src/HOL/Limits.thy
 author huffman Mon May 03 17:39:46 2010 -0700 (2010-05-03) changeset 36655 88f0125c3bd2 parent 36654 7c8eb32724ce child 36656 fec55067ae9b permissions -rw-r--r--
remove unneeded premise
1 (*  Title       : Limits.thy
2     Author      : Brian Huffman
3 *)
5 header {* Filters and Limits *}
7 theory Limits
8 imports RealVector RComplete
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
41   eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where
42   [code del]: "eventually P net \<longleftrightarrow> Rep_net net P"
44 lemma eventually_Abs_net:
45   assumes "is_filter net" shows "eventually P (Abs_net net) = net P"
46 unfolding eventually_def using assms by (simp add: Abs_net_inverse)
48 lemma expand_net_eq:
49   shows "net = net' \<longleftrightarrow> (\<forall>P. eventually P net = eventually P net')"
50 unfolding Rep_net_inject [symmetric] expand_fun_eq eventually_def ..
52 lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
53 unfolding eventually_def
54 by (rule is_filter.True [OF is_filter_Rep_net])
56 lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P net"
57 proof -
58   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
59   thus "eventually P net" by simp
60 qed
62 lemma eventually_mono:
63   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
64 unfolding eventually_def
65 by (rule is_filter.mono [OF is_filter_Rep_net])
67 lemma eventually_conj:
68   assumes P: "eventually (\<lambda>x. P x) net"
69   assumes Q: "eventually (\<lambda>x. Q x) net"
70   shows "eventually (\<lambda>x. P x \<and> Q x) net"
71 using assms unfolding eventually_def
72 by (rule is_filter.conj [OF is_filter_Rep_net])
74 lemma eventually_mp:
75   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
76   assumes "eventually (\<lambda>x. P x) net"
77   shows "eventually (\<lambda>x. Q x) net"
78 proof (rule eventually_mono)
79   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
80   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) net"
81     using assms by (rule eventually_conj)
82 qed
84 lemma eventually_rev_mp:
85   assumes "eventually (\<lambda>x. P x) net"
86   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
87   shows "eventually (\<lambda>x. Q x) net"
88 using assms(2) assms(1) by (rule eventually_mp)
90 lemma eventually_conj_iff:
91   "eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
92 by (auto intro: eventually_conj elim: eventually_rev_mp)
94 lemma eventually_elim1:
95   assumes "eventually (\<lambda>i. P i) net"
96   assumes "\<And>i. P i \<Longrightarrow> Q i"
97   shows "eventually (\<lambda>i. Q i) net"
98 using assms by (auto elim!: eventually_rev_mp)
100 lemma eventually_elim2:
101   assumes "eventually (\<lambda>i. P i) net"
102   assumes "eventually (\<lambda>i. Q i) net"
103   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
104   shows "eventually (\<lambda>i. R i) net"
105 using assms by (auto elim!: eventually_rev_mp)
108 subsection {* Finer-than relation *}
110 text {* @{term "net \<le> net'"} means that @{term net} is finer than
111 @{term net'}. *}
113 instantiation net :: (type) complete_lattice
114 begin
116 definition
117   le_net_def [code del]:
118     "net \<le> net' \<longleftrightarrow> (\<forall>P. eventually P net' \<longrightarrow> eventually P net)"
120 definition
121   less_net_def [code del]:
122     "(net :: 'a net) < net' \<longleftrightarrow> net \<le> net' \<and> \<not> net' \<le> net"
124 definition
125   top_net_def [code del]:
126     "top = Abs_net (\<lambda>P. \<forall>x. P x)"
128 definition
129   bot_net_def [code del]:
130     "bot = Abs_net (\<lambda>P. True)"
132 definition
133   sup_net_def [code del]:
134     "sup net net' = Abs_net (\<lambda>P. eventually P net \<and> eventually P net')"
136 definition
137   inf_net_def [code del]:
138     "inf a b = Abs_net
139       (\<lambda>P. \<exists>Q R. eventually Q a \<and> eventually R b \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
141 definition
142   Sup_net_def [code del]:
143     "Sup A = Abs_net (\<lambda>P. \<forall>net\<in>A. eventually P net)"
145 definition
146   Inf_net_def [code del]:
147     "Inf A = Sup {x::'a net. \<forall>y\<in>A. x \<le> y}"
149 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
150 unfolding top_net_def
151 by (rule eventually_Abs_net, rule is_filter.intro, auto)
153 lemma eventually_bot [simp]: "eventually P bot"
154 unfolding bot_net_def
155 by (subst eventually_Abs_net, rule is_filter.intro, auto)
157 lemma eventually_sup:
158   "eventually P (sup net net') \<longleftrightarrow> eventually P net \<and> eventually P net'"
159 unfolding sup_net_def
160 by (rule eventually_Abs_net, rule is_filter.intro)
161    (auto elim!: eventually_rev_mp)
163 lemma eventually_inf:
164   "eventually P (inf a b) \<longleftrightarrow>
165    (\<exists>Q R. eventually Q a \<and> eventually R b \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
166 unfolding inf_net_def
167 apply (rule eventually_Abs_net, rule is_filter.intro)
168 apply (fast intro: eventually_True)
169 apply clarify
170 apply (intro exI conjI)
171 apply (erule (1) eventually_conj)
172 apply (erule (1) eventually_conj)
173 apply simp
174 apply auto
175 done
177 lemma eventually_Sup:
178   "eventually P (Sup A) \<longleftrightarrow> (\<forall>net\<in>A. eventually P net)"
179 unfolding Sup_net_def
180 apply (rule eventually_Abs_net, rule is_filter.intro)
181 apply (auto intro: eventually_conj elim!: eventually_rev_mp)
182 done
184 instance proof
185   fix x y :: "'a net" show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
186     by (rule less_net_def)
187 next
188   fix x :: "'a net" show "x \<le> x"
189     unfolding le_net_def by simp
190 next
191   fix x y z :: "'a net" assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
192     unfolding le_net_def by simp
193 next
194   fix x y :: "'a net" assume "x \<le> y" and "y \<le> x" thus "x = y"
195     unfolding le_net_def expand_net_eq by fast
196 next
197   fix x :: "'a net" show "x \<le> top"
198     unfolding le_net_def eventually_top by (simp add: always_eventually)
199 next
200   fix x :: "'a net" show "bot \<le> x"
201     unfolding le_net_def by simp
202 next
203   fix x y :: "'a net" show "x \<le> sup x y" and "y \<le> sup x y"
204     unfolding le_net_def eventually_sup by simp_all
205 next
206   fix x y z :: "'a net" assume "x \<le> z" and "y \<le> z" thus "sup x y \<le> z"
207     unfolding le_net_def eventually_sup by simp
208 next
209   fix x y :: "'a net" show "inf x y \<le> x" and "inf x y \<le> y"
210     unfolding le_net_def eventually_inf by (auto intro: eventually_True)
211 next
212   fix x y z :: "'a net" assume "x \<le> y" and "x \<le> z" thus "x \<le> inf y z"
213     unfolding le_net_def eventually_inf
214     by (auto elim!: eventually_mono intro: eventually_conj)
215 next
216   fix x :: "'a net" and A assume "x \<in> A" thus "x \<le> Sup A"
217     unfolding le_net_def eventually_Sup by simp
218 next
219   fix A and y :: "'a net" assume "\<And>x. x \<in> A \<Longrightarrow> x \<le> y" thus "Sup A \<le> y"
220     unfolding le_net_def eventually_Sup by simp
221 next
222   fix z :: "'a net" and A assume "z \<in> A" thus "Inf A \<le> z"
223     unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp
224 next
225   fix A and x :: "'a net" assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" thus "x \<le> Inf A"
226     unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp
227 qed
229 end
231 lemma net_leD:
232   "net \<le> net' \<Longrightarrow> eventually P net' \<Longrightarrow> eventually P net"
233 unfolding le_net_def by simp
235 lemma net_leI:
236   "(\<And>P. eventually P net' \<Longrightarrow> eventually P net) \<Longrightarrow> net \<le> net'"
237 unfolding le_net_def by simp
239 lemma eventually_False:
240   "eventually (\<lambda>x. False) net \<longleftrightarrow> net = bot"
241 unfolding expand_net_eq by (auto elim: eventually_rev_mp)
244 subsection {* Map function for nets *}
246 definition
247   netmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a net \<Rightarrow> 'b net"
248 where [code del]:
249   "netmap f net = Abs_net (\<lambda>P. eventually (\<lambda>x. P (f x)) net)"
251 lemma eventually_netmap:
252   "eventually P (netmap f net) = eventually (\<lambda>x. P (f x)) net"
253 unfolding netmap_def
254 apply (rule eventually_Abs_net)
255 apply (rule is_filter.intro)
256 apply (auto elim!: eventually_rev_mp)
257 done
259 lemma netmap_ident: "netmap (\<lambda>x. x) net = net"
260 by (simp add: expand_net_eq eventually_netmap)
262 lemma netmap_netmap: "netmap f (netmap g net) = netmap (\<lambda>x. f (g x)) net"
263 by (simp add: expand_net_eq eventually_netmap)
265 lemma netmap_mono: "net \<le> net' \<Longrightarrow> netmap f net \<le> netmap f net'"
266 unfolding le_net_def eventually_netmap by simp
268 lemma netmap_bot [simp]: "netmap f bot = bot"
269 by (simp add: expand_net_eq eventually_netmap)
272 subsection {* Standard Nets *}
274 definition
275   sequentially :: "nat net"
276 where [code del]:
277   "sequentially = Abs_net (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
279 definition
280   within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70)
281 where [code del]:
282   "net within S = Abs_net (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net)"
284 definition
285   nhds :: "'a::topological_space \<Rightarrow> 'a net"
286 where [code del]:
287   "nhds a = Abs_net (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
289 definition
290   at :: "'a::topological_space \<Rightarrow> 'a net"
291 where [code del]:
292   "at a = nhds a within - {a}"
294 lemma eventually_sequentially:
295   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
296 unfolding sequentially_def
297 proof (rule eventually_Abs_net, rule is_filter.intro)
298   fix P Q :: "nat \<Rightarrow> bool"
299   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
300   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
301   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
302   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
303 qed auto
305 lemma eventually_within:
306   "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
307 unfolding within_def
308 by (rule eventually_Abs_net, rule is_filter.intro)
309    (auto elim!: eventually_rev_mp)
311 lemma within_UNIV: "net within UNIV = net"
312   unfolding expand_net_eq eventually_within by simp
314 lemma eventually_nhds:
315   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
316 unfolding nhds_def
317 proof (rule eventually_Abs_net, rule is_filter.intro)
318   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
319   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
320 next
321   fix P Q
322   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
323      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
324   then obtain S T where
325     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
326     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
327   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
328     by (simp add: open_Int)
329   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
330 qed auto
332 lemma eventually_at_topological:
333   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
334 unfolding at_def eventually_within eventually_nhds by simp
336 lemma eventually_at:
337   fixes a :: "'a::metric_space"
338   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
339 unfolding eventually_at_topological open_dist
340 apply safe
341 apply fast
342 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
343 apply clarsimp
344 apply (rule_tac x="d - dist x a" in exI, clarsimp)
345 apply (simp only: less_diff_eq)
346 apply (erule le_less_trans [OF dist_triangle])
347 done
350 subsection {* Boundedness *}
352 definition
353   Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
354   [code del]: "Bfun f net = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) net)"
356 lemma BfunI:
357   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) net" shows "Bfun f net"
358 unfolding Bfun_def
359 proof (intro exI conjI allI)
360   show "0 < max K 1" by simp
361 next
362   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
363     using K by (rule eventually_elim1, simp)
364 qed
366 lemma BfunE:
367   assumes "Bfun f net"
368   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
369 using assms unfolding Bfun_def by fast
372 subsection {* Convergence to Zero *}
374 definition
375   Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
376   [code del]: "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)"
378 lemma ZfunI:
379   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net"
380 unfolding Zfun_def by simp
382 lemma ZfunD:
383   "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
384 unfolding Zfun_def by simp
386 lemma Zfun_ssubst:
387   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
388 unfolding Zfun_def by (auto elim!: eventually_rev_mp)
390 lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
391 unfolding Zfun_def by simp
393 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net"
394 unfolding Zfun_def by simp
396 lemma Zfun_imp_Zfun:
397   assumes f: "Zfun f net"
398   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
399   shows "Zfun (\<lambda>x. g x) net"
400 proof (cases)
401   assume K: "0 < K"
402   show ?thesis
403   proof (rule ZfunI)
404     fix r::real assume "0 < r"
405     hence "0 < r / K"
406       using K by (rule divide_pos_pos)
407     then have "eventually (\<lambda>x. norm (f x) < r / K) net"
408       using ZfunD [OF f] by fast
409     with g show "eventually (\<lambda>x. norm (g x) < r) net"
410     proof (rule eventually_elim2)
411       fix x
412       assume *: "norm (g x) \<le> norm (f x) * K"
413       assume "norm (f x) < r / K"
414       hence "norm (f x) * K < r"
415         by (simp add: pos_less_divide_eq K)
416       thus "norm (g x) < r"
417         by (simp add: order_le_less_trans [OF *])
418     qed
419   qed
420 next
421   assume "\<not> 0 < K"
422   hence K: "K \<le> 0" by (simp only: not_less)
423   show ?thesis
424   proof (rule ZfunI)
425     fix r :: real
426     assume "0 < r"
427     from g show "eventually (\<lambda>x. norm (g x) < r) net"
428     proof (rule eventually_elim1)
429       fix x
430       assume "norm (g x) \<le> norm (f x) * K"
431       also have "\<dots> \<le> norm (f x) * 0"
432         using K norm_ge_zero by (rule mult_left_mono)
433       finally show "norm (g x) < r"
434         using `0 < r` by simp
435     qed
436   qed
437 qed
439 lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net"
440 by (erule_tac K="1" in Zfun_imp_Zfun, simp)
443   assumes f: "Zfun f net" and g: "Zfun g net"
444   shows "Zfun (\<lambda>x. f x + g x) net"
445 proof (rule ZfunI)
446   fix r::real assume "0 < r"
447   hence r: "0 < r / 2" by simp
448   have "eventually (\<lambda>x. norm (f x) < r/2) net"
449     using f r by (rule ZfunD)
450   moreover
451   have "eventually (\<lambda>x. norm (g x) < r/2) net"
452     using g r by (rule ZfunD)
453   ultimately
454   show "eventually (\<lambda>x. norm (f x + g x) < r) net"
455   proof (rule eventually_elim2)
456     fix x
457     assume *: "norm (f x) < r/2" "norm (g x) < r/2"
458     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
459       by (rule norm_triangle_ineq)
460     also have "\<dots> < r/2 + r/2"
461       using * by (rule add_strict_mono)
462     finally show "norm (f x + g x) < r"
463       by simp
464   qed
465 qed
467 lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. - f x) net"
468 unfolding Zfun_def by simp
470 lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) net"
471 by (simp only: diff_minus Zfun_add Zfun_minus)
473 lemma (in bounded_linear) Zfun:
474   assumes g: "Zfun g net"
475   shows "Zfun (\<lambda>x. f (g x)) net"
476 proof -
477   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
478     using bounded by fast
479   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
480     by simp
481   with g show ?thesis
482     by (rule Zfun_imp_Zfun)
483 qed
485 lemma (in bounded_bilinear) Zfun:
486   assumes f: "Zfun f net"
487   assumes g: "Zfun g net"
488   shows "Zfun (\<lambda>x. f x ** g x) net"
489 proof (rule ZfunI)
490   fix r::real assume r: "0 < r"
491   obtain K where K: "0 < K"
492     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
493     using pos_bounded by fast
494   from K have K': "0 < inverse K"
495     by (rule positive_imp_inverse_positive)
496   have "eventually (\<lambda>x. norm (f x) < r) net"
497     using f r by (rule ZfunD)
498   moreover
499   have "eventually (\<lambda>x. norm (g x) < inverse K) net"
500     using g K' by (rule ZfunD)
501   ultimately
502   show "eventually (\<lambda>x. norm (f x ** g x) < r) net"
503   proof (rule eventually_elim2)
504     fix x
505     assume *: "norm (f x) < r" "norm (g x) < inverse K"
506     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
507       by (rule norm_le)
508     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
509       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
510     also from K have "r * inverse K * K = r"
511       by simp
512     finally show "norm (f x ** g x) < r" .
513   qed
514 qed
516 lemma (in bounded_bilinear) Zfun_left:
517   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net"
518 by (rule bounded_linear_left [THEN bounded_linear.Zfun])
520 lemma (in bounded_bilinear) Zfun_right:
521   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net"
522 by (rule bounded_linear_right [THEN bounded_linear.Zfun])
524 lemmas Zfun_mult = mult.Zfun
525 lemmas Zfun_mult_right = mult.Zfun_right
526 lemmas Zfun_mult_left = mult.Zfun_left
529 subsection {* Limits *}
531 definition
532   tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
533     (infixr "--->" 55)
534 where [code del]:
535   "(f ---> l) net \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
537 ML {*
538 structure Tendsto_Intros = Named_Thms
539 (
540   val name = "tendsto_intros"
541   val description = "introduction rules for tendsto"
542 )
543 *}
545 setup Tendsto_Intros.setup
547 lemma topological_tendstoI:
548   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net)
549     \<Longrightarrow> (f ---> l) net"
550   unfolding tendsto_def by auto
552 lemma topological_tendstoD:
553   "(f ---> l) net \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
554   unfolding tendsto_def by auto
556 lemma tendstoI:
557   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
558   shows "(f ---> l) net"
559 apply (rule topological_tendstoI)
560 apply (simp add: open_dist)
561 apply (drule (1) bspec, clarify)
562 apply (drule assms)
563 apply (erule eventually_elim1, simp)
564 done
566 lemma tendstoD:
567   "(f ---> l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
568 apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
569 apply (clarsimp simp add: open_dist)
570 apply (rule_tac x="e - dist x l" in exI, clarsimp)
571 apply (simp only: less_diff_eq)
572 apply (erule le_less_trans [OF dist_triangle])
573 apply simp
574 apply simp
575 done
577 lemma tendsto_iff:
578   "(f ---> l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
579 using tendstoI tendstoD by fast
581 lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
582 by (simp only: tendsto_iff Zfun_def dist_norm)
584 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
585 unfolding tendsto_def eventually_at_topological by auto
587 lemma tendsto_ident_at_within [tendsto_intros]:
588   "((\<lambda>x. x) ---> a) (at a within S)"
589 unfolding tendsto_def eventually_within eventually_at_topological by auto
591 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) net"
592 by (simp add: tendsto_def)
594 lemma tendsto_dist [tendsto_intros]:
595   assumes f: "(f ---> l) net" and g: "(g ---> m) net"
596   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
597 proof (rule tendstoI)
598   fix e :: real assume "0 < e"
599   hence e2: "0 < e/2" by simp
600   from tendstoD [OF f e2] tendstoD [OF g e2]
601   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
602   proof (rule eventually_elim2)
603     fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
604     then show "dist (dist (f x) (g x)) (dist l m) < e"
605       unfolding dist_real_def
606       using dist_triangle2 [of "f x" "g x" "l"]
607       using dist_triangle2 [of "g x" "l" "m"]
608       using dist_triangle3 [of "l" "m" "f x"]
609       using dist_triangle [of "f x" "m" "g x"]
610       by arith
611   qed
612 qed
614 lemma tendsto_norm [tendsto_intros]:
615   "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
616 apply (simp add: tendsto_iff dist_norm, safe)
617 apply (drule_tac x="e" in spec, safe)
618 apply (erule eventually_elim1)
619 apply (erule order_le_less_trans [OF norm_triangle_ineq3])
620 done
623   fixes a b c d :: "'a::ab_group_add"
624   shows "(a + c) - (b + d) = (a - b) + (c - d)"
625 by simp
627 lemma minus_diff_minus:
628   fixes a b :: "'a::ab_group_add"
629   shows "(- a) - (- b) = - (a - b)"
630 by simp
632 lemma tendsto_add [tendsto_intros]:
633   fixes a b :: "'a::real_normed_vector"
634   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
637 lemma tendsto_minus [tendsto_intros]:
638   fixes a :: "'a::real_normed_vector"
639   shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
640 by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
642 lemma tendsto_minus_cancel:
643   fixes a :: "'a::real_normed_vector"
644   shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
645 by (drule tendsto_minus, simp)
647 lemma tendsto_diff [tendsto_intros]:
648   fixes a b :: "'a::real_normed_vector"
649   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
650 by (simp add: diff_minus tendsto_add tendsto_minus)
652 lemma tendsto_setsum [tendsto_intros]:
653   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
654   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) net"
655   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) net"
656 proof (cases "finite S")
657   assume "finite S" thus ?thesis using assms
658   proof (induct set: finite)
659     case empty show ?case
660       by (simp add: tendsto_const)
661   next
662     case (insert i F) thus ?case
664   qed
665 next
666   assume "\<not> finite S" thus ?thesis
667     by (simp add: tendsto_const)
668 qed
670 lemma (in bounded_linear) tendsto [tendsto_intros]:
671   "(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
672 by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
674 lemma (in bounded_bilinear) tendsto [tendsto_intros]:
675   "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
676 by (simp only: tendsto_Zfun_iff prod_diff_prod
677                Zfun_add Zfun Zfun_left Zfun_right)
680 subsection {* Continuity of Inverse *}
682 lemma (in bounded_bilinear) Zfun_prod_Bfun:
683   assumes f: "Zfun f net"
684   assumes g: "Bfun g net"
685   shows "Zfun (\<lambda>x. f x ** g x) net"
686 proof -
687   obtain K where K: "0 \<le> K"
688     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
689     using nonneg_bounded by fast
690   obtain B where B: "0 < B"
691     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net"
692     using g by (rule BfunE)
693   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net"
694   using norm_g proof (rule eventually_elim1)
695     fix x
696     assume *: "norm (g x) \<le> B"
697     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
698       by (rule norm_le)
699     also have "\<dots> \<le> norm (f x) * B * K"
700       by (intro mult_mono' order_refl norm_g norm_ge_zero
701                 mult_nonneg_nonneg K *)
702     also have "\<dots> = norm (f x) * (B * K)"
703       by (rule mult_assoc)
704     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
705   qed
706   with f show ?thesis
707     by (rule Zfun_imp_Zfun)
708 qed
710 lemma (in bounded_bilinear) flip:
711   "bounded_bilinear (\<lambda>x y. y ** x)"
712 apply default
713 apply (rule add_right)
714 apply (rule add_left)
715 apply (rule scaleR_right)
716 apply (rule scaleR_left)
717 apply (subst mult_commute)
718 using bounded by fast
720 lemma (in bounded_bilinear) Bfun_prod_Zfun:
721   assumes f: "Bfun f net"
722   assumes g: "Zfun g net"
723   shows "Zfun (\<lambda>x. f x ** g x) net"
724 using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
726 lemma inverse_diff_inverse:
727   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
728    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
729 by (simp add: algebra_simps)
731 lemma Bfun_inverse_lemma:
732   fixes x :: "'a::real_normed_div_algebra"
733   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
734 apply (subst nonzero_norm_inverse, clarsimp)
735 apply (erule (1) le_imp_inverse_le)
736 done
738 lemma Bfun_inverse:
739   fixes a :: "'a::real_normed_div_algebra"
740   assumes f: "(f ---> a) net"
741   assumes a: "a \<noteq> 0"
742   shows "Bfun (\<lambda>x. inverse (f x)) net"
743 proof -
744   from a have "0 < norm a" by simp
745   hence "\<exists>r>0. r < norm a" by (rule dense)
746   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
747   have "eventually (\<lambda>x. dist (f x) a < r) net"
748     using tendstoD [OF f r1] by fast
749   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
750   proof (rule eventually_elim1)
751     fix x
752     assume "dist (f x) a < r"
753     hence 1: "norm (f x - a) < r"
754       by (simp add: dist_norm)
755     hence 2: "f x \<noteq> 0" using r2 by auto
756     hence "norm (inverse (f x)) = inverse (norm (f x))"
757       by (rule nonzero_norm_inverse)
758     also have "\<dots> \<le> inverse (norm a - r)"
759     proof (rule le_imp_inverse_le)
760       show "0 < norm a - r" using r2 by simp
761     next
762       have "norm a - norm (f x) \<le> norm (a - f x)"
763         by (rule norm_triangle_ineq2)
764       also have "\<dots> = norm (f x - a)"
765         by (rule norm_minus_commute)
766       also have "\<dots> < r" using 1 .
767       finally show "norm a - r \<le> norm (f x)" by simp
768     qed
769     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
770   qed
771   thus ?thesis by (rule BfunI)
772 qed
774 lemma tendsto_inverse_lemma:
775   fixes a :: "'a::real_normed_div_algebra"
776   shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
777          \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
778 apply (subst tendsto_Zfun_iff)
779 apply (rule Zfun_ssubst)
780 apply (erule eventually_elim1)
781 apply (erule (1) inverse_diff_inverse)
782 apply (rule Zfun_minus)
783 apply (rule Zfun_mult_left)
784 apply (rule mult.Bfun_prod_Zfun)
785 apply (erule (1) Bfun_inverse)
786 apply (simp add: tendsto_Zfun_iff)
787 done
789 lemma tendsto_inverse [tendsto_intros]:
790   fixes a :: "'a::real_normed_div_algebra"
791   assumes f: "(f ---> a) net"
792   assumes a: "a \<noteq> 0"
793   shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
794 proof -
795   from a have "0 < norm a" by simp
796   with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
797     by (rule tendstoD)
798   then have "eventually (\<lambda>x. f x \<noteq> 0) net"
799     unfolding dist_norm by (auto elim!: eventually_elim1)
800   with f a show ?thesis
801     by (rule tendsto_inverse_lemma)
802 qed
804 lemma tendsto_divide [tendsto_intros]:
805   fixes a b :: "'a::real_normed_field"
806   shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
807     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
808 by (simp add: mult.tendsto tendsto_inverse divide_inverse)
810 end