1(* Title: HOL/Transcendental.thy 2 Author: Jacques D. Fleuriot, University of Cambridge, University of Edinburgh 3 Author: Lawrence C Paulson 4 Author: Jeremy Avigad 5*) 6 7section \<open>Power Series, Transcendental Functions etc.\<close> 8 9theory Transcendental 10imports Series Deriv NthRoot 11begin 12 13text \<open>A theorem about the factcorial function on the reals.\<close> 14 15lemma square_fact_le_2_fact: "fact n * fact n \<le> (fact (2 * n) :: real)" 16proof (induct n) 17 case 0 18 then show ?case by simp 19next 20 case (Suc n) 21 have "(fact (Suc n)) * (fact (Suc n)) = of_nat (Suc n) * of_nat (Suc n) * (fact n * fact n :: real)" 22 by (simp add: field_simps) 23 also have "\<dots> \<le> of_nat (Suc n) * of_nat (Suc n) * fact (2 * n)" 24 by (rule mult_left_mono [OF Suc]) simp 25 also have "\<dots> \<le> of_nat (Suc (Suc (2 * n))) * of_nat (Suc (2 * n)) * fact (2 * n)" 26 by (rule mult_right_mono)+ (auto simp: field_simps) 27 also have "\<dots> = fact (2 * Suc n)" by (simp add: field_simps) 28 finally show ?case . 29qed 30 31lemma fact_in_Reals: "fact n \<in> \<real>" 32 by (induction n) auto 33 34lemma of_real_fact [simp]: "of_real (fact n) = fact n" 35 by (metis of_nat_fact of_real_of_nat_eq) 36 37lemma pochhammer_of_real: "pochhammer (of_real x) n = of_real (pochhammer x n)" 38 by (simp add: pochhammer_prod) 39 40lemma norm_fact [simp]: "norm (fact n :: 'a::real_normed_algebra_1) = fact n" 41proof - 42 have "(fact n :: 'a) = of_real (fact n)" 43 by simp 44 also have "norm \<dots> = fact n" 45 by (subst norm_of_real) simp 46 finally show ?thesis . 47qed 48 49lemma root_test_convergence: 50 fixes f :: "nat \<Rightarrow> 'a::banach" 51 assumes f: "(\<lambda>n. root n (norm (f n))) \<longlonglongrightarrow> x" \<comment> \<open>could be weakened to lim sup\<close> 52 and "x < 1" 53 shows "summable f" 54proof - 55 have "0 \<le> x" 56 by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1]) 57 from \<open>x < 1\<close> obtain z where z: "x < z" "z < 1" 58 by (metis dense) 59 from f \<open>x < z\<close> have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially" 60 by (rule order_tendstoD) 61 then have "eventually (\<lambda>n. norm (f n) \<le> z^n) sequentially" 62 using eventually_ge_at_top 63 proof eventually_elim 64 fix n 65 assume less: "root n (norm (f n)) < z" and n: "1 \<le> n" 66 from power_strict_mono[OF less, of n] n show "norm (f n) \<le> z ^ n" 67 by simp 68 qed 69 then show "summable f" 70 unfolding eventually_sequentially 71 using z \<open>0 \<le> x\<close> by (auto intro!: summable_comparison_test[OF _ summable_geometric]) 72qed 73 74subsection \<open>More facts about binomial coefficients\<close> 75 76text \<open> 77 These facts could have been proven before, but having real numbers 78 makes the proofs a lot easier. 79\<close> 80 81lemma central_binomial_odd: 82 "odd n \<Longrightarrow> n choose (Suc (n div 2)) = n choose (n div 2)" 83proof - 84 assume "odd n" 85 hence "Suc (n div 2) \<le> n" by presburger 86 hence "n choose (Suc (n div 2)) = n choose (n - Suc (n div 2))" 87 by (rule binomial_symmetric) 88 also from \<open>odd n\<close> have "n - Suc (n div 2) = n div 2" by presburger 89 finally show ?thesis . 90qed 91 92lemma binomial_less_binomial_Suc: 93 assumes k: "k < n div 2" 94 shows "n choose k < n choose (Suc k)" 95proof - 96 from k have k': "k \<le> n" "Suc k \<le> n" by simp_all 97 from k' have "real (n choose k) = fact n / (fact k * fact (n - k))" 98 by (simp add: binomial_fact) 99 also from k' have "n - k = Suc (n - Suc k)" by simp 100 also from k' have "fact \<dots> = (real n - real k) * fact (n - Suc k)" 101 by (subst fact_Suc) (simp_all add: of_nat_diff) 102 also from k have "fact k = fact (Suc k) / (real k + 1)" by (simp add: field_simps) 103 also have "fact n / (fact (Suc k) / (real k + 1) * ((real n - real k) * fact (n - Suc k))) = 104 (n choose (Suc k)) * ((real k + 1) / (real n - real k))" 105 using k by (simp add: field_split_simps binomial_fact) 106 also from assms have "(real k + 1) / (real n - real k) < 1" by simp 107 finally show ?thesis using k by (simp add: mult_less_cancel_left) 108qed 109 110lemma binomial_strict_mono: 111 assumes "k < k'" "2*k' \<le> n" 112 shows "n choose k < n choose k'" 113proof - 114 from assms have "k \<le> k' - 1" by simp 115 thus ?thesis 116 proof (induction rule: inc_induct) 117 case base 118 with assms binomial_less_binomial_Suc[of "k' - 1" n] 119 show ?case by simp 120 next 121 case (step k) 122 from step.prems step.hyps assms have "n choose k < n choose (Suc k)" 123 by (intro binomial_less_binomial_Suc) simp_all 124 also have "\<dots> < n choose k'" by (rule step.IH) 125 finally show ?case . 126 qed 127qed 128 129lemma binomial_mono: 130 assumes "k \<le> k'" "2*k' \<le> n" 131 shows "n choose k \<le> n choose k'" 132 using assms binomial_strict_mono[of k k' n] by (cases "k = k'") simp_all 133 134lemma binomial_strict_antimono: 135 assumes "k < k'" "2 * k \<ge> n" "k' \<le> n" 136 shows "n choose k > n choose k'" 137proof - 138 from assms have "n choose (n - k) > n choose (n - k')" 139 by (intro binomial_strict_mono) (simp_all add: algebra_simps) 140 with assms show ?thesis by (simp add: binomial_symmetric [symmetric]) 141qed 142 143lemma binomial_antimono: 144 assumes "k \<le> k'" "k \<ge> n div 2" "k' \<le> n" 145 shows "n choose k \<ge> n choose k'" 146proof (cases "k = k'") 147 case False 148 note not_eq = False 149 show ?thesis 150 proof (cases "k = n div 2 \<and> odd n") 151 case False 152 with assms(2) have "2*k \<ge> n" by presburger 153 with not_eq assms binomial_strict_antimono[of k k' n] 154 show ?thesis by simp 155 next 156 case True 157 have "n choose k' \<le> n choose (Suc (n div 2))" 158 proof (cases "k' = Suc (n div 2)") 159 case False 160 with assms True not_eq have "Suc (n div 2) < k'" by simp 161 with assms binomial_strict_antimono[of "Suc (n div 2)" k' n] True 162 show ?thesis by auto 163 qed simp_all 164 also from True have "\<dots> = n choose k" by (simp add: central_binomial_odd) 165 finally show ?thesis . 166 qed 167qed simp_all 168 169lemma binomial_maximum: "n choose k \<le> n choose (n div 2)" 170proof - 171 have "k \<le> n div 2 \<longleftrightarrow> 2*k \<le> n" by linarith 172 consider "2*k \<le> n" | "2*k \<ge> n" "k \<le> n" | "k > n" by linarith 173 thus ?thesis 174 proof cases 175 case 1 176 thus ?thesis by (intro binomial_mono) linarith+ 177 next 178 case 2 179 thus ?thesis by (intro binomial_antimono) simp_all 180 qed (simp_all add: binomial_eq_0) 181qed 182 183lemma binomial_maximum': "(2*n) choose k \<le> (2*n) choose n" 184 using binomial_maximum[of "2*n"] by simp 185 186lemma central_binomial_lower_bound: 187 assumes "n > 0" 188 shows "4^n / (2*real n) \<le> real ((2*n) choose n)" 189proof - 190 from binomial[of 1 1 "2*n"] 191 have "4 ^ n = (\<Sum>k\<le>2*n. (2*n) choose k)" 192 by (simp add: power_mult power2_eq_square One_nat_def [symmetric] del: One_nat_def) 193 also have "{..2*n} = {0<..<2*n} \<union> {0,2*n}" by auto 194 also have "(\<Sum>k\<in>\<dots>. (2*n) choose k) = 195 (\<Sum>k\<in>{0<..<2*n}. (2*n) choose k) + (\<Sum>k\<in>{0,2*n}. (2*n) choose k)" 196 by (subst sum.union_disjoint) auto 197 also have "(\<Sum>k\<in>{0,2*n}. (2*n) choose k) \<le> (\<Sum>k\<le>1. (n choose k)\<^sup>2)" 198 by (cases n) simp_all 199 also from assms have "\<dots> \<le> (\<Sum>k\<le>n. (n choose k)\<^sup>2)" 200 by (intro sum_mono2) auto 201 also have "\<dots> = (2*n) choose n" by (rule choose_square_sum) 202 also have "(\<Sum>k\<in>{0<..<2*n}. (2*n) choose k) \<le> (\<Sum>k\<in>{0<..<2*n}. (2*n) choose n)" 203 by (intro sum_mono binomial_maximum') 204 also have "\<dots> = card {0<..<2*n} * ((2*n) choose n)" by simp 205 also have "card {0<..<2*n} \<le> 2*n - 1" by (cases n) simp_all 206 also have "(2 * n - 1) * (2 * n choose n) + (2 * n choose n) = ((2*n) choose n) * (2*n)" 207 using assms by (simp add: algebra_simps) 208 finally have "4 ^ n \<le> (2 * n choose n) * (2 * n)" by simp_all 209 hence "real (4 ^ n) \<le> real ((2 * n choose n) * (2 * n))" 210 by (subst of_nat_le_iff) 211 with assms show ?thesis by (simp add: field_simps) 212qed 213 214 215subsection \<open>Properties of Power Series\<close> 216 217lemma powser_zero [simp]: "(\<Sum>n. f n * 0 ^ n) = f 0" 218 for f :: "nat \<Rightarrow> 'a::real_normed_algebra_1" 219proof - 220 have "(\<Sum>n<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)" 221 by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left) 222 then show ?thesis by simp 223qed 224 225lemma powser_sums_zero: "(\<lambda>n. a n * 0^n) sums a 0" 226 for a :: "nat \<Rightarrow> 'a::real_normed_div_algebra" 227 using sums_finite [of "{0}" "\<lambda>n. a n * 0 ^ n"] 228 by simp 229 230lemma powser_sums_zero_iff [simp]: "(\<lambda>n. a n * 0^n) sums x \<longleftrightarrow> a 0 = x" 231 for a :: "nat \<Rightarrow> 'a::real_normed_div_algebra" 232 using powser_sums_zero sums_unique2 by blast 233 234text \<open> 235 Power series has a circle or radius of convergence: if it sums for \<open>x\<close>, 236 then it sums absolutely for \<open>z\<close> with \<^term>\<open>\<bar>z\<bar> < \<bar>x\<bar>\<close>.\<close> 237 238lemma powser_insidea: 239 fixes x z :: "'a::real_normed_div_algebra" 240 assumes 1: "summable (\<lambda>n. f n * x^n)" 241 and 2: "norm z < norm x" 242 shows "summable (\<lambda>n. norm (f n * z ^ n))" 243proof - 244 from 2 have x_neq_0: "x \<noteq> 0" by clarsimp 245 from 1 have "(\<lambda>n. f n * x^n) \<longlonglongrightarrow> 0" 246 by (rule summable_LIMSEQ_zero) 247 then have "convergent (\<lambda>n. f n * x^n)" 248 by (rule convergentI) 249 then have "Cauchy (\<lambda>n. f n * x^n)" 250 by (rule convergent_Cauchy) 251 then have "Bseq (\<lambda>n. f n * x^n)" 252 by (rule Cauchy_Bseq) 253 then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x^n) \<le> K" 254 by (auto simp: Bseq_def) 255 have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le> K * norm (z ^ n) * inverse (norm (x^n))" 256 proof (intro exI allI impI) 257 fix n :: nat 258 assume "0 \<le> n" 259 have "norm (norm (f n * z ^ n)) * norm (x^n) = 260 norm (f n * x^n) * norm (z ^ n)" 261 by (simp add: norm_mult abs_mult) 262 also have "\<dots> \<le> K * norm (z ^ n)" 263 by (simp only: mult_right_mono 4 norm_ge_zero) 264 also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x^n)) * norm (x^n))" 265 by (simp add: x_neq_0) 266 also have "\<dots> = K * norm (z ^ n) * inverse (norm (x^n)) * norm (x^n)" 267 by (simp only: mult.assoc) 268 finally show "norm (norm (f n * z ^ n)) \<le> K * norm (z ^ n) * inverse (norm (x^n))" 269 by (simp add: mult_le_cancel_right x_neq_0) 270 qed 271 moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))" 272 proof - 273 from 2 have "norm (norm (z * inverse x)) < 1" 274 using x_neq_0 275 by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric]) 276 then have "summable (\<lambda>n. norm (z * inverse x) ^ n)" 277 by (rule summable_geometric) 278 then have "summable (\<lambda>n. K * norm (z * inverse x) ^ n)" 279 by (rule summable_mult) 280 then show "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x^n)))" 281 using x_neq_0 282 by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib 283 power_inverse norm_power mult.assoc) 284 qed 285 ultimately show "summable (\<lambda>n. norm (f n * z ^ n))" 286 by (rule summable_comparison_test) 287qed 288 289lemma powser_inside: 290 fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}" 291 shows 292 "summable (\<lambda>n. f n * (x^n)) \<Longrightarrow> norm z < norm x \<Longrightarrow> 293 summable (\<lambda>n. f n * (z ^ n))" 294 by (rule powser_insidea [THEN summable_norm_cancel]) 295 296lemma powser_times_n_limit_0: 297 fixes x :: "'a::{real_normed_div_algebra,banach}" 298 assumes "norm x < 1" 299 shows "(\<lambda>n. of_nat n * x ^ n) \<longlonglongrightarrow> 0" 300proof - 301 have "norm x / (1 - norm x) \<ge> 0" 302 using assms by (auto simp: field_split_simps) 303 moreover obtain N where N: "norm x / (1 - norm x) < of_int N" 304 using ex_le_of_int by (meson ex_less_of_int) 305 ultimately have N0: "N>0" 306 by auto 307 then have *: "real_of_int (N + 1) * norm x / real_of_int N < 1" 308 using N assms by (auto simp: field_simps) 309 have **: "real_of_int N * (norm x * (real_of_nat (Suc n) * norm (x ^ n))) \<le> 310 real_of_nat n * (norm x * ((1 + N) * norm (x ^ n)))" if "N \<le> int n" for n :: nat 311 proof - 312 from that have "real_of_int N * real_of_nat (Suc n) \<le> real_of_nat n * real_of_int (1 + N)" 313 by (simp add: algebra_simps) 314 then have "(real_of_int N * real_of_nat (Suc n)) * (norm x * norm (x ^ n)) \<le> 315 (real_of_nat n * (1 + N)) * (norm x * norm (x ^ n))" 316 using N0 mult_mono by fastforce 317 then show ?thesis 318 by (simp add: algebra_simps) 319 qed 320 show ?thesis using * 321 by (rule summable_LIMSEQ_zero [OF summable_ratio_test, where N1="nat N"]) 322 (simp add: N0 norm_mult field_simps ** del: of_nat_Suc of_int_add) 323qed 324 325corollary lim_n_over_pown: 326 fixes x :: "'a::{real_normed_field,banach}" 327 shows "1 < norm x \<Longrightarrow> ((\<lambda>n. of_nat n / x^n) \<longlongrightarrow> 0) sequentially" 328 using powser_times_n_limit_0 [of "inverse x"] 329 by (simp add: norm_divide field_split_simps) 330 331lemma sum_split_even_odd: 332 fixes f :: "nat \<Rightarrow> real" 333 shows "(\<Sum>i<2 * n. if even i then f i else g i) = (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))" 334proof (induct n) 335 case 0 336 then show ?case by simp 337next 338 case (Suc n) 339 have "(\<Sum>i<2 * Suc n. if even i then f i else g i) = 340 (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))" 341 using Suc.hyps unfolding One_nat_def by auto 342 also have "\<dots> = (\<Sum>i<Suc n. f (2 * i)) + (\<Sum>i<Suc n. g (2 * i + 1))" 343 by auto 344 finally show ?case . 345qed 346 347lemma sums_if': 348 fixes g :: "nat \<Rightarrow> real" 349 assumes "g sums x" 350 shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" 351 unfolding sums_def 352proof (rule LIMSEQ_I) 353 fix r :: real 354 assume "0 < r" 355 from \<open>g sums x\<close>[unfolded sums_def, THEN LIMSEQ_D, OF this] 356 obtain no where no_eq: "\<And>n. n \<ge> no \<Longrightarrow> (norm (sum g {..<n} - x) < r)" 357 by blast 358 359 let ?SUM = "\<lambda> m. \<Sum>i<m. if even i then 0 else g ((i - 1) div 2)" 360 have "(norm (?SUM m - x) < r)" if "m \<ge> 2 * no" for m 361 proof - 362 from that have "m div 2 \<ge> no" by auto 363 have sum_eq: "?SUM (2 * (m div 2)) = sum g {..< m div 2}" 364 using sum_split_even_odd by auto 365 then have "(norm (?SUM (2 * (m div 2)) - x) < r)" 366 using no_eq unfolding sum_eq using \<open>m div 2 \<ge> no\<close> by auto 367 moreover 368 have "?SUM (2 * (m div 2)) = ?SUM m" 369 proof (cases "even m") 370 case True 371 then show ?thesis 372 by (auto simp: even_two_times_div_two) 373 next 374 case False 375 then have eq: "Suc (2 * (m div 2)) = m" by simp 376 then have "even (2 * (m div 2))" using \<open>odd m\<close> by auto 377 have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq .. 378 also have "\<dots> = ?SUM (2 * (m div 2))" using \<open>even (2 * (m div 2))\<close> by auto 379 finally show ?thesis by auto 380 qed 381 ultimately show ?thesis by auto 382 qed 383 then show "\<exists>no. \<forall> m \<ge> no. norm (?SUM m - x) < r" 384 by blast 385qed 386 387lemma sums_if: 388 fixes g :: "nat \<Rightarrow> real" 389 assumes "g sums x" and "f sums y" 390 shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)" 391proof - 392 let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)" 393 have if_sum: "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)" 394 for B T E 395 by (cases B) auto 396 have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" 397 using sums_if'[OF \<open>g sums x\<close>] . 398 have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" 399 by auto 400 have "?s sums y" using sums_if'[OF \<open>f sums y\<close>] . 401 from this[unfolded sums_def, THEN LIMSEQ_Suc] 402 have "(\<lambda>n. if even n then f (n div 2) else 0) sums y" 403 by (simp add: lessThan_Suc_eq_insert_0 sum.atLeast1_atMost_eq image_Suc_lessThan 404 if_eq sums_def cong del: if_weak_cong) 405 from sums_add[OF g_sums this] show ?thesis 406 by (simp only: if_sum) 407qed 408 409subsection \<open>Alternating series test / Leibniz formula\<close> 410(* FIXME: generalise these results from the reals via type classes? *) 411 412lemma sums_alternating_upper_lower: 413 fixes a :: "nat \<Rightarrow> real" 414 assumes mono: "\<And>n. a (Suc n) \<le> a n" 415 and a_pos: "\<And>n. 0 \<le> a n" 416 and "a \<longlonglongrightarrow> 0" 417 shows "\<exists>l. ((\<forall>n. (\<Sum>i<2*n. (- 1)^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i<2*n. (- 1)^i*a i) \<longlonglongrightarrow> l) \<and> 418 ((\<forall>n. l \<le> (\<Sum>i<2*n + 1. (- 1)^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. (- 1)^i*a i) \<longlonglongrightarrow> l)" 419 (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)") 420proof (rule nested_sequence_unique) 421 have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" by auto 422 423 show "\<forall>n. ?f n \<le> ?f (Suc n)" 424 proof 425 show "?f n \<le> ?f (Suc n)" for n 426 using mono[of "2*n"] by auto 427 qed 428 show "\<forall>n. ?g (Suc n) \<le> ?g n" 429 proof 430 show "?g (Suc n) \<le> ?g n" for n 431 using mono[of "Suc (2*n)"] by auto 432 qed 433 show "\<forall>n. ?f n \<le> ?g n" 434 proof 435 show "?f n \<le> ?g n" for n 436 using fg_diff a_pos by auto 437 qed 438 show "(\<lambda>n. ?f n - ?g n) \<longlonglongrightarrow> 0" 439 unfolding fg_diff 440 proof (rule LIMSEQ_I) 441 fix r :: real 442 assume "0 < r" 443 with \<open>a \<longlonglongrightarrow> 0\<close>[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" 444 by auto 445 then have "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" 446 by auto 447 then show "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" 448 by auto 449 qed 450qed 451 452lemma summable_Leibniz': 453 fixes a :: "nat \<Rightarrow> real" 454 assumes a_zero: "a \<longlonglongrightarrow> 0" 455 and a_pos: "\<And>n. 0 \<le> a n" 456 and a_monotone: "\<And>n. a (Suc n) \<le> a n" 457 shows summable: "summable (\<lambda> n. (-1)^n * a n)" 458 and "\<And>n. (\<Sum>i<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)" 459 and "(\<lambda>n. \<Sum>i<2*n. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)" 460 and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i<2*n+1. (-1)^i*a i)" 461 and "(\<lambda>n. \<Sum>i<2*n+1. (-1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (-1)^i*a i)" 462proof - 463 let ?S = "\<lambda>n. (-1)^n * a n" 464 let ?P = "\<lambda>n. \<Sum>i<n. ?S i" 465 let ?f = "\<lambda>n. ?P (2 * n)" 466 let ?g = "\<lambda>n. ?P (2 * n + 1)" 467 obtain l :: real 468 where below_l: "\<forall> n. ?f n \<le> l" 469 and "?f \<longlonglongrightarrow> l" 470 and above_l: "\<forall> n. l \<le> ?g n" 471 and "?g \<longlonglongrightarrow> l" 472 using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast 473 474 let ?Sa = "\<lambda>m. \<Sum>n<m. ?S n" 475 have "?Sa \<longlonglongrightarrow> l" 476 proof (rule LIMSEQ_I) 477 fix r :: real 478 assume "0 < r" 479 with \<open>?f \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D] 480 obtain f_no where f: "\<And>n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" 481 by auto 482 from \<open>0 < r\<close> \<open>?g \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D] 483 obtain g_no where g: "\<And>n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" 484 by auto 485 have "norm (?Sa n - l) < r" if "n \<ge> (max (2 * f_no) (2 * g_no))" for n 486 proof - 487 from that have "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto 488 show ?thesis 489 proof (cases "even n") 490 case True 491 then have n_eq: "2 * (n div 2) = n" 492 by (simp add: even_two_times_div_two) 493 with \<open>n \<ge> 2 * f_no\<close> have "n div 2 \<ge> f_no" 494 by auto 495 from f[OF this] show ?thesis 496 unfolding n_eq atLeastLessThanSuc_atLeastAtMost . 497 next 498 case False 499 then have "even (n - 1)" by simp 500 then have n_eq: "2 * ((n - 1) div 2) = n - 1" 501 by (simp add: even_two_times_div_two) 502 then have range_eq: "n - 1 + 1 = n" 503 using odd_pos[OF False] by auto 504 from n_eq \<open>n \<ge> 2 * g_no\<close> have "(n - 1) div 2 \<ge> g_no" 505 by auto 506 from g[OF this] show ?thesis 507 by (simp only: n_eq range_eq) 508 qed 509 qed 510 then show "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast 511 qed 512 then have sums_l: "(\<lambda>i. (-1)^i * a i) sums l" 513 by (simp only: sums_def) 514 then show "summable ?S" 515 by (auto simp: summable_def) 516 517 have "l = suminf ?S" by (rule sums_unique[OF sums_l]) 518 519 fix n 520 show "suminf ?S \<le> ?g n" 521 unfolding sums_unique[OF sums_l, symmetric] using above_l by auto 522 show "?f n \<le> suminf ?S" 523 unfolding sums_unique[OF sums_l, symmetric] using below_l by auto 524 show "?g \<longlonglongrightarrow> suminf ?S" 525 using \<open>?g \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto 526 show "?f \<longlonglongrightarrow> suminf ?S" 527 using \<open>?f \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto 528qed 529 530theorem summable_Leibniz: 531 fixes a :: "nat \<Rightarrow> real" 532 assumes a_zero: "a \<longlonglongrightarrow> 0" 533 and "monoseq a" 534 shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable") 535 and "0 < a 0 \<longrightarrow> 536 (\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n. (- 1)^i * a i .. \<Sum>i<2*n+1. (- 1)^i * a i})" (is "?pos") 537 and "a 0 < 0 \<longrightarrow> 538 (\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n+1. (- 1)^i * a i .. \<Sum>i<2*n. (- 1)^i * a i})" (is "?neg") 539 and "(\<lambda>n. \<Sum>i<2*n. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?f") 540 and "(\<lambda>n. \<Sum>i<2*n+1. (- 1)^i*a i) \<longlonglongrightarrow> (\<Sum>i. (- 1)^i*a i)" (is "?g") 541proof - 542 have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g" 543 proof (cases "(\<forall>n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)") 544 case True 545 then have ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" 546 and ge0: "\<And>n. 0 \<le> a n" 547 by auto 548 have mono: "a (Suc n) \<le> a n" for n 549 using ord[where n="Suc n" and m=n] by auto 550 note leibniz = summable_Leibniz'[OF \<open>a \<longlonglongrightarrow> 0\<close> ge0] 551 from leibniz[OF mono] 552 show ?thesis using \<open>0 \<le> a 0\<close> by auto 553 next 554 let ?a = "\<lambda>n. - a n" 555 case False 556 with monoseq_le[OF \<open>monoseq a\<close> \<open>a \<longlonglongrightarrow> 0\<close>] 557 have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto 558 then have ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" 559 by auto 560 have monotone: "?a (Suc n) \<le> ?a n" for n 561 using ord[where n="Suc n" and m=n] by auto 562 note leibniz = 563 summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", 564 OF tendsto_minus[OF \<open>a \<longlonglongrightarrow> 0\<close>, unfolded minus_zero] monotone] 565 have "summable (\<lambda> n. (-1)^n * ?a n)" 566 using leibniz(1) by auto 567 then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" 568 unfolding summable_def by auto 569 from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l" 570 by auto 571 then have ?summable by (auto simp: summable_def) 572 moreover 573 have "\<bar>- a - - b\<bar> = \<bar>a - b\<bar>" for a b :: real 574 unfolding minus_diff_minus by auto 575 576 from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus] 577 have move_minus: "(\<Sum>n. - ((- 1) ^ n * a n)) = - (\<Sum>n. (- 1) ^ n * a n)" 578 by auto 579 580 have ?pos using \<open>0 \<le> ?a 0\<close> by auto 581 moreover have ?neg 582 using leibniz(2,4) 583 unfolding mult_minus_right sum_negf move_minus neg_le_iff_le 584 by auto 585 moreover have ?f and ?g 586 using leibniz(3,5)[unfolded mult_minus_right sum_negf move_minus, THEN tendsto_minus_cancel] 587 by auto 588 ultimately show ?thesis by auto 589 qed 590 then show ?summable and ?pos and ?neg and ?f and ?g 591 by safe 592qed 593 594 595subsection \<open>Term-by-Term Differentiability of Power Series\<close> 596 597definition diffs :: "(nat \<Rightarrow> 'a::ring_1) \<Rightarrow> nat \<Rightarrow> 'a" 598 where "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))" 599 600text \<open>Lemma about distributing negation over it.\<close> 601lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)" 602 by (simp add: diffs_def) 603 604lemma diffs_equiv: 605 fixes x :: "'a::{real_normed_vector,ring_1}" 606 shows "summable (\<lambda>n. diffs c n * x^n) \<Longrightarrow> 607 (\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)" 608 unfolding diffs_def 609 by (simp add: summable_sums sums_Suc_imp) 610 611lemma lemma_termdiff1: 612 fixes z :: "'a :: {monoid_mult,comm_ring}" 613 shows "(\<Sum>p<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) = 614 (\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))" 615 by (auto simp: algebra_simps power_add [symmetric]) 616 617lemma sumr_diff_mult_const2: "sum f {..<n} - of_nat n * r = (\<Sum>i<n. f i - r)" 618 for r :: "'a::ring_1" 619 by (simp add: sum_subtractf) 620 621lemma lemma_realpow_rev_sumr: 622 "(\<Sum>p<Suc n. (x ^ p) * (y ^ (n - p))) = (\<Sum>p<Suc n. (x ^ (n - p)) * (y ^ p))" 623 by (subst sum.nat_diff_reindex[symmetric]) simp 624 625lemma lemma_termdiff2: 626 fixes h :: "'a::field" 627 assumes h: "h \<noteq> 0" 628 shows "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) = 629 h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))" 630 (is "?lhs = ?rhs") 631proof (cases n) 632 case (Suc n) 633 have 0: "\<And>x k. (\<Sum>n<Suc k. h * (z ^ x * (z ^ (k - n) * (h + z) ^ n))) = 634 (\<Sum>j<Suc k. h * ((h + z) ^ j * z ^ (x + k - j)))" 635 apply (rule sum.cong [OF refl]) 636 by (simp add: power_add [symmetric] mult.commute) 637 have *: "(\<Sum>i<n. z ^ i * ((z + h) ^ (n - i) - z ^ (n - i))) = 638 (\<Sum>i<n. \<Sum>j<n - i. h * ((z + h) ^ j * z ^ (n - Suc j)))" 639 apply (rule sum.cong [OF refl]) 640 apply (clarsimp simp add: less_iff_Suc_add sum_distrib_left diff_power_eq_sum ac_simps 0 641 simp del: sum.lessThan_Suc power_Suc) 642 done 643 have "h * ?lhs = h * ?rhs" 644 apply (simp add: right_diff_distrib diff_divide_distrib h mult.assoc [symmetric]) 645 using Suc 646 apply (simp add: diff_power_eq_sum h right_diff_distrib [symmetric] mult.assoc 647 del: power_Suc sum.lessThan_Suc of_nat_Suc) 648 apply (subst lemma_realpow_rev_sumr) 649 apply (subst sumr_diff_mult_const2) 650 apply (simp add: lemma_termdiff1 sum_distrib_left *) 651 done 652 then show ?thesis 653 by (simp add: h) 654qed auto 655 656 657lemma real_sum_nat_ivl_bounded2: 658 fixes K :: "'a::linordered_semidom" 659 assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K" 660 and K: "0 \<le> K" 661 shows "sum f {..<n-k} \<le> of_nat n * K" 662 apply (rule order_trans [OF sum_mono [OF f]]) 663 apply (auto simp: mult_right_mono K) 664 done 665 666lemma lemma_termdiff3: 667 fixes h z :: "'a::real_normed_field" 668 assumes 1: "h \<noteq> 0" 669 and 2: "norm z \<le> K" 670 and 3: "norm (z + h) \<le> K" 671 shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) \<le> 672 of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h" 673proof - 674 have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) = 675 norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) * norm h" 676 by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult) 677 also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h" 678 proof (rule mult_right_mono [OF _ norm_ge_zero]) 679 from norm_ge_zero 2 have K: "0 \<le> K" 680 by (rule order_trans) 681 have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n" 682 apply (erule subst) 683 apply (simp only: norm_mult norm_power power_add) 684 apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K) 685 done 686 show "norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) \<le> 687 of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))" 688 apply (intro 689 order_trans [OF norm_sum] 690 real_sum_nat_ivl_bounded2 691 mult_nonneg_nonneg 692 of_nat_0_le_iff 693 zero_le_power K) 694 apply (rule le_Kn, simp) 695 done 696 qed 697 also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h" 698 by (simp only: mult.assoc) 699 finally show ?thesis . 700qed 701 702lemma lemma_termdiff4: 703 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" 704 and k :: real 705 assumes k: "0 < k" 706 and le: "\<And>h. h \<noteq> 0 \<Longrightarrow> norm h < k \<Longrightarrow> norm (f h) \<le> K * norm h" 707 shows "f \<midarrow>0\<rightarrow> 0" 708proof (rule tendsto_norm_zero_cancel) 709 show "(\<lambda>h. norm (f h)) \<midarrow>0\<rightarrow> 0" 710 proof (rule real_tendsto_sandwich) 711 show "eventually (\<lambda>h. 0 \<le> norm (f h)) (at 0)" 712 by simp 713 show "eventually (\<lambda>h. norm (f h) \<le> K * norm h) (at 0)" 714 using k by (auto simp: eventually_at dist_norm le) 715 show "(\<lambda>h. 0) \<midarrow>(0::'a)\<rightarrow> (0::real)" 716 by (rule tendsto_const) 717 have "(\<lambda>h. K * norm h) \<midarrow>(0::'a)\<rightarrow> K * norm (0::'a)" 718 by (intro tendsto_intros) 719 then show "(\<lambda>h. K * norm h) \<midarrow>(0::'a)\<rightarrow> 0" 720 by simp 721 qed 722qed 723 724lemma lemma_termdiff5: 725 fixes g :: "'a::real_normed_vector \<Rightarrow> nat \<Rightarrow> 'b::banach" 726 and k :: real 727 assumes k: "0 < k" 728 and f: "summable f" 729 and le: "\<And>h n. h \<noteq> 0 \<Longrightarrow> norm h < k \<Longrightarrow> norm (g h n) \<le> f n * norm h" 730 shows "(\<lambda>h. suminf (g h)) \<midarrow>0\<rightarrow> 0" 731proof (rule lemma_termdiff4 [OF k]) 732 fix h :: 'a 733 assume "h \<noteq> 0" and "norm h < k" 734 then have 1: "\<forall>n. norm (g h n) \<le> f n * norm h" 735 by (simp add: le) 736 then have "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h" 737 by simp 738 moreover from f have 2: "summable (\<lambda>n. f n * norm h)" 739 by (rule summable_mult2) 740 ultimately have 3: "summable (\<lambda>n. norm (g h n))" 741 by (rule summable_comparison_test) 742 then have "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))" 743 by (rule summable_norm) 744 also from 1 3 2 have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)" 745 by (rule suminf_le) 746 also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h" 747 by (rule suminf_mult2 [symmetric]) 748 finally show "norm (suminf (g h)) \<le> suminf f * norm h" . 749qed 750 751 752(* FIXME: Long proofs *) 753 754lemma termdiffs_aux: 755 fixes x :: "'a::{real_normed_field,banach}" 756 assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)" 757 and 2: "norm x < norm K" 758 shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0" 759proof - 760 from dense [OF 2] obtain r where r1: "norm x < r" and r2: "r < norm K" 761 by fast 762 from norm_ge_zero r1 have r: "0 < r" 763 by (rule order_le_less_trans) 764 then have r_neq_0: "r \<noteq> 0" by simp 765 show ?thesis 766 proof (rule lemma_termdiff5) 767 show "0 < r - norm x" 768 using r1 by simp 769 from r r2 have "norm (of_real r::'a) < norm K" 770 by simp 771 with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))" 772 by (rule powser_insidea) 773 then have "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)" 774 using r by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc) 775 then have "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))" 776 by (rule diffs_equiv [THEN sums_summable]) 777 also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) = 778 (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))" 779 apply (rule ext) 780 apply (case_tac n) 781 apply (simp_all add: diffs_def r_neq_0) 782 done 783 finally have "summable 784 (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))" 785 by (rule diffs_equiv [THEN sums_summable]) 786 also have 787 "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0)) = 788 (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" 789 apply (rule ext) 790 apply (case_tac n, simp) 791 apply (rename_tac nat) 792 apply (case_tac nat, simp) 793 apply (simp add: r_neq_0) 794 done 795 finally show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" . 796 next 797 fix h :: 'a 798 fix n :: nat 799 assume h: "h \<noteq> 0" 800 assume "norm h < r - norm x" 801 then have "norm x + norm h < r" by simp 802 with norm_triangle_ineq have xh: "norm (x + h) < r" 803 by (rule order_le_less_trans) 804 show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<le> 805 norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h" 806 apply (simp only: norm_mult mult.assoc) 807 apply (rule mult_left_mono [OF _ norm_ge_zero]) 808 apply (simp add: mult.assoc [symmetric]) 809 apply (metis h lemma_termdiff3 less_eq_real_def r1 xh) 810 done 811 qed 812qed 813 814lemma termdiffs: 815 fixes K x :: "'a::{real_normed_field,banach}" 816 assumes 1: "summable (\<lambda>n. c n * K ^ n)" 817 and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)" 818 and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)" 819 and 4: "norm x < norm K" 820 shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. (diffs c) n * x^n)" 821 unfolding DERIV_def 822proof (rule LIM_zero_cancel) 823 show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x^n)) / h 824 - suminf (\<lambda>n. diffs c n * x^n)) \<midarrow>0\<rightarrow> 0" 825 proof (rule LIM_equal2) 826 show "0 < norm K - norm x" 827 using 4 by (simp add: less_diff_eq) 828 next 829 fix h :: 'a 830 assume "norm (h - 0) < norm K - norm x" 831 then have "norm x + norm h < norm K" by simp 832 then have 5: "norm (x + h) < norm K" 833 by (rule norm_triangle_ineq [THEN order_le_less_trans]) 834 have "summable (\<lambda>n. c n * x^n)" 835 and "summable (\<lambda>n. c n * (x + h) ^ n)" 836 and "summable (\<lambda>n. diffs c n * x^n)" 837 using 1 2 4 5 by (auto elim: powser_inside) 838 then have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) = 839 (\<Sum>n. (c n * (x + h) ^ n - c n * x^n) / h - of_nat n * c n * x ^ (n - Suc 0))" 840 by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums) 841 then show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x^n)) / h - (\<Sum>n. diffs c n * x^n) = 842 (\<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))" 843 by (simp add: algebra_simps) 844 next 845 show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0" 846 by (rule termdiffs_aux [OF 3 4]) 847 qed 848qed 849 850subsection \<open>The Derivative of a Power Series Has the Same Radius of Convergence\<close> 851 852lemma termdiff_converges: 853 fixes x :: "'a::{real_normed_field,banach}" 854 assumes K: "norm x < K" 855 and sm: "\<And>x. norm x < K \<Longrightarrow> summable(\<lambda>n. c n * x ^ n)" 856 shows "summable (\<lambda>n. diffs c n * x ^ n)" 857proof (cases "x = 0") 858 case True 859 then show ?thesis 860 using powser_sums_zero sums_summable by auto 861next 862 case False 863 then have "K > 0" 864 using K less_trans zero_less_norm_iff by blast 865 then obtain r :: real where r: "norm x < norm r" "norm r < K" "r > 0" 866 using K False 867 by (auto simp: field_simps abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"]) 868 have to0: "(\<lambda>n. of_nat n * (x / of_real r) ^ n) \<longlonglongrightarrow> 0" 869 using r by (simp add: norm_divide powser_times_n_limit_0 [of "x / of_real r"]) 870 obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> real_of_nat n * norm x ^ n < r ^ n" 871 using r LIMSEQ_D [OF to0, of 1] 872 by (auto simp: norm_divide norm_mult norm_power field_simps) 873 have "summable (\<lambda>n. (of_nat n * c n) * x ^ n)" 874 proof (rule summable_comparison_test') 875 show "summable (\<lambda>n. norm (c n * of_real r ^ n))" 876 apply (rule powser_insidea [OF sm [of "of_real ((r+K)/2)"]]) 877 using N r norm_of_real [of "r + K", where 'a = 'a] by auto 878 show "\<And>n. N \<le> n \<Longrightarrow> norm (of_nat n * c n * x ^ n) \<le> norm (c n * of_real r ^ n)" 879 using N r by (fastforce simp add: norm_mult norm_power less_eq_real_def) 880 qed 881 then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ Suc n)" 882 using summable_iff_shift [of "\<lambda>n. of_nat n * c n * x ^ n" 1] 883 by simp 884 then have "summable (\<lambda>n. (of_nat (Suc n) * c(Suc n)) * x ^ n)" 885 using False summable_mult2 [of "\<lambda>n. (of_nat (Suc n) * c(Suc n) * x ^ n) * x" "inverse x"] 886 by (simp add: mult.assoc) (auto simp: ac_simps) 887 then show ?thesis 888 by (simp add: diffs_def) 889qed 890 891lemma termdiff_converges_all: 892 fixes x :: "'a::{real_normed_field,banach}" 893 assumes "\<And>x. summable (\<lambda>n. c n * x^n)" 894 shows "summable (\<lambda>n. diffs c n * x^n)" 895 by (rule termdiff_converges [where K = "1 + norm x"]) (use assms in auto) 896 897lemma termdiffs_strong: 898 fixes K x :: "'a::{real_normed_field,banach}" 899 assumes sm: "summable (\<lambda>n. c n * K ^ n)" 900 and K: "norm x < norm K" 901 shows "DERIV (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. diffs c n * x^n)" 902proof - 903 have K2: "norm ((of_real (norm K) + of_real (norm x)) / 2 :: 'a) < norm K" 904 using K 905 apply (auto simp: norm_divide field_simps) 906 apply (rule le_less_trans [of _ "of_real (norm K) + of_real (norm x)"]) 907 apply (auto simp: mult_2_right norm_triangle_mono) 908 done 909 then have [simp]: "norm ((of_real (norm K) + of_real (norm x)) :: 'a) < norm K * 2" 910 by simp 911 have "summable (\<lambda>n. c n * (of_real (norm x + norm K) / 2) ^ n)" 912 by (metis K2 summable_norm_cancel [OF powser_insidea [OF sm]] add.commute of_real_add) 913 moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs c n * x ^ n)" 914 by (blast intro: sm termdiff_converges powser_inside) 915 moreover have "\<And>x. norm x < norm K \<Longrightarrow> summable (\<lambda>n. diffs(diffs c) n * x ^ n)" 916 by (blast intro: sm termdiff_converges powser_inside) 917 ultimately show ?thesis 918 apply (rule termdiffs [where K = "of_real (norm x + norm K) / 2"]) 919 using K 920 apply (auto simp: field_simps) 921 apply (simp flip: of_real_add) 922 done 923qed 924 925lemma termdiffs_strong_converges_everywhere: 926 fixes K x :: "'a::{real_normed_field,banach}" 927 assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)" 928 shows "((\<lambda>x. \<Sum>n. c n * x^n) has_field_derivative (\<Sum>n. diffs c n * x^n)) (at x)" 929 using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x] 930 by (force simp del: of_real_add) 931 932lemma termdiffs_strong': 933 fixes z :: "'a :: {real_normed_field,banach}" 934 assumes "\<And>z. norm z < K \<Longrightarrow> summable (\<lambda>n. c n * z ^ n)" 935 assumes "norm z < K" 936 shows "((\<lambda>z. \<Sum>n. c n * z^n) has_field_derivative (\<Sum>n. diffs c n * z^n)) (at z)" 937proof (rule termdiffs_strong) 938 define L :: real where "L = (norm z + K) / 2" 939 have "0 \<le> norm z" by simp 940 also note \<open>norm z < K\<close> 941 finally have K: "K \<ge> 0" by simp 942 from assms K have L: "L \<ge> 0" "norm z < L" "L < K" by (simp_all add: L_def) 943 from L show "norm z < norm (of_real L :: 'a)" by simp 944 from L show "summable (\<lambda>n. c n * of_real L ^ n)" by (intro assms(1)) simp_all 945qed 946 947lemma termdiffs_sums_strong: 948 fixes z :: "'a :: {banach,real_normed_field}" 949 assumes sums: "\<And>z. norm z < K \<Longrightarrow> (\<lambda>n. c n * z ^ n) sums f z" 950 assumes deriv: "(f has_field_derivative f') (at z)" 951 assumes norm: "norm z < K" 952 shows "(\<lambda>n. diffs c n * z ^ n) sums f'" 953proof - 954 have summable: "summable (\<lambda>n. diffs c n * z^n)" 955 by (intro termdiff_converges[OF norm] sums_summable[OF sums]) 956 from norm have "eventually (\<lambda>z. z \<in> norm -` {..<K}) (nhds z)" 957 by (intro eventually_nhds_in_open open_vimage) 958 (simp_all add: continuous_on_norm) 959 hence eq: "eventually (\<lambda>z. (\<Sum>n. c n * z^n) = f z) (nhds z)" 960 by eventually_elim (insert sums, simp add: sums_iff) 961 962 have "((\<lambda>z. \<Sum>n. c n * z^n) has_field_derivative (\<Sum>n. diffs c n * z^n)) (at z)" 963 by (intro termdiffs_strong'[OF _ norm] sums_summable[OF sums]) 964 hence "(f has_field_derivative (\<Sum>n. diffs c n * z^n)) (at z)" 965 by (subst (asm) DERIV_cong_ev[OF refl eq refl]) 966 from this and deriv have "(\<Sum>n. diffs c n * z^n) = f'" by (rule DERIV_unique) 967 with summable show ?thesis by (simp add: sums_iff) 968qed 969 970lemma isCont_powser: 971 fixes K x :: "'a::{real_normed_field,banach}" 972 assumes "summable (\<lambda>n. c n * K ^ n)" 973 assumes "norm x < norm K" 974 shows "isCont (\<lambda>x. \<Sum>n. c n * x^n) x" 975 using termdiffs_strong[OF assms] by (blast intro!: DERIV_isCont) 976 977lemmas isCont_powser' = isCont_o2[OF _ isCont_powser] 978 979lemma isCont_powser_converges_everywhere: 980 fixes K x :: "'a::{real_normed_field,banach}" 981 assumes "\<And>y. summable (\<lambda>n. c n * y ^ n)" 982 shows "isCont (\<lambda>x. \<Sum>n. c n * x^n) x" 983 using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x] 984 by (force intro!: DERIV_isCont simp del: of_real_add) 985 986lemma powser_limit_0: 987 fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" 988 assumes s: "0 < s" 989 and sm: "\<And>x. norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)" 990 shows "(f \<longlongrightarrow> a 0) (at 0)" 991proof - 992 have "norm (of_real s / 2 :: 'a) < s" 993 using s by (auto simp: norm_divide) 994 then have "summable (\<lambda>n. a n * (of_real s / 2) ^ n)" 995 by (rule sums_summable [OF sm]) 996 then have "((\<lambda>x. \<Sum>n. a n * x ^ n) has_field_derivative (\<Sum>n. diffs a n * 0 ^ n)) (at 0)" 997 by (rule termdiffs_strong) (use s in \<open>auto simp: norm_divide\<close>) 998 then have "isCont (\<lambda>x. \<Sum>n. a n * x ^ n) 0" 999 by (blast intro: DERIV_continuous) 1000 then have "((\<lambda>x. \<Sum>n. a n * x ^ n) \<longlongrightarrow> a 0) (at 0)" 1001 by (simp add: continuous_within) 1002 then show ?thesis 1003 apply (rule Lim_transform) 1004 apply (clarsimp simp: LIM_eq) 1005 apply (rule_tac x=s in exI) 1006 using s sm sums_unique by fastforce 1007qed 1008 1009lemma powser_limit_0_strong: 1010 fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" 1011 assumes s: "0 < s" 1012 and sm: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)" 1013 shows "(f \<longlongrightarrow> a 0) (at 0)" 1014proof - 1015 have *: "((\<lambda>x. if x = 0 then a 0 else f x) \<longlongrightarrow> a 0) (at 0)" 1016 by (rule powser_limit_0 [OF s]) (auto simp: powser_sums_zero sm) 1017 show ?thesis 1018 apply (subst LIM_equal [where g = "(\<lambda>x. if x = 0 then a 0 else f x)"]) 1019 apply (simp_all add: *) 1020 done 1021qed 1022 1023 1024subsection \<open>Derivability of power series\<close> 1025 1026lemma DERIV_series': 1027 fixes f :: "real \<Rightarrow> nat \<Rightarrow> real" 1028 assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)" 1029 and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" 1030 and x0_in_I: "x0 \<in> {a <..< b}" 1031 and "summable (f' x0)" 1032 and "summable L" 1033 and L_def: "\<And>n x y. x \<in> {a <..< b} \<Longrightarrow> y \<in> {a <..< b} \<Longrightarrow> \<bar>f x n - f y n\<bar> \<le> L n * \<bar>x - y\<bar>" 1034 shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))" 1035 unfolding DERIV_def 1036proof (rule LIM_I) 1037 fix r :: real 1038 assume "0 < r" then have "0 < r/3" by auto 1039 1040 obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3" 1041 using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable L\<close>] by auto 1042 1043 obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3" 1044 using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable (f' x0)\<close>] by auto 1045 1046 let ?N = "Suc (max N_L N_f')" 1047 have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") 1048 and L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" 1049 using N_L[of "?N"] and N_f' [of "?N"] by auto 1050 1051 let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x" 1052 1053 let ?r = "r / (3 * real ?N)" 1054 from \<open>0 < r\<close> have "0 < ?r" by simp 1055 1056 let ?s = "\<lambda>n. SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)" 1057 define S' where "S' = Min (?s ` {..< ?N })" 1058 1059 have "0 < S'" 1060 unfolding S'_def 1061 proof (rule iffD2[OF Min_gr_iff]) 1062 show "\<forall>x \<in> (?s ` {..< ?N }). 0 < x" 1063 proof 1064 fix x 1065 assume "x \<in> ?s ` {..<?N}" 1066 then obtain n where "x = ?s n" and "n \<in> {..<?N}" 1067 using image_iff[THEN iffD1] by blast 1068 from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>, unfolded real_norm_def] 1069 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)" 1070 by auto 1071 have "0 < ?s n" 1072 by (rule someI2[where a=s]) (auto simp: s_bound simp del: of_nat_Suc) 1073 then show "0 < x" by (simp only: \<open>x = ?s n\<close>) 1074 qed 1075 qed auto 1076 1077 define S where "S = min (min (x0 - a) (b - x0)) S'" 1078 then have "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0" 1079 and "S \<le> S'" using x0_in_I and \<open>0 < S'\<close> 1080 by auto 1081 1082 have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r" 1083 if "x \<noteq> 0" and "\<bar>x\<bar> < S" for x 1084 proof - 1085 from that have x_in_I: "x0 + x \<in> {a <..< b}" 1086 using S_a S_b by auto 1087 1088 note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]] 1089 note div_smbl = summable_divide[OF diff_smbl] 1090 note all_smbl = summable_diff[OF div_smbl \<open>summable (f' x0)\<close>] 1091 note ign = summable_ignore_initial_segment[where k="?N"] 1092 note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]] 1093 note div_shft_smbl = summable_divide[OF diff_shft_smbl] 1094 note all_shft_smbl = summable_diff[OF div_smbl ign[OF \<open>summable (f' x0)\<close>]] 1095 1096 have 1: "\<bar>(\<bar>?diff (n + ?N) x\<bar>)\<bar> \<le> L (n + ?N)" for n 1097 proof - 1098 have "\<bar>?diff (n + ?N) x\<bar> \<le> L (n + ?N) * \<bar>(x0 + x) - x0\<bar> / \<bar>x\<bar>" 1099 using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] 1100 by (simp only: abs_divide) 1101 with \<open>x \<noteq> 0\<close> show ?thesis by auto 1102 qed 1103 note 2 = summable_rabs_comparison_test[OF _ ign[OF \<open>summable L\<close>]] 1104 from 1 have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" 1105 by (metis (lifting) abs_idempotent 1106 order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF \<open>summable L\<close>]]]) 1107 then have "\<bar>\<Sum>i. ?diff (i + ?N) x\<bar> \<le> r / 3" (is "?L_part \<le> r/3") 1108 using L_estimate by auto 1109 1110 have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n\<bar> \<le> (\<Sum>n<?N. \<bar>?diff n x - f' x0 n\<bar>)" .. 1111 also have "\<dots> < (\<Sum>n<?N. ?r)" 1112 proof (rule sum_strict_mono) 1113 fix n 1114 assume "n \<in> {..< ?N}" 1115 have "\<bar>x\<bar> < S" using \<open>\<bar>x\<bar> < S\<close> . 1116 also have "S \<le> S'" using \<open>S \<le> S'\<close> . 1117 also have "S' \<le> ?s n" 1118 unfolding S'_def 1119 proof (rule Min_le_iff[THEN iffD2]) 1120 have "?s n \<in> (?s ` {..<?N}) \<and> ?s n \<le> ?s n" 1121 using \<open>n \<in> {..< ?N}\<close> by auto 1122 then show "\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n" 1123 by blast 1124 qed auto 1125 finally have "\<bar>x\<bar> < ?s n" . 1126 1127 from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>, 1128 unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2] 1129 have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" . 1130 with \<open>x \<noteq> 0\<close> and \<open>\<bar>x\<bar> < ?s n\<close> show "\<bar>?diff n x - f' x0 n\<bar> < ?r" 1131 by blast 1132 qed auto 1133 also have "\<dots> = of_nat (card {..<?N}) * ?r" 1134 by (rule sum_constant) 1135 also have "\<dots> = real ?N * ?r" 1136 by simp 1137 also have "\<dots> = r/3" 1138 by (auto simp del: of_nat_Suc) 1139 finally have "\<bar>\<Sum>n<?N. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") . 1140 1141 from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]] 1142 have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> = 1143 \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>" 1144 unfolding suminf_diff[OF div_smbl \<open>summable (f' x0)\<close>, symmetric] 1145 using suminf_divide[OF diff_smbl, symmetric] by auto 1146 also have "\<dots> \<le> ?diff_part + \<bar>(\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N))\<bar>" 1147 unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] 1148 unfolding suminf_diff[OF div_shft_smbl ign[OF \<open>summable (f' x0)\<close>]] 1149 apply (simp only: add.commute) 1150 using abs_triangle_ineq by blast 1151 also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" 1152 using abs_triangle_ineq4 by auto 1153 also have "\<dots> < r /3 + r/3 + r/3" 1154 using \<open>?diff_part < r/3\<close> \<open>?L_part \<le> r/3\<close> and \<open>?f'_part < r/3\<close> 1155 by (rule add_strict_mono [OF add_less_le_mono]) 1156 finally show ?thesis 1157 by auto 1158 qed 1159 then show "\<exists>s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow> 1160 norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" 1161 using \<open>0 < S\<close> by auto 1162qed 1163 1164lemma DERIV_power_series': 1165 fixes f :: "nat \<Rightarrow> real" 1166 assumes converges: "\<And>x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda>n. f n * real (Suc n) * x^n)" 1167 and x0_in_I: "x0 \<in> {-R <..< R}" 1168 and "0 < R" 1169 shows "DERIV (\<lambda>x. (\<Sum>n. f n * x^(Suc n))) x0 :> (\<Sum>n. f n * real (Suc n) * x0^n)" 1170 (is "DERIV (\<lambda>x. suminf (?f x)) x0 :> suminf (?f' x0)") 1171proof - 1172 have for_subinterval: "DERIV (\<lambda>x. suminf (?f x)) x0 :> suminf (?f' x0)" 1173 if "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'" for R' 1174 proof - 1175 from that have "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" 1176 by auto 1177 show ?thesis 1178 proof (rule DERIV_series') 1179 show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)" 1180 proof - 1181 have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" 1182 using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps) 1183 then have in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" 1184 using \<open>R' < R\<close> by auto 1185 have "norm R' < norm ((R' + R) / 2)" 1186 using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps) 1187 from powser_insidea[OF converges[OF in_Rball] this] show ?thesis 1188 by auto 1189 qed 1190 next 1191 fix n x y 1192 assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}" 1193 show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>" 1194 proof - 1195 have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> = 1196 (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>" 1197 unfolding right_diff_distrib[symmetric] diff_power_eq_sum abs_mult 1198 by auto 1199 also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)" 1200 proof (rule mult_left_mono) 1201 have "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)" 1202 by (rule sum_abs) 1203 also have "\<dots> \<le> (\<Sum>p<Suc n. R' ^ n)" 1204 proof (rule sum_mono) 1205 fix p 1206 assume "p \<in> {..<Suc n}" 1207 then have "p \<le> n" by auto 1208 have "\<bar>x^n\<bar> \<le> R'^n" if "x \<in> {-R'<..<R'}" for n and x :: real 1209 proof - 1210 from that have "\<bar>x\<bar> \<le> R'" by auto 1211 then show ?thesis 1212 unfolding power_abs by (rule power_mono) auto 1213 qed 1214 from mult_mono[OF this[OF \<open>x \<in> {-R'<..<R'}\<close>, of p] this[OF \<open>y \<in> {-R'<..<R'}\<close>, of "n-p"]] 1215 and \<open>0 < R'\<close> 1216 have "\<bar>x^p * y^(n - p)\<bar> \<le> R'^p * R'^(n - p)" 1217 unfolding abs_mult by auto 1218 then show "\<bar>x^p * y^(n - p)\<bar> \<le> R'^n" 1219 unfolding power_add[symmetric] using \<open>p \<le> n\<close> by auto 1220 qed 1221 also have "\<dots> = real (Suc n) * R' ^ n" 1222 unfolding sum_constant card_atLeastLessThan by auto 1223 finally show "\<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>" 1224 unfolding abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF \<open>0 < R'\<close>]]] 1225 by linarith 1226 show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" 1227 unfolding abs_mult[symmetric] by auto 1228 qed 1229 also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" 1230 unfolding abs_mult mult.assoc[symmetric] by algebra 1231 finally show ?thesis . 1232 qed 1233 next 1234 show "DERIV (\<lambda>x. ?f x n) x0 :> ?f' x0 n" for n 1235 by (auto intro!: derivative_eq_intros simp del: power_Suc) 1236 next 1237 fix x 1238 assume "x \<in> {-R' <..< R'}" 1239 then have "R' \<in> {-R <..< R}" and "norm x < norm R'" 1240 using assms \<open>R' < R\<close> by auto 1241 have "summable (\<lambda>n. f n * x^n)" 1242 proof (rule summable_comparison_test, intro exI allI impI) 1243 fix n 1244 have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" 1245 by (rule mult_left_mono) auto 1246 show "norm (f n * x^n) \<le> norm (f n * real (Suc n) * x^n)" 1247 unfolding real_norm_def abs_mult 1248 using le mult_right_mono by fastforce 1249 qed (rule powser_insidea[OF converges[OF \<open>R' \<in> {-R <..< R}\<close>] \<open>norm x < norm R'\<close>]) 1250 from this[THEN summable_mult2[where c=x], simplified mult.assoc, simplified mult.commute] 1251 show "summable (?f x)" by auto 1252 next 1253 show "summable (?f' x0)" 1254 using converges[OF \<open>x0 \<in> {-R <..< R}\<close>] . 1255 show "x0 \<in> {-R' <..< R'}" 1256 using \<open>x0 \<in> {-R' <..< R'}\<close> . 1257 qed 1258 qed 1259 let ?R = "(R + \<bar>x0\<bar>) / 2" 1260 have "\<bar>x0\<bar> < ?R" 1261 using assms by (auto simp: field_simps) 1262 then have "- ?R < x0" 1263 proof (cases "x0 < 0") 1264 case True 1265 then have "- x0 < ?R" 1266 using \<open>\<bar>x0\<bar> < ?R\<close> by auto 1267 then show ?thesis 1268 unfolding neg_less_iff_less[symmetric, of "- x0"] by auto 1269 next 1270 case False 1271 have "- ?R < 0" using assms by auto 1272 also have "\<dots> \<le> x0" using False by auto 1273 finally show ?thesis . 1274 qed 1275 then have "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" 1276 using assms by (auto simp: field_simps) 1277 from for_subinterval[OF this] show ?thesis . 1278qed 1279 1280lemma geometric_deriv_sums: 1281 fixes z :: "'a :: {real_normed_field,banach}" 1282 assumes "norm z < 1" 1283 shows "(\<lambda>n. of_nat (Suc n) * z ^ n) sums (1 / (1 - z)^2)" 1284proof - 1285 have "(\<lambda>n. diffs (\<lambda>n. 1) n * z^n) sums (1 / (1 - z)^2)" 1286 proof (rule termdiffs_sums_strong) 1287 fix z :: 'a assume "norm z < 1" 1288 thus "(\<lambda>n. 1 * z^n) sums (1 / (1 - z))" by (simp add: geometric_sums) 1289 qed (insert assms, auto intro!: derivative_eq_intros simp: power2_eq_square) 1290 thus ?thesis unfolding diffs_def by simp 1291qed 1292 1293lemma isCont_pochhammer [continuous_intros]: "isCont (\<lambda>z. pochhammer z n) z" 1294 for z :: "'a::real_normed_field" 1295 by (induct n) (auto simp: pochhammer_rec') 1296 1297lemma continuous_on_pochhammer [continuous_intros]: "continuous_on A (\<lambda>z. pochhammer z n)" 1298 for A :: "'a::real_normed_field set" 1299 by (intro continuous_at_imp_continuous_on ballI isCont_pochhammer) 1300 1301lemmas continuous_on_pochhammer' [continuous_intros] = 1302 continuous_on_compose2[OF continuous_on_pochhammer _ subset_UNIV] 1303 1304 1305subsection \<open>Exponential Function\<close> 1306 1307definition exp :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}" 1308 where "exp = (\<lambda>x. \<Sum>n. x^n /\<^sub>R fact n)" 1309 1310lemma summable_exp_generic: 1311 fixes x :: "'a::{real_normed_algebra_1,banach}" 1312 defines S_def: "S \<equiv> \<lambda>n. x^n /\<^sub>R fact n" 1313 shows "summable S" 1314proof - 1315 have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R (Suc n)" 1316 unfolding S_def by (simp del: mult_Suc) 1317 obtain r :: real where r0: "0 < r" and r1: "r < 1" 1318 using dense [OF zero_less_one] by fast 1319 obtain N :: nat where N: "norm x < real N * r" 1320 using ex_less_of_nat_mult r0 by auto 1321 from r1 show ?thesis 1322 proof (rule summable_ratio_test [rule_format]) 1323 fix n :: nat 1324 assume n: "N \<le> n" 1325 have "norm x \<le> real N * r" 1326 using N by (rule order_less_imp_le) 1327 also have "real N * r \<le> real (Suc n) * r" 1328 using r0 n by (simp add: mult_right_mono) 1329 finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)" 1330 using norm_ge_zero by (rule mult_right_mono) 1331 then have "norm (x * S n) \<le> real (Suc n) * r * norm (S n)" 1332 by (rule order_trans [OF norm_mult_ineq]) 1333 then have "norm (x * S n) / real (Suc n) \<le> r * norm (S n)" 1334 by (simp add: pos_divide_le_eq ac_simps) 1335 then show "norm (S (Suc n)) \<le> r * norm (S n)" 1336 by (simp add: S_Suc inverse_eq_divide) 1337 qed 1338qed 1339 1340lemma summable_norm_exp: "summable (\<lambda>n. norm (x^n /\<^sub>R fact n))" 1341 for x :: "'a::{real_normed_algebra_1,banach}" 1342proof (rule summable_norm_comparison_test [OF exI, rule_format]) 1343 show "summable (\<lambda>n. norm x^n /\<^sub>R fact n)" 1344 by (rule summable_exp_generic) 1345 show "norm (x^n /\<^sub>R fact n) \<le> norm x^n /\<^sub>R fact n" for n 1346 by (simp add: norm_power_ineq) 1347qed 1348 1349lemma summable_exp: "summable (\<lambda>n. inverse (fact n) * x^n)" 1350 for x :: "'a::{real_normed_field,banach}" 1351 using summable_exp_generic [where x=x] 1352 by (simp add: scaleR_conv_of_real nonzero_of_real_inverse) 1353 1354lemma exp_converges: "(\<lambda>n. x^n /\<^sub>R fact n) sums exp x" 1355 unfolding exp_def by (rule summable_exp_generic [THEN summable_sums]) 1356 1357lemma exp_fdiffs: 1358 "diffs (\<lambda>n. inverse (fact n)) = (\<lambda>n. inverse (fact n :: 'a::{real_normed_field,banach}))" 1359 by (simp add: diffs_def mult_ac nonzero_inverse_mult_distrib nonzero_of_real_inverse 1360 del: mult_Suc of_nat_Suc) 1361 1362lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))" 1363 by (simp add: diffs_def) 1364 1365lemma DERIV_exp [simp]: "DERIV exp x :> exp x" 1366 unfolding exp_def scaleR_conv_of_real 1367proof (rule DERIV_cong) 1368 have sinv: "summable (\<lambda>n. of_real (inverse (fact n)) * x ^ n)" for x::'a 1369 by (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real]) 1370 note xx = exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real] 1371 show "((\<lambda>x. \<Sum>n. of_real (inverse (fact n)) * x ^ n) has_field_derivative 1372 (\<Sum>n. diffs (\<lambda>n. of_real (inverse (fact n))) n * x ^ n)) (at x)" 1373 by (rule termdiffs [where K="of_real (1 + norm x)"]) (simp_all only: diffs_of_real exp_fdiffs sinv norm_of_real) 1374 show "(\<Sum>n. diffs (\<lambda>n. of_real (inverse (fact n))) n * x ^ n) = (\<Sum>n. of_real (inverse (fact n)) * x ^ n)" 1375 by (simp add: diffs_of_real exp_fdiffs) 1376qed 1377 1378declare DERIV_exp[THEN DERIV_chain2, derivative_intros] 1379 and DERIV_exp[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 1380 1381lemmas has_derivative_exp[derivative_intros] = DERIV_exp[THEN DERIV_compose_FDERIV] 1382 1383lemma norm_exp: "norm (exp x) \<le> exp (norm x)" 1384proof - 1385 from summable_norm[OF summable_norm_exp, of x] 1386 have "norm (exp x) \<le> (\<Sum>n. inverse (fact n) * norm (x^n))" 1387 by (simp add: exp_def) 1388 also have "\<dots> \<le> exp (norm x)" 1389 using summable_exp_generic[of "norm x"] summable_norm_exp[of x] 1390 by (auto simp: exp_def intro!: suminf_le norm_power_ineq) 1391 finally show ?thesis . 1392qed 1393 1394lemma isCont_exp: "isCont exp x" 1395 for x :: "'a::{real_normed_field,banach}" 1396 by (rule DERIV_exp [THEN DERIV_isCont]) 1397 1398lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a" 1399 for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 1400 by (rule isCont_o2 [OF _ isCont_exp]) 1401 1402lemma tendsto_exp [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) \<longlongrightarrow> exp a) F" 1403 for f:: "_ \<Rightarrow>'a::{real_normed_field,banach}" 1404 by (rule isCont_tendsto_compose [OF isCont_exp]) 1405 1406lemma continuous_exp [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))" 1407 for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 1408 unfolding continuous_def by (rule tendsto_exp) 1409 1410lemma continuous_on_exp [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))" 1411 for f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 1412 unfolding continuous_on_def by (auto intro: tendsto_exp) 1413 1414 1415subsubsection \<open>Properties of the Exponential Function\<close> 1416 1417lemma exp_zero [simp]: "exp 0 = 1" 1418 unfolding exp_def by (simp add: scaleR_conv_of_real) 1419 1420lemma exp_series_add_commuting: 1421 fixes x y :: "'a::{real_normed_algebra_1,banach}" 1422 defines S_def: "S \<equiv> \<lambda>x n. x^n /\<^sub>R fact n" 1423 assumes comm: "x * y = y * x" 1424 shows "S (x + y) n = (\<Sum>i\<le>n. S x i * S y (n - i))" 1425proof (induct n) 1426 case 0 1427 show ?case 1428 unfolding S_def by simp 1429next 1430 case (Suc n) 1431 have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)" 1432 unfolding S_def by (simp del: mult_Suc) 1433 then have times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)" 1434 by simp 1435 have S_comm: "\<And>n. S x n * y = y * S x n" 1436 by (simp add: power_commuting_commutes comm S_def) 1437 1438 have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n" 1439 by (simp only: times_S) 1440 also have "\<dots> = (x + y) * (\<Sum>i\<le>n. S x i * S y (n - i))" 1441 by (simp only: Suc) 1442 also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n - i)) + y * (\<Sum>i\<le>n. S x i * S y (n - i))" 1443 by (rule distrib_right) 1444 also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n - i)) + (\<Sum>i\<le>n. S x i * y * S y (n - i))" 1445 by (simp add: sum_distrib_left ac_simps S_comm) 1446 also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n - i)) + (\<Sum>i\<le>n. S x i * (y * S y (n - i)))" 1447 by (simp add: ac_simps) 1448 also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i))) + 1449 (\<Sum>i\<le>n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))" 1450 by (simp add: times_S Suc_diff_le) 1451 also have "(\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i))) = 1452 (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n - i)))" 1453 by (subst sum.atMost_Suc_shift) simp 1454 also have "(\<Sum>i\<le>n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i))) = 1455 (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))" 1456 by simp 1457 also have "(\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n - i))) + 1458 (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i))) = 1459 (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n - i)))" 1460 by (simp only: sum.distrib [symmetric] scaleR_left_distrib [symmetric] 1461 of_nat_add [symmetric]) simp 1462 also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))" 1463 by (simp only: scaleR_right.sum) 1464 finally show "S (x + y) (Suc n) = (\<Sum>i\<le>Suc n. S x i * S y (Suc n - i))" 1465 by (simp del: sum.cl_ivl_Suc) 1466qed 1467 1468lemma exp_add_commuting: "x * y = y * x \<Longrightarrow> exp (x + y) = exp x * exp y" 1469 by (simp only: exp_def Cauchy_product summable_norm_exp exp_series_add_commuting) 1470 1471lemma exp_times_arg_commute: "exp A * A = A * exp A" 1472 by (simp add: exp_def suminf_mult[symmetric] summable_exp_generic power_commutes suminf_mult2) 1473 1474lemma exp_add: "exp (x + y) = exp x * exp y" 1475 for x y :: "'a::{real_normed_field,banach}" 1476 by (rule exp_add_commuting) (simp add: ac_simps) 1477 1478lemma exp_double: "exp(2 * z) = exp z ^ 2" 1479 by (simp add: exp_add_commuting mult_2 power2_eq_square) 1480 1481lemmas mult_exp_exp = exp_add [symmetric] 1482 1483lemma exp_of_real: "exp (of_real x) = of_real (exp x)" 1484 unfolding exp_def 1485 apply (subst suminf_of_real [OF summable_exp_generic]) 1486 apply (simp add: scaleR_conv_of_real) 1487 done 1488 1489lemmas of_real_exp = exp_of_real[symmetric] 1490 1491corollary exp_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> exp z \<in> \<real>" 1492 by (metis Reals_cases Reals_of_real exp_of_real) 1493 1494lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0" 1495proof 1496 have "exp x * exp (- x) = 1" 1497 by (simp add: exp_add_commuting[symmetric]) 1498 also assume "exp x = 0" 1499 finally show False by simp 1500qed 1501 1502lemma exp_minus_inverse: "exp x * exp (- x) = 1" 1503 by (simp add: exp_add_commuting[symmetric]) 1504 1505lemma exp_minus: "exp (- x) = inverse (exp x)" 1506 for x :: "'a::{real_normed_field,banach}" 1507 by (intro inverse_unique [symmetric] exp_minus_inverse) 1508 1509lemma exp_diff: "exp (x - y) = exp x / exp y" 1510 for x :: "'a::{real_normed_field,banach}" 1511 using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse) 1512 1513lemma exp_of_nat_mult: "exp (of_nat n * x) = exp x ^ n" 1514 for x :: "'a::{real_normed_field,banach}" 1515 by (induct n) (auto simp: distrib_left exp_add mult.commute) 1516 1517corollary exp_of_nat2_mult: "exp (x * of_nat n) = exp x ^ n" 1518 for x :: "'a::{real_normed_field,banach}" 1519 by (metis exp_of_nat_mult mult_of_nat_commute) 1520 1521lemma exp_sum: "finite I \<Longrightarrow> exp (sum f I) = prod (\<lambda>x. exp (f x)) I" 1522 by (induct I rule: finite_induct) (auto simp: exp_add_commuting mult.commute) 1523 1524lemma exp_divide_power_eq: 1525 fixes x :: "'a::{real_normed_field,banach}" 1526 assumes "n > 0" 1527 shows "exp (x / of_nat n) ^ n = exp x" 1528 using assms 1529proof (induction n arbitrary: x) 1530 case (Suc n) 1531 show ?case 1532 proof (cases "n = 0") 1533 case True 1534 then show ?thesis by simp 1535 next 1536 case False 1537 have [simp]: "1 + (of_nat n * of_nat n + of_nat n * 2) \<noteq> (0::'a)" 1538 using of_nat_eq_iff [of "1 + n * n + n * 2" "0"] 1539 by simp 1540 from False have [simp]: "x * of_nat n / (1 + of_nat n) / of_nat n = x / (1 + of_nat n)" 1541 by simp 1542 have [simp]: "x / (1 + of_nat n) + x * of_nat n / (1 + of_nat n) = x" 1543 using of_nat_neq_0 1544 by (auto simp add: field_split_simps) 1545 show ?thesis 1546 using Suc.IH [of "x * of_nat n / (1 + of_nat n)"] False 1547 by (simp add: exp_add [symmetric]) 1548 qed 1549qed simp 1550 1551 1552subsubsection \<open>Properties of the Exponential Function on Reals\<close> 1553 1554text \<open>Comparisons of \<^term>\<open>exp x\<close> with zero.\<close> 1555 1556text \<open>Proof: because every exponential can be seen as a square.\<close> 1557lemma exp_ge_zero [simp]: "0 \<le> exp x" 1558 for x :: real 1559proof - 1560 have "0 \<le> exp (x/2) * exp (x/2)" 1561 by simp 1562 then show ?thesis 1563 by (simp add: exp_add [symmetric]) 1564qed 1565 1566lemma exp_gt_zero [simp]: "0 < exp x" 1567 for x :: real 1568 by (simp add: order_less_le) 1569 1570lemma not_exp_less_zero [simp]: "\<not> exp x < 0" 1571 for x :: real 1572 by (simp add: not_less) 1573 1574lemma not_exp_le_zero [simp]: "\<not> exp x \<le> 0" 1575 for x :: real 1576 by (simp add: not_le) 1577 1578lemma abs_exp_cancel [simp]: "\<bar>exp x\<bar> = exp x" 1579 for x :: real 1580 by simp 1581 1582text \<open>Strict monotonicity of exponential.\<close> 1583 1584lemma exp_ge_add_one_self_aux: 1585 fixes x :: real 1586 assumes "0 \<le> x" 1587 shows "1 + x \<le> exp x" 1588 using order_le_imp_less_or_eq [OF assms] 1589proof 1590 assume "0 < x" 1591 have "1 + x \<le> (\<Sum>n<2. inverse (fact n) * x^n)" 1592 by (auto simp: numeral_2_eq_2) 1593 also have "\<dots> \<le> (\<Sum>n. inverse (fact n) * x^n)" 1594 apply (rule sum_le_suminf [OF summable_exp]) 1595 using \<open>0 < x\<close> 1596 apply (auto simp add: zero_le_mult_iff) 1597 done 1598 finally show "1 + x \<le> exp x" 1599 by (simp add: exp_def) 1600qed auto 1601 1602lemma exp_gt_one: "0 < x \<Longrightarrow> 1 < exp x" 1603 for x :: real 1604proof - 1605 assume x: "0 < x" 1606 then have "1 < 1 + x" by simp 1607 also from x have "1 + x \<le> exp x" 1608 by (simp add: exp_ge_add_one_self_aux) 1609 finally show ?thesis . 1610qed 1611 1612lemma exp_less_mono: 1613 fixes x y :: real 1614 assumes "x < y" 1615 shows "exp x < exp y" 1616proof - 1617 from \<open>x < y\<close> have "0 < y - x" by simp 1618 then have "1 < exp (y - x)" by (rule exp_gt_one) 1619 then have "1 < exp y / exp x" by (simp only: exp_diff) 1620 then show "exp x < exp y" by simp 1621qed 1622 1623lemma exp_less_cancel: "exp x < exp y \<Longrightarrow> x < y" 1624 for x y :: real 1625 unfolding linorder_not_le [symmetric] 1626 by (auto simp: order_le_less exp_less_mono) 1627 1628lemma exp_less_cancel_iff [iff]: "exp x < exp y \<longleftrightarrow> x < y" 1629 for x y :: real 1630 by (auto intro: exp_less_mono exp_less_cancel) 1631 1632lemma exp_le_cancel_iff [iff]: "exp x \<le> exp y \<longleftrightarrow> x \<le> y" 1633 for x y :: real 1634 by (auto simp: linorder_not_less [symmetric]) 1635 1636lemma exp_inj_iff [iff]: "exp x = exp y \<longleftrightarrow> x = y" 1637 for x y :: real 1638 by (simp add: order_eq_iff) 1639 1640text \<open>Comparisons of \<^term>\<open>exp x\<close> with one.\<close> 1641 1642lemma one_less_exp_iff [simp]: "1 < exp x \<longleftrightarrow> 0 < x" 1643 for x :: real 1644 using exp_less_cancel_iff [where x = 0 and y = x] by simp 1645 1646lemma exp_less_one_iff [simp]: "exp x < 1 \<longleftrightarrow> x < 0" 1647 for x :: real 1648 using exp_less_cancel_iff [where x = x and y = 0] by simp 1649 1650lemma one_le_exp_iff [simp]: "1 \<le> exp x \<longleftrightarrow> 0 \<le> x" 1651 for x :: real 1652 using exp_le_cancel_iff [where x = 0 and y = x] by simp 1653 1654lemma exp_le_one_iff [simp]: "exp x \<le> 1 \<longleftrightarrow> x \<le> 0" 1655 for x :: real 1656 using exp_le_cancel_iff [where x = x and y = 0] by simp 1657 1658lemma exp_eq_one_iff [simp]: "exp x = 1 \<longleftrightarrow> x = 0" 1659 for x :: real 1660 using exp_inj_iff [where x = x and y = 0] by simp 1661 1662lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x \<and> x \<le> y - 1 \<and> exp x = y" 1663 for y :: real 1664proof (rule IVT) 1665 assume "1 \<le> y" 1666 then have "0 \<le> y - 1" by simp 1667 then have "1 + (y - 1) \<le> exp (y - 1)" 1668 by (rule exp_ge_add_one_self_aux) 1669 then show "y \<le> exp (y - 1)" by simp 1670qed (simp_all add: le_diff_eq) 1671 1672lemma exp_total: "0 < y \<Longrightarrow> \<exists>x. exp x = y" 1673 for y :: real 1674proof (rule linorder_le_cases [of 1 y]) 1675 assume "1 \<le> y" 1676 then show "\<exists>x. exp x = y" 1677 by (fast dest: lemma_exp_total) 1678next 1679 assume "0 < y" and "y \<le> 1" 1680 then have "1 \<le> inverse y" 1681 by (simp add: one_le_inverse_iff) 1682 then obtain x where "exp x = inverse y" 1683 by (fast dest: lemma_exp_total) 1684 then have "exp (- x) = y" 1685 by (simp add: exp_minus) 1686 then show "\<exists>x. exp x = y" .. 1687qed 1688 1689 1690subsection \<open>Natural Logarithm\<close> 1691 1692class ln = real_normed_algebra_1 + banach + 1693 fixes ln :: "'a \<Rightarrow> 'a" 1694 assumes ln_one [simp]: "ln 1 = 0" 1695 1696definition powr :: "'a \<Rightarrow> 'a \<Rightarrow> 'a::ln" (infixr "powr" 80) 1697 \<comment> \<open>exponentation via ln and exp\<close> 1698 where "x powr a \<equiv> if x = 0 then 0 else exp (a * ln x)" 1699 1700lemma powr_0 [simp]: "0 powr z = 0" 1701 by (simp add: powr_def) 1702 1703 1704instantiation real :: ln 1705begin 1706 1707definition ln_real :: "real \<Rightarrow> real" 1708 where "ln_real x = (THE u. exp u = x)" 1709 1710instance 1711 by intro_classes (simp add: ln_real_def) 1712 1713end 1714 1715lemma powr_eq_0_iff [simp]: "w powr z = 0 \<longleftrightarrow> w = 0" 1716 by (simp add: powr_def) 1717 1718lemma ln_exp [simp]: "ln (exp x) = x" 1719 for x :: real 1720 by (simp add: ln_real_def) 1721 1722lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x" 1723 for x :: real 1724 by (auto dest: exp_total) 1725 1726lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x" 1727 for x :: real 1728 by (metis exp_gt_zero exp_ln) 1729 1730lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y" 1731 for x :: real 1732 by (erule subst) (rule ln_exp) 1733 1734lemma ln_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y" 1735 for x :: real 1736 by (rule ln_unique) (simp add: exp_add) 1737 1738lemma ln_prod: "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> f i > 0) \<Longrightarrow> ln (prod f I) = sum (\<lambda>x. ln(f x)) I" 1739 for f :: "'a \<Rightarrow> real" 1740 by (induct I rule: finite_induct) (auto simp: ln_mult prod_pos) 1741 1742lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x" 1743 for x :: real 1744 by (rule ln_unique) (simp add: exp_minus) 1745 1746lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y" 1747 for x :: real 1748 by (rule ln_unique) (simp add: exp_diff) 1749 1750lemma ln_realpow: "0 < x \<Longrightarrow> ln (x^n) = real n * ln x" 1751 by (rule ln_unique) (simp add: exp_of_nat_mult) 1752 1753lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y" 1754 for x :: real 1755 by (subst exp_less_cancel_iff [symmetric]) simp 1756 1757lemma ln_le_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y" 1758 for x :: real 1759 by (simp add: linorder_not_less [symmetric]) 1760 1761lemma ln_inj_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y" 1762 for x :: real 1763 by (simp add: order_eq_iff) 1764 1765lemma ln_add_one_self_le_self: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x" 1766 for x :: real 1767 by (rule exp_le_cancel_iff [THEN iffD1]) (simp add: exp_ge_add_one_self_aux) 1768 1769lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x" 1770 for x :: real 1771 by (rule order_less_le_trans [where y = "ln (1 + x)"]) (simp_all add: ln_add_one_self_le_self) 1772 1773lemma ln_ge_iff: "\<And>x::real. 0 < x \<Longrightarrow> y \<le> ln x \<longleftrightarrow> exp y \<le> x" 1774 using exp_le_cancel_iff exp_total by force 1775 1776lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x" 1777 for x :: real 1778 using ln_le_cancel_iff [of 1 x] by simp 1779 1780lemma ln_ge_zero_imp_ge_one: "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x" 1781 for x :: real 1782 using ln_le_cancel_iff [of 1 x] by simp 1783 1784lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x" 1785 for x :: real 1786 using ln_le_cancel_iff [of 1 x] by simp 1787 1788lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1" 1789 for x :: real 1790 using ln_less_cancel_iff [of x 1] by simp 1791 1792lemma ln_le_zero_iff [simp]: "0 < x \<Longrightarrow> ln x \<le> 0 \<longleftrightarrow> x \<le> 1" 1793 for x :: real 1794 by (metis less_numeral_extra(1) ln_le_cancel_iff ln_one) 1795 1796lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x" 1797 for x :: real 1798 using ln_less_cancel_iff [of 1 x] by simp 1799 1800lemma ln_gt_zero_imp_gt_one: "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x" 1801 for x :: real 1802 using ln_less_cancel_iff [of 1 x] by simp 1803 1804lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x" 1805 for x :: real 1806 using ln_less_cancel_iff [of 1 x] by simp 1807 1808lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1" 1809 for x :: real 1810 using ln_inj_iff [of x 1] by simp 1811 1812lemma ln_less_zero: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0" 1813 for x :: real 1814 by simp 1815 1816lemma ln_neg_is_const: "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)" 1817 for x :: real 1818 by (auto simp: ln_real_def intro!: arg_cong[where f = The]) 1819 1820lemma powr_eq_one_iff [simp]: 1821 "a powr x = 1 \<longleftrightarrow> x = 0" if "a > 1" for a x :: real 1822 using that by (auto simp: powr_def split: if_splits) 1823 1824lemma isCont_ln: 1825 fixes x :: real 1826 assumes "x \<noteq> 0" 1827 shows "isCont ln x" 1828proof (cases "0 < x") 1829 case True 1830 then have "isCont ln (exp (ln x))" 1831 by (intro isCont_inverse_function[where d = "\<bar>x\<bar>" and f = exp]) auto 1832 with True show ?thesis 1833 by simp 1834next 1835 case False 1836 with \<open>x \<noteq> 0\<close> show "isCont ln x" 1837 unfolding isCont_def 1838 by (subst filterlim_cong[OF _ refl, of _ "nhds (ln 0)" _ "\<lambda>_. ln 0"]) 1839 (auto simp: ln_neg_is_const not_less eventually_at dist_real_def 1840 intro!: exI[of _ "\<bar>x\<bar>"]) 1841qed 1842 1843lemma tendsto_ln [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) \<longlongrightarrow> ln a) F" 1844 for a :: real 1845 by (rule isCont_tendsto_compose [OF isCont_ln]) 1846 1847lemma continuous_ln: 1848 "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. ln (f x :: real))" 1849 unfolding continuous_def by (rule tendsto_ln) 1850 1851lemma isCont_ln' [continuous_intros]: 1852 "continuous (at x) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x :: real))" 1853 unfolding continuous_at by (rule tendsto_ln) 1854 1855lemma continuous_within_ln [continuous_intros]: 1856 "continuous (at x within s) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x :: real))" 1857 unfolding continuous_within by (rule tendsto_ln) 1858 1859lemma continuous_on_ln [continuous_intros]: 1860 "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. f x \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x :: real))" 1861 unfolding continuous_on_def by (auto intro: tendsto_ln) 1862 1863lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x" 1864 for x :: real 1865 by (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"]) 1866 (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln) 1867 1868lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x" 1869 for x :: real 1870 by (rule DERIV_ln[THEN DERIV_cong]) (simp_all add: divide_inverse) 1871 1872declare DERIV_ln_divide[THEN DERIV_chain2, derivative_intros] 1873 and DERIV_ln_divide[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 1874 1875lemmas has_derivative_ln[derivative_intros] = DERIV_ln[THEN DERIV_compose_FDERIV] 1876 1877lemma ln_series: 1878 assumes "0 < x" and "x < 2" 1879 shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" 1880 (is "ln x = suminf (?f (x - 1))") 1881proof - 1882 let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n" 1883 1884 have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))" 1885 proof (rule DERIV_isconst3 [where x = x]) 1886 fix x :: real 1887 assume "x \<in> {0 <..< 2}" 1888 then have "0 < x" and "x < 2" by auto 1889 have "norm (1 - x) < 1" 1890 using \<open>0 < x\<close> and \<open>x < 2\<close> by auto 1891 have "1 / x = 1 / (1 - (1 - x))" by auto 1892 also have "\<dots> = (\<Sum> n. (1 - x)^n)" 1893 using geometric_sums[OF \<open>norm (1 - x) < 1\<close>] by (rule sums_unique) 1894 also have "\<dots> = suminf (?f' x)" 1895 unfolding power_mult_distrib[symmetric] 1896 by (rule arg_cong[where f=suminf], rule arg_cong[where f="(^)"], auto) 1897 finally have "DERIV ln x :> suminf (?f' x)" 1898 using DERIV_ln[OF \<open>0 < x\<close>] unfolding divide_inverse by auto 1899 moreover 1900 have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto 1901 have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> 1902 (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)" 1903 proof (rule DERIV_power_series') 1904 show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" 1905 using \<open>0 < x\<close> \<open>x < 2\<close> by auto 1906 next 1907 fix x :: real 1908 assume "x \<in> {- 1<..<1}" 1909 then have "norm (-x) < 1" by auto 1910 show "summable (\<lambda>n. (- 1) ^ n * (1 / real (n + 1)) * real (Suc n) * x^n)" 1911 unfolding One_nat_def 1912 by (auto simp: power_mult_distrib[symmetric] summable_geometric[OF \<open>norm (-x) < 1\<close>]) 1913 qed 1914 then have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" 1915 unfolding One_nat_def by auto 1916 then have "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" 1917 unfolding DERIV_def repos . 1918 ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> suminf (?f' x) - suminf (?f' x)" 1919 by (rule DERIV_diff) 1920 then show "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto 1921 qed (auto simp: assms) 1922 then show ?thesis by auto 1923qed 1924 1925lemma exp_first_terms: 1926 fixes x :: "'a::{real_normed_algebra_1,banach}" 1927 shows "exp x = (\<Sum>n<k. inverse(fact n) *\<^sub>R (x ^ n)) + (\<Sum>n. inverse(fact (n + k)) *\<^sub>R (x ^ (n + k)))" 1928proof - 1929 have "exp x = suminf (\<lambda>n. inverse(fact n) *\<^sub>R (x^n))" 1930 by (simp add: exp_def) 1931 also from summable_exp_generic have "\<dots> = (\<Sum> n. inverse(fact(n+k)) *\<^sub>R (x ^ (n + k))) + 1932 (\<Sum> n::nat<k. inverse(fact n) *\<^sub>R (x^n))" (is "_ = _ + ?a") 1933 by (rule suminf_split_initial_segment) 1934 finally show ?thesis by simp 1935qed 1936 1937lemma exp_first_term: "exp x = 1 + (\<Sum>n. inverse (fact (Suc n)) *\<^sub>R (x ^ Suc n))" 1938 for x :: "'a::{real_normed_algebra_1,banach}" 1939 using exp_first_terms[of x 1] by simp 1940 1941lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum>n. inverse (fact (n + 2)) *\<^sub>R (x ^ (n + 2)))" 1942 for x :: "'a::{real_normed_algebra_1,banach}" 1943 using exp_first_terms[of x 2] by (simp add: eval_nat_numeral) 1944 1945lemma exp_bound: 1946 fixes x :: real 1947 assumes a: "0 \<le> x" 1948 and b: "x \<le> 1" 1949 shows "exp x \<le> 1 + x + x\<^sup>2" 1950proof - 1951 have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n + 2))) \<le> x\<^sup>2" 1952 proof - 1953 have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))" 1954 by (intro sums_mult geometric_sums) simp 1955 then have sumsx: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2" 1956 by simp 1957 have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n + 2))) \<le> suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))" 1958 proof (intro suminf_le allI) 1959 show "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)" for n :: nat 1960 proof - 1961 have "(2::nat) * 2 ^ n \<le> fact (n + 2)" 1962 by (induct n) simp_all 1963 then have "real ((2::nat) * 2 ^ n) \<le> real_of_nat (fact (n + 2))" 1964 by (simp only: of_nat_le_iff) 1965 then have "((2::real) * 2 ^ n) \<le> fact (n + 2)" 1966 unfolding of_nat_fact by simp 1967 then have "inverse (fact (n + 2)) \<le> inverse ((2::real) * 2 ^ n)" 1968 by (rule le_imp_inverse_le) simp 1969 then have "inverse (fact (n + 2)) \<le> 1/(2::real) * (1/2)^n" 1970 by (simp add: power_inverse [symmetric]) 1971 then have "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)" 1972 by (rule mult_mono) (rule mult_mono, simp_all add: power_le_one a b) 1973 then show ?thesis 1974 unfolding power_add by (simp add: ac_simps del: fact_Suc) 1975 qed 1976 show "summable (\<lambda>n. inverse (fact (n + 2)) * x ^ (n + 2))" 1977 by (rule summable_exp [THEN summable_ignore_initial_segment]) 1978 show "summable (\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n)" 1979 by (rule sums_summable [OF sumsx]) 1980 qed 1981 also have "\<dots> = x\<^sup>2" 1982 by (rule sums_unique [THEN sym]) (rule sumsx) 1983 finally show ?thesis . 1984 qed 1985 then show ?thesis 1986 unfolding exp_first_two_terms by auto 1987qed 1988 1989corollary exp_half_le2: "exp(1/2) \<le> (2::real)" 1990 using exp_bound [of "1/2"] 1991 by (simp add: field_simps) 1992 1993corollary exp_le: "exp 1 \<le> (3::real)" 1994 using exp_bound [of 1] 1995 by (simp add: field_simps) 1996 1997lemma exp_bound_half: "norm z \<le> 1/2 \<Longrightarrow> norm (exp z) \<le> 2" 1998 by (blast intro: order_trans intro!: exp_half_le2 norm_exp) 1999 2000lemma exp_bound_lemma: 2001 assumes "norm z \<le> 1/2" 2002 shows "norm (exp z) \<le> 1 + 2 * norm z" 2003proof - 2004 have *: "(norm z)\<^sup>2 \<le> norm z * 1" 2005 unfolding power2_eq_square 2006 by (rule mult_left_mono) (use assms in auto) 2007 have "norm (exp z) \<le> exp (norm z)" 2008 by (rule norm_exp) 2009 also have "\<dots> \<le> 1 + (norm z) + (norm z)\<^sup>2" 2010 using assms exp_bound by auto 2011 also have "\<dots> \<le> 1 + 2 * norm z" 2012 using * by auto 2013 finally show ?thesis . 2014qed 2015 2016lemma real_exp_bound_lemma: "0 \<le> x \<Longrightarrow> x \<le> 1/2 \<Longrightarrow> exp x \<le> 1 + 2 * x" 2017 for x :: real 2018 using exp_bound_lemma [of x] by simp 2019 2020lemma ln_one_minus_pos_upper_bound: 2021 fixes x :: real 2022 assumes a: "0 \<le> x" and b: "x < 1" 2023 shows "ln (1 - x) \<le> - x" 2024proof - 2025 have "(1 - x) * (1 + x + x\<^sup>2) = 1 - x^3" 2026 by (simp add: algebra_simps power2_eq_square power3_eq_cube) 2027 also have "\<dots> \<le> 1" 2028 by (auto simp: a) 2029 finally have "(1 - x) * (1 + x + x\<^sup>2) \<le> 1" . 2030 moreover have c: "0 < 1 + x + x\<^sup>2" 2031 by (simp add: add_pos_nonneg a) 2032 ultimately have "1 - x \<le> 1 / (1 + x + x\<^sup>2)" 2033 by (elim mult_imp_le_div_pos) 2034 also have "\<dots> \<le> 1 / exp x" 2035 by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs 2036 real_sqrt_pow2_iff real_sqrt_power) 2037 also have "\<dots> = exp (- x)" 2038 by (auto simp: exp_minus divide_inverse) 2039 finally have "1 - x \<le> exp (- x)" . 2040 also have "1 - x = exp (ln (1 - x))" 2041 by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq) 2042 finally have "exp (ln (1 - x)) \<le> exp (- x)" . 2043 then show ?thesis 2044 by (auto simp only: exp_le_cancel_iff) 2045qed 2046 2047lemma exp_ge_add_one_self [simp]: "1 + x \<le> exp x" 2048 for x :: real 2049proof (cases "0 \<le> x \<or> x \<le> -1") 2050 case True 2051 then show ?thesis 2052 apply (rule disjE) 2053 apply (simp add: exp_ge_add_one_self_aux) 2054 using exp_ge_zero order_trans real_add_le_0_iff by blast 2055next 2056 case False 2057 then have ln1: "ln (1 + x) \<le> x" 2058 using ln_one_minus_pos_upper_bound [of "-x"] by simp 2059 have "1 + x = exp (ln (1 + x))" 2060 using False by auto 2061 also have "\<dots> \<le> exp x" 2062 by (simp add: ln1) 2063 finally show ?thesis . 2064qed 2065 2066lemma ln_one_plus_pos_lower_bound: 2067 fixes x :: real 2068 assumes a: "0 \<le> x" and b: "x \<le> 1" 2069 shows "x - x\<^sup>2 \<le> ln (1 + x)" 2070proof - 2071 have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)" 2072 by (rule exp_diff) 2073 also have "\<dots> \<le> (1 + x + x\<^sup>2) / exp (x \<^sup>2)" 2074 by (metis a b divide_right_mono exp_bound exp_ge_zero) 2075 also have "\<dots> \<le> (1 + x + x\<^sup>2) / (1 + x\<^sup>2)" 2076 by (simp add: a divide_left_mono add_pos_nonneg) 2077 also from a have "\<dots> \<le> 1 + x" 2078 by (simp add: field_simps add_strict_increasing zero_le_mult_iff) 2079 finally have "exp (x - x\<^sup>2) \<le> 1 + x" . 2080 also have "\<dots> = exp (ln (1 + x))" 2081 proof - 2082 from a have "0 < 1 + x" by auto 2083 then show ?thesis 2084 by (auto simp only: exp_ln_iff [THEN sym]) 2085 qed 2086 finally have "exp (x - x\<^sup>2) \<le> exp (ln (1 + x))" . 2087 then show ?thesis 2088 by (metis exp_le_cancel_iff) 2089qed 2090 2091lemma ln_one_minus_pos_lower_bound: 2092 fixes x :: real 2093 assumes a: "0 \<le> x" and b: "x \<le> 1 / 2" 2094 shows "- x - 2 * x\<^sup>2 \<le> ln (1 - x)" 2095proof - 2096 from b have c: "x < 1" by auto 2097 then have "ln (1 - x) = - ln (1 + x / (1 - x))" 2098 by (auto simp: ln_inverse [symmetric] field_simps intro: arg_cong [where f=ln]) 2099 also have "- (x / (1 - x)) \<le> \<dots>" 2100 proof - 2101 have "ln (1 + x / (1 - x)) \<le> x / (1 - x)" 2102 using a c by (intro ln_add_one_self_le_self) auto 2103 then show ?thesis 2104 by auto 2105 qed 2106 also have "- (x / (1 - x)) = - x / (1 - x)" 2107 by auto 2108 finally have d: "- x / (1 - x) \<le> ln (1 - x)" . 2109 have "0 < 1 - x" using a b by simp 2110 then have e: "- x - 2 * x\<^sup>2 \<le> - x / (1 - x)" 2111 using mult_right_le_one_le[of "x * x" "2 * x"] a b 2112 by (simp add: field_simps power2_eq_square) 2113 from e d show "- x - 2 * x\<^sup>2 \<le> ln (1 - x)" 2114 by (rule order_trans) 2115qed 2116 2117lemma ln_add_one_self_le_self2: 2118 fixes x :: real 2119 shows "-1 < x \<Longrightarrow> ln (1 + x) \<le> x" 2120 by (metis diff_gt_0_iff_gt diff_minus_eq_add exp_ge_add_one_self exp_le_cancel_iff exp_ln minus_less_iff) 2121 2122lemma abs_ln_one_plus_x_minus_x_bound_nonneg: 2123 fixes x :: real 2124 assumes x: "0 \<le> x" and x1: "x \<le> 1" 2125 shows "\<bar>ln (1 + x) - x\<bar> \<le> x\<^sup>2" 2126proof - 2127 from x have "ln (1 + x) \<le> x" 2128 by (rule ln_add_one_self_le_self) 2129 then have "ln (1 + x) - x \<le> 0" 2130 by simp 2131 then have "\<bar>ln(1 + x) - x\<bar> = - (ln(1 + x) - x)" 2132 by (rule abs_of_nonpos) 2133 also have "\<dots> = x - ln (1 + x)" 2134 by simp 2135 also have "\<dots> \<le> x\<^sup>2" 2136 proof - 2137 from x x1 have "x - x\<^sup>2 \<le> ln (1 + x)" 2138 by (intro ln_one_plus_pos_lower_bound) 2139 then show ?thesis 2140 by simp 2141 qed 2142 finally show ?thesis . 2143qed 2144 2145lemma abs_ln_one_plus_x_minus_x_bound_nonpos: 2146 fixes x :: real 2147 assumes a: "-(1 / 2) \<le> x" and b: "x \<le> 0" 2148 shows "\<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2" 2149proof - 2150 have *: "- (-x) - 2 * (-x)\<^sup>2 \<le> ln (1 - (- x))" 2151 by (metis a b diff_zero ln_one_minus_pos_lower_bound minus_diff_eq neg_le_iff_le) 2152 have "\<bar>ln (1 + x) - x\<bar> = x - ln (1 - (- x))" 2153 using a ln_add_one_self_le_self2 [of x] by (simp add: abs_if) 2154 also have "\<dots> \<le> 2 * x\<^sup>2" 2155 using * by (simp add: algebra_simps) 2156 finally show ?thesis . 2157qed 2158 2159lemma abs_ln_one_plus_x_minus_x_bound: 2160 fixes x :: real 2161 assumes "\<bar>x\<bar> \<le> 1 / 2" 2162 shows "\<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2" 2163proof (cases "0 \<le> x") 2164 case True 2165 then show ?thesis 2166 using abs_ln_one_plus_x_minus_x_bound_nonneg assms by fastforce 2167next 2168 case False 2169 then show ?thesis 2170 using abs_ln_one_plus_x_minus_x_bound_nonpos assms by auto 2171qed 2172 2173lemma ln_x_over_x_mono: 2174 fixes x :: real 2175 assumes x: "exp 1 \<le> x" "x \<le> y" 2176 shows "ln y / y \<le> ln x / x" 2177proof - 2178 note x 2179 moreover have "0 < exp (1::real)" by simp 2180 ultimately have a: "0 < x" and b: "0 < y" 2181 by (fast intro: less_le_trans order_trans)+ 2182 have "x * ln y - x * ln x = x * (ln y - ln x)" 2183 by (simp add: algebra_simps) 2184 also have "\<dots> = x * ln (y / x)" 2185 by (simp only: ln_div a b) 2186 also have "y / x = (x + (y - x)) / x" 2187 by simp 2188 also have "\<dots> = 1 + (y - x) / x" 2189 using x a by (simp add: field_simps) 2190 also have "x * ln (1 + (y - x) / x) \<le> x * ((y - x) / x)" 2191 using x a 2192 by (intro mult_left_mono ln_add_one_self_le_self) simp_all 2193 also have "\<dots> = y - x" 2194 using a by simp 2195 also have "\<dots> = (y - x) * ln (exp 1)" by simp 2196 also have "\<dots> \<le> (y - x) * ln x" 2197 using a x exp_total of_nat_1 x(1) by (fastforce intro: mult_left_mono) 2198 also have "\<dots> = y * ln x - x * ln x" 2199 by (rule left_diff_distrib) 2200 finally have "x * ln y \<le> y * ln x" 2201 by arith 2202 then have "ln y \<le> (y * ln x) / x" 2203 using a by (simp add: field_simps) 2204 also have "\<dots> = y * (ln x / x)" by simp 2205 finally show ?thesis 2206 using b by (simp add: field_simps) 2207qed 2208 2209lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1" 2210 for x :: real 2211 using exp_ge_add_one_self[of "ln x"] by simp 2212 2213corollary ln_diff_le: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x - ln y \<le> (x - y) / y" 2214 for x :: real 2215 by (simp add: ln_div [symmetric] diff_divide_distrib ln_le_minus_one) 2216 2217lemma ln_eq_minus_one: 2218 fixes x :: real 2219 assumes "0 < x" "ln x = x - 1" 2220 shows "x = 1" 2221proof - 2222 let ?l = "\<lambda>y. ln y - y + 1" 2223 have D: "\<And>x::real. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)" 2224 by (auto intro!: derivative_eq_intros) 2225 2226 show ?thesis 2227 proof (cases rule: linorder_cases) 2228 assume "x < 1" 2229 from dense[OF \<open>x < 1\<close>] obtain a where "x < a" "a < 1" by blast 2230 from \<open>x < a\<close> have "?l x < ?l a" 2231 proof (rule DERIV_pos_imp_increasing) 2232 fix y 2233 assume "x \<le> y" "y \<le> a" 2234 with \<open>0 < x\<close> \<open>a < 1\<close> have "0 < 1 / y - 1" "0 < y" 2235 by (auto simp: field_simps) 2236 with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z" by blast 2237 qed 2238 also have "\<dots> \<le> 0" 2239 using ln_le_minus_one \<open>0 < x\<close> \<open>x < a\<close> by (auto simp: field_simps) 2240 finally show "x = 1" using assms by auto 2241 next 2242 assume "1 < x" 2243 from dense[OF this] obtain a where "1 < a" "a < x" by blast 2244 from \<open>a < x\<close> have "?l x < ?l a" 2245 proof (rule DERIV_neg_imp_decreasing) 2246 fix y 2247 assume "a \<le> y" "y \<le> x" 2248 with \<open>1 < a\<close> have "1 / y - 1 < 0" "0 < y" 2249 by (auto simp: field_simps) 2250 with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0" 2251 by blast 2252 qed 2253 also have "\<dots> \<le> 0" 2254 using ln_le_minus_one \<open>1 < a\<close> by (auto simp: field_simps) 2255 finally show "x = 1" using assms by auto 2256 next 2257 assume "x = 1" 2258 then show ?thesis by simp 2259 qed 2260qed 2261 2262lemma ln_x_over_x_tendsto_0: "((\<lambda>x::real. ln x / x) \<longlongrightarrow> 0) at_top" 2263proof (rule lhospital_at_top_at_top[where f' = inverse and g' = "\<lambda>_. 1"]) 2264 from eventually_gt_at_top[of "0::real"] 2265 show "\<forall>\<^sub>F x in at_top. (ln has_real_derivative inverse x) (at x)" 2266 by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps) 2267qed (use tendsto_inverse_0 in 2268 \<open>auto simp: filterlim_ident dest!: tendsto_mono[OF at_top_le_at_infinity]\<close>) 2269 2270lemma exp_ge_one_plus_x_over_n_power_n: 2271 assumes "x \<ge> - real n" "n > 0" 2272 shows "(1 + x / of_nat n) ^ n \<le> exp x" 2273proof (cases "x = - of_nat n") 2274 case False 2275 from assms False have "(1 + x / of_nat n) ^ n = exp (of_nat n * ln (1 + x / of_nat n))" 2276 by (subst exp_of_nat_mult, subst exp_ln) (simp_all add: field_simps) 2277 also from assms False have "ln (1 + x / real n) \<le> x / real n" 2278 by (intro ln_add_one_self_le_self2) (simp_all add: field_simps) 2279 with assms have "exp (of_nat n * ln (1 + x / of_nat n)) \<le> exp x" 2280 by (simp add: field_simps) 2281 finally show ?thesis . 2282next 2283 case True 2284 then show ?thesis by (simp add: zero_power) 2285qed 2286 2287lemma exp_ge_one_minus_x_over_n_power_n: 2288 assumes "x \<le> real n" "n > 0" 2289 shows "(1 - x / of_nat n) ^ n \<le> exp (-x)" 2290 using exp_ge_one_plus_x_over_n_power_n[of n "-x"] assms by simp 2291 2292lemma exp_at_bot: "(exp \<longlongrightarrow> (0::real)) at_bot" 2293 unfolding tendsto_Zfun_iff 2294proof (rule ZfunI, simp add: eventually_at_bot_dense) 2295 fix r :: real 2296 assume "0 < r" 2297 have "exp x < r" if "x < ln r" for x 2298 by (metis \<open>0 < r\<close> exp_less_mono exp_ln that) 2299 then show "\<exists>k. \<forall>n<k. exp n < r" by auto 2300qed 2301 2302lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top" 2303 by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g=ln]) 2304 (auto intro: eventually_gt_at_top) 2305 2306lemma lim_exp_minus_1: "((\<lambda>z::'a. (exp(z) - 1) / z) \<longlongrightarrow> 1) (at 0)" 2307 for x :: "'a::{real_normed_field,banach}" 2308proof - 2309 have "((\<lambda>z::'a. exp(z) - 1) has_field_derivative 1) (at 0)" 2310 by (intro derivative_eq_intros | simp)+ 2311 then show ?thesis 2312 by (simp add: Deriv.has_field_derivative_iff) 2313qed 2314 2315lemma ln_at_0: "LIM x at_right 0. ln (x::real) :> at_bot" 2316 by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g=exp]) 2317 (auto simp: eventually_at_filter) 2318 2319lemma ln_at_top: "LIM x at_top. ln (x::real) :> at_top" 2320 by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g=exp]) 2321 (auto intro: eventually_gt_at_top) 2322 2323lemma filtermap_ln_at_top: "filtermap (ln::real \<Rightarrow> real) at_top = at_top" 2324 by (intro filtermap_fun_inverse[of exp] exp_at_top ln_at_top) auto 2325 2326lemma filtermap_exp_at_top: "filtermap (exp::real \<Rightarrow> real) at_top = at_top" 2327 by (intro filtermap_fun_inverse[of ln] exp_at_top ln_at_top) 2328 (auto simp: eventually_at_top_dense) 2329 2330lemma filtermap_ln_at_right: "filtermap ln (at_right (0::real)) = at_bot" 2331 by (auto intro!: filtermap_fun_inverse[where g="\<lambda>x. exp x"] ln_at_0 2332 simp: filterlim_at exp_at_bot) 2333 2334lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) \<longlongrightarrow> (0::real)) at_top" 2335proof (induct k) 2336 case 0 2337 show "((\<lambda>x. x ^ 0 / exp x) \<longlongrightarrow> (0::real)) at_top" 2338 by (simp add: inverse_eq_divide[symmetric]) 2339 (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono 2340 at_top_le_at_infinity order_refl) 2341next 2342 case (Suc k) 2343 show ?case 2344 proof (rule lhospital_at_top_at_top) 2345 show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top" 2346 by eventually_elim (intro derivative_eq_intros, auto) 2347 show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top" 2348 by eventually_elim auto 2349 show "eventually (\<lambda>x. exp x \<noteq> 0) at_top" 2350 by auto 2351 from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"] 2352 show "((\<lambda>x. real (Suc k) * x ^ k / exp x) \<longlongrightarrow> 0) at_top" 2353 by simp 2354 qed (rule exp_at_top) 2355qed 2356 2357subsubsection\<open> A couple of simple bounds\<close> 2358 2359lemma exp_plus_inverse_exp: 2360 fixes x::real 2361 shows "2 \<le> exp x + inverse (exp x)" 2362proof - 2363 have "2 \<le> exp x + exp (-x)" 2364 using exp_ge_add_one_self [of x] exp_ge_add_one_self [of "-x"] 2365 by linarith 2366 then show ?thesis 2367 by (simp add: exp_minus) 2368qed 2369 2370lemma real_le_x_sinh: 2371 fixes x::real 2372 assumes "0 \<le> x" 2373 shows "x \<le> (exp x - inverse(exp x)) / 2" 2374proof - 2375 have *: "exp a - inverse(exp a) - 2*a \<le> exp b - inverse(exp b) - 2*b" if "a \<le> b" for a b::real 2376 using exp_plus_inverse_exp 2377 by (fastforce intro: derivative_eq_intros DERIV_nonneg_imp_nondecreasing [OF that]) 2378 show ?thesis 2379 using*[OF assms] by simp 2380qed 2381 2382lemma real_le_abs_sinh: 2383 fixes x::real 2384 shows "abs x \<le> abs((exp x - inverse(exp x)) / 2)" 2385proof (cases "0 \<le> x") 2386 case True 2387 show ?thesis 2388 using real_le_x_sinh [OF True] True by (simp add: abs_if) 2389next 2390 case False 2391 have "-x \<le> (exp(-x) - inverse(exp(-x))) / 2" 2392 by (meson False linear neg_le_0_iff_le real_le_x_sinh) 2393 also have "\<dots> \<le> \<bar>(exp x - inverse (exp x)) / 2\<bar>" 2394 by (metis (no_types, hide_lams) abs_divide abs_le_iff abs_minus_cancel 2395 add.inverse_inverse exp_minus minus_diff_eq order_refl) 2396 finally show ?thesis 2397 using False by linarith 2398qed 2399 2400subsection\<open>The general logarithm\<close> 2401 2402definition log :: "real \<Rightarrow> real \<Rightarrow> real" 2403 \<comment> \<open>logarithm of \<^term>\<open>x\<close> to base \<^term>\<open>a\<close>\<close> 2404 where "log a x = ln x / ln a" 2405 2406lemma tendsto_log [tendsto_intros]: 2407 "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> 0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> 2408 ((\<lambda>x. log (f x) (g x)) \<longlongrightarrow> log a b) F" 2409 unfolding log_def by (intro tendsto_intros) auto 2410 2411lemma continuous_log: 2412 assumes "continuous F f" 2413 and "continuous F g" 2414 and "0 < f (Lim F (\<lambda>x. x))" 2415 and "f (Lim F (\<lambda>x. x)) \<noteq> 1" 2416 and "0 < g (Lim F (\<lambda>x. x))" 2417 shows "continuous F (\<lambda>x. log (f x) (g x))" 2418 using assms unfolding continuous_def by (rule tendsto_log) 2419 2420lemma continuous_at_within_log[continuous_intros]: 2421 assumes "continuous (at a within s) f" 2422 and "continuous (at a within s) g" 2423 and "0 < f a" 2424 and "f a \<noteq> 1" 2425 and "0 < g a" 2426 shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))" 2427 using assms unfolding continuous_within by (rule tendsto_log) 2428 2429lemma isCont_log[continuous_intros, simp]: 2430 assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a" 2431 shows "isCont (\<lambda>x. log (f x) (g x)) a" 2432 using assms unfolding continuous_at by (rule tendsto_log) 2433 2434lemma continuous_on_log[continuous_intros]: 2435 assumes "continuous_on s f" "continuous_on s g" 2436 and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x" 2437 shows "continuous_on s (\<lambda>x. log (f x) (g x))" 2438 using assms unfolding continuous_on_def by (fast intro: tendsto_log) 2439 2440lemma powr_one_eq_one [simp]: "1 powr a = 1" 2441 by (simp add: powr_def) 2442 2443lemma powr_zero_eq_one [simp]: "x powr 0 = (if x = 0 then 0 else 1)" 2444 by (simp add: powr_def) 2445 2446lemma powr_one_gt_zero_iff [simp]: "x powr 1 = x \<longleftrightarrow> 0 \<le> x" 2447 for x :: real 2448 by (auto simp: powr_def) 2449declare powr_one_gt_zero_iff [THEN iffD2, simp] 2450 2451lemma powr_diff: 2452 fixes w:: "'a::{ln,real_normed_field}" shows "w powr (z1 - z2) = w powr z1 / w powr z2" 2453 by (simp add: powr_def algebra_simps exp_diff) 2454 2455lemma powr_mult: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)" 2456 for a x y :: real 2457 by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left) 2458 2459lemma powr_ge_pzero [simp]: "0 \<le> x powr y" 2460 for x y :: real 2461 by (simp add: powr_def) 2462 2463lemma powr_non_neg[simp]: "\<not>a powr x < 0" for a x::real 2464 using powr_ge_pzero[of a x] by arith 2465 2466lemma powr_divide: "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)" 2467 for a b x :: real 2468 apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult) 2469 apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse) 2470 done 2471 2472lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)" 2473 for a b x :: "'a::{ln,real_normed_field}" 2474 by (simp add: powr_def exp_add [symmetric] distrib_right) 2475 2476lemma powr_mult_base: "0 \<le> x \<Longrightarrow>x * x powr y = x powr (1 + y)" 2477 for x :: real 2478 by (auto simp: powr_add) 2479 2480lemma powr_powr: "(x powr a) powr b = x powr (a * b)" 2481 for a b x :: real 2482 by (simp add: powr_def) 2483 2484lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a" 2485 for a b x :: real 2486 by (simp add: powr_powr mult.commute) 2487 2488lemma powr_minus: "x powr (- a) = inverse (x powr a)" 2489 for a x :: "'a::{ln,real_normed_field}" 2490 by (simp add: powr_def exp_minus [symmetric]) 2491 2492lemma powr_minus_divide: "x powr (- a) = 1/(x powr a)" 2493 for a x :: "'a::{ln,real_normed_field}" 2494 by (simp add: divide_inverse powr_minus) 2495 2496lemma divide_powr_uminus: "a / b powr c = a * b powr (- c)" 2497 for a b c :: real 2498 by (simp add: powr_minus_divide) 2499 2500lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b" 2501 for a b x :: real 2502 by (simp add: powr_def) 2503 2504lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b" 2505 for a b x :: real 2506 by (simp add: powr_def) 2507 2508lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> x powr a < x powr b \<longleftrightarrow> a < b" 2509 for a b x :: real 2510 by (blast intro: powr_less_cancel powr_less_mono) 2511 2512lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> x powr a \<le> x powr b \<longleftrightarrow> a \<le> b" 2513 for a b x :: real 2514 by (simp add: linorder_not_less [symmetric]) 2515 2516lemma powr_realpow: "0 < x \<Longrightarrow> x powr (real n) = x^n" 2517by (induction n) (simp_all add: ac_simps powr_add) 2518 2519lemma log_ln: "ln x = log (exp(1)) x" 2520 by (simp add: log_def) 2521 2522lemma DERIV_log: 2523 assumes "x > 0" 2524 shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)" 2525proof - 2526 define lb where "lb = 1 / ln b" 2527 moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x" 2528 using \<open>x > 0\<close> by (auto intro!: derivative_eq_intros) 2529 ultimately show ?thesis 2530 by (simp add: log_def) 2531qed 2532 2533lemmas DERIV_log[THEN DERIV_chain2, derivative_intros] 2534 and DERIV_log[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 2535 2536lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x" 2537 by (simp add: powr_def log_def) 2538 2539lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y" 2540 by (simp add: log_def powr_def) 2541 2542lemma log_mult: 2543 "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> 2544 log a (x * y) = log a x + log a y" 2545 by (simp add: log_def ln_mult divide_inverse distrib_right) 2546 2547lemma log_eq_div_ln_mult_log: 2548 "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 2549 log a x = (ln b/ln a) * log b x" 2550 by (simp add: log_def divide_inverse) 2551 2552text\<open>Base 10 logarithms\<close> 2553lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x" 2554 by (simp add: log_def) 2555 2556lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x" 2557 by (simp add: log_def) 2558 2559lemma log_one [simp]: "log a 1 = 0" 2560 by (simp add: log_def) 2561 2562lemma log_eq_one [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a a = 1" 2563 by (simp add: log_def) 2564 2565lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x" 2566 using ln_inverse log_def by auto 2567 2568lemma log_divide: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a (x/y) = log a x - log a y" 2569 by (simp add: log_mult divide_inverse log_inverse) 2570 2571lemma powr_gt_zero [simp]: "0 < x powr a \<longleftrightarrow> x \<noteq> 0" 2572 for a x :: real 2573 by (simp add: powr_def) 2574 2575lemma powr_nonneg_iff[simp]: "a powr x \<le> 0 \<longleftrightarrow> a = 0" 2576 for a x::real 2577 by (meson not_less powr_gt_zero) 2578 2579lemma log_add_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log b x + y = log b (x * b powr y)" 2580 and add_log_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> y + log b x = log b (b powr y * x)" 2581 and log_minus_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log b x - y = log b (x * b powr -y)" 2582 and minus_log_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> y - log b x = log b (b powr y / x)" 2583 by (simp_all add: log_mult log_divide) 2584 2585lemma log_less_cancel_iff [simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y" 2586 using powr_less_cancel_iff [of a] powr_log_cancel [of a x] powr_log_cancel [of a y] 2587 by (metis less_eq_real_def less_trans not_le zero_less_one) 2588 2589lemma log_inj: 2590 assumes "1 < b" 2591 shows "inj_on (log b) {0 <..}" 2592proof (rule inj_onI, simp) 2593 fix x y 2594 assume pos: "0 < x" "0 < y" and *: "log b x = log b y" 2595 show "x = y" 2596 proof (cases rule: linorder_cases) 2597 assume "x = y" 2598 then show ?thesis by simp 2599 next 2600 assume "x < y" 2601 then have "log b x < log b y" 2602 using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp 2603 then show ?thesis using * by simp 2604 next 2605 assume "y < x" 2606 then have "log b y < log b x" 2607 using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp 2608 then show ?thesis using * by simp 2609 qed 2610qed 2611 2612lemma log_le_cancel_iff [simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x \<le> log a y \<longleftrightarrow> x \<le> y" 2613 by (simp add: linorder_not_less [symmetric]) 2614 2615lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x" 2616 using log_less_cancel_iff[of a 1 x] by simp 2617 2618lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x" 2619 using log_le_cancel_iff[of a 1 x] by simp 2620 2621lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1" 2622 using log_less_cancel_iff[of a x 1] by simp 2623 2624lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1" 2625 using log_le_cancel_iff[of a x 1] by simp 2626 2627lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x" 2628 using log_less_cancel_iff[of a a x] by simp 2629 2630lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x" 2631 using log_le_cancel_iff[of a a x] by simp 2632 2633lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a" 2634 using log_less_cancel_iff[of a x a] by simp 2635 2636lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a" 2637 using log_le_cancel_iff[of a x a] by simp 2638 2639lemma le_log_iff: 2640 fixes b x y :: real 2641 assumes "1 < b" "x > 0" 2642 shows "y \<le> log b x \<longleftrightarrow> b powr y \<le> x" 2643 using assms 2644 by (metis less_irrefl less_trans powr_le_cancel_iff powr_log_cancel zero_less_one) 2645 2646lemma less_log_iff: 2647 assumes "1 < b" "x > 0" 2648 shows "y < log b x \<longleftrightarrow> b powr y < x" 2649 by (metis assms dual_order.strict_trans less_irrefl powr_less_cancel_iff 2650 powr_log_cancel zero_less_one) 2651 2652lemma 2653 assumes "1 < b" "x > 0" 2654 shows log_less_iff: "log b x < y \<longleftrightarrow> x < b powr y" 2655 and log_le_iff: "log b x \<le> y \<longleftrightarrow> x \<le> b powr y" 2656 using le_log_iff[OF assms, of y] less_log_iff[OF assms, of y] 2657 by auto 2658 2659lemmas powr_le_iff = le_log_iff[symmetric] 2660 and powr_less_iff = less_log_iff[symmetric] 2661 and less_powr_iff = log_less_iff[symmetric] 2662 and le_powr_iff = log_le_iff[symmetric] 2663 2664lemma le_log_of_power: 2665 assumes "b ^ n \<le> m" "1 < b" 2666 shows "n \<le> log b m" 2667proof - 2668 from assms have "0 < m" by (metis less_trans zero_less_power less_le_trans zero_less_one) 2669 thus ?thesis using assms by (simp add: le_log_iff powr_realpow) 2670qed 2671 2672lemma le_log2_of_power: "2 ^ n \<le> m \<Longrightarrow> n \<le> log 2 m" for m n :: nat 2673using le_log_of_power[of 2] by simp 2674 2675lemma log_of_power_le: "\<lbrakk> m \<le> b ^ n; b > 1; m > 0 \<rbrakk> \<Longrightarrow> log b (real m) \<le> n" 2676by (simp add: log_le_iff powr_realpow) 2677 2678lemma log2_of_power_le: "\<lbrakk> m \<le> 2 ^ n; m > 0 \<rbrakk> \<Longrightarrow> log 2 m \<le> n" for m n :: nat 2679using log_of_power_le[of _ 2] by simp 2680 2681lemma log_of_power_less: "\<lbrakk> m < b ^ n; b > 1; m > 0 \<rbrakk> \<Longrightarrow> log b (real m) < n" 2682by (simp add: log_less_iff powr_realpow) 2683 2684lemma log2_of_power_less: "\<lbrakk> m < 2 ^ n; m > 0 \<rbrakk> \<Longrightarrow> log 2 m < n" for m n :: nat 2685using log_of_power_less[of _ 2] by simp 2686 2687lemma less_log_of_power: 2688 assumes "b ^ n < m" "1 < b" 2689 shows "n < log b m" 2690proof - 2691 have "0 < m" by (metis assms less_trans zero_less_power zero_less_one) 2692 thus ?thesis using assms by (simp add: less_log_iff powr_realpow) 2693qed 2694 2695lemma less_log2_of_power: "2 ^ n < m \<Longrightarrow> n < log 2 m" for m n :: nat 2696using less_log_of_power[of 2] by simp 2697 2698lemma gr_one_powr[simp]: 2699 fixes x y :: real shows "\<lbrakk> x > 1; y > 0 \<rbrakk> \<Longrightarrow> 1 < x powr y" 2700by(simp add: less_powr_iff) 2701 2702lemma log_pow_cancel [simp]: 2703 "a > 0 \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a ^ b) = b" 2704 by (simp add: ln_realpow log_def) 2705 2706lemma floor_log_eq_powr_iff: "x > 0 \<Longrightarrow> b > 1 \<Longrightarrow> \<lfloor>log b x\<rfloor> = k \<longleftrightarrow> b powr k \<le> x \<and> x < b powr (k + 1)" 2707 by (auto simp: floor_eq_iff powr_le_iff less_powr_iff) 2708 2709lemma floor_log_nat_eq_powr_iff: fixes b n k :: nat 2710 shows "\<lbrakk> b \<ge> 2; k > 0 \<rbrakk> \<Longrightarrow> 2711 floor (log b (real k)) = n \<longleftrightarrow> b^n \<le> k \<and> k < b^(n+1)" 2712by (auto simp: floor_log_eq_powr_iff powr_add powr_realpow 2713 of_nat_power[symmetric] of_nat_mult[symmetric] ac_simps 2714 simp del: of_nat_power of_nat_mult) 2715 2716lemma floor_log_nat_eq_if: fixes b n k :: nat 2717 assumes "b^n \<le> k" "k < b^(n+1)" "b \<ge> 2" 2718 shows "floor (log b (real k)) = n" 2719proof - 2720 have "k \<ge> 1" using assms(1,3) one_le_power[of b n] by linarith 2721 with assms show ?thesis by(simp add: floor_log_nat_eq_powr_iff) 2722qed 2723 2724lemma ceiling_log_eq_powr_iff: "\<lbrakk> x > 0; b > 1 \<rbrakk> 2725 \<Longrightarrow> \<lceil>log b x\<rceil> = int k + 1 \<longleftrightarrow> b powr k < x \<and> x \<le> b powr (k + 1)" 2726by (auto simp: ceiling_eq_iff powr_less_iff le_powr_iff) 2727 2728lemma ceiling_log_nat_eq_powr_iff: fixes b n k :: nat 2729 shows "\<lbrakk> b \<ge> 2; k > 0 \<rbrakk> \<Longrightarrow> 2730 ceiling (log b (real k)) = int n + 1 \<longleftrightarrow> (b^n < k \<and> k \<le> b^(n+1))" 2731using ceiling_log_eq_powr_iff 2732by (auto simp: powr_add powr_realpow of_nat_power[symmetric] of_nat_mult[symmetric] ac_simps 2733 simp del: of_nat_power of_nat_mult) 2734 2735lemma ceiling_log_nat_eq_if: fixes b n k :: nat 2736 assumes "b^n < k" "k \<le> b^(n+1)" "b \<ge> 2" 2737 shows "ceiling (log b (real k)) = int n + 1" 2738proof - 2739 have "k \<ge> 1" using assms(1,3) one_le_power[of b n] by linarith 2740 with assms show ?thesis by(simp add: ceiling_log_nat_eq_powr_iff) 2741qed 2742 2743lemma floor_log2_div2: fixes n :: nat assumes "n \<ge> 2" 2744shows "floor(log 2 n) = floor(log 2 (n div 2)) + 1" 2745proof cases 2746 assume "n=2" thus ?thesis by simp 2747next 2748 let ?m = "n div 2" 2749 assume "n\<noteq>2" 2750 hence "1 \<le> ?m" using assms by arith 2751 then obtain i where i: "2 ^ i \<le> ?m" "?m < 2 ^ (i + 1)" 2752 using ex_power_ivl1[of 2 ?m] by auto 2753 have "2^(i+1) \<le> 2*?m" using i(1) by simp 2754 also have "2*?m \<le> n" by arith 2755 finally have *: "2^(i+1) \<le> \<dots>" . 2756 have "n < 2^(i+1+1)" using i(2) by simp 2757 from floor_log_nat_eq_if[OF * this] floor_log_nat_eq_if[OF i] 2758 show ?thesis by simp 2759qed 2760 2761lemma ceiling_log2_div2: assumes "n \<ge> 2" 2762shows "ceiling(log 2 (real n)) = ceiling(log 2 ((n-1) div 2 + 1)) + 1" 2763proof cases 2764 assume "n=2" thus ?thesis by simp 2765next 2766 let ?m = "(n-1) div 2 + 1" 2767 assume "n\<noteq>2" 2768 hence "2 \<le> ?m" using assms by arith 2769 then obtain i where i: "2 ^ i < ?m" "?m \<le> 2 ^ (i + 1)" 2770 using ex_power_ivl2[of 2 ?m] by auto 2771 have "n \<le> 2*?m" by arith 2772 also have "2*?m \<le> 2 ^ ((i+1)+1)" using i(2) by simp 2773 finally have *: "n \<le> \<dots>" . 2774 have "2^(i+1) < n" using i(1) by (auto simp: less_Suc_eq_0_disj) 2775 from ceiling_log_nat_eq_if[OF this *] ceiling_log_nat_eq_if[OF i] 2776 show ?thesis by simp 2777qed 2778 2779lemma powr_real_of_int: 2780 "x > 0 \<Longrightarrow> x powr real_of_int n = (if n \<ge> 0 then x ^ nat n else inverse (x ^ nat (- n)))" 2781 using powr_realpow[of x "nat n"] powr_realpow[of x "nat (-n)"] 2782 by (auto simp: field_simps powr_minus) 2783 2784lemma powr_numeral [simp]: "0 \<le> x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)" 2785 by (metis less_le power_zero_numeral powr_0 of_nat_numeral powr_realpow) 2786 2787lemma powr_int: 2788 assumes "x > 0" 2789 shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))" 2790proof (cases "i < 0") 2791 case True 2792 have r: "x powr i = 1 / x powr (- i)" 2793 by (simp add: powr_minus field_simps) 2794 show ?thesis using \<open>i < 0\<close> \<open>x > 0\<close> 2795 by (simp add: r field_simps powr_realpow[symmetric]) 2796next 2797 case False 2798 then show ?thesis 2799 by (simp add: assms powr_realpow[symmetric]) 2800qed 2801 2802definition powr_real :: "real \<Rightarrow> real \<Rightarrow> real" 2803 where [code_abbrev, simp]: "powr_real = Transcendental.powr" 2804 2805lemma compute_powr_real [code]: 2806 "powr_real b i = 2807 (if b \<le> 0 then Code.abort (STR ''powr_real with nonpositive base'') (\<lambda>_. powr_real b i) 2808 else if \<lfloor>i\<rfloor> = i then (if 0 \<le> i then b ^ nat \<lfloor>i\<rfloor> else 1 / b ^ nat \<lfloor>- i\<rfloor>) 2809 else Code.abort (STR ''powr_real with non-integer exponent'') (\<lambda>_. powr_real b i))" 2810 for b i :: real 2811 by (auto simp: powr_int) 2812 2813lemma powr_one: "0 \<le> x \<Longrightarrow> x powr 1 = x" 2814 for x :: real 2815 using powr_realpow [of x 1] by simp 2816 2817lemma powr_neg_one: "0 < x \<Longrightarrow> x powr - 1 = 1 / x" 2818 for x :: real 2819 using powr_int [of x "- 1"] by simp 2820 2821lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n" 2822 for x :: real 2823 using powr_int [of x "- numeral n"] by simp 2824 2825lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)" 2826 by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr) 2827 2828lemma ln_powr: "x \<noteq> 0 \<Longrightarrow> ln (x powr y) = y * ln x" 2829 for x :: real 2830 by (simp add: powr_def) 2831 2832lemma ln_root: "n > 0 \<Longrightarrow> b > 0 \<Longrightarrow> ln (root n b) = ln b / n" 2833 by (simp add: root_powr_inverse ln_powr) 2834 2835lemma ln_sqrt: "0 < x \<Longrightarrow> ln (sqrt x) = ln x / 2" 2836 by (simp add: ln_powr ln_powr[symmetric] mult.commute) 2837 2838lemma log_root: "n > 0 \<Longrightarrow> a > 0 \<Longrightarrow> log b (root n a) = log b a / n" 2839 by (simp add: log_def ln_root) 2840 2841lemma log_powr: "x \<noteq> 0 \<Longrightarrow> log b (x powr y) = y * log b x" 2842 by (simp add: log_def ln_powr) 2843 2844(* [simp] is not worth it, interferes with some proofs *) 2845lemma log_nat_power: "0 < x \<Longrightarrow> log b (x^n) = real n * log b x" 2846 by (simp add: log_powr powr_realpow [symmetric]) 2847 2848lemma log_of_power_eq: 2849 assumes "m = b ^ n" "b > 1" 2850 shows "n = log b (real m)" 2851proof - 2852 have "n = log b (b ^ n)" using assms(2) by (simp add: log_nat_power) 2853 also have "\<dots> = log b m" using assms by simp 2854 finally show ?thesis . 2855qed 2856 2857lemma log2_of_power_eq: "m = 2 ^ n \<Longrightarrow> n = log 2 m" for m n :: nat 2858using log_of_power_eq[of _ 2] by simp 2859 2860lemma log_base_change: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log b x = log a x / log a b" 2861 by (simp add: log_def) 2862 2863lemma log_base_pow: "0 < a \<Longrightarrow> log (a ^ n) x = log a x / n" 2864 by (simp add: log_def ln_realpow) 2865 2866lemma log_base_powr: "a \<noteq> 0 \<Longrightarrow> log (a powr b) x = log a x / b" 2867 by (simp add: log_def ln_powr) 2868 2869lemma log_base_root: "n > 0 \<Longrightarrow> b > 0 \<Longrightarrow> log (root n b) x = n * (log b x)" 2870 by (simp add: log_def ln_root) 2871 2872lemma ln_bound: "0 < x \<Longrightarrow> ln x \<le> x" for x :: real 2873 using ln_le_minus_one by force 2874 2875lemma powr_mono: 2876 fixes x :: real 2877 assumes "a \<le> b" and "1 \<le> x" shows "x powr a \<le> x powr b" 2878 using assms less_eq_real_def by auto 2879 2880lemma ge_one_powr_ge_zero: "1 \<le> x \<Longrightarrow> 0 \<le> a \<Longrightarrow> 1 \<le> x powr a" 2881 for x :: real 2882 using powr_mono by fastforce 2883 2884lemma powr_less_mono2: "0 < a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> x powr a < y powr a" 2885 for x :: real 2886 by (simp add: powr_def) 2887 2888lemma powr_less_mono2_neg: "a < 0 \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> y powr a < x powr a" 2889 for x :: real 2890 by (simp add: powr_def) 2891 2892lemma powr_mono2: "x powr a \<le> y powr a" if "0 \<le> a" "0 \<le> x" "x \<le> y" 2893 for x :: real 2894 using less_eq_real_def powr_less_mono2 that by auto 2895 2896lemma powr_le1: "0 \<le> a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> x powr a \<le> 1" 2897 for x :: real 2898 using powr_mono2 by fastforce 2899 2900lemma powr_mono2': 2901 fixes a x y :: real 2902 assumes "a \<le> 0" "x > 0" "x \<le> y" 2903 shows "x powr a \<ge> y powr a" 2904proof - 2905 from assms have "x powr - a \<le> y powr - a" 2906 by (intro powr_mono2) simp_all 2907 with assms show ?thesis 2908 by (auto simp: powr_minus field_simps) 2909qed 2910 2911lemma powr_mono_both: 2912 fixes x :: real 2913 assumes "0 \<le> a" "a \<le> b" "1 \<le> x" "x \<le> y" 2914 shows "x powr a \<le> y powr b" 2915 by (meson assms order.trans powr_mono powr_mono2 zero_le_one) 2916 2917lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y" 2918 for x :: real 2919 unfolding powr_def exp_inj_iff by simp 2920 2921lemma powr_half_sqrt: "0 \<le> x \<Longrightarrow> x powr (1/2) = sqrt x" 2922 by (simp add: powr_def root_powr_inverse sqrt_def) 2923 2924lemma square_powr_half [simp]: 2925 fixes x::real shows "x\<^sup>2 powr (1/2) = \<bar>x\<bar>" 2926 by (simp add: powr_half_sqrt) 2927 2928lemma ln_powr_bound: "1 \<le> x \<Longrightarrow> 0 < a \<Longrightarrow> ln x \<le> (x powr a) / a" 2929 for x :: real 2930 by (metis exp_gt_zero linear ln_eq_zero_iff ln_exp ln_less_self ln_powr mult.commute 2931 mult_imp_le_div_pos not_less powr_gt_zero) 2932 2933lemma ln_powr_bound2: 2934 fixes x :: real 2935 assumes "1 < x" and "0 < a" 2936 shows "(ln x) powr a \<le> (a powr a) * x" 2937proof - 2938 from assms have "ln x \<le> (x powr (1 / a)) / (1 / a)" 2939 by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff) 2940 also have "\<dots> = a * (x powr (1 / a))" 2941 by simp 2942 finally have "(ln x) powr a \<le> (a * (x powr (1 / a))) powr a" 2943 by (metis assms less_imp_le ln_gt_zero powr_mono2) 2944 also have "\<dots> = (a powr a) * ((x powr (1 / a)) powr a)" 2945 using assms powr_mult by auto 2946 also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)" 2947 by (rule powr_powr) 2948 also have "\<dots> = x" using assms 2949 by auto 2950 finally show ?thesis . 2951qed 2952 2953lemma tendsto_powr: 2954 fixes a b :: real 2955 assumes f: "(f \<longlongrightarrow> a) F" 2956 and g: "(g \<longlongrightarrow> b) F" 2957 and a: "a \<noteq> 0" 2958 shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F" 2959 unfolding powr_def 2960proof (rule filterlim_If) 2961 from f show "((\<lambda>x. 0) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))" 2962 by simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_mono dest: t1_space_nhds) 2963 from f g a show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) 2964 (inf F (principal {x. f x \<noteq> 0}))" 2965 by (auto intro!: tendsto_intros intro: tendsto_mono inf_le1) 2966qed 2967 2968lemma tendsto_powr'[tendsto_intros]: 2969 fixes a :: real 2970 assumes f: "(f \<longlongrightarrow> a) F" 2971 and g: "(g \<longlongrightarrow> b) F" 2972 and a: "a \<noteq> 0 \<or> (b > 0 \<and> eventually (\<lambda>x. f x \<ge> 0) F)" 2973 shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F" 2974proof - 2975 from a consider "a \<noteq> 0" | "a = 0" "b > 0" "eventually (\<lambda>x. f x \<ge> 0) F" 2976 by auto 2977 then show ?thesis 2978 proof cases 2979 case 1 2980 with f g show ?thesis by (rule tendsto_powr) 2981 next 2982 case 2 2983 have "((\<lambda>x. if f x = 0 then 0 else exp (g x * ln (f x))) \<longlongrightarrow> 0) F" 2984 proof (intro filterlim_If) 2985 have "filterlim f (principal {0<..}) (inf F (principal {z. f z \<noteq> 0}))" 2986 using \<open>eventually (\<lambda>x. f x \<ge> 0) F\<close> 2987 by (auto simp: filterlim_iff eventually_inf_principal 2988 eventually_principal elim: eventually_mono) 2989 moreover have "filterlim f (nhds a) (inf F (principal {z. f z \<noteq> 0}))" 2990 by (rule tendsto_mono[OF _ f]) simp_all 2991 ultimately have f: "filterlim f (at_right 0) (inf F (principal {x. f x \<noteq> 0}))" 2992 by (simp add: at_within_def filterlim_inf \<open>a = 0\<close>) 2993 have g: "(g \<longlongrightarrow> b) (inf F (principal {z. f z \<noteq> 0}))" 2994 by (rule tendsto_mono[OF _ g]) simp_all 2995 show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> 0) (inf F (principal {x. f x \<noteq> 0}))" 2996 by (rule filterlim_compose[OF exp_at_bot] filterlim_tendsto_pos_mult_at_bot 2997 filterlim_compose[OF ln_at_0] f g \<open>b > 0\<close>)+ 2998 qed simp_all 2999 with \<open>a = 0\<close> show ?thesis 3000 by (simp add: powr_def) 3001 qed 3002qed 3003 3004lemma continuous_powr: 3005 assumes "continuous F f" 3006 and "continuous F g" 3007 and "f (Lim F (\<lambda>x. x)) \<noteq> 0" 3008 shows "continuous F (\<lambda>x. (f x) powr (g x :: real))" 3009 using assms unfolding continuous_def by (rule tendsto_powr) 3010 3011lemma continuous_at_within_powr[continuous_intros]: 3012 fixes f g :: "_ \<Rightarrow> real" 3013 assumes "continuous (at a within s) f" 3014 and "continuous (at a within s) g" 3015 and "f a \<noteq> 0" 3016 shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))" 3017 using assms unfolding continuous_within by (rule tendsto_powr) 3018 3019lemma isCont_powr[continuous_intros, simp]: 3020 fixes f g :: "_ \<Rightarrow> real" 3021 assumes "isCont f a" "isCont g a" "f a \<noteq> 0" 3022 shows "isCont (\<lambda>x. (f x) powr g x) a" 3023 using assms unfolding continuous_at by (rule tendsto_powr) 3024 3025lemma continuous_on_powr[continuous_intros]: 3026 fixes f g :: "_ \<Rightarrow> real" 3027 assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> 0" 3028 shows "continuous_on s (\<lambda>x. (f x) powr (g x))" 3029 using assms unfolding continuous_on_def by (fast intro: tendsto_powr) 3030 3031lemma tendsto_powr2: 3032 fixes a :: real 3033 assumes f: "(f \<longlongrightarrow> a) F" 3034 and g: "(g \<longlongrightarrow> b) F" 3035 and "\<forall>\<^sub>F x in F. 0 \<le> f x" 3036 and b: "0 < b" 3037 shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F" 3038 using tendsto_powr'[of f a F g b] assms by auto 3039 3040lemma has_derivative_powr[derivative_intros]: 3041 assumes g[derivative_intros]: "(g has_derivative g') (at x within X)" 3042 and f[derivative_intros]:"(f has_derivative f') (at x within X)" 3043 assumes pos: "0 < g x" and "x \<in> X" 3044 shows "((\<lambda>x. g x powr f x::real) has_derivative (\<lambda>h. (g x powr f x) * (f' h * ln (g x) + g' h * f x / g x))) (at x within X)" 3045proof - 3046 have "\<forall>\<^sub>F x in at x within X. g x > 0" 3047 by (rule order_tendstoD[OF _ pos]) 3048 (rule has_derivative_continuous[OF g, unfolded continuous_within]) 3049 then obtain d where "d > 0" and pos': "\<And>x'. x' \<in> X \<Longrightarrow> dist x' x < d \<Longrightarrow> 0 < g x'" 3050 using pos unfolding eventually_at by force 3051 have "((\<lambda>x. exp (f x * ln (g x))) has_derivative 3052 (\<lambda>h. (g x powr f x) * (f' h * ln (g x) + g' h * f x / g x))) (at x within X)" 3053 using pos 3054 by (auto intro!: derivative_eq_intros simp: field_split_simps powr_def) 3055 then show ?thesis 3056 by (rule has_derivative_transform_within[OF _ \<open>d > 0\<close> \<open>x \<in> X\<close>]) (auto simp: powr_def dest: pos') 3057qed 3058 3059lemma DERIV_powr: 3060 fixes r :: real 3061 assumes g: "DERIV g x :> m" 3062 and pos: "g x > 0" 3063 and f: "DERIV f x :> r" 3064 shows "DERIV (\<lambda>x. g x powr f x) x :> (g x powr f x) * (r * ln (g x) + m * f x / g x)" 3065 using assms 3066 by (auto intro!: derivative_eq_intros ext simp: has_field_derivative_def algebra_simps) 3067 3068lemma DERIV_fun_powr: 3069 fixes r :: real 3070 assumes g: "DERIV g x :> m" 3071 and pos: "g x > 0" 3072 shows "DERIV (\<lambda>x. (g x) powr r) x :> r * (g x) powr (r - of_nat 1) * m" 3073 using DERIV_powr[OF g pos DERIV_const, of r] pos 3074 by (simp add: powr_diff field_simps) 3075 3076lemma has_real_derivative_powr: 3077 assumes "z > 0" 3078 shows "((\<lambda>z. z powr r) has_real_derivative r * z powr (r - 1)) (at z)" 3079proof (subst DERIV_cong_ev[OF refl _ refl]) 3080 from assms have "eventually (\<lambda>z. z \<noteq> 0) (nhds z)" 3081 by (intro t1_space_nhds) auto 3082 then show "eventually (\<lambda>z. z powr r = exp (r * ln z)) (nhds z)" 3083 unfolding powr_def by eventually_elim simp 3084 from assms show "((\<lambda>z. exp (r * ln z)) has_real_derivative r * z powr (r - 1)) (at z)" 3085 by (auto intro!: derivative_eq_intros simp: powr_def field_simps exp_diff) 3086qed 3087 3088declare has_real_derivative_powr[THEN DERIV_chain2, derivative_intros] 3089 3090lemma tendsto_zero_powrI: 3091 assumes "(f \<longlongrightarrow> (0::real)) F" "(g \<longlongrightarrow> b) F" "\<forall>\<^sub>F x in F. 0 \<le> f x" "0 < b" 3092 shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> 0) F" 3093 using tendsto_powr2[OF assms] by simp 3094 3095lemma continuous_on_powr': 3096 fixes f g :: "_ \<Rightarrow> real" 3097 assumes "continuous_on s f" "continuous_on s g" 3098 and "\<forall>x\<in>s. f x \<ge> 0 \<and> (f x = 0 \<longrightarrow> g x > 0)" 3099 shows "continuous_on s (\<lambda>x. (f x) powr (g x))" 3100 unfolding continuous_on_def 3101proof 3102 fix x 3103 assume x: "x \<in> s" 3104 from assms x show "((\<lambda>x. f x powr g x) \<longlongrightarrow> f x powr g x) (at x within s)" 3105 proof (cases "f x = 0") 3106 case True 3107 from assms(3) have "eventually (\<lambda>x. f x \<ge> 0) (at x within s)" 3108 by (auto simp: at_within_def eventually_inf_principal) 3109 with True x assms show ?thesis 3110 by (auto intro!: tendsto_zero_powrI[of f _ g "g x"] simp: continuous_on_def) 3111 next 3112 case False 3113 with assms x show ?thesis 3114 by (auto intro!: tendsto_powr' simp: continuous_on_def) 3115 qed 3116qed 3117 3118lemma tendsto_neg_powr: 3119 assumes "s < 0" 3120 and f: "LIM x F. f x :> at_top" 3121 shows "((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F" 3122proof - 3123 have "((\<lambda>x. exp (s * ln (f x))) \<longlongrightarrow> (0::real)) F" (is "?X") 3124 by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_top] 3125 filterlim_tendsto_neg_mult_at_bot assms) 3126 also have "?X \<longleftrightarrow> ((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F" 3127 using f filterlim_at_top_dense[of f F] 3128 by (intro filterlim_cong[OF refl refl]) (auto simp: neq_iff powr_def elim: eventually_mono) 3129 finally show ?thesis . 3130qed 3131 3132lemma tendsto_exp_limit_at_right: "((\<lambda>y. (1 + x * y) powr (1 / y)) \<longlongrightarrow> exp x) (at_right 0)" 3133 for x :: real 3134proof (cases "x = 0") 3135 case True 3136 then show ?thesis by simp 3137next 3138 case False 3139 have "((\<lambda>y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)" 3140 by (auto intro!: derivative_eq_intros) 3141 then have "((\<lambda>y. ln (1 + x * y) / y) \<longlongrightarrow> x) (at 0)" 3142 by (auto simp: has_field_derivative_def field_has_derivative_at) 3143 then have *: "((\<lambda>y. exp (ln (1 + x * y) / y)) \<longlongrightarrow> exp x) (at 0)" 3144 by (rule tendsto_intros) 3145 then show ?thesis 3146 proof (rule filterlim_mono_eventually) 3147 show "eventually (\<lambda>xa. exp (ln (1 + x * xa) / xa) = (1 + x * xa) powr (1 / xa)) (at_right 0)" 3148 unfolding eventually_at_right[OF zero_less_one] 3149 using False 3150 by (intro exI[of _ "1 / \<bar>x\<bar>"]) (auto simp: field_simps powr_def abs_if add_nonneg_eq_0_iff) 3151 qed (simp_all add: at_eq_sup_left_right) 3152qed 3153 3154lemma tendsto_exp_limit_at_top: "((\<lambda>y. (1 + x / y) powr y) \<longlongrightarrow> exp x) at_top" 3155 for x :: real 3156 by (simp add: filterlim_at_top_to_right inverse_eq_divide tendsto_exp_limit_at_right) 3157 3158lemma tendsto_exp_limit_sequentially: "(\<lambda>n. (1 + x / n) ^ n) \<longlonglongrightarrow> exp x" 3159 for x :: real 3160proof (rule filterlim_mono_eventually) 3161 from reals_Archimedean2 [of "\<bar>x\<bar>"] obtain n :: nat where *: "real n > \<bar>x\<bar>" .. 3162 then have "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top" 3163 by (intro eventually_sequentiallyI [of n]) (auto simp: field_split_simps) 3164 then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top" 3165 by (rule eventually_mono) (erule powr_realpow) 3166 show "(\<lambda>n. (1 + x / real n) powr real n) \<longlonglongrightarrow> exp x" 3167 by (rule filterlim_compose [OF tendsto_exp_limit_at_top filterlim_real_sequentially]) 3168qed auto 3169 3170 3171subsection \<open>Sine and Cosine\<close> 3172 3173definition sin_coeff :: "nat \<Rightarrow> real" 3174 where "sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / (fact n))" 3175 3176definition cos_coeff :: "nat \<Rightarrow> real" 3177 where "cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / (fact n) else 0)" 3178 3179definition sin :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}" 3180 where "sin = (\<lambda>x. \<Sum>n. sin_coeff n *\<^sub>R x^n)" 3181 3182definition cos :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}" 3183 where "cos = (\<lambda>x. \<Sum>n. cos_coeff n *\<^sub>R x^n)" 3184 3185lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0" 3186 unfolding sin_coeff_def by simp 3187 3188lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1" 3189 unfolding cos_coeff_def by simp 3190 3191lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)" 3192 unfolding cos_coeff_def sin_coeff_def 3193 by (simp del: mult_Suc) 3194 3195lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)" 3196 unfolding cos_coeff_def sin_coeff_def 3197 by (simp del: mult_Suc) (auto elim: oddE) 3198 3199lemma summable_norm_sin: "summable (\<lambda>n. norm (sin_coeff n *\<^sub>R x^n))" 3200 for x :: "'a::{real_normed_algebra_1,banach}" 3201 unfolding sin_coeff_def 3202 apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]]) 3203 apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) 3204 done 3205 3206lemma summable_norm_cos: "summable (\<lambda>n. norm (cos_coeff n *\<^sub>R x^n))" 3207 for x :: "'a::{real_normed_algebra_1,banach}" 3208 unfolding cos_coeff_def 3209 apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]]) 3210 apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) 3211 done 3212 3213lemma sin_converges: "(\<lambda>n. sin_coeff n *\<^sub>R x^n) sums sin x" 3214 unfolding sin_def 3215 by (metis (full_types) summable_norm_cancel summable_norm_sin summable_sums) 3216 3217lemma cos_converges: "(\<lambda>n. cos_coeff n *\<^sub>R x^n) sums cos x" 3218 unfolding cos_def 3219 by (metis (full_types) summable_norm_cancel summable_norm_cos summable_sums) 3220 3221lemma sin_of_real: "sin (of_real x) = of_real (sin x)" 3222 for x :: real 3223proof - 3224 have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) = (\<lambda>n. sin_coeff n *\<^sub>R (of_real x)^n)" 3225 proof 3226 show "of_real (sin_coeff n *\<^sub>R x^n) = sin_coeff n *\<^sub>R of_real x^n" for n 3227 by (simp add: scaleR_conv_of_real) 3228 qed 3229 also have "\<dots> sums (sin (of_real x))" 3230 by (rule sin_converges) 3231 finally have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) sums (sin (of_real x))" . 3232 then show ?thesis 3233 using sums_unique2 sums_of_real [OF sin_converges] 3234 by blast 3235qed 3236 3237corollary sin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> sin z \<in> \<real>" 3238 by (metis Reals_cases Reals_of_real sin_of_real) 3239 3240lemma cos_of_real: "cos (of_real x) = of_real (cos x)" 3241 for x :: real 3242proof - 3243 have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) = (\<lambda>n. cos_coeff n *\<^sub>R (of_real x)^n)" 3244 proof 3245 show "of_real (cos_coeff n *\<^sub>R x^n) = cos_coeff n *\<^sub>R of_real x^n" for n 3246 by (simp add: scaleR_conv_of_real) 3247 qed 3248 also have "\<dots> sums (cos (of_real x))" 3249 by (rule cos_converges) 3250 finally have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) sums (cos (of_real x))" . 3251 then show ?thesis 3252 using sums_unique2 sums_of_real [OF cos_converges] 3253 by blast 3254qed 3255 3256corollary cos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cos z \<in> \<real>" 3257 by (metis Reals_cases Reals_of_real cos_of_real) 3258 3259lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff" 3260 by (simp add: diffs_def sin_coeff_Suc del: of_nat_Suc) 3261 3262lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)" 3263 by (simp add: diffs_def cos_coeff_Suc del: of_nat_Suc) 3264 3265lemma sin_int_times_real: "sin (of_int m * of_real x) = of_real (sin (of_int m * x))" 3266 by (metis sin_of_real of_real_mult of_real_of_int_eq) 3267 3268lemma cos_int_times_real: "cos (of_int m * of_real x) = of_real (cos (of_int m * x))" 3269 by (metis cos_of_real of_real_mult of_real_of_int_eq) 3270 3271text \<open>Now at last we can get the derivatives of exp, sin and cos.\<close> 3272 3273lemma DERIV_sin [simp]: "DERIV sin x :> cos x" 3274 for x :: "'a::{real_normed_field,banach}" 3275 unfolding sin_def cos_def scaleR_conv_of_real 3276 apply (rule DERIV_cong) 3277 apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"]) 3278 apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff 3279 summable_minus_iff scaleR_conv_of_real [symmetric] 3280 summable_norm_sin [THEN summable_norm_cancel] 3281 summable_norm_cos [THEN summable_norm_cancel]) 3282 done 3283 3284declare DERIV_sin[THEN DERIV_chain2, derivative_intros] 3285 and DERIV_sin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 3286 3287lemmas has_derivative_sin[derivative_intros] = DERIV_sin[THEN DERIV_compose_FDERIV] 3288 3289lemma DERIV_cos [simp]: "DERIV cos x :> - sin x" 3290 for x :: "'a::{real_normed_field,banach}" 3291 unfolding sin_def cos_def scaleR_conv_of_real 3292 apply (rule DERIV_cong) 3293 apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"]) 3294 apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus 3295 diffs_sin_coeff diffs_cos_coeff 3296 summable_minus_iff scaleR_conv_of_real [symmetric] 3297 summable_norm_sin [THEN summable_norm_cancel] 3298 summable_norm_cos [THEN summable_norm_cancel]) 3299 done 3300 3301declare DERIV_cos[THEN DERIV_chain2, derivative_intros] 3302 and DERIV_cos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 3303 3304lemmas has_derivative_cos[derivative_intros] = DERIV_cos[THEN DERIV_compose_FDERIV] 3305 3306lemma isCont_sin: "isCont sin x" 3307 for x :: "'a::{real_normed_field,banach}" 3308 by (rule DERIV_sin [THEN DERIV_isCont]) 3309 3310lemma continuous_on_sin_real: "continuous_on {a..b} sin" for a::real 3311 using continuous_at_imp_continuous_on isCont_sin by blast 3312 3313lemma isCont_cos: "isCont cos x" 3314 for x :: "'a::{real_normed_field,banach}" 3315 by (rule DERIV_cos [THEN DERIV_isCont]) 3316 3317lemma continuous_on_cos_real: "continuous_on {a..b} cos" for a::real 3318 using continuous_at_imp_continuous_on isCont_cos by blast 3319 3320lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a" 3321 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3322 by (rule isCont_o2 [OF _ isCont_sin]) 3323 3324(* FIXME a context for f would be better *) 3325 3326lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a" 3327 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3328 by (rule isCont_o2 [OF _ isCont_cos]) 3329 3330lemma tendsto_sin [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) \<longlongrightarrow> sin a) F" 3331 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3332 by (rule isCont_tendsto_compose [OF isCont_sin]) 3333 3334lemma tendsto_cos [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) \<longlongrightarrow> cos a) F" 3335 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3336 by (rule isCont_tendsto_compose [OF isCont_cos]) 3337 3338lemma continuous_sin [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))" 3339 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3340 unfolding continuous_def by (rule tendsto_sin) 3341 3342lemma continuous_on_sin [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))" 3343 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3344 unfolding continuous_on_def by (auto intro: tendsto_sin) 3345 3346lemma continuous_within_sin: "continuous (at z within s) sin" 3347 for z :: "'a::{real_normed_field,banach}" 3348 by (simp add: continuous_within tendsto_sin) 3349 3350lemma continuous_cos [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))" 3351 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3352 unfolding continuous_def by (rule tendsto_cos) 3353 3354lemma continuous_on_cos [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))" 3355 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3356 unfolding continuous_on_def by (auto intro: tendsto_cos) 3357 3358lemma continuous_within_cos: "continuous (at z within s) cos" 3359 for z :: "'a::{real_normed_field,banach}" 3360 by (simp add: continuous_within tendsto_cos) 3361 3362 3363subsection \<open>Properties of Sine and Cosine\<close> 3364 3365lemma sin_zero [simp]: "sin 0 = 0" 3366 by (simp add: sin_def sin_coeff_def scaleR_conv_of_real) 3367 3368lemma cos_zero [simp]: "cos 0 = 1" 3369 by (simp add: cos_def cos_coeff_def scaleR_conv_of_real) 3370 3371lemma DERIV_fun_sin: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. sin (g x)) x :> cos (g x) * m" 3372 by (auto intro!: derivative_intros) 3373 3374lemma DERIV_fun_cos: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. cos(g x)) x :> - sin (g x) * m" 3375 by (auto intro!: derivative_eq_intros) 3376 3377 3378subsection \<open>Deriving the Addition Formulas\<close> 3379 3380text \<open>The product of two cosine series.\<close> 3381lemma cos_x_cos_y: 3382 fixes x :: "'a::{real_normed_field,banach}" 3383 shows 3384 "(\<lambda>p. \<Sum>n\<le>p. 3385 if even p \<and> even n 3386 then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) 3387 sums (cos x * cos y)" 3388proof - 3389 have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p - n)) = 3390 (if even p \<and> even n then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p - n) 3391 else 0)" 3392 if "n \<le> p" for n p :: nat 3393 proof - 3394 from that have *: "even n \<Longrightarrow> even p \<Longrightarrow> 3395 (-1) ^ (n div 2) * (-1) ^ ((p - n) div 2) = (-1 :: real) ^ (p div 2)" 3396 by (metis div_add power_add le_add_diff_inverse odd_add) 3397 with that show ?thesis 3398 by (auto simp: algebra_simps cos_coeff_def binomial_fact) 3399 qed 3400 then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> even n 3401 then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) = 3402 (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))" 3403 by simp 3404 also have "\<dots> = (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n *\<^sub>R x^n) * (cos_coeff (p - n) *\<^sub>R y^(p-n)))" 3405 by (simp add: algebra_simps) 3406 also have "\<dots> sums (cos x * cos y)" 3407 using summable_norm_cos 3408 by (auto simp: cos_def scaleR_conv_of_real intro!: Cauchy_product_sums) 3409 finally show ?thesis . 3410qed 3411 3412text \<open>The product of two sine series.\<close> 3413lemma sin_x_sin_y: 3414 fixes x :: "'a::{real_normed_field,banach}" 3415 shows 3416 "(\<lambda>p. \<Sum>n\<le>p. 3417 if even p \<and> odd n 3418 then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) 3419 else 0) 3420 sums (sin x * sin y)" 3421proof - 3422 have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) = 3423 (if even p \<and> odd n 3424 then -((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) 3425 else 0)" 3426 if "n \<le> p" for n p :: nat 3427 proof - 3428 have "(-1) ^ ((n - Suc 0) div 2) * (-1) ^ ((p - Suc n) div 2) = - ((-1 :: real) ^ (p div 2))" 3429 if np: "odd n" "even p" 3430 proof - 3431 from \<open>n \<le> p\<close> np have *: "n - Suc 0 + (p - Suc n) = p - Suc (Suc 0)" "Suc (Suc 0) \<le> p" 3432 by arith+ 3433 have "(p - Suc (Suc 0)) div 2 = p div 2 - Suc 0" 3434 by simp 3435 with \<open>n \<le> p\<close> np * show ?thesis 3436 apply (simp add: power_add [symmetric] div_add [symmetric] del: div_add) 3437 apply (metis (no_types) One_nat_def Suc_1 le_div_geq minus_minus 3438 mult.left_neutral mult_minus_left power.simps(2) zero_less_Suc) 3439 done 3440 qed 3441 then show ?thesis 3442 using \<open>n\<le>p\<close> by (auto simp: algebra_simps sin_coeff_def binomial_fact) 3443 qed 3444 then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> odd n 3445 then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) = 3446 (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))" 3447 by simp 3448 also have "\<dots> = (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n *\<^sub>R x^n) * (sin_coeff (p - n) *\<^sub>R y^(p-n)))" 3449 by (simp add: algebra_simps) 3450 also have "\<dots> sums (sin x * sin y)" 3451 using summable_norm_sin 3452 by (auto simp: sin_def scaleR_conv_of_real intro!: Cauchy_product_sums) 3453 finally show ?thesis . 3454qed 3455 3456lemma sums_cos_x_plus_y: 3457 fixes x :: "'a::{real_normed_field,banach}" 3458 shows 3459 "(\<lambda>p. \<Sum>n\<le>p. 3460 if even p 3461 then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) 3462 else 0) 3463 sums cos (x + y)" 3464proof - 3465 have 3466 "(\<Sum>n\<le>p. 3467 if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) 3468 else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)" 3469 for p :: nat 3470 proof - 3471 have 3472 "(\<Sum>n\<le>p. if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) = 3473 (if even p then \<Sum>n\<le>p. ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)" 3474 by simp 3475 also have "\<dots> = 3476 (if even p 3477 then of_real ((-1) ^ (p div 2) / (fact p)) * (\<Sum>n\<le>p. (p choose n) *\<^sub>R (x^n) * y^(p-n)) 3478 else 0)" 3479 by (auto simp: sum_distrib_left field_simps scaleR_conv_of_real nonzero_of_real_divide) 3480 also have "\<dots> = cos_coeff p *\<^sub>R ((x + y) ^ p)" 3481 by (simp add: cos_coeff_def binomial_ring [of x y] scaleR_conv_of_real atLeast0AtMost) 3482 finally show ?thesis . 3483 qed 3484 then have 3485 "(\<lambda>p. \<Sum>n\<le>p. 3486 if even p 3487 then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) 3488 else 0) = (\<lambda>p. cos_coeff p *\<^sub>R ((x+y)^p))" 3489 by simp 3490 also have "\<dots> sums cos (x + y)" 3491 by (rule cos_converges) 3492 finally show ?thesis . 3493qed 3494 3495theorem cos_add: 3496 fixes x :: "'a::{real_normed_field,banach}" 3497 shows "cos (x + y) = cos x * cos y - sin x * sin y" 3498proof - 3499 have 3500 "(if even p \<and> even n 3501 then ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) - 3502 (if even p \<and> odd n 3503 then - ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) = 3504 (if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)" 3505 if "n \<le> p" for n p :: nat 3506 by simp 3507 then have 3508 "(\<lambda>p. \<Sum>n\<le>p. (if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)) 3509 sums (cos x * cos y - sin x * sin y)" 3510 using sums_diff [OF cos_x_cos_y [of x y] sin_x_sin_y [of x y]] 3511 by (simp add: sum_subtractf [symmetric]) 3512 then show ?thesis 3513 by (blast intro: sums_cos_x_plus_y sums_unique2) 3514qed 3515 3516lemma sin_minus_converges: "(\<lambda>n. - (sin_coeff n *\<^sub>R (-x)^n)) sums sin x" 3517proof - 3518 have [simp]: "\<And>n. - (sin_coeff n *\<^sub>R (-x)^n) = (sin_coeff n *\<^sub>R x^n)" 3519 by (auto simp: sin_coeff_def elim!: oddE) 3520 show ?thesis 3521 by (simp add: sin_def summable_norm_sin [THEN summable_norm_cancel, THEN summable_sums]) 3522qed 3523 3524lemma sin_minus [simp]: "sin (- x) = - sin x" 3525 for x :: "'a::{real_normed_algebra_1,banach}" 3526 using sin_minus_converges [of x] 3527 by (auto simp: sin_def summable_norm_sin [THEN summable_norm_cancel] 3528 suminf_minus sums_iff equation_minus_iff) 3529 3530lemma cos_minus_converges: "(\<lambda>n. (cos_coeff n *\<^sub>R (-x)^n)) sums cos x" 3531proof - 3532 have [simp]: "\<And>n. (cos_coeff n *\<^sub>R (-x)^n) = (cos_coeff n *\<^sub>R x^n)" 3533 by (auto simp: Transcendental.cos_coeff_def elim!: evenE) 3534 show ?thesis 3535 by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel, THEN summable_sums]) 3536qed 3537 3538lemma cos_minus [simp]: "cos (-x) = cos x" 3539 for x :: "'a::{real_normed_algebra_1,banach}" 3540 using cos_minus_converges [of x] 3541 by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel] 3542 suminf_minus sums_iff equation_minus_iff) 3543 3544lemma sin_cos_squared_add [simp]: "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1" 3545 for x :: "'a::{real_normed_field,banach}" 3546 using cos_add [of x "-x"] 3547 by (simp add: power2_eq_square algebra_simps) 3548 3549lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1" 3550 for x :: "'a::{real_normed_field,banach}" 3551 by (subst add.commute, rule sin_cos_squared_add) 3552 3553lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1" 3554 for x :: "'a::{real_normed_field,banach}" 3555 using sin_cos_squared_add2 [unfolded power2_eq_square] . 3556 3557lemma sin_squared_eq: "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2" 3558 for x :: "'a::{real_normed_field,banach}" 3559 unfolding eq_diff_eq by (rule sin_cos_squared_add) 3560 3561lemma cos_squared_eq: "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2" 3562 for x :: "'a::{real_normed_field,banach}" 3563 unfolding eq_diff_eq by (rule sin_cos_squared_add2) 3564 3565lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1" 3566 for x :: real 3567 by (rule power2_le_imp_le) (simp_all add: sin_squared_eq) 3568 3569lemma sin_ge_minus_one [simp]: "- 1 \<le> sin x" 3570 for x :: real 3571 using abs_sin_le_one [of x] by (simp add: abs_le_iff) 3572 3573lemma sin_le_one [simp]: "sin x \<le> 1" 3574 for x :: real 3575 using abs_sin_le_one [of x] by (simp add: abs_le_iff) 3576 3577lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1" 3578 for x :: real 3579 by (rule power2_le_imp_le) (simp_all add: cos_squared_eq) 3580 3581lemma cos_ge_minus_one [simp]: "- 1 \<le> cos x" 3582 for x :: real 3583 using abs_cos_le_one [of x] by (simp add: abs_le_iff) 3584 3585lemma cos_le_one [simp]: "cos x \<le> 1" 3586 for x :: real 3587 using abs_cos_le_one [of x] by (simp add: abs_le_iff) 3588 3589lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y" 3590 for x :: "'a::{real_normed_field,banach}" 3591 using cos_add [of x "- y"] by simp 3592 3593lemma cos_double: "cos(2*x) = (cos x)\<^sup>2 - (sin x)\<^sup>2" 3594 for x :: "'a::{real_normed_field,banach}" 3595 using cos_add [where x=x and y=x] by (simp add: power2_eq_square) 3596 3597lemma sin_cos_le1: "\<bar>sin x * sin y + cos x * cos y\<bar> \<le> 1" 3598 for x :: real 3599 using cos_diff [of x y] by (metis abs_cos_le_one add.commute) 3600 3601lemma DERIV_fun_pow: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m" 3602 by (auto intro!: derivative_eq_intros simp:) 3603 3604lemma DERIV_fun_exp: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. exp (g x)) x :> exp (g x) * m" 3605 by (auto intro!: derivative_intros) 3606 3607 3608subsection \<open>The Constant Pi\<close> 3609 3610definition pi :: real 3611 where "pi = 2 * (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)" 3612 3613text \<open>Show that there's a least positive \<^term>\<open>x\<close> with \<^term>\<open>cos x = 0\<close>; 3614 hence define pi.\<close> 3615 3616lemma sin_paired: "(\<lambda>n. (- 1) ^ n / (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums sin x" 3617 for x :: real 3618proof - 3619 have "(\<lambda>n. \<Sum>k = n*2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x" 3620 by (rule sums_group) (use sin_converges [of x, unfolded scaleR_conv_of_real] in auto) 3621 then show ?thesis 3622 by (simp add: sin_coeff_def ac_simps) 3623qed 3624 3625lemma sin_gt_zero_02: 3626 fixes x :: real 3627 assumes "0 < x" and "x < 2" 3628 shows "0 < sin x" 3629proof - 3630 let ?f = "\<lambda>n::nat. \<Sum>k = n*2..<n*2+2. (- 1) ^ k / (fact (2*k+1)) * x^(2*k+1)" 3631 have pos: "\<forall>n. 0 < ?f n" 3632 proof 3633 fix n :: nat 3634 let ?k2 = "real (Suc (Suc (4 * n)))" 3635 let ?k3 = "real (Suc (Suc (Suc (4 * n))))" 3636 have "x * x < ?k2 * ?k3" 3637 using assms by (intro mult_strict_mono', simp_all) 3638 then have "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)" 3639 by (intro mult_strict_right_mono zero_less_power \<open>0 < x\<close>) 3640 then show "0 < ?f n" 3641 by (simp add: ac_simps divide_less_eq) 3642qed 3643 have sums: "?f sums sin x" 3644 by (rule sin_paired [THEN sums_group]) simp 3645 show "0 < sin x" 3646 unfolding sums_unique [OF sums] 3647 using sums_summable [OF sums] pos 3648 by (rule suminf_pos) 3649qed 3650 3651lemma cos_double_less_one: "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1" 3652 for x :: real 3653 using sin_gt_zero_02 [where x = x] by (auto simp: cos_squared_eq cos_double) 3654 3655lemma cos_paired: "(\<lambda>n. (- 1) ^ n / (fact (2 * n)) * x ^ (2 * n)) sums cos x" 3656 for x :: real 3657proof - 3658 have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x" 3659 by (rule sums_group) (use cos_converges [of x, unfolded scaleR_conv_of_real] in auto) 3660 then show ?thesis 3661 by (simp add: cos_coeff_def ac_simps) 3662qed 3663 3664lemma sum_pos_lt_pair: 3665 fixes f :: "nat \<Rightarrow> real" 3666 assumes f: "summable f" and fplus: "\<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc (Suc 0) * d) + 1))" 3667 shows "sum f {..<k} < suminf f" 3668proof - 3669 have "(\<lambda>n. \<Sum>n = n * Suc (Suc 0)..<n * Suc (Suc 0) + Suc (Suc 0). f (n + k)) 3670 sums (\<Sum>n. f (n + k))" 3671 proof (rule sums_group) 3672 show "(\<lambda>n. f (n + k)) sums (\<Sum>n. f (n + k))" 3673 by (simp add: f summable_iff_shift summable_sums) 3674 qed auto 3675 with fplus have "0 < (\<Sum>n. f (n + k))" 3676 apply (simp add: add.commute) 3677 apply (metis (no_types, lifting) suminf_pos summable_def sums_unique) 3678 done 3679 then show ?thesis 3680 by (simp add: f suminf_minus_initial_segment) 3681qed 3682 3683lemma cos_two_less_zero [simp]: "cos 2 < (0::real)" 3684proof - 3685 note fact_Suc [simp del] 3686 from sums_minus [OF cos_paired] 3687 have *: "(\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / fact (2 * n))) sums - cos (2::real)" 3688 by simp 3689 then have sm: "summable (\<lambda>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" 3690 by (rule sums_summable) 3691 have "0 < (\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" 3692 by (simp add: fact_num_eq_if power_eq_if) 3693 moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n)))) < 3694 (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" 3695 proof - 3696 { 3697 fix d 3698 let ?six4d = "Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))" 3699 have "(4::real) * (fact (?six4d)) < (Suc (Suc (?six4d)) * fact (Suc (?six4d)))" 3700 unfolding of_nat_mult by (rule mult_strict_mono) (simp_all add: fact_less_mono) 3701 then have "(4::real) * (fact (?six4d)) < (fact (Suc (Suc (?six4d))))" 3702 by (simp only: fact_Suc [of "Suc (?six4d)"] of_nat_mult of_nat_fact) 3703 then have "(4::real) * inverse (fact (Suc (Suc (?six4d)))) < inverse (fact (?six4d))" 3704 by (simp add: inverse_eq_divide less_divide_eq) 3705 } 3706 then show ?thesis 3707 by (force intro!: sum_pos_lt_pair [OF sm] simp add: divide_inverse algebra_simps) 3708 qed 3709 ultimately have "0 < (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" 3710 by (rule order_less_trans) 3711 moreover from * have "- cos 2 = (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" 3712 by (rule sums_unique) 3713 ultimately have "(0::real) < - cos 2" by simp 3714 then show ?thesis by simp 3715qed 3716 3717lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq] 3718lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le] 3719 3720lemma cos_is_zero: "\<exists>!x::real. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0" 3721proof (rule ex_ex1I) 3722 show "\<exists>x::real. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0" 3723 by (rule IVT2) simp_all 3724next 3725 fix a b :: real 3726 assume ab: "0 \<le> a \<and> a \<le> 2 \<and> cos a = 0" "0 \<le> b \<and> b \<le> 2 \<and> cos b = 0" 3727 have cosd: "\<And>x::real. cos differentiable (at x)" 3728 unfolding real_differentiable_def by (auto intro: DERIV_cos) 3729 show "a = b" 3730 proof (cases a b rule: linorder_cases) 3731 case less 3732 then obtain z where "a < z" "z < b" "(cos has_real_derivative 0) (at z)" 3733 using Rolle by (metis cosd continuous_on_cos_real ab) 3734 then have "sin z = 0" 3735 using DERIV_cos DERIV_unique neg_equal_0_iff_equal by blast 3736 then show ?thesis 3737 by (metis \<open>a < z\<close> \<open>z < b\<close> ab order_less_le_trans less_le sin_gt_zero_02) 3738 next 3739 case greater 3740 then obtain z where "b < z" "z < a" "(cos has_real_derivative 0) (at z)" 3741 using Rolle by (metis cosd continuous_on_cos_real ab) 3742 then have "sin z = 0" 3743 using DERIV_cos DERIV_unique neg_equal_0_iff_equal by blast 3744 then show ?thesis 3745 by (metis \<open>b < z\<close> \<open>z < a\<close> ab order_less_le_trans less_le sin_gt_zero_02) 3746 qed auto 3747qed 3748 3749lemma pi_half: "pi/2 = (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)" 3750 by (simp add: pi_def) 3751 3752lemma cos_pi_half [simp]: "cos (pi/2) = 0" 3753 by (simp add: pi_half cos_is_zero [THEN theI']) 3754 3755lemma cos_of_real_pi_half [simp]: "cos ((of_real pi/2) :: 'a) = 0" 3756 if "SORT_CONSTRAINT('a::{real_field,banach,real_normed_algebra_1})" 3757 by (metis cos_pi_half cos_of_real eq_numeral_simps(4) 3758 nonzero_of_real_divide of_real_0 of_real_numeral) 3759 3760lemma pi_half_gt_zero [simp]: "0 < pi/2" 3761proof - 3762 have "0 \<le> pi/2" 3763 by (simp add: pi_half cos_is_zero [THEN theI']) 3764 then show ?thesis 3765 by (metis cos_pi_half cos_zero less_eq_real_def one_neq_zero) 3766qed 3767 3768lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric] 3769lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le] 3770 3771lemma pi_half_less_two [simp]: "pi/2 < 2" 3772proof - 3773 have "pi/2 \<le> 2" 3774 by (simp add: pi_half cos_is_zero [THEN theI']) 3775 then show ?thesis 3776 by (metis cos_pi_half cos_two_neq_zero le_less) 3777qed 3778 3779lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq] 3780lemmas pi_half_le_two [simp] = pi_half_less_two [THEN order_less_imp_le] 3781 3782lemma pi_gt_zero [simp]: "0 < pi" 3783 using pi_half_gt_zero by simp 3784 3785lemma pi_ge_zero [simp]: "0 \<le> pi" 3786 by (rule pi_gt_zero [THEN order_less_imp_le]) 3787 3788lemma pi_neq_zero [simp]: "pi \<noteq> 0" 3789 by (rule pi_gt_zero [THEN less_imp_neq, symmetric]) 3790 3791lemma pi_not_less_zero [simp]: "\<not> pi < 0" 3792 by (simp add: linorder_not_less) 3793 3794lemma minus_pi_half_less_zero: "-(pi/2) < 0" 3795 by simp 3796 3797lemma m2pi_less_pi: "- (2*pi) < pi" 3798 by simp 3799 3800lemma sin_pi_half [simp]: "sin(pi/2) = 1" 3801 using sin_cos_squared_add2 [where x = "pi/2"] 3802 using sin_gt_zero_02 [OF pi_half_gt_zero pi_half_less_two] 3803 by (simp add: power2_eq_1_iff) 3804 3805lemma sin_of_real_pi_half [simp]: "sin ((of_real pi/2) :: 'a) = 1" 3806 if "SORT_CONSTRAINT('a::{real_field,banach,real_normed_algebra_1})" 3807 using sin_pi_half 3808 by (metis sin_pi_half eq_numeral_simps(4) nonzero_of_real_divide of_real_1 of_real_numeral sin_of_real) 3809 3810lemma sin_cos_eq: "sin x = cos (of_real pi/2 - x)" 3811 for x :: "'a::{real_normed_field,banach}" 3812 by (simp add: cos_diff) 3813 3814lemma minus_sin_cos_eq: "- sin x = cos (x + of_real pi/2)" 3815 for x :: "'a::{real_normed_field,banach}" 3816 by (simp add: cos_add nonzero_of_real_divide) 3817 3818lemma cos_sin_eq: "cos x = sin (of_real pi/2 - x)" 3819 for x :: "'a::{real_normed_field,banach}" 3820 using sin_cos_eq [of "of_real pi/2 - x"] by simp 3821 3822lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y" 3823 for x :: "'a::{real_normed_field,banach}" 3824 using cos_add [of "of_real pi/2 - x" "-y"] 3825 by (simp add: cos_sin_eq) (simp add: sin_cos_eq) 3826 3827lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y" 3828 for x :: "'a::{real_normed_field,banach}" 3829 using sin_add [of x "- y"] by simp 3830 3831lemma sin_double: "sin(2 * x) = 2 * sin x * cos x" 3832 for x :: "'a::{real_normed_field,banach}" 3833 using sin_add [where x=x and y=x] by simp 3834 3835lemma cos_of_real_pi [simp]: "cos (of_real pi) = -1" 3836 using cos_add [where x = "pi/2" and y = "pi/2"] 3837 by (simp add: cos_of_real) 3838 3839lemma sin_of_real_pi [simp]: "sin (of_real pi) = 0" 3840 using sin_add [where x = "pi/2" and y = "pi/2"] 3841 by (simp add: sin_of_real) 3842 3843lemma cos_pi [simp]: "cos pi = -1" 3844 using cos_add [where x = "pi/2" and y = "pi/2"] by simp 3845 3846lemma sin_pi [simp]: "sin pi = 0" 3847 using sin_add [where x = "pi/2" and y = "pi/2"] by simp 3848 3849lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x" 3850 by (simp add: sin_add) 3851 3852lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x" 3853 by (simp add: sin_add) 3854 3855lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x" 3856 by (simp add: cos_add) 3857 3858lemma cos_periodic_pi2 [simp]: "cos (pi + x) = - cos x" 3859 by (simp add: cos_add) 3860 3861lemma sin_periodic [simp]: "sin (x + 2 * pi) = sin x" 3862 by (simp add: sin_add sin_double cos_double) 3863 3864lemma cos_periodic [simp]: "cos (x + 2 * pi) = cos x" 3865 by (simp add: cos_add sin_double cos_double) 3866 3867lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n" 3868 by (induct n) (auto simp: distrib_right) 3869 3870lemma cos_npi2 [simp]: "cos (pi * real n) = (- 1) ^ n" 3871 by (metis cos_npi mult.commute) 3872 3873lemma sin_npi [simp]: "sin (real n * pi) = 0" 3874 for n :: nat 3875 by (induct n) (auto simp: distrib_right) 3876 3877lemma sin_npi2 [simp]: "sin (pi * real n) = 0" 3878 for n :: nat 3879 by (simp add: mult.commute [of pi]) 3880 3881lemma cos_two_pi [simp]: "cos (2 * pi) = 1" 3882 by (simp add: cos_double) 3883 3884lemma sin_two_pi [simp]: "sin (2 * pi) = 0" 3885 by (simp add: sin_double) 3886 3887lemma sin_times_sin: "sin w * sin z = (cos (w - z) - cos (w + z)) / 2" 3888 for w :: "'a::{real_normed_field,banach}" 3889 by (simp add: cos_diff cos_add) 3890 3891lemma sin_times_cos: "sin w * cos z = (sin (w + z) + sin (w - z)) / 2" 3892 for w :: "'a::{real_normed_field,banach}" 3893 by (simp add: sin_diff sin_add) 3894 3895lemma cos_times_sin: "cos w * sin z = (sin (w + z) - sin (w - z)) / 2" 3896 for w :: "'a::{real_normed_field,banach}" 3897 by (simp add: sin_diff sin_add) 3898 3899lemma cos_times_cos: "cos w * cos z = (cos (w - z) + cos (w + z)) / 2" 3900 for w :: "'a::{real_normed_field,banach}" 3901 by (simp add: cos_diff cos_add) 3902 3903lemma sin_plus_sin: "sin w + sin z = 2 * sin ((w + z) / 2) * cos ((w - z) / 2)" 3904 for w :: "'a::{real_normed_field,banach}" 3905 apply (simp add: mult.assoc sin_times_cos) 3906 apply (simp add: field_simps) 3907 done 3908 3909lemma sin_diff_sin: "sin w - sin z = 2 * sin ((w - z) / 2) * cos ((w + z) / 2)" 3910 for w :: "'a::{real_normed_field,banach}" 3911 apply (simp add: mult.assoc sin_times_cos) 3912 apply (simp add: field_simps) 3913 done 3914 3915lemma cos_plus_cos: "cos w + cos z = 2 * cos ((w + z) / 2) * cos ((w - z) / 2)" 3916 for w :: "'a::{real_normed_field,banach,field}" 3917 apply (simp add: mult.assoc cos_times_cos) 3918 apply (simp add: field_simps) 3919 done 3920 3921lemma cos_diff_cos: "cos w - cos z = 2 * sin ((w + z) / 2) * sin ((z - w) / 2)" 3922 for w :: "'a::{real_normed_field,banach,field}" 3923 apply (simp add: mult.assoc sin_times_sin) 3924 apply (simp add: field_simps) 3925 done 3926 3927lemma cos_double_cos: "cos (2 * z) = 2 * cos z ^ 2 - 1" 3928 for z :: "'a::{real_normed_field,banach}" 3929 by (simp add: cos_double sin_squared_eq) 3930 3931lemma cos_double_sin: "cos (2 * z) = 1 - 2 * sin z ^ 2" 3932 for z :: "'a::{real_normed_field,banach}" 3933 by (simp add: cos_double sin_squared_eq) 3934 3935lemma sin_pi_minus [simp]: "sin (pi - x) = sin x" 3936 by (metis sin_minus sin_periodic_pi minus_minus uminus_add_conv_diff) 3937 3938lemma cos_pi_minus [simp]: "cos (pi - x) = - (cos x)" 3939 by (metis cos_minus cos_periodic_pi uminus_add_conv_diff) 3940 3941lemma sin_minus_pi [simp]: "sin (x - pi) = - (sin x)" 3942 by (simp add: sin_diff) 3943 3944lemma cos_minus_pi [simp]: "cos (x - pi) = - (cos x)" 3945 by (simp add: cos_diff) 3946 3947lemma sin_2pi_minus [simp]: "sin (2 * pi - x) = - (sin x)" 3948 by (metis sin_periodic_pi2 add_diff_eq mult_2 sin_pi_minus) 3949 3950lemma cos_2pi_minus [simp]: "cos (2 * pi - x) = cos x" 3951 by (metis (no_types, hide_lams) cos_add cos_minus cos_two_pi sin_minus sin_two_pi 3952 diff_0_right minus_diff_eq mult_1 mult_zero_left uminus_add_conv_diff) 3953 3954lemma sin_gt_zero2: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < sin x" 3955 by (metis sin_gt_zero_02 order_less_trans pi_half_less_two) 3956 3957lemma sin_less_zero: 3958 assumes "- pi/2 < x" and "x < 0" 3959 shows "sin x < 0" 3960proof - 3961 have "0 < sin (- x)" 3962 using assms by (simp only: sin_gt_zero2) 3963 then show ?thesis by simp 3964qed 3965 3966lemma pi_less_4: "pi < 4" 3967 using pi_half_less_two by auto 3968 3969lemma cos_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cos x" 3970 by (simp add: cos_sin_eq sin_gt_zero2) 3971 3972lemma cos_gt_zero_pi: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cos x" 3973 using cos_gt_zero [of x] cos_gt_zero [of "-x"] 3974 by (cases rule: linorder_cases [of x 0]) auto 3975 3976lemma cos_ge_zero: "-(pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> 0 \<le> cos x" 3977 by (auto simp: order_le_less cos_gt_zero_pi) 3978 (metis cos_pi_half eq_divide_eq eq_numeral_simps(4)) 3979 3980lemma sin_gt_zero: "0 < x \<Longrightarrow> x < pi \<Longrightarrow> 0 < sin x" 3981 by (simp add: sin_cos_eq cos_gt_zero_pi) 3982 3983lemma sin_lt_zero: "pi < x \<Longrightarrow> x < 2 * pi \<Longrightarrow> sin x < 0" 3984 using sin_gt_zero [of "x - pi"] 3985 by (simp add: sin_diff) 3986 3987lemma pi_ge_two: "2 \<le> pi" 3988proof (rule ccontr) 3989 assume "\<not> ?thesis" 3990 then have "pi < 2" by auto 3991 have "\<exists>y > pi. y < 2 \<and> y < 2 * pi" 3992 proof (cases "2 < 2 * pi") 3993 case True 3994 with dense[OF \<open>pi < 2\<close>] show ?thesis by auto 3995 next 3996 case False 3997 have "pi < 2 * pi" by auto 3998 from dense[OF this] and False show ?thesis by auto 3999 qed 4000 then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" 4001 by blast 4002 then have "0 < sin y" 4003 using sin_gt_zero_02 by auto 4004 moreover have "sin y < 0" 4005 using sin_gt_zero[of "y - pi"] \<open>pi < y\<close> and \<open>y < 2 * pi\<close> sin_periodic_pi[of "y - pi"] 4006 by auto 4007 ultimately show False by auto 4008qed 4009 4010lemma sin_ge_zero: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> sin x" 4011 by (auto simp: order_le_less sin_gt_zero) 4012 4013lemma sin_le_zero: "pi \<le> x \<Longrightarrow> x < 2 * pi \<Longrightarrow> sin x \<le> 0" 4014 using sin_ge_zero [of "x - pi"] by (simp add: sin_diff) 4015 4016lemma sin_pi_divide_n_ge_0 [simp]: 4017 assumes "n \<noteq> 0" 4018 shows "0 \<le> sin (pi / real n)" 4019 by (rule sin_ge_zero) (use assms in \<open>simp_all add: field_split_simps\<close>) 4020 4021lemma sin_pi_divide_n_gt_0: 4022 assumes "2 \<le> n" 4023 shows "0 < sin (pi / real n)" 4024 by (rule sin_gt_zero) (use assms in \<open>simp_all add: field_split_simps\<close>) 4025 4026text\<open>Proof resembles that of \<open>cos_is_zero\<close> but with \<^term>\<open>pi\<close> for the upper bound\<close> 4027lemma cos_total: 4028 assumes y: "-1 \<le> y" "y \<le> 1" 4029 shows "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y" 4030proof (rule ex_ex1I) 4031 show "\<exists>x::real. 0 \<le> x \<and> x \<le> pi \<and> cos x = y" 4032 by (rule IVT2) (simp_all add: y) 4033next 4034 fix a b :: real 4035 assume ab: "0 \<le> a \<and> a \<le> pi \<and> cos a = y" "0 \<le> b \<and> b \<le> pi \<and> cos b = y" 4036 have cosd: "\<And>x::real. cos differentiable (at x)" 4037 unfolding real_differentiable_def by (auto intro: DERIV_cos) 4038 show "a = b" 4039 proof (cases a b rule: linorder_cases) 4040 case less 4041 then obtain z where "a < z" "z < b" "(cos has_real_derivative 0) (at z)" 4042 using Rolle by (metis cosd continuous_on_cos_real ab) 4043 then have "sin z = 0" 4044 using DERIV_cos DERIV_unique neg_equal_0_iff_equal by blast 4045 then show ?thesis 4046 by (metis \<open>a < z\<close> \<open>z < b\<close> ab order_less_le_trans less_le sin_gt_zero) 4047 next 4048 case greater 4049 then obtain z where "b < z" "z < a" "(cos has_real_derivative 0) (at z)" 4050 using Rolle by (metis cosd continuous_on_cos_real ab) 4051 then have "sin z = 0" 4052 using DERIV_cos DERIV_unique neg_equal_0_iff_equal by blast 4053 then show ?thesis 4054 by (metis \<open>b < z\<close> \<open>z < a\<close> ab order_less_le_trans less_le sin_gt_zero) 4055 qed auto 4056qed 4057 4058lemma sin_total: 4059 assumes y: "-1 \<le> y" "y \<le> 1" 4060 shows "\<exists>!x. - (pi/2) \<le> x \<and> x \<le> pi/2 \<and> sin x = y" 4061proof - 4062 from cos_total [OF y] 4063 obtain x where x: "0 \<le> x" "x \<le> pi" "cos x = y" 4064 and uniq: "\<And>x'. 0 \<le> x' \<Longrightarrow> x' \<le> pi \<Longrightarrow> cos x' = y \<Longrightarrow> x' = x " 4065 by blast 4066 show ?thesis 4067 unfolding sin_cos_eq 4068 proof (rule ex1I [where a="pi/2 - x"]) 4069 show "- (pi/2) \<le> z \<and> z \<le> pi/2 \<and> cos (of_real pi/2 - z) = y \<Longrightarrow> 4070 z = pi/2 - x" for z 4071 using uniq [of "pi/2 -z"] by auto 4072 qed (use x in auto) 4073qed 4074 4075lemma cos_zero_lemma: 4076 assumes "0 \<le> x" "cos x = 0" 4077 shows "\<exists>n. odd n \<and> x = of_nat n * (pi/2) \<and> n > 0" 4078proof - 4079 have xle: "x < (1 + real_of_int \<lfloor>x/pi\<rfloor>) * pi" 4080 using floor_correct [of "x/pi"] 4081 by (simp add: add.commute divide_less_eq) 4082 obtain n where "real n * pi \<le> x" "x < real (Suc n) * pi" 4083 proof 4084 show "real (nat \<lfloor>x / pi\<rfloor>) * pi \<le> x" 4085 using assms floor_divide_lower [of pi x] by auto 4086 show "x < real (Suc (nat \<lfloor>x / pi\<rfloor>)) * pi" 4087 using assms floor_divide_upper [of pi x] by (simp add: xle) 4088 qed 4089 then have x: "0 \<le> x - n * pi" "(x - n * pi) \<le> pi" "cos (x - n * pi) = 0" 4090 by (auto simp: algebra_simps cos_diff assms) 4091 then have "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = 0" 4092 by (auto simp: intro!: cos_total) 4093 then obtain \<theta> where \<theta>: "0 \<le> \<theta>" "\<theta> \<le> pi" "cos \<theta> = 0" 4094 and uniq: "\<And>\<phi>. 0 \<le> \<phi> \<Longrightarrow> \<phi> \<le> pi \<Longrightarrow> cos \<phi> = 0 \<Longrightarrow> \<phi> = \<theta>" 4095 by blast 4096 then have "x - real n * pi = \<theta>" 4097 using x by blast 4098 moreover have "pi/2 = \<theta>" 4099 using pi_half_ge_zero uniq by fastforce 4100 ultimately show ?thesis 4101 by (rule_tac x = "Suc (2 * n)" in exI) (simp add: algebra_simps) 4102qed 4103 4104lemma sin_zero_lemma: "0 \<le> x \<Longrightarrow> sin x = 0 \<Longrightarrow> \<exists>n::nat. even n \<and> x = real n * (pi/2)" 4105 using cos_zero_lemma [of "x + pi/2"] 4106 apply (clarsimp simp add: cos_add) 4107 apply (rule_tac x = "n - 1" in exI) 4108 apply (simp add: algebra_simps of_nat_diff) 4109 done 4110 4111lemma cos_zero_iff: 4112 "cos x = 0 \<longleftrightarrow> ((\<exists>n. odd n \<and> x = real n * (pi/2)) \<or> (\<exists>n. odd n \<and> x = - (real n * (pi/2))))" 4113 (is "?lhs = ?rhs") 4114proof - 4115 have *: "cos (real n * pi/2) = 0" if "odd n" for n :: nat 4116 proof - 4117 from that obtain m where "n = 2 * m + 1" .. 4118 then show ?thesis 4119 by (simp add: field_simps) (simp add: cos_add add_divide_distrib) 4120 qed 4121 show ?thesis 4122 proof 4123 show ?rhs if ?lhs 4124 using that cos_zero_lemma [of x] cos_zero_lemma [of "-x"] by force 4125 show ?lhs if ?rhs 4126 using that by (auto dest: * simp del: eq_divide_eq_numeral1) 4127 qed 4128qed 4129 4130lemma sin_zero_iff: 4131 "sin x = 0 \<longleftrightarrow> ((\<exists>n. even n \<and> x = real n * (pi/2)) \<or> (\<exists>n. even n \<and> x = - (real n * (pi/2))))" 4132 (is "?lhs = ?rhs") 4133proof 4134 show ?rhs if ?lhs 4135 using that sin_zero_lemma [of x] sin_zero_lemma [of "-x"] by force 4136 show ?lhs if ?rhs 4137 using that by (auto elim: evenE) 4138qed 4139 4140lemma sin_zero_pi_iff: 4141 fixes x::real 4142 assumes "\<bar>x\<bar> < pi" 4143 shows "sin x = 0 \<longleftrightarrow> x = 0" 4144proof 4145 show "x = 0" if "sin x = 0" 4146 using that assms by (auto simp: sin_zero_iff) 4147qed auto 4148 4149lemma cos_zero_iff_int: "cos x = 0 \<longleftrightarrow> (\<exists>n. odd n \<and> x = of_int n * (pi/2))" 4150proof - 4151 have 1: "\<And>n. odd n \<Longrightarrow> \<exists>i. odd i \<and> real n = real_of_int i" 4152 by (metis even_of_nat of_int_of_nat_eq) 4153 have 2: "\<And>n. odd n \<Longrightarrow> \<exists>i. odd i \<and> - (real n * pi) = real_of_int i * pi" 4154 by (metis even_minus even_of_nat mult.commute mult_minus_right of_int_minus of_int_of_nat_eq) 4155 have 3: "\<lbrakk>odd i; \<forall>n. even n \<or> real_of_int i \<noteq> - (real n)\<rbrakk> 4156 \<Longrightarrow> \<exists>n. odd n \<and> real_of_int i = real n" for i 4157 by (cases i rule: int_cases2) auto 4158 show ?thesis 4159 by (force simp: cos_zero_iff intro!: 1 2 3) 4160qed 4161 4162lemma sin_zero_iff_int: "sin x = 0 \<longleftrightarrow> (\<exists>n. even n \<and> x = of_int n * (pi/2))" 4163proof safe 4164 assume "sin x = 0" 4165 then show "\<exists>n. even n \<and> x = of_int n * (pi/2)" 4166 apply (simp add: sin_zero_iff, safe) 4167 apply (metis even_of_nat of_int_of_nat_eq) 4168 apply (rule_tac x="- (int n)" in exI) 4169 apply simp 4170 done 4171next 4172 fix i :: int 4173 assume "even i" 4174 then show "sin (of_int i * (pi/2)) = 0" 4175 by (cases i rule: int_cases2, simp_all add: sin_zero_iff) 4176qed 4177 4178lemma sin_zero_iff_int2: "sin x = 0 \<longleftrightarrow> (\<exists>n::int. x = of_int n * pi)" 4179 apply (simp only: sin_zero_iff_int) 4180 apply (safe elim!: evenE) 4181 apply (simp_all add: field_simps) 4182 using dvd_triv_left apply fastforce 4183 done 4184 4185lemma sin_npi_int [simp]: "sin (pi * of_int n) = 0" 4186 by (simp add: sin_zero_iff_int2) 4187 4188lemma cos_monotone_0_pi: 4189 assumes "0 \<le> y" and "y < x" and "x \<le> pi" 4190 shows "cos x < cos y" 4191proof - 4192 have "- (x - y) < 0" using assms by auto 4193 from MVT2[OF \<open>y < x\<close> DERIV_cos] 4194 obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" 4195 by auto 4196 then have "0 < z" and "z < pi" 4197 using assms by auto 4198 then have "0 < sin z" 4199 using sin_gt_zero by auto 4200 then have "cos x - cos y < 0" 4201 unfolding cos_diff minus_mult_commute[symmetric] 4202 using \<open>- (x - y) < 0\<close> by (rule mult_pos_neg2) 4203 then show ?thesis by auto 4204qed 4205 4206lemma cos_monotone_0_pi_le: 4207 assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" 4208 shows "cos x \<le> cos y" 4209proof (cases "y < x") 4210 case True 4211 show ?thesis 4212 using cos_monotone_0_pi[OF \<open>0 \<le> y\<close> True \<open>x \<le> pi\<close>] by auto 4213next 4214 case False 4215 then have "y = x" using \<open>y \<le> x\<close> by auto 4216 then show ?thesis by auto 4217qed 4218 4219lemma cos_monotone_minus_pi_0: 4220 assumes "- pi \<le> y" and "y < x" and "x \<le> 0" 4221 shows "cos y < cos x" 4222proof - 4223 have "0 \<le> - x" and "- x < - y" and "- y \<le> pi" 4224 using assms by auto 4225 from cos_monotone_0_pi[OF this] show ?thesis 4226 unfolding cos_minus . 4227qed 4228 4229lemma cos_monotone_minus_pi_0': 4230 assumes "- pi \<le> y" and "y \<le> x" and "x \<le> 0" 4231 shows "cos y \<le> cos x" 4232proof (cases "y < x") 4233 case True 4234 show ?thesis using cos_monotone_minus_pi_0[OF \<open>-pi \<le> y\<close> True \<open>x \<le> 0\<close>] 4235 by auto 4236next 4237 case False 4238 then have "y = x" using \<open>y \<le> x\<close> by auto 4239 then show ?thesis by auto 4240qed 4241 4242lemma sin_monotone_2pi: 4243 assumes "- (pi/2) \<le> y" and "y < x" and "x \<le> pi/2" 4244 shows "sin y < sin x" 4245 unfolding sin_cos_eq 4246 using assms by (auto intro: cos_monotone_0_pi) 4247 4248lemma sin_monotone_2pi_le: 4249 assumes "- (pi/2) \<le> y" and "y \<le> x" and "x \<le> pi/2" 4250 shows "sin y \<le> sin x" 4251 by (metis assms le_less sin_monotone_2pi) 4252 4253lemma sin_x_le_x: 4254 fixes x :: real 4255 assumes x: "x \<ge> 0" 4256 shows "sin x \<le> x" 4257proof - 4258 let ?f = "\<lambda>x. x - sin x" 4259 from x have "?f x \<ge> ?f 0" 4260 apply (rule DERIV_nonneg_imp_nondecreasing) 4261 apply (intro allI impI exI[of _ "1 - cos x" for x]) 4262 apply (auto intro!: derivative_eq_intros simp: field_simps) 4263 done 4264 then show "sin x \<le> x" by simp 4265qed 4266 4267lemma sin_x_ge_neg_x: 4268 fixes x :: real 4269 assumes x: "x \<ge> 0" 4270 shows "sin x \<ge> - x" 4271proof - 4272 let ?f = "\<lambda>x. x + sin x" 4273 from x have "?f x \<ge> ?f 0" 4274 apply (rule DERIV_nonneg_imp_nondecreasing) 4275 apply (intro allI impI exI[of _ "1 + cos x" for x]) 4276 apply (auto intro!: derivative_eq_intros simp: field_simps real_0_le_add_iff) 4277 done 4278 then show "sin x \<ge> -x" by simp 4279qed 4280 4281lemma abs_sin_x_le_abs_x: "\<bar>sin x\<bar> \<le> \<bar>x\<bar>" 4282 for x :: real 4283 using sin_x_ge_neg_x [of x] sin_x_le_x [of x] sin_x_ge_neg_x [of "-x"] sin_x_le_x [of "-x"] 4284 by (auto simp: abs_real_def) 4285 4286 4287subsection \<open>More Corollaries about Sine and Cosine\<close> 4288 4289lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi/2) = (-1) ^ n" 4290proof - 4291 have "sin ((real n + 1/2) * pi) = cos (real n * pi)" 4292 by (auto simp: algebra_simps sin_add) 4293 then show ?thesis 4294 by (simp add: distrib_right add_divide_distrib add.commute mult.commute [of pi]) 4295qed 4296 4297lemma cos_2npi [simp]: "cos (2 * real n * pi) = 1" 4298 for n :: nat 4299 by (cases "even n") (simp_all add: cos_double mult.assoc) 4300 4301lemma cos_3over2_pi [simp]: "cos (3/2*pi) = 0" 4302proof - 4303 have "cos (3/2*pi) = cos (pi + pi/2)" 4304 by simp 4305 also have "... = 0" 4306 by (subst cos_add, simp) 4307 finally show ?thesis . 4308qed 4309 4310lemma sin_2npi [simp]: "sin (2 * real n * pi) = 0" 4311 for n :: nat 4312 by (auto simp: mult.assoc sin_double) 4313 4314lemma sin_3over2_pi [simp]: "sin (3/2*pi) = - 1" 4315proof - 4316 have "sin (3/2*pi) = sin (pi + pi/2)" 4317 by simp 4318 also have "... = -1" 4319 by (subst sin_add, simp) 4320 finally show ?thesis . 4321qed 4322 4323lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0" 4324 by (simp only: cos_add sin_add of_nat_Suc distrib_right distrib_left add_divide_distrib, auto) 4325 4326lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)" 4327 by (auto intro!: derivative_eq_intros) 4328 4329lemma sin_zero_norm_cos_one: 4330 fixes x :: "'a::{real_normed_field,banach}" 4331 assumes "sin x = 0" 4332 shows "norm (cos x) = 1" 4333 using sin_cos_squared_add [of x, unfolded assms] 4334 by (simp add: square_norm_one) 4335 4336lemma sin_zero_abs_cos_one: "sin x = 0 \<Longrightarrow> \<bar>cos x\<bar> = (1::real)" 4337 using sin_zero_norm_cos_one by fastforce 4338 4339lemma cos_one_sin_zero: 4340 fixes x :: "'a::{real_normed_field,banach}" 4341 assumes "cos x = 1" 4342 shows "sin x = 0" 4343 using sin_cos_squared_add [of x, unfolded assms] 4344 by simp 4345 4346lemma sin_times_pi_eq_0: "sin (x * pi) = 0 \<longleftrightarrow> x \<in> \<int>" 4347 by (simp add: sin_zero_iff_int2) (metis Ints_cases Ints_of_int) 4348 4349lemma cos_one_2pi: "cos x = 1 \<longleftrightarrow> (\<exists>n::nat. x = n * 2 * pi) \<or> (\<exists>n::nat. x = - (n * 2 * pi))" 4350 (is "?lhs = ?rhs") 4351proof 4352 assume ?lhs 4353 then have "sin x = 0" 4354 by (simp add: cos_one_sin_zero) 4355 then show ?rhs 4356 proof (simp only: sin_zero_iff, elim exE disjE conjE) 4357 fix n :: nat 4358 assume n: "even n" "x = real n * (pi/2)" 4359 then obtain m where m: "n = 2 * m" 4360 using dvdE by blast 4361 then have me: "even m" using \<open>?lhs\<close> n 4362 by (auto simp: field_simps) (metis one_neq_neg_one power_minus_odd power_one) 4363 show ?rhs 4364 using m me n 4365 by (auto simp: field_simps elim!: evenE) 4366 next 4367 fix n :: nat 4368 assume n: "even n" "x = - (real n * (pi/2))" 4369 then obtain m where m: "n = 2 * m" 4370 using dvdE by blast 4371 then have me: "even m" using \<open>?lhs\<close> n 4372 by (auto simp: field_simps) (metis one_neq_neg_one power_minus_odd power_one) 4373 show ?rhs 4374 using m me n 4375 by (auto simp: field_simps elim!: evenE) 4376 qed 4377next 4378 assume ?rhs 4379 then show "cos x = 1" 4380 by (metis cos_2npi cos_minus mult.assoc mult.left_commute) 4381qed 4382 4383lemma cos_one_2pi_int: "cos x = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2 * pi)" (is "?lhs = ?rhs") 4384proof 4385 assume "cos x = 1" 4386 then show ?rhs 4387 by (metis cos_one_2pi mult.commute mult_minus_right of_int_minus of_int_of_nat_eq) 4388next 4389 assume ?rhs 4390 then show "cos x = 1" 4391 by (clarsimp simp add: cos_one_2pi) (metis mult_minus_right of_int_of_nat) 4392qed 4393 4394lemma cos_npi_int [simp]: 4395 fixes n::int shows "cos (pi * of_int n) = (if even n then 1 else -1)" 4396 by (auto simp: algebra_simps cos_one_2pi_int elim!: oddE evenE) 4397 4398lemma sin_cos_sqrt: "0 \<le> sin x \<Longrightarrow> sin x = sqrt (1 - (cos(x) ^ 2))" 4399 using sin_squared_eq real_sqrt_unique by fastforce 4400 4401lemma sin_eq_0_pi: "- pi < x \<Longrightarrow> x < pi \<Longrightarrow> sin x = 0 \<Longrightarrow> x = 0" 4402 by (metis sin_gt_zero sin_minus minus_less_iff neg_0_less_iff_less not_less_iff_gr_or_eq) 4403 4404lemma cos_treble_cos: "cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x" 4405 for x :: "'a::{real_normed_field,banach}" 4406proof - 4407 have *: "(sin x * (sin x * 3)) = 3 - (cos x * (cos x * 3))" 4408 by (simp add: mult.assoc [symmetric] sin_squared_eq [unfolded power2_eq_square]) 4409 have "cos(3 * x) = cos(2*x + x)" 4410 by simp 4411 also have "\<dots> = 4 * cos x ^ 3 - 3 * cos x" 4412 apply (simp only: cos_add cos_double sin_double) 4413 apply (simp add: * field_simps power2_eq_square power3_eq_cube) 4414 done 4415 finally show ?thesis . 4416qed 4417 4418lemma cos_45: "cos (pi / 4) = sqrt 2 / 2" 4419proof - 4420 let ?c = "cos (pi / 4)" 4421 let ?s = "sin (pi / 4)" 4422 have nonneg: "0 \<le> ?c" 4423 by (simp add: cos_ge_zero) 4424 have "0 = cos (pi / 4 + pi / 4)" 4425 by simp 4426 also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2" 4427 by (simp only: cos_add power2_eq_square) 4428 also have "\<dots> = 2 * ?c\<^sup>2 - 1" 4429 by (simp add: sin_squared_eq) 4430 finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2" 4431 by (simp add: power_divide) 4432 then show ?thesis 4433 using nonneg by (rule power2_eq_imp_eq) simp 4434qed 4435 4436lemma cos_30: "cos (pi / 6) = sqrt 3/2" 4437proof - 4438 let ?c = "cos (pi / 6)" 4439 let ?s = "sin (pi / 6)" 4440 have pos_c: "0 < ?c" 4441 by (rule cos_gt_zero) simp_all 4442 have "0 = cos (pi / 6 + pi / 6 + pi / 6)" 4443 by simp 4444 also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s" 4445 by (simp only: cos_add sin_add) 4446 also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)" 4447 by (simp add: algebra_simps power2_eq_square) 4448 finally have "?c\<^sup>2 = (sqrt 3/2)\<^sup>2" 4449 using pos_c by (simp add: sin_squared_eq power_divide) 4450 then show ?thesis 4451 using pos_c [THEN order_less_imp_le] 4452 by (rule power2_eq_imp_eq) simp 4453qed 4454 4455lemma sin_45: "sin (pi / 4) = sqrt 2 / 2" 4456 by (simp add: sin_cos_eq cos_45) 4457 4458lemma sin_60: "sin (pi / 3) = sqrt 3/2" 4459 by (simp add: sin_cos_eq cos_30) 4460 4461lemma cos_60: "cos (pi / 3) = 1 / 2" 4462proof - 4463 have "0 \<le> cos (pi / 3)" 4464 by (rule cos_ge_zero) (use pi_half_ge_zero in \<open>linarith+\<close>) 4465 then show ?thesis 4466 by (simp add: cos_squared_eq sin_60 power_divide power2_eq_imp_eq) 4467qed 4468 4469lemma sin_30: "sin (pi / 6) = 1 / 2" 4470 by (simp add: sin_cos_eq cos_60) 4471 4472lemma cos_integer_2pi: "n \<in> \<int> \<Longrightarrow> cos(2 * pi * n) = 1" 4473 by (metis Ints_cases cos_one_2pi_int mult.assoc mult.commute) 4474 4475lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2 * pi * n) = 0" 4476 by (metis sin_two_pi Ints_mult mult.assoc mult.commute sin_times_pi_eq_0) 4477 4478lemma cos_int_2pin [simp]: "cos ((2 * pi) * of_int n) = 1" 4479 by (simp add: cos_one_2pi_int) 4480 4481lemma sin_int_2pin [simp]: "sin ((2 * pi) * of_int n) = 0" 4482 by (metis Ints_of_int sin_integer_2pi) 4483 4484lemma sincos_principal_value: "\<exists>y. (- pi < y \<and> y \<le> pi) \<and> (sin y = sin x \<and> cos y = cos x)" 4485 apply (rule exI [where x="pi - (2 * pi) * frac ((pi - x) / (2 * pi))"]) 4486 apply (auto simp: field_simps frac_lt_1) 4487 apply (simp_all add: frac_def field_simps) 4488 apply (simp_all add: add_divide_distrib diff_divide_distrib) 4489 apply (simp_all add: sin_add cos_add mult.assoc [symmetric]) 4490 done 4491 4492 4493subsection \<open>Tangent\<close> 4494 4495definition tan :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4496 where "tan = (\<lambda>x. sin x / cos x)" 4497 4498lemma tan_of_real: "of_real (tan x) = (tan (of_real x) :: 'a::{real_normed_field,banach})" 4499 by (simp add: tan_def sin_of_real cos_of_real) 4500 4501lemma tan_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> tan z \<in> \<real>" 4502 for z :: "'a::{real_normed_field,banach}" 4503 by (simp add: tan_def) 4504 4505lemma tan_zero [simp]: "tan 0 = 0" 4506 by (simp add: tan_def) 4507 4508lemma tan_pi [simp]: "tan pi = 0" 4509 by (simp add: tan_def) 4510 4511lemma tan_npi [simp]: "tan (real n * pi) = 0" 4512 for n :: nat 4513 by (simp add: tan_def) 4514 4515lemma tan_minus [simp]: "tan (- x) = - tan x" 4516 by (simp add: tan_def) 4517 4518lemma tan_periodic [simp]: "tan (x + 2 * pi) = tan x" 4519 by (simp add: tan_def) 4520 4521lemma lemma_tan_add1: "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)" 4522 by (simp add: tan_def cos_add field_simps) 4523 4524lemma add_tan_eq: "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)" 4525 for x :: "'a::{real_normed_field,banach}" 4526 by (simp add: tan_def sin_add field_simps) 4527 4528lemma tan_add: 4529 "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> cos (x + y) \<noteq> 0 \<Longrightarrow> tan (x + y) = (tan x + tan y)/(1 - tan x * tan y)" 4530 for x :: "'a::{real_normed_field,banach}" 4531 by (simp add: add_tan_eq lemma_tan_add1 field_simps) (simp add: tan_def) 4532 4533lemma tan_double: "cos x \<noteq> 0 \<Longrightarrow> cos (2 * x) \<noteq> 0 \<Longrightarrow> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)" 4534 for x :: "'a::{real_normed_field,banach}" 4535 using tan_add [of x x] by (simp add: power2_eq_square) 4536 4537lemma tan_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < tan x" 4538 by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) 4539 4540lemma tan_less_zero: 4541 assumes "- pi/2 < x" and "x < 0" 4542 shows "tan x < 0" 4543proof - 4544 have "0 < tan (- x)" 4545 using assms by (simp only: tan_gt_zero) 4546 then show ?thesis by simp 4547qed 4548 4549lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)" 4550 for x :: "'a::{real_normed_field,banach,field}" 4551 unfolding tan_def sin_double cos_double sin_squared_eq 4552 by (simp add: power2_eq_square) 4553 4554lemma tan_30: "tan (pi / 6) = 1 / sqrt 3" 4555 unfolding tan_def by (simp add: sin_30 cos_30) 4556 4557lemma tan_45: "tan (pi / 4) = 1" 4558 unfolding tan_def by (simp add: sin_45 cos_45) 4559 4560lemma tan_60: "tan (pi / 3) = sqrt 3" 4561 unfolding tan_def by (simp add: sin_60 cos_60) 4562 4563lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)" 4564 for x :: "'a::{real_normed_field,banach}" 4565 unfolding tan_def 4566 by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square) 4567 4568declare DERIV_tan[THEN DERIV_chain2, derivative_intros] 4569 and DERIV_tan[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 4570 4571lemmas has_derivative_tan[derivative_intros] = DERIV_tan[THEN DERIV_compose_FDERIV] 4572 4573lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x" 4574 for x :: "'a::{real_normed_field,banach}" 4575 by (rule DERIV_tan [THEN DERIV_isCont]) 4576 4577lemma isCont_tan' [simp,continuous_intros]: 4578 fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a" 4579 shows "isCont f a \<Longrightarrow> cos (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a" 4580 by (rule isCont_o2 [OF _ isCont_tan]) 4581 4582lemma tendsto_tan [tendsto_intros]: 4583 fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4584 shows "(f \<longlongrightarrow> a) F \<Longrightarrow> cos a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. tan (f x)) \<longlongrightarrow> tan a) F" 4585 by (rule isCont_tendsto_compose [OF isCont_tan]) 4586 4587lemma continuous_tan: 4588 fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4589 shows "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))" 4590 unfolding continuous_def by (rule tendsto_tan) 4591 4592lemma continuous_on_tan [continuous_intros]: 4593 fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4594 shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))" 4595 unfolding continuous_on_def by (auto intro: tendsto_tan) 4596 4597lemma continuous_within_tan [continuous_intros]: 4598 fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4599 shows "continuous (at x within s) f \<Longrightarrow> 4600 cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))" 4601 unfolding continuous_within by (rule tendsto_tan) 4602 4603lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) \<midarrow>pi/2\<rightarrow> 0" 4604 by (rule tendsto_cong_limit, (rule tendsto_intros)+, simp_all) 4605 4606lemma lemma_tan_total: 4607 assumes "0 < y" shows "\<exists>x. 0 < x \<and> x < pi/2 \<and> y < tan x" 4608proof - 4609 obtain s where "0 < s" 4610 and s: "\<And>x. \<lbrakk>x \<noteq> pi/2; norm (x - pi/2) < s\<rbrakk> \<Longrightarrow> norm (cos x / sin x - 0) < inverse y" 4611 using LIM_D [OF LIM_cos_div_sin, of "inverse y"] that assms by force 4612 obtain e where e: "0 < e" "e < s" "e < pi/2" 4613 using \<open>0 < s\<close> field_lbound_gt_zero pi_half_gt_zero by blast 4614 show ?thesis 4615 proof (intro exI conjI) 4616 have "0 < sin e" "0 < cos e" 4617 using e by (auto intro: cos_gt_zero sin_gt_zero2 simp: mult.commute) 4618 then 4619 show "y < tan (pi/2 - e)" 4620 using s [of "pi/2 - e"] e assms 4621 by (simp add: tan_def sin_diff cos_diff) (simp add: field_simps split: if_split_asm) 4622 qed (use e in auto) 4623qed 4624 4625lemma tan_total_pos: 4626 assumes "0 \<le> y" shows "\<exists>x. 0 \<le> x \<and> x < pi/2 \<and> tan x = y" 4627proof (cases "y = 0") 4628 case True 4629 then show ?thesis 4630 using pi_half_gt_zero tan_zero by blast 4631next 4632 case False 4633 with assms have "y > 0" 4634 by linarith 4635 obtain x where x: "0 < x" "x < pi/2" "y < tan x" 4636 using lemma_tan_total \<open>0 < y\<close> by blast 4637 have "\<exists>u\<ge>0. u \<le> x \<and> tan u = y" 4638 proof (intro IVT allI impI) 4639 show "isCont tan u" if "0 \<le> u \<and> u \<le> x" for u 4640 proof - 4641 have "cos u \<noteq> 0" 4642 using antisym_conv2 cos_gt_zero that x(2) by fastforce 4643 with assms show ?thesis 4644 by (auto intro!: DERIV_tan [THEN DERIV_isCont]) 4645 qed 4646 qed (use assms x in auto) 4647 then show ?thesis 4648 using x(2) by auto 4649qed 4650 4651lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y" 4652proof (cases "0::real" y rule: le_cases) 4653 case le 4654 then show ?thesis 4655 by (meson less_le_trans minus_pi_half_less_zero tan_total_pos) 4656next 4657 case ge 4658 with tan_total_pos [of "-y"] obtain x where "0 \<le> x" "x < pi / 2" "tan x = - y" 4659 by force 4660 then show ?thesis 4661 by (rule_tac x="-x" in exI) auto 4662qed 4663 4664proposition tan_total: "\<exists>! x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y" 4665proof - 4666 have "u = v" if u: "- (pi / 2) < u" "u < pi / 2" and v: "- (pi / 2) < v" "v < pi / 2" 4667 and eq: "tan u = tan v" for u v 4668 proof (cases u v rule: linorder_cases) 4669 case less 4670 have "\<And>x. u \<le> x \<and> x \<le> v \<longrightarrow> isCont tan x" 4671 by (metis cos_gt_zero_pi isCont_tan le_less_trans less_irrefl less_le_trans u(1) v(2)) 4672 then have "continuous_on {u..v} tan" 4673 by (simp add: continuous_at_imp_continuous_on) 4674 moreover have "\<And>x. u < x \<and> x < v \<Longrightarrow> tan differentiable (at x)" 4675 by (metis DERIV_tan cos_gt_zero_pi real_differentiable_def less_numeral_extra(3) order.strict_trans u(1) v(2)) 4676 ultimately obtain z where "u < z" "z < v" "DERIV tan z :> 0" 4677 by (metis less Rolle eq) 4678 moreover have "cos z \<noteq> 0" 4679 by (metis (no_types) \<open>u < z\<close> \<open>z < v\<close> cos_gt_zero_pi less_le_trans linorder_not_less not_less_iff_gr_or_eq u(1) v(2)) 4680 ultimately show ?thesis 4681 using DERIV_unique [OF _ DERIV_tan] by fastforce 4682 next 4683 case greater 4684 have "\<And>x. v \<le> x \<and> x \<le> u \<Longrightarrow> isCont tan x" 4685 by (metis cos_gt_zero_pi isCont_tan le_less_trans less_irrefl less_le_trans u(2) v(1)) 4686 then have "continuous_on {v..u} tan" 4687 by (simp add: continuous_at_imp_continuous_on) 4688 moreover have "\<And>x. v < x \<and> x < u \<Longrightarrow> tan differentiable (at x)" 4689 by (metis DERIV_tan cos_gt_zero_pi real_differentiable_def less_numeral_extra(3) order.strict_trans u(2) v(1)) 4690 ultimately obtain z where "v < z" "z < u" "DERIV tan z :> 0" 4691 by (metis greater Rolle eq) 4692 moreover have "cos z \<noteq> 0" 4693 by (metis \<open>v < z\<close> \<open>z < u\<close> cos_gt_zero_pi less_eq_real_def less_le_trans order_less_irrefl u(2) v(1)) 4694 ultimately show ?thesis 4695 using DERIV_unique [OF _ DERIV_tan] by fastforce 4696 qed auto 4697 then have "\<exists>!x. - (pi / 2) < x \<and> x < pi / 2 \<and> tan x = y" 4698 if x: "- (pi / 2) < x" "x < pi / 2" "tan x = y" for x 4699 using that by auto 4700 then show ?thesis 4701 using lemma_tan_total1 [where y = y] 4702 by auto 4703qed 4704 4705lemma tan_monotone: 4706 assumes "- (pi/2) < y" and "y < x" and "x < pi/2" 4707 shows "tan y < tan x" 4708proof - 4709 have "DERIV tan x' :> inverse ((cos x')\<^sup>2)" if "y \<le> x'" "x' \<le> x" for x' 4710 proof - 4711 have "-(pi/2) < x'" and "x' < pi/2" 4712 using that assms by auto 4713 with cos_gt_zero_pi have "cos x' \<noteq> 0" by force 4714 then show "DERIV tan x' :> inverse ((cos x')\<^sup>2)" 4715 by (rule DERIV_tan) 4716 qed 4717 from MVT2[OF \<open>y < x\<close> this] 4718 obtain z where "y < z" and "z < x" 4719 and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto 4720 then have "- (pi/2) < z" and "z < pi/2" 4721 using assms by auto 4722 then have "0 < cos z" 4723 using cos_gt_zero_pi by auto 4724 then have inv_pos: "0 < inverse ((cos z)\<^sup>2)" 4725 by auto 4726 have "0 < x - y" using \<open>y < x\<close> by auto 4727 with inv_pos have "0 < tan x - tan y" 4728 unfolding tan_diff by auto 4729 then show ?thesis by auto 4730qed 4731 4732lemma tan_monotone': 4733 assumes "- (pi/2) < y" 4734 and "y < pi/2" 4735 and "- (pi/2) < x" 4736 and "x < pi/2" 4737 shows "y < x \<longleftrightarrow> tan y < tan x" 4738proof 4739 assume "y < x" 4740 then show "tan y < tan x" 4741 using tan_monotone and \<open>- (pi/2) < y\<close> and \<open>x < pi/2\<close> by auto 4742next 4743 assume "tan y < tan x" 4744 show "y < x" 4745 proof (rule ccontr) 4746 assume "\<not> ?thesis" 4747 then have "x \<le> y" by auto 4748 then have "tan x \<le> tan y" 4749 proof (cases "x = y") 4750 case True 4751 then show ?thesis by auto 4752 next 4753 case False 4754 then have "x < y" using \<open>x \<le> y\<close> by auto 4755 from tan_monotone[OF \<open>- (pi/2) < x\<close> this \<open>y < pi/2\<close>] show ?thesis 4756 by auto 4757 qed 4758 then show False 4759 using \<open>tan y < tan x\<close> by auto 4760 qed 4761qed 4762 4763lemma tan_inverse: "1 / (tan y) = tan (pi/2 - y)" 4764 unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto 4765 4766lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x" 4767 by (simp add: tan_def) 4768 4769lemma tan_periodic_nat[simp]: "tan (x + real n * pi) = tan x" 4770 for n :: nat 4771proof (induct n arbitrary: x) 4772 case 0 4773 then show ?case by simp 4774next 4775 case (Suc n) 4776 have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" 4777 unfolding Suc_eq_plus1 of_nat_add distrib_right by auto 4778 show ?case 4779 unfolding split_pi_off using Suc by auto 4780qed 4781 4782lemma tan_periodic_int[simp]: "tan (x + of_int i * pi) = tan x" 4783proof (cases "0 \<le> i") 4784 case True 4785 then have i_nat: "of_int i = of_int (nat i)" by auto 4786 show ?thesis unfolding i_nat 4787 by (metis of_int_of_nat_eq tan_periodic_nat) 4788next 4789 case False 4790 then have i_nat: "of_int i = - of_int (nat (- i))" by auto 4791 have "tan x = tan (x + of_int i * pi - of_int i * pi)" 4792 by auto 4793 also have "\<dots> = tan (x + of_int i * pi)" 4794 unfolding i_nat mult_minus_left diff_minus_eq_add 4795 by (metis of_int_of_nat_eq tan_periodic_nat) 4796 finally show ?thesis by auto 4797qed 4798 4799lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x" 4800 using tan_periodic_int[of _ "numeral n" ] by simp 4801 4802lemma tan_minus_45: "tan (-(pi/4)) = -1" 4803 unfolding tan_def by (simp add: sin_45 cos_45) 4804 4805lemma tan_diff: 4806 "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> cos (x - y) \<noteq> 0 \<Longrightarrow> tan (x - y) = (tan x - tan y)/(1 + tan x * tan y)" 4807 for x :: "'a::{real_normed_field,banach}" 4808 using tan_add [of x "-y"] by simp 4809 4810lemma tan_pos_pi2_le: "0 \<le> x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 \<le> tan x" 4811 using less_eq_real_def tan_gt_zero by auto 4812 4813lemma cos_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> cos x = 1 / sqrt (1 + tan x ^ 2)" 4814 using cos_gt_zero_pi [of x] 4815 by (simp add: field_split_simps tan_def real_sqrt_divide abs_if split: if_split_asm) 4816 4817lemma sin_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> sin x = tan x / sqrt (1 + tan x ^ 2)" 4818 using cos_gt_zero [of "x"] cos_gt_zero [of "-x"] 4819 by (force simp: field_split_simps tan_def real_sqrt_divide abs_if split: if_split_asm) 4820 4821lemma tan_mono_le: "-(pi/2) < x \<Longrightarrow> x \<le> y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x \<le> tan y" 4822 using less_eq_real_def tan_monotone by auto 4823 4824lemma tan_mono_lt_eq: 4825 "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> -(pi/2) < y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x < tan y \<longleftrightarrow> x < y" 4826 using tan_monotone' by blast 4827 4828lemma tan_mono_le_eq: 4829 "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> -(pi/2) < y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x \<le> tan y \<longleftrightarrow> x \<le> y" 4830 by (meson tan_mono_le not_le tan_monotone) 4831 4832lemma tan_bound_pi2: "\<bar>x\<bar> < pi/4 \<Longrightarrow> \<bar>tan x\<bar> < 1" 4833 using tan_45 tan_monotone [of x "pi/4"] tan_monotone [of "-x" "pi/4"] 4834 by (auto simp: abs_if split: if_split_asm) 4835 4836lemma tan_cot: "tan(pi/2 - x) = inverse(tan x)" 4837 by (simp add: tan_def sin_diff cos_diff) 4838 4839 4840subsection \<open>Cotangent\<close> 4841 4842definition cot :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4843 where "cot = (\<lambda>x. cos x / sin x)" 4844 4845lemma cot_of_real: "of_real (cot x) = (cot (of_real x) :: 'a::{real_normed_field,banach})" 4846 by (simp add: cot_def sin_of_real cos_of_real) 4847 4848lemma cot_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cot z \<in> \<real>" 4849 for z :: "'a::{real_normed_field,banach}" 4850 by (simp add: cot_def) 4851 4852lemma cot_zero [simp]: "cot 0 = 0" 4853 by (simp add: cot_def) 4854 4855lemma cot_pi [simp]: "cot pi = 0" 4856 by (simp add: cot_def) 4857 4858lemma cot_npi [simp]: "cot (real n * pi) = 0" 4859 for n :: nat 4860 by (simp add: cot_def) 4861 4862lemma cot_minus [simp]: "cot (- x) = - cot x" 4863 by (simp add: cot_def) 4864 4865lemma cot_periodic [simp]: "cot (x + 2 * pi) = cot x" 4866 by (simp add: cot_def) 4867 4868lemma cot_altdef: "cot x = inverse (tan x)" 4869 by (simp add: cot_def tan_def) 4870 4871lemma tan_altdef: "tan x = inverse (cot x)" 4872 by (simp add: cot_def tan_def) 4873 4874lemma tan_cot': "tan (pi/2 - x) = cot x" 4875 by (simp add: tan_cot cot_altdef) 4876 4877lemma cot_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cot x" 4878 by (simp add: cot_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) 4879 4880lemma cot_less_zero: 4881 assumes lb: "- pi/2 < x" and "x < 0" 4882 shows "cot x < 0" 4883proof - 4884 have "0 < cot (- x)" 4885 using assms by (simp only: cot_gt_zero) 4886 then show ?thesis by simp 4887qed 4888 4889lemma DERIV_cot [simp]: "sin x \<noteq> 0 \<Longrightarrow> DERIV cot x :> -inverse ((sin x)\<^sup>2)" 4890 for x :: "'a::{real_normed_field,banach}" 4891 unfolding cot_def using cos_squared_eq[of x] 4892 by (auto intro!: derivative_eq_intros) (simp add: divide_inverse power2_eq_square) 4893 4894lemma isCont_cot: "sin x \<noteq> 0 \<Longrightarrow> isCont cot x" 4895 for x :: "'a::{real_normed_field,banach}" 4896 by (rule DERIV_cot [THEN DERIV_isCont]) 4897 4898lemma isCont_cot' [simp,continuous_intros]: 4899 "isCont f a \<Longrightarrow> sin (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. cot (f x)) a" 4900 for a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a" 4901 by (rule isCont_o2 [OF _ isCont_cot]) 4902 4903lemma tendsto_cot [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> sin a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. cot (f x)) \<longlongrightarrow> cot a) F" 4904 for f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4905 by (rule isCont_tendsto_compose [OF isCont_cot]) 4906 4907lemma continuous_cot: 4908 "continuous F f \<Longrightarrow> sin (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. cot (f x))" 4909 for f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4910 unfolding continuous_def by (rule tendsto_cot) 4911 4912lemma continuous_on_cot [continuous_intros]: 4913 fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4914 shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. sin (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. cot (f x))" 4915 unfolding continuous_on_def by (auto intro: tendsto_cot) 4916 4917lemma continuous_within_cot [continuous_intros]: 4918 fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4919 shows "continuous (at x within s) f \<Longrightarrow> sin (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. cot (f x))" 4920 unfolding continuous_within by (rule tendsto_cot) 4921 4922 4923subsection \<open>Inverse Trigonometric Functions\<close> 4924 4925definition arcsin :: "real \<Rightarrow> real" 4926 where "arcsin y = (THE x. -(pi/2) \<le> x \<and> x \<le> pi/2 \<and> sin x = y)" 4927 4928definition arccos :: "real \<Rightarrow> real" 4929 where "arccos y = (THE x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y)" 4930 4931definition arctan :: "real \<Rightarrow> real" 4932 where "arctan y = (THE x. -(pi/2) < x \<and> x < pi/2 \<and> tan x = y)" 4933 4934lemma arcsin: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi/2 \<and> sin (arcsin y) = y" 4935 unfolding arcsin_def by (rule theI' [OF sin_total]) 4936 4937lemma arcsin_pi: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi \<and> sin (arcsin y) = y" 4938 by (drule (1) arcsin) (force intro: order_trans) 4939 4940lemma sin_arcsin [simp]: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin (arcsin y) = y" 4941 by (blast dest: arcsin) 4942 4943lemma arcsin_bounded: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi/2" 4944 by (blast dest: arcsin) 4945 4946lemma arcsin_lbound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y" 4947 by (blast dest: arcsin) 4948 4949lemma arcsin_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2" 4950 by (blast dest: arcsin) 4951 4952lemma arcsin_lt_bounded: 4953 assumes "- 1 < y" "y < 1" 4954 shows "- (pi/2) < arcsin y \<and> arcsin y < pi/2" 4955proof - 4956 have "arcsin y \<noteq> pi/2" 4957 by (metis arcsin assms not_less not_less_iff_gr_or_eq sin_pi_half) 4958 moreover have "arcsin y \<noteq> - pi/2" 4959 by (metis arcsin assms minus_divide_left not_less not_less_iff_gr_or_eq sin_minus sin_pi_half) 4960 ultimately show ?thesis 4961 using arcsin_bounded [of y] assms by auto 4962qed 4963 4964lemma arcsin_sin: "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> arcsin (sin x) = x" 4965 unfolding arcsin_def 4966 using the1_equality [OF sin_total] by simp 4967 4968lemma arcsin_0 [simp]: "arcsin 0 = 0" 4969 using arcsin_sin [of 0] by simp 4970 4971lemma arcsin_1 [simp]: "arcsin 1 = pi/2" 4972 using arcsin_sin [of "pi/2"] by simp 4973 4974lemma arcsin_minus_1 [simp]: "arcsin (- 1) = - (pi/2)" 4975 using arcsin_sin [of "- pi/2"] by simp 4976 4977lemma arcsin_minus: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin (- x) = - arcsin x" 4978 by (metis (no_types, hide_lams) arcsin arcsin_sin minus_minus neg_le_iff_le sin_minus) 4979 4980lemma arcsin_eq_iff: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x = arcsin y \<longleftrightarrow> x = y" 4981 by (metis abs_le_iff arcsin minus_le_iff) 4982 4983lemma cos_arcsin_nonzero: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> cos (arcsin x) \<noteq> 0" 4984 using arcsin_lt_bounded cos_gt_zero_pi by force 4985 4986lemma arccos: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi \<and> cos (arccos y) = y" 4987 unfolding arccos_def by (rule theI' [OF cos_total]) 4988 4989lemma cos_arccos [simp]: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> cos (arccos y) = y" 4990 by (blast dest: arccos) 4991 4992lemma arccos_bounded: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi" 4993 by (blast dest: arccos) 4994 4995lemma arccos_lbound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y" 4996 by (blast dest: arccos) 4997 4998lemma arccos_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> pi" 4999 by (blast dest: arccos) 5000 5001lemma arccos_lt_bounded: 5002 assumes "- 1 < y" "y < 1" 5003 shows "0 < arccos y \<and> arccos y < pi" 5004proof - 5005 have "arccos y \<noteq> 0" 5006 by (metis (no_types) arccos assms(1) assms(2) cos_zero less_eq_real_def less_irrefl) 5007 moreover have "arccos y \<noteq> -pi" 5008 by (metis arccos assms(1) assms(2) cos_minus cos_pi not_less not_less_iff_gr_or_eq) 5009 ultimately show ?thesis 5010 using arccos_bounded [of y] assms 5011 by (metis arccos cos_pi not_less not_less_iff_gr_or_eq) 5012qed 5013 5014lemma arccos_cos: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> arccos (cos x) = x" 5015 by (auto simp: arccos_def intro!: the1_equality cos_total) 5016 5017lemma arccos_cos2: "x \<le> 0 \<Longrightarrow> - pi \<le> x \<Longrightarrow> arccos (cos x) = -x" 5018 by (auto simp: arccos_def intro!: the1_equality cos_total) 5019 5020lemma cos_arcsin: 5021 assumes "- 1 \<le> x" "x \<le> 1" 5022 shows "cos (arcsin x) = sqrt (1 - x\<^sup>2)" 5023proof (rule power2_eq_imp_eq) 5024 show "(cos (arcsin x))\<^sup>2 = (sqrt (1 - x\<^sup>2))\<^sup>2" 5025 by (simp add: square_le_1 assms cos_squared_eq) 5026 show "0 \<le> cos (arcsin x)" 5027 using arcsin assms cos_ge_zero by blast 5028 show "0 \<le> sqrt (1 - x\<^sup>2)" 5029 by (simp add: square_le_1 assms) 5030qed 5031 5032lemma sin_arccos: 5033 assumes "- 1 \<le> x" "x \<le> 1" 5034 shows "sin (arccos x) = sqrt (1 - x\<^sup>2)" 5035proof (rule power2_eq_imp_eq) 5036 show "(sin (arccos x))\<^sup>2 = (sqrt (1 - x\<^sup>2))\<^sup>2" 5037 by (simp add: square_le_1 assms sin_squared_eq) 5038 show "0 \<le> sin (arccos x)" 5039 by (simp add: arccos_bounded assms sin_ge_zero) 5040 show "0 \<le> sqrt (1 - x\<^sup>2)" 5041 by (simp add: square_le_1 assms) 5042qed 5043 5044lemma arccos_0 [simp]: "arccos 0 = pi/2" 5045 by (metis arccos_cos cos_gt_zero cos_pi cos_pi_half pi_gt_zero 5046 pi_half_ge_zero not_le not_zero_less_neg_numeral numeral_One) 5047 5048lemma arccos_1 [simp]: "arccos 1 = 0" 5049 using arccos_cos by force 5050 5051lemma arccos_minus_1 [simp]: "arccos (- 1) = pi" 5052 by (metis arccos_cos cos_pi order_refl pi_ge_zero) 5053 5054lemma arccos_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos (- x) = pi - arccos x" 5055 by (metis arccos_cos arccos_cos2 cos_minus_pi cos_total diff_le_0_iff_le le_add_same_cancel1 5056 minus_diff_eq uminus_add_conv_diff) 5057 5058corollary arccos_minus_abs: 5059 assumes "\<bar>x\<bar> \<le> 1" 5060 shows "arccos (- x) = pi - arccos x" 5061using assms by (simp add: arccos_minus) 5062 5063lemma sin_arccos_nonzero: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> sin (arccos x) \<noteq> 0" 5064 using arccos_lt_bounded sin_gt_zero by force 5065 5066lemma arctan: "- (pi/2) < arctan y \<and> arctan y < pi/2 \<and> tan (arctan y) = y" 5067 unfolding arctan_def by (rule theI' [OF tan_total]) 5068 5069lemma tan_arctan: "tan (arctan y) = y" 5070 by (simp add: arctan) 5071 5072lemma arctan_bounded: "- (pi/2) < arctan y \<and> arctan y < pi/2" 5073 by (auto simp only: arctan) 5074 5075lemma arctan_lbound: "- (pi/2) < arctan y" 5076 by (simp add: arctan) 5077 5078lemma arctan_ubound: "arctan y < pi/2" 5079 by (auto simp only: arctan) 5080 5081lemma arctan_unique: 5082 assumes "-(pi/2) < x" 5083 and "x < pi/2" 5084 and "tan x = y" 5085 shows "arctan y = x" 5086 using assms arctan [of y] tan_total [of y] by (fast elim: ex1E) 5087 5088lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x" 5089 by (rule arctan_unique) simp_all 5090 5091lemma arctan_zero_zero [simp]: "arctan 0 = 0" 5092 by (rule arctan_unique) simp_all 5093 5094lemma arctan_minus: "arctan (- x) = - arctan x" 5095 using arctan [of "x"] by (auto simp: arctan_unique) 5096 5097lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0" 5098 by (intro less_imp_neq [symmetric] cos_gt_zero_pi arctan_lbound arctan_ubound) 5099 5100lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)" 5101proof (rule power2_eq_imp_eq) 5102 have "0 < 1 + x\<^sup>2" by (simp add: add_pos_nonneg) 5103 show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp 5104 show "0 \<le> cos (arctan x)" 5105 by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound) 5106 have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1" 5107 unfolding tan_def by (simp add: distrib_left power_divide) 5108 then show "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2" 5109 using \<open>0 < 1 + x\<^sup>2\<close> by (simp add: arctan power_divide eq_divide_eq) 5110qed 5111 5112lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)" 5113 using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]] 5114 using tan_arctan [of x] unfolding tan_def cos_arctan 5115 by (simp add: eq_divide_eq) 5116 5117lemma tan_sec: "cos x \<noteq> 0 \<Longrightarrow> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2" 5118 for x :: "'a::{real_normed_field,banach,field}" 5119 by (simp add: add_divide_eq_iff inverse_eq_divide power2_eq_square tan_def) 5120 5121lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y" 5122 by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan) 5123 5124lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y" 5125 by (simp only: not_less [symmetric] arctan_less_iff) 5126 5127lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y" 5128 by (simp only: eq_iff [where 'a=real] arctan_le_iff) 5129 5130lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x" 5131 using arctan_less_iff [of 0 x] by simp 5132 5133lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0" 5134 using arctan_less_iff [of x 0] by simp 5135 5136lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x" 5137 using arctan_le_iff [of 0 x] by simp 5138 5139lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0" 5140 using arctan_le_iff [of x 0] by simp 5141 5142lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0" 5143 using arctan_eq_iff [of x 0] by simp 5144 5145lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin" 5146proof - 5147 have "continuous_on (sin ` {- pi/2 .. pi/2}) arcsin" 5148 by (rule continuous_on_inv) (auto intro: continuous_intros simp: arcsin_sin) 5149 also have "sin ` {- pi/2 .. pi/2} = {-1 .. 1}" 5150 proof safe 5151 fix x :: real 5152 assume "x \<in> {-1..1}" 5153 then show "x \<in> sin ` {- pi/2..pi/2}" 5154 using arcsin_lbound arcsin_ubound 5155 by (intro image_eqI[where x="arcsin x"]) auto 5156 qed simp 5157 finally show ?thesis . 5158qed 5159 5160lemma continuous_on_arcsin [continuous_intros]: 5161 "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. -1 \<le> f x \<and> f x \<le> 1) \<Longrightarrow> continuous_on s (\<lambda>x. arcsin (f x))" 5162 using continuous_on_compose[of s f, OF _ continuous_on_subset[OF continuous_on_arcsin']] 5163 by (auto simp: comp_def subset_eq) 5164 5165lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x" 5166 using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"] 5167 by (auto simp: continuous_on_eq_continuous_at subset_eq) 5168 5169lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos" 5170proof - 5171 have "continuous_on (cos ` {0 .. pi}) arccos" 5172 by (rule continuous_on_inv) (auto intro: continuous_intros simp: arccos_cos) 5173 also have "cos ` {0 .. pi} = {-1 .. 1}" 5174 proof safe 5175 fix x :: real 5176 assume "x \<in> {-1..1}" 5177 then show "x \<in> cos ` {0..pi}" 5178 using arccos_lbound arccos_ubound 5179 by (intro image_eqI[where x="arccos x"]) auto 5180 qed simp 5181 finally show ?thesis . 5182qed 5183 5184lemma continuous_on_arccos [continuous_intros]: 5185 "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. -1 \<le> f x \<and> f x \<le> 1) \<Longrightarrow> continuous_on s (\<lambda>x. arccos (f x))" 5186 using continuous_on_compose[of s f, OF _ continuous_on_subset[OF continuous_on_arccos']] 5187 by (auto simp: comp_def subset_eq) 5188 5189lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x" 5190 using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"] 5191 by (auto simp: continuous_on_eq_continuous_at subset_eq) 5192 5193lemma isCont_arctan: "isCont arctan x" 5194proof - 5195 obtain u where u: "- (pi / 2) < u" "u < arctan x" 5196 by (meson arctan arctan_less_iff linordered_field_no_lb) 5197 obtain v where v: "arctan x < v" "v < pi / 2" 5198 by (meson arctan_less_iff arctan_ubound linordered_field_no_ub) 5199 have "isCont arctan (tan (arctan x))" 5200 proof (rule isCont_inverse_function2 [of u "arctan x" v]) 5201 show "\<And>z. \<lbrakk>u \<le> z; z \<le> v\<rbrakk> \<Longrightarrow> arctan (tan z) = z" 5202 using arctan_unique u(1) v(2) by auto 5203 then show "\<And>z. \<lbrakk>u \<le> z; z \<le> v\<rbrakk> \<Longrightarrow> isCont tan z" 5204 by (metis arctan cos_gt_zero_pi isCont_tan less_irrefl) 5205 qed (use u v in auto) 5206 then show ?thesis 5207 by (simp add: arctan) 5208qed 5209 5210lemma tendsto_arctan [tendsto_intros]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) \<longlongrightarrow> arctan x) F" 5211 by (rule isCont_tendsto_compose [OF isCont_arctan]) 5212 5213lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))" 5214 unfolding continuous_def by (rule tendsto_arctan) 5215 5216lemma continuous_on_arctan [continuous_intros]: 5217 "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))" 5218 unfolding continuous_on_def by (auto intro: tendsto_arctan) 5219 5220lemma DERIV_arcsin: 5221 assumes "- 1 < x" "x < 1" 5222 shows "DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))" 5223proof (rule DERIV_inverse_function) 5224 show "(sin has_real_derivative sqrt (1 - x\<^sup>2)) (at (arcsin x))" 5225 by (rule derivative_eq_intros | use assms cos_arcsin in force)+ 5226 show "sqrt (1 - x\<^sup>2) \<noteq> 0" 5227 using abs_square_eq_1 assms by force 5228qed (use assms isCont_arcsin in auto) 5229 5230lemma DERIV_arccos: 5231 assumes "- 1 < x" "x < 1" 5232 shows "DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))" 5233proof (rule DERIV_inverse_function) 5234 show "(cos has_real_derivative - sqrt (1 - x\<^sup>2)) (at (arccos x))" 5235 by (rule derivative_eq_intros | use assms sin_arccos in force)+ 5236 show "- sqrt (1 - x\<^sup>2) \<noteq> 0" 5237 using abs_square_eq_1 assms by force 5238qed (use assms isCont_arccos in auto) 5239 5240lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)" 5241proof (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"]) 5242 show "(tan has_real_derivative 1 + x\<^sup>2) (at (arctan x))" 5243 apply (rule derivative_eq_intros | simp)+ 5244 by (metis arctan cos_arctan_not_zero power_inverse tan_sec) 5245 show "\<And>y. \<lbrakk>x - 1 < y; y < x + 1\<rbrakk> \<Longrightarrow> tan (arctan y) = y" 5246 using tan_arctan by blast 5247 show "1 + x\<^sup>2 \<noteq> 0" 5248 by (metis power_one sum_power2_eq_zero_iff zero_neq_one) 5249qed (use isCont_arctan in auto) 5250 5251declare 5252 DERIV_arcsin[THEN DERIV_chain2, derivative_intros] 5253 DERIV_arcsin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 5254 DERIV_arccos[THEN DERIV_chain2, derivative_intros] 5255 DERIV_arccos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 5256 DERIV_arctan[THEN DERIV_chain2, derivative_intros] 5257 DERIV_arctan[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 5258 5259lemmas has_derivative_arctan[derivative_intros] = DERIV_arctan[THEN DERIV_compose_FDERIV] 5260 and has_derivative_arccos[derivative_intros] = DERIV_arccos[THEN DERIV_compose_FDERIV] 5261 and has_derivative_arcsin[derivative_intros] = DERIV_arcsin[THEN DERIV_compose_FDERIV] 5262 5263lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- (pi/2)))" 5264 by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. - pi/2 < x \<and> x < pi/2" and P="\<lambda>x. True" and g=arctan]) 5265 (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1 5266 intro!: tan_monotone exI[of _ "pi/2"]) 5267 5268lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))" 5269 by (rule filterlim_at_top_at_left[where Q="\<lambda>x. - pi/2 < x \<and> x < pi/2" and P="\<lambda>x. True" and g=arctan]) 5270 (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1 5271 intro!: tan_monotone exI[of _ "pi/2"]) 5272 5273lemma tendsto_arctan_at_top: "(arctan \<longlongrightarrow> (pi/2)) at_top" 5274proof (rule tendstoI) 5275 fix e :: real 5276 assume "0 < e" 5277 define y where "y = pi/2 - min (pi/2) e" 5278 then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y" 5279 using \<open>0 < e\<close> by auto 5280 show "eventually (\<lambda>x. dist (arctan x) (pi/2) < e) at_top" 5281 proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI) 5282 fix x 5283 assume "tan y < x" 5284 then have "arctan (tan y) < arctan x" 5285 by (simp add: arctan_less_iff) 5286 with y have "y < arctan x" 5287 by (subst (asm) arctan_tan) simp_all 5288 with arctan_ubound[of x, arith] y \<open>0 < e\<close> 5289 show "dist (arctan x) (pi/2) < e" 5290 by (simp add: dist_real_def) 5291 qed 5292qed 5293 5294lemma tendsto_arctan_at_bot: "(arctan \<longlongrightarrow> - (pi/2)) at_bot" 5295 unfolding filterlim_at_bot_mirror arctan_minus 5296 by (intro tendsto_minus tendsto_arctan_at_top) 5297 5298 5299subsection \<open>Prove Totality of the Trigonometric Functions\<close> 5300 5301lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y" 5302 by (simp add: abs_le_iff) 5303 5304lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)" 5305 by (simp add: sin_arccos abs_le_iff) 5306 5307lemma sin_mono_less_eq: 5308 "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x < sin y \<longleftrightarrow> x < y" 5309 by (metis not_less_iff_gr_or_eq sin_monotone_2pi) 5310 5311lemma sin_mono_le_eq: 5312 "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x \<le> sin y \<longleftrightarrow> x \<le> y" 5313 by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le) 5314 5315lemma sin_inj_pi: 5316 "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x = sin y \<Longrightarrow> x = y" 5317 by (metis arcsin_sin) 5318 5319lemma arcsin_le_iff: 5320 assumes "x \<ge> -1" "x \<le> 1" "y \<ge> -pi/2" "y \<le> pi/2" 5321 shows "arcsin x \<le> y \<longleftrightarrow> x \<le> sin y" 5322proof - 5323 have "arcsin x \<le> y \<longleftrightarrow> sin (arcsin x) \<le> sin y" 5324 using arcsin_bounded[of x] assms by (subst sin_mono_le_eq) auto 5325 also from assms have "sin (arcsin x) = x" by simp 5326 finally show ?thesis . 5327qed 5328 5329lemma le_arcsin_iff: 5330 assumes "x \<ge> -1" "x \<le> 1" "y \<ge> -pi/2" "y \<le> pi/2" 5331 shows "arcsin x \<ge> y \<longleftrightarrow> x \<ge> sin y" 5332proof - 5333 have "arcsin x \<ge> y \<longleftrightarrow> sin (arcsin x) \<ge> sin y" 5334 using arcsin_bounded[of x] assms by (subst sin_mono_le_eq) auto 5335 also from assms have "sin (arcsin x) = x" by simp 5336 finally show ?thesis . 5337qed 5338 5339lemma cos_mono_less_eq: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x < cos y \<longleftrightarrow> y < x" 5340 by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear) 5341 5342lemma cos_mono_le_eq: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x \<le> cos y \<longleftrightarrow> y \<le> x" 5343 by (metis arccos_cos cos_monotone_0_pi_le eq_iff linear) 5344 5345lemma cos_inj_pi: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x = cos y \<Longrightarrow> x = y" 5346 by (metis arccos_cos) 5347 5348lemma arccos_le_pi2: "\<lbrakk>0 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi/2" 5349 by (metis (mono_tags) arccos_0 arccos cos_le_one cos_monotone_0_pi_le 5350 cos_pi cos_pi_half pi_half_ge_zero antisym_conv less_eq_neg_nonpos linear minus_minus order.trans order_refl) 5351 5352lemma sincos_total_pi_half: 5353 assumes "0 \<le> x" "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1" 5354 shows "\<exists>t. 0 \<le> t \<and> t \<le> pi/2 \<and> x = cos t \<and> y = sin t" 5355proof - 5356 have x1: "x \<le> 1" 5357 using assms by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2) 5358 with assms have *: "0 \<le> arccos x" "cos (arccos x) = x" 5359 by (auto simp: arccos) 5360 from assms have "y = sqrt (1 - x\<^sup>2)" 5361 by (metis abs_of_nonneg add.commute add_diff_cancel real_sqrt_abs) 5362 with x1 * assms arccos_le_pi2 [of x] show ?thesis 5363 by (rule_tac x="arccos x" in exI) (auto simp: sin_arccos) 5364qed 5365 5366lemma sincos_total_pi: 5367 assumes "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1" 5368 shows "\<exists>t. 0 \<le> t \<and> t \<le> pi \<and> x = cos t \<and> y = sin t" 5369proof (cases rule: le_cases [of 0 x]) 5370 case le 5371 from sincos_total_pi_half [OF le] show ?thesis 5372 by (metis pi_ge_two pi_half_le_two add.commute add_le_cancel_left add_mono assms) 5373next 5374 case ge 5375 then have "0 \<le> -x" 5376 by simp 5377 then obtain t where t: "t\<ge>0" "t \<le> pi/2" "-x = cos t" "y = sin t" 5378 using sincos_total_pi_half assms 5379 by auto (metis \<open>0 \<le> - x\<close> power2_minus) 5380 show ?thesis 5381 by (rule exI [where x = "pi -t"]) (use t in auto) 5382qed 5383 5384lemma sincos_total_2pi_le: 5385 assumes "x\<^sup>2 + y\<^sup>2 = 1" 5386 shows "\<exists>t. 0 \<le> t \<and> t \<le> 2 * pi \<and> x = cos t \<and> y = sin t" 5387proof (cases rule: le_cases [of 0 y]) 5388 case le 5389 from sincos_total_pi [OF le] show ?thesis 5390 by (metis assms le_add_same_cancel1 mult.commute mult_2_right order.trans) 5391next 5392 case ge 5393 then have "0 \<le> -y" 5394 by simp 5395 then obtain t where t: "t\<ge>0" "t \<le> pi" "x = cos t" "-y = sin t" 5396 using sincos_total_pi assms 5397 by auto (metis \<open>0 \<le> - y\<close> power2_minus) 5398 show ?thesis 5399 by (rule exI [where x = "2 * pi - t"]) (use t in auto) 5400qed 5401 5402lemma sincos_total_2pi: 5403 assumes "x\<^sup>2 + y\<^sup>2 = 1" 5404 obtains t where "0 \<le> t" "t < 2*pi" "x = cos t" "y = sin t" 5405proof - 5406 from sincos_total_2pi_le [OF assms] 5407 obtain t where t: "0 \<le> t" "t \<le> 2*pi" "x = cos t" "y = sin t" 5408 by blast 5409 show ?thesis 5410 by (cases "t = 2 * pi") (use t that in \<open>force+\<close>) 5411qed 5412 5413lemma arcsin_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x < arcsin y \<longleftrightarrow> x < y" 5414 by (rule trans [OF sin_mono_less_eq [symmetric]]) (use arcsin_ubound arcsin_lbound in auto) 5415 5416lemma arcsin_le_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y \<longleftrightarrow> x \<le> y" 5417 using arcsin_less_mono not_le by blast 5418 5419lemma arcsin_less_arcsin: "- 1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x < arcsin y" 5420 using arcsin_less_mono by auto 5421 5422lemma arcsin_le_arcsin: "- 1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y" 5423 using arcsin_le_mono by auto 5424 5425lemma arccos_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x < arccos y \<longleftrightarrow> y < x" 5426 by (rule trans [OF cos_mono_less_eq [symmetric]]) (use arccos_ubound arccos_lbound in auto) 5427 5428lemma arccos_le_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x \<le> arccos y \<longleftrightarrow> y \<le> x" 5429 using arccos_less_mono [of y x] by (simp add: not_le [symmetric]) 5430 5431lemma arccos_less_arccos: "- 1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y < arccos x" 5432 using arccos_less_mono by auto 5433 5434lemma arccos_le_arccos: "- 1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> arccos x" 5435 using arccos_le_mono by auto 5436 5437lemma arccos_eq_iff: "\<bar>x\<bar> \<le> 1 \<and> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x = arccos y \<longleftrightarrow> x = y" 5438 using cos_arccos_abs by fastforce 5439 5440 5441lemma arccos_cos_eq_abs: 5442 assumes "\<bar>\<theta>\<bar> \<le> pi" 5443 shows "arccos (cos \<theta>) = \<bar>\<theta>\<bar>" 5444 unfolding arccos_def 5445proof (intro the_equality conjI; clarify?) 5446 show "cos \<bar>\<theta>\<bar> = cos \<theta>" 5447 by (simp add: abs_real_def) 5448 show "x = \<bar>\<theta>\<bar>" if "cos x = cos \<theta>" "0 \<le> x" "x \<le> pi" for x 5449 by (simp add: \<open>cos \<bar>\<theta>\<bar> = cos \<theta>\<close> assms cos_inj_pi that) 5450qed (use assms in auto) 5451 5452lemma arccos_cos_eq_abs_2pi: 5453 obtains k where "arccos (cos \<theta>) = \<bar>\<theta> - of_int k * (2 * pi)\<bar>" 5454proof - 5455 define k where "k \<equiv> \<lfloor>(\<theta> + pi) / (2 * pi)\<rfloor>" 5456 have lepi: "\<bar>\<theta> - of_int k * (2 * pi)\<bar> \<le> pi" 5457 using floor_divide_lower [of "2*pi" "\<theta> + pi"] floor_divide_upper [of "2*pi" "\<theta> + pi"] 5458 by (auto simp: k_def abs_if algebra_simps) 5459 have "arccos (cos \<theta>) = arccos (cos (\<theta> - of_int k * (2 * pi)))" 5460 using cos_int_2pin sin_int_2pin by (simp add: cos_diff mult.commute) 5461 also have "\<dots> = \<bar>\<theta> - of_int k * (2 * pi)\<bar>" 5462 using arccos_cos_eq_abs lepi by blast 5463 finally show ?thesis 5464 using that by metis 5465qed 5466 5467lemma cos_limit_1: 5468 assumes "(\<lambda>j. cos (\<theta> j)) \<longlonglongrightarrow> 1" 5469 shows "\<exists>k. (\<lambda>j. \<theta> j - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> 0" 5470proof - 5471 have "\<forall>\<^sub>F j in sequentially. cos (\<theta> j) \<in> {- 1..1}" 5472 by auto 5473 then have "(\<lambda>j. arccos (cos (\<theta> j))) \<longlonglongrightarrow> arccos 1" 5474 using continuous_on_tendsto_compose [OF continuous_on_arccos' assms] by auto 5475 moreover have "\<And>j. \<exists>k. arccos (cos (\<theta> j)) = \<bar>\<theta> j - of_int k * (2 * pi)\<bar>" 5476 using arccos_cos_eq_abs_2pi by metis 5477 then have "\<exists>k. \<forall>j. arccos (cos (\<theta> j)) = \<bar>\<theta> j - of_int (k j) * (2 * pi)\<bar>" 5478 by metis 5479 ultimately have "\<exists>k. (\<lambda>j. \<bar>\<theta> j - of_int (k j) * (2 * pi)\<bar>) \<longlonglongrightarrow> 0" 5480 by auto 5481 then show ?thesis 5482 by (simp add: tendsto_rabs_zero_iff) 5483qed 5484 5485lemma cos_diff_limit_1: 5486 assumes "(\<lambda>j. cos (\<theta> j - \<Theta>)) \<longlonglongrightarrow> 1" 5487 obtains k where "(\<lambda>j. \<theta> j - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> \<Theta>" 5488proof - 5489 obtain k where "(\<lambda>j. (\<theta> j - \<Theta>) - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> 0" 5490 using cos_limit_1 [OF assms] by auto 5491 then have "(\<lambda>j. \<Theta> + ((\<theta> j - \<Theta>) - of_int (k j) * (2 * pi))) \<longlonglongrightarrow> \<Theta> + 0" 5492 by (rule tendsto_add [OF tendsto_const]) 5493 with that show ?thesis 5494 by auto 5495qed 5496 5497subsection \<open>Machin's formula\<close> 5498 5499lemma arctan_one: "arctan 1 = pi / 4" 5500 by (rule arctan_unique) (simp_all add: tan_45 m2pi_less_pi) 5501 5502lemma tan_total_pi4: 5503 assumes "\<bar>x\<bar> < 1" 5504 shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x" 5505proof 5506 show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x" 5507 unfolding arctan_one [symmetric] arctan_minus [symmetric] 5508 unfolding arctan_less_iff 5509 using assms by (auto simp: arctan) 5510qed 5511 5512lemma arctan_add: 5513 assumes "\<bar>x\<bar> \<le> 1" "\<bar>y\<bar> < 1" 5514 shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))" 5515proof (rule arctan_unique [symmetric]) 5516 have "- (pi / 4) \<le> arctan x" "- (pi / 4) < arctan y" 5517 unfolding arctan_one [symmetric] arctan_minus [symmetric] 5518 unfolding arctan_le_iff arctan_less_iff 5519 using assms by auto 5520 from add_le_less_mono [OF this] show 1: "- (pi/2) < arctan x + arctan y" 5521 by simp 5522 have "arctan x \<le> pi / 4" "arctan y < pi / 4" 5523 unfolding arctan_one [symmetric] 5524 unfolding arctan_le_iff arctan_less_iff 5525 using assms by auto 5526 from add_le_less_mono [OF this] show 2: "arctan x + arctan y < pi/2" 5527 by simp 5528 show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)" 5529 using cos_gt_zero_pi [OF 1 2] by (simp add: arctan tan_add) 5530qed 5531 5532lemma arctan_double: "\<bar>x\<bar> < 1 \<Longrightarrow> 2 * arctan x = arctan ((2 * x) / (1 - x\<^sup>2))" 5533 by (metis arctan_add linear mult_2 not_less power2_eq_square) 5534 5535theorem machin: "pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)" 5536proof - 5537 have "\<bar>1 / 5\<bar> < (1 :: real)" 5538 by auto 5539 from arctan_add[OF less_imp_le[OF this] this] have "2 * arctan (1 / 5) = arctan (5 / 12)" 5540 by auto 5541 moreover 5542 have "\<bar>5 / 12\<bar> < (1 :: real)" 5543 by auto 5544 from arctan_add[OF less_imp_le[OF this] this] have "2 * arctan (5 / 12) = arctan (120 / 119)" 5545 by auto 5546 moreover 5547 have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" 5548 by auto 5549 from arctan_add[OF this] have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" 5550 by auto 5551 ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" 5552 by auto 5553 then show ?thesis 5554 unfolding arctan_one by algebra 5555qed 5556 5557lemma machin_Euler: "5 * arctan (1 / 7) + 2 * arctan (3 / 79) = pi / 4" 5558proof - 5559 have 17: "\<bar>1 / 7\<bar> < (1 :: real)" by auto 5560 with arctan_double have "2 * arctan (1 / 7) = arctan (7 / 24)" 5561 by simp (simp add: field_simps) 5562 moreover 5563 have "\<bar>7 / 24\<bar> < (1 :: real)" by auto 5564 with arctan_double have "2 * arctan (7 / 24) = arctan (336 / 527)" 5565 by simp (simp add: field_simps) 5566 moreover 5567 have "\<bar>336 / 527\<bar> < (1 :: real)" by auto 5568 from arctan_add[OF less_imp_le[OF 17] this] 5569 have "arctan(1/7) + arctan (336 / 527) = arctan (2879 / 3353)" 5570 by auto 5571 ultimately have I: "5 * arctan (1 / 7) = arctan (2879 / 3353)" by auto 5572 have 379: "\<bar>3 / 79\<bar> < (1 :: real)" by auto 5573 with arctan_double have II: "2 * arctan (3 / 79) = arctan (237 / 3116)" 5574 by simp (simp add: field_simps) 5575 have *: "\<bar>2879 / 3353\<bar> < (1 :: real)" by auto 5576 have "\<bar>237 / 3116\<bar> < (1 :: real)" by auto 5577 from arctan_add[OF less_imp_le[OF *] this] have "arctan (2879/3353) + arctan (237/3116) = pi/4" 5578 by (simp add: arctan_one) 5579 with I II show ?thesis by auto 5580qed 5581 5582(*But could also prove MACHIN_GAUSS: 5583 12 * arctan(1/18) + 8 * arctan(1/57) - 5 * arctan(1/239) = pi/4*) 5584 5585 5586subsection \<open>Introducing the inverse tangent power series\<close> 5587 5588lemma monoseq_arctan_series: 5589 fixes x :: real 5590 assumes "\<bar>x\<bar> \<le> 1" 5591 shows "monoseq (\<lambda>n. 1 / real (n * 2 + 1) * x^(n * 2 + 1))" 5592 (is "monoseq ?a") 5593proof (cases "x = 0") 5594 case True 5595 then show ?thesis by (auto simp: monoseq_def) 5596next 5597 case False 5598 have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" 5599 using assms by auto 5600 show "monoseq ?a" 5601 proof - 5602 have mono: "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le> 5603 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" 5604 if "0 \<le> x" and "x \<le> 1" for n and x :: real 5605 proof (rule mult_mono) 5606 show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))" 5607 by (rule frac_le) simp_all 5608 show "0 \<le> 1 / real (Suc (n * 2))" 5609 by auto 5610 show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)" 5611 by (rule power_decreasing) (simp_all add: \<open>0 \<le> x\<close> \<open>x \<le> 1\<close>) 5612 show "0 \<le> x ^ Suc (Suc n * 2)" 5613 by (rule zero_le_power) (simp add: \<open>0 \<le> x\<close>) 5614 qed 5615 show ?thesis 5616 proof (cases "0 \<le> x") 5617 case True 5618 from mono[OF this \<open>x \<le> 1\<close>, THEN allI] 5619 show ?thesis 5620 unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI2) 5621 next 5622 case False 5623 then have "0 \<le> - x" and "- x \<le> 1" 5624 using \<open>-1 \<le> x\<close> by auto 5625 from mono[OF this] 5626 have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge> 5627 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" for n 5628 using \<open>0 \<le> -x\<close> by auto 5629 then show ?thesis 5630 unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI]) 5631 qed 5632 qed 5633qed 5634 5635lemma zeroseq_arctan_series: 5636 fixes x :: real 5637 assumes "\<bar>x\<bar> \<le> 1" 5638 shows "(\<lambda>n. 1 / real (n * 2 + 1) * x^(n * 2 + 1)) \<longlonglongrightarrow> 0" 5639 (is "?a \<longlonglongrightarrow> 0") 5640proof (cases "x = 0") 5641 case True 5642 then show ?thesis by simp 5643next 5644 case False 5645 have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" 5646 using assms by auto 5647 show "?a \<longlonglongrightarrow> 0" 5648 proof (cases "\<bar>x\<bar> < 1") 5649 case True 5650 then have "norm x < 1" by auto 5651 from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF \<open>norm x < 1\<close>, THEN LIMSEQ_Suc]] 5652 have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) \<longlonglongrightarrow> 0" 5653 unfolding inverse_eq_divide Suc_eq_plus1 by simp 5654 then show ?thesis 5655 using pos2 by (rule LIMSEQ_linear) 5656 next 5657 case False 5658 then have "x = -1 \<or> x = 1" 5659 using \<open>\<bar>x\<bar> \<le> 1\<close> by auto 5660 then have n_eq: "\<And> n. x ^ (n * 2 + 1) = x" 5661 unfolding One_nat_def by auto 5662 from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]] 5663 show ?thesis 5664 unfolding n_eq Suc_eq_plus1 by auto 5665 qed 5666qed 5667 5668lemma summable_arctan_series: 5669 fixes n :: nat 5670 assumes "\<bar>x\<bar> \<le> 1" 5671 shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" 5672 (is "summable (?c x)") 5673 by (rule summable_Leibniz(1), 5674 rule zeroseq_arctan_series[OF assms], 5675 rule monoseq_arctan_series[OF assms]) 5676 5677lemma DERIV_arctan_series: 5678 assumes "\<bar>x\<bar> < 1" 5679 shows "DERIV (\<lambda>x'. \<Sum>k. (-1)^k * (1 / real (k * 2 + 1) * x' ^ (k * 2 + 1))) x :> 5680 (\<Sum>k. (-1)^k * x^(k * 2))" 5681 (is "DERIV ?arctan _ :> ?Int") 5682proof - 5683 let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0" 5684 5685 have n_even: "even n \<Longrightarrow> 2 * (n div 2) = n" for n :: nat 5686 by presburger 5687 then have if_eq: "?f n * real (Suc n) * x'^n = 5688 (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)" 5689 for n x' 5690 by auto 5691 5692 have summable_Integral: "summable (\<lambda> n. (- 1) ^ n * x^(2 * n))" if "\<bar>x\<bar> < 1" for x :: real 5693 proof - 5694 from that have "x\<^sup>2 < 1" 5695 by (simp add: abs_square_less_1) 5696 have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)" 5697 by (rule summable_Leibniz(1)) 5698 (auto intro!: LIMSEQ_realpow_zero monoseq_realpow \<open>x\<^sup>2 < 1\<close> order_less_imp_le[OF \<open>x\<^sup>2 < 1\<close>]) 5699 then show ?thesis 5700 by (simp only: power_mult) 5701 qed 5702 5703 have sums_even: "(sums) f = (sums) (\<lambda> n. if even n then f (n div 2) else 0)" 5704 for f :: "nat \<Rightarrow> real" 5705 proof - 5706 have "f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x" for x :: real 5707 proof 5708 assume "f sums x" 5709 from sums_if[OF sums_zero this] show "(\<lambda>n. if even n then f (n div 2) else 0) sums x" 5710 by auto 5711 next 5712 assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x" 5713 from LIMSEQ_linear[OF this[simplified sums_def] pos2, simplified sum_split_even_odd[simplified mult.commute]] 5714 show "f sums x" 5715 unfolding sums_def by auto 5716 qed 5717 then show ?thesis .. 5718 qed 5719 5720 have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int" 5721 unfolding if_eq mult.commute[of _ 2] 5722 suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * x ^ (2 * n)", symmetric] 5723 by auto 5724 5725 have arctan_eq: "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x" for x 5726 proof - 5727 have if_eq': "\<And>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n = 5728 (if even n then (- 1) ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)" 5729 using n_even by auto 5730 have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)" 5731 by auto 5732 then show ?thesis 5733 unfolding if_eq' idx_eq suminf_def 5734 sums_even[of "\<lambda> n. (- 1) ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric] 5735 by auto 5736 qed 5737 5738 have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum>n. ?f n * real (Suc n) * x^n)" 5739 proof (rule DERIV_power_series') 5740 show "x \<in> {- 1 <..< 1}" 5741 using \<open>\<bar> x \<bar> < 1\<close> by auto 5742 show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" 5743 if x'_bounds: "x' \<in> {- 1 <..< 1}" for x' :: real 5744 proof - 5745 from that have "\<bar>x'\<bar> < 1" by auto 5746 then show ?thesis 5747 using that sums_summable sums_if [OF sums_0 [of "\<lambda>x. 0"] summable_sums [OF summable_Integral]] 5748 by (auto simp add: if_distrib [of "\<lambda>x. x * y" for y] cong: if_cong) 5749 qed 5750 qed auto 5751 then show ?thesis 5752 by (simp only: Int_eq arctan_eq) 5753qed 5754 5755lemma arctan_series: 5756 assumes "\<bar>x\<bar> \<le> 1" 5757 shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))" 5758 (is "_ = suminf (\<lambda> n. ?c x n)") 5759proof - 5760 let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)" 5761 5762 have DERIV_arctan_suminf: "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" 5763 if "0 < r" and "r < 1" and "\<bar>x\<bar> < r" for r x :: real 5764 proof (rule DERIV_arctan_series) 5765 from that show "\<bar>x\<bar> < 1" 5766 using \<open>r < 1\<close> and \<open>\<bar>x\<bar> < r\<close> by auto 5767 qed 5768 5769 { 5770 fix x :: real 5771 assume "\<bar>x\<bar> \<le> 1" 5772 note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] 5773 } note arctan_series_borders = this 5774 5775 have when_less_one: "arctan x = (\<Sum>k. ?c x k)" if "\<bar>x\<bar> < 1" for x :: real 5776 proof - 5777 obtain r where "\<bar>x\<bar> < r" and "r < 1" 5778 using dense[OF \<open>\<bar>x\<bar> < 1\<close>] by blast 5779 then have "0 < r" and "- r < x" and "x < r" by auto 5780 5781 have suminf_eq_arctan_bounded: "suminf (?c x) - arctan x = suminf (?c a) - arctan a" 5782 if "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b" for x a b 5783 proof - 5784 from that have "\<bar>x\<bar> < r" by auto 5785 show "suminf (?c x) - arctan x = suminf (?c a) - arctan a" 5786 proof (rule DERIV_isconst2[of "a" "b"]) 5787 show "a < b" and "a \<le> x" and "x \<le> b" 5788 using \<open>a < b\<close> \<open>a \<le> x\<close> \<open>x \<le> b\<close> by auto 5789 have "\<forall>x. - r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0" 5790 proof (rule allI, rule impI) 5791 fix x 5792 assume "-r < x \<and> x < r" 5793 then have "\<bar>x\<bar> < r" by auto 5794 with \<open>r < 1\<close> have "\<bar>x\<bar> < 1" by auto 5795 have "\<bar>- (x\<^sup>2)\<bar> < 1" using abs_square_less_1 \<open>\<bar>x\<bar> < 1\<close> by auto 5796 then have "(\<lambda>n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))" 5797 unfolding real_norm_def[symmetric] by (rule geometric_sums) 5798 then have "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))" 5799 unfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2] by auto 5800 then have suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)" 5801 using sums_unique unfolding inverse_eq_divide by auto 5802 have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))" 5803 unfolding suminf_c'_eq_geom 5804 by (rule DERIV_arctan_suminf[OF \<open>0 < r\<close> \<open>r < 1\<close> \<open>\<bar>x\<bar> < r\<close>]) 5805 from DERIV_diff [OF this DERIV_arctan] show "DERIV (\<lambda>x. suminf (?c x) - arctan x) x :> 0" 5806 by auto 5807 qed 5808 then have DERIV_in_rball: "\<forall>y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda>x. suminf (?c x) - arctan x) y :> 0" 5809 using \<open>-r < a\<close> \<open>b < r\<close> by auto 5810 then show "\<And>y. \<lbrakk>a < y; y < b\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. suminf (?c x) - arctan x) y :> 0" 5811 using \<open>\<bar>x\<bar> < r\<close> by auto 5812 show "continuous_on {a..b} (\<lambda>x. suminf (?c x) - arctan x)" 5813 using DERIV_in_rball DERIV_atLeastAtMost_imp_continuous_on by blast 5814 qed 5815 qed 5816 5817 have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0" 5818 unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero 5819 by auto 5820 5821 have "suminf (?c x) - arctan x = 0" 5822 proof (cases "x = 0") 5823 case True 5824 then show ?thesis 5825 using suminf_arctan_zero by auto 5826 next 5827 case False 5828 then have "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" 5829 by auto 5830 have "suminf (?c (- \<bar>x\<bar>)) - arctan (- \<bar>x\<bar>) = suminf (?c 0) - arctan 0" 5831 by (rule suminf_eq_arctan_bounded[where x1=0 and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>", symmetric]) 5832 (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>-\<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less) 5833 moreover 5834 have "suminf (?c x) - arctan x = suminf (?c (- \<bar>x\<bar>)) - arctan (- \<bar>x\<bar>)" 5835 by (rule suminf_eq_arctan_bounded[where x1=x and a1="- \<bar>x\<bar>" and b1="\<bar>x\<bar>"]) 5836 (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>- \<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less) 5837 ultimately show ?thesis 5838 using suminf_arctan_zero by auto 5839 qed 5840 then show ?thesis by auto 5841 qed 5842 5843 show "arctan x = suminf (\<lambda>n. ?c x n)" 5844 proof (cases "\<bar>x\<bar> < 1") 5845 case True 5846 then show ?thesis by (rule when_less_one) 5847 next 5848 case False 5849 then have "\<bar>x\<bar> = 1" using \<open>\<bar>x\<bar> \<le> 1\<close> by auto 5850 let ?a = "\<lambda>x n. \<bar>1 / real (n * 2 + 1) * x^(n * 2 + 1)\<bar>" 5851 let ?diff = "\<lambda>x n. \<bar>arctan x - (\<Sum>i<n. ?c x i)\<bar>" 5852 have "?diff 1 n \<le> ?a 1 n" for n :: nat 5853 proof - 5854 have "0 < (1 :: real)" by auto 5855 moreover 5856 have "?diff x n \<le> ?a x n" if "0 < x" and "x < 1" for x :: real 5857 proof - 5858 from that have "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" 5859 by auto 5860 from \<open>0 < x\<close> have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)" 5861 by auto 5862 note bounds = mp[OF arctan_series_borders(2)[OF \<open>\<bar>x\<bar> \<le> 1\<close>] this, unfolded when_less_one[OF \<open>\<bar>x\<bar> < 1\<close>, symmetric], THEN spec] 5863 have "0 < 1 / real (n*2+1) * x^(n*2+1)" 5864 by (rule mult_pos_pos) (simp_all only: zero_less_power[OF \<open>0 < x\<close>], auto) 5865 then have a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)" 5866 by (rule abs_of_pos) 5867 show ?thesis 5868 proof (cases "even n") 5869 case True 5870 then have sgn_pos: "(-1)^n = (1::real)" by auto 5871 from \<open>even n\<close> obtain m where "n = 2 * m" .. 5872 then have "2 * m = n" .. 5873 from bounds[of m, unfolded this atLeastAtMost_iff] 5874 have "\<bar>arctan x - (\<Sum>i<n. (?c x i))\<bar> \<le> (\<Sum>i<n + 1. (?c x i)) - (\<Sum>i<n. (?c x i))" 5875 by auto 5876 also have "\<dots> = ?c x n" by auto 5877 also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto 5878 finally show ?thesis . 5879 next 5880 case False 5881 then have sgn_neg: "(-1)^n = (-1::real)" by auto 5882 from \<open>odd n\<close> obtain m where "n = 2 * m + 1" .. 5883 then have m_def: "2 * m + 1 = n" .. 5884 then have m_plus: "2 * (m + 1) = n + 1" by auto 5885 from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2] 5886 have "\<bar>arctan x - (\<Sum>i<n. (?c x i))\<bar> \<le> (\<Sum>i<n. (?c x i)) - (\<Sum>i<n+1. (?c x i))" by auto 5887 also have "\<dots> = - ?c x n" by auto 5888 also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto 5889 finally show ?thesis . 5890 qed 5891 qed 5892 hence "\<forall>x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto 5893 moreover have "isCont (\<lambda> x. ?a x n - ?diff x n) x" for x 5894 unfolding diff_conv_add_uminus divide_inverse 5895 by (auto intro!: isCont_add isCont_rabs continuous_ident isCont_minus isCont_arctan 5896 continuous_at_within_inverse isCont_mult isCont_power continuous_const isCont_sum 5897 simp del: add_uminus_conv_diff) 5898 ultimately have "0 \<le> ?a 1 n - ?diff 1 n" 5899 by (rule LIM_less_bound) 5900 then show ?thesis by auto 5901 qed 5902 have "?a 1 \<longlonglongrightarrow> 0" 5903 unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def 5904 by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat simp del: of_nat_Suc) 5905 have "?diff 1 \<longlonglongrightarrow> 0" 5906 proof (rule LIMSEQ_I) 5907 fix r :: real 5908 assume "0 < r" 5909 obtain N :: nat where N_I: "N \<le> n \<Longrightarrow> ?a 1 n < r" for n 5910 using LIMSEQ_D[OF \<open>?a 1 \<longlonglongrightarrow> 0\<close> \<open>0 < r\<close>] by auto 5911 have "norm (?diff 1 n - 0) < r" if "N \<le> n" for n 5912 using \<open>?diff 1 n \<le> ?a 1 n\<close> N_I[OF that] by auto 5913 then show "\<exists>N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast 5914 qed 5915 from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus] 5916 have "(?c 1) sums (arctan 1)" unfolding sums_def by auto 5917 then have "arctan 1 = (\<Sum>i. ?c 1 i)" by (rule sums_unique) 5918 5919 show ?thesis 5920 proof (cases "x = 1") 5921 case True 5922 then show ?thesis by (simp add: \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close>) 5923 next 5924 case False 5925 then have "x = -1" using \<open>\<bar>x\<bar> = 1\<close> by auto 5926 5927 have "- (pi/2) < 0" using pi_gt_zero by auto 5928 have "- (2 * pi) < 0" using pi_gt_zero by auto 5929 5930 have c_minus_minus: "?c (- 1) i = - ?c 1 i" for i by auto 5931 5932 have "arctan (- 1) = arctan (tan (-(pi / 4)))" 5933 unfolding tan_45 tan_minus .. 5934 also have "\<dots> = - (pi / 4)" 5935 by (rule arctan_tan) (auto simp: order_less_trans[OF \<open>- (pi/2) < 0\<close> pi_gt_zero]) 5936 also have "\<dots> = - (arctan (tan (pi / 4)))" 5937 unfolding neg_equal_iff_equal 5938 by (rule arctan_tan[symmetric]) (auto simp: order_less_trans[OF \<open>- (2 * pi) < 0\<close> pi_gt_zero]) 5939 also have "\<dots> = - (arctan 1)" 5940 unfolding tan_45 .. 5941 also have "\<dots> = - (\<Sum> i. ?c 1 i)" 5942 using \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close> by auto 5943 also have "\<dots> = (\<Sum> i. ?c (- 1) i)" 5944 using suminf_minus[OF sums_summable[OF \<open>(?c 1) sums (arctan 1)\<close>]] 5945 unfolding c_minus_minus by auto 5946 finally show ?thesis using \<open>x = -1\<close> by auto 5947 qed 5948 qed 5949qed 5950 5951lemma arctan_half: "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))" 5952 for x :: real 5953proof - 5954 obtain y where low: "- (pi/2) < y" and high: "y < pi/2" and y_eq: "tan y = x" 5955 using tan_total by blast 5956 then have low2: "- (pi/2) < y / 2" and high2: "y / 2 < pi/2" 5957 by auto 5958 5959 have "0 < cos y" by (rule cos_gt_zero_pi[OF low high]) 5960 then have "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y" 5961 by auto 5962 5963 have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2" 5964 unfolding tan_def power_divide .. 5965 also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2" 5966 using \<open>cos y \<noteq> 0\<close> by auto 5967 also have "\<dots> = 1 / (cos y)\<^sup>2" 5968 unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 .. 5969 finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" . 5970 5971 have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)" 5972 unfolding tan_def using \<open>cos y \<noteq> 0\<close> by (simp add: field_simps) 5973 also have "\<dots> = tan y / (1 + 1 / cos y)" 5974 using \<open>cos y \<noteq> 0\<close> unfolding add_divide_distrib by auto 5975 also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))" 5976 unfolding cos_sqrt .. 5977 also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))" 5978 unfolding real_sqrt_divide by auto 5979 finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))" 5980 unfolding \<open>1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2\<close> . 5981 5982 have "arctan x = y" 5983 using arctan_tan low high y_eq by auto 5984 also have "\<dots> = 2 * (arctan (tan (y/2)))" 5985 using arctan_tan[OF low2 high2] by auto 5986 also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" 5987 unfolding tan_half by auto 5988 finally show ?thesis 5989 unfolding eq \<open>tan y = x\<close> . 5990qed 5991 5992lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y" 5993 by (simp only: arctan_less_iff) 5994 5995lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y" 5996 by (simp only: arctan_le_iff) 5997 5998lemma arctan_inverse: 5999 assumes "x \<noteq> 0" 6000 shows "arctan (1 / x) = sgn x * pi/2 - arctan x" 6001proof (rule arctan_unique) 6002 show "- (pi/2) < sgn x * pi/2 - arctan x" 6003 using arctan_bounded [of x] assms 6004 unfolding sgn_real_def 6005 apply (auto simp: arctan algebra_simps) 6006 apply (drule zero_less_arctan_iff [THEN iffD2], arith) 6007 done 6008 show "sgn x * pi/2 - arctan x < pi/2" 6009 using arctan_bounded [of "- x"] assms 6010 unfolding sgn_real_def arctan_minus 6011 by (auto simp: algebra_simps) 6012 show "tan (sgn x * pi/2 - arctan x) = 1 / x" 6013 unfolding tan_inverse [of "arctan x", unfolded tan_arctan] 6014 unfolding sgn_real_def 6015 by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff) 6016qed 6017 6018theorem pi_series: "pi / 4 = (\<Sum>k. (-1)^k * 1 / real (k * 2 + 1))" 6019 (is "_ = ?SUM") 6020proof - 6021 have "pi / 4 = arctan 1" 6022 using arctan_one by auto 6023 also have "\<dots> = ?SUM" 6024 using arctan_series[of 1] by auto 6025 finally show ?thesis by auto 6026qed 6027 6028 6029subsection \<open>Existence of Polar Coordinates\<close> 6030 6031lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1" 6032 by (rule power2_le_imp_le [OF _ zero_le_one]) 6033 (simp add: power_divide divide_le_eq not_sum_power2_lt_zero) 6034 6035lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one] 6036 6037lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one] 6038 6039lemma polar_Ex: "\<exists>r::real. \<exists>a. x = r * cos a \<and> y = r * sin a" 6040proof - 6041 have polar_ex1: "0 < y \<Longrightarrow> \<exists>r a. x = r * cos a \<and> y = r * sin a" for y 6042 apply (rule exI [where x = "sqrt (x\<^sup>2 + y\<^sup>2)"]) 6043 apply (rule exI [where x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))"]) 6044 apply (simp add: cos_arccos_lemma1 sin_arccos_lemma1 power_divide 6045 real_sqrt_mult [symmetric] right_diff_distrib) 6046 done 6047 show ?thesis 6048 proof (cases "0::real" y rule: linorder_cases) 6049 case less 6050 then show ?thesis 6051 by (rule polar_ex1) 6052 next 6053 case equal 6054 then show ?thesis 6055 by (force simp: intro!: cos_zero sin_zero) 6056 next 6057 case greater 6058 with polar_ex1 [where y="-y"] show ?thesis 6059 by auto (metis cos_minus minus_minus minus_mult_right sin_minus) 6060 qed 6061qed 6062 6063 6064subsection \<open>Basics about polynomial functions: products, extremal behaviour and root counts\<close> 6065 6066lemma pairs_le_eq_Sigma: "{(i, j). i + j \<le> m} = Sigma (atMost m) (\<lambda>r. atMost (m - r))" 6067 for m :: nat 6068 by auto 6069 6070lemma sum_up_index_split: "(\<Sum>k\<le>m + n. f k) = (\<Sum>k\<le>m. f k) + (\<Sum>k = Suc m..m + n. f k)" 6071 by (metis atLeast0AtMost Suc_eq_plus1 le0 sum.ub_add_nat) 6072 6073lemma Sigma_interval_disjoint: "(SIGMA i:A. {..v i}) \<inter> (SIGMA i:A.{v i<..w}) = {}" 6074 for w :: "'a::order" 6075 by auto 6076 6077lemma product_atMost_eq_Un: "A \<times> {..m} = (SIGMA i:A.{..m - i}) \<union> (SIGMA i:A.{m - i<..m})" 6078 for m :: nat 6079 by auto 6080 6081lemma polynomial_product: (*with thanks to Chaitanya Mangla*) 6082 fixes x :: "'a::idom" 6083 assumes m: "\<And>i. i > m \<Longrightarrow> a i = 0" 6084 and n: "\<And>j. j > n \<Longrightarrow> b j = 0" 6085 shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) = 6086 (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)" 6087proof - 6088 have "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) = (\<Sum>i\<le>m. \<Sum>j\<le>n. (a i * x ^ i) * (b j * x ^ j))" 6089 by (rule sum_product) 6090 also have "\<dots> = (\<Sum>i\<le>m + n. \<Sum>j\<le>n + m. a i * x ^ i * (b j * x ^ j))" 6091 using assms by (auto simp: sum_up_index_split) 6092 also have "\<dots> = (\<Sum>r\<le>m + n. \<Sum>j\<le>m + n - r. a r * x ^ r * (b j * x ^ j))" 6093 apply (simp add: add_ac sum.Sigma product_atMost_eq_Un) 6094 apply (clarsimp simp add: sum_Un Sigma_interval_disjoint intro!: sum.neutral) 6095 apply (metis add_diff_assoc2 add.commute add_lessD1 leD m n nat_le_linear neqE) 6096 done 6097 also have "\<dots> = (\<Sum>(i,j)\<in>{(i,j). i+j \<le> m+n}. (a i * x ^ i) * (b j * x ^ j))" 6098 by (auto simp: pairs_le_eq_Sigma sum.Sigma) 6099 also have "\<dots> = (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)" 6100 apply (subst sum.triangle_reindex_eq) 6101 apply (auto simp: algebra_simps sum_distrib_left intro!: sum.cong) 6102 apply (metis le_add_diff_inverse power_add) 6103 done 6104 finally show ?thesis . 6105qed 6106 6107lemma polynomial_product_nat: 6108 fixes x :: nat 6109 assumes m: "\<And>i. i > m \<Longrightarrow> a i = 0" 6110 and n: "\<And>j. j > n \<Longrightarrow> b j = 0" 6111 shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) = 6112 (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)" 6113 using polynomial_product [of m a n b x] assms 6114 by (simp only: of_nat_mult [symmetric] of_nat_power [symmetric] 6115 of_nat_eq_iff Int.int_sum [symmetric]) 6116 6117lemma polyfun_diff: (*COMPLEX_SUB_POLYFUN in HOL Light*) 6118 fixes x :: "'a::idom" 6119 assumes "1 \<le> n" 6120 shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) = 6121 (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)" 6122proof - 6123 have h: "bij_betw (\<lambda>(i,j). (j,i)) ((SIGMA i : atMost n. lessThan i)) (SIGMA j : lessThan n. {Suc j..n})" 6124 by (auto simp: bij_betw_def inj_on_def) 6125 have "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) = (\<Sum>i\<le>n. a i * (x^i - y^i))" 6126 by (simp add: right_diff_distrib sum_subtractf) 6127 also have "\<dots> = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))" 6128 by (simp add: power_diff_sumr2 mult.assoc) 6129 also have "\<dots> = (\<Sum>i\<le>n. \<Sum>j<i. a i * (x - y) * (y^(i - Suc j) * x^j))" 6130 by (simp add: sum_distrib_left) 6131 also have "\<dots> = (\<Sum>(i,j) \<in> (SIGMA i : atMost n. lessThan i). a i * (x - y) * (y^(i - Suc j) * x^j))" 6132 by (simp add: sum.Sigma) 6133 also have "\<dots> = (\<Sum>(j,i) \<in> (SIGMA j : lessThan n. {Suc j..n}). a i * (x - y) * (y^(i - Suc j) * x^j))" 6134 by (auto simp: sum.reindex_bij_betw [OF h, symmetric] intro: sum.cong_simp) 6135 also have "\<dots> = (\<Sum>j<n. \<Sum>i=Suc j..n. a i * (x - y) * (y^(i - Suc j) * x^j))" 6136 by (simp add: sum.Sigma) 6137 also have "\<dots> = (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)" 6138 by (simp add: sum_distrib_left mult_ac) 6139 finally show ?thesis . 6140qed 6141 6142lemma polyfun_diff_alt: (*COMPLEX_SUB_POLYFUN_ALT in HOL Light*) 6143 fixes x :: "'a::idom" 6144 assumes "1 \<le> n" 6145 shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) = 6146 (x - y) * ((\<Sum>j<n. \<Sum>k<n-j. a(j + k + 1) * y^k * x^j))" 6147proof - 6148 have "(\<Sum>i=Suc j..n. a i * y^(i - j - 1)) = (\<Sum>k<n-j. a(j+k+1) * y^k)" 6149 if "j < n" for j :: nat 6150 proof - 6151 have h: "bij_betw (\<lambda>i. i - (j + 1)) {Suc j..n} (lessThan (n-j))" 6152 apply (auto simp: bij_betw_def inj_on_def) 6153 apply (rule_tac x="x + Suc j" in image_eqI, auto) 6154 done 6155 then show ?thesis 6156 by (auto simp: sum.reindex_bij_betw [OF h, symmetric] intro: sum.cong_simp) 6157 qed 6158 then show ?thesis 6159 by (simp add: polyfun_diff [OF assms] sum_distrib_right) 6160qed 6161 6162lemma polyfun_linear_factor: (*COMPLEX_POLYFUN_LINEAR_FACTOR in HOL Light*) 6163 fixes a :: "'a::idom" 6164 shows "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c(i) * z^i) = (z - a) * (\<Sum>i<n. b(i) * z^i) + (\<Sum>i\<le>n. c(i) * a^i)" 6165proof (cases "n = 0") 6166 case True then show ?thesis 6167 by simp 6168next 6169 case False 6170 have "(\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) = (z - a) * (\<Sum>i<n. b i * z^i) + (\<Sum>i\<le>n. c i * a^i)) \<longleftrightarrow> 6171 (\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) - (\<Sum>i\<le>n. c i * a^i) = (z - a) * (\<Sum>i<n. b i * z^i))" 6172 by (simp add: algebra_simps) 6173 also have "\<dots> \<longleftrightarrow> 6174 (\<exists>b. \<forall>z. (z - a) * (\<Sum>j<n. (\<Sum>i = Suc j..n. c i * a^(i - Suc j)) * z^j) = 6175 (z - a) * (\<Sum>i<n. b i * z^i))" 6176 using False by (simp add: polyfun_diff) 6177 also have "\<dots> = True" by auto 6178 finally show ?thesis 6179 by simp 6180qed 6181 6182lemma polyfun_linear_factor_root: (*COMPLEX_POLYFUN_LINEAR_FACTOR_ROOT in HOL Light*) 6183 fixes a :: "'a::idom" 6184 assumes "(\<Sum>i\<le>n. c(i) * a^i) = 0" 6185 obtains b where "\<And>z. (\<Sum>i\<le>n. c i * z^i) = (z - a) * (\<Sum>i<n. b i * z^i)" 6186 using polyfun_linear_factor [of c n a] assms by auto 6187 6188(*The material of this section, up until this point, could go into a new theory of polynomials 6189 based on Main alone. The remaining material involves limits, continuity, series, etc.*) 6190 6191lemma isCont_polynom: "isCont (\<lambda>w. \<Sum>i\<le>n. c i * w^i) a" 6192 for c :: "nat \<Rightarrow> 'a::real_normed_div_algebra" 6193 by simp 6194 6195lemma zero_polynom_imp_zero_coeffs: 6196 fixes c :: "nat \<Rightarrow> 'a::{ab_semigroup_mult,real_normed_div_algebra}" 6197 assumes "\<And>w. (\<Sum>i\<le>n. c i * w^i) = 0" "k \<le> n" 6198 shows "c k = 0" 6199 using assms 6200proof (induction n arbitrary: c k) 6201 case 0 6202 then show ?case 6203 by simp 6204next 6205 case (Suc n c k) 6206 have [simp]: "c 0 = 0" using Suc.prems(1) [of 0] 6207 by simp 6208 have "(\<Sum>i\<le>Suc n. c i * w^i) = w * (\<Sum>i\<le>n. c (Suc i) * w^i)" for w 6209 proof - 6210 have "(\<Sum>i\<le>Suc n. c i * w^i) = (\<Sum>i\<le>n. c (Suc i) * w ^ Suc i)" 6211 unfolding Set_Interval.sum.atMost_Suc_shift 6212 by simp 6213 also have "\<dots> = w * (\<Sum>i\<le>n. c (Suc i) * w^i)" 6214 by (simp add: sum_distrib_left ac_simps) 6215 finally show ?thesis . 6216 qed 6217 then have w: "\<And>w. w \<noteq> 0 \<Longrightarrow> (\<Sum>i\<le>n. c (Suc i) * w^i) = 0" 6218 using Suc by auto 6219 then have "(\<lambda>h. \<Sum>i\<le>n. c (Suc i) * h^i) \<midarrow>0\<rightarrow> 0" 6220 by (simp cong: LIM_cong) \<comment> \<open>the case \<open>w = 0\<close> by continuity\<close> 6221 then have "(\<Sum>i\<le>n. c (Suc i) * 0^i) = 0" 6222 using isCont_polynom [of 0 "\<lambda>i. c (Suc i)" n] LIM_unique 6223 by (force simp: Limits.isCont_iff) 6224 then have "\<And>w. (\<Sum>i\<le>n. c (Suc i) * w^i) = 0" 6225 using w by metis 6226 then have "\<And>i. i \<le> n \<Longrightarrow> c (Suc i) = 0" 6227 using Suc.IH [of "\<lambda>i. c (Suc i)"] by blast 6228 then show ?case using \<open>k \<le> Suc n\<close> 6229 by (cases k) auto 6230qed 6231 6232lemma polyfun_rootbound: (*COMPLEX_POLYFUN_ROOTBOUND in HOL Light*) 6233 fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}" 6234 assumes "c k \<noteq> 0" "k\<le>n" 6235 shows "finite {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<and> card {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<le> n" 6236 using assms 6237proof (induction n arbitrary: c k) 6238 case 0 6239 then show ?case 6240 by simp 6241next 6242 case (Suc m c k) 6243 let ?succase = ?case 6244 show ?case 6245 proof (cases "{z. (\<Sum>i\<le>Suc m. c(i) * z^i) = 0} = {}") 6246 case True 6247 then show ?succase 6248 by simp 6249 next 6250 case False 6251 then obtain z0 where z0: "(\<Sum>i\<le>Suc m. c(i) * z0^i) = 0" 6252 by blast 6253 then obtain b where b: "\<And>w. (\<Sum>i\<le>Suc m. c i * w^i) = (w - z0) * (\<Sum>i\<le>m. b i * w^i)" 6254 using polyfun_linear_factor_root [OF z0, unfolded lessThan_Suc_atMost] 6255 by blast 6256 then have eq: "{z. (\<Sum>i\<le>Suc m. c i * z^i) = 0} = insert z0 {z. (\<Sum>i\<le>m. b i * z^i) = 0}" 6257 by auto 6258 have "\<not> (\<forall>k\<le>m. b k = 0)" 6259 proof 6260 assume [simp]: "\<forall>k\<le>m. b k = 0" 6261 then have "\<And>w. (\<Sum>i\<le>m. b i * w^i) = 0" 6262 by simp 6263 then have "\<And>w. (\<Sum>i\<le>Suc m. c i * w^i) = 0" 6264 using b by simp 6265 then have "\<And>k. k \<le> Suc m \<Longrightarrow> c k = 0" 6266 using zero_polynom_imp_zero_coeffs by blast 6267 then show False using Suc.prems by blast 6268 qed 6269 then obtain k' where bk': "b k' \<noteq> 0" "k' \<le> m" 6270 by blast 6271 show ?succase 6272 using Suc.IH [of b k'] bk' 6273 by (simp add: eq card_insert_if del: sum.atMost_Suc) 6274 qed 6275qed 6276 6277lemma 6278 fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}" 6279 assumes "c k \<noteq> 0" "k\<le>n" 6280 shows polyfun_roots_finite: "finite {z. (\<Sum>i\<le>n. c(i) * z^i) = 0}" 6281 and polyfun_roots_card: "card {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<le> n" 6282 using polyfun_rootbound assms by auto 6283 6284lemma polyfun_finite_roots: (*COMPLEX_POLYFUN_FINITE_ROOTS in HOL Light*) 6285 fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}" 6286 shows "finite {x. (\<Sum>i\<le>n. c i * x^i) = 0} \<longleftrightarrow> (\<exists>i\<le>n. c i \<noteq> 0)" 6287 (is "?lhs = ?rhs") 6288proof 6289 assume ?lhs 6290 moreover have "\<not> finite {x. (\<Sum>i\<le>n. c i * x^i) = 0}" if "\<forall>i\<le>n. c i = 0" 6291 proof - 6292 from that have "\<And>x. (\<Sum>i\<le>n. c i * x^i) = 0" 6293 by simp 6294 then show ?thesis 6295 using ex_new_if_finite [OF infinite_UNIV_char_0 [where 'a='a]] 6296 by auto 6297 qed 6298 ultimately show ?rhs by metis 6299next 6300 assume ?rhs 6301 with polyfun_rootbound show ?lhs by blast 6302qed 6303 6304lemma polyfun_eq_0: "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = 0) \<longleftrightarrow> (\<forall>i\<le>n. c i = 0)" 6305 for c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}" 6306 (*COMPLEX_POLYFUN_EQ_0 in HOL Light*) 6307 using zero_polynom_imp_zero_coeffs by auto 6308 6309lemma polyfun_eq_coeffs: "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = (\<Sum>i\<le>n. d i * x^i)) \<longleftrightarrow> (\<forall>i\<le>n. c i = d i)" 6310 for c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}" 6311proof - 6312 have "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = (\<Sum>i\<le>n. d i * x^i)) \<longleftrightarrow> (\<forall>x. (\<Sum>i\<le>n. (c i - d i) * x^i) = 0)" 6313 by (simp add: left_diff_distrib Groups_Big.sum_subtractf) 6314 also have "\<dots> \<longleftrightarrow> (\<forall>i\<le>n. c i - d i = 0)" 6315 by (rule polyfun_eq_0) 6316 finally show ?thesis 6317 by simp 6318qed 6319 6320lemma polyfun_eq_const: (*COMPLEX_POLYFUN_EQ_CONST in HOL Light*) 6321 fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}" 6322 shows "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = k) \<longleftrightarrow> c 0 = k \<and> (\<forall>i \<in> {1..n}. c i = 0)" 6323 (is "?lhs = ?rhs") 6324proof - 6325 have *: "\<forall>x. (\<Sum>i\<le>n. (if i=0 then k else 0) * x^i) = k" 6326 by (induct n) auto 6327 show ?thesis 6328 proof 6329 assume ?lhs 6330 with * have "(\<forall>i\<le>n. c i = (if i=0 then k else 0))" 6331 by (simp add: polyfun_eq_coeffs [symmetric]) 6332 then show ?rhs by simp 6333 next 6334 assume ?rhs 6335 then show ?lhs by (induct n) auto 6336 qed 6337qed 6338 6339lemma root_polyfun: 6340 fixes z :: "'a::idom" 6341 assumes "1 \<le> n" 6342 shows "z^n = a \<longleftrightarrow> (\<Sum>i\<le>n. (if i = 0 then -a else if i=n then 1 else 0) * z^i) = 0" 6343 using assms by (cases n) (simp_all add: sum.atLeast_Suc_atMost atLeast0AtMost [symmetric]) 6344 6345lemma 6346 assumes "SORT_CONSTRAINT('a::{idom,real_normed_div_algebra})" 6347 and "1 \<le> n" 6348 shows finite_roots_unity: "finite {z::'a. z^n = 1}" 6349 and card_roots_unity: "card {z::'a. z^n = 1} \<le> n" 6350 using polyfun_rootbound [of "\<lambda>i. if i = 0 then -1 else if i=n then 1 else 0" n n] assms(2) 6351 by (auto simp: root_polyfun [OF assms(2)]) 6352 6353 6354subsection \<open>Hyperbolic functions\<close> 6355 6356definition sinh :: "'a :: {banach, real_normed_algebra_1} \<Rightarrow> 'a" where 6357 "sinh x = (exp x - exp (-x)) /\<^sub>R 2" 6358 6359definition cosh :: "'a :: {banach, real_normed_algebra_1} \<Rightarrow> 'a" where 6360 "cosh x = (exp x + exp (-x)) /\<^sub>R 2" 6361 6362definition tanh :: "'a :: {banach, real_normed_field} \<Rightarrow> 'a" where 6363 "tanh x = sinh x / cosh x" 6364 6365definition arsinh :: "'a :: {banach, real_normed_algebra_1, ln} \<Rightarrow> 'a" where 6366 "arsinh x = ln (x + (x^2 + 1) powr of_real (1/2))" 6367 6368definition arcosh :: "'a :: {banach, real_normed_algebra_1, ln} \<Rightarrow> 'a" where 6369 "arcosh x = ln (x + (x^2 - 1) powr of_real (1/2))" 6370 6371definition artanh :: "'a :: {banach, real_normed_field, ln} \<Rightarrow> 'a" where 6372 "artanh x = ln ((1 + x) / (1 - x)) / 2" 6373 6374lemma arsinh_0 [simp]: "arsinh 0 = 0" 6375 by (simp add: arsinh_def) 6376 6377lemma arcosh_1 [simp]: "arcosh 1 = 0" 6378 by (simp add: arcosh_def) 6379 6380lemma artanh_0 [simp]: "artanh 0 = 0" 6381 by (simp add: artanh_def) 6382 6383lemma tanh_altdef: 6384 "tanh x = (exp x - exp (-x)) / (exp x + exp (-x))" 6385proof - 6386 have "tanh x = (2 *\<^sub>R sinh x) / (2 *\<^sub>R cosh x)" 6387 by (simp add: tanh_def scaleR_conv_of_real) 6388 also have "2 *\<^sub>R sinh x = exp x - exp (-x)" 6389 by (simp add: sinh_def) 6390 also have "2 *\<^sub>R cosh x = exp x + exp (-x)" 6391 by (simp add: cosh_def) 6392 finally show ?thesis . 6393qed 6394 6395lemma tanh_real_altdef: "tanh (x::real) = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))" 6396proof - 6397 have [simp]: "exp (2 * x) = exp x * exp x" "exp (x * 2) = exp x * exp x" 6398 by (subst exp_add [symmetric]; simp)+ 6399 have "tanh x = (2 * exp (-x) * sinh x) / (2 * exp (-x) * cosh x)" 6400 by (simp add: tanh_def) 6401 also have "2 * exp (-x) * sinh x = 1 - exp (-2*x)" 6402 by (simp add: exp_minus field_simps sinh_def) 6403 also have "2 * exp (-x) * cosh x = 1 + exp (-2*x)" 6404 by (simp add: exp_minus field_simps cosh_def) 6405 finally show ?thesis . 6406qed 6407 6408 6409lemma sinh_converges: "(\<lambda>n. if even n then 0 else x ^ n /\<^sub>R fact n) sums sinh x" 6410proof - 6411 have "(\<lambda>n. (x ^ n /\<^sub>R fact n - (-x) ^ n /\<^sub>R fact n) /\<^sub>R 2) sums sinh x" 6412 unfolding sinh_def by (intro sums_scaleR_right sums_diff exp_converges) 6413 also have "(\<lambda>n. (x ^ n /\<^sub>R fact n - (-x) ^ n /\<^sub>R fact n) /\<^sub>R 2) = 6414 (\<lambda>n. if even n then 0 else x ^ n /\<^sub>R fact n)" by auto 6415 finally show ?thesis . 6416qed 6417 6418lemma cosh_converges: "(\<lambda>n. if even n then x ^ n /\<^sub>R fact n else 0) sums cosh x" 6419proof - 6420 have "(\<lambda>n. (x ^ n /\<^sub>R fact n + (-x) ^ n /\<^sub>R fact n) /\<^sub>R 2) sums cosh x" 6421 unfolding cosh_def by (intro sums_scaleR_right sums_add exp_converges) 6422 also have "(\<lambda>n. (x ^ n /\<^sub>R fact n + (-x) ^ n /\<^sub>R fact n) /\<^sub>R 2) = 6423 (\<lambda>n. if even n then x ^ n /\<^sub>R fact n else 0)" by auto 6424 finally show ?thesis . 6425qed 6426 6427lemma sinh_0 [simp]: "sinh 0 = 0" 6428 by (simp add: sinh_def) 6429 6430lemma cosh_0 [simp]: "cosh 0 = 1" 6431proof - 6432 have "cosh 0 = (1/2) *\<^sub>R (1 + 1)" by (simp add: cosh_def) 6433 also have "\<dots> = 1" by (rule scaleR_half_double) 6434 finally show ?thesis . 6435qed 6436 6437lemma tanh_0 [simp]: "tanh 0 = 0" 6438 by (simp add: tanh_def) 6439 6440lemma sinh_minus [simp]: "sinh (- x) = -sinh x" 6441 by (simp add: sinh_def algebra_simps) 6442 6443lemma cosh_minus [simp]: "cosh (- x) = cosh x" 6444 by (simp add: cosh_def algebra_simps) 6445 6446lemma tanh_minus [simp]: "tanh (-x) = -tanh x" 6447 by (simp add: tanh_def) 6448 6449lemma sinh_ln_real: "x > 0 \<Longrightarrow> sinh (ln x :: real) = (x - inverse x) / 2" 6450 by (simp add: sinh_def exp_minus) 6451 6452lemma cosh_ln_real: "x > 0 \<Longrightarrow> cosh (ln x :: real) = (x + inverse x) / 2" 6453 by (simp add: cosh_def exp_minus) 6454 6455lemma tanh_ln_real: 6456 "tanh (ln x :: real) = (x ^ 2 - 1) / (x ^ 2 + 1)" if "x > 0" 6457proof - 6458 from that have "(x * 2 - inverse x * 2) * (x\<^sup>2 + 1) = 6459 (x\<^sup>2 - 1) * (2 * x + 2 * inverse x)" 6460 by (simp add: field_simps power2_eq_square) 6461 moreover have "x\<^sup>2 + 1 > 0" 6462 using that by (simp add: ac_simps add_pos_nonneg) 6463 moreover have "2 * x + 2 * inverse x > 0" 6464 using that by (simp add: add_pos_pos) 6465 ultimately have "(x * 2 - inverse x * 2) / 6466 (2 * x + 2 * inverse x) = 6467 (x\<^sup>2 - 1) / (x\<^sup>2 + 1)" 6468 by (simp add: frac_eq_eq) 6469 with that show ?thesis 6470 by (simp add: tanh_def sinh_ln_real cosh_ln_real) 6471qed 6472 6473lemma has_field_derivative_scaleR_right [derivative_intros]: 6474 "(f has_field_derivative D) F \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_field_derivative (c *\<^sub>R D)) F" 6475 unfolding has_field_derivative_def 6476 using has_derivative_scaleR_right[of f "\<lambda>x. D * x" F c] 6477 by (simp add: mult_scaleR_left [symmetric] del: mult_scaleR_left) 6478 6479lemma has_field_derivative_sinh [THEN DERIV_chain2, derivative_intros]: 6480 "(sinh has_field_derivative cosh x) (at (x :: 'a :: {banach, real_normed_field}))" 6481 unfolding sinh_def cosh_def by (auto intro!: derivative_eq_intros) 6482 6483lemma has_field_derivative_cosh [THEN DERIV_chain2, derivative_intros]: 6484 "(cosh has_field_derivative sinh x) (at (x :: 'a :: {banach, real_normed_field}))" 6485 unfolding sinh_def cosh_def by (auto intro!: derivative_eq_intros) 6486 6487lemma has_field_derivative_tanh [THEN DERIV_chain2, derivative_intros]: 6488 "cosh x \<noteq> 0 \<Longrightarrow> (tanh has_field_derivative 1 - tanh x ^ 2) 6489 (at (x :: 'a :: {banach, real_normed_field}))" 6490 unfolding tanh_def by (auto intro!: derivative_eq_intros simp: power2_eq_square field_split_simps) 6491 6492lemma has_derivative_sinh [derivative_intros]: 6493 fixes g :: "'a \<Rightarrow> ('a :: {banach, real_normed_field})" 6494 assumes "(g has_derivative (\<lambda>x. Db * x)) (at x within s)" 6495 shows "((\<lambda>x. sinh (g x)) has_derivative (\<lambda>y. (cosh (g x) * Db) * y)) (at x within s)" 6496proof - 6497 have "((\<lambda>x. - g x) has_derivative (\<lambda>y. -(Db * y))) (at x within s)" 6498 using assms by (intro derivative_intros) 6499 also have "(\<lambda>y. -(Db * y)) = (\<lambda>x. (-Db) * x)" by (simp add: fun_eq_iff) 6500 finally have "((\<lambda>x. sinh (g x)) has_derivative 6501 (\<lambda>y. (exp (g x) * Db * y - exp (-g x) * (-Db) * y) /\<^sub>R 2)) (at x within s)" 6502 unfolding sinh_def by (intro derivative_intros assms) 6503 also have "(\<lambda>y. (exp (g x) * Db * y - exp (-g x) * (-Db) * y) /\<^sub>R 2) = (\<lambda>y. (cosh (g x) * Db) * y)" 6504 by (simp add: fun_eq_iff cosh_def algebra_simps) 6505 finally show ?thesis . 6506qed 6507 6508lemma has_derivative_cosh [derivative_intros]: 6509 fixes g :: "'a \<Rightarrow> ('a :: {banach, real_normed_field})" 6510 assumes "(g has_derivative (\<lambda>y. Db * y)) (at x within s)" 6511 shows "((\<lambda>x. cosh (g x)) has_derivative (\<lambda>y. (sinh (g x) * Db) * y)) (at x within s)" 6512proof - 6513 have "((\<lambda>x. - g x) has_derivative (\<lambda>y. -(Db * y))) (at x within s)" 6514 using assms by (intro derivative_intros) 6515 also have "(\<lambda>y. -(Db * y)) = (\<lambda>y. (-Db) * y)" by (simp add: fun_eq_iff) 6516 finally have "((\<lambda>x. cosh (g x)) has_derivative 6517 (\<lambda>y. (exp (g x) * Db * y + exp (-g x) * (-Db) * y) /\<^sub>R 2)) (at x within s)" 6518 unfolding cosh_def by (intro derivative_intros assms) 6519 also have "(\<lambda>y. (exp (g x) * Db * y + exp (-g x) * (-Db) * y) /\<^sub>R 2) = (\<lambda>y. (sinh (g x) * Db) * y)" 6520 by (simp add: fun_eq_iff sinh_def algebra_simps) 6521 finally show ?thesis . 6522qed 6523 6524lemma sinh_plus_cosh: "sinh x + cosh x = exp x" 6525proof - 6526 have "sinh x + cosh x = (1 / 2) *\<^sub>R (exp x + exp x)" 6527 by (simp add: sinh_def cosh_def algebra_simps) 6528 also have "\<dots> = exp x" by (rule scaleR_half_double) 6529 finally show ?thesis . 6530qed 6531 6532lemma cosh_plus_sinh: "cosh x + sinh x = exp x" 6533 by (subst add.commute) (rule sinh_plus_cosh) 6534 6535lemma cosh_minus_sinh: "cosh x - sinh x = exp (-x)" 6536proof - 6537 have "cosh x - sinh x = (1 / 2) *\<^sub>R (exp (-x) + exp (-x))" 6538 by (simp add: sinh_def cosh_def algebra_simps) 6539 also have "\<dots> = exp (-x)" by (rule scaleR_half_double) 6540 finally show ?thesis . 6541qed 6542 6543lemma sinh_minus_cosh: "sinh x - cosh x = -exp (-x)" 6544 using cosh_minus_sinh[of x] by (simp add: algebra_simps) 6545 6546 6547context 6548 fixes x :: "'a :: {real_normed_field, banach}" 6549begin 6550 6551lemma sinh_zero_iff: "sinh x = 0 \<longleftrightarrow> exp x \<in> {1, -1}" 6552 by (auto simp: sinh_def field_simps exp_minus power2_eq_square square_eq_1_iff) 6553 6554lemma cosh_zero_iff: "cosh x = 0 \<longleftrightarrow> exp x ^ 2 = -1" 6555 by (auto simp: cosh_def exp_minus field_simps power2_eq_square eq_neg_iff_add_eq_0) 6556 6557lemma cosh_square_eq: "cosh x ^ 2 = sinh x ^ 2 + 1" 6558 by (simp add: cosh_def sinh_def algebra_simps power2_eq_square exp_add [symmetric] 6559 scaleR_conv_of_real) 6560 6561lemma sinh_square_eq: "sinh x ^ 2 = cosh x ^ 2 - 1" 6562 by (simp add: cosh_square_eq) 6563 6564lemma hyperbolic_pythagoras: "cosh x ^ 2 - sinh x ^ 2 = 1" 6565 by (simp add: cosh_square_eq) 6566 6567lemma sinh_add: "sinh (x + y) = sinh x * cosh y + cosh x * sinh y" 6568 by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric]) 6569 6570lemma sinh_diff: "sinh (x - y) = sinh x * cosh y - cosh x * sinh y" 6571 by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric]) 6572 6573lemma cosh_add: "cosh (x + y) = cosh x * cosh y + sinh x * sinh y" 6574 by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric]) 6575 6576lemma cosh_diff: "cosh (x - y) = cosh x * cosh y - sinh x * sinh y" 6577 by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric]) 6578 6579lemma tanh_add: 6580 "tanh (x + y) = (tanh x + tanh y) / (1 + tanh x * tanh y)" 6581 if "cosh x \<noteq> 0" "cosh y \<noteq> 0" 6582proof - 6583 have "(sinh x * cosh y + cosh x * sinh y) * (1 + sinh x * sinh y / (cosh x * cosh y)) = 6584 (cosh x * cosh y + sinh x * sinh y) * ((sinh x * cosh y + sinh y * cosh x) / (cosh y * cosh x))" 6585 using that by (simp add: field_split_simps) 6586 also have "(sinh x * cosh y + sinh y * cosh x) / (cosh y * cosh x) = sinh x / cosh x + sinh y / cosh y" 6587 using that by (simp add: field_split_simps) 6588 finally have "(sinh x * cosh y + cosh x * sinh y) * (1 + sinh x * sinh y / (cosh x * cosh y)) = 6589 (sinh x / cosh x + sinh y / cosh y) * (cosh x * cosh y + sinh x * sinh y)" 6590 by simp 6591 then show ?thesis 6592 using that by (auto simp add: tanh_def sinh_add cosh_add eq_divide_eq) 6593 (simp_all add: field_split_simps) 6594qed 6595 6596lemma sinh_double: "sinh (2 * x) = 2 * sinh x * cosh x" 6597 using sinh_add[of x] by simp 6598 6599lemma cosh_double: "cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2" 6600 using cosh_add[of x] by (simp add: power2_eq_square) 6601 6602end 6603 6604lemma sinh_field_def: "sinh z = (exp z - exp (-z)) / (2 :: 'a :: {banach, real_normed_field})" 6605 by (simp add: sinh_def scaleR_conv_of_real) 6606 6607lemma cosh_field_def: "cosh z = (exp z + exp (-z)) / (2 :: 'a :: {banach, real_normed_field})" 6608 by (simp add: cosh_def scaleR_conv_of_real) 6609 6610 6611subsubsection \<open>More specific properties of the real functions\<close> 6612 6613lemma sinh_real_zero_iff [simp]: "sinh (x::real) = 0 \<longleftrightarrow> x = 0" 6614proof - 6615 have "(-1 :: real) < 0" by simp 6616 also have "0 < exp x" by simp 6617 finally have "exp x \<noteq> -1" by (intro notI) simp 6618 thus ?thesis by (subst sinh_zero_iff) simp 6619qed 6620 6621lemma plus_inverse_ge_2: 6622 fixes x :: real 6623 assumes "x > 0" 6624 shows "x + inverse x \<ge> 2" 6625proof - 6626 have "0 \<le> (x - 1) ^ 2" by simp 6627 also have "\<dots> = x^2 - 2*x + 1" by (simp add: power2_eq_square algebra_simps) 6628 finally show ?thesis using assms by (simp add: field_simps power2_eq_square) 6629qed 6630 6631lemma sinh_real_nonneg_iff [simp]: "sinh (x :: real) \<ge> 0 \<longleftrightarrow> x \<ge> 0" 6632 by (simp add: sinh_def) 6633 6634lemma sinh_real_pos_iff [simp]: "sinh (x :: real) > 0 \<longleftrightarrow> x > 0" 6635 by (simp add: sinh_def) 6636 6637lemma sinh_real_nonpos_iff [simp]: "sinh (x :: real) \<le> 0 \<longleftrightarrow> x \<le> 0" 6638 by (simp add: sinh_def) 6639 6640lemma sinh_real_neg_iff [simp]: "sinh (x :: real) < 0 \<longleftrightarrow> x < 0" 6641 by (simp add: sinh_def) 6642 6643lemma cosh_real_ge_1: "cosh (x :: real) \<ge> 1" 6644 using plus_inverse_ge_2[of "exp x"] by (simp add: cosh_def exp_minus) 6645 6646lemma cosh_real_pos [simp]: "cosh (x :: real) > 0" 6647 using cosh_real_ge_1[of x] by simp 6648 6649lemma cosh_real_nonneg[simp]: "cosh (x :: real) \<ge> 0" 6650 using cosh_real_ge_1[of x] by simp 6651 6652lemma cosh_real_nonzero [simp]: "cosh (x :: real) \<noteq> 0" 6653 using cosh_real_ge_1[of x] by simp 6654 6655lemma tanh_real_nonneg_iff [simp]: "tanh (x :: real) \<ge> 0 \<longleftrightarrow> x \<ge> 0" 6656 by (simp add: tanh_def field_simps) 6657 6658lemma tanh_real_pos_iff [simp]: "tanh (x :: real) > 0 \<longleftrightarrow> x > 0" 6659 by (simp add: tanh_def field_simps) 6660 6661lemma tanh_real_nonpos_iff [simp]: "tanh (x :: real) \<le> 0 \<longleftrightarrow> x \<le> 0" 6662 by (simp add: tanh_def field_simps) 6663 6664lemma tanh_real_neg_iff [simp]: "tanh (x :: real) < 0 \<longleftrightarrow> x < 0" 6665 by (simp add: tanh_def field_simps) 6666 6667lemma tanh_real_zero_iff [simp]: "tanh (x :: real) = 0 \<longleftrightarrow> x = 0" 6668 by (simp add: tanh_def field_simps) 6669 6670lemma arsinh_real_def: "arsinh (x::real) = ln (x + sqrt (x^2 + 1))" 6671 by (simp add: arsinh_def powr_half_sqrt) 6672 6673lemma arcosh_real_def: "x \<ge> 1 \<Longrightarrow> arcosh (x::real) = ln (x + sqrt (x^2 - 1))" 6674 by (simp add: arcosh_def powr_half_sqrt) 6675 6676lemma arsinh_real_aux: "0 < x + sqrt (x ^ 2 + 1 :: real)" 6677proof (cases "x < 0") 6678 case True 6679 have "(-x) ^ 2 = x ^ 2" by simp 6680 also have "x ^ 2 < x ^ 2 + 1" by simp 6681 finally have "sqrt ((-x) ^ 2) < sqrt (x ^ 2 + 1)" 6682 by (rule real_sqrt_less_mono) 6683 thus ?thesis using True by simp 6684qed (auto simp: add_nonneg_pos) 6685 6686lemma arsinh_minus_real [simp]: "arsinh (-x::real) = -arsinh x" 6687proof - 6688 have "arsinh (-x) = ln (sqrt (x\<^sup>2 + 1) - x)" 6689 by (simp add: arsinh_real_def) 6690 also have "sqrt (x^2 + 1) - x = inverse (sqrt (x^2 + 1) + x)" 6691 using arsinh_real_aux[of x] by (simp add: field_split_simps algebra_simps power2_eq_square) 6692 also have "ln \<dots> = -arsinh x" 6693 using arsinh_real_aux[of x] by (simp add: arsinh_real_def ln_inverse) 6694 finally show ?thesis . 6695qed 6696 6697lemma artanh_minus_real [simp]: 6698 assumes "abs x < 1" 6699 shows "artanh (-x::real) = -artanh x" 6700 using assms by (simp add: artanh_def ln_div field_simps) 6701 6702lemma sinh_less_cosh_real: "sinh (x :: real) < cosh x" 6703 by (simp add: sinh_def cosh_def) 6704 6705lemma sinh_le_cosh_real: "sinh (x :: real) \<le> cosh x" 6706 by (simp add: sinh_def cosh_def) 6707 6708lemma tanh_real_lt_1: "tanh (x :: real) < 1" 6709 by (simp add: tanh_def sinh_less_cosh_real) 6710 6711lemma tanh_real_gt_neg1: "tanh (x :: real) > -1" 6712proof - 6713 have "- cosh x < sinh x" by (simp add: sinh_def cosh_def field_split_simps) 6714 thus ?thesis by (simp add: tanh_def field_simps) 6715qed 6716 6717lemma tanh_real_bounds: "tanh (x :: real) \<in> {-1<..<1}" 6718 using tanh_real_lt_1 tanh_real_gt_neg1 by simp 6719 6720context 6721 fixes x :: real 6722begin 6723 6724lemma arsinh_sinh_real: "arsinh (sinh x) = x" 6725 by (simp add: arsinh_real_def powr_def sinh_square_eq sinh_plus_cosh) 6726 6727lemma arcosh_cosh_real: "x \<ge> 0 \<Longrightarrow> arcosh (cosh x) = x" 6728 by (simp add: arcosh_real_def powr_def cosh_square_eq cosh_real_ge_1 cosh_plus_sinh) 6729 6730lemma artanh_tanh_real: "artanh (tanh x) = x" 6731proof - 6732 have "artanh (tanh x) = ln (cosh x * (cosh x + sinh x) / (cosh x * (cosh x - sinh x))) / 2" 6733 by (simp add: artanh_def tanh_def field_split_simps) 6734 also have "cosh x * (cosh x + sinh x) / (cosh x * (cosh x - sinh x)) = 6735 (cosh x + sinh x) / (cosh x - sinh x)" by simp 6736 also have "\<dots> = (exp x)^2" 6737 by (simp add: cosh_plus_sinh cosh_minus_sinh exp_minus field_simps power2_eq_square) 6738 also have "ln ((exp x)^2) / 2 = x" by (simp add: ln_realpow) 6739 finally show ?thesis . 6740qed 6741 6742end 6743 6744lemma sinh_real_strict_mono: "strict_mono (sinh :: real \<Rightarrow> real)" 6745 by (rule pos_deriv_imp_strict_mono derivative_intros)+ auto 6746 6747lemma cosh_real_strict_mono: 6748 assumes "0 \<le> x" and "x < (y::real)" 6749 shows "cosh x < cosh y" 6750proof - 6751 from assms have "\<exists>z>x. z < y \<and> cosh y - cosh x = (y - x) * sinh z" 6752 by (intro MVT2) (auto dest: connectedD_interval intro!: derivative_eq_intros) 6753 then obtain z where z: "z > x" "z < y" "cosh y - cosh x = (y - x) * sinh z" by blast 6754 note \<open>cosh y - cosh x = (y - x) * sinh z\<close> 6755 also from \<open>z > x\<close> and assms have "(y - x) * sinh z > 0" by (intro mult_pos_pos) auto 6756 finally show "cosh x < cosh y" by simp 6757qed 6758 6759lemma tanh_real_strict_mono: "strict_mono (tanh :: real \<Rightarrow> real)" 6760proof - 6761 have *: "tanh x ^ 2 < 1" for x :: real 6762 using tanh_real_bounds[of x] by (simp add: abs_square_less_1 abs_if) 6763 show ?thesis 6764 by (rule pos_deriv_imp_strict_mono) (insert *, auto intro!: derivative_intros) 6765qed 6766 6767lemma sinh_real_abs [simp]: "sinh (abs x :: real) = abs (sinh x)" 6768 by (simp add: abs_if) 6769 6770lemma cosh_real_abs [simp]: "cosh (abs x :: real) = cosh x" 6771 by (simp add: abs_if) 6772 6773lemma tanh_real_abs [simp]: "tanh (abs x :: real) = abs (tanh x)" 6774 by (auto simp: abs_if) 6775 6776lemma sinh_real_eq_iff [simp]: "sinh x = sinh y \<longleftrightarrow> x = (y :: real)" 6777 using sinh_real_strict_mono by (simp add: strict_mono_eq) 6778 6779lemma tanh_real_eq_iff [simp]: "tanh x = tanh y \<longleftrightarrow> x = (y :: real)" 6780 using tanh_real_strict_mono by (simp add: strict_mono_eq) 6781 6782lemma cosh_real_eq_iff [simp]: "cosh x = cosh y \<longleftrightarrow> abs x = abs (y :: real)" 6783proof - 6784 have "cosh x = cosh y \<longleftrightarrow> x = y" if "x \<ge> 0" "y \<ge> 0" for x y :: real 6785 using cosh_real_strict_mono[of x y] cosh_real_strict_mono[of y x] that 6786 by (cases x y rule: linorder_cases) auto 6787 from this[of "abs x" "abs y"] show ?thesis by simp 6788qed 6789 6790lemma sinh_real_le_iff [simp]: "sinh x \<le> sinh y \<longleftrightarrow> x \<le> (y::real)" 6791 using sinh_real_strict_mono by (simp add: strict_mono_less_eq) 6792 6793lemma cosh_real_nonneg_le_iff: "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> cosh x \<le> cosh y \<longleftrightarrow> x \<le> (y::real)" 6794 using cosh_real_strict_mono[of x y] cosh_real_strict_mono[of y x] 6795 by (cases x y rule: linorder_cases) auto 6796 6797lemma cosh_real_nonpos_le_iff: "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> cosh x \<le> cosh y \<longleftrightarrow> x \<ge> (y::real)" 6798 using cosh_real_nonneg_le_iff[of "-x" "-y"] by simp 6799 6800lemma tanh_real_le_iff [simp]: "tanh x \<le> tanh y \<longleftrightarrow> x \<le> (y::real)" 6801 using tanh_real_strict_mono by (simp add: strict_mono_less_eq) 6802 6803 6804lemma sinh_real_less_iff [simp]: "sinh x < sinh y \<longleftrightarrow> x < (y::real)" 6805 using sinh_real_strict_mono by (simp add: strict_mono_less) 6806 6807lemma cosh_real_nonneg_less_iff: "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> cosh x < cosh y \<longleftrightarrow> x < (y::real)" 6808 using cosh_real_strict_mono[of x y] cosh_real_strict_mono[of y x] 6809 by (cases x y rule: linorder_cases) auto 6810 6811lemma cosh_real_nonpos_less_iff: "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> cosh x < cosh y \<longleftrightarrow> x > (y::real)" 6812 using cosh_real_nonneg_less_iff[of "-x" "-y"] by simp 6813 6814lemma tanh_real_less_iff [simp]: "tanh x < tanh y \<longleftrightarrow> x < (y::real)" 6815 using tanh_real_strict_mono by (simp add: strict_mono_less) 6816 6817 6818subsubsection \<open>Limits\<close> 6819 6820lemma sinh_real_at_top: "filterlim (sinh :: real \<Rightarrow> real) at_top at_top" 6821proof - 6822 have *: "((\<lambda>x. - exp (- x)) \<longlongrightarrow> (-0::real)) at_top" 6823 by (intro tendsto_minus filterlim_compose[OF exp_at_bot] filterlim_uminus_at_bot_at_top) 6824 have "filterlim (\<lambda>x. (1 / 2) * (-exp (-x) + exp x) :: real) at_top at_top" 6825 by (rule filterlim_tendsto_pos_mult_at_top[OF _ _ 6826 filterlim_tendsto_add_at_top[OF *]] tendsto_const)+ (auto simp: exp_at_top) 6827 also have "(\<lambda>x. (1 / 2) * (-exp (-x) + exp x) :: real) = sinh" 6828 by (simp add: fun_eq_iff sinh_def) 6829 finally show ?thesis . 6830qed 6831 6832lemma sinh_real_at_bot: "filterlim (sinh :: real \<Rightarrow> real) at_bot at_bot" 6833proof - 6834 have "filterlim (\<lambda>x. -sinh x :: real) at_bot at_top" 6835 by (simp add: filterlim_uminus_at_top [symmetric] sinh_real_at_top) 6836 also have "(\<lambda>x. -sinh x :: real) = (\<lambda>x. sinh (-x))" by simp 6837 finally show ?thesis by (subst filterlim_at_bot_mirror) 6838qed 6839 6840lemma cosh_real_at_top: "filterlim (cosh :: real \<Rightarrow> real) at_top at_top" 6841proof - 6842 have *: "((\<lambda>x. exp (- x)) \<longlongrightarrow> (0::real)) at_top" 6843 by (intro filterlim_compose[OF exp_at_bot] filterlim_uminus_at_bot_at_top) 6844 have "filterlim (\<lambda>x. (1 / 2) * (exp (-x) + exp x) :: real) at_top at_top" 6845 by (rule filterlim_tendsto_pos_mult_at_top[OF _ _ 6846 filterlim_tendsto_add_at_top[OF *]] tendsto_const)+ (auto simp: exp_at_top) 6847 also have "(\<lambda>x. (1 / 2) * (exp (-x) + exp x) :: real) = cosh" 6848 by (simp add: fun_eq_iff cosh_def) 6849 finally show ?thesis . 6850qed 6851 6852lemma cosh_real_at_bot: "filterlim (cosh :: real \<Rightarrow> real) at_top at_bot" 6853proof - 6854 have "filterlim (\<lambda>x. cosh (-x) :: real) at_top at_top" 6855 by (simp add: cosh_real_at_top) 6856 thus ?thesis by (subst filterlim_at_bot_mirror) 6857qed 6858 6859lemma tanh_real_at_top: "(tanh \<longlongrightarrow> (1::real)) at_top" 6860proof - 6861 have "((\<lambda>x::real. (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))) \<longlongrightarrow> (1 - 0) / (1 + 0)) at_top" 6862 by (intro tendsto_intros filterlim_compose[OF exp_at_bot] 6863 filterlim_tendsto_neg_mult_at_bot[OF tendsto_const] filterlim_ident) auto 6864 also have "(\<lambda>x::real. (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))) = tanh" 6865 by (rule ext) (simp add: tanh_real_altdef) 6866 finally show ?thesis by simp 6867qed 6868 6869lemma tanh_real_at_bot: "(tanh \<longlongrightarrow> (-1::real)) at_bot" 6870proof - 6871 have "((\<lambda>x::real. -tanh x) \<longlongrightarrow> -1) at_top" 6872 by (intro tendsto_minus tanh_real_at_top) 6873 also have "(\<lambda>x. -tanh x :: real) = (\<lambda>x. tanh (-x))" by simp 6874 finally show ?thesis by (subst filterlim_at_bot_mirror) 6875qed 6876 6877 6878subsubsection \<open>Properties of the inverse hyperbolic functions\<close> 6879 6880lemma isCont_sinh: "isCont sinh (x :: 'a :: {real_normed_field, banach})" 6881 unfolding sinh_def [abs_def] by (auto intro!: continuous_intros) 6882 6883lemma isCont_cosh: "isCont cosh (x :: 'a :: {real_normed_field, banach})" 6884 unfolding cosh_def [abs_def] by (auto intro!: continuous_intros) 6885 6886lemma isCont_tanh: "cosh x \<noteq> 0 \<Longrightarrow> isCont tanh (x :: 'a :: {real_normed_field, banach})" 6887 unfolding tanh_def [abs_def] 6888 by (auto intro!: continuous_intros isCont_divide isCont_sinh isCont_cosh) 6889 6890lemma continuous_on_sinh [continuous_intros]: 6891 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6892 assumes "continuous_on A f" 6893 shows "continuous_on A (\<lambda>x. sinh (f x))" 6894 unfolding sinh_def using assms by (intro continuous_intros) 6895 6896lemma continuous_on_cosh [continuous_intros]: 6897 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6898 assumes "continuous_on A f" 6899 shows "continuous_on A (\<lambda>x. cosh (f x))" 6900 unfolding cosh_def using assms by (intro continuous_intros) 6901 6902lemma continuous_sinh [continuous_intros]: 6903 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6904 assumes "continuous F f" 6905 shows "continuous F (\<lambda>x. sinh (f x))" 6906 unfolding sinh_def using assms by (intro continuous_intros) 6907 6908lemma continuous_cosh [continuous_intros]: 6909 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6910 assumes "continuous F f" 6911 shows "continuous F (\<lambda>x. cosh (f x))" 6912 unfolding cosh_def using assms by (intro continuous_intros) 6913 6914lemma continuous_on_tanh [continuous_intros]: 6915 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6916 assumes "continuous_on A f" "\<And>x. x \<in> A \<Longrightarrow> cosh (f x) \<noteq> 0" 6917 shows "continuous_on A (\<lambda>x. tanh (f x))" 6918 unfolding tanh_def using assms by (intro continuous_intros) auto 6919 6920lemma continuous_at_within_tanh [continuous_intros]: 6921 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6922 assumes "continuous (at x within A) f" "cosh (f x) \<noteq> 0" 6923 shows "continuous (at x within A) (\<lambda>x. tanh (f x))" 6924 unfolding tanh_def using assms by (intro continuous_intros continuous_divide) auto 6925 6926lemma continuous_tanh [continuous_intros]: 6927 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6928 assumes "continuous F f" "cosh (f (Lim F (\<lambda>x. x))) \<noteq> 0" 6929 shows "continuous F (\<lambda>x. tanh (f x))" 6930 unfolding tanh_def using assms by (intro continuous_intros continuous_divide) auto 6931 6932lemma tendsto_sinh [tendsto_intros]: 6933 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6934 shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. sinh (f x)) \<longlongrightarrow> sinh a) F" 6935 by (rule isCont_tendsto_compose [OF isCont_sinh]) 6936 6937lemma tendsto_cosh [tendsto_intros]: 6938 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6939 shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. cosh (f x)) \<longlongrightarrow> cosh a) F" 6940 by (rule isCont_tendsto_compose [OF isCont_cosh]) 6941 6942lemma tendsto_tanh [tendsto_intros]: 6943 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6944 shows "(f \<longlongrightarrow> a) F \<Longrightarrow> cosh a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. tanh (f x)) \<longlongrightarrow> tanh a) F" 6945 by (rule isCont_tendsto_compose [OF isCont_tanh]) 6946 6947 6948lemma arsinh_real_has_field_derivative [derivative_intros]: 6949 fixes x :: real 6950 shows "(arsinh has_field_derivative (1 / (sqrt (x ^ 2 + 1)))) (at x within A)" 6951proof - 6952 have pos: "1 + x ^ 2 > 0" by (intro add_pos_nonneg) auto 6953 from pos arsinh_real_aux[of x] show ?thesis unfolding arsinh_def [abs_def] 6954 by (auto intro!: derivative_eq_intros simp: powr_minus powr_half_sqrt field_split_simps) 6955qed 6956 6957lemma arcosh_real_has_field_derivative [derivative_intros]: 6958 fixes x :: real 6959 assumes "x > 1" 6960 shows "(arcosh has_field_derivative (1 / (sqrt (x ^ 2 - 1)))) (at x within A)" 6961proof - 6962 from assms have "x + sqrt (x\<^sup>2 - 1) > 0" by (simp add: add_pos_pos) 6963 thus ?thesis using assms unfolding arcosh_def [abs_def] 6964 by (auto intro!: derivative_eq_intros 6965 simp: powr_minus powr_half_sqrt field_split_simps power2_eq_1_iff) 6966qed 6967 6968lemma artanh_real_has_field_derivative [derivative_intros]: 6969 "(artanh has_field_derivative (1 / (1 - x ^ 2))) (at x within A)" if 6970 "\<bar>x\<bar> < 1" for x :: real 6971proof - 6972 from that have "- 1 < x" "x < 1" by linarith+ 6973 hence "(artanh has_field_derivative (4 - 4 * x) / ((1 + x) * (1 - x) * (1 - x) * 4)) 6974 (at x within A)" unfolding artanh_def [abs_def] 6975 by (auto intro!: derivative_eq_intros simp: powr_minus powr_half_sqrt) 6976 also have "(4 - 4 * x) / ((1 + x) * (1 - x) * (1 - x) * 4) = 1 / ((1 + x) * (1 - x))" 6977 using \<open>-1 < x\<close> \<open>x < 1\<close> by (simp add: frac_eq_eq) 6978 also have "(1 + x) * (1 - x) = 1 - x ^ 2" 6979 by (simp add: algebra_simps power2_eq_square) 6980 finally show ?thesis . 6981qed 6982 6983lemma continuous_on_arsinh [continuous_intros]: "continuous_on A (arsinh :: real \<Rightarrow> real)" 6984 by (rule DERIV_continuous_on derivative_intros)+ 6985 6986lemma continuous_on_arcosh [continuous_intros]: 6987 assumes "A \<subseteq> {1..}" 6988 shows "continuous_on A (arcosh :: real \<Rightarrow> real)" 6989proof - 6990 have pos: "x + sqrt (x ^ 2 - 1) > 0" if "x \<ge> 1" for x 6991 using that by (intro add_pos_nonneg) auto 6992 show ?thesis 6993 unfolding arcosh_def [abs_def] 6994 by (intro continuous_on_subset [OF _ assms] continuous_on_ln continuous_on_add 6995 continuous_on_id continuous_on_powr') 6996 (auto dest: pos simp: powr_half_sqrt intro!: continuous_intros) 6997qed 6998 6999lemma continuous_on_artanh [continuous_intros]: 7000 assumes "A \<subseteq> {-1<..<1}" 7001 shows "continuous_on A (artanh :: real \<Rightarrow> real)" 7002 unfolding artanh_def [abs_def] 7003 by (intro continuous_on_subset [OF _ assms]) (auto intro!: continuous_intros) 7004 7005lemma continuous_on_arsinh' [continuous_intros]: 7006 fixes f :: "real \<Rightarrow> real" 7007 assumes "continuous_on A f" 7008 shows "continuous_on A (\<lambda>x. arsinh (f x))" 7009 by (rule continuous_on_compose2[OF continuous_on_arsinh assms]) auto 7010 7011lemma continuous_on_arcosh' [continuous_intros]: 7012 fixes f :: "real \<Rightarrow> real" 7013 assumes "continuous_on A f" "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 1" 7014 shows "continuous_on A (\<lambda>x. arcosh (f x))" 7015 by (rule continuous_on_compose2[OF continuous_on_arcosh assms(1) order.refl]) 7016 (use assms(2) in auto) 7017 7018lemma continuous_on_artanh' [continuous_intros]: 7019 fixes f :: "real \<Rightarrow> real" 7020 assumes "continuous_on A f" "\<And>x. x \<in> A \<Longrightarrow> f x \<in> {-1<..<1}" 7021 shows "continuous_on A (\<lambda>x. artanh (f x))" 7022 by (rule continuous_on_compose2[OF continuous_on_artanh assms(1) order.refl]) 7023 (use assms(2) in auto) 7024 7025lemma isCont_arsinh [continuous_intros]: "isCont arsinh (x :: real)" 7026 using continuous_on_arsinh[of UNIV] by (auto simp: continuous_on_eq_continuous_at) 7027 7028lemma isCont_arcosh [continuous_intros]: 7029 assumes "x > 1" 7030 shows "isCont arcosh (x :: real)" 7031proof - 7032 have "continuous_on {1::real<..} arcosh" 7033 by (rule continuous_on_arcosh) auto 7034 with assms show ?thesis by (auto simp: continuous_on_eq_continuous_at) 7035qed 7036 7037lemma isCont_artanh [continuous_intros]: 7038 assumes "x > -1" "x < 1" 7039 shows "isCont artanh (x :: real)" 7040proof - 7041 have "continuous_on {-1<..<(1::real)} artanh" 7042 by (rule continuous_on_artanh) auto 7043 with assms show ?thesis by (auto simp: continuous_on_eq_continuous_at) 7044qed 7045 7046lemma tendsto_arsinh [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. arsinh (f x)) \<longlongrightarrow> arsinh a) F" 7047 for f :: "_ \<Rightarrow> real" 7048 by (rule isCont_tendsto_compose [OF isCont_arsinh]) 7049 7050lemma tendsto_arcosh_strong [tendsto_intros]: 7051 fixes f :: "_ \<Rightarrow> real" 7052 assumes "(f \<longlongrightarrow> a) F" "a \<ge> 1" "eventually (\<lambda>x. f x \<ge> 1) F" 7053 shows "((\<lambda>x. arcosh (f x)) \<longlongrightarrow> arcosh a) F" 7054 by (rule continuous_on_tendsto_compose[OF continuous_on_arcosh[OF order.refl]]) 7055 (use assms in auto) 7056 7057lemma tendsto_arcosh: 7058 fixes f :: "_ \<Rightarrow> real" 7059 assumes "(f \<longlongrightarrow> a) F" "a > 1" 7060 shows "((\<lambda>x. arcosh (f x)) \<longlongrightarrow> arcosh a) F" 7061 by (rule isCont_tendsto_compose [OF isCont_arcosh]) (use assms in auto) 7062 7063lemma tendsto_arcosh_at_left_1: "(arcosh \<longlongrightarrow> 0) (at_right (1::real))" 7064proof - 7065 have "(arcosh \<longlongrightarrow> arcosh 1) (at_right (1::real))" 7066 by (rule tendsto_arcosh_strong) (auto simp: eventually_at intro!: exI[of _ 1]) 7067 thus ?thesis by simp 7068qed 7069 7070lemma tendsto_artanh [tendsto_intros]: 7071 fixes f :: "'a \<Rightarrow> real" 7072 assumes "(f \<longlongrightarrow> a) F" "a > -1" "a < 1" 7073 shows "((\<lambda>x. artanh (f x)) \<longlongrightarrow> artanh a) F" 7074 by (rule isCont_tendsto_compose [OF isCont_artanh]) (use assms in auto) 7075 7076lemma continuous_arsinh [continuous_intros]: 7077 "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arsinh (f x :: real))" 7078 unfolding continuous_def by (rule tendsto_arsinh) 7079 7080(* TODO: This rule does not work for one-sided continuity at 1 *) 7081lemma continuous_arcosh_strong [continuous_intros]: 7082 assumes "continuous F f" "eventually (\<lambda>x. f x \<ge> 1) F" 7083 shows "continuous F (\<lambda>x. arcosh (f x :: real))" 7084proof (cases "F = bot") 7085 case False 7086 show ?thesis 7087 unfolding continuous_def 7088 proof (intro tendsto_arcosh_strong) 7089 show "1 \<le> f (Lim F (\<lambda>x. x))" 7090 using assms False unfolding continuous_def by (rule tendsto_lowerbound) 7091 qed (insert assms, auto simp: continuous_def) 7092qed auto 7093 7094lemma continuous_arcosh: 7095 "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) > 1 \<Longrightarrow> continuous F (\<lambda>x. arcosh (f x :: real))" 7096 unfolding continuous_def by (rule tendsto_arcosh) auto 7097 7098lemma continuous_artanh [continuous_intros]: 7099 "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<in> {-1<..<1} \<Longrightarrow> continuous F (\<lambda>x. artanh (f x :: real))" 7100 unfolding continuous_def by (rule tendsto_artanh) auto 7101 7102lemma arsinh_real_at_top: 7103 "filterlim (arsinh :: real \<Rightarrow> real) at_top at_top" 7104proof (subst filterlim_cong[OF refl refl]) 7105 show "filterlim (\<lambda>x. ln (x + sqrt (1 + x\<^sup>2))) at_top at_top" 7106 by (intro filterlim_compose[OF ln_at_top filterlim_at_top_add_at_top] filterlim_ident 7107 filterlim_compose[OF sqrt_at_top] filterlim_tendsto_add_at_top[OF tendsto_const] 7108 filterlim_pow_at_top) auto 7109qed (auto intro!: eventually_mono[OF eventually_ge_at_top[of 1]] simp: arsinh_real_def add_ac) 7110 7111lemma arsinh_real_at_bot: 7112 "filterlim (arsinh :: real \<Rightarrow> real) at_bot at_bot" 7113proof - 7114 have "filterlim (\<lambda>x::real. -arsinh x) at_bot at_top" 7115 by (subst filterlim_uminus_at_top [symmetric]) (rule arsinh_real_at_top) 7116 also have "(\<lambda>x::real. -arsinh x) = (\<lambda>x. arsinh (-x))" by simp 7117 finally show ?thesis 7118 by (subst filterlim_at_bot_mirror) 7119qed 7120 7121lemma arcosh_real_at_top: 7122 "filterlim (arcosh :: real \<Rightarrow> real) at_top at_top" 7123proof (subst filterlim_cong[OF refl refl]) 7124 show "filterlim (\<lambda>x. ln (x + sqrt (-1 + x\<^sup>2))) at_top at_top" 7125 by (intro filterlim_compose[OF ln_at_top filterlim_at_top_add_at_top] filterlim_ident 7126 filterlim_compose[OF sqrt_at_top] filterlim_tendsto_add_at_top[OF tendsto_const] 7127 filterlim_pow_at_top) auto 7128qed (auto intro!: eventually_mono[OF eventually_ge_at_top[of 1]] simp: arcosh_real_def) 7129 7130lemma artanh_real_at_left_1: 7131 "filterlim (artanh :: real \<Rightarrow> real) at_top (at_left 1)" 7132proof - 7133 have *: "filterlim (\<lambda>x::real. (1 + x) / (1 - x)) at_top (at_left 1)" 7134 by (rule LIM_at_top_divide) 7135 (auto intro!: tendsto_eq_intros eventually_mono[OF eventually_at_left_real[of 0]]) 7136 have "filterlim (\<lambda>x::real. (1/2) * ln ((1 + x) / (1 - x))) at_top (at_left 1)" 7137 by (intro filterlim_tendsto_pos_mult_at_top[OF tendsto_const] * 7138 filterlim_compose[OF ln_at_top]) auto 7139 also have "(\<lambda>x::real. (1/2) * ln ((1 + x) / (1 - x))) = artanh" 7140 by (simp add: artanh_def [abs_def]) 7141 finally show ?thesis . 7142qed 7143 7144lemma artanh_real_at_right_1: 7145 "filterlim (artanh :: real \<Rightarrow> real) at_bot (at_right (-1))" 7146proof - 7147 have "?thesis \<longleftrightarrow> filterlim (\<lambda>x::real. -artanh x) at_top (at_right (-1))" 7148 by (simp add: filterlim_uminus_at_bot) 7149 also have "\<dots> \<longleftrightarrow> filterlim (\<lambda>x::real. artanh (-x)) at_top (at_right (-1))" 7150 by (intro filterlim_cong refl eventually_mono[OF eventually_at_right_real[of "-1" "1"]]) auto 7151 also have "\<dots> \<longleftrightarrow> filterlim (artanh :: real \<Rightarrow> real) at_top (at_left 1)" 7152 by (simp add: filterlim_at_left_to_right) 7153 also have \<dots> by (rule artanh_real_at_left_1) 7154 finally show ?thesis . 7155qed 7156 7157 7158subsection \<open>Simprocs for root and power literals\<close> 7159 7160lemma numeral_powr_numeral_real [simp]: 7161 "numeral m powr numeral n = (numeral m ^ numeral n :: real)" 7162 by (simp add: powr_numeral) 7163 7164context 7165begin 7166 7167private lemma sqrt_numeral_simproc_aux: 7168 assumes "m * m \<equiv> n" 7169 shows "sqrt (numeral n :: real) \<equiv> numeral m" 7170proof - 7171 have "numeral n \<equiv> numeral m * (numeral m :: real)" by (simp add: assms [symmetric]) 7172 moreover have "sqrt \<dots> \<equiv> numeral m" by (subst real_sqrt_abs2) simp 7173 ultimately show "sqrt (numeral n :: real) \<equiv> numeral m" by simp 7174qed 7175 7176private lemma root_numeral_simproc_aux: 7177 assumes "Num.pow m n \<equiv> x" 7178 shows "root (numeral n) (numeral x :: real) \<equiv> numeral m" 7179 by (subst assms [symmetric], subst numeral_pow, subst real_root_pos2) simp_all 7180 7181private lemma powr_numeral_simproc_aux: 7182 assumes "Num.pow y n = x" 7183 shows "numeral x powr (m / numeral n :: real) \<equiv> numeral y powr m" 7184 by (subst assms [symmetric], subst numeral_pow, subst powr_numeral [symmetric]) 7185 (simp, subst powr_powr, simp_all) 7186 7187private lemma numeral_powr_inverse_eq: 7188 "numeral x powr (inverse (numeral n)) = numeral x powr (1 / numeral n :: real)" 7189 by simp 7190 7191 7192ML \<open> 7193 7194signature ROOT_NUMERAL_SIMPROC = sig 7195 7196val sqrt : int option -> int -> int option 7197val sqrt' : int option -> int -> int option 7198val nth_root : int option -> int -> int -> int option 7199val nth_root' : int option -> int -> int -> int option 7200val sqrt_simproc : Proof.context -> cterm -> thm option 7201val root_simproc : int * int -> Proof.context -> cterm -> thm option 7202val powr_simproc : int * int -> Proof.context -> cterm -> thm option 7203 7204end 7205 7206structure Root_Numeral_Simproc : ROOT_NUMERAL_SIMPROC = struct 7207 7208fun iterate NONE p f x = 7209 let 7210 fun go x = if p x then x else go (f x) 7211 in 7212 SOME (go x) 7213 end 7214 | iterate (SOME threshold) p f x = 7215 let 7216 fun go (threshold, x) = 7217 if p x then SOME x else if threshold = 0 then NONE else go (threshold - 1, f x) 7218 in 7219 go (threshold, x) 7220 end 7221 7222 7223fun nth_root _ 1 x = SOME x 7224 | nth_root _ _ 0 = SOME 0 7225 | nth_root _ _ 1 = SOME 1 7226 | nth_root threshold n x = 7227 let 7228 fun newton_step y = ((n - 1) * y + x div Integer.pow (n - 1) y) div n 7229 fun is_root y = Integer.pow n y <= x andalso x < Integer.pow n (y + 1) 7230 in 7231 if x < n then 7232 SOME 1 7233 else if x < Integer.pow n 2 then 7234 SOME 1 7235 else 7236 let 7237 val y = Real.floor (Math.pow (Real.fromInt x, Real.fromInt 1 / Real.fromInt n)) 7238 in 7239 if is_root y then 7240 SOME y 7241 else 7242 iterate threshold is_root newton_step ((x + n - 1) div n) 7243 end 7244 end 7245 7246fun nth_root' _ 1 x = SOME x 7247 | nth_root' _ _ 0 = SOME 0 7248 | nth_root' _ _ 1 = SOME 1 7249 | nth_root' threshold n x = if x < n then NONE else if x < Integer.pow n 2 then NONE else 7250 case nth_root threshold n x of 7251 NONE => NONE 7252 | SOME y => if Integer.pow n y = x then SOME y else NONE 7253 7254fun sqrt _ 0 = SOME 0 7255 | sqrt _ 1 = SOME 1 7256 | sqrt threshold n = 7257 let 7258 fun aux (a, b) = if n >= b * b then aux (b, b * b) else (a, b) 7259 val (lower_root, lower_n) = aux (1, 2) 7260 fun newton_step x = (x + n div x) div 2 7261 fun is_sqrt r = r*r <= n andalso n < (r+1)*(r+1) 7262 val y = Real.floor (Math.sqrt (Real.fromInt n)) 7263 in 7264 if is_sqrt y then 7265 SOME y 7266 else 7267 Option.mapPartial (iterate threshold is_sqrt newton_step o (fn x => x * lower_root)) 7268 (sqrt threshold (n div lower_n)) 7269 end 7270 7271fun sqrt' threshold x = 7272 case sqrt threshold x of 7273 NONE => NONE 7274 | SOME y => if y * y = x then SOME y else NONE 7275 7276fun sqrt_simproc ctxt ct = 7277 let 7278 val n = ct |> Thm.term_of |> dest_comb |> snd |> dest_comb |> snd |> HOLogic.dest_numeral 7279 in 7280 case sqrt' (SOME 10000) n of 7281 NONE => NONE 7282 | SOME m => 7283 SOME (Thm.instantiate' [] (map (SOME o Thm.cterm_of ctxt o HOLogic.mk_numeral) [m, n]) 7284 @{thm sqrt_numeral_simproc_aux}) 7285 end 7286 handle TERM _ => NONE 7287 7288fun root_simproc (threshold1, threshold2) ctxt ct = 7289 let 7290 val [n, x] = 7291 ct |> Thm.term_of |> strip_comb |> snd |> map (dest_comb #> snd #> HOLogic.dest_numeral) 7292 in 7293 if n > threshold1 orelse x > threshold2 then NONE else 7294 case nth_root' (SOME 100) n x of 7295 NONE => NONE 7296 | SOME m => 7297 SOME (Thm.instantiate' [] (map (SOME o Thm.cterm_of ctxt o HOLogic.mk_numeral) [m, n, x]) 7298 @{thm root_numeral_simproc_aux}) 7299 end 7300 handle TERM _ => NONE 7301 | Match => NONE 7302 7303fun powr_simproc (threshold1, threshold2) ctxt ct = 7304 let 7305 val eq_thm = Conv.try_conv (Conv.rewr_conv @{thm numeral_powr_inverse_eq}) ct 7306 val ct = Thm.dest_equals_rhs (Thm.cprop_of eq_thm) 7307 val (_, [x, t]) = strip_comb (Thm.term_of ct) 7308 val (_, [m, n]) = strip_comb t 7309 val [x, n] = map (dest_comb #> snd #> HOLogic.dest_numeral) [x, n] 7310 in 7311 if n > threshold1 orelse x > threshold2 then NONE else 7312 case nth_root' (SOME 100) n x of 7313 NONE => NONE 7314 | SOME y => 7315 let 7316 val [y, n, x] = map HOLogic.mk_numeral [y, n, x] 7317 val thm = Thm.instantiate' [] (map (SOME o Thm.cterm_of ctxt) [y, n, x, m]) 7318 @{thm powr_numeral_simproc_aux} 7319 in 7320 SOME (@{thm transitive} OF [eq_thm, thm]) 7321 end 7322 end 7323 handle TERM _ => NONE 7324 | Match => NONE 7325 7326end 7327\<close> 7328 7329end 7330 7331simproc_setup sqrt_numeral ("sqrt (numeral n)") = 7332 \<open>K Root_Numeral_Simproc.sqrt_simproc\<close> 7333 7334simproc_setup root_numeral ("root (numeral n) (numeral x)") = 7335 \<open>K (Root_Numeral_Simproc.root_simproc (200, Integer.pow 200 2))\<close> 7336 7337simproc_setup powr_divide_numeral 7338 ("numeral x powr (m / numeral n :: real)" | "numeral x powr (inverse (numeral n) :: real)") = 7339 \<open>K (Root_Numeral_Simproc.powr_simproc (200, Integer.pow 200 2))\<close> 7340 7341 7342lemma "root 100 1267650600228229401496703205376 = 2" 7343 by simp 7344 7345lemma "sqrt 196 = 14" 7346 by simp 7347 7348lemma "256 powr (7 / 4 :: real) = 16384" 7349 by simp 7350 7351lemma "27 powr (inverse 3) = (3::real)" 7352 by simp 7353 7354end 7355