2)"
2627 by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)
2629 lemma DERIV_sin_add [simp]: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)"
2630 apply (rule lemma_DERIV_subst)
2631 apply (rule_tac f = sin and g = "%x. x + k" in DERIV_chain2)
2632 apply (best intro!: DERIV_intros intro: DERIV_chain2)+
2633 apply (simp (no_asm))
2636 lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
2638 have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
2639 by (auto simp add: algebra_simps sin_add)
2641 by (simp add: real_of_nat_Suc left_distrib add_divide_distrib
2642 mult_commute [of pi])
2645 lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
2646 by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
2648 lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
2649 apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
2650 apply (subst cos_add, simp)
2653 lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
2654 by (auto simp add: mult_assoc)
2656 lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
2657 apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
2658 apply (subst sin_add, simp)
2663 "cos(x + 1 / 2 * real(Suc m) * pi) = -sin (x + 1 / 2 * real m * pi)"
2664 apply (simp only: cos_add sin_add real_of_nat_Suc right_distrib left_distrib minus_mult_right, auto)
2668 lemma [simp]: "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
2669 by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto)
2671 lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
2672 by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto)
2674 lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
2675 apply (rule lemma_DERIV_subst)
2676 apply (rule_tac f = cos and g = "%x. x + k" in DERIV_chain2)
2677 apply (best intro!: DERIV_intros intro: DERIV_chain2)+
2678 apply (simp (no_asm))
2681 lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
2682 by (auto simp add: sin_zero_iff even_mult_two_ex)
2684 lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
2685 by (cut_tac x = x in sin_cos_squared_add3, auto)
2687 subsection {* Machins formula *}
2689 lemma tan_total_pi4: assumes "\<bar>x\<bar> < 1"
2690 shows "\<exists> z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
2692 obtain z where "- (pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast
2693 have "tan (pi / 4) = 1" and "tan (- (pi / 4)) = - 1" using tan_45 tan_minus by auto
2694 have "z \<noteq> pi / 4"
2696 assume "\<not> (z \<noteq> pi / 4)" hence "z = pi / 4" by auto
2697 have "tan z = 1" unfolding `z = pi / 4` `tan (pi / 4) = 1` ..
2698 thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto
2700 have "z \<noteq> - (pi / 4)"
2702 assume "\<not> (z \<noteq> - (pi / 4))" hence "z = - (pi / 4)" by auto
2703 have "tan z = - 1" unfolding `z = - (pi / 4)` `tan (- (pi / 4)) = - 1` ..
2704 thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto
2709 assume "\<not> (z < pi / 4)" hence "pi / 4 < z" using `z \<noteq> pi / 4` by auto
2710 have "- (pi / 2) < pi / 4" using m2pi_less_pi by auto
2711 from tan_monotone[OF this `pi / 4 < z` `z < pi / 2`]
2712 have "1 < x" unfolding `tan z = x` `tan (pi / 4) = 1` .
2713 thus False using `\<bar>x\<bar> < 1` by auto
2716 have "-(pi / 4) < z"
2718 assume "\<not> (-(pi / 4) < z)" hence "z < - (pi / 4)" using `z \<noteq> - (pi / 4)` by auto
2719 have "-(pi / 4) < pi / 2" using m2pi_less_pi by auto
2720 from tan_monotone[OF `-(pi / 2) < z` `z < -(pi / 4)` this]
2721 have "x < - 1" unfolding `tan z = x` `tan (-(pi / 4)) = - 1` .
2722 thus False using `\<bar>x\<bar> < 1` by auto
2724 ultimately show ?thesis using `tan z = x` by auto
2727 lemma arctan_add: assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
2728 shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
2730 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
2732 have "pi / 4 < pi / 2" by auto
2734 have "\<exists> x'. -(pi/4) \<le> x' \<and> x' \<le> pi/4 \<and> tan x' = x"
2735 proof (cases "\<bar>x\<bar> < 1")
2736 case True from tan_total_pi4[OF this] obtain x' where "-(pi/4) < x'" and "x' < pi/4" and "tan x' = x" by blast
2737 hence "-(pi/4) \<le> x'" and "x' \<le> pi/4" and "tan x' = x" by auto
2738 thus ?thesis by auto
2742 proof (cases "x = 1")
2743 case True hence "tan (pi/4) = x" using tan_45 by auto
2745 have "- pi \<le> pi" unfolding minus_le_self_iff by auto
2746 hence "-(pi/4) \<le> pi/4" and "pi/4 \<le> pi/4" by auto
2747 ultimately show ?thesis by blast
2749 case False hence "x = -1" using `\<not> \<bar>x\<bar> < 1` and `\<bar>x\<bar> \<le> 1` by auto
2750 hence "tan (-(pi/4)) = x" using tan_45 tan_minus by auto
2752 have "- pi \<le> pi" unfolding minus_le_self_iff by auto
2753 hence "-(pi/4) \<le> pi/4" and "-(pi/4) \<le> -(pi/4)" by auto
2754 ultimately show ?thesis by blast
2757 then obtain x' where "-(pi/4) \<le> x'" and "x' \<le> pi/4" and "tan x' = x" by blast
2758 hence "-(pi/2) < x'" and "x' < pi/2" using order_le_less_trans[OF `x' \<le> pi/4` `pi / 4 < pi / 2`] by auto
2760 have "cos x' \<noteq> 0" using cos_gt_zero_pi[THEN less_imp_neq] and `-(pi/2) < x'` and `x' < pi/2` by auto
2761 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
2762 ultimately have "cos x' * cos y' \<noteq> 0" by auto
2764 have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> (A / C) / (B / C) = A / B" by auto
2765 have divide_mult_commute: "\<And> A B C D :: real. A * B / (C * D) = (A / C) * (B / D)" by auto
2767 have "tan (x' + y') = sin (x' + y') / (cos x' * cos y' - sin x' * sin y')" unfolding tan_def cos_add ..
2768 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`] ..
2769 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 ..
2770 finally have tan_eq: "tan (x' + y') = (x + y) / (1 - x * y)" unfolding `tan y' = y` `tan x' = x` .
2772 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)
2773 moreover have "arctan (tan (x')) = x'" using `-(pi/2) < x'` and `x' < pi/2` by (auto intro!: arctan_tan)
2774 moreover have "arctan (tan (y')) = y'" using `-(pi/4) < y'` and `y' < pi/4` by (auto intro!: arctan_tan)
2775 ultimately have "arctan x + arctan y = arctan (tan (x' + y'))" unfolding `tan y' = y` [symmetric] `tan x' = x`[symmetric] by auto
2776 thus "arctan x + arctan y = arctan ((x + y) / (1 - x * y))" unfolding tan_eq .
2779 lemma arctan1_eq_pi4: "arctan 1 = pi / 4" unfolding tan_45[symmetric] by (rule arctan_tan, auto simp add: m2pi_less_pi)
2781 theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
2783 have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
2784 from arctan_add[OF less_imp_le[OF this] this]
2785 have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
2787 have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
2788 from arctan_add[OF less_imp_le[OF this] this]
2789 have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
2791 have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
2792 from arctan_add[OF this]
2793 have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
2794 ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
2795 thus ?thesis unfolding arctan1_eq_pi4 by algebra
2797 subsection {* Introducing the arcus tangens power series *}
2799 lemma monoseq_arctan_series: fixes x :: real
2800 assumes "\<bar>x\<bar> \<le> 1" shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
2801 proof (cases "x = 0") case True thus ?thesis unfolding monoseq_def One_nat_def by auto
2804 have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
2807 { fix n fix x :: real assume "0 \<le> x" and "x \<le> 1"
2808 have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le> 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
2809 proof (rule mult_mono)
2810 show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))" by (rule frac_le) simp_all
2811 show "0 \<le> 1 / real (Suc (n * 2))" by auto
2812 show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)" by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`)
2813 show "0 \<le> x ^ Suc (Suc n * 2)" by (rule zero_le_power) (simp add: `0 \<le> x`)
2818 proof (cases "0 \<le> x")
2819 case True from mono[OF this `x \<le> 1`, THEN allI]
2820 show ?thesis unfolding Suc_plus1[symmetric] by (rule mono_SucI2)
2822 case False hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto
2824 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
2825 thus ?thesis unfolding Suc_plus1[symmetric] by (rule mono_SucI1[OF allI])
2830 lemma zeroseq_arctan_series: fixes x :: real
2831 assumes "\<bar>x\<bar> \<le> 1" shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
2832 proof (cases "x = 0") case True thus ?thesis unfolding One_nat_def by (auto simp add: LIMSEQ_const)
2835 have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
2837 proof (cases "\<bar>x\<bar> < 1")
2838 case True hence "norm x < 1" by auto
2839 from LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]]
2840 have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"
2841 unfolding inverse_eq_divide Suc_plus1 by simp
2842 then show ?thesis using pos2 by (rule LIMSEQ_linear)
2844 case False hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto
2845 hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x" unfolding One_nat_def by auto
2846 from LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] LIMSEQ_const[of x]]
2847 show ?thesis unfolding n_eq Suc_plus1 by auto
2851 lemma summable_arctan_series: fixes x :: real and n :: nat
2852 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)")
2853 by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
2855 lemma less_one_imp_sqr_less_one: fixes x :: real assumes "\<bar>x\<bar> < 1" shows "x^2 < 1"
2857 from mult_mono1[OF less_imp_le[OF `\<bar>x\<bar> < 1`] abs_ge_zero[of x]]
2858 have "\<bar> x^2 \<bar> < 1" using `\<bar> x \<bar> < 1` unfolding numeral_2_eq_2 power_Suc2 by auto
2859 thus ?thesis using zero_le_power2 by auto
2862 lemma DERIV_arctan_series: assumes "\<bar> x \<bar> < 1"
2863 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")
2865 let "?f n" = "if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
2867 { fix n :: nat assume "even n" hence "2 * (n div 2) = n" by presburger } note n_even=this
2868 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
2870 { fix x :: real assume "\<bar>x\<bar> < 1" hence "x^2 < 1" by (rule less_one_imp_sqr_less_one)
2871 have "summable (\<lambda> n. -1 ^ n * (x^2) ^n)"
2872 by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x^2 < 1` order_less_imp_le[OF `x^2 < 1`])
2873 hence "summable (\<lambda> n. -1 ^ n * x^(2*n))" unfolding power_mult .
2874 } note summable_Integral = this
2876 { fix f :: "nat \<Rightarrow> real"
2877 have "\<And> x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
2879 fix x :: real assume "f sums x"
2880 from sums_if[OF sums_zero this]
2881 show "(\<lambda> n. if even n then f (n div 2) else 0) sums x" by auto
2883 fix x :: real assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
2884 from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult_commute]]
2885 show "f sums x" unfolding sums_def by auto
2887 hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
2888 } note sums_even = this
2890 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]
2894 have if_eq': "\<And> n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
2895 (if even n then -1 ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
2896 using n_even by auto
2897 have idx_eq: "\<And> n. n * 2 + 1 = Suc (2 * n)" by auto
2898 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]
2900 } note arctan_eq = this
2902 have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
2903 proof (rule DERIV_power_series')
2904 show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto
2905 { fix x' :: real assume x'_bounds: "x' \<in> {- 1 <..< 1}"
2906 hence "\<bar>x'\<bar> < 1" by auto
2908 let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
2909 show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
2910 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`])
2913 thus ?thesis unfolding Int_eq arctan_eq .
2916 lemma arctan_series: assumes "\<bar> x \<bar> \<le> 1"
2917 shows "arctan x = (\<Sum> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "_ = suminf (\<lambda> n. ?c x n)")
2919 let "?c' x n" = "(-1)^n * x^(n*2)"
2921 { fix r x :: real assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
2922 have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto
2923 from DERIV_arctan_series[OF this]
2924 have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
2925 } note DERIV_arctan_suminf = this
2927 { fix x :: real assume "\<bar>x\<bar> \<le> 1" note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] }
2928 note arctan_series_borders = this
2930 { fix x :: real assume "\<bar>x\<bar> < 1" have "arctan x = (\<Sum> k. ?c x k)"
2932 obtain r where "\<bar>x\<bar> < r" and "r < 1" using dense[OF `\<bar>x\<bar> < 1`] by blast
2933 hence "0 < r" and "-r < x" and "x < r" by auto
2935 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"
2937 fix x a b assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
2938 hence "\<bar>x\<bar> < r" by auto
2939 show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
2940 proof (rule DERIV_isconst2[of "a" "b"])
2941 show "a < b" and "a \<le> x" and "x \<le> b" using `a < b` `a \<le> x` `x \<le> b` by auto
2942 have "\<forall> x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
2943 proof (rule allI, rule impI)
2944 fix x assume "-r < x \<and> x < r" hence "\<bar>x\<bar> < r" by auto
2945 hence "\<bar>x\<bar> < 1" using `r < 1` by auto
2946 have "\<bar> - (x^2) \<bar> < 1" using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
2947 hence "(\<lambda> n. (- (x^2)) ^ n) sums (1 / (1 - (- (x^2))))" unfolding real_norm_def[symmetric] by (rule geometric_sums)
2948 hence "(?c' x) sums (1 / (1 - (- (x^2))))" unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto
2949 hence suminf_c'_eq_geom: "inverse (1 + x^2) = suminf (?c' x)" using sums_unique unfolding inverse_eq_divide by auto
2950 have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x^2))" unfolding suminf_c'_eq_geom
2951 by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
2952 from DERIV_add_minus[OF this DERIV_arctan]
2953 show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0" unfolding diff_minus by auto
2955 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
2956 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
2957 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
2961 have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
2962 unfolding Suc_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero by auto
2964 have "suminf (?c x) - arctan x = 0"
2965 proof (cases "x = 0")
2966 case True thus ?thesis using suminf_arctan_zero by auto
2968 case False hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
2969 have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
2970 by (rule suminf_eq_arctan_bounded[where x="0" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>", symmetric], auto simp add: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>`)
2972 have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
2973 by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"], auto simp add: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>`)
2975 show ?thesis using suminf_arctan_zero by auto
2977 thus ?thesis by auto
2978 qed } note when_less_one = this
2980 show "arctan x = suminf (\<lambda> n. ?c x n)"
2981 proof (cases "\<bar>x\<bar> < 1")
2982 case True thus ?thesis by (rule when_less_one)
2983 next case False hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto
2984 let "?a x n" = "\<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
2985 let "?diff x n" = "\<bar> arctan x - (\<Sum> i = 0..<n. ?c x i)\<bar>"
2987 have "0 < (1 :: real)" by auto
2989 { fix x :: real assume "0 < x" and "x < 1" hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
2990 from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)" by auto
2991 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]
2992 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)
2993 hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)" by (rule abs_of_pos)
2994 have "?diff x n \<le> ?a x n"
2995 proof (cases "even n")
2996 case True hence sgn_pos: "(-1)^n = (1::real)" by auto
2997 from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
2998 from bounds[of m, unfolded this atLeastAtMost_iff]
2999 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
3000 also have "\<dots> = ?c x n" unfolding One_nat_def by auto
3001 also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
3002 finally show ?thesis .
3004 case False hence sgn_neg: "(-1)^n = (-1::real)" by auto
3005 from `odd n` obtain m where m_def: "2 * m + 1 = n" unfolding odd_Suc_mult_two_ex by auto
3006 hence m_plus: "2 * (m + 1) = n + 1" by auto
3007 from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
3008 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
3009 also have "\<dots> = - ?c x n" unfolding One_nat_def by auto
3010 also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
3011 finally show ?thesis .
3013 hence "0 \<le> ?a x n - ?diff x n" by auto
3015 hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
3016 moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
3017 unfolding real_diff_def divide_inverse
3018 by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum)
3019 ultimately have "0 \<le> ?a 1 n - ?diff 1 n" by (rule LIM_less_bound)
3020 hence "?diff 1 n \<le> ?a 1 n" by auto
3023 unfolding LIMSEQ_rabs_zero power_one divide_inverse One_nat_def
3024 by (auto intro!: LIMSEQ_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)
3025 have "?diff 1 ----> 0"
3026 proof (rule LIMSEQ_I)
3027 fix r :: real assume "0 < r"
3028 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
3029 { fix n assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this]
3030 have "norm (?diff 1 n - 0) < r" by auto }
3031 thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
3033 from this[unfolded LIMSEQ_rabs_zero real_diff_def add_commute[of "arctan 1"], THEN LIMSEQ_add_const, of "- arctan 1", THEN LIMSEQ_minus]
3034 have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
3035 hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
3038 proof (cases "x = 1", simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
3039 assume "x \<noteq> 1" hence "x = -1" using `\<bar>x\<bar> = 1` by auto
3041 have "- (pi / 2) < 0" using pi_gt_zero by auto
3042 have "- (2 * pi) < 0" using pi_gt_zero by auto
3044 have c_minus_minus: "\<And> i. ?c (- 1) i = - ?c 1 i" unfolding One_nat_def by auto
3046 have "arctan (- 1) = arctan (tan (-(pi / 4)))" unfolding tan_45 tan_minus ..
3047 also have "\<dots> = - (pi / 4)" by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
3048 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])
3049 also have "\<dots> = - (arctan 1)" unfolding tan_45 ..
3050 also have "\<dots> = - (\<Sum> i. ?c 1 i)" using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto
3051 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
3052 finally show ?thesis using `x = -1` by auto
3057 lemma arctan_half: fixes x :: real
3058 shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x^2)))"
3060 obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x" using tan_total by blast
3061 hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2" by auto
3063 have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> A / B = (A / C) / (B / C)" by auto
3065 have "0 < cos y" using cos_gt_zero_pi[OF low high] .
3066 hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y) ^ 2) = cos y" by auto
3068 have "1 + (tan y)^2 = 1 + sin y^2 / cos y^2" unfolding tan_def power_divide ..
3069 also have "\<dots> = cos y^2 / cos y^2 + sin y^2 / cos y^2" using `cos y \<noteq> 0` by auto
3070 also have "\<dots> = 1 / cos y^2" unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
3071 finally have "1 + (tan y)^2 = 1 / cos y^2" .
3073 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] ..
3074 also have "\<dots> = tan y / (1 + 1 / cos y)" using `cos y \<noteq> 0` unfolding add_divide_distrib by auto
3075 also have "\<dots> = tan y / (1 + 1 / sqrt(cos y^2))" unfolding cos_sqrt ..
3076 also have "\<dots> = tan y / (1 + sqrt(1 / cos y^2))" unfolding real_sqrt_divide by auto
3077 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` .
3079 have "arctan x = y" using arctan_tan low high y_eq by auto
3080 also have "\<dots> = 2 * (arctan (tan (y/2)))" using arctan_tan[OF low2 high2] by auto
3081 also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" unfolding tan_half[OF low2 high2] by auto
3082 finally show ?thesis unfolding eq `tan y = x` .
3085 lemma arctan_monotone: assumes "x < y"
3086 shows "arctan x < arctan y"
3088 obtain z where "-(pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast
3089 obtain w where "-(pi / 2) < w" and "w < pi / 2" and "tan w = y" using tan_total by blast
3090 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` .
3092 unfolding `tan z = x`[symmetric] arctan_tan[OF `-(pi / 2) < z` `z < pi / 2`]
3093 unfolding `tan w = y`[symmetric] arctan_tan[OF `-(pi / 2) < w` `w < pi / 2`] .
3096 lemma arctan_monotone': assumes "x \<le> y" shows "arctan x \<le> arctan y"
3097 proof (cases "x = y")
3098 case False hence "x < y" using `x \<le> y` by auto from arctan_monotone[OF this] show ?thesis by auto
3101 lemma arctan_minus: "arctan (- x) = - arctan x"
3103 obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast
3104 thus ?thesis unfolding `tan y = x`[symmetric] tan_minus[symmetric] using arctan_tan[of y] arctan_tan[of "-y"] by auto
3107 lemma arctan_inverse: assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
3109 obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast
3110 hence "y = arctan x" unfolding `tan y = x`[symmetric] using arctan_tan by auto
3112 { 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
3113 have "tan y > 0" using tan_monotone'[OF _ _ `- (pi / 2) < y` `y < pi / 2`, of 0] tan_zero `0 < y` by auto
3114 hence "x > 0" using `tan y = x` by auto
3116 have "- (pi / 2) < pi / 2 - y" using `y > 0` `y < pi / 2` by auto
3117 moreover have "pi / 2 - y < pi / 2" using `y > 0` `y < pi / 2` by auto
3118 ultimately have "arctan (1 / x) = pi / 2 - y" unfolding `tan y = x`[symmetric] tan_inverse using arctan_tan by auto
3119 hence "arctan (1 / x) = sgn x * pi / 2 - arctan x" unfolding `y = arctan x` real_sgn_pos[OF `x > 0`] by auto
3123 proof (cases "y > 0")
3124 case True from pos_y[OF this `y < pi / 2` `y = arctan x` `tan y = x`] show ?thesis .
3126 case False hence "y \<le> 0" by auto
3127 moreover have "y \<noteq> 0"
3129 assume "\<not> y \<noteq> 0" hence "y = 0" by auto
3130 have "x = 0" unfolding `tan y = x`[symmetric] `y = 0` tan_zero ..
3131 thus False using `x \<noteq> 0` by auto
3133 ultimately have "y < 0" by auto
3134 hence "0 < - y" and "-y < pi / 2" using `- (pi / 2) < y` by auto
3135 moreover have "-y = arctan (-x)" unfolding arctan_minus `y = arctan x` ..
3136 moreover have "tan (-y) = -x" unfolding tan_minus `tan y = x` ..
3137 ultimately have "arctan (1 / -x) = sgn (-x) * pi / 2 - arctan (-x)" using pos_y by blast
3138 hence "arctan (- (1 / x)) = - (sgn x * pi / 2 - arctan x)" unfolding arctan_minus[of x] divide_minus_right sgn_minus by auto
3139 thus ?thesis unfolding arctan_minus neg_equal_iff_equal .
3143 theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
3145 have "pi / 4 = arctan 1" using arctan1_eq_pi4 by auto
3146 also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
3147 finally show ?thesis by auto
3150 subsection {* Existence of Polar Coordinates *}
3152 lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<twosuperior> + y\<twosuperior>)\<bar> \<le> 1"
3153 apply (rule power2_le_imp_le [OF _ zero_le_one])
3154 apply (simp add: abs_divide power_divide divide_le_eq not_sum_power2_lt_zero)
3157 lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
3158 by (simp add: abs_le_iff)
3160 lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<twosuperior>)"
3161 by (simp add: sin_arccos abs_le_iff)
3163 lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
3165 lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
3168 "0 < y ==> \<exists>r a. x = r * cos a & y = r * sin a"
3169 apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>)" in exI)
3170 apply (rule_tac x = "arccos (x / sqrt (x\<twosuperior> + y\<twosuperior>))" in exI)
3171 apply (simp add: cos_arccos_lemma1)
3172 apply (simp add: sin_arccos_lemma1)
3173 apply (simp add: power_divide)
3174 apply (simp add: real_sqrt_mult [symmetric])
3175 apply (simp add: right_diff_distrib)
3179 "y < 0 ==> \<exists>r a. x = r * cos a & y = r * sin a"
3180 apply (insert polar_ex1 [where x=x and y="-y"], simp, clarify)
3181 apply (rule_tac x = r in exI)
3182 apply (rule_tac x = "-a" in exI, simp)
3185 lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
3186 apply (rule_tac x=0 and y=y in linorder_cases)
3187 apply (erule polar_ex1)
3188 apply (rule_tac x=x in exI, rule_tac x=0 in exI, simp)
3189 apply (erule polar_ex2)