src/HOL/Transcendental.thy
 author huffman Tue Jan 17 16:30:54 2012 +0100 (2012-01-17) changeset 46240 933f35c4e126 parent 45915 0e5a87b772f9 child 47108 2a1953f0d20d permissions -rw-r--r--
factor-cancellation simprocs now call the full simplifier to prove that factors are non-zero
 wenzelm@32960 ` 1` ```(* Title: HOL/Transcendental.thy ``` wenzelm@32960 ` 2` ``` Author: Jacques D. Fleuriot, University of Cambridge, University of Edinburgh ``` wenzelm@32960 ` 3` ``` Author: Lawrence C Paulson ``` paulson@12196 ` 4` ```*) ``` paulson@12196 ` 5` paulson@15077 ` 6` ```header{*Power Series, Transcendental Functions etc.*} ``` paulson@15077 ` 7` nipkow@15131 ` 8` ```theory Transcendental ``` haftmann@25600 ` 9` ```imports Fact Series Deriv NthRoot ``` nipkow@15131 ` 10` ```begin ``` paulson@15077 ` 11` huffman@29164 ` 12` ```subsection {* Properties of Power Series *} ``` paulson@15077 ` 13` huffman@23082 ` 14` ```lemma lemma_realpow_diff: ``` haftmann@31017 ` 15` ``` fixes y :: "'a::monoid_mult" ``` huffman@23082 ` 16` ``` shows "p \ n \ y ^ (Suc n - p) = (y ^ (n - p)) * y" ``` huffman@23082 ` 17` ```proof - ``` huffman@23082 ` 18` ``` assume "p \ n" ``` huffman@23082 ` 19` ``` hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le) ``` huffman@30273 ` 20` ``` thus ?thesis by (simp add: power_commutes) ``` huffman@23082 ` 21` ```qed ``` paulson@15077 ` 22` paulson@15077 ` 23` ```lemma lemma_realpow_diff_sumr: ``` haftmann@31017 ` 24` ``` fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows ``` hoelzl@41970 ` 25` ``` "(\p=0..p=0..p=0..p=0..p=0..z\ < \x\"}.*} ``` paulson@15077 ` 55` paulson@15077 ` 56` ```lemma powser_insidea: ``` huffman@44726 ` 57` ``` fixes x z :: "'a::real_normed_field" ``` huffman@20849 ` 58` ``` assumes 1: "summable (\n. f n * x ^ n)" ``` huffman@23082 ` 59` ``` assumes 2: "norm z < norm x" ``` huffman@23082 ` 60` ``` shows "summable (\n. norm (f n * z ^ n))" ``` huffman@20849 ` 61` ```proof - ``` huffman@20849 ` 62` ``` from 2 have x_neq_0: "x \ 0" by clarsimp ``` huffman@20849 ` 63` ``` from 1 have "(\n. f n * x ^ n) ----> 0" ``` huffman@20849 ` 64` ``` by (rule summable_LIMSEQ_zero) ``` huffman@20849 ` 65` ``` hence "convergent (\n. f n * x ^ n)" ``` huffman@20849 ` 66` ``` by (rule convergentI) ``` huffman@20849 ` 67` ``` hence "Cauchy (\n. f n * x ^ n)" ``` huffman@44726 ` 68` ``` by (rule convergent_Cauchy) ``` huffman@20849 ` 69` ``` hence "Bseq (\n. f n * x ^ n)" ``` huffman@20849 ` 70` ``` by (rule Cauchy_Bseq) ``` huffman@23082 ` 71` ``` then obtain K where 3: "0 < K" and 4: "\n. norm (f n * x ^ n) \ K" ``` huffman@20849 ` 72` ``` by (simp add: Bseq_def, safe) ``` huffman@23082 ` 73` ``` have "\N. \n\N. norm (norm (f n * z ^ n)) \ ``` huffman@23082 ` 74` ``` K * norm (z ^ n) * inverse (norm (x ^ n))" ``` huffman@20849 ` 75` ``` proof (intro exI allI impI) ``` huffman@20849 ` 76` ``` fix n::nat assume "0 \ n" ``` huffman@23082 ` 77` ``` have "norm (norm (f n * z ^ n)) * norm (x ^ n) = ``` huffman@23082 ` 78` ``` norm (f n * x ^ n) * norm (z ^ n)" ``` huffman@23082 ` 79` ``` by (simp add: norm_mult abs_mult) ``` huffman@23082 ` 80` ``` also have "\ \ K * norm (z ^ n)" ``` huffman@23082 ` 81` ``` by (simp only: mult_right_mono 4 norm_ge_zero) ``` huffman@23082 ` 82` ``` also have "\ = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))" ``` huffman@20849 ` 83` ``` by (simp add: x_neq_0) ``` huffman@23082 ` 84` ``` also have "\ = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)" ``` huffman@20849 ` 85` ``` by (simp only: mult_assoc) ``` huffman@23082 ` 86` ``` finally show "norm (norm (f n * z ^ n)) \ ``` huffman@23082 ` 87` ``` K * norm (z ^ n) * inverse (norm (x ^ n))" ``` huffman@20849 ` 88` ``` by (simp add: mult_le_cancel_right x_neq_0) ``` huffman@20849 ` 89` ``` qed ``` huffman@23082 ` 90` ``` moreover have "summable (\n. K * norm (z ^ n) * inverse (norm (x ^ n)))" ``` huffman@20849 ` 91` ``` proof - ``` huffman@23082 ` 92` ``` from 2 have "norm (norm (z * inverse x)) < 1" ``` huffman@23082 ` 93` ``` using x_neq_0 ``` huffman@23082 ` 94` ``` by (simp add: nonzero_norm_divide divide_inverse [symmetric]) ``` huffman@23082 ` 95` ``` hence "summable (\n. norm (z * inverse x) ^ n)" ``` huffman@20849 ` 96` ``` by (rule summable_geometric) ``` huffman@23082 ` 97` ``` hence "summable (\n. K * norm (z * inverse x) ^ n)" ``` huffman@20849 ` 98` ``` by (rule summable_mult) ``` huffman@23082 ` 99` ``` thus "summable (\n. K * norm (z ^ n) * inverse (norm (x ^ n)))" ``` huffman@23082 ` 100` ``` using x_neq_0 ``` huffman@23082 ` 101` ``` by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib ``` huffman@23082 ` 102` ``` power_inverse norm_power mult_assoc) ``` huffman@20849 ` 103` ``` qed ``` huffman@23082 ` 104` ``` ultimately show "summable (\n. norm (f n * z ^ n))" ``` huffman@20849 ` 105` ``` by (rule summable_comparison_test) ``` huffman@20849 ` 106` ```qed ``` paulson@15077 ` 107` paulson@15229 ` 108` ```lemma powser_inside: ``` haftmann@31017 ` 109` ``` fixes f :: "nat \ 'a::{real_normed_field,banach}" shows ``` hoelzl@41970 ` 110` ``` "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |] ``` paulson@15077 ` 111` ``` ==> summable (%n. f(n) * (z ^ n))" ``` huffman@23082 ` 112` ```by (rule powser_insidea [THEN summable_norm_cancel]) ``` paulson@15077 ` 113` hoelzl@29803 ` 114` ```lemma sum_split_even_odd: fixes f :: "nat \ real" shows ``` hoelzl@41970 ` 115` ``` "(\ i = 0 ..< 2 * n. if even i then f i else g i) = ``` hoelzl@29803 ` 116` ``` (\ i = 0 ..< n. f (2 * i)) + (\ i = 0 ..< n. g (2 * i + 1))" ``` hoelzl@29803 ` 117` ```proof (induct n) ``` hoelzl@29803 ` 118` ``` case (Suc n) ``` hoelzl@41970 ` 119` ``` have "(\ i = 0 ..< 2 * Suc n. if even i then f i else g i) = ``` hoelzl@29803 ` 120` ``` (\ i = 0 ..< n. f (2 * i)) + (\ i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))" ``` huffman@30082 ` 121` ``` using Suc.hyps unfolding One_nat_def by auto ``` hoelzl@29803 ` 122` ``` also have "\ = (\ i = 0 ..< Suc n. f (2 * i)) + (\ i = 0 ..< Suc n. g (2 * i + 1))" by auto ``` hoelzl@29803 ` 123` ``` finally show ?case . ``` hoelzl@29803 ` 124` ```qed auto ``` hoelzl@29803 ` 125` hoelzl@29803 ` 126` ```lemma sums_if': fixes g :: "nat \ real" assumes "g sums x" ``` hoelzl@29803 ` 127` ``` shows "(\ n. if even n then 0 else g ((n - 1) div 2)) sums x" ``` hoelzl@29803 ` 128` ``` unfolding sums_def ``` hoelzl@29803 ` 129` ```proof (rule LIMSEQ_I) ``` hoelzl@29803 ` 130` ``` fix r :: real assume "0 < r" ``` hoelzl@29803 ` 131` ``` from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this] ``` hoelzl@29803 ` 132` ``` obtain no where no_eq: "\ n. n \ no \ (norm (setsum g { 0.. 2 * no" hence "m div 2 \ no" by auto ``` hoelzl@41970 ` 136` ``` have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }" ``` hoelzl@29803 ` 137` ``` using sum_split_even_odd by auto ``` hoelzl@29803 ` 138` ``` hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 \ no` by auto ``` hoelzl@29803 ` 139` ``` moreover ``` hoelzl@29803 ` 140` ``` have "?SUM (2 * (m div 2)) = ?SUM m" ``` hoelzl@29803 ` 141` ``` proof (cases "even m") ``` hoelzl@29803 ` 142` ``` case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] .. ``` hoelzl@29803 ` 143` ``` next ``` hoelzl@29803 ` 144` ``` case False hence "even (Suc m)" by auto ``` hoelzl@29803 ` 145` ``` from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]] odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]] ``` hoelzl@29803 ` 146` ``` have eq: "Suc (2 * (m div 2)) = m" by auto ``` hoelzl@29803 ` 147` ``` hence "even (2 * (m div 2))" using `odd m` by auto ``` hoelzl@29803 ` 148` ``` have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq .. ``` hoelzl@29803 ` 149` ``` also have "\ = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto ``` hoelzl@29803 ` 150` ``` finally show ?thesis by auto ``` hoelzl@29803 ` 151` ``` qed ``` hoelzl@29803 ` 152` ``` ultimately have "(norm (?SUM m - x) < r)" by auto ``` hoelzl@29803 ` 153` ``` } ``` hoelzl@29803 ` 154` ``` thus "\ no. \ m \ no. norm (?SUM m - x) < r" by blast ``` hoelzl@29803 ` 155` ```qed ``` hoelzl@29803 ` 156` hoelzl@29803 ` 157` ```lemma sums_if: fixes g :: "nat \ real" assumes "g sums x" and "f sums y" ``` hoelzl@29803 ` 158` ``` shows "(\ n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)" ``` hoelzl@29803 ` 159` ```proof - ``` hoelzl@29803 ` 160` ``` let ?s = "\ n. if even n then 0 else f ((n - 1) div 2)" ``` hoelzl@29803 ` 161` ``` { fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)" ``` hoelzl@29803 ` 162` ``` by (cases B) auto } note if_sum = this ``` hoelzl@29803 ` 163` ``` have g_sums: "(\ n. if even n then 0 else g ((n - 1) div 2)) sums x" using sums_if'[OF `g sums x`] . ``` hoelzl@41970 ` 164` ``` { ``` hoelzl@29803 ` 165` ``` have "?s 0 = 0" by auto ``` hoelzl@29803 ` 166` ``` have Suc_m1: "\ n. Suc n - 1 = n" by auto ``` wenzelm@41550 ` 167` ``` have if_eq: "\B T E. (if \ B then T else E) = (if B then E else T)" by auto ``` hoelzl@29803 ` 168` hoelzl@29803 ` 169` ``` have "?s sums y" using sums_if'[OF `f sums y`] . ``` hoelzl@41970 ` 170` ``` from this[unfolded sums_def, THEN LIMSEQ_Suc] ``` hoelzl@29803 ` 171` ``` have "(\ n. if even n then f (n div 2) else 0) sums y" ``` hoelzl@29803 ` 172` ``` unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric] ``` hoelzl@29803 ` 173` ``` image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def] ``` nipkow@31148 ` 174` ``` even_Suc Suc_m1 if_eq . ``` hoelzl@29803 ` 175` ``` } from sums_add[OF g_sums this] ``` hoelzl@29803 ` 176` ``` show ?thesis unfolding if_sum . ``` hoelzl@29803 ` 177` ```qed ``` hoelzl@29803 ` 178` hoelzl@29803 ` 179` ```subsection {* Alternating series test / Leibniz formula *} ``` hoelzl@29803 ` 180` hoelzl@29803 ` 181` ```lemma sums_alternating_upper_lower: ``` hoelzl@29803 ` 182` ``` fixes a :: "nat \ real" ``` hoelzl@29803 ` 183` ``` assumes mono: "\n. a (Suc n) \ a n" and a_pos: "\n. 0 \ a n" and "a ----> 0" ``` hoelzl@41970 ` 184` ``` shows "\l. ((\n. (\i=0..<2*n. -1^i*a i) \ l) \ (\ n. \i=0..<2*n. -1^i*a i) ----> l) \ ``` hoelzl@29803 ` 185` ``` ((\n. l \ (\i=0..<2*n + 1. -1^i*a i)) \ (\ n. \i=0..<2*n + 1. -1^i*a i) ----> l)" ``` hoelzl@29803 ` 186` ``` (is "\l. ((\n. ?f n \ l) \ _) \ ((\n. l \ ?g n) \ _)") ``` hoelzl@29803 ` 187` ```proof - ``` huffman@30082 ` 188` ``` have fg_diff: "\n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto ``` hoelzl@29803 ` 189` hoelzl@29803 ` 190` ``` have "\ n. ?f n \ ?f (Suc n)" ``` hoelzl@29803 ` 191` ``` proof fix n show "?f n \ ?f (Suc n)" using mono[of "2*n"] by auto qed ``` hoelzl@29803 ` 192` ``` moreover ``` hoelzl@29803 ` 193` ``` have "\ n. ?g (Suc n) \ ?g n" ``` huffman@30082 ` 194` ``` proof fix n show "?g (Suc n) \ ?g n" using mono[of "Suc (2*n)"] ``` huffman@30082 ` 195` ``` unfolding One_nat_def by auto qed ``` hoelzl@29803 ` 196` ``` moreover ``` hoelzl@41970 ` 197` ``` have "\ n. ?f n \ ?g n" ``` huffman@30082 ` 198` ``` proof fix n show "?f n \ ?g n" using fg_diff a_pos ``` huffman@30082 ` 199` ``` unfolding One_nat_def by auto qed ``` hoelzl@29803 ` 200` ``` moreover ``` hoelzl@29803 ` 201` ``` have "(\ n. ?f n - ?g n) ----> 0" unfolding fg_diff ``` hoelzl@29803 ` 202` ``` proof (rule LIMSEQ_I) ``` hoelzl@29803 ` 203` ``` fix r :: real assume "0 < r" ``` hoelzl@41970 ` 204` ``` with `a ----> 0`[THEN LIMSEQ_D] ``` hoelzl@29803 ` 205` ``` obtain N where "\ n. n \ N \ norm (a n - 0) < r" by auto ``` hoelzl@29803 ` 206` ``` hence "\ n \ N. norm (- a (2 * n) - 0) < r" by auto ``` hoelzl@29803 ` 207` ``` thus "\ N. \ n \ N. norm (- a (2 * n) - 0) < r" by auto ``` hoelzl@29803 ` 208` ``` qed ``` hoelzl@29803 ` 209` ``` ultimately ``` hoelzl@29803 ` 210` ``` show ?thesis by (rule lemma_nest_unique) ``` hoelzl@41970 ` 211` ```qed ``` hoelzl@29803 ` 212` hoelzl@29803 ` 213` ```lemma summable_Leibniz': fixes a :: "nat \ real" ``` hoelzl@29803 ` 214` ``` assumes a_zero: "a ----> 0" and a_pos: "\ n. 0 \ a n" ``` hoelzl@29803 ` 215` ``` and a_monotone: "\ n. a (Suc n) \ a n" ``` hoelzl@29803 ` 216` ``` shows summable: "summable (\ n. (-1)^n * a n)" ``` hoelzl@29803 ` 217` ``` and "\n. (\i=0..<2*n. (-1)^i*a i) \ (\i. (-1)^i*a i)" ``` hoelzl@29803 ` 218` ``` and "(\n. \i=0..<2*n. (-1)^i*a i) ----> (\i. (-1)^i*a i)" ``` hoelzl@29803 ` 219` ``` and "\n. (\i. (-1)^i*a i) \ (\i=0..<2*n+1. (-1)^i*a i)" ``` hoelzl@29803 ` 220` ``` and "(\n. \i=0..<2*n+1. (-1)^i*a i) ----> (\i. (-1)^i*a i)" ``` hoelzl@29803 ` 221` ```proof - ``` hoelzl@29803 ` 222` ``` let "?S n" = "(-1)^n * a n" ``` hoelzl@29803 ` 223` ``` let "?P n" = "\i=0.. n. ?f n \ l" and "?f ----> l" and above_l: "\ n. l \ ?g n" and "?g ----> l" ``` hoelzl@29803 ` 227` ``` using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast ``` hoelzl@41970 ` 228` hoelzl@29803 ` 229` ``` let ?Sa = "\ m. \ n = 0.. l" ``` hoelzl@29803 ` 231` ``` proof (rule LIMSEQ_I) ``` hoelzl@29803 ` 232` ``` fix r :: real assume "0 < r" ``` hoelzl@29803 ` 233` hoelzl@41970 ` 234` ``` with `?f ----> l`[THEN LIMSEQ_D] ``` hoelzl@29803 ` 235` ``` obtain f_no where f: "\ n. n \ f_no \ norm (?f n - l) < r" by auto ``` hoelzl@29803 ` 236` hoelzl@41970 ` 237` ``` from `0 < r` `?g ----> l`[THEN LIMSEQ_D] ``` hoelzl@29803 ` 238` ``` obtain g_no where g: "\ n. n \ g_no \ norm (?g n - l) < r" by auto ``` hoelzl@29803 ` 239` hoelzl@29803 ` 240` ``` { fix n :: nat ``` hoelzl@29803 ` 241` ``` assume "n \ (max (2 * f_no) (2 * g_no))" hence "n \ 2 * f_no" and "n \ 2 * g_no" by auto ``` hoelzl@29803 ` 242` ``` have "norm (?Sa n - l) < r" ``` hoelzl@29803 ` 243` ``` proof (cases "even n") ``` wenzelm@32960 ` 244` ``` case True from even_nat_div_two_times_two[OF this] ``` wenzelm@32960 ` 245` ``` have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto ``` wenzelm@32960 ` 246` ``` with `n \ 2 * f_no` have "n div 2 \ f_no" by auto ``` wenzelm@32960 ` 247` ``` from f[OF this] ``` wenzelm@32960 ` 248` ``` show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost . ``` hoelzl@29803 ` 249` ``` next ``` huffman@35213 ` 250` ``` case False hence "even (n - 1)" by simp ``` wenzelm@32960 ` 251` ``` from even_nat_div_two_times_two[OF this] ``` wenzelm@32960 ` 252` ``` have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto ``` wenzelm@32960 ` 253` ``` hence range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto ``` wenzelm@32960 ` 254` wenzelm@32960 ` 255` ``` from n_eq `n \ 2 * g_no` have "(n - 1) div 2 \ g_no" by auto ``` wenzelm@32960 ` 256` ``` from g[OF this] ``` wenzelm@32960 ` 257` ``` show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq . ``` hoelzl@29803 ` 258` ``` qed ``` hoelzl@29803 ` 259` ``` } ``` hoelzl@29803 ` 260` ``` thus "\ no. \ n \ no. norm (?Sa n - l) < r" by blast ``` hoelzl@29803 ` 261` ``` qed ``` hoelzl@29803 ` 262` ``` hence sums_l: "(\i. (-1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] . ``` hoelzl@29803 ` 263` ``` thus "summable ?S" using summable_def by auto ``` hoelzl@29803 ` 264` hoelzl@29803 ` 265` ``` have "l = suminf ?S" using sums_unique[OF sums_l] . ``` hoelzl@29803 ` 266` hoelzl@29803 ` 267` ``` { fix n show "suminf ?S \ ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto } ``` hoelzl@29803 ` 268` ``` { fix n show "?f n \ suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_l by auto } ``` hoelzl@29803 ` 269` ``` show "?g ----> suminf ?S" using `?g ----> l` `l = suminf ?S` by auto ``` hoelzl@29803 ` 270` ``` show "?f ----> suminf ?S" using `?f ----> l` `l = suminf ?S` by auto ``` hoelzl@29803 ` 271` ```qed ``` hoelzl@29803 ` 272` hoelzl@29803 ` 273` ```theorem summable_Leibniz: fixes a :: "nat \ real" ``` hoelzl@29803 ` 274` ``` assumes a_zero: "a ----> 0" and "monoseq a" ``` hoelzl@29803 ` 275` ``` shows "summable (\ n. (-1)^n * a n)" (is "?summable") ``` hoelzl@29803 ` 276` ``` and "0 < a 0 \ (\n. (\i. -1^i*a i) \ { \i=0..<2*n. -1^i * a i .. \i=0..<2*n+1. -1^i * a i})" (is "?pos") ``` hoelzl@29803 ` 277` ``` and "a 0 < 0 \ (\n. (\i. -1^i*a i) \ { \i=0..<2*n+1. -1^i * a i .. \i=0..<2*n. -1^i * a i})" (is "?neg") ``` hoelzl@29803 ` 278` ``` and "(\n. \i=0..<2*n. -1^i*a i) ----> (\i. -1^i*a i)" (is "?f") ``` hoelzl@29803 ` 279` ``` and "(\n. \i=0..<2*n+1. -1^i*a i) ----> (\i. -1^i*a i)" (is "?g") ``` hoelzl@29803 ` 280` ```proof - ``` hoelzl@29803 ` 281` ``` have "?summable \ ?pos \ ?neg \ ?f \ ?g" ``` hoelzl@29803 ` 282` ``` proof (cases "(\ n. 0 \ a n) \ (\m. \n\m. a n \ a m)") ``` hoelzl@29803 ` 283` ``` case True ``` hoelzl@29803 ` 284` ``` hence ord: "\n m. m \ n \ a n \ a m" and ge0: "\ n. 0 \ a n" by auto ``` hoelzl@29803 ` 285` ``` { fix n have "a (Suc n) \ a n" using ord[where n="Suc n" and m=n] by auto } ``` hoelzl@29803 ` 286` ``` note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] and mono = this ``` hoelzl@29803 ` 287` ``` from leibniz[OF mono] ``` hoelzl@29803 ` 288` ``` show ?thesis using `0 \ a 0` by auto ``` hoelzl@29803 ` 289` ``` next ``` hoelzl@29803 ` 290` ``` let ?a = "\ n. - a n" ``` hoelzl@29803 ` 291` ``` case False ``` hoelzl@29803 ` 292` ``` with monoseq_le[OF `monoseq a` `a ----> 0`] ``` hoelzl@29803 ` 293` ``` have "(\ n. a n \ 0) \ (\m. \n\m. a m \ a n)" by auto ``` hoelzl@29803 ` 294` ``` hence ord: "\n m. m \ n \ ?a n \ ?a m" and ge0: "\ n. 0 \ ?a n" by auto ``` hoelzl@29803 ` 295` ``` { fix n have "?a (Suc n) \ ?a n" using ord[where n="Suc n" and m=n] by auto } ``` hoelzl@29803 ` 296` ``` note monotone = this ``` huffman@44568 ` 297` ``` note leibniz = summable_Leibniz'[OF _ ge0, of "\x. x", OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone] ``` hoelzl@29803 ` 298` ``` have "summable (\ n. (-1)^n * ?a n)" using leibniz(1) by auto ``` hoelzl@29803 ` 299` ``` then obtain l where "(\ n. (-1)^n * ?a n) sums l" unfolding summable_def by auto ``` hoelzl@29803 ` 300` ``` from this[THEN sums_minus] ``` hoelzl@29803 ` 301` ``` have "(\ n. (-1)^n * a n) sums -l" by auto ``` hoelzl@29803 ` 302` ``` hence ?summable unfolding summable_def by auto ``` hoelzl@29803 ` 303` ``` moreover ``` hoelzl@29803 ` 304` ``` have "\ a b :: real. \ - a - - b \ = \a - b\" unfolding minus_diff_minus by auto ``` hoelzl@41970 ` 305` hoelzl@29803 ` 306` ``` from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus] ``` hoelzl@29803 ` 307` ``` have move_minus: "(\n. - (-1 ^ n * a n)) = - (\n. -1 ^ n * a n)" by auto ``` hoelzl@29803 ` 308` hoelzl@29803 ` 309` ``` have ?pos using `0 \ ?a 0` by auto ``` hoelzl@29803 ` 310` ``` moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto ``` huffman@44568 ` 311` ``` moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel] by auto ``` hoelzl@29803 ` 312` ``` ultimately show ?thesis by auto ``` hoelzl@29803 ` 313` ``` qed ``` hoelzl@29803 ` 314` ``` from this[THEN conjunct1] this[THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1] ``` hoelzl@29803 ` 315` ``` this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2] ``` hoelzl@29803 ` 316` ``` show ?summable and ?pos and ?neg and ?f and ?g . ``` hoelzl@29803 ` 317` ```qed ``` paulson@15077 ` 318` huffman@29164 ` 319` ```subsection {* Term-by-Term Differentiability of Power Series *} ``` huffman@23043 ` 320` huffman@23043 ` 321` ```definition ``` huffman@23082 ` 322` ``` diffs :: "(nat => 'a::ring_1) => nat => 'a" where ``` huffman@23082 ` 323` ``` "diffs c = (%n. of_nat (Suc n) * c(Suc n))" ``` paulson@15077 ` 324` paulson@15077 ` 325` ```text{*Lemma about distributing negation over it*} ``` paulson@15077 ` 326` ```lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)" ``` paulson@15077 ` 327` ```by (simp add: diffs_def) ``` paulson@15077 ` 328` huffman@29163 ` 329` ```lemma sums_Suc_imp: ``` huffman@29163 ` 330` ``` assumes f: "f 0 = 0" ``` huffman@29163 ` 331` ``` shows "(\n. f (Suc n)) sums s \ (\n. f n) sums s" ``` huffman@29163 ` 332` ```unfolding sums_def ``` huffman@29163 ` 333` ```apply (rule LIMSEQ_imp_Suc) ``` huffman@29163 ` 334` ```apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric]) ``` huffman@29163 ` 335` ```apply (simp only: setsum_shift_bounds_Suc_ivl) ``` paulson@15077 ` 336` ```done ``` paulson@15077 ` 337` paulson@15229 ` 338` ```lemma diffs_equiv: ``` hoelzl@41970 ` 339` ``` fixes x :: "'a::{real_normed_vector, ring_1}" ``` hoelzl@41970 ` 340` ``` shows "summable (%n. (diffs c)(n) * (x ^ n)) ==> ``` hoelzl@41970 ` 341` ``` (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums ``` nipkow@15546 ` 342` ``` (\n. (diffs c)(n) * (x ^ n))" ``` huffman@29163 ` 343` ```unfolding diffs_def ``` huffman@29163 ` 344` ```apply (drule summable_sums) ``` huffman@29163 ` 345` ```apply (rule sums_Suc_imp, simp_all) ``` paulson@15077 ` 346` ```done ``` paulson@15077 ` 347` paulson@15077 ` 348` ```lemma lemma_termdiff1: ``` haftmann@31017 ` 349` ``` fixes z :: "'a :: {monoid_mult,comm_ring}" shows ``` hoelzl@41970 ` 350` ``` "(\p=0..p=0..i = 0.. 0" shows ``` huffman@23082 ` 361` ``` "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) = ``` huffman@20860 ` 362` ``` h * (\p=0..< n - Suc 0. \q=0..< n - Suc 0 - p. ``` huffman@23082 ` 363` ``` (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs") ``` huffman@23082 ` 364` ```apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h) ``` huffman@20860 ` 365` ```apply (simp add: right_diff_distrib diff_divide_distrib h) ``` paulson@15077 ` 366` ```apply (simp add: mult_assoc [symmetric]) ``` huffman@20860 ` 367` ```apply (cases "n", simp) ``` huffman@20860 ` 368` ```apply (simp add: lemma_realpow_diff_sumr2 h ``` huffman@20860 ` 369` ``` right_diff_distrib [symmetric] mult_assoc ``` huffman@30273 ` 370` ``` del: power_Suc setsum_op_ivl_Suc of_nat_Suc) ``` huffman@20860 ` 371` ```apply (subst lemma_realpow_rev_sumr) ``` huffman@23082 ` 372` ```apply (subst sumr_diff_mult_const2) ``` huffman@20860 ` 373` ```apply simp ``` huffman@20860 ` 374` ```apply (simp only: lemma_termdiff1 setsum_right_distrib) ``` huffman@20860 ` 375` ```apply (rule setsum_cong [OF refl]) ``` nipkow@15539 ` 376` ```apply (simp add: diff_minus [symmetric] less_iff_Suc_add) ``` huffman@20860 ` 377` ```apply (clarify) ``` huffman@20860 ` 378` ```apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac ``` huffman@30273 ` 379` ``` del: setsum_op_ivl_Suc power_Suc) ``` huffman@20860 ` 380` ```apply (subst mult_assoc [symmetric], subst power_add [symmetric]) ``` huffman@20860 ` 381` ```apply (simp add: mult_ac) ``` huffman@20860 ` 382` ```done ``` huffman@20860 ` 383` huffman@20860 ` 384` ```lemma real_setsum_nat_ivl_bounded2: ``` haftmann@35028 ` 385` ``` fixes K :: "'a::linordered_semidom" ``` huffman@23082 ` 386` ``` assumes f: "\p::nat. p < n \ f p \ K" ``` huffman@23082 ` 387` ``` assumes K: "0 \ K" ``` huffman@23082 ` 388` ``` shows "setsum f {0.. of_nat n * K" ``` huffman@23082 ` 389` ```apply (rule order_trans [OF setsum_mono]) ``` huffman@23082 ` 390` ```apply (rule f, simp) ``` huffman@23082 ` 391` ```apply (simp add: mult_right_mono K) ``` paulson@15077 ` 392` ```done ``` paulson@15077 ` 393` paulson@15229 ` 394` ```lemma lemma_termdiff3: ``` haftmann@31017 ` 395` ``` fixes h z :: "'a::{real_normed_field}" ``` huffman@20860 ` 396` ``` assumes 1: "h \ 0" ``` huffman@23082 ` 397` ``` assumes 2: "norm z \ K" ``` huffman@23082 ` 398` ``` assumes 3: "norm (z + h) \ K" ``` huffman@23082 ` 399` ``` shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) ``` huffman@23082 ` 400` ``` \ of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h" ``` huffman@20860 ` 401` ```proof - ``` huffman@23082 ` 402` ``` have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) = ``` huffman@23082 ` 403` ``` norm (\p = 0..q = 0.. \ of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h" ``` huffman@23082 ` 410` ``` proof (rule mult_right_mono [OF _ norm_ge_zero]) ``` huffman@23082 ` 411` ``` from norm_ge_zero 2 have K: "0 \ K" by (rule order_trans) ``` huffman@23082 ` 412` ``` have le_Kn: "\i j n. i + j = n \ norm ((z + h) ^ i * z ^ j) \ K ^ n" ``` huffman@20860 ` 413` ``` apply (erule subst) ``` huffman@23082 ` 414` ``` apply (simp only: norm_mult norm_power power_add) ``` huffman@23082 ` 415` ``` apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K) ``` huffman@20860 ` 416` ``` done ``` huffman@23082 ` 417` ``` show "norm (\p = 0..q = 0.. of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))" ``` huffman@20860 ` 420` ``` apply (intro ``` huffman@23082 ` 421` ``` order_trans [OF norm_setsum] ``` huffman@20860 ` 422` ``` real_setsum_nat_ivl_bounded2 ``` huffman@20860 ` 423` ``` mult_nonneg_nonneg ``` huffman@23082 ` 424` ``` zero_le_imp_of_nat ``` huffman@20860 ` 425` ``` zero_le_power K) ``` huffman@20860 ` 426` ``` apply (rule le_Kn, simp) ``` huffman@20860 ` 427` ``` done ``` huffman@20860 ` 428` ``` qed ``` huffman@23082 ` 429` ``` also have "\ = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h" ``` huffman@20860 ` 430` ``` by (simp only: mult_assoc) ``` huffman@20860 ` 431` ``` finally show ?thesis . ``` huffman@20860 ` 432` ```qed ``` paulson@15077 ` 433` huffman@20860 ` 434` ```lemma lemma_termdiff4: ``` haftmann@31017 ` 435` ``` fixes f :: "'a::{real_normed_field} \ ``` huffman@23082 ` 436` ``` 'b::real_normed_vector" ``` huffman@20860 ` 437` ``` assumes k: "0 < (k::real)" ``` huffman@23082 ` 438` ``` assumes le: "\h. \h \ 0; norm h < k\ \ norm (f h) \ K * norm h" ``` huffman@20860 ` 439` ``` shows "f -- 0 --> 0" ``` huffman@31338 ` 440` ```unfolding LIM_eq diff_0_right ``` huffman@29163 ` 441` ```proof (safe) ``` huffman@29163 ` 442` ``` let ?h = "of_real (k / 2)::'a" ``` huffman@29163 ` 443` ``` have "?h \ 0" and "norm ?h < k" using k by simp_all ``` huffman@29163 ` 444` ``` hence "norm (f ?h) \ K * norm ?h" by (rule le) ``` huffman@29163 ` 445` ``` hence "0 \ K * norm ?h" by (rule order_trans [OF norm_ge_zero]) ``` huffman@29163 ` 446` ``` hence zero_le_K: "0 \ K" using k by (simp add: zero_le_mult_iff) ``` huffman@29163 ` 447` huffman@20860 ` 448` ``` fix r::real assume r: "0 < r" ``` huffman@23082 ` 449` ``` show "\s. 0 < s \ (\x. x \ 0 \ norm x < s \ norm (f x) < r)" ``` huffman@20860 ` 450` ``` proof (cases) ``` huffman@20860 ` 451` ``` assume "K = 0" ``` huffman@23082 ` 452` ``` with k r le have "0 < k \ (\x. x \ 0 \ norm x < k \ norm (f x) < r)" ``` huffman@20860 ` 453` ``` by simp ``` huffman@23082 ` 454` ``` thus "\s. 0 < s \ (\x. x \ 0 \ norm x < s \ norm (f x) < r)" .. ``` huffman@20860 ` 455` ``` next ``` huffman@20860 ` 456` ``` assume K_neq_zero: "K \ 0" ``` huffman@20860 ` 457` ``` with zero_le_K have K: "0 < K" by simp ``` huffman@23082 ` 458` ``` show "\s. 0 < s \ (\x. x \ 0 \ norm x < s \ norm (f x) < r)" ``` huffman@20860 ` 459` ``` proof (rule exI, safe) ``` huffman@20860 ` 460` ``` from k r K show "0 < min k (r * inverse K / 2)" ``` huffman@20860 ` 461` ``` by (simp add: mult_pos_pos positive_imp_inverse_positive) ``` huffman@20860 ` 462` ``` next ``` huffman@23082 ` 463` ``` fix x::'a ``` huffman@23082 ` 464` ``` assume x1: "x \ 0" and x2: "norm x < min k (r * inverse K / 2)" ``` huffman@23082 ` 465` ``` from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2" ``` huffman@20860 ` 466` ``` by simp_all ``` huffman@23082 ` 467` ``` from x1 x3 le have "norm (f x) \ K * norm x" by simp ``` huffman@23082 ` 468` ``` also from x4 K have "K * norm x < K * (r * inverse K / 2)" ``` huffman@20860 ` 469` ``` by (rule mult_strict_left_mono) ``` huffman@20860 ` 470` ``` also have "\ = r / 2" ``` huffman@20860 ` 471` ``` using K_neq_zero by simp ``` huffman@20860 ` 472` ``` also have "r / 2 < r" ``` huffman@20860 ` 473` ``` using r by simp ``` huffman@23082 ` 474` ``` finally show "norm (f x) < r" . ``` huffman@20860 ` 475` ``` qed ``` huffman@20860 ` 476` ``` qed ``` huffman@20860 ` 477` ```qed ``` paulson@15077 ` 478` paulson@15229 ` 479` ```lemma lemma_termdiff5: ``` haftmann@31017 ` 480` ``` fixes g :: "'a::{real_normed_field} \ ``` huffman@23082 ` 481` ``` nat \ 'b::banach" ``` huffman@20860 ` 482` ``` assumes k: "0 < (k::real)" ``` huffman@20860 ` 483` ``` assumes f: "summable f" ``` huffman@23082 ` 484` ``` assumes le: "\h n. \h \ 0; norm h < k\ \ norm (g h n) \ f n * norm h" ``` huffman@20860 ` 485` ``` shows "(\h. suminf (g h)) -- 0 --> 0" ``` huffman@20860 ` 486` ```proof (rule lemma_termdiff4 [OF k]) ``` huffman@23082 ` 487` ``` fix h::'a assume "h \ 0" and "norm h < k" ``` huffman@23082 ` 488` ``` hence A: "\n. norm (g h n) \ f n * norm h" ``` huffman@20860 ` 489` ``` by (simp add: le) ``` huffman@23082 ` 490` ``` hence "\N. \n\N. norm (norm (g h n)) \ f n * norm h" ``` huffman@20860 ` 491` ``` by simp ``` huffman@23082 ` 492` ``` moreover from f have B: "summable (\n. f n * norm h)" ``` huffman@20860 ` 493` ``` by (rule summable_mult2) ``` huffman@23082 ` 494` ``` ultimately have C: "summable (\n. norm (g h n))" ``` huffman@20860 ` 495` ``` by (rule summable_comparison_test) ``` huffman@23082 ` 496` ``` hence "norm (suminf (g h)) \ (\n. norm (g h n))" ``` huffman@23082 ` 497` ``` by (rule summable_norm) ``` huffman@23082 ` 498` ``` also from A C B have "(\n. norm (g h n)) \ (\n. f n * norm h)" ``` huffman@20860 ` 499` ``` by (rule summable_le) ``` huffman@23082 ` 500` ``` also from f have "(\n. f n * norm h) = suminf f * norm h" ``` huffman@20860 ` 501` ``` by (rule suminf_mult2 [symmetric]) ``` huffman@23082 ` 502` ``` finally show "norm (suminf (g h)) \ suminf f * norm h" . ``` huffman@20860 ` 503` ```qed ``` paulson@15077 ` 504` paulson@15077 ` 505` paulson@15077 ` 506` ```text{* FIXME: Long proofs*} ``` paulson@15077 ` 507` paulson@15077 ` 508` ```lemma termdiffs_aux: ``` haftmann@31017 ` 509` ``` fixes x :: "'a::{real_normed_field,banach}" ``` huffman@20849 ` 510` ``` assumes 1: "summable (\n. diffs (diffs c) n * K ^ n)" ``` huffman@23082 ` 511` ``` assumes 2: "norm x < norm K" ``` huffman@20860 ` 512` ``` shows "(\h. \n. c n * (((x + h) ^ n - x ^ n) / h ``` huffman@23082 ` 513` ``` - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0" ``` huffman@20849 ` 514` ```proof - ``` huffman@20860 ` 515` ``` from dense [OF 2] ``` huffman@23082 ` 516` ``` obtain r where r1: "norm x < r" and r2: "r < norm K" by fast ``` huffman@23082 ` 517` ``` from norm_ge_zero r1 have r: "0 < r" ``` huffman@20860 ` 518` ``` by (rule order_le_less_trans) ``` huffman@20860 ` 519` ``` hence r_neq_0: "r \ 0" by simp ``` huffman@20860 ` 520` ``` show ?thesis ``` huffman@20849 ` 521` ``` proof (rule lemma_termdiff5) ``` huffman@23082 ` 522` ``` show "0 < r - norm x" using r1 by simp ``` huffman@20849 ` 523` ``` next ``` huffman@23082 ` 524` ``` from r r2 have "norm (of_real r::'a) < norm K" ``` huffman@23082 ` 525` ``` by simp ``` huffman@23082 ` 526` ``` with 1 have "summable (\n. norm (diffs (diffs c) n * (of_real r ^ n)))" ``` huffman@20860 ` 527` ``` by (rule powser_insidea) ``` huffman@23082 ` 528` ``` hence "summable (\n. diffs (diffs (\n. norm (c n))) n * r ^ n)" ``` huffman@23082 ` 529` ``` using r ``` huffman@23082 ` 530` ``` by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc) ``` huffman@23082 ` 531` ``` hence "summable (\n. of_nat n * diffs (\n. norm (c n)) n * r ^ (n - Suc 0))" ``` huffman@20860 ` 532` ``` by (rule diffs_equiv [THEN sums_summable]) ``` huffman@23082 ` 533` ``` also have "(\n. of_nat n * diffs (\n. norm (c n)) n * r ^ (n - Suc 0)) ``` huffman@23082 ` 534` ``` = (\n. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))" ``` huffman@20849 ` 535` ``` apply (rule ext) ``` huffman@20849 ` 536` ``` apply (simp add: diffs_def) ``` huffman@20849 ` 537` ``` apply (case_tac n, simp_all add: r_neq_0) ``` huffman@20849 ` 538` ``` done ``` hoelzl@41970 ` 539` ``` finally have "summable ``` huffman@23082 ` 540` ``` (\n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))" ``` huffman@20860 ` 541` ``` by (rule diffs_equiv [THEN sums_summable]) ``` huffman@20860 ` 542` ``` also have ``` huffman@23082 ` 543` ``` "(\n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * ``` huffman@20860 ` 544` ``` r ^ (n - Suc 0)) = ``` huffman@23082 ` 545` ``` (\n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" ``` huffman@20849 ` 546` ``` apply (rule ext) ``` huffman@20849 ` 547` ``` apply (case_tac "n", simp) ``` huffman@20849 ` 548` ``` apply (case_tac "nat", simp) ``` huffman@20849 ` 549` ``` apply (simp add: r_neq_0) ``` huffman@20849 ` 550` ``` done ``` huffman@20860 ` 551` ``` finally show ``` huffman@23082 ` 552` ``` "summable (\n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" . ``` huffman@20849 ` 553` ``` next ``` huffman@23082 ` 554` ``` fix h::'a and n::nat ``` huffman@20860 ` 555` ``` assume h: "h \ 0" ``` huffman@23082 ` 556` ``` assume "norm h < r - norm x" ``` huffman@23082 ` 557` ``` hence "norm x + norm h < r" by simp ``` huffman@23082 ` 558` ``` with norm_triangle_ineq have xh: "norm (x + h) < r" ``` huffman@20860 ` 559` ``` by (rule order_le_less_trans) ``` huffman@23082 ` 560` ``` show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) ``` huffman@23082 ` 561` ``` \ norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h" ``` huffman@23082 ` 562` ``` apply (simp only: norm_mult mult_assoc) ``` huffman@23082 ` 563` ``` apply (rule mult_left_mono [OF _ norm_ge_zero]) ``` huffman@20860 ` 564` ``` apply (simp (no_asm) add: mult_assoc [symmetric]) ``` huffman@20860 ` 565` ``` apply (rule lemma_termdiff3) ``` huffman@20860 ` 566` ``` apply (rule h) ``` huffman@20860 ` 567` ``` apply (rule r1 [THEN order_less_imp_le]) ``` huffman@20860 ` 568` ``` apply (rule xh [THEN order_less_imp_le]) ``` huffman@20860 ` 569` ``` done ``` huffman@20849 ` 570` ``` qed ``` huffman@20849 ` 571` ```qed ``` webertj@20217 ` 572` huffman@20860 ` 573` ```lemma termdiffs: ``` haftmann@31017 ` 574` ``` fixes K x :: "'a::{real_normed_field,banach}" ``` huffman@20860 ` 575` ``` assumes 1: "summable (\n. c n * K ^ n)" ``` huffman@20860 ` 576` ``` assumes 2: "summable (\n. (diffs c) n * K ^ n)" ``` huffman@20860 ` 577` ``` assumes 3: "summable (\n. (diffs (diffs c)) n * K ^ n)" ``` huffman@23082 ` 578` ``` assumes 4: "norm x < norm K" ``` huffman@20860 ` 579` ``` shows "DERIV (\x. \n. c n * x ^ n) x :> (\n. (diffs c) n * x ^ n)" ``` huffman@29163 ` 580` ```unfolding deriv_def ``` huffman@29163 ` 581` ```proof (rule LIM_zero_cancel) ``` huffman@20860 ` 582` ``` show "(\h. (suminf (\n. c n * (x + h) ^ n) - suminf (\n. c n * x ^ n)) / h ``` huffman@20860 ` 583` ``` - suminf (\n. diffs c n * x ^ n)) -- 0 --> 0" ``` huffman@20860 ` 584` ``` proof (rule LIM_equal2) ``` huffman@29163 ` 585` ``` show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq) ``` huffman@20860 ` 586` ``` next ``` huffman@23082 ` 587` ``` fix h :: 'a ``` huffman@20860 ` 588` ``` assume "h \ 0" ``` huffman@23082 ` 589` ``` assume "norm (h - 0) < norm K - norm x" ``` huffman@23082 ` 590` ``` hence "norm x + norm h < norm K" by simp ``` huffman@23082 ` 591` ``` hence 5: "norm (x + h) < norm K" ``` huffman@23082 ` 592` ``` by (rule norm_triangle_ineq [THEN order_le_less_trans]) ``` huffman@20860 ` 593` ``` have A: "summable (\n. c n * x ^ n)" ``` huffman@20860 ` 594` ``` by (rule powser_inside [OF 1 4]) ``` huffman@20860 ` 595` ``` have B: "summable (\n. c n * (x + h) ^ n)" ``` huffman@20860 ` 596` ``` by (rule powser_inside [OF 1 5]) ``` huffman@20860 ` 597` ``` have C: "summable (\n. diffs c n * x ^ n)" ``` huffman@20860 ` 598` ``` by (rule powser_inside [OF 2 4]) ``` huffman@20860 ` 599` ``` show "((\n. c n * (x + h) ^ n) - (\n. c n * x ^ n)) / h ``` hoelzl@41970 ` 600` ``` - (\n. diffs c n * x ^ n) = ``` huffman@23082 ` 601` ``` (\n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))" ``` huffman@20860 ` 602` ``` apply (subst sums_unique [OF diffs_equiv [OF C]]) ``` huffman@20860 ` 603` ``` apply (subst suminf_diff [OF B A]) ``` huffman@20860 ` 604` ``` apply (subst suminf_divide [symmetric]) ``` huffman@20860 ` 605` ``` apply (rule summable_diff [OF B A]) ``` huffman@20860 ` 606` ``` apply (subst suminf_diff) ``` huffman@20860 ` 607` ``` apply (rule summable_divide) ``` huffman@20860 ` 608` ``` apply (rule summable_diff [OF B A]) ``` huffman@20860 ` 609` ``` apply (rule sums_summable [OF diffs_equiv [OF C]]) ``` huffman@29163 ` 610` ``` apply (rule arg_cong [where f="suminf"], rule ext) ``` nipkow@29667 ` 611` ``` apply (simp add: algebra_simps) ``` huffman@20860 ` 612` ``` done ``` huffman@20860 ` 613` ``` next ``` huffman@20860 ` 614` ``` show "(\h. \n. c n * (((x + h) ^ n - x ^ n) / h - ``` huffman@23082 ` 615` ``` of_nat n * x ^ (n - Suc 0))) -- 0 --> 0" ``` huffman@20860 ` 616` ``` by (rule termdiffs_aux [OF 3 4]) ``` huffman@20860 ` 617` ``` qed ``` huffman@20860 ` 618` ```qed ``` huffman@20860 ` 619` paulson@15077 ` 620` hoelzl@29803 ` 621` ```subsection {* Derivability of power series *} ``` hoelzl@29803 ` 622` hoelzl@29803 ` 623` ```lemma DERIV_series': fixes f :: "real \ nat \ real" ``` hoelzl@29803 ` 624` ``` assumes DERIV_f: "\ n. DERIV (\ x. f x n) x0 :> (f' x0 n)" ``` hoelzl@29803 ` 625` ``` and allf_summable: "\ x. x \ {a <..< b} \ summable (f x)" and x0_in_I: "x0 \ {a <..< b}" ``` hoelzl@29803 ` 626` ``` and "summable (f' x0)" ``` hoelzl@29803 ` 627` ``` and "summable L" and L_def: "\ n x y. \ x \ { a <..< b} ; y \ { a <..< b} \ \ \ f x n - f y n \ \ L n * \ x - y \" ``` hoelzl@29803 ` 628` ``` shows "DERIV (\ x. suminf (f x)) x0 :> (suminf (f' x0))" ``` hoelzl@29803 ` 629` ``` unfolding deriv_def ``` hoelzl@29803 ` 630` ```proof (rule LIM_I) ``` hoelzl@29803 ` 631` ``` fix r :: real assume "0 < r" hence "0 < r/3" by auto ``` hoelzl@29803 ` 632` hoelzl@41970 ` 633` ``` obtain N_L where N_L: "\ n. N_L \ n \ \ \ i. L (i + n) \ < r/3" ``` hoelzl@29803 ` 634` ``` using suminf_exist_split[OF `0 < r/3` `summable L`] by auto ``` hoelzl@29803 ` 635` hoelzl@41970 ` 636` ``` obtain N_f' where N_f': "\ n. N_f' \ n \ \ \ i. f' x0 (i + n) \ < r/3" ``` hoelzl@29803 ` 637` ``` using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto ``` hoelzl@29803 ` 638` hoelzl@29803 ` 639` ``` let ?N = "Suc (max N_L N_f')" ``` hoelzl@29803 ` 640` ``` have "\ \ i. f' x0 (i + ?N) \ < r/3" (is "?f'_part < r/3") and ``` hoelzl@29803 ` 641` ``` L_estimate: "\ \ i. L (i + ?N) \ < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto ``` hoelzl@29803 ` 642` hoelzl@29803 ` 643` ``` let "?diff i x" = "(f (x0 + x) i - f x0 i) / x" ``` hoelzl@29803 ` 644` hoelzl@29803 ` 645` ``` let ?r = "r / (3 * real ?N)" ``` hoelzl@29803 ` 646` ``` have "0 < 3 * real ?N" by auto ``` hoelzl@29803 ` 647` ``` from divide_pos_pos[OF `0 < r` this] ``` hoelzl@29803 ` 648` ``` have "0 < ?r" . ``` hoelzl@29803 ` 649` hoelzl@29803 ` 650` ``` let "?s n" = "SOME s. 0 < s \ (\ x. x \ 0 \ \ x \ < s \ \ ?diff n x - f' x0 n \ < ?r)" ``` hoelzl@29803 ` 651` ``` def S' \ "Min (?s ` { 0 ..< ?N })" ``` hoelzl@29803 ` 652` hoelzl@29803 ` 653` ``` have "0 < S'" unfolding S'_def ``` hoelzl@29803 ` 654` ``` proof (rule iffD2[OF Min_gr_iff]) ``` hoelzl@29803 ` 655` ``` show "\ x \ (?s ` { 0 ..< ?N }). 0 < x" ``` hoelzl@29803 ` 656` ``` proof (rule ballI) ``` hoelzl@29803 ` 657` ``` fix x assume "x \ ?s ` {0.. {0.. (\x. x \ 0 \ \x\ < s \ \?diff n x - f' x0 n\ < ?r)" by auto ``` hoelzl@29803 ` 661` ``` have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound) ``` hoelzl@29803 ` 662` ``` thus "0 < x" unfolding `x = ?s n` . ``` hoelzl@29803 ` 663` ``` qed ``` hoelzl@29803 ` 664` ``` qed auto ``` hoelzl@29803 ` 665` hoelzl@29803 ` 666` ``` def S \ "min (min (x0 - a) (b - x0)) S'" ``` hoelzl@29803 ` 667` ``` hence "0 < S" and S_a: "S \ x0 - a" and S_b: "S \ b - x0" and "S \ S'" using x0_in_I and `0 < S'` ``` hoelzl@29803 ` 668` ``` by auto ``` hoelzl@29803 ` 669` hoelzl@29803 ` 670` ``` { fix x assume "x \ 0" and "\ x \ < S" ``` hoelzl@29803 ` 671` ``` hence x_in_I: "x0 + x \ { a <..< b }" using S_a S_b by auto ``` hoelzl@41970 ` 672` hoelzl@29803 ` 673` ``` note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]] ``` hoelzl@29803 ` 674` ``` note div_smbl = summable_divide[OF diff_smbl] ``` hoelzl@29803 ` 675` ``` note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`] ``` hoelzl@29803 ` 676` ``` note ign = summable_ignore_initial_segment[where k="?N"] ``` hoelzl@29803 ` 677` ``` note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]] ``` hoelzl@29803 ` 678` ``` note div_shft_smbl = summable_divide[OF diff_shft_smbl] ``` hoelzl@29803 ` 679` ``` note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]] ``` hoelzl@29803 ` 680` hoelzl@29803 ` 681` ``` { fix n ``` hoelzl@41970 ` 682` ``` have "\ ?diff (n + ?N) x \ \ L (n + ?N) * \ (x0 + x) - x0 \ / \ x \" ``` wenzelm@32960 ` 683` ``` using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide . ``` hoelzl@29803 ` 684` ``` hence "\ ( \ ?diff (n + ?N) x \) \ \ L (n + ?N)" using `x \ 0` by auto ``` hoelzl@29803 ` 685` ``` } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]] ``` hoelzl@29803 ` 686` ``` from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]] ``` hoelzl@29803 ` 687` ``` have "\ \ i. ?diff (i + ?N) x \ \ (\ i. L (i + ?N))" . ``` hoelzl@29803 ` 688` ``` hence "\ \ i. ?diff (i + ?N) x \ \ r / 3" (is "?L_part \ r/3") using L_estimate by auto ``` hoelzl@29803 ` 689` hoelzl@29803 ` 690` ``` have "\\n \ { 0 ..< ?N}. ?diff n x - f' x0 n \ \ (\n \ { 0 ..< ?N}. \?diff n x - f' x0 n \)" .. ``` hoelzl@29803 ` 691` ``` also have "\ < (\n \ { 0 ..< ?N}. ?r)" ``` hoelzl@29803 ` 692` ``` proof (rule setsum_strict_mono) ``` hoelzl@29803 ` 693` ``` fix n assume "n \ { 0 ..< ?N}" ``` hoelzl@29803 ` 694` ``` have "\ x \ < S" using `\ x \ < S` . ``` hoelzl@29803 ` 695` ``` also have "S \ S'" using `S \ S'` . ``` hoelzl@41970 ` 696` ``` also have "S' \ ?s n" unfolding S'_def ``` hoelzl@29803 ` 697` ``` proof (rule Min_le_iff[THEN iffD2]) ``` wenzelm@32960 ` 698` ``` have "?s n \ (?s ` {0.. ?s n \ ?s n" using `n \ { 0 ..< ?N}` by auto ``` wenzelm@32960 ` 699` ``` thus "\ a \ (?s ` {0.. ?s n" by blast ``` hoelzl@29803 ` 700` ``` qed auto ``` hoelzl@29803 ` 701` ``` finally have "\ x \ < ?s n" . ``` hoelzl@29803 ` 702` hoelzl@29803 ` 703` ``` from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2] ``` hoelzl@29803 ` 704` ``` have "\x. x \ 0 \ \x\ < ?s n \ \?diff n x - f' x0 n\ < ?r" . ``` hoelzl@29803 ` 705` ``` with `x \ 0` and `\x\ < ?s n` ``` hoelzl@29803 ` 706` ``` show "\?diff n x - f' x0 n\ < ?r" by blast ``` hoelzl@29803 ` 707` ``` qed auto ``` hoelzl@29803 ` 708` ``` also have "\ = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant) ``` hoelzl@29803 ` 709` ``` also have "\ = real ?N * ?r" unfolding real_eq_of_nat by auto ``` hoelzl@29803 ` 710` ``` also have "\ = r/3" by auto ``` hoelzl@29803 ` 711` ``` finally have "\\n \ { 0 ..< ?N}. ?diff n x - f' x0 n \ < r / 3" (is "?diff_part < r / 3") . ``` hoelzl@29803 ` 712` hoelzl@29803 ` 713` ``` from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]] ``` hoelzl@41970 ` 714` ``` have "\ (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \ = ``` hoelzl@29803 ` 715` ``` \ \n. ?diff n x - f' x0 n \" unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] using suminf_divide[OF diff_smbl, symmetric] by auto ``` hoelzl@29803 ` 716` ``` also have "\ \ ?diff_part + \ (\n. ?diff (n + ?N) x) - (\ n. f' x0 (n + ?N)) \" unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]] by (rule abs_triangle_ineq) ``` hoelzl@29803 ` 717` ``` also have "\ \ ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto ``` hoelzl@41970 ` 718` ``` also have "\ < r /3 + r/3 + r/3" ``` huffman@36842 ` 719` ``` using `?diff_part < r/3` `?L_part \ r/3` and `?f'_part < r/3` ``` huffman@36842 ` 720` ``` by (rule add_strict_mono [OF add_less_le_mono]) ``` hoelzl@29803 ` 721` ``` finally have "\ (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \ < r" ``` hoelzl@29803 ` 722` ``` by auto ``` hoelzl@41970 ` 723` ``` } thus "\ s > 0. \ x. x \ 0 \ norm (x - 0) < s \ ``` hoelzl@29803 ` 724` ``` norm (((\n. f (x0 + x) n) - (\n. f x0 n)) / x - (\n. f' x0 n)) < r" using `0 < S` ``` hoelzl@29803 ` 725` ``` unfolding real_norm_def diff_0_right by blast ``` hoelzl@29803 ` 726` ```qed ``` hoelzl@29803 ` 727` hoelzl@29803 ` 728` ```lemma DERIV_power_series': fixes f :: "nat \ real" ``` hoelzl@29803 ` 729` ``` assumes converges: "\ x. x \ {-R <..< R} \ summable (\ n. f n * real (Suc n) * x^n)" ``` hoelzl@29803 ` 730` ``` and x0_in_I: "x0 \ {-R <..< R}" and "0 < R" ``` hoelzl@29803 ` 731` ``` shows "DERIV (\ x. (\ n. f n * x^(Suc n))) x0 :> (\ n. f n * real (Suc n) * x0^n)" ``` hoelzl@29803 ` 732` ``` (is "DERIV (\ x. (suminf (?f x))) x0 :> (suminf (?f' x0))") ``` hoelzl@29803 ` 733` ```proof - ``` hoelzl@29803 ` 734` ``` { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'" ``` hoelzl@29803 ` 735` ``` hence "x0 \ {-R' <..< R'}" and "R' \ {-R <..< R}" and "x0 \ {-R <..< R}" by auto ``` hoelzl@29803 ` 736` ``` have "DERIV (\ x. (suminf (?f x))) x0 :> (suminf (?f' x0))" ``` hoelzl@29803 ` 737` ``` proof (rule DERIV_series') ``` hoelzl@29803 ` 738` ``` show "summable (\ n. \f n * real (Suc n) * R'^n\)" ``` hoelzl@29803 ` 739` ``` proof - ``` wenzelm@32960 ` 740` ``` have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto ``` wenzelm@32960 ` 741` ``` hence in_Rball: "(R' + R) / 2 \ {-R <..< R}" using `R' < R` by auto ``` wenzelm@32960 ` 742` ``` have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto ``` wenzelm@32960 ` 743` ``` from powser_insidea[OF converges[OF in_Rball] this] show ?thesis by auto ``` hoelzl@29803 ` 744` ``` qed ``` hoelzl@29803 ` 745` ``` { fix n x y assume "x \ {-R' <..< R'}" and "y \ {-R' <..< R'}" ``` wenzelm@32960 ` 746` ``` show "\?f x n - ?f y n\ \ \f n * real (Suc n) * R'^n\ * \x-y\" ``` wenzelm@32960 ` 747` ``` proof - ``` hoelzl@41970 ` 748` ``` have "\f n * x ^ (Suc n) - f n * y ^ (Suc n)\ = (\f n\ * \x-y\) * \\p = 0.." ``` wenzelm@32960 ` 749` ``` unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto ``` hoelzl@41970 ` 750` ``` also have "\ \ (\f n\ * \x-y\) * (\real (Suc n)\ * \R' ^ n\)" ``` wenzelm@32960 ` 751` ``` proof (rule mult_left_mono) ``` wenzelm@32960 ` 752` ``` have "\\p = 0.. \ (\p = 0..x ^ p * y ^ (n - p)\)" by (rule setsum_abs) ``` wenzelm@32960 ` 753` ``` also have "\ \ (\p = 0.. {0.. n" by auto ``` wenzelm@32960 ` 756` ``` { fix n fix x :: real assume "x \ {-R'<..x\ \ R'" by auto ``` wenzelm@32960 ` 758` ``` hence "\x^n\ \ R'^n" unfolding power_abs by (rule power_mono, auto) ``` wenzelm@32960 ` 759` ``` } from mult_mono[OF this[OF `x \ {-R'<.. {-R'<..x^p * y^(n-p)\ \ R'^p * R'^(n-p)" unfolding abs_mult by auto ``` wenzelm@32960 ` 761` ``` thus "\x^p * y^(n-p)\ \ R'^n" unfolding power_add[symmetric] using `p \ n` by auto ``` wenzelm@32960 ` 762` ``` qed ``` wenzelm@32960 ` 763` ``` also have "\ = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto ``` wenzelm@32960 ` 764` ``` finally show "\\p = 0.. \ \real (Suc n)\ * \R' ^ n\" unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] . ``` wenzelm@32960 ` 765` ``` show "0 \ \f n\ * \x - y\" unfolding abs_mult[symmetric] by auto ``` wenzelm@32960 ` 766` ``` qed ``` huffman@36777 ` 767` ``` also have "\ = \f n * real (Suc n) * R' ^ n\ * \x - y\" unfolding abs_mult mult_assoc[symmetric] by algebra ``` wenzelm@32960 ` 768` ``` finally show ?thesis . ``` wenzelm@32960 ` 769` ``` qed } ``` hoelzl@31881 ` 770` ``` { fix n show "DERIV (\ x. ?f x n) x0 :> (?f' x0 n)" ``` wenzelm@32960 ` 771` ``` by (auto intro!: DERIV_intros simp del: power_Suc) } ``` hoelzl@29803 ` 772` ``` { fix x assume "x \ {-R' <..< R'}" hence "R' \ {-R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto ``` wenzelm@32960 ` 773` ``` have "summable (\ n. f n * x^n)" ``` wenzelm@32960 ` 774` ``` proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \ {-R <..< R}`] `norm x < norm R'`]], rule allI) ``` wenzelm@32960 ` 775` ``` fix n ``` wenzelm@32960 ` 776` ``` have le: "\f n\ * 1 \ \f n\ * real (Suc n)" by (rule mult_left_mono, auto) ``` wenzelm@32960 ` 777` ``` show "\f n * x ^ n\ \ norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult ``` wenzelm@32960 ` 778` ``` by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right]) ``` wenzelm@32960 ` 779` ``` qed ``` huffman@36777 ` 780` ``` from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute] ``` wenzelm@32960 ` 781` ``` show "summable (?f x)" by auto } ``` hoelzl@29803 ` 782` ``` show "summable (?f' x0)" using converges[OF `x0 \ {-R <..< R}`] . ``` hoelzl@29803 ` 783` ``` show "x0 \ {-R' <..< R'}" using `x0 \ {-R' <..< R'}` . ``` hoelzl@29803 ` 784` ``` qed ``` hoelzl@29803 ` 785` ``` } note for_subinterval = this ``` hoelzl@29803 ` 786` ``` let ?R = "(R + \x0\) / 2" ``` hoelzl@29803 ` 787` ``` have "\x0\ < ?R" using assms by auto ``` hoelzl@29803 ` 788` ``` hence "- ?R < x0" ``` hoelzl@29803 ` 789` ``` proof (cases "x0 < 0") ``` hoelzl@29803 ` 790` ``` case True ``` hoelzl@29803 ` 791` ``` hence "- x0 < ?R" using `\x0\ < ?R` by auto ``` hoelzl@29803 ` 792` ``` thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto ``` hoelzl@29803 ` 793` ``` next ``` hoelzl@29803 ` 794` ``` case False ``` hoelzl@29803 ` 795` ``` have "- ?R < 0" using assms by auto ``` hoelzl@41970 ` 796` ``` also have "\ \ x0" using False by auto ``` hoelzl@29803 ` 797` ``` finally show ?thesis . ``` hoelzl@29803 ` 798` ``` qed ``` hoelzl@29803 ` 799` ``` hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto ``` hoelzl@29803 ` 800` ``` from for_subinterval[OF this] ``` hoelzl@29803 ` 801` ``` show ?thesis . ``` hoelzl@29803 ` 802` ```qed ``` chaieb@29695 ` 803` huffman@29164 ` 804` ```subsection {* Exponential Function *} ``` huffman@23043 ` 805` huffman@44308 ` 806` ```definition exp :: "'a \ 'a::{real_normed_field,banach}" where ``` huffman@44308 ` 807` ``` "exp = (\x. \n. x ^ n /\<^sub>R real (fact n))" ``` huffman@23043 ` 808` huffman@23115 ` 809` ```lemma summable_exp_generic: ``` haftmann@31017 ` 810` ``` fixes x :: "'a::{real_normed_algebra_1,banach}" ``` haftmann@25062 ` 811` ``` defines S_def: "S \ \n. x ^ n /\<^sub>R real (fact n)" ``` huffman@23115 ` 812` ``` shows "summable S" ``` huffman@23115 ` 813` ```proof - ``` haftmann@25062 ` 814` ``` have S_Suc: "\n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)" ``` huffman@30273 ` 815` ``` unfolding S_def by (simp del: mult_Suc) ``` huffman@23115 ` 816` ``` obtain r :: real where r0: "0 < r" and r1: "r < 1" ``` huffman@23115 ` 817` ``` using dense [OF zero_less_one] by fast ``` huffman@23115 ` 818` ``` obtain N :: nat where N: "norm x < real N * r" ``` huffman@23115 ` 819` ``` using reals_Archimedean3 [OF r0] by fast ``` huffman@23115 ` 820` ``` from r1 show ?thesis ``` huffman@23115 ` 821` ``` proof (rule ratio_test [rule_format]) ``` huffman@23115 ` 822` ``` fix n :: nat ``` huffman@23115 ` 823` ``` assume n: "N \ n" ``` huffman@23115 ` 824` ``` have "norm x \ real N * r" ``` huffman@23115 ` 825` ``` using N by (rule order_less_imp_le) ``` huffman@23115 ` 826` ``` also have "real N * r \ real (Suc n) * r" ``` huffman@23115 ` 827` ``` using r0 n by (simp add: mult_right_mono) ``` huffman@23115 ` 828` ``` finally have "norm x * norm (S n) \ real (Suc n) * r * norm (S n)" ``` huffman@23115 ` 829` ``` using norm_ge_zero by (rule mult_right_mono) ``` huffman@23115 ` 830` ``` hence "norm (x * S n) \ real (Suc n) * r * norm (S n)" ``` huffman@23115 ` 831` ``` by (rule order_trans [OF norm_mult_ineq]) ``` huffman@23115 ` 832` ``` hence "norm (x * S n) / real (Suc n) \ r * norm (S n)" ``` huffman@23115 ` 833` ``` by (simp add: pos_divide_le_eq mult_ac) ``` huffman@23115 ` 834` ``` thus "norm (S (Suc n)) \ r * norm (S n)" ``` huffman@35216 ` 835` ``` by (simp add: S_Suc inverse_eq_divide) ``` huffman@23115 ` 836` ``` qed ``` huffman@23115 ` 837` ```qed ``` huffman@23115 ` 838` huffman@23115 ` 839` ```lemma summable_norm_exp: ``` haftmann@31017 ` 840` ``` fixes x :: "'a::{real_normed_algebra_1,banach}" ``` haftmann@25062 ` 841` ``` shows "summable (\n. norm (x ^ n /\<^sub>R real (fact n)))" ``` huffman@23115 ` 842` ```proof (rule summable_norm_comparison_test [OF exI, rule_format]) ``` haftmann@25062 ` 843` ``` show "summable (\n. norm x ^ n /\<^sub>R real (fact n))" ``` huffman@23115 ` 844` ``` by (rule summable_exp_generic) ``` huffman@23115 ` 845` ```next ``` haftmann@25062 ` 846` ``` fix n show "norm (x ^ n /\<^sub>R real (fact n)) \ norm x ^ n /\<^sub>R real (fact n)" ``` huffman@35216 ` 847` ``` by (simp add: norm_power_ineq) ``` huffman@23115 ` 848` ```qed ``` huffman@23115 ` 849` huffman@23043 ` 850` ```lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)" ``` huffman@23115 ` 851` ```by (insert summable_exp_generic [where x=x], simp) ``` huffman@23043 ` 852` haftmann@25062 ` 853` ```lemma exp_converges: "(\n. x ^ n /\<^sub>R real (fact n)) sums exp x" ``` huffman@23115 ` 854` ```unfolding exp_def by (rule summable_exp_generic [THEN summable_sums]) ``` huffman@23043 ` 855` huffman@23043 ` 856` hoelzl@41970 ` 857` ```lemma exp_fdiffs: ``` paulson@15077 ` 858` ``` "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))" ``` huffman@23431 ` 859` ```by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult ``` huffman@23082 ` 860` ``` del: mult_Suc of_nat_Suc) ``` paulson@15077 ` 861` huffman@23115 ` 862` ```lemma diffs_of_real: "diffs (\n. of_real (f n)) = (\n. of_real (diffs f n))" ``` huffman@23115 ` 863` ```by (simp add: diffs_def) ``` huffman@23115 ` 864` paulson@15077 ` 865` ```lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)" ``` huffman@44308 ` 866` ```unfolding exp_def scaleR_conv_of_real ``` huffman@44308 ` 867` ```apply (rule DERIV_cong) ``` huffman@44308 ` 868` ```apply (rule termdiffs [where K="of_real (1 + norm x)"]) ``` huffman@23115 ` 869` ```apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs) ``` huffman@23115 ` 870` ```apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+ ``` huffman@23115 ` 871` ```apply (simp del: of_real_add) ``` paulson@15077 ` 872` ```done ``` paulson@15077 ` 873` huffman@44311 ` 874` ```lemma isCont_exp: "isCont exp x" ``` huffman@44311 ` 875` ``` by (rule DERIV_exp [THEN DERIV_isCont]) ``` huffman@44311 ` 876` huffman@44311 ` 877` ```lemma isCont_exp' [simp]: "isCont f a \ isCont (\x. exp (f x)) a" ``` huffman@44311 ` 878` ``` by (rule isCont_o2 [OF _ isCont_exp]) ``` huffman@44311 ` 879` huffman@44311 ` 880` ```lemma tendsto_exp [tendsto_intros]: ``` huffman@44311 ` 881` ``` "(f ---> a) F \ ((\x. exp (f x)) ---> exp a) F" ``` huffman@44311 ` 882` ``` by (rule isCont_tendsto_compose [OF isCont_exp]) ``` huffman@23045 ` 883` huffman@23045 ` 884` huffman@29167 ` 885` ```subsubsection {* Properties of the Exponential Function *} ``` paulson@15077 ` 886` huffman@23278 ` 887` ```lemma powser_zero: ``` haftmann@31017 ` 888` ``` fixes f :: "nat \ 'a::{real_normed_algebra_1}" ``` huffman@23278 ` 889` ``` shows "(\n. f n * 0 ^ n) = f 0" ``` paulson@15077 ` 890` ```proof - ``` huffman@23278 ` 891` ``` have "(\n = 0..<1. f n * 0 ^ n) = (\n. f n * 0 ^ n)" ``` huffman@23115 ` 892` ``` by (rule sums_unique [OF series_zero], simp add: power_0_left) ``` huffman@30082 ` 893` ``` thus ?thesis unfolding One_nat_def by simp ``` paulson@15077 ` 894` ```qed ``` paulson@15077 ` 895` huffman@23278 ` 896` ```lemma exp_zero [simp]: "exp 0 = 1" ``` huffman@23278 ` 897` ```unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero) ``` huffman@23278 ` 898` huffman@23115 ` 899` ```lemma setsum_cl_ivl_Suc2: ``` huffman@23115 ` 900` ``` "(\i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\i=m..n. f (Suc i)))" ``` nipkow@28069 ` 901` ```by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl ``` huffman@23115 ` 902` ``` del: setsum_cl_ivl_Suc) ``` huffman@23115 ` 903` huffman@23115 ` 904` ```lemma exp_series_add: ``` haftmann@31017 ` 905` ``` fixes x y :: "'a::{real_field}" ``` haftmann@25062 ` 906` ``` defines S_def: "S \ \x n. x ^ n /\<^sub>R real (fact n)" ``` huffman@23115 ` 907` ``` shows "S (x + y) n = (\i=0..n. S x i * S y (n - i))" ``` huffman@23115 ` 908` ```proof (induct n) ``` huffman@23115 ` 909` ``` case 0 ``` huffman@23115 ` 910` ``` show ?case ``` huffman@23115 ` 911` ``` unfolding S_def by simp ``` huffman@23115 ` 912` ```next ``` huffman@23115 ` 913` ``` case (Suc n) ``` haftmann@25062 ` 914` ``` have S_Suc: "\x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)" ``` huffman@30273 ` 915` ``` unfolding S_def by (simp del: mult_Suc) ``` haftmann@25062 ` 916` ``` hence times_S: "\x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)" ``` huffman@23115 ` 917` ``` by simp ``` huffman@23115 ` 918` haftmann@25062 ` 919` ``` have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n" ``` huffman@23115 ` 920` ``` by (simp only: times_S) ``` huffman@23115 ` 921` ``` also have "\ = (x + y) * (\i=0..n. S x i * S y (n-i))" ``` huffman@23115 ` 922` ``` by (simp only: Suc) ``` huffman@23115 ` 923` ``` also have "\ = x * (\i=0..n. S x i * S y (n-i)) ``` huffman@23115 ` 924` ``` + y * (\i=0..n. S x i * S y (n-i))" ``` huffman@23115 ` 925` ``` by (rule left_distrib) ``` huffman@23115 ` 926` ``` also have "\ = (\i=0..n. (x * S x i) * S y (n-i)) ``` huffman@23115 ` 927` ``` + (\i=0..n. S x i * (y * S y (n-i)))" ``` huffman@23115 ` 928` ``` by (simp only: setsum_right_distrib mult_ac) ``` haftmann@25062 ` 929` ``` also have "\ = (\i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) ``` haftmann@25062 ` 930` ``` + (\i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))" ``` huffman@23115 ` 931` ``` by (simp add: times_S Suc_diff_le) ``` haftmann@25062 ` 932` ``` also have "(\i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) = ``` haftmann@25062 ` 933` ``` (\i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))" ``` huffman@23115 ` 934` ``` by (subst setsum_cl_ivl_Suc2, simp) ``` haftmann@25062 ` 935` ``` also have "(\i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) = ``` haftmann@25062 ` 936` ``` (\i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))" ``` huffman@23115 ` 937` ``` by (subst setsum_cl_ivl_Suc, simp) ``` haftmann@25062 ` 938` ``` also have "(\i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) + ``` haftmann@25062 ` 939` ``` (\i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) = ``` haftmann@25062 ` 940` ``` (\i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))" ``` huffman@23115 ` 941` ``` by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric] ``` huffman@23115 ` 942` ``` real_of_nat_add [symmetric], simp) ``` haftmann@25062 ` 943` ``` also have "\ = real (Suc n) *\<^sub>R (\i=0..Suc n. S x i * S y (Suc n-i))" ``` huffman@23127 ` 944` ``` by (simp only: scaleR_right.setsum) ``` huffman@23115 ` 945` ``` finally show ``` huffman@23115 ` 946` ``` "S (x + y) (Suc n) = (\i=0..Suc n. S x i * S y (Suc n - i))" ``` huffman@35216 ` 947` ``` by (simp del: setsum_cl_ivl_Suc) ``` huffman@23115 ` 948` ```qed ``` huffman@23115 ` 949` huffman@23115 ` 950` ```lemma exp_add: "exp (x + y) = exp x * exp y" ``` huffman@23115 ` 951` ```unfolding exp_def ``` huffman@23115 ` 952` ```by (simp only: Cauchy_product summable_norm_exp exp_series_add) ``` huffman@23115 ` 953` huffman@29170 ` 954` ```lemma mult_exp_exp: "exp x * exp y = exp (x + y)" ``` huffman@29170 ` 955` ```by (rule exp_add [symmetric]) ``` huffman@29170 ` 956` huffman@23241 ` 957` ```lemma exp_of_real: "exp (of_real x) = of_real (exp x)" ``` huffman@23241 ` 958` ```unfolding exp_def ``` huffman@44282 ` 959` ```apply (subst suminf_of_real) ``` huffman@23241 ` 960` ```apply (rule summable_exp_generic) ``` huffman@23241 ` 961` ```apply (simp add: scaleR_conv_of_real) ``` huffman@23241 ` 962` ```done ``` huffman@23241 ` 963` huffman@29170 ` 964` ```lemma exp_not_eq_zero [simp]: "exp x \ 0" ``` huffman@29170 ` 965` ```proof ``` huffman@29170 ` 966` ``` have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp) ``` huffman@29170 ` 967` ``` also assume "exp x = 0" ``` huffman@29170 ` 968` ``` finally show "False" by simp ``` paulson@15077 ` 969` ```qed ``` paulson@15077 ` 970` huffman@29170 ` 971` ```lemma exp_minus: "exp (- x) = inverse (exp x)" ``` huffman@29170 ` 972` ```by (rule inverse_unique [symmetric], simp add: mult_exp_exp) ``` paulson@15077 ` 973` huffman@29170 ` 974` ```lemma exp_diff: "exp (x - y) = exp x / exp y" ``` huffman@29170 ` 975` ``` unfolding diff_minus divide_inverse ``` huffman@29170 ` 976` ``` by (simp add: exp_add exp_minus) ``` paulson@15077 ` 977` huffman@29167 ` 978` huffman@29167 ` 979` ```subsubsection {* Properties of the Exponential Function on Reals *} ``` huffman@29167 ` 980` huffman@29170 ` 981` ```text {* Comparisons of @{term "exp x"} with zero. *} ``` huffman@29167 ` 982` huffman@29167 ` 983` ```text{*Proof: because every exponential can be seen as a square.*} ``` huffman@29167 ` 984` ```lemma exp_ge_zero [simp]: "0 \ exp (x::real)" ``` huffman@29167 ` 985` ```proof - ``` huffman@29167 ` 986` ``` have "0 \ exp (x/2) * exp (x/2)" by simp ``` huffman@29167 ` 987` ``` thus ?thesis by (simp add: exp_add [symmetric]) ``` huffman@29167 ` 988` ```qed ``` huffman@29167 ` 989` huffman@23115 ` 990` ```lemma exp_gt_zero [simp]: "0 < exp (x::real)" ``` paulson@15077 ` 991` ```by (simp add: order_less_le) ``` paulson@15077 ` 992` huffman@29170 ` 993` ```lemma not_exp_less_zero [simp]: "\ exp (x::real) < 0" ``` huffman@29170 ` 994` ```by (simp add: not_less) ``` huffman@29170 ` 995` huffman@29170 ` 996` ```lemma not_exp_le_zero [simp]: "\ exp (x::real) \ 0" ``` huffman@29170 ` 997` ```by (simp add: not_le) ``` paulson@15077 ` 998` huffman@23115 ` 999` ```lemma abs_exp_cancel [simp]: "\exp x::real\ = exp x" ``` huffman@29165 ` 1000` ```by simp ``` paulson@15077 ` 1001` paulson@15077 ` 1002` ```lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n" ``` paulson@15251 ` 1003` ```apply (induct "n") ``` paulson@15077 ` 1004` ```apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute) ``` paulson@15077 ` 1005` ```done ``` paulson@15077 ` 1006` huffman@29170 ` 1007` ```text {* Strict monotonicity of exponential. *} ``` huffman@29170 ` 1008` huffman@29170 ` 1009` ```lemma exp_ge_add_one_self_aux: "0 \ (x::real) ==> (1 + x) \ exp(x)" ``` huffman@29170 ` 1010` ```apply (drule order_le_imp_less_or_eq, auto) ``` huffman@29170 ` 1011` ```apply (simp add: exp_def) ``` huffman@36777 ` 1012` ```apply (rule order_trans) ``` huffman@29170 ` 1013` ```apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le) ``` huffman@29170 ` 1014` ```apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff) ``` huffman@29170 ` 1015` ```done ``` huffman@29170 ` 1016` huffman@29170 ` 1017` ```lemma exp_gt_one: "0 < (x::real) \ 1 < exp x" ``` huffman@29170 ` 1018` ```proof - ``` huffman@29170 ` 1019` ``` assume x: "0 < x" ``` huffman@29170 ` 1020` ``` hence "1 < 1 + x" by simp ``` huffman@29170 ` 1021` ``` also from x have "1 + x \ exp x" ``` huffman@29170 ` 1022` ``` by (simp add: exp_ge_add_one_self_aux) ``` huffman@29170 ` 1023` ``` finally show ?thesis . ``` huffman@29170 ` 1024` ```qed ``` huffman@29170 ` 1025` paulson@15077 ` 1026` ```lemma exp_less_mono: ``` huffman@23115 ` 1027` ``` fixes x y :: real ``` huffman@29165 ` 1028` ``` assumes "x < y" shows "exp x < exp y" ``` paulson@15077 ` 1029` ```proof - ``` huffman@29165 ` 1030` ``` from `x < y` have "0 < y - x" by simp ``` huffman@29165 ` 1031` ``` hence "1 < exp (y - x)" by (rule exp_gt_one) ``` huffman@29165 ` 1032` ``` hence "1 < exp y / exp x" by (simp only: exp_diff) ``` huffman@29165 ` 1033` ``` thus "exp x < exp y" by simp ``` paulson@15077 ` 1034` ```qed ``` paulson@15077 ` 1035` huffman@23115 ` 1036` ```lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y" ``` huffman@29170 ` 1037` ```apply (simp add: linorder_not_le [symmetric]) ``` huffman@29170 ` 1038` ```apply (auto simp add: order_le_less exp_less_mono) ``` paulson@15077 ` 1039` ```done ``` paulson@15077 ` 1040` huffman@29170 ` 1041` ```lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \ x < y" ``` paulson@15077 ` 1042` ```by (auto intro: exp_less_mono exp_less_cancel) ``` paulson@15077 ` 1043` huffman@29170 ` 1044` ```lemma exp_le_cancel_iff [iff]: "exp (x::real) \ exp y \ x \ y" ``` paulson@15077 ` 1045` ```by (auto simp add: linorder_not_less [symmetric]) ``` paulson@15077 ` 1046` huffman@29170 ` 1047` ```lemma exp_inj_iff [iff]: "exp (x::real) = exp y \ x = y" ``` paulson@15077 ` 1048` ```by (simp add: order_eq_iff) ``` paulson@15077 ` 1049` huffman@29170 ` 1050` ```text {* Comparisons of @{term "exp x"} with one. *} ``` huffman@29170 ` 1051` huffman@29170 ` 1052` ```lemma one_less_exp_iff [simp]: "1 < exp (x::real) \ 0 < x" ``` huffman@29170 ` 1053` ``` using exp_less_cancel_iff [where x=0 and y=x] by simp ``` huffman@29170 ` 1054` huffman@29170 ` 1055` ```lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \ x < 0" ``` huffman@29170 ` 1056` ``` using exp_less_cancel_iff [where x=x and y=0] by simp ``` huffman@29170 ` 1057` huffman@29170 ` 1058` ```lemma one_le_exp_iff [simp]: "1 \ exp (x::real) \ 0 \ x" ``` huffman@29170 ` 1059` ``` using exp_le_cancel_iff [where x=0 and y=x] by simp ``` huffman@29170 ` 1060` huffman@29170 ` 1061` ```lemma exp_le_one_iff [simp]: "exp (x::real) \ 1 \ x \ 0" ``` huffman@29170 ` 1062` ``` using exp_le_cancel_iff [where x=x and y=0] by simp ``` huffman@29170 ` 1063` huffman@29170 ` 1064` ```lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \ x = 0" ``` huffman@29170 ` 1065` ``` using exp_inj_iff [where x=x and y=0] by simp ``` huffman@29170 ` 1066` huffman@23115 ` 1067` ```lemma lemma_exp_total: "1 \ y ==> \x. 0 \ x & x \ y - 1 & exp(x::real) = y" ``` huffman@44755 ` 1068` ```proof (rule IVT) ``` huffman@44755 ` 1069` ``` assume "1 \ y" ``` huffman@44755 ` 1070` ``` hence "0 \ y - 1" by simp ``` huffman@44755 ` 1071` ``` hence "1 + (y - 1) \ exp (y - 1)" by (rule exp_ge_add_one_self_aux) ``` huffman@44755 ` 1072` ``` thus "y \ exp (y - 1)" by simp ``` huffman@44755 ` 1073` ```qed (simp_all add: le_diff_eq) ``` paulson@15077 ` 1074` huffman@23115 ` 1075` ```lemma exp_total: "0 < (y::real) ==> \x. exp x = y" ``` huffman@44755 ` 1076` ```proof (rule linorder_le_cases [of 1 y]) ``` huffman@44755 ` 1077` ``` assume "1 \ y" thus "\x. exp x = y" ``` huffman@44755 ` 1078` ``` by (fast dest: lemma_exp_total) ``` huffman@44755 ` 1079` ```next ``` huffman@44755 ` 1080` ``` assume "0 < y" and "y \ 1" ``` huffman@44755 ` 1081` ``` hence "1 \ inverse y" by (simp add: one_le_inverse_iff) ``` huffman@44755 ` 1082` ``` then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total) ``` huffman@44755 ` 1083` ``` hence "exp (- x) = y" by (simp add: exp_minus) ``` huffman@44755 ` 1084` ``` thus "\x. exp x = y" .. ``` huffman@44755 ` 1085` ```qed ``` paulson@15077 ` 1086` paulson@15077 ` 1087` huffman@29164 ` 1088` ```subsection {* Natural Logarithm *} ``` paulson@15077 ` 1089` huffman@44308 ` 1090` ```definition ln :: "real \ real" where ``` huffman@23043 ` 1091` ``` "ln x = (THE u. exp u = x)" ``` huffman@23043 ` 1092` huffman@23043 ` 1093` ```lemma ln_exp [simp]: "ln (exp x) = x" ``` huffman@44308 ` 1094` ``` by (simp add: ln_def) ``` paulson@15077 ` 1095` huffman@22654 ` 1096` ```lemma exp_ln [simp]: "0 < x \ exp (ln x) = x" ``` huffman@44308 ` 1097` ``` by (auto dest: exp_total) ``` huffman@22654 ` 1098` huffman@29171 ` 1099` ```lemma exp_ln_iff [simp]: "exp (ln x) = x \ 0 < x" ``` huffman@44308 ` 1100` ``` by (metis exp_gt_zero exp_ln) ``` paulson@15077 ` 1101` huffman@29171 ` 1102` ```lemma ln_unique: "exp y = x \ ln x = y" ``` huffman@44308 ` 1103` ``` by (erule subst, rule ln_exp) ``` huffman@29171 ` 1104` huffman@29171 ` 1105` ```lemma ln_one [simp]: "ln 1 = 0" ``` huffman@44308 ` 1106` ``` by (rule ln_unique, simp) ``` huffman@29171 ` 1107` huffman@29171 ` 1108` ```lemma ln_mult: "\0 < x; 0 < y\ \ ln (x * y) = ln x + ln y" ``` huffman@44308 ` 1109` ``` by (rule ln_unique, simp add: exp_add) ``` huffman@29171 ` 1110` huffman@29171 ` 1111` ```lemma ln_inverse: "0 < x \ ln (inverse x) = - ln x" ``` huffman@44308 ` 1112` ``` by (rule ln_unique, simp add: exp_minus) ``` huffman@29171 ` 1113` huffman@29171 ` 1114` ```lemma ln_div: "\0 < x; 0 < y\ \ ln (x / y) = ln x - ln y" ``` huffman@44308 ` 1115` ``` by (rule ln_unique, simp add: exp_diff) ``` paulson@15077 ` 1116` huffman@29171 ` 1117` ```lemma ln_realpow: "0 < x \ ln (x ^ n) = real n * ln x" ``` huffman@44308 ` 1118` ``` by (rule ln_unique, simp add: exp_real_of_nat_mult) ``` huffman@29171 ` 1119` huffman@29171 ` 1120` ```lemma ln_less_cancel_iff [simp]: "\0 < x; 0 < y\ \ ln x < ln y \ x < y" ``` huffman@44308 ` 1121` ``` by (subst exp_less_cancel_iff [symmetric], simp) ``` huffman@29171 ` 1122` huffman@29171 ` 1123` ```lemma ln_le_cancel_iff [simp]: "\0 < x; 0 < y\ \ ln x \ ln y \ x \ y" ``` huffman@44308 ` 1124` ``` by (simp add: linorder_not_less [symmetric]) ``` huffman@29171 ` 1125` huffman@29171 ` 1126` ```lemma ln_inj_iff [simp]: "\0 < x; 0 < y\ \ ln x = ln y \ x = y" ``` huffman@44308 ` 1127` ``` by (simp add: order_eq_iff) ``` huffman@29171 ` 1128` huffman@29171 ` 1129` ```lemma ln_add_one_self_le_self [simp]: "0 \ x \ ln (1 + x) \ x" ``` huffman@44308 ` 1130` ``` apply (rule exp_le_cancel_iff [THEN iffD1]) ``` huffman@44308 ` 1131` ``` apply (simp add: exp_ge_add_one_self_aux) ``` huffman@44308 ` 1132` ``` done ``` paulson@15077 ` 1133` huffman@29171 ` 1134` ```lemma ln_less_self [simp]: "0 < x \ ln x < x" ``` huffman@44308 ` 1135` ``` by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all ``` huffman@44308 ` 1136` huffman@44308 ` 1137` ```lemma ln_ge_zero [simp]: "1 \ x \ 0 \ ln x" ``` huffman@44308 ` 1138` ``` using ln_le_cancel_iff [of 1 x] by simp ``` huffman@44308 ` 1139` huffman@44308 ` 1140` ```lemma ln_ge_zero_imp_ge_one: "\0 \ ln x; 0 < x\ \ 1 \ x" ``` huffman@44308 ` 1141` ``` using ln_le_cancel_iff [of 1 x] by simp ``` huffman@44308 ` 1142` huffman@44308 ` 1143` ```lemma ln_ge_zero_iff [simp]: "0 < x \ (0 \ ln x) = (1 \ x)" ``` huffman@44308 ` 1144` ``` using ln_le_cancel_iff [of 1 x] by simp ``` huffman@44308 ` 1145` huffman@44308 ` 1146` ```lemma ln_less_zero_iff [simp]: "0 < x \ (ln x < 0) = (x < 1)" ``` huffman@44308 ` 1147` ``` using ln_less_cancel_iff [of x 1] by simp ``` huffman@44308 ` 1148` huffman@44308 ` 1149` ```lemma ln_gt_zero: "1 < x \ 0 < ln x" ``` huffman@44308 ` 1150` ``` using ln_less_cancel_iff [of 1 x] by simp ``` huffman@44308 ` 1151` huffman@44308 ` 1152` ```lemma ln_gt_zero_imp_gt_one: "\0 < ln x; 0 < x\ \ 1 < x" ``` huffman@44308 ` 1153` ``` using ln_less_cancel_iff [of 1 x] by simp ``` huffman@44308 ` 1154` huffman@44308 ` 1155` ```lemma ln_gt_zero_iff [simp]: "0 < x \ (0 < ln x) = (1 < x)" ``` huffman@44308 ` 1156` ``` using ln_less_cancel_iff [of 1 x] by simp ``` huffman@44308 ` 1157` huffman@44308 ` 1158` ```lemma ln_eq_zero_iff [simp]: "0 < x \ (ln x = 0) = (x = 1)" ``` huffman@44308 ` 1159` ``` using ln_inj_iff [of x 1] by simp ``` huffman@44308 ` 1160` huffman@44308 ` 1161` ```lemma ln_less_zero: "\0 < x; x < 1\ \ ln x < 0" ``` huffman@44308 ` 1162` ``` by simp ``` paulson@15077 ` 1163` huffman@23045 ` 1164` ```lemma isCont_ln: "0 < x \ isCont ln x" ``` huffman@44308 ` 1165` ``` apply (subgoal_tac "isCont ln (exp (ln x))", simp) ``` huffman@44308 ` 1166` ``` apply (rule isCont_inverse_function [where f=exp], simp_all) ``` huffman@44308 ` 1167` ``` done ``` huffman@23045 ` 1168` huffman@45915 ` 1169` ```lemma tendsto_ln [tendsto_intros]: ``` huffman@45915 ` 1170` ``` "\(f ---> a) F; 0 < a\ \ ((\x. ln (f x)) ---> ln a) F" ``` huffman@45915 ` 1171` ``` by (rule isCont_tendsto_compose [OF isCont_ln]) ``` huffman@45915 ` 1172` huffman@23045 ` 1173` ```lemma DERIV_ln: "0 < x \ DERIV ln x :> inverse x" ``` huffman@44308 ` 1174` ``` apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"]) ``` huffman@44317 ` 1175` ``` apply (erule DERIV_cong [OF DERIV_exp exp_ln]) ``` huffman@44308 ` 1176` ``` apply (simp_all add: abs_if isCont_ln) ``` huffman@44308 ` 1177` ``` done ``` huffman@23045 ` 1178` paulson@33667 ` 1179` ```lemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x" ``` paulson@33667 ` 1180` ``` by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse) ``` paulson@33667 ` 1181` hoelzl@29803 ` 1182` ```lemma ln_series: assumes "0 < x" and "x < 2" ``` hoelzl@29803 ` 1183` ``` shows "ln x = (\ n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))") ``` hoelzl@29803 ` 1184` ```proof - ``` hoelzl@29803 ` 1185` ``` let "?f' x n" = "(-1)^n * (x - 1)^n" ``` hoelzl@29803 ` 1186` hoelzl@29803 ` 1187` ``` have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))" ``` hoelzl@29803 ` 1188` ``` proof (rule DERIV_isconst3[where x=x]) ``` hoelzl@29803 ` 1189` ``` fix x :: real assume "x \ {0 <..< 2}" hence "0 < x" and "x < 2" by auto ``` hoelzl@29803 ` 1190` ``` have "norm (1 - x) < 1" using `0 < x` and `x < 2` by auto ``` hoelzl@29803 ` 1191` ``` have "1 / x = 1 / (1 - (1 - x))" by auto ``` hoelzl@29803 ` 1192` ``` also have "\ = (\ n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique) ``` hoelzl@29803 ` 1193` ``` also have "\ = suminf (?f' x)" unfolding power_mult_distrib[symmetric] by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto) ``` huffman@36777 ` 1194` ``` finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto ``` hoelzl@29803 ` 1195` ``` moreover ``` hoelzl@29803 ` 1196` ``` have repos: "\ h x :: real. h - 1 + x = h + x - 1" by auto ``` hoelzl@29803 ` 1197` ``` have "DERIV (\x. suminf (?f x)) (x - 1) :> (\n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)" ``` hoelzl@29803 ` 1198` ``` proof (rule DERIV_power_series') ``` hoelzl@29803 ` 1199` ``` show "x - 1 \ {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto ``` hoelzl@29803 ` 1200` ``` { fix x :: real assume "x \ {- 1<..<1}" hence "norm (-x) < 1" by auto ``` wenzelm@32960 ` 1201` ``` show "summable (\n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)" ``` huffman@30082 ` 1202` ``` unfolding One_nat_def ``` huffman@35216 ` 1203` ``` by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`]) ``` hoelzl@29803 ` 1204` ``` } ``` hoelzl@29803 ` 1205` ``` qed ``` huffman@30082 ` 1206` ``` hence "DERIV (\x. suminf (?f x)) (x - 1) :> suminf (?f' x)" unfolding One_nat_def by auto ``` hoelzl@29803 ` 1207` ``` hence "DERIV (\x. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos . ``` hoelzl@29803 ` 1208` ``` ultimately have "DERIV (\x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))" ``` hoelzl@29803 ` 1209` ``` by (rule DERIV_diff) ``` hoelzl@29803 ` 1210` ``` thus "DERIV (\x. ln x - suminf (?f (x - 1))) x :> 0" by auto ``` hoelzl@29803 ` 1211` ``` qed (auto simp add: assms) ``` huffman@44289 ` 1212` ``` thus ?thesis by auto ``` hoelzl@29803 ` 1213` ```qed ``` paulson@15077 ` 1214` huffman@29164 ` 1215` ```subsection {* Sine and Cosine *} ``` huffman@29164 ` 1216` huffman@44308 ` 1217` ```definition sin_coeff :: "nat \ real" where ``` huffman@31271 ` 1218` ``` "sin_coeff = (\n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))" ``` huffman@31271 ` 1219` huffman@44308 ` 1220` ```definition cos_coeff :: "nat \ real" where ``` huffman@31271 ` 1221` ``` "cos_coeff = (\n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)" ``` huffman@31271 ` 1222` huffman@44308 ` 1223` ```definition sin :: "real \ real" where ``` huffman@44308 ` 1224` ``` "sin = (\x. \n. sin_coeff n * x ^ n)" ``` huffman@44308 ` 1225` huffman@44308 ` 1226` ```definition cos :: "real \ real" where ``` huffman@44308 ` 1227` ``` "cos = (\x. \n. cos_coeff n * x ^ n)" ``` huffman@31271 ` 1228` huffman@44319 ` 1229` ```lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0" ``` huffman@44319 ` 1230` ``` unfolding sin_coeff_def by simp ``` huffman@44319 ` 1231` huffman@44319 ` 1232` ```lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1" ``` huffman@44319 ` 1233` ``` unfolding cos_coeff_def by simp ``` huffman@44319 ` 1234` huffman@44319 ` 1235` ```lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)" ``` huffman@44319 ` 1236` ``` unfolding cos_coeff_def sin_coeff_def ``` huffman@44319 ` 1237` ``` by (simp del: mult_Suc) ``` huffman@44319 ` 1238` huffman@44319 ` 1239` ```lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)" ``` huffman@44319 ` 1240` ``` unfolding cos_coeff_def sin_coeff_def ``` huffman@44319 ` 1241` ``` by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex) ``` huffman@44319 ` 1242` huffman@31271 ` 1243` ```lemma summable_sin: "summable (\n. sin_coeff n * x ^ n)" ``` huffman@31271 ` 1244` ```unfolding sin_coeff_def ``` huffman@44308 ` 1245` ```apply (rule summable_comparison_test [OF _ summable_exp [where x="\x\"]]) ``` huffman@29164 ` 1246` ```apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) ``` huffman@29164 ` 1247` ```done ``` huffman@29164 ` 1248` huffman@31271 ` 1249` ```lemma summable_cos: "summable (\n. cos_coeff n * x ^ n)" ``` huffman@31271 ` 1250` ```unfolding cos_coeff_def ``` huffman@44308 ` 1251` ```apply (rule summable_comparison_test [OF _ summable_exp [where x="\x\"]]) ``` huffman@29164 ` 1252` ```apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) ``` huffman@29164 ` 1253` ```done ``` huffman@29164 ` 1254` huffman@31271 ` 1255` ```lemma sin_converges: "(\n. sin_coeff n * x ^ n) sums sin(x)" ``` huffman@29164 ` 1256` ```unfolding sin_def by (rule summable_sin [THEN summable_sums]) ``` huffman@29164 ` 1257` huffman@31271 ` 1258` ```lemma cos_converges: "(\n. cos_coeff n * x ^ n) sums cos(x)" ``` huffman@29164 ` 1259` ```unfolding cos_def by (rule summable_cos [THEN summable_sums]) ``` huffman@29164 ` 1260` huffman@44319 ` 1261` ```lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff" ``` huffman@44319 ` 1262` ``` by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc) ``` huffman@44319 ` 1263` huffman@44319 ` 1264` ```lemma diffs_cos_coeff: "diffs cos_coeff = (\n. - sin_coeff n)" ``` huffman@44319 ` 1265` ``` by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc) ``` huffman@29164 ` 1266` huffman@29164 ` 1267` ```text{*Now at last we can get the derivatives of exp, sin and cos*} ``` huffman@29164 ` 1268` huffman@29164 ` 1269` ```lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)" ``` huffman@44319 ` 1270` ``` unfolding sin_def cos_def ``` huffman@44319 ` 1271` ``` apply (rule DERIV_cong, rule termdiffs [where K="1 + \x\"]) ``` huffman@44319 ` 1272` ``` apply (simp_all add: diffs_sin_coeff diffs_cos_coeff ``` huffman@44319 ` 1273` ``` summable_minus summable_sin summable_cos) ``` huffman@44319 ` 1274` ``` done ``` huffman@29164 ` 1275` huffman@29164 ` 1276` ```lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)" ``` huffman@44319 ` 1277` ``` unfolding cos_def sin_def ``` huffman@44319 ` 1278` ``` apply (rule DERIV_cong, rule termdiffs [where K="1 + \x\"]) ``` huffman@44319 ` 1279` ``` apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus ``` huffman@44319 ` 1280` ``` summable_minus summable_sin summable_cos suminf_minus) ``` huffman@44319 ` 1281` ``` done ``` huffman@29164 ` 1282` huffman@44311 ` 1283` ```lemma isCont_sin: "isCont sin x" ``` huffman@44311 ` 1284` ``` by (rule DERIV_sin [THEN DERIV_isCont]) ``` huffman@44311 ` 1285` huffman@44311 ` 1286` ```lemma isCont_cos: "isCont cos x" ``` huffman@44311 ` 1287` ``` by (rule DERIV_cos [THEN DERIV_isCont]) ``` huffman@44311 ` 1288` huffman@44311 ` 1289` ```lemma isCont_sin' [simp]: "isCont f a \ isCont (\x. sin (f x)) a" ``` huffman@44311 ` 1290` ``` by (rule isCont_o2 [OF _ isCont_sin]) ``` huffman@44311 ` 1291` huffman@44311 ` 1292` ```lemma isCont_cos' [simp]: "isCont f a \ isCont (\x. cos (f x)) a" ``` huffman@44311 ` 1293` ``` by (rule isCont_o2 [OF _ isCont_cos]) ``` huffman@44311 ` 1294` huffman@44311 ` 1295` ```lemma tendsto_sin [tendsto_intros]: ``` huffman@44311 ` 1296` ``` "(f ---> a) F \ ((\x. sin (f x)) ---> sin a) F" ``` huffman@44311 ` 1297` ``` by (rule isCont_tendsto_compose [OF isCont_sin]) ``` huffman@44311 ` 1298` huffman@44311 ` 1299` ```lemma tendsto_cos [tendsto_intros]: ``` huffman@44311 ` 1300` ``` "(f ---> a) F \ ((\x. cos (f x)) ---> cos a) F" ``` huffman@44311 ` 1301` ``` by (rule isCont_tendsto_compose [OF isCont_cos]) ``` huffman@29164 ` 1302` hoelzl@31880 ` 1303` ```declare ``` hoelzl@31880 ` 1304` ``` DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] ``` hoelzl@43335 ` 1305` ``` DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 1306` ``` DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 1307` ``` DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 1308` huffman@29164 ` 1309` ```subsection {* Properties of Sine and Cosine *} ``` paulson@15077 ` 1310` paulson@15077 ` 1311` ```lemma sin_zero [simp]: "sin 0 = 0" ``` huffman@44311 ` 1312` ``` unfolding sin_def sin_coeff_def by (simp add: powser_zero) ``` paulson@15077 ` 1313` paulson@15077 ` 1314` ```lemma cos_zero [simp]: "cos 0 = 1" ``` huffman@44311 ` 1315` ``` unfolding cos_def cos_coeff_def by (simp add: powser_zero) ``` paulson@15077 ` 1316` huffman@44308 ` 1317` ```lemma sin_cos_squared_add [simp]: "(sin x)\ + (cos x)\ = 1" ``` huffman@44308 ` 1318` ```proof - ``` huffman@44308 ` 1319` ``` have "\x. DERIV (\x. (sin x)\ + (cos x)\) x :> 0" ``` huffman@44308 ` 1320` ``` by (auto intro!: DERIV_intros) ``` huffman@44308 ` 1321` ``` hence "(sin x)\ + (cos x)\ = (sin 0)\ + (cos 0)\" ``` huffman@44308 ` 1322` ``` by (rule DERIV_isconst_all) ``` huffman@44308 ` 1323` ``` thus "(sin x)\ + (cos x)\ = 1" by simp ``` huffman@44308 ` 1324` ```qed ``` huffman@44308 ` 1325` huffman@44308 ` 1326` ```lemma sin_cos_squared_add2 [simp]: "(cos x)\ + (sin x)\ = 1" ``` huffman@44308 ` 1327` ``` by (subst add_commute, rule sin_cos_squared_add) ``` paulson@15077 ` 1328` paulson@15077 ` 1329` ```lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1" ``` huffman@44308 ` 1330` ``` using sin_cos_squared_add2 [unfolded power2_eq_square] . ``` paulson@15077 ` 1331` paulson@15077 ` 1332` ```lemma sin_squared_eq: "(sin x)\ = 1 - (cos x)\" ``` huffman@44308 ` 1333` ``` unfolding eq_diff_eq by (rule sin_cos_squared_add) ``` paulson@15077 ` 1334` paulson@15077 ` 1335` ```lemma cos_squared_eq: "(cos x)\ = 1 - (sin x)\" ``` huffman@44308 ` 1336` ``` unfolding eq_diff_eq by (rule sin_cos_squared_add2) ``` paulson@15077 ` 1337` paulson@15081 ` 1338` ```lemma abs_sin_le_one [simp]: "\sin x\ \ 1" ``` huffman@44308 ` 1339` ``` by (rule power2_le_imp_le, simp_all add: sin_squared_eq) ``` paulson@15077 ` 1340` paulson@15077 ` 1341` ```lemma sin_ge_minus_one [simp]: "-1 \ sin x" ``` huffman@44308 ` 1342` ``` using abs_sin_le_one [of x] unfolding abs_le_iff by simp ``` paulson@15077 ` 1343` paulson@15077 ` 1344` ```lemma sin_le_one [simp]: "sin x \ 1" ``` huffman@44308 ` 1345` ``` using abs_sin_le_one [of x] unfolding abs_le_iff by simp ``` paulson@15077 ` 1346` paulson@15081 ` 1347` ```lemma abs_cos_le_one [simp]: "\cos x\ \ 1" ``` huffman@44308 ` 1348` ``` by (rule power2_le_imp_le, simp_all add: cos_squared_eq) ``` paulson@15077 ` 1349` paulson@15077 ` 1350` ```lemma cos_ge_minus_one [simp]: "-1 \ cos x" ``` huffman@44308 ` 1351` ``` using abs_cos_le_one [of x] unfolding abs_le_iff by simp ``` paulson@15077 ` 1352` paulson@15077 ` 1353` ```lemma cos_le_one [simp]: "cos x \ 1" ``` huffman@44308 ` 1354` ``` using abs_cos_le_one [of x] unfolding abs_le_iff by simp ``` paulson@15077 ` 1355` hoelzl@41970 ` 1356` ```lemma DERIV_fun_pow: "DERIV g x :> m ==> ``` paulson@15077 ` 1357` ``` DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m" ``` huffman@44311 ` 1358` ``` by (auto intro!: DERIV_intros) ``` paulson@15077 ` 1359` paulson@15229 ` 1360` ```lemma DERIV_fun_exp: ``` paulson@15229 ` 1361` ``` "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m" ``` huffman@44311 ` 1362` ``` by (auto intro!: DERIV_intros) ``` paulson@15077 ` 1363` paulson@15229 ` 1364` ```lemma DERIV_fun_sin: ``` paulson@15229 ` 1365` ``` "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m" ``` huffman@44311 ` 1366` ``` by (auto intro!: DERIV_intros) ``` paulson@15077 ` 1367` paulson@15229 ` 1368` ```lemma DERIV_fun_cos: ``` paulson@15229 ` 1369` ``` "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m" ``` huffman@44311 ` 1370` ``` by (auto intro!: DERIV_intros) ``` paulson@15077 ` 1371` huffman@44308 ` 1372` ```lemma sin_cos_add_lemma: ``` hoelzl@41970 ` 1373` ``` "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 + ``` paulson@15077 ` 1374` ``` (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0" ``` huffman@44308 ` 1375` ``` (is "?f x = 0") ``` huffman@44308 ` 1376` ```proof - ``` huffman@44308 ` 1377` ``` have "\x. DERIV (\x. ?f x) x :> 0" ``` huffman@44308 ` 1378` ``` by (auto intro!: DERIV_intros simp add: algebra_simps) ``` huffman@44308 ` 1379` ``` hence "?f x = ?f 0" ``` huffman@44308 ` 1380` ``` by (rule DERIV_isconst_all) ``` huffman@44308 ` 1381` ``` thus ?thesis by simp ``` huffman@44308 ` 1382` ```qed ``` paulson@15077 ` 1383` paulson@15077 ` 1384` ```lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y" ``` huffman@44308 ` 1385` ``` using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp ``` paulson@15077 ` 1386` paulson@15077 ` 1387` ```lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y" ``` huffman@44308 ` 1388` ``` using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp ``` huffman@44308 ` 1389` huffman@44308 ` 1390` ```lemma sin_cos_minus_lemma: ``` huffman@44308 ` 1391` ``` "(sin(-x) + sin(x))\ + (cos(-x) - cos(x))\ = 0" (is "?f x = 0") ``` huffman@44308 ` 1392` ```proof - ``` huffman@44308 ` 1393` ``` have "\x. DERIV (\x. ?f x) x :> 0" ``` huffman@44308 ` 1394` ``` by (auto intro!: DERIV_intros simp add: algebra_simps) ``` huffman@44308 ` 1395` ``` hence "?f x = ?f 0" ``` huffman@44308 ` 1396` ``` by (rule DERIV_isconst_all) ``` huffman@44308 ` 1397` ``` thus ?thesis by simp ``` huffman@44308 ` 1398` ```qed ``` paulson@15077 ` 1399` paulson@15077 ` 1400` ```lemma sin_minus [simp]: "sin (-x) = -sin(x)" ``` huffman@44308 ` 1401` ``` using sin_cos_minus_lemma [where x=x] by simp ``` paulson@15077 ` 1402` paulson@15077 ` 1403` ```lemma cos_minus [simp]: "cos (-x) = cos(x)" ``` huffman@44308 ` 1404` ``` using sin_cos_minus_lemma [where x=x] by simp ``` paulson@15077 ` 1405` paulson@15077 ` 1406` ```lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y" ``` huffman@44308 ` 1407` ``` by (simp add: diff_minus sin_add) ``` paulson@15077 ` 1408` paulson@15077 ` 1409` ```lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x" ``` huffman@44308 ` 1410` ``` by (simp add: sin_diff mult_commute) ``` paulson@15077 ` 1411` paulson@15077 ` 1412` ```lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y" ``` huffman@44308 ` 1413` ``` by (simp add: diff_minus cos_add) ``` paulson@15077 ` 1414` paulson@15077 ` 1415` ```lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x" ``` huffman@44308 ` 1416` ``` by (simp add: cos_diff mult_commute) ``` paulson@15077 ` 1417` paulson@15077 ` 1418` ```lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x" ``` huffman@29165 ` 1419` ``` using sin_add [where x=x and y=x] by simp ``` paulson@15077 ` 1420` paulson@15077 ` 1421` ```lemma cos_double: "cos(2* x) = ((cos x)\) - ((sin x)\)" ``` huffman@29165 ` 1422` ``` using cos_add [where x=x and y=x] ``` huffman@29165 ` 1423` ``` by (simp add: power2_eq_square) ``` paulson@15077 ` 1424` paulson@15077 ` 1425` huffman@29164 ` 1426` ```subsection {* The Constant Pi *} ``` paulson@15077 ` 1427` huffman@44308 ` 1428` ```definition pi :: "real" where ``` huffman@23053 ` 1429` ``` "pi = 2 * (THE x. 0 \ (x::real) & x \ 2 & cos x = 0)" ``` huffman@23043 ` 1430` hoelzl@41970 ` 1431` ```text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"}; ``` paulson@15077 ` 1432` ``` hence define pi.*} ``` paulson@15077 ` 1433` paulson@15077 ` 1434` ```lemma sin_paired: ``` hoelzl@41970 ` 1435` ``` "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) ``` paulson@15077 ` 1436` ``` sums sin x" ``` paulson@15077 ` 1437` ```proof - ``` huffman@31271 ` 1438` ``` have "(\n. \k = n * 2..n. 0 < ?f n" ``` huffman@44728 ` 1448` ``` proof ``` huffman@44728 ` 1449` ``` fix n :: nat ``` huffman@44728 ` 1450` ``` let ?k2 = "real (Suc (Suc (4 * n)))" ``` huffman@44728 ` 1451` ``` let ?k3 = "real (Suc (Suc (Suc (4 * n))))" ``` huffman@44728 ` 1452` ``` have "x * x < ?k2 * ?k3" ``` huffman@44728 ` 1453` ``` using assms by (intro mult_strict_mono', simp_all) ``` huffman@44728 ` 1454` ``` hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)" ``` huffman@44728 ` 1455` ``` by (intro mult_strict_right_mono zero_less_power `0 < x`) ``` huffman@44728 ` 1456` ``` thus "0 < ?f n" ``` huffman@44728 ` 1457` ``` by (simp del: mult_Suc, ``` huffman@44728 ` 1458` ``` simp add: less_divide_eq mult_pos_pos field_simps del: mult_Suc) ``` huffman@44728 ` 1459` ``` qed ``` huffman@44728 ` 1460` ``` have sums: "?f sums sin x" ``` huffman@44728 ` 1461` ``` by (rule sin_paired [THEN sums_group], simp) ``` huffman@44728 ` 1462` ``` show "0 < sin x" ``` huffman@44728 ` 1463` ``` unfolding sums_unique [OF sums] ``` huffman@44728 ` 1464` ``` using sums_summable [OF sums] pos ``` huffman@44728 ` 1465` ``` by (rule suminf_gt_zero) ``` huffman@44728 ` 1466` ```qed ``` paulson@15077 ` 1467` paulson@15077 ` 1468` ```lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1" ``` huffman@44311 ` 1469` ```apply (cut_tac x = x in sin_gt_zero) ``` paulson@15077 ` 1470` ```apply (auto simp add: cos_squared_eq cos_double) ``` paulson@15077 ` 1471` ```done ``` paulson@15077 ` 1472` paulson@15077 ` 1473` ```lemma cos_paired: ``` huffman@23177 ` 1474` ``` "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x" ``` paulson@15077 ` 1475` ```proof - ``` huffman@31271 ` 1476` ``` have "(\n. \k = n * 2.. inverse x * y < inverse x1 * u" ``` hoelzl@41970 ` 1484` ```apply (rule_tac c=x in mult_less_imp_less_left) ``` huffman@36824 ` 1485` ```apply (auto simp add: mult_assoc [symmetric]) ``` huffman@36824 ` 1486` ```apply (simp (no_asm) add: mult_ac) ``` hoelzl@41970 ` 1487` ```apply (rule_tac c=x1 in mult_less_imp_less_right) ``` huffman@36824 ` 1488` ```apply (auto simp add: mult_ac) ``` huffman@36824 ` 1489` ```done ``` huffman@36824 ` 1490` huffman@36824 ` 1491` ```lemma real_mult_inverse_cancel2: ``` huffman@36824 ` 1492` ``` "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1" ``` huffman@36824 ` 1493` ```apply (auto dest: real_mult_inverse_cancel simp add: mult_ac) ``` huffman@36824 ` 1494` ```done ``` huffman@36824 ` 1495` huffman@36824 ` 1496` ```lemma realpow_num_eq_if: ``` huffman@36824 ` 1497` ``` fixes m :: "'a::power" ``` huffman@36824 ` 1498` ``` shows "m ^ n = (if n=0 then 1 else m * m ^ (n - 1))" ``` huffman@36824 ` 1499` ```by (cases n, auto) ``` huffman@36824 ` 1500` huffman@23053 ` 1501` ```lemma cos_two_less_zero [simp]: "cos (2) < 0" ``` paulson@15077 ` 1502` ```apply (cut_tac x = 2 in cos_paired) ``` paulson@15077 ` 1503` ```apply (drule sums_minus) ``` hoelzl@41970 ` 1504` ```apply (rule neg_less_iff_less [THEN iffD1]) ``` nipkow@15539 ` 1505` ```apply (frule sums_unique, auto) ``` nipkow@15539 ` 1506` ```apply (rule_tac y = ``` huffman@23177 ` 1507` ``` "\n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))" ``` paulson@15481 ` 1508` ``` in order_less_trans) ``` avigad@32047 ` 1509` ```apply (simp (no_asm) add: fact_num_eq_if_nat realpow_num_eq_if del: fact_Suc) ``` nipkow@15561 ` 1510` ```apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc) ``` paulson@15077 ` 1511` ```apply (rule sumr_pos_lt_pair) ``` paulson@15077 ` 1512` ```apply (erule sums_summable, safe) ``` huffman@30082 ` 1513` ```unfolding One_nat_def ``` hoelzl@41970 ` 1514` ```apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric] ``` avigad@32047 ` 1515` ``` del: fact_Suc) ``` huffman@46240 ` 1516` ```apply (simp add: inverse_eq_divide less_divide_eq del: fact_Suc) ``` avigad@32047 ` 1517` ```apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"]) ``` paulson@15481 ` 1518` ```apply (simp only: real_of_nat_mult) ``` huffman@23007 ` 1519` ```apply (rule mult_strict_mono, force) ``` huffman@27483 ` 1520` ``` apply (rule_tac [3] real_of_nat_ge_zero) ``` paulson@15481 ` 1521` ``` prefer 2 apply force ``` paulson@15077 ` 1522` ```apply (rule real_of_nat_less_iff [THEN iffD2]) ``` avigad@32036 ` 1523` ```apply (rule fact_less_mono_nat, auto) ``` paulson@15077 ` 1524` ```done ``` huffman@23053 ` 1525` huffman@23053 ` 1526` ```lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq] ``` huffman@23053 ` 1527` ```lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le] ``` paulson@15077 ` 1528` paulson@15077 ` 1529` ```lemma cos_is_zero: "EX! x. 0 \ x & x \ 2 & cos x = 0" ``` huffman@44730 ` 1530` ```proof (rule ex_ex1I) ``` huffman@44730 ` 1531` ``` show "\x. 0 \ x & x \ 2 & cos x = 0" ``` huffman@44730 ` 1532` ``` by (rule IVT2, simp_all) ``` huffman@44730 ` 1533` ```next ``` huffman@44730 ` 1534` ``` fix x y ``` huffman@44730 ` 1535` ``` assume x: "0 \ x \ x \ 2 \ cos x = 0" ``` huffman@44730 ` 1536` ``` assume y: "0 \ y \ y \ 2 \ cos y = 0" ``` huffman@44730 ` 1537` ``` have [simp]: "\x. cos differentiable x" ``` huffman@44730 ` 1538` ``` unfolding differentiable_def by (auto intro: DERIV_cos) ``` huffman@44730 ` 1539` ``` from x y show "x = y" ``` huffman@44730 ` 1540` ``` apply (cut_tac less_linear [of x y], auto) ``` huffman@44730 ` 1541` ``` apply (drule_tac f = cos in Rolle) ``` huffman@44730 ` 1542` ``` apply (drule_tac [5] f = cos in Rolle) ``` huffman@44730 ` 1543` ``` apply (auto dest!: DERIV_cos [THEN DERIV_unique]) ``` huffman@44730 ` 1544` ``` apply (metis order_less_le_trans less_le sin_gt_zero) ``` huffman@44730 ` 1545` ``` apply (metis order_less_le_trans less_le sin_gt_zero) ``` huffman@44730 ` 1546` ``` done ``` huffman@44730 ` 1547` ```qed ``` hoelzl@31880 ` 1548` huffman@23053 ` 1549` ```lemma pi_half: "pi/2 = (THE x. 0 \ x & x \ 2 & cos x = 0)" ``` paulson@15077 ` 1550` ```by (simp add: pi_def) ``` paulson@15077 ` 1551` paulson@15077 ` 1552` ```lemma cos_pi_half [simp]: "cos (pi / 2) = 0" ``` huffman@23053 ` 1553` ```by (simp add: pi_half cos_is_zero [THEN theI']) ``` huffman@23053 ` 1554` huffman@23053 ` 1555` ```lemma pi_half_gt_zero [simp]: "0 < pi / 2" ``` huffman@23053 ` 1556` ```apply (rule order_le_neq_trans) ``` huffman@23053 ` 1557` ```apply (simp add: pi_half cos_is_zero [THEN theI']) ``` huffman@23053 ` 1558` ```apply (rule notI, drule arg_cong [where f=cos], simp) ``` paulson@15077 ` 1559` ```done ``` paulson@15077 ` 1560` huffman@23053 ` 1561` ```lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric] ``` huffman@23053 ` 1562` ```lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le] ``` paulson@15077 ` 1563` huffman@23053 ` 1564` ```lemma pi_half_less_two [simp]: "pi / 2 < 2" ``` huffman@23053 ` 1565` ```apply (rule order_le_neq_trans) ``` huffman@23053 ` 1566` ```apply (simp add: pi_half cos_is_zero [THEN theI']) ``` huffman@23053 ` 1567` ```apply (rule notI, drule arg_cong [where f=cos], simp) ``` paulson@15077 ` 1568` ```done ``` huffman@23053 ` 1569` huffman@23053 ` 1570` ```lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq] ``` huffman@23053 ` 1571` ```lemmas pi_half_le_two [simp] = pi_half_less_two [THEN order_less_imp_le] ``` paulson@15077 ` 1572` paulson@15077 ` 1573` ```lemma pi_gt_zero [simp]: "0 < pi" ``` huffman@23053 ` 1574` ```by (insert pi_half_gt_zero, simp) ``` huffman@23053 ` 1575` huffman@23053 ` 1576` ```lemma pi_ge_zero [simp]: "0 \ pi" ``` huffman@23053 ` 1577` ```by (rule pi_gt_zero [THEN order_less_imp_le]) ``` paulson@15077 ` 1578` paulson@15077 ` 1579` ```lemma pi_neq_zero [simp]: "pi \ 0" ``` huffman@22998 ` 1580` ```by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym]) ``` paulson@15077 ` 1581` huffman@23053 ` 1582` ```lemma pi_not_less_zero [simp]: "\ pi < 0" ``` huffman@23053 ` 1583` ```by (simp add: linorder_not_less) ``` paulson@15077 ` 1584` huffman@29165 ` 1585` ```lemma minus_pi_half_less_zero: "-(pi/2) < 0" ``` huffman@29165 ` 1586` ```by simp ``` paulson@15077 ` 1587` hoelzl@29803 ` 1588` ```lemma m2pi_less_pi: "- (2 * pi) < pi" ``` huffman@45308 ` 1589` ```by simp ``` hoelzl@29803 ` 1590` paulson@15077 ` 1591` ```lemma sin_pi_half [simp]: "sin(pi/2) = 1" ``` paulson@15077 ` 1592` ```apply (cut_tac x = "pi/2" in sin_cos_squared_add2) ``` paulson@15077 ` 1593` ```apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two]) ``` huffman@36970 ` 1594` ```apply (simp add: power2_eq_1_iff) ``` paulson@15077 ` 1595` ```done ``` paulson@15077 ` 1596` paulson@15077 ` 1597` ```lemma cos_pi [simp]: "cos pi = -1" ``` nipkow@15539 ` 1598` ```by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp) ``` paulson@15077 ` 1599` paulson@15077 ` 1600` ```lemma sin_pi [simp]: "sin pi = 0" ``` nipkow@15539 ` 1601` ```by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp) ``` paulson@15077 ` 1602` paulson@15077 ` 1603` ```lemma sin_cos_eq: "sin x = cos (pi/2 - x)" ``` huffman@45309 ` 1604` ```by (simp add: cos_diff) ``` paulson@15077 ` 1605` paulson@15077 ` 1606` ```lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)" ``` paulson@15229 ` 1607` ```by (simp add: cos_add) ``` paulson@15077 ` 1608` paulson@15077 ` 1609` ```lemma cos_sin_eq: "cos x = sin (pi/2 - x)" ``` huffman@45309 ` 1610` ```by (simp add: sin_diff) ``` paulson@15077 ` 1611` paulson@15077 ` 1612` ```lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x" ``` paulson@15229 ` 1613` ```by (simp add: sin_add) ``` paulson@15077 ` 1614` paulson@15077 ` 1615` ```lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x" ``` paulson@15229 ` 1616` ```by (simp add: sin_add) ``` paulson@15077 ` 1617` paulson@15077 ` 1618` ```lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x" ``` paulson@15229 ` 1619` ```by (simp add: cos_add) ``` paulson@15077 ` 1620` paulson@15077 ` 1621` ```lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x" ``` paulson@15077 ` 1622` ```by (simp add: sin_add cos_double) ``` paulson@15077 ` 1623` paulson@15077 ` 1624` ```lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x" ``` paulson@15077 ` 1625` ```by (simp add: cos_add cos_double) ``` paulson@15077 ` 1626` paulson@15077 ` 1627` ```lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n" ``` paulson@15251 ` 1628` ```apply (induct "n") ``` paulson@15077 ` 1629` ```apply (auto simp add: real_of_nat_Suc left_distrib) ``` paulson@15077 ` 1630` ```done ``` paulson@15077 ` 1631` paulson@15383 ` 1632` ```lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n" ``` paulson@15383 ` 1633` ```proof - ``` paulson@15383 ` 1634` ``` have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute) ``` hoelzl@41970 ` 1635` ``` also have "... = -1 ^ n" by (rule cos_npi) ``` paulson@15383 ` 1636` ``` finally show ?thesis . ``` paulson@15383 ` 1637` ```qed ``` paulson@15383 ` 1638` paulson@15077 ` 1639` ```lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0" ``` paulson@15251 ` 1640` ```apply (induct "n") ``` paulson@15077 ` 1641` ```apply (auto simp add: real_of_nat_Suc left_distrib) ``` paulson@15077 ` 1642` ```done ``` paulson@15077 ` 1643` paulson@15077 ` 1644` ```lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0" ``` hoelzl@41970 ` 1645` ```by (simp add: mult_commute [of pi]) ``` paulson@15077 ` 1646` paulson@15077 ` 1647` ```lemma cos_two_pi [simp]: "cos (2 * pi) = 1" ``` paulson@15077 ` 1648` ```by (simp add: cos_double) ``` paulson@15077 ` 1649` paulson@15077 ` 1650` ```lemma sin_two_pi [simp]: "sin (2 * pi) = 0" ``` paulson@15229 ` 1651` ```by simp ``` paulson@15077 ` 1652` paulson@15077 ` 1653` ```lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x" ``` paulson@15077 ` 1654` ```apply (rule sin_gt_zero, assumption) ``` paulson@15077 ` 1655` ```apply (rule order_less_trans, assumption) ``` paulson@15077 ` 1656` ```apply (rule pi_half_less_two) ``` paulson@15077 ` 1657` ```done ``` paulson@15077 ` 1658` hoelzl@41970 ` 1659` ```lemma sin_less_zero: ``` paulson@15077 ` 1660` ``` assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0" ``` paulson@15077 ` 1661` ```proof - ``` hoelzl@41970 ` 1662` ``` have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2) ``` paulson@15077 ` 1663` ``` thus ?thesis by simp ``` paulson@15077 ` 1664` ```qed ``` paulson@15077 ` 1665` paulson@15077 ` 1666` ```lemma pi_less_4: "pi < 4" ``` paulson@15077 ` 1667` ```by (cut_tac pi_half_less_two, auto) ``` paulson@15077 ` 1668` paulson@15077 ` 1669` ```lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x" ``` paulson@15077 ` 1670` ```apply (cut_tac pi_less_4) ``` paulson@15077 ` 1671` ```apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all) ``` paulson@15077 ` 1672` ```apply (cut_tac cos_is_zero, safe) ``` paulson@15077 ` 1673` ```apply (rename_tac y z) ``` paulson@15077 ` 1674` ```apply (drule_tac x = y in spec) ``` hoelzl@41970 ` 1675` ```apply (drule_tac x = "pi/2" in spec, simp) ``` paulson@15077 ` 1676` ```done ``` paulson@15077 ` 1677` paulson@15077 ` 1678` ```lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x" ``` paulson@15077 ` 1679` ```apply (rule_tac x = x and y = 0 in linorder_cases) ``` paulson@15077 ` 1680` ```apply (rule cos_minus [THEN subst]) ``` paulson@15077 ` 1681` ```apply (rule cos_gt_zero) ``` paulson@15077 ` 1682` ```apply (auto intro: cos_gt_zero) ``` paulson@15077 ` 1683` ```done ``` hoelzl@41970 ` 1684` paulson@15077 ` 1685` ```lemma cos_ge_zero: "[| -(pi/2) \ x; x \ pi/2 |] ==> 0 \ cos x" ``` paulson@15077 ` 1686` ```apply (auto simp add: order_le_less cos_gt_zero_pi) ``` hoelzl@41970 ` 1687` ```apply (subgoal_tac "x = pi/2", auto) ``` paulson@15077 ` 1688` ```done ``` paulson@15077 ` 1689` paulson@15077 ` 1690` ```lemma sin_gt_zero_pi: "[| 0 < x; x < pi |] ==> 0 < sin x" ``` huffman@45309 ` 1691` ```by (simp add: sin_cos_eq cos_gt_zero_pi) ``` hoelzl@29803 ` 1692` hoelzl@29803 ` 1693` ```lemma pi_ge_two: "2 \ pi" ``` hoelzl@29803 ` 1694` ```proof (rule ccontr) ``` hoelzl@29803 ` 1695` ``` assume "\ 2 \ pi" hence "pi < 2" by auto ``` hoelzl@29803 ` 1696` ``` have "\y > pi. y < 2 \ y < 2 * pi" ``` hoelzl@29803 ` 1697` ``` proof (cases "2 < 2 * pi") ``` hoelzl@29803 ` 1698` ``` case True with dense[OF `pi < 2`] show ?thesis by auto ``` hoelzl@29803 ` 1699` ``` next ``` hoelzl@29803 ` 1700` ``` case False have "pi < 2 * pi" by auto ``` hoelzl@29803 ` 1701` ``` from dense[OF this] and False show ?thesis by auto ``` hoelzl@29803 ` 1702` ``` qed ``` hoelzl@29803 ` 1703` ``` then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast ``` hoelzl@29803 ` 1704` ``` hence "0 < sin y" using sin_gt_zero by auto ``` hoelzl@41970 ` 1705` ``` moreover ``` hoelzl@29803 ` 1706` ``` have "sin y < 0" using sin_gt_zero_pi[of "y - pi"] `pi < y` and `y < 2 * pi` sin_periodic_pi[of "y - pi"] by auto ``` hoelzl@29803 ` 1707` ``` ultimately show False by auto ``` hoelzl@29803 ` 1708` ```qed ``` hoelzl@29803 ` 1709` paulson@15077 ` 1710` ```lemma sin_ge_zero: "[| 0 \ x; x \ pi |] ==> 0 \ sin x" ``` paulson@15077 ` 1711` ```by (auto simp add: order_le_less sin_gt_zero_pi) ``` paulson@15077 ` 1712` huffman@44745 ` 1713` ```text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}. ``` huffman@44745 ` 1714` ``` It should be possible to factor out some of the common parts. *} ``` huffman@44745 ` 1715` paulson@15077 ` 1716` ```lemma cos_total: "[| -1 \ y; y \ 1 |] ==> EX! x. 0 \ x & x \ pi & (cos x = y)" ``` huffman@44745 ` 1717` ```proof (rule ex_ex1I) ``` huffman@44745 ` 1718` ``` assume y: "-1 \ y" "y \ 1" ``` huffman@44745 ` 1719` ``` show "\x. 0 \ x & x \ pi & cos x = y" ``` huffman@44745 ` 1720` ``` by (rule IVT2, simp_all add: y) ``` huffman@44745 ` 1721` ```next ``` huffman@44745 ` 1722` ``` fix a b ``` huffman@44745 ` 1723` ``` assume a: "0 \ a \ a \ pi \ cos a = y" ``` huffman@44745 ` 1724` ``` assume b: "0 \ b \ b \ pi \ cos b = y" ``` huffman@44745 ` 1725` ``` have [simp]: "\x. cos differentiable x" ``` huffman@44745 ` 1726` ``` unfolding differentiable_def by (auto intro: DERIV_cos) ``` huffman@44745 ` 1727` ``` from a b show "a = b" ``` huffman@44745 ` 1728` ``` apply (cut_tac less_linear [of a b], auto) ``` huffman@44745 ` 1729` ``` apply (drule_tac f = cos in Rolle) ``` huffman@44745 ` 1730` ``` apply (drule_tac [5] f = cos in Rolle) ``` huffman@44745 ` 1731` ``` apply (auto dest!: DERIV_cos [THEN DERIV_unique]) ``` huffman@44745 ` 1732` ``` apply (metis order_less_le_trans less_le sin_gt_zero_pi) ``` huffman@44745 ` 1733` ``` apply (metis order_less_le_trans less_le sin_gt_zero_pi) ``` huffman@44745 ` 1734` ``` done ``` huffman@44745 ` 1735` ```qed ``` paulson@15077 ` 1736` paulson@15077 ` 1737` ```lemma sin_total: ``` paulson@15077 ` 1738` ``` "[| -1 \ y; y \ 1 |] ==> EX! x. -(pi/2) \ x & x \ pi/2 & (sin x = y)" ``` paulson@15077 ` 1739` ```apply (rule ccontr) ``` paulson@15077 ` 1740` ```apply (subgoal_tac "\x. (- (pi/2) \ x & x \ pi/2 & (sin x = y)) = (0 \ (x + pi/2) & (x + pi/2) \ pi & (cos (x + pi/2) = -y))") ``` wenzelm@18585 ` 1741` ```apply (erule contrapos_np) ``` huffman@45309 ` 1742` ```apply simp ``` hoelzl@41970 ` 1743` ```apply (cut_tac y="-y" in cos_total, simp) apply simp ``` paulson@15077 ` 1744` ```apply (erule ex1E) ``` paulson@15229 ` 1745` ```apply (rule_tac a = "x - (pi/2)" in ex1I) ``` huffman@23286 ` 1746` ```apply (simp (no_asm) add: add_assoc) ``` paulson@15077 ` 1747` ```apply (rotate_tac 3) ``` huffman@45309 ` 1748` ```apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all add: cos_add) ``` paulson@15077 ` 1749` ```done ``` paulson@15077 ` 1750` paulson@15077 ` 1751` ```lemma reals_Archimedean4: ``` paulson@15077 ` 1752` ``` "[| 0 < y; 0 \ x |] ==> \n. real n * y \ x & x < real (Suc n) * y" ``` paulson@15077 ` 1753` ```apply (auto dest!: reals_Archimedean3) ``` hoelzl@41970 ` 1754` ```apply (drule_tac x = x in spec, clarify) ``` paulson@15077 ` 1755` ```apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y") ``` hoelzl@41970 ` 1756` ``` prefer 2 apply (erule LeastI) ``` hoelzl@41970 ` 1757` ```apply (case_tac "LEAST m::nat. x < real m * y", simp) ``` paulson@15077 ` 1758` ```apply (subgoal_tac "~ x < real nat * y") ``` hoelzl@41970 ` 1759` ``` prefer 2 apply (rule not_less_Least, simp, force) ``` paulson@15077 ` 1760` ```done ``` paulson@15077 ` 1761` hoelzl@41970 ` 1762` ```(* Pre Isabelle99-2 proof was simpler- numerals arithmetic ``` paulson@15077 ` 1763` ``` now causes some unwanted re-arrangements of literals! *) ``` paulson@15229 ` 1764` ```lemma cos_zero_lemma: ``` hoelzl@41970 ` 1765` ``` "[| 0 \ x; cos x = 0 |] ==> ``` paulson@15077 ` 1766` ``` \n::nat. ~even n & x = real n * (pi/2)" ``` paulson@15077 ` 1767` ```apply (drule pi_gt_zero [THEN reals_Archimedean4], safe) ``` hoelzl@41970 ` 1768` ```apply (subgoal_tac "0 \ x - real n * pi & ``` paulson@15086 ` 1769` ``` (x - real n * pi) \ pi & (cos (x - real n * pi) = 0) ") ``` nipkow@29667 ` 1770` ```apply (auto simp add: algebra_simps real_of_nat_Suc) ``` nipkow@29667 ` 1771` ``` prefer 2 apply (simp add: cos_diff) ``` paulson@15077 ` 1772` ```apply (simp add: cos_diff) ``` paulson@15077 ` 1773` ```apply (subgoal_tac "EX! x. 0 \ x & x \ pi & cos x = 0") ``` paulson@15077 ` 1774` ```apply (rule_tac [2] cos_total, safe) ``` paulson@15077 ` 1775` ```apply (drule_tac x = "x - real n * pi" in spec) ``` paulson@15077 ` 1776` ```apply (drule_tac x = "pi/2" in spec) ``` paulson@15077 ` 1777` ```apply (simp add: cos_diff) ``` paulson@15229 ` 1778` ```apply (rule_tac x = "Suc (2 * n)" in exI) ``` nipkow@29667 ` 1779` ```apply (simp add: real_of_nat_Suc algebra_simps, auto) ``` paulson@15077 ` 1780` ```done ``` paulson@15077 ` 1781` paulson@15229 ` 1782` ```lemma sin_zero_lemma: ``` hoelzl@41970 ` 1783` ``` "[| 0 \ x; sin x = 0 |] ==> ``` paulson@15077 ` 1784` ``` \n::nat. even n & x = real n * (pi/2)" ``` paulson@15077 ` 1785` ```apply (subgoal_tac "\n::nat. ~ even n & x + pi/2 = real n * (pi/2) ") ``` paulson@15077 ` 1786` ``` apply (clarify, rule_tac x = "n - 1" in exI) ``` paulson@15077 ` 1787` ``` apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) ``` paulson@15085 ` 1788` ```apply (rule cos_zero_lemma) ``` huffman@45309 ` 1789` ```apply (simp_all add: cos_add) ``` paulson@15077 ` 1790` ```done ``` paulson@15077 ` 1791` paulson@15077 ` 1792` paulson@15229 ` 1793` ```lemma cos_zero_iff: ``` hoelzl@41970 ` 1794` ``` "(cos x = 0) = ``` hoelzl@41970 ` 1795` ``` ((\n::nat. ~even n & (x = real n * (pi/2))) | ``` paulson@15077 ` 1796` ``` (\n::nat. ~even n & (x = -(real n * (pi/2)))))" ``` paulson@15077 ` 1797` ```apply (rule iffI) ``` paulson@15077 ` 1798` ```apply (cut_tac linorder_linear [of 0 x], safe) ``` paulson@15077 ` 1799` ```apply (drule cos_zero_lemma, assumption+) ``` hoelzl@41970 ` 1800` ```apply (cut_tac x="-x" in cos_zero_lemma, simp, simp) ``` hoelzl@41970 ` 1801` ```apply (force simp add: minus_equation_iff [of x]) ``` hoelzl@41970 ` 1802` ```apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) ``` nipkow@15539 ` 1803` ```apply (auto simp add: cos_add) ``` paulson@15077 ` 1804` ```done ``` paulson@15077 ` 1805` paulson@15077 ` 1806` ```(* ditto: but to a lesser extent *) ``` paulson@15229 ` 1807` ```lemma sin_zero_iff: ``` hoelzl@41970 ` 1808` ``` "(sin x = 0) = ``` hoelzl@41970 ` 1809` ``` ((\n::nat. even n & (x = real n * (pi/2))) | ``` paulson@15077 ` 1810` ``` (\n::nat. even n & (x = -(real n * (pi/2)))))" ``` paulson@15077 ` 1811` ```apply (rule iffI) ``` paulson@15077 ` 1812` ```apply (cut_tac linorder_linear [of 0 x], safe) ``` paulson@15077 ` 1813` ```apply (drule sin_zero_lemma, assumption+) ``` paulson@15077 ` 1814` ```apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe) ``` hoelzl@41970 ` 1815` ```apply (force simp add: minus_equation_iff [of x]) ``` nipkow@15539 ` 1816` ```apply (auto simp add: even_mult_two_ex) ``` paulson@15077 ` 1817` ```done ``` paulson@15077 ` 1818` hoelzl@29803 ` 1819` ```lemma cos_monotone_0_pi: assumes "0 \ y" and "y < x" and "x \ pi" ``` hoelzl@29803 ` 1820` ``` shows "cos x < cos y" ``` hoelzl@29803 ` 1821` ```proof - ``` wenzelm@33549 ` 1822` ``` have "- (x - y) < 0" using assms by auto ``` hoelzl@29803 ` 1823` hoelzl@29803 ` 1824` ``` from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]] ``` hoelzl@29803 ` 1825` ``` obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" by auto ``` wenzelm@33549 ` 1826` ``` hence "0 < z" and "z < pi" using assms by auto ``` hoelzl@29803 ` 1827` ``` hence "0 < sin z" using sin_gt_zero_pi by auto ``` hoelzl@29803 ` 1828` ``` hence "cos x - cos y < 0" unfolding cos_diff minus_mult_commute[symmetric] using `- (x - y) < 0` by (rule mult_pos_neg2) ``` hoelzl@29803 ` 1829` ``` thus ?thesis by auto ``` hoelzl@29803 ` 1830` ```qed ``` hoelzl@29803 ` 1831` hoelzl@29803 ` 1832` ```lemma cos_monotone_0_pi': assumes "0 \ y" and "y \ x" and "x \ pi" shows "cos x \ cos y" ``` hoelzl@29803 ` 1833` ```proof (cases "y < x") ``` hoelzl@29803 ` 1834` ``` case True show ?thesis using cos_monotone_0_pi[OF `0 \ y` True `x \ pi`] by auto ``` hoelzl@29803 ` 1835` ```next ``` hoelzl@29803 ` 1836` ``` case False hence "y = x" using `y \ x` by auto ``` hoelzl@29803 ` 1837` ``` thus ?thesis by auto ``` hoelzl@29803 ` 1838` ```qed ``` hoelzl@29803 ` 1839` hoelzl@29803 ` 1840` ```lemma cos_monotone_minus_pi_0: assumes "-pi \ y" and "y < x" and "x \ 0" ``` hoelzl@29803 ` 1841` ``` shows "cos y < cos x" ``` hoelzl@29803 ` 1842` ```proof - ``` wenzelm@33549 ` 1843` ``` have "0 \ -x" and "-x < -y" and "-y \ pi" using assms by auto ``` hoelzl@29803 ` 1844` ``` from cos_monotone_0_pi[OF this] ``` hoelzl@29803 ` 1845` ``` show ?thesis unfolding cos_minus . ``` hoelzl@29803 ` 1846` ```qed ``` hoelzl@29803 ` 1847` hoelzl@29803 ` 1848` ```lemma cos_monotone_minus_pi_0': assumes "-pi \ y" and "y \ x" and "x \ 0" shows "cos y \ cos x" ``` hoelzl@29803 ` 1849` ```proof (cases "y < x") ``` hoelzl@29803 ` 1850` ``` case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi \ y` True `x \ 0`] by auto ``` hoelzl@29803 ` 1851` ```next ``` hoelzl@29803 ` 1852` ``` case False hence "y = x" using `y \ x` by auto ``` hoelzl@29803 ` 1853` ``` thus ?thesis by auto ``` hoelzl@29803 ` 1854` ```qed ``` hoelzl@29803 ` 1855` hoelzl@29803 ` 1856` ```lemma sin_monotone_2pi': assumes "- (pi / 2) \ y" and "y \ x" and "x \ pi / 2" shows "sin y \ sin x" ``` hoelzl@29803 ` 1857` ```proof - ``` wenzelm@33549 ` 1858` ``` have "0 \ y + pi / 2" and "y + pi / 2 \ x + pi / 2" and "x + pi /2 \ pi" ``` wenzelm@33549 ` 1859` ``` using pi_ge_two and assms by auto ``` hoelzl@29803 ` 1860` ``` from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] by auto ``` hoelzl@29803 ` 1861` ```qed ``` paulson@15077 ` 1862` huffman@29164 ` 1863` ```subsection {* Tangent *} ``` paulson@15077 ` 1864` huffman@44311 ` 1865` ```definition tan :: "real \ real" where ``` huffman@44311 ` 1866` ``` "tan = (\x. sin x / cos x)" ``` huffman@23043 ` 1867` paulson@15077 ` 1868` ```lemma tan_zero [simp]: "tan 0 = 0" ``` huffman@44311 ` 1869` ``` by (simp add: tan_def) ``` paulson@15077 ` 1870` paulson@15077 ` 1871` ```lemma tan_pi [simp]: "tan pi = 0" ``` huffman@44311 ` 1872` ``` by (simp add: tan_def) ``` paulson@15077 ` 1873` paulson@15077 ` 1874` ```lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0" ``` huffman@44311 ` 1875` ``` by (simp add: tan_def) ``` paulson@15077 ` 1876` paulson@15077 ` 1877` ```lemma tan_minus [simp]: "tan (-x) = - tan x" ``` huffman@44311 ` 1878` ``` by (simp add: tan_def) ``` paulson@15077 ` 1879` paulson@15077 ` 1880` ```lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x" ``` huffman@44311 ` 1881` ``` by (simp add: tan_def) ``` paulson@15077 ` 1882` hoelzl@41970 ` 1883` ```lemma lemma_tan_add1: ``` huffman@44311 ` 1884` ``` "\cos x \ 0; cos y \ 0\ \ 1 - tan x * tan y = cos (x + y)/(cos x * cos y)" ``` huffman@44311 ` 1885` ``` by (simp add: tan_def cos_add field_simps) ``` paulson@15077 ` 1886` hoelzl@41970 ` 1887` ```lemma add_tan_eq: ``` huffman@44311 ` 1888` ``` "\cos x \ 0; cos y \ 0\ \ tan x + tan y = sin(x + y)/(cos x * cos y)" ``` huffman@44311 ` 1889` ``` by (simp add: tan_def sin_add field_simps) ``` paulson@15077 ` 1890` paulson@15229 ` 1891` ```lemma tan_add: ``` hoelzl@41970 ` 1892` ``` "[| cos x \ 0; cos y \ 0; cos (x + y) \ 0 |] ``` paulson@15077 ` 1893` ``` ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))" ``` huffman@44311 ` 1894` ``` by (simp add: add_tan_eq lemma_tan_add1, simp add: tan_def) ``` paulson@15077 ` 1895` paulson@15229 ` 1896` ```lemma tan_double: ``` hoelzl@41970 ` 1897` ``` "[| cos x \ 0; cos (2 * x) \ 0 |] ``` paulson@15077 ` 1898` ``` ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))" ``` huffman@44311 ` 1899` ``` using tan_add [of x x] by (simp add: power2_eq_square) ``` paulson@15077 ` 1900` paulson@15077 ` 1901` ```lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x" ``` hoelzl@41970 ` 1902` ```by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) ``` hoelzl@41970 ` 1903` hoelzl@41970 ` 1904` ```lemma tan_less_zero: ``` paulson@15077 ` 1905` ``` assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0" ``` paulson@15077 ` 1906` ```proof - ``` hoelzl@41970 ` 1907` ``` have "0 < tan (- x)" using assms by (simp only: tan_gt_zero) ``` paulson@15077 ` 1908` ``` thus ?thesis by simp ``` paulson@15077 ` 1909` ```qed ``` paulson@15077 ` 1910` huffman@44756 ` 1911` ```lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)" ``` huffman@44756 ` 1912` ``` unfolding tan_def sin_double cos_double sin_squared_eq ``` huffman@44756 ` 1913` ``` by (simp add: power2_eq_square) ``` hoelzl@29803 ` 1914` huffman@44311 ` 1915` ```lemma DERIV_tan [simp]: "cos x \ 0 \ DERIV tan x :> inverse ((cos x)\)" ``` huffman@44311 ` 1916` ``` unfolding tan_def ``` huffman@44311 ` 1917` ``` by (auto intro!: DERIV_intros, simp add: divide_inverse power2_eq_square) ``` huffman@44311 ` 1918` huffman@44311 ` 1919` ```lemma isCont_tan: "cos x \ 0 \ isCont tan x" ``` huffman@44311 ` 1920` ``` by (rule DERIV_tan [THEN DERIV_isCont]) ``` huffman@44311 ` 1921` huffman@44311 ` 1922` ```lemma isCont_tan' [simp]: ``` huffman@44311 ` 1923` ``` "\isCont f a; cos (f a) \ 0\ \ isCont (\x. tan (f x)) a" ``` huffman@44311 ` 1924` ``` by (rule isCont_o2 [OF _ isCont_tan]) ``` huffman@44311 ` 1925` huffman@44311 ` 1926` ```lemma tendsto_tan [tendsto_intros]: ``` huffman@44311 ` 1927` ``` "\(f ---> a) F; cos a \ 0\ \ ((\x. tan (f x)) ---> tan a) F" ``` huffman@44311 ` 1928` ``` by (rule isCont_tendsto_compose [OF isCont_tan]) ``` huffman@44311 ` 1929` huffman@44311 ` 1930` ```lemma LIM_cos_div_sin: "(%x. cos(x)/sin(x)) -- pi/2 --> 0" ``` huffman@44311 ` 1931` ``` by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all) ``` paulson@15077 ` 1932` paulson@15077 ` 1933` ```lemma lemma_tan_total: "0 < y ==> \x. 0 < x & x < pi/2 & y < tan x" ``` paulson@15077 ` 1934` ```apply (cut_tac LIM_cos_div_sin) ``` huffman@31338 ` 1935` ```apply (simp only: LIM_eq) ``` paulson@15077 ` 1936` ```apply (drule_tac x = "inverse y" in spec, safe, force) ``` paulson@15077 ` 1937` ```apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe) ``` paulson@15229 ` 1938` ```apply (rule_tac x = "(pi/2) - e" in exI) ``` paulson@15077 ` 1939` ```apply (simp (no_asm_simp)) ``` paulson@15229 ` 1940` ```apply (drule_tac x = "(pi/2) - e" in spec) ``` huffman@45309 ` 1941` ```apply (auto simp add: tan_def sin_diff cos_diff) ``` paulson@15077 ` 1942` ```apply (rule inverse_less_iff_less [THEN iffD1]) ``` paulson@15079 ` 1943` ```apply (auto simp add: divide_inverse) ``` huffman@36777 ` 1944` ```apply (rule mult_pos_pos) ``` paulson@15229 ` 1945` ```apply (subgoal_tac [3] "0 < sin e & 0 < cos e") ``` huffman@36777 ` 1946` ```apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute) ``` paulson@15077 ` 1947` ```done ``` paulson@15077 ` 1948` paulson@15077 ` 1949` ```lemma tan_total_pos: "0 \ y ==> \x. 0 \ x & x < pi/2 & tan x = y" ``` huffman@22998 ` 1950` ```apply (frule order_le_imp_less_or_eq, safe) ``` paulson@15077 ` 1951` ``` prefer 2 apply force ``` paulson@15077 ` 1952` ```apply (drule lemma_tan_total, safe) ``` paulson@15077 ` 1953` ```apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl) ``` paulson@15077 ` 1954` ```apply (auto intro!: DERIV_tan [THEN DERIV_isCont]) ``` paulson@15077 ` 1955` ```apply (drule_tac y = xa in order_le_imp_less_or_eq) ``` paulson@15077 ` 1956` ```apply (auto dest: cos_gt_zero) ``` paulson@15077 ` 1957` ```done ``` paulson@15077 ` 1958` paulson@15077 ` 1959` ```lemma lemma_tan_total1: "\x. -(pi/2) < x & x < (pi/2) & tan x = y" ``` paulson@15077 ` 1960` ```apply (cut_tac linorder_linear [of 0 y], safe) ``` paulson@15077 ` 1961` ```apply (drule tan_total_pos) ``` paulson@15077 ` 1962` ```apply (cut_tac [2] y="-y" in tan_total_pos, safe) ``` paulson@15077 ` 1963` ```apply (rule_tac [3] x = "-x" in exI) ``` huffman@44710 ` 1964` ```apply (auto del: exI intro!: exI) ``` paulson@15077 ` 1965` ```done ``` paulson@15077 ` 1966` paulson@15077 ` 1967` ```lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y" ``` paulson@15077 ` 1968` ```apply (cut_tac y = y in lemma_tan_total1, auto) ``` paulson@15077 ` 1969` ```apply (cut_tac x = xa and y = y in linorder_less_linear, auto) ``` paulson@15077 ` 1970` ```apply (subgoal_tac [2] "\z. y < z & z < xa & DERIV tan z :> 0") ``` paulson@15077 ` 1971` ```apply (subgoal_tac "\z. xa < z & z < y & DERIV tan z :> 0") ``` paulson@15077 ` 1972` ```apply (rule_tac [4] Rolle) ``` paulson@15077 ` 1973` ```apply (rule_tac [2] Rolle) ``` huffman@44710 ` 1974` ```apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI ``` paulson@15077 ` 1975` ``` simp add: differentiable_def) ``` paulson@15077 ` 1976` ```txt{*Now, simulate TRYALL*} ``` paulson@15077 ` 1977` ```apply (rule_tac [!] DERIV_tan asm_rl) ``` paulson@15077 ` 1978` ```apply (auto dest!: DERIV_unique [OF _ DERIV_tan] ``` hoelzl@41970 ` 1979` ``` simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym]) ``` paulson@15077 ` 1980` ```done ``` paulson@15077 ` 1981` hoelzl@29803 ` 1982` ```lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2" ``` hoelzl@29803 ` 1983` ``` shows "tan y < tan x" ``` hoelzl@29803 ` 1984` ```proof - ``` hoelzl@29803 ` 1985` ``` have "\ x'. y \ x' \ x' \ x \ DERIV tan x' :> inverse (cos x'^2)" ``` hoelzl@29803 ` 1986` ``` proof (rule allI, rule impI) ``` hoelzl@29803 ` 1987` ``` fix x' :: real assume "y \ x' \ x' \ x" ``` wenzelm@33549 ` 1988` ``` hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto ``` hoelzl@29803 ` 1989` ``` from cos_gt_zero_pi[OF this] ``` hoelzl@29803 ` 1990` ``` have "cos x' \ 0" by auto ``` hoelzl@29803 ` 1991` ``` thus "DERIV tan x' :> inverse (cos x'^2)" by (rule DERIV_tan) ``` hoelzl@29803 ` 1992` ``` qed ``` hoelzl@41970 ` 1993` ``` from MVT2[OF `y < x` this] ``` hoelzl@29803 ` 1994` ``` obtain z where "y < z" and "z < x" and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\)" by auto ``` wenzelm@33549 ` 1995` ``` hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto ``` hoelzl@29803 ` 1996` ``` hence "0 < cos z" using cos_gt_zero_pi by auto ``` hoelzl@29803 ` 1997` ``` hence inv_pos: "0 < inverse ((cos z)\)" by auto ``` hoelzl@29803 ` 1998` ``` have "0 < x - y" using `y < x` by auto ``` huffman@36777 ` 1999` ``` from mult_pos_pos [OF this inv_pos] ``` hoelzl@29803 ` 2000` ``` have "0 < tan x - tan y" unfolding tan_diff by auto ``` hoelzl@29803 ` 2001` ``` thus ?thesis by auto ``` hoelzl@29803 ` 2002` ```qed ``` hoelzl@29803 ` 2003` hoelzl@29803 ` 2004` ```lemma tan_monotone': assumes "- (pi / 2) < y" and "y < pi / 2" and "- (pi / 2) < x" and "x < pi / 2" ``` hoelzl@29803 ` 2005` ``` shows "(y < x) = (tan y < tan x)" ``` hoelzl@29803 ` 2006` ```proof ``` hoelzl@29803 ` 2007` ``` assume "y < x" thus "tan y < tan x" using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto ``` hoelzl@29803 ` 2008` ```next ``` hoelzl@29803 ` 2009` ``` assume "tan y < tan x" ``` hoelzl@29803 ` 2010` ``` show "y < x" ``` hoelzl@29803 ` 2011` ``` proof (rule ccontr) ``` hoelzl@29803 ` 2012` ``` assume "\ y < x" hence "x \ y" by auto ``` hoelzl@41970 ` 2013` ``` hence "tan x \ tan y" ``` hoelzl@29803 ` 2014` ``` proof (cases "x = y") ``` hoelzl@29803 ` 2015` ``` case True thus ?thesis by auto ``` hoelzl@29803 ` 2016` ``` next ``` hoelzl@29803 ` 2017` ``` case False hence "x < y" using `x \ y` by auto ``` hoelzl@29803 ` 2018` ``` from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto ``` hoelzl@29803 ` 2019` ``` qed ``` hoelzl@29803 ` 2020` ``` thus False using `tan y < tan x` by auto ``` hoelzl@29803 ` 2021` ``` qed ``` hoelzl@29803 ` 2022` ```qed ``` hoelzl@29803 ` 2023` hoelzl@29803 ` 2024` ```lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)" unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto ``` hoelzl@29803 ` 2025` hoelzl@41970 ` 2026` ```lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x" ``` hoelzl@29803 ` 2027` ``` by (simp add: tan_def) ``` hoelzl@29803 ` 2028` hoelzl@41970 ` 2029` ```lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x" ``` hoelzl@29803 ` 2030` ```proof (induct n arbitrary: x) ``` hoelzl@29803 ` 2031` ``` case (Suc n) ``` huffman@36777 ` 2032` ``` have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" unfolding Suc_eq_plus1 real_of_nat_add real_of_one left_distrib by auto ``` hoelzl@29803 ` 2033` ``` show ?case unfolding split_pi_off using Suc by auto ``` hoelzl@29803 ` 2034` ```qed auto ``` hoelzl@29803 ` 2035` hoelzl@29803 ` 2036` ```lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x" ``` hoelzl@29803 ` 2037` ```proof (cases "0 \ i") ``` hoelzl@29803 ` 2038` ``` case True hence i_nat: "real i = real (nat i)" by auto ``` hoelzl@29803 ` 2039` ``` show ?thesis unfolding i_nat by auto ``` hoelzl@29803 ` 2040` ```next ``` hoelzl@29803 ` 2041` ``` case False hence i_nat: "real i = - real (nat (-i))" by auto ``` hoelzl@29803 ` 2042` ``` have "tan x = tan (x + real i * pi - real i * pi)" by auto ``` hoelzl@29803 ` 2043` ``` also have "\ = tan (x + real i * pi)" unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat) ``` hoelzl@29803 ` 2044` ``` finally show ?thesis by auto ``` hoelzl@29803 ` 2045` ```qed ``` hoelzl@29803 ` 2046` hoelzl@29803 ` 2047` ```lemma tan_periodic_n[simp]: "tan (x + number_of n * pi) = tan x" ``` hoelzl@29803 ` 2048` ``` using tan_periodic_int[of _ "number_of n" ] unfolding real_number_of . ``` huffman@23043 ` 2049` huffman@23043 ` 2050` ```subsection {* Inverse Trigonometric Functions *} ``` huffman@23043 ` 2051` huffman@23043 ` 2052` ```definition ``` huffman@23043 ` 2053` ``` arcsin :: "real => real" where ``` huffman@23043 ` 2054` ``` "arcsin y = (THE x. -(pi/2) \ x & x \ pi/2 & sin x = y)" ``` huffman@23043 ` 2055` huffman@23043 ` 2056` ```definition ``` huffman@23043 ` 2057` ``` arccos :: "real => real" where ``` huffman@23043 ` 2058` ``` "arccos y = (THE x. 0 \ x & x \ pi & cos x = y)" ``` huffman@23043 ` 2059` hoelzl@41970 ` 2060` ```definition ``` huffman@23043 ` 2061` ``` arctan :: "real => real" where ``` huffman@23043 ` 2062` ``` "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)" ``` huffman@23043 ` 2063` paulson@15229 ` 2064` ```lemma arcsin: ``` hoelzl@41970 ` 2065` ``` "[| -1 \ y; y \ 1 |] ``` hoelzl@41970 ` 2066` ``` ==> -(pi/2) \ arcsin y & ``` paulson@15077 ` 2067` ``` arcsin y \ pi/2 & sin(arcsin y) = y" ``` huffman@23011 ` 2068` ```unfolding arcsin_def by (rule theI' [OF sin_total]) ``` huffman@23011 ` 2069` huffman@23011 ` 2070` ```lemma arcsin_pi: ``` hoelzl@41970 ` 2071` ``` "[| -1 \ y; y \ 1 |] ``` huffman@23011 ` 2072` ``` ==> -(pi/2) \ arcsin y & arcsin y \ pi & sin(arcsin y) = y" ``` huffman@23011 ` 2073` ```apply (drule (1) arcsin) ``` huffman@23011 ` 2074` ```apply (force intro: order_trans) ``` paulson@15077 ` 2075` ```done ``` paulson@15077 ` 2076` paulson@15077 ` 2077` ```lemma sin_arcsin [simp]: "[| -1 \ y; y \ 1 |] ==> sin(arcsin y) = y" ``` paulson@15077 ` 2078` ```by (blast dest: arcsin) ``` hoelzl@41970 ` 2079` paulson@15077 ` 2080` ```lemma arcsin_bounded: ``` paulson@15077 ` 2081` ``` "[| -1 \ y; y \ 1 |] ==> -(pi/2) \ arcsin y & arcsin y \ pi/2" ``` paulson@15077 ` 2082` ```by (blast dest: arcsin) ``` paulson@15077 ` 2083` paulson@15077 ` 2084` ```lemma arcsin_lbound: "[| -1 \ y; y \ 1 |] ==> -(pi/2) \ arcsin y" ``` paulson@15077 ` 2085` ```by (blast dest: arcsin) ``` paulson@15077 ` 2086` paulson@15077 ` 2087` ```lemma arcsin_ubound: "[| -1 \ y; y \ 1 |] ==> arcsin y \ pi/2" ``` paulson@15077 ` 2088` ```by (blast dest: arcsin) ``` paulson@15077 ` 2089` paulson@15077 ` 2090` ```lemma arcsin_lt_bounded: ``` paulson@15077 ` 2091` ``` "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2" ``` paulson@15077 ` 2092` ```apply (frule order_less_imp_le) ``` paulson@15077 ` 2093` ```apply (frule_tac y = y in order_less_imp_le) ``` paulson@15077 ` 2094` ```apply (frule arcsin_bounded) ``` paulson@15077 ` 2095` ```apply (safe, simp) ``` paulson@15077 ` 2096` ```apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq) ``` paulson@15077 ` 2097` ```apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe) ``` paulson@15077 ` 2098` ```apply (drule_tac [!] f = sin in arg_cong, auto) ``` paulson@15077 ` 2099` ```done ``` paulson@15077 ` 2100` paulson@15077 ` 2101` ```lemma arcsin_sin: "[|-(pi/2) \ x; x \ pi/2 |] ==> arcsin(sin x) = x" ``` paulson@15077 ` 2102` ```apply (unfold arcsin_def) ``` huffman@23011 ` 2103` ```apply (rule the1_equality) ``` paulson@15077 ` 2104` ```apply (rule sin_total, auto) ``` paulson@15077 ` 2105` ```done ``` paulson@15077 ` 2106` huffman@22975 ` 2107` ```lemma arccos: ``` hoelzl@41970 ` 2108` ``` "[| -1 \ y; y \ 1 |] ``` huffman@22975 ` 2109` ``` ==> 0 \ arccos y & arccos y \ pi & cos(arccos y) = y" ``` huffman@23011 ` 2110` ```unfolding arccos_def by (rule theI' [OF cos_total]) ``` paulson@15077 ` 2111` huffman@22975 ` 2112` ```lemma cos_arccos [simp]: "[| -1 \ y; y \ 1 |] ==> cos(arccos y) = y" ``` huffman@22975 ` 2113` ```by (blast dest: arccos) ``` hoelzl@41970 ` 2114` huffman@22975 ` 2115` ```lemma arccos_bounded: "[| -1 \ y; y \ 1 |] ==> 0 \ arccos y & arccos y \ pi" ``` huffman@22975 ` 2116` ```by (blast dest: arccos) ``` paulson@15077 ` 2117` huffman@22975 ` 2118` ```lemma arccos_lbound: "[| -1 \ y; y \ 1 |] ==> 0 \ arccos y" ``` huffman@22975 ` 2119` ```by (blast dest: arccos) ``` paulson@15077 ` 2120` huffman@22975 ` 2121` ```lemma arccos_ubound: "[| -1 \ y; y \ 1 |] ==> arccos y \ pi" ``` huffman@22975 ` 2122` ```by (blast dest: arccos) ``` paulson@15077 ` 2123` huffman@22975 ` 2124` ```lemma arccos_lt_bounded: ``` hoelzl@41970 ` 2125` ``` "[| -1 < y; y < 1 |] ``` huffman@22975 ` 2126` ``` ==> 0 < arccos y & arccos y < pi" ``` paulson@15077 ` 2127` ```apply (frule order_less_imp_le) ``` paulson@15077 ` 2128` ```apply (frule_tac y = y in order_less_imp_le) ``` huffman@22975 ` 2129` ```apply (frule arccos_bounded, auto) ``` huffman@22975 ` 2130` ```apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq) ``` paulson@15077 ` 2131` ```apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto) ``` paulson@15077 ` 2132` ```apply (drule_tac [!] f = cos in arg_cong, auto) ``` paulson@15077 ` 2133` ```done ``` paulson@15077 ` 2134` huffman@22975 ` 2135` ```lemma arccos_cos: "[|0 \ x; x \ pi |] ==> arccos(cos x) = x" ``` huffman@22975 ` 2136` ```apply (simp add: arccos_def) ``` huffman@23011 ` 2137` ```apply (auto intro!: the1_equality cos_total) ``` paulson@15077 ` 2138` ```done ``` paulson@15077 ` 2139` huffman@22975 ` 2140` ```lemma arccos_cos2: "[|x \ 0; -pi \ x |] ==> arccos(cos x) = -x" ``` huffman@22975 ` 2141` ```apply (simp add: arccos_def) ``` huffman@23011 ` 2142` ```apply (auto intro!: the1_equality cos_total) ``` paulson@15077 ` 2143` ```done ``` paulson@15077 ` 2144` huffman@23045 ` 2145` ```lemma cos_arcsin: "\-1 \ x; x \ 1\ \ cos (arcsin x) = sqrt (1 - x\)" ``` huffman@23045 ` 2146` ```apply (subgoal_tac "x\ \ 1") ``` huffman@23052 ` 2147` ```apply (rule power2_eq_imp_eq) ``` huffman@23045 ` 2148` ```apply (simp add: cos_squared_eq) ``` huffman@23045 ` 2149` ```apply (rule cos_ge_zero) ``` huffman@23045 ` 2150` ```apply (erule (1) arcsin_lbound) ``` huffman@23045 ` 2151` ```apply (erule (1) arcsin_ubound) ``` huffman@23045 ` 2152` ```apply simp ``` huffman@23045 ` 2153` ```apply (subgoal_tac "\x\\ \ 1\", simp) ``` huffman@23045 ` 2154` ```apply (rule power_mono, simp, simp) ``` huffman@23045 ` 2155` ```done ``` huffman@23045 ` 2156` huffman@23045 ` 2157` ```lemma sin_arccos: "\-1 \ x; x \ 1\ \ sin (arccos x) = sqrt (1 - x\)" ``` huffman@23045 ` 2158` ```apply (subgoal_tac "x\ \ 1") ``` huffman@23052 ` 2159` ```apply (rule power2_eq_imp_eq) ``` huffman@23045 ` 2160` ```apply (simp add: sin_squared_eq) ``` huffman@23045 ` 2161` ```apply (rule sin_ge_zero) ``` huffman@23045 ` 2162` ```apply (erule (1) arccos_lbound) ``` huffman@23045 ` 2163` ```apply (erule (1) arccos_ubound) ``` huffman@23045 ` 2164` ```apply simp ``` huffman@23045 ` 2165` ```apply (subgoal_tac "\x\\ \ 1\", simp) ``` huffman@23045 ` 2166` ```apply (rule power_mono, simp, simp) ``` huffman@23045 ` 2167` ```done ``` huffman@23045 ` 2168` paulson@15077 ` 2169` ```lemma arctan [simp]: ``` paulson@15077 ` 2170` ``` "- (pi/2) < arctan y & arctan y < pi/2 & tan (arctan y) = y" ``` huffman@23011 ` 2171` ```unfolding arctan_def by (rule theI' [OF tan_total]) ``` paulson@15077 ` 2172` paulson@15077 ` 2173` ```lemma tan_arctan: "tan(arctan y) = y" ``` paulson@15077 ` 2174` ```by auto ``` paulson@15077 ` 2175` paulson@15077 ` 2176` ```lemma arctan_bounded: "- (pi/2) < arctan y & arctan y < pi/2" ``` paulson@15077 ` 2177` ```by (auto simp only: arctan) ``` paulson@15077 ` 2178` paulson@15077 ` 2179` ```lemma arctan_lbound: "- (pi/2) < arctan y" ``` paulson@15077 ` 2180` ```by auto ``` paulson@15077 ` 2181` paulson@15077 ` 2182` ```lemma arctan_ubound: "arctan y < pi/2" ``` paulson@15077 ` 2183` ```by (auto simp only: arctan) ``` paulson@15077 ` 2184` huffman@44746 ` 2185` ```lemma arctan_unique: ``` huffman@44746 ` 2186` ``` assumes "-(pi/2) < x" and "x < pi/2" and "tan x = y" ``` huffman@44746 ` 2187` ``` shows "arctan y = x" ``` huffman@44746 ` 2188` ``` using assms arctan [of y] tan_total [of y] by (fast elim: ex1E) ``` huffman@44746 ` 2189` hoelzl@41970 ` 2190` ```lemma arctan_tan: ``` paulson@15077 ` 2191` ``` "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x" ``` huffman@44746 ` 2192` ``` by (rule arctan_unique, simp_all) ``` paulson@15077 ` 2193` paulson@15077 ` 2194` ```lemma arctan_zero_zero [simp]: "arctan 0 = 0" ``` huffman@44746 ` 2195` ``` by (rule arctan_unique, simp_all) ``` huffman@44746 ` 2196` huffman@44746 ` 2197` ```lemma arctan_minus: "arctan (- x) = - arctan x" ``` huffman@44746 ` 2198` ``` apply (rule arctan_unique) ``` huffman@44746 ` 2199` ``` apply (simp only: neg_less_iff_less arctan_ubound) ``` huffman@44746 ` 2200` ``` apply (metis minus_less_iff arctan_lbound) ``` huffman@44746 ` 2201` ``` apply simp ``` huffman@44746 ` 2202` ``` done ``` paulson@15077 ` 2203` huffman@44725 ` 2204` ```lemma cos_arctan_not_zero [simp]: "cos (arctan x) \ 0" ``` huffman@44725 ` 2205` ``` by (intro less_imp_neq [symmetric] cos_gt_zero_pi ``` huffman@44725 ` 2206` ``` arctan_lbound arctan_ubound) ``` huffman@44725 ` 2207` huffman@44725 ` 2208` ```lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\)" ``` huffman@44725 ` 2209` ```proof (rule power2_eq_imp_eq) ``` huffman@44725 ` 2210` ``` have "0 < 1 + x\" by (simp add: add_pos_nonneg) ``` huffman@44725 ` 2211` ``` show "0 \ 1 / sqrt (1 + x\)" by simp ``` huffman@44725 ` 2212` ``` show "0 \ cos (arctan x)" ``` huffman@44725 ` 2213` ``` by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound) ``` huffman@44725 ` 2214` ``` have "(cos (arctan x))\ * (1 + (tan (arctan x))\) = 1" ``` huffman@44725 ` 2215` ``` unfolding tan_def by (simp add: right_distrib power_divide) ``` huffman@44725 ` 2216` ``` thus "(cos (arctan x))\ = (1 / sqrt (1 + x\))\" ``` huffman@44725 ` 2217` ``` using `0 < 1 + x\` by (simp add: power_divide eq_divide_eq) ``` huffman@44725 ` 2218` ```qed ``` huffman@44725 ` 2219` huffman@44725 ` 2220` ```lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\)" ``` huffman@44725 ` 2221` ``` using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]] ``` huffman@44725 ` 2222` ``` using tan_arctan [of x] unfolding tan_def cos_arctan ``` huffman@44725 ` 2223` ``` by (simp add: eq_divide_eq) ``` paulson@15077 ` 2224` paulson@15077 ` 2225` ```lemma tan_sec: "cos x \ 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2" ``` paulson@15077 ` 2226` ```apply (rule power_inverse [THEN subst]) ``` paulson@15077 ` 2227` ```apply (rule_tac c1 = "(cos x)\" in real_mult_right_cancel [THEN iffD1]) ``` huffman@22960 ` 2228` ```apply (auto dest: field_power_not_zero ``` hoelzl@41970 ` 2229` ``` simp add: power_mult_distrib left_distrib power_divide tan_def ``` huffman@30273 ` 2230` ``` mult_assoc power_inverse [symmetric]) ``` paulson@15077 ` 2231` ```done ``` paulson@15077 ` 2232` huffman@44746 ` 2233` ```lemma arctan_less_iff: "arctan x < arctan y \ x < y" ``` huffman@44746 ` 2234` ``` by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan) ``` huffman@44746 ` 2235` huffman@44746 ` 2236` ```lemma arctan_le_iff: "arctan x \ arctan y \ x \ y" ``` huffman@44746 ` 2237` ``` by (simp only: not_less [symmetric] arctan_less_iff) ``` huffman@44746 ` 2238` huffman@44746 ` 2239` ```lemma arctan_eq_iff: "arctan x = arctan y \ x = y" ``` huffman@44746 ` 2240` ``` by (simp only: eq_iff [where 'a=real] arctan_le_iff) ``` huffman@44746 ` 2241` huffman@44746 ` 2242` ```lemma zero_less_arctan_iff [simp]: "0 < arctan x \ 0 < x" ``` huffman@44746 ` 2243` ``` using arctan_less_iff [of 0 x] by simp ``` huffman@44746 ` 2244` huffman@44746 ` 2245` ```lemma arctan_less_zero_iff [simp]: "arctan x < 0 \ x < 0" ``` huffman@44746 ` 2246` ``` using arctan_less_iff [of x 0] by simp ``` huffman@44746 ` 2247` huffman@44746 ` 2248` ```lemma zero_le_arctan_iff [simp]: "0 \ arctan x \ 0 \ x" ``` huffman@44746 ` 2249` ``` using arctan_le_iff [of 0 x] by simp ``` huffman@44746 ` 2250` huffman@44746 ` 2251` ```lemma arctan_le_zero_iff [simp]: "arctan x \ 0 \ x \ 0" ``` huffman@44746 ` 2252` ``` using arctan_le_iff [of x 0] by simp ``` huffman@44746 ` 2253` huffman@44746 ` 2254` ```lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \ x = 0" ``` huffman@44746 ` 2255` ``` using arctan_eq_iff [of x 0] by simp ``` huffman@44746 ` 2256` huffman@23045 ` 2257` ```lemma isCont_inverse_function2: ``` huffman@23045 ` 2258` ``` fixes f g :: "real \ real" shows ``` huffman@23045 ` 2259` ``` "\a < x; x < b; ``` huffman@23045 ` 2260` ``` \z. a \ z \ z \ b \ g (f z) = z; ``` huffman@23045 ` 2261` ``` \z. a \ z \ z \ b \ isCont f z\ ``` huffman@23045 ` 2262` ``` \ isCont g (f x)" ``` huffman@23045 ` 2263` ```apply (rule isCont_inverse_function ``` huffman@23045 ` 2264` ``` [where f=f and d="min (x - a) (b - x)"]) ``` huffman@23045 ` 2265` ```apply (simp_all add: abs_le_iff) ``` huffman@23045 ` 2266` ```done ``` huffman@23045 ` 2267` huffman@23045 ` 2268` ```lemma isCont_arcsin: "\-1 < x; x < 1\ \ isCont arcsin x" ``` huffman@23045 ` 2269` ```apply (subgoal_tac "isCont arcsin (sin (arcsin x))", simp) ``` huffman@23045 ` 2270` ```apply (rule isCont_inverse_function2 [where f=sin]) ``` huffman@23045 ` 2271` ```apply (erule (1) arcsin_lt_bounded [THEN conjunct1]) ``` huffman@23045 ` 2272` ```apply (erule (1) arcsin_lt_bounded [THEN conjunct2]) ``` huffman@23045 ` 2273` ```apply (fast intro: arcsin_sin, simp) ``` huffman@23045 ` 2274` ```done ``` huffman@23045 ` 2275` huffman@23045 ` 2276` ```lemma isCont_arccos: "\-1 < x; x < 1\ \ isCont arccos x" ``` huffman@23045 ` 2277` ```apply (subgoal_tac "isCont arccos (cos (arccos x))", simp) ``` huffman@23045 ` 2278` ```apply (rule isCont_inverse_function2 [where f=cos]) ``` huffman@23045 ` 2279` ```apply (erule (1) arccos_lt_bounded [THEN conjunct1]) ``` huffman@23045 ` 2280` ```apply (erule (1) arccos_lt_bounded [THEN conjunct2]) ``` huffman@23045 ` 2281` ```apply (fast intro: arccos_cos, simp) ``` huffman@23045 ` 2282` ```done ``` huffman@23045 ` 2283` huffman@23045 ` 2284` ```lemma isCont_arctan: "isCont arctan x" ``` huffman@23045 ` 2285` ```apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify) ``` huffman@23045 ` 2286` ```apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify) ``` huffman@23045 ` 2287` ```apply (subgoal_tac "isCont arctan (tan (arctan x))", simp) ``` huffman@23045 ` 2288` ```apply (erule (1) isCont_inverse_function2 [where f=tan]) ``` paulson@33667 ` 2289` ```apply (metis arctan_tan order_le_less_trans order_less_le_trans) ``` huffman@36777 ` 2290` ```apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le) ``` huffman@23045 ` 2291` ```done ``` huffman@23045 ` 2292` huffman@23045 ` 2293` ```lemma DERIV_arcsin: ``` huffman@23045 ` 2294` ``` "\-1 < x; x < 1\ \ DERIV arcsin x :> inverse (sqrt (1 - x\))" ``` huffman@23045 ` 2295` ```apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"]) ``` huffman@44308 ` 2296` ```apply (rule DERIV_cong [OF DERIV_sin]) ``` huffman@23045 ` 2297` ```apply (simp add: cos_arcsin) ``` huffman@23045 ` 2298` ```apply (subgoal_tac "\x\\ < 1\", simp) ``` huffman@23045 ` 2299` ```apply (rule power_strict_mono, simp, simp, simp) ``` huffman@23045 ` 2300` ```apply assumption ``` huffman@23045 ` 2301` ```apply assumption ``` huffman@23045 ` 2302` ```apply simp ``` huffman@23045 ` 2303` ```apply (erule (1) isCont_arcsin) ``` huffman@23045 ` 2304` ```done ``` huffman@23045 ` 2305` huffman@23045 ` 2306` ```lemma DERIV_arccos: ``` huffman@23045 ` 2307` ``` "\-1 < x; x < 1\ \ DERIV arccos x :> inverse (- sqrt (1 - x\))" ``` huffman@23045 ` 2308` ```apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"]) ``` huffman@44308 ` 2309` ```apply (rule DERIV_cong [OF DERIV_cos]) ``` huffman@23045 ` 2310` ```apply (simp add: sin_arccos) ``` huffman@23045 ` 2311` ```apply (subgoal_tac "\x\\ < 1\", simp) ``` huffman@23045 ` 2312` ```apply (rule power_strict_mono, simp, simp, simp) ``` huffman@23045 ` 2313` ```apply assumption ``` huffman@23045 ` 2314` ```apply assumption ``` huffman@23045 ` 2315` ```apply simp ``` huffman@23045 ` 2316` ```apply (erule (1) isCont_arccos) ``` huffman@23045 ` 2317` ```done ``` huffman@23045 ` 2318` huffman@23045 ` 2319` ```lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\)" ``` huffman@23045 ` 2320` ```apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"]) ``` huffman@44308 ` 2321` ```apply (rule DERIV_cong [OF DERIV_tan]) ``` huffman@23045 ` 2322` ```apply (rule cos_arctan_not_zero) ``` huffman@23045 ` 2323` ```apply (simp add: power_inverse tan_sec [symmetric]) ``` huffman@23045 ` 2324` ```apply (subgoal_tac "0 < 1 + x\", simp) ``` huffman@23045 ` 2325` ```apply (simp add: add_pos_nonneg) ``` huffman@23045 ` 2326` ```apply (simp, simp, simp, rule isCont_arctan) ``` huffman@23045 ` 2327` ```done ``` huffman@23045 ` 2328` hoelzl@31880 ` 2329` ```declare ``` hoelzl@31880 ` 2330` ``` DERIV_arcsin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 2331` ``` DERIV_arccos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 2332` ``` DERIV_arctan[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros] ``` hoelzl@31880 ` 2333` huffman@23043 ` 2334` ```subsection {* More Theorems about Sin and Cos *} ``` huffman@23043 ` 2335` huffman@23052 ` 2336` ```lemma cos_45: "cos (pi / 4) = sqrt 2 / 2" ``` huffman@23052 ` 2337` ```proof - ``` huffman@23052 ` 2338` ``` let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)" ``` huffman@23052 ` 2339` ``` have nonneg: "0 \ ?c" ``` huffman@45308 ` 2340` ``` by (simp add: cos_ge_zero) ``` huffman@23052 ` 2341` ``` have "0 = cos (pi / 4 + pi / 4)" ``` huffman@23052 ` 2342` ``` by simp ``` huffman@23052 ` 2343` ``` also have "cos (pi / 4 + pi / 4) = ?c\ - ?s\" ``` huffman@23052 ` 2344` ``` by (simp only: cos_add power2_eq_square) ``` huffman@23052 ` 2345` ``` also have "\ = 2 * ?c\ - 1" ``` huffman@23052 ` 2346` ``` by (simp add: sin_squared_eq) ``` huffman@23052 ` 2347` ``` finally have "?c\ = (sqrt 2 / 2)\" ``` huffman@23052 ` 2348` ``` by (simp add: power_divide) ``` huffman@23052 ` 2349` ``` thus ?thesis ``` huffman@23052 ` 2350` ``` using nonneg by (rule power2_eq_imp_eq) simp ``` huffman@23052 ` 2351` ```qed ``` huffman@23052 ` 2352` huffman@23052 ` 2353` ```lemma cos_30: "cos (pi / 6) = sqrt 3 / 2" ``` huffman@23052 ` 2354` ```proof - ``` huffman@23052 ` 2355` ``` let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)" ``` huffman@23052 ` 2356` ``` have pos_c: "0 < ?c" ``` huffman@23052 ` 2357` ``` by (rule cos_gt_zero, simp, simp) ``` huffman@23052 ` 2358` ``` have "0 = cos (pi / 6 + pi / 6 + pi / 6)" ``` huffman@23066 ` 2359` ``` by simp ``` huffman@23052 ` 2360` ``` also have "\ = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s" ``` huffman@23052 ` 2361` ``` by (simp only: cos_add sin_add) ``` huffman@23052 ` 2362` ``` also have "\ = ?c * (?c\ - 3 * ?s\)" ``` nipkow@29667 ` 2363` ``` by (simp add: algebra_simps power2_eq_square) ``` huffman@23052 ` 2364` ``` finally have "?c\ = (sqrt 3 / 2)\" ``` huffman@23052 ` 2365` ``` using pos_c by (simp add: sin_squared_eq power_divide) ``` huffman@23052 ` 2366` ``` thus ?thesis ``` huffman@23052 ` 2367` ``` using pos_c [THEN order_less_imp_le] ``` huffman@23052 ` 2368` ``` by (rule power2_eq_imp_eq) simp ``` huffman@23052 ` 2369` ```qed ``` huffman@23052 ` 2370` huffman@23052 ` 2371` ```lemma sin_45: "sin (pi / 4) = sqrt 2 / 2" ``` huffman@45309 ` 2372` ```by (simp add: sin_cos_eq cos_45) ``` huffman@23052 ` 2373` huffman@23052 ` 2374` ```lemma sin_60: "sin (pi / 3) = sqrt 3 / 2" ``` huffman@45309 ` 2375` ```by (simp add: sin_cos_eq cos_30) ``` huffman@23052 ` 2376` huffman@23052 ` 2377` ```lemma cos_60: "cos (pi / 3) = 1 / 2" ```