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: divide_simps binomial_fact) 106 also from assms have "(real k + 1) / (real n - real k) < 1" by simp 107 finally show ?thesis using k by (simp add: mult_less_cancel_left) 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 "\<bar>z\<bar> < \<bar>x\<bar>"}.\<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: divide_simps) 303 moreover obtain N where N: "norm x / (1 - norm x) < of_int N" 304 using ex_le_of_int by (meson ex_less_of_int) 305 ultimately have N0: "N>0" 306 by auto 307 then have *: "real_of_int (N + 1) * norm x / real_of_int N < 1" 308 using N assms by (auto simp: field_simps) 309 have **: "real_of_int N * (norm x * (real_of_nat (Suc n) * norm (x ^ n))) \<le> 310 real_of_nat n * (norm x * ((1 + N) * norm (x ^ n)))" if "N \<le> int n" for n :: nat 311 proof - 312 from that have "real_of_int N * real_of_nat (Suc n) \<le> real_of_nat n * real_of_int (1 + N)" 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 divide_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 nat_diff_sum_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 continuous_on_id) 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 then have [simp]: "x * of_nat n / (1 + of_nat n) / of_nat n = x / (1 + of_nat n)" 1538 by simp 1539 have [simp]: "x / (1 + of_nat n) + x * of_nat n / (1 + of_nat n) = x" 1540 apply (simp add: divide_simps) 1541 using of_nat_eq_0_iff apply (fastforce simp: distrib_left) 1542 done 1543 show ?thesis 1544 using Suc.IH [of "x * of_nat n / (1 + of_nat n)"] False 1545 by (simp add: exp_add [symmetric]) 1546 qed 1547qed simp 1548 1549 1550subsubsection \<open>Properties of the Exponential Function on Reals\<close> 1551 1552text \<open>Comparisons of @{term "exp x"} with zero.\<close> 1553 1554text \<open>Proof: because every exponential can be seen as a square.\<close> 1555lemma exp_ge_zero [simp]: "0 \<le> exp x" 1556 for x :: real 1557proof - 1558 have "0 \<le> exp (x/2) * exp (x/2)" 1559 by simp 1560 then show ?thesis 1561 by (simp add: exp_add [symmetric]) 1562qed 1563 1564lemma exp_gt_zero [simp]: "0 < exp x" 1565 for x :: real 1566 by (simp add: order_less_le) 1567 1568lemma not_exp_less_zero [simp]: "\<not> exp x < 0" 1569 for x :: real 1570 by (simp add: not_less) 1571 1572lemma not_exp_le_zero [simp]: "\<not> exp x \<le> 0" 1573 for x :: real 1574 by (simp add: not_le) 1575 1576lemma abs_exp_cancel [simp]: "\<bar>exp x\<bar> = exp x" 1577 for x :: real 1578 by simp 1579 1580text \<open>Strict monotonicity of exponential.\<close> 1581 1582lemma exp_ge_add_one_self_aux: 1583 fixes x :: real 1584 assumes "0 \<le> x" 1585 shows "1 + x \<le> exp x" 1586 using order_le_imp_less_or_eq [OF assms] 1587proof 1588 assume "0 < x" 1589 have "1 + x \<le> (\<Sum>n<2. inverse (fact n) * x^n)" 1590 by (auto simp: numeral_2_eq_2) 1591 also have "\<dots> \<le> (\<Sum>n. inverse (fact n) * x^n)" 1592 apply (rule sum_le_suminf [OF summable_exp]) 1593 using \<open>0 < x\<close> 1594 apply (auto simp add: zero_le_mult_iff) 1595 done 1596 finally show "1 + x \<le> exp x" 1597 by (simp add: exp_def) 1598qed auto 1599 1600lemma exp_gt_one: "0 < x \<Longrightarrow> 1 < exp x" 1601 for x :: real 1602proof - 1603 assume x: "0 < x" 1604 then have "1 < 1 + x" by simp 1605 also from x have "1 + x \<le> exp x" 1606 by (simp add: exp_ge_add_one_self_aux) 1607 finally show ?thesis . 1608qed 1609 1610lemma exp_less_mono: 1611 fixes x y :: real 1612 assumes "x < y" 1613 shows "exp x < exp y" 1614proof - 1615 from \<open>x < y\<close> have "0 < y - x" by simp 1616 then have "1 < exp (y - x)" by (rule exp_gt_one) 1617 then have "1 < exp y / exp x" by (simp only: exp_diff) 1618 then show "exp x < exp y" by simp 1619qed 1620 1621lemma exp_less_cancel: "exp x < exp y \<Longrightarrow> x < y" 1622 for x y :: real 1623 unfolding linorder_not_le [symmetric] 1624 by (auto simp: order_le_less exp_less_mono) 1625 1626lemma exp_less_cancel_iff [iff]: "exp x < exp y \<longleftrightarrow> x < y" 1627 for x y :: real 1628 by (auto intro: exp_less_mono exp_less_cancel) 1629 1630lemma exp_le_cancel_iff [iff]: "exp x \<le> exp y \<longleftrightarrow> x \<le> y" 1631 for x y :: real 1632 by (auto simp: linorder_not_less [symmetric]) 1633 1634lemma exp_inj_iff [iff]: "exp x = exp y \<longleftrightarrow> x = y" 1635 for x y :: real 1636 by (simp add: order_eq_iff) 1637 1638text \<open>Comparisons of @{term "exp x"} with one.\<close> 1639 1640lemma one_less_exp_iff [simp]: "1 < exp x \<longleftrightarrow> 0 < x" 1641 for x :: real 1642 using exp_less_cancel_iff [where x = 0 and y = x] by simp 1643 1644lemma exp_less_one_iff [simp]: "exp x < 1 \<longleftrightarrow> x < 0" 1645 for x :: real 1646 using exp_less_cancel_iff [where x = x and y = 0] by simp 1647 1648lemma one_le_exp_iff [simp]: "1 \<le> exp x \<longleftrightarrow> 0 \<le> x" 1649 for x :: real 1650 using exp_le_cancel_iff [where x = 0 and y = x] by simp 1651 1652lemma exp_le_one_iff [simp]: "exp x \<le> 1 \<longleftrightarrow> x \<le> 0" 1653 for x :: real 1654 using exp_le_cancel_iff [where x = x and y = 0] by simp 1655 1656lemma exp_eq_one_iff [simp]: "exp x = 1 \<longleftrightarrow> x = 0" 1657 for x :: real 1658 using exp_inj_iff [where x = x and y = 0] by simp 1659 1660lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x \<and> x \<le> y - 1 \<and> exp x = y" 1661 for y :: real 1662proof (rule IVT) 1663 assume "1 \<le> y" 1664 then have "0 \<le> y - 1" by simp 1665 then have "1 + (y - 1) \<le> exp (y - 1)" 1666 by (rule exp_ge_add_one_self_aux) 1667 then show "y \<le> exp (y - 1)" by simp 1668qed (simp_all add: le_diff_eq) 1669 1670lemma exp_total: "0 < y \<Longrightarrow> \<exists>x. exp x = y" 1671 for y :: real 1672proof (rule linorder_le_cases [of 1 y]) 1673 assume "1 \<le> y" 1674 then show "\<exists>x. exp x = y" 1675 by (fast dest: lemma_exp_total) 1676next 1677 assume "0 < y" and "y \<le> 1" 1678 then have "1 \<le> inverse y" 1679 by (simp add: one_le_inverse_iff) 1680 then obtain x where "exp x = inverse y" 1681 by (fast dest: lemma_exp_total) 1682 then have "exp (- x) = y" 1683 by (simp add: exp_minus) 1684 then show "\<exists>x. exp x = y" .. 1685qed 1686 1687 1688subsection \<open>Natural Logarithm\<close> 1689 1690class ln = real_normed_algebra_1 + banach + 1691 fixes ln :: "'a \<Rightarrow> 'a" 1692 assumes ln_one [simp]: "ln 1 = 0" 1693 1694definition powr :: "'a \<Rightarrow> 'a \<Rightarrow> 'a::ln" (infixr "powr" 80) 1695 \<comment> \<open>exponentation via ln and exp\<close> 1696 where [code del]: "x powr a \<equiv> if x = 0 then 0 else exp (a * ln x)" 1697 1698lemma powr_0 [simp]: "0 powr z = 0" 1699 by (simp add: powr_def) 1700 1701 1702instantiation real :: ln 1703begin 1704 1705definition ln_real :: "real \<Rightarrow> real" 1706 where "ln_real x = (THE u. exp u = x)" 1707 1708instance 1709 by intro_classes (simp add: ln_real_def) 1710 1711end 1712 1713lemma powr_eq_0_iff [simp]: "w powr z = 0 \<longleftrightarrow> w = 0" 1714 by (simp add: powr_def) 1715 1716lemma ln_exp [simp]: "ln (exp x) = x" 1717 for x :: real 1718 by (simp add: ln_real_def) 1719 1720lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x" 1721 for x :: real 1722 by (auto dest: exp_total) 1723 1724lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x" 1725 for x :: real 1726 by (metis exp_gt_zero exp_ln) 1727 1728lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y" 1729 for x :: real 1730 by (erule subst) (rule ln_exp) 1731 1732lemma ln_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y" 1733 for x :: real 1734 by (rule ln_unique) (simp add: exp_add) 1735 1736lemma 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" 1737 for f :: "'a \<Rightarrow> real" 1738 by (induct I rule: finite_induct) (auto simp: ln_mult prod_pos) 1739 1740lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x" 1741 for x :: real 1742 by (rule ln_unique) (simp add: exp_minus) 1743 1744lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y" 1745 for x :: real 1746 by (rule ln_unique) (simp add: exp_diff) 1747 1748lemma ln_realpow: "0 < x \<Longrightarrow> ln (x^n) = real n * ln x" 1749 by (rule ln_unique) (simp add: exp_of_nat_mult) 1750 1751lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y" 1752 for x :: real 1753 by (subst exp_less_cancel_iff [symmetric]) simp 1754 1755lemma ln_le_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y" 1756 for x :: real 1757 by (simp add: linorder_not_less [symmetric]) 1758 1759lemma ln_inj_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y" 1760 for x :: real 1761 by (simp add: order_eq_iff) 1762 1763lemma ln_add_one_self_le_self: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x" 1764 for x :: real 1765 by (rule exp_le_cancel_iff [THEN iffD1]) (simp add: exp_ge_add_one_self_aux) 1766 1767lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x" 1768 for x :: real 1769 by (rule order_less_le_trans [where y = "ln (1 + x)"]) (simp_all add: ln_add_one_self_le_self) 1770 1771lemma ln_ge_iff: "\<And>x::real. 0 < x \<Longrightarrow> y \<le> ln x \<longleftrightarrow> exp y \<le> x" 1772 using exp_le_cancel_iff exp_total by force 1773 1774lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x" 1775 for x :: real 1776 using ln_le_cancel_iff [of 1 x] by simp 1777 1778lemma ln_ge_zero_imp_ge_one: "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x" 1779 for x :: real 1780 using ln_le_cancel_iff [of 1 x] by simp 1781 1782lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x" 1783 for x :: real 1784 using ln_le_cancel_iff [of 1 x] by simp 1785 1786lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1" 1787 for x :: real 1788 using ln_less_cancel_iff [of x 1] by simp 1789 1790lemma ln_le_zero_iff [simp]: "0 < x \<Longrightarrow> ln x \<le> 0 \<longleftrightarrow> x \<le> 1" 1791 for x :: real 1792 by (metis less_numeral_extra(1) ln_le_cancel_iff ln_one) 1793 1794lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x" 1795 for x :: real 1796 using ln_less_cancel_iff [of 1 x] by simp 1797 1798lemma ln_gt_zero_imp_gt_one: "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x" 1799 for x :: real 1800 using ln_less_cancel_iff [of 1 x] by simp 1801 1802lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x" 1803 for x :: real 1804 using ln_less_cancel_iff [of 1 x] by simp 1805 1806lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1" 1807 for x :: real 1808 using ln_inj_iff [of x 1] by simp 1809 1810lemma ln_less_zero: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0" 1811 for x :: real 1812 by simp 1813 1814lemma ln_neg_is_const: "x \<le> 0 \<Longrightarrow> ln x = (THE x. False)" 1815 for x :: real 1816 by (auto simp: ln_real_def intro!: arg_cong[where f = The]) 1817 1818lemma isCont_ln: 1819 fixes x :: real 1820 assumes "x \<noteq> 0" 1821 shows "isCont ln x" 1822proof (cases "0 < x") 1823 case True 1824 then have "isCont ln (exp (ln x))" 1825 by (intro isCont_inverse_function[where d = "\<bar>x\<bar>" and f = exp]) auto 1826 with True show ?thesis 1827 by simp 1828next 1829 case False 1830 with \<open>x \<noteq> 0\<close> show "isCont ln x" 1831 unfolding isCont_def 1832 by (subst filterlim_cong[OF _ refl, of _ "nhds (ln 0)" _ "\<lambda>_. ln 0"]) 1833 (auto simp: ln_neg_is_const not_less eventually_at dist_real_def 1834 intro!: exI[of _ "\<bar>x\<bar>"]) 1835qed 1836 1837lemma tendsto_ln [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. ln (f x)) \<longlongrightarrow> ln a) F" 1838 for a :: real 1839 by (rule isCont_tendsto_compose [OF isCont_ln]) 1840 1841lemma continuous_ln: 1842 "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. ln (f x :: real))" 1843 unfolding continuous_def by (rule tendsto_ln) 1844 1845lemma isCont_ln' [continuous_intros]: 1846 "continuous (at x) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x :: real))" 1847 unfolding continuous_at by (rule tendsto_ln) 1848 1849lemma continuous_within_ln [continuous_intros]: 1850 "continuous (at x within s) f \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x :: real))" 1851 unfolding continuous_within by (rule tendsto_ln) 1852 1853lemma continuous_on_ln [continuous_intros]: 1854 "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. f x \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x :: real))" 1855 unfolding continuous_on_def by (auto intro: tendsto_ln) 1856 1857lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x" 1858 for x :: real 1859 by (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"]) 1860 (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln) 1861 1862lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x" 1863 for x :: real 1864 by (rule DERIV_ln[THEN DERIV_cong]) (simp_all add: divide_inverse) 1865 1866declare DERIV_ln_divide[THEN DERIV_chain2, derivative_intros] 1867 and DERIV_ln_divide[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 1868 1869lemmas has_derivative_ln[derivative_intros] = DERIV_ln[THEN DERIV_compose_FDERIV] 1870 1871lemma ln_series: 1872 assumes "0 < x" and "x < 2" 1873 shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" 1874 (is "ln x = suminf (?f (x - 1))") 1875proof - 1876 let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n" 1877 1878 have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))" 1879 proof (rule DERIV_isconst3 [where x = x]) 1880 fix x :: real 1881 assume "x \<in> {0 <..< 2}" 1882 then have "0 < x" and "x < 2" by auto 1883 have "norm (1 - x) < 1" 1884 using \<open>0 < x\<close> and \<open>x < 2\<close> by auto 1885 have "1 / x = 1 / (1 - (1 - x))" by auto 1886 also have "\<dots> = (\<Sum> n. (1 - x)^n)" 1887 using geometric_sums[OF \<open>norm (1 - x) < 1\<close>] by (rule sums_unique) 1888 also have "\<dots> = suminf (?f' x)" 1889 unfolding power_mult_distrib[symmetric] 1890 by (rule arg_cong[where f=suminf], rule arg_cong[where f="(^)"], auto) 1891 finally have "DERIV ln x :> suminf (?f' x)" 1892 using DERIV_ln[OF \<open>0 < x\<close>] unfolding divide_inverse by auto 1893 moreover 1894 have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto 1895 have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> 1896 (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)" 1897 proof (rule DERIV_power_series') 1898 show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" 1899 using \<open>0 < x\<close> \<open>x < 2\<close> by auto 1900 next 1901 fix x :: real 1902 assume "x \<in> {- 1<..<1}" 1903 then have "norm (-x) < 1" by auto 1904 show "summable (\<lambda>n. (- 1) ^ n * (1 / real (n + 1)) * real (Suc n) * x^n)" 1905 unfolding One_nat_def 1906 by (auto simp: power_mult_distrib[symmetric] summable_geometric[OF \<open>norm (-x) < 1\<close>]) 1907 qed 1908 then have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" 1909 unfolding One_nat_def by auto 1910 then have "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" 1911 unfolding DERIV_def repos . 1912 ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> suminf (?f' x) - suminf (?f' x)" 1913 by (rule DERIV_diff) 1914 then show "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto 1915 qed (auto simp: assms) 1916 then show ?thesis by auto 1917qed 1918 1919lemma exp_first_terms: 1920 fixes x :: "'a::{real_normed_algebra_1,banach}" 1921 shows "exp x = (\<Sum>n<k. inverse(fact n) *\<^sub>R (x ^ n)) + (\<Sum>n. inverse(fact (n + k)) *\<^sub>R (x ^ (n + k)))" 1922proof - 1923 have "exp x = suminf (\<lambda>n. inverse(fact n) *\<^sub>R (x^n))" 1924 by (simp add: exp_def) 1925 also from summable_exp_generic have "\<dots> = (\<Sum> n. inverse(fact(n+k)) *\<^sub>R (x ^ (n + k))) + 1926 (\<Sum> n::nat<k. inverse(fact n) *\<^sub>R (x^n))" (is "_ = _ + ?a") 1927 by (rule suminf_split_initial_segment) 1928 finally show ?thesis by simp 1929qed 1930 1931lemma exp_first_term: "exp x = 1 + (\<Sum>n. inverse (fact (Suc n)) *\<^sub>R (x ^ Suc n))" 1932 for x :: "'a::{real_normed_algebra_1,banach}" 1933 using exp_first_terms[of x 1] by simp 1934 1935lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum>n. inverse (fact (n + 2)) *\<^sub>R (x ^ (n + 2)))" 1936 for x :: "'a::{real_normed_algebra_1,banach}" 1937 using exp_first_terms[of x 2] by (simp add: eval_nat_numeral) 1938 1939lemma exp_bound: 1940 fixes x :: real 1941 assumes a: "0 \<le> x" 1942 and b: "x \<le> 1" 1943 shows "exp x \<le> 1 + x + x\<^sup>2" 1944proof - 1945 have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n + 2))) \<le> x\<^sup>2" 1946 proof - 1947 have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))" 1948 by (intro sums_mult geometric_sums) simp 1949 then have sumsx: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2" 1950 by simp 1951 have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n + 2))) \<le> suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))" 1952 proof (intro suminf_le allI) 1953 show "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)" for n :: nat 1954 proof - 1955 have "(2::nat) * 2 ^ n \<le> fact (n + 2)" 1956 by (induct n) simp_all 1957 then have "real ((2::nat) * 2 ^ n) \<le> real_of_nat (fact (n + 2))" 1958 by (simp only: of_nat_le_iff) 1959 then have "((2::real) * 2 ^ n) \<le> fact (n + 2)" 1960 unfolding of_nat_fact by simp 1961 then have "inverse (fact (n + 2)) \<le> inverse ((2::real) * 2 ^ n)" 1962 by (rule le_imp_inverse_le) simp 1963 then have "inverse (fact (n + 2)) \<le> 1/(2::real) * (1/2)^n" 1964 by (simp add: power_inverse [symmetric]) 1965 then have "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)" 1966 by (rule mult_mono) (rule mult_mono, simp_all add: power_le_one a b) 1967 then show ?thesis 1968 unfolding power_add by (simp add: ac_simps del: fact_Suc) 1969 qed 1970 show "summable (\<lambda>n. inverse (fact (n + 2)) * x ^ (n + 2))" 1971 by (rule summable_exp [THEN summable_ignore_initial_segment]) 1972 show "summable (\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n)" 1973 by (rule sums_summable [OF sumsx]) 1974 qed 1975 also have "\<dots> = x\<^sup>2" 1976 by (rule sums_unique [THEN sym]) (rule sumsx) 1977 finally show ?thesis . 1978 qed 1979 then show ?thesis 1980 unfolding exp_first_two_terms by auto 1981qed 1982 1983corollary exp_half_le2: "exp(1/2) \<le> (2::real)" 1984 using exp_bound [of "1/2"] 1985 by (simp add: field_simps) 1986 1987corollary exp_le: "exp 1 \<le> (3::real)" 1988 using exp_bound [of 1] 1989 by (simp add: field_simps) 1990 1991lemma exp_bound_half: "norm z \<le> 1/2 \<Longrightarrow> norm (exp z) \<le> 2" 1992 by (blast intro: order_trans intro!: exp_half_le2 norm_exp) 1993 1994lemma exp_bound_lemma: 1995 assumes "norm z \<le> 1/2" 1996 shows "norm (exp z) \<le> 1 + 2 * norm z" 1997proof - 1998 have *: "(norm z)\<^sup>2 \<le> norm z * 1" 1999 unfolding power2_eq_square 2000 by (rule mult_left_mono) (use assms in auto) 2001 have "norm (exp z) \<le> exp (norm z)" 2002 by (rule norm_exp) 2003 also have "\<dots> \<le> 1 + (norm z) + (norm z)\<^sup>2" 2004 using assms exp_bound by auto 2005 also have "\<dots> \<le> 1 + 2 * norm z" 2006 using * by auto 2007 finally show ?thesis . 2008qed 2009 2010lemma real_exp_bound_lemma: "0 \<le> x \<Longrightarrow> x \<le> 1/2 \<Longrightarrow> exp x \<le> 1 + 2 * x" 2011 for x :: real 2012 using exp_bound_lemma [of x] by simp 2013 2014lemma ln_one_minus_pos_upper_bound: 2015 fixes x :: real 2016 assumes a: "0 \<le> x" and b: "x < 1" 2017 shows "ln (1 - x) \<le> - x" 2018proof - 2019 have "(1 - x) * (1 + x + x\<^sup>2) = 1 - x^3" 2020 by (simp add: algebra_simps power2_eq_square power3_eq_cube) 2021 also have "\<dots> \<le> 1" 2022 by (auto simp: a) 2023 finally have "(1 - x) * (1 + x + x\<^sup>2) \<le> 1" . 2024 moreover have c: "0 < 1 + x + x\<^sup>2" 2025 by (simp add: add_pos_nonneg a) 2026 ultimately have "1 - x \<le> 1 / (1 + x + x\<^sup>2)" 2027 by (elim mult_imp_le_div_pos) 2028 also have "\<dots> \<le> 1 / exp x" 2029 by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs 2030 real_sqrt_pow2_iff real_sqrt_power) 2031 also have "\<dots> = exp (- x)" 2032 by (auto simp: exp_minus divide_inverse) 2033 finally have "1 - x \<le> exp (- x)" . 2034 also have "1 - x = exp (ln (1 - x))" 2035 by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq) 2036 finally have "exp (ln (1 - x)) \<le> exp (- x)" . 2037 then show ?thesis 2038 by (auto simp only: exp_le_cancel_iff) 2039qed 2040 2041lemma exp_ge_add_one_self [simp]: "1 + x \<le> exp x" 2042 for x :: real 2043proof (cases "0 \<le> x \<or> x \<le> -1") 2044 case True 2045 then show ?thesis 2046 apply (rule disjE) 2047 apply (simp add: exp_ge_add_one_self_aux) 2048 using exp_ge_zero order_trans real_add_le_0_iff by blast 2049next 2050 case False 2051 then have ln1: "ln (1 + x) \<le> x" 2052 using ln_one_minus_pos_upper_bound [of "-x"] by simp 2053 have "1 + x = exp (ln (1 + x))" 2054 using False by auto 2055 also have "\<dots> \<le> exp x" 2056 by (simp add: ln1) 2057 finally show ?thesis . 2058qed 2059 2060lemma ln_one_plus_pos_lower_bound: 2061 fixes x :: real 2062 assumes a: "0 \<le> x" and b: "x \<le> 1" 2063 shows "x - x\<^sup>2 \<le> ln (1 + x)" 2064proof - 2065 have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)" 2066 by (rule exp_diff) 2067 also have "\<dots> \<le> (1 + x + x\<^sup>2) / exp (x \<^sup>2)" 2068 by (metis a b divide_right_mono exp_bound exp_ge_zero) 2069 also have "\<dots> \<le> (1 + x + x\<^sup>2) / (1 + x\<^sup>2)" 2070 by (simp add: a divide_left_mono add_pos_nonneg) 2071 also from a have "\<dots> \<le> 1 + x" 2072 by (simp add: field_simps add_strict_increasing zero_le_mult_iff) 2073 finally have "exp (x - x\<^sup>2) \<le> 1 + x" . 2074 also have "\<dots> = exp (ln (1 + x))" 2075 proof - 2076 from a have "0 < 1 + x" by auto 2077 then show ?thesis 2078 by (auto simp only: exp_ln_iff [THEN sym]) 2079 qed 2080 finally have "exp (x - x\<^sup>2) \<le> exp (ln (1 + x))" . 2081 then show ?thesis 2082 by (metis exp_le_cancel_iff) 2083qed 2084 2085lemma ln_one_minus_pos_lower_bound: 2086 fixes x :: real 2087 assumes a: "0 \<le> x" and b: "x \<le> 1 / 2" 2088 shows "- x - 2 * x\<^sup>2 \<le> ln (1 - x)" 2089proof - 2090 from b have c: "x < 1" by auto 2091 then have "ln (1 - x) = - ln (1 + x / (1 - x))" 2092 by (auto simp: ln_inverse [symmetric] field_simps intro: arg_cong [where f=ln]) 2093 also have "- (x / (1 - x)) \<le> \<dots>" 2094 proof - 2095 have "ln (1 + x / (1 - x)) \<le> x / (1 - x)" 2096 using a c by (intro ln_add_one_self_le_self) auto 2097 then show ?thesis 2098 by auto 2099 qed 2100 also have "- (x / (1 - x)) = - x / (1 - x)" 2101 by auto 2102 finally have d: "- x / (1 - x) \<le> ln (1 - x)" . 2103 have "0 < 1 - x" using a b by simp 2104 then have e: "- x - 2 * x\<^sup>2 \<le> - x / (1 - x)" 2105 using mult_right_le_one_le[of "x * x" "2 * x"] a b 2106 by (simp add: field_simps power2_eq_square) 2107 from e d show "- x - 2 * x\<^sup>2 \<le> ln (1 - x)" 2108 by (rule order_trans) 2109qed 2110 2111lemma ln_add_one_self_le_self2: 2112 fixes x :: real 2113 shows "-1 < x \<Longrightarrow> ln (1 + x) \<le> x" 2114 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) 2115 2116lemma abs_ln_one_plus_x_minus_x_bound_nonneg: 2117 fixes x :: real 2118 assumes x: "0 \<le> x" and x1: "x \<le> 1" 2119 shows "\<bar>ln (1 + x) - x\<bar> \<le> x\<^sup>2" 2120proof - 2121 from x have "ln (1 + x) \<le> x" 2122 by (rule ln_add_one_self_le_self) 2123 then have "ln (1 + x) - x \<le> 0" 2124 by simp 2125 then have "\<bar>ln(1 + x) - x\<bar> = - (ln(1 + x) - x)" 2126 by (rule abs_of_nonpos) 2127 also have "\<dots> = x - ln (1 + x)" 2128 by simp 2129 also have "\<dots> \<le> x\<^sup>2" 2130 proof - 2131 from x x1 have "x - x\<^sup>2 \<le> ln (1 + x)" 2132 by (intro ln_one_plus_pos_lower_bound) 2133 then show ?thesis 2134 by simp 2135 qed 2136 finally show ?thesis . 2137qed 2138 2139lemma abs_ln_one_plus_x_minus_x_bound_nonpos: 2140 fixes x :: real 2141 assumes a: "-(1 / 2) \<le> x" and b: "x \<le> 0" 2142 shows "\<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2" 2143proof - 2144 have *: "- (-x) - 2 * (-x)\<^sup>2 \<le> ln (1 - (- x))" 2145 by (metis a b diff_zero ln_one_minus_pos_lower_bound minus_diff_eq neg_le_iff_le) 2146 have "\<bar>ln (1 + x) - x\<bar> = x - ln (1 - (- x))" 2147 using a ln_add_one_self_le_self2 [of x] by (simp add: abs_if) 2148 also have "\<dots> \<le> 2 * x\<^sup>2" 2149 using * by (simp add: algebra_simps) 2150 finally show ?thesis . 2151qed 2152 2153lemma abs_ln_one_plus_x_minus_x_bound: 2154 fixes x :: real 2155 assumes "\<bar>x\<bar> \<le> 1 / 2" 2156 shows "\<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2" 2157proof (cases "0 \<le> x") 2158 case True 2159 then show ?thesis 2160 using abs_ln_one_plus_x_minus_x_bound_nonneg assms by fastforce 2161next 2162 case False 2163 then show ?thesis 2164 using abs_ln_one_plus_x_minus_x_bound_nonpos assms by auto 2165qed 2166 2167lemma ln_x_over_x_mono: 2168 fixes x :: real 2169 assumes x: "exp 1 \<le> x" "x \<le> y" 2170 shows "ln y / y \<le> ln x / x" 2171proof - 2172 note x 2173 moreover have "0 < exp (1::real)" by simp 2174 ultimately have a: "0 < x" and b: "0 < y" 2175 by (fast intro: less_le_trans order_trans)+ 2176 have "x * ln y - x * ln x = x * (ln y - ln x)" 2177 by (simp add: algebra_simps) 2178 also have "\<dots> = x * ln (y / x)" 2179 by (simp only: ln_div a b) 2180 also have "y / x = (x + (y - x)) / x" 2181 by simp 2182 also have "\<dots> = 1 + (y - x) / x" 2183 using x a by (simp add: field_simps) 2184 also have "x * ln (1 + (y - x) / x) \<le> x * ((y - x) / x)" 2185 using x a 2186 by (intro mult_left_mono ln_add_one_self_le_self) simp_all 2187 also have "\<dots> = y - x" 2188 using a by simp 2189 also have "\<dots> = (y - x) * ln (exp 1)" by simp 2190 also have "\<dots> \<le> (y - x) * ln x" 2191 using a x exp_total of_nat_1 x(1) by (fastforce intro: mult_left_mono) 2192 also have "\<dots> = y * ln x - x * ln x" 2193 by (rule left_diff_distrib) 2194 finally have "x * ln y \<le> y * ln x" 2195 by arith 2196 then have "ln y \<le> (y * ln x) / x" 2197 using a by (simp add: field_simps) 2198 also have "\<dots> = y * (ln x / x)" by simp 2199 finally show ?thesis 2200 using b by (simp add: field_simps) 2201qed 2202 2203lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1" 2204 for x :: real 2205 using exp_ge_add_one_self[of "ln x"] by simp 2206 2207corollary ln_diff_le: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x - ln y \<le> (x - y) / y" 2208 for x :: real 2209 by (simp add: ln_div [symmetric] diff_divide_distrib ln_le_minus_one) 2210 2211lemma ln_eq_minus_one: 2212 fixes x :: real 2213 assumes "0 < x" "ln x = x - 1" 2214 shows "x = 1" 2215proof - 2216 let ?l = "\<lambda>y. ln y - y + 1" 2217 have D: "\<And>x::real. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)" 2218 by (auto intro!: derivative_eq_intros) 2219 2220 show ?thesis 2221 proof (cases rule: linorder_cases) 2222 assume "x < 1" 2223 from dense[OF \<open>x < 1\<close>] obtain a where "x < a" "a < 1" by blast 2224 from \<open>x < a\<close> have "?l x < ?l a" 2225 proof (rule DERIV_pos_imp_increasing, safe) 2226 fix y 2227 assume "x \<le> y" "y \<le> a" 2228 with \<open>0 < x\<close> \<open>a < 1\<close> have "0 < 1 / y - 1" "0 < y" 2229 by (auto simp: field_simps) 2230 with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z" by blast 2231 qed 2232 also have "\<dots> \<le> 0" 2233 using ln_le_minus_one \<open>0 < x\<close> \<open>x < a\<close> by (auto simp: field_simps) 2234 finally show "x = 1" using assms by auto 2235 next 2236 assume "1 < x" 2237 from dense[OF this] obtain a where "1 < a" "a < x" by blast 2238 from \<open>a < x\<close> have "?l x < ?l a" 2239 proof (rule DERIV_neg_imp_decreasing) 2240 fix y 2241 assume "a \<le> y" "y \<le> x" 2242 with \<open>1 < a\<close> have "1 / y - 1 < 0" "0 < y" 2243 by (auto simp: field_simps) 2244 with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0" 2245 by blast 2246 qed 2247 also have "\<dots> \<le> 0" 2248 using ln_le_minus_one \<open>1 < a\<close> by (auto simp: field_simps) 2249 finally show "x = 1" using assms by auto 2250 next 2251 assume "x = 1" 2252 then show ?thesis by simp 2253 qed 2254qed 2255 2256lemma ln_x_over_x_tendsto_0: "((\<lambda>x::real. ln x / x) \<longlongrightarrow> 0) at_top" 2257proof (rule lhospital_at_top_at_top[where f' = inverse and g' = "\<lambda>_. 1"]) 2258 from eventually_gt_at_top[of "0::real"] 2259 show "\<forall>\<^sub>F x in at_top. (ln has_real_derivative inverse x) (at x)" 2260 by eventually_elim (auto intro!: derivative_eq_intros simp: field_simps) 2261qed (use tendsto_inverse_0 in 2262 \<open>auto simp: filterlim_ident dest!: tendsto_mono[OF at_top_le_at_infinity]\<close>) 2263 2264lemma exp_ge_one_plus_x_over_n_power_n: 2265 assumes "x \<ge> - real n" "n > 0" 2266 shows "(1 + x / of_nat n) ^ n \<le> exp x" 2267proof (cases "x = - of_nat n") 2268 case False 2269 from assms False have "(1 + x / of_nat n) ^ n = exp (of_nat n * ln (1 + x / of_nat n))" 2270 by (subst exp_of_nat_mult, subst exp_ln) (simp_all add: field_simps) 2271 also from assms False have "ln (1 + x / real n) \<le> x / real n" 2272 by (intro ln_add_one_self_le_self2) (simp_all add: field_simps) 2273 with assms have "exp (of_nat n * ln (1 + x / of_nat n)) \<le> exp x" 2274 by (simp add: field_simps) 2275 finally show ?thesis . 2276next 2277 case True 2278 then show ?thesis by (simp add: zero_power) 2279qed 2280 2281lemma exp_ge_one_minus_x_over_n_power_n: 2282 assumes "x \<le> real n" "n > 0" 2283 shows "(1 - x / of_nat n) ^ n \<le> exp (-x)" 2284 using exp_ge_one_plus_x_over_n_power_n[of n "-x"] assms by simp 2285 2286lemma exp_at_bot: "(exp \<longlongrightarrow> (0::real)) at_bot" 2287 unfolding tendsto_Zfun_iff 2288proof (rule ZfunI, simp add: eventually_at_bot_dense) 2289 fix r :: real 2290 assume "0 < r" 2291 have "exp x < r" if "x < ln r" for x 2292 by (metis \<open>0 < r\<close> exp_less_mono exp_ln that) 2293 then show "\<exists>k. \<forall>n<k. exp n < r" by auto 2294qed 2295 2296lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top" 2297 by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g=ln]) 2298 (auto intro: eventually_gt_at_top) 2299 2300lemma lim_exp_minus_1: "((\<lambda>z::'a. (exp(z) - 1) / z) \<longlongrightarrow> 1) (at 0)" 2301 for x :: "'a::{real_normed_field,banach}" 2302proof - 2303 have "((\<lambda>z::'a. exp(z) - 1) has_field_derivative 1) (at 0)" 2304 by (intro derivative_eq_intros | simp)+ 2305 then show ?thesis 2306 by (simp add: Deriv.has_field_derivative_iff) 2307qed 2308 2309lemma ln_at_0: "LIM x at_right 0. ln (x::real) :> at_bot" 2310 by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g=exp]) 2311 (auto simp: eventually_at_filter) 2312 2313lemma ln_at_top: "LIM x at_top. ln (x::real) :> at_top" 2314 by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g=exp]) 2315 (auto intro: eventually_gt_at_top) 2316 2317lemma filtermap_ln_at_top: "filtermap (ln::real \<Rightarrow> real) at_top = at_top" 2318 by (intro filtermap_fun_inverse[of exp] exp_at_top ln_at_top) auto 2319 2320lemma filtermap_exp_at_top: "filtermap (exp::real \<Rightarrow> real) at_top = at_top" 2321 by (intro filtermap_fun_inverse[of ln] exp_at_top ln_at_top) 2322 (auto simp: eventually_at_top_dense) 2323 2324lemma filtermap_ln_at_right: "filtermap ln (at_right (0::real)) = at_bot" 2325 by (auto intro!: filtermap_fun_inverse[where g="\<lambda>x. exp x"] ln_at_0 2326 simp: filterlim_at exp_at_bot) 2327 2328lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) \<longlongrightarrow> (0::real)) at_top" 2329proof (induct k) 2330 case 0 2331 show "((\<lambda>x. x ^ 0 / exp x) \<longlongrightarrow> (0::real)) at_top" 2332 by (simp add: inverse_eq_divide[symmetric]) 2333 (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono 2334 at_top_le_at_infinity order_refl) 2335next 2336 case (Suc k) 2337 show ?case 2338 proof (rule lhospital_at_top_at_top) 2339 show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top" 2340 by eventually_elim (intro derivative_eq_intros, auto) 2341 show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top" 2342 by eventually_elim auto 2343 show "eventually (\<lambda>x. exp x \<noteq> 0) at_top" 2344 by auto 2345 from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"] 2346 show "((\<lambda>x. real (Suc k) * x ^ k / exp x) \<longlongrightarrow> 0) at_top" 2347 by simp 2348 qed (rule exp_at_top) 2349qed 2350 2351subsubsection\<open> A couple of simple bounds\<close> 2352 2353lemma exp_plus_inverse_exp: 2354 fixes x::real 2355 shows "2 \<le> exp x + inverse (exp x)" 2356proof - 2357 have "2 \<le> exp x + exp (-x)" 2358 using exp_ge_add_one_self [of x] exp_ge_add_one_self [of "-x"] 2359 by linarith 2360 then show ?thesis 2361 by (simp add: exp_minus) 2362qed 2363 2364lemma real_le_x_sinh: 2365 fixes x::real 2366 assumes "0 \<le> x" 2367 shows "x \<le> (exp x - inverse(exp x)) / 2" 2368proof - 2369 have *: "exp a - inverse(exp a) - 2*a \<le> exp b - inverse(exp b) - 2*b" if "a \<le> b" for a b::real 2370 using exp_plus_inverse_exp 2371 by (fastforce intro: derivative_eq_intros DERIV_nonneg_imp_nondecreasing [OF that]) 2372 show ?thesis 2373 using*[OF assms] by simp 2374qed 2375 2376lemma real_le_abs_sinh: 2377 fixes x::real 2378 shows "abs x \<le> abs((exp x - inverse(exp x)) / 2)" 2379proof (cases "0 \<le> x") 2380 case True 2381 show ?thesis 2382 using real_le_x_sinh [OF True] True by (simp add: abs_if) 2383next 2384 case False 2385 have "-x \<le> (exp(-x) - inverse(exp(-x))) / 2" 2386 by (meson False linear neg_le_0_iff_le real_le_x_sinh) 2387 also have "\<dots> \<le> \<bar>(exp x - inverse (exp x)) / 2\<bar>" 2388 by (metis (no_types, hide_lams) abs_divide abs_le_iff abs_minus_cancel 2389 add.inverse_inverse exp_minus minus_diff_eq order_refl) 2390 finally show ?thesis 2391 using False by linarith 2392qed 2393 2394subsection\<open>The general logarithm\<close> 2395 2396definition log :: "real \<Rightarrow> real \<Rightarrow> real" 2397 \<comment> \<open>logarithm of @{term x} to base @{term a}\<close> 2398 where "log a x = ln x / ln a" 2399 2400lemma tendsto_log [tendsto_intros]: 2401 "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> 0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> 2402 ((\<lambda>x. log (f x) (g x)) \<longlongrightarrow> log a b) F" 2403 unfolding log_def by (intro tendsto_intros) auto 2404 2405lemma continuous_log: 2406 assumes "continuous F f" 2407 and "continuous F g" 2408 and "0 < f (Lim F (\<lambda>x. x))" 2409 and "f (Lim F (\<lambda>x. x)) \<noteq> 1" 2410 and "0 < g (Lim F (\<lambda>x. x))" 2411 shows "continuous F (\<lambda>x. log (f x) (g x))" 2412 using assms unfolding continuous_def by (rule tendsto_log) 2413 2414lemma continuous_at_within_log[continuous_intros]: 2415 assumes "continuous (at a within s) f" 2416 and "continuous (at a within s) g" 2417 and "0 < f a" 2418 and "f a \<noteq> 1" 2419 and "0 < g a" 2420 shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))" 2421 using assms unfolding continuous_within by (rule tendsto_log) 2422 2423lemma isCont_log[continuous_intros, simp]: 2424 assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a" 2425 shows "isCont (\<lambda>x. log (f x) (g x)) a" 2426 using assms unfolding continuous_at by (rule tendsto_log) 2427 2428lemma continuous_on_log[continuous_intros]: 2429 assumes "continuous_on s f" "continuous_on s g" 2430 and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x" 2431 shows "continuous_on s (\<lambda>x. log (f x) (g x))" 2432 using assms unfolding continuous_on_def by (fast intro: tendsto_log) 2433 2434lemma powr_one_eq_one [simp]: "1 powr a = 1" 2435 by (simp add: powr_def) 2436 2437lemma powr_zero_eq_one [simp]: "x powr 0 = (if x = 0 then 0 else 1)" 2438 by (simp add: powr_def) 2439 2440lemma powr_one_gt_zero_iff [simp]: "x powr 1 = x \<longleftrightarrow> 0 \<le> x" 2441 for x :: real 2442 by (auto simp: powr_def) 2443declare powr_one_gt_zero_iff [THEN iffD2, simp] 2444 2445lemma powr_diff: 2446 fixes w:: "'a::{ln,real_normed_field}" shows "w powr (z1 - z2) = w powr z1 / w powr z2" 2447 by (simp add: powr_def algebra_simps exp_diff) 2448 2449lemma powr_mult: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)" 2450 for a x y :: real 2451 by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left) 2452 2453lemma powr_ge_pzero [simp]: "0 \<le> x powr y" 2454 for x y :: real 2455 by (simp add: powr_def) 2456 2457lemma powr_non_neg[simp]: "\<not>a powr x < 0" for a x::real 2458 using powr_ge_pzero[of a x] by arith 2459 2460lemma powr_divide: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)" 2461 for a b x :: real 2462 apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult) 2463 apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse) 2464 done 2465 2466lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)" 2467 for a b x :: "'a::{ln,real_normed_field}" 2468 by (simp add: powr_def exp_add [symmetric] distrib_right) 2469 2470lemma powr_mult_base: "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)" 2471 for x :: real 2472 by (auto simp: powr_add) 2473 2474lemma powr_powr: "(x powr a) powr b = x powr (a * b)" 2475 for a b x :: real 2476 by (simp add: powr_def) 2477 2478lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a" 2479 for a b x :: real 2480 by (simp add: powr_powr mult.commute) 2481 2482lemma powr_minus: "x powr (- a) = inverse (x powr a)" 2483 for a x :: "'a::{ln,real_normed_field}" 2484 by (simp add: powr_def exp_minus [symmetric]) 2485 2486lemma powr_minus_divide: "x powr (- a) = 1/(x powr a)" 2487 for a x :: "'a::{ln,real_normed_field}" 2488 by (simp add: divide_inverse powr_minus) 2489 2490lemma divide_powr_uminus: "a / b powr c = a * b powr (- c)" 2491 for a b c :: real 2492 by (simp add: powr_minus_divide) 2493 2494lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b" 2495 for a b x :: real 2496 by (simp add: powr_def) 2497 2498lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b" 2499 for a b x :: real 2500 by (simp add: powr_def) 2501 2502lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> x powr a < x powr b \<longleftrightarrow> a < b" 2503 for a b x :: real 2504 by (blast intro: powr_less_cancel powr_less_mono) 2505 2506lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> x powr a \<le> x powr b \<longleftrightarrow> a \<le> b" 2507 for a b x :: real 2508 by (simp add: linorder_not_less [symmetric]) 2509 2510lemma powr_realpow: "0 < x \<Longrightarrow> x powr (real n) = x^n" 2511by (induction n) (simp_all add: ac_simps powr_add) 2512 2513lemma log_ln: "ln x = log (exp(1)) x" 2514 by (simp add: log_def) 2515 2516lemma DERIV_log: 2517 assumes "x > 0" 2518 shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)" 2519proof - 2520 define lb where "lb = 1 / ln b" 2521 moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x" 2522 using \<open>x > 0\<close> by (auto intro!: derivative_eq_intros) 2523 ultimately show ?thesis 2524 by (simp add: log_def) 2525qed 2526 2527lemmas DERIV_log[THEN DERIV_chain2, derivative_intros] 2528 and DERIV_log[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 2529 2530lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x" 2531 by (simp add: powr_def log_def) 2532 2533lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y" 2534 by (simp add: log_def powr_def) 2535 2536lemma log_mult: 2537 "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> 2538 log a (x * y) = log a x + log a y" 2539 by (simp add: log_def ln_mult divide_inverse distrib_right) 2540 2541lemma log_eq_div_ln_mult_log: 2542 "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 2543 log a x = (ln b/ln a) * log b x" 2544 by (simp add: log_def divide_inverse) 2545 2546text\<open>Base 10 logarithms\<close> 2547lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x" 2548 by (simp add: log_def) 2549 2550lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x" 2551 by (simp add: log_def) 2552 2553lemma log_one [simp]: "log a 1 = 0" 2554 by (simp add: log_def) 2555 2556lemma log_eq_one [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a a = 1" 2557 by (simp add: log_def) 2558 2559lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x" 2560 using ln_inverse log_def by auto 2561 2562lemma 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" 2563 by (simp add: log_mult divide_inverse log_inverse) 2564 2565lemma powr_gt_zero [simp]: "0 < x powr a \<longleftrightarrow> x \<noteq> 0" 2566 for a x :: real 2567 by (simp add: powr_def) 2568 2569lemma powr_nonneg_iff[simp]: "a powr x \<le> 0 \<longleftrightarrow> a = 0" 2570 for a x::real 2571 by (meson not_less powr_gt_zero) 2572 2573lemma log_add_eq_powr: "0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log b x + y = log b (x * b powr y)" 2574 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)" 2575 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)" 2576 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)" 2577 by (simp_all add: log_mult log_divide) 2578 2579lemma log_less_cancel_iff [simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y" 2580 using powr_less_cancel_iff [of a] powr_log_cancel [of a x] powr_log_cancel [of a y] 2581 by (metis less_eq_real_def less_trans not_le zero_less_one) 2582 2583lemma log_inj: 2584 assumes "1 < b" 2585 shows "inj_on (log b) {0 <..}" 2586proof (rule inj_onI, simp) 2587 fix x y 2588 assume pos: "0 < x" "0 < y" and *: "log b x = log b y" 2589 show "x = y" 2590 proof (cases rule: linorder_cases) 2591 assume "x = y" 2592 then show ?thesis by simp 2593 next 2594 assume "x < y" 2595 then have "log b x < log b y" 2596 using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp 2597 then show ?thesis using * by simp 2598 next 2599 assume "y < x" 2600 then have "log b y < log b x" 2601 using log_less_cancel_iff[OF \<open>1 < b\<close>] pos by simp 2602 then show ?thesis using * by simp 2603 qed 2604qed 2605 2606lemma 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" 2607 by (simp add: linorder_not_less [symmetric]) 2608 2609lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x" 2610 using log_less_cancel_iff[of a 1 x] by simp 2611 2612lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x" 2613 using log_le_cancel_iff[of a 1 x] by simp 2614 2615lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1" 2616 using log_less_cancel_iff[of a x 1] by simp 2617 2618lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1" 2619 using log_le_cancel_iff[of a x 1] by simp 2620 2621lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x" 2622 using log_less_cancel_iff[of a a x] by simp 2623 2624lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x" 2625 using log_le_cancel_iff[of a a x] by simp 2626 2627lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a" 2628 using log_less_cancel_iff[of a x a] by simp 2629 2630lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a" 2631 using log_le_cancel_iff[of a x a] by simp 2632 2633lemma le_log_iff: 2634 fixes b x y :: real 2635 assumes "1 < b" "x > 0" 2636 shows "y \<le> log b x \<longleftrightarrow> b powr y \<le> x" 2637 using assms 2638 by (metis less_irrefl less_trans powr_le_cancel_iff powr_log_cancel zero_less_one) 2639 2640lemma less_log_iff: 2641 assumes "1 < b" "x > 0" 2642 shows "y < log b x \<longleftrightarrow> b powr y < x" 2643 by (metis assms dual_order.strict_trans less_irrefl powr_less_cancel_iff 2644 powr_log_cancel zero_less_one) 2645 2646lemma 2647 assumes "1 < b" "x > 0" 2648 shows log_less_iff: "log b x < y \<longleftrightarrow> x < b powr y" 2649 and log_le_iff: "log b x \<le> y \<longleftrightarrow> x \<le> b powr y" 2650 using le_log_iff[OF assms, of y] less_log_iff[OF assms, of y] 2651 by auto 2652 2653lemmas powr_le_iff = le_log_iff[symmetric] 2654 and powr_less_iff = less_log_iff[symmetric] 2655 and less_powr_iff = log_less_iff[symmetric] 2656 and le_powr_iff = log_le_iff[symmetric] 2657 2658lemma le_log_of_power: 2659 assumes "b ^ n \<le> m" "1 < b" 2660 shows "n \<le> log b m" 2661proof - 2662 from assms have "0 < m" by (metis less_trans zero_less_power less_le_trans zero_less_one) 2663 thus ?thesis using assms by (simp add: le_log_iff powr_realpow) 2664qed 2665 2666lemma le_log2_of_power: "2 ^ n \<le> m \<Longrightarrow> n \<le> log 2 m" for m n :: nat 2667using le_log_of_power[of 2] by simp 2668 2669lemma log_of_power_le: "\<lbrakk> m \<le> b ^ n; b > 1; m > 0 \<rbrakk> \<Longrightarrow> log b (real m) \<le> n" 2670by (simp add: log_le_iff powr_realpow) 2671 2672lemma log2_of_power_le: "\<lbrakk> m \<le> 2 ^ n; m > 0 \<rbrakk> \<Longrightarrow> log 2 m \<le> n" for m n :: nat 2673using log_of_power_le[of _ 2] by simp 2674 2675lemma log_of_power_less: "\<lbrakk> m < b ^ n; b > 1; m > 0 \<rbrakk> \<Longrightarrow> log b (real m) < n" 2676by (simp add: log_less_iff powr_realpow) 2677 2678lemma log2_of_power_less: "\<lbrakk> m < 2 ^ n; m > 0 \<rbrakk> \<Longrightarrow> log 2 m < n" for m n :: nat 2679using log_of_power_less[of _ 2] by simp 2680 2681lemma less_log_of_power: 2682 assumes "b ^ n < m" "1 < b" 2683 shows "n < log b m" 2684proof - 2685 have "0 < m" by (metis assms less_trans zero_less_power zero_less_one) 2686 thus ?thesis using assms by (simp add: less_log_iff powr_realpow) 2687qed 2688 2689lemma less_log2_of_power: "2 ^ n < m \<Longrightarrow> n < log 2 m" for m n :: nat 2690using less_log_of_power[of 2] by simp 2691 2692lemma gr_one_powr[simp]: 2693 fixes x y :: real shows "\<lbrakk> x > 1; y > 0 \<rbrakk> \<Longrightarrow> 1 < x powr y" 2694by(simp add: less_powr_iff) 2695 2696lemma 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)" 2697 by (auto simp: floor_eq_iff powr_le_iff less_powr_iff) 2698 2699lemma floor_log_nat_eq_powr_iff: fixes b n k :: nat 2700 shows "\<lbrakk> b \<ge> 2; k > 0 \<rbrakk> \<Longrightarrow> 2701 floor (log b (real k)) = n \<longleftrightarrow> b^n \<le> k \<and> k < b^(n+1)" 2702by (auto simp: floor_log_eq_powr_iff powr_add powr_realpow 2703 of_nat_power[symmetric] of_nat_mult[symmetric] ac_simps 2704 simp del: of_nat_power of_nat_mult) 2705 2706lemma floor_log_nat_eq_if: fixes b n k :: nat 2707 assumes "b^n \<le> k" "k < b^(n+1)" "b \<ge> 2" 2708 shows "floor (log b (real k)) = n" 2709proof - 2710 have "k \<ge> 1" using assms(1,3) one_le_power[of b n] by linarith 2711 with assms show ?thesis by(simp add: floor_log_nat_eq_powr_iff) 2712qed 2713 2714lemma ceiling_log_eq_powr_iff: "\<lbrakk> x > 0; b > 1 \<rbrakk> 2715 \<Longrightarrow> \<lceil>log b x\<rceil> = int k + 1 \<longleftrightarrow> b powr k < x \<and> x \<le> b powr (k + 1)" 2716by (auto simp: ceiling_eq_iff powr_less_iff le_powr_iff) 2717 2718lemma ceiling_log_nat_eq_powr_iff: fixes b n k :: nat 2719 shows "\<lbrakk> b \<ge> 2; k > 0 \<rbrakk> \<Longrightarrow> 2720 ceiling (log b (real k)) = int n + 1 \<longleftrightarrow> (b^n < k \<and> k \<le> b^(n+1))" 2721using ceiling_log_eq_powr_iff 2722by (auto simp: powr_add powr_realpow of_nat_power[symmetric] of_nat_mult[symmetric] ac_simps 2723 simp del: of_nat_power of_nat_mult) 2724 2725lemma ceiling_log_nat_eq_if: fixes b n k :: nat 2726 assumes "b^n < k" "k \<le> b^(n+1)" "b \<ge> 2" 2727 shows "ceiling (log b (real k)) = int n + 1" 2728proof - 2729 have "k \<ge> 1" using assms(1,3) one_le_power[of b n] by linarith 2730 with assms show ?thesis by(simp add: ceiling_log_nat_eq_powr_iff) 2731qed 2732 2733lemma floor_log2_div2: fixes n :: nat assumes "n \<ge> 2" 2734shows "floor(log 2 n) = floor(log 2 (n div 2)) + 1" 2735proof cases 2736 assume "n=2" thus ?thesis by simp 2737next 2738 let ?m = "n div 2" 2739 assume "n\<noteq>2" 2740 hence "1 \<le> ?m" using assms by arith 2741 then obtain i where i: "2 ^ i \<le> ?m" "?m < 2 ^ (i + 1)" 2742 using ex_power_ivl1[of 2 ?m] by auto 2743 have "2^(i+1) \<le> 2*?m" using i(1) by simp 2744 also have "2*?m \<le> n" by arith 2745 finally have *: "2^(i+1) \<le> \<dots>" . 2746 have "n < 2^(i+1+1)" using i(2) by simp 2747 from floor_log_nat_eq_if[OF * this] floor_log_nat_eq_if[OF i] 2748 show ?thesis by simp 2749qed 2750 2751lemma ceiling_log2_div2: assumes "n \<ge> 2" 2752shows "ceiling(log 2 (real n)) = ceiling(log 2 ((n-1) div 2 + 1)) + 1" 2753proof cases 2754 assume "n=2" thus ?thesis by simp 2755next 2756 let ?m = "(n-1) div 2 + 1" 2757 assume "n\<noteq>2" 2758 hence "2 \<le> ?m" using assms by arith 2759 then obtain i where i: "2 ^ i < ?m" "?m \<le> 2 ^ (i + 1)" 2760 using ex_power_ivl2[of 2 ?m] by auto 2761 have "n \<le> 2*?m" by arith 2762 also have "2*?m \<le> 2 ^ ((i+1)+1)" using i(2) by simp 2763 finally have *: "n \<le> \<dots>" . 2764 have "2^(i+1) < n" using i(1) by (auto simp: less_Suc_eq_0_disj) 2765 from ceiling_log_nat_eq_if[OF this *] ceiling_log_nat_eq_if[OF i] 2766 show ?thesis by simp 2767qed 2768 2769lemma powr_real_of_int: 2770 "x > 0 \<Longrightarrow> x powr real_of_int n = (if n \<ge> 0 then x ^ nat n else inverse (x ^ nat (- n)))" 2771 using powr_realpow[of x "nat n"] powr_realpow[of x "nat (-n)"] 2772 by (auto simp: field_simps powr_minus) 2773 2774lemma powr_numeral [simp]: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)" 2775 by (metis of_nat_numeral powr_realpow) 2776 2777lemma powr_int: 2778 assumes "x > 0" 2779 shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))" 2780proof (cases "i < 0") 2781 case True 2782 have r: "x powr i = 1 / x powr (- i)" 2783 by (simp add: powr_minus field_simps) 2784 show ?thesis using \<open>i < 0\<close> \<open>x > 0\<close> 2785 by (simp add: r field_simps powr_realpow[symmetric]) 2786next 2787 case False 2788 then show ?thesis 2789 by (simp add: assms powr_realpow[symmetric]) 2790qed 2791 2792lemma compute_powr[code]: 2793 fixes i :: real 2794 shows "b powr i = 2795 (if b \<le> 0 then Code.abort (STR ''op powr with nonpositive base'') (\<lambda>_. b powr i) 2796 else if \<lfloor>i\<rfloor> = i then (if 0 \<le> i then b ^ nat \<lfloor>i\<rfloor> else 1 / b ^ nat \<lfloor>- i\<rfloor>) 2797 else Code.abort (STR ''op powr with non-integer exponent'') (\<lambda>_. b powr i))" 2798 by (auto simp: powr_int) 2799 2800lemma powr_one: "0 \<le> x \<Longrightarrow> x powr 1 = x" 2801 for x :: real 2802 using powr_realpow [of x 1] by simp 2803 2804lemma powr_neg_one: "0 < x \<Longrightarrow> x powr - 1 = 1 / x" 2805 for x :: real 2806 using powr_int [of x "- 1"] by simp 2807 2808lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n" 2809 for x :: real 2810 using powr_int [of x "- numeral n"] by simp 2811 2812lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)" 2813 by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr) 2814 2815lemma ln_powr: "x \<noteq> 0 \<Longrightarrow> ln (x powr y) = y * ln x" 2816 for x :: real 2817 by (simp add: powr_def) 2818 2819lemma ln_root: "n > 0 \<Longrightarrow> b > 0 \<Longrightarrow> ln (root n b) = ln b / n" 2820 by (simp add: root_powr_inverse ln_powr) 2821 2822lemma ln_sqrt: "0 < x \<Longrightarrow> ln (sqrt x) = ln x / 2" 2823 by (simp add: ln_powr ln_powr[symmetric] mult.commute) 2824 2825lemma log_root: "n > 0 \<Longrightarrow> a > 0 \<Longrightarrow> log b (root n a) = log b a / n" 2826 by (simp add: log_def ln_root) 2827 2828lemma log_powr: "x \<noteq> 0 \<Longrightarrow> log b (x powr y) = y * log b x" 2829 by (simp add: log_def ln_powr) 2830 2831(* [simp] is not worth it, interferes with some proofs *) 2832lemma log_nat_power: "0 < x \<Longrightarrow> log b (x^n) = real n * log b x" 2833 by (simp add: log_powr powr_realpow [symmetric]) 2834 2835lemma log_of_power_eq: 2836 assumes "m = b ^ n" "b > 1" 2837 shows "n = log b (real m)" 2838proof - 2839 have "n = log b (b ^ n)" using assms(2) by (simp add: log_nat_power) 2840 also have "\<dots> = log b m" using assms by simp 2841 finally show ?thesis . 2842qed 2843 2844lemma log2_of_power_eq: "m = 2 ^ n \<Longrightarrow> n = log 2 m" for m n :: nat 2845using log_of_power_eq[of _ 2] by simp 2846 2847lemma log_base_change: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log b x = log a x / log a b" 2848 by (simp add: log_def) 2849 2850lemma log_base_pow: "0 < a \<Longrightarrow> log (a ^ n) x = log a x / n" 2851 by (simp add: log_def ln_realpow) 2852 2853lemma log_base_powr: "a \<noteq> 0 \<Longrightarrow> log (a powr b) x = log a x / b" 2854 by (simp add: log_def ln_powr) 2855 2856lemma log_base_root: "n > 0 \<Longrightarrow> b > 0 \<Longrightarrow> log (root n b) x = n * (log b x)" 2857 by (simp add: log_def ln_root) 2858 2859lemma ln_bound: "0 < x \<Longrightarrow> ln x \<le> x" for x :: real 2860 using ln_le_minus_one by force 2861 2862lemma powr_mono: 2863 fixes x :: real 2864 assumes "a \<le> b" and "1 \<le> x" shows "x powr a \<le> x powr b" 2865 using assms less_eq_real_def by auto 2866 2867lemma ge_one_powr_ge_zero: "1 \<le> x \<Longrightarrow> 0 \<le> a \<Longrightarrow> 1 \<le> x powr a" 2868 for x :: real 2869 using powr_mono by fastforce 2870 2871lemma powr_less_mono2: "0 < a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> x powr a < y powr a" 2872 for x :: real 2873 by (simp add: powr_def) 2874 2875lemma powr_less_mono2_neg: "a < 0 \<Longrightarrow> 0 < x \<Longrightarrow> x < y \<Longrightarrow> y powr a < x powr a" 2876 for x :: real 2877 by (simp add: powr_def) 2878 2879lemma powr_mono2: "x powr a \<le> y powr a" if "0 \<le> a" "0 \<le> x" "x \<le> y" 2880 for x :: real 2881 using less_eq_real_def powr_less_mono2 that by auto 2882 2883lemma powr_le1: "0 \<le> a \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> x powr a \<le> 1" 2884 for x :: real 2885 using powr_mono2 by fastforce 2886 2887lemma powr_mono2': 2888 fixes a x y :: real 2889 assumes "a \<le> 0" "x > 0" "x \<le> y" 2890 shows "x powr a \<ge> y powr a" 2891proof - 2892 from assms have "x powr - a \<le> y powr - a" 2893 by (intro powr_mono2) simp_all 2894 with assms show ?thesis 2895 by (auto simp: powr_minus field_simps) 2896qed 2897 2898lemma powr_mono_both: 2899 fixes x :: real 2900 assumes "0 \<le> a" "a \<le> b" "1 \<le> x" "x \<le> y" 2901 shows "x powr a \<le> y powr b" 2902 by (meson assms order.trans powr_mono powr_mono2 zero_le_one) 2903 2904lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y" 2905 for x :: real 2906 unfolding powr_def exp_inj_iff by simp 2907 2908lemma powr_half_sqrt: "0 \<le> x \<Longrightarrow> x powr (1/2) = sqrt x" 2909 by (simp add: powr_def root_powr_inverse sqrt_def) 2910 2911lemma ln_powr_bound: "1 \<le> x \<Longrightarrow> 0 < a \<Longrightarrow> ln x \<le> (x powr a) / a" 2912 for x :: real 2913 by (metis exp_gt_zero linear ln_eq_zero_iff ln_exp ln_less_self ln_powr mult.commute 2914 mult_imp_le_div_pos not_less powr_gt_zero) 2915 2916lemma ln_powr_bound2: 2917 fixes x :: real 2918 assumes "1 < x" and "0 < a" 2919 shows "(ln x) powr a \<le> (a powr a) * x" 2920proof - 2921 from assms have "ln x \<le> (x powr (1 / a)) / (1 / a)" 2922 by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff) 2923 also have "\<dots> = a * (x powr (1 / a))" 2924 by simp 2925 finally have "(ln x) powr a \<le> (a * (x powr (1 / a))) powr a" 2926 by (metis assms less_imp_le ln_gt_zero powr_mono2) 2927 also have "\<dots> = (a powr a) * ((x powr (1 / a)) powr a)" 2928 using assms powr_mult by auto 2929 also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)" 2930 by (rule powr_powr) 2931 also have "\<dots> = x" using assms 2932 by auto 2933 finally show ?thesis . 2934qed 2935 2936lemma tendsto_powr: 2937 fixes a b :: real 2938 assumes f: "(f \<longlongrightarrow> a) F" 2939 and g: "(g \<longlongrightarrow> b) F" 2940 and a: "a \<noteq> 0" 2941 shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F" 2942 unfolding powr_def 2943proof (rule filterlim_If) 2944 from f show "((\<lambda>x. 0) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) (inf F (principal {x. f x = 0}))" 2945 by simp (auto simp: filterlim_iff eventually_inf_principal elim: eventually_mono dest: t1_space_nhds) 2946 from f g a show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> (if a = 0 then 0 else exp (b * ln a))) 2947 (inf F (principal {x. f x \<noteq> 0}))" 2948 by (auto intro!: tendsto_intros intro: tendsto_mono inf_le1) 2949qed 2950 2951lemma tendsto_powr'[tendsto_intros]: 2952 fixes a :: real 2953 assumes f: "(f \<longlongrightarrow> a) F" 2954 and g: "(g \<longlongrightarrow> b) F" 2955 and a: "a \<noteq> 0 \<or> (b > 0 \<and> eventually (\<lambda>x. f x \<ge> 0) F)" 2956 shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F" 2957proof - 2958 from a consider "a \<noteq> 0" | "a = 0" "b > 0" "eventually (\<lambda>x. f x \<ge> 0) F" 2959 by auto 2960 then show ?thesis 2961 proof cases 2962 case 1 2963 with f g show ?thesis by (rule tendsto_powr) 2964 next 2965 case 2 2966 have "((\<lambda>x. if f x = 0 then 0 else exp (g x * ln (f x))) \<longlongrightarrow> 0) F" 2967 proof (intro filterlim_If) 2968 have "filterlim f (principal {0<..}) (inf F (principal {z. f z \<noteq> 0}))" 2969 using \<open>eventually (\<lambda>x. f x \<ge> 0) F\<close> 2970 by (auto simp: filterlim_iff eventually_inf_principal 2971 eventually_principal elim: eventually_mono) 2972 moreover have "filterlim f (nhds a) (inf F (principal {z. f z \<noteq> 0}))" 2973 by (rule tendsto_mono[OF _ f]) simp_all 2974 ultimately have f: "filterlim f (at_right 0) (inf F (principal {x. f x \<noteq> 0}))" 2975 by (simp add: at_within_def filterlim_inf \<open>a = 0\<close>) 2976 have g: "(g \<longlongrightarrow> b) (inf F (principal {z. f z \<noteq> 0}))" 2977 by (rule tendsto_mono[OF _ g]) simp_all 2978 show "((\<lambda>x. exp (g x * ln (f x))) \<longlongrightarrow> 0) (inf F (principal {x. f x \<noteq> 0}))" 2979 by (rule filterlim_compose[OF exp_at_bot] filterlim_tendsto_pos_mult_at_bot 2980 filterlim_compose[OF ln_at_0] f g \<open>b > 0\<close>)+ 2981 qed simp_all 2982 with \<open>a = 0\<close> show ?thesis 2983 by (simp add: powr_def) 2984 qed 2985qed 2986 2987lemma continuous_powr: 2988 assumes "continuous F f" 2989 and "continuous F g" 2990 and "f (Lim F (\<lambda>x. x)) \<noteq> 0" 2991 shows "continuous F (\<lambda>x. (f x) powr (g x :: real))" 2992 using assms unfolding continuous_def by (rule tendsto_powr) 2993 2994lemma continuous_at_within_powr[continuous_intros]: 2995 fixes f g :: "_ \<Rightarrow> real" 2996 assumes "continuous (at a within s) f" 2997 and "continuous (at a within s) g" 2998 and "f a \<noteq> 0" 2999 shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))" 3000 using assms unfolding continuous_within by (rule tendsto_powr) 3001 3002lemma isCont_powr[continuous_intros, simp]: 3003 fixes f g :: "_ \<Rightarrow> real" 3004 assumes "isCont f a" "isCont g a" "f a \<noteq> 0" 3005 shows "isCont (\<lambda>x. (f x) powr g x) a" 3006 using assms unfolding continuous_at by (rule tendsto_powr) 3007 3008lemma continuous_on_powr[continuous_intros]: 3009 fixes f g :: "_ \<Rightarrow> real" 3010 assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. f x \<noteq> 0" 3011 shows "continuous_on s (\<lambda>x. (f x) powr (g x))" 3012 using assms unfolding continuous_on_def by (fast intro: tendsto_powr) 3013 3014lemma tendsto_powr2: 3015 fixes a :: real 3016 assumes f: "(f \<longlongrightarrow> a) F" 3017 and g: "(g \<longlongrightarrow> b) F" 3018 and "\<forall>\<^sub>F x in F. 0 \<le> f x" 3019 and b: "0 < b" 3020 shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F" 3021 using tendsto_powr'[of f a F g b] assms by auto 3022 3023lemma has_derivative_powr[derivative_intros]: 3024 assumes g[derivative_intros]: "(g has_derivative g') (at x within X)" 3025 and f[derivative_intros]:"(f has_derivative f') (at x within X)" 3026 assumes pos: "0 < g x" and "x \<in> X" 3027 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)" 3028proof - 3029 have "\<forall>\<^sub>F x in at x within X. g x > 0" 3030 by (rule order_tendstoD[OF _ pos]) 3031 (rule has_derivative_continuous[OF g, unfolded continuous_within]) 3032 then obtain d where "d > 0" and pos': "\<And>x'. x' \<in> X \<Longrightarrow> dist x' x < d \<Longrightarrow> 0 < g x'" 3033 using pos unfolding eventually_at by force 3034 have "((\<lambda>x. exp (f x * ln (g x))) has_derivative 3035 (\<lambda>h. (g x powr f x) * (f' h * ln (g x) + g' h * f x / g x))) (at x within X)" 3036 using pos 3037 by (auto intro!: derivative_eq_intros simp: divide_simps powr_def) 3038 then show ?thesis 3039 by (rule has_derivative_transform_within[OF _ \<open>d > 0\<close> \<open>x \<in> X\<close>]) (auto simp: powr_def dest: pos') 3040qed 3041 3042lemma DERIV_powr: 3043 fixes r :: real 3044 assumes g: "DERIV g x :> m" 3045 and pos: "g x > 0" 3046 and f: "DERIV f x :> r" 3047 shows "DERIV (\<lambda>x. g x powr f x) x :> (g x powr f x) * (r * ln (g x) + m * f x / g x)" 3048 using assms 3049 by (auto intro!: derivative_eq_intros ext simp: has_field_derivative_def algebra_simps) 3050 3051lemma DERIV_fun_powr: 3052 fixes r :: real 3053 assumes g: "DERIV g x :> m" 3054 and pos: "g x > 0" 3055 shows "DERIV (\<lambda>x. (g x) powr r) x :> r * (g x) powr (r - of_nat 1) * m" 3056 using DERIV_powr[OF g pos DERIV_const, of r] pos 3057 by (simp add: powr_diff field_simps) 3058 3059lemma has_real_derivative_powr: 3060 assumes "z > 0" 3061 shows "((\<lambda>z. z powr r) has_real_derivative r * z powr (r - 1)) (at z)" 3062proof (subst DERIV_cong_ev[OF refl _ refl]) 3063 from assms have "eventually (\<lambda>z. z \<noteq> 0) (nhds z)" 3064 by (intro t1_space_nhds) auto 3065 then show "eventually (\<lambda>z. z powr r = exp (r * ln z)) (nhds z)" 3066 unfolding powr_def by eventually_elim simp 3067 from assms show "((\<lambda>z. exp (r * ln z)) has_real_derivative r * z powr (r - 1)) (at z)" 3068 by (auto intro!: derivative_eq_intros simp: powr_def field_simps exp_diff) 3069qed 3070 3071declare has_real_derivative_powr[THEN DERIV_chain2, derivative_intros] 3072 3073lemma tendsto_zero_powrI: 3074 assumes "(f \<longlongrightarrow> (0::real)) F" "(g \<longlongrightarrow> b) F" "\<forall>\<^sub>F x in F. 0 \<le> f x" "0 < b" 3075 shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> 0) F" 3076 using tendsto_powr2[OF assms] by simp 3077 3078lemma continuous_on_powr': 3079 fixes f g :: "_ \<Rightarrow> real" 3080 assumes "continuous_on s f" "continuous_on s g" 3081 and "\<forall>x\<in>s. f x \<ge> 0 \<and> (f x = 0 \<longrightarrow> g x > 0)" 3082 shows "continuous_on s (\<lambda>x. (f x) powr (g x))" 3083 unfolding continuous_on_def 3084proof 3085 fix x 3086 assume x: "x \<in> s" 3087 from assms x show "((\<lambda>x. f x powr g x) \<longlongrightarrow> f x powr g x) (at x within s)" 3088 proof (cases "f x = 0") 3089 case True 3090 from assms(3) have "eventually (\<lambda>x. f x \<ge> 0) (at x within s)" 3091 by (auto simp: at_within_def eventually_inf_principal) 3092 with True x assms show ?thesis 3093 by (auto intro!: tendsto_zero_powrI[of f _ g "g x"] simp: continuous_on_def) 3094 next 3095 case False 3096 with assms x show ?thesis 3097 by (auto intro!: tendsto_powr' simp: continuous_on_def) 3098 qed 3099qed 3100 3101lemma tendsto_neg_powr: 3102 assumes "s < 0" 3103 and f: "LIM x F. f x :> at_top" 3104 shows "((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F" 3105proof - 3106 have "((\<lambda>x. exp (s * ln (f x))) \<longlongrightarrow> (0::real)) F" (is "?X") 3107 by (auto intro!: filterlim_compose[OF exp_at_bot] filterlim_compose[OF ln_at_top] 3108 filterlim_tendsto_neg_mult_at_bot assms) 3109 also have "?X \<longleftrightarrow> ((\<lambda>x. f x powr s) \<longlongrightarrow> (0::real)) F" 3110 using f filterlim_at_top_dense[of f F] 3111 by (intro filterlim_cong[OF refl refl]) (auto simp: neq_iff powr_def elim: eventually_mono) 3112 finally show ?thesis . 3113qed 3114 3115lemma tendsto_exp_limit_at_right: "((\<lambda>y. (1 + x * y) powr (1 / y)) \<longlongrightarrow> exp x) (at_right 0)" 3116 for x :: real 3117proof (cases "x = 0") 3118 case True 3119 then show ?thesis by simp 3120next 3121 case False 3122 have "((\<lambda>y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)" 3123 by (auto intro!: derivative_eq_intros) 3124 then have "((\<lambda>y. ln (1 + x * y) / y) \<longlongrightarrow> x) (at 0)" 3125 by (auto simp: has_field_derivative_def field_has_derivative_at) 3126 then have *: "((\<lambda>y. exp (ln (1 + x * y) / y)) \<longlongrightarrow> exp x) (at 0)" 3127 by (rule tendsto_intros) 3128 then show ?thesis 3129 proof (rule filterlim_mono_eventually) 3130 show "eventually (\<lambda>xa. exp (ln (1 + x * xa) / xa) = (1 + x * xa) powr (1 / xa)) (at_right 0)" 3131 unfolding eventually_at_right[OF zero_less_one] 3132 using False 3133 by (intro exI[of _ "1 / \<bar>x\<bar>"]) (auto simp: field_simps powr_def abs_if add_nonneg_eq_0_iff) 3134 qed (simp_all add: at_eq_sup_left_right) 3135qed 3136 3137lemma tendsto_exp_limit_at_top: "((\<lambda>y. (1 + x / y) powr y) \<longlongrightarrow> exp x) at_top" 3138 for x :: real 3139 by (simp add: filterlim_at_top_to_right inverse_eq_divide tendsto_exp_limit_at_right) 3140 3141lemma tendsto_exp_limit_sequentially: "(\<lambda>n. (1 + x / n) ^ n) \<longlonglongrightarrow> exp x" 3142 for x :: real 3143proof (rule filterlim_mono_eventually) 3144 from reals_Archimedean2 [of "\<bar>x\<bar>"] obtain n :: nat where *: "real n > \<bar>x\<bar>" .. 3145 then have "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top" 3146 by (intro eventually_sequentiallyI [of n]) (auto simp: divide_simps) 3147 then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top" 3148 by (rule eventually_mono) (erule powr_realpow) 3149 show "(\<lambda>n. (1 + x / real n) powr real n) \<longlonglongrightarrow> exp x" 3150 by (rule filterlim_compose [OF tendsto_exp_limit_at_top filterlim_real_sequentially]) 3151qed auto 3152 3153 3154subsection \<open>Sine and Cosine\<close> 3155 3156definition sin_coeff :: "nat \<Rightarrow> real" 3157 where "sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / (fact n))" 3158 3159definition cos_coeff :: "nat \<Rightarrow> real" 3160 where "cos_coeff = (\<lambda>n. if even n then ((- 1) ^ (n div 2)) / (fact n) else 0)" 3161 3162definition sin :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}" 3163 where "sin = (\<lambda>x. \<Sum>n. sin_coeff n *\<^sub>R x^n)" 3164 3165definition cos :: "'a \<Rightarrow> 'a::{real_normed_algebra_1,banach}" 3166 where "cos = (\<lambda>x. \<Sum>n. cos_coeff n *\<^sub>R x^n)" 3167 3168lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0" 3169 unfolding sin_coeff_def by simp 3170 3171lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1" 3172 unfolding cos_coeff_def by simp 3173 3174lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)" 3175 unfolding cos_coeff_def sin_coeff_def 3176 by (simp del: mult_Suc) 3177 3178lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)" 3179 unfolding cos_coeff_def sin_coeff_def 3180 by (simp del: mult_Suc) (auto elim: oddE) 3181 3182lemma summable_norm_sin: "summable (\<lambda>n. norm (sin_coeff n *\<^sub>R x^n))" 3183 for x :: "'a::{real_normed_algebra_1,banach}" 3184 unfolding sin_coeff_def 3185 apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]]) 3186 apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) 3187 done 3188 3189lemma summable_norm_cos: "summable (\<lambda>n. norm (cos_coeff n *\<^sub>R x^n))" 3190 for x :: "'a::{real_normed_algebra_1,banach}" 3191 unfolding cos_coeff_def 3192 apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]]) 3193 apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) 3194 done 3195 3196lemma sin_converges: "(\<lambda>n. sin_coeff n *\<^sub>R x^n) sums sin x" 3197 unfolding sin_def 3198 by (metis (full_types) summable_norm_cancel summable_norm_sin summable_sums) 3199 3200lemma cos_converges: "(\<lambda>n. cos_coeff n *\<^sub>R x^n) sums cos x" 3201 unfolding cos_def 3202 by (metis (full_types) summable_norm_cancel summable_norm_cos summable_sums) 3203 3204lemma sin_of_real: "sin (of_real x) = of_real (sin x)" 3205 for x :: real 3206proof - 3207 have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) = (\<lambda>n. sin_coeff n *\<^sub>R (of_real x)^n)" 3208 proof 3209 show "of_real (sin_coeff n *\<^sub>R x^n) = sin_coeff n *\<^sub>R of_real x^n" for n 3210 by (simp add: scaleR_conv_of_real) 3211 qed 3212 also have "\<dots> sums (sin (of_real x))" 3213 by (rule sin_converges) 3214 finally have "(\<lambda>n. of_real (sin_coeff n *\<^sub>R x^n)) sums (sin (of_real x))" . 3215 then show ?thesis 3216 using sums_unique2 sums_of_real [OF sin_converges] 3217 by blast 3218qed 3219 3220corollary sin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> sin z \<in> \<real>" 3221 by (metis Reals_cases Reals_of_real sin_of_real) 3222 3223lemma cos_of_real: "cos (of_real x) = of_real (cos x)" 3224 for x :: real 3225proof - 3226 have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) = (\<lambda>n. cos_coeff n *\<^sub>R (of_real x)^n)" 3227 proof 3228 show "of_real (cos_coeff n *\<^sub>R x^n) = cos_coeff n *\<^sub>R of_real x^n" for n 3229 by (simp add: scaleR_conv_of_real) 3230 qed 3231 also have "\<dots> sums (cos (of_real x))" 3232 by (rule cos_converges) 3233 finally have "(\<lambda>n. of_real (cos_coeff n *\<^sub>R x^n)) sums (cos (of_real x))" . 3234 then show ?thesis 3235 using sums_unique2 sums_of_real [OF cos_converges] 3236 by blast 3237qed 3238 3239corollary cos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cos z \<in> \<real>" 3240 by (metis Reals_cases Reals_of_real cos_of_real) 3241 3242lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff" 3243 by (simp add: diffs_def sin_coeff_Suc del: of_nat_Suc) 3244 3245lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)" 3246 by (simp add: diffs_def cos_coeff_Suc del: of_nat_Suc) 3247 3248lemma sin_int_times_real: "sin (of_int m * of_real x) = of_real (sin (of_int m * x))" 3249 by (metis sin_of_real of_real_mult of_real_of_int_eq) 3250 3251lemma cos_int_times_real: "cos (of_int m * of_real x) = of_real (cos (of_int m * x))" 3252 by (metis cos_of_real of_real_mult of_real_of_int_eq) 3253 3254text \<open>Now at last we can get the derivatives of exp, sin and cos.\<close> 3255 3256lemma DERIV_sin [simp]: "DERIV sin x :> cos x" 3257 for x :: "'a::{real_normed_field,banach}" 3258 unfolding sin_def cos_def scaleR_conv_of_real 3259 apply (rule DERIV_cong) 3260 apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"]) 3261 apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff 3262 summable_minus_iff scaleR_conv_of_real [symmetric] 3263 summable_norm_sin [THEN summable_norm_cancel] 3264 summable_norm_cos [THEN summable_norm_cancel]) 3265 done 3266 3267declare DERIV_sin[THEN DERIV_chain2, derivative_intros] 3268 and DERIV_sin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 3269 3270lemmas has_derivative_sin[derivative_intros] = DERIV_sin[THEN DERIV_compose_FDERIV] 3271 3272lemma DERIV_cos [simp]: "DERIV cos x :> - sin x" 3273 for x :: "'a::{real_normed_field,banach}" 3274 unfolding sin_def cos_def scaleR_conv_of_real 3275 apply (rule DERIV_cong) 3276 apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"]) 3277 apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus 3278 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_cos[THEN DERIV_chain2, derivative_intros] 3285 and DERIV_cos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 3286 3287lemmas has_derivative_cos[derivative_intros] = DERIV_cos[THEN DERIV_compose_FDERIV] 3288 3289lemma isCont_sin: "isCont sin x" 3290 for x :: "'a::{real_normed_field,banach}" 3291 by (rule DERIV_sin [THEN DERIV_isCont]) 3292 3293lemma isCont_cos: "isCont cos x" 3294 for x :: "'a::{real_normed_field,banach}" 3295 by (rule DERIV_cos [THEN DERIV_isCont]) 3296 3297lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a" 3298 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3299 by (rule isCont_o2 [OF _ isCont_sin]) 3300 3301(* FIXME a context for f would be better *) 3302 3303lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a" 3304 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3305 by (rule isCont_o2 [OF _ isCont_cos]) 3306 3307lemma tendsto_sin [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) \<longlongrightarrow> sin a) F" 3308 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3309 by (rule isCont_tendsto_compose [OF isCont_sin]) 3310 3311lemma tendsto_cos [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) \<longlongrightarrow> cos a) F" 3312 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3313 by (rule isCont_tendsto_compose [OF isCont_cos]) 3314 3315lemma continuous_sin [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))" 3316 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3317 unfolding continuous_def by (rule tendsto_sin) 3318 3319lemma continuous_on_sin [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))" 3320 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3321 unfolding continuous_on_def by (auto intro: tendsto_sin) 3322 3323lemma continuous_within_sin: "continuous (at z within s) sin" 3324 for z :: "'a::{real_normed_field,banach}" 3325 by (simp add: continuous_within tendsto_sin) 3326 3327lemma continuous_cos [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))" 3328 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3329 unfolding continuous_def by (rule tendsto_cos) 3330 3331lemma continuous_on_cos [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))" 3332 for f :: "_ \<Rightarrow> 'a::{real_normed_field,banach}" 3333 unfolding continuous_on_def by (auto intro: tendsto_cos) 3334 3335lemma continuous_within_cos: "continuous (at z within s) cos" 3336 for z :: "'a::{real_normed_field,banach}" 3337 by (simp add: continuous_within tendsto_cos) 3338 3339 3340subsection \<open>Properties of Sine and Cosine\<close> 3341 3342lemma sin_zero [simp]: "sin 0 = 0" 3343 by (simp add: sin_def sin_coeff_def scaleR_conv_of_real) 3344 3345lemma cos_zero [simp]: "cos 0 = 1" 3346 by (simp add: cos_def cos_coeff_def scaleR_conv_of_real) 3347 3348lemma DERIV_fun_sin: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. sin (g x)) x :> cos (g x) * m" 3349 by (auto intro!: derivative_intros) 3350 3351lemma DERIV_fun_cos: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. cos(g x)) x :> - sin (g x) * m" 3352 by (auto intro!: derivative_eq_intros) 3353 3354 3355subsection \<open>Deriving the Addition Formulas\<close> 3356 3357text \<open>The product of two cosine series.\<close> 3358lemma cos_x_cos_y: 3359 fixes x :: "'a::{real_normed_field,banach}" 3360 shows 3361 "(\<lambda>p. \<Sum>n\<le>p. 3362 if even p \<and> even n 3363 then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) 3364 sums (cos x * cos y)" 3365proof - 3366 have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p - n)) = 3367 (if even p \<and> even n then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p - n) 3368 else 0)" 3369 if "n \<le> p" for n p :: nat 3370 proof - 3371 from that have *: "even n \<Longrightarrow> even p \<Longrightarrow> 3372 (-1) ^ (n div 2) * (-1) ^ ((p - n) div 2) = (-1 :: real) ^ (p div 2)" 3373 by (metis div_add power_add le_add_diff_inverse odd_add) 3374 with that show ?thesis 3375 by (auto simp: algebra_simps cos_coeff_def binomial_fact) 3376 qed 3377 then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> even n 3378 then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) = 3379 (\<lambda>p. \<Sum>n\<le>p. (cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))" 3380 by simp 3381 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)))" 3382 by (simp add: algebra_simps) 3383 also have "\<dots> sums (cos x * cos y)" 3384 using summable_norm_cos 3385 by (auto simp: cos_def scaleR_conv_of_real intro!: Cauchy_product_sums) 3386 finally show ?thesis . 3387qed 3388 3389text \<open>The product of two sine series.\<close> 3390lemma sin_x_sin_y: 3391 fixes x :: "'a::{real_normed_field,banach}" 3392 shows 3393 "(\<lambda>p. \<Sum>n\<le>p. 3394 if even p \<and> odd n 3395 then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) 3396 else 0) 3397 sums (sin x * sin y)" 3398proof - 3399 have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) = 3400 (if even p \<and> odd n 3401 then -((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) 3402 else 0)" 3403 if "n \<le> p" for n p :: nat 3404 proof - 3405 have "(-1) ^ ((n - Suc 0) div 2) * (-1) ^ ((p - Suc n) div 2) = - ((-1 :: real) ^ (p div 2))" 3406 if np: "odd n" "even p" 3407 proof - 3408 from \<open>n \<le> p\<close> np have *: "n - Suc 0 + (p - Suc n) = p - Suc (Suc 0)" "Suc (Suc 0) \<le> p" 3409 by arith+ 3410 have "(p - Suc (Suc 0)) div 2 = p div 2 - Suc 0" 3411 by simp 3412 with \<open>n \<le> p\<close> np * show ?thesis 3413 apply (simp add: power_add [symmetric] div_add [symmetric] del: div_add) 3414 apply (metis (no_types) One_nat_def Suc_1 le_div_geq minus_minus 3415 mult.left_neutral mult_minus_left power.simps(2) zero_less_Suc) 3416 done 3417 qed 3418 then show ?thesis 3419 using \<open>n\<le>p\<close> by (auto simp: algebra_simps sin_coeff_def binomial_fact) 3420 qed 3421 then have "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> odd n 3422 then - ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) = 3423 (\<lambda>p. \<Sum>n\<le>p. (sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))" 3424 by simp 3425 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)))" 3426 by (simp add: algebra_simps) 3427 also have "\<dots> sums (sin x * sin y)" 3428 using summable_norm_sin 3429 by (auto simp: sin_def scaleR_conv_of_real intro!: Cauchy_product_sums) 3430 finally show ?thesis . 3431qed 3432 3433lemma sums_cos_x_plus_y: 3434 fixes x :: "'a::{real_normed_field,banach}" 3435 shows 3436 "(\<lambda>p. \<Sum>n\<le>p. 3437 if even p 3438 then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) 3439 else 0) 3440 sums cos (x + y)" 3441proof - 3442 have 3443 "(\<Sum>n\<le>p. 3444 if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) 3445 else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)" 3446 for p :: nat 3447 proof - 3448 have 3449 "(\<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) = 3450 (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)" 3451 by simp 3452 also have "\<dots> = 3453 (if even p 3454 then of_real ((-1) ^ (p div 2) / (fact p)) * (\<Sum>n\<le>p. (p choose n) *\<^sub>R (x^n) * y^(p-n)) 3455 else 0)" 3456 by (auto simp: sum_distrib_left field_simps scaleR_conv_of_real nonzero_of_real_divide) 3457 also have "\<dots> = cos_coeff p *\<^sub>R ((x + y) ^ p)" 3458 by (simp add: cos_coeff_def binomial_ring [of x y] scaleR_conv_of_real atLeast0AtMost) 3459 finally show ?thesis . 3460 qed 3461 then have 3462 "(\<lambda>p. \<Sum>n\<le>p. 3463 if even p 3464 then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) 3465 else 0) = (\<lambda>p. cos_coeff p *\<^sub>R ((x+y)^p))" 3466 by simp 3467 also have "\<dots> sums cos (x + y)" 3468 by (rule cos_converges) 3469 finally show ?thesis . 3470qed 3471 3472theorem cos_add: 3473 fixes x :: "'a::{real_normed_field,banach}" 3474 shows "cos (x + y) = cos x * cos y - sin x * sin y" 3475proof - 3476 have 3477 "(if even p \<and> even n 3478 then ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) - 3479 (if even p \<and> odd n 3480 then - ((- 1) ^ (p div 2) * int (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) = 3481 (if even p then ((-1) ^ (p div 2) * (p choose n) / (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)" 3482 if "n \<le> p" for n p :: nat 3483 by simp 3484 then have 3485 "(\<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)) 3486 sums (cos x * cos y - sin x * sin y)" 3487 using sums_diff [OF cos_x_cos_y [of x y] sin_x_sin_y [of x y]] 3488 by (simp add: sum_subtractf [symmetric]) 3489 then show ?thesis 3490 by (blast intro: sums_cos_x_plus_y sums_unique2) 3491qed 3492 3493lemma sin_minus_converges: "(\<lambda>n. - (sin_coeff n *\<^sub>R (-x)^n)) sums sin x" 3494proof - 3495 have [simp]: "\<And>n. - (sin_coeff n *\<^sub>R (-x)^n) = (sin_coeff n *\<^sub>R x^n)" 3496 by (auto simp: sin_coeff_def elim!: oddE) 3497 show ?thesis 3498 by (simp add: sin_def summable_norm_sin [THEN summable_norm_cancel, THEN summable_sums]) 3499qed 3500 3501lemma sin_minus [simp]: "sin (- x) = - sin x" 3502 for x :: "'a::{real_normed_algebra_1,banach}" 3503 using sin_minus_converges [of x] 3504 by (auto simp: sin_def summable_norm_sin [THEN summable_norm_cancel] 3505 suminf_minus sums_iff equation_minus_iff) 3506 3507lemma cos_minus_converges: "(\<lambda>n. (cos_coeff n *\<^sub>R (-x)^n)) sums cos x" 3508proof - 3509 have [simp]: "\<And>n. (cos_coeff n *\<^sub>R (-x)^n) = (cos_coeff n *\<^sub>R x^n)" 3510 by (auto simp: Transcendental.cos_coeff_def elim!: evenE) 3511 show ?thesis 3512 by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel, THEN summable_sums]) 3513qed 3514 3515lemma cos_minus [simp]: "cos (-x) = cos x" 3516 for x :: "'a::{real_normed_algebra_1,banach}" 3517 using cos_minus_converges [of x] 3518 by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel] 3519 suminf_minus sums_iff equation_minus_iff) 3520 3521lemma sin_cos_squared_add [simp]: "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1" 3522 for x :: "'a::{real_normed_field,banach}" 3523 using cos_add [of x "-x"] 3524 by (simp add: power2_eq_square algebra_simps) 3525 3526lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1" 3527 for x :: "'a::{real_normed_field,banach}" 3528 by (subst add.commute, rule sin_cos_squared_add) 3529 3530lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1" 3531 for x :: "'a::{real_normed_field,banach}" 3532 using sin_cos_squared_add2 [unfolded power2_eq_square] . 3533 3534lemma sin_squared_eq: "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2" 3535 for x :: "'a::{real_normed_field,banach}" 3536 unfolding eq_diff_eq by (rule sin_cos_squared_add) 3537 3538lemma cos_squared_eq: "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2" 3539 for x :: "'a::{real_normed_field,banach}" 3540 unfolding eq_diff_eq by (rule sin_cos_squared_add2) 3541 3542lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1" 3543 for x :: real 3544 by (rule power2_le_imp_le) (simp_all add: sin_squared_eq) 3545 3546lemma sin_ge_minus_one [simp]: "- 1 \<le> sin x" 3547 for x :: real 3548 using abs_sin_le_one [of x] by (simp add: abs_le_iff) 3549 3550lemma sin_le_one [simp]: "sin x \<le> 1" 3551 for x :: real 3552 using abs_sin_le_one [of x] by (simp add: abs_le_iff) 3553 3554lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1" 3555 for x :: real 3556 by (rule power2_le_imp_le) (simp_all add: cos_squared_eq) 3557 3558lemma cos_ge_minus_one [simp]: "- 1 \<le> cos x" 3559 for x :: real 3560 using abs_cos_le_one [of x] by (simp add: abs_le_iff) 3561 3562lemma cos_le_one [simp]: "cos x \<le> 1" 3563 for x :: real 3564 using abs_cos_le_one [of x] by (simp add: abs_le_iff) 3565 3566lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y" 3567 for x :: "'a::{real_normed_field,banach}" 3568 using cos_add [of x "- y"] by simp 3569 3570lemma cos_double: "cos(2*x) = (cos x)\<^sup>2 - (sin x)\<^sup>2" 3571 for x :: "'a::{real_normed_field,banach}" 3572 using cos_add [where x=x and y=x] by (simp add: power2_eq_square) 3573 3574lemma sin_cos_le1: "\<bar>sin x * sin y + cos x * cos y\<bar> \<le> 1" 3575 for x :: real 3576 using cos_diff [of x y] by (metis abs_cos_le_one add.commute) 3577 3578lemma DERIV_fun_pow: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m" 3579 by (auto intro!: derivative_eq_intros simp:) 3580 3581lemma DERIV_fun_exp: "DERIV g x :> m \<Longrightarrow> DERIV (\<lambda>x. exp (g x)) x :> exp (g x) * m" 3582 by (auto intro!: derivative_intros) 3583 3584 3585subsection \<open>The Constant Pi\<close> 3586 3587definition pi :: real 3588 where "pi = 2 * (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)" 3589 3590text \<open>Show that there's a least positive @{term x} with @{term "cos x = 0"}; 3591 hence define pi.\<close> 3592 3593lemma sin_paired: "(\<lambda>n. (- 1) ^ n / (fact (2 * n + 1)) * x ^ (2 * n + 1)) sums sin x" 3594 for x :: real 3595proof - 3596 have "(\<lambda>n. \<Sum>k = n*2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x" 3597 by (rule sums_group) (use sin_converges [of x, unfolded scaleR_conv_of_real] in auto) 3598 then show ?thesis 3599 by (simp add: sin_coeff_def ac_simps) 3600qed 3601 3602lemma sin_gt_zero_02: 3603 fixes x :: real 3604 assumes "0 < x" and "x < 2" 3605 shows "0 < sin x" 3606proof - 3607 let ?f = "\<lambda>n::nat. \<Sum>k = n*2..<n*2+2. (- 1) ^ k / (fact (2*k+1)) * x^(2*k+1)" 3608 have pos: "\<forall>n. 0 < ?f n" 3609 proof 3610 fix n :: nat 3611 let ?k2 = "real (Suc (Suc (4 * n)))" 3612 let ?k3 = "real (Suc (Suc (Suc (4 * n))))" 3613 have "x * x < ?k2 * ?k3" 3614 using assms by (intro mult_strict_mono', simp_all) 3615 then have "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)" 3616 by (intro mult_strict_right_mono zero_less_power \<open>0 < x\<close>) 3617 then show "0 < ?f n" 3618 by (simp add: divide_simps mult_ac del: mult_Suc) 3619qed 3620 have sums: "?f sums sin x" 3621 by (rule sin_paired [THEN sums_group]) simp 3622 show "0 < sin x" 3623 unfolding sums_unique [OF sums] 3624 using sums_summable [OF sums] pos 3625 by (rule suminf_pos) 3626qed 3627 3628lemma cos_double_less_one: "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1" 3629 for x :: real 3630 using sin_gt_zero_02 [where x = x] by (auto simp: cos_squared_eq cos_double) 3631 3632lemma cos_paired: "(\<lambda>n. (- 1) ^ n / (fact (2 * n)) * x ^ (2 * n)) sums cos x" 3633 for x :: real 3634proof - 3635 have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x" 3636 by (rule sums_group) (use cos_converges [of x, unfolded scaleR_conv_of_real] in auto) 3637 then show ?thesis 3638 by (simp add: cos_coeff_def ac_simps) 3639qed 3640 3641lemma sum_pos_lt_pair: 3642 fixes f :: "nat \<Rightarrow> real" 3643 assumes f: "summable f" and fplus: "\<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc (Suc 0) * d) + 1))" 3644 shows "sum f {..<k} < suminf f" 3645proof - 3646 have "(\<lambda>n. \<Sum>n = n * Suc (Suc 0)..<n * Suc (Suc 0) + Suc (Suc 0). f (n + k)) 3647 sums (\<Sum>n. f (n + k))" 3648 proof (rule sums_group) 3649 show "(\<lambda>n. f (n + k)) sums (\<Sum>n. f (n + k))" 3650 by (simp add: f summable_iff_shift summable_sums) 3651 qed auto 3652 with fplus have "0 < (\<Sum>n. f (n + k))" 3653 apply (simp add: add.commute) 3654 apply (metis (no_types, lifting) suminf_pos summable_def sums_unique) 3655 done 3656 then show ?thesis 3657 by (simp add: f suminf_minus_initial_segment) 3658qed 3659 3660lemma cos_two_less_zero [simp]: "cos 2 < (0::real)" 3661proof - 3662 note fact_Suc [simp del] 3663 from sums_minus [OF cos_paired] 3664 have *: "(\<lambda>n. - ((- 1) ^ n * 2 ^ (2 * n) / fact (2 * n))) sums - cos (2::real)" 3665 by simp 3666 then have sm: "summable (\<lambda>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" 3667 by (rule sums_summable) 3668 have "0 < (\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" 3669 by (simp add: fact_num_eq_if power_eq_if) 3670 moreover have "(\<Sum>n<Suc (Suc (Suc 0)). - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n)))) < 3671 (\<Sum>n. - ((- 1) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" 3672 proof - 3673 { 3674 fix d 3675 let ?six4d = "Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))" 3676 have "(4::real) * (fact (?six4d)) < (Suc (Suc (?six4d)) * fact (Suc (?six4d)))" 3677 unfolding of_nat_mult by (rule mult_strict_mono) (simp_all add: fact_less_mono) 3678 then have "(4::real) * (fact (?six4d)) < (fact (Suc (Suc (?six4d))))" 3679 by (simp only: fact_Suc [of "Suc (?six4d)"] of_nat_mult of_nat_fact) 3680 then have "(4::real) * inverse (fact (Suc (Suc (?six4d)))) < inverse (fact (?six4d))" 3681 by (simp add: inverse_eq_divide less_divide_eq) 3682 } 3683 then show ?thesis 3684 by (force intro!: sum_pos_lt_pair [OF sm] simp add: divide_inverse algebra_simps) 3685 qed 3686 ultimately have "0 < (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" 3687 by (rule order_less_trans) 3688 moreover from * have "- cos 2 = (\<Sum>n. - ((- 1::real) ^ n * 2 ^ (2 * n) / (fact (2 * n))))" 3689 by (rule sums_unique) 3690 ultimately have "(0::real) < - cos 2" by simp 3691 then show ?thesis by simp 3692qed 3693 3694lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq] 3695lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le] 3696 3697lemma cos_is_zero: "\<exists>!x::real. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0" 3698proof (rule ex_ex1I) 3699 show "\<exists>x::real. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0" 3700 by (rule IVT2) simp_all 3701next 3702 fix a b :: real 3703 assume ab: "0 \<le> a \<and> a \<le> 2 \<and> cos a = 0" "0 \<le> b \<and> b \<le> 2 \<and> cos b = 0" 3704 have cosd: "\<And>x::real. cos differentiable (at x)" 3705 unfolding real_differentiable_def by (auto intro: DERIV_cos) 3706 show "a = b" 3707 proof (cases a b rule: linorder_cases) 3708 case less 3709 then obtain z where "a < z" "z < b" "(cos has_real_derivative 0) (at z)" 3710 using Rolle by (metis cosd isCont_cos ab) 3711 then have "sin z = 0" 3712 using DERIV_cos DERIV_unique neg_equal_0_iff_equal by blast 3713 then show ?thesis 3714 by (metis \<open>a < z\<close> \<open>z < b\<close> ab order_less_le_trans less_le sin_gt_zero_02) 3715 next 3716 case greater 3717 then obtain z where "b < z" "z < a" "(cos has_real_derivative 0) (at z)" 3718 using Rolle by (metis cosd isCont_cos ab) 3719 then have "sin z = 0" 3720 using DERIV_cos DERIV_unique neg_equal_0_iff_equal by blast 3721 then show ?thesis 3722 by (metis \<open>b < z\<close> \<open>z < a\<close> ab order_less_le_trans less_le sin_gt_zero_02) 3723 qed auto 3724qed 3725 3726lemma pi_half: "pi/2 = (THE x. 0 \<le> x \<and> x \<le> 2 \<and> cos x = 0)" 3727 by (simp add: pi_def) 3728 3729lemma cos_pi_half [simp]: "cos (pi/2) = 0" 3730 by (simp add: pi_half cos_is_zero [THEN theI']) 3731 3732lemma cos_of_real_pi_half [simp]: "cos ((of_real pi/2) :: 'a) = 0" 3733 if "SORT_CONSTRAINT('a::{real_field,banach,real_normed_algebra_1})" 3734 by (metis cos_pi_half cos_of_real eq_numeral_simps(4) 3735 nonzero_of_real_divide of_real_0 of_real_numeral) 3736 3737lemma pi_half_gt_zero [simp]: "0 < pi/2" 3738proof - 3739 have "0 \<le> pi/2" 3740 by (simp add: pi_half cos_is_zero [THEN theI']) 3741 then show ?thesis 3742 by (metis cos_pi_half cos_zero less_eq_real_def one_neq_zero) 3743qed 3744 3745lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric] 3746lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le] 3747 3748lemma pi_half_less_two [simp]: "pi/2 < 2" 3749proof - 3750 have "pi/2 \<le> 2" 3751 by (simp add: pi_half cos_is_zero [THEN theI']) 3752 then show ?thesis 3753 by (metis cos_pi_half cos_two_neq_zero le_less) 3754qed 3755 3756lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq] 3757lemmas pi_half_le_two [simp] = pi_half_less_two [THEN order_less_imp_le] 3758 3759lemma pi_gt_zero [simp]: "0 < pi" 3760 using pi_half_gt_zero by simp 3761 3762lemma pi_ge_zero [simp]: "0 \<le> pi" 3763 by (rule pi_gt_zero [THEN order_less_imp_le]) 3764 3765lemma pi_neq_zero [simp]: "pi \<noteq> 0" 3766 by (rule pi_gt_zero [THEN less_imp_neq, symmetric]) 3767 3768lemma pi_not_less_zero [simp]: "\<not> pi < 0" 3769 by (simp add: linorder_not_less) 3770 3771lemma minus_pi_half_less_zero: "-(pi/2) < 0" 3772 by simp 3773 3774lemma m2pi_less_pi: "- (2*pi) < pi" 3775 by simp 3776 3777lemma sin_pi_half [simp]: "sin(pi/2) = 1" 3778 using sin_cos_squared_add2 [where x = "pi/2"] 3779 using sin_gt_zero_02 [OF pi_half_gt_zero pi_half_less_two] 3780 by (simp add: power2_eq_1_iff) 3781 3782lemma sin_of_real_pi_half [simp]: "sin ((of_real pi/2) :: 'a) = 1" 3783 if "SORT_CONSTRAINT('a::{real_field,banach,real_normed_algebra_1})" 3784 using sin_pi_half 3785 by (metis sin_pi_half eq_numeral_simps(4) nonzero_of_real_divide of_real_1 of_real_numeral sin_of_real) 3786 3787lemma sin_cos_eq: "sin x = cos (of_real pi/2 - x)" 3788 for x :: "'a::{real_normed_field,banach}" 3789 by (simp add: cos_diff) 3790 3791lemma minus_sin_cos_eq: "- sin x = cos (x + of_real pi/2)" 3792 for x :: "'a::{real_normed_field,banach}" 3793 by (simp add: cos_add nonzero_of_real_divide) 3794 3795lemma cos_sin_eq: "cos x = sin (of_real pi/2 - x)" 3796 for x :: "'a::{real_normed_field,banach}" 3797 using sin_cos_eq [of "of_real pi/2 - x"] by simp 3798 3799lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y" 3800 for x :: "'a::{real_normed_field,banach}" 3801 using cos_add [of "of_real pi/2 - x" "-y"] 3802 by (simp add: cos_sin_eq) (simp add: sin_cos_eq) 3803 3804lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y" 3805 for x :: "'a::{real_normed_field,banach}" 3806 using sin_add [of x "- y"] by simp 3807 3808lemma sin_double: "sin(2 * x) = 2 * sin x * cos x" 3809 for x :: "'a::{real_normed_field,banach}" 3810 using sin_add [where x=x and y=x] by simp 3811 3812lemma cos_of_real_pi [simp]: "cos (of_real pi) = -1" 3813 using cos_add [where x = "pi/2" and y = "pi/2"] 3814 by (simp add: cos_of_real) 3815 3816lemma sin_of_real_pi [simp]: "sin (of_real pi) = 0" 3817 using sin_add [where x = "pi/2" and y = "pi/2"] 3818 by (simp add: sin_of_real) 3819 3820lemma cos_pi [simp]: "cos pi = -1" 3821 using cos_add [where x = "pi/2" and y = "pi/2"] by simp 3822 3823lemma sin_pi [simp]: "sin pi = 0" 3824 using sin_add [where x = "pi/2" and y = "pi/2"] by simp 3825 3826lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x" 3827 by (simp add: sin_add) 3828 3829lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x" 3830 by (simp add: sin_add) 3831 3832lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x" 3833 by (simp add: cos_add) 3834 3835lemma cos_periodic_pi2 [simp]: "cos (pi + x) = - cos x" 3836 by (simp add: cos_add) 3837 3838lemma sin_periodic [simp]: "sin (x + 2 * pi) = sin x" 3839 by (simp add: sin_add sin_double cos_double) 3840 3841lemma cos_periodic [simp]: "cos (x + 2 * pi) = cos x" 3842 by (simp add: cos_add sin_double cos_double) 3843 3844lemma cos_npi [simp]: "cos (real n * pi) = (- 1) ^ n" 3845 by (induct n) (auto simp: distrib_right) 3846 3847lemma cos_npi2 [simp]: "cos (pi * real n) = (- 1) ^ n" 3848 by (metis cos_npi mult.commute) 3849 3850lemma sin_npi [simp]: "sin (real n * pi) = 0" 3851 for n :: nat 3852 by (induct n) (auto simp: distrib_right) 3853 3854lemma sin_npi2 [simp]: "sin (pi * real n) = 0" 3855 for n :: nat 3856 by (simp add: mult.commute [of pi]) 3857 3858lemma cos_two_pi [simp]: "cos (2 * pi) = 1" 3859 by (simp add: cos_double) 3860 3861lemma sin_two_pi [simp]: "sin (2 * pi) = 0" 3862 by (simp add: sin_double) 3863 3864lemma sin_times_sin: "sin w * sin z = (cos (w - z) - cos (w + z)) / 2" 3865 for w :: "'a::{real_normed_field,banach}" 3866 by (simp add: cos_diff cos_add) 3867 3868lemma sin_times_cos: "sin w * cos z = (sin (w + z) + sin (w - z)) / 2" 3869 for w :: "'a::{real_normed_field,banach}" 3870 by (simp add: sin_diff sin_add) 3871 3872lemma cos_times_sin: "cos w * sin z = (sin (w + z) - sin (w - z)) / 2" 3873 for w :: "'a::{real_normed_field,banach}" 3874 by (simp add: sin_diff sin_add) 3875 3876lemma cos_times_cos: "cos w * cos z = (cos (w - z) + cos (w + z)) / 2" 3877 for w :: "'a::{real_normed_field,banach}" 3878 by (simp add: cos_diff cos_add) 3879 3880lemma sin_plus_sin: "sin w + sin z = 2 * sin ((w + z) / 2) * cos ((w - z) / 2)" 3881 for w :: "'a::{real_normed_field,banach}" 3882 apply (simp add: mult.assoc sin_times_cos) 3883 apply (simp add: field_simps) 3884 done 3885 3886lemma sin_diff_sin: "sin w - sin z = 2 * sin ((w - z) / 2) * cos ((w + z) / 2)" 3887 for w :: "'a::{real_normed_field,banach}" 3888 apply (simp add: mult.assoc sin_times_cos) 3889 apply (simp add: field_simps) 3890 done 3891 3892lemma cos_plus_cos: "cos w + cos z = 2 * cos ((w + z) / 2) * cos ((w - z) / 2)" 3893 for w :: "'a::{real_normed_field,banach,field}" 3894 apply (simp add: mult.assoc cos_times_cos) 3895 apply (simp add: field_simps) 3896 done 3897 3898lemma cos_diff_cos: "cos w - cos z = 2 * sin ((w + z) / 2) * sin ((z - w) / 2)" 3899 for w :: "'a::{real_normed_field,banach,field}" 3900 apply (simp add: mult.assoc sin_times_sin) 3901 apply (simp add: field_simps) 3902 done 3903 3904lemma cos_double_cos: "cos (2 * z) = 2 * cos z ^ 2 - 1" 3905 for z :: "'a::{real_normed_field,banach}" 3906 by (simp add: cos_double sin_squared_eq) 3907 3908lemma cos_double_sin: "cos (2 * z) = 1 - 2 * sin z ^ 2" 3909 for z :: "'a::{real_normed_field,banach}" 3910 by (simp add: cos_double sin_squared_eq) 3911 3912lemma sin_pi_minus [simp]: "sin (pi - x) = sin x" 3913 by (metis sin_minus sin_periodic_pi minus_minus uminus_add_conv_diff) 3914 3915lemma cos_pi_minus [simp]: "cos (pi - x) = - (cos x)" 3916 by (metis cos_minus cos_periodic_pi uminus_add_conv_diff) 3917 3918lemma sin_minus_pi [simp]: "sin (x - pi) = - (sin x)" 3919 by (simp add: sin_diff) 3920 3921lemma cos_minus_pi [simp]: "cos (x - pi) = - (cos x)" 3922 by (simp add: cos_diff) 3923 3924lemma sin_2pi_minus [simp]: "sin (2 * pi - x) = - (sin x)" 3925 by (metis sin_periodic_pi2 add_diff_eq mult_2 sin_pi_minus) 3926 3927lemma cos_2pi_minus [simp]: "cos (2 * pi - x) = cos x" 3928 by (metis (no_types, hide_lams) cos_add cos_minus cos_two_pi sin_minus sin_two_pi 3929 diff_0_right minus_diff_eq mult_1 mult_zero_left uminus_add_conv_diff) 3930 3931lemma sin_gt_zero2: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < sin x" 3932 by (metis sin_gt_zero_02 order_less_trans pi_half_less_two) 3933 3934lemma sin_less_zero: 3935 assumes "- pi/2 < x" and "x < 0" 3936 shows "sin x < 0" 3937proof - 3938 have "0 < sin (- x)" 3939 using assms by (simp only: sin_gt_zero2) 3940 then show ?thesis by simp 3941qed 3942 3943lemma pi_less_4: "pi < 4" 3944 using pi_half_less_two by auto 3945 3946lemma cos_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cos x" 3947 by (simp add: cos_sin_eq sin_gt_zero2) 3948 3949lemma cos_gt_zero_pi: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cos x" 3950 using cos_gt_zero [of x] cos_gt_zero [of "-x"] 3951 by (cases rule: linorder_cases [of x 0]) auto 3952 3953lemma cos_ge_zero: "-(pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> 0 \<le> cos x" 3954 by (auto simp: order_le_less cos_gt_zero_pi) 3955 (metis cos_pi_half eq_divide_eq eq_numeral_simps(4)) 3956 3957lemma sin_gt_zero: "0 < x \<Longrightarrow> x < pi \<Longrightarrow> 0 < sin x" 3958 by (simp add: sin_cos_eq cos_gt_zero_pi) 3959 3960lemma sin_lt_zero: "pi < x \<Longrightarrow> x < 2 * pi \<Longrightarrow> sin x < 0" 3961 using sin_gt_zero [of "x - pi"] 3962 by (simp add: sin_diff) 3963 3964lemma pi_ge_two: "2 \<le> pi" 3965proof (rule ccontr) 3966 assume "\<not> ?thesis" 3967 then have "pi < 2" by auto 3968 have "\<exists>y > pi. y < 2 \<and> y < 2 * pi" 3969 proof (cases "2 < 2 * pi") 3970 case True 3971 with dense[OF \<open>pi < 2\<close>] show ?thesis by auto 3972 next 3973 case False 3974 have "pi < 2 * pi" by auto 3975 from dense[OF this] and False show ?thesis by auto 3976 qed 3977 then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" 3978 by blast 3979 then have "0 < sin y" 3980 using sin_gt_zero_02 by auto 3981 moreover have "sin y < 0" 3982 using sin_gt_zero[of "y - pi"] \<open>pi < y\<close> and \<open>y < 2 * pi\<close> sin_periodic_pi[of "y - pi"] 3983 by auto 3984 ultimately show False by auto 3985qed 3986 3987lemma sin_ge_zero: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> sin x" 3988 by (auto simp: order_le_less sin_gt_zero) 3989 3990lemma sin_le_zero: "pi \<le> x \<Longrightarrow> x < 2 * pi \<Longrightarrow> sin x \<le> 0" 3991 using sin_ge_zero [of "x - pi"] by (simp add: sin_diff) 3992 3993lemma sin_pi_divide_n_ge_0 [simp]: 3994 assumes "n \<noteq> 0" 3995 shows "0 \<le> sin (pi / real n)" 3996 by (rule sin_ge_zero) (use assms in \<open>simp_all add: divide_simps\<close>) 3997 3998lemma sin_pi_divide_n_gt_0: 3999 assumes "2 \<le> n" 4000 shows "0 < sin (pi / real n)" 4001 by (rule sin_gt_zero) (use assms in \<open>simp_all add: divide_simps\<close>) 4002 4003text\<open>Proof resembles that of @{text cos_is_zero} but with @{term pi} for the upper bound\<close> 4004lemma cos_total: 4005 assumes y: "-1 \<le> y" "y \<le> 1" 4006 shows "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y" 4007proof (rule ex_ex1I) 4008 show "\<exists>x::real. 0 \<le> x \<and> x \<le> pi \<and> cos x = y" 4009 by (rule IVT2) (simp_all add: y) 4010next 4011 fix a b :: real 4012 assume ab: "0 \<le> a \<and> a \<le> pi \<and> cos a = y" "0 \<le> b \<and> b \<le> pi \<and> cos b = y" 4013 have cosd: "\<And>x::real. cos differentiable (at x)" 4014 unfolding real_differentiable_def by (auto intro: DERIV_cos) 4015 show "a = b" 4016 proof (cases a b rule: linorder_cases) 4017 case less 4018 then obtain z where "a < z" "z < b" "(cos has_real_derivative 0) (at z)" 4019 using Rolle by (metis cosd isCont_cos ab) 4020 then have "sin z = 0" 4021 using DERIV_cos DERIV_unique neg_equal_0_iff_equal by blast 4022 then show ?thesis 4023 by (metis \<open>a < z\<close> \<open>z < b\<close> ab order_less_le_trans less_le sin_gt_zero) 4024 next 4025 case greater 4026 then obtain z where "b < z" "z < a" "(cos has_real_derivative 0) (at z)" 4027 using Rolle by (metis cosd isCont_cos ab) 4028 then have "sin z = 0" 4029 using DERIV_cos DERIV_unique neg_equal_0_iff_equal by blast 4030 then show ?thesis 4031 by (metis \<open>b < z\<close> \<open>z < a\<close> ab order_less_le_trans less_le sin_gt_zero) 4032 qed auto 4033qed 4034 4035lemma sin_total: 4036 assumes y: "-1 \<le> y" "y \<le> 1" 4037 shows "\<exists>!x. - (pi/2) \<le> x \<and> x \<le> pi/2 \<and> sin x = y" 4038proof - 4039 from cos_total [OF y] 4040 obtain x where x: "0 \<le> x" "x \<le> pi" "cos x = y" 4041 and uniq: "\<And>x'. 0 \<le> x' \<Longrightarrow> x' \<le> pi \<Longrightarrow> cos x' = y \<Longrightarrow> x' = x " 4042 by blast 4043 show ?thesis 4044 unfolding sin_cos_eq 4045 proof (rule ex1I [where a="pi/2 - x"]) 4046 show "- (pi/2) \<le> z \<and> z \<le> pi/2 \<and> cos (of_real pi/2 - z) = y \<Longrightarrow> 4047 z = pi/2 - x" for z 4048 using uniq [of "pi/2 -z"] by auto 4049 qed (use x in auto) 4050qed 4051 4052lemma cos_zero_lemma: 4053 assumes "0 \<le> x" "cos x = 0" 4054 shows "\<exists>n. odd n \<and> x = of_nat n * (pi/2) \<and> n > 0" 4055proof - 4056 have xle: "x < (1 + real_of_int \<lfloor>x/pi\<rfloor>) * pi" 4057 using floor_correct [of "x/pi"] 4058 by (simp add: add.commute divide_less_eq) 4059 obtain n where "real n * pi \<le> x" "x < real (Suc n) * pi" 4060 proof 4061 show "real (nat \<lfloor>x / pi\<rfloor>) * pi \<le> x" 4062 using assms floor_divide_lower [of pi x] by auto 4063 show "x < real (Suc (nat \<lfloor>x / pi\<rfloor>)) * pi" 4064 using assms floor_divide_upper [of pi x] by (simp add: xle) 4065 qed 4066 then have x: "0 \<le> x - n * pi" "(x - n * pi) \<le> pi" "cos (x - n * pi) = 0" 4067 by (auto simp: algebra_simps cos_diff assms) 4068 then have "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = 0" 4069 by (auto simp: intro!: cos_total) 4070 then obtain \<theta> where \<theta>: "0 \<le> \<theta>" "\<theta> \<le> pi" "cos \<theta> = 0" 4071 and uniq: "\<And>\<phi>. 0 \<le> \<phi> \<Longrightarrow> \<phi> \<le> pi \<Longrightarrow> cos \<phi> = 0 \<Longrightarrow> \<phi> = \<theta>" 4072 by blast 4073 then have "x - real n * pi = \<theta>" 4074 using x by blast 4075 moreover have "pi/2 = \<theta>" 4076 using pi_half_ge_zero uniq by fastforce 4077 ultimately show ?thesis 4078 by (rule_tac x = "Suc (2 * n)" in exI) (simp add: algebra_simps) 4079qed 4080 4081lemma sin_zero_lemma: "0 \<le> x \<Longrightarrow> sin x = 0 \<Longrightarrow> \<exists>n::nat. even n \<and> x = real n * (pi/2)" 4082 using cos_zero_lemma [of "x + pi/2"] 4083 apply (clarsimp simp add: cos_add) 4084 apply (rule_tac x = "n - 1" in exI) 4085 apply (simp add: algebra_simps of_nat_diff) 4086 done 4087 4088lemma cos_zero_iff: 4089 "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))))" 4090 (is "?lhs = ?rhs") 4091proof - 4092 have *: "cos (real n * pi/2) = 0" if "odd n" for n :: nat 4093 proof - 4094 from that obtain m where "n = 2 * m + 1" .. 4095 then show ?thesis 4096 by (simp add: field_simps) (simp add: cos_add add_divide_distrib) 4097 qed 4098 show ?thesis 4099 proof 4100 show ?rhs if ?lhs 4101 using that cos_zero_lemma [of x] cos_zero_lemma [of "-x"] by force 4102 show ?lhs if ?rhs 4103 using that by (auto dest: * simp del: eq_divide_eq_numeral1) 4104 qed 4105qed 4106 4107lemma sin_zero_iff: 4108 "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))))" 4109 (is "?lhs = ?rhs") 4110proof 4111 show ?rhs if ?lhs 4112 using that sin_zero_lemma [of x] sin_zero_lemma [of "-x"] by force 4113 show ?lhs if ?rhs 4114 using that by (auto elim: evenE) 4115qed 4116 4117lemma cos_zero_iff_int: "cos x = 0 \<longleftrightarrow> (\<exists>n. odd n \<and> x = of_int n * (pi/2))" 4118proof - 4119 have 1: "\<And>n. odd n \<Longrightarrow> \<exists>i. odd i \<and> real n = real_of_int i" 4120 by (metis even_of_nat of_int_of_nat_eq) 4121 have 2: "\<And>n. odd n \<Longrightarrow> \<exists>i. odd i \<and> - (real n * pi) = real_of_int i * pi" 4122 by (metis even_minus even_of_nat mult.commute mult_minus_right of_int_minus of_int_of_nat_eq) 4123 have 3: "\<lbrakk>odd i; \<forall>n. even n \<or> real_of_int i \<noteq> - (real n)\<rbrakk> 4124 \<Longrightarrow> \<exists>n. odd n \<and> real_of_int i = real n" for i 4125 by (cases i rule: int_cases2) auto 4126 show ?thesis 4127 by (force simp: cos_zero_iff intro!: 1 2 3) 4128qed 4129 4130lemma sin_zero_iff_int: "sin x = 0 \<longleftrightarrow> (\<exists>n. even n \<and> x = of_int n * (pi/2))" 4131proof safe 4132 assume "sin x = 0" 4133 then show "\<exists>n. even n \<and> x = of_int n * (pi/2)" 4134 apply (simp add: sin_zero_iff, safe) 4135 apply (metis even_of_nat of_int_of_nat_eq) 4136 apply (rule_tac x="- (int n)" in exI) 4137 apply simp 4138 done 4139next 4140 fix i :: int 4141 assume "even i" 4142 then show "sin (of_int i * (pi/2)) = 0" 4143 by (cases i rule: int_cases2, simp_all add: sin_zero_iff) 4144qed 4145 4146lemma sin_zero_iff_int2: "sin x = 0 \<longleftrightarrow> (\<exists>n::int. x = of_int n * pi)" 4147 apply (simp only: sin_zero_iff_int) 4148 apply (safe elim!: evenE) 4149 apply (simp_all add: field_simps) 4150 using dvd_triv_left apply fastforce 4151 done 4152 4153lemma sin_npi_int [simp]: "sin (pi * of_int n) = 0" 4154 by (simp add: sin_zero_iff_int2) 4155 4156lemma cos_monotone_0_pi: 4157 assumes "0 \<le> y" and "y < x" and "x \<le> pi" 4158 shows "cos x < cos y" 4159proof - 4160 have "- (x - y) < 0" using assms by auto 4161 from MVT2[OF \<open>y < x\<close> DERIV_cos] 4162 obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" 4163 by auto 4164 then have "0 < z" and "z < pi" 4165 using assms by auto 4166 then have "0 < sin z" 4167 using sin_gt_zero by auto 4168 then have "cos x - cos y < 0" 4169 unfolding cos_diff minus_mult_commute[symmetric] 4170 using \<open>- (x - y) < 0\<close> by (rule mult_pos_neg2) 4171 then show ?thesis by auto 4172qed 4173 4174lemma cos_monotone_0_pi_le: 4175 assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" 4176 shows "cos x \<le> cos y" 4177proof (cases "y < x") 4178 case True 4179 show ?thesis 4180 using cos_monotone_0_pi[OF \<open>0 \<le> y\<close> True \<open>x \<le> pi\<close>] by auto 4181next 4182 case False 4183 then have "y = x" using \<open>y \<le> x\<close> by auto 4184 then show ?thesis by auto 4185qed 4186 4187lemma cos_monotone_minus_pi_0: 4188 assumes "- pi \<le> y" and "y < x" and "x \<le> 0" 4189 shows "cos y < cos x" 4190proof - 4191 have "0 \<le> - x" and "- x < - y" and "- y \<le> pi" 4192 using assms by auto 4193 from cos_monotone_0_pi[OF this] show ?thesis 4194 unfolding cos_minus . 4195qed 4196 4197lemma cos_monotone_minus_pi_0': 4198 assumes "- pi \<le> y" and "y \<le> x" and "x \<le> 0" 4199 shows "cos y \<le> cos x" 4200proof (cases "y < x") 4201 case True 4202 show ?thesis using cos_monotone_minus_pi_0[OF \<open>-pi \<le> y\<close> True \<open>x \<le> 0\<close>] 4203 by auto 4204next 4205 case False 4206 then have "y = x" using \<open>y \<le> x\<close> by auto 4207 then show ?thesis by auto 4208qed 4209 4210lemma sin_monotone_2pi: 4211 assumes "- (pi/2) \<le> y" and "y < x" and "x \<le> pi/2" 4212 shows "sin y < sin x" 4213 unfolding sin_cos_eq 4214 using assms by (auto intro: cos_monotone_0_pi) 4215 4216lemma sin_monotone_2pi_le: 4217 assumes "- (pi/2) \<le> y" and "y \<le> x" and "x \<le> pi/2" 4218 shows "sin y \<le> sin x" 4219 by (metis assms le_less sin_monotone_2pi) 4220 4221lemma sin_x_le_x: 4222 fixes x :: real 4223 assumes x: "x \<ge> 0" 4224 shows "sin x \<le> x" 4225proof - 4226 let ?f = "\<lambda>x. x - sin x" 4227 from x have "?f x \<ge> ?f 0" 4228 apply (rule DERIV_nonneg_imp_nondecreasing) 4229 apply (intro allI impI exI[of _ "1 - cos x" for x]) 4230 apply (auto intro!: derivative_eq_intros simp: field_simps) 4231 done 4232 then show "sin x \<le> x" by simp 4233qed 4234 4235lemma sin_x_ge_neg_x: 4236 fixes x :: real 4237 assumes x: "x \<ge> 0" 4238 shows "sin x \<ge> - x" 4239proof - 4240 let ?f = "\<lambda>x. x + sin x" 4241 from x have "?f x \<ge> ?f 0" 4242 apply (rule DERIV_nonneg_imp_nondecreasing) 4243 apply (intro allI impI exI[of _ "1 + cos x" for x]) 4244 apply (auto intro!: derivative_eq_intros simp: field_simps real_0_le_add_iff) 4245 done 4246 then show "sin x \<ge> -x" by simp 4247qed 4248 4249lemma abs_sin_x_le_abs_x: "\<bar>sin x\<bar> \<le> \<bar>x\<bar>" 4250 for x :: real 4251 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"] 4252 by (auto simp: abs_real_def) 4253 4254 4255subsection \<open>More Corollaries about Sine and Cosine\<close> 4256 4257lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi/2) = (-1) ^ n" 4258proof - 4259 have "sin ((real n + 1/2) * pi) = cos (real n * pi)" 4260 by (auto simp: algebra_simps sin_add) 4261 then show ?thesis 4262 by (simp add: distrib_right add_divide_distrib add.commute mult.commute [of pi]) 4263qed 4264 4265lemma cos_2npi [simp]: "cos (2 * real n * pi) = 1" 4266 for n :: nat 4267 by (cases "even n") (simp_all add: cos_double mult.assoc) 4268 4269lemma cos_3over2_pi [simp]: "cos (3/2*pi) = 0" 4270proof - 4271 have "cos (3/2*pi) = cos (pi + pi/2)" 4272 by simp 4273 also have "... = 0" 4274 by (subst cos_add, simp) 4275 finally show ?thesis . 4276qed 4277 4278lemma sin_2npi [simp]: "sin (2 * real n * pi) = 0" 4279 for n :: nat 4280 by (auto simp: mult.assoc sin_double) 4281 4282lemma sin_3over2_pi [simp]: "sin (3/2*pi) = - 1" 4283proof - 4284 have "sin (3/2*pi) = sin (pi + pi/2)" 4285 by simp 4286 also have "... = -1" 4287 by (subst sin_add, simp) 4288 finally show ?thesis . 4289qed 4290 4291lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0" 4292 by (simp only: cos_add sin_add of_nat_Suc distrib_right distrib_left add_divide_distrib, auto) 4293 4294lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)" 4295 by (auto intro!: derivative_eq_intros) 4296 4297lemma sin_zero_norm_cos_one: 4298 fixes x :: "'a::{real_normed_field,banach}" 4299 assumes "sin x = 0" 4300 shows "norm (cos x) = 1" 4301 using sin_cos_squared_add [of x, unfolded assms] 4302 by (simp add: square_norm_one) 4303 4304lemma sin_zero_abs_cos_one: "sin x = 0 \<Longrightarrow> \<bar>cos x\<bar> = (1::real)" 4305 using sin_zero_norm_cos_one by fastforce 4306 4307lemma cos_one_sin_zero: 4308 fixes x :: "'a::{real_normed_field,banach}" 4309 assumes "cos x = 1" 4310 shows "sin x = 0" 4311 using sin_cos_squared_add [of x, unfolded assms] 4312 by simp 4313 4314lemma sin_times_pi_eq_0: "sin (x * pi) = 0 \<longleftrightarrow> x \<in> \<int>" 4315 by (simp add: sin_zero_iff_int2) (metis Ints_cases Ints_of_int) 4316 4317lemma cos_one_2pi: "cos x = 1 \<longleftrightarrow> (\<exists>n::nat. x = n * 2 * pi) \<or> (\<exists>n::nat. x = - (n * 2 * pi))" 4318 (is "?lhs = ?rhs") 4319proof 4320 assume ?lhs 4321 then have "sin x = 0" 4322 by (simp add: cos_one_sin_zero) 4323 then show ?rhs 4324 proof (simp only: sin_zero_iff, elim exE disjE conjE) 4325 fix n :: nat 4326 assume n: "even n" "x = real n * (pi/2)" 4327 then obtain m where m: "n = 2 * m" 4328 using dvdE by blast 4329 then have me: "even m" using \<open>?lhs\<close> n 4330 by (auto simp: field_simps) (metis one_neq_neg_one power_minus_odd power_one) 4331 show ?rhs 4332 using m me n 4333 by (auto simp: field_simps elim!: evenE) 4334 next 4335 fix n :: nat 4336 assume n: "even n" "x = - (real n * (pi/2))" 4337 then obtain m where m: "n = 2 * m" 4338 using dvdE by blast 4339 then have me: "even m" using \<open>?lhs\<close> n 4340 by (auto simp: field_simps) (metis one_neq_neg_one power_minus_odd power_one) 4341 show ?rhs 4342 using m me n 4343 by (auto simp: field_simps elim!: evenE) 4344 qed 4345next 4346 assume ?rhs 4347 then show "cos x = 1" 4348 by (metis cos_2npi cos_minus mult.assoc mult.left_commute) 4349qed 4350 4351lemma cos_one_2pi_int: "cos x = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2 * pi)" (is "?lhs = ?rhs") 4352proof 4353 assume "cos x = 1" 4354 then show ?rhs 4355 by (metis cos_one_2pi mult.commute mult_minus_right of_int_minus of_int_of_nat_eq) 4356next 4357 assume ?rhs 4358 then show "cos x = 1" 4359 by (clarsimp simp add: cos_one_2pi) (metis mult_minus_right of_int_of_nat) 4360qed 4361 4362lemma cos_npi_int [simp]: 4363 fixes n::int shows "cos (pi * of_int n) = (if even n then 1 else -1)" 4364 by (auto simp: algebra_simps cos_one_2pi_int elim!: oddE evenE) 4365 4366lemma sin_cos_sqrt: "0 \<le> sin x \<Longrightarrow> sin x = sqrt (1 - (cos(x) ^ 2))" 4367 using sin_squared_eq real_sqrt_unique by fastforce 4368 4369lemma sin_eq_0_pi: "- pi < x \<Longrightarrow> x < pi \<Longrightarrow> sin x = 0 \<Longrightarrow> x = 0" 4370 by (metis sin_gt_zero sin_minus minus_less_iff neg_0_less_iff_less not_less_iff_gr_or_eq) 4371 4372lemma cos_treble_cos: "cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x" 4373 for x :: "'a::{real_normed_field,banach}" 4374proof - 4375 have *: "(sin x * (sin x * 3)) = 3 - (cos x * (cos x * 3))" 4376 by (simp add: mult.assoc [symmetric] sin_squared_eq [unfolded power2_eq_square]) 4377 have "cos(3 * x) = cos(2*x + x)" 4378 by simp 4379 also have "\<dots> = 4 * cos x ^ 3 - 3 * cos x" 4380 apply (simp only: cos_add cos_double sin_double) 4381 apply (simp add: * field_simps power2_eq_square power3_eq_cube) 4382 done 4383 finally show ?thesis . 4384qed 4385 4386lemma cos_45: "cos (pi / 4) = sqrt 2 / 2" 4387proof - 4388 let ?c = "cos (pi / 4)" 4389 let ?s = "sin (pi / 4)" 4390 have nonneg: "0 \<le> ?c" 4391 by (simp add: cos_ge_zero) 4392 have "0 = cos (pi / 4 + pi / 4)" 4393 by simp 4394 also have "cos (pi / 4 + pi / 4) = ?c\<^sup>2 - ?s\<^sup>2" 4395 by (simp only: cos_add power2_eq_square) 4396 also have "\<dots> = 2 * ?c\<^sup>2 - 1" 4397 by (simp add: sin_squared_eq) 4398 finally have "?c\<^sup>2 = (sqrt 2 / 2)\<^sup>2" 4399 by (simp add: power_divide) 4400 then show ?thesis 4401 using nonneg by (rule power2_eq_imp_eq) simp 4402qed 4403 4404lemma cos_30: "cos (pi / 6) = sqrt 3/2" 4405proof - 4406 let ?c = "cos (pi / 6)" 4407 let ?s = "sin (pi / 6)" 4408 have pos_c: "0 < ?c" 4409 by (rule cos_gt_zero) simp_all 4410 have "0 = cos (pi / 6 + pi / 6 + pi / 6)" 4411 by simp 4412 also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s" 4413 by (simp only: cos_add sin_add) 4414 also have "\<dots> = ?c * (?c\<^sup>2 - 3 * ?s\<^sup>2)" 4415 by (simp add: algebra_simps power2_eq_square) 4416 finally have "?c\<^sup>2 = (sqrt 3/2)\<^sup>2" 4417 using pos_c by (simp add: sin_squared_eq power_divide) 4418 then show ?thesis 4419 using pos_c [THEN order_less_imp_le] 4420 by (rule power2_eq_imp_eq) simp 4421qed 4422 4423lemma sin_45: "sin (pi / 4) = sqrt 2 / 2" 4424 by (simp add: sin_cos_eq cos_45) 4425 4426lemma sin_60: "sin (pi / 3) = sqrt 3/2" 4427 by (simp add: sin_cos_eq cos_30) 4428 4429lemma cos_60: "cos (pi / 3) = 1 / 2" 4430proof - 4431 have "0 \<le> cos (pi / 3)" 4432 by (rule cos_ge_zero) (use pi_half_ge_zero in \<open>linarith+\<close>) 4433 then show ?thesis 4434 by (simp add: cos_squared_eq sin_60 power_divide power2_eq_imp_eq) 4435qed 4436 4437lemma sin_30: "sin (pi / 6) = 1 / 2" 4438 by (simp add: sin_cos_eq cos_60) 4439 4440lemma cos_integer_2pi: "n \<in> \<int> \<Longrightarrow> cos(2 * pi * n) = 1" 4441 by (metis Ints_cases cos_one_2pi_int mult.assoc mult.commute) 4442 4443lemma sin_integer_2pi: "n \<in> \<int> \<Longrightarrow> sin(2 * pi * n) = 0" 4444 by (metis sin_two_pi Ints_mult mult.assoc mult.commute sin_times_pi_eq_0) 4445 4446lemma cos_int_2pin [simp]: "cos ((2 * pi) * of_int n) = 1" 4447 by (simp add: cos_one_2pi_int) 4448 4449lemma sin_int_2pin [simp]: "sin ((2 * pi) * of_int n) = 0" 4450 by (metis Ints_of_int sin_integer_2pi) 4451 4452lemma sincos_principal_value: "\<exists>y. (- pi < y \<and> y \<le> pi) \<and> (sin y = sin x \<and> cos y = cos x)" 4453 apply (rule exI [where x="pi - (2 * pi) * frac ((pi - x) / (2 * pi))"]) 4454 apply (auto simp: field_simps frac_lt_1) 4455 apply (simp_all add: frac_def divide_simps) 4456 apply (simp_all add: add_divide_distrib diff_divide_distrib) 4457 apply (simp_all add: sin_diff cos_diff mult.assoc [symmetric] cos_integer_2pi sin_integer_2pi) 4458 done 4459 4460 4461subsection \<open>Tangent\<close> 4462 4463definition tan :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4464 where "tan = (\<lambda>x. sin x / cos x)" 4465 4466lemma tan_of_real: "of_real (tan x) = (tan (of_real x) :: 'a::{real_normed_field,banach})" 4467 by (simp add: tan_def sin_of_real cos_of_real) 4468 4469lemma tan_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> tan z \<in> \<real>" 4470 for z :: "'a::{real_normed_field,banach}" 4471 by (simp add: tan_def) 4472 4473lemma tan_zero [simp]: "tan 0 = 0" 4474 by (simp add: tan_def) 4475 4476lemma tan_pi [simp]: "tan pi = 0" 4477 by (simp add: tan_def) 4478 4479lemma tan_npi [simp]: "tan (real n * pi) = 0" 4480 for n :: nat 4481 by (simp add: tan_def) 4482 4483lemma tan_minus [simp]: "tan (- x) = - tan x" 4484 by (simp add: tan_def) 4485 4486lemma tan_periodic [simp]: "tan (x + 2 * pi) = tan x" 4487 by (simp add: tan_def) 4488 4489lemma 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)" 4490 by (simp add: tan_def cos_add field_simps) 4491 4492lemma add_tan_eq: "cos x \<noteq> 0 \<Longrightarrow> cos y \<noteq> 0 \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)" 4493 for x :: "'a::{real_normed_field,banach}" 4494 by (simp add: tan_def sin_add field_simps) 4495 4496lemma tan_add: 4497 "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)" 4498 for x :: "'a::{real_normed_field,banach}" 4499 by (simp add: add_tan_eq lemma_tan_add1 field_simps) (simp add: tan_def) 4500 4501lemma tan_double: "cos x \<noteq> 0 \<Longrightarrow> cos (2 * x) \<noteq> 0 \<Longrightarrow> tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)" 4502 for x :: "'a::{real_normed_field,banach}" 4503 using tan_add [of x x] by (simp add: power2_eq_square) 4504 4505lemma tan_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < tan x" 4506 by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) 4507 4508lemma tan_less_zero: 4509 assumes "- pi/2 < x" and "x < 0" 4510 shows "tan x < 0" 4511proof - 4512 have "0 < tan (- x)" 4513 using assms by (simp only: tan_gt_zero) 4514 then show ?thesis by simp 4515qed 4516 4517lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)" 4518 for x :: "'a::{real_normed_field,banach,field}" 4519 unfolding tan_def sin_double cos_double sin_squared_eq 4520 by (simp add: power2_eq_square) 4521 4522lemma tan_30: "tan (pi / 6) = 1 / sqrt 3" 4523 unfolding tan_def by (simp add: sin_30 cos_30) 4524 4525lemma tan_45: "tan (pi / 4) = 1" 4526 unfolding tan_def by (simp add: sin_45 cos_45) 4527 4528lemma tan_60: "tan (pi / 3) = sqrt 3" 4529 unfolding tan_def by (simp add: sin_60 cos_60) 4530 4531lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<^sup>2)" 4532 for x :: "'a::{real_normed_field,banach}" 4533 unfolding tan_def 4534 by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square) 4535 4536declare DERIV_tan[THEN DERIV_chain2, derivative_intros] 4537 and DERIV_tan[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 4538 4539lemmas has_derivative_tan[derivative_intros] = DERIV_tan[THEN DERIV_compose_FDERIV] 4540 4541lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x" 4542 for x :: "'a::{real_normed_field,banach}" 4543 by (rule DERIV_tan [THEN DERIV_isCont]) 4544 4545lemma isCont_tan' [simp,continuous_intros]: 4546 fixes a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a" 4547 shows "isCont f a \<Longrightarrow> cos (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a" 4548 by (rule isCont_o2 [OF _ isCont_tan]) 4549 4550lemma tendsto_tan [tendsto_intros]: 4551 fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4552 shows "(f \<longlongrightarrow> a) F \<Longrightarrow> cos a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. tan (f x)) \<longlongrightarrow> tan a) F" 4553 by (rule isCont_tendsto_compose [OF isCont_tan]) 4554 4555lemma continuous_tan: 4556 fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4557 shows "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))" 4558 unfolding continuous_def by (rule tendsto_tan) 4559 4560lemma continuous_on_tan [continuous_intros]: 4561 fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4562 shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))" 4563 unfolding continuous_on_def by (auto intro: tendsto_tan) 4564 4565lemma continuous_within_tan [continuous_intros]: 4566 fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4567 shows "continuous (at x within s) f \<Longrightarrow> 4568 cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))" 4569 unfolding continuous_within by (rule tendsto_tan) 4570 4571lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) \<midarrow>pi/2\<rightarrow> 0" 4572 by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all) 4573 4574lemma lemma_tan_total: 4575 assumes "0 < y" shows "\<exists>x. 0 < x \<and> x < pi/2 \<and> y < tan x" 4576proof - 4577 obtain s where "0 < s" 4578 and s: "\<And>x. \<lbrakk>x \<noteq> pi/2; norm (x - pi/2) < s\<rbrakk> \<Longrightarrow> norm (cos x / sin x - 0) < inverse y" 4579 using LIM_D [OF LIM_cos_div_sin, of "inverse y"] that assms by force 4580 obtain e where e: "0 < e" "e < s" "e < pi/2" 4581 using \<open>0 < s\<close> field_lbound_gt_zero pi_half_gt_zero by blast 4582 show ?thesis 4583 proof (intro exI conjI) 4584 have "0 < sin e" "0 < cos e" 4585 using e by (auto intro: cos_gt_zero sin_gt_zero2 simp: mult.commute) 4586 then 4587 show "y < tan (pi/2 - e)" 4588 using s [of "pi/2 - e"] e assms 4589 by (simp add: tan_def sin_diff cos_diff) (simp add: field_simps split: if_split_asm) 4590 qed (use e in auto) 4591qed 4592 4593lemma tan_total_pos: 4594 assumes "0 \<le> y" shows "\<exists>x. 0 \<le> x \<and> x < pi/2 \<and> tan x = y" 4595proof (cases "y = 0") 4596 case True 4597 then show ?thesis 4598 using pi_half_gt_zero tan_zero by blast 4599next 4600 case False 4601 with assms have "y > 0" 4602 by linarith 4603 obtain x where x: "0 < x" "x < pi/2" "y < tan x" 4604 using lemma_tan_total \<open>0 < y\<close> by blast 4605 have "\<exists>u\<ge>0. u \<le> x \<and> tan u = y" 4606 proof (intro IVT allI impI) 4607 show "isCont tan u" if "0 \<le> u \<and> u \<le> x" for u 4608 proof - 4609 have "cos u \<noteq> 0" 4610 using antisym_conv2 cos_gt_zero that x(2) by fastforce 4611 with assms show ?thesis 4612 by (auto intro!: DERIV_tan [THEN DERIV_isCont]) 4613 qed 4614 qed (use assms x in auto) 4615 then show ?thesis 4616 using x(2) by auto 4617qed 4618 4619lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y" 4620proof (cases "0::real" y rule: le_cases) 4621 case le 4622 then show ?thesis 4623 by (meson less_le_trans minus_pi_half_less_zero tan_total_pos) 4624next 4625 case ge 4626 with tan_total_pos [of "-y"] obtain x where "0 \<le> x" "x < pi / 2" "tan x = - y" 4627 by force 4628 then show ?thesis 4629 by (rule_tac x="-x" in exI) auto 4630qed 4631 4632proposition tan_total: "\<exists>! x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y" 4633proof - 4634 have "u = v" if u: "- (pi / 2) < u" "u < pi / 2" and v: "- (pi / 2) < v" "v < pi / 2" 4635 and eq: "tan u = tan v" for u v 4636 proof (cases u v rule: linorder_cases) 4637 case less 4638 have "\<And>x. u \<le> x \<and> x \<le> v \<longrightarrow> isCont tan x" 4639 by (metis cos_gt_zero_pi isCont_tan less_numeral_extra(3) less_trans order.not_eq_order_implies_strict u v) 4640 moreover have "\<And>x. u < x \<and> x < v \<Longrightarrow> tan differentiable (at x)" 4641 by (metis DERIV_tan cos_gt_zero_pi differentiableI less_numeral_extra(3) order.strict_trans u(1) v(2)) 4642 ultimately obtain z where "u < z" "z < v" "DERIV tan z :> 0" 4643 by (metis less Rolle eq) 4644 moreover have "cos z \<noteq> 0" 4645 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)) 4646 ultimately show ?thesis 4647 using DERIV_unique [OF _ DERIV_tan] by fastforce 4648 next 4649 case greater 4650 have "\<And>x. v \<le> x \<and> x \<le> u \<Longrightarrow> isCont tan x" 4651 by (metis cos_gt_zero_pi isCont_tan less_numeral_extra(3) less_trans order.not_eq_order_implies_strict u v) 4652 moreover have "\<And>x. v < x \<and> x < u \<Longrightarrow> tan differentiable (at x)" 4653 by (metis DERIV_tan cos_gt_zero_pi differentiableI less_numeral_extra(3) order.strict_trans u(2) v(1)) 4654 ultimately obtain z where "v < z" "z < u" "DERIV tan z :> 0" 4655 by (metis greater Rolle eq) 4656 moreover have "cos z \<noteq> 0" 4657 by (metis \<open>v < z\<close> \<open>z < u\<close> cos_gt_zero_pi less_le_trans linorder_not_less not_less_iff_gr_or_eq u(2) v(1)) 4658 ultimately show ?thesis 4659 using DERIV_unique [OF _ DERIV_tan] by fastforce 4660 qed auto 4661 then have "\<exists>!x. - (pi / 2) < x \<and> x < pi / 2 \<and> tan x = y" 4662 if x: "- (pi / 2) < x" "x < pi / 2" "tan x = y" for x 4663 using that by auto 4664 then show ?thesis 4665 using lemma_tan_total1 [where y = y] 4666 by auto 4667qed 4668 4669lemma tan_monotone: 4670 assumes "- (pi/2) < y" and "y < x" and "x < pi/2" 4671 shows "tan y < tan x" 4672proof - 4673 have "DERIV tan x' :> inverse ((cos x')\<^sup>2)" if "y \<le> x'" "x' \<le> x" for x' 4674 proof - 4675 have "-(pi/2) < x'" and "x' < pi/2" 4676 using that assms by auto 4677 with cos_gt_zero_pi have "cos x' \<noteq> 0" by force 4678 then show "DERIV tan x' :> inverse ((cos x')\<^sup>2)" 4679 by (rule DERIV_tan) 4680 qed 4681 from MVT2[OF \<open>y < x\<close> this] 4682 obtain z where "y < z" and "z < x" 4683 and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto 4684 then have "- (pi/2) < z" and "z < pi/2" 4685 using assms by auto 4686 then have "0 < cos z" 4687 using cos_gt_zero_pi by auto 4688 then have inv_pos: "0 < inverse ((cos z)\<^sup>2)" 4689 by auto 4690 have "0 < x - y" using \<open>y < x\<close> by auto 4691 with inv_pos have "0 < tan x - tan y" 4692 unfolding tan_diff by auto 4693 then show ?thesis by auto 4694qed 4695 4696lemma tan_monotone': 4697 assumes "- (pi/2) < y" 4698 and "y < pi/2" 4699 and "- (pi/2) < x" 4700 and "x < pi/2" 4701 shows "y < x \<longleftrightarrow> tan y < tan x" 4702proof 4703 assume "y < x" 4704 then show "tan y < tan x" 4705 using tan_monotone and \<open>- (pi/2) < y\<close> and \<open>x < pi/2\<close> by auto 4706next 4707 assume "tan y < tan x" 4708 show "y < x" 4709 proof (rule ccontr) 4710 assume "\<not> ?thesis" 4711 then have "x \<le> y" by auto 4712 then have "tan x \<le> tan y" 4713 proof (cases "x = y") 4714 case True 4715 then show ?thesis by auto 4716 next 4717 case False 4718 then have "x < y" using \<open>x \<le> y\<close> by auto 4719 from tan_monotone[OF \<open>- (pi/2) < x\<close> this \<open>y < pi/2\<close>] show ?thesis 4720 by auto 4721 qed 4722 then show False 4723 using \<open>tan y < tan x\<close> by auto 4724 qed 4725qed 4726 4727lemma tan_inverse: "1 / (tan y) = tan (pi/2 - y)" 4728 unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto 4729 4730lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x" 4731 by (simp add: tan_def) 4732 4733lemma tan_periodic_nat[simp]: "tan (x + real n * pi) = tan x" 4734 for n :: nat 4735proof (induct n arbitrary: x) 4736 case 0 4737 then show ?case by simp 4738next 4739 case (Suc n) 4740 have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" 4741 unfolding Suc_eq_plus1 of_nat_add distrib_right by auto 4742 show ?case 4743 unfolding split_pi_off using Suc by auto 4744qed 4745 4746lemma tan_periodic_int[simp]: "tan (x + of_int i * pi) = tan x" 4747proof (cases "0 \<le> i") 4748 case True 4749 then have i_nat: "of_int i = of_int (nat i)" by auto 4750 show ?thesis unfolding i_nat 4751 by (metis of_int_of_nat_eq tan_periodic_nat) 4752next 4753 case False 4754 then have i_nat: "of_int i = - of_int (nat (- i))" by auto 4755 have "tan x = tan (x + of_int i * pi - of_int i * pi)" 4756 by auto 4757 also have "\<dots> = tan (x + of_int i * pi)" 4758 unfolding i_nat mult_minus_left diff_minus_eq_add 4759 by (metis of_int_of_nat_eq tan_periodic_nat) 4760 finally show ?thesis by auto 4761qed 4762 4763lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x" 4764 using tan_periodic_int[of _ "numeral n" ] by simp 4765 4766lemma tan_minus_45: "tan (-(pi/4)) = -1" 4767 unfolding tan_def by (simp add: sin_45 cos_45) 4768 4769lemma tan_diff: 4770 "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)" 4771 for x :: "'a::{real_normed_field,banach}" 4772 using tan_add [of x "-y"] by simp 4773 4774lemma tan_pos_pi2_le: "0 \<le> x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 \<le> tan x" 4775 using less_eq_real_def tan_gt_zero by auto 4776 4777lemma cos_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> cos x = 1 / sqrt (1 + tan x ^ 2)" 4778 using cos_gt_zero_pi [of x] 4779 by (simp add: divide_simps tan_def real_sqrt_divide abs_if split: if_split_asm) 4780 4781lemma sin_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> sin x = tan x / sqrt (1 + tan x ^ 2)" 4782 using cos_gt_zero [of "x"] cos_gt_zero [of "-x"] 4783 by (force simp: divide_simps tan_def real_sqrt_divide abs_if split: if_split_asm) 4784 4785lemma tan_mono_le: "-(pi/2) < x \<Longrightarrow> x \<le> y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x \<le> tan y" 4786 using less_eq_real_def tan_monotone by auto 4787 4788lemma tan_mono_lt_eq: 4789 "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> -(pi/2) < y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x < tan y \<longleftrightarrow> x < y" 4790 using tan_monotone' by blast 4791 4792lemma tan_mono_le_eq: 4793 "-(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" 4794 by (meson tan_mono_le not_le tan_monotone) 4795 4796lemma tan_bound_pi2: "\<bar>x\<bar> < pi/4 \<Longrightarrow> \<bar>tan x\<bar> < 1" 4797 using tan_45 tan_monotone [of x "pi/4"] tan_monotone [of "-x" "pi/4"] 4798 by (auto simp: abs_if split: if_split_asm) 4799 4800lemma tan_cot: "tan(pi/2 - x) = inverse(tan x)" 4801 by (simp add: tan_def sin_diff cos_diff) 4802 4803 4804subsection \<open>Cotangent\<close> 4805 4806definition cot :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4807 where "cot = (\<lambda>x. cos x / sin x)" 4808 4809lemma cot_of_real: "of_real (cot x) = (cot (of_real x) :: 'a::{real_normed_field,banach})" 4810 by (simp add: cot_def sin_of_real cos_of_real) 4811 4812lemma cot_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cot z \<in> \<real>" 4813 for z :: "'a::{real_normed_field,banach}" 4814 by (simp add: cot_def) 4815 4816lemma cot_zero [simp]: "cot 0 = 0" 4817 by (simp add: cot_def) 4818 4819lemma cot_pi [simp]: "cot pi = 0" 4820 by (simp add: cot_def) 4821 4822lemma cot_npi [simp]: "cot (real n * pi) = 0" 4823 for n :: nat 4824 by (simp add: cot_def) 4825 4826lemma cot_minus [simp]: "cot (- x) = - cot x" 4827 by (simp add: cot_def) 4828 4829lemma cot_periodic [simp]: "cot (x + 2 * pi) = cot x" 4830 by (simp add: cot_def) 4831 4832lemma cot_altdef: "cot x = inverse (tan x)" 4833 by (simp add: cot_def tan_def) 4834 4835lemma tan_altdef: "tan x = inverse (cot x)" 4836 by (simp add: cot_def tan_def) 4837 4838lemma tan_cot': "tan (pi/2 - x) = cot x" 4839 by (simp add: tan_cot cot_altdef) 4840 4841lemma cot_gt_zero: "0 < x \<Longrightarrow> x < pi/2 \<Longrightarrow> 0 < cot x" 4842 by (simp add: cot_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) 4843 4844lemma cot_less_zero: 4845 assumes lb: "- pi/2 < x" and "x < 0" 4846 shows "cot x < 0" 4847proof - 4848 have "0 < cot (- x)" 4849 using assms by (simp only: cot_gt_zero) 4850 then show ?thesis by simp 4851qed 4852 4853lemma DERIV_cot [simp]: "sin x \<noteq> 0 \<Longrightarrow> DERIV cot x :> -inverse ((sin x)\<^sup>2)" 4854 for x :: "'a::{real_normed_field,banach}" 4855 unfolding cot_def using cos_squared_eq[of x] 4856 by (auto intro!: derivative_eq_intros) (simp add: divide_inverse power2_eq_square) 4857 4858lemma isCont_cot: "sin x \<noteq> 0 \<Longrightarrow> isCont cot x" 4859 for x :: "'a::{real_normed_field,banach}" 4860 by (rule DERIV_cot [THEN DERIV_isCont]) 4861 4862lemma isCont_cot' [simp,continuous_intros]: 4863 "isCont f a \<Longrightarrow> sin (f a) \<noteq> 0 \<Longrightarrow> isCont (\<lambda>x. cot (f x)) a" 4864 for a :: "'a::{real_normed_field,banach}" and f :: "'a \<Rightarrow> 'a" 4865 by (rule isCont_o2 [OF _ isCont_cot]) 4866 4867lemma tendsto_cot [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> sin a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. cot (f x)) \<longlongrightarrow> cot a) F" 4868 for f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4869 by (rule isCont_tendsto_compose [OF isCont_cot]) 4870 4871lemma continuous_cot: 4872 "continuous F f \<Longrightarrow> sin (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. cot (f x))" 4873 for f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4874 unfolding continuous_def by (rule tendsto_cot) 4875 4876lemma continuous_on_cot [continuous_intros]: 4877 fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4878 shows "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. sin (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. cot (f x))" 4879 unfolding continuous_on_def by (auto intro: tendsto_cot) 4880 4881lemma continuous_within_cot [continuous_intros]: 4882 fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" 4883 shows "continuous (at x within s) f \<Longrightarrow> sin (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. cot (f x))" 4884 unfolding continuous_within by (rule tendsto_cot) 4885 4886 4887subsection \<open>Inverse Trigonometric Functions\<close> 4888 4889definition arcsin :: "real \<Rightarrow> real" 4890 where "arcsin y = (THE x. -(pi/2) \<le> x \<and> x \<le> pi/2 \<and> sin x = y)" 4891 4892definition arccos :: "real \<Rightarrow> real" 4893 where "arccos y = (THE x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y)" 4894 4895definition arctan :: "real \<Rightarrow> real" 4896 where "arctan y = (THE x. -(pi/2) < x \<and> x < pi/2 \<and> tan x = y)" 4897 4898lemma 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" 4899 unfolding arcsin_def by (rule theI' [OF sin_total]) 4900 4901lemma 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" 4902 by (drule (1) arcsin) (force intro: order_trans) 4903 4904lemma sin_arcsin [simp]: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin (arcsin y) = y" 4905 by (blast dest: arcsin) 4906 4907lemma arcsin_bounded: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y \<and> arcsin y \<le> pi/2" 4908 by (blast dest: arcsin) 4909 4910lemma arcsin_lbound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> - (pi/2) \<le> arcsin y" 4911 by (blast dest: arcsin) 4912 4913lemma arcsin_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2" 4914 by (blast dest: arcsin) 4915 4916lemma arcsin_lt_bounded: 4917 assumes "- 1 < y" "y < 1" 4918 shows "- (pi/2) < arcsin y \<and> arcsin y < pi/2" 4919proof - 4920 have "arcsin y \<noteq> pi/2" 4921 by (metis arcsin assms not_less not_less_iff_gr_or_eq sin_pi_half) 4922 moreover have "arcsin y \<noteq> - pi/2" 4923 by (metis arcsin assms minus_divide_left not_less not_less_iff_gr_or_eq sin_minus sin_pi_half) 4924 ultimately show ?thesis 4925 using arcsin_bounded [of y] assms by auto 4926qed 4927 4928lemma arcsin_sin: "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> arcsin (sin x) = x" 4929 unfolding arcsin_def 4930 using the1_equality [OF sin_total] by simp 4931 4932lemma arcsin_0 [simp]: "arcsin 0 = 0" 4933 using arcsin_sin [of 0] by simp 4934 4935lemma arcsin_1 [simp]: "arcsin 1 = pi/2" 4936 using arcsin_sin [of "pi/2"] by simp 4937 4938lemma arcsin_minus_1 [simp]: "arcsin (- 1) = - (pi/2)" 4939 using arcsin_sin [of "- pi/2"] by simp 4940 4941lemma arcsin_minus: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin (- x) = - arcsin x" 4942 by (metis (no_types, hide_lams) arcsin arcsin_sin minus_minus neg_le_iff_le sin_minus) 4943 4944lemma arcsin_eq_iff: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x = arcsin y \<longleftrightarrow> x = y" 4945 by (metis abs_le_iff arcsin minus_le_iff) 4946 4947lemma cos_arcsin_nonzero: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> cos (arcsin x) \<noteq> 0" 4948 using arcsin_lt_bounded cos_gt_zero_pi by force 4949 4950lemma arccos: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi \<and> cos (arccos y) = y" 4951 unfolding arccos_def by (rule theI' [OF cos_total]) 4952 4953lemma cos_arccos [simp]: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> cos (arccos y) = y" 4954 by (blast dest: arccos) 4955 4956lemma arccos_bounded: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y \<and> arccos y \<le> pi" 4957 by (blast dest: arccos) 4958 4959lemma arccos_lbound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> 0 \<le> arccos y" 4960 by (blast dest: arccos) 4961 4962lemma arccos_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> pi" 4963 by (blast dest: arccos) 4964 4965lemma arccos_lt_bounded: 4966 assumes "- 1 < y" "y < 1" 4967 shows "0 < arccos y \<and> arccos y < pi" 4968proof - 4969 have "arccos y \<noteq> 0" 4970 by (metis (no_types) arccos assms(1) assms(2) cos_zero less_eq_real_def less_irrefl) 4971 moreover have "arccos y \<noteq> -pi" 4972 by (metis arccos assms(1) assms(2) cos_minus cos_pi not_less not_less_iff_gr_or_eq) 4973 ultimately show ?thesis 4974 using arccos_bounded [of y] assms 4975 by (metis arccos cos_pi not_less not_less_iff_gr_or_eq) 4976qed 4977 4978lemma arccos_cos: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> arccos (cos x) = x" 4979 by (auto simp: arccos_def intro!: the1_equality cos_total) 4980 4981lemma arccos_cos2: "x \<le> 0 \<Longrightarrow> - pi \<le> x \<Longrightarrow> arccos (cos x) = -x" 4982 by (auto simp: arccos_def intro!: the1_equality cos_total) 4983 4984lemma cos_arcsin: 4985 assumes "- 1 \<le> x" "x \<le> 1" 4986 shows "cos (arcsin x) = sqrt (1 - x\<^sup>2)" 4987proof (rule power2_eq_imp_eq) 4988 show "(cos (arcsin x))\<^sup>2 = (sqrt (1 - x\<^sup>2))\<^sup>2" 4989 by (simp add: square_le_1 assms cos_squared_eq) 4990 show "0 \<le> cos (arcsin x)" 4991 using arcsin assms cos_ge_zero by blast 4992 show "0 \<le> sqrt (1 - x\<^sup>2)" 4993 by (simp add: square_le_1 assms) 4994qed 4995 4996lemma sin_arccos: 4997 assumes "- 1 \<le> x" "x \<le> 1" 4998 shows "sin (arccos x) = sqrt (1 - x\<^sup>2)" 4999proof (rule power2_eq_imp_eq) 5000 show "(sin (arccos x))\<^sup>2 = (sqrt (1 - x\<^sup>2))\<^sup>2" 5001 by (simp add: square_le_1 assms sin_squared_eq) 5002 show "0 \<le> sin (arccos x)" 5003 by (simp add: arccos_bounded assms sin_ge_zero) 5004 show "0 \<le> sqrt (1 - x\<^sup>2)" 5005 by (simp add: square_le_1 assms) 5006qed 5007 5008lemma arccos_0 [simp]: "arccos 0 = pi/2" 5009 by (metis arccos_cos cos_gt_zero cos_pi cos_pi_half pi_gt_zero 5010 pi_half_ge_zero not_le not_zero_less_neg_numeral numeral_One) 5011 5012lemma arccos_1 [simp]: "arccos 1 = 0" 5013 using arccos_cos by force 5014 5015lemma arccos_minus_1 [simp]: "arccos (- 1) = pi" 5016 by (metis arccos_cos cos_pi order_refl pi_ge_zero) 5017 5018lemma arccos_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos (- x) = pi - arccos x" 5019 by (metis arccos_cos arccos_cos2 cos_minus_pi cos_total diff_le_0_iff_le le_add_same_cancel1 5020 minus_diff_eq uminus_add_conv_diff) 5021 5022corollary arccos_minus_abs: 5023 assumes "\<bar>x\<bar> \<le> 1" 5024 shows "arccos (- x) = pi - arccos x" 5025using assms by (simp add: arccos_minus) 5026 5027lemma sin_arccos_nonzero: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> sin (arccos x) \<noteq> 0" 5028 using arccos_lt_bounded sin_gt_zero by force 5029 5030lemma arctan: "- (pi/2) < arctan y \<and> arctan y < pi/2 \<and> tan (arctan y) = y" 5031 unfolding arctan_def by (rule theI' [OF tan_total]) 5032 5033lemma tan_arctan: "tan (arctan y) = y" 5034 by (simp add: arctan) 5035 5036lemma arctan_bounded: "- (pi/2) < arctan y \<and> arctan y < pi/2" 5037 by (auto simp only: arctan) 5038 5039lemma arctan_lbound: "- (pi/2) < arctan y" 5040 by (simp add: arctan) 5041 5042lemma arctan_ubound: "arctan y < pi/2" 5043 by (auto simp only: arctan) 5044 5045lemma arctan_unique: 5046 assumes "-(pi/2) < x" 5047 and "x < pi/2" 5048 and "tan x = y" 5049 shows "arctan y = x" 5050 using assms arctan [of y] tan_total [of y] by (fast elim: ex1E) 5051 5052lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x" 5053 by (rule arctan_unique) simp_all 5054 5055lemma arctan_zero_zero [simp]: "arctan 0 = 0" 5056 by (rule arctan_unique) simp_all 5057 5058lemma arctan_minus: "arctan (- x) = - arctan x" 5059 using arctan [of "x"] by (auto simp: arctan_unique) 5060 5061lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0" 5062 by (intro less_imp_neq [symmetric] cos_gt_zero_pi arctan_lbound arctan_ubound) 5063 5064lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)" 5065proof (rule power2_eq_imp_eq) 5066 have "0 < 1 + x\<^sup>2" by (simp add: add_pos_nonneg) 5067 show "0 \<le> 1 / sqrt (1 + x\<^sup>2)" by simp 5068 show "0 \<le> cos (arctan x)" 5069 by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound) 5070 have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1" 5071 unfolding tan_def by (simp add: distrib_left power_divide) 5072 then show "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2" 5073 using \<open>0 < 1 + x\<^sup>2\<close> by (simp add: arctan power_divide eq_divide_eq) 5074qed 5075 5076lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)" 5077 using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]] 5078 using tan_arctan [of x] unfolding tan_def cos_arctan 5079 by (simp add: eq_divide_eq) 5080 5081lemma tan_sec: "cos x \<noteq> 0 \<Longrightarrow> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2" 5082 for x :: "'a::{real_normed_field,banach,field}" 5083 by (simp add: add_divide_eq_iff inverse_eq_divide power2_eq_square tan_def) 5084 5085lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y" 5086 by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan) 5087 5088lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y" 5089 by (simp only: not_less [symmetric] arctan_less_iff) 5090 5091lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y" 5092 by (simp only: eq_iff [where 'a=real] arctan_le_iff) 5093 5094lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x" 5095 using arctan_less_iff [of 0 x] by simp 5096 5097lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0" 5098 using arctan_less_iff [of x 0] by simp 5099 5100lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x" 5101 using arctan_le_iff [of 0 x] by simp 5102 5103lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0" 5104 using arctan_le_iff [of x 0] by simp 5105 5106lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0" 5107 using arctan_eq_iff [of x 0] by simp 5108 5109lemma continuous_on_arcsin': "continuous_on {-1 .. 1} arcsin" 5110proof - 5111 have "continuous_on (sin ` {- pi/2 .. pi/2}) arcsin" 5112 by (rule continuous_on_inv) (auto intro: continuous_intros simp: arcsin_sin) 5113 also have "sin ` {- pi/2 .. pi/2} = {-1 .. 1}" 5114 proof safe 5115 fix x :: real 5116 assume "x \<in> {-1..1}" 5117 then show "x \<in> sin ` {- pi/2..pi/2}" 5118 using arcsin_lbound arcsin_ubound 5119 by (intro image_eqI[where x="arcsin x"]) auto 5120 qed simp 5121 finally show ?thesis . 5122qed 5123 5124lemma continuous_on_arcsin [continuous_intros]: 5125 "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))" 5126 using continuous_on_compose[of s f, OF _ continuous_on_subset[OF continuous_on_arcsin']] 5127 by (auto simp: comp_def subset_eq) 5128 5129lemma isCont_arcsin: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arcsin x" 5130 using continuous_on_arcsin'[THEN continuous_on_subset, of "{ -1 <..< 1 }"] 5131 by (auto simp: continuous_on_eq_continuous_at subset_eq) 5132 5133lemma continuous_on_arccos': "continuous_on {-1 .. 1} arccos" 5134proof - 5135 have "continuous_on (cos ` {0 .. pi}) arccos" 5136 by (rule continuous_on_inv) (auto intro: continuous_intros simp: arccos_cos) 5137 also have "cos ` {0 .. pi} = {-1 .. 1}" 5138 proof safe 5139 fix x :: real 5140 assume "x \<in> {-1..1}" 5141 then show "x \<in> cos ` {0..pi}" 5142 using arccos_lbound arccos_ubound 5143 by (intro image_eqI[where x="arccos x"]) auto 5144 qed simp 5145 finally show ?thesis . 5146qed 5147 5148lemma continuous_on_arccos [continuous_intros]: 5149 "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))" 5150 using continuous_on_compose[of s f, OF _ continuous_on_subset[OF continuous_on_arccos']] 5151 by (auto simp: comp_def subset_eq) 5152 5153lemma isCont_arccos: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> isCont arccos x" 5154 using continuous_on_arccos'[THEN continuous_on_subset, of "{ -1 <..< 1 }"] 5155 by (auto simp: continuous_on_eq_continuous_at subset_eq) 5156 5157lemma isCont_arctan: "isCont arctan x" 5158proof - 5159 obtain u where u: "- (pi / 2) < u" "u < arctan x" 5160 by (meson arctan arctan_less_iff linordered_field_no_lb) 5161 obtain v where v: "arctan x < v" "v < pi / 2" 5162 by (meson arctan_less_iff arctan_ubound linordered_field_no_ub) 5163 have "isCont arctan (tan (arctan x))" 5164 proof (rule isCont_inverse_function2 [of u "arctan x" v]) 5165 show "\<And>z. \<lbrakk>u \<le> z; z \<le> v\<rbrakk> \<Longrightarrow> arctan (tan z) = z" 5166 using arctan_unique u(1) v(2) by auto 5167 then show "\<And>z. \<lbrakk>u \<le> z; z \<le> v\<rbrakk> \<Longrightarrow> isCont tan z" 5168 by (metis arctan cos_gt_zero_pi isCont_tan less_irrefl) 5169 qed (use u v in auto) 5170 then show ?thesis 5171 by (simp add: arctan) 5172qed 5173 5174lemma tendsto_arctan [tendsto_intros]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) \<longlongrightarrow> arctan x) F" 5175 by (rule isCont_tendsto_compose [OF isCont_arctan]) 5176 5177lemma continuous_arctan [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arctan (f x))" 5178 unfolding continuous_def by (rule tendsto_arctan) 5179 5180lemma continuous_on_arctan [continuous_intros]: 5181 "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))" 5182 unfolding continuous_on_def by (auto intro: tendsto_arctan) 5183 5184lemma DERIV_arcsin: 5185 assumes "- 1 < x" "x < 1" 5186 shows "DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))" 5187proof (rule DERIV_inverse_function) 5188 show "(sin has_real_derivative sqrt (1 - x\<^sup>2)) (at (arcsin x))" 5189 by (rule derivative_eq_intros | use assms cos_arcsin in force)+ 5190 show "sqrt (1 - x\<^sup>2) \<noteq> 0" 5191 using abs_square_eq_1 assms by force 5192qed (use assms isCont_arcsin in auto) 5193 5194lemma DERIV_arccos: 5195 assumes "- 1 < x" "x < 1" 5196 shows "DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))" 5197proof (rule DERIV_inverse_function) 5198 show "(cos has_real_derivative - sqrt (1 - x\<^sup>2)) (at (arccos x))" 5199 by (rule derivative_eq_intros | use assms sin_arccos in force)+ 5200 show "- sqrt (1 - x\<^sup>2) \<noteq> 0" 5201 using abs_square_eq_1 assms by force 5202qed (use assms isCont_arccos in auto) 5203 5204lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)" 5205proof (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"]) 5206 show "(tan has_real_derivative 1 + x\<^sup>2) (at (arctan x))" 5207 apply (rule derivative_eq_intros | simp)+ 5208 by (metis arctan cos_arctan_not_zero power_inverse tan_sec) 5209 show "\<And>y. \<lbrakk>x - 1 < y; y < x + 1\<rbrakk> \<Longrightarrow> tan (arctan y) = y" 5210 using tan_arctan by blast 5211 show "1 + x\<^sup>2 \<noteq> 0" 5212 by (metis power_one sum_power2_eq_zero_iff zero_neq_one) 5213qed (use isCont_arctan in auto) 5214 5215declare 5216 DERIV_arcsin[THEN DERIV_chain2, derivative_intros] 5217 DERIV_arcsin[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 5218 DERIV_arccos[THEN DERIV_chain2, derivative_intros] 5219 DERIV_arccos[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 5220 DERIV_arctan[THEN DERIV_chain2, derivative_intros] 5221 DERIV_arctan[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros] 5222 5223lemmas has_derivative_arctan[derivative_intros] = DERIV_arctan[THEN DERIV_compose_FDERIV] 5224 and has_derivative_arccos[derivative_intros] = DERIV_arccos[THEN DERIV_compose_FDERIV] 5225 and has_derivative_arcsin[derivative_intros] = DERIV_arcsin[THEN DERIV_compose_FDERIV] 5226 5227lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- (pi/2)))" 5228 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]) 5229 (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1 5230 intro!: tan_monotone exI[of _ "pi/2"]) 5231 5232lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))" 5233 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]) 5234 (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1 5235 intro!: tan_monotone exI[of _ "pi/2"]) 5236 5237lemma tendsto_arctan_at_top: "(arctan \<longlongrightarrow> (pi/2)) at_top" 5238proof (rule tendstoI) 5239 fix e :: real 5240 assume "0 < e" 5241 define y where "y = pi/2 - min (pi/2) e" 5242 then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y" 5243 using \<open>0 < e\<close> by auto 5244 show "eventually (\<lambda>x. dist (arctan x) (pi/2) < e) at_top" 5245 proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI) 5246 fix x 5247 assume "tan y < x" 5248 then have "arctan (tan y) < arctan x" 5249 by (simp add: arctan_less_iff) 5250 with y have "y < arctan x" 5251 by (subst (asm) arctan_tan) simp_all 5252 with arctan_ubound[of x, arith] y \<open>0 < e\<close> 5253 show "dist (arctan x) (pi/2) < e" 5254 by (simp add: dist_real_def) 5255 qed 5256qed 5257 5258lemma tendsto_arctan_at_bot: "(arctan \<longlongrightarrow> - (pi/2)) at_bot" 5259 unfolding filterlim_at_bot_mirror arctan_minus 5260 by (intro tendsto_minus tendsto_arctan_at_top) 5261 5262 5263subsection \<open>Prove Totality of the Trigonometric Functions\<close> 5264 5265lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y" 5266 by (simp add: abs_le_iff) 5267 5268lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)" 5269 by (simp add: sin_arccos abs_le_iff) 5270 5271lemma sin_mono_less_eq: 5272 "- (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" 5273 by (metis not_less_iff_gr_or_eq sin_monotone_2pi) 5274 5275lemma sin_mono_le_eq: 5276 "- (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" 5277 by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le) 5278 5279lemma sin_inj_pi: 5280 "- (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" 5281 by (metis arcsin_sin) 5282 5283lemma 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" 5284 by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear) 5285 5286lemma 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" 5287 by (metis arccos_cos cos_monotone_0_pi_le eq_iff linear) 5288 5289lemma 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" 5290 by (metis arccos_cos) 5291 5292lemma arccos_le_pi2: "\<lbrakk>0 \<le> y; y \<le> 1\<rbrakk> \<Longrightarrow> arccos y \<le> pi/2" 5293 by (metis (mono_tags) arccos_0 arccos cos_le_one cos_monotone_0_pi_le 5294 cos_pi cos_pi_half pi_half_ge_zero antisym_conv less_eq_neg_nonpos linear minus_minus order.trans order_refl) 5295 5296lemma sincos_total_pi_half: 5297 assumes "0 \<le> x" "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1" 5298 shows "\<exists>t. 0 \<le> t \<and> t \<le> pi/2 \<and> x = cos t \<and> y = sin t" 5299proof - 5300 have x1: "x \<le> 1" 5301 using assms by (metis le_add_same_cancel1 power2_le_imp_le power_one zero_le_power2) 5302 with assms have *: "0 \<le> arccos x" "cos (arccos x) = x" 5303 by (auto simp: arccos) 5304 from assms have "y = sqrt (1 - x\<^sup>2)" 5305 by (metis abs_of_nonneg add.commute add_diff_cancel real_sqrt_abs) 5306 with x1 * assms arccos_le_pi2 [of x] show ?thesis 5307 by (rule_tac x="arccos x" in exI) (auto simp: sin_arccos) 5308qed 5309 5310lemma sincos_total_pi: 5311 assumes "0 \<le> y" "x\<^sup>2 + y\<^sup>2 = 1" 5312 shows "\<exists>t. 0 \<le> t \<and> t \<le> pi \<and> x = cos t \<and> y = sin t" 5313proof (cases rule: le_cases [of 0 x]) 5314 case le 5315 from sincos_total_pi_half [OF le] show ?thesis 5316 by (metis pi_ge_two pi_half_le_two add.commute add_le_cancel_left add_mono assms) 5317next 5318 case ge 5319 then have "0 \<le> -x" 5320 by simp 5321 then obtain t where t: "t\<ge>0" "t \<le> pi/2" "-x = cos t" "y = sin t" 5322 using sincos_total_pi_half assms 5323 by auto (metis \<open>0 \<le> - x\<close> power2_minus) 5324 show ?thesis 5325 by (rule exI [where x = "pi -t"]) (use t in auto) 5326qed 5327 5328lemma sincos_total_2pi_le: 5329 assumes "x\<^sup>2 + y\<^sup>2 = 1" 5330 shows "\<exists>t. 0 \<le> t \<and> t \<le> 2 * pi \<and> x = cos t \<and> y = sin t" 5331proof (cases rule: le_cases [of 0 y]) 5332 case le 5333 from sincos_total_pi [OF le] show ?thesis 5334 by (metis assms le_add_same_cancel1 mult.commute mult_2_right order.trans) 5335next 5336 case ge 5337 then have "0 \<le> -y" 5338 by simp 5339 then obtain t where t: "t\<ge>0" "t \<le> pi" "x = cos t" "-y = sin t" 5340 using sincos_total_pi assms 5341 by auto (metis \<open>0 \<le> - y\<close> power2_minus) 5342 show ?thesis 5343 by (rule exI [where x = "2 * pi - t"]) (use t in auto) 5344qed 5345 5346lemma sincos_total_2pi: 5347 assumes "x\<^sup>2 + y\<^sup>2 = 1" 5348 obtains t where "0 \<le> t" "t < 2*pi" "x = cos t" "y = sin t" 5349proof - 5350 from sincos_total_2pi_le [OF assms] 5351 obtain t where t: "0 \<le> t" "t \<le> 2*pi" "x = cos t" "y = sin t" 5352 by blast 5353 show ?thesis 5354 by (cases "t = 2 * pi") (use t that in \<open>force+\<close>) 5355qed 5356 5357lemma arcsin_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x < arcsin y \<longleftrightarrow> x < y" 5358 by (rule trans [OF sin_mono_less_eq [symmetric]]) (use arcsin_ubound arcsin_lbound in auto) 5359 5360lemma arcsin_le_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y \<longleftrightarrow> x \<le> y" 5361 using arcsin_less_mono not_le by blast 5362 5363lemma arcsin_less_arcsin: "- 1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x < arcsin y" 5364 using arcsin_less_mono by auto 5365 5366lemma arcsin_le_arcsin: "- 1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y" 5367 using arcsin_le_mono by auto 5368 5369lemma arccos_less_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x < arccos y \<longleftrightarrow> y < x" 5370 by (rule trans [OF cos_mono_less_eq [symmetric]]) (use arccos_ubound arccos_lbound in auto) 5371 5372lemma arccos_le_mono: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x \<le> arccos y \<longleftrightarrow> y \<le> x" 5373 using arccos_less_mono [of y x] by (simp add: not_le [symmetric]) 5374 5375lemma arccos_less_arccos: "- 1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y < arccos x" 5376 using arccos_less_mono by auto 5377 5378lemma arccos_le_arccos: "- 1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> arccos x" 5379 using arccos_le_mono by auto 5380 5381lemma arccos_eq_iff: "\<bar>x\<bar> \<le> 1 \<and> \<bar>y\<bar> \<le> 1 \<Longrightarrow> arccos x = arccos y \<longleftrightarrow> x = y" 5382 using cos_arccos_abs by fastforce 5383 5384 5385lemma arccos_cos_eq_abs: 5386 assumes "\<bar>\<theta>\<bar> \<le> pi" 5387 shows "arccos (cos \<theta>) = \<bar>\<theta>\<bar>" 5388 unfolding arccos_def 5389proof (intro the_equality conjI; clarify?) 5390 show "cos \<bar>\<theta>\<bar> = cos \<theta>" 5391 by (simp add: abs_real_def) 5392 show "x = \<bar>\<theta>\<bar>" if "cos x = cos \<theta>" "0 \<le> x" "x \<le> pi" for x 5393 by (simp add: \<open>cos \<bar>\<theta>\<bar> = cos \<theta>\<close> assms cos_inj_pi that) 5394qed (use assms in auto) 5395 5396lemma arccos_cos_eq_abs_2pi: 5397 obtains k where "arccos (cos \<theta>) = \<bar>\<theta> - of_int k * (2 * pi)\<bar>" 5398proof - 5399 define k where "k \<equiv> \<lfloor>(\<theta> + pi) / (2 * pi)\<rfloor>" 5400 have lepi: "\<bar>\<theta> - of_int k * (2 * pi)\<bar> \<le> pi" 5401 using floor_divide_lower [of "2*pi" "\<theta> + pi"] floor_divide_upper [of "2*pi" "\<theta> + pi"] 5402 by (auto simp: k_def abs_if algebra_simps) 5403 have "arccos (cos \<theta>) = arccos (cos (\<theta> - of_int k * (2 * pi)))" 5404 using cos_int_2pin sin_int_2pin by (simp add: cos_diff mult.commute) 5405 also have "\<dots> = \<bar>\<theta> - of_int k * (2 * pi)\<bar>" 5406 using arccos_cos_eq_abs lepi by blast 5407 finally show ?thesis 5408 using that by metis 5409qed 5410 5411lemma cos_limit_1: 5412 assumes "(\<lambda>j. cos (\<theta> j)) \<longlonglongrightarrow> 1" 5413 shows "\<exists>k. (\<lambda>j. \<theta> j - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> 0" 5414proof - 5415 have "\<forall>\<^sub>F j in sequentially. cos (\<theta> j) \<in> {- 1..1}" 5416 by auto 5417 then have "(\<lambda>j. arccos (cos (\<theta> j))) \<longlonglongrightarrow> arccos 1" 5418 using continuous_on_tendsto_compose [OF continuous_on_arccos' assms] by auto 5419 moreover have "\<And>j. \<exists>k. arccos (cos (\<theta> j)) = \<bar>\<theta> j - of_int k * (2 * pi)\<bar>" 5420 using arccos_cos_eq_abs_2pi by metis 5421 then have "\<exists>k. \<forall>j. arccos (cos (\<theta> j)) = \<bar>\<theta> j - of_int (k j) * (2 * pi)\<bar>" 5422 by metis 5423 ultimately have "\<exists>k. (\<lambda>j. \<bar>\<theta> j - of_int (k j) * (2 * pi)\<bar>) \<longlonglongrightarrow> 0" 5424 by auto 5425 then show ?thesis 5426 by (simp add: tendsto_rabs_zero_iff) 5427qed 5428 5429lemma cos_diff_limit_1: 5430 assumes "(\<lambda>j. cos (\<theta> j - \<Theta>)) \<longlonglongrightarrow> 1" 5431 obtains k where "(\<lambda>j. \<theta> j - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> \<Theta>" 5432proof - 5433 obtain k where "(\<lambda>j. (\<theta> j - \<Theta>) - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> 0" 5434 using cos_limit_1 [OF assms] by auto 5435 then have "(\<lambda>j. \<Theta> + ((\<theta> j - \<Theta>) - of_int (k j) * (2 * pi))) \<longlonglongrightarrow> \<Theta> + 0" 5436 by (rule tendsto_add [OF tendsto_const]) 5437 with that show ?thesis 5438 by auto 5439qed 5440 5441subsection \<open>Machin's formula\<close> 5442 5443lemma arctan_one: "arctan 1 = pi / 4" 5444 by (rule arctan_unique) (simp_all add: tan_45 m2pi_less_pi) 5445 5446lemma tan_total_pi4: 5447 assumes "\<bar>x\<bar> < 1" 5448 shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x" 5449proof 5450 show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x" 5451 unfolding arctan_one [symmetric] arctan_minus [symmetric] 5452 unfolding arctan_less_iff 5453 using assms by (auto simp: arctan) 5454qed 5455 5456lemma arctan_add: 5457 assumes "\<bar>x\<bar> \<le> 1" "\<bar>y\<bar> < 1" 5458 shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))" 5459proof (rule arctan_unique [symmetric]) 5460 have "- (pi / 4) \<le> arctan x" "- (pi / 4) < arctan y" 5461 unfolding arctan_one [symmetric] arctan_minus [symmetric] 5462 unfolding arctan_le_iff arctan_less_iff 5463 using assms by auto 5464 from add_le_less_mono [OF this] show 1: "- (pi/2) < arctan x + arctan y" 5465 by simp 5466 have "arctan x \<le> pi / 4" "arctan y < pi / 4" 5467 unfolding arctan_one [symmetric] 5468 unfolding arctan_le_iff arctan_less_iff 5469 using assms by auto 5470 from add_le_less_mono [OF this] show 2: "arctan x + arctan y < pi/2" 5471 by simp 5472 show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)" 5473 using cos_gt_zero_pi [OF 1 2] by (simp add: arctan tan_add) 5474qed 5475 5476lemma arctan_double: "\<bar>x\<bar> < 1 \<Longrightarrow> 2 * arctan x = arctan ((2 * x) / (1 - x\<^sup>2))" 5477 by (metis arctan_add linear mult_2 not_less power2_eq_square) 5478 5479theorem machin: "pi / 4 = 4 * arctan (1 / 5) - arctan (1 / 239)" 5480proof - 5481 have "\<bar>1 / 5\<bar> < (1 :: real)" 5482 by auto 5483 from arctan_add[OF less_imp_le[OF this] this] have "2 * arctan (1 / 5) = arctan (5 / 12)" 5484 by auto 5485 moreover 5486 have "\<bar>5 / 12\<bar> < (1 :: real)" 5487 by auto 5488 from arctan_add[OF less_imp_le[OF this] this] have "2 * arctan (5 / 12) = arctan (120 / 119)" 5489 by auto 5490 moreover 5491 have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" 5492 by auto 5493 from arctan_add[OF this] have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" 5494 by auto 5495 ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" 5496 by auto 5497 then show ?thesis 5498 unfolding arctan_one by algebra 5499qed 5500 5501lemma machin_Euler: "5 * arctan (1 / 7) + 2 * arctan (3 / 79) = pi / 4" 5502proof - 5503 have 17: "\<bar>1 / 7\<bar> < (1 :: real)" by auto 5504 with arctan_double have "2 * arctan (1 / 7) = arctan (7 / 24)" 5505 by simp (simp add: field_simps) 5506 moreover 5507 have "\<bar>7 / 24\<bar> < (1 :: real)" by auto 5508 with arctan_double have "2 * arctan (7 / 24) = arctan (336 / 527)" 5509 by simp (simp add: field_simps) 5510 moreover 5511 have "\<bar>336 / 527\<bar> < (1 :: real)" by auto 5512 from arctan_add[OF less_imp_le[OF 17] this] 5513 have "arctan(1/7) + arctan (336 / 527) = arctan (2879 / 3353)" 5514 by auto 5515 ultimately have I: "5 * arctan (1 / 7) = arctan (2879 / 3353)" by auto 5516 have 379: "\<bar>3 / 79\<bar> < (1 :: real)" by auto 5517 with arctan_double have II: "2 * arctan (3 / 79) = arctan (237 / 3116)" 5518 by simp (simp add: field_simps) 5519 have *: "\<bar>2879 / 3353\<bar> < (1 :: real)" by auto 5520 have "\<bar>237 / 3116\<bar> < (1 :: real)" by auto 5521 from arctan_add[OF less_imp_le[OF *] this] have "arctan (2879/3353) + arctan (237/3116) = pi/4" 5522 by (simp add: arctan_one) 5523 with I II show ?thesis by auto 5524qed 5525 5526(*But could also prove MACHIN_GAUSS: 5527 12 * arctan(1/18) + 8 * arctan(1/57) - 5 * arctan(1/239) = pi/4*) 5528 5529 5530subsection \<open>Introducing the inverse tangent power series\<close> 5531 5532lemma monoseq_arctan_series: 5533 fixes x :: real 5534 assumes "\<bar>x\<bar> \<le> 1" 5535 shows "monoseq (\<lambda>n. 1 / real (n * 2 + 1) * x^(n * 2 + 1))" 5536 (is "monoseq ?a") 5537proof (cases "x = 0") 5538 case True 5539 then show ?thesis by (auto simp: monoseq_def) 5540next 5541 case False 5542 have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" 5543 using assms by auto 5544 show "monoseq ?a" 5545 proof - 5546 have mono: "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le> 5547 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" 5548 if "0 \<le> x" and "x \<le> 1" for n and x :: real 5549 proof (rule mult_mono) 5550 show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))" 5551 by (rule frac_le) simp_all 5552 show "0 \<le> 1 / real (Suc (n * 2))" 5553 by auto 5554 show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)" 5555 by (rule power_decreasing) (simp_all add: \<open>0 \<le> x\<close> \<open>x \<le> 1\<close>) 5556 show "0 \<le> x ^ Suc (Suc n * 2)" 5557 by (rule zero_le_power) (simp add: \<open>0 \<le> x\<close>) 5558 qed 5559 show ?thesis 5560 proof (cases "0 \<le> x") 5561 case True 5562 from mono[OF this \<open>x \<le> 1\<close>, THEN allI] 5563 show ?thesis 5564 unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI2) 5565 next 5566 case False 5567 then have "0 \<le> - x" and "- x \<le> 1" 5568 using \<open>-1 \<le> x\<close> by auto 5569 from mono[OF this] 5570 have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge> 5571 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" for n 5572 using \<open>0 \<le> -x\<close> by auto 5573 then show ?thesis 5574 unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI]) 5575 qed 5576 qed 5577qed 5578 5579lemma zeroseq_arctan_series: 5580 fixes x :: real 5581 assumes "\<bar>x\<bar> \<le> 1" 5582 shows "(\<lambda>n. 1 / real (n * 2 + 1) * x^(n * 2 + 1)) \<longlonglongrightarrow> 0" 5583 (is "?a \<longlonglongrightarrow> 0") 5584proof (cases "x = 0") 5585 case True 5586 then show ?thesis by simp 5587next 5588 case False 5589 have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" 5590 using assms by auto 5591 show "?a \<longlonglongrightarrow> 0" 5592 proof (cases "\<bar>x\<bar> < 1") 5593 case True 5594 then have "norm x < 1" by auto 5595 from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF \<open>norm x < 1\<close>, THEN LIMSEQ_Suc]] 5596 have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) \<longlonglongrightarrow> 0" 5597 unfolding inverse_eq_divide Suc_eq_plus1 by simp 5598 then show ?thesis 5599 using pos2 by (rule LIMSEQ_linear) 5600 next 5601 case False 5602 then have "x = -1 \<or> x = 1" 5603 using \<open>\<bar>x\<bar> \<le> 1\<close> by auto 5604 then have n_eq: "\<And> n. x ^ (n * 2 + 1) = x" 5605 unfolding One_nat_def by auto 5606 from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]] 5607 show ?thesis 5608 unfolding n_eq Suc_eq_plus1 by auto 5609 qed 5610qed 5611 5612lemma summable_arctan_series: 5613 fixes n :: nat 5614 assumes "\<bar>x\<bar> \<le> 1" 5615 shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" 5616 (is "summable (?c x)") 5617 by (rule summable_Leibniz(1), 5618 rule zeroseq_arctan_series[OF assms], 5619 rule monoseq_arctan_series[OF assms]) 5620 5621lemma DERIV_arctan_series: 5622 assumes "\<bar>x\<bar> < 1" 5623 shows "DERIV (\<lambda>x'. \<Sum>k. (-1)^k * (1 / real (k * 2 + 1) * x' ^ (k * 2 + 1))) x :> 5624 (\<Sum>k. (-1)^k * x^(k * 2))" 5625 (is "DERIV ?arctan _ :> ?Int") 5626proof - 5627 let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0" 5628 5629 have n_even: "even n \<Longrightarrow> 2 * (n div 2) = n" for n :: nat 5630 by presburger 5631 then have if_eq: "?f n * real (Suc n) * x'^n = 5632 (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)" 5633 for n x' 5634 by auto 5635 5636 have summable_Integral: "summable (\<lambda> n. (- 1) ^ n * x^(2 * n))" if "\<bar>x\<bar> < 1" for x :: real 5637 proof - 5638 from that have "x\<^sup>2 < 1" 5639 by (simp add: abs_square_less_1) 5640 have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)" 5641 by (rule summable_Leibniz(1)) 5642 (auto intro!: LIMSEQ_realpow_zero monoseq_realpow \<open>x\<^sup>2 < 1\<close> order_less_imp_le[OF \<open>x\<^sup>2 < 1\<close>]) 5643 then show ?thesis 5644 by (simp only: power_mult) 5645 qed 5646 5647 have sums_even: "(sums) f = (sums) (\<lambda> n. if even n then f (n div 2) else 0)" 5648 for f :: "nat \<Rightarrow> real" 5649 proof - 5650 have "f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x" for x :: real 5651 proof 5652 assume "f sums x" 5653 from sums_if[OF sums_zero this] show "(\<lambda>n. if even n then f (n div 2) else 0) sums x" 5654 by auto 5655 next 5656 assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x" 5657 from LIMSEQ_linear[OF this[simplified sums_def] pos2, simplified sum_split_even_odd[simplified mult.commute]] 5658 show "f sums x" 5659 unfolding sums_def by auto 5660 qed 5661 then show ?thesis .. 5662 qed 5663 5664 have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int" 5665 unfolding if_eq mult.commute[of _ 2] 5666 suminf_def sums_even[of "\<lambda> n. (- 1) ^ n * x ^ (2 * n)", symmetric] 5667 by auto 5668 5669 have arctan_eq: "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x" for x 5670 proof - 5671 have if_eq': "\<And>n. (if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n = 5672 (if even n then (- 1) ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)" 5673 using n_even by auto 5674 have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)" 5675 by auto 5676 then show ?thesis 5677 unfolding if_eq' idx_eq suminf_def 5678 sums_even[of "\<lambda> n. (- 1) ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric] 5679 by auto 5680 qed 5681 5682 have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum>n. ?f n * real (Suc n) * x^n)" 5683 proof (rule DERIV_power_series') 5684 show "x \<in> {- 1 <..< 1}" 5685 using \<open>\<bar> x \<bar> < 1\<close> by auto 5686 show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" 5687 if x'_bounds: "x' \<in> {- 1 <..< 1}" for x' :: real 5688 proof - 5689 from that have "\<bar>x'\<bar> < 1" by auto 5690 then show ?thesis 5691 using that sums_summable sums_if [OF sums_0 [of "\<lambda>x. 0"] summable_sums [OF summable_Integral]] 5692 by (auto simp add: if_distrib [of "\<lambda>x. x * y" for y] cong: if_cong) 5693 qed 5694 qed auto 5695 then show ?thesis 5696 by (simp only: Int_eq arctan_eq) 5697qed 5698 5699lemma arctan_series: 5700 assumes "\<bar>x\<bar> \<le> 1" 5701 shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))" 5702 (is "_ = suminf (\<lambda> n. ?c x n)") 5703proof - 5704 let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)" 5705 5706 have DERIV_arctan_suminf: "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" 5707 if "0 < r" and "r < 1" and "\<bar>x\<bar> < r" for r x :: real 5708 proof (rule DERIV_arctan_series) 5709 from that show "\<bar>x\<bar> < 1" 5710 using \<open>r < 1\<close> and \<open>\<bar>x\<bar> < r\<close> by auto 5711 qed 5712 5713 { 5714 fix x :: real 5715 assume "\<bar>x\<bar> \<le> 1" 5716 note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] 5717 } note arctan_series_borders = this 5718 5719 have when_less_one: "arctan x = (\<Sum>k. ?c x k)" if "\<bar>x\<bar> < 1" for x :: real 5720 proof - 5721 obtain r where "\<bar>x\<bar> < r" and "r < 1" 5722 using dense[OF \<open>\<bar>x\<bar> < 1\<close>] by blast 5723 then have "0 < r" and "- r < x" and "x < r" by auto 5724 5725 have suminf_eq_arctan_bounded: "suminf (?c x) - arctan x = suminf (?c a) - arctan a" 5726 if "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b" for x a b 5727 proof - 5728 from that have "\<bar>x\<bar> < r" by auto 5729 show "suminf (?c x) - arctan x = suminf (?c a) - arctan a" 5730 proof (rule DERIV_isconst2[of "a" "b"]) 5731 show "a < b" and "a \<le> x" and "x \<le> b" 5732 using \<open>a < b\<close> \<open>a \<le> x\<close> \<open>x \<le> b\<close> by auto 5733 have "\<forall>x. - r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0" 5734 proof (rule allI, rule impI) 5735 fix x 5736 assume "-r < x \<and> x < r" 5737 then have "\<bar>x\<bar> < r" by auto 5738 with \<open>r < 1\<close> have "\<bar>x\<bar> < 1" by auto 5739 have "\<bar>- (x\<^sup>2)\<bar> < 1" using abs_square_less_1 \<open>\<bar>x\<bar> < 1\<close> by auto 5740 then have "(\<lambda>n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))" 5741 unfolding real_norm_def[symmetric] by (rule geometric_sums) 5742 then have "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))" 5743 unfolding power_mult_distrib[symmetric] power_mult mult.commute[of _ 2] by auto 5744 then have suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)" 5745 using sums_unique unfolding inverse_eq_divide by auto 5746 have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))" 5747 unfolding suminf_c'_eq_geom 5748 by (rule DERIV_arctan_suminf[OF \<open>0 < r\<close> \<open>r < 1\<close> \<open>\<bar>x\<bar> < r\<close>]) 5749 from DERIV_diff [OF this DERIV_arctan] show "DERIV (\<lambda>x. suminf (?c x) - arctan x) x :> 0" 5750 by auto 5751 qed 5752 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" 5753 using \<open>-r < a\<close> \<open>b < r\<close> by auto 5754 then show "\<And>y. \<lbrakk>a < y; y < b\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. suminf (?c x) - arctan x) y :> 0" 5755 using \<open>\<bar>x\<bar> < r\<close> by auto 5756 show "\<And>y. \<lbrakk>a \<le> y; y \<le> b\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. suminf (?c x) - arctan x) y" 5757 using DERIV_in_rball DERIV_isCont by auto 5758 qed 5759 qed 5760 5761 have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0" 5762 unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero 5763 by auto 5764 5765 have "suminf (?c x) - arctan x = 0" 5766 proof (cases "x = 0") 5767 case True 5768 then show ?thesis 5769 using suminf_arctan_zero by auto 5770 next 5771 case False 5772 then have "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" 5773 by auto 5774 have "suminf (?c (- \<bar>x\<bar>)) - arctan (- \<bar>x\<bar>) = suminf (?c 0) - arctan 0" 5775 by (rule suminf_eq_arctan_bounded[where x1=0 and a1="-\<bar>x\<bar>" and b1="\<bar>x\<bar>", symmetric]) 5776 (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>-\<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less) 5777 moreover 5778 have "suminf (?c x) - arctan x = suminf (?c (- \<bar>x\<bar>)) - arctan (- \<bar>x\<bar>)" 5779 by (rule suminf_eq_arctan_bounded[where x1=x and a1="- \<bar>x\<bar>" and b1="\<bar>x\<bar>"]) 5780 (simp_all only: \<open>\<bar>x\<bar> < r\<close> \<open>- \<bar>x\<bar> < \<bar>x\<bar>\<close> neg_less_iff_less) 5781 ultimately show ?thesis 5782 using suminf_arctan_zero by auto 5783 qed 5784 then show ?thesis by auto 5785 qed 5786 5787 show "arctan x = suminf (\<lambda>n. ?c x n)" 5788 proof (cases "\<bar>x\<bar> < 1") 5789 case True 5790 then show ?thesis by (rule when_less_one) 5791 next 5792 case False 5793 then have "\<bar>x\<bar> = 1" using \<open>\<bar>x\<bar> \<le> 1\<close> by auto 5794 let ?a = "\<lambda>x n. \<bar>1 / real (n * 2 + 1) * x^(n * 2 + 1)\<bar>" 5795 let ?diff = "\<lambda>x n. \<bar>arctan x - (\<Sum>i<n. ?c x i)\<bar>" 5796 have "?diff 1 n \<le> ?a 1 n" for n :: nat 5797 proof - 5798 have "0 < (1 :: real)" by auto 5799 moreover 5800 have "?diff x n \<le> ?a x n" if "0 < x" and "x < 1" for x :: real 5801 proof - 5802 from that have "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" 5803 by auto 5804 from \<open>0 < x\<close> have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)" 5805 by auto 5806 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] 5807 have "0 < 1 / real (n*2+1) * x^(n*2+1)" 5808 by (rule mult_pos_pos) (simp_all only: zero_less_power[OF \<open>0 < x\<close>], auto) 5809 then have a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)" 5810 by (rule abs_of_pos) 5811 show ?thesis 5812 proof (cases "even n") 5813 case True 5814 then have sgn_pos: "(-1)^n = (1::real)" by auto 5815 from \<open>even n\<close> obtain m where "n = 2 * m" .. 5816 then have "2 * m = n" .. 5817 from bounds[of m, unfolded this atLeastAtMost_iff] 5818 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))" 5819 by auto 5820 also have "\<dots> = ?c x n" by auto 5821 also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto 5822 finally show ?thesis . 5823 next 5824 case False 5825 then have sgn_neg: "(-1)^n = (-1::real)" by auto 5826 from \<open>odd n\<close> obtain m where "n = 2 * m + 1" .. 5827 then have m_def: "2 * m + 1 = n" .. 5828 then have m_plus: "2 * (m + 1) = n + 1" by auto 5829 from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2] 5830 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 5831 also have "\<dots> = - ?c x n" by auto 5832 also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto 5833 finally show ?thesis . 5834 qed 5835 qed 5836 hence "\<forall>x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto 5837 moreover have "isCont (\<lambda> x. ?a x n - ?diff x n) x" for x 5838 unfolding diff_conv_add_uminus divide_inverse 5839 by (auto intro!: isCont_add isCont_rabs continuous_ident isCont_minus isCont_arctan 5840 continuous_at_within_inverse isCont_mult isCont_power continuous_const isCont_sum 5841 simp del: add_uminus_conv_diff) 5842 ultimately have "0 \<le> ?a 1 n - ?diff 1 n" 5843 by (rule LIM_less_bound) 5844 then show ?thesis by auto 5845 qed 5846 have "?a 1 \<longlonglongrightarrow> 0" 5847 unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def 5848 by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat simp del: of_nat_Suc) 5849 have "?diff 1 \<longlonglongrightarrow> 0" 5850 proof (rule LIMSEQ_I) 5851 fix r :: real 5852 assume "0 < r" 5853 obtain N :: nat where N_I: "N \<le> n \<Longrightarrow> ?a 1 n < r" for n 5854 using LIMSEQ_D[OF \<open>?a 1 \<longlonglongrightarrow> 0\<close> \<open>0 < r\<close>] by auto 5855 have "norm (?diff 1 n - 0) < r" if "N \<le> n" for n 5856 using \<open>?diff 1 n \<le> ?a 1 n\<close> N_I[OF that] by auto 5857 then show "\<exists>N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast 5858 qed 5859 from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus] 5860 have "(?c 1) sums (arctan 1)" unfolding sums_def by auto 5861 then have "arctan 1 = (\<Sum>i. ?c 1 i)" by (rule sums_unique) 5862 5863 show ?thesis 5864 proof (cases "x = 1") 5865 case True 5866 then show ?thesis by (simp add: \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close>) 5867 next 5868 case False 5869 then have "x = -1" using \<open>\<bar>x\<bar> = 1\<close> by auto 5870 5871 have "- (pi/2) < 0" using pi_gt_zero by auto 5872 have "- (2 * pi) < 0" using pi_gt_zero by auto 5873 5874 have c_minus_minus: "?c (- 1) i = - ?c 1 i" for i by auto 5875 5876 have "arctan (- 1) = arctan (tan (-(pi / 4)))" 5877 unfolding tan_45 tan_minus .. 5878 also have "\<dots> = - (pi / 4)" 5879 by (rule arctan_tan) (auto simp: order_less_trans[OF \<open>- (pi/2) < 0\<close> pi_gt_zero]) 5880 also have "\<dots> = - (arctan (tan (pi / 4)))" 5881 unfolding neg_equal_iff_equal 5882 by (rule arctan_tan[symmetric]) (auto simp: order_less_trans[OF \<open>- (2 * pi) < 0\<close> pi_gt_zero]) 5883 also have "\<dots> = - (arctan 1)" 5884 unfolding tan_45 .. 5885 also have "\<dots> = - (\<Sum> i. ?c 1 i)" 5886 using \<open>arctan 1 = (\<Sum> i. ?c 1 i)\<close> by auto 5887 also have "\<dots> = (\<Sum> i. ?c (- 1) i)" 5888 using suminf_minus[OF sums_summable[OF \<open>(?c 1) sums (arctan 1)\<close>]] 5889 unfolding c_minus_minus by auto 5890 finally show ?thesis using \<open>x = -1\<close> by auto 5891 qed 5892 qed 5893qed 5894 5895lemma arctan_half: "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))" 5896 for x :: real 5897proof - 5898 obtain y where low: "- (pi/2) < y" and high: "y < pi/2" and y_eq: "tan y = x" 5899 using tan_total by blast 5900 then have low2: "- (pi/2) < y / 2" and high2: "y / 2 < pi/2" 5901 by auto 5902 5903 have "0 < cos y" by (rule cos_gt_zero_pi[OF low high]) 5904 then have "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y" 5905 by auto 5906 5907 have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2" 5908 unfolding tan_def power_divide .. 5909 also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2" 5910 using \<open>cos y \<noteq> 0\<close> by auto 5911 also have "\<dots> = 1 / (cos y)\<^sup>2" 5912 unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 .. 5913 finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" . 5914 5915 have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)" 5916 unfolding tan_def using \<open>cos y \<noteq> 0\<close> by (simp add: field_simps) 5917 also have "\<dots> = tan y / (1 + 1 / cos y)" 5918 using \<open>cos y \<noteq> 0\<close> unfolding add_divide_distrib by auto 5919 also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))" 5920 unfolding cos_sqrt .. 5921 also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))" 5922 unfolding real_sqrt_divide by auto 5923 finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))" 5924 unfolding \<open>1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2\<close> . 5925 5926 have "arctan x = y" 5927 using arctan_tan low high y_eq by auto 5928 also have "\<dots> = 2 * (arctan (tan (y/2)))" 5929 using arctan_tan[OF low2 high2] by auto 5930 also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" 5931 unfolding tan_half by auto 5932 finally show ?thesis 5933 unfolding eq \<open>tan y = x\<close> . 5934qed 5935 5936lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y" 5937 by (simp only: arctan_less_iff) 5938 5939lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y" 5940 by (simp only: arctan_le_iff) 5941 5942lemma arctan_inverse: 5943 assumes "x \<noteq> 0" 5944 shows "arctan (1 / x) = sgn x * pi/2 - arctan x" 5945proof (rule arctan_unique) 5946 show "- (pi/2) < sgn x * pi/2 - arctan x" 5947 using arctan_bounded [of x] assms 5948 unfolding sgn_real_def 5949 apply (auto simp: arctan algebra_simps) 5950 apply (drule zero_less_arctan_iff [THEN iffD2], arith) 5951 done 5952 show "sgn x * pi/2 - arctan x < pi/2" 5953 using arctan_bounded [of "- x"] assms 5954 unfolding sgn_real_def arctan_minus 5955 by (auto simp: algebra_simps) 5956 show "tan (sgn x * pi/2 - arctan x) = 1 / x" 5957 unfolding tan_inverse [of "arctan x", unfolded tan_arctan] 5958 unfolding sgn_real_def 5959 by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff) 5960qed 5961 5962theorem pi_series: "pi / 4 = (\<Sum>k. (-1)^k * 1 / real (k * 2 + 1))" 5963 (is "_ = ?SUM") 5964proof - 5965 have "pi / 4 = arctan 1" 5966 using arctan_one by auto 5967 also have "\<dots> = ?SUM" 5968 using arctan_series[of 1] by auto 5969 finally show ?thesis by auto 5970qed 5971 5972 5973subsection \<open>Existence of Polar Coordinates\<close> 5974 5975lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1" 5976 by (rule power2_le_imp_le [OF _ zero_le_one]) 5977 (simp add: power_divide divide_le_eq not_sum_power2_lt_zero) 5978 5979lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one] 5980 5981lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one] 5982 5983lemma polar_Ex: "\<exists>r::real. \<exists>a. x = r * cos a \<and> y = r * sin a" 5984proof - 5985 have polar_ex1: "0 < y \<Longrightarrow> \<exists>r a. x = r * cos a \<and> y = r * sin a" for y 5986 apply (rule exI [where x = "sqrt (x\<^sup>2 + y\<^sup>2)"]) 5987 apply (rule exI [where x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))"]) 5988 apply (simp add: cos_arccos_lemma1 sin_arccos_lemma1 power_divide 5989 real_sqrt_mult [symmetric] right_diff_distrib) 5990 done 5991 show ?thesis 5992 proof (cases "0::real" y rule: linorder_cases) 5993 case less 5994 then show ?thesis 5995 by (rule polar_ex1) 5996 next 5997 case equal 5998 then show ?thesis 5999 by (force simp: intro!: cos_zero sin_zero) 6000 next 6001 case greater 6002 with polar_ex1 [where y="-y"] show ?thesis 6003 by auto (metis cos_minus minus_minus minus_mult_right sin_minus) 6004 qed 6005qed 6006 6007 6008subsection \<open>Basics about polynomial functions: products, extremal behaviour and root counts\<close> 6009 6010lemma pairs_le_eq_Sigma: "{(i, j). i + j \<le> m} = Sigma (atMost m) (\<lambda>r. atMost (m - r))" 6011 for m :: nat 6012 by auto 6013 6014lemma 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)" 6015 by (metis atLeast0AtMost Suc_eq_plus1 le0 sum_ub_add_nat) 6016 6017lemma Sigma_interval_disjoint: "(SIGMA i:A. {..v i}) \<inter> (SIGMA i:A.{v i<..w}) = {}" 6018 for w :: "'a::order" 6019 by auto 6020 6021lemma product_atMost_eq_Un: "A \<times> {..m} = (SIGMA i:A.{..m - i}) \<union> (SIGMA i:A.{m - i<..m})" 6022 for m :: nat 6023 by auto 6024 6025lemma polynomial_product: (*with thanks to Chaitanya Mangla*) 6026 fixes x :: "'a::idom" 6027 assumes m: "\<And>i. i > m \<Longrightarrow> a i = 0" 6028 and n: "\<And>j. j > n \<Longrightarrow> b j = 0" 6029 shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) = 6030 (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)" 6031proof - 6032 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))" 6033 by (rule sum_product) 6034 also have "\<dots> = (\<Sum>i\<le>m + n. \<Sum>j\<le>n + m. a i * x ^ i * (b j * x ^ j))" 6035 using assms by (auto simp: sum_up_index_split) 6036 also have "\<dots> = (\<Sum>r\<le>m + n. \<Sum>j\<le>m + n - r. a r * x ^ r * (b j * x ^ j))" 6037 apply (simp add: add_ac sum.Sigma product_atMost_eq_Un) 6038 apply (clarsimp simp add: sum_Un Sigma_interval_disjoint intro!: sum.neutral) 6039 apply (metis add_diff_assoc2 add.commute add_lessD1 leD m n nat_le_linear neqE) 6040 done 6041 also have "\<dots> = (\<Sum>(i,j)\<in>{(i,j). i+j \<le> m+n}. (a i * x ^ i) * (b j * x ^ j))" 6042 by (auto simp: pairs_le_eq_Sigma sum.Sigma) 6043 also have "\<dots> = (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)" 6044 apply (subst sum_triangle_reindex_eq) 6045 apply (auto simp: algebra_simps sum_distrib_left intro!: sum.cong) 6046 apply (metis le_add_diff_inverse power_add) 6047 done 6048 finally show ?thesis . 6049qed 6050 6051lemma polynomial_product_nat: 6052 fixes x :: nat 6053 assumes m: "\<And>i. i > m \<Longrightarrow> a i = 0" 6054 and n: "\<And>j. j > n \<Longrightarrow> b j = 0" 6055 shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) = 6056 (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)" 6057 using polynomial_product [of m a n b x] assms 6058 by (simp only: of_nat_mult [symmetric] of_nat_power [symmetric] 6059 of_nat_eq_iff Int.int_sum [symmetric]) 6060 6061lemma polyfun_diff: (*COMPLEX_SUB_POLYFUN in HOL Light*) 6062 fixes x :: "'a::idom" 6063 assumes "1 \<le> n" 6064 shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) = 6065 (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)" 6066proof - 6067 have h: "bij_betw (\<lambda>(i,j). (j,i)) ((SIGMA i : atMost n. lessThan i)) (SIGMA j : lessThan n. {Suc j..n})" 6068 by (auto simp: bij_betw_def inj_on_def) 6069 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))" 6070 by (simp add: right_diff_distrib sum_subtractf) 6071 also have "\<dots> = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))" 6072 by (simp add: power_diff_sumr2 mult.assoc) 6073 also have "\<dots> = (\<Sum>i\<le>n. \<Sum>j<i. a i * (x - y) * (y^(i - Suc j) * x^j))" 6074 by (simp add: sum_distrib_left) 6075 also have "\<dots> = (\<Sum>(i,j) \<in> (SIGMA i : atMost n. lessThan i). a i * (x - y) * (y^(i - Suc j) * x^j))" 6076 by (simp add: sum.Sigma) 6077 also have "\<dots> = (\<Sum>(j,i) \<in> (SIGMA j : lessThan n. {Suc j..n}). a i * (x - y) * (y^(i - Suc j) * x^j))" 6078 by (auto simp: sum.reindex_bij_betw [OF h, symmetric] intro: sum.strong_cong) 6079 also have "\<dots> = (\<Sum>j<n. \<Sum>i=Suc j..n. a i * (x - y) * (y^(i - Suc j) * x^j))" 6080 by (simp add: sum.Sigma) 6081 also have "\<dots> = (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)" 6082 by (simp add: sum_distrib_left mult_ac) 6083 finally show ?thesis . 6084qed 6085 6086lemma polyfun_diff_alt: (*COMPLEX_SUB_POLYFUN_ALT in HOL Light*) 6087 fixes x :: "'a::idom" 6088 assumes "1 \<le> n" 6089 shows "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) = 6090 (x - y) * ((\<Sum>j<n. \<Sum>k<n-j. a(j + k + 1) * y^k * x^j))" 6091proof - 6092 have "(\<Sum>i=Suc j..n. a i * y^(i - j - 1)) = (\<Sum>k<n-j. a(j+k+1) * y^k)" 6093 if "j < n" for j :: nat 6094 proof - 6095 have h: "bij_betw (\<lambda>i. i - (j + 1)) {Suc j..n} (lessThan (n-j))" 6096 apply (auto simp: bij_betw_def inj_on_def) 6097 apply (rule_tac x="x + Suc j" in image_eqI, auto) 6098 done 6099 then show ?thesis 6100 by (auto simp: sum.reindex_bij_betw [OF h, symmetric] intro: sum.strong_cong) 6101 qed 6102 then show ?thesis 6103 by (simp add: polyfun_diff [OF assms] sum_distrib_right) 6104qed 6105 6106lemma polyfun_linear_factor: (*COMPLEX_POLYFUN_LINEAR_FACTOR in HOL Light*) 6107 fixes a :: "'a::idom" 6108 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)" 6109proof (cases "n = 0") 6110 case True then show ?thesis 6111 by simp 6112next 6113 case False 6114 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> 6115 (\<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))" 6116 by (simp add: algebra_simps) 6117 also have "\<dots> \<longleftrightarrow> 6118 (\<exists>b. \<forall>z. (z - a) * (\<Sum>j<n. (\<Sum>i = Suc j..n. c i * a^(i - Suc j)) * z^j) = 6119 (z - a) * (\<Sum>i<n. b i * z^i))" 6120 using False by (simp add: polyfun_diff) 6121 also have "\<dots> = True" by auto 6122 finally show ?thesis 6123 by simp 6124qed 6125 6126lemma polyfun_linear_factor_root: (*COMPLEX_POLYFUN_LINEAR_FACTOR_ROOT in HOL Light*) 6127 fixes a :: "'a::idom" 6128 assumes "(\<Sum>i\<le>n. c(i) * a^i) = 0" 6129 obtains b where "\<And>z. (\<Sum>i\<le>n. c i * z^i) = (z - a) * (\<Sum>i<n. b i * z^i)" 6130 using polyfun_linear_factor [of c n a] assms by auto 6131 6132(*The material of this section, up until this point, could go into a new theory of polynomials 6133 based on Main alone. The remaining material involves limits, continuity, series, etc.*) 6134 6135lemma isCont_polynom: "isCont (\<lambda>w. \<Sum>i\<le>n. c i * w^i) a" 6136 for c :: "nat \<Rightarrow> 'a::real_normed_div_algebra" 6137 by simp 6138 6139lemma zero_polynom_imp_zero_coeffs: 6140 fixes c :: "nat \<Rightarrow> 'a::{ab_semigroup_mult,real_normed_div_algebra}" 6141 assumes "\<And>w. (\<Sum>i\<le>n. c i * w^i) = 0" "k \<le> n" 6142 shows "c k = 0" 6143 using assms 6144proof (induction n arbitrary: c k) 6145 case 0 6146 then show ?case 6147 by simp 6148next 6149 case (Suc n c k) 6150 have [simp]: "c 0 = 0" using Suc.prems(1) [of 0] 6151 by simp 6152 have "(\<Sum>i\<le>Suc n. c i * w^i) = w * (\<Sum>i\<le>n. c (Suc i) * w^i)" for w 6153 proof - 6154 have "(\<Sum>i\<le>Suc n. c i * w^i) = (\<Sum>i\<le>n. c (Suc i) * w ^ Suc i)" 6155 unfolding Set_Interval.sum_atMost_Suc_shift 6156 by simp 6157 also have "\<dots> = w * (\<Sum>i\<le>n. c (Suc i) * w^i)" 6158 by (simp add: sum_distrib_left ac_simps) 6159 finally show ?thesis . 6160 qed 6161 then have w: "\<And>w. w \<noteq> 0 \<Longrightarrow> (\<Sum>i\<le>n. c (Suc i) * w^i) = 0" 6162 using Suc by auto 6163 then have "(\<lambda>h. \<Sum>i\<le>n. c (Suc i) * h^i) \<midarrow>0\<rightarrow> 0" 6164 by (simp cong: LIM_cong) \<comment> \<open>the case \<open>w = 0\<close> by continuity\<close> 6165 then have "(\<Sum>i\<le>n. c (Suc i) * 0^i) = 0" 6166 using isCont_polynom [of 0 "\<lambda>i. c (Suc i)" n] LIM_unique 6167 by (force simp: Limits.isCont_iff) 6168 then have "\<And>w. (\<Sum>i\<le>n. c (Suc i) * w^i) = 0" 6169 using w by metis 6170 then have "\<And>i. i \<le> n \<Longrightarrow> c (Suc i) = 0" 6171 using Suc.IH [of "\<lambda>i. c (Suc i)"] by blast 6172 then show ?case using \<open>k \<le> Suc n\<close> 6173 by (cases k) auto 6174qed 6175 6176lemma polyfun_rootbound: (*COMPLEX_POLYFUN_ROOTBOUND in HOL Light*) 6177 fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}" 6178 assumes "c k \<noteq> 0" "k\<le>n" 6179 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" 6180 using assms 6181proof (induction n arbitrary: c k) 6182 case 0 6183 then show ?case 6184 by simp 6185next 6186 case (Suc m c k) 6187 let ?succase = ?case 6188 show ?case 6189 proof (cases "{z. (\<Sum>i\<le>Suc m. c(i) * z^i) = 0} = {}") 6190 case True 6191 then show ?succase 6192 by simp 6193 next 6194 case False 6195 then obtain z0 where z0: "(\<Sum>i\<le>Suc m. c(i) * z0^i) = 0" 6196 by blast 6197 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)" 6198 using polyfun_linear_factor_root [OF z0, unfolded lessThan_Suc_atMost] 6199 by blast 6200 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}" 6201 by auto 6202 have "\<not> (\<forall>k\<le>m. b k = 0)" 6203 proof 6204 assume [simp]: "\<forall>k\<le>m. b k = 0" 6205 then have "\<And>w. (\<Sum>i\<le>m. b i * w^i) = 0" 6206 by simp 6207 then have "\<And>w. (\<Sum>i\<le>Suc m. c i * w^i) = 0" 6208 using b by simp 6209 then have "\<And>k. k \<le> Suc m \<Longrightarrow> c k = 0" 6210 using zero_polynom_imp_zero_coeffs by blast 6211 then show False using Suc.prems by blast 6212 qed 6213 then obtain k' where bk': "b k' \<noteq> 0" "k' \<le> m" 6214 by blast 6215 show ?succase 6216 using Suc.IH [of b k'] bk' 6217 by (simp add: eq card_insert_if del: sum_atMost_Suc) 6218 qed 6219qed 6220 6221lemma 6222 fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}" 6223 assumes "c k \<noteq> 0" "k\<le>n" 6224 shows polyfun_roots_finite: "finite {z. (\<Sum>i\<le>n. c(i) * z^i) = 0}" 6225 and polyfun_roots_card: "card {z. (\<Sum>i\<le>n. c(i) * z^i) = 0} \<le> n" 6226 using polyfun_rootbound assms by auto 6227 6228lemma polyfun_finite_roots: (*COMPLEX_POLYFUN_FINITE_ROOTS in HOL Light*) 6229 fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}" 6230 shows "finite {x. (\<Sum>i\<le>n. c i * x^i) = 0} \<longleftrightarrow> (\<exists>i\<le>n. c i \<noteq> 0)" 6231 (is "?lhs = ?rhs") 6232proof 6233 assume ?lhs 6234 moreover have "\<not> finite {x. (\<Sum>i\<le>n. c i * x^i) = 0}" if "\<forall>i\<le>n. c i = 0" 6235 proof - 6236 from that have "\<And>x. (\<Sum>i\<le>n. c i * x^i) = 0" 6237 by simp 6238 then show ?thesis 6239 using ex_new_if_finite [OF infinite_UNIV_char_0 [where 'a='a]] 6240 by auto 6241 qed 6242 ultimately show ?rhs by metis 6243next 6244 assume ?rhs 6245 with polyfun_rootbound show ?lhs by blast 6246qed 6247 6248lemma polyfun_eq_0: "(\<forall>x. (\<Sum>i\<le>n. c i * x^i) = 0) \<longleftrightarrow> (\<forall>i\<le>n. c i = 0)" 6249 for c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}" 6250 (*COMPLEX_POLYFUN_EQ_0 in HOL Light*) 6251 using zero_polynom_imp_zero_coeffs by auto 6252 6253lemma 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)" 6254 for c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}" 6255proof - 6256 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)" 6257 by (simp add: left_diff_distrib Groups_Big.sum_subtractf) 6258 also have "\<dots> \<longleftrightarrow> (\<forall>i\<le>n. c i - d i = 0)" 6259 by (rule polyfun_eq_0) 6260 finally show ?thesis 6261 by simp 6262qed 6263 6264lemma polyfun_eq_const: (*COMPLEX_POLYFUN_EQ_CONST in HOL Light*) 6265 fixes c :: "nat \<Rightarrow> 'a::{idom,real_normed_div_algebra}" 6266 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)" 6267 (is "?lhs = ?rhs") 6268proof - 6269 have *: "\<forall>x. (\<Sum>i\<le>n. (if i=0 then k else 0) * x^i) = k" 6270 by (induct n) auto 6271 show ?thesis 6272 proof 6273 assume ?lhs 6274 with * have "(\<forall>i\<le>n. c i = (if i=0 then k else 0))" 6275 by (simp add: polyfun_eq_coeffs [symmetric]) 6276 then show ?rhs by simp 6277 next 6278 assume ?rhs 6279 then show ?lhs by (induct n) auto 6280 qed 6281qed 6282 6283lemma root_polyfun: 6284 fixes z :: "'a::idom" 6285 assumes "1 \<le> n" 6286 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" 6287 using assms by (cases n) (simp_all add: sum_head_Suc atLeast0AtMost [symmetric]) 6288 6289lemma 6290 assumes "SORT_CONSTRAINT('a::{idom,real_normed_div_algebra})" 6291 and "1 \<le> n" 6292 shows finite_roots_unity: "finite {z::'a. z^n = 1}" 6293 and card_roots_unity: "card {z::'a. z^n = 1} \<le> n" 6294 using polyfun_rootbound [of "\<lambda>i. if i = 0 then -1 else if i=n then 1 else 0" n n] assms(2) 6295 by (auto simp: root_polyfun [OF assms(2)]) 6296 6297 6298subsection \<open>Hyperbolic functions\<close> 6299 6300definition sinh :: "'a :: {banach, real_normed_algebra_1} \<Rightarrow> 'a" where 6301 "sinh x = (exp x - exp (-x)) /\<^sub>R 2" 6302 6303definition cosh :: "'a :: {banach, real_normed_algebra_1} \<Rightarrow> 'a" where 6304 "cosh x = (exp x + exp (-x)) /\<^sub>R 2" 6305 6306definition tanh :: "'a :: {banach, real_normed_field} \<Rightarrow> 'a" where 6307 "tanh x = sinh x / cosh x" 6308 6309definition arsinh :: "'a :: {banach, real_normed_algebra_1, ln} \<Rightarrow> 'a" where 6310 "arsinh x = ln (x + (x^2 + 1) powr of_real (1/2))" 6311 6312definition arcosh :: "'a :: {banach, real_normed_algebra_1, ln} \<Rightarrow> 'a" where 6313 "arcosh x = ln (x + (x^2 - 1) powr of_real (1/2))" 6314 6315definition artanh :: "'a :: {banach, real_normed_field, ln} \<Rightarrow> 'a" where 6316 "artanh x = ln ((1 + x) / (1 - x)) / 2" 6317 6318lemma arsinh_0 [simp]: "arsinh 0 = 0" 6319 by (simp add: arsinh_def) 6320 6321lemma arcosh_1 [simp]: "arcosh 1 = 0" 6322 by (simp add: arcosh_def) 6323 6324lemma artanh_0 [simp]: "artanh 0 = 0" 6325 by (simp add: artanh_def) 6326 6327lemma tanh_altdef: 6328 "tanh x = (exp x - exp (-x)) / (exp x + exp (-x))" 6329proof - 6330 have "tanh x = (2 *\<^sub>R sinh x) / (2 *\<^sub>R cosh x)" 6331 by (simp add: tanh_def scaleR_conv_of_real) 6332 also have "2 *\<^sub>R sinh x = exp x - exp (-x)" 6333 by (simp add: sinh_def) 6334 also have "2 *\<^sub>R cosh x = exp x + exp (-x)" 6335 by (simp add: cosh_def) 6336 finally show ?thesis . 6337qed 6338 6339lemma tanh_real_altdef: "tanh (x::real) = (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))" 6340proof - 6341 have [simp]: "exp (2 * x) = exp x * exp x" "exp (x * 2) = exp x * exp x" 6342 by (subst exp_add [symmetric]; simp)+ 6343 have "tanh x = (2 * exp (-x) * sinh x) / (2 * exp (-x) * cosh x)" 6344 by (simp add: tanh_def) 6345 also have "2 * exp (-x) * sinh x = 1 - exp (-2*x)" 6346 by (simp add: exp_minus field_simps sinh_def) 6347 also have "2 * exp (-x) * cosh x = 1 + exp (-2*x)" 6348 by (simp add: exp_minus field_simps cosh_def) 6349 finally show ?thesis . 6350qed 6351 6352 6353lemma sinh_converges: "(\<lambda>n. if even n then 0 else x ^ n /\<^sub>R fact n) sums sinh x" 6354proof - 6355 have "(\<lambda>n. (x ^ n /\<^sub>R fact n - (-x) ^ n /\<^sub>R fact n) /\<^sub>R 2) sums sinh x" 6356 unfolding sinh_def by (intro sums_scaleR_right sums_diff exp_converges) 6357 also have "(\<lambda>n. (x ^ n /\<^sub>R fact n - (-x) ^ n /\<^sub>R fact n) /\<^sub>R 2) = 6358 (\<lambda>n. if even n then 0 else x ^ n /\<^sub>R fact n)" by auto 6359 finally show ?thesis . 6360qed 6361 6362lemma cosh_converges: "(\<lambda>n. if even n then x ^ n /\<^sub>R fact n else 0) sums cosh x" 6363proof - 6364 have "(\<lambda>n. (x ^ n /\<^sub>R fact n + (-x) ^ n /\<^sub>R fact n) /\<^sub>R 2) sums cosh x" 6365 unfolding cosh_def by (intro sums_scaleR_right sums_add exp_converges) 6366 also have "(\<lambda>n. (x ^ n /\<^sub>R fact n + (-x) ^ n /\<^sub>R fact n) /\<^sub>R 2) = 6367 (\<lambda>n. if even n then x ^ n /\<^sub>R fact n else 0)" by auto 6368 finally show ?thesis . 6369qed 6370 6371lemma sinh_0 [simp]: "sinh 0 = 0" 6372 by (simp add: sinh_def) 6373 6374lemma cosh_0 [simp]: "cosh 0 = 1" 6375proof - 6376 have "cosh 0 = (1/2) *\<^sub>R (1 + 1)" by (simp add: cosh_def) 6377 also have "\<dots> = 1" by (rule scaleR_half_double) 6378 finally show ?thesis . 6379qed 6380 6381lemma tanh_0 [simp]: "tanh 0 = 0" 6382 by (simp add: tanh_def) 6383 6384lemma sinh_minus [simp]: "sinh (- x) = -sinh x" 6385 by (simp add: sinh_def algebra_simps) 6386 6387lemma cosh_minus [simp]: "cosh (- x) = cosh x" 6388 by (simp add: cosh_def algebra_simps) 6389 6390lemma tanh_minus [simp]: "tanh (-x) = -tanh x" 6391 by (simp add: tanh_def) 6392 6393lemma sinh_ln_real: "x > 0 \<Longrightarrow> sinh (ln x :: real) = (x - inverse x) / 2" 6394 by (simp add: sinh_def exp_minus) 6395 6396lemma cosh_ln_real: "x > 0 \<Longrightarrow> cosh (ln x :: real) = (x + inverse x) / 2" 6397 by (simp add: cosh_def exp_minus) 6398 6399lemma tanh_ln_real: "x > 0 \<Longrightarrow> tanh (ln x :: real) = (x ^ 2 - 1) / (x ^ 2 + 1)" 6400 by (simp add: tanh_def sinh_ln_real cosh_ln_real divide_simps power2_eq_square) 6401 6402lemma has_field_derivative_scaleR_right [derivative_intros]: 6403 "(f has_field_derivative D) F \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_field_derivative (c *\<^sub>R D)) F" 6404 unfolding has_field_derivative_def 6405 using has_derivative_scaleR_right[of f "\<lambda>x. D * x" F c] 6406 by (simp add: mult_scaleR_left [symmetric] del: mult_scaleR_left) 6407 6408lemma has_field_derivative_sinh [THEN DERIV_chain2, derivative_intros]: 6409 "(sinh has_field_derivative cosh x) (at (x :: 'a :: {banach, real_normed_field}))" 6410 unfolding sinh_def cosh_def by (auto intro!: derivative_eq_intros) 6411 6412lemma has_field_derivative_cosh [THEN DERIV_chain2, derivative_intros]: 6413 "(cosh has_field_derivative sinh x) (at (x :: 'a :: {banach, real_normed_field}))" 6414 unfolding sinh_def cosh_def by (auto intro!: derivative_eq_intros) 6415 6416lemma has_field_derivative_tanh [THEN DERIV_chain2, derivative_intros]: 6417 "cosh x \<noteq> 0 \<Longrightarrow> (tanh has_field_derivative 1 - tanh x ^ 2) 6418 (at (x :: 'a :: {banach, real_normed_field}))" 6419 unfolding tanh_def by (auto intro!: derivative_eq_intros simp: power2_eq_square divide_simps) 6420 6421lemma has_derivative_sinh [derivative_intros]: 6422 fixes g :: "'a \<Rightarrow> ('a :: {banach, real_normed_field})" 6423 assumes "(g has_derivative (\<lambda>x. Db * x)) (at x within s)" 6424 shows "((\<lambda>x. sinh (g x)) has_derivative (\<lambda>y. (cosh (g x) * Db) * y)) (at x within s)" 6425proof - 6426 have "((\<lambda>x. - g x) has_derivative (\<lambda>y. -(Db * y))) (at x within s)" 6427 using assms by (intro derivative_intros) 6428 also have "(\<lambda>y. -(Db * y)) = (\<lambda>x. (-Db) * x)" by (simp add: fun_eq_iff) 6429 finally have "((\<lambda>x. sinh (g x)) has_derivative 6430 (\<lambda>y. (exp (g x) * Db * y - exp (-g x) * (-Db) * y) /\<^sub>R 2)) (at x within s)" 6431 unfolding sinh_def by (intro derivative_intros assms) 6432 also have "(\<lambda>y. (exp (g x) * Db * y - exp (-g x) * (-Db) * y) /\<^sub>R 2) = (\<lambda>y. (cosh (g x) * Db) * y)" 6433 by (simp add: fun_eq_iff cosh_def algebra_simps) 6434 finally show ?thesis . 6435qed 6436 6437lemma has_derivative_cosh [derivative_intros]: 6438 fixes g :: "'a \<Rightarrow> ('a :: {banach, real_normed_field})" 6439 assumes "(g has_derivative (\<lambda>y. Db * y)) (at x within s)" 6440 shows "((\<lambda>x. cosh (g x)) has_derivative (\<lambda>y. (sinh (g x) * Db) * y)) (at x within s)" 6441proof - 6442 have "((\<lambda>x. - g x) has_derivative (\<lambda>y. -(Db * y))) (at x within s)" 6443 using assms by (intro derivative_intros) 6444 also have "(\<lambda>y. -(Db * y)) = (\<lambda>y. (-Db) * y)" by (simp add: fun_eq_iff) 6445 finally have "((\<lambda>x. cosh (g x)) has_derivative 6446 (\<lambda>y. (exp (g x) * Db * y + exp (-g x) * (-Db) * y) /\<^sub>R 2)) (at x within s)" 6447 unfolding cosh_def by (intro derivative_intros assms) 6448 also have "(\<lambda>y. (exp (g x) * Db * y + exp (-g x) * (-Db) * y) /\<^sub>R 2) = (\<lambda>y. (sinh (g x) * Db) * y)" 6449 by (simp add: fun_eq_iff sinh_def algebra_simps) 6450 finally show ?thesis . 6451qed 6452 6453lemma sinh_plus_cosh: "sinh x + cosh x = exp x" 6454proof - 6455 have "sinh x + cosh x = (1 / 2) *\<^sub>R (exp x + exp x)" 6456 by (simp add: sinh_def cosh_def algebra_simps) 6457 also have "\<dots> = exp x" by (rule scaleR_half_double) 6458 finally show ?thesis . 6459qed 6460 6461lemma cosh_plus_sinh: "cosh x + sinh x = exp x" 6462 by (subst add.commute) (rule sinh_plus_cosh) 6463 6464lemma cosh_minus_sinh: "cosh x - sinh x = exp (-x)" 6465proof - 6466 have "cosh x - sinh x = (1 / 2) *\<^sub>R (exp (-x) + exp (-x))" 6467 by (simp add: sinh_def cosh_def algebra_simps) 6468 also have "\<dots> = exp (-x)" by (rule scaleR_half_double) 6469 finally show ?thesis . 6470qed 6471 6472lemma sinh_minus_cosh: "sinh x - cosh x = -exp (-x)" 6473 using cosh_minus_sinh[of x] by (simp add: algebra_simps) 6474 6475 6476context 6477 fixes x :: "'a :: {real_normed_field, banach}" 6478begin 6479 6480lemma sinh_zero_iff: "sinh x = 0 \<longleftrightarrow> exp x \<in> {1, -1}" 6481 by (auto simp: sinh_def field_simps exp_minus power2_eq_square square_eq_1_iff) 6482 6483lemma cosh_zero_iff: "cosh x = 0 \<longleftrightarrow> exp x ^ 2 = -1" 6484 by (auto simp: cosh_def exp_minus field_simps power2_eq_square eq_neg_iff_add_eq_0) 6485 6486lemma cosh_square_eq: "cosh x ^ 2 = sinh x ^ 2 + 1" 6487 by (simp add: cosh_def sinh_def algebra_simps power2_eq_square exp_add [symmetric] 6488 scaleR_conv_of_real) 6489 6490lemma sinh_square_eq: "sinh x ^ 2 = cosh x ^ 2 - 1" 6491 by (simp add: cosh_square_eq) 6492 6493lemma hyperbolic_pythagoras: "cosh x ^ 2 - sinh x ^ 2 = 1" 6494 by (simp add: cosh_square_eq) 6495 6496lemma sinh_add: "sinh (x + y) = sinh x * cosh y + cosh x * sinh y" 6497 by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric]) 6498 6499lemma sinh_diff: "sinh (x - y) = sinh x * cosh y - cosh x * sinh y" 6500 by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric]) 6501 6502lemma cosh_add: "cosh (x + y) = cosh x * cosh y + sinh x * sinh y" 6503 by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric]) 6504 6505lemma cosh_diff: "cosh (x - y) = cosh x * cosh y - sinh x * sinh y" 6506 by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric]) 6507 6508lemma tanh_add: 6509 "cosh x \<noteq> 0 \<Longrightarrow> cosh y \<noteq> 0 \<Longrightarrow> tanh (x + y) = (tanh x + tanh y) / (1 + tanh x * tanh y)" 6510 by (simp add: tanh_def sinh_add cosh_add divide_simps) 6511 6512lemma sinh_double: "sinh (2 * x) = 2 * sinh x * cosh x" 6513 using sinh_add[of x] by simp 6514 6515lemma cosh_double: "cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2" 6516 using cosh_add[of x] by (simp add: power2_eq_square) 6517 6518end 6519 6520lemma sinh_field_def: "sinh z = (exp z - exp (-z)) / (2 :: 'a :: {banach, real_normed_field})" 6521 by (simp add: sinh_def scaleR_conv_of_real) 6522 6523lemma cosh_field_def: "cosh z = (exp z + exp (-z)) / (2 :: 'a :: {banach, real_normed_field})" 6524 by (simp add: cosh_def scaleR_conv_of_real) 6525 6526 6527subsubsection \<open>More specific properties of the real functions\<close> 6528 6529lemma sinh_real_zero_iff [simp]: "sinh (x::real) = 0 \<longleftrightarrow> x = 0" 6530proof - 6531 have "(-1 :: real) < 0" by simp 6532 also have "0 < exp x" by simp 6533 finally have "exp x \<noteq> -1" by (intro notI) simp 6534 thus ?thesis by (subst sinh_zero_iff) simp 6535qed 6536 6537lemma plus_inverse_ge_2: 6538 fixes x :: real 6539 assumes "x > 0" 6540 shows "x + inverse x \<ge> 2" 6541proof - 6542 have "0 \<le> (x - 1) ^ 2" by simp 6543 also have "\<dots> = x^2 - 2*x + 1" by (simp add: power2_eq_square algebra_simps) 6544 finally show ?thesis using assms by (simp add: field_simps power2_eq_square) 6545qed 6546 6547lemma sinh_real_nonneg_iff [simp]: "sinh (x :: real) \<ge> 0 \<longleftrightarrow> x \<ge> 0" 6548 by (simp add: sinh_def) 6549 6550lemma sinh_real_pos_iff [simp]: "sinh (x :: real) > 0 \<longleftrightarrow> x > 0" 6551 by (simp add: sinh_def) 6552 6553lemma sinh_real_nonpos_iff [simp]: "sinh (x :: real) \<le> 0 \<longleftrightarrow> x \<le> 0" 6554 by (simp add: sinh_def) 6555 6556lemma sinh_real_neg_iff [simp]: "sinh (x :: real) < 0 \<longleftrightarrow> x < 0" 6557 by (simp add: sinh_def) 6558 6559lemma cosh_real_ge_1: "cosh (x :: real) \<ge> 1" 6560 using plus_inverse_ge_2[of "exp x"] by (simp add: cosh_def exp_minus) 6561 6562lemma cosh_real_pos [simp]: "cosh (x :: real) > 0" 6563 using cosh_real_ge_1[of x] by simp 6564 6565lemma cosh_real_nonneg[simp]: "cosh (x :: real) \<ge> 0" 6566 using cosh_real_ge_1[of x] by simp 6567 6568lemma cosh_real_nonzero [simp]: "cosh (x :: real) \<noteq> 0" 6569 using cosh_real_ge_1[of x] by simp 6570 6571lemma tanh_real_nonneg_iff [simp]: "tanh (x :: real) \<ge> 0 \<longleftrightarrow> x \<ge> 0" 6572 by (simp add: tanh_def field_simps) 6573 6574lemma tanh_real_pos_iff [simp]: "tanh (x :: real) > 0 \<longleftrightarrow> x > 0" 6575 by (simp add: tanh_def field_simps) 6576 6577lemma tanh_real_nonpos_iff [simp]: "tanh (x :: real) \<le> 0 \<longleftrightarrow> x \<le> 0" 6578 by (simp add: tanh_def field_simps) 6579 6580lemma tanh_real_neg_iff [simp]: "tanh (x :: real) < 0 \<longleftrightarrow> x < 0" 6581 by (simp add: tanh_def field_simps) 6582 6583lemma tanh_real_zero_iff [simp]: "tanh (x :: real) = 0 \<longleftrightarrow> x = 0" 6584 by (simp add: tanh_def field_simps) 6585 6586lemma arsinh_real_def: "arsinh (x::real) = ln (x + sqrt (x^2 + 1))" 6587 by (simp add: arsinh_def powr_half_sqrt) 6588 6589lemma arcosh_real_def: "x \<ge> 1 \<Longrightarrow> arcosh (x::real) = ln (x + sqrt (x^2 - 1))" 6590 by (simp add: arcosh_def powr_half_sqrt) 6591 6592lemma arsinh_real_aux: "0 < x + sqrt (x ^ 2 + 1 :: real)" 6593proof (cases "x < 0") 6594 case True 6595 have "(-x) ^ 2 = x ^ 2" by simp 6596 also have "x ^ 2 < x ^ 2 + 1" by simp 6597 finally have "sqrt ((-x) ^ 2) < sqrt (x ^ 2 + 1)" 6598 by (rule real_sqrt_less_mono) 6599 thus ?thesis using True by simp 6600qed (auto simp: add_nonneg_pos) 6601 6602lemma arsinh_minus_real [simp]: "arsinh (-x::real) = -arsinh x" 6603proof - 6604 have "arsinh (-x) = ln (sqrt (x\<^sup>2 + 1) - x)" 6605 by (simp add: arsinh_real_def) 6606 also have "sqrt (x^2 + 1) - x = inverse (sqrt (x^2 + 1) + x)" 6607 using arsinh_real_aux[of x] by (simp add: divide_simps algebra_simps power2_eq_square) 6608 also have "ln \<dots> = -arsinh x" 6609 using arsinh_real_aux[of x] by (simp add: arsinh_real_def ln_inverse) 6610 finally show ?thesis . 6611qed 6612 6613lemma artanh_minus_real [simp]: 6614 assumes "abs x < 1" 6615 shows "artanh (-x::real) = -artanh x" 6616 using assms by (simp add: artanh_def ln_div field_simps) 6617 6618lemma sinh_less_cosh_real: "sinh (x :: real) < cosh x" 6619 by (simp add: sinh_def cosh_def) 6620 6621lemma sinh_le_cosh_real: "sinh (x :: real) \<le> cosh x" 6622 by (simp add: sinh_def cosh_def) 6623 6624lemma tanh_real_lt_1: "tanh (x :: real) < 1" 6625 by (simp add: tanh_def sinh_less_cosh_real) 6626 6627lemma tanh_real_gt_neg1: "tanh (x :: real) > -1" 6628proof - 6629 have "- cosh x < sinh x" by (simp add: sinh_def cosh_def divide_simps) 6630 thus ?thesis by (simp add: tanh_def field_simps) 6631qed 6632 6633lemma tanh_real_bounds: "tanh (x :: real) \<in> {-1<..<1}" 6634 using tanh_real_lt_1 tanh_real_gt_neg1 by simp 6635 6636context 6637 fixes x :: real 6638begin 6639 6640lemma arsinh_sinh_real: "arsinh (sinh x) = x" 6641 by (simp add: arsinh_real_def powr_def sinh_square_eq sinh_plus_cosh) 6642 6643lemma arcosh_cosh_real: "x \<ge> 0 \<Longrightarrow> arcosh (cosh x) = x" 6644 by (simp add: arcosh_real_def powr_def cosh_square_eq cosh_real_ge_1 cosh_plus_sinh) 6645 6646lemma artanh_tanh_real: "artanh (tanh x) = x" 6647proof - 6648 have "artanh (tanh x) = ln (cosh x * (cosh x + sinh x) / (cosh x * (cosh x - sinh x))) / 2" 6649 by (simp add: artanh_def tanh_def divide_simps) 6650 also have "cosh x * (cosh x + sinh x) / (cosh x * (cosh x - sinh x)) = 6651 (cosh x + sinh x) / (cosh x - sinh x)" by simp 6652 also have "\<dots> = (exp x)^2" 6653 by (simp add: cosh_plus_sinh cosh_minus_sinh exp_minus field_simps power2_eq_square) 6654 also have "ln ((exp x)^2) / 2 = x" by (simp add: ln_realpow) 6655 finally show ?thesis . 6656qed 6657 6658end 6659 6660lemma sinh_real_strict_mono: "strict_mono (sinh :: real \<Rightarrow> real)" 6661 by (rule pos_deriv_imp_strict_mono derivative_intros)+ auto 6662 6663lemma cosh_real_strict_mono: 6664 assumes "0 \<le> x" and "x < (y::real)" 6665 shows "cosh x < cosh y" 6666proof - 6667 from assms have "\<exists>z>x. z < y \<and> cosh y - cosh x = (y - x) * sinh z" 6668 by (intro MVT2) (auto dest: connectedD_interval intro!: derivative_eq_intros) 6669 then obtain z where z: "z > x" "z < y" "cosh y - cosh x = (y - x) * sinh z" by blast 6670 note \<open>cosh y - cosh x = (y - x) * sinh z\<close> 6671 also from \<open>z > x\<close> and assms have "(y - x) * sinh z > 0" by (intro mult_pos_pos) auto 6672 finally show "cosh x < cosh y" by simp 6673qed 6674 6675lemma tanh_real_strict_mono: "strict_mono (tanh :: real \<Rightarrow> real)" 6676proof - 6677 have *: "tanh x ^ 2 < 1" for x :: real 6678 using tanh_real_bounds[of x] by (simp add: abs_square_less_1 abs_if) 6679 show ?thesis 6680 by (rule pos_deriv_imp_strict_mono) (insert *, auto intro!: derivative_intros) 6681qed 6682 6683lemma sinh_real_abs [simp]: "sinh (abs x :: real) = abs (sinh x)" 6684 by (simp add: abs_if) 6685 6686lemma cosh_real_abs [simp]: "cosh (abs x :: real) = cosh x" 6687 by (simp add: abs_if) 6688 6689lemma tanh_real_abs [simp]: "tanh (abs x :: real) = abs (tanh x)" 6690 by (auto simp: abs_if) 6691 6692lemma sinh_real_eq_iff [simp]: "sinh x = sinh y \<longleftrightarrow> x = (y :: real)" 6693 using sinh_real_strict_mono by (simp add: strict_mono_eq) 6694 6695lemma tanh_real_eq_iff [simp]: "tanh x = tanh y \<longleftrightarrow> x = (y :: real)" 6696 using tanh_real_strict_mono by (simp add: strict_mono_eq) 6697 6698lemma cosh_real_eq_iff [simp]: "cosh x = cosh y \<longleftrightarrow> abs x = abs (y :: real)" 6699proof - 6700 have "cosh x = cosh y \<longleftrightarrow> x = y" if "x \<ge> 0" "y \<ge> 0" for x y :: real 6701 using cosh_real_strict_mono[of x y] cosh_real_strict_mono[of y x] that 6702 by (cases x y rule: linorder_cases) auto 6703 from this[of "abs x" "abs y"] show ?thesis by simp 6704qed 6705 6706lemma sinh_real_le_iff [simp]: "sinh x \<le> sinh y \<longleftrightarrow> x \<le> (y::real)" 6707 using sinh_real_strict_mono by (simp add: strict_mono_less_eq) 6708 6709lemma cosh_real_nonneg_le_iff: "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> cosh x \<le> cosh y \<longleftrightarrow> x \<le> (y::real)" 6710 using cosh_real_strict_mono[of x y] cosh_real_strict_mono[of y x] 6711 by (cases x y rule: linorder_cases) auto 6712 6713lemma cosh_real_nonpos_le_iff: "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> cosh x \<le> cosh y \<longleftrightarrow> x \<ge> (y::real)" 6714 using cosh_real_nonneg_le_iff[of "-x" "-y"] by simp 6715 6716lemma tanh_real_le_iff [simp]: "tanh x \<le> tanh y \<longleftrightarrow> x \<le> (y::real)" 6717 using tanh_real_strict_mono by (simp add: strict_mono_less_eq) 6718 6719 6720lemma sinh_real_less_iff [simp]: "sinh x < sinh y \<longleftrightarrow> x < (y::real)" 6721 using sinh_real_strict_mono by (simp add: strict_mono_less) 6722 6723lemma cosh_real_nonneg_less_iff: "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> cosh x < cosh y \<longleftrightarrow> x < (y::real)" 6724 using cosh_real_strict_mono[of x y] cosh_real_strict_mono[of y x] 6725 by (cases x y rule: linorder_cases) auto 6726 6727lemma cosh_real_nonpos_less_iff: "x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> cosh x < cosh y \<longleftrightarrow> x > (y::real)" 6728 using cosh_real_nonneg_less_iff[of "-x" "-y"] by simp 6729 6730lemma tanh_real_less_iff [simp]: "tanh x < tanh y \<longleftrightarrow> x < (y::real)" 6731 using tanh_real_strict_mono by (simp add: strict_mono_less) 6732 6733 6734subsubsection \<open>Limits\<close> 6735 6736lemma sinh_real_at_top: "filterlim (sinh :: real \<Rightarrow> real) at_top at_top" 6737proof - 6738 have *: "((\<lambda>x. - exp (- x)) \<longlongrightarrow> (-0::real)) at_top" 6739 by (intro tendsto_minus filterlim_compose[OF exp_at_bot] filterlim_uminus_at_bot_at_top) 6740 have "filterlim (\<lambda>x. (1 / 2) * (-exp (-x) + exp x) :: real) at_top at_top" 6741 by (rule filterlim_tendsto_pos_mult_at_top[OF _ _ 6742 filterlim_tendsto_add_at_top[OF *]] tendsto_const)+ (auto simp: exp_at_top) 6743 also have "(\<lambda>x. (1 / 2) * (-exp (-x) + exp x) :: real) = sinh" 6744 by (simp add: fun_eq_iff sinh_def) 6745 finally show ?thesis . 6746qed 6747 6748lemma sinh_real_at_bot: "filterlim (sinh :: real \<Rightarrow> real) at_bot at_bot" 6749proof - 6750 have "filterlim (\<lambda>x. -sinh x :: real) at_bot at_top" 6751 by (simp add: filterlim_uminus_at_top [symmetric] sinh_real_at_top) 6752 also have "(\<lambda>x. -sinh x :: real) = (\<lambda>x. sinh (-x))" by simp 6753 finally show ?thesis by (subst filterlim_at_bot_mirror) 6754qed 6755 6756lemma cosh_real_at_top: "filterlim (cosh :: real \<Rightarrow> real) at_top at_top" 6757proof - 6758 have *: "((\<lambda>x. exp (- x)) \<longlongrightarrow> (0::real)) at_top" 6759 by (intro filterlim_compose[OF exp_at_bot] filterlim_uminus_at_bot_at_top) 6760 have "filterlim (\<lambda>x. (1 / 2) * (exp (-x) + exp x) :: real) at_top at_top" 6761 by (rule filterlim_tendsto_pos_mult_at_top[OF _ _ 6762 filterlim_tendsto_add_at_top[OF *]] tendsto_const)+ (auto simp: exp_at_top) 6763 also have "(\<lambda>x. (1 / 2) * (exp (-x) + exp x) :: real) = cosh" 6764 by (simp add: fun_eq_iff cosh_def) 6765 finally show ?thesis . 6766qed 6767 6768lemma cosh_real_at_bot: "filterlim (cosh :: real \<Rightarrow> real) at_top at_bot" 6769proof - 6770 have "filterlim (\<lambda>x. cosh (-x) :: real) at_top at_top" 6771 by (simp add: cosh_real_at_top) 6772 thus ?thesis by (subst filterlim_at_bot_mirror) 6773qed 6774 6775lemma tanh_real_at_top: "(tanh \<longlongrightarrow> (1::real)) at_top" 6776proof - 6777 have "((\<lambda>x::real. (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))) \<longlongrightarrow> (1 - 0) / (1 + 0)) at_top" 6778 by (intro tendsto_intros filterlim_compose[OF exp_at_bot] 6779 filterlim_tendsto_neg_mult_at_bot[OF tendsto_const] filterlim_ident) auto 6780 also have "(\<lambda>x::real. (1 - exp (- 2 * x)) / (1 + exp (- 2 * x))) = tanh" 6781 by (rule ext) (simp add: tanh_real_altdef) 6782 finally show ?thesis by simp 6783qed 6784 6785lemma tanh_real_at_bot: "(tanh \<longlongrightarrow> (-1::real)) at_bot" 6786proof - 6787 have "((\<lambda>x::real. -tanh x) \<longlongrightarrow> -1) at_top" 6788 by (intro tendsto_minus tanh_real_at_top) 6789 also have "(\<lambda>x. -tanh x :: real) = (\<lambda>x. tanh (-x))" by simp 6790 finally show ?thesis by (subst filterlim_at_bot_mirror) 6791qed 6792 6793 6794subsubsection \<open>Properties of the inverse hyperbolic functions\<close> 6795 6796lemma isCont_sinh: "isCont sinh (x :: 'a :: {real_normed_field, banach})" 6797 unfolding sinh_def [abs_def] by (auto intro!: continuous_intros) 6798 6799lemma isCont_cosh: "isCont cosh (x :: 'a :: {real_normed_field, banach})" 6800 unfolding cosh_def [abs_def] by (auto intro!: continuous_intros) 6801 6802lemma isCont_tanh: "cosh x \<noteq> 0 \<Longrightarrow> isCont tanh (x :: 'a :: {real_normed_field, banach})" 6803 unfolding tanh_def [abs_def] 6804 by (auto intro!: continuous_intros isCont_divide isCont_sinh isCont_cosh) 6805 6806lemma continuous_on_sinh [continuous_intros]: 6807 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6808 assumes "continuous_on A f" 6809 shows "continuous_on A (\<lambda>x. sinh (f x))" 6810 unfolding sinh_def using assms by (intro continuous_intros) 6811 6812lemma continuous_on_cosh [continuous_intros]: 6813 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6814 assumes "continuous_on A f" 6815 shows "continuous_on A (\<lambda>x. cosh (f x))" 6816 unfolding cosh_def using assms by (intro continuous_intros) 6817 6818lemma continuous_sinh [continuous_intros]: 6819 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6820 assumes "continuous F f" 6821 shows "continuous F (\<lambda>x. sinh (f x))" 6822 unfolding sinh_def using assms by (intro continuous_intros) 6823 6824lemma continuous_cosh [continuous_intros]: 6825 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6826 assumes "continuous F f" 6827 shows "continuous F (\<lambda>x. cosh (f x))" 6828 unfolding cosh_def using assms by (intro continuous_intros) 6829 6830lemma continuous_on_tanh [continuous_intros]: 6831 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6832 assumes "continuous_on A f" "\<And>x. x \<in> A \<Longrightarrow> cosh (f x) \<noteq> 0" 6833 shows "continuous_on A (\<lambda>x. tanh (f x))" 6834 unfolding tanh_def using assms by (intro continuous_intros) auto 6835 6836lemma continuous_at_within_tanh [continuous_intros]: 6837 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6838 assumes "continuous (at x within A) f" "cosh (f x) \<noteq> 0" 6839 shows "continuous (at x within A) (\<lambda>x. tanh (f x))" 6840 unfolding tanh_def using assms by (intro continuous_intros continuous_divide) auto 6841 6842lemma continuous_tanh [continuous_intros]: 6843 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6844 assumes "continuous F f" "cosh (f (Lim F (\<lambda>x. x))) \<noteq> 0" 6845 shows "continuous F (\<lambda>x. tanh (f x))" 6846 unfolding tanh_def using assms by (intro continuous_intros continuous_divide) auto 6847 6848lemma tendsto_sinh [tendsto_intros]: 6849 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6850 shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. sinh (f x)) \<longlongrightarrow> sinh a) F" 6851 by (rule isCont_tendsto_compose [OF isCont_sinh]) 6852 6853lemma tendsto_cosh [tendsto_intros]: 6854 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6855 shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. cosh (f x)) \<longlongrightarrow> cosh a) F" 6856 by (rule isCont_tendsto_compose [OF isCont_cosh]) 6857 6858lemma tendsto_tanh [tendsto_intros]: 6859 fixes f :: "_ \<Rightarrow>'a::{real_normed_field,banach}" 6860 shows "(f \<longlongrightarrow> a) F \<Longrightarrow> cosh a \<noteq> 0 \<Longrightarrow> ((\<lambda>x. tanh (f x)) \<longlongrightarrow> tanh a) F" 6861 by (rule isCont_tendsto_compose [OF isCont_tanh]) 6862 6863 6864lemma arsinh_real_has_field_derivative [derivative_intros]: 6865 fixes x :: real 6866 shows "(arsinh has_field_derivative (1 / (sqrt (x ^ 2 + 1)))) (at x within A)" 6867proof - 6868 have pos: "1 + x ^ 2 > 0" by (intro add_pos_nonneg) auto 6869 from pos arsinh_real_aux[of x] show ?thesis unfolding arsinh_def [abs_def] 6870 by (auto intro!: derivative_eq_intros simp: powr_minus powr_half_sqrt divide_simps) 6871qed 6872 6873lemma arcosh_real_has_field_derivative [derivative_intros]: 6874 fixes x :: real 6875 assumes "x > 1" 6876 shows "(arcosh has_field_derivative (1 / (sqrt (x ^ 2 - 1)))) (at x within A)" 6877proof - 6878 from assms have "x + sqrt (x\<^sup>2 - 1) > 0" by (simp add: add_pos_pos) 6879 thus ?thesis using assms unfolding arcosh_def [abs_def] 6880 by (auto intro!: derivative_eq_intros 6881 simp: powr_minus powr_half_sqrt divide_simps power2_eq_1_iff) 6882qed 6883 6884lemma artanh_real_has_field_derivative [derivative_intros]: 6885 fixes x :: real 6886 assumes "abs x < 1" 6887 shows "(artanh has_field_derivative (1 / (1 - x ^ 2))) (at x within A)" 6888proof - 6889 from assms have "x > -1" "x < 1" by linarith+ 6890 hence "(artanh has_field_derivative (4 - 4 * x) / ((1 + x) * (1 - x) * (1 - x) * 4)) 6891 (at x within A)" unfolding artanh_def [abs_def] 6892 by (auto intro!: derivative_eq_intros simp: powr_minus powr_half_sqrt) 6893 also have "(4 - 4 * x) / ((1 + x) * (1 - x) * (1 - x) * 4) = 1 / ((1 + x) * (1 - x))" 6894 by (simp add: divide_simps) 6895 also have "(1 + x) * (1 - x) = 1 - x ^ 2" by (simp add: algebra_simps power2_eq_square) 6896 finally show ?thesis . 6897qed 6898 6899lemma continuous_on_arsinh [continuous_intros]: "continuous_on A (arsinh :: real \<Rightarrow> real)" 6900 by (rule DERIV_continuous_on derivative_intros)+ 6901 6902lemma continuous_on_arcosh [continuous_intros]: 6903 assumes "A \<subseteq> {1..}" 6904 shows "continuous_on A (arcosh :: real \<Rightarrow> real)" 6905proof - 6906 have pos: "x + sqrt (x ^ 2 - 1) > 0" if "x \<ge> 1" for x 6907 using that by (intro add_pos_nonneg) auto 6908 show ?thesis 6909 unfolding arcosh_def [abs_def] 6910 by (intro continuous_on_subset [OF _ assms] continuous_on_ln continuous_on_add 6911 continuous_on_id continuous_on_powr') 6912 (auto dest: pos simp: powr_half_sqrt intro!: continuous_intros) 6913qed 6914 6915lemma continuous_on_artanh [continuous_intros]: 6916 assumes "A \<subseteq> {-1<..<1}" 6917 shows "continuous_on A (artanh :: real \<Rightarrow> real)" 6918 unfolding artanh_def [abs_def] 6919 by (intro continuous_on_subset [OF _ assms]) (auto intro!: continuous_intros) 6920 6921lemma continuous_on_arsinh' [continuous_intros]: 6922 fixes f :: "real \<Rightarrow> real" 6923 assumes "continuous_on A f" 6924 shows "continuous_on A (\<lambda>x. arsinh (f x))" 6925 by (rule continuous_on_compose2[OF continuous_on_arsinh assms]) auto 6926 6927lemma continuous_on_arcosh' [continuous_intros]: 6928 fixes f :: "real \<Rightarrow> real" 6929 assumes "continuous_on A f" "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 1" 6930 shows "continuous_on A (\<lambda>x. arcosh (f x))" 6931 by (rule continuous_on_compose2[OF continuous_on_arcosh assms(1) order.refl]) 6932 (use assms(2) in auto) 6933 6934lemma continuous_on_artanh' [continuous_intros]: 6935 fixes f :: "real \<Rightarrow> real" 6936 assumes "continuous_on A f" "\<And>x. x \<in> A \<Longrightarrow> f x \<in> {-1<..<1}" 6937 shows "continuous_on A (\<lambda>x. artanh (f x))" 6938 by (rule continuous_on_compose2[OF continuous_on_artanh assms(1) order.refl]) 6939 (use assms(2) in auto) 6940 6941lemma isCont_arsinh [continuous_intros]: "isCont arsinh (x :: real)" 6942 using continuous_on_arsinh[of UNIV] by (auto simp: continuous_on_eq_continuous_at) 6943 6944lemma isCont_arcosh [continuous_intros]: 6945 assumes "x > 1" 6946 shows "isCont arcosh (x :: real)" 6947proof - 6948 have "continuous_on {1::real<..} arcosh" 6949 by (rule continuous_on_arcosh) auto 6950 with assms show ?thesis by (auto simp: continuous_on_eq_continuous_at) 6951qed 6952 6953lemma isCont_artanh [continuous_intros]: 6954 assumes "x > -1" "x < 1" 6955 shows "isCont artanh (x :: real)" 6956proof - 6957 have "continuous_on {-1<..<(1::real)} artanh" 6958 by (rule continuous_on_artanh) auto 6959 with assms show ?thesis by (auto simp: continuous_on_eq_continuous_at) 6960qed 6961 6962lemma tendsto_arsinh [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. arsinh (f x)) \<longlongrightarrow> arsinh a) F" 6963 for f :: "_ \<Rightarrow> real" 6964 by (rule isCont_tendsto_compose [OF isCont_arsinh]) 6965 6966lemma tendsto_arcosh_strong [tendsto_intros]: 6967 fixes f :: "_ \<Rightarrow> real" 6968 assumes "(f \<longlongrightarrow> a) F" "a \<ge> 1" "eventually (\<lambda>x. f x \<ge> 1) F" 6969 shows "((\<lambda>x. arcosh (f x)) \<longlongrightarrow> arcosh a) F" 6970 by (rule continuous_on_tendsto_compose[OF continuous_on_arcosh[OF order.refl]]) 6971 (use assms in auto) 6972 6973lemma tendsto_arcosh: 6974 fixes f :: "_ \<Rightarrow> real" 6975 assumes "(f \<longlongrightarrow> a) F" "a > 1" 6976 shows "((\<lambda>x. arcosh (f x)) \<longlongrightarrow> arcosh a) F" 6977 by (rule isCont_tendsto_compose [OF isCont_arcosh]) (use assms in auto) 6978 6979lemma tendsto_arcosh_at_left_1: "(arcosh \<longlongrightarrow> 0) (at_right (1::real))" 6980proof - 6981 have "(arcosh \<longlongrightarrow> arcosh 1) (at_right (1::real))" 6982 by (rule tendsto_arcosh_strong) (auto simp: eventually_at intro!: exI[of _ 1]) 6983 thus ?thesis by simp 6984qed 6985 6986lemma tendsto_artanh [tendsto_intros]: 6987 fixes f :: "'a \<Rightarrow> real" 6988 assumes "(f \<longlongrightarrow> a) F" "a > -1" "a < 1" 6989 shows "((\<lambda>x. artanh (f x)) \<longlongrightarrow> artanh a) F" 6990 by (rule isCont_tendsto_compose [OF isCont_artanh]) (use assms in auto) 6991 6992lemma continuous_arsinh [continuous_intros]: 6993 "continuous F f \<Longrightarrow> continuous F (\<lambda>x. arsinh (f x :: real))" 6994 unfolding continuous_def by (rule tendsto_arsinh) 6995 6996(* TODO: This rule does not work for one-sided continuity at 1 *) 6997lemma continuous_arcosh_strong [continuous_intros]: 6998 assumes "continuous F f" "eventually (\<lambda>x. f x \<ge> 1) F" 6999 shows "continuous F (\<lambda>x. arcosh (f x :: real))" 7000proof (cases "F = bot") 7001 case False 7002 show ?thesis 7003 unfolding continuous_def 7004 proof (intro tendsto_arcosh_strong) 7005 show "1 \<le> f (Lim F (\<lambda>x. x))" 7006 using assms False unfolding continuous_def by (rule tendsto_lowerbound) 7007 qed (insert assms, auto simp: continuous_def) 7008qed auto 7009 7010lemma continuous_arcosh: 7011 "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) > 1 \<Longrightarrow> continuous F (\<lambda>x. arcosh (f x :: real))" 7012 unfolding continuous_def by (rule tendsto_arcosh) auto 7013 7014lemma continuous_artanh [continuous_intros]: 7015 "continuous F f \<Longrightarrow> f (Lim F (\<lambda>x. x)) \<in> {-1<..<1} \<Longrightarrow> continuous F (\<lambda>x. artanh (f x :: real))" 7016 unfolding continuous_def by (rule tendsto_artanh) auto 7017 7018lemma arsinh_real_at_top: 7019 "filterlim (arsinh :: real \<Rightarrow> real) at_top at_top" 7020proof (subst filterlim_cong[OF refl refl]) 7021 show "filterlim (\<lambda>x. ln (x + sqrt (1 + x\<^sup>2))) at_top at_top" 7022 by (intro filterlim_compose[OF ln_at_top filterlim_at_top_add_at_top] filterlim_ident 7023 filterlim_compose[OF sqrt_at_top] filterlim_tendsto_add_at_top[OF tendsto_const] 7024 filterlim_pow_at_top) auto 7025qed (auto intro!: eventually_mono[OF eventually_ge_at_top[of 1]] simp: arsinh_real_def add_ac) 7026 7027lemma arsinh_real_at_bot: 7028 "filterlim (arsinh :: real \<Rightarrow> real) at_bot at_bot" 7029proof - 7030 have "filterlim (\<lambda>x::real. -arsinh x) at_bot at_top" 7031 by (subst filterlim_uminus_at_top [symmetric]) (rule arsinh_real_at_top) 7032 also have "(\<lambda>x::real. -arsinh x) = (\<lambda>x. arsinh (-x))" by simp 7033 finally show ?thesis 7034 by (subst filterlim_at_bot_mirror) 7035qed 7036 7037lemma arcosh_real_at_top: 7038 "filterlim (arcosh :: real \<Rightarrow> real) at_top at_top" 7039proof (subst filterlim_cong[OF refl refl]) 7040 show "filterlim (\<lambda>x. ln (x + sqrt (-1 + x\<^sup>2))) at_top at_top" 7041 by (intro filterlim_compose[OF ln_at_top filterlim_at_top_add_at_top] filterlim_ident 7042 filterlim_compose[OF sqrt_at_top] filterlim_tendsto_add_at_top[OF tendsto_const] 7043 filterlim_pow_at_top) auto 7044qed (auto intro!: eventually_mono[OF eventually_ge_at_top[of 1]] simp: arcosh_real_def) 7045 7046lemma artanh_real_at_left_1: 7047 "filterlim (artanh :: real \<Rightarrow> real) at_top (at_left 1)" 7048proof - 7049 have *: "filterlim (\<lambda>x::real. (1 + x) / (1 - x)) at_top (at_left 1)" 7050 by (rule LIM_at_top_divide) 7051 (auto intro!: tendsto_eq_intros eventually_mono[OF eventually_at_left_real[of 0]]) 7052 have "filterlim (\<lambda>x::real. (1/2) * ln ((1 + x) / (1 - x))) at_top (at_left 1)" 7053 by (intro filterlim_tendsto_pos_mult_at_top[OF tendsto_const] * 7054 filterlim_compose[OF ln_at_top]) auto 7055 also have "(\<lambda>x::real. (1/2) * ln ((1 + x) / (1 - x))) = artanh" 7056 by (simp add: artanh_def [abs_def]) 7057 finally show ?thesis . 7058qed 7059 7060lemma artanh_real_at_right_1: 7061 "filterlim (artanh :: real \<Rightarrow> real) at_bot (at_right (-1))" 7062proof - 7063 have "?thesis \<longleftrightarrow> filterlim (\<lambda>x::real. -artanh x) at_top (at_right (-1))" 7064 by (simp add: filterlim_uminus_at_bot) 7065 also have "\<dots> \<longleftrightarrow> filterlim (\<lambda>x::real. artanh (-x)) at_top (at_right (-1))" 7066 by (intro filterlim_cong refl eventually_mono[OF eventually_at_right_real[of "-1" "1"]]) auto 7067 also have "\<dots> \<longleftrightarrow> filterlim (artanh :: real \<Rightarrow> real) at_top (at_left 1)" 7068 by (simp add: filterlim_at_left_to_right) 7069 also have \<dots> by (rule artanh_real_at_left_1) 7070 finally show ?thesis . 7071qed 7072 7073 7074subsection \<open>Simprocs for root and power literals\<close> 7075 7076lemma numeral_powr_numeral_real [simp]: 7077 "numeral m powr numeral n = (numeral m ^ numeral n :: real)" 7078 by (simp add: powr_numeral) 7079 7080context 7081begin 7082 7083private lemma sqrt_numeral_simproc_aux: 7084 assumes "m * m \<equiv> n" 7085 shows "sqrt (numeral n :: real) \<equiv> numeral m" 7086proof - 7087 have "numeral n \<equiv> numeral m * (numeral m :: real)" by (simp add: assms [symmetric]) 7088 moreover have "sqrt \<dots> \<equiv> numeral m" by (subst real_sqrt_abs2) simp 7089 ultimately show "sqrt (numeral n :: real) \<equiv> numeral m" by simp 7090qed 7091 7092private lemma root_numeral_simproc_aux: 7093 assumes "Num.pow m n \<equiv> x" 7094 shows "root (numeral n) (numeral x :: real) \<equiv> numeral m" 7095 by (subst assms [symmetric], subst numeral_pow, subst real_root_pos2) simp_all 7096 7097private lemma powr_numeral_simproc_aux: 7098 assumes "Num.pow y n = x" 7099 shows "numeral x powr (m / numeral n :: real) \<equiv> numeral y powr m" 7100 by (subst assms [symmetric], subst numeral_pow, subst powr_numeral [symmetric]) 7101 (simp, subst powr_powr, simp_all) 7102 7103private lemma numeral_powr_inverse_eq: 7104 "numeral x powr (inverse (numeral n)) = numeral x powr (1 / numeral n :: real)" 7105 by simp 7106 7107 7108ML \<open> 7109 7110signature ROOT_NUMERAL_SIMPROC = sig 7111 7112val sqrt : int option -> int -> int option 7113val sqrt' : int option -> int -> int option 7114val nth_root : int option -> int -> int -> int option 7115val nth_root' : int option -> int -> int -> int option 7116val sqrt_simproc : Proof.context -> cterm -> thm option 7117val root_simproc : int * int -> Proof.context -> cterm -> thm option 7118val powr_simproc : int * int -> Proof.context -> cterm -> thm option 7119 7120end 7121 7122structure Root_Numeral_Simproc : ROOT_NUMERAL_SIMPROC = struct 7123 7124fun iterate NONE p f x = 7125 let 7126 fun go x = if p x then x else go (f x) 7127 in 7128 SOME (go x) 7129 end 7130 | iterate (SOME threshold) p f x = 7131 let 7132 fun go (threshold, x) = 7133 if p x then SOME x else if threshold = 0 then NONE else go (threshold - 1, f x) 7134 in 7135 go (threshold, x) 7136 end 7137 7138 7139fun nth_root _ 1 x = SOME x 7140 | nth_root _ _ 0 = SOME 0 7141 | nth_root _ _ 1 = SOME 1 7142 | nth_root threshold n x = 7143 let 7144 fun newton_step y = ((n - 1) * y + x div Integer.pow (n - 1) y) div n 7145 fun is_root y = Integer.pow n y <= x andalso x < Integer.pow n (y + 1) 7146 in 7147 if x < n then 7148 SOME 1 7149 else if x < Integer.pow n 2 then 7150 SOME 1 7151 else 7152 let 7153 val y = Real.floor (Math.pow (Real.fromInt x, Real.fromInt 1 / Real.fromInt n)) 7154 in 7155 if is_root y then 7156 SOME y 7157 else 7158 iterate threshold is_root newton_step ((x + n - 1) div n) 7159 end 7160 end 7161 7162fun nth_root' _ 1 x = SOME x 7163 | nth_root' _ _ 0 = SOME 0 7164 | nth_root' _ _ 1 = SOME 1 7165 | nth_root' threshold n x = if x < n then NONE else if x < Integer.pow n 2 then NONE else 7166 case nth_root threshold n x of 7167 NONE => NONE 7168 | SOME y => if Integer.pow n y = x then SOME y else NONE 7169 7170fun sqrt _ 0 = SOME 0 7171 | sqrt _ 1 = SOME 1 7172 | sqrt threshold n = 7173 let 7174 fun aux (a, b) = if n >= b * b then aux (b, b * b) else (a, b) 7175 val (lower_root, lower_n) = aux (1, 2) 7176 fun newton_step x = (x + n div x) div 2 7177 fun is_sqrt r = r*r <= n andalso n < (r+1)*(r+1) 7178 val y = Real.floor (Math.sqrt (Real.fromInt n)) 7179 in 7180 if is_sqrt y then 7181 SOME y 7182 else 7183 Option.mapPartial (iterate threshold is_sqrt newton_step o (fn x => x * lower_root)) 7184 (sqrt threshold (n div lower_n)) 7185 end 7186 7187fun sqrt' threshold x = 7188 case sqrt threshold x of 7189 NONE => NONE 7190 | SOME y => if y * y = x then SOME y else NONE 7191 7192fun sqrt_simproc ctxt ct = 7193 let 7194 val n = ct |> Thm.term_of |> dest_comb |> snd |> dest_comb |> snd |> HOLogic.dest_numeral 7195 in 7196 case sqrt' (SOME 10000) n of 7197 NONE => NONE 7198 | SOME m => 7199 SOME (Thm.instantiate' [] (map (SOME o Thm.cterm_of ctxt o HOLogic.mk_numeral) [m, n]) 7200 @{thm sqrt_numeral_simproc_aux}) 7201 end 7202 handle TERM _ => NONE 7203 7204fun root_simproc (threshold1, threshold2) ctxt ct = 7205 let 7206 val [n, x] = 7207 ct |> Thm.term_of |> strip_comb |> snd |> map (dest_comb #> snd #> HOLogic.dest_numeral) 7208 in 7209 if n > threshold1 orelse x > threshold2 then NONE else 7210 case nth_root' (SOME 100) n x of 7211 NONE => NONE 7212 | SOME m => 7213 SOME (Thm.instantiate' [] (map (SOME o Thm.cterm_of ctxt o HOLogic.mk_numeral) [m, n, x]) 7214 @{thm root_numeral_simproc_aux}) 7215 end 7216 handle TERM _ => NONE 7217 | Match => NONE 7218 7219fun powr_simproc (threshold1, threshold2) ctxt ct = 7220 let 7221 val eq_thm = Conv.try_conv (Conv.rewr_conv @{thm numeral_powr_inverse_eq}) ct 7222 val ct = Thm.dest_equals_rhs (Thm.cprop_of eq_thm) 7223 val (_, [x, t]) = strip_comb (Thm.term_of ct) 7224 val (_, [m, n]) = strip_comb t 7225 val [x, n] = map (dest_comb #> snd #> HOLogic.dest_numeral) [x, n] 7226 in 7227 if n > threshold1 orelse x > threshold2 then NONE else 7228 case nth_root' (SOME 100) n x of 7229 NONE => NONE 7230 | SOME y => 7231 let 7232 val [y, n, x] = map HOLogic.mk_numeral [y, n, x] 7233 val thm = Thm.instantiate' [] (map (SOME o Thm.cterm_of ctxt) [y, n, x, m]) 7234 @{thm powr_numeral_simproc_aux} 7235 in 7236 SOME (@{thm transitive} OF [eq_thm, thm]) 7237 end 7238 end 7239 handle TERM _ => NONE 7240 | Match => NONE 7241 7242end 7243\<close> 7244 7245end 7246 7247simproc_setup sqrt_numeral ("sqrt (numeral n)") = 7248 \<open>K Root_Numeral_Simproc.sqrt_simproc\<close> 7249 7250simproc_setup root_numeral ("root (numeral n) (numeral x)") = 7251 \<open>K (Root_Numeral_Simproc.root_simproc (200, Integer.pow 200 2))\<close> 7252 7253simproc_setup powr_divide_numeral 7254 ("numeral x powr (m / numeral n :: real)" | "numeral x powr (inverse (numeral n) :: real)") = 7255 \<open>K (Root_Numeral_Simproc.powr_simproc (200, Integer.pow 200 2))\<close> 7256 7257 7258lemma "root 100 1267650600228229401496703205376 = 2" 7259 by simp 7260 7261lemma "sqrt 196 = 14" 7262 by simp 7263 7264lemma "256 powr (7 / 4 :: real) = 16384" 7265 by simp 7266 7267lemma "27 powr (inverse 3) = (3::real)" 7268 by simp 7269 7270end 7271