src/HOL/Limits.thy
 author huffman Sat May 01 11:46:47 2010 -0700 (2010-05-01) changeset 36630 aa1f8acdcc1c parent 36629 de62713aec6e child 36654 7c8eb32724ce permissions -rw-r--r--
complete_lattice instance for net type
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 {* Standard Nets *}
246 definition
247   sequentially :: "nat net"
248 where [code del]:
249   "sequentially = Abs_net (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
251 definition
252   within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70)
253 where [code del]:
254   "net within S = Abs_net (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net)"
256 definition
257   at :: "'a::topological_space \<Rightarrow> 'a net"
258 where [code del]:
259   "at a = Abs_net (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
261 lemma eventually_sequentially:
262   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
263 unfolding sequentially_def
264 proof (rule eventually_Abs_net, rule is_filter.intro)
265   fix P Q :: "nat \<Rightarrow> bool"
266   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
267   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
268   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
269   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
270 qed auto
272 lemma eventually_within:
273   "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
274 unfolding within_def
275 by (rule eventually_Abs_net, rule is_filter.intro)
276    (auto elim!: eventually_rev_mp)
278 lemma within_UNIV: "net within UNIV = net"
279   unfolding expand_net_eq eventually_within by simp
281 lemma eventually_at_topological:
282   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
283 unfolding at_def
284 proof (rule eventually_Abs_net, rule is_filter.intro)
285   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. x \<noteq> a \<longrightarrow> True)" by simp
286   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> True)" by - rule
287 next
288   fix P Q
289   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x)"
290      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> Q x)"
291   then obtain S T where
292     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x)"
293     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> Q x)" by auto
294   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). x \<noteq> a \<longrightarrow> P x \<and> Q x)"
295     by (simp add: open_Int)
296   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x \<and> Q x)" by - rule
297 qed auto
299 lemma eventually_at:
300   fixes a :: "'a::metric_space"
301   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
302 unfolding eventually_at_topological open_dist
303 apply safe
304 apply fast
305 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
306 apply clarsimp
307 apply (rule_tac x="d - dist x a" in exI, clarsimp)
308 apply (simp only: less_diff_eq)
309 apply (erule le_less_trans [OF dist_triangle])
310 done
313 subsection {* Boundedness *}
315 definition
316   Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
317   [code del]: "Bfun f net = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) net)"
319 lemma BfunI:
320   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) net" shows "Bfun f net"
321 unfolding Bfun_def
322 proof (intro exI conjI allI)
323   show "0 < max K 1" by simp
324 next
325   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
326     using K by (rule eventually_elim1, simp)
327 qed
329 lemma BfunE:
330   assumes "Bfun f net"
331   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
332 using assms unfolding Bfun_def by fast
335 subsection {* Convergence to Zero *}
337 definition
338   Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
339   [code del]: "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)"
341 lemma ZfunI:
342   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net"
343 unfolding Zfun_def by simp
345 lemma ZfunD:
346   "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
347 unfolding Zfun_def by simp
349 lemma Zfun_ssubst:
350   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
351 unfolding Zfun_def by (auto elim!: eventually_rev_mp)
353 lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
354 unfolding Zfun_def by simp
356 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net"
357 unfolding Zfun_def by simp
359 lemma Zfun_imp_Zfun:
360   assumes f: "Zfun f net"
361   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
362   shows "Zfun (\<lambda>x. g x) net"
363 proof (cases)
364   assume K: "0 < K"
365   show ?thesis
366   proof (rule ZfunI)
367     fix r::real assume "0 < r"
368     hence "0 < r / K"
369       using K by (rule divide_pos_pos)
370     then have "eventually (\<lambda>x. norm (f x) < r / K) net"
371       using ZfunD [OF f] by fast
372     with g show "eventually (\<lambda>x. norm (g x) < r) net"
373     proof (rule eventually_elim2)
374       fix x
375       assume *: "norm (g x) \<le> norm (f x) * K"
376       assume "norm (f x) < r / K"
377       hence "norm (f x) * K < r"
378         by (simp add: pos_less_divide_eq K)
379       thus "norm (g x) < r"
380         by (simp add: order_le_less_trans [OF *])
381     qed
382   qed
383 next
384   assume "\<not> 0 < K"
385   hence K: "K \<le> 0" by (simp only: not_less)
386   show ?thesis
387   proof (rule ZfunI)
388     fix r :: real
389     assume "0 < r"
390     from g show "eventually (\<lambda>x. norm (g x) < r) net"
391     proof (rule eventually_elim1)
392       fix x
393       assume "norm (g x) \<le> norm (f x) * K"
394       also have "\<dots> \<le> norm (f x) * 0"
395         using K norm_ge_zero by (rule mult_left_mono)
396       finally show "norm (g x) < r"
397         using `0 < r` by simp
398     qed
399   qed
400 qed
402 lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net"
403 by (erule_tac K="1" in Zfun_imp_Zfun, simp)
406   assumes f: "Zfun f net" and g: "Zfun g net"
407   shows "Zfun (\<lambda>x. f x + g x) net"
408 proof (rule ZfunI)
409   fix r::real assume "0 < r"
410   hence r: "0 < r / 2" by simp
411   have "eventually (\<lambda>x. norm (f x) < r/2) net"
412     using f r by (rule ZfunD)
413   moreover
414   have "eventually (\<lambda>x. norm (g x) < r/2) net"
415     using g r by (rule ZfunD)
416   ultimately
417   show "eventually (\<lambda>x. norm (f x + g x) < r) net"
418   proof (rule eventually_elim2)
419     fix x
420     assume *: "norm (f x) < r/2" "norm (g x) < r/2"
421     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
422       by (rule norm_triangle_ineq)
423     also have "\<dots> < r/2 + r/2"
424       using * by (rule add_strict_mono)
425     finally show "norm (f x + g x) < r"
426       by simp
427   qed
428 qed
430 lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. - f x) net"
431 unfolding Zfun_def by simp
433 lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) net"
434 by (simp only: diff_minus Zfun_add Zfun_minus)
436 lemma (in bounded_linear) Zfun:
437   assumes g: "Zfun g net"
438   shows "Zfun (\<lambda>x. f (g x)) net"
439 proof -
440   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
441     using bounded by fast
442   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
443     by simp
444   with g show ?thesis
445     by (rule Zfun_imp_Zfun)
446 qed
448 lemma (in bounded_bilinear) Zfun:
449   assumes f: "Zfun f net"
450   assumes g: "Zfun g net"
451   shows "Zfun (\<lambda>x. f x ** g x) net"
452 proof (rule ZfunI)
453   fix r::real assume r: "0 < r"
454   obtain K where K: "0 < K"
455     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
456     using pos_bounded by fast
457   from K have K': "0 < inverse K"
458     by (rule positive_imp_inverse_positive)
459   have "eventually (\<lambda>x. norm (f x) < r) net"
460     using f r by (rule ZfunD)
461   moreover
462   have "eventually (\<lambda>x. norm (g x) < inverse K) net"
463     using g K' by (rule ZfunD)
464   ultimately
465   show "eventually (\<lambda>x. norm (f x ** g x) < r) net"
466   proof (rule eventually_elim2)
467     fix x
468     assume *: "norm (f x) < r" "norm (g x) < inverse K"
469     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
470       by (rule norm_le)
471     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
472       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
473     also from K have "r * inverse K * K = r"
474       by simp
475     finally show "norm (f x ** g x) < r" .
476   qed
477 qed
479 lemma (in bounded_bilinear) Zfun_left:
480   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net"
481 by (rule bounded_linear_left [THEN bounded_linear.Zfun])
483 lemma (in bounded_bilinear) Zfun_right:
484   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net"
485 by (rule bounded_linear_right [THEN bounded_linear.Zfun])
487 lemmas Zfun_mult = mult.Zfun
488 lemmas Zfun_mult_right = mult.Zfun_right
489 lemmas Zfun_mult_left = mult.Zfun_left
492 subsection {* Limits *}
494 definition
495   tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
496     (infixr "--->" 55)
497 where [code del]:
498   "(f ---> l) net \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
500 ML {*
501 structure Tendsto_Intros = Named_Thms
502 (
503   val name = "tendsto_intros"
504   val description = "introduction rules for tendsto"
505 )
506 *}
508 setup Tendsto_Intros.setup
510 lemma topological_tendstoI:
511   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net)
512     \<Longrightarrow> (f ---> l) net"
513   unfolding tendsto_def by auto
515 lemma topological_tendstoD:
516   "(f ---> l) net \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
517   unfolding tendsto_def by auto
519 lemma tendstoI:
520   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
521   shows "(f ---> l) net"
522 apply (rule topological_tendstoI)
523 apply (simp add: open_dist)
524 apply (drule (1) bspec, clarify)
525 apply (drule assms)
526 apply (erule eventually_elim1, simp)
527 done
529 lemma tendstoD:
530   "(f ---> l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
531 apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
532 apply (clarsimp simp add: open_dist)
533 apply (rule_tac x="e - dist x l" in exI, clarsimp)
534 apply (simp only: less_diff_eq)
535 apply (erule le_less_trans [OF dist_triangle])
536 apply simp
537 apply simp
538 done
540 lemma tendsto_iff:
541   "(f ---> l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
542 using tendstoI tendstoD by fast
544 lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
545 by (simp only: tendsto_iff Zfun_def dist_norm)
547 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
548 unfolding tendsto_def eventually_at_topological by auto
550 lemma tendsto_ident_at_within [tendsto_intros]:
551   "a \<in> S \<Longrightarrow> ((\<lambda>x. x) ---> a) (at a within S)"
552 unfolding tendsto_def eventually_within eventually_at_topological by auto
554 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) net"
555 by (simp add: tendsto_def)
557 lemma tendsto_dist [tendsto_intros]:
558   assumes f: "(f ---> l) net" and g: "(g ---> m) net"
559   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
560 proof (rule tendstoI)
561   fix e :: real assume "0 < e"
562   hence e2: "0 < e/2" by simp
563   from tendstoD [OF f e2] tendstoD [OF g e2]
564   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
565   proof (rule eventually_elim2)
566     fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
567     then show "dist (dist (f x) (g x)) (dist l m) < e"
568       unfolding dist_real_def
569       using dist_triangle2 [of "f x" "g x" "l"]
570       using dist_triangle2 [of "g x" "l" "m"]
571       using dist_triangle3 [of "l" "m" "f x"]
572       using dist_triangle [of "f x" "m" "g x"]
573       by arith
574   qed
575 qed
577 lemma tendsto_norm [tendsto_intros]:
578   "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
579 apply (simp add: tendsto_iff dist_norm, safe)
580 apply (drule_tac x="e" in spec, safe)
581 apply (erule eventually_elim1)
582 apply (erule order_le_less_trans [OF norm_triangle_ineq3])
583 done
586   fixes a b c d :: "'a::ab_group_add"
587   shows "(a + c) - (b + d) = (a - b) + (c - d)"
588 by simp
590 lemma minus_diff_minus:
591   fixes a b :: "'a::ab_group_add"
592   shows "(- a) - (- b) = - (a - b)"
593 by simp
595 lemma tendsto_add [tendsto_intros]:
596   fixes a b :: "'a::real_normed_vector"
597   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
600 lemma tendsto_minus [tendsto_intros]:
601   fixes a :: "'a::real_normed_vector"
602   shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
603 by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
605 lemma tendsto_minus_cancel:
606   fixes a :: "'a::real_normed_vector"
607   shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
608 by (drule tendsto_minus, simp)
610 lemma tendsto_diff [tendsto_intros]:
611   fixes a b :: "'a::real_normed_vector"
612   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
613 by (simp add: diff_minus tendsto_add tendsto_minus)
615 lemma tendsto_setsum [tendsto_intros]:
616   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
617   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) net"
618   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) net"
619 proof (cases "finite S")
620   assume "finite S" thus ?thesis using assms
621   proof (induct set: finite)
622     case empty show ?case
623       by (simp add: tendsto_const)
624   next
625     case (insert i F) thus ?case
627   qed
628 next
629   assume "\<not> finite S" thus ?thesis
630     by (simp add: tendsto_const)
631 qed
633 lemma (in bounded_linear) tendsto [tendsto_intros]:
634   "(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
635 by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
637 lemma (in bounded_bilinear) tendsto [tendsto_intros]:
638   "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
639 by (simp only: tendsto_Zfun_iff prod_diff_prod
640                Zfun_add Zfun Zfun_left Zfun_right)
643 subsection {* Continuity of Inverse *}
645 lemma (in bounded_bilinear) Zfun_prod_Bfun:
646   assumes f: "Zfun f net"
647   assumes g: "Bfun g net"
648   shows "Zfun (\<lambda>x. f x ** g x) net"
649 proof -
650   obtain K where K: "0 \<le> K"
651     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
652     using nonneg_bounded by fast
653   obtain B where B: "0 < B"
654     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net"
655     using g by (rule BfunE)
656   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net"
657   using norm_g proof (rule eventually_elim1)
658     fix x
659     assume *: "norm (g x) \<le> B"
660     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
661       by (rule norm_le)
662     also have "\<dots> \<le> norm (f x) * B * K"
663       by (intro mult_mono' order_refl norm_g norm_ge_zero
664                 mult_nonneg_nonneg K *)
665     also have "\<dots> = norm (f x) * (B * K)"
666       by (rule mult_assoc)
667     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
668   qed
669   with f show ?thesis
670     by (rule Zfun_imp_Zfun)
671 qed
673 lemma (in bounded_bilinear) flip:
674   "bounded_bilinear (\<lambda>x y. y ** x)"
675 apply default
676 apply (rule add_right)
677 apply (rule add_left)
678 apply (rule scaleR_right)
679 apply (rule scaleR_left)
680 apply (subst mult_commute)
681 using bounded by fast
683 lemma (in bounded_bilinear) Bfun_prod_Zfun:
684   assumes f: "Bfun f net"
685   assumes g: "Zfun g net"
686   shows "Zfun (\<lambda>x. f x ** g x) net"
687 using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
689 lemma inverse_diff_inverse:
690   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
691    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
692 by (simp add: algebra_simps)
694 lemma Bfun_inverse_lemma:
695   fixes x :: "'a::real_normed_div_algebra"
696   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
697 apply (subst nonzero_norm_inverse, clarsimp)
698 apply (erule (1) le_imp_inverse_le)
699 done
701 lemma Bfun_inverse:
702   fixes a :: "'a::real_normed_div_algebra"
703   assumes f: "(f ---> a) net"
704   assumes a: "a \<noteq> 0"
705   shows "Bfun (\<lambda>x. inverse (f x)) net"
706 proof -
707   from a have "0 < norm a" by simp
708   hence "\<exists>r>0. r < norm a" by (rule dense)
709   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
710   have "eventually (\<lambda>x. dist (f x) a < r) net"
711     using tendstoD [OF f r1] by fast
712   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
713   proof (rule eventually_elim1)
714     fix x
715     assume "dist (f x) a < r"
716     hence 1: "norm (f x - a) < r"
717       by (simp add: dist_norm)
718     hence 2: "f x \<noteq> 0" using r2 by auto
719     hence "norm (inverse (f x)) = inverse (norm (f x))"
720       by (rule nonzero_norm_inverse)
721     also have "\<dots> \<le> inverse (norm a - r)"
722     proof (rule le_imp_inverse_le)
723       show "0 < norm a - r" using r2 by simp
724     next
725       have "norm a - norm (f x) \<le> norm (a - f x)"
726         by (rule norm_triangle_ineq2)
727       also have "\<dots> = norm (f x - a)"
728         by (rule norm_minus_commute)
729       also have "\<dots> < r" using 1 .
730       finally show "norm a - r \<le> norm (f x)" by simp
731     qed
732     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
733   qed
734   thus ?thesis by (rule BfunI)
735 qed
737 lemma tendsto_inverse_lemma:
738   fixes a :: "'a::real_normed_div_algebra"
739   shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
740          \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
741 apply (subst tendsto_Zfun_iff)
742 apply (rule Zfun_ssubst)
743 apply (erule eventually_elim1)
744 apply (erule (1) inverse_diff_inverse)
745 apply (rule Zfun_minus)
746 apply (rule Zfun_mult_left)
747 apply (rule mult.Bfun_prod_Zfun)
748 apply (erule (1) Bfun_inverse)
749 apply (simp add: tendsto_Zfun_iff)
750 done
752 lemma tendsto_inverse [tendsto_intros]:
753   fixes a :: "'a::real_normed_div_algebra"
754   assumes f: "(f ---> a) net"
755   assumes a: "a \<noteq> 0"
756   shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
757 proof -
758   from a have "0 < norm a" by simp
759   with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
760     by (rule tendstoD)
761   then have "eventually (\<lambda>x. f x \<noteq> 0) net"
762     unfolding dist_norm by (auto elim!: eventually_elim1)
763   with f a show ?thesis
764     by (rule tendsto_inverse_lemma)
765 qed
767 lemma tendsto_divide [tendsto_intros]:
768   fixes a b :: "'a::real_normed_field"
769   shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
770     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
771 by (simp add: mult.tendsto tendsto_inverse divide_inverse)
773 end