src/HOL/Limits.thy
 author noschinl Mon Mar 12 21:34:43 2012 +0100 (2012-03-12) changeset 46887 cb891d9a23c1 parent 46886 4cd29473c65d child 47432 e1576d13e933 permissions -rw-r--r--
use eventually_elim method
1 (*  Title       : Limits.thy
2     Author      : Brian Huffman
3 *)
5 header {* Filters and Limits *}
7 theory Limits
8 imports RealVector
9 begin
11 subsection {* Filters *}
13 text {*
14   This definition also allows non-proper filters.
15 *}
17 locale is_filter =
18   fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
19   assumes True: "F (\<lambda>x. True)"
20   assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
21   assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
23 typedef (open) 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
24 proof
25   show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
26 qed
28 lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
29   using Rep_filter [of F] by simp
31 lemma Abs_filter_inverse':
32   assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
33   using assms by (simp add: Abs_filter_inverse)
36 subsection {* Eventually *}
38 definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
39   where "eventually P F \<longleftrightarrow> Rep_filter F P"
41 lemma eventually_Abs_filter:
42   assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
43   unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
45 lemma filter_eq_iff:
46   shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
47   unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
49 lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
50   unfolding eventually_def
51   by (rule is_filter.True [OF is_filter_Rep_filter])
53 lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
54 proof -
55   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
56   thus "eventually P F" by simp
57 qed
59 lemma eventually_mono:
60   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
61   unfolding eventually_def
62   by (rule is_filter.mono [OF is_filter_Rep_filter])
64 lemma eventually_conj:
65   assumes P: "eventually (\<lambda>x. P x) F"
66   assumes Q: "eventually (\<lambda>x. Q x) F"
67   shows "eventually (\<lambda>x. P x \<and> Q x) F"
68   using assms unfolding eventually_def
69   by (rule is_filter.conj [OF is_filter_Rep_filter])
71 lemma eventually_mp:
72   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
73   assumes "eventually (\<lambda>x. P x) F"
74   shows "eventually (\<lambda>x. Q x) F"
75 proof (rule eventually_mono)
76   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
77   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
78     using assms by (rule eventually_conj)
79 qed
81 lemma eventually_rev_mp:
82   assumes "eventually (\<lambda>x. P x) F"
83   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
84   shows "eventually (\<lambda>x. Q x) F"
85 using assms(2) assms(1) by (rule eventually_mp)
87 lemma eventually_conj_iff:
88   "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
89   by (auto intro: eventually_conj elim: eventually_rev_mp)
91 lemma eventually_elim1:
92   assumes "eventually (\<lambda>i. P i) F"
93   assumes "\<And>i. P i \<Longrightarrow> Q i"
94   shows "eventually (\<lambda>i. Q i) F"
95   using assms by (auto elim!: eventually_rev_mp)
97 lemma eventually_elim2:
98   assumes "eventually (\<lambda>i. P i) F"
99   assumes "eventually (\<lambda>i. Q i) F"
100   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
101   shows "eventually (\<lambda>i. R i) F"
102   using assms by (auto elim!: eventually_rev_mp)
104 lemma eventually_subst:
105   assumes "eventually (\<lambda>n. P n = Q n) F"
106   shows "eventually P F = eventually Q F" (is "?L = ?R")
107 proof -
108   from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
109       and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
110     by (auto elim: eventually_elim1)
111   then show ?thesis by (auto elim: eventually_elim2)
112 qed
114 ML {*
115   fun ev_elim_tac ctxt thms thm = let
116       val thy = Proof_Context.theory_of ctxt
117       val mp_thms = thms RL [@{thm eventually_rev_mp}]
118       val raw_elim_thm =
119         (@{thm allI} RS @{thm always_eventually})
120         |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
121         |> fold (fn _ => fn thm => @{thm impI} RS thm) thms
122       val cases_prop = prop_of (raw_elim_thm RS thm)
123       val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])
124     in
125       CASES cases (rtac raw_elim_thm 1) thm
126     end
128   fun eventually_elim_setup name =
129     Method.setup name (Scan.succeed (fn ctxt => METHOD_CASES (ev_elim_tac ctxt)))
130       "elimination of eventually quantifiers"
131 *}
133 setup {* eventually_elim_setup @{binding "eventually_elim"} *}
136 subsection {* Finer-than relation *}
138 text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
139 filter @{term F'}. *}
141 instantiation filter :: (type) complete_lattice
142 begin
144 definition le_filter_def:
145   "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
147 definition
148   "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
150 definition
151   "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
153 definition
154   "bot = Abs_filter (\<lambda>P. True)"
156 definition
157   "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
159 definition
160   "inf F F' = Abs_filter
161       (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
163 definition
164   "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
166 definition
167   "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
169 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
170   unfolding top_filter_def
171   by (rule eventually_Abs_filter, rule is_filter.intro, auto)
173 lemma eventually_bot [simp]: "eventually P bot"
174   unfolding bot_filter_def
175   by (subst eventually_Abs_filter, rule is_filter.intro, auto)
177 lemma eventually_sup:
178   "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
179   unfolding sup_filter_def
180   by (rule eventually_Abs_filter, rule is_filter.intro)
181      (auto elim!: eventually_rev_mp)
183 lemma eventually_inf:
184   "eventually P (inf F F') \<longleftrightarrow>
185    (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
186   unfolding inf_filter_def
187   apply (rule eventually_Abs_filter, rule is_filter.intro)
188   apply (fast intro: eventually_True)
189   apply clarify
190   apply (intro exI conjI)
191   apply (erule (1) eventually_conj)
192   apply (erule (1) eventually_conj)
193   apply simp
194   apply auto
195   done
197 lemma eventually_Sup:
198   "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
199   unfolding Sup_filter_def
200   apply (rule eventually_Abs_filter, rule is_filter.intro)
201   apply (auto intro: eventually_conj elim!: eventually_rev_mp)
202   done
204 instance proof
205   fix F F' F'' :: "'a filter" and S :: "'a filter set"
206   { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
207     by (rule less_filter_def) }
208   { show "F \<le> F"
209     unfolding le_filter_def by simp }
210   { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
211     unfolding le_filter_def by simp }
212   { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
213     unfolding le_filter_def filter_eq_iff by fast }
214   { show "F \<le> top"
215     unfolding le_filter_def eventually_top by (simp add: always_eventually) }
216   { show "bot \<le> F"
217     unfolding le_filter_def by simp }
218   { show "F \<le> sup F F'" and "F' \<le> sup F F'"
219     unfolding le_filter_def eventually_sup by simp_all }
220   { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
221     unfolding le_filter_def eventually_sup by simp }
222   { show "inf F F' \<le> F" and "inf F F' \<le> F'"
223     unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
224   { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
225     unfolding le_filter_def eventually_inf
226     by (auto elim!: eventually_mono intro: eventually_conj) }
227   { assume "F \<in> S" thus "F \<le> Sup S"
228     unfolding le_filter_def eventually_Sup by simp }
229   { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
230     unfolding le_filter_def eventually_Sup by simp }
231   { assume "F'' \<in> S" thus "Inf S \<le> F''"
232     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
233   { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
234     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
235 qed
237 end
239 lemma filter_leD:
240   "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
241   unfolding le_filter_def by simp
243 lemma filter_leI:
244   "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
245   unfolding le_filter_def by simp
247 lemma eventually_False:
248   "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
249   unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
251 abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
252   where "trivial_limit F \<equiv> F = bot"
254 lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
255   by (rule eventually_False [symmetric])
258 subsection {* Map function for filters *}
260 definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
261   where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
263 lemma eventually_filtermap:
264   "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
265   unfolding filtermap_def
266   apply (rule eventually_Abs_filter)
267   apply (rule is_filter.intro)
268   apply (auto elim!: eventually_rev_mp)
269   done
271 lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
272   by (simp add: filter_eq_iff eventually_filtermap)
274 lemma filtermap_filtermap:
275   "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
276   by (simp add: filter_eq_iff eventually_filtermap)
278 lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
279   unfolding le_filter_def eventually_filtermap by simp
281 lemma filtermap_bot [simp]: "filtermap f bot = bot"
282   by (simp add: filter_eq_iff eventually_filtermap)
285 subsection {* Sequentially *}
287 definition sequentially :: "nat filter"
288   where "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
290 lemma eventually_sequentially:
291   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
292 unfolding sequentially_def
293 proof (rule eventually_Abs_filter, rule is_filter.intro)
294   fix P Q :: "nat \<Rightarrow> bool"
295   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
296   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
297   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
298   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
299 qed auto
301 lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
302   unfolding filter_eq_iff eventually_sequentially by auto
304 lemmas trivial_limit_sequentially = sequentially_bot
306 lemma eventually_False_sequentially [simp]:
307   "\<not> eventually (\<lambda>n. False) sequentially"
308   by (simp add: eventually_False)
310 lemma le_sequentially:
311   "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
312   unfolding le_filter_def eventually_sequentially
313   by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
315 lemma eventually_sequentiallyI:
316   assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
317   shows "eventually P sequentially"
318 using assms by (auto simp: eventually_sequentially)
321 subsection {* Standard filters *}
323 definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
324   where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)"
326 definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
327   where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
329 definition (in topological_space) at :: "'a \<Rightarrow> 'a filter"
330   where "at a = nhds a within - {a}"
332 lemma eventually_within:
333   "eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F"
334   unfolding within_def
335   by (rule eventually_Abs_filter, rule is_filter.intro)
336      (auto elim!: eventually_rev_mp)
338 lemma within_UNIV [simp]: "F within UNIV = F"
339   unfolding filter_eq_iff eventually_within by simp
341 lemma within_empty [simp]: "F within {} = bot"
342   unfolding filter_eq_iff eventually_within by simp
344 lemma eventually_nhds:
345   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
346 unfolding nhds_def
347 proof (rule eventually_Abs_filter, rule is_filter.intro)
348   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
349   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
350 next
351   fix P Q
352   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
353      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
354   then obtain S T where
355     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
356     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
357   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
358     by (simp add: open_Int)
359   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
360 qed auto
362 lemma eventually_nhds_metric:
363   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
364 unfolding eventually_nhds open_dist
365 apply safe
366 apply fast
367 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
368 apply clarsimp
369 apply (rule_tac x="d - dist x a" in exI, clarsimp)
370 apply (simp only: less_diff_eq)
371 apply (erule le_less_trans [OF dist_triangle])
372 done
374 lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
375   unfolding trivial_limit_def eventually_nhds by simp
377 lemma eventually_at_topological:
378   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
379 unfolding at_def eventually_within eventually_nhds by simp
381 lemma eventually_at:
382   fixes a :: "'a::metric_space"
383   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
384 unfolding at_def eventually_within eventually_nhds_metric by auto
386 lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
387   unfolding trivial_limit_def eventually_at_topological
388   by (safe, case_tac "S = {a}", simp, fast, fast)
390 lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
391   by (simp add: at_eq_bot_iff not_open_singleton)
394 subsection {* Boundedness *}
396 definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
397   where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
399 lemma BfunI:
400   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
401 unfolding Bfun_def
402 proof (intro exI conjI allI)
403   show "0 < max K 1" by simp
404 next
405   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
406     using K by (rule eventually_elim1, simp)
407 qed
409 lemma BfunE:
410   assumes "Bfun f F"
411   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
412 using assms unfolding Bfun_def by fast
415 subsection {* Convergence to Zero *}
417 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
418   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
420 lemma ZfunI:
421   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
422   unfolding Zfun_def by simp
424 lemma ZfunD:
425   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
426   unfolding Zfun_def by simp
428 lemma Zfun_ssubst:
429   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
430   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
432 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
433   unfolding Zfun_def by simp
435 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
436   unfolding Zfun_def by simp
438 lemma Zfun_imp_Zfun:
439   assumes f: "Zfun f F"
440   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
441   shows "Zfun (\<lambda>x. g x) F"
442 proof (cases)
443   assume K: "0 < K"
444   show ?thesis
445   proof (rule ZfunI)
446     fix r::real assume "0 < r"
447     hence "0 < r / K"
448       using K by (rule divide_pos_pos)
449     then have "eventually (\<lambda>x. norm (f x) < r / K) F"
450       using ZfunD [OF f] by fast
451     with g show "eventually (\<lambda>x. norm (g x) < r) F"
452     proof eventually_elim
453       case (elim x)
454       hence "norm (f x) * K < r"
455         by (simp add: pos_less_divide_eq K)
456       thus ?case
457         by (simp add: order_le_less_trans [OF elim(1)])
458     qed
459   qed
460 next
461   assume "\<not> 0 < K"
462   hence K: "K \<le> 0" by (simp only: not_less)
463   show ?thesis
464   proof (rule ZfunI)
465     fix r :: real
466     assume "0 < r"
467     from g show "eventually (\<lambda>x. norm (g x) < r) F"
468     proof eventually_elim
469       case (elim x)
470       also have "norm (f x) * K \<le> norm (f x) * 0"
471         using K norm_ge_zero by (rule mult_left_mono)
472       finally show ?case
473         using `0 < r` by simp
474     qed
475   qed
476 qed
478 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
479   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
482   assumes f: "Zfun f F" and g: "Zfun g F"
483   shows "Zfun (\<lambda>x. f x + g x) F"
484 proof (rule ZfunI)
485   fix r::real assume "0 < r"
486   hence r: "0 < r / 2" by simp
487   have "eventually (\<lambda>x. norm (f x) < r/2) F"
488     using f r by (rule ZfunD)
489   moreover
490   have "eventually (\<lambda>x. norm (g x) < r/2) F"
491     using g r by (rule ZfunD)
492   ultimately
493   show "eventually (\<lambda>x. norm (f x + g x) < r) F"
494   proof eventually_elim
495     case (elim x)
496     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
497       by (rule norm_triangle_ineq)
498     also have "\<dots> < r/2 + r/2"
499       using elim by (rule add_strict_mono)
500     finally show ?case
501       by simp
502   qed
503 qed
505 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
506   unfolding Zfun_def by simp
508 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
509   by (simp only: diff_minus Zfun_add Zfun_minus)
511 lemma (in bounded_linear) Zfun:
512   assumes g: "Zfun g F"
513   shows "Zfun (\<lambda>x. f (g x)) F"
514 proof -
515   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
516     using bounded by fast
517   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
518     by simp
519   with g show ?thesis
520     by (rule Zfun_imp_Zfun)
521 qed
523 lemma (in bounded_bilinear) Zfun:
524   assumes f: "Zfun f F"
525   assumes g: "Zfun g F"
526   shows "Zfun (\<lambda>x. f x ** g x) F"
527 proof (rule ZfunI)
528   fix r::real assume r: "0 < r"
529   obtain K where K: "0 < K"
530     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
531     using pos_bounded by fast
532   from K have K': "0 < inverse K"
533     by (rule positive_imp_inverse_positive)
534   have "eventually (\<lambda>x. norm (f x) < r) F"
535     using f r by (rule ZfunD)
536   moreover
537   have "eventually (\<lambda>x. norm (g x) < inverse K) F"
538     using g K' by (rule ZfunD)
539   ultimately
540   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
541   proof eventually_elim
542     case (elim x)
543     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
544       by (rule norm_le)
545     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
546       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
547     also from K have "r * inverse K * K = r"
548       by simp
549     finally show ?case .
550   qed
551 qed
553 lemma (in bounded_bilinear) Zfun_left:
554   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
555   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
557 lemma (in bounded_bilinear) Zfun_right:
558   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
559   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
561 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
562 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
563 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
566 subsection {* Limits *}
568 definition (in topological_space)
569   tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
570   "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
572 definition real_tendsto_inf :: "('a \<Rightarrow> real) \<Rightarrow> 'a filter \<Rightarrow> bool" where
573   "real_tendsto_inf f F \<equiv> \<forall>x. eventually (\<lambda>y. x < f y) F"
575 ML {*
576 structure Tendsto_Intros = Named_Thms
577 (
578   val name = @{binding tendsto_intros}
579   val description = "introduction rules for tendsto"
580 )
581 *}
583 setup Tendsto_Intros.setup
585 lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
586   unfolding tendsto_def le_filter_def by fast
588 lemma topological_tendstoI:
589   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F)
590     \<Longrightarrow> (f ---> l) F"
591   unfolding tendsto_def by auto
593 lemma topological_tendstoD:
594   "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
595   unfolding tendsto_def by auto
597 lemma tendstoI:
598   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
599   shows "(f ---> l) F"
600   apply (rule topological_tendstoI)
601   apply (simp add: open_dist)
602   apply (drule (1) bspec, clarify)
603   apply (drule assms)
604   apply (erule eventually_elim1, simp)
605   done
607 lemma tendstoD:
608   "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
609   apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
610   apply (clarsimp simp add: open_dist)
611   apply (rule_tac x="e - dist x l" in exI, clarsimp)
612   apply (simp only: less_diff_eq)
613   apply (erule le_less_trans [OF dist_triangle])
614   apply simp
615   apply simp
616   done
618 lemma tendsto_iff:
619   "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
620   using tendstoI tendstoD by fast
622 lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
623   by (simp only: tendsto_iff Zfun_def dist_norm)
625 lemma tendsto_bot [simp]: "(f ---> a) bot"
626   unfolding tendsto_def by simp
628 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
629   unfolding tendsto_def eventually_at_topological by auto
631 lemma tendsto_ident_at_within [tendsto_intros]:
632   "((\<lambda>x. x) ---> a) (at a within S)"
633   unfolding tendsto_def eventually_within eventually_at_topological by auto
635 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
636   by (simp add: tendsto_def)
638 lemma tendsto_unique:
639   fixes f :: "'a \<Rightarrow> 'b::t2_space"
640   assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
641   shows "a = b"
642 proof (rule ccontr)
643   assume "a \<noteq> b"
644   obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
645     using hausdorff [OF `a \<noteq> b`] by fast
646   have "eventually (\<lambda>x. f x \<in> U) F"
647     using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
648   moreover
649   have "eventually (\<lambda>x. f x \<in> V) F"
650     using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
651   ultimately
652   have "eventually (\<lambda>x. False) F"
653   proof eventually_elim
654     case (elim x)
655     hence "f x \<in> U \<inter> V" by simp
656     with `U \<inter> V = {}` show ?case by simp
657   qed
658   with `\<not> trivial_limit F` show "False"
659     by (simp add: trivial_limit_def)
660 qed
662 lemma tendsto_const_iff:
663   fixes a b :: "'a::t2_space"
664   assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
665   by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
667 lemma tendsto_compose:
668   assumes g: "(g ---> g l) (at l)"
669   assumes f: "(f ---> l) F"
670   shows "((\<lambda>x. g (f x)) ---> g l) F"
671 proof (rule topological_tendstoI)
672   fix B assume B: "open B" "g l \<in> B"
673   obtain A where A: "open A" "l \<in> A"
674     and gB: "\<forall>y. y \<in> A \<longrightarrow> g y \<in> B"
675     using topological_tendstoD [OF g B] B(2)
676     unfolding eventually_at_topological by fast
677   hence "\<forall>x. f x \<in> A \<longrightarrow> g (f x) \<in> B" by simp
678   from this topological_tendstoD [OF f A]
679   show "eventually (\<lambda>x. g (f x) \<in> B) F"
680     by (rule eventually_mono)
681 qed
683 lemma tendsto_compose_eventually:
684   assumes g: "(g ---> m) (at l)"
685   assumes f: "(f ---> l) F"
686   assumes inj: "eventually (\<lambda>x. f x \<noteq> l) F"
687   shows "((\<lambda>x. g (f x)) ---> m) F"
688 proof (rule topological_tendstoI)
689   fix B assume B: "open B" "m \<in> B"
690   obtain A where A: "open A" "l \<in> A"
691     and gB: "\<And>y. y \<in> A \<Longrightarrow> y \<noteq> l \<Longrightarrow> g y \<in> B"
692     using topological_tendstoD [OF g B]
693     unfolding eventually_at_topological by fast
694   show "eventually (\<lambda>x. g (f x) \<in> B) F"
695     using topological_tendstoD [OF f A] inj
696     by (rule eventually_elim2) (simp add: gB)
697 qed
699 lemma metric_tendsto_imp_tendsto:
700   assumes f: "(f ---> a) F"
701   assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
702   shows "(g ---> b) F"
703 proof (rule tendstoI)
704   fix e :: real assume "0 < e"
705   with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
706   with le show "eventually (\<lambda>x. dist (g x) b < e) F"
707     using le_less_trans by (rule eventually_elim2)
708 qed
710 lemma real_tendsto_inf_real: "real_tendsto_inf real sequentially"
711 proof (unfold real_tendsto_inf_def, rule allI)
712   fix x show "eventually (\<lambda>y. x < real y) sequentially"
713     by (rule eventually_sequentiallyI[of "natceiling (x + 1)"])
714         (simp add: natceiling_le_eq)
715 qed
719 subsubsection {* Distance and norms *}
721 lemma tendsto_dist [tendsto_intros]:
722   assumes f: "(f ---> l) F" and g: "(g ---> m) F"
723   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
724 proof (rule tendstoI)
725   fix e :: real assume "0 < e"
726   hence e2: "0 < e/2" by simp
727   from tendstoD [OF f e2] tendstoD [OF g e2]
728   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
729   proof (eventually_elim)
730     case (elim x)
731     then show "dist (dist (f x) (g x)) (dist l m) < e"
732       unfolding dist_real_def
733       using dist_triangle2 [of "f x" "g x" "l"]
734       using dist_triangle2 [of "g x" "l" "m"]
735       using dist_triangle3 [of "l" "m" "f x"]
736       using dist_triangle [of "f x" "m" "g x"]
737       by arith
738   qed
739 qed
741 lemma norm_conv_dist: "norm x = dist x 0"
742   unfolding dist_norm by simp
744 lemma tendsto_norm [tendsto_intros]:
745   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
746   unfolding norm_conv_dist by (intro tendsto_intros)
748 lemma tendsto_norm_zero:
749   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
750   by (drule tendsto_norm, simp)
752 lemma tendsto_norm_zero_cancel:
753   "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
754   unfolding tendsto_iff dist_norm by simp
756 lemma tendsto_norm_zero_iff:
757   "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
758   unfolding tendsto_iff dist_norm by simp
760 lemma tendsto_rabs [tendsto_intros]:
761   "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
762   by (fold real_norm_def, rule tendsto_norm)
764 lemma tendsto_rabs_zero:
765   "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
766   by (fold real_norm_def, rule tendsto_norm_zero)
768 lemma tendsto_rabs_zero_cancel:
769   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
770   by (fold real_norm_def, rule tendsto_norm_zero_cancel)
772 lemma tendsto_rabs_zero_iff:
773   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
774   by (fold real_norm_def, rule tendsto_norm_zero_iff)
776 subsubsection {* Addition and subtraction *}
778 lemma tendsto_add [tendsto_intros]:
779   fixes a b :: "'a::real_normed_vector"
780   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
784   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
785   shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
786   by (drule (1) tendsto_add, simp)
788 lemma tendsto_minus [tendsto_intros]:
789   fixes a :: "'a::real_normed_vector"
790   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
791   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
793 lemma tendsto_minus_cancel:
794   fixes a :: "'a::real_normed_vector"
795   shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
796   by (drule tendsto_minus, simp)
798 lemma tendsto_diff [tendsto_intros]:
799   fixes a b :: "'a::real_normed_vector"
800   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
801   by (simp add: diff_minus tendsto_add tendsto_minus)
803 lemma tendsto_setsum [tendsto_intros]:
804   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
805   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
806   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
807 proof (cases "finite S")
808   assume "finite S" thus ?thesis using assms
810 next
811   assume "\<not> finite S" thus ?thesis
812     by (simp add: tendsto_const)
813 qed
815 lemma real_tendsto_sandwich:
816   fixes f g h :: "'a \<Rightarrow> real"
817   assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
818   assumes lim: "(f ---> c) net" "(h ---> c) net"
819   shows "(g ---> c) net"
820 proof -
821   have "((\<lambda>n. g n - f n) ---> 0) net"
822   proof (rule metric_tendsto_imp_tendsto)
823     show "eventually (\<lambda>n. dist (g n - f n) 0 \<le> dist (h n - f n) 0) net"
824       using ev by (rule eventually_elim2) (simp add: dist_real_def)
825     show "((\<lambda>n. h n - f n) ---> 0) net"
826       using tendsto_diff[OF lim(2,1)] by simp
827   qed
828   from tendsto_add[OF this lim(1)] show ?thesis by simp
829 qed
831 subsubsection {* Linear operators and multiplication *}
833 lemma (in bounded_linear) tendsto:
834   "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
835   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
837 lemma (in bounded_linear) tendsto_zero:
838   "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
839   by (drule tendsto, simp only: zero)
841 lemma (in bounded_bilinear) tendsto:
842   "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
843   by (simp only: tendsto_Zfun_iff prod_diff_prod
844                  Zfun_add Zfun Zfun_left Zfun_right)
846 lemma (in bounded_bilinear) tendsto_zero:
847   assumes f: "(f ---> 0) F"
848   assumes g: "(g ---> 0) F"
849   shows "((\<lambda>x. f x ** g x) ---> 0) F"
850   using tendsto [OF f g] by (simp add: zero_left)
852 lemma (in bounded_bilinear) tendsto_left_zero:
853   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
854   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
856 lemma (in bounded_bilinear) tendsto_right_zero:
857   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
858   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
860 lemmas tendsto_of_real [tendsto_intros] =
861   bounded_linear.tendsto [OF bounded_linear_of_real]
863 lemmas tendsto_scaleR [tendsto_intros] =
864   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
866 lemmas tendsto_mult [tendsto_intros] =
867   bounded_bilinear.tendsto [OF bounded_bilinear_mult]
869 lemmas tendsto_mult_zero =
870   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
872 lemmas tendsto_mult_left_zero =
873   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
875 lemmas tendsto_mult_right_zero =
876   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
878 lemma tendsto_power [tendsto_intros]:
879   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
880   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
881   by (induct n) (simp_all add: tendsto_const tendsto_mult)
883 lemma tendsto_setprod [tendsto_intros]:
884   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
885   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
886   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
887 proof (cases "finite S")
888   assume "finite S" thus ?thesis using assms
889     by (induct, simp add: tendsto_const, simp add: tendsto_mult)
890 next
891   assume "\<not> finite S" thus ?thesis
892     by (simp add: tendsto_const)
893 qed
895 subsubsection {* Inverse and division *}
897 lemma (in bounded_bilinear) Zfun_prod_Bfun:
898   assumes f: "Zfun f F"
899   assumes g: "Bfun g F"
900   shows "Zfun (\<lambda>x. f x ** g x) F"
901 proof -
902   obtain K where K: "0 \<le> K"
903     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
904     using nonneg_bounded by fast
905   obtain B where B: "0 < B"
906     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
907     using g by (rule BfunE)
908   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
909   using norm_g proof eventually_elim
910     case (elim x)
911     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
912       by (rule norm_le)
913     also have "\<dots> \<le> norm (f x) * B * K"
914       by (intro mult_mono' order_refl norm_g norm_ge_zero
915                 mult_nonneg_nonneg K elim)
916     also have "\<dots> = norm (f x) * (B * K)"
917       by (rule mult_assoc)
918     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
919   qed
920   with f show ?thesis
921     by (rule Zfun_imp_Zfun)
922 qed
924 lemma (in bounded_bilinear) flip:
925   "bounded_bilinear (\<lambda>x y. y ** x)"
926   apply default
927   apply (rule add_right)
928   apply (rule add_left)
929   apply (rule scaleR_right)
930   apply (rule scaleR_left)
931   apply (subst mult_commute)
932   using bounded by fast
934 lemma (in bounded_bilinear) Bfun_prod_Zfun:
935   assumes f: "Bfun f F"
936   assumes g: "Zfun g F"
937   shows "Zfun (\<lambda>x. f x ** g x) F"
938   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
940 lemma Bfun_inverse_lemma:
941   fixes x :: "'a::real_normed_div_algebra"
942   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
943   apply (subst nonzero_norm_inverse, clarsimp)
944   apply (erule (1) le_imp_inverse_le)
945   done
947 lemma Bfun_inverse:
948   fixes a :: "'a::real_normed_div_algebra"
949   assumes f: "(f ---> a) F"
950   assumes a: "a \<noteq> 0"
951   shows "Bfun (\<lambda>x. inverse (f x)) F"
952 proof -
953   from a have "0 < norm a" by simp
954   hence "\<exists>r>0. r < norm a" by (rule dense)
955   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
956   have "eventually (\<lambda>x. dist (f x) a < r) F"
957     using tendstoD [OF f r1] by fast
958   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
959   proof eventually_elim
960     case (elim x)
961     hence 1: "norm (f x - a) < r"
962       by (simp add: dist_norm)
963     hence 2: "f x \<noteq> 0" using r2 by auto
964     hence "norm (inverse (f x)) = inverse (norm (f x))"
965       by (rule nonzero_norm_inverse)
966     also have "\<dots> \<le> inverse (norm a - r)"
967     proof (rule le_imp_inverse_le)
968       show "0 < norm a - r" using r2 by simp
969     next
970       have "norm a - norm (f x) \<le> norm (a - f x)"
971         by (rule norm_triangle_ineq2)
972       also have "\<dots> = norm (f x - a)"
973         by (rule norm_minus_commute)
974       also have "\<dots> < r" using 1 .
975       finally show "norm a - r \<le> norm (f x)" by simp
976     qed
977     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
978   qed
979   thus ?thesis by (rule BfunI)
980 qed
982 lemma tendsto_inverse [tendsto_intros]:
983   fixes a :: "'a::real_normed_div_algebra"
984   assumes f: "(f ---> a) F"
985   assumes a: "a \<noteq> 0"
986   shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
987 proof -
988   from a have "0 < norm a" by simp
989   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
990     by (rule tendstoD)
991   then have "eventually (\<lambda>x. f x \<noteq> 0) F"
992     unfolding dist_norm by (auto elim!: eventually_elim1)
993   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
994     - (inverse (f x) * (f x - a) * inverse a)) F"
995     by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
996   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
997     by (intro Zfun_minus Zfun_mult_left
998       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
999       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
1000   ultimately show ?thesis
1001     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
1002 qed
1004 lemma tendsto_divide [tendsto_intros]:
1005   fixes a b :: "'a::real_normed_field"
1006   shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
1007     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
1008   by (simp add: tendsto_mult tendsto_inverse divide_inverse)
1010 lemma tendsto_sgn [tendsto_intros]:
1011   fixes l :: "'a::real_normed_vector"
1012   shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
1013   unfolding sgn_div_norm by (simp add: tendsto_intros)
1015 end