src/HOL/Limits.thy
 author huffman Fri Mar 30 12:32:35 2012 +0200 (2012-03-30) changeset 47220 52426c62b5d0 parent 46887 cb891d9a23c1 child 47432 e1576d13e933 permissions -rw-r--r--
replace lemmas eval_nat_numeral with a simpler reformulation
 huffman@31349 ` 1` ```(* Title : Limits.thy ``` huffman@31349 ` 2` ``` Author : Brian Huffman ``` huffman@31349 ` 3` ```*) ``` huffman@31349 ` 4` huffman@31349 ` 5` ```header {* Filters and Limits *} ``` huffman@31349 ` 6` huffman@31349 ` 7` ```theory Limits ``` huffman@36822 ` 8` ```imports RealVector ``` huffman@31349 ` 9` ```begin ``` huffman@31349 ` 10` huffman@44081 ` 11` ```subsection {* Filters *} ``` huffman@31392 ` 12` huffman@31392 ` 13` ```text {* ``` huffman@44081 ` 14` ``` This definition also allows non-proper filters. ``` huffman@31392 ` 15` ```*} ``` huffman@31392 ` 16` huffman@36358 ` 17` ```locale is_filter = ``` huffman@44081 ` 18` ``` fixes F :: "('a \ bool) \ bool" ``` huffman@44081 ` 19` ``` assumes True: "F (\x. True)" ``` huffman@44081 ` 20` ``` assumes conj: "F (\x. P x) \ F (\x. Q x) \ F (\x. P x \ Q x)" ``` huffman@44081 ` 21` ``` assumes mono: "\x. P x \ Q x \ F (\x. P x) \ F (\x. Q x)" ``` huffman@36358 ` 22` huffman@44081 ` 23` ```typedef (open) 'a filter = "{F :: ('a \ bool) \ bool. is_filter F}" ``` huffman@31392 ` 24` ```proof ``` huffman@44081 ` 25` ``` show "(\x. True) \ ?filter" by (auto intro: is_filter.intro) ``` huffman@31392 ` 26` ```qed ``` huffman@31349 ` 27` huffman@44195 ` 28` ```lemma is_filter_Rep_filter: "is_filter (Rep_filter F)" ``` huffman@44195 ` 29` ``` using Rep_filter [of F] by simp ``` huffman@31392 ` 30` huffman@44081 ` 31` ```lemma Abs_filter_inverse': ``` huffman@44081 ` 32` ``` assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F" ``` huffman@44081 ` 33` ``` using assms by (simp add: Abs_filter_inverse) ``` huffman@31392 ` 34` huffman@31392 ` 35` huffman@31392 ` 36` ```subsection {* Eventually *} ``` huffman@31349 ` 37` huffman@44081 ` 38` ```definition eventually :: "('a \ bool) \ 'a filter \ bool" ``` huffman@44195 ` 39` ``` where "eventually P F \ Rep_filter F P" ``` huffman@36358 ` 40` huffman@44081 ` 41` ```lemma eventually_Abs_filter: ``` huffman@44081 ` 42` ``` assumes "is_filter F" shows "eventually P (Abs_filter F) = F P" ``` huffman@44081 ` 43` ``` unfolding eventually_def using assms by (simp add: Abs_filter_inverse) ``` huffman@31349 ` 44` huffman@44081 ` 45` ```lemma filter_eq_iff: ``` huffman@44195 ` 46` ``` shows "F = F' \ (\P. eventually P F = eventually P F')" ``` huffman@44081 ` 47` ``` unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def .. ``` huffman@36360 ` 48` huffman@44195 ` 49` ```lemma eventually_True [simp]: "eventually (\x. True) F" ``` huffman@44081 ` 50` ``` unfolding eventually_def ``` huffman@44081 ` 51` ``` by (rule is_filter.True [OF is_filter_Rep_filter]) ``` huffman@31349 ` 52` huffman@44195 ` 53` ```lemma always_eventually: "\x. P x \ eventually P F" ``` huffman@36630 ` 54` ```proof - ``` huffman@36630 ` 55` ``` assume "\x. P x" hence "P = (\x. True)" by (simp add: ext) ``` huffman@44195 ` 56` ``` thus "eventually P F" by simp ``` huffman@36630 ` 57` ```qed ``` huffman@36630 ` 58` huffman@31349 ` 59` ```lemma eventually_mono: ``` huffman@44195 ` 60` ``` "(\x. P x \ Q x) \ eventually P F \ eventually Q F" ``` huffman@44081 ` 61` ``` unfolding eventually_def ``` huffman@44081 ` 62` ``` by (rule is_filter.mono [OF is_filter_Rep_filter]) ``` huffman@31349 ` 63` huffman@31349 ` 64` ```lemma eventually_conj: ``` huffman@44195 ` 65` ``` assumes P: "eventually (\x. P x) F" ``` huffman@44195 ` 66` ``` assumes Q: "eventually (\x. Q x) F" ``` huffman@44195 ` 67` ``` shows "eventually (\x. P x \ Q x) F" ``` huffman@44081 ` 68` ``` using assms unfolding eventually_def ``` huffman@44081 ` 69` ``` by (rule is_filter.conj [OF is_filter_Rep_filter]) ``` huffman@31349 ` 70` huffman@31349 ` 71` ```lemma eventually_mp: ``` huffman@44195 ` 72` ``` assumes "eventually (\x. P x \ Q x) F" ``` huffman@44195 ` 73` ``` assumes "eventually (\x. P x) F" ``` huffman@44195 ` 74` ``` shows "eventually (\x. Q x) F" ``` huffman@31349 ` 75` ```proof (rule eventually_mono) ``` huffman@31349 ` 76` ``` show "\x. (P x \ Q x) \ P x \ Q x" by simp ``` huffman@44195 ` 77` ``` show "eventually (\x. (P x \ Q x) \ P x) F" ``` huffman@31349 ` 78` ``` using assms by (rule eventually_conj) ``` huffman@31349 ` 79` ```qed ``` huffman@31349 ` 80` huffman@31349 ` 81` ```lemma eventually_rev_mp: ``` huffman@44195 ` 82` ``` assumes "eventually (\x. P x) F" ``` huffman@44195 ` 83` ``` assumes "eventually (\x. P x \ Q x) F" ``` huffman@44195 ` 84` ``` shows "eventually (\x. Q x) F" ``` huffman@31349 ` 85` ```using assms(2) assms(1) by (rule eventually_mp) ``` huffman@31349 ` 86` huffman@31349 ` 87` ```lemma eventually_conj_iff: ``` huffman@44195 ` 88` ``` "eventually (\x. P x \ Q x) F \ eventually P F \ eventually Q F" ``` huffman@44081 ` 89` ``` by (auto intro: eventually_conj elim: eventually_rev_mp) ``` huffman@31349 ` 90` huffman@31349 ` 91` ```lemma eventually_elim1: ``` huffman@44195 ` 92` ``` assumes "eventually (\i. P i) F" ``` huffman@31349 ` 93` ``` assumes "\i. P i \ Q i" ``` huffman@44195 ` 94` ``` shows "eventually (\i. Q i) F" ``` huffman@44081 ` 95` ``` using assms by (auto elim!: eventually_rev_mp) ``` huffman@31349 ` 96` huffman@31349 ` 97` ```lemma eventually_elim2: ``` huffman@44195 ` 98` ``` assumes "eventually (\i. P i) F" ``` huffman@44195 ` 99` ``` assumes "eventually (\i. Q i) F" ``` huffman@31349 ` 100` ``` assumes "\i. P i \ Q i \ R i" ``` huffman@44195 ` 101` ``` shows "eventually (\i. R i) F" ``` huffman@44081 ` 102` ``` using assms by (auto elim!: eventually_rev_mp) ``` huffman@31349 ` 103` noschinl@45892 ` 104` ```lemma eventually_subst: ``` noschinl@45892 ` 105` ``` assumes "eventually (\n. P n = Q n) F" ``` noschinl@45892 ` 106` ``` shows "eventually P F = eventually Q F" (is "?L = ?R") ``` noschinl@45892 ` 107` ```proof - ``` noschinl@45892 ` 108` ``` from assms have "eventually (\x. P x \ Q x) F" ``` noschinl@45892 ` 109` ``` and "eventually (\x. Q x \ P x) F" ``` noschinl@45892 ` 110` ``` by (auto elim: eventually_elim1) ``` noschinl@45892 ` 111` ``` then show ?thesis by (auto elim: eventually_elim2) ``` noschinl@45892 ` 112` ```qed ``` noschinl@45892 ` 113` noschinl@46886 ` 114` ```ML {* ``` noschinl@46886 ` 115` ``` fun ev_elim_tac ctxt thms thm = let ``` noschinl@46886 ` 116` ``` val thy = Proof_Context.theory_of ctxt ``` noschinl@46886 ` 117` ``` val mp_thms = thms RL [@{thm eventually_rev_mp}] ``` noschinl@46886 ` 118` ``` val raw_elim_thm = ``` noschinl@46886 ` 119` ``` (@{thm allI} RS @{thm always_eventually}) ``` noschinl@46886 ` 120` ``` |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms ``` noschinl@46886 ` 121` ``` |> fold (fn _ => fn thm => @{thm impI} RS thm) thms ``` noschinl@46886 ` 122` ``` val cases_prop = prop_of (raw_elim_thm RS thm) ``` noschinl@46886 ` 123` ``` val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])]) ``` noschinl@46886 ` 124` ``` in ``` noschinl@46886 ` 125` ``` CASES cases (rtac raw_elim_thm 1) thm ``` noschinl@46886 ` 126` ``` end ``` noschinl@46886 ` 127` noschinl@46886 ` 128` ``` fun eventually_elim_setup name = ``` noschinl@46886 ` 129` ``` Method.setup name (Scan.succeed (fn ctxt => METHOD_CASES (ev_elim_tac ctxt))) ``` noschinl@46886 ` 130` ``` "elimination of eventually quantifiers" ``` noschinl@46886 ` 131` ```*} ``` noschinl@46886 ` 132` noschinl@46886 ` 133` ```setup {* eventually_elim_setup @{binding "eventually_elim"} *} ``` noschinl@45892 ` 134` noschinl@45892 ` 135` huffman@36360 ` 136` ```subsection {* Finer-than relation *} ``` huffman@36360 ` 137` huffman@44195 ` 138` ```text {* @{term "F \ F'"} means that filter @{term F} is finer than ``` huffman@44195 ` 139` ```filter @{term F'}. *} ``` huffman@36360 ` 140` huffman@44081 ` 141` ```instantiation filter :: (type) complete_lattice ``` huffman@36360 ` 142` ```begin ``` huffman@36360 ` 143` huffman@44081 ` 144` ```definition le_filter_def: ``` huffman@44195 ` 145` ``` "F \ F' \ (\P. eventually P F' \ eventually P F)" ``` huffman@36360 ` 146` huffman@36360 ` 147` ```definition ``` huffman@44195 ` 148` ``` "(F :: 'a filter) < F' \ F \ F' \ \ F' \ F" ``` huffman@36360 ` 149` huffman@36360 ` 150` ```definition ``` huffman@44081 ` 151` ``` "top = Abs_filter (\P. \x. P x)" ``` huffman@36630 ` 152` huffman@36630 ` 153` ```definition ``` huffman@44081 ` 154` ``` "bot = Abs_filter (\P. True)" ``` huffman@36360 ` 155` huffman@36630 ` 156` ```definition ``` huffman@44195 ` 157` ``` "sup F F' = Abs_filter (\P. eventually P F \ eventually P F')" ``` huffman@36630 ` 158` huffman@36630 ` 159` ```definition ``` huffman@44195 ` 160` ``` "inf F F' = Abs_filter ``` huffman@44195 ` 161` ``` (\P. \Q R. eventually Q F \ eventually R F' \ (\x. Q x \ R x \ P x))" ``` huffman@36630 ` 162` huffman@36630 ` 163` ```definition ``` huffman@44195 ` 164` ``` "Sup S = Abs_filter (\P. \F\S. eventually P F)" ``` huffman@36630 ` 165` huffman@36630 ` 166` ```definition ``` huffman@44195 ` 167` ``` "Inf S = Sup {F::'a filter. \F'\S. F \ F'}" ``` huffman@36630 ` 168` huffman@36630 ` 169` ```lemma eventually_top [simp]: "eventually P top \ (\x. P x)" ``` huffman@44081 ` 170` ``` unfolding top_filter_def ``` huffman@44081 ` 171` ``` by (rule eventually_Abs_filter, rule is_filter.intro, auto) ``` huffman@36630 ` 172` huffman@36629 ` 173` ```lemma eventually_bot [simp]: "eventually P bot" ``` huffman@44081 ` 174` ``` unfolding bot_filter_def ``` huffman@44081 ` 175` ``` by (subst eventually_Abs_filter, rule is_filter.intro, auto) ``` huffman@36360 ` 176` huffman@36630 ` 177` ```lemma eventually_sup: ``` huffman@44195 ` 178` ``` "eventually P (sup F F') \ eventually P F \ eventually P F'" ``` huffman@44081 ` 179` ``` unfolding sup_filter_def ``` huffman@44081 ` 180` ``` by (rule eventually_Abs_filter, rule is_filter.intro) ``` huffman@44081 ` 181` ``` (auto elim!: eventually_rev_mp) ``` huffman@36630 ` 182` huffman@36630 ` 183` ```lemma eventually_inf: ``` huffman@44195 ` 184` ``` "eventually P (inf F F') \ ``` huffman@44195 ` 185` ``` (\Q R. eventually Q F \ eventually R F' \ (\x. Q x \ R x \ P x))" ``` huffman@44081 ` 186` ``` unfolding inf_filter_def ``` huffman@44081 ` 187` ``` apply (rule eventually_Abs_filter, rule is_filter.intro) ``` huffman@44081 ` 188` ``` apply (fast intro: eventually_True) ``` huffman@44081 ` 189` ``` apply clarify ``` huffman@44081 ` 190` ``` apply (intro exI conjI) ``` huffman@44081 ` 191` ``` apply (erule (1) eventually_conj) ``` huffman@44081 ` 192` ``` apply (erule (1) eventually_conj) ``` huffman@44081 ` 193` ``` apply simp ``` huffman@44081 ` 194` ``` apply auto ``` huffman@44081 ` 195` ``` done ``` huffman@36630 ` 196` huffman@36630 ` 197` ```lemma eventually_Sup: ``` huffman@44195 ` 198` ``` "eventually P (Sup S) \ (\F\S. eventually P F)" ``` huffman@44081 ` 199` ``` unfolding Sup_filter_def ``` huffman@44081 ` 200` ``` apply (rule eventually_Abs_filter, rule is_filter.intro) ``` huffman@44081 ` 201` ``` apply (auto intro: eventually_conj elim!: eventually_rev_mp) ``` huffman@44081 ` 202` ``` done ``` huffman@36630 ` 203` huffman@36360 ` 204` ```instance proof ``` huffman@44195 ` 205` ``` fix F F' F'' :: "'a filter" and S :: "'a filter set" ``` huffman@44195 ` 206` ``` { show "F < F' \ F \ F' \ \ F' \ F" ``` huffman@44195 ` 207` ``` by (rule less_filter_def) } ``` huffman@44195 ` 208` ``` { show "F \ F" ``` huffman@44195 ` 209` ``` unfolding le_filter_def by simp } ``` huffman@44195 ` 210` ``` { assume "F \ F'" and "F' \ F''" thus "F \ F''" ``` huffman@44195 ` 211` ``` unfolding le_filter_def by simp } ``` huffman@44195 ` 212` ``` { assume "F \ F'" and "F' \ F" thus "F = F'" ``` huffman@44195 ` 213` ``` unfolding le_filter_def filter_eq_iff by fast } ``` huffman@44195 ` 214` ``` { show "F \ top" ``` huffman@44195 ` 215` ``` unfolding le_filter_def eventually_top by (simp add: always_eventually) } ``` huffman@44195 ` 216` ``` { show "bot \ F" ``` huffman@44195 ` 217` ``` unfolding le_filter_def by simp } ``` huffman@44195 ` 218` ``` { show "F \ sup F F'" and "F' \ sup F F'" ``` huffman@44195 ` 219` ``` unfolding le_filter_def eventually_sup by simp_all } ``` huffman@44195 ` 220` ``` { assume "F \ F''" and "F' \ F''" thus "sup F F' \ F''" ``` huffman@44195 ` 221` ``` unfolding le_filter_def eventually_sup by simp } ``` huffman@44195 ` 222` ``` { show "inf F F' \ F" and "inf F F' \ F'" ``` huffman@44195 ` 223` ``` unfolding le_filter_def eventually_inf by (auto intro: eventually_True) } ``` huffman@44195 ` 224` ``` { assume "F \ F'" and "F \ F''" thus "F \ inf F' F''" ``` huffman@44081 ` 225` ``` unfolding le_filter_def eventually_inf ``` huffman@44195 ` 226` ``` by (auto elim!: eventually_mono intro: eventually_conj) } ``` huffman@44195 ` 227` ``` { assume "F \ S" thus "F \ Sup S" ``` huffman@44195 ` 228` ``` unfolding le_filter_def eventually_Sup by simp } ``` huffman@44195 ` 229` ``` { assume "\F. F \ S \ F \ F'" thus "Sup S \ F'" ``` huffman@44195 ` 230` ``` unfolding le_filter_def eventually_Sup by simp } ``` huffman@44195 ` 231` ``` { assume "F'' \ S" thus "Inf S \ F''" ``` huffman@44195 ` 232` ``` unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } ``` huffman@44195 ` 233` ``` { assume "\F'. F' \ S \ F \ F'" thus "F \ Inf S" ``` huffman@44195 ` 234` ``` unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } ``` huffman@36360 ` 235` ```qed ``` huffman@36360 ` 236` huffman@36360 ` 237` ```end ``` huffman@36360 ` 238` huffman@44081 ` 239` ```lemma filter_leD: ``` huffman@44195 ` 240` ``` "F \ F' \ eventually P F' \ eventually P F" ``` huffman@44081 ` 241` ``` unfolding le_filter_def by simp ``` huffman@36360 ` 242` huffman@44081 ` 243` ```lemma filter_leI: ``` huffman@44195 ` 244` ``` "(\P. eventually P F' \ eventually P F) \ F \ F'" ``` huffman@44081 ` 245` ``` unfolding le_filter_def by simp ``` huffman@36360 ` 246` huffman@36360 ` 247` ```lemma eventually_False: ``` huffman@44195 ` 248` ``` "eventually (\x. False) F \ F = bot" ``` huffman@44081 ` 249` ``` unfolding filter_eq_iff by (auto elim: eventually_rev_mp) ``` huffman@36360 ` 250` huffman@44342 ` 251` ```abbreviation (input) trivial_limit :: "'a filter \ bool" ``` huffman@44342 ` 252` ``` where "trivial_limit F \ F = bot" ``` huffman@44342 ` 253` huffman@44342 ` 254` ```lemma trivial_limit_def: "trivial_limit F \ eventually (\x. False) F" ``` huffman@44342 ` 255` ``` by (rule eventually_False [symmetric]) ``` huffman@44342 ` 256` huffman@44342 ` 257` huffman@44081 ` 258` ```subsection {* Map function for filters *} ``` huffman@36654 ` 259` huffman@44081 ` 260` ```definition filtermap :: "('a \ 'b) \ 'a filter \ 'b filter" ``` huffman@44195 ` 261` ``` where "filtermap f F = Abs_filter (\P. eventually (\x. P (f x)) F)" ``` huffman@36654 ` 262` huffman@44081 ` 263` ```lemma eventually_filtermap: ``` huffman@44195 ` 264` ``` "eventually P (filtermap f F) = eventually (\x. P (f x)) F" ``` huffman@44081 ` 265` ``` unfolding filtermap_def ``` huffman@44081 ` 266` ``` apply (rule eventually_Abs_filter) ``` huffman@44081 ` 267` ``` apply (rule is_filter.intro) ``` huffman@44081 ` 268` ``` apply (auto elim!: eventually_rev_mp) ``` huffman@44081 ` 269` ``` done ``` huffman@36654 ` 270` huffman@44195 ` 271` ```lemma filtermap_ident: "filtermap (\x. x) F = F" ``` huffman@44081 ` 272` ``` by (simp add: filter_eq_iff eventually_filtermap) ``` huffman@36654 ` 273` huffman@44081 ` 274` ```lemma filtermap_filtermap: ``` huffman@44195 ` 275` ``` "filtermap f (filtermap g F) = filtermap (\x. f (g x)) F" ``` huffman@44081 ` 276` ``` by (simp add: filter_eq_iff eventually_filtermap) ``` huffman@36654 ` 277` huffman@44195 ` 278` ```lemma filtermap_mono: "F \ F' \ filtermap f F \ filtermap f F'" ``` huffman@44081 ` 279` ``` unfolding le_filter_def eventually_filtermap by simp ``` huffman@44081 ` 280` huffman@44081 ` 281` ```lemma filtermap_bot [simp]: "filtermap f bot = bot" ``` huffman@44081 ` 282` ``` by (simp add: filter_eq_iff eventually_filtermap) ``` huffman@36654 ` 283` huffman@36654 ` 284` huffman@36662 ` 285` ```subsection {* Sequentially *} ``` huffman@31392 ` 286` huffman@44081 ` 287` ```definition sequentially :: "nat filter" ``` huffman@44081 ` 288` ``` where "sequentially = Abs_filter (\P. \k. \n\k. P n)" ``` huffman@31392 ` 289` huffman@36662 ` 290` ```lemma eventually_sequentially: ``` huffman@36662 ` 291` ``` "eventually P sequentially \ (\N. \n\N. P n)" ``` huffman@36662 ` 292` ```unfolding sequentially_def ``` huffman@44081 ` 293` ```proof (rule eventually_Abs_filter, rule is_filter.intro) ``` huffman@36662 ` 294` ``` fix P Q :: "nat \ bool" ``` huffman@36662 ` 295` ``` assume "\i. \n\i. P n" and "\j. \n\j. Q n" ``` huffman@36662 ` 296` ``` then obtain i j where "\n\i. P n" and "\n\j. Q n" by auto ``` huffman@36662 ` 297` ``` then have "\n\max i j. P n \ Q n" by simp ``` huffman@36662 ` 298` ``` then show "\k. \n\k. P n \ Q n" .. ``` huffman@36662 ` 299` ```qed auto ``` huffman@36662 ` 300` huffman@44342 ` 301` ```lemma sequentially_bot [simp, intro]: "sequentially \ bot" ``` huffman@44081 ` 302` ``` unfolding filter_eq_iff eventually_sequentially by auto ``` huffman@36662 ` 303` huffman@44342 ` 304` ```lemmas trivial_limit_sequentially = sequentially_bot ``` huffman@44342 ` 305` huffman@36662 ` 306` ```lemma eventually_False_sequentially [simp]: ``` huffman@36662 ` 307` ``` "\ eventually (\n. False) sequentially" ``` huffman@44081 ` 308` ``` by (simp add: eventually_False) ``` huffman@36662 ` 309` huffman@36662 ` 310` ```lemma le_sequentially: ``` huffman@44195 ` 311` ``` "F \ sequentially \ (\N. eventually (\n. N \ n) F)" ``` huffman@44081 ` 312` ``` unfolding le_filter_def eventually_sequentially ``` huffman@44081 ` 313` ``` by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp) ``` huffman@36662 ` 314` noschinl@45892 ` 315` ```lemma eventually_sequentiallyI: ``` noschinl@45892 ` 316` ``` assumes "\x. c \ x \ P x" ``` noschinl@45892 ` 317` ``` shows "eventually P sequentially" ``` noschinl@45892 ` 318` ```using assms by (auto simp: eventually_sequentially) ``` noschinl@45892 ` 319` huffman@36662 ` 320` huffman@44081 ` 321` ```subsection {* Standard filters *} ``` huffman@36662 ` 322` huffman@44081 ` 323` ```definition within :: "'a filter \ 'a set \ 'a filter" (infixr "within" 70) ``` huffman@44195 ` 324` ``` where "F within S = Abs_filter (\P. eventually (\x. x \ S \ P x) F)" ``` huffman@31392 ` 325` huffman@44206 ` 326` ```definition (in topological_space) nhds :: "'a \ 'a filter" ``` huffman@44081 ` 327` ``` where "nhds a = Abs_filter (\P. \S. open S \ a \ S \ (\x\S. P x))" ``` huffman@36654 ` 328` huffman@44206 ` 329` ```definition (in topological_space) at :: "'a \ 'a filter" ``` huffman@44081 ` 330` ``` where "at a = nhds a within - {a}" ``` huffman@31447 ` 331` huffman@31392 ` 332` ```lemma eventually_within: ``` huffman@44195 ` 333` ``` "eventually P (F within S) = eventually (\x. x \ S \ P x) F" ``` huffman@44081 ` 334` ``` unfolding within_def ``` huffman@44081 ` 335` ``` by (rule eventually_Abs_filter, rule is_filter.intro) ``` huffman@44081 ` 336` ``` (auto elim!: eventually_rev_mp) ``` huffman@31392 ` 337` huffman@45031 ` 338` ```lemma within_UNIV [simp]: "F within UNIV = F" ``` huffman@45031 ` 339` ``` unfolding filter_eq_iff eventually_within by simp ``` huffman@45031 ` 340` huffman@45031 ` 341` ```lemma within_empty [simp]: "F within {} = bot" ``` huffman@44081 ` 342` ``` unfolding filter_eq_iff eventually_within by simp ``` huffman@36360 ` 343` huffman@36654 ` 344` ```lemma eventually_nhds: ``` huffman@36654 ` 345` ``` "eventually P (nhds a) \ (\S. open S \ a \ S \ (\x\S. P x))" ``` huffman@36654 ` 346` ```unfolding nhds_def ``` huffman@44081 ` 347` ```proof (rule eventually_Abs_filter, rule is_filter.intro) ``` huffman@36654 ` 348` ``` have "open UNIV \ a \ UNIV \ (\x\UNIV. True)" by simp ``` huffman@36654 ` 349` ``` thus "\S. open S \ a \ S \ (\x\S. True)" by - rule ``` huffman@36358 ` 350` ```next ``` huffman@36358 ` 351` ``` fix P Q ``` huffman@36654 ` 352` ``` assume "\S. open S \ a \ S \ (\x\S. P x)" ``` huffman@36654 ` 353` ``` and "\T. open T \ a \ T \ (\x\T. Q x)" ``` huffman@36358 ` 354` ``` then obtain S T where ``` huffman@36654 ` 355` ``` "open S \ a \ S \ (\x\S. P x)" ``` huffman@36654 ` 356` ``` "open T \ a \ T \ (\x\T. Q x)" by auto ``` huffman@36654 ` 357` ``` hence "open (S \ T) \ a \ S \ T \ (\x\(S \ T). P x \ Q x)" ``` huffman@36358 ` 358` ``` by (simp add: open_Int) ``` huffman@36654 ` 359` ``` thus "\S. open S \ a \ S \ (\x\S. P x \ Q x)" by - rule ``` huffman@36358 ` 360` ```qed auto ``` huffman@31447 ` 361` huffman@36656 ` 362` ```lemma eventually_nhds_metric: ``` huffman@36656 ` 363` ``` "eventually P (nhds a) \ (\d>0. \x. dist x a < d \ P x)" ``` huffman@36656 ` 364` ```unfolding eventually_nhds open_dist ``` huffman@31447 ` 365` ```apply safe ``` huffman@31447 ` 366` ```apply fast ``` huffman@31492 ` 367` ```apply (rule_tac x="{x. dist x a < d}" in exI, simp) ``` huffman@31447 ` 368` ```apply clarsimp ``` huffman@31447 ` 369` ```apply (rule_tac x="d - dist x a" in exI, clarsimp) ``` huffman@31447 ` 370` ```apply (simp only: less_diff_eq) ``` huffman@31447 ` 371` ```apply (erule le_less_trans [OF dist_triangle]) ``` huffman@31447 ` 372` ```done ``` huffman@31447 ` 373` huffman@44571 ` 374` ```lemma nhds_neq_bot [simp]: "nhds a \ bot" ``` huffman@44571 ` 375` ``` unfolding trivial_limit_def eventually_nhds by simp ``` huffman@44571 ` 376` huffman@36656 ` 377` ```lemma eventually_at_topological: ``` huffman@36656 ` 378` ``` "eventually P (at a) \ (\S. open S \ a \ S \ (\x\S. x \ a \ P x))" ``` huffman@36656 ` 379` ```unfolding at_def eventually_within eventually_nhds by simp ``` huffman@36656 ` 380` huffman@36656 ` 381` ```lemma eventually_at: ``` huffman@36656 ` 382` ``` fixes a :: "'a::metric_space" ``` huffman@36656 ` 383` ``` shows "eventually P (at a) \ (\d>0. \x. x \ a \ dist x a < d \ P x)" ``` huffman@36656 ` 384` ```unfolding at_def eventually_within eventually_nhds_metric by auto ``` huffman@36656 ` 385` huffman@44571 ` 386` ```lemma at_eq_bot_iff: "at a = bot \ open {a}" ``` huffman@44571 ` 387` ``` unfolding trivial_limit_def eventually_at_topological ``` huffman@44571 ` 388` ``` by (safe, case_tac "S = {a}", simp, fast, fast) ``` huffman@44571 ` 389` huffman@44571 ` 390` ```lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \ bot" ``` huffman@44571 ` 391` ``` by (simp add: at_eq_bot_iff not_open_singleton) ``` huffman@44571 ` 392` huffman@31392 ` 393` huffman@31355 ` 394` ```subsection {* Boundedness *} ``` huffman@31355 ` 395` huffman@44081 ` 396` ```definition Bfun :: "('a \ 'b::real_normed_vector) \ 'a filter \ bool" ``` huffman@44195 ` 397` ``` where "Bfun f F = (\K>0. eventually (\x. norm (f x) \ K) F)" ``` huffman@31355 ` 398` huffman@31487 ` 399` ```lemma BfunI: ``` huffman@44195 ` 400` ``` assumes K: "eventually (\x. norm (f x) \ K) F" shows "Bfun f F" ``` huffman@31355 ` 401` ```unfolding Bfun_def ``` huffman@31355 ` 402` ```proof (intro exI conjI allI) ``` huffman@31355 ` 403` ``` show "0 < max K 1" by simp ``` huffman@31355 ` 404` ```next ``` huffman@44195 ` 405` ``` show "eventually (\x. norm (f x) \ max K 1) F" ``` huffman@31355 ` 406` ``` using K by (rule eventually_elim1, simp) ``` huffman@31355 ` 407` ```qed ``` huffman@31355 ` 408` huffman@31355 ` 409` ```lemma BfunE: ``` huffman@44195 ` 410` ``` assumes "Bfun f F" ``` huffman@44195 ` 411` ``` obtains B where "0 < B" and "eventually (\x. norm (f x) \ B) F" ``` huffman@31355 ` 412` ```using assms unfolding Bfun_def by fast ``` huffman@31355 ` 413` huffman@31355 ` 414` huffman@31349 ` 415` ```subsection {* Convergence to Zero *} ``` huffman@31349 ` 416` huffman@44081 ` 417` ```definition Zfun :: "('a \ 'b::real_normed_vector) \ 'a filter \ bool" ``` huffman@44195 ` 418` ``` where "Zfun f F = (\r>0. eventually (\x. norm (f x) < r) F)" ``` huffman@31349 ` 419` huffman@31349 ` 420` ```lemma ZfunI: ``` huffman@44195 ` 421` ``` "(\r. 0 < r \ eventually (\x. norm (f x) < r) F) \ Zfun f F" ``` huffman@44081 ` 422` ``` unfolding Zfun_def by simp ``` huffman@31349 ` 423` huffman@31349 ` 424` ```lemma ZfunD: ``` huffman@44195 ` 425` ``` "\Zfun f F; 0 < r\ \ eventually (\x. norm (f x) < r) F" ``` huffman@44081 ` 426` ``` unfolding Zfun_def by simp ``` huffman@31349 ` 427` huffman@31355 ` 428` ```lemma Zfun_ssubst: ``` huffman@44195 ` 429` ``` "eventually (\x. f x = g x) F \ Zfun g F \ Zfun f F" ``` huffman@44081 ` 430` ``` unfolding Zfun_def by (auto elim!: eventually_rev_mp) ``` huffman@31355 ` 431` huffman@44195 ` 432` ```lemma Zfun_zero: "Zfun (\x. 0) F" ``` huffman@44081 ` 433` ``` unfolding Zfun_def by simp ``` huffman@31349 ` 434` huffman@44195 ` 435` ```lemma Zfun_norm_iff: "Zfun (\x. norm (f x)) F = Zfun (\x. f x) F" ``` huffman@44081 ` 436` ``` unfolding Zfun_def by simp ``` huffman@31349 ` 437` huffman@31349 ` 438` ```lemma Zfun_imp_Zfun: ``` huffman@44195 ` 439` ``` assumes f: "Zfun f F" ``` huffman@44195 ` 440` ``` assumes g: "eventually (\x. norm (g x) \ norm (f x) * K) F" ``` huffman@44195 ` 441` ``` shows "Zfun (\x. g x) F" ``` huffman@31349 ` 442` ```proof (cases) ``` huffman@31349 ` 443` ``` assume K: "0 < K" ``` huffman@31349 ` 444` ``` show ?thesis ``` huffman@31349 ` 445` ``` proof (rule ZfunI) ``` huffman@31349 ` 446` ``` fix r::real assume "0 < r" ``` huffman@31349 ` 447` ``` hence "0 < r / K" ``` huffman@31349 ` 448` ``` using K by (rule divide_pos_pos) ``` huffman@44195 ` 449` ``` then have "eventually (\x. norm (f x) < r / K) F" ``` huffman@31487 ` 450` ``` using ZfunD [OF f] by fast ``` huffman@44195 ` 451` ``` with g show "eventually (\x. norm (g x) < r) F" ``` noschinl@46887 ` 452` ``` proof eventually_elim ``` noschinl@46887 ` 453` ``` case (elim x) ``` huffman@31487 ` 454` ``` hence "norm (f x) * K < r" ``` huffman@31349 ` 455` ``` by (simp add: pos_less_divide_eq K) ``` noschinl@46887 ` 456` ``` thus ?case ``` noschinl@46887 ` 457` ``` by (simp add: order_le_less_trans [OF elim(1)]) ``` huffman@31349 ` 458` ``` qed ``` huffman@31349 ` 459` ``` qed ``` huffman@31349 ` 460` ```next ``` huffman@31349 ` 461` ``` assume "\ 0 < K" ``` huffman@31349 ` 462` ``` hence K: "K \ 0" by (simp only: not_less) ``` huffman@31355 ` 463` ``` show ?thesis ``` huffman@31355 ` 464` ``` proof (rule ZfunI) ``` huffman@31355 ` 465` ``` fix r :: real ``` huffman@31355 ` 466` ``` assume "0 < r" ``` huffman@44195 ` 467` ``` from g show "eventually (\x. norm (g x) < r) F" ``` noschinl@46887 ` 468` ``` proof eventually_elim ``` noschinl@46887 ` 469` ``` case (elim x) ``` noschinl@46887 ` 470` ``` also have "norm (f x) * K \ norm (f x) * 0" ``` huffman@31355 ` 471` ``` using K norm_ge_zero by (rule mult_left_mono) ``` noschinl@46887 ` 472` ``` finally show ?case ``` huffman@31355 ` 473` ``` using `0 < r` by simp ``` huffman@31355 ` 474` ``` qed ``` huffman@31355 ` 475` ``` qed ``` huffman@31349 ` 476` ```qed ``` huffman@31349 ` 477` huffman@44195 ` 478` ```lemma Zfun_le: "\Zfun g F; \x. norm (f x) \ norm (g x)\ \ Zfun f F" ``` huffman@44081 ` 479` ``` by (erule_tac K="1" in Zfun_imp_Zfun, simp) ``` huffman@31349 ` 480` huffman@31349 ` 481` ```lemma Zfun_add: ``` huffman@44195 ` 482` ``` assumes f: "Zfun f F" and g: "Zfun g F" ``` huffman@44195 ` 483` ``` shows "Zfun (\x. f x + g x) F" ``` huffman@31349 ` 484` ```proof (rule ZfunI) ``` huffman@31349 ` 485` ``` fix r::real assume "0 < r" ``` huffman@31349 ` 486` ``` hence r: "0 < r / 2" by simp ``` huffman@44195 ` 487` ``` have "eventually (\x. norm (f x) < r/2) F" ``` huffman@31487 ` 488` ``` using f r by (rule ZfunD) ``` huffman@31349 ` 489` ``` moreover ``` huffman@44195 ` 490` ``` have "eventually (\x. norm (g x) < r/2) F" ``` huffman@31487 ` 491` ``` using g r by (rule ZfunD) ``` huffman@31349 ` 492` ``` ultimately ``` huffman@44195 ` 493` ``` show "eventually (\x. norm (f x + g x) < r) F" ``` noschinl@46887 ` 494` ``` proof eventually_elim ``` noschinl@46887 ` 495` ``` case (elim x) ``` huffman@31487 ` 496` ``` have "norm (f x + g x) \ norm (f x) + norm (g x)" ``` huffman@31349 ` 497` ``` by (rule norm_triangle_ineq) ``` huffman@31349 ` 498` ``` also have "\ < r/2 + r/2" ``` noschinl@46887 ` 499` ``` using elim by (rule add_strict_mono) ``` noschinl@46887 ` 500` ``` finally show ?case ``` huffman@31349 ` 501` ``` by simp ``` huffman@31349 ` 502` ``` qed ``` huffman@31349 ` 503` ```qed ``` huffman@31349 ` 504` huffman@44195 ` 505` ```lemma Zfun_minus: "Zfun f F \ Zfun (\x. - f x) F" ``` huffman@44081 ` 506` ``` unfolding Zfun_def by simp ``` huffman@31349 ` 507` huffman@44195 ` 508` ```lemma Zfun_diff: "\Zfun f F; Zfun g F\ \ Zfun (\x. f x - g x) F" ``` huffman@44081 ` 509` ``` by (simp only: diff_minus Zfun_add Zfun_minus) ``` huffman@31349 ` 510` huffman@31349 ` 511` ```lemma (in bounded_linear) Zfun: ``` huffman@44195 ` 512` ``` assumes g: "Zfun g F" ``` huffman@44195 ` 513` ``` shows "Zfun (\x. f (g x)) F" ``` huffman@31349 ` 514` ```proof - ``` huffman@31349 ` 515` ``` obtain K where "\x. norm (f x) \ norm x * K" ``` huffman@31349 ` 516` ``` using bounded by fast ``` huffman@44195 ` 517` ``` then have "eventually (\x. norm (f (g x)) \ norm (g x) * K) F" ``` huffman@31355 ` 518` ``` by simp ``` huffman@31487 ` 519` ``` with g show ?thesis ``` huffman@31349 ` 520` ``` by (rule Zfun_imp_Zfun) ``` huffman@31349 ` 521` ```qed ``` huffman@31349 ` 522` huffman@31349 ` 523` ```lemma (in bounded_bilinear) Zfun: ``` huffman@44195 ` 524` ``` assumes f: "Zfun f F" ``` huffman@44195 ` 525` ``` assumes g: "Zfun g F" ``` huffman@44195 ` 526` ``` shows "Zfun (\x. f x ** g x) F" ``` huffman@31349 ` 527` ```proof (rule ZfunI) ``` huffman@31349 ` 528` ``` fix r::real assume r: "0 < r" ``` huffman@31349 ` 529` ``` obtain K where K: "0 < K" ``` huffman@31349 ` 530` ``` and norm_le: "\x y. norm (x ** y) \ norm x * norm y * K" ``` huffman@31349 ` 531` ``` using pos_bounded by fast ``` huffman@31349 ` 532` ``` from K have K': "0 < inverse K" ``` huffman@31349 ` 533` ``` by (rule positive_imp_inverse_positive) ``` huffman@44195 ` 534` ``` have "eventually (\x. norm (f x) < r) F" ``` huffman@31487 ` 535` ``` using f r by (rule ZfunD) ``` huffman@31349 ` 536` ``` moreover ``` huffman@44195 ` 537` ``` have "eventually (\x. norm (g x) < inverse K) F" ``` huffman@31487 ` 538` ``` using g K' by (rule ZfunD) ``` huffman@31349 ` 539` ``` ultimately ``` huffman@44195 ` 540` ``` show "eventually (\x. norm (f x ** g x) < r) F" ``` noschinl@46887 ` 541` ``` proof eventually_elim ``` noschinl@46887 ` 542` ``` case (elim x) ``` huffman@31487 ` 543` ``` have "norm (f x ** g x) \ norm (f x) * norm (g x) * K" ``` huffman@31349 ` 544` ``` by (rule norm_le) ``` huffman@31487 ` 545` ``` also have "norm (f x) * norm (g x) * K < r * inverse K * K" ``` noschinl@46887 ` 546` ``` by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K) ``` huffman@31349 ` 547` ``` also from K have "r * inverse K * K = r" ``` huffman@31349 ` 548` ``` by simp ``` noschinl@46887 ` 549` ``` finally show ?case . ``` huffman@31349 ` 550` ``` qed ``` huffman@31349 ` 551` ```qed ``` huffman@31349 ` 552` huffman@31349 ` 553` ```lemma (in bounded_bilinear) Zfun_left: ``` huffman@44195 ` 554` ``` "Zfun f F \ Zfun (\x. f x ** a) F" ``` huffman@44081 ` 555` ``` by (rule bounded_linear_left [THEN bounded_linear.Zfun]) ``` huffman@31349 ` 556` huffman@31349 ` 557` ```lemma (in bounded_bilinear) Zfun_right: ``` huffman@44195 ` 558` ``` "Zfun f F \ Zfun (\x. a ** f x) F" ``` huffman@44081 ` 559` ``` by (rule bounded_linear_right [THEN bounded_linear.Zfun]) ``` huffman@31349 ` 560` huffman@44282 ` 561` ```lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult] ``` huffman@44282 ` 562` ```lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult] ``` huffman@44282 ` 563` ```lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult] ``` huffman@31349 ` 564` huffman@31349 ` 565` wenzelm@31902 ` 566` ```subsection {* Limits *} ``` huffman@31349 ` 567` huffman@44206 ` 568` ```definition (in topological_space) ``` huffman@44206 ` 569` ``` tendsto :: "('b \ 'a) \ 'a \ 'b filter \ bool" (infixr "--->" 55) where ``` huffman@44195 ` 570` ``` "(f ---> l) F \ (\S. open S \ l \ S \ eventually (\x. f x \ S) F)" ``` huffman@31349 ` 571` noschinl@45892 ` 572` ```definition real_tendsto_inf :: "('a \ real) \ 'a filter \ bool" where ``` noschinl@45892 ` 573` ``` "real_tendsto_inf f F \ \x. eventually (\y. x < f y) F" ``` noschinl@45892 ` 574` wenzelm@31902 ` 575` ```ML {* ``` wenzelm@31902 ` 576` ```structure Tendsto_Intros = Named_Thms ``` wenzelm@31902 ` 577` ```( ``` wenzelm@45294 ` 578` ``` val name = @{binding tendsto_intros} ``` wenzelm@31902 ` 579` ``` val description = "introduction rules for tendsto" ``` wenzelm@31902 ` 580` ```) ``` huffman@31565 ` 581` ```*} ``` huffman@31565 ` 582` wenzelm@31902 ` 583` ```setup Tendsto_Intros.setup ``` huffman@31565 ` 584` huffman@44195 ` 585` ```lemma tendsto_mono: "F \ F' \ (f ---> l) F' \ (f ---> l) F" ``` huffman@44081 ` 586` ``` unfolding tendsto_def le_filter_def by fast ``` huffman@36656 ` 587` huffman@31488 ` 588` ```lemma topological_tendstoI: ``` huffman@44195 ` 589` ``` "(\S. open S \ l \ S \ eventually (\x. f x \ S) F) ``` huffman@44195 ` 590` ``` \ (f ---> l) F" ``` huffman@31349 ` 591` ``` unfolding tendsto_def by auto ``` huffman@31349 ` 592` huffman@31488 ` 593` ```lemma topological_tendstoD: ``` huffman@44195 ` 594` ``` "(f ---> l) F \ open S \ l \ S \ eventually (\x. f x \ S) F" ``` huffman@31488 ` 595` ``` unfolding tendsto_def by auto ``` huffman@31488 ` 596` huffman@31488 ` 597` ```lemma tendstoI: ``` huffman@44195 ` 598` ``` assumes "\e. 0 < e \ eventually (\x. dist (f x) l < e) F" ``` huffman@44195 ` 599` ``` shows "(f ---> l) F" ``` huffman@44081 ` 600` ``` apply (rule topological_tendstoI) ``` huffman@44081 ` 601` ``` apply (simp add: open_dist) ``` huffman@44081 ` 602` ``` apply (drule (1) bspec, clarify) ``` huffman@44081 ` 603` ``` apply (drule assms) ``` huffman@44081 ` 604` ``` apply (erule eventually_elim1, simp) ``` huffman@44081 ` 605` ``` done ``` huffman@31488 ` 606` huffman@31349 ` 607` ```lemma tendstoD: ``` huffman@44195 ` 608` ``` "(f ---> l) F \ 0 < e \ eventually (\x. dist (f x) l < e) F" ``` huffman@44081 ` 609` ``` apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD) ``` huffman@44081 ` 610` ``` apply (clarsimp simp add: open_dist) ``` huffman@44081 ` 611` ``` apply (rule_tac x="e - dist x l" in exI, clarsimp) ``` huffman@44081 ` 612` ``` apply (simp only: less_diff_eq) ``` huffman@44081 ` 613` ``` apply (erule le_less_trans [OF dist_triangle]) ``` huffman@44081 ` 614` ``` apply simp ``` huffman@44081 ` 615` ``` apply simp ``` huffman@44081 ` 616` ``` done ``` huffman@31488 ` 617` huffman@31488 ` 618` ```lemma tendsto_iff: ``` huffman@44195 ` 619` ``` "(f ---> l) F \ (\e>0. eventually (\x. dist (f x) l < e) F)" ``` huffman@44081 ` 620` ``` using tendstoI tendstoD by fast ``` huffman@31349 ` 621` huffman@44195 ` 622` ```lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\x. f x - a) F" ``` huffman@44081 ` 623` ``` by (simp only: tendsto_iff Zfun_def dist_norm) ``` huffman@31349 ` 624` huffman@45031 ` 625` ```lemma tendsto_bot [simp]: "(f ---> a) bot" ``` huffman@45031 ` 626` ``` unfolding tendsto_def by simp ``` huffman@45031 ` 627` huffman@31565 ` 628` ```lemma tendsto_ident_at [tendsto_intros]: "((\x. x) ---> a) (at a)" ``` huffman@44081 ` 629` ``` unfolding tendsto_def eventually_at_topological by auto ``` huffman@31565 ` 630` huffman@31565 ` 631` ```lemma tendsto_ident_at_within [tendsto_intros]: ``` huffman@36655 ` 632` ``` "((\x. x) ---> a) (at a within S)" ``` huffman@44081 ` 633` ``` unfolding tendsto_def eventually_within eventually_at_topological by auto ``` huffman@31565 ` 634` huffman@44195 ` 635` ```lemma tendsto_const [tendsto_intros]: "((\x. k) ---> k) F" ``` huffman@44081 ` 636` ``` by (simp add: tendsto_def) ``` huffman@31349 ` 637` huffman@44205 ` 638` ```lemma tendsto_unique: ``` huffman@44205 ` 639` ``` fixes f :: "'a \ 'b::t2_space" ``` huffman@44205 ` 640` ``` assumes "\ trivial_limit F" and "(f ---> a) F" and "(f ---> b) F" ``` huffman@44205 ` 641` ``` shows "a = b" ``` huffman@44205 ` 642` ```proof (rule ccontr) ``` huffman@44205 ` 643` ``` assume "a \ b" ``` huffman@44205 ` 644` ``` obtain U V where "open U" "open V" "a \ U" "b \ V" "U \ V = {}" ``` huffman@44205 ` 645` ``` using hausdorff [OF `a \ b`] by fast ``` huffman@44205 ` 646` ``` have "eventually (\x. f x \ U) F" ``` huffman@44205 ` 647` ``` using `(f ---> a) F` `open U` `a \ U` by (rule topological_tendstoD) ``` huffman@44205 ` 648` ``` moreover ``` huffman@44205 ` 649` ``` have "eventually (\x. f x \ V) F" ``` huffman@44205 ` 650` ``` using `(f ---> b) F` `open V` `b \ V` by (rule topological_tendstoD) ``` huffman@44205 ` 651` ``` ultimately ``` huffman@44205 ` 652` ``` have "eventually (\x. False) F" ``` noschinl@46887 ` 653` ``` proof eventually_elim ``` noschinl@46887 ` 654` ``` case (elim x) ``` huffman@44205 ` 655` ``` hence "f x \ U \ V" by simp ``` noschinl@46887 ` 656` ``` with `U \ V = {}` show ?case by simp ``` huffman@44205 ` 657` ``` qed ``` huffman@44205 ` 658` ``` with `\ trivial_limit F` show "False" ``` huffman@44205 ` 659` ``` by (simp add: trivial_limit_def) ``` huffman@44205 ` 660` ```qed ``` huffman@44205 ` 661` huffman@36662 ` 662` ```lemma tendsto_const_iff: ``` huffman@44205 ` 663` ``` fixes a b :: "'a::t2_space" ``` huffman@44205 ` 664` ``` assumes "\ trivial_limit F" shows "((\x. a) ---> b) F \ a = b" ``` huffman@44205 ` 665` ``` by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const]) ``` huffman@44205 ` 666` huffman@44218 ` 667` ```lemma tendsto_compose: ``` huffman@44218 ` 668` ``` assumes g: "(g ---> g l) (at l)" ``` huffman@44218 ` 669` ``` assumes f: "(f ---> l) F" ``` huffman@44218 ` 670` ``` shows "((\x. g (f x)) ---> g l) F" ``` huffman@44218 ` 671` ```proof (rule topological_tendstoI) ``` huffman@44218 ` 672` ``` fix B assume B: "open B" "g l \ B" ``` huffman@44218 ` 673` ``` obtain A where A: "open A" "l \ A" ``` huffman@44218 ` 674` ``` and gB: "\y. y \ A \ g y \ B" ``` huffman@44218 ` 675` ``` using topological_tendstoD [OF g B] B(2) ``` huffman@44218 ` 676` ``` unfolding eventually_at_topological by fast ``` huffman@44218 ` 677` ``` hence "\x. f x \ A \ g (f x) \ B" by simp ``` huffman@44218 ` 678` ``` from this topological_tendstoD [OF f A] ``` huffman@44218 ` 679` ``` show "eventually (\x. g (f x) \ B) F" ``` huffman@44218 ` 680` ``` by (rule eventually_mono) ``` huffman@44218 ` 681` ```qed ``` huffman@44218 ` 682` huffman@44253 ` 683` ```lemma tendsto_compose_eventually: ``` huffman@44253 ` 684` ``` assumes g: "(g ---> m) (at l)" ``` huffman@44253 ` 685` ``` assumes f: "(f ---> l) F" ``` huffman@44253 ` 686` ``` assumes inj: "eventually (\x. f x \ l) F" ``` huffman@44253 ` 687` ``` shows "((\x. g (f x)) ---> m) F" ``` huffman@44253 ` 688` ```proof (rule topological_tendstoI) ``` huffman@44253 ` 689` ``` fix B assume B: "open B" "m \ B" ``` huffman@44253 ` 690` ``` obtain A where A: "open A" "l \ A" ``` huffman@44253 ` 691` ``` and gB: "\y. y \ A \ y \ l \ g y \ B" ``` huffman@44253 ` 692` ``` using topological_tendstoD [OF g B] ``` huffman@44253 ` 693` ``` unfolding eventually_at_topological by fast ``` huffman@44253 ` 694` ``` show "eventually (\x. g (f x) \ B) F" ``` huffman@44253 ` 695` ``` using topological_tendstoD [OF f A] inj ``` huffman@44253 ` 696` ``` by (rule eventually_elim2) (simp add: gB) ``` huffman@44253 ` 697` ```qed ``` huffman@44253 ` 698` huffman@44251 ` 699` ```lemma metric_tendsto_imp_tendsto: ``` huffman@44251 ` 700` ``` assumes f: "(f ---> a) F" ``` huffman@44251 ` 701` ``` assumes le: "eventually (\x. dist (g x) b \ dist (f x) a) F" ``` huffman@44251 ` 702` ``` shows "(g ---> b) F" ``` huffman@44251 ` 703` ```proof (rule tendstoI) ``` huffman@44251 ` 704` ``` fix e :: real assume "0 < e" ``` huffman@44251 ` 705` ``` with f have "eventually (\x. dist (f x) a < e) F" by (rule tendstoD) ``` huffman@44251 ` 706` ``` with le show "eventually (\x. dist (g x) b < e) F" ``` huffman@44251 ` 707` ``` using le_less_trans by (rule eventually_elim2) ``` huffman@44251 ` 708` ```qed ``` huffman@44251 ` 709` noschinl@45892 ` 710` ```lemma real_tendsto_inf_real: "real_tendsto_inf real sequentially" ``` noschinl@45892 ` 711` ```proof (unfold real_tendsto_inf_def, rule allI) ``` noschinl@45892 ` 712` ``` fix x show "eventually (\y. x < real y) sequentially" ``` noschinl@45892 ` 713` ``` by (rule eventually_sequentiallyI[of "natceiling (x + 1)"]) ``` noschinl@45892 ` 714` ``` (simp add: natceiling_le_eq) ``` noschinl@45892 ` 715` ```qed ``` noschinl@45892 ` 716` noschinl@45892 ` 717` noschinl@45892 ` 718` huffman@44205 ` 719` ```subsubsection {* Distance and norms *} ``` huffman@36662 ` 720` huffman@31565 ` 721` ```lemma tendsto_dist [tendsto_intros]: ``` huffman@44195 ` 722` ``` assumes f: "(f ---> l) F" and g: "(g ---> m) F" ``` huffman@44195 ` 723` ``` shows "((\x. dist (f x) (g x)) ---> dist l m) F" ``` huffman@31565 ` 724` ```proof (rule tendstoI) ``` huffman@31565 ` 725` ``` fix e :: real assume "0 < e" ``` huffman@31565 ` 726` ``` hence e2: "0 < e/2" by simp ``` huffman@31565 ` 727` ``` from tendstoD [OF f e2] tendstoD [OF g e2] ``` huffman@44195 ` 728` ``` show "eventually (\x. dist (dist (f x) (g x)) (dist l m) < e) F" ``` noschinl@46887 ` 729` ``` proof (eventually_elim) ``` noschinl@46887 ` 730` ``` case (elim x) ``` huffman@31565 ` 731` ``` then show "dist (dist (f x) (g x)) (dist l m) < e" ``` huffman@31565 ` 732` ``` unfolding dist_real_def ``` huffman@31565 ` 733` ``` using dist_triangle2 [of "f x" "g x" "l"] ``` huffman@31565 ` 734` ``` using dist_triangle2 [of "g x" "l" "m"] ``` huffman@31565 ` 735` ``` using dist_triangle3 [of "l" "m" "f x"] ``` huffman@31565 ` 736` ``` using dist_triangle [of "f x" "m" "g x"] ``` huffman@31565 ` 737` ``` by arith ``` huffman@31565 ` 738` ``` qed ``` huffman@31565 ` 739` ```qed ``` huffman@31565 ` 740` huffman@36662 ` 741` ```lemma norm_conv_dist: "norm x = dist x 0" ``` huffman@44081 ` 742` ``` unfolding dist_norm by simp ``` huffman@36662 ` 743` huffman@31565 ` 744` ```lemma tendsto_norm [tendsto_intros]: ``` huffman@44195 ` 745` ``` "(f ---> a) F \ ((\x. norm (f x)) ---> norm a) F" ``` huffman@44081 ` 746` ``` unfolding norm_conv_dist by (intro tendsto_intros) ``` huffman@36662 ` 747` huffman@36662 ` 748` ```lemma tendsto_norm_zero: ``` huffman@44195 ` 749` ``` "(f ---> 0) F \ ((\x. norm (f x)) ---> 0) F" ``` huffman@44081 ` 750` ``` by (drule tendsto_norm, simp) ``` huffman@36662 ` 751` huffman@36662 ` 752` ```lemma tendsto_norm_zero_cancel: ``` huffman@44195 ` 753` ``` "((\x. norm (f x)) ---> 0) F \ (f ---> 0) F" ``` huffman@44081 ` 754` ``` unfolding tendsto_iff dist_norm by simp ``` huffman@36662 ` 755` huffman@36662 ` 756` ```lemma tendsto_norm_zero_iff: ``` huffman@44195 ` 757` ``` "((\x. norm (f x)) ---> 0) F \ (f ---> 0) F" ``` huffman@44081 ` 758` ``` unfolding tendsto_iff dist_norm by simp ``` huffman@31349 ` 759` huffman@44194 ` 760` ```lemma tendsto_rabs [tendsto_intros]: ``` huffman@44195 ` 761` ``` "(f ---> (l::real)) F \ ((\x. \f x\) ---> \l\) F" ``` huffman@44194 ` 762` ``` by (fold real_norm_def, rule tendsto_norm) ``` huffman@44194 ` 763` huffman@44194 ` 764` ```lemma tendsto_rabs_zero: ``` huffman@44195 ` 765` ``` "(f ---> (0::real)) F \ ((\x. \f x\) ---> 0) F" ``` huffman@44194 ` 766` ``` by (fold real_norm_def, rule tendsto_norm_zero) ``` huffman@44194 ` 767` huffman@44194 ` 768` ```lemma tendsto_rabs_zero_cancel: ``` huffman@44195 ` 769` ``` "((\x. \f x\) ---> (0::real)) F \ (f ---> 0) F" ``` huffman@44194 ` 770` ``` by (fold real_norm_def, rule tendsto_norm_zero_cancel) ``` huffman@44194 ` 771` huffman@44194 ` 772` ```lemma tendsto_rabs_zero_iff: ``` huffman@44195 ` 773` ``` "((\x. \f x\) ---> (0::real)) F \ (f ---> 0) F" ``` huffman@44194 ` 774` ``` by (fold real_norm_def, rule tendsto_norm_zero_iff) ``` huffman@44194 ` 775` huffman@44194 ` 776` ```subsubsection {* Addition and subtraction *} ``` huffman@44194 ` 777` huffman@31565 ` 778` ```lemma tendsto_add [tendsto_intros]: ``` huffman@31349 ` 779` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@44195 ` 780` ``` shows "\(f ---> a) F; (g ---> b) F\ \ ((\x. f x + g x) ---> a + b) F" ``` huffman@44081 ` 781` ``` by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add) ``` huffman@31349 ` 782` huffman@44194 ` 783` ```lemma tendsto_add_zero: ``` huffman@44194 ` 784` ``` fixes f g :: "'a::type \ 'b::real_normed_vector" ``` huffman@44195 ` 785` ``` shows "\(f ---> 0) F; (g ---> 0) F\ \ ((\x. f x + g x) ---> 0) F" ``` huffman@44194 ` 786` ``` by (drule (1) tendsto_add, simp) ``` huffman@44194 ` 787` huffman@31565 ` 788` ```lemma tendsto_minus [tendsto_intros]: ``` huffman@31349 ` 789` ``` fixes a :: "'a::real_normed_vector" ``` huffman@44195 ` 790` ``` shows "(f ---> a) F \ ((\x. - f x) ---> - a) F" ``` huffman@44081 ` 791` ``` by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus) ``` huffman@31349 ` 792` huffman@31349 ` 793` ```lemma tendsto_minus_cancel: ``` huffman@31349 ` 794` ``` fixes a :: "'a::real_normed_vector" ``` huffman@44195 ` 795` ``` shows "((\x. - f x) ---> - a) F \ (f ---> a) F" ``` huffman@44081 ` 796` ``` by (drule tendsto_minus, simp) ``` huffman@31349 ` 797` huffman@31565 ` 798` ```lemma tendsto_diff [tendsto_intros]: ``` huffman@31349 ` 799` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@44195 ` 800` ``` shows "\(f ---> a) F; (g ---> b) F\ \ ((\x. f x - g x) ---> a - b) F" ``` huffman@44081 ` 801` ``` by (simp add: diff_minus tendsto_add tendsto_minus) ``` huffman@31349 ` 802` huffman@31588 ` 803` ```lemma tendsto_setsum [tendsto_intros]: ``` huffman@31588 ` 804` ``` fixes f :: "'a \ 'b \ 'c::real_normed_vector" ``` huffman@44195 ` 805` ``` assumes "\i. i \ S \ (f i ---> a i) F" ``` huffman@44195 ` 806` ``` shows "((\x. \i\S. f i x) ---> (\i\S. a i)) F" ``` huffman@31588 ` 807` ```proof (cases "finite S") ``` huffman@31588 ` 808` ``` assume "finite S" thus ?thesis using assms ``` huffman@44194 ` 809` ``` by (induct, simp add: tendsto_const, simp add: tendsto_add) ``` huffman@31588 ` 810` ```next ``` huffman@31588 ` 811` ``` assume "\ finite S" thus ?thesis ``` huffman@31588 ` 812` ``` by (simp add: tendsto_const) ``` huffman@31588 ` 813` ```qed ``` huffman@31588 ` 814` noschinl@45892 ` 815` ```lemma real_tendsto_sandwich: ``` noschinl@45892 ` 816` ``` fixes f g h :: "'a \ real" ``` noschinl@45892 ` 817` ``` assumes ev: "eventually (\n. f n \ g n) net" "eventually (\n. g n \ h n) net" ``` noschinl@45892 ` 818` ``` assumes lim: "(f ---> c) net" "(h ---> c) net" ``` noschinl@45892 ` 819` ``` shows "(g ---> c) net" ``` noschinl@45892 ` 820` ```proof - ``` noschinl@45892 ` 821` ``` have "((\n. g n - f n) ---> 0) net" ``` noschinl@45892 ` 822` ``` proof (rule metric_tendsto_imp_tendsto) ``` noschinl@45892 ` 823` ``` show "eventually (\n. dist (g n - f n) 0 \ dist (h n - f n) 0) net" ``` noschinl@45892 ` 824` ``` using ev by (rule eventually_elim2) (simp add: dist_real_def) ``` noschinl@45892 ` 825` ``` show "((\n. h n - f n) ---> 0) net" ``` noschinl@45892 ` 826` ``` using tendsto_diff[OF lim(2,1)] by simp ``` noschinl@45892 ` 827` ``` qed ``` noschinl@45892 ` 828` ``` from tendsto_add[OF this lim(1)] show ?thesis by simp ``` noschinl@45892 ` 829` ```qed ``` noschinl@45892 ` 830` huffman@44194 ` 831` ```subsubsection {* Linear operators and multiplication *} ``` huffman@44194 ` 832` huffman@44282 ` 833` ```lemma (in bounded_linear) tendsto: ``` huffman@44195 ` 834` ``` "(g ---> a) F \ ((\x. f (g x)) ---> f a) F" ``` huffman@44081 ` 835` ``` by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun) ``` huffman@31349 ` 836` huffman@44194 ` 837` ```lemma (in bounded_linear) tendsto_zero: ``` huffman@44195 ` 838` ``` "(g ---> 0) F \ ((\x. f (g x)) ---> 0) F" ``` huffman@44194 ` 839` ``` by (drule tendsto, simp only: zero) ``` huffman@44194 ` 840` huffman@44282 ` 841` ```lemma (in bounded_bilinear) tendsto: ``` huffman@44195 ` 842` ``` "\(f ---> a) F; (g ---> b) F\ \ ((\x. f x ** g x) ---> a ** b) F" ``` huffman@44081 ` 843` ``` by (simp only: tendsto_Zfun_iff prod_diff_prod ``` huffman@44081 ` 844` ``` Zfun_add Zfun Zfun_left Zfun_right) ``` huffman@31349 ` 845` huffman@44194 ` 846` ```lemma (in bounded_bilinear) tendsto_zero: ``` huffman@44195 ` 847` ``` assumes f: "(f ---> 0) F" ``` huffman@44195 ` 848` ``` assumes g: "(g ---> 0) F" ``` huffman@44195 ` 849` ``` shows "((\x. f x ** g x) ---> 0) F" ``` huffman@44194 ` 850` ``` using tendsto [OF f g] by (simp add: zero_left) ``` huffman@31355 ` 851` huffman@44194 ` 852` ```lemma (in bounded_bilinear) tendsto_left_zero: ``` huffman@44195 ` 853` ``` "(f ---> 0) F \ ((\x. f x ** c) ---> 0) F" ``` huffman@44194 ` 854` ``` by (rule bounded_linear.tendsto_zero [OF bounded_linear_left]) ``` huffman@44194 ` 855` huffman@44194 ` 856` ```lemma (in bounded_bilinear) tendsto_right_zero: ``` huffman@44195 ` 857` ``` "(f ---> 0) F \ ((\x. c ** f x) ---> 0) F" ``` huffman@44194 ` 858` ``` by (rule bounded_linear.tendsto_zero [OF bounded_linear_right]) ``` huffman@44194 ` 859` huffman@44282 ` 860` ```lemmas tendsto_of_real [tendsto_intros] = ``` huffman@44282 ` 861` ``` bounded_linear.tendsto [OF bounded_linear_of_real] ``` huffman@44282 ` 862` huffman@44282 ` 863` ```lemmas tendsto_scaleR [tendsto_intros] = ``` huffman@44282 ` 864` ``` bounded_bilinear.tendsto [OF bounded_bilinear_scaleR] ``` huffman@44282 ` 865` huffman@44282 ` 866` ```lemmas tendsto_mult [tendsto_intros] = ``` huffman@44282 ` 867` ``` bounded_bilinear.tendsto [OF bounded_bilinear_mult] ``` huffman@44194 ` 868` huffman@44568 ` 869` ```lemmas tendsto_mult_zero = ``` huffman@44568 ` 870` ``` bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult] ``` huffman@44568 ` 871` huffman@44568 ` 872` ```lemmas tendsto_mult_left_zero = ``` huffman@44568 ` 873` ``` bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult] ``` huffman@44568 ` 874` huffman@44568 ` 875` ```lemmas tendsto_mult_right_zero = ``` huffman@44568 ` 876` ``` bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult] ``` huffman@44568 ` 877` huffman@44194 ` 878` ```lemma tendsto_power [tendsto_intros]: ``` huffman@44194 ` 879` ``` fixes f :: "'a \ 'b::{power,real_normed_algebra}" ``` huffman@44195 ` 880` ``` shows "(f ---> a) F \ ((\x. f x ^ n) ---> a ^ n) F" ``` huffman@44194 ` 881` ``` by (induct n) (simp_all add: tendsto_const tendsto_mult) ``` huffman@44194 ` 882` huffman@44194 ` 883` ```lemma tendsto_setprod [tendsto_intros]: ``` huffman@44194 ` 884` ``` fixes f :: "'a \ 'b \ 'c::{real_normed_algebra,comm_ring_1}" ``` huffman@44195 ` 885` ``` assumes "\i. i \ S \ (f i ---> L i) F" ``` huffman@44195 ` 886` ``` shows "((\x. \i\S. f i x) ---> (\i\S. L i)) F" ``` huffman@44194 ` 887` ```proof (cases "finite S") ``` huffman@44194 ` 888` ``` assume "finite S" thus ?thesis using assms ``` huffman@44194 ` 889` ``` by (induct, simp add: tendsto_const, simp add: tendsto_mult) ``` huffman@44194 ` 890` ```next ``` huffman@44194 ` 891` ``` assume "\ finite S" thus ?thesis ``` huffman@44194 ` 892` ``` by (simp add: tendsto_const) ``` huffman@44194 ` 893` ```qed ``` huffman@44194 ` 894` huffman@44194 ` 895` ```subsubsection {* Inverse and division *} ``` huffman@31355 ` 896` huffman@31355 ` 897` ```lemma (in bounded_bilinear) Zfun_prod_Bfun: ``` huffman@44195 ` 898` ``` assumes f: "Zfun f F" ``` huffman@44195 ` 899` ``` assumes g: "Bfun g F" ``` huffman@44195 ` 900` ``` shows "Zfun (\x. f x ** g x) F" ``` huffman@31355 ` 901` ```proof - ``` huffman@31355 ` 902` ``` obtain K where K: "0 \ K" ``` huffman@31355 ` 903` ``` and norm_le: "\x y. norm (x ** y) \ norm x * norm y * K" ``` huffman@31355 ` 904` ``` using nonneg_bounded by fast ``` huffman@31355 ` 905` ``` obtain B where B: "0 < B" ``` huffman@44195 ` 906` ``` and norm_g: "eventually (\x. norm (g x) \ B) F" ``` huffman@31487 ` 907` ``` using g by (rule BfunE) ``` huffman@44195 ` 908` ``` have "eventually (\x. norm (f x ** g x) \ norm (f x) * (B * K)) F" ``` noschinl@46887 ` 909` ``` using norm_g proof eventually_elim ``` noschinl@46887 ` 910` ``` case (elim x) ``` huffman@31487 ` 911` ``` have "norm (f x ** g x) \ norm (f x) * norm (g x) * K" ``` huffman@31355 ` 912` ``` by (rule norm_le) ``` huffman@31487 ` 913` ``` also have "\ \ norm (f x) * B * K" ``` huffman@31487 ` 914` ``` by (intro mult_mono' order_refl norm_g norm_ge_zero ``` noschinl@46887 ` 915` ``` mult_nonneg_nonneg K elim) ``` huffman@31487 ` 916` ``` also have "\ = norm (f x) * (B * K)" ``` huffman@31355 ` 917` ``` by (rule mult_assoc) ``` huffman@31487 ` 918` ``` finally show "norm (f x ** g x) \ norm (f x) * (B * K)" . ``` huffman@31355 ` 919` ``` qed ``` huffman@31487 ` 920` ``` with f show ?thesis ``` huffman@31487 ` 921` ``` by (rule Zfun_imp_Zfun) ``` huffman@31355 ` 922` ```qed ``` huffman@31355 ` 923` huffman@31355 ` 924` ```lemma (in bounded_bilinear) flip: ``` huffman@31355 ` 925` ``` "bounded_bilinear (\x y. y ** x)" ``` huffman@44081 ` 926` ``` apply default ``` huffman@44081 ` 927` ``` apply (rule add_right) ``` huffman@44081 ` 928` ``` apply (rule add_left) ``` huffman@44081 ` 929` ``` apply (rule scaleR_right) ``` huffman@44081 ` 930` ``` apply (rule scaleR_left) ``` huffman@44081 ` 931` ``` apply (subst mult_commute) ``` huffman@44081 ` 932` ``` using bounded by fast ``` huffman@31355 ` 933` huffman@31355 ` 934` ```lemma (in bounded_bilinear) Bfun_prod_Zfun: ``` huffman@44195 ` 935` ``` assumes f: "Bfun f F" ``` huffman@44195 ` 936` ``` assumes g: "Zfun g F" ``` huffman@44195 ` 937` ``` shows "Zfun (\x. f x ** g x) F" ``` huffman@44081 ` 938` ``` using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun) ``` huffman@31355 ` 939` huffman@31355 ` 940` ```lemma Bfun_inverse_lemma: ``` huffman@31355 ` 941` ``` fixes x :: "'a::real_normed_div_algebra" ``` huffman@31355 ` 942` ``` shows "\r \ norm x; 0 < r\ \ norm (inverse x) \ inverse r" ``` huffman@44081 ` 943` ``` apply (subst nonzero_norm_inverse, clarsimp) ``` huffman@44081 ` 944` ``` apply (erule (1) le_imp_inverse_le) ``` huffman@44081 ` 945` ``` done ``` huffman@31355 ` 946` huffman@31355 ` 947` ```lemma Bfun_inverse: ``` huffman@31355 ` 948` ``` fixes a :: "'a::real_normed_div_algebra" ``` huffman@44195 ` 949` ``` assumes f: "(f ---> a) F" ``` huffman@31355 ` 950` ``` assumes a: "a \ 0" ``` huffman@44195 ` 951` ``` shows "Bfun (\x. inverse (f x)) F" ``` huffman@31355 ` 952` ```proof - ``` huffman@31355 ` 953` ``` from a have "0 < norm a" by simp ``` huffman@31355 ` 954` ``` hence "\r>0. r < norm a" by (rule dense) ``` huffman@31355 ` 955` ``` then obtain r where r1: "0 < r" and r2: "r < norm a" by fast ``` huffman@44195 ` 956` ``` have "eventually (\x. dist (f x) a < r) F" ``` huffman@31487 ` 957` ``` using tendstoD [OF f r1] by fast ``` huffman@44195 ` 958` ``` hence "eventually (\x. norm (inverse (f x)) \ inverse (norm a - r)) F" ``` noschinl@46887 ` 959` ``` proof eventually_elim ``` noschinl@46887 ` 960` ``` case (elim x) ``` huffman@31487 ` 961` ``` hence 1: "norm (f x - a) < r" ``` huffman@31355 ` 962` ``` by (simp add: dist_norm) ``` huffman@31487 ` 963` ``` hence 2: "f x \ 0" using r2 by auto ``` huffman@31487 ` 964` ``` hence "norm (inverse (f x)) = inverse (norm (f x))" ``` huffman@31355 ` 965` ``` by (rule nonzero_norm_inverse) ``` huffman@31355 ` 966` ``` also have "\ \ inverse (norm a - r)" ``` huffman@31355 ` 967` ``` proof (rule le_imp_inverse_le) ``` huffman@31355 ` 968` ``` show "0 < norm a - r" using r2 by simp ``` huffman@31355 ` 969` ``` next ``` huffman@31487 ` 970` ``` have "norm a - norm (f x) \ norm (a - f x)" ``` huffman@31355 ` 971` ``` by (rule norm_triangle_ineq2) ``` huffman@31487 ` 972` ``` also have "\ = norm (f x - a)" ``` huffman@31355 ` 973` ``` by (rule norm_minus_commute) ``` huffman@31355 ` 974` ``` also have "\ < r" using 1 . ``` huffman@31487 ` 975` ``` finally show "norm a - r \ norm (f x)" by simp ``` huffman@31355 ` 976` ``` qed ``` huffman@31487 ` 977` ``` finally show "norm (inverse (f x)) \ inverse (norm a - r)" . ``` huffman@31355 ` 978` ``` qed ``` huffman@31355 ` 979` ``` thus ?thesis by (rule BfunI) ``` huffman@31355 ` 980` ```qed ``` huffman@31355 ` 981` huffman@31565 ` 982` ```lemma tendsto_inverse [tendsto_intros]: ``` huffman@31355 ` 983` ``` fixes a :: "'a::real_normed_div_algebra" ``` huffman@44195 ` 984` ``` assumes f: "(f ---> a) F" ``` huffman@31355 ` 985` ``` assumes a: "a \ 0" ``` huffman@44195 ` 986` ``` shows "((\x. inverse (f x)) ---> inverse a) F" ``` huffman@31355 ` 987` ```proof - ``` huffman@31355 ` 988` ``` from a have "0 < norm a" by simp ``` huffman@44195 ` 989` ``` with f have "eventually (\x. dist (f x) a < norm a) F" ``` huffman@31355 ` 990` ``` by (rule tendstoD) ``` huffman@44195 ` 991` ``` then have "eventually (\x. f x \ 0) F" ``` huffman@31355 ` 992` ``` unfolding dist_norm by (auto elim!: eventually_elim1) ``` huffman@44627 ` 993` ``` with a have "eventually (\x. inverse (f x) - inverse a = ``` huffman@44627 ` 994` ``` - (inverse (f x) * (f x - a) * inverse a)) F" ``` huffman@44627 ` 995` ``` by (auto elim!: eventually_elim1 simp: inverse_diff_inverse) ``` huffman@44627 ` 996` ``` moreover have "Zfun (\x. - (inverse (f x) * (f x - a) * inverse a)) F" ``` huffman@44627 ` 997` ``` by (intro Zfun_minus Zfun_mult_left ``` huffman@44627 ` 998` ``` bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult] ``` huffman@44627 ` 999` ``` Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff]) ``` huffman@44627 ` 1000` ``` ultimately show ?thesis ``` huffman@44627 ` 1001` ``` unfolding tendsto_Zfun_iff by (rule Zfun_ssubst) ``` huffman@31355 ` 1002` ```qed ``` huffman@31355 ` 1003` huffman@31565 ` 1004` ```lemma tendsto_divide [tendsto_intros]: ``` huffman@31355 ` 1005` ``` fixes a b :: "'a::real_normed_field" ``` huffman@44195 ` 1006` ``` shows "\(f ---> a) F; (g ---> b) F; b \ 0\ ``` huffman@44195 ` 1007` ``` \ ((\x. f x / g x) ---> a / b) F" ``` huffman@44282 ` 1008` ``` by (simp add: tendsto_mult tendsto_inverse divide_inverse) ``` huffman@31355 ` 1009` huffman@44194 ` 1010` ```lemma tendsto_sgn [tendsto_intros]: ``` huffman@44194 ` 1011` ``` fixes l :: "'a::real_normed_vector" ``` huffman@44195 ` 1012` ``` shows "\(f ---> l) F; l \ 0\ \ ((\x. sgn (f x)) ---> sgn l) F" ``` huffman@44194 ` 1013` ``` unfolding sgn_div_norm by (simp add: tendsto_intros) ``` huffman@44194 ` 1014` huffman@31349 ` 1015` ```end ```