src/HOL/Limits.thy
 author hoelzl Mon Mar 14 14:37:35 2011 +0100 (2011-03-14) changeset 41970 47d6e13d1710 parent 39302 d7728f65b353 child 44079 bcc60791b7b9 permissions -rw-r--r--
generalize infinite sums
1 (*  Title       : Limits.thy
2     Author      : Brian Huffman
3 *)
5 header {* Filters and Limits *}
7 theory Limits
8 imports RealVector
9 begin
11 subsection {* Nets *}
13 text {*
14   A net is now defined simply as a filter on a set.
15   The definition also allows non-proper filters.
16 *}
18 locale is_filter =
19   fixes net :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
20   assumes True: "net (\<lambda>x. True)"
21   assumes conj: "net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x) \<Longrightarrow> net (\<lambda>x. P x \<and> Q x)"
22   assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x)"
24 typedef (open) 'a net =
25   "{net :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter net}"
26 proof
27   show "(\<lambda>x. True) \<in> ?net" by (auto intro: is_filter.intro)
28 qed
30 lemma is_filter_Rep_net: "is_filter (Rep_net net)"
31 using Rep_net [of net] by simp
33 lemma Abs_net_inverse':
34   assumes "is_filter net" shows "Rep_net (Abs_net net) = net"
35 using assms by (simp add: Abs_net_inverse)
38 subsection {* Eventually *}
40 definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where
41   "eventually P net \<longleftrightarrow> Rep_net net P"
43 lemma eventually_Abs_net:
44   assumes "is_filter net" shows "eventually P (Abs_net net) = net P"
45 unfolding eventually_def using assms by (simp add: Abs_net_inverse)
47 lemma expand_net_eq:
48   shows "net = net' \<longleftrightarrow> (\<forall>P. eventually P net = eventually P net')"
49 unfolding Rep_net_inject [symmetric] fun_eq_iff eventually_def ..
51 lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
52 unfolding eventually_def
53 by (rule is_filter.True [OF is_filter_Rep_net])
55 lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P net"
56 proof -
57   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
58   thus "eventually P net" by simp
59 qed
61 lemma eventually_mono:
62   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
63 unfolding eventually_def
64 by (rule is_filter.mono [OF is_filter_Rep_net])
66 lemma eventually_conj:
67   assumes P: "eventually (\<lambda>x. P x) net"
68   assumes Q: "eventually (\<lambda>x. Q x) net"
69   shows "eventually (\<lambda>x. P x \<and> Q x) net"
70 using assms unfolding eventually_def
71 by (rule is_filter.conj [OF is_filter_Rep_net])
73 lemma eventually_mp:
74   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
75   assumes "eventually (\<lambda>x. P x) net"
76   shows "eventually (\<lambda>x. Q x) net"
77 proof (rule eventually_mono)
78   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
79   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) net"
80     using assms by (rule eventually_conj)
81 qed
83 lemma eventually_rev_mp:
84   assumes "eventually (\<lambda>x. P x) net"
85   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
86   shows "eventually (\<lambda>x. Q x) net"
87 using assms(2) assms(1) by (rule eventually_mp)
89 lemma eventually_conj_iff:
90   "eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
91 by (auto intro: eventually_conj elim: eventually_rev_mp)
93 lemma eventually_elim1:
94   assumes "eventually (\<lambda>i. P i) net"
95   assumes "\<And>i. P i \<Longrightarrow> Q i"
96   shows "eventually (\<lambda>i. Q i) net"
97 using assms by (auto elim!: eventually_rev_mp)
99 lemma eventually_elim2:
100   assumes "eventually (\<lambda>i. P i) net"
101   assumes "eventually (\<lambda>i. Q i) net"
102   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
103   shows "eventually (\<lambda>i. R i) net"
104 using assms by (auto elim!: eventually_rev_mp)
106 subsection {* Finer-than relation *}
108 text {* @{term "net \<le> net'"} means that @{term net} is finer than
109 @{term net'}. *}
111 instantiation net :: (type) complete_lattice
112 begin
114 definition
115   le_net_def: "net \<le> net' \<longleftrightarrow> (\<forall>P. eventually P net' \<longrightarrow> eventually P net)"
117 definition
118   less_net_def: "(net :: 'a net) < net' \<longleftrightarrow> net \<le> net' \<and> \<not> net' \<le> net"
120 definition
121   top_net_def: "top = Abs_net (\<lambda>P. \<forall>x. P x)"
123 definition
124   bot_net_def: "bot = Abs_net (\<lambda>P. True)"
126 definition
127   sup_net_def: "sup net net' = Abs_net (\<lambda>P. eventually P net \<and> eventually P net')"
129 definition
130   inf_net_def: "inf a b = Abs_net
131       (\<lambda>P. \<exists>Q R. eventually Q a \<and> eventually R b \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
133 definition
134   Sup_net_def: "Sup A = Abs_net (\<lambda>P. \<forall>net\<in>A. eventually P net)"
136 definition
137   Inf_net_def: "Inf A = Sup {x::'a net. \<forall>y\<in>A. x \<le> y}"
139 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
140 unfolding top_net_def
141 by (rule eventually_Abs_net, rule is_filter.intro, auto)
143 lemma eventually_bot [simp]: "eventually P bot"
144 unfolding bot_net_def
145 by (subst eventually_Abs_net, rule is_filter.intro, auto)
147 lemma eventually_sup:
148   "eventually P (sup net net') \<longleftrightarrow> eventually P net \<and> eventually P net'"
149 unfolding sup_net_def
150 by (rule eventually_Abs_net, rule is_filter.intro)
151    (auto elim!: eventually_rev_mp)
153 lemma eventually_inf:
154   "eventually P (inf a b) \<longleftrightarrow>
155    (\<exists>Q R. eventually Q a \<and> eventually R b \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
156 unfolding inf_net_def
157 apply (rule eventually_Abs_net, rule is_filter.intro)
158 apply (fast intro: eventually_True)
159 apply clarify
160 apply (intro exI conjI)
161 apply (erule (1) eventually_conj)
162 apply (erule (1) eventually_conj)
163 apply simp
164 apply auto
165 done
167 lemma eventually_Sup:
168   "eventually P (Sup A) \<longleftrightarrow> (\<forall>net\<in>A. eventually P net)"
169 unfolding Sup_net_def
170 apply (rule eventually_Abs_net, rule is_filter.intro)
171 apply (auto intro: eventually_conj elim!: eventually_rev_mp)
172 done
174 instance proof
175   fix x y :: "'a net" show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
176     by (rule less_net_def)
177 next
178   fix x :: "'a net" show "x \<le> x"
179     unfolding le_net_def by simp
180 next
181   fix x y z :: "'a net" assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
182     unfolding le_net_def by simp
183 next
184   fix x y :: "'a net" assume "x \<le> y" and "y \<le> x" thus "x = y"
185     unfolding le_net_def expand_net_eq by fast
186 next
187   fix x :: "'a net" show "x \<le> top"
188     unfolding le_net_def eventually_top by (simp add: always_eventually)
189 next
190   fix x :: "'a net" show "bot \<le> x"
191     unfolding le_net_def by simp
192 next
193   fix x y :: "'a net" show "x \<le> sup x y" and "y \<le> sup x y"
194     unfolding le_net_def eventually_sup by simp_all
195 next
196   fix x y z :: "'a net" assume "x \<le> z" and "y \<le> z" thus "sup x y \<le> z"
197     unfolding le_net_def eventually_sup by simp
198 next
199   fix x y :: "'a net" show "inf x y \<le> x" and "inf x y \<le> y"
200     unfolding le_net_def eventually_inf by (auto intro: eventually_True)
201 next
202   fix x y z :: "'a net" assume "x \<le> y" and "x \<le> z" thus "x \<le> inf y z"
203     unfolding le_net_def eventually_inf
204     by (auto elim!: eventually_mono intro: eventually_conj)
205 next
206   fix x :: "'a net" and A assume "x \<in> A" thus "x \<le> Sup A"
207     unfolding le_net_def eventually_Sup by simp
208 next
209   fix A and y :: "'a net" assume "\<And>x. x \<in> A \<Longrightarrow> x \<le> y" thus "Sup A \<le> y"
210     unfolding le_net_def eventually_Sup by simp
211 next
212   fix z :: "'a net" and A assume "z \<in> A" thus "Inf A \<le> z"
213     unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp
214 next
215   fix A and x :: "'a net" assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" thus "x \<le> Inf A"
216     unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp
217 qed
219 end
221 lemma net_leD:
222   "net \<le> net' \<Longrightarrow> eventually P net' \<Longrightarrow> eventually P net"
223 unfolding le_net_def by simp
225 lemma net_leI:
226   "(\<And>P. eventually P net' \<Longrightarrow> eventually P net) \<Longrightarrow> net \<le> net'"
227 unfolding le_net_def by simp
229 lemma eventually_False:
230   "eventually (\<lambda>x. False) net \<longleftrightarrow> net = bot"
231 unfolding expand_net_eq by (auto elim: eventually_rev_mp)
233 subsection {* Map function for nets *}
235 definition netmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a net \<Rightarrow> 'b net" where
236   "netmap f net = Abs_net (\<lambda>P. eventually (\<lambda>x. P (f x)) net)"
238 lemma eventually_netmap:
239   "eventually P (netmap f net) = eventually (\<lambda>x. P (f x)) net"
240 unfolding netmap_def
241 apply (rule eventually_Abs_net)
242 apply (rule is_filter.intro)
243 apply (auto elim!: eventually_rev_mp)
244 done
246 lemma netmap_ident: "netmap (\<lambda>x. x) net = net"
247 by (simp add: expand_net_eq eventually_netmap)
249 lemma netmap_netmap: "netmap f (netmap g net) = netmap (\<lambda>x. f (g x)) net"
250 by (simp add: expand_net_eq eventually_netmap)
252 lemma netmap_mono: "net \<le> net' \<Longrightarrow> netmap f net \<le> netmap f net'"
253 unfolding le_net_def eventually_netmap by simp
255 lemma netmap_bot [simp]: "netmap f bot = bot"
256 by (simp add: expand_net_eq eventually_netmap)
259 subsection {* Sequentially *}
261 definition sequentially :: "nat net" where
262   "sequentially = Abs_net (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
264 lemma eventually_sequentially:
265   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
266 unfolding sequentially_def
267 proof (rule eventually_Abs_net, rule is_filter.intro)
268   fix P Q :: "nat \<Rightarrow> bool"
269   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
270   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
271   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
272   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
273 qed auto
275 lemma sequentially_bot [simp]: "sequentially \<noteq> bot"
276 unfolding expand_net_eq eventually_sequentially by auto
278 lemma eventually_False_sequentially [simp]:
279   "\<not> eventually (\<lambda>n. False) sequentially"
280 by (simp add: eventually_False)
282 lemma le_sequentially:
283   "net \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) net)"
284 unfolding le_net_def eventually_sequentially
285 by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
288 definition
289   trivial_limit :: "'a net \<Rightarrow> bool" where
290   "trivial_limit net \<longleftrightarrow> eventually (\<lambda>x. False) net"
292 lemma trivial_limit_sequentially[intro]: "\<not> trivial_limit sequentially"
293   by (auto simp add: trivial_limit_def eventually_sequentially)
295 subsection {* Standard Nets *}
297 definition within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
298   "net within S = Abs_net (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net)"
300 definition nhds :: "'a::topological_space \<Rightarrow> 'a net" where
301   "nhds a = Abs_net (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
303 definition at :: "'a::topological_space \<Rightarrow> 'a net" where
304   "at a = nhds a within - {a}"
306 lemma eventually_within:
307   "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
308 unfolding within_def
309 by (rule eventually_Abs_net, rule is_filter.intro)
310    (auto elim!: eventually_rev_mp)
312 lemma within_UNIV: "net within UNIV = net"
313   unfolding expand_net_eq eventually_within by simp
315 lemma eventually_nhds:
316   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
317 unfolding nhds_def
318 proof (rule eventually_Abs_net, rule is_filter.intro)
319   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
320   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
321 next
322   fix P Q
323   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
324      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
325   then obtain S T where
326     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
327     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
328   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
329     by (simp add: open_Int)
330   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
331 qed auto
333 lemma eventually_nhds_metric:
334   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
335 unfolding eventually_nhds open_dist
336 apply safe
337 apply fast
338 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
339 apply clarsimp
340 apply (rule_tac x="d - dist x a" in exI, clarsimp)
341 apply (simp only: less_diff_eq)
342 apply (erule le_less_trans [OF dist_triangle])
343 done
345 lemma eventually_at_topological:
346   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
347 unfolding at_def eventually_within eventually_nhds by simp
349 lemma eventually_at:
350   fixes a :: "'a::metric_space"
351   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
352 unfolding at_def eventually_within eventually_nhds_metric by auto
355 subsection {* Boundedness *}
357 definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
358   "Bfun f net = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) net)"
360 lemma BfunI:
361   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) net" shows "Bfun f net"
362 unfolding Bfun_def
363 proof (intro exI conjI allI)
364   show "0 < max K 1" by simp
365 next
366   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
367     using K by (rule eventually_elim1, simp)
368 qed
370 lemma BfunE:
371   assumes "Bfun f net"
372   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
373 using assms unfolding Bfun_def by fast
376 subsection {* Convergence to Zero *}
378 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
379   "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)"
381 lemma ZfunI:
382   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net"
383 unfolding Zfun_def by simp
385 lemma ZfunD:
386   "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
387 unfolding Zfun_def by simp
389 lemma Zfun_ssubst:
390   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
391 unfolding Zfun_def by (auto elim!: eventually_rev_mp)
393 lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
394 unfolding Zfun_def by simp
396 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net"
397 unfolding Zfun_def by simp
399 lemma Zfun_imp_Zfun:
400   assumes f: "Zfun f net"
401   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
402   shows "Zfun (\<lambda>x. g x) net"
403 proof (cases)
404   assume K: "0 < K"
405   show ?thesis
406   proof (rule ZfunI)
407     fix r::real assume "0 < r"
408     hence "0 < r / K"
409       using K by (rule divide_pos_pos)
410     then have "eventually (\<lambda>x. norm (f x) < r / K) net"
411       using ZfunD [OF f] by fast
412     with g show "eventually (\<lambda>x. norm (g x) < r) net"
413     proof (rule eventually_elim2)
414       fix x
415       assume *: "norm (g x) \<le> norm (f x) * K"
416       assume "norm (f x) < r / K"
417       hence "norm (f x) * K < r"
418         by (simp add: pos_less_divide_eq K)
419       thus "norm (g x) < r"
420         by (simp add: order_le_less_trans [OF *])
421     qed
422   qed
423 next
424   assume "\<not> 0 < K"
425   hence K: "K \<le> 0" by (simp only: not_less)
426   show ?thesis
427   proof (rule ZfunI)
428     fix r :: real
429     assume "0 < r"
430     from g show "eventually (\<lambda>x. norm (g x) < r) net"
431     proof (rule eventually_elim1)
432       fix x
433       assume "norm (g x) \<le> norm (f x) * K"
434       also have "\<dots> \<le> norm (f x) * 0"
435         using K norm_ge_zero by (rule mult_left_mono)
436       finally show "norm (g x) < r"
437         using `0 < r` by simp
438     qed
439   qed
440 qed
442 lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net"
443 by (erule_tac K="1" in Zfun_imp_Zfun, simp)
446   assumes f: "Zfun f net" and g: "Zfun g net"
447   shows "Zfun (\<lambda>x. f x + g x) net"
448 proof (rule ZfunI)
449   fix r::real assume "0 < r"
450   hence r: "0 < r / 2" by simp
451   have "eventually (\<lambda>x. norm (f x) < r/2) net"
452     using f r by (rule ZfunD)
453   moreover
454   have "eventually (\<lambda>x. norm (g x) < r/2) net"
455     using g r by (rule ZfunD)
456   ultimately
457   show "eventually (\<lambda>x. norm (f x + g x) < r) net"
458   proof (rule eventually_elim2)
459     fix x
460     assume *: "norm (f x) < r/2" "norm (g x) < r/2"
461     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
462       by (rule norm_triangle_ineq)
463     also have "\<dots> < r/2 + r/2"
464       using * by (rule add_strict_mono)
465     finally show "norm (f x + g x) < r"
466       by simp
467   qed
468 qed
470 lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. - f x) net"
471 unfolding Zfun_def by simp
473 lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) net"
474 by (simp only: diff_minus Zfun_add Zfun_minus)
476 lemma (in bounded_linear) Zfun:
477   assumes g: "Zfun g net"
478   shows "Zfun (\<lambda>x. f (g x)) net"
479 proof -
480   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
481     using bounded by fast
482   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
483     by simp
484   with g show ?thesis
485     by (rule Zfun_imp_Zfun)
486 qed
488 lemma (in bounded_bilinear) Zfun:
489   assumes f: "Zfun f net"
490   assumes g: "Zfun g net"
491   shows "Zfun (\<lambda>x. f x ** g x) net"
492 proof (rule ZfunI)
493   fix r::real assume r: "0 < r"
494   obtain K where K: "0 < K"
495     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
496     using pos_bounded by fast
497   from K have K': "0 < inverse K"
498     by (rule positive_imp_inverse_positive)
499   have "eventually (\<lambda>x. norm (f x) < r) net"
500     using f r by (rule ZfunD)
501   moreover
502   have "eventually (\<lambda>x. norm (g x) < inverse K) net"
503     using g K' by (rule ZfunD)
504   ultimately
505   show "eventually (\<lambda>x. norm (f x ** g x) < r) net"
506   proof (rule eventually_elim2)
507     fix x
508     assume *: "norm (f x) < r" "norm (g x) < inverse K"
509     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
510       by (rule norm_le)
511     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
512       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
513     also from K have "r * inverse K * K = r"
514       by simp
515     finally show "norm (f x ** g x) < r" .
516   qed
517 qed
519 lemma (in bounded_bilinear) Zfun_left:
520   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net"
521 by (rule bounded_linear_left [THEN bounded_linear.Zfun])
523 lemma (in bounded_bilinear) Zfun_right:
524   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net"
525 by (rule bounded_linear_right [THEN bounded_linear.Zfun])
527 lemmas Zfun_mult = mult.Zfun
528 lemmas Zfun_mult_right = mult.Zfun_right
529 lemmas Zfun_mult_left = mult.Zfun_left
532 subsection {* Limits *}
534 definition tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
535     (infixr "--->" 55) where
536   "(f ---> l) net \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
538 ML {*
539 structure Tendsto_Intros = Named_Thms
540 (
541   val name = "tendsto_intros"
542   val description = "introduction rules for tendsto"
543 )
544 *}
546 setup Tendsto_Intros.setup
548 lemma tendsto_mono: "net \<le> net' \<Longrightarrow> (f ---> l) net' \<Longrightarrow> (f ---> l) net"
549 unfolding tendsto_def le_net_def by fast
551 lemma topological_tendstoI:
552   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net)
553     \<Longrightarrow> (f ---> l) net"
554   unfolding tendsto_def by auto
556 lemma topological_tendstoD:
557   "(f ---> l) net \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
558   unfolding tendsto_def by auto
560 lemma tendstoI:
561   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
562   shows "(f ---> l) net"
563 apply (rule topological_tendstoI)
564 apply (simp add: open_dist)
565 apply (drule (1) bspec, clarify)
566 apply (drule assms)
567 apply (erule eventually_elim1, simp)
568 done
570 lemma tendstoD:
571   "(f ---> l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
572 apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
573 apply (clarsimp simp add: open_dist)
574 apply (rule_tac x="e - dist x l" in exI, clarsimp)
575 apply (simp only: less_diff_eq)
576 apply (erule le_less_trans [OF dist_triangle])
577 apply simp
578 apply simp
579 done
581 lemma tendsto_iff:
582   "(f ---> l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
583 using tendstoI tendstoD by fast
585 lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
586 by (simp only: tendsto_iff Zfun_def dist_norm)
588 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
589 unfolding tendsto_def eventually_at_topological by auto
591 lemma tendsto_ident_at_within [tendsto_intros]:
592   "((\<lambda>x. x) ---> a) (at a within S)"
593 unfolding tendsto_def eventually_within eventually_at_topological by auto
595 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) net"
596 by (simp add: tendsto_def)
598 lemma tendsto_const_iff:
599   fixes k l :: "'a::metric_space"
600   assumes "net \<noteq> bot" shows "((\<lambda>n. k) ---> l) net \<longleftrightarrow> k = l"
601 apply (safe intro!: tendsto_const)
602 apply (rule ccontr)
603 apply (drule_tac e="dist k l" in tendstoD)
604 apply (simp add: zero_less_dist_iff)
605 apply (simp add: eventually_False assms)
606 done
608 lemma tendsto_dist [tendsto_intros]:
609   assumes f: "(f ---> l) net" and g: "(g ---> m) net"
610   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
611 proof (rule tendstoI)
612   fix e :: real assume "0 < e"
613   hence e2: "0 < e/2" by simp
614   from tendstoD [OF f e2] tendstoD [OF g e2]
615   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
616   proof (rule eventually_elim2)
617     fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
618     then show "dist (dist (f x) (g x)) (dist l m) < e"
619       unfolding dist_real_def
620       using dist_triangle2 [of "f x" "g x" "l"]
621       using dist_triangle2 [of "g x" "l" "m"]
622       using dist_triangle3 [of "l" "m" "f x"]
623       using dist_triangle [of "f x" "m" "g x"]
624       by arith
625   qed
626 qed
628 lemma norm_conv_dist: "norm x = dist x 0"
629 unfolding dist_norm by simp
631 lemma tendsto_norm [tendsto_intros]:
632   "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
633 unfolding norm_conv_dist by (intro tendsto_intros)
635 lemma tendsto_norm_zero:
636   "(f ---> 0) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) net"
637 by (drule tendsto_norm, simp)
639 lemma tendsto_norm_zero_cancel:
640   "((\<lambda>x. norm (f x)) ---> 0) net \<Longrightarrow> (f ---> 0) net"
641 unfolding tendsto_iff dist_norm by simp
643 lemma tendsto_norm_zero_iff:
644   "((\<lambda>x. norm (f x)) ---> 0) net \<longleftrightarrow> (f ---> 0) net"
645 unfolding tendsto_iff dist_norm by simp
648   fixes a b c d :: "'a::ab_group_add"
649   shows "(a + c) - (b + d) = (a - b) + (c - d)"
650 by simp
652 lemma minus_diff_minus:
653   fixes a b :: "'a::ab_group_add"
654   shows "(- a) - (- b) = - (a - b)"
655 by simp
657 lemma tendsto_add [tendsto_intros]:
658   fixes a b :: "'a::real_normed_vector"
659   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
662 lemma tendsto_minus [tendsto_intros]:
663   fixes a :: "'a::real_normed_vector"
664   shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
665 by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
667 lemma tendsto_minus_cancel:
668   fixes a :: "'a::real_normed_vector"
669   shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
670 by (drule tendsto_minus, simp)
672 lemma tendsto_diff [tendsto_intros]:
673   fixes a b :: "'a::real_normed_vector"
674   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
675 by (simp add: diff_minus tendsto_add tendsto_minus)
677 lemma tendsto_setsum [tendsto_intros]:
678   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
679   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) net"
680   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) net"
681 proof (cases "finite S")
682   assume "finite S" thus ?thesis using assms
683   proof (induct set: finite)
684     case empty show ?case
685       by (simp add: tendsto_const)
686   next
687     case (insert i F) thus ?case
689   qed
690 next
691   assume "\<not> finite S" thus ?thesis
692     by (simp add: tendsto_const)
693 qed
695 lemma (in bounded_linear) tendsto [tendsto_intros]:
696   "(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
697 by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
699 lemma (in bounded_bilinear) tendsto [tendsto_intros]:
700   "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
701 by (simp only: tendsto_Zfun_iff prod_diff_prod
702                Zfun_add Zfun Zfun_left Zfun_right)
705 subsection {* Continuity of Inverse *}
707 lemma (in bounded_bilinear) Zfun_prod_Bfun:
708   assumes f: "Zfun f net"
709   assumes g: "Bfun g net"
710   shows "Zfun (\<lambda>x. f x ** g x) net"
711 proof -
712   obtain K where K: "0 \<le> K"
713     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
714     using nonneg_bounded by fast
715   obtain B where B: "0 < B"
716     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net"
717     using g by (rule BfunE)
718   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net"
719   using norm_g proof (rule eventually_elim1)
720     fix x
721     assume *: "norm (g x) \<le> B"
722     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
723       by (rule norm_le)
724     also have "\<dots> \<le> norm (f x) * B * K"
725       by (intro mult_mono' order_refl norm_g norm_ge_zero
726                 mult_nonneg_nonneg K *)
727     also have "\<dots> = norm (f x) * (B * K)"
728       by (rule mult_assoc)
729     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
730   qed
731   with f show ?thesis
732     by (rule Zfun_imp_Zfun)
733 qed
735 lemma (in bounded_bilinear) flip:
736   "bounded_bilinear (\<lambda>x y. y ** x)"
737 apply default
738 apply (rule add_right)
739 apply (rule add_left)
740 apply (rule scaleR_right)
741 apply (rule scaleR_left)
742 apply (subst mult_commute)
743 using bounded by fast
745 lemma (in bounded_bilinear) Bfun_prod_Zfun:
746   assumes f: "Bfun f net"
747   assumes g: "Zfun g net"
748   shows "Zfun (\<lambda>x. f x ** g x) net"
749 using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
751 lemma inverse_diff_inverse:
752   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
753    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
754 by (simp add: algebra_simps)
756 lemma Bfun_inverse_lemma:
757   fixes x :: "'a::real_normed_div_algebra"
758   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
759 apply (subst nonzero_norm_inverse, clarsimp)
760 apply (erule (1) le_imp_inverse_le)
761 done
763 lemma Bfun_inverse:
764   fixes a :: "'a::real_normed_div_algebra"
765   assumes f: "(f ---> a) net"
766   assumes a: "a \<noteq> 0"
767   shows "Bfun (\<lambda>x. inverse (f x)) net"
768 proof -
769   from a have "0 < norm a" by simp
770   hence "\<exists>r>0. r < norm a" by (rule dense)
771   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
772   have "eventually (\<lambda>x. dist (f x) a < r) net"
773     using tendstoD [OF f r1] by fast
774   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
775   proof (rule eventually_elim1)
776     fix x
777     assume "dist (f x) a < r"
778     hence 1: "norm (f x - a) < r"
779       by (simp add: dist_norm)
780     hence 2: "f x \<noteq> 0" using r2 by auto
781     hence "norm (inverse (f x)) = inverse (norm (f x))"
782       by (rule nonzero_norm_inverse)
783     also have "\<dots> \<le> inverse (norm a - r)"
784     proof (rule le_imp_inverse_le)
785       show "0 < norm a - r" using r2 by simp
786     next
787       have "norm a - norm (f x) \<le> norm (a - f x)"
788         by (rule norm_triangle_ineq2)
789       also have "\<dots> = norm (f x - a)"
790         by (rule norm_minus_commute)
791       also have "\<dots> < r" using 1 .
792       finally show "norm a - r \<le> norm (f x)" by simp
793     qed
794     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
795   qed
796   thus ?thesis by (rule BfunI)
797 qed
799 lemma tendsto_inverse_lemma:
800   fixes a :: "'a::real_normed_div_algebra"
801   shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
802          \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
803 apply (subst tendsto_Zfun_iff)
804 apply (rule Zfun_ssubst)
805 apply (erule eventually_elim1)
806 apply (erule (1) inverse_diff_inverse)
807 apply (rule Zfun_minus)
808 apply (rule Zfun_mult_left)
809 apply (rule mult.Bfun_prod_Zfun)
810 apply (erule (1) Bfun_inverse)
811 apply (simp add: tendsto_Zfun_iff)
812 done
814 lemma tendsto_inverse [tendsto_intros]:
815   fixes a :: "'a::real_normed_div_algebra"
816   assumes f: "(f ---> a) net"
817   assumes a: "a \<noteq> 0"
818   shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
819 proof -
820   from a have "0 < norm a" by simp
821   with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
822     by (rule tendstoD)
823   then have "eventually (\<lambda>x. f x \<noteq> 0) net"
824     unfolding dist_norm by (auto elim!: eventually_elim1)
825   with f a show ?thesis
826     by (rule tendsto_inverse_lemma)
827 qed
829 lemma tendsto_divide [tendsto_intros]:
830   fixes a b :: "'a::real_normed_field"
831   shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
832     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
833 by (simp add: mult.tendsto tendsto_inverse divide_inverse)
835 lemma tendsto_unique:
836   fixes f :: "'a \<Rightarrow> 'b::t2_space"
837   assumes "\<not> trivial_limit net"  "(f ---> l) net"  "(f ---> l') net"
838   shows "l = l'"
839 proof (rule ccontr)
840   assume "l \<noteq> l'"
841   obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
842     using hausdorff [OF `l \<noteq> l'`] by fast
843   have "eventually (\<lambda>x. f x \<in> U) net"
844     using `(f ---> l) net` `open U` `l \<in> U` by (rule topological_tendstoD)
845   moreover
846   have "eventually (\<lambda>x. f x \<in> V) net"
847     using `(f ---> l') net` `open V` `l' \<in> V` by (rule topological_tendstoD)
848   ultimately
849   have "eventually (\<lambda>x. False) net"
850   proof (rule eventually_elim2)
851     fix x
852     assume "f x \<in> U" "f x \<in> V"
853     hence "f x \<in> U \<inter> V" by simp
854     with `U \<inter> V = {}` show "False" by simp
855   qed
856   with `\<not> trivial_limit net` show "False"
857     by (simp add: trivial_limit_def)
858 qed
860 end