src/HOL/Transcendental.thy
 author huffman Wed Dec 24 08:06:27 2008 -0800 (2008-12-24) changeset 29165 562f95f06244 parent 29164 0d49c5b55046 child 29166 c23b2d108612 permissions -rw-r--r--
cleaned up some proofs; removed redundant simp rules
 paulson@12196 ` 1` ```(* Title : Transcendental.thy ``` paulson@12196 ` 2` ``` Author : Jacques D. Fleuriot ``` paulson@12196 ` 3` ``` Copyright : 1998,1999 University of Cambridge ``` paulson@13958 ` 4` ``` 1999,2001 University of Edinburgh ``` paulson@15077 ` 5` ``` Conversion to Isar and new proofs by Lawrence C Paulson, 2004 ``` paulson@12196 ` 6` ```*) ``` paulson@12196 ` 7` paulson@15077 ` 8` ```header{*Power Series, Transcendental Functions etc.*} ``` paulson@15077 ` 9` nipkow@15131 ` 10` ```theory Transcendental ``` haftmann@25600 ` 11` ```imports Fact Series Deriv NthRoot ``` nipkow@15131 ` 12` ```begin ``` paulson@15077 ` 13` huffman@29164 ` 14` ```subsection {* Properties of Power Series *} ``` paulson@15077 ` 15` huffman@23082 ` 16` ```lemma lemma_realpow_diff: ``` huffman@23082 ` 17` ``` fixes y :: "'a::recpower" ``` huffman@23082 ` 18` ``` shows "p \ n \ y ^ (Suc n - p) = (y ^ (n - p)) * y" ``` huffman@23082 ` 19` ```proof - ``` huffman@23082 ` 20` ``` assume "p \ n" ``` huffman@23082 ` 21` ``` hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le) ``` huffman@23082 ` 22` ``` thus ?thesis by (simp add: power_Suc power_commutes) ``` huffman@23082 ` 23` ```qed ``` paulson@15077 ` 24` paulson@15077 ` 25` ```lemma lemma_realpow_diff_sumr: ``` huffman@23082 ` 26` ``` fixes y :: "'a::{recpower,comm_semiring_0}" shows ``` huffman@23082 ` 27` ``` "(\p=0..p=0..p=0..p=0..p=0..z\ < \x\"}.*} ``` paulson@15077 ` 57` paulson@15077 ` 58` ```lemma powser_insidea: ``` huffman@23082 ` 59` ``` fixes x z :: "'a::{real_normed_field,banach,recpower}" ``` huffman@20849 ` 60` ``` assumes 1: "summable (\n. f n * x ^ n)" ``` huffman@23082 ` 61` ``` assumes 2: "norm z < norm x" ``` huffman@23082 ` 62` ``` shows "summable (\n. norm (f n * z ^ n))" ``` huffman@20849 ` 63` ```proof - ``` huffman@20849 ` 64` ``` from 2 have x_neq_0: "x \ 0" by clarsimp ``` huffman@20849 ` 65` ``` from 1 have "(\n. f n * x ^ n) ----> 0" ``` huffman@20849 ` 66` ``` by (rule summable_LIMSEQ_zero) ``` huffman@20849 ` 67` ``` hence "convergent (\n. f n * x ^ n)" ``` huffman@20849 ` 68` ``` by (rule convergentI) ``` huffman@20849 ` 69` ``` hence "Cauchy (\n. f n * x ^ n)" ``` huffman@20849 ` 70` ``` by (simp add: Cauchy_convergent_iff) ``` huffman@20849 ` 71` ``` hence "Bseq (\n. f n * x ^ n)" ``` huffman@20849 ` 72` ``` by (rule Cauchy_Bseq) ``` huffman@23082 ` 73` ``` then obtain K where 3: "0 < K" and 4: "\n. norm (f n * x ^ n) \ K" ``` huffman@20849 ` 74` ``` by (simp add: Bseq_def, safe) ``` huffman@23082 ` 75` ``` have "\N. \n\N. norm (norm (f n * z ^ n)) \ ``` huffman@23082 ` 76` ``` K * norm (z ^ n) * inverse (norm (x ^ n))" ``` huffman@20849 ` 77` ``` proof (intro exI allI impI) ``` huffman@20849 ` 78` ``` fix n::nat assume "0 \ n" ``` huffman@23082 ` 79` ``` have "norm (norm (f n * z ^ n)) * norm (x ^ n) = ``` huffman@23082 ` 80` ``` norm (f n * x ^ n) * norm (z ^ n)" ``` huffman@23082 ` 81` ``` by (simp add: norm_mult abs_mult) ``` huffman@23082 ` 82` ``` also have "\ \ K * norm (z ^ n)" ``` huffman@23082 ` 83` ``` by (simp only: mult_right_mono 4 norm_ge_zero) ``` huffman@23082 ` 84` ``` also have "\ = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))" ``` huffman@20849 ` 85` ``` by (simp add: x_neq_0) ``` huffman@23082 ` 86` ``` also have "\ = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)" ``` huffman@20849 ` 87` ``` by (simp only: mult_assoc) ``` huffman@23082 ` 88` ``` finally show "norm (norm (f n * z ^ n)) \ ``` huffman@23082 ` 89` ``` K * norm (z ^ n) * inverse (norm (x ^ n))" ``` huffman@20849 ` 90` ``` by (simp add: mult_le_cancel_right x_neq_0) ``` huffman@20849 ` 91` ``` qed ``` huffman@23082 ` 92` ``` moreover have "summable (\n. K * norm (z ^ n) * inverse (norm (x ^ n)))" ``` huffman@20849 ` 93` ``` proof - ``` huffman@23082 ` 94` ``` from 2 have "norm (norm (z * inverse x)) < 1" ``` huffman@23082 ` 95` ``` using x_neq_0 ``` huffman@23082 ` 96` ``` by (simp add: nonzero_norm_divide divide_inverse [symmetric]) ``` huffman@23082 ` 97` ``` hence "summable (\n. norm (z * inverse x) ^ n)" ``` huffman@20849 ` 98` ``` by (rule summable_geometric) ``` huffman@23082 ` 99` ``` hence "summable (\n. K * norm (z * inverse x) ^ n)" ``` huffman@20849 ` 100` ``` by (rule summable_mult) ``` huffman@23082 ` 101` ``` thus "summable (\n. K * norm (z ^ n) * inverse (norm (x ^ n)))" ``` huffman@23082 ` 102` ``` using x_neq_0 ``` huffman@23082 ` 103` ``` by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib ``` huffman@23082 ` 104` ``` power_inverse norm_power mult_assoc) ``` huffman@20849 ` 105` ``` qed ``` huffman@23082 ` 106` ``` ultimately show "summable (\n. norm (f n * z ^ n))" ``` huffman@20849 ` 107` ``` by (rule summable_comparison_test) ``` huffman@20849 ` 108` ```qed ``` paulson@15077 ` 109` paulson@15229 ` 110` ```lemma powser_inside: ``` huffman@23082 ` 111` ``` fixes f :: "nat \ 'a::{real_normed_field,banach,recpower}" shows ``` huffman@23082 ` 112` ``` "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |] ``` paulson@15077 ` 113` ``` ==> summable (%n. f(n) * (z ^ n))" ``` huffman@23082 ` 114` ```by (rule powser_insidea [THEN summable_norm_cancel]) ``` paulson@15077 ` 115` paulson@15077 ` 116` huffman@29164 ` 117` ```subsection {* Term-by-Term Differentiability of Power Series *} ``` huffman@23043 ` 118` huffman@23043 ` 119` ```definition ``` huffman@23082 ` 120` ``` diffs :: "(nat => 'a::ring_1) => nat => 'a" where ``` huffman@23082 ` 121` ``` "diffs c = (%n. of_nat (Suc n) * c(Suc n))" ``` paulson@15077 ` 122` paulson@15077 ` 123` ```text{*Lemma about distributing negation over it*} ``` paulson@15077 ` 124` ```lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)" ``` paulson@15077 ` 125` ```by (simp add: diffs_def) ``` paulson@15077 ` 126` huffman@29163 ` 127` ```lemma sums_Suc_imp: ``` huffman@29163 ` 128` ``` assumes f: "f 0 = 0" ``` huffman@29163 ` 129` ``` shows "(\n. f (Suc n)) sums s \ (\n. f n) sums s" ``` huffman@29163 ` 130` ```unfolding sums_def ``` huffman@29163 ` 131` ```apply (rule LIMSEQ_imp_Suc) ``` huffman@29163 ` 132` ```apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric]) ``` huffman@29163 ` 133` ```apply (simp only: setsum_shift_bounds_Suc_ivl) ``` paulson@15077 ` 134` ```done ``` paulson@15077 ` 135` paulson@15229 ` 136` ```lemma diffs_equiv: ``` paulson@15229 ` 137` ``` "summable (%n. (diffs c)(n) * (x ^ n)) ==> ``` huffman@23082 ` 138` ``` (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums ``` nipkow@15546 ` 139` ``` (\n. (diffs c)(n) * (x ^ n))" ``` huffman@29163 ` 140` ```unfolding diffs_def ``` huffman@29163 ` 141` ```apply (drule summable_sums) ``` huffman@29163 ` 142` ```apply (rule sums_Suc_imp, simp_all) ``` paulson@15077 ` 143` ```done ``` paulson@15077 ` 144` paulson@15077 ` 145` ```lemma lemma_termdiff1: ``` huffman@23082 ` 146` ``` fixes z :: "'a :: {recpower,comm_ring}" shows ``` nipkow@15539 ` 147` ``` "(\p=0..p=0..i = 0.. 0" shows ``` huffman@23082 ` 159` ``` "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) = ``` huffman@20860 ` 160` ``` h * (\p=0..< n - Suc 0. \q=0..< n - Suc 0 - p. ``` huffman@23082 ` 161` ``` (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs") ``` huffman@23082 ` 162` ```apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h) ``` huffman@20860 ` 163` ```apply (simp add: right_diff_distrib diff_divide_distrib h) ``` paulson@15077 ` 164` ```apply (simp add: mult_assoc [symmetric]) ``` huffman@20860 ` 165` ```apply (cases "n", simp) ``` huffman@20860 ` 166` ```apply (simp add: lemma_realpow_diff_sumr2 h ``` huffman@20860 ` 167` ``` right_diff_distrib [symmetric] mult_assoc ``` huffman@23082 ` 168` ``` del: realpow_Suc setsum_op_ivl_Suc of_nat_Suc) ``` huffman@20860 ` 169` ```apply (subst lemma_realpow_rev_sumr) ``` huffman@23082 ` 170` ```apply (subst sumr_diff_mult_const2) ``` huffman@20860 ` 171` ```apply simp ``` huffman@20860 ` 172` ```apply (simp only: lemma_termdiff1 setsum_right_distrib) ``` huffman@20860 ` 173` ```apply (rule setsum_cong [OF refl]) ``` nipkow@15539 ` 174` ```apply (simp add: diff_minus [symmetric] less_iff_Suc_add) ``` huffman@20860 ` 175` ```apply (clarify) ``` huffman@20860 ` 176` ```apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac ``` huffman@20860 ` 177` ``` del: setsum_op_ivl_Suc realpow_Suc) ``` huffman@20860 ` 178` ```apply (subst mult_assoc [symmetric], subst power_add [symmetric]) ``` huffman@20860 ` 179` ```apply (simp add: mult_ac) ``` huffman@20860 ` 180` ```done ``` huffman@20860 ` 181` huffman@20860 ` 182` ```lemma real_setsum_nat_ivl_bounded2: ``` huffman@23082 ` 183` ``` fixes K :: "'a::ordered_semidom" ``` huffman@23082 ` 184` ``` assumes f: "\p::nat. p < n \ f p \ K" ``` huffman@23082 ` 185` ``` assumes K: "0 \ K" ``` huffman@23082 ` 186` ``` shows "setsum f {0.. of_nat n * K" ``` huffman@23082 ` 187` ```apply (rule order_trans [OF setsum_mono]) ``` huffman@23082 ` 188` ```apply (rule f, simp) ``` huffman@23082 ` 189` ```apply (simp add: mult_right_mono K) ``` paulson@15077 ` 190` ```done ``` paulson@15077 ` 191` paulson@15229 ` 192` ```lemma lemma_termdiff3: ``` huffman@23112 ` 193` ``` fixes h z :: "'a::{real_normed_field,recpower}" ``` huffman@20860 ` 194` ``` assumes 1: "h \ 0" ``` huffman@23082 ` 195` ``` assumes 2: "norm z \ K" ``` huffman@23082 ` 196` ``` assumes 3: "norm (z + h) \ K" ``` huffman@23082 ` 197` ``` shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) ``` huffman@23082 ` 198` ``` \ of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h" ``` huffman@20860 ` 199` ```proof - ``` huffman@23082 ` 200` ``` have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) = ``` huffman@23082 ` 201` ``` norm (\p = 0..q = 0.. \ of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h" ``` huffman@23082 ` 208` ``` proof (rule mult_right_mono [OF _ norm_ge_zero]) ``` huffman@23082 ` 209` ``` from norm_ge_zero 2 have K: "0 \ K" by (rule order_trans) ``` huffman@23082 ` 210` ``` have le_Kn: "\i j n. i + j = n \ norm ((z + h) ^ i * z ^ j) \ K ^ n" ``` huffman@20860 ` 211` ``` apply (erule subst) ``` huffman@23082 ` 212` ``` apply (simp only: norm_mult norm_power power_add) ``` huffman@23082 ` 213` ``` apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K) ``` huffman@20860 ` 214` ``` done ``` huffman@23082 ` 215` ``` show "norm (\p = 0..q = 0.. of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))" ``` huffman@20860 ` 218` ``` apply (intro ``` huffman@23082 ` 219` ``` order_trans [OF norm_setsum] ``` huffman@20860 ` 220` ``` real_setsum_nat_ivl_bounded2 ``` huffman@20860 ` 221` ``` mult_nonneg_nonneg ``` huffman@23082 ` 222` ``` zero_le_imp_of_nat ``` huffman@20860 ` 223` ``` zero_le_power K) ``` huffman@20860 ` 224` ``` apply (rule le_Kn, simp) ``` huffman@20860 ` 225` ``` done ``` huffman@20860 ` 226` ``` qed ``` huffman@23082 ` 227` ``` also have "\ = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h" ``` huffman@20860 ` 228` ``` by (simp only: mult_assoc) ``` huffman@20860 ` 229` ``` finally show ?thesis . ``` huffman@20860 ` 230` ```qed ``` paulson@15077 ` 231` huffman@20860 ` 232` ```lemma lemma_termdiff4: ``` huffman@23112 ` 233` ``` fixes f :: "'a::{real_normed_field,recpower} \ ``` huffman@23082 ` 234` ``` 'b::real_normed_vector" ``` huffman@20860 ` 235` ``` assumes k: "0 < (k::real)" ``` huffman@23082 ` 236` ``` assumes le: "\h. \h \ 0; norm h < k\ \ norm (f h) \ K * norm h" ``` huffman@20860 ` 237` ``` shows "f -- 0 --> 0" ``` huffman@29163 ` 238` ```unfolding LIM_def diff_0_right ``` huffman@29163 ` 239` ```proof (safe) ``` huffman@29163 ` 240` ``` let ?h = "of_real (k / 2)::'a" ``` huffman@29163 ` 241` ``` have "?h \ 0" and "norm ?h < k" using k by simp_all ``` huffman@29163 ` 242` ``` hence "norm (f ?h) \ K * norm ?h" by (rule le) ``` huffman@29163 ` 243` ``` hence "0 \ K * norm ?h" by (rule order_trans [OF norm_ge_zero]) ``` huffman@29163 ` 244` ``` hence zero_le_K: "0 \ K" using k by (simp add: zero_le_mult_iff) ``` huffman@29163 ` 245` huffman@20860 ` 246` ``` fix r::real assume r: "0 < r" ``` huffman@23082 ` 247` ``` show "\s. 0 < s \ (\x. x \ 0 \ norm x < s \ norm (f x) < r)" ``` huffman@20860 ` 248` ``` proof (cases) ``` huffman@20860 ` 249` ``` assume "K = 0" ``` huffman@23082 ` 250` ``` with k r le have "0 < k \ (\x. x \ 0 \ norm x < k \ norm (f x) < r)" ``` huffman@20860 ` 251` ``` by simp ``` huffman@23082 ` 252` ``` thus "\s. 0 < s \ (\x. x \ 0 \ norm x < s \ norm (f x) < r)" .. ``` huffman@20860 ` 253` ``` next ``` huffman@20860 ` 254` ``` assume K_neq_zero: "K \ 0" ``` huffman@20860 ` 255` ``` with zero_le_K have K: "0 < K" by simp ``` huffman@23082 ` 256` ``` show "\s. 0 < s \ (\x. x \ 0 \ norm x < s \ norm (f x) < r)" ``` huffman@20860 ` 257` ``` proof (rule exI, safe) ``` huffman@20860 ` 258` ``` from k r K show "0 < min k (r * inverse K / 2)" ``` huffman@20860 ` 259` ``` by (simp add: mult_pos_pos positive_imp_inverse_positive) ``` huffman@20860 ` 260` ``` next ``` huffman@23082 ` 261` ``` fix x::'a ``` huffman@23082 ` 262` ``` assume x1: "x \ 0" and x2: "norm x < min k (r * inverse K / 2)" ``` huffman@23082 ` 263` ``` from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2" ``` huffman@20860 ` 264` ``` by simp_all ``` huffman@23082 ` 265` ``` from x1 x3 le have "norm (f x) \ K * norm x" by simp ``` huffman@23082 ` 266` ``` also from x4 K have "K * norm x < K * (r * inverse K / 2)" ``` huffman@20860 ` 267` ``` by (rule mult_strict_left_mono) ``` huffman@20860 ` 268` ``` also have "\ = r / 2" ``` huffman@20860 ` 269` ``` using K_neq_zero by simp ``` huffman@20860 ` 270` ``` also have "r / 2 < r" ``` huffman@20860 ` 271` ``` using r by simp ``` huffman@23082 ` 272` ``` finally show "norm (f x) < r" . ``` huffman@20860 ` 273` ``` qed ``` huffman@20860 ` 274` ``` qed ``` huffman@20860 ` 275` ```qed ``` paulson@15077 ` 276` paulson@15229 ` 277` ```lemma lemma_termdiff5: ``` huffman@23112 ` 278` ``` fixes g :: "'a::{recpower,real_normed_field} \ ``` huffman@23082 ` 279` ``` nat \ 'b::banach" ``` huffman@20860 ` 280` ``` assumes k: "0 < (k::real)" ``` huffman@20860 ` 281` ``` assumes f: "summable f" ``` huffman@23082 ` 282` ``` assumes le: "\h n. \h \ 0; norm h < k\ \ norm (g h n) \ f n * norm h" ``` huffman@20860 ` 283` ``` shows "(\h. suminf (g h)) -- 0 --> 0" ``` huffman@20860 ` 284` ```proof (rule lemma_termdiff4 [OF k]) ``` huffman@23082 ` 285` ``` fix h::'a assume "h \ 0" and "norm h < k" ``` huffman@23082 ` 286` ``` hence A: "\n. norm (g h n) \ f n * norm h" ``` huffman@20860 ` 287` ``` by (simp add: le) ``` huffman@23082 ` 288` ``` hence "\N. \n\N. norm (norm (g h n)) \ f n * norm h" ``` huffman@20860 ` 289` ``` by simp ``` huffman@23082 ` 290` ``` moreover from f have B: "summable (\n. f n * norm h)" ``` huffman@20860 ` 291` ``` by (rule summable_mult2) ``` huffman@23082 ` 292` ``` ultimately have C: "summable (\n. norm (g h n))" ``` huffman@20860 ` 293` ``` by (rule summable_comparison_test) ``` huffman@23082 ` 294` ``` hence "norm (suminf (g h)) \ (\n. norm (g h n))" ``` huffman@23082 ` 295` ``` by (rule summable_norm) ``` huffman@23082 ` 296` ``` also from A C B have "(\n. norm (g h n)) \ (\n. f n * norm h)" ``` huffman@20860 ` 297` ``` by (rule summable_le) ``` huffman@23082 ` 298` ``` also from f have "(\n. f n * norm h) = suminf f * norm h" ``` huffman@20860 ` 299` ``` by (rule suminf_mult2 [symmetric]) ``` huffman@23082 ` 300` ``` finally show "norm (suminf (g h)) \ suminf f * norm h" . ``` huffman@20860 ` 301` ```qed ``` paulson@15077 ` 302` paulson@15077 ` 303` paulson@15077 ` 304` ```text{* FIXME: Long proofs*} ``` paulson@15077 ` 305` paulson@15077 ` 306` ```lemma termdiffs_aux: ``` huffman@23112 ` 307` ``` fixes x :: "'a::{recpower,real_normed_field,banach}" ``` huffman@20849 ` 308` ``` assumes 1: "summable (\n. diffs (diffs c) n * K ^ n)" ``` huffman@23082 ` 309` ``` assumes 2: "norm x < norm K" ``` huffman@20860 ` 310` ``` shows "(\h. \n. c n * (((x + h) ^ n - x ^ n) / h ``` huffman@23082 ` 311` ``` - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0" ``` huffman@20849 ` 312` ```proof - ``` huffman@20860 ` 313` ``` from dense [OF 2] ``` huffman@23082 ` 314` ``` obtain r where r1: "norm x < r" and r2: "r < norm K" by fast ``` huffman@23082 ` 315` ``` from norm_ge_zero r1 have r: "0 < r" ``` huffman@20860 ` 316` ``` by (rule order_le_less_trans) ``` huffman@20860 ` 317` ``` hence r_neq_0: "r \ 0" by simp ``` huffman@20860 ` 318` ``` show ?thesis ``` huffman@20849 ` 319` ``` proof (rule lemma_termdiff5) ``` huffman@23082 ` 320` ``` show "0 < r - norm x" using r1 by simp ``` huffman@20849 ` 321` ``` next ``` huffman@23082 ` 322` ``` from r r2 have "norm (of_real r::'a) < norm K" ``` huffman@23082 ` 323` ``` by simp ``` huffman@23082 ` 324` ``` with 1 have "summable (\n. norm (diffs (diffs c) n * (of_real r ^ n)))" ``` huffman@20860 ` 325` ``` by (rule powser_insidea) ``` huffman@23082 ` 326` ``` hence "summable (\n. diffs (diffs (\n. norm (c n))) n * r ^ n)" ``` huffman@23082 ` 327` ``` using r ``` huffman@23082 ` 328` ``` by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc) ``` huffman@23082 ` 329` ``` hence "summable (\n. of_nat n * diffs (\n. norm (c n)) n * r ^ (n - Suc 0))" ``` huffman@20860 ` 330` ``` by (rule diffs_equiv [THEN sums_summable]) ``` huffman@23082 ` 331` ``` also have "(\n. of_nat n * diffs (\n. norm (c n)) n * r ^ (n - Suc 0)) ``` huffman@23082 ` 332` ``` = (\n. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))" ``` huffman@20849 ` 333` ``` apply (rule ext) ``` huffman@20849 ` 334` ``` apply (simp add: diffs_def) ``` huffman@20849 ` 335` ``` apply (case_tac n, simp_all add: r_neq_0) ``` huffman@20849 ` 336` ``` done ``` huffman@20860 ` 337` ``` finally have "summable ``` huffman@23082 ` 338` ``` (\n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))" ``` huffman@20860 ` 339` ``` by (rule diffs_equiv [THEN sums_summable]) ``` huffman@20860 ` 340` ``` also have ``` huffman@23082 ` 341` ``` "(\n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * ``` huffman@20860 ` 342` ``` r ^ (n - Suc 0)) = ``` huffman@23082 ` 343` ``` (\n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" ``` huffman@20849 ` 344` ``` apply (rule ext) ``` huffman@20849 ` 345` ``` apply (case_tac "n", simp) ``` huffman@20849 ` 346` ``` apply (case_tac "nat", simp) ``` huffman@20849 ` 347` ``` apply (simp add: r_neq_0) ``` huffman@20849 ` 348` ``` done ``` huffman@20860 ` 349` ``` finally show ``` huffman@23082 ` 350` ``` "summable (\n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" . ``` huffman@20849 ` 351` ``` next ``` huffman@23082 ` 352` ``` fix h::'a and n::nat ``` huffman@20860 ` 353` ``` assume h: "h \ 0" ``` huffman@23082 ` 354` ``` assume "norm h < r - norm x" ``` huffman@23082 ` 355` ``` hence "norm x + norm h < r" by simp ``` huffman@23082 ` 356` ``` with norm_triangle_ineq have xh: "norm (x + h) < r" ``` huffman@20860 ` 357` ``` by (rule order_le_less_trans) ``` huffman@23082 ` 358` ``` show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) ``` huffman@23082 ` 359` ``` \ norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h" ``` huffman@23082 ` 360` ``` apply (simp only: norm_mult mult_assoc) ``` huffman@23082 ` 361` ``` apply (rule mult_left_mono [OF _ norm_ge_zero]) ``` huffman@20860 ` 362` ``` apply (simp (no_asm) add: mult_assoc [symmetric]) ``` huffman@20860 ` 363` ``` apply (rule lemma_termdiff3) ``` huffman@20860 ` 364` ``` apply (rule h) ``` huffman@20860 ` 365` ``` apply (rule r1 [THEN order_less_imp_le]) ``` huffman@20860 ` 366` ``` apply (rule xh [THEN order_less_imp_le]) ``` huffman@20860 ` 367` ``` done ``` huffman@20849 ` 368` ``` qed ``` huffman@20849 ` 369` ```qed ``` webertj@20217 ` 370` huffman@20860 ` 371` ```lemma termdiffs: ``` huffman@23112 ` 372` ``` fixes K x :: "'a::{recpower,real_normed_field,banach}" ``` huffman@20860 ` 373` ``` assumes 1: "summable (\n. c n * K ^ n)" ``` huffman@20860 ` 374` ``` assumes 2: "summable (\n. (diffs c) n * K ^ n)" ``` huffman@20860 ` 375` ``` assumes 3: "summable (\n. (diffs (diffs c)) n * K ^ n)" ``` huffman@23082 ` 376` ``` assumes 4: "norm x < norm K" ``` huffman@20860 ` 377` ``` shows "DERIV (\x. \n. c n * x ^ n) x :> (\n. (diffs c) n * x ^ n)" ``` huffman@29163 ` 378` ```unfolding deriv_def ``` huffman@29163 ` 379` ```proof (rule LIM_zero_cancel) ``` huffman@20860 ` 380` ``` show "(\h. (suminf (\n. c n * (x + h) ^ n) - suminf (\n. c n * x ^ n)) / h ``` huffman@20860 ` 381` ``` - suminf (\n. diffs c n * x ^ n)) -- 0 --> 0" ``` huffman@20860 ` 382` ``` proof (rule LIM_equal2) ``` huffman@29163 ` 383` ``` show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq) ``` huffman@20860 ` 384` ``` next ``` huffman@23082 ` 385` ``` fix h :: 'a ``` huffman@20860 ` 386` ``` assume "h \ 0" ``` huffman@23082 ` 387` ``` assume "norm (h - 0) < norm K - norm x" ``` huffman@23082 ` 388` ``` hence "norm x + norm h < norm K" by simp ``` huffman@23082 ` 389` ``` hence 5: "norm (x + h) < norm K" ``` huffman@23082 ` 390` ``` by (rule norm_triangle_ineq [THEN order_le_less_trans]) ``` huffman@20860 ` 391` ``` have A: "summable (\n. c n * x ^ n)" ``` huffman@20860 ` 392` ``` by (rule powser_inside [OF 1 4]) ``` huffman@20860 ` 393` ``` have B: "summable (\n. c n * (x + h) ^ n)" ``` huffman@20860 ` 394` ``` by (rule powser_inside [OF 1 5]) ``` huffman@20860 ` 395` ``` have C: "summable (\n. diffs c n * x ^ n)" ``` huffman@20860 ` 396` ``` by (rule powser_inside [OF 2 4]) ``` huffman@20860 ` 397` ``` show "((\n. c n * (x + h) ^ n) - (\n. c n * x ^ n)) / h ``` huffman@20860 ` 398` ``` - (\n. diffs c n * x ^ n) = ``` huffman@23082 ` 399` ``` (\n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))" ``` huffman@20860 ` 400` ``` apply (subst sums_unique [OF diffs_equiv [OF C]]) ``` huffman@20860 ` 401` ``` apply (subst suminf_diff [OF B A]) ``` huffman@20860 ` 402` ``` apply (subst suminf_divide [symmetric]) ``` huffman@20860 ` 403` ``` apply (rule summable_diff [OF B A]) ``` huffman@20860 ` 404` ``` apply (subst suminf_diff) ``` huffman@20860 ` 405` ``` apply (rule summable_divide) ``` huffman@20860 ` 406` ``` apply (rule summable_diff [OF B A]) ``` huffman@20860 ` 407` ``` apply (rule sums_summable [OF diffs_equiv [OF C]]) ``` huffman@29163 ` 408` ``` apply (rule arg_cong [where f="suminf"], rule ext) ``` nipkow@23477 ` 409` ``` apply (simp add: ring_simps) ``` huffman@20860 ` 410` ``` done ``` huffman@20860 ` 411` ``` next ``` huffman@20860 ` 412` ``` show "(\h. \n. c n * (((x + h) ^ n - x ^ n) / h - ``` huffman@23082 ` 413` ``` of_nat n * x ^ (n - Suc 0))) -- 0 --> 0" ``` huffman@20860 ` 414` ``` by (rule termdiffs_aux [OF 3 4]) ``` huffman@20860 ` 415` ``` qed ``` huffman@20860 ` 416` ```qed ``` huffman@20860 ` 417` paulson@15077 ` 418` huffman@29164 ` 419` ```subsection {* Exponential Function *} ``` huffman@23043 ` 420` huffman@23043 ` 421` ```definition ``` huffman@23115 ` 422` ``` exp :: "'a \ 'a::{recpower,real_normed_field,banach}" where ``` haftmann@25062 ` 423` ``` "exp x = (\n. x ^ n /\<^sub>R real (fact n))" ``` huffman@23043 ` 424` huffman@23115 ` 425` ```lemma summable_exp_generic: ``` huffman@23115 ` 426` ``` fixes x :: "'a::{real_normed_algebra_1,recpower,banach}" ``` haftmann@25062 ` 427` ``` defines S_def: "S \ \n. x ^ n /\<^sub>R real (fact n)" ``` huffman@23115 ` 428` ``` shows "summable S" ``` huffman@23115 ` 429` ```proof - ``` haftmann@25062 ` 430` ``` have S_Suc: "\n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)" ``` huffman@23115 ` 431` ``` unfolding S_def by (simp add: power_Suc del: mult_Suc) ``` huffman@23115 ` 432` ``` obtain r :: real where r0: "0 < r" and r1: "r < 1" ``` huffman@23115 ` 433` ``` using dense [OF zero_less_one] by fast ``` huffman@23115 ` 434` ``` obtain N :: nat where N: "norm x < real N * r" ``` huffman@23115 ` 435` ``` using reals_Archimedean3 [OF r0] by fast ``` huffman@23115 ` 436` ``` from r1 show ?thesis ``` huffman@23115 ` 437` ``` proof (rule ratio_test [rule_format]) ``` huffman@23115 ` 438` ``` fix n :: nat ``` huffman@23115 ` 439` ``` assume n: "N \ n" ``` huffman@23115 ` 440` ``` have "norm x \ real N * r" ``` huffman@23115 ` 441` ``` using N by (rule order_less_imp_le) ``` huffman@23115 ` 442` ``` also have "real N * r \ real (Suc n) * r" ``` huffman@23115 ` 443` ``` using r0 n by (simp add: mult_right_mono) ``` huffman@23115 ` 444` ``` finally have "norm x * norm (S n) \ real (Suc n) * r * norm (S n)" ``` huffman@23115 ` 445` ``` using norm_ge_zero by (rule mult_right_mono) ``` huffman@23115 ` 446` ``` hence "norm (x * S n) \ real (Suc n) * r * norm (S n)" ``` huffman@23115 ` 447` ``` by (rule order_trans [OF norm_mult_ineq]) ``` huffman@23115 ` 448` ``` hence "norm (x * S n) / real (Suc n) \ r * norm (S n)" ``` huffman@23115 ` 449` ``` by (simp add: pos_divide_le_eq mult_ac) ``` huffman@23115 ` 450` ``` thus "norm (S (Suc n)) \ r * norm (S n)" ``` huffman@23115 ` 451` ``` by (simp add: S_Suc norm_scaleR inverse_eq_divide) ``` huffman@23115 ` 452` ``` qed ``` huffman@23115 ` 453` ```qed ``` huffman@23115 ` 454` huffman@23115 ` 455` ```lemma summable_norm_exp: ``` huffman@23115 ` 456` ``` fixes x :: "'a::{real_normed_algebra_1,recpower,banach}" ``` haftmann@25062 ` 457` ``` shows "summable (\n. norm (x ^ n /\<^sub>R real (fact n)))" ``` huffman@23115 ` 458` ```proof (rule summable_norm_comparison_test [OF exI, rule_format]) ``` haftmann@25062 ` 459` ``` show "summable (\n. norm x ^ n /\<^sub>R real (fact n))" ``` huffman@23115 ` 460` ``` by (rule summable_exp_generic) ``` huffman@23115 ` 461` ```next ``` haftmann@25062 ` 462` ``` fix n show "norm (x ^ n /\<^sub>R real (fact n)) \ norm x ^ n /\<^sub>R real (fact n)" ``` huffman@23115 ` 463` ``` by (simp add: norm_scaleR norm_power_ineq) ``` huffman@23115 ` 464` ```qed ``` huffman@23115 ` 465` huffman@23043 ` 466` ```lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)" ``` huffman@23115 ` 467` ```by (insert summable_exp_generic [where x=x], simp) ``` huffman@23043 ` 468` haftmann@25062 ` 469` ```lemma exp_converges: "(\n. x ^ n /\<^sub>R real (fact n)) sums exp x" ``` huffman@23115 ` 470` ```unfolding exp_def by (rule summable_exp_generic [THEN summable_sums]) ``` huffman@23043 ` 471` huffman@23043 ` 472` huffman@29164 ` 473` ```subsection {* Formal Derivatives of Exp, Sin, and Cos Series *} ``` paulson@15077 ` 474` paulson@15077 ` 475` ```lemma exp_fdiffs: ``` paulson@15077 ` 476` ``` "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))" ``` huffman@23431 ` 477` ```by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult ``` huffman@23082 ` 478` ``` del: mult_Suc of_nat_Suc) ``` paulson@15077 ` 479` huffman@23115 ` 480` ```lemma diffs_of_real: "diffs (\n. of_real (f n)) = (\n. of_real (diffs f n))" ``` huffman@23115 ` 481` ```by (simp add: diffs_def) ``` huffman@23115 ` 482` haftmann@25062 ` 483` ```lemma lemma_exp_ext: "exp = (\x. \n. x ^ n /\<^sub>R real (fact n))" ``` paulson@15077 ` 484` ```by (auto intro!: ext simp add: exp_def) ``` paulson@15077 ` 485` paulson@15077 ` 486` ```lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)" ``` paulson@15229 ` 487` ```apply (simp add: exp_def) ``` paulson@15077 ` 488` ```apply (subst lemma_exp_ext) ``` huffman@23115 ` 489` ```apply (subgoal_tac "DERIV (\u. \n. of_real (inverse (real (fact n))) * u ^ n) x :> (\n. diffs (\n. of_real (inverse (real (fact n)))) n * x ^ n)") ``` huffman@23115 ` 490` ```apply (rule_tac [2] K = "of_real (1 + norm x)" in termdiffs) ``` huffman@23115 ` 491` ```apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs) ``` huffman@23115 ` 492` ```apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+ ``` huffman@23115 ` 493` ```apply (simp del: of_real_add) ``` paulson@15077 ` 494` ```done ``` paulson@15077 ` 495` huffman@23045 ` 496` ```lemma isCont_exp [simp]: "isCont exp x" ``` huffman@23045 ` 497` ```by (rule DERIV_exp [THEN DERIV_isCont]) ``` huffman@23045 ` 498` huffman@23045 ` 499` huffman@29164 ` 500` ```subsection {* Properties of the Exponential Function *} ``` paulson@15077 ` 501` huffman@23278 ` 502` ```lemma powser_zero: ``` huffman@23278 ` 503` ``` fixes f :: "nat \ 'a::{real_normed_algebra_1,recpower}" ``` huffman@23278 ` 504` ``` shows "(\n. f n * 0 ^ n) = f 0" ``` paulson@15077 ` 505` ```proof - ``` huffman@23278 ` 506` ``` have "(\n = 0..<1. f n * 0 ^ n) = (\n. f n * 0 ^ n)" ``` huffman@23115 ` 507` ``` by (rule sums_unique [OF series_zero], simp add: power_0_left) ``` huffman@23278 ` 508` ``` thus ?thesis by simp ``` paulson@15077 ` 509` ```qed ``` paulson@15077 ` 510` huffman@23278 ` 511` ```lemma exp_zero [simp]: "exp 0 = 1" ``` huffman@23278 ` 512` ```unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero) ``` huffman@23278 ` 513` huffman@23115 ` 514` ```lemma setsum_cl_ivl_Suc2: ``` huffman@23115 ` 515` ``` "(\i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\i=m..n. f (Suc i)))" ``` nipkow@28069 ` 516` ```by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl ``` huffman@23115 ` 517` ``` del: setsum_cl_ivl_Suc) ``` huffman@23115 ` 518` huffman@23115 ` 519` ```lemma exp_series_add: ``` huffman@23115 ` 520` ``` fixes x y :: "'a::{real_field,recpower}" ``` haftmann@25062 ` 521` ``` defines S_def: "S \ \x n. x ^ n /\<^sub>R real (fact n)" ``` huffman@23115 ` 522` ``` shows "S (x + y) n = (\i=0..n. S x i * S y (n - i))" ``` huffman@23115 ` 523` ```proof (induct n) ``` huffman@23115 ` 524` ``` case 0 ``` huffman@23115 ` 525` ``` show ?case ``` huffman@23115 ` 526` ``` unfolding S_def by simp ``` huffman@23115 ` 527` ```next ``` huffman@23115 ` 528` ``` case (Suc n) ``` haftmann@25062 ` 529` ``` have S_Suc: "\x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)" ``` huffman@23115 ` 530` ``` unfolding S_def by (simp add: power_Suc del: mult_Suc) ``` haftmann@25062 ` 531` ``` hence times_S: "\x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)" ``` huffman@23115 ` 532` ``` by simp ``` huffman@23115 ` 533` haftmann@25062 ` 534` ``` have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n" ``` huffman@23115 ` 535` ``` by (simp only: times_S) ``` huffman@23115 ` 536` ``` also have "\ = (x + y) * (\i=0..n. S x i * S y (n-i))" ``` huffman@23115 ` 537` ``` by (simp only: Suc) ``` huffman@23115 ` 538` ``` also have "\ = x * (\i=0..n. S x i * S y (n-i)) ``` huffman@23115 ` 539` ``` + y * (\i=0..n. S x i * S y (n-i))" ``` huffman@23115 ` 540` ``` by (rule left_distrib) ``` huffman@23115 ` 541` ``` also have "\ = (\i=0..n. (x * S x i) * S y (n-i)) ``` huffman@23115 ` 542` ``` + (\i=0..n. S x i * (y * S y (n-i)))" ``` huffman@23115 ` 543` ``` by (simp only: setsum_right_distrib mult_ac) ``` haftmann@25062 ` 544` ``` also have "\ = (\i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) ``` haftmann@25062 ` 545` ``` + (\i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))" ``` huffman@23115 ` 546` ``` by (simp add: times_S Suc_diff_le) ``` haftmann@25062 ` 547` ``` also have "(\i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) = ``` haftmann@25062 ` 548` ``` (\i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))" ``` huffman@23115 ` 549` ``` by (subst setsum_cl_ivl_Suc2, simp) ``` haftmann@25062 ` 550` ``` also have "(\i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) = ``` haftmann@25062 ` 551` ``` (\i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))" ``` huffman@23115 ` 552` ``` by (subst setsum_cl_ivl_Suc, simp) ``` haftmann@25062 ` 553` ``` also have "(\i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) + ``` haftmann@25062 ` 554` ``` (\i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) = ``` haftmann@25062 ` 555` ``` (\i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))" ``` huffman@23115 ` 556` ``` by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric] ``` huffman@23115 ` 557` ``` real_of_nat_add [symmetric], simp) ``` haftmann@25062 ` 558` ``` also have "\ = real (Suc n) *\<^sub>R (\i=0..Suc n. S x i * S y (Suc n-i))" ``` huffman@23127 ` 559` ``` by (simp only: scaleR_right.setsum) ``` huffman@23115 ` 560` ``` finally show ``` huffman@23115 ` 561` ``` "S (x + y) (Suc n) = (\i=0..Suc n. S x i * S y (Suc n - i))" ``` huffman@23115 ` 562` ``` by (simp add: scaleR_cancel_left del: setsum_cl_ivl_Suc) ``` huffman@23115 ` 563` ```qed ``` huffman@23115 ` 564` huffman@23115 ` 565` ```lemma exp_add: "exp (x + y) = exp x * exp y" ``` huffman@23115 ` 566` ```unfolding exp_def ``` huffman@23115 ` 567` ```by (simp only: Cauchy_product summable_norm_exp exp_series_add) ``` huffman@23115 ` 568` huffman@23241 ` 569` ```lemma exp_of_real: "exp (of_real x) = of_real (exp x)" ``` huffman@23241 ` 570` ```unfolding exp_def ``` huffman@23241 ` 571` ```apply (subst of_real.suminf) ``` huffman@23241 ` 572` ```apply (rule summable_exp_generic) ``` huffman@23241 ` 573` ```apply (simp add: scaleR_conv_of_real) ``` huffman@23241 ` 574` ```done ``` huffman@23241 ` 575` huffman@23115 ` 576` ```lemma exp_ge_add_one_self_aux: "0 \ (x::real) ==> (1 + x) \ exp(x)" ``` huffman@22998 ` 577` ```apply (drule order_le_imp_less_or_eq, auto) ``` paulson@15229 ` 578` ```apply (simp add: exp_def) ``` paulson@15077 ` 579` ```apply (rule real_le_trans) ``` paulson@15229 ` 580` ```apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le) ``` nipkow@25875 ` 581` ```apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff) ``` paulson@15077 ` 582` ```done ``` paulson@15077 ` 583` huffman@23115 ` 584` ```lemma exp_gt_one [simp]: "0 < (x::real) ==> 1 < exp x" ``` paulson@15077 ` 585` ```apply (rule order_less_le_trans) ``` avigad@17014 ` 586` ```apply (rule_tac [2] exp_ge_add_one_self_aux, auto) ``` paulson@15077 ` 587` ```done ``` paulson@15077 ` 588` paulson@15077 ` 589` ```lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)" ``` paulson@15077 ` 590` ```proof - ``` paulson@15077 ` 591` ``` have "DERIV (exp \ (\x. x + y)) x :> exp (x + y) * (1+0)" ``` huffman@23069 ` 592` ``` by (fast intro: DERIV_chain DERIV_add DERIV_exp DERIV_ident DERIV_const) ``` paulson@15077 ` 593` ``` thus ?thesis by (simp add: o_def) ``` paulson@15077 ` 594` ```qed ``` paulson@15077 ` 595` paulson@15077 ` 596` ```lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)" ``` paulson@15077 ` 597` ```proof - ``` paulson@15077 ` 598` ``` have "DERIV (exp \ uminus) x :> exp (- x) * - 1" ``` huffman@23069 ` 599` ``` by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_ident) ``` paulson@15077 ` 600` ``` thus ?thesis by (simp add: o_def) ``` paulson@15077 ` 601` ```qed ``` paulson@15077 ` 602` paulson@15077 ` 603` ```lemma DERIV_exp_exp_zero [simp]: "DERIV (%x. exp (x + y) * exp (- x)) x :> 0" ``` paulson@15077 ` 604` ```proof - ``` paulson@15077 ` 605` ``` have "DERIV (\x. exp (x + y) * exp (- x)) x ``` paulson@15077 ` 606` ``` :> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)" ``` paulson@15077 ` 607` ``` by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult) ``` huffman@23115 ` 608` ``` thus ?thesis by (simp add: mult_commute) ``` paulson@15077 ` 609` ```qed ``` paulson@15077 ` 610` huffman@23115 ` 611` ```lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y::real)" ``` paulson@15077 ` 612` ```proof - ``` paulson@15077 ` 613` ``` have "\x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0" by simp ``` paulson@15077 ` 614` ``` hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)" ``` paulson@15077 ` 615` ``` by (rule DERIV_isconst_all) ``` paulson@15077 ` 616` ``` thus ?thesis by simp ``` paulson@15077 ` 617` ```qed ``` paulson@15077 ` 618` paulson@15077 ` 619` ```lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1" ``` huffman@23115 ` 620` ```by (simp add: exp_add [symmetric]) ``` paulson@15077 ` 621` paulson@15077 ` 622` ```lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1" ``` paulson@15077 ` 623` ```by (simp add: mult_commute) ``` paulson@15077 ` 624` paulson@15077 ` 625` paulson@15077 ` 626` ```lemma exp_minus: "exp(-x) = inverse(exp(x))" ``` paulson@15077 ` 627` ```by (auto intro: inverse_unique [symmetric]) ``` paulson@15077 ` 628` paulson@15077 ` 629` ```text{*Proof: because every exponential can be seen as a square.*} ``` huffman@23115 ` 630` ```lemma exp_ge_zero [simp]: "0 \ exp (x::real)" ``` paulson@15077 ` 631` ```apply (rule_tac t = x in real_sum_of_halves [THEN subst]) ``` paulson@15077 ` 632` ```apply (subst exp_add, auto) ``` paulson@15077 ` 633` ```done ``` paulson@15077 ` 634` paulson@15077 ` 635` ```lemma exp_not_eq_zero [simp]: "exp x \ 0" ``` paulson@15077 ` 636` ```apply (cut_tac x = x in exp_mult_minus2) ``` paulson@15077 ` 637` ```apply (auto simp del: exp_mult_minus2) ``` paulson@15077 ` 638` ```done ``` paulson@15077 ` 639` huffman@23115 ` 640` ```lemma exp_gt_zero [simp]: "0 < exp (x::real)" ``` paulson@15077 ` 641` ```by (simp add: order_less_le) ``` paulson@15077 ` 642` huffman@29165 ` 643` ```lemma inv_exp_gt_zero: "0 < inverse(exp x::real)" ``` huffman@29165 ` 644` ```by simp ``` paulson@15077 ` 645` huffman@23115 ` 646` ```lemma abs_exp_cancel [simp]: "\exp x::real\ = exp x" ``` huffman@29165 ` 647` ```by simp ``` paulson@15077 ` 648` paulson@15077 ` 649` ```lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n" ``` paulson@15251 ` 650` ```apply (induct "n") ``` paulson@15077 ` 651` ```apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute) ``` paulson@15077 ` 652` ```done ``` paulson@15077 ` 653` paulson@15077 ` 654` ```lemma exp_diff: "exp(x - y) = exp(x)/(exp y)" ``` paulson@15229 ` 655` ```apply (simp add: diff_minus divide_inverse) ``` huffman@29165 ` 656` ```apply (simp add: exp_add exp_minus) ``` paulson@15077 ` 657` ```done ``` paulson@15077 ` 658` paulson@15077 ` 659` ```lemma exp_less_mono: ``` huffman@23115 ` 660` ``` fixes x y :: real ``` huffman@29165 ` 661` ``` assumes "x < y" shows "exp x < exp y" ``` paulson@15077 ` 662` ```proof - ``` huffman@29165 ` 663` ``` from `x < y` have "0 < y - x" by simp ``` huffman@29165 ` 664` ``` hence "1 < exp (y - x)" by (rule exp_gt_one) ``` huffman@29165 ` 665` ``` hence "1 < exp y / exp x" by (simp only: exp_diff) ``` huffman@29165 ` 666` ``` thus "exp x < exp y" by simp ``` paulson@15077 ` 667` ```qed ``` paulson@15077 ` 668` huffman@23115 ` 669` ```lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y" ``` paulson@15228 ` 670` ```apply (simp add: linorder_not_le [symmetric]) ``` paulson@15228 ` 671` ```apply (auto simp add: order_le_less exp_less_mono) ``` paulson@15077 ` 672` ```done ``` paulson@15077 ` 673` huffman@23115 ` 674` ```lemma exp_less_cancel_iff [iff]: "(exp(x::real) < exp(y)) = (x < y)" ``` paulson@15077 ` 675` ```by (auto intro: exp_less_mono exp_less_cancel) ``` paulson@15077 ` 676` huffman@23115 ` 677` ```lemma exp_le_cancel_iff [iff]: "(exp(x::real) \ exp(y)) = (x \ y)" ``` paulson@15077 ` 678` ```by (auto simp add: linorder_not_less [symmetric]) ``` paulson@15077 ` 679` huffman@23115 ` 680` ```lemma exp_inj_iff [iff]: "(exp (x::real) = exp y) = (x = y)" ``` paulson@15077 ` 681` ```by (simp add: order_eq_iff) ``` paulson@15077 ` 682` huffman@23115 ` 683` ```lemma lemma_exp_total: "1 \ y ==> \x. 0 \ x & x \ y - 1 & exp(x::real) = y" ``` paulson@15077 ` 684` ```apply (rule IVT) ``` huffman@23045 ` 685` ```apply (auto intro: isCont_exp simp add: le_diff_eq) ``` paulson@15077 ` 686` ```apply (subgoal_tac "1 + (y - 1) \ exp (y - 1)") ``` huffman@29165 ` 687` ```apply simp ``` avigad@17014 ` 688` ```apply (rule exp_ge_add_one_self_aux, simp) ``` paulson@15077 ` 689` ```done ``` paulson@15077 ` 690` huffman@23115 ` 691` ```lemma exp_total: "0 < (y::real) ==> \x. exp x = y" ``` paulson@15077 ` 692` ```apply (rule_tac x = 1 and y = y in linorder_cases) ``` paulson@15077 ` 693` ```apply (drule order_less_imp_le [THEN lemma_exp_total]) ``` paulson@15077 ` 694` ```apply (rule_tac [2] x = 0 in exI) ``` paulson@15077 ` 695` ```apply (frule_tac [3] real_inverse_gt_one) ``` paulson@15077 ` 696` ```apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto) ``` paulson@15077 ` 697` ```apply (rule_tac x = "-x" in exI) ``` paulson@15077 ` 698` ```apply (simp add: exp_minus) ``` paulson@15077 ` 699` ```done ``` paulson@15077 ` 700` paulson@15077 ` 701` huffman@29164 ` 702` ```subsection {* Natural Logarithm *} ``` paulson@15077 ` 703` huffman@23043 ` 704` ```definition ``` huffman@23043 ` 705` ``` ln :: "real => real" where ``` huffman@23043 ` 706` ``` "ln x = (THE u. exp u = x)" ``` huffman@23043 ` 707` huffman@23043 ` 708` ```lemma ln_exp [simp]: "ln (exp x) = x" ``` paulson@15077 ` 709` ```by (simp add: ln_def) ``` paulson@15077 ` 710` huffman@22654 ` 711` ```lemma exp_ln [simp]: "0 < x \ exp (ln x) = x" ``` huffman@22654 ` 712` ```by (auto dest: exp_total) ``` huffman@22654 ` 713` huffman@23043 ` 714` ```lemma exp_ln_iff [simp]: "(exp (ln x) = x) = (0 < x)" ``` paulson@15077 ` 715` ```apply (auto dest: exp_total) ``` paulson@15077 ` 716` ```apply (erule subst, simp) ``` paulson@15077 ` 717` ```done ``` paulson@15077 ` 718` paulson@15077 ` 719` ```lemma ln_mult: "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)" ``` paulson@15077 ` 720` ```apply (rule exp_inj_iff [THEN iffD1]) ``` huffman@22654 ` 721` ```apply (simp add: exp_add exp_ln mult_pos_pos) ``` paulson@15077 ` 722` ```done ``` paulson@15077 ` 723` paulson@15077 ` 724` ```lemma ln_inj_iff[simp]: "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)" ``` paulson@15077 ` 725` ```apply (simp only: exp_ln_iff [symmetric]) ``` paulson@15077 ` 726` ```apply (erule subst)+ ``` paulson@15077 ` 727` ```apply simp ``` paulson@15077 ` 728` ```done ``` paulson@15077 ` 729` paulson@15077 ` 730` ```lemma ln_one[simp]: "ln 1 = 0" ``` paulson@15077 ` 731` ```by (rule exp_inj_iff [THEN iffD1], auto) ``` paulson@15077 ` 732` paulson@15077 ` 733` ```lemma ln_inverse: "0 < x ==> ln(inverse x) = - ln x" ``` paulson@15077 ` 734` ```apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1]) ``` paulson@15077 ` 735` ```apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric]) ``` paulson@15077 ` 736` ```done ``` paulson@15077 ` 737` paulson@15077 ` 738` ```lemma ln_div: ``` paulson@15077 ` 739` ``` "[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y" ``` paulson@15229 ` 740` ```apply (simp add: divide_inverse) ``` paulson@15077 ` 741` ```apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse) ``` paulson@15077 ` 742` ```done ``` paulson@15077 ` 743` paulson@15077 ` 744` ```lemma ln_less_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)" ``` paulson@15077 ` 745` ```apply (simp only: exp_ln_iff [symmetric]) ``` paulson@15077 ` 746` ```apply (erule subst)+ ``` paulson@15077 ` 747` ```apply simp ``` paulson@15077 ` 748` ```done ``` paulson@15077 ` 749` paulson@15077 ` 750` ```lemma ln_le_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x \ ln y) = (x \ y)" ``` paulson@15077 ` 751` ```by (auto simp add: linorder_not_less [symmetric]) ``` paulson@15077 ` 752` paulson@15077 ` 753` ```lemma ln_realpow: "0 < x ==> ln(x ^ n) = real n * ln(x)" ``` paulson@15077 ` 754` ```by (auto dest!: exp_total simp add: exp_real_of_nat_mult [symmetric]) ``` paulson@15077 ` 755` paulson@15077 ` 756` ```lemma ln_add_one_self_le_self [simp]: "0 \ x ==> ln(1 + x) \ x" ``` paulson@15077 ` 757` ```apply (rule ln_exp [THEN subst]) ``` avigad@17014 ` 758` ```apply (rule ln_le_cancel_iff [THEN iffD2]) ``` avigad@17014 ` 759` ```apply (auto simp add: exp_ge_add_one_self_aux) ``` paulson@15077 ` 760` ```done ``` paulson@15077 ` 761` paulson@15077 ` 762` ```lemma ln_less_self [simp]: "0 < x ==> ln x < x" ``` paulson@15077 ` 763` ```apply (rule order_less_le_trans) ``` paulson@15077 ` 764` ```apply (rule_tac [2] ln_add_one_self_le_self) ``` paulson@15077 ` 765` ```apply (rule ln_less_cancel_iff [THEN iffD2], auto) ``` paulson@15077 ` 766` ```done ``` paulson@15077 ` 767` paulson@15234 ` 768` ```lemma ln_ge_zero [simp]: ``` paulson@15077 ` 769` ``` assumes x: "1 \ x" shows "0 \ ln x" ``` paulson@15077 ` 770` ```proof - ``` paulson@15077 ` 771` ``` have "0 < x" using x by arith ``` paulson@15077 ` 772` ``` hence "exp 0 \ exp (ln x)" ``` huffman@22915 ` 773` ``` by (simp add: x) ``` paulson@15077 ` 774` ``` thus ?thesis by (simp only: exp_le_cancel_iff) ``` paulson@15077 ` 775` ```qed ``` paulson@15077 ` 776` paulson@15077 ` 777` ```lemma ln_ge_zero_imp_ge_one: ``` paulson@15077 ` 778` ``` assumes ln: "0 \ ln x" ``` paulson@15077 ` 779` ``` and x: "0 < x" ``` paulson@15077 ` 780` ``` shows "1 \ x" ``` paulson@15077 ` 781` ```proof - ``` paulson@15077 ` 782` ``` from ln have "ln 1 \ ln x" by simp ``` paulson@15077 ` 783` ``` thus ?thesis by (simp add: x del: ln_one) ``` paulson@15077 ` 784` ```qed ``` paulson@15077 ` 785` paulson@15077 ` 786` ```lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \ ln x) = (1 \ x)" ``` paulson@15077 ` 787` ```by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one) ``` paulson@15077 ` 788` paulson@15234 ` 789` ```lemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)" ``` paulson@15234 ` 790` ```by (insert ln_ge_zero_iff [of x], arith) ``` paulson@15234 ` 791` paulson@15077 ` 792` ```lemma ln_gt_zero: ``` paulson@15077 ` 793` ``` assumes x: "1 < x" shows "0 < ln x" ``` paulson@15077 ` 794` ```proof - ``` paulson@15077 ` 795` ``` have "0 < x" using x by arith ``` huffman@22915 ` 796` ``` hence "exp 0 < exp (ln x)" by (simp add: x) ``` paulson@15077 ` 797` ``` thus ?thesis by (simp only: exp_less_cancel_iff) ``` paulson@15077 ` 798` ```qed ``` paulson@15077 ` 799` paulson@15077 ` 800` ```lemma ln_gt_zero_imp_gt_one: ``` paulson@15077 ` 801` ``` assumes ln: "0 < ln x" ``` paulson@15077 ` 802` ``` and x: "0 < x" ``` paulson@15077 ` 803` ``` shows "1 < x" ``` paulson@15077 ` 804` ```proof - ``` paulson@15077 ` 805` ``` from ln have "ln 1 < ln x" by simp ``` paulson@15077 ` 806` ``` thus ?thesis by (simp add: x del: ln_one) ``` paulson@15077 ` 807` ```qed ``` paulson@15077 ` 808` paulson@15077 ` 809` ```lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)" ``` paulson@15077 ` 810` ```by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one) ``` paulson@15077 ` 811` paulson@15234 ` 812` ```lemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)" ``` paulson@15234 ` 813` ```by (insert ln_less_zero_iff [of x] ln_gt_zero_iff [of x], arith) ``` paulson@15077 ` 814` paulson@15077 ` 815` ```lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0" ``` paulson@15234 ` 816` ```by simp ``` paulson@15077 ` 817` paulson@15077 ` 818` ```lemma exp_ln_eq: "exp u = x ==> ln x = u" ``` paulson@15077 ` 819` ```by auto ``` paulson@15077 ` 820` huffman@23045 ` 821` ```lemma isCont_ln: "0 < x \ isCont ln x" ``` huffman@23045 ` 822` ```apply (subgoal_tac "isCont ln (exp (ln x))", simp) ``` huffman@23045 ` 823` ```apply (rule isCont_inverse_function [where f=exp], simp_all) ``` huffman@23045 ` 824` ```done ``` huffman@23045 ` 825` huffman@23045 ` 826` ```lemma DERIV_ln: "0 < x \ DERIV ln x :> inverse x" ``` huffman@23045 ` 827` ```apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"]) ``` huffman@23045 ` 828` ```apply (erule lemma_DERIV_subst [OF DERIV_exp exp_ln]) ``` huffman@23045 ` 829` ```apply (simp_all add: abs_if isCont_ln) ``` huffman@23045 ` 830` ```done ``` huffman@23045 ` 831` paulson@15077 ` 832` huffman@29164 ` 833` ```subsection {* Sine and Cosine *} ``` huffman@29164 ` 834` huffman@29164 ` 835` ```definition ``` huffman@29164 ` 836` ``` sin :: "real => real" where ``` huffman@29164 ` 837` ``` "sin x = (\n. (if even(n) then 0 else ``` huffman@29164 ` 838` ``` (-1 ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n)" ``` huffman@29164 ` 839` ``` ``` huffman@29164 ` 840` ```definition ``` huffman@29164 ` 841` ``` cos :: "real => real" where ``` huffman@29164 ` 842` ``` "cos x = (\n. (if even(n) then (-1 ^ (n div 2))/(real (fact n)) ``` huffman@29164 ` 843` ``` else 0) * x ^ n)" ``` huffman@29164 ` 844` huffman@29164 ` 845` ```lemma summable_sin: ``` huffman@29164 ` 846` ``` "summable (%n. ``` huffman@29164 ` 847` ``` (if even n then 0 ``` huffman@29164 ` 848` ``` else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * ``` huffman@29164 ` 849` ``` x ^ n)" ``` huffman@29164 ` 850` ```apply (rule_tac g = "(%n. inverse (real (fact n)) * \x\ ^ n)" in summable_comparison_test) ``` huffman@29164 ` 851` ```apply (rule_tac [2] summable_exp) ``` huffman@29164 ` 852` ```apply (rule_tac x = 0 in exI) ``` huffman@29164 ` 853` ```apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) ``` huffman@29164 ` 854` ```done ``` huffman@29164 ` 855` huffman@29164 ` 856` ```lemma summable_cos: ``` huffman@29164 ` 857` ``` "summable (%n. ``` huffman@29164 ` 858` ``` (if even n then ``` huffman@29164 ` 859` ``` -1 ^ (n div 2)/(real (fact n)) else 0) * x ^ n)" ``` huffman@29164 ` 860` ```apply (rule_tac g = "(%n. inverse (real (fact n)) * \x\ ^ n)" in summable_comparison_test) ``` huffman@29164 ` 861` ```apply (rule_tac [2] summable_exp) ``` huffman@29164 ` 862` ```apply (rule_tac x = 0 in exI) ``` huffman@29164 ` 863` ```apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) ``` huffman@29164 ` 864` ```done ``` huffman@29164 ` 865` huffman@29164 ` 866` ```lemma lemma_STAR_sin: ``` huffman@29164 ` 867` ``` "(if even n then 0 ``` huffman@29164 ` 868` ``` else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0" ``` huffman@29164 ` 869` ```by (induct "n", auto) ``` huffman@29164 ` 870` huffman@29164 ` 871` ```lemma lemma_STAR_cos: ``` huffman@29164 ` 872` ``` "0 < n --> ``` huffman@29164 ` 873` ``` -1 ^ (n div 2)/(real (fact n)) * 0 ^ n = 0" ``` huffman@29164 ` 874` ```by (induct "n", auto) ``` huffman@29164 ` 875` huffman@29164 ` 876` ```lemma lemma_STAR_cos1: ``` huffman@29164 ` 877` ``` "0 < n --> ``` huffman@29164 ` 878` ``` (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0" ``` huffman@29164 ` 879` ```by (induct "n", auto) ``` huffman@29164 ` 880` huffman@29164 ` 881` ```lemma lemma_STAR_cos2: ``` huffman@29164 ` 882` ``` "(\n=1..n. - ((if even n then 0 ``` huffman@29164 ` 939` ``` else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))" ``` huffman@29164 ` 940` ```by (auto intro!: sums_unique sums_minus sin_converges) ``` huffman@29164 ` 941` huffman@29164 ` 942` ```lemma lemma_sin_ext: ``` huffman@29164 ` 943` ``` "sin = (%x. \n. ``` huffman@29164 ` 944` ``` (if even n then 0 ``` huffman@29164 ` 945` ``` else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * ``` huffman@29164 ` 946` ``` x ^ n)" ``` huffman@29164 ` 947` ```by (auto intro!: ext simp add: sin_def) ``` huffman@29164 ` 948` huffman@29164 ` 949` ```lemma lemma_cos_ext: ``` huffman@29164 ` 950` ``` "cos = (%x. \n. ``` huffman@29164 ` 951` ``` (if even n then -1 ^ (n div 2)/(real (fact n)) else 0) * ``` huffman@29164 ` 952` ``` x ^ n)" ``` huffman@29164 ` 953` ```by (auto intro!: ext simp add: cos_def) ``` huffman@29164 ` 954` huffman@29164 ` 955` ```lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)" ``` huffman@29164 ` 956` ```apply (simp add: cos_def) ``` huffman@29164 ` 957` ```apply (subst lemma_sin_ext) ``` huffman@29164 ` 958` ```apply (auto simp add: sin_fdiffs2 [symmetric]) ``` huffman@29164 ` 959` ```apply (rule_tac K = "1 + \x\" in termdiffs) ``` huffman@29164 ` 960` ```apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs) ``` huffman@29164 ` 961` ```done ``` huffman@29164 ` 962` huffman@29164 ` 963` ```lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)" ``` huffman@29164 ` 964` ```apply (subst lemma_cos_ext) ``` huffman@29164 ` 965` ```apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left) ``` huffman@29164 ` 966` ```apply (rule_tac K = "1 + \x\" in termdiffs) ``` huffman@29164 ` 967` ```apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus) ``` huffman@29164 ` 968` ```done ``` huffman@29164 ` 969` huffman@29164 ` 970` ```lemma isCont_sin [simp]: "isCont sin x" ``` huffman@29164 ` 971` ```by (rule DERIV_sin [THEN DERIV_isCont]) ``` huffman@29164 ` 972` huffman@29164 ` 973` ```lemma isCont_cos [simp]: "isCont cos x" ``` huffman@29164 ` 974` ```by (rule DERIV_cos [THEN DERIV_isCont]) ``` huffman@29164 ` 975` huffman@29164 ` 976` huffman@29164 ` 977` ```subsection {* Properties of Sine and Cosine *} ``` paulson@15077 ` 978` paulson@15077 ` 979` ```lemma sin_zero [simp]: "sin 0 = 0" ``` huffman@23278 ` 980` ```unfolding sin_def by (simp add: powser_zero) ``` paulson@15077 ` 981` paulson@15077 ` 982` ```lemma cos_zero [simp]: "cos 0 = 1" ``` huffman@23278 ` 983` ```unfolding cos_def by (simp add: powser_zero) ``` paulson@15077 ` 984` paulson@15077 ` 985` ```lemma DERIV_sin_sin_mult [simp]: ``` paulson@15077 ` 986` ``` "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)" ``` paulson@15077 ` 987` ```by (rule DERIV_mult, auto) ``` paulson@15077 ` 988` paulson@15077 ` 989` ```lemma DERIV_sin_sin_mult2 [simp]: ``` paulson@15077 ` 990` ``` "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)" ``` paulson@15077 ` 991` ```apply (cut_tac x = x in DERIV_sin_sin_mult) ``` paulson@15077 ` 992` ```apply (auto simp add: mult_assoc) ``` paulson@15077 ` 993` ```done ``` paulson@15077 ` 994` paulson@15077 ` 995` ```lemma DERIV_sin_realpow2 [simp]: ``` paulson@15077 ` 996` ``` "DERIV (%x. (sin x)\) x :> cos(x) * sin(x) + cos(x) * sin(x)" ``` paulson@15077 ` 997` ```by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric]) ``` paulson@15077 ` 998` paulson@15077 ` 999` ```lemma DERIV_sin_realpow2a [simp]: ``` paulson@15077 ` 1000` ``` "DERIV (%x. (sin x)\) x :> 2 * cos(x) * sin(x)" ``` paulson@15077 ` 1001` ```by (auto simp add: numeral_2_eq_2) ``` paulson@15077 ` 1002` paulson@15077 ` 1003` ```lemma DERIV_cos_cos_mult [simp]: ``` paulson@15077 ` 1004` ``` "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)" ``` paulson@15077 ` 1005` ```by (rule DERIV_mult, auto) ``` paulson@15077 ` 1006` paulson@15077 ` 1007` ```lemma DERIV_cos_cos_mult2 [simp]: ``` paulson@15077 ` 1008` ``` "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)" ``` paulson@15077 ` 1009` ```apply (cut_tac x = x in DERIV_cos_cos_mult) ``` paulson@15077 ` 1010` ```apply (auto simp add: mult_ac) ``` paulson@15077 ` 1011` ```done ``` paulson@15077 ` 1012` paulson@15077 ` 1013` ```lemma DERIV_cos_realpow2 [simp]: ``` paulson@15077 ` 1014` ``` "DERIV (%x. (cos x)\) x :> -sin(x) * cos(x) + -sin(x) * cos(x)" ``` paulson@15077 ` 1015` ```by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric]) ``` paulson@15077 ` 1016` paulson@15077 ` 1017` ```lemma DERIV_cos_realpow2a [simp]: ``` paulson@15077 ` 1018` ``` "DERIV (%x. (cos x)\) x :> -2 * cos(x) * sin(x)" ``` paulson@15077 ` 1019` ```by (auto simp add: numeral_2_eq_2) ``` paulson@15077 ` 1020` paulson@15077 ` 1021` ```lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E" ``` paulson@15077 ` 1022` ```by auto ``` paulson@15077 ` 1023` paulson@15077 ` 1024` ```lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\) x :> -(2 * cos(x) * sin(x))" ``` paulson@15077 ` 1025` ```apply (rule lemma_DERIV_subst) ``` paulson@15077 ` 1026` ```apply (rule DERIV_cos_realpow2a, auto) ``` paulson@15077 ` 1027` ```done ``` paulson@15077 ` 1028` paulson@15077 ` 1029` ```(* most useful *) ``` paulson@15229 ` 1030` ```lemma DERIV_cos_cos_mult3 [simp]: ``` paulson@15229 ` 1031` ``` "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))" ``` paulson@15077 ` 1032` ```apply (rule lemma_DERIV_subst) ``` paulson@15077 ` 1033` ```apply (rule DERIV_cos_cos_mult2, auto) ``` paulson@15077 ` 1034` ```done ``` paulson@15077 ` 1035` paulson@15077 ` 1036` ```lemma DERIV_sin_circle_all: ``` paulson@15077 ` 1037` ``` "\x. DERIV (%x. (sin x)\ + (cos x)\) x :> ``` paulson@15077 ` 1038` ``` (2*cos(x)*sin(x) - 2*cos(x)*sin(x))" ``` paulson@15229 ` 1039` ```apply (simp only: diff_minus, safe) ``` paulson@15229 ` 1040` ```apply (rule DERIV_add) ``` paulson@15077 ` 1041` ```apply (auto simp add: numeral_2_eq_2) ``` paulson@15077 ` 1042` ```done ``` paulson@15077 ` 1043` paulson@15229 ` 1044` ```lemma DERIV_sin_circle_all_zero [simp]: ``` paulson@15229 ` 1045` ``` "\x. DERIV (%x. (sin x)\ + (cos x)\) x :> 0" ``` paulson@15077 ` 1046` ```by (cut_tac DERIV_sin_circle_all, auto) ``` paulson@15077 ` 1047` paulson@15077 ` 1048` ```lemma sin_cos_squared_add [simp]: "((sin x)\) + ((cos x)\) = 1" ``` paulson@15077 ` 1049` ```apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all]) ``` paulson@15077 ` 1050` ```apply (auto simp add: numeral_2_eq_2) ``` paulson@15077 ` 1051` ```done ``` paulson@15077 ` 1052` paulson@15077 ` 1053` ```lemma sin_cos_squared_add2 [simp]: "((cos x)\) + ((sin x)\) = 1" ``` huffman@23286 ` 1054` ```apply (subst add_commute) ``` paulson@15077 ` 1055` ```apply (simp (no_asm) del: realpow_Suc) ``` paulson@15077 ` 1056` ```done ``` paulson@15077 ` 1057` paulson@15077 ` 1058` ```lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1" ``` paulson@15077 ` 1059` ```apply (cut_tac x = x in sin_cos_squared_add2) ``` paulson@15077 ` 1060` ```apply (auto simp add: numeral_2_eq_2) ``` paulson@15077 ` 1061` ```done ``` paulson@15077 ` 1062` paulson@15077 ` 1063` ```lemma sin_squared_eq: "(sin x)\ = 1 - (cos x)\" ``` paulson@15229 ` 1064` ```apply (rule_tac a1 = "(cos x)\" in add_right_cancel [THEN iffD1]) ``` paulson@15077 ` 1065` ```apply (simp del: realpow_Suc) ``` paulson@15077 ` 1066` ```done ``` paulson@15077 ` 1067` paulson@15077 ` 1068` ```lemma cos_squared_eq: "(cos x)\ = 1 - (sin x)\" ``` paulson@15077 ` 1069` ```apply (rule_tac a1 = "(sin x)\" in add_right_cancel [THEN iffD1]) ``` paulson@15077 ` 1070` ```apply (simp del: realpow_Suc) ``` paulson@15077 ` 1071` ```done ``` paulson@15077 ` 1072` paulson@15081 ` 1073` ```lemma abs_sin_le_one [simp]: "\sin x\ \ 1" ``` huffman@23097 ` 1074` ```by (rule power2_le_imp_le, simp_all add: sin_squared_eq) ``` paulson@15077 ` 1075` paulson@15077 ` 1076` ```lemma sin_ge_minus_one [simp]: "-1 \ sin x" ``` paulson@15077 ` 1077` ```apply (insert abs_sin_le_one [of x]) ``` huffman@22998 ` 1078` ```apply (simp add: abs_le_iff del: abs_sin_le_one) ``` paulson@15077 ` 1079` ```done ``` paulson@15077 ` 1080` paulson@15077 ` 1081` ```lemma sin_le_one [simp]: "sin x \ 1" ``` paulson@15077 ` 1082` ```apply (insert abs_sin_le_one [of x]) ``` huffman@22998 ` 1083` ```apply (simp add: abs_le_iff del: abs_sin_le_one) ``` paulson@15077 ` 1084` ```done ``` paulson@15077 ` 1085` paulson@15081 ` 1086` ```lemma abs_cos_le_one [simp]: "\cos x\ \ 1" ``` huffman@23097 ` 1087` ```by (rule power2_le_imp_le, simp_all add: cos_squared_eq) ``` paulson@15077 ` 1088` paulson@15077 ` 1089` ```lemma cos_ge_minus_one [simp]: "-1 \ cos x" ``` paulson@15077 ` 1090` ```apply (insert abs_cos_le_one [of x]) ``` huffman@22998 ` 1091` ```apply (simp add: abs_le_iff del: abs_cos_le_one) ``` paulson@15077 ` 1092` ```done ``` paulson@15077 ` 1093` paulson@15077 ` 1094` ```lemma cos_le_one [simp]: "cos x \ 1" ``` paulson@15077 ` 1095` ```apply (insert abs_cos_le_one [of x]) ``` huffman@22998 ` 1096` ```apply (simp add: abs_le_iff del: abs_cos_le_one) ``` paulson@15077 ` 1097` ```done ``` paulson@15077 ` 1098` paulson@15077 ` 1099` ```lemma DERIV_fun_pow: "DERIV g x :> m ==> ``` paulson@15077 ` 1100` ``` DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m" ``` paulson@15077 ` 1101` ```apply (rule lemma_DERIV_subst) ``` paulson@15229 ` 1102` ```apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2) ``` paulson@15077 ` 1103` ```apply (rule DERIV_pow, auto) ``` paulson@15077 ` 1104` ```done ``` paulson@15077 ` 1105` paulson@15229 ` 1106` ```lemma DERIV_fun_exp: ``` paulson@15229 ` 1107` ``` "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m" ``` paulson@15077 ` 1108` ```apply (rule lemma_DERIV_subst) ``` paulson@15077 ` 1109` ```apply (rule_tac f = exp in DERIV_chain2) ``` paulson@15077 ` 1110` ```apply (rule DERIV_exp, auto) ``` paulson@15077 ` 1111` ```done ``` paulson@15077 ` 1112` paulson@15229 ` 1113` ```lemma DERIV_fun_sin: ``` paulson@15229 ` 1114` ``` "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m" ``` paulson@15077 ` 1115` ```apply (rule lemma_DERIV_subst) ``` paulson@15077 ` 1116` ```apply (rule_tac f = sin in DERIV_chain2) ``` paulson@15077 ` 1117` ```apply (rule DERIV_sin, auto) ``` paulson@15077 ` 1118` ```done ``` paulson@15077 ` 1119` paulson@15229 ` 1120` ```lemma DERIV_fun_cos: ``` paulson@15229 ` 1121` ``` "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m" ``` paulson@15077 ` 1122` ```apply (rule lemma_DERIV_subst) ``` paulson@15077 ` 1123` ```apply (rule_tac f = cos in DERIV_chain2) ``` paulson@15077 ` 1124` ```apply (rule DERIV_cos, auto) ``` paulson@15077 ` 1125` ```done ``` paulson@15077 ` 1126` huffman@23069 ` 1127` ```lemmas DERIV_intros = DERIV_ident DERIV_const DERIV_cos DERIV_cmult ``` paulson@15077 ` 1128` ``` DERIV_sin DERIV_exp DERIV_inverse DERIV_pow ``` paulson@15077 ` 1129` ``` DERIV_add DERIV_diff DERIV_mult DERIV_minus ``` paulson@15077 ` 1130` ``` DERIV_inverse_fun DERIV_quotient DERIV_fun_pow ``` paulson@15077 ` 1131` ``` DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos ``` paulson@15077 ` 1132` paulson@15077 ` 1133` ```(* lemma *) ``` paulson@15229 ` 1134` ```lemma lemma_DERIV_sin_cos_add: ``` paulson@15229 ` 1135` ``` "\x. ``` paulson@15077 ` 1136` ``` DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 + ``` paulson@15077 ` 1137` ``` (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0" ``` paulson@15077 ` 1138` ```apply (safe, rule lemma_DERIV_subst) ``` paulson@15077 ` 1139` ```apply (best intro!: DERIV_intros intro: DERIV_chain2) ``` paulson@15077 ` 1140` ``` --{*replaces the old @{text DERIV_tac}*} ``` paulson@15229 ` 1141` ```apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac) ``` paulson@15077 ` 1142` ```done ``` paulson@15077 ` 1143` paulson@15077 ` 1144` ```lemma sin_cos_add [simp]: ``` paulson@15077 ` 1145` ``` "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 + ``` paulson@15077 ` 1146` ``` (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0" ``` paulson@15077 ` 1147` ```apply (cut_tac y = 0 and x = x and y7 = y ``` paulson@15077 ` 1148` ``` in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all]) ``` paulson@15077 ` 1149` ```apply (auto simp add: numeral_2_eq_2) ``` paulson@15077 ` 1150` ```done ``` paulson@15077 ` 1151` paulson@15077 ` 1152` ```lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y" ``` paulson@15077 ` 1153` ```apply (cut_tac x = x and y = y in sin_cos_add) ``` huffman@22969 ` 1154` ```apply (simp del: sin_cos_add) ``` paulson@15077 ` 1155` ```done ``` paulson@15077 ` 1156` paulson@15077 ` 1157` ```lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y" ``` paulson@15077 ` 1158` ```apply (cut_tac x = x and y = y in sin_cos_add) ``` huffman@22969 ` 1159` ```apply (simp del: sin_cos_add) ``` paulson@15077 ` 1160` ```done ``` paulson@15077 ` 1161` paulson@15085 ` 1162` ```lemma lemma_DERIV_sin_cos_minus: ``` paulson@15085 ` 1163` ``` "\x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0" ``` paulson@15077 ` 1164` ```apply (safe, rule lemma_DERIV_subst) ``` paulson@15077 ` 1165` ```apply (best intro!: DERIV_intros intro: DERIV_chain2) ``` paulson@15229 ` 1166` ```apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac) ``` paulson@15077 ` 1167` ```done ``` paulson@15077 ` 1168` huffman@29165 ` 1169` ```lemma sin_cos_minus: ``` paulson@15085 ` 1170` ``` "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0" ``` paulson@15085 ` 1171` ```apply (cut_tac y = 0 and x = x ``` paulson@15085 ` 1172` ``` in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all]) ``` huffman@22969 ` 1173` ```apply simp ``` paulson@15077 ` 1174` ```done ``` paulson@15077 ` 1175` paulson@15077 ` 1176` ```lemma sin_minus [simp]: "sin (-x) = -sin(x)" ``` huffman@29165 ` 1177` ``` using sin_cos_minus [where x=x] by simp ``` paulson@15077 ` 1178` paulson@15077 ` 1179` ```lemma cos_minus [simp]: "cos (-x) = cos(x)" ``` huffman@29165 ` 1180` ``` using sin_cos_minus [where x=x] by simp ``` paulson@15077 ` 1181` paulson@15077 ` 1182` ```lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y" ``` huffman@22969 ` 1183` ```by (simp add: diff_minus sin_add) ``` paulson@15077 ` 1184` paulson@15077 ` 1185` ```lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x" ``` paulson@15077 ` 1186` ```by (simp add: sin_diff mult_commute) ``` paulson@15077 ` 1187` paulson@15077 ` 1188` ```lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y" ``` huffman@22969 ` 1189` ```by (simp add: diff_minus cos_add) ``` paulson@15077 ` 1190` paulson@15077 ` 1191` ```lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x" ``` paulson@15077 ` 1192` ```by (simp add: cos_diff mult_commute) ``` paulson@15077 ` 1193` paulson@15077 ` 1194` ```lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x" ``` huffman@29165 ` 1195` ``` using sin_add [where x=x and y=x] by simp ``` paulson@15077 ` 1196` paulson@15077 ` 1197` ```lemma cos_double: "cos(2* x) = ((cos x)\) - ((sin x)\)" ``` huffman@29165 ` 1198` ``` using cos_add [where x=x and y=x] ``` huffman@29165 ` 1199` ``` by (simp add: power2_eq_square) ``` paulson@15077 ` 1200` paulson@15077 ` 1201` huffman@29164 ` 1202` ```subsection {* The Constant Pi *} ``` paulson@15077 ` 1203` huffman@23043 ` 1204` ```definition ``` huffman@23043 ` 1205` ``` pi :: "real" where ``` huffman@23053 ` 1206` ``` "pi = 2 * (THE x. 0 \ (x::real) & x \ 2 & cos x = 0)" ``` huffman@23043 ` 1207` paulson@15077 ` 1208` ```text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"}; ``` paulson@15077 ` 1209` ``` hence define pi.*} ``` paulson@15077 ` 1210` paulson@15077 ` 1211` ```lemma sin_paired: ``` huffman@23177 ` 1212` ``` "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) ``` paulson@15077 ` 1213` ``` sums sin x" ``` paulson@15077 ` 1214` ```proof - ``` paulson@15077 ` 1215` ``` have "(\n. \k = n * 2.. 0 < sin x" ``` paulson@15077 ` 1226` ```apply (subgoal_tac ``` paulson@15077 ` 1227` ``` "(\n. \k = n * 2..n. -1 ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))") ``` paulson@15077 ` 1230` ``` prefer 2 ``` paulson@15077 ` 1231` ``` apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp) ``` paulson@15077 ` 1232` ```apply (rotate_tac 2) ``` paulson@15077 ` 1233` ```apply (drule sin_paired [THEN sums_unique, THEN ssubst]) ``` paulson@15077 ` 1234` ```apply (auto simp del: fact_Suc realpow_Suc) ``` paulson@15077 ` 1235` ```apply (frule sums_unique) ``` paulson@15077 ` 1236` ```apply (auto simp del: fact_Suc realpow_Suc) ``` paulson@15077 ` 1237` ```apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans]) ``` paulson@15077 ` 1238` ```apply (auto simp del: fact_Suc realpow_Suc) ``` paulson@15077 ` 1239` ```apply (erule sums_summable) ``` paulson@15077 ` 1240` ```apply (case_tac "m=0") ``` paulson@15077 ` 1241` ```apply (simp (no_asm_simp)) ``` paulson@15234 ` 1242` ```apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x") ``` nipkow@15539 ` 1243` ```apply (simp only: mult_less_cancel_left, simp) ``` nipkow@15539 ` 1244` ```apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric]) ``` paulson@15077 ` 1245` ```apply (subgoal_tac "x*x < 2*3", simp) ``` paulson@15077 ` 1246` ```apply (rule mult_strict_mono) ``` paulson@15085 ` 1247` ```apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc) ``` paulson@15077 ` 1248` ```apply (subst fact_Suc) ``` paulson@15077 ` 1249` ```apply (subst fact_Suc) ``` paulson@15077 ` 1250` ```apply (subst fact_Suc) ``` paulson@15077 ` 1251` ```apply (subst fact_Suc) ``` paulson@15077 ` 1252` ```apply (subst real_of_nat_mult) ``` paulson@15077 ` 1253` ```apply (subst real_of_nat_mult) ``` paulson@15077 ` 1254` ```apply (subst real_of_nat_mult) ``` paulson@15077 ` 1255` ```apply (subst real_of_nat_mult) ``` nipkow@15539 ` 1256` ```apply (simp (no_asm) add: divide_inverse del: fact_Suc) ``` paulson@15077 ` 1257` ```apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc) ``` paulson@15077 ` 1258` ```apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right) ``` paulson@15077 ` 1259` ```apply (auto simp add: mult_assoc simp del: fact_Suc) ``` paulson@15077 ` 1260` ```apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right) ``` paulson@15077 ` 1261` ```apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc) ``` paulson@15077 ` 1262` ```apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)") ``` paulson@15077 ` 1263` ```apply (erule ssubst)+ ``` paulson@15077 ` 1264` ```apply (auto simp del: fact_Suc) ``` paulson@15077 ` 1265` ```apply (subgoal_tac "0 < x ^ (4 * m) ") ``` paulson@15077 ` 1266` ``` prefer 2 apply (simp only: zero_less_power) ``` paulson@15077 ` 1267` ```apply (simp (no_asm_simp) add: mult_less_cancel_left) ``` paulson@15077 ` 1268` ```apply (rule mult_strict_mono) ``` paulson@15077 ` 1269` ```apply (simp_all (no_asm_simp)) ``` paulson@15077 ` 1270` ```done ``` paulson@15077 ` 1271` paulson@15077 ` 1272` ```lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x" ``` paulson@15077 ` 1273` ```by (auto intro: sin_gt_zero) ``` paulson@15077 ` 1274` paulson@15077 ` 1275` ```lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1" ``` paulson@15077 ` 1276` ```apply (cut_tac x = x in sin_gt_zero1) ``` paulson@15077 ` 1277` ```apply (auto simp add: cos_squared_eq cos_double) ``` paulson@15077 ` 1278` ```done ``` paulson@15077 ` 1279` paulson@15077 ` 1280` ```lemma cos_paired: ``` huffman@23177 ` 1281` ``` "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x" ``` paulson@15077 ` 1282` ```proof - ``` paulson@15077 ` 1283` ``` have "(\n. \k = n * 2..n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))" ``` paulson@15481 ` 1302` ``` in order_less_trans) ``` paulson@15077 ` 1303` ```apply (simp (no_asm) add: fact_num_eq_if realpow_num_eq_if del: fact_Suc realpow_Suc) ``` nipkow@15561 ` 1304` ```apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc) ``` paulson@15077 ` 1305` ```apply (rule sumr_pos_lt_pair) ``` paulson@15077 ` 1306` ```apply (erule sums_summable, safe) ``` paulson@15085 ` 1307` ```apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric] ``` paulson@15085 ` 1308` ``` del: fact_Suc) ``` paulson@15077 ` 1309` ```apply (rule real_mult_inverse_cancel2) ``` paulson@15077 ` 1310` ```apply (rule real_of_nat_fact_gt_zero)+ ``` paulson@15077 ` 1311` ```apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc) ``` paulson@15077 ` 1312` ```apply (subst fact_lemma) ``` paulson@15481 ` 1313` ```apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"]) ``` paulson@15481 ` 1314` ```apply (simp only: real_of_nat_mult) ``` huffman@23007 ` 1315` ```apply (rule mult_strict_mono, force) ``` huffman@27483 ` 1316` ``` apply (rule_tac [3] real_of_nat_ge_zero) ``` paulson@15481 ` 1317` ``` prefer 2 apply force ``` paulson@15077 ` 1318` ```apply (rule real_of_nat_less_iff [THEN iffD2]) ``` paulson@15077 ` 1319` ```apply (rule fact_less_mono, auto) ``` paulson@15077 ` 1320` ```done ``` huffman@23053 ` 1321` huffman@23053 ` 1322` ```lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq] ``` huffman@23053 ` 1323` ```lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le] ``` paulson@15077 ` 1324` paulson@15077 ` 1325` ```lemma cos_is_zero: "EX! x. 0 \ x & x \ 2 & cos x = 0" ``` paulson@15077 ` 1326` ```apply (subgoal_tac "\x. 0 \ x & x \ 2 & cos x = 0") ``` paulson@15077 ` 1327` ```apply (rule_tac [2] IVT2) ``` paulson@15077 ` 1328` ```apply (auto intro: DERIV_isCont DERIV_cos) ``` paulson@15077 ` 1329` ```apply (cut_tac x = xa and y = y in linorder_less_linear) ``` paulson@15077 ` 1330` ```apply (rule ccontr) ``` paulson@15077 ` 1331` ```apply (subgoal_tac " (\x. cos differentiable x) & (\x. isCont cos x) ") ``` paulson@15077 ` 1332` ```apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def) ``` paulson@15077 ` 1333` ```apply (drule_tac f = cos in Rolle) ``` paulson@15077 ` 1334` ```apply (drule_tac [5] f = cos in Rolle) ``` paulson@15077 ` 1335` ```apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def) ``` paulson@15077 ` 1336` ```apply (drule_tac y1 = xa in order_le_less_trans [THEN sin_gt_zero]) ``` paulson@15077 ` 1337` ```apply (assumption, rule_tac y=y in order_less_le_trans, simp_all) ``` paulson@15077 ` 1338` ```apply (drule_tac y1 = y in order_le_less_trans [THEN sin_gt_zero], assumption, simp_all) ``` paulson@15077 ` 1339` ```done ``` paulson@15077 ` 1340` ``` ``` huffman@23053 ` 1341` ```lemma pi_half: "pi/2 = (THE x. 0 \ x & x \ 2 & cos x = 0)" ``` paulson@15077 ` 1342` ```by (simp add: pi_def) ``` paulson@15077 ` 1343` paulson@15077 ` 1344` ```lemma cos_pi_half [simp]: "cos (pi / 2) = 0" ``` huffman@23053 ` 1345` ```by (simp add: pi_half cos_is_zero [THEN theI']) ``` huffman@23053 ` 1346` huffman@23053 ` 1347` ```lemma pi_half_gt_zero [simp]: "0 < pi / 2" ``` huffman@23053 ` 1348` ```apply (rule order_le_neq_trans) ``` huffman@23053 ` 1349` ```apply (simp add: pi_half cos_is_zero [THEN theI']) ``` huffman@23053 ` 1350` ```apply (rule notI, drule arg_cong [where f=cos], simp) ``` paulson@15077 ` 1351` ```done ``` paulson@15077 ` 1352` huffman@23053 ` 1353` ```lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric] ``` huffman@23053 ` 1354` ```lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le] ``` paulson@15077 ` 1355` huffman@23053 ` 1356` ```lemma pi_half_less_two [simp]: "pi / 2 < 2" ``` huffman@23053 ` 1357` ```apply (rule order_le_neq_trans) ``` huffman@23053 ` 1358` ```apply (simp add: pi_half cos_is_zero [THEN theI']) ``` huffman@23053 ` 1359` ```apply (rule notI, drule arg_cong [where f=cos], simp) ``` paulson@15077 ` 1360` ```done ``` huffman@23053 ` 1361` huffman@23053 ` 1362` ```lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq] ``` huffman@23053 ` 1363` ```lemmas pi_half_le_two [simp] = pi_half_less_two [THEN order_less_imp_le] ``` paulson@15077 ` 1364` paulson@15077 ` 1365` ```lemma pi_gt_zero [simp]: "0 < pi" ``` huffman@23053 ` 1366` ```by (insert pi_half_gt_zero, simp) ``` huffman@23053 ` 1367` huffman@23053 ` 1368` ```lemma pi_ge_zero [simp]: "0 \ pi" ``` huffman@23053 ` 1369` ```by (rule pi_gt_zero [THEN order_less_imp_le]) ``` paulson@15077 ` 1370` paulson@15077 ` 1371` ```lemma pi_neq_zero [simp]: "pi \ 0" ``` huffman@22998 ` 1372` ```by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym]) ``` paulson@15077 ` 1373` huffman@23053 ` 1374` ```lemma pi_not_less_zero [simp]: "\ pi < 0" ``` huffman@23053 ` 1375` ```by (simp add: linorder_not_less) ``` paulson@15077 ` 1376` huffman@29165 ` 1377` ```lemma minus_pi_half_less_zero: "-(pi/2) < 0" ``` huffman@29165 ` 1378` ```by simp ``` paulson@15077 ` 1379` paulson@15077 ` 1380` ```lemma sin_pi_half [simp]: "sin(pi/2) = 1" ``` paulson@15077 ` 1381` ```apply (cut_tac x = "pi/2" in sin_cos_squared_add2) ``` paulson@15077 ` 1382` ```apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two]) ``` huffman@23053 ` 1383` ```apply (simp add: power2_eq_square) ``` paulson@15077 ` 1384` ```done ``` paulson@15077 ` 1385` paulson@15077 ` 1386` ```lemma cos_pi [simp]: "cos pi = -1" ``` nipkow@15539 ` 1387` ```by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp) ``` paulson@15077 ` 1388` paulson@15077 ` 1389` ```lemma sin_pi [simp]: "sin pi = 0" ``` nipkow@15539 ` 1390` ```by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp) ``` paulson@15077 ` 1391` paulson@15077 ` 1392` ```lemma sin_cos_eq: "sin x = cos (pi/2 - x)" ``` paulson@15229 ` 1393` ```by (simp add: diff_minus cos_add) ``` huffman@23053 ` 1394` ```declare sin_cos_eq [symmetric, simp] ``` paulson@15077 ` 1395` paulson@15077 ` 1396` ```lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)" ``` paulson@15229 ` 1397` ```by (simp add: cos_add) ``` paulson@15077 ` 1398` ```declare minus_sin_cos_eq [symmetric, simp] ``` paulson@15077 ` 1399` paulson@15077 ` 1400` ```lemma cos_sin_eq: "cos x = sin (pi/2 - x)" ``` paulson@15229 ` 1401` ```by (simp add: diff_minus sin_add) ``` huffman@23053 ` 1402` ```declare cos_sin_eq [symmetric, simp] ``` paulson@15077 ` 1403` paulson@15077 ` 1404` ```lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x" ``` paulson@15229 ` 1405` ```by (simp add: sin_add) ``` paulson@15077 ` 1406` paulson@15077 ` 1407` ```lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x" ``` paulson@15229 ` 1408` ```by (simp add: sin_add) ``` paulson@15077 ` 1409` paulson@15077 ` 1410` ```lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x" ``` paulson@15229 ` 1411` ```by (simp add: cos_add) ``` paulson@15077 ` 1412` paulson@15077 ` 1413` ```lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x" ``` paulson@15077 ` 1414` ```by (simp add: sin_add cos_double) ``` paulson@15077 ` 1415` paulson@15077 ` 1416` ```lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x" ``` paulson@15077 ` 1417` ```by (simp add: cos_add cos_double) ``` paulson@15077 ` 1418` paulson@15077 ` 1419` ```lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n" ``` paulson@15251 ` 1420` ```apply (induct "n") ``` paulson@15077 ` 1421` ```apply (auto simp add: real_of_nat_Suc left_distrib) ``` paulson@15077 ` 1422` ```done ``` paulson@15077 ` 1423` paulson@15383 ` 1424` ```lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n" ``` paulson@15383 ` 1425` ```proof - ``` paulson@15383 ` 1426` ``` have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute) ``` paulson@15383 ` 1427` ``` also have "... = -1 ^ n" by (rule cos_npi) ``` paulson@15383 ` 1428` ``` finally show ?thesis . ``` paulson@15383 ` 1429` ```qed ``` paulson@15383 ` 1430` paulson@15077 ` 1431` ```lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0" ``` paulson@15251 ` 1432` ```apply (induct "n") ``` paulson@15077 ` 1433` ```apply (auto simp add: real_of_nat_Suc left_distrib) ``` paulson@15077 ` 1434` ```done ``` paulson@15077 ` 1435` paulson@15077 ` 1436` ```lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0" ``` paulson@15383 ` 1437` ```by (simp add: mult_commute [of pi]) ``` paulson@15077 ` 1438` paulson@15077 ` 1439` ```lemma cos_two_pi [simp]: "cos (2 * pi) = 1" ``` paulson@15077 ` 1440` ```by (simp add: cos_double) ``` paulson@15077 ` 1441` paulson@15077 ` 1442` ```lemma sin_two_pi [simp]: "sin (2 * pi) = 0" ``` paulson@15229 ` 1443` ```by simp ``` paulson@15077 ` 1444` paulson@15077 ` 1445` ```lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x" ``` paulson@15077 ` 1446` ```apply (rule sin_gt_zero, assumption) ``` paulson@15077 ` 1447` ```apply (rule order_less_trans, assumption) ``` paulson@15077 ` 1448` ```apply (rule pi_half_less_two) ``` paulson@15077 ` 1449` ```done ``` paulson@15077 ` 1450` paulson@15077 ` 1451` ```lemma sin_less_zero: ``` paulson@15077 ` 1452` ``` assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0" ``` paulson@15077 ` 1453` ```proof - ``` paulson@15077 ` 1454` ``` have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2) ``` paulson@15077 ` 1455` ``` thus ?thesis by simp ``` paulson@15077 ` 1456` ```qed ``` paulson@15077 ` 1457` paulson@15077 ` 1458` ```lemma pi_less_4: "pi < 4" ``` paulson@15077 ` 1459` ```by (cut_tac pi_half_less_two, auto) ``` paulson@15077 ` 1460` paulson@15077 ` 1461` ```lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x" ``` paulson@15077 ` 1462` ```apply (cut_tac pi_less_4) ``` paulson@15077 ` 1463` ```apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all) ``` paulson@15077 ` 1464` ```apply (cut_tac cos_is_zero, safe) ``` paulson@15077 ` 1465` ```apply (rename_tac y z) ``` paulson@15077 ` 1466` ```apply (drule_tac x = y in spec) ``` paulson@15077 ` 1467` ```apply (drule_tac x = "pi/2" in spec, simp) ``` paulson@15077 ` 1468` ```done ``` paulson@15077 ` 1469` paulson@15077 ` 1470` ```lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x" ``` paulson@15077 ` 1471` ```apply (rule_tac x = x and y = 0 in linorder_cases) ``` paulson@15077 ` 1472` ```apply (rule cos_minus [THEN subst]) ``` paulson@15077 ` 1473` ```apply (rule cos_gt_zero) ``` paulson@15077 ` 1474` ```apply (auto intro: cos_gt_zero) ``` paulson@15077 ` 1475` ```done ``` paulson@15077 ` 1476` ``` ``` paulson@15077 ` 1477` ```lemma cos_ge_zero: "[| -(pi/2) \ x; x \ pi/2 |] ==> 0 \ cos x" ``` paulson@15077 ` 1478` ```apply (auto simp add: order_le_less cos_gt_zero_pi) ``` paulson@15077 ` 1479` ```apply (subgoal_tac "x = pi/2", auto) ``` paulson@15077 ` 1480` ```done ``` paulson@15077 ` 1481` paulson@15077 ` 1482` ```lemma sin_gt_zero_pi: "[| 0 < x; x < pi |] ==> 0 < sin x" ``` paulson@15077 ` 1483` ```apply (subst sin_cos_eq) ``` paulson@15077 ` 1484` ```apply (rotate_tac 1) ``` paulson@15077 ` 1485` ```apply (drule real_sum_of_halves [THEN ssubst]) ``` paulson@15077 ` 1486` ```apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric]) ``` paulson@15077 ` 1487` ```done ``` paulson@15077 ` 1488` paulson@15077 ` 1489` ```lemma sin_ge_zero: "[| 0 \ x; x \ pi |] ==> 0 \ sin x" ``` paulson@15077 ` 1490` ```by (auto simp add: order_le_less sin_gt_zero_pi) ``` paulson@15077 ` 1491` paulson@15077 ` 1492` ```lemma cos_total: "[| -1 \ y; y \ 1 |] ==> EX! x. 0 \ x & x \ pi & (cos x = y)" ``` paulson@15077 ` 1493` ```apply (subgoal_tac "\x. 0 \ x & x \ pi & cos x = y") ``` paulson@15077 ` 1494` ```apply (rule_tac [2] IVT2) ``` paulson@15077 ` 1495` ```apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos) ``` paulson@15077 ` 1496` ```apply (cut_tac x = xa and y = y in linorder_less_linear) ``` paulson@15077 ` 1497` ```apply (rule ccontr, auto) ``` paulson@15077 ` 1498` ```apply (drule_tac f = cos in Rolle) ``` paulson@15077 ` 1499` ```apply (drule_tac [5] f = cos in Rolle) ``` paulson@15077 ` 1500` ```apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos ``` paulson@15077 ` 1501` ``` dest!: DERIV_cos [THEN DERIV_unique] ``` paulson@15077 ` 1502` ``` simp add: differentiable_def) ``` paulson@15077 ` 1503` ```apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans]) ``` paulson@15077 ` 1504` ```done ``` paulson@15077 ` 1505` paulson@15077 ` 1506` ```lemma sin_total: ``` paulson@15077 ` 1507` ``` "[| -1 \ y; y \ 1 |] ==> EX! x. -(pi/2) \ x & x \ pi/2 & (sin x = y)" ``` paulson@15077 ` 1508` ```apply (rule ccontr) ``` paulson@15077 ` 1509` ```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 ` 1510` ```apply (erule contrapos_np) ``` paulson@15077 ` 1511` ```apply (simp del: minus_sin_cos_eq [symmetric]) ``` paulson@15077 ` 1512` ```apply (cut_tac y="-y" in cos_total, simp) apply simp ``` paulson@15077 ` 1513` ```apply (erule ex1E) ``` paulson@15229 ` 1514` ```apply (rule_tac a = "x - (pi/2)" in ex1I) ``` huffman@23286 ` 1515` ```apply (simp (no_asm) add: add_assoc) ``` paulson@15077 ` 1516` ```apply (rotate_tac 3) ``` paulson@15077 ` 1517` ```apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all) ``` paulson@15077 ` 1518` ```done ``` paulson@15077 ` 1519` paulson@15077 ` 1520` ```lemma reals_Archimedean4: ``` paulson@15077 ` 1521` ``` "[| 0 < y; 0 \ x |] ==> \n. real n * y \ x & x < real (Suc n) * y" ``` paulson@15077 ` 1522` ```apply (auto dest!: reals_Archimedean3) ``` paulson@15077 ` 1523` ```apply (drule_tac x = x in spec, clarify) ``` paulson@15077 ` 1524` ```apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y") ``` paulson@15077 ` 1525` ``` prefer 2 apply (erule LeastI) ``` paulson@15077 ` 1526` ```apply (case_tac "LEAST m::nat. x < real m * y", simp) ``` paulson@15077 ` 1527` ```apply (subgoal_tac "~ x < real nat * y") ``` paulson@15077 ` 1528` ``` prefer 2 apply (rule not_less_Least, simp, force) ``` paulson@15077 ` 1529` ```done ``` paulson@15077 ` 1530` paulson@15077 ` 1531` ```(* Pre Isabelle99-2 proof was simpler- numerals arithmetic ``` paulson@15077 ` 1532` ``` now causes some unwanted re-arrangements of literals! *) ``` paulson@15229 ` 1533` ```lemma cos_zero_lemma: ``` paulson@15229 ` 1534` ``` "[| 0 \ x; cos x = 0 |] ==> ``` paulson@15077 ` 1535` ``` \n::nat. ~even n & x = real n * (pi/2)" ``` paulson@15077 ` 1536` ```apply (drule pi_gt_zero [THEN reals_Archimedean4], safe) ``` paulson@15086 ` 1537` ```apply (subgoal_tac "0 \ x - real n * pi & ``` paulson@15086 ` 1538` ``` (x - real n * pi) \ pi & (cos (x - real n * pi) = 0) ") ``` paulson@15086 ` 1539` ```apply (auto simp add: compare_rls) ``` paulson@15077 ` 1540` ``` prefer 3 apply (simp add: cos_diff) ``` paulson@15077 ` 1541` ``` prefer 2 apply (simp add: real_of_nat_Suc left_distrib) ``` paulson@15077 ` 1542` ```apply (simp add: cos_diff) ``` paulson@15077 ` 1543` ```apply (subgoal_tac "EX! x. 0 \ x & x \ pi & cos x = 0") ``` paulson@15077 ` 1544` ```apply (rule_tac [2] cos_total, safe) ``` paulson@15077 ` 1545` ```apply (drule_tac x = "x - real n * pi" in spec) ``` paulson@15077 ` 1546` ```apply (drule_tac x = "pi/2" in spec) ``` paulson@15077 ` 1547` ```apply (simp add: cos_diff) ``` paulson@15229 ` 1548` ```apply (rule_tac x = "Suc (2 * n)" in exI) ``` paulson@15077 ` 1549` ```apply (simp add: real_of_nat_Suc left_distrib, auto) ``` paulson@15077 ` 1550` ```done ``` paulson@15077 ` 1551` paulson@15229 ` 1552` ```lemma sin_zero_lemma: ``` paulson@15229 ` 1553` ``` "[| 0 \ x; sin x = 0 |] ==> ``` paulson@15077 ` 1554` ``` \n::nat. even n & x = real n * (pi/2)" ``` paulson@15077 ` 1555` ```apply (subgoal_tac "\n::nat. ~ even n & x + pi/2 = real n * (pi/2) ") ``` paulson@15077 ` 1556` ``` apply (clarify, rule_tac x = "n - 1" in exI) ``` paulson@15077 ` 1557` ``` apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) ``` paulson@15085 ` 1558` ```apply (rule cos_zero_lemma) ``` paulson@15085 ` 1559` ```apply (simp_all add: add_increasing) ``` paulson@15077 ` 1560` ```done ``` paulson@15077 ` 1561` paulson@15077 ` 1562` paulson@15229 ` 1563` ```lemma cos_zero_iff: ``` paulson@15229 ` 1564` ``` "(cos x = 0) = ``` paulson@15077 ` 1565` ``` ((\n::nat. ~even n & (x = real n * (pi/2))) | ``` paulson@15077 ` 1566` ``` (\n::nat. ~even n & (x = -(real n * (pi/2)))))" ``` paulson@15077 ` 1567` ```apply (rule iffI) ``` paulson@15077 ` 1568` ```apply (cut_tac linorder_linear [of 0 x], safe) ``` paulson@15077 ` 1569` ```apply (drule cos_zero_lemma, assumption+) ``` paulson@15077 ` 1570` ```apply (cut_tac x="-x" in cos_zero_lemma, simp, simp) ``` paulson@15077 ` 1571` ```apply (force simp add: minus_equation_iff [of x]) ``` paulson@15077 ` 1572` ```apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) ``` nipkow@15539 ` 1573` ```apply (auto simp add: cos_add) ``` paulson@15077 ` 1574` ```done ``` paulson@15077 ` 1575` paulson@15077 ` 1576` ```(* ditto: but to a lesser extent *) ``` paulson@15229 ` 1577` ```lemma sin_zero_iff: ``` paulson@15229 ` 1578` ``` "(sin x = 0) = ``` paulson@15077 ` 1579` ``` ((\n::nat. even n & (x = real n * (pi/2))) | ``` paulson@15077 ` 1580` ``` (\n::nat. even n & (x = -(real n * (pi/2)))))" ``` paulson@15077 ` 1581` ```apply (rule iffI) ``` paulson@15077 ` 1582` ```apply (cut_tac linorder_linear [of 0 x], safe) ``` paulson@15077 ` 1583` ```apply (drule sin_zero_lemma, assumption+) ``` paulson@15077 ` 1584` ```apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe) ``` paulson@15077 ` 1585` ```apply (force simp add: minus_equation_iff [of x]) ``` nipkow@15539 ` 1586` ```apply (auto simp add: even_mult_two_ex) ``` paulson@15077 ` 1587` ```done ``` paulson@15077 ` 1588` paulson@15077 ` 1589` huffman@29164 ` 1590` ```subsection {* Tangent *} ``` paulson@15077 ` 1591` huffman@23043 ` 1592` ```definition ``` huffman@23043 ` 1593` ``` tan :: "real => real" where ``` huffman@23043 ` 1594` ``` "tan x = (sin x)/(cos x)" ``` huffman@23043 ` 1595` paulson@15077 ` 1596` ```lemma tan_zero [simp]: "tan 0 = 0" ``` paulson@15077 ` 1597` ```by (simp add: tan_def) ``` paulson@15077 ` 1598` paulson@15077 ` 1599` ```lemma tan_pi [simp]: "tan pi = 0" ``` paulson@15077 ` 1600` ```by (simp add: tan_def) ``` paulson@15077 ` 1601` paulson@15077 ` 1602` ```lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0" ``` paulson@15077 ` 1603` ```by (simp add: tan_def) ``` paulson@15077 ` 1604` paulson@15077 ` 1605` ```lemma tan_minus [simp]: "tan (-x) = - tan x" ``` paulson@15077 ` 1606` ```by (simp add: tan_def minus_mult_left) ``` paulson@15077 ` 1607` paulson@15077 ` 1608` ```lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x" ``` paulson@15077 ` 1609` ```by (simp add: tan_def) ``` paulson@15077 ` 1610` paulson@15077 ` 1611` ```lemma lemma_tan_add1: ``` paulson@15077 ` 1612` ``` "[| cos x \ 0; cos y \ 0 |] ``` paulson@15077 ` 1613` ``` ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)" ``` paulson@15229 ` 1614` ```apply (simp add: tan_def divide_inverse) ``` paulson@15229 ` 1615` ```apply (auto simp del: inverse_mult_distrib ``` paulson@15229 ` 1616` ``` simp add: inverse_mult_distrib [symmetric] mult_ac) ``` paulson@15077 ` 1617` ```apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst]) ``` paulson@15229 ` 1618` ```apply (auto simp del: inverse_mult_distrib ``` paulson@15229 ` 1619` ``` simp add: mult_assoc left_diff_distrib cos_add) ``` paulson@15234 ` 1620` ```done ``` paulson@15077 ` 1621` paulson@15077 ` 1622` ```lemma add_tan_eq: ``` paulson@15077 ` 1623` ``` "[| cos x \ 0; cos y \ 0 |] ``` paulson@15077 ` 1624` ``` ==> tan x + tan y = sin(x + y)/(cos x * cos y)" ``` paulson@15229 ` 1625` ```apply (simp add: tan_def) ``` paulson@15077 ` 1626` ```apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst]) ``` paulson@15077 ` 1627` ```apply (auto simp add: mult_assoc left_distrib) ``` nipkow@15539 ` 1628` ```apply (simp add: sin_add) ``` paulson@15077 ` 1629` ```done ``` paulson@15077 ` 1630` paulson@15229 ` 1631` ```lemma tan_add: ``` paulson@15229 ` 1632` ``` "[| cos x \ 0; cos y \ 0; cos (x + y) \ 0 |] ``` paulson@15077 ` 1633` ``` ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))" ``` paulson@15077 ` 1634` ```apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1) ``` paulson@15077 ` 1635` ```apply (simp add: tan_def) ``` paulson@15077 ` 1636` ```done ``` paulson@15077 ` 1637` paulson@15229 ` 1638` ```lemma tan_double: ``` paulson@15229 ` 1639` ``` "[| cos x \ 0; cos (2 * x) \ 0 |] ``` paulson@15077 ` 1640` ``` ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))" ``` paulson@15077 ` 1641` ```apply (insert tan_add [of x x]) ``` paulson@15077 ` 1642` ```apply (simp add: mult_2 [symmetric]) ``` paulson@15077 ` 1643` ```apply (auto simp add: numeral_2_eq_2) ``` paulson@15077 ` 1644` ```done ``` paulson@15077 ` 1645` paulson@15077 ` 1646` ```lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x" ``` paulson@15229 ` 1647` ```by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) ``` paulson@15077 ` 1648` paulson@15077 ` 1649` ```lemma tan_less_zero: ``` paulson@15077 ` 1650` ``` assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0" ``` paulson@15077 ` 1651` ```proof - ``` paulson@15077 ` 1652` ``` have "0 < tan (- x)" using prems by (simp only: tan_gt_zero) ``` paulson@15077 ` 1653` ``` thus ?thesis by simp ``` paulson@15077 ` 1654` ```qed ``` paulson@15077 ` 1655` paulson@15077 ` 1656` ```lemma lemma_DERIV_tan: ``` paulson@15077 ` 1657` ``` "cos x \ 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\)" ``` paulson@15077 ` 1658` ```apply (rule lemma_DERIV_subst) ``` paulson@15077 ` 1659` ```apply (best intro!: DERIV_intros intro: DERIV_chain2) ``` paulson@15079 ` 1660` ```apply (auto simp add: divide_inverse numeral_2_eq_2) ``` paulson@15077 ` 1661` ```done ``` paulson@15077 ` 1662` paulson@15077 ` 1663` ```lemma DERIV_tan [simp]: "cos x \ 0 ==> DERIV tan x :> inverse((cos x)\)" ``` paulson@15077 ` 1664` ```by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric]) ``` paulson@15077 ` 1665` huffman@23045 ` 1666` ```lemma isCont_tan [simp]: "cos x \ 0 ==> isCont tan x" ``` huffman@23045 ` 1667` ```by (rule DERIV_tan [THEN DERIV_isCont]) ``` huffman@23045 ` 1668` paulson@15077 ` 1669` ```lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0" ``` paulson@15077 ` 1670` ```apply (subgoal_tac "(\x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1") ``` paulson@15229 ` 1671` ```apply (simp add: divide_inverse [symmetric]) ``` huffman@22613 ` 1672` ```apply (rule LIM_mult) ``` paulson@15077 ` 1673` ```apply (rule_tac [2] inverse_1 [THEN subst]) ``` paulson@15077 ` 1674` ```apply (rule_tac [2] LIM_inverse) ``` paulson@15077 ` 1675` ```apply (simp_all add: divide_inverse [symmetric]) ``` paulson@15077 ` 1676` ```apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric]) ``` paulson@15077 ` 1677` ```apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+ ``` paulson@15077 ` 1678` ```done ``` paulson@15077 ` 1679` paulson@15077 ` 1680` ```lemma lemma_tan_total: "0 < y ==> \x. 0 < x & x < pi/2 & y < tan x" ``` paulson@15077 ` 1681` ```apply (cut_tac LIM_cos_div_sin) ``` paulson@15077 ` 1682` ```apply (simp only: LIM_def) ``` paulson@15077 ` 1683` ```apply (drule_tac x = "inverse y" in spec, safe, force) ``` paulson@15077 ` 1684` ```apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe) ``` paulson@15229 ` 1685` ```apply (rule_tac x = "(pi/2) - e" in exI) ``` paulson@15077 ` 1686` ```apply (simp (no_asm_simp)) ``` paulson@15229 ` 1687` ```apply (drule_tac x = "(pi/2) - e" in spec) ``` paulson@15229 ` 1688` ```apply (auto simp add: tan_def) ``` paulson@15077 ` 1689` ```apply (rule inverse_less_iff_less [THEN iffD1]) ``` paulson@15079 ` 1690` ```apply (auto simp add: divide_inverse) ``` paulson@15229 ` 1691` ```apply (rule real_mult_order) ``` paulson@15229 ` 1692` ```apply (subgoal_tac [3] "0 < sin e & 0 < cos e") ``` paulson@15229 ` 1693` ```apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute) ``` paulson@15077 ` 1694` ```done ``` paulson@15077 ` 1695` paulson@15077 ` 1696` ```lemma tan_total_pos: "0 \ y ==> \x. 0 \ x & x < pi/2 & tan x = y" ``` huffman@22998 ` 1697` ```apply (frule order_le_imp_less_or_eq, safe) ``` paulson@15077 ` 1698` ``` prefer 2 apply force ``` paulson@15077 ` 1699` ```apply (drule lemma_tan_total, safe) ``` paulson@15077 ` 1700` ```apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl) ``` paulson@15077 ` 1701` ```apply (auto intro!: DERIV_tan [THEN DERIV_isCont]) ``` paulson@15077 ` 1702` ```apply (drule_tac y = xa in order_le_imp_less_or_eq) ``` paulson@15077 ` 1703` ```apply (auto dest: cos_gt_zero) ``` paulson@15077 ` 1704` ```done ``` paulson@15077 ` 1705` paulson@15077 ` 1706` ```lemma lemma_tan_total1: "\x. -(pi/2) < x & x < (pi/2) & tan x = y" ``` paulson@15077 ` 1707` ```apply (cut_tac linorder_linear [of 0 y], safe) ``` paulson@15077 ` 1708` ```apply (drule tan_total_pos) ``` paulson@15077 ` 1709` ```apply (cut_tac [2] y="-y" in tan_total_pos, safe) ``` paulson@15077 ` 1710` ```apply (rule_tac [3] x = "-x" in exI) ``` paulson@15077 ` 1711` ```apply (auto intro!: exI) ``` paulson@15077 ` 1712` ```done ``` paulson@15077 ` 1713` paulson@15077 ` 1714` ```lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y" ``` paulson@15077 ` 1715` ```apply (cut_tac y = y in lemma_tan_total1, auto) ``` paulson@15077 ` 1716` ```apply (cut_tac x = xa and y = y in linorder_less_linear, auto) ``` paulson@15077 ` 1717` ```apply (subgoal_tac [2] "\z. y < z & z < xa & DERIV tan z :> 0") ``` paulson@15077 ` 1718` ```apply (subgoal_tac "\z. xa < z & z < y & DERIV tan z :> 0") ``` paulson@15077 ` 1719` ```apply (rule_tac [4] Rolle) ``` paulson@15077 ` 1720` ```apply (rule_tac [2] Rolle) ``` paulson@15077 ` 1721` ```apply (auto intro!: DERIV_tan DERIV_isCont exI ``` paulson@15077 ` 1722` ``` simp add: differentiable_def) ``` paulson@15077 ` 1723` ```txt{*Now, simulate TRYALL*} ``` paulson@15077 ` 1724` ```apply (rule_tac [!] DERIV_tan asm_rl) ``` paulson@15077 ` 1725` ```apply (auto dest!: DERIV_unique [OF _ DERIV_tan] ``` huffman@22998 ` 1726` ``` simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym]) ``` paulson@15077 ` 1727` ```done ``` paulson@15077 ` 1728` huffman@23043 ` 1729` huffman@23043 ` 1730` ```subsection {* Inverse Trigonometric Functions *} ``` huffman@23043 ` 1731` huffman@23043 ` 1732` ```definition ``` huffman@23043 ` 1733` ``` arcsin :: "real => real" where ``` huffman@23043 ` 1734` ``` "arcsin y = (THE x. -(pi/2) \ x & x \ pi/2 & sin x = y)" ``` huffman@23043 ` 1735` huffman@23043 ` 1736` ```definition ``` huffman@23043 ` 1737` ``` arccos :: "real => real" where ``` huffman@23043 ` 1738` ``` "arccos y = (THE x. 0 \ x & x \ pi & cos x = y)" ``` huffman@23043 ` 1739` huffman@23043 ` 1740` ```definition ``` huffman@23043 ` 1741` ``` arctan :: "real => real" where ``` huffman@23043 ` 1742` ``` "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)" ``` huffman@23043 ` 1743` paulson@15229 ` 1744` ```lemma arcsin: ``` paulson@15229 ` 1745` ``` "[| -1 \ y; y \ 1 |] ``` paulson@15077 ` 1746` ``` ==> -(pi/2) \ arcsin y & ``` paulson@15077 ` 1747` ``` arcsin y \ pi/2 & sin(arcsin y) = y" ``` huffman@23011 ` 1748` ```unfolding arcsin_def by (rule theI' [OF sin_total]) ``` huffman@23011 ` 1749` huffman@23011 ` 1750` ```lemma arcsin_pi: ``` huffman@23011 ` 1751` ``` "[| -1 \ y; y \ 1 |] ``` huffman@23011 ` 1752` ``` ==> -(pi/2) \ arcsin y & arcsin y \ pi & sin(arcsin y) = y" ``` huffman@23011 ` 1753` ```apply (drule (1) arcsin) ``` huffman@23011 ` 1754` ```apply (force intro: order_trans) ``` paulson@15077 ` 1755` ```done ``` paulson@15077 ` 1756` paulson@15077 ` 1757` ```lemma sin_arcsin [simp]: "[| -1 \ y; y \ 1 |] ==> sin(arcsin y) = y" ``` paulson@15077 ` 1758` ```by (blast dest: arcsin) ``` paulson@15077 ` 1759` ``` ``` paulson@15077 ` 1760` ```lemma arcsin_bounded: ``` paulson@15077 ` 1761` ``` "[| -1 \ y; y \ 1 |] ==> -(pi/2) \ arcsin y & arcsin y \ pi/2" ``` paulson@15077 ` 1762` ```by (blast dest: arcsin) ``` paulson@15077 ` 1763` paulson@15077 ` 1764` ```lemma arcsin_lbound: "[| -1 \ y; y \ 1 |] ==> -(pi/2) \ arcsin y" ``` paulson@15077 ` 1765` ```by (blast dest: arcsin) ``` paulson@15077 ` 1766` paulson@15077 ` 1767` ```lemma arcsin_ubound: "[| -1 \ y; y \ 1 |] ==> arcsin y \ pi/2" ``` paulson@15077 ` 1768` ```by (blast dest: arcsin) ``` paulson@15077 ` 1769` paulson@15077 ` 1770` ```lemma arcsin_lt_bounded: ``` paulson@15077 ` 1771` ``` "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2" ``` paulson@15077 ` 1772` ```apply (frule order_less_imp_le) ``` paulson@15077 ` 1773` ```apply (frule_tac y = y in order_less_imp_le) ``` paulson@15077 ` 1774` ```apply (frule arcsin_bounded) ``` paulson@15077 ` 1775` ```apply (safe, simp) ``` paulson@15077 ` 1776` ```apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq) ``` paulson@15077 ` 1777` ```apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe) ``` paulson@15077 ` 1778` ```apply (drule_tac [!] f = sin in arg_cong, auto) ``` paulson@15077 ` 1779` ```done ``` paulson@15077 ` 1780` paulson@15077 ` 1781` ```lemma arcsin_sin: "[|-(pi/2) \ x; x \ pi/2 |] ==> arcsin(sin x) = x" ``` paulson@15077 ` 1782` ```apply (unfold arcsin_def) ``` huffman@23011 ` 1783` ```apply (rule the1_equality) ``` paulson@15077 ` 1784` ```apply (rule sin_total, auto) ``` paulson@15077 ` 1785` ```done ``` paulson@15077 ` 1786` huffman@22975 ` 1787` ```lemma arccos: ``` paulson@15229 ` 1788` ``` "[| -1 \ y; y \ 1 |] ``` huffman@22975 ` 1789` ``` ==> 0 \ arccos y & arccos y \ pi & cos(arccos y) = y" ``` huffman@23011 ` 1790` ```unfolding arccos_def by (rule theI' [OF cos_total]) ``` paulson@15077 ` 1791` huffman@22975 ` 1792` ```lemma cos_arccos [simp]: "[| -1 \ y; y \ 1 |] ==> cos(arccos y) = y" ``` huffman@22975 ` 1793` ```by (blast dest: arccos) ``` paulson@15077 ` 1794` ``` ``` huffman@22975 ` 1795` ```lemma arccos_bounded: "[| -1 \ y; y \ 1 |] ==> 0 \ arccos y & arccos y \ pi" ``` huffman@22975 ` 1796` ```by (blast dest: arccos) ``` paulson@15077 ` 1797` huffman@22975 ` 1798` ```lemma arccos_lbound: "[| -1 \ y; y \ 1 |] ==> 0 \ arccos y" ``` huffman@22975 ` 1799` ```by (blast dest: arccos) ``` paulson@15077 ` 1800` huffman@22975 ` 1801` ```lemma arccos_ubound: "[| -1 \ y; y \ 1 |] ==> arccos y \ pi" ``` huffman@22975 ` 1802` ```by (blast dest: arccos) ``` paulson@15077 ` 1803` huffman@22975 ` 1804` ```lemma arccos_lt_bounded: ``` paulson@15229 ` 1805` ``` "[| -1 < y; y < 1 |] ``` huffman@22975 ` 1806` ``` ==> 0 < arccos y & arccos y < pi" ``` paulson@15077 ` 1807` ```apply (frule order_less_imp_le) ``` paulson@15077 ` 1808` ```apply (frule_tac y = y in order_less_imp_le) ``` huffman@22975 ` 1809` ```apply (frule arccos_bounded, auto) ``` huffman@22975 ` 1810` ```apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq) ``` paulson@15077 ` 1811` ```apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto) ``` paulson@15077 ` 1812` ```apply (drule_tac [!] f = cos in arg_cong, auto) ``` paulson@15077 ` 1813` ```done ``` paulson@15077 ` 1814` huffman@22975 ` 1815` ```lemma arccos_cos: "[|0 \ x; x \ pi |] ==> arccos(cos x) = x" ``` huffman@22975 ` 1816` ```apply (simp add: arccos_def) ``` huffman@23011 ` 1817` ```apply (auto intro!: the1_equality cos_total) ``` paulson@15077 ` 1818` ```done ``` paulson@15077 ` 1819` huffman@22975 ` 1820` ```lemma arccos_cos2: "[|x \ 0; -pi \ x |] ==> arccos(cos x) = -x" ``` huffman@22975 ` 1821` ```apply (simp add: arccos_def) ``` huffman@23011 ` 1822` ```apply (auto intro!: the1_equality cos_total) ``` paulson@15077 ` 1823` ```done ``` paulson@15077 ` 1824` huffman@23045 ` 1825` ```lemma cos_arcsin: "\-1 \ x; x \ 1\ \ cos (arcsin x) = sqrt (1 - x\)" ``` huffman@23045 ` 1826` ```apply (subgoal_tac "x\ \ 1") ``` huffman@23052 ` 1827` ```apply (rule power2_eq_imp_eq) ``` huffman@23045 ` 1828` ```apply (simp add: cos_squared_eq) ``` huffman@23045 ` 1829` ```apply (rule cos_ge_zero) ``` huffman@23045 ` 1830` ```apply (erule (1) arcsin_lbound) ``` huffman@23045 ` 1831` ```apply (erule (1) arcsin_ubound) ``` huffman@23045 ` 1832` ```apply simp ``` huffman@23045 ` 1833` ```apply (subgoal_tac "\x\\ \ 1\", simp) ``` huffman@23045 ` 1834` ```apply (rule power_mono, simp, simp) ``` huffman@23045 ` 1835` ```done ``` huffman@23045 ` 1836` huffman@23045 ` 1837` ```lemma sin_arccos: "\-1 \ x; x \ 1\ \ sin (arccos x) = sqrt (1 - x\)" ``` huffman@23045 ` 1838` ```apply (subgoal_tac "x\ \ 1") ``` huffman@23052 ` 1839` ```apply (rule power2_eq_imp_eq) ``` huffman@23045 ` 1840` ```apply (simp add: sin_squared_eq) ``` huffman@23045 ` 1841` ```apply (rule sin_ge_zero) ``` huffman@23045 ` 1842` ```apply (erule (1) arccos_lbound) ``` huffman@23045 ` 1843` ```apply (erule (1) arccos_ubound) ``` huffman@23045 ` 1844` ```apply simp ``` huffman@23045 ` 1845` ```apply (subgoal_tac "\x\\ \ 1\", simp) ``` huffman@23045 ` 1846` ```apply (rule power_mono, simp, simp) ``` huffman@23045 ` 1847` ```done ``` huffman@23045 ` 1848` paulson@15077 ` 1849` ```lemma arctan [simp]: ``` paulson@15077 ` 1850` ``` "- (pi/2) < arctan y & arctan y < pi/2 & tan (arctan y) = y" ``` huffman@23011 ` 1851` ```unfolding arctan_def by (rule theI' [OF tan_total]) ``` paulson@15077 ` 1852` paulson@15077 ` 1853` ```lemma tan_arctan: "tan(arctan y) = y" ``` paulson@15077 ` 1854` ```by auto ``` paulson@15077 ` 1855` paulson@15077 ` 1856` ```lemma arctan_bounded: "- (pi/2) < arctan y & arctan y < pi/2" ``` paulson@15077 ` 1857` ```by (auto simp only: arctan) ``` paulson@15077 ` 1858` paulson@15077 ` 1859` ```lemma arctan_lbound: "- (pi/2) < arctan y" ``` paulson@15077 ` 1860` ```by auto ``` paulson@15077 ` 1861` paulson@15077 ` 1862` ```lemma arctan_ubound: "arctan y < pi/2" ``` paulson@15077 ` 1863` ```by (auto simp only: arctan) ``` paulson@15077 ` 1864` paulson@15077 ` 1865` ```lemma arctan_tan: ``` paulson@15077 ` 1866` ``` "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x" ``` paulson@15077 ` 1867` ```apply (unfold arctan_def) ``` huffman@23011 ` 1868` ```apply (rule the1_equality) ``` paulson@15077 ` 1869` ```apply (rule tan_total, auto) ``` paulson@15077 ` 1870` ```done ``` paulson@15077 ` 1871` paulson@15077 ` 1872` ```lemma arctan_zero_zero [simp]: "arctan 0 = 0" ``` paulson@15077 ` 1873` ```by (insert arctan_tan [of 0], simp) ``` paulson@15077 ` 1874` paulson@15077 ` 1875` ```lemma cos_arctan_not_zero [simp]: "cos(arctan x) \ 0" ``` paulson@15077 ` 1876` ```apply (auto simp add: cos_zero_iff) ``` paulson@15077 ` 1877` ```apply (case_tac "n") ``` paulson@15077 ` 1878` ```apply (case_tac [3] "n") ``` paulson@15077 ` 1879` ```apply (cut_tac [2] y = x in arctan_ubound) ``` paulson@15077 ` 1880` ```apply (cut_tac [4] y = x in arctan_lbound) ``` paulson@15077 ` 1881` ```apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff) ``` paulson@15077 ` 1882` ```done ``` paulson@15077 ` 1883` paulson@15077 ` 1884` ```lemma tan_sec: "cos x \ 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2" ``` paulson@15077 ` 1885` ```apply (rule power_inverse [THEN subst]) ``` paulson@15077 ` 1886` ```apply (rule_tac c1 = "(cos x)\" in real_mult_right_cancel [THEN iffD1]) ``` huffman@22960 ` 1887` ```apply (auto dest: field_power_not_zero ``` huffman@20516 ` 1888` ``` simp add: power_mult_distrib left_distrib power_divide tan_def ``` paulson@15077 ` 1889` ``` mult_assoc power_inverse [symmetric] ``` paulson@15077 ` 1890` ``` simp del: realpow_Suc) ``` paulson@15077 ` 1891` ```done ``` paulson@15077 ` 1892` huffman@23045 ` 1893` ```lemma isCont_inverse_function2: ``` huffman@23045 ` 1894` ``` fixes f g :: "real \ real" shows ``` huffman@23045 ` 1895` ``` "\a < x; x < b; ``` huffman@23045 ` 1896` ``` \z. a \ z \ z \ b \ g (f z) = z; ``` huffman@23045 ` 1897` ``` \z. a \ z \ z \ b \ isCont f z\ ``` huffman@23045 ` 1898` ``` \ isCont g (f x)" ``` huffman@23045 ` 1899` ```apply (rule isCont_inverse_function ``` huffman@23045 ` 1900` ``` [where f=f and d="min (x - a) (b - x)"]) ``` huffman@23045 ` 1901` ```apply (simp_all add: abs_le_iff) ``` huffman@23045 ` 1902` ```done ``` huffman@23045 ` 1903` huffman@23045 ` 1904` ```lemma isCont_arcsin: "\-1 < x; x < 1\ \ isCont arcsin x" ``` huffman@23045 ` 1905` ```apply (subgoal_tac "isCont arcsin (sin (arcsin x))", simp) ``` huffman@23045 ` 1906` ```apply (rule isCont_inverse_function2 [where f=sin]) ``` huffman@23045 ` 1907` ```apply (erule (1) arcsin_lt_bounded [THEN conjunct1]) ``` huffman@23045 ` 1908` ```apply (erule (1) arcsin_lt_bounded [THEN conjunct2]) ``` huffman@23045 ` 1909` ```apply (fast intro: arcsin_sin, simp) ``` huffman@23045 ` 1910` ```done ``` huffman@23045 ` 1911` huffman@23045 ` 1912` ```lemma isCont_arccos: "\-1 < x; x < 1\ \ isCont arccos x" ``` huffman@23045 ` 1913` ```apply (subgoal_tac "isCont arccos (cos (arccos x))", simp) ``` huffman@23045 ` 1914` ```apply (rule isCont_inverse_function2 [where f=cos]) ``` huffman@23045 ` 1915` ```apply (erule (1) arccos_lt_bounded [THEN conjunct1]) ``` huffman@23045 ` 1916` ```apply (erule (1) arccos_lt_bounded [THEN conjunct2]) ``` huffman@23045 ` 1917` ```apply (fast intro: arccos_cos, simp) ``` huffman@23045 ` 1918` ```done ``` huffman@23045 ` 1919` huffman@23045 ` 1920` ```lemma isCont_arctan: "isCont arctan x" ``` huffman@23045 ` 1921` ```apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify) ``` huffman@23045 ` 1922` ```apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify) ``` huffman@23045 ` 1923` ```apply (subgoal_tac "isCont arctan (tan (arctan x))", simp) ``` huffman@23045 ` 1924` ```apply (erule (1) isCont_inverse_function2 [where f=tan]) ``` huffman@23045 ` 1925` ```apply (clarify, rule arctan_tan) ``` huffman@23045 ` 1926` ```apply (erule (1) order_less_le_trans) ``` huffman@23045 ` 1927` ```apply (erule (1) order_le_less_trans) ``` huffman@23045 ` 1928` ```apply (clarify, rule isCont_tan) ``` huffman@23045 ` 1929` ```apply (rule less_imp_neq [symmetric]) ``` huffman@23045 ` 1930` ```apply (rule cos_gt_zero_pi) ``` huffman@23045 ` 1931` ```apply (erule (1) order_less_le_trans) ``` huffman@23045 ` 1932` ```apply (erule (1) order_le_less_trans) ``` huffman@23045 ` 1933` ```done ``` huffman@23045 ` 1934` huffman@23045 ` 1935` ```lemma DERIV_arcsin: ``` huffman@23045 ` 1936` ``` "\-1 < x; x < 1\ \ DERIV arcsin x :> inverse (sqrt (1 - x\))" ``` huffman@23045 ` 1937` ```apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"]) ``` huffman@23045 ` 1938` ```apply (rule lemma_DERIV_subst [OF DERIV_sin]) ``` huffman@23045 ` 1939` ```apply (simp add: cos_arcsin) ``` huffman@23045 ` 1940` ```apply (subgoal_tac "\x\\ < 1\", simp) ``` huffman@23045 ` 1941` ```apply (rule power_strict_mono, simp, simp, simp) ``` huffman@23045 ` 1942` ```apply assumption ``` huffman@23045 ` 1943` ```apply assumption ``` huffman@23045 ` 1944` ```apply simp ``` huffman@23045 ` 1945` ```apply (erule (1) isCont_arcsin) ``` huffman@23045 ` 1946` ```done ``` huffman@23045 ` 1947` huffman@23045 ` 1948` ```lemma DERIV_arccos: ``` huffman@23045 ` 1949` ``` "\-1 < x; x < 1\ \ DERIV arccos x :> inverse (- sqrt (1 - x\))" ``` huffman@23045 ` 1950` ```apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"]) ``` huffman@23045 ` 1951` ```apply (rule lemma_DERIV_subst [OF DERIV_cos]) ``` huffman@23045 ` 1952` ```apply (simp add: sin_arccos) ``` huffman@23045 ` 1953` ```apply (subgoal_tac "\x\\ < 1\", simp) ``` huffman@23045 ` 1954` ```apply (rule power_strict_mono, simp, simp, simp) ``` huffman@23045 ` 1955` ```apply assumption ``` huffman@23045 ` 1956` ```apply assumption ``` huffman@23045 ` 1957` ```apply simp ``` huffman@23045 ` 1958` ```apply (erule (1) isCont_arccos) ``` huffman@23045 ` 1959` ```done ``` huffman@23045 ` 1960` huffman@23045 ` 1961` ```lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\)" ``` huffman@23045 ` 1962` ```apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"]) ``` huffman@23045 ` 1963` ```apply (rule lemma_DERIV_subst [OF DERIV_tan]) ``` huffman@23045 ` 1964` ```apply (rule cos_arctan_not_zero) ``` huffman@23045 ` 1965` ```apply (simp add: power_inverse tan_sec [symmetric]) ``` huffman@23045 ` 1966` ```apply (subgoal_tac "0 < 1 + x\", simp) ``` huffman@23045 ` 1967` ```apply (simp add: add_pos_nonneg) ``` huffman@23045 ` 1968` ```apply (simp, simp, simp, rule isCont_arctan) ``` huffman@23045 ` 1969` ```done ``` huffman@23045 ` 1970` huffman@23045 ` 1971` huffman@23043 ` 1972` ```subsection {* More Theorems about Sin and Cos *} ``` huffman@23043 ` 1973` huffman@23052 ` 1974` ```lemma cos_45: "cos (pi / 4) = sqrt 2 / 2" ``` huffman@23052 ` 1975` ```proof - ``` huffman@23052 ` 1976` ``` let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)" ``` huffman@23052 ` 1977` ``` have nonneg: "0 \ ?c" ``` huffman@23052 ` 1978` ``` by (rule cos_ge_zero, rule order_trans [where y=0], simp_all) ``` huffman@23052 ` 1979` ``` have "0 = cos (pi / 4 + pi / 4)" ``` huffman@23052 ` 1980` ``` by simp ``` huffman@23052 ` 1981` ``` also have "cos (pi / 4 + pi / 4) = ?c\ - ?s\" ``` huffman@23052 ` 1982` ``` by (simp only: cos_add power2_eq_square) ``` huffman@23052 ` 1983` ``` also have "\ = 2 * ?c\ - 1" ``` huffman@23052 ` 1984` ``` by (simp add: sin_squared_eq) ``` huffman@23052 ` 1985` ``` finally have "?c\ = (sqrt 2 / 2)\" ``` huffman@23052 ` 1986` ``` by (simp add: power_divide) ``` huffman@23052 ` 1987` ``` thus ?thesis ``` huffman@23052 ` 1988` ``` using nonneg by (rule power2_eq_imp_eq) simp ``` huffman@23052 ` 1989` ```qed ``` huffman@23052 ` 1990` huffman@23052 ` 1991` ```lemma cos_30: "cos (pi / 6) = sqrt 3 / 2" ``` huffman@23052 ` 1992` ```proof - ``` huffman@23052 ` 1993` ``` let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)" ``` huffman@23052 ` 1994` ``` have pos_c: "0 < ?c" ``` huffman@23052 ` 1995` ``` by (rule cos_gt_zero, simp, simp) ``` huffman@23052 ` 1996` ``` have "0 = cos (pi / 6 + pi / 6 + pi / 6)" ``` huffman@23066 ` 1997` ``` by simp ``` huffman@23052 ` 1998` ``` also have "\ = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s" ``` huffman@23052 ` 1999` ``` by (simp only: cos_add sin_add) ``` huffman@23052 ` 2000` ``` also have "\ = ?c * (?c\ - 3 * ?s\)" ``` nipkow@23477 ` 2001` ``` by (simp add: ring_simps power2_eq_square) ``` huffman@23052 ` 2002` ``` finally have "?c\ = (sqrt 3 / 2)\" ``` huffman@23052 ` 2003` ``` using pos_c by (simp add: sin_squared_eq power_divide) ``` huffman@23052 ` 2004` ``` thus ?thesis ``` huffman@23052 ` 2005` ``` using pos_c [THEN order_less_imp_le] ``` huffman@23052 ` 2006` ``` by (rule power2_eq_imp_eq) simp ``` huffman@23052 ` 2007` ```qed ``` huffman@23052 ` 2008` huffman@23052 ` 2009` ```lemma sin_45: "sin (pi / 4) = sqrt 2 / 2" ``` huffman@23052 ` 2010` ```proof - ``` huffman@23052 ` 2011` ``` have "sin (pi / 4) = cos (pi / 2 - pi / 4)" by (rule sin_cos_eq) ``` huffman@23052 ` 2012` ``` also have "pi / 2 - pi / 4 = pi / 4" by simp ``` huffman@23052 ` 2013` ``` also have "cos (pi / 4) = sqrt 2 / 2" by (rule cos_45) ``` huffman@23052 ` 2014` ``` finally show ?thesis . ``` huffman@23052 ` 2015` ```qed ``` huffman@23052 ` 2016` huffman@23052 ` 2017` ```lemma sin_60: "sin (pi / 3) = sqrt 3 / 2" ``` huffman@23052 ` 2018` ```proof - ``` huffman@23052 ` 2019` ``` have "sin (pi / 3) = cos (pi / 2 - pi / 3)" by (rule sin_cos_eq) ``` huffman@23052 ` 2020` ``` also have "pi / 2 - pi / 3 = pi / 6" by simp ``` huffman@23052 ` 2021` ``` also have "cos (pi / 6) = sqrt 3 / 2" by (rule cos_30) ``` huffman@23052 ` 2022` ``` finally show ?thesis . ``` huffman@23052 ` 2023` ```qed ``` huffman@23052 ` 2024` huffman@23052 ` 2025` ```lemma cos_60: "cos (pi / 3) = 1 / 2" ``` huffman@23052 ` 2026` ```apply (rule power2_eq_imp_eq) ``` huffman@23052 ` 2027` ```apply (simp add: cos_squared_eq sin_60 power_divide) ``` huffman@23052 ` 2028` ```apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all) ``` huffman@23052 ` 2029` ```done ``` huffman@23052 ` 2030` huffman@23052 ` 2031` ```lemma sin_30: "sin (pi / 6) = 1 / 2" ``` huffman@23052 ` 2032` ```proof - ``` huffman@23052 ` 2033` ``` have "sin (pi / 6) = cos (pi / 2 - pi / 6)" by (rule sin_cos_eq) ``` huffman@23066 ` 2034` ``` also have "pi / 2 - pi / 6 = pi / 3" by simp ``` huffman@23052 ` 2035` ``` also have "cos (pi / 3) = 1 / 2" by (rule cos_60) ``` huffman@23052 ` 2036` ``` finally show ?thesis . ``` huffman@23052 ` 2037` ```qed ``` huffman@23052 ` 2038` huffman@23052 ` 2039` ```lemma tan_30: "tan (pi / 6) = 1 / sqrt 3" ``` huffman@23052 ` 2040` ```unfolding tan_def by (simp add: sin_30 cos_30) ``` huffman@23052 ` 2041` huffman@23052 ` 2042` ```lemma tan_45: "tan (pi / 4) = 1" ``` huffman@23052 ` 2043` ```unfolding tan_def by (simp add: sin_45 cos_45) ``` huffman@23052 ` 2044` huffman@23052 ` 2045` ```lemma tan_60: "tan (pi / 3) = sqrt 3" ``` huffman@23052 ` 2046` ```unfolding tan_def by (simp add: sin_60 cos_60) ``` huffman@23052 ` 2047` paulson@15085 ` 2048` ```text{*NEEDED??*} ``` paulson@15229 ` 2049` ```lemma [simp]: ``` paulson@15229 ` 2050` ``` "sin (x + 1 / 2 * real (Suc m) * pi) = ``` paulson@15229 ` 2051` ``` cos (x + 1 / 2 * real (m) * pi)" ``` paulson@15229 ` 2052` ```by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib, auto) ``` paulson@15077 ` 2053` paulson@15085 ` 2054` ```text{*NEEDED??*} ``` paulson@15229 ` 2055` ```lemma [simp]: ``` paulson@15229 ` 2056` ``` "sin (x + real (Suc m) * pi / 2) = ``` paulson@15229 ` 2057` ``` cos (x + real (m) * pi / 2)" ``` paulson@15229 ` 2058` ```by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto) ``` paulson@15077 ` 2059` paulson@15077 ` 2060` ```lemma DERIV_sin_add [simp]: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)" ``` paulson@15077 ` 2061` ```apply (rule lemma_DERIV_subst) ``` paulson@15077 ` 2062` ```apply (rule_tac f = sin and g = "%x. x + k" in DERIV_chain2) ``` paulson@15077 ` 2063` ```apply (best intro!: DERIV_intros intro: DERIV_chain2)+ ``` paulson@15077 ` 2064` ```apply (simp (no_asm)) ``` paulson@15077 ` 2065` ```done ``` paulson@15077 ` 2066` paulson@15383 ` 2067` ```lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n" ``` paulson@15383 ` 2068` ```proof - ``` paulson@15383 ` 2069` ``` have "sin ((real n + 1/2) * pi) = cos (real n * pi)" ``` paulson@15383 ` 2070` ``` by (auto simp add: right_distrib sin_add left_distrib mult_ac) ``` paulson@15383 ` 2071` ``` thus ?thesis ``` paulson@15383 ` 2072` ``` by (simp add: real_of_nat_Suc left_distrib add_divide_distrib ``` paulson@15383 ` 2073` ``` mult_commute [of pi]) ``` paulson@15383 ` 2074` ```qed ``` paulson@15077 ` 2075` paulson@15077 ` 2076` ```lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1" ``` paulson@15077 ` 2077` ```by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2) ``` paulson@15077 ` 2078` paulson@15077 ` 2079` ```lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0" ``` huffman@23066 ` 2080` ```apply (subgoal_tac "cos (pi + pi/2) = 0", simp) ``` huffman@23066 ` 2081` ```apply (subst cos_add, simp) ``` paulson@15077 ` 2082` ```done ``` paulson@15077 ` 2083` paulson@15077 ` 2084` ```lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0" ``` paulson@15077 ` 2085` ```by (auto simp add: mult_assoc) ``` paulson@15077 ` 2086` paulson@15077 ` 2087` ```lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1" ``` huffman@23066 ` 2088` ```apply (subgoal_tac "sin (pi + pi/2) = - 1", simp) ``` huffman@23066 ` 2089` ```apply (subst sin_add, simp) ``` paulson@15077 ` 2090` ```done ``` paulson@15077 ` 2091` paulson@15077 ` 2092` ```(*NEEDED??*) ``` paulson@15229 ` 2093` ```lemma [simp]: ``` paulson@15229 ` 2094` ``` "cos(x + 1 / 2 * real(Suc m) * pi) = -sin (x + 1 / 2 * real m * pi)" ``` paulson@15077 ` 2095` ```apply (simp only: cos_add sin_add real_of_nat_Suc right_distrib left_distrib minus_mult_right, auto) ``` paulson@15077 ` 2096` ```done ``` paulson@15077 ` 2097` paulson@15077 ` 2098` ```(*NEEDED??*) ``` paulson@15077 ` 2099` ```lemma [simp]: "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)" ``` paulson@15229 ` 2100` ```by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto) ``` paulson@15077 ` 2101` paulson@15077 ` 2102` ```lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0" ``` paulson@15229 ` 2103` ```by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto) ``` paulson@15077 ` 2104` paulson@15077 ` 2105` ```lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)" ``` paulson@15077 ` 2106` ```apply (rule lemma_DERIV_subst) ``` paulson@15077 ` 2107` ```apply (rule_tac f = cos and g = "%x. x + k" in DERIV_chain2) ``` paulson@15077 ` 2108` ```apply (best intro!: DERIV_intros intro: DERIV_chain2)+ ``` paulson@15077 ` 2109` ```apply (simp (no_asm)) ``` paulson@15077 ` 2110` ```done ``` paulson@15077 ` 2111` paulson@15081 ` 2112` ```lemma sin_zero_abs_cos_one: "sin x = 0 ==> \cos x\ = 1" ``` nipkow@15539 ` 2113` ```by (auto simp add: sin_zero_iff even_mult_two_ex) ``` paulson@15077 ` 2114` huffman@23115 ` 2115` ```lemma exp_eq_one_iff [simp]: "(exp (x::real) = 1) = (x = 0)" ``` paulson@15077 ` 2116` ```apply auto ``` paulson@15077 ` 2117` ```apply (drule_tac f = ln in arg_cong, auto) ``` paulson@15077 ` 2118` ```done ``` paulson@15077 ` 2119` paulson@15077 ` 2120` ```lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0" ``` paulson@15077 ` 2121` ```by (cut_tac x = x in sin_cos_squared_add3, auto) ``` paulson@15077 ` 2122` paulson@15077 ` 2123` huffman@22978 ` 2124` ```subsection {* Existence of Polar Coordinates *} ``` paulson@15077 ` 2125` huffman@22978 ` 2126` ```lemma cos_x_y_le_one: "\x / sqrt (x\ + y\)\ \ 1" ``` huffman@22978 ` 2127` ```apply (rule power2_le_imp_le [OF _ zero_le_one]) ``` huffman@22978 ` 2128` ```apply (simp add: abs_divide power_divide divide_le_eq not_sum_power2_lt_zero) ``` paulson@15077 ` 2129` ```done ``` paulson@15077 ` 2130` huffman@22978 ` 2131` ```lemma cos_arccos_abs: "\y\ \ 1 \ cos (arccos y) = y" ``` huffman@22978 ` 2132` ```by (simp add: abs_le_iff) ``` paulson@15077 ` 2133` huffman@23045 ` 2134` ```lemma sin_arccos_abs: "\y\ \ 1 \ sin (arccos y) = sqrt (1 - y\)" ``` huffman@23045 ` 2135` ```by (simp add: sin_arccos abs_le_iff) ``` paulson@15077 ` 2136` huffman@22978 ` 2137` ```lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one] ``` paulson@15228 ` 2138` huffman@23045 ` 2139` ```lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one] ``` paulson@15077 ` 2140` paulson@15229 ` 2141` ```lemma polar_ex1: ``` huffman@22978 ` 2142` ``` "0 < y ==> \r a. x = r * cos a & y = r * sin a" ``` paulson@15229 ` 2143` ```apply (rule_tac x = "sqrt (x\ + y\)" in exI) ``` huffman@22978 ` 2144` ```apply (rule_tac x = "arccos (x / sqrt (x\ + y\))" in exI) ``` huffman@22978 ` 2145` ```apply (simp add: cos_arccos_lemma1) ``` huffman@23045 ` 2146` ```apply (simp add: sin_arccos_lemma1) ``` huffman@23045 ` 2147` ```apply (simp add: power_divide) ``` huffman@23045 ` 2148` ```apply (simp add: real_sqrt_mult [symmetric]) ``` huffman@23045 ` 2149` ```apply (simp add: right_diff_distrib) ``` paulson@15077 ` 2150` ```done ``` paulson@15077 ` 2151` paulson@15229 ` 2152` ```lemma polar_ex2: ``` huffman@22978 ` 2153` ``` "y < 0 ==> \r a. x = r * cos a & y = r * sin a" ``` huffman@22978 ` 2154` ```apply (insert polar_ex1 [where x=x and y="-y"], simp, clarify) ``` paulson@15077 ` 2155` ```apply (rule_tac x = r in exI) ``` huffman@22978 ` 2156` ```apply (rule_tac x = "-a" in exI, simp) ``` paulson@15077 ` 2157` ```done ``` paulson@15077 ` 2158` paulson@15077 ` 2159` ```lemma polar_Ex: "\r a. x = r * cos a & y = r * sin a" ``` huffman@22978 ` 2160` ```apply (rule_tac x=0 and y=y in linorder_cases) ``` huffman@22978 ` 2161` ```apply (erule polar_ex1) ``` huffman@22978 ` 2162` ```apply (rule_tac x=x in exI, rule_tac x=0 in exI, simp) ``` huffman@22978 ` 2163` ```apply (erule polar_ex2) ``` paulson@15077 ` 2164` ```done ``` paulson@15077 ` 2165` paulson@15077 ` 2166` huffman@23043 ` 2167` ```subsection {* Theorems about Limits *} ``` huffman@23043 ` 2168` paulson@15077 ` 2169` ```(* need to rename second isCont_inverse *) ``` paulson@15077 ` 2170` paulson@15229 ` 2171` ```lemma isCont_inv_fun: ``` huffman@20561 ` 2172` ``` fixes f g :: "real \ real" ``` huffman@20561 ` 2173` ``` shows "[| 0 < d; \z. \z - x\ \ d --> g(f(z)) = z; ``` paulson@15077 ` 2174` ``` \z. \z - x\ \ d --> isCont f z |] ``` paulson@15077 ` 2175` ``` ==> isCont g (f x)" ``` huffman@22722 ` 2176` ```by (rule isCont_inverse_function) ``` paulson@15077 ` 2177` paulson@15077 ` 2178` ```lemma isCont_inv_fun_inv: ``` huffman@20552 ` 2179` ``` fixes f g :: "real \ real" ``` huffman@20552 ` 2180` ``` shows "[| 0 < d; ``` paulson@15077 ` 2181` ``` \z. \z - x\ \ d --> g(f(z)) = z; ``` paulson@15077 ` 2182` ``` \z. \z - x\ \ d --> isCont f z |] ``` paulson@15077 ` 2183` ``` ==> \e. 0 < e & ``` paulson@15081 ` 2184` ``` (\y. 0 < \y - f(x)\ & \y - f(x)\ < e --> f(g(y)) = y)" ``` paulson@15077 ` 2185` ```apply (drule isCont_inj_range) ``` paulson@15077 ` 2186` ```prefer 2 apply (assumption, assumption, auto) ``` paulson@15077 ` 2187` ```apply (rule_tac x = e in exI, auto) ``` paulson@15077 ` 2188` ```apply (rotate_tac 2) ``` paulson@15077 ` 2189` ```apply (drule_tac x = y in spec, auto) ``` paulson@15077 ` 2190` ```done ``` paulson@15077 ` 2191` paulson@15077 ` 2192` paulson@15077 ` 2193` ```text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*} ``` paulson@15229 ` 2194` ```lemma LIM_fun_gt_zero: ``` huffman@20552 ` 2195` ``` "[| f -- c --> (l::real); 0 < l |] ``` huffman@20561 ` 2196` ``` ==> \r. 0 < r & (\x::real. x \ c & \c - x\ < r --> 0 < f x)" ``` paulson@15077 ` 2197` ```apply (auto simp add: LIM_def) ``` paulson@15077 ` 2198` ```apply (drule_tac x = "l/2" in spec, safe, force) ``` paulson@15077 ` 2199` ```apply (rule_tac x = s in exI) ``` huffman@22998 ` 2200` ```apply (auto simp only: abs_less_iff) ``` paulson@15077 ` 2201` ```done ``` paulson@15077 ` 2202` paulson@15229 ` 2203` ```lemma LIM_fun_less_zero: ``` huffman@20552 ` 2204` ``` "[| f -- c --> (l::real); l < 0 |] ``` huffman@20561 ` 2205` ``` ==> \r. 0 < r & (\x::real. x \ c & \c - x\ < r --> f x < 0)" ``` paulson@15077 ` 2206` ```apply (auto simp add: LIM_def) ``` paulson@15077 ` 2207` ```apply (drule_tac x = "-l/2" in spec, safe, force) ``` paulson@15077 ` 2208` ```apply (rule_tac x = s in exI) ``` huffman@22998 ` 2209` ```apply (auto simp only: abs_less_iff) ``` paulson@15077 ` 2210` ```done ``` paulson@15077 ` 2211` paulson@15077 ` 2212` paulson@15077 ` 2213` ```lemma LIM_fun_not_zero: ``` huffman@20552 ` 2214` ``` "[| f -- c --> (l::real); l \ 0 |] ``` huffman@20561 ` 2215` ``` ==> \r. 0 < r & (\x::real. x \ c & \c - x\ < r --> f x \ 0)" ``` paulson@15077 ` 2216` ```apply (cut_tac x = l and y = 0 in linorder_less_linear, auto) ``` paulson@15077 ` 2217` ```apply (drule LIM_fun_less_zero) ``` paulson@15241 ` 2218` ```apply (drule_tac [3] LIM_fun_gt_zero) ``` paulson@15241 ` 2219` ```apply force+ ``` paulson@15077 ` 2220` ```done ``` webertj@20432 ` 2221` ``` ``` paulson@12196 ` 2222` ```end ```