1 (* title: HOL/Library/Topology_Euclidian_Space.thy
2 Author: Amine Chaieb, University of Cambridge
3 Author: Robert Himmelmann, TU Muenchen
4 Author: Brian Huffman, Portland State University
7 header {* Elementary topology in Euclidean space. *}
9 theory Topology_Euclidean_Space
10 imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs" Norm_Arith
13 subsection {* General notion of a topology as a value *}
15 definition "istopology L \<longleftrightarrow> L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))"
16 typedef (open) 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
17 morphisms "openin" "topology"
18 unfolding istopology_def by blast
20 lemma istopology_open_in[intro]: "istopology(openin U)"
21 using openin[of U] by blast
23 lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
24 using topology_inverse[unfolded mem_Collect_eq] .
26 lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
27 using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
29 lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
31 {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}
33 {assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
34 hence "openin T1 = openin T2" by (simp add: fun_eq_iff)
35 hence "topology (openin T1) = topology (openin T2)" by simp
36 hence "T1 = T2" unfolding openin_inverse .}
37 ultimately show ?thesis by blast
40 text{* Infer the "universe" from union of all sets in the topology. *}
42 definition "topspace T = \<Union>{S. openin T S}"
44 subsubsection {* Main properties of open sets *}
47 fixes U :: "'a topology"
49 "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
50 "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
51 using openin[of U] unfolding istopology_def mem_Collect_eq
54 lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
55 unfolding topspace_def by blast
56 lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
58 lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
59 using openin_clauses by simp
61 lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"
62 using openin_clauses by simp
64 lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
65 using openin_Union[of "{S,T}" U] by auto
67 lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
69 lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)" (is "?lhs \<longleftrightarrow> ?rhs")
71 assume ?lhs then show ?rhs by auto
74 let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
75 have "openin U ?t" by (simp add: openin_Union)
76 also have "?t = S" using H by auto
77 finally show "openin U S" .
80 subsubsection {* Closed sets *}
82 definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
84 lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
85 lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
86 lemma closedin_topspace[intro,simp]:
87 "closedin U (topspace U)" by (simp add: closedin_def)
88 lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
89 by (auto simp add: Diff_Un closedin_def)
91 lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
92 lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
93 shows "closedin U (\<Inter> K)" using Ke Kc unfolding closedin_def Diff_Inter by auto
95 lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
96 using closedin_Inter[of "{S,T}" U] by auto
98 lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
99 lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
100 apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
101 apply (metis openin_subset subset_eq)
104 lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
105 by (simp add: openin_closedin_eq)
107 lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
109 have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S] oS cT
110 by (auto simp add: topspace_def openin_subset)
111 then show ?thesis using oS cT by (auto simp add: closedin_def)
114 lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
116 have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S] oS cT
117 by (auto simp add: topspace_def )
118 then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
121 subsubsection {* Subspace topology *}
123 definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
125 lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
128 have "?L {}" by blast
129 {fix A B assume A: "?L A" and B: "?L B"
130 from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V" by blast
131 have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)" using Sa Sb by blast+
132 then have "?L (A \<inter> B)" by blast}
134 {fix K assume K: "K \<subseteq> Collect ?L"
135 have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
137 apply (simp add: Ball_def image_iff)
139 from K[unfolded th0 subset_image_iff]
140 obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
141 have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
142 moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq)
143 ultimately have "?L (\<Union>K)" by blast}
144 ultimately show ?thesis
145 unfolding subset_eq mem_Collect_eq istopology_def by blast
148 lemma openin_subtopology:
149 "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
150 unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
153 lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
154 by (auto simp add: topspace_def openin_subtopology)
156 lemma closedin_subtopology:
157 "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
158 unfolding closedin_def topspace_subtopology
159 apply (simp add: openin_subtopology)
162 apply (rule_tac x="topspace U - T" in exI)
165 lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
166 unfolding openin_subtopology
167 apply (rule iffI, clarify)
168 apply (frule openin_subset[of U]) apply blast
169 apply (rule exI[where x="topspace U"])
172 lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"
173 shows "subtopology U V = U"
176 {fix T assume T: "openin U T" "S = T \<inter> V"
177 from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
178 have "openin U S" unfolding eq using T by blast}
180 {assume S: "openin U S"
181 hence "\<exists>T. openin U T \<and> S = T \<inter> V"
182 using openin_subset[OF S] UV by auto}
183 ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
184 then show ?thesis unfolding topology_eq openin_subtopology by blast
187 lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
188 by (simp add: subtopology_superset)
190 lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
191 by (simp add: subtopology_superset)
193 subsubsection {* The standard Euclidean topology *}
196 euclidean :: "'a::topological_space topology" where
197 "euclidean = topology open"
199 lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
200 unfolding euclidean_def
201 apply (rule cong[where x=S and y=S])
202 apply (rule topology_inverse[symmetric])
203 apply (auto simp add: istopology_def)
206 lemma topspace_euclidean: "topspace euclidean = UNIV"
207 apply (simp add: topspace_def)
209 by (auto simp add: open_openin[symmetric])
211 lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
212 by (simp add: topspace_euclidean topspace_subtopology)
214 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
215 by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
217 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
218 by (simp add: open_openin openin_subopen[symmetric])
220 text {* Basic "localization" results are handy for connectedness. *}
222 lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
223 by (auto simp add: openin_subtopology open_openin[symmetric])
225 lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
226 by (auto simp add: openin_open)
228 lemma open_openin_trans[trans]:
229 "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
230 by (metis Int_absorb1 openin_open_Int)
232 lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
233 by (auto simp add: openin_open)
235 lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
236 by (simp add: closedin_subtopology closed_closedin Int_ac)
238 lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
239 by (metis closedin_closed)
241 lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
242 apply (subgoal_tac "S \<inter> T = T" )
244 apply (frule closedin_closed_Int[of T S])
247 lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
248 by (auto simp add: closedin_closed)
250 lemma openin_euclidean_subtopology_iff:
251 fixes S U :: "'a::metric_space set"
252 shows "openin (subtopology euclidean U) S
253 \<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")
255 assume ?lhs thus ?rhs unfolding openin_open open_dist by blast
257 def T \<equiv> "{x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
258 have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
261 apply (rule_tac x="d - dist x a" in exI)
262 apply (clarsimp simp add: less_diff_eq)
263 apply (erule rev_bexI)
264 apply (rule_tac x=d in exI, clarify)
265 apply (erule le_less_trans [OF dist_triangle])
267 assume ?rhs hence 2: "S = U \<inter> T"
270 apply (drule (1) bspec, erule rev_bexI)
274 unfolding openin_open open_dist by fast
277 text {* These "transitivity" results are handy too *}
279 lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
280 \<Longrightarrow> openin (subtopology euclidean U) S"
281 unfolding open_openin openin_open by blast
283 lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
284 by (auto simp add: openin_open intro: openin_trans)
286 lemma closedin_trans[trans]:
287 "closedin (subtopology euclidean T) S \<Longrightarrow>
288 closedin (subtopology euclidean U) T
289 ==> closedin (subtopology euclidean U) S"
290 by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
292 lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
293 by (auto simp add: closedin_closed intro: closedin_trans)
296 subsection {* Open and closed balls *}
299 ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
300 "ball x e = {y. dist x y < e}"
303 cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
304 "cball x e = {y. dist x y \<le> e}"
306 lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
307 lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
309 lemma mem_ball_0 [simp]:
310 fixes x :: "'a::real_normed_vector"
311 shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
312 by (simp add: dist_norm)
314 lemma mem_cball_0 [simp]:
315 fixes x :: "'a::real_normed_vector"
316 shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
317 by (simp add: dist_norm)
319 lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e" by simp
320 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
321 lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
322 lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
323 lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
324 by (simp add: set_eq_iff) arith
326 lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
327 by (simp add: set_eq_iff)
329 lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
330 "(a::real) - b < 0 \<longleftrightarrow> a < b"
331 "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
332 lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
333 "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b" by arith+
335 lemma open_ball[intro, simp]: "open (ball x e)"
336 unfolding open_dist ball_def mem_Collect_eq Ball_def
337 unfolding dist_commute
339 apply (rule_tac x="e - dist xa x" in exI)
340 using dist_triangle_alt[where z=x]
341 apply (clarsimp simp add: diff_less_iff)
343 apply (erule_tac x="y" in allE)
344 apply (erule_tac x="xa" in allE)
347 lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
348 lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
349 unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
352 assumes "open S" "x\<in>S"
353 obtains e where "e>0" "ball x e \<subseteq> S"
354 using assms unfolding open_contains_ball by auto
356 lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
357 by (metis open_contains_ball subset_eq centre_in_ball)
359 lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
360 unfolding mem_ball set_eq_iff
361 apply (simp add: not_less)
362 by (metis zero_le_dist order_trans dist_self)
364 lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
367 subsection{* Connectedness *}
369 definition "connected S \<longleftrightarrow>
370 ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
371 \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
373 lemma connected_local:
374 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
375 openin (subtopology euclidean S) e1 \<and>
376 openin (subtopology euclidean S) e2 \<and>
377 S \<subseteq> e1 \<union> e2 \<and>
378 e1 \<inter> e2 = {} \<and>
381 unfolding connected_def openin_open by (safe, blast+)
384 fixes P :: "'a set \<Rightarrow> bool"
385 shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
387 {assume "?lhs" hence ?rhs by blast }
389 {fix S assume H: "P S"
390 have "S = - (- S)" by auto
391 with H have "P (- (- S))" by metis }
392 ultimately show ?thesis by metis
395 lemma connected_clopen: "connected S \<longleftrightarrow>
396 (\<forall>T. openin (subtopology euclidean S) T \<and>
397 closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
399 have " \<not> connected S \<longleftrightarrow> (\<exists>e1 e2. open e1 \<and> open (- e2) \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
400 unfolding connected_def openin_open closedin_closed
401 apply (subst exists_diff) by blast
402 hence th0: "connected S \<longleftrightarrow> \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
403 (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis
405 have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
406 (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
407 unfolding connected_def openin_open closedin_closed by auto
409 {fix e1 have "?P e2 e1 \<longleftrightarrow> (\<exists>t. closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t\<noteq>S)"
411 then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
412 then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
413 then show ?thesis unfolding th0 th1 by simp
416 lemma connected_empty[simp, intro]: "connected {}"
417 by (simp add: connected_def)
420 subsection{* Limit points *}
422 definition (in topological_space)
423 islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where
424 "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
427 assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
429 using assms unfolding islimpt_def by auto
432 assumes "x islimpt S" and "x \<in> T" and "open T"
433 obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
434 using assms unfolding islimpt_def by auto
436 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
437 unfolding islimpt_def eventually_at_topological by auto
439 lemma islimpt_subset: "\<lbrakk>x islimpt S; S \<subseteq> T\<rbrakk> \<Longrightarrow> x islimpt T"
440 unfolding islimpt_def by fast
442 lemma islimpt_approachable:
443 fixes x :: "'a::metric_space"
444 shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
445 unfolding islimpt_iff_eventually eventually_at by fast
447 lemma islimpt_approachable_le:
448 fixes x :: "'a::metric_space"
449 shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
450 unfolding islimpt_approachable
451 using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
452 THEN arg_cong [where f=Not]]
453 by (simp add: Bex_def conj_commute conj_left_commute)
455 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
456 unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
458 text {* A perfect space has no isolated points. *}
460 lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"
461 unfolding islimpt_UNIV_iff by (rule not_open_singleton)
463 lemma perfect_choose_dist:
464 fixes x :: "'a::{perfect_space, metric_space}"
465 shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
466 using islimpt_UNIV [of x]
467 by (simp add: islimpt_approachable)
469 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
471 apply (subst open_subopen)
472 apply (simp add: islimpt_def subset_eq)
473 by (metis ComplE ComplI)
475 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
476 unfolding islimpt_def by auto
478 lemma finite_set_avoid:
479 fixes a :: "'a::metric_space"
480 assumes fS: "finite S" shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
481 proof(induct rule: finite_induct[OF fS])
482 case 1 thus ?case by (auto intro: zero_less_one)
485 from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
486 {assume "x = a" hence ?case using d by auto }
488 {assume xa: "x\<noteq>a"
489 let ?d = "min d (dist a x)"
490 have dp: "?d > 0" using xa d(1) using dist_nz by auto
491 from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
492 with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
493 ultimately show ?case by blast
496 lemma islimpt_finite:
497 fixes S :: "'a::metric_space set"
498 assumes fS: "finite S" shows "\<not> a islimpt S"
499 unfolding islimpt_approachable
500 using finite_set_avoid[OF fS, of a] by (metis dist_commute not_le)
502 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
505 apply (metis Un_upper1 Un_upper2 islimpt_subset)
506 unfolding islimpt_def
507 apply (rule ccontr, clarsimp, rename_tac A B)
508 apply (drule_tac x="A \<inter> B" in spec)
509 apply (auto simp add: open_Int)
512 lemma discrete_imp_closed:
513 fixes S :: "'a::metric_space set"
514 assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
517 {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
518 from e have e2: "e/2 > 0" by arith
519 from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
520 let ?m = "min (e/2) (dist x y) "
521 from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
522 from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
523 have th: "dist z y < e" using z y
524 by (intro dist_triangle_lt [where z=x], simp)
525 from d[rule_format, OF y(1) z(1) th] y z
526 have False by (auto simp add: dist_commute)}
527 then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
531 subsection {* Interior of a Set *}
533 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
535 lemma interiorI [intro?]:
536 assumes "open T" and "x \<in> T" and "T \<subseteq> S"
537 shows "x \<in> interior S"
538 using assms unfolding interior_def by fast
540 lemma interiorE [elim?]:
541 assumes "x \<in> interior S"
542 obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
543 using assms unfolding interior_def by fast
545 lemma open_interior [simp, intro]: "open (interior S)"
546 by (simp add: interior_def open_Union)
548 lemma interior_subset: "interior S \<subseteq> S"
549 by (auto simp add: interior_def)
551 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
552 by (auto simp add: interior_def)
554 lemma interior_open: "open S \<Longrightarrow> interior S = S"
555 by (intro equalityI interior_subset interior_maximal subset_refl)
557 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
558 by (metis open_interior interior_open)
560 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
561 by (metis interior_maximal interior_subset subset_trans)
563 lemma interior_empty [simp]: "interior {} = {}"
564 using open_empty by (rule interior_open)
566 lemma interior_UNIV [simp]: "interior UNIV = UNIV"
567 using open_UNIV by (rule interior_open)
569 lemma interior_interior [simp]: "interior (interior S) = interior S"
570 using open_interior by (rule interior_open)
572 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
573 by (auto simp add: interior_def)
575 lemma interior_unique:
576 assumes "T \<subseteq> S" and "open T"
577 assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
578 shows "interior S = T"
579 by (intro equalityI assms interior_subset open_interior interior_maximal)
581 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
582 by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
583 Int_lower2 interior_maximal interior_subset open_Int open_interior)
585 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
586 using open_contains_ball_eq [where S="interior S"]
587 by (simp add: open_subset_interior)
589 lemma interior_limit_point [intro]:
590 fixes x :: "'a::perfect_space"
591 assumes x: "x \<in> interior S" shows "x islimpt S"
592 using x islimpt_UNIV [of x]
593 unfolding interior_def islimpt_def
594 apply (clarsimp, rename_tac T T')
595 apply (drule_tac x="T \<inter> T'" in spec)
596 apply (auto simp add: open_Int)
599 lemma interior_closed_Un_empty_interior:
600 assumes cS: "closed S" and iT: "interior T = {}"
601 shows "interior (S \<union> T) = interior S"
603 show "interior S \<subseteq> interior (S \<union> T)"
604 by (rule interior_mono, rule Un_upper1)
606 show "interior (S \<union> T) \<subseteq> interior S"
608 fix x assume "x \<in> interior (S \<union> T)"
609 then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
610 show "x \<in> interior S"
612 assume "x \<notin> interior S"
613 with `x \<in> R` `open R` obtain y where "y \<in> R - S"
614 unfolding interior_def by fast
615 from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
616 from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
617 from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
618 show "False" unfolding interior_def by fast
623 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
624 proof (rule interior_unique)
625 show "interior A \<times> interior B \<subseteq> A \<times> B"
626 by (intro Sigma_mono interior_subset)
627 show "open (interior A \<times> interior B)"
628 by (intro open_Times open_interior)
629 fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"
631 fix x y assume "(x, y) \<in> T"
632 then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
633 using `open T` unfolding open_prod_def by fast
634 hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
635 using `T \<subseteq> A \<times> B` by auto
636 thus "x \<in> interior A" and "y \<in> interior B"
637 by (auto intro: interiorI)
642 subsection {* Closure of a Set *}
644 definition "closure S = S \<union> {x | x. x islimpt S}"
646 lemma interior_closure: "interior S = - (closure (- S))"
647 unfolding interior_def closure_def islimpt_def by auto
649 lemma closure_interior: "closure S = - interior (- S)"
650 unfolding interior_closure by simp
652 lemma closed_closure[simp, intro]: "closed (closure S)"
653 unfolding closure_interior by (simp add: closed_Compl)
655 lemma closure_subset: "S \<subseteq> closure S"
656 unfolding closure_def by simp
658 lemma closure_hull: "closure S = closed hull S"
659 unfolding hull_def closure_interior interior_def by auto
661 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
662 unfolding closure_hull using closed_Inter by (rule hull_eq)
664 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
665 unfolding closure_eq .
667 lemma closure_closure [simp]: "closure (closure S) = closure S"
668 unfolding closure_hull by (rule hull_hull)
670 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
671 unfolding closure_hull by (rule hull_mono)
673 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
674 unfolding closure_hull by (rule hull_minimal)
676 lemma closure_unique:
677 assumes "S \<subseteq> T" and "closed T"
678 assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
679 shows "closure S = T"
680 using assms unfolding closure_hull by (rule hull_unique)
682 lemma closure_empty [simp]: "closure {} = {}"
683 using closed_empty by (rule closure_closed)
685 lemma closure_UNIV [simp]: "closure UNIV = UNIV"
686 using closed_UNIV by (rule closure_closed)
688 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"
689 unfolding closure_interior by simp
691 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
692 using closure_empty closure_subset[of S]
695 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
696 using closure_eq[of S] closure_subset[of S]
699 lemma open_inter_closure_eq_empty:
700 "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
701 using open_subset_interior[of S "- T"]
702 using interior_subset[of "- T"]
703 unfolding closure_interior
706 lemma open_inter_closure_subset:
707 "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
710 assume as: "open S" "x \<in> S \<inter> closure T"
711 { assume *:"x islimpt T"
712 have "x islimpt (S \<inter> T)"
713 proof (rule islimptI)
715 assume "x \<in> A" "open A"
716 with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
717 by (simp_all add: open_Int)
718 with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
720 hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
722 thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
725 then show "x \<in> closure (S \<inter> T)" using as
726 unfolding closure_def
730 lemma closure_complement: "closure (- S) = - interior S"
731 unfolding closure_interior by simp
733 lemma interior_complement: "interior (- S) = - closure S"
734 unfolding closure_interior by simp
736 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
737 proof (rule closure_unique)
738 show "A \<times> B \<subseteq> closure A \<times> closure B"
739 by (intro Sigma_mono closure_subset)
740 show "closed (closure A \<times> closure B)"
741 by (intro closed_Times closed_closure)
742 fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"
743 apply (simp add: closed_def open_prod_def, clarify)
745 apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
746 apply (simp add: closure_interior interior_def)
747 apply (drule_tac x=C in spec)
748 apply (drule_tac x=D in spec)
754 subsection {* Frontier (aka boundary) *}
756 definition "frontier S = closure S - interior S"
758 lemma frontier_closed: "closed(frontier S)"
759 by (simp add: frontier_def closed_Diff)
761 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
762 by (auto simp add: frontier_def interior_closure)
764 lemma frontier_straddle:
765 fixes a :: "'a::metric_space"
766 shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))"
767 unfolding frontier_def closure_interior
768 by (auto simp add: mem_interior subset_eq ball_def)
770 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
771 by (metis frontier_def closure_closed Diff_subset)
773 lemma frontier_empty[simp]: "frontier {} = {}"
774 by (simp add: frontier_def)
776 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
778 { assume "frontier S \<subseteq> S"
779 hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
780 hence "closed S" using closure_subset_eq by auto
782 thus ?thesis using frontier_subset_closed[of S] ..
785 lemma frontier_complement: "frontier(- S) = frontier S"
786 by (auto simp add: frontier_def closure_complement interior_complement)
788 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
789 using frontier_complement frontier_subset_eq[of "- S"]
790 unfolding open_closed by auto
793 subsection {* Filters and the ``eventually true'' quantifier *}
796 at_infinity :: "'a::real_normed_vector filter" where
797 "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
800 indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
801 (infixr "indirection" 70) where
802 "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
804 text{* Prove That They are all filters. *}
806 lemma eventually_at_infinity:
807 "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
808 unfolding at_infinity_def
809 proof (rule eventually_Abs_filter, rule is_filter.intro)
810 fix P Q :: "'a \<Rightarrow> bool"
811 assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
812 then obtain r s where
813 "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
814 then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
815 then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
818 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
820 lemma trivial_limit_within:
821 shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
823 assume "trivial_limit (at a within S)"
824 thus "\<not> a islimpt S"
825 unfolding trivial_limit_def
826 unfolding eventually_within eventually_at_topological
827 unfolding islimpt_def
828 apply (clarsimp simp add: set_eq_iff)
829 apply (rename_tac T, rule_tac x=T in exI)
830 apply (clarsimp, drule_tac x=y in bspec, simp_all)
833 assume "\<not> a islimpt S"
834 thus "trivial_limit (at a within S)"
835 unfolding trivial_limit_def
836 unfolding eventually_within eventually_at_topological
837 unfolding islimpt_def
839 apply (rule_tac x=T in exI)
844 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
845 using trivial_limit_within [of a UNIV] by simp
847 lemma trivial_limit_at:
848 fixes a :: "'a::perfect_space"
849 shows "\<not> trivial_limit (at a)"
852 lemma trivial_limit_at_infinity:
853 "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
854 unfolding trivial_limit_def eventually_at_infinity
856 apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
857 apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
858 apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
859 apply (drule_tac x=UNIV in spec, simp)
862 text {* Some property holds "sufficiently close" to the limit point. *}
864 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
865 "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
866 unfolding eventually_at dist_nz by auto
868 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
869 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
870 unfolding eventually_within eventually_at dist_nz by auto
872 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
873 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
874 unfolding eventually_within
875 by auto (metis dense order_le_less_trans)
877 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
878 unfolding trivial_limit_def
879 by (auto elim: eventually_rev_mp)
881 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
884 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
885 by (simp add: filter_eq_iff)
887 text{* Combining theorems for "eventually" *}
889 lemma eventually_rev_mono:
890 "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
891 using eventually_mono [of P Q] by fast
893 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
894 by (simp add: eventually_False)
897 subsection {* Limits *}
899 text{* Notation Lim to avoid collition with lim defined in analysis *}
901 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"
902 where "Lim A f = (THE l. (f ---> l) A)"
905 "(f ---> l) net \<longleftrightarrow>
906 trivial_limit net \<or>
907 (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
908 unfolding tendsto_iff trivial_limit_eq by auto
910 text{* Show that they yield usual definitions in the various cases. *}
912 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
913 (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> dist (f x) l < e)"
914 by (auto simp add: tendsto_iff eventually_within_le)
916 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
917 (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
918 by (auto simp add: tendsto_iff eventually_within)
920 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
921 (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
922 by (auto simp add: tendsto_iff eventually_at)
924 lemma Lim_at_infinity:
925 "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
926 by (auto simp add: tendsto_iff eventually_at_infinity)
928 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
929 by (rule topological_tendstoI, auto elim: eventually_rev_mono)
931 text{* The expected monotonicity property. *}
933 lemma Lim_within_empty: "(f ---> l) (net within {})"
934 unfolding tendsto_def Limits.eventually_within by simp
936 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
937 unfolding tendsto_def Limits.eventually_within
938 by (auto elim!: eventually_elim1)
940 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
941 shows "(f ---> l) (net within (S \<union> T))"
942 using assms unfolding tendsto_def Limits.eventually_within
944 apply (drule spec, drule (1) mp, drule (1) mp)
945 apply (drule spec, drule (1) mp, drule (1) mp)
946 apply (auto elim: eventually_elim2)
950 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow> S \<union> T = UNIV
952 by (metis Lim_Un within_UNIV)
954 text{* Interrelations between restricted and unrestricted limits. *}
956 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
958 unfolding tendsto_def Limits.eventually_within
959 apply (clarify, drule spec, drule (1) mp, drule (1) mp)
960 by (auto elim!: eventually_elim1)
962 lemma eventually_within_interior:
963 assumes "x \<in> interior S"
964 shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
966 from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
968 then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
969 unfolding Limits.eventually_within Limits.eventually_at_topological
971 with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
974 unfolding Limits.eventually_at_topological by auto
976 { assume "?rhs" hence "?lhs"
977 unfolding Limits.eventually_within
978 by (auto elim: eventually_elim1)
983 lemma at_within_interior:
984 "x \<in> interior S \<Longrightarrow> at x within S = at x"
985 by (simp add: filter_eq_iff eventually_within_interior)
987 lemma at_within_open:
988 "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"
989 by (simp only: at_within_interior interior_open)
991 lemma Lim_within_open:
992 fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
993 assumes"a \<in> S" "open S"
994 shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
995 using assms by (simp only: at_within_open)
997 lemma Lim_within_LIMSEQ:
998 fixes a :: "'a::metric_space"
999 assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
1000 shows "(X ---> L) (at a within T)"
1001 using assms unfolding tendsto_def [where l=L]
1002 by (simp add: sequentially_imp_eventually_within)
1004 lemma Lim_right_bound:
1005 fixes f :: "real \<Rightarrow> real"
1006 assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
1007 assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
1008 shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
1010 assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
1012 assume [simp]: "{x<..} \<inter> I \<noteq> {}"
1014 proof (rule Lim_within_LIMSEQ, safe)
1015 fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"
1017 show "(\<lambda>n. f (S n)) ----> Inf (f ` ({x<..} \<inter> I))"
1018 proof (rule LIMSEQ_I, rule ccontr)
1019 fix r :: real assume "0 < r"
1020 with Inf_close[of "f ` ({x<..} \<inter> I)" r]
1021 obtain y where y: "x < y" "y \<in> I" "f y < Inf (f ` ({x <..} \<inter> I)) + r" by auto
1022 from `x < y` have "0 < y - x" by auto
1023 from S(2)[THEN LIMSEQ_D, OF this]
1024 obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto
1026 assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f ` ({x<..} \<inter> I))) < r)"
1027 moreover have "\<And>n. Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
1028 using S bnd by (intro Inf_lower[where z=K]) auto
1029 ultimately obtain n where n: "N \<le> n" "r + Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
1030 by (auto simp: not_less field_simps)
1031 with N[OF n(1)] mono[OF _ `y \<in> I`, of "S n"] S(1)[THEN spec, of n] y
1037 text{* Another limit point characterization. *}
1039 lemma islimpt_sequential:
1040 fixes x :: "'a::metric_space"
1041 shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
1045 then obtain f where f:"\<forall>y. y>0 \<longrightarrow> f y \<in> S \<and> f y \<noteq> x \<and> dist (f y) x < y"
1046 unfolding islimpt_approachable
1047 using choice[of "\<lambda>e y. e>0 \<longrightarrow> y\<in>S \<and> y\<noteq>x \<and> dist y x < e"] by auto
1048 let ?I = "\<lambda>n. inverse (real (Suc n))"
1049 have "\<forall>n. f (?I n) \<in> S - {x}" using f by simp
1050 moreover have "(\<lambda>n. f (?I n)) ----> x"
1051 proof (rule metric_tendsto_imp_tendsto)
1053 by (rule LIMSEQ_inverse_real_of_nat)
1054 show "eventually (\<lambda>n. dist (f (?I n)) x \<le> dist (?I n) 0) sequentially"
1055 by (simp add: norm_conv_dist [symmetric] less_imp_le f)
1057 ultimately show ?rhs by fast
1060 then obtain f::"nat\<Rightarrow>'a" where f:"(\<forall>n. f n \<in> S - {x})" "(\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f n) x < e)" unfolding LIMSEQ_def by auto
1061 { fix e::real assume "e>0"
1062 then obtain N where "dist (f N) x < e" using f(2) by auto
1063 moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
1064 ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
1066 thus ?lhs unfolding islimpt_approachable by auto
1069 lemma Lim_inv: (* TODO: delete *)
1070 fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
1071 assumes "(f ---> l) A" and "l \<noteq> 0"
1072 shows "((inverse o f) ---> inverse l) A"
1073 unfolding o_def using assms by (rule tendsto_inverse)
1076 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1077 shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
1078 by (simp add: Lim dist_norm)
1080 lemma Lim_null_comparison:
1081 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1082 assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
1083 shows "(f ---> 0) net"
1084 proof (rule metric_tendsto_imp_tendsto)
1085 show "(g ---> 0) net" by fact
1086 show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
1087 using assms(1) by (rule eventually_elim1, simp add: dist_norm)
1090 lemma Lim_transform_bound:
1091 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1092 fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
1093 assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net" "(g ---> 0) net"
1094 shows "(f ---> 0) net"
1095 using assms(1) tendsto_norm_zero [OF assms(2)]
1096 by (rule Lim_null_comparison)
1098 text{* Deducing things about the limit from the elements. *}
1100 lemma Lim_in_closed_set:
1101 assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
1104 assume "l \<notin> S"
1105 with `closed S` have "open (- S)" "l \<in> - S"
1106 by (simp_all add: open_Compl)
1107 with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
1108 by (rule topological_tendstoD)
1109 with assms(2) have "eventually (\<lambda>x. False) net"
1110 by (rule eventually_elim2) simp
1111 with assms(3) show "False"
1112 by (simp add: eventually_False)
1115 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
1117 lemma Lim_dist_ubound:
1118 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
1119 shows "dist a l <= e"
1121 have "dist a l \<in> {..e}"
1122 proof (rule Lim_in_closed_set)
1123 show "closed {..e}" by simp
1124 show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)
1125 show "\<not> trivial_limit net" by fact
1126 show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)
1128 thus ?thesis by simp
1131 lemma Lim_norm_ubound:
1132 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1133 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
1134 shows "norm(l) <= e"
1136 have "norm l \<in> {..e}"
1137 proof (rule Lim_in_closed_set)
1138 show "closed {..e}" by simp
1139 show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)
1140 show "\<not> trivial_limit net" by fact
1141 show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
1143 thus ?thesis by simp
1146 lemma Lim_norm_lbound:
1147 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1148 assumes "\<not> (trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. e <= norm(f x)) net"
1149 shows "e \<le> norm l"
1151 have "norm l \<in> {e..}"
1152 proof (rule Lim_in_closed_set)
1153 show "closed {e..}" by simp
1154 show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)
1155 show "\<not> trivial_limit net" by fact
1156 show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
1158 thus ?thesis by simp
1161 text{* Uniqueness of the limit, when nontrivial. *}
1164 fixes f :: "'a \<Rightarrow> 'b::t2_space"
1165 shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
1166 unfolding Lim_def using tendsto_unique[of net f] by auto
1168 text{* Limit under bilinear function *}
1171 assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
1172 shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
1173 using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
1174 by (rule bounded_bilinear.tendsto)
1176 text{* These are special for limits out of the same vector space. *}
1178 lemma Lim_within_id: "(id ---> a) (at a within s)"
1179 unfolding id_def by (rule tendsto_ident_at_within)
1181 lemma Lim_at_id: "(id ---> a) (at a)"
1182 unfolding id_def by (rule tendsto_ident_at)
1185 fixes a :: "'a::real_normed_vector"
1186 fixes l :: "'b::topological_space"
1187 shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
1188 using LIM_offset_zero LIM_offset_zero_cancel ..
1190 text{* It's also sometimes useful to extract the limit point from the filter. *}
1193 netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
1194 "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
1196 lemma netlimit_within:
1197 assumes "\<not> trivial_limit (at a within S)"
1198 shows "netlimit (at a within S) = a"
1199 unfolding netlimit_def
1200 apply (rule some_equality)
1201 apply (rule Lim_at_within)
1202 apply (rule tendsto_ident_at)
1203 apply (erule tendsto_unique [OF assms])
1204 apply (rule Lim_at_within)
1205 apply (rule tendsto_ident_at)
1209 fixes a :: "'a::{perfect_space,t2_space}"
1210 shows "netlimit (at a) = a"
1211 using netlimit_within [of a UNIV] by simp
1213 lemma lim_within_interior:
1214 "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
1215 by (simp add: at_within_interior)
1217 lemma netlimit_within_interior:
1218 fixes x :: "'a::{t2_space,perfect_space}"
1219 assumes "x \<in> interior S"
1220 shows "netlimit (at x within S) = x"
1221 using assms by (simp add: at_within_interior netlimit_at)
1223 text{* Transformation of limit. *}
1225 lemma Lim_transform:
1226 fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
1227 assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
1228 shows "(g ---> l) net"
1229 using tendsto_diff [OF assms(2) assms(1)] by simp
1231 lemma Lim_transform_eventually:
1232 "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
1233 apply (rule topological_tendstoI)
1234 apply (drule (2) topological_tendstoD)
1235 apply (erule (1) eventually_elim2, simp)
1238 lemma Lim_transform_within:
1239 assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
1240 and "(f ---> l) (at x within S)"
1241 shows "(g ---> l) (at x within S)"
1242 proof (rule Lim_transform_eventually)
1243 show "eventually (\<lambda>x. f x = g x) (at x within S)"
1244 unfolding eventually_within
1245 using assms(1,2) by auto
1246 show "(f ---> l) (at x within S)" by fact
1249 lemma Lim_transform_at:
1250 assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
1251 and "(f ---> l) (at x)"
1252 shows "(g ---> l) (at x)"
1253 proof (rule Lim_transform_eventually)
1254 show "eventually (\<lambda>x. f x = g x) (at x)"
1255 unfolding eventually_at
1256 using assms(1,2) by auto
1257 show "(f ---> l) (at x)" by fact
1260 text{* Common case assuming being away from some crucial point like 0. *}
1262 lemma Lim_transform_away_within:
1263 fixes a b :: "'a::t1_space"
1264 assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1265 and "(f ---> l) (at a within S)"
1266 shows "(g ---> l) (at a within S)"
1267 proof (rule Lim_transform_eventually)
1268 show "(f ---> l) (at a within S)" by fact
1269 show "eventually (\<lambda>x. f x = g x) (at a within S)"
1270 unfolding Limits.eventually_within eventually_at_topological
1271 by (rule exI [where x="- {b}"], simp add: open_Compl assms)
1274 lemma Lim_transform_away_at:
1275 fixes a b :: "'a::t1_space"
1276 assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1277 and fl: "(f ---> l) (at a)"
1278 shows "(g ---> l) (at a)"
1279 using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
1282 text{* Alternatively, within an open set. *}
1284 lemma Lim_transform_within_open:
1285 assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
1286 and "(f ---> l) (at a)"
1287 shows "(g ---> l) (at a)"
1288 proof (rule Lim_transform_eventually)
1289 show "eventually (\<lambda>x. f x = g x) (at a)"
1290 unfolding eventually_at_topological
1291 using assms(1,2,3) by auto
1292 show "(f ---> l) (at a)" by fact
1295 text{* A congruence rule allowing us to transform limits assuming not at point. *}
1297 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
1299 lemma Lim_cong_within(*[cong add]*):
1300 assumes "a = b" "x = y" "S = T"
1301 assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
1302 shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
1303 unfolding tendsto_def Limits.eventually_within eventually_at_topological
1306 lemma Lim_cong_at(*[cong add]*):
1307 assumes "a = b" "x = y"
1308 assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
1309 shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
1310 unfolding tendsto_def eventually_at_topological
1313 text{* Useful lemmas on closure and set of possible sequential limits.*}
1315 lemma closure_sequential:
1316 fixes l :: "'a::metric_space"
1317 shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
1319 assume "?lhs" moreover
1320 { assume "l \<in> S"
1321 hence "?rhs" using tendsto_const[of l sequentially] by auto
1323 { assume "l islimpt S"
1324 hence "?rhs" unfolding islimpt_sequential by auto
1326 show "?rhs" unfolding closure_def by auto
1329 thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
1332 lemma closed_sequential_limits:
1333 fixes S :: "'a::metric_space set"
1334 shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
1335 unfolding closed_limpt
1336 using closure_sequential [where 'a='a] closure_closed [where 'a='a] closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]
1339 lemma closure_approachable:
1340 fixes S :: "'a::metric_space set"
1341 shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
1342 apply (auto simp add: closure_def islimpt_approachable)
1343 by (metis dist_self)
1345 lemma closed_approachable:
1346 fixes S :: "'a::metric_space set"
1347 shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
1348 by (metis closure_closed closure_approachable)
1350 text{* Some other lemmas about sequences. *}
1352 lemma sequentially_offset:
1353 assumes "eventually (\<lambda>i. P i) sequentially"
1354 shows "eventually (\<lambda>i. P (i + k)) sequentially"
1355 using assms unfolding eventually_sequentially by (metis trans_le_add1)
1358 assumes "(f ---> l) sequentially"
1359 shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
1360 using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)
1362 lemma seq_offset_neg:
1363 "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
1364 apply (rule topological_tendstoI)
1365 apply (drule (2) topological_tendstoD)
1366 apply (simp only: eventually_sequentially)
1367 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
1371 lemma seq_offset_rev:
1372 "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
1373 by (rule LIMSEQ_offset) (* FIXME: redundant *)
1375 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
1376 using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)
1378 subsection {* More properties of closed balls *}
1380 lemma closed_cball: "closed (cball x e)"
1381 unfolding cball_def closed_def
1382 unfolding Collect_neg_eq [symmetric] not_le
1383 apply (clarsimp simp add: open_dist, rename_tac y)
1384 apply (rule_tac x="dist x y - e" in exI, clarsimp)
1385 apply (rename_tac x')
1386 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
1390 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
1392 { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
1393 hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
1395 { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
1396 hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
1398 show ?thesis unfolding open_contains_ball by auto
1401 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
1402 by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
1404 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
1405 apply (simp add: interior_def, safe)
1406 apply (force simp add: open_contains_cball)
1407 apply (rule_tac x="ball x e" in exI)
1408 apply (simp add: subset_trans [OF ball_subset_cball])
1412 fixes x y :: "'a::{real_normed_vector,perfect_space}"
1413 shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
1416 { assume "e \<le> 0"
1417 hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
1418 have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
1420 hence "e > 0" by (metis not_less)
1422 have "y \<in> cball x e" using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"] ball_subset_cball[of x e] `?lhs` unfolding closed_limpt by auto
1423 ultimately show "?rhs" by auto
1425 assume "?rhs" hence "e>0" by auto
1426 { fix d::real assume "d>0"
1427 have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1428 proof(cases "d \<le> dist x y")
1429 case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1431 case True hence False using `d \<le> dist x y` `d>0` by auto
1432 thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
1436 have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
1437 = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1438 unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
1439 also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
1440 using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
1441 unfolding scaleR_minus_left scaleR_one
1442 by (auto simp add: norm_minus_commute)
1443 also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
1444 unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
1445 unfolding left_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
1446 also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
1447 finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
1451 have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
1452 using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
1454 have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
1455 using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
1456 unfolding dist_norm by auto
1457 ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by (rule_tac x="y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto
1460 case False hence "d > dist x y" by auto
1461 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1464 obtain z where **: "z \<noteq> y" "dist z y < min e d"
1465 using perfect_choose_dist[of "min e d" y]
1466 using `d > 0` `e>0` by auto
1467 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1469 using `z \<noteq> y` **
1470 by (rule_tac x=z in bexI, auto simp add: dist_commute)
1472 case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1473 using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
1476 thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
1479 lemma closure_ball_lemma:
1480 fixes x y :: "'a::real_normed_vector"
1481 assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
1482 proof (rule islimptI)
1483 fix T assume "y \<in> T" "open T"
1484 then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
1485 unfolding open_dist by fast
1486 (* choose point between x and y, within distance r of y. *)
1487 def k \<equiv> "min 1 (r / (2 * dist x y))"
1488 def z \<equiv> "y + scaleR k (x - y)"
1489 have z_def2: "z = x + scaleR (1 - k) (y - x)"
1490 unfolding z_def by (simp add: algebra_simps)
1492 unfolding z_def k_def using `0 < r`
1493 by (simp add: dist_norm min_def)
1494 hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
1495 have "dist x z < dist x y"
1496 unfolding z_def2 dist_norm
1497 apply (simp add: norm_minus_commute)
1498 apply (simp only: dist_norm [symmetric])
1499 apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
1500 apply (rule mult_strict_right_mono)
1501 apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
1502 apply (simp add: zero_less_dist_iff `x \<noteq> y`)
1504 hence "z \<in> ball x (dist x y)" by simp
1506 unfolding z_def k_def using `x \<noteq> y` `0 < r`
1507 by (simp add: min_def)
1508 show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
1509 using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
1514 fixes x :: "'a::real_normed_vector"
1515 shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
1516 apply (rule equalityI)
1517 apply (rule closure_minimal)
1518 apply (rule ball_subset_cball)
1519 apply (rule closed_cball)
1520 apply (rule subsetI, rename_tac y)
1521 apply (simp add: le_less [where 'a=real])
1523 apply (rule subsetD [OF closure_subset], simp)
1524 apply (simp add: closure_def)
1526 apply (rule closure_ball_lemma)
1527 apply (simp add: zero_less_dist_iff)
1530 (* In a trivial vector space, this fails for e = 0. *)
1531 lemma interior_cball:
1532 fixes x :: "'a::{real_normed_vector, perfect_space}"
1533 shows "interior (cball x e) = ball x e"
1534 proof(cases "e\<ge>0")
1535 case False note cs = this
1536 from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
1537 { fix y assume "y \<in> cball x e"
1538 hence False unfolding mem_cball using dist_nz[of x y] cs by auto }
1539 hence "cball x e = {}" by auto
1540 hence "interior (cball x e) = {}" using interior_empty by auto
1541 ultimately show ?thesis by blast
1543 case True note cs = this
1544 have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
1545 { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
1546 then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
1548 then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
1549 using perfect_choose_dist [of d] by auto
1550 have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
1551 hence xa_cball:"xa \<in> cball x e" using as(1) by auto
1553 hence "y \<in> ball x e" proof(cases "x = y")
1555 hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
1556 thus "y \<in> ball x e" using `x = y ` by simp
1559 have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
1560 using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
1561 hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
1562 have "y - x \<noteq> 0" using `x \<noteq> y` by auto
1563 hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
1564 using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
1566 have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x = norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
1567 by (auto simp add: dist_norm algebra_simps)
1568 also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1569 by (auto simp add: algebra_simps)
1570 also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
1572 also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
1573 finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
1574 thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
1576 hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
1577 ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
1580 lemma frontier_ball:
1581 fixes a :: "'a::real_normed_vector"
1582 shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
1583 apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
1584 apply (simp add: set_eq_iff)
1587 lemma frontier_cball:
1588 fixes a :: "'a::{real_normed_vector, perfect_space}"
1589 shows "frontier(cball a e) = {x. dist a x = e}"
1590 apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
1591 apply (simp add: set_eq_iff)
1594 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
1595 apply (simp add: set_eq_iff not_le)
1596 by (metis zero_le_dist dist_self order_less_le_trans)
1597 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
1599 lemma cball_eq_sing:
1600 fixes x :: "'a::{metric_space,perfect_space}"
1601 shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
1602 proof (rule linorder_cases)
1604 obtain a where "a \<noteq> x" "dist a x < e"
1605 using perfect_choose_dist [OF e] by auto
1606 hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
1607 with e show ?thesis by (auto simp add: set_eq_iff)
1611 fixes x :: "'a::metric_space"
1612 shows "e = 0 ==> cball x e = {x}"
1613 by (auto simp add: set_eq_iff)
1616 subsection {* Boundedness *}
1618 (* FIXME: This has to be unified with BSEQ!! *)
1619 definition (in metric_space)
1620 bounded :: "'a set \<Rightarrow> bool" where
1621 "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
1623 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
1624 unfolding bounded_def
1626 apply (rule_tac x="dist a x + e" in exI, clarify)
1627 apply (drule (1) bspec)
1628 apply (erule order_trans [OF dist_triangle add_left_mono])
1632 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
1633 unfolding bounded_any_center [where a=0]
1634 by (simp add: dist_norm)
1636 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
1637 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
1638 by (metis bounded_def subset_eq)
1640 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
1641 by (metis bounded_subset interior_subset)
1643 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
1645 from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
1646 { fix y assume "y \<in> closure S"
1647 then obtain f where f: "\<forall>n. f n \<in> S" "(f ---> y) sequentially"
1648 unfolding closure_sequential by auto
1649 have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
1650 hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
1651 by (rule eventually_mono, simp add: f(1))
1652 have "dist x y \<le> a"
1653 apply (rule Lim_dist_ubound [of sequentially f])
1654 apply (rule trivial_limit_sequentially)
1659 thus ?thesis unfolding bounded_def by auto
1662 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
1663 apply (simp add: bounded_def)
1664 apply (rule_tac x=x in exI)
1665 apply (rule_tac x=e in exI)
1669 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
1670 by (metis ball_subset_cball bounded_cball bounded_subset)
1672 lemma finite_imp_bounded[intro]:
1673 fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"
1675 { fix a and F :: "'a set" assume as:"bounded F"
1676 then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
1677 hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
1678 hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
1680 thus ?thesis using finite_induct[of S bounded] using bounded_empty assms by auto
1683 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
1684 apply (auto simp add: bounded_def)
1685 apply (rename_tac x y r s)
1686 apply (rule_tac x=x in exI)
1687 apply (rule_tac x="max r (dist x y + s)" in exI)
1688 apply (rule ballI, rename_tac z, safe)
1689 apply (drule (1) bspec, simp)
1690 apply (drule (1) bspec)
1691 apply (rule min_max.le_supI2)
1692 apply (erule order_trans [OF dist_triangle add_left_mono])
1695 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
1696 by (induct rule: finite_induct[of F], auto)
1698 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
1699 apply (simp add: bounded_iff)
1700 apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
1703 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
1704 by (metis Int_lower1 Int_lower2 bounded_subset)
1706 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
1707 apply (metis Diff_subset bounded_subset)
1710 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
1711 by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
1713 lemma not_bounded_UNIV[simp, intro]:
1714 "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
1715 proof(auto simp add: bounded_pos not_le)
1716 obtain x :: 'a where "x \<noteq> 0"
1717 using perfect_choose_dist [OF zero_less_one] by fast
1718 fix b::real assume b: "b >0"
1719 have b1: "b +1 \<ge> 0" using b by simp
1720 with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
1721 by (simp add: norm_sgn)
1722 then show "\<exists>x::'a. b < norm x" ..
1725 lemma bounded_linear_image:
1726 assumes "bounded S" "bounded_linear f"
1727 shows "bounded(f ` S)"
1729 from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
1730 from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x" using bounded_linear.pos_bounded by (auto simp add: mult_ac)
1731 { fix x assume "x\<in>S"
1732 hence "norm x \<le> b" using b by auto
1733 hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
1734 by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
1736 thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
1737 using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
1740 lemma bounded_scaling:
1741 fixes S :: "'a::real_normed_vector set"
1742 shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
1743 apply (rule bounded_linear_image, assumption)
1744 apply (rule bounded_linear_scaleR_right)
1747 lemma bounded_translation:
1748 fixes S :: "'a::real_normed_vector set"
1749 assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
1751 from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
1752 { fix x assume "x\<in>S"
1753 hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
1755 thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
1756 by (auto intro!: add exI[of _ "b + norm a"])
1760 text{* Some theorems on sups and infs using the notion "bounded". *}
1763 fixes S :: "real set"
1764 shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"
1765 by (simp add: bounded_iff)
1767 lemma bounded_has_Sup:
1768 fixes S :: "real set"
1769 assumes "bounded S" "S \<noteq> {}"
1770 shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
1772 fix x assume "x\<in>S"
1773 thus "x \<le> Sup S"
1774 by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
1776 show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
1777 by (metis SupInf.Sup_least)
1781 fixes S :: "real set"
1782 shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
1783 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal)
1785 lemma Sup_insert_finite:
1786 fixes S :: "real set"
1787 shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
1788 apply (rule Sup_insert)
1789 apply (rule finite_imp_bounded)
1792 lemma bounded_has_Inf:
1793 fixes S :: "real set"
1794 assumes "bounded S" "S \<noteq> {}"
1795 shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
1797 fix x assume "x\<in>S"
1798 from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
1799 thus "x \<ge> Inf S" using `x\<in>S`
1800 by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
1802 show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
1803 by (metis SupInf.Inf_greatest)
1807 fixes S :: "real set"
1808 shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
1809 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)
1810 lemma Inf_insert_finite:
1811 fixes S :: "real set"
1812 shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
1813 by (rule Inf_insert, rule finite_imp_bounded, simp)
1815 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
1816 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
1817 apply (frule isGlb_isLb)
1818 apply (frule_tac x = y in isGlb_isLb)
1819 apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
1823 subsection {* Equivalent versions of compactness *}
1825 subsubsection{* Sequential compactness *}
1828 compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
1829 "compact S \<longleftrightarrow>
1830 (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
1831 (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
1834 assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
1836 unfolding compact_def using assms by fast
1839 assumes "compact S" "\<forall>n. f n \<in> S"
1840 obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
1841 using assms unfolding compact_def by fast
1844 A metric space (or topological vector space) is said to have the
1845 Heine-Borel property if every closed and bounded subset is compact.
1848 class heine_borel = metric_space +
1849 assumes bounded_imp_convergent_subsequence:
1850 "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
1851 \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
1853 lemma bounded_closed_imp_compact:
1854 fixes s::"'a::heine_borel set"
1855 assumes "bounded s" and "closed s" shows "compact s"
1856 proof (unfold compact_def, clarify)
1857 fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
1858 obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
1859 using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
1860 from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
1861 have "l \<in> s" using `closed s` fr l
1862 unfolding closed_sequential_limits by blast
1863 show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
1864 using `l \<in> s` r l by blast
1867 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
1869 show "0 \<le> r 0" by auto
1871 fix n assume "n \<le> r n"
1872 moreover have "r n < r (Suc n)"
1873 using assms [unfolded subseq_def] by auto
1874 ultimately show "Suc n \<le> r (Suc n)" by auto
1877 lemma eventually_subseq:
1878 assumes r: "subseq r"
1879 shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
1880 unfolding eventually_sequentially
1881 by (metis subseq_bigger [OF r] le_trans)
1884 "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
1885 unfolding tendsto_def eventually_sequentially o_def
1886 by (metis subseq_bigger le_trans)
1888 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
1890 apply (rule_tac x="nat_rec e f" in exI)
1892 apply (rule def_nat_rec_0, simp)
1893 apply (rule allI, rule def_nat_rec_Suc, simp)
1894 apply (rule allI, rule impI, rule ext)
1896 apply (induct_tac x)
1898 apply (erule_tac x="n" in allE)
1902 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
1903 assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
1904 shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N. abs(s n - l) < e"
1906 have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
1907 then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
1908 { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
1910 obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
1911 have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
1912 with n have "s N \<le> t - e" using `e>0` by auto
1913 hence "s n \<le> t - e" using assms(1)[unfolded incseq_def, THEN spec[where x=n], THEN spec[where x=N]] using `n\<le>N` by auto }
1914 hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
1915 hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto }
1916 thus ?thesis by blast
1919 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
1920 assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
1921 shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
1922 using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
1923 unfolding monoseq_def incseq_def
1924 apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
1925 unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
1927 (* TODO: merge this lemma with the ones above *)
1928 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"
1929 assumes "bounded {s n| n::nat. True}" "\<forall>n. (s n) \<le>(s(Suc n))"
1930 shows "\<exists>l. (s ---> l) sequentially"
1932 obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le> a" using assms(1)[unfolded bounded_iff] by auto
1934 have "\<And> n. n\<ge>m \<longrightarrow> (s m) \<le> (s n)"
1935 apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)
1936 apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq) }
1937 hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto
1938 then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar> (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a]
1939 unfolding monoseq_def by auto
1940 thus ?thesis unfolding LIMSEQ_def apply(rule_tac x="l" in exI)
1941 unfolding dist_norm by auto
1944 lemma compact_real_lemma:
1945 assumes "\<forall>n::nat. abs(s n) \<le> b"
1946 shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
1948 obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
1949 using seq_monosub[of s] by auto
1950 thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
1951 unfolding tendsto_iff dist_norm eventually_sequentially by auto
1954 instance real :: heine_borel
1956 fix s :: "real set" and f :: "nat \<Rightarrow> real"
1957 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
1958 then obtain b where b: "\<forall>n. abs (f n) \<le> b"
1959 unfolding bounded_iff by auto
1960 obtain l :: real and r :: "nat \<Rightarrow> nat" where
1961 r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
1962 using compact_real_lemma [OF b] by auto
1963 thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
1967 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $$ i) ` s)"
1968 apply (erule bounded_linear_image)
1969 apply (rule bounded_linear_euclidean_component)
1972 lemma compact_lemma:
1973 fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
1974 assumes "bounded s" and "\<forall>n. f n \<in> s"
1975 shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>
1976 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
1978 fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"
1979 have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
1980 hence "\<exists>l::'a. \<exists>r. subseq r \<and>
1981 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
1982 proof(induct d) case empty thus ?case unfolding subseq_def by auto
1983 next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto
1984 have s': "bounded ((\<lambda>x. x $$ k) ` s)" using `bounded s` by (rule bounded_component)
1985 obtain l1::"'a" and r1 where r1:"subseq r1" and
1986 lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially"
1987 using insert(3) using insert(4) by auto
1988 have f': "\<forall>n. f (r1 n) $$ k \<in> (\<lambda>x. x $$ k) ` s" using `\<forall>n. f n \<in> s` by simp
1989 obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $$ k) ---> l2) sequentially"
1990 using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
1991 def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
1992 using r1 and r2 unfolding r_def o_def subseq_def by auto
1994 def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"
1995 { fix e::real assume "e>0"
1996 from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially" by blast
1997 from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $$ k) l2 < e) sequentially" by (rule tendstoD)
1998 from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$ i) (l1 $$ i) < e) sequentially"
1999 by (rule eventually_subseq)
2000 have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$ i) (l $$ i) < e) sequentially"
2001 using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def
2002 using insert.prems by auto
2004 ultimately show ?case by auto
2006 thus "\<exists>l::'a. \<exists>r. subseq r \<and>
2007 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2008 apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe
2009 apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe
2010 apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)
2011 apply(erule_tac x=i in ballE)
2012 proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a
2013 assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"
2014 hence *:"i\<ge>DIM('a)" by auto
2015 thus "dist (f (r n) $$ i) (l $$ i) < e" using e by auto
2019 instance euclidean_space \<subseteq> heine_borel
2021 fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"
2022 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2023 then obtain l::'a and r where r: "subseq r"
2024 and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $$ i) (l $$ i) < e) sequentially"
2025 using compact_lemma [OF s f] by blast
2026 let ?d = "{..<DIM('a)}"
2027 { fix e::real assume "e>0"
2028 hence "0 < e / (real_of_nat (card ?d))"
2029 using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
2030 with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))) sequentially"
2033 { fix n assume n: "\<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))"
2034 have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $$ i) (l $$ i))"
2035 apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)
2036 also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
2037 apply(rule setsum_strict_mono) using n by auto
2038 finally have "dist (f (r n)) l < e" unfolding setsum_constant
2039 using DIM_positive[where 'a='a] by auto
2041 ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
2042 by (rule eventually_elim1)
2044 hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
2045 with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
2048 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
2049 unfolding bounded_def
2051 apply (rule_tac x="a" in exI)
2052 apply (rule_tac x="e" in exI)
2054 apply (drule (1) bspec)
2055 apply (simp add: dist_Pair_Pair)
2056 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
2059 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
2060 unfolding bounded_def
2062 apply (rule_tac x="b" in exI)
2063 apply (rule_tac x="e" in exI)
2065 apply (drule (1) bspec)
2066 apply (simp add: dist_Pair_Pair)
2067 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
2070 instance prod :: (heine_borel, heine_borel) heine_borel
2072 fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
2073 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2074 from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
2075 from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
2076 obtain l1 r1 where r1: "subseq r1"
2077 and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
2078 using bounded_imp_convergent_subsequence [OF s1 f1]
2079 unfolding o_def by fast
2080 from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
2081 from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
2082 obtain l2 r2 where r2: "subseq r2"
2083 and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
2084 using bounded_imp_convergent_subsequence [OF s2 f2]
2085 unfolding o_def by fast
2086 have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
2087 using lim_subseq [OF r2 l1] unfolding o_def .
2088 have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
2089 using tendsto_Pair [OF l1' l2] unfolding o_def by simp
2090 have r: "subseq (r1 \<circ> r2)"
2091 using r1 r2 unfolding subseq_def by simp
2092 show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2096 subsubsection{* Completeness *}
2099 "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
2100 unfolding Cauchy_def by blast
2103 complete :: "'a::metric_space set \<Rightarrow> bool" where
2104 "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
2105 --> (\<exists>l \<in> s. (f ---> l) sequentially))"
2107 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
2112 with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
2113 by (erule_tac x="e/2" in allE) auto
2115 assume nm:"N \<le> m \<and> N \<le> n"
2116 hence "dist (s m) (s n) < e" using N
2117 using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
2120 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
2124 unfolding cauchy_def
2128 unfolding cauchy_def
2129 using dist_triangle_half_l
2133 lemma convergent_imp_cauchy:
2134 "(s ---> l) sequentially ==> Cauchy s"
2135 proof(simp only: cauchy_def, rule, rule)
2136 fix e::real assume "e>0" "(s ---> l) sequentially"
2137 then obtain N::nat where N:"\<forall>n\<ge>N. dist (s n) l < e/2" unfolding LIMSEQ_def by(erule_tac x="e/2" in allE) auto
2138 thus "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e" using dist_triangle_half_l[of _ l e _] by (rule_tac x=N in exI) auto
2141 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
2143 from assms obtain N::nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1" unfolding cauchy_def apply(erule_tac x= 1 in allE) by auto
2144 hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
2146 have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
2147 then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
2148 unfolding bounded_any_center [where a="s N"] by auto
2149 ultimately show "?thesis"
2150 unfolding bounded_any_center [where a="s N"]
2151 apply(rule_tac x="max a 1" in exI) apply auto
2152 apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto
2155 lemma compact_imp_complete: assumes "compact s" shows "complete s"
2157 { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
2158 from as(1) obtain l r where lr: "l\<in>s" "subseq r" "((f \<circ> r) ---> l) sequentially" using assms unfolding compact_def by blast
2160 note lr' = subseq_bigger [OF lr(2)]
2162 { fix e::real assume "e>0"
2163 from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2" unfolding cauchy_def using `e>0` apply (erule_tac x="e/2" in allE) by auto
2164 from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2" using `e>0` by auto
2165 { fix n::nat assume n:"n \<ge> max N M"
2166 have "dist ((f \<circ> r) n) l < e/2" using n M by auto
2167 moreover have "r n \<ge> N" using lr'[of n] n by auto
2168 hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
2169 ultimately have "dist (f n) l < e" using dist_triangle_half_r[of "f (r n)" "f n" e l] by (auto simp add: dist_commute) }
2170 hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast }
2171 hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding LIMSEQ_def by auto }
2172 thus ?thesis unfolding complete_def by auto
2175 instance heine_borel < complete_space
2177 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2178 hence "bounded (range f)"
2179 by (rule cauchy_imp_bounded)
2180 hence "compact (closure (range f))"
2181 using bounded_closed_imp_compact [of "closure (range f)"] by auto
2182 hence "complete (closure (range f))"
2183 by (rule compact_imp_complete)
2184 moreover have "\<forall>n. f n \<in> closure (range f)"
2185 using closure_subset [of "range f"] by auto
2186 ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
2187 using `Cauchy f` unfolding complete_def by auto
2188 then show "convergent f"
2189 unfolding convergent_def by auto
2192 instance euclidean_space \<subseteq> banach ..
2194 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
2195 proof(simp add: complete_def, rule, rule)
2196 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2197 hence "convergent f" by (rule Cauchy_convergent)
2198 thus "\<exists>l. f ----> l" unfolding convergent_def .
2201 lemma complete_imp_closed: assumes "complete s" shows "closed s"
2203 { fix x assume "x islimpt s"
2204 then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
2205 unfolding islimpt_sequential by auto
2206 then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
2207 using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
2208 hence "x \<in> s" using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
2210 thus "closed s" unfolding closed_limpt by auto
2213 lemma complete_eq_closed:
2214 fixes s :: "'a::complete_space set"
2215 shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
2217 assume ?lhs thus ?rhs by (rule complete_imp_closed)
2220 { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
2221 then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
2222 hence "\<exists>l\<in>s. (f ---> l) sequentially" using `?rhs`[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]] using as(1) by auto }
2223 thus ?lhs unfolding complete_def by auto
2226 lemma convergent_eq_cauchy:
2227 fixes s :: "nat \<Rightarrow> 'a::complete_space"
2228 shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"
2229 unfolding Cauchy_convergent_iff convergent_def ..
2231 lemma convergent_imp_bounded:
2232 fixes s :: "nat \<Rightarrow> 'a::metric_space"
2233 shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"
2234 by (intro cauchy_imp_bounded convergent_imp_cauchy)
2236 subsubsection{* Total boundedness *}
2238 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
2239 "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
2240 declare helper_1.simps[simp del]
2242 lemma compact_imp_totally_bounded:
2244 shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
2245 proof(rule, rule, rule ccontr)
2246 fix e::real assume "e>0" and assm:"\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k)"
2247 def x \<equiv> "helper_1 s e"
2249 have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
2250 proof(induct_tac rule:nat_less_induct)
2251 fix n def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
2252 assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
2253 have "\<not> s \<subseteq> (\<Union>x\<in>x ` {0..<n}. ball x e)" using assm apply simp apply(erule_tac x="x ` {0 ..< n}" in allE) using as by auto
2254 then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
2255 have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
2256 apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
2257 thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
2259 hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
2260 then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded compact_def, THEN spec[where x=x]] by auto
2261 from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
2262 then obtain N::nat where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e" unfolding cauchy_def using `e>0` by auto
2264 using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
2265 using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
2266 using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
2269 subsubsection{* Heine-Borel theorem *}
2271 text {* Following Burkill \& Burkill vol. 2. *}
2273 lemma heine_borel_lemma: fixes s::"'a::metric_space set"
2274 assumes "compact s" "s \<subseteq> (\<Union> t)" "\<forall>b \<in> t. open b"
2275 shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
2277 assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"
2278 hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto
2280 have "1 / real (n + 1) > 0" by auto
2281 hence "\<exists>x. x\<in>s \<and> (\<forall>xa\<in>t. \<not> (ball x (inverse (real (n+1))) \<subseteq> xa))" using cont unfolding Bex_def by auto }
2282 hence "\<forall>n::nat. \<exists>x. x \<in> s \<and> (\<forall>xa\<in>t. \<not> ball x (inverse (real (n + 1))) \<subseteq> xa)" by auto
2283 then obtain f where f:"\<forall>n::nat. f n \<in> s \<and> (\<forall>xa\<in>t. \<not> ball (f n) (inverse (real (n + 1))) \<subseteq> xa)"
2284 using choice[of "\<lambda>n::nat. \<lambda>x. x\<in>s \<and> (\<forall>xa\<in>t. \<not> ball x (inverse (real (n + 1))) \<subseteq> xa)"] by auto
2286 then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
2287 using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
2289 obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
2290 then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"
2291 using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto
2293 then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"
2294 using lr[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] by auto
2296 obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
2297 have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
2298 apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
2299 using subseq_bigger[OF r, of "N1 + N2"] by auto
2301 def x \<equiv> "(f (r (N1 + N2)))"
2302 have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
2303 using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto
2304 have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto
2305 then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto
2307 have "dist x l < e/2" using N1 unfolding x_def o_def by auto
2308 hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)
2310 thus False using e and `y\<notin>b` by auto
2313 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2314 \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"
2316 fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"
2317 then obtain e::real where "e>0" and "\<forall>x\<in>s. \<exists>b\<in>f. ball x e \<subseteq> b" using heine_borel_lemma[of s f] by auto
2318 hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto
2319 hence "\<exists>bb. \<forall>x\<in>s. bb x \<in>f \<and> ball x e \<subseteq> bb x" using bchoice[of s "\<lambda>x b. b\<in>f \<and> ball x e \<subseteq> b"] by auto
2320 then obtain bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast
2322 from `compact s` have "\<exists> k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" using compact_imp_totally_bounded[of s] `e>0` by auto
2323 then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto
2325 have "finite (bb ` k)" using k(1) by auto
2327 { fix x assume "x\<in>s"
2328 hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3) unfolding subset_eq by auto
2329 hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast
2330 hence "x \<in> \<Union>(bb ` k)" using Union_iff[of x "bb ` k"] by auto
2332 ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f'" using bb k(2) by (rule_tac x="bb ` k" in exI) auto
2335 subsubsection {* Bolzano-Weierstrass property *}
2337 lemma heine_borel_imp_bolzano_weierstrass:
2338 assumes "\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f) --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f'))"
2339 "infinite t" "t \<subseteq> s"
2340 shows "\<exists>x \<in> s. x islimpt t"
2342 assume "\<not> (\<exists>x \<in> s. x islimpt t)"
2343 then obtain f where f:"\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)" unfolding islimpt_def
2344 using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"] by auto
2345 obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
2346 using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
2347 from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
2348 { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
2349 hence "x \<in> f x" "y \<in> f x \<longrightarrow> y = x" using f[THEN bspec[where x=x]] and `t\<subseteq>s` by auto
2350 hence "x = y" using `f x = f y` and f[THEN bspec[where x=y]] and `y\<in>t` and `t\<subseteq>s` by auto }
2351 hence "inj_on f t" unfolding inj_on_def by simp
2352 hence "infinite (f ` t)" using assms(2) using finite_imageD by auto
2354 { fix x assume "x\<in>t" "f x \<notin> g"
2355 from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
2356 then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
2357 hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
2358 hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto }
2359 hence "f ` t \<subseteq> g" by auto
2360 ultimately show False using g(2) using finite_subset by auto
2363 subsubsection {* Complete the chain of compactness variants *}
2365 lemma islimpt_range_imp_convergent_subsequence:
2366 fixes f :: "nat \<Rightarrow> 'a::metric_space"
2367 assumes "l islimpt (range f)"
2368 shows "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2369 proof (intro exI conjI)
2370 have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"
2371 using assms unfolding islimpt_def
2372 by (drule_tac x="ball l e" in spec)
2373 (auto simp add: zero_less_dist_iff dist_commute)
2375 def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"
2376 have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"
2377 unfolding t_def by (rule LeastI2_ex [OF * conjunct1])
2378 have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"
2379 unfolding t_def by (rule LeastI2_ex [OF * conjunct2])
2380 have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"
2381 unfolding t_def by (simp add: Least_le)
2382 have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"
2383 unfolding t_def by (drule not_less_Least) simp
2384 have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"
2386 apply (erule f_t_neq)
2387 apply (erule (1) less_le_trans [OF f_t_closer])
2389 have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"
2390 by (drule f_t_closer) auto
2391 have t_less: "\<And>e. 0 < e \<Longrightarrow> t e < t (dist (f (t e)) l)"
2392 apply (subst less_le)
2394 apply (rule t_antimono)
2395 apply (erule f_t_neq)
2396 apply (erule f_t_closer [THEN less_imp_le])
2397 apply (rule t_dist_f_neq [symmetric])
2398 apply (erule f_t_neq)
2400 have dist_f_t_less':
2401 "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"
2402 apply (simp add: le_less)
2404 apply (rule less_trans)
2405 apply (erule f_t_closer)
2406 apply (rule le_less_trans)
2407 apply (erule less_tD)
2408 apply (erule f_t_neq)
2409 apply (erule f_t_closer)
2411 apply (erule f_t_closer)
2414 def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"
2415 have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"
2416 unfolding r_def by simp_all
2417 have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"
2418 by (induct_tac n) (simp_all add: r_simps f_t_neq)
2421 unfolding subseq_Suc_iff
2424 apply (simp_all add: r_simps)
2425 apply (rule t_less, rule zero_less_one)
2426 apply (rule t_less, rule f_r_neq)
2428 show "((f \<circ> r) ---> l) sequentially"
2429 unfolding LIMSEQ_def o_def
2430 apply (clarify, rename_tac e, rule_tac x="t e" in exI, clarify)
2431 apply (drule le_trans, rule seq_suble [OF `subseq r`])
2432 apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)
2436 lemma finite_range_imp_infinite_repeats:
2437 fixes f :: "nat \<Rightarrow> 'a"
2438 assumes "finite (range f)"
2439 shows "\<exists>k. infinite {n. f n = k}"
2441 { fix A :: "'a set" assume "finite A"
2442 hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"
2444 case empty thus ?case by simp
2448 proof (cases "finite {n. f n = x}")
2450 with `infinite {n. f n \<in> insert x A}`
2451 have "infinite {n. f n \<in> A}" by simp
2452 thus "\<exists>k. infinite {n. f n = k}" by (rule insert)
2454 case False thus "\<exists>k. infinite {n. f n = k}" ..
2458 from assms show "\<exists>k. infinite {n. f n = k}"
2462 lemma bolzano_weierstrass_imp_compact:
2463 fixes s :: "'a::metric_space set"
2464 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2467 { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
2468 have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2469 proof (cases "finite (range f)")
2471 hence "\<exists>l. infinite {n. f n = l}"
2472 by (rule finite_range_imp_infinite_repeats)
2473 then obtain l where "infinite {n. f n = l}" ..
2474 hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"
2475 by (rule infinite_enumerate)
2476 then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto
2477 hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2478 unfolding o_def by (simp add: fr tendsto_const)
2479 hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2481 from f have "\<forall>n. f (r n) \<in> s" by simp
2482 hence "l \<in> s" by (simp add: fr)
2483 thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2484 by (rule rev_bexI) fact
2487 with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto
2488 then obtain l where "l \<in> s" "l islimpt (range f)" ..
2489 have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2490 using `l islimpt (range f)`
2491 by (rule islimpt_range_imp_convergent_subsequence)
2492 with `l \<in> s` show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
2495 thus ?thesis unfolding compact_def by auto
2498 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where
2499 "helper_2 beyond 0 = beyond 0" |
2500 "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"
2502 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"
2503 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2506 assume "\<not> bounded s"
2507 then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"
2508 unfolding bounded_any_center [where a=undefined]
2509 apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto
2510 hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"
2511 unfolding linorder_not_le by auto
2512 def x \<equiv> "helper_2 beyond"
2514 { fix m n ::nat assume "m<n"
2515 hence "dist undefined (x m) + 1 < dist undefined (x n)"
2517 case 0 thus ?case by auto
2520 have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"
2521 unfolding x_def and helper_2.simps
2522 using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto
2523 thus ?case proof(cases "m < n")
2524 case True thus ?thesis using Suc and * by auto
2526 case False hence "m = n" using Suc(2) by auto
2527 thus ?thesis using * by auto
2530 { fix m n ::nat assume "m\<noteq>n"
2531 have "1 < dist (x m) (x n)"
2534 hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto
2535 thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith
2537 case False hence "n<m" using `m\<noteq>n` by auto
2538 hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto
2539 thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith
2540 qed } note ** = this
2541 { fix a b assume "x a = x b" "a \<noteq> b"
2542 hence False using **[of a b] by auto }
2543 hence "inj x" unfolding inj_on_def by auto
2547 proof(cases "n = 0")
2548 case True thus ?thesis unfolding x_def using beyond by auto
2550 case False then obtain z where "n = Suc z" using not0_implies_Suc by auto
2551 thus ?thesis unfolding x_def using beyond by auto
2553 ultimately have "infinite (range x) \<and> range x \<subseteq> s" unfolding x_def using range_inj_infinite[of "helper_2 beyond"] using beyond(1) by auto
2555 then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto
2556 then obtain y where "x y \<noteq> l" and y:"dist (x y) l < 1/2" unfolding islimpt_approachable apply(erule_tac x="1/2" in allE) by auto
2557 then obtain z where "x z \<noteq> l" and z:"dist (x z) l < dist (x y) l" using l[unfolded islimpt_approachable, THEN spec[where x="dist (x y) l"]]
2558 unfolding dist_nz by auto
2559 show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto
2562 lemma sequence_infinite_lemma:
2563 fixes f :: "nat \<Rightarrow> 'a::t1_space"
2564 assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"
2565 shows "infinite (range f)"
2567 assume "finite (range f)"
2568 hence "closed (range f)" by (rule finite_imp_closed)
2569 hence "open (- range f)" by (rule open_Compl)
2570 from assms(1) have "l \<in> - range f" by auto
2571 from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
2572 using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)
2573 thus False unfolding eventually_sequentially by auto
2576 lemma closure_insert:
2577 fixes x :: "'a::t1_space"
2578 shows "closure (insert x s) = insert x (closure s)"
2579 apply (rule closure_unique)
2580 apply (rule insert_mono [OF closure_subset])
2581 apply (rule closed_insert [OF closed_closure])
2582 apply (simp add: closure_minimal)
2585 lemma islimpt_insert:
2586 fixes x :: "'a::t1_space"
2587 shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
2589 assume *: "x islimpt (insert a s)"
2591 proof (rule islimptI)
2592 fix t assume t: "x \<in> t" "open t"
2593 show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
2594 proof (cases "x = a")
2596 obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
2597 using * t by (rule islimptE)
2598 with `x = a` show ?thesis by auto
2601 with t have t': "x \<in> t - {a}" "open (t - {a})"
2602 by (simp_all add: open_Diff)
2603 obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
2604 using * t' by (rule islimptE)
2605 thus ?thesis by auto
2609 assume "x islimpt s" thus "x islimpt (insert a s)"
2610 by (rule islimpt_subset) auto
2613 lemma islimpt_union_finite:
2614 fixes x :: "'a::t1_space"
2615 shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
2616 by (induct set: finite, simp_all add: islimpt_insert)
2618 lemma sequence_unique_limpt:
2619 fixes f :: "nat \<Rightarrow> 'a::t2_space"
2620 assumes "(f ---> l) sequentially" and "l' islimpt (range f)"
2623 assume "l' \<noteq> l"
2624 obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
2625 using hausdorff [OF `l' \<noteq> l`] by auto
2626 have "eventually (\<lambda>n. f n \<in> t) sequentially"
2627 using assms(1) `open t` `l \<in> t` by (rule topological_tendstoD)
2628 then obtain N where "\<forall>n\<ge>N. f n \<in> t"
2629 unfolding eventually_sequentially by auto
2631 have "UNIV = {..<N} \<union> {N..}" by auto
2632 hence "l' islimpt (f ` ({..<N} \<union> {N..}))" using assms(2) by simp
2633 hence "l' islimpt (f ` {..<N} \<union> f ` {N..})" by (simp add: image_Un)
2634 hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite)
2635 then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
2636 using `l' \<in> s` `open s` by (rule islimptE)
2637 then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto
2638 with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t" by simp
2639 with `s \<inter> t = {}` show False by simp
2642 lemma bolzano_weierstrass_imp_closed:
2643 fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)
2644 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2647 { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
2649 proof(cases "\<forall>n. x n \<noteq> l")
2650 case False thus "l\<in>s" using as(1) by auto
2652 case True note cas = this
2653 with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto
2654 then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto
2655 thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
2657 thus ?thesis unfolding closed_sequential_limits by fast
2660 text {* Hence express everything as an equivalence. *}
2662 lemma compact_eq_heine_borel:
2663 fixes s :: "'a::metric_space set"
2664 shows "compact s \<longleftrightarrow>
2665 (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2666 --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")
2668 assume ?lhs thus ?rhs by (rule compact_imp_heine_borel)
2671 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
2672 by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])
2673 thus ?lhs by (rule bolzano_weierstrass_imp_compact)
2676 lemma compact_eq_bolzano_weierstrass:
2677 fixes s :: "'a::metric_space set"
2678 shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
2680 assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto
2682 assume ?rhs thus ?lhs by (rule bolzano_weierstrass_imp_compact)
2685 lemma compact_eq_bounded_closed:
2686 fixes s :: "'a::heine_borel set"
2687 shows "compact s \<longleftrightarrow> bounded s \<and> closed s" (is "?lhs = ?rhs")
2689 assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto
2691 assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto
2694 lemma compact_imp_bounded:
2695 fixes s :: "'a::metric_space set"
2696 shows "compact s ==> bounded s"
2699 hence "\<forall>f. (\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f \<longrightarrow> (\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')"
2700 by (rule compact_imp_heine_borel)
2701 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
2702 using heine_borel_imp_bolzano_weierstrass[of s] by auto
2704 by (rule bolzano_weierstrass_imp_bounded)
2707 lemma compact_imp_closed:
2708 fixes s :: "'a::metric_space set"
2709 shows "compact s ==> closed s"
2712 hence "\<forall>f. (\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f \<longrightarrow> (\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')"
2713 by (rule compact_imp_heine_borel)
2714 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
2715 using heine_borel_imp_bolzano_weierstrass[of s] by auto
2717 by (rule bolzano_weierstrass_imp_closed)
2720 text{* In particular, some common special cases. *}
2722 lemma compact_empty[simp]:
2724 unfolding compact_def
2727 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
2728 unfolding subseq_def by simp (* TODO: move somewhere else *)
2730 lemma compact_union [intro]:
2731 assumes "compact s" and "compact t"
2732 shows "compact (s \<union> t)"
2733 proof (rule compactI)
2734 fix f :: "nat \<Rightarrow> 'a"
2735 assume "\<forall>n. f n \<in> s \<union> t"
2736 hence "infinite {n. f n \<in> s \<union> t}" by simp
2737 hence "infinite {n. f n \<in> s} \<or> infinite {n. f n \<in> t}" by simp
2738 thus "\<exists>l\<in>s \<union> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2740 assume "infinite {n. f n \<in> s}"
2741 from infinite_enumerate [OF this]
2742 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> s" by auto
2743 obtain r l where "l \<in> s" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
2744 using `compact s` `\<forall>n. (f \<circ> q) n \<in> s` by (rule compactE)
2745 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
2746 using `subseq q` by (simp_all add: subseq_o o_assoc)
2747 thus ?thesis by auto
2749 assume "infinite {n. f n \<in> t}"
2750 from infinite_enumerate [OF this]
2751 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> t" by auto
2752 obtain r l where "l \<in> t" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
2753 using `compact t` `\<forall>n. (f \<circ> q) n \<in> t` by (rule compactE)
2754 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
2755 using `subseq q` by (simp_all add: subseq_o o_assoc)
2756 thus ?thesis by auto
2760 lemma compact_inter_closed [intro]:
2761 assumes "compact s" and "closed t"
2762 shows "compact (s \<inter> t)"
2763 proof (rule compactI)
2764 fix f :: "nat \<Rightarrow> 'a"
2765 assume "\<forall>n. f n \<in> s \<inter> t"
2766 hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t" by simp_all
2767 obtain l r where "l \<in> s" "subseq r" "((f \<circ> r) ---> l) sequentially"
2768 using `compact s` `\<forall>n. f n \<in> s` by (rule compactE)
2770 from `closed t` `\<forall>n. f n \<in> t` `((f \<circ> r) ---> l) sequentially` have "l \<in> t"
2771 unfolding closed_sequential_limits o_def by fast
2772 ultimately show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2776 lemma closed_inter_compact [intro]:
2777 assumes "closed s" and "compact t"
2778 shows "compact (s \<inter> t)"
2779 using compact_inter_closed [of t s] assms
2780 by (simp add: Int_commute)
2782 lemma compact_inter [intro]:
2783 assumes "compact s" and "compact t"
2784 shows "compact (s \<inter> t)"
2785 using assms by (intro compact_inter_closed compact_imp_closed)
2787 lemma compact_sing [simp]: "compact {a}"
2788 unfolding compact_def o_def subseq_def
2789 by (auto simp add: tendsto_const)
2791 lemma compact_insert [simp]:
2792 assumes "compact s" shows "compact (insert x s)"
2794 have "compact ({x} \<union> s)"
2795 using compact_sing assms by (rule compact_union)
2796 thus ?thesis by simp
2799 lemma finite_imp_compact:
2800 shows "finite s \<Longrightarrow> compact s"
2801 by (induct set: finite) simp_all
2803 lemma compact_cball[simp]:
2804 fixes x :: "'a::heine_borel"
2805 shows "compact(cball x e)"
2806 using compact_eq_bounded_closed bounded_cball closed_cball
2809 lemma compact_frontier_bounded[intro]:
2810 fixes s :: "'a::heine_borel set"
2811 shows "bounded s ==> compact(frontier s)"
2812 unfolding frontier_def
2813 using compact_eq_bounded_closed
2816 lemma compact_frontier[intro]:
2817 fixes s :: "'a::heine_borel set"
2818 shows "compact s ==> compact (frontier s)"
2819 using compact_eq_bounded_closed compact_frontier_bounded
2822 lemma frontier_subset_compact:
2823 fixes s :: "'a::heine_borel set"
2824 shows "compact s ==> frontier s \<subseteq> s"
2825 using frontier_subset_closed compact_eq_bounded_closed
2829 fixes s :: "'a::t1_space set"
2830 shows "open s \<Longrightarrow> open (s - {x})"
2831 by (simp add: open_Diff)
2833 text{* Finite intersection property. I could make it an equivalence in fact. *}
2835 lemma compact_imp_fip:
2836 assumes "compact s" "\<forall>t \<in> f. closed t"
2837 "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"
2838 shows "s \<inter> (\<Inter> f) \<noteq> {}"
2840 assume as:"s \<inter> (\<Inter> f) = {}"
2841 hence "s \<subseteq> \<Union> uminus ` f" by auto
2842 moreover have "Ball (uminus ` f) open" using open_Diff closed_Diff using assms(2) by auto
2843 ultimately obtain f' where f':"f' \<subseteq> uminus ` f" "finite f'" "s \<subseteq> \<Union>f'" using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="(\<lambda>t. - t) ` f"]] by auto
2844 hence "finite (uminus ` f') \<and> uminus ` f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)
2845 hence "s \<inter> \<Inter>uminus ` f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus ` f'"]] by auto
2846 thus False using f'(3) unfolding subset_eq and Union_iff by blast
2850 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}
2852 lemma bounded_closed_nest:
2853 assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
2854 "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)" "bounded(s 0)"
2855 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
2857 from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n" using choice[of "\<lambda>n x. x\<in> s n"] by auto
2858 from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
2860 then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
2861 unfolding compact_def apply(erule_tac x=x in allE) using x using assms(3) by blast
2864 { fix e::real assume "e>0"
2865 with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding LIMSEQ_def by auto
2866 hence "dist ((x \<circ> r) (max N n)) l < e" by auto
2868 have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
2869 hence "(x \<circ> r) (max N n) \<in> s n"
2870 using x apply(erule_tac x=n in allE)
2871 using x apply(erule_tac x="r (max N n)" in allE)
2872 using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto
2873 ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
2875 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
2877 thus ?thesis by auto
2880 text {* Decreasing case does not even need compactness, just completeness. *}
2882 lemma decreasing_closed_nest:
2883 assumes "\<forall>n. closed(s n)"
2884 "\<forall>n. (s n \<noteq> {})"
2885 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
2886 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
2887 shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"
2889 have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
2890 hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
2891 then obtain t where t: "\<forall>n. t n \<in> s n" by auto
2892 { fix e::real assume "e>0"
2893 then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
2894 { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
2895 hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding subset_eq t by blast+
2896 hence "dist (t m) (t n) < e" using N by auto
2898 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
2900 hence "Cauchy t" unfolding cauchy_def by auto
2901 then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
2903 { fix e::real assume "e>0"
2904 then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded LIMSEQ_def] by auto
2905 have "t (max n N) \<in> s n" using assms(3) unfolding subset_eq apply(erule_tac x=n in allE) apply (erule_tac x="max n N" in allE) using t by auto
2906 hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
2908 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
2910 then show ?thesis by auto
2913 text {* Strengthen it to the intersection actually being a singleton. *}
2915 lemma decreasing_closed_nest_sing:
2916 fixes s :: "nat \<Rightarrow> 'a::complete_space set"
2917 assumes "\<forall>n. closed(s n)"
2918 "\<forall>n. s n \<noteq> {}"
2919 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
2920 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
2921 shows "\<exists>a. \<Inter>(range s) = {a}"
2923 obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
2924 { fix b assume b:"b \<in> \<Inter>(range s)"
2925 { fix e::real assume "e>0"
2926 hence "dist a b < e" using assms(4 )using b using a by blast
2928 hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)
2930 with a have "\<Inter>(range s) = {a}" unfolding image_def by auto
2934 text{* Cauchy-type criteria for uniform convergence. *}
2936 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows
2937 "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
2938 (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x --> dist (s m x) (s n x) < e)" (is "?lhs = ?rhs")
2941 then obtain l where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e" by auto
2942 { fix e::real assume "e>0"
2943 then obtain N::nat where N:"\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2" using l[THEN spec[where x="e/2"]] by auto
2944 { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
2945 hence "dist (s m x) (s n x) < e"
2946 using N[THEN spec[where x=m], THEN spec[where x=x]]
2947 using N[THEN spec[where x=n], THEN spec[where x=x]]
2948 using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto }
2949 hence "\<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x --> dist (s m x) (s n x) < e" by auto }
2953 hence "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)" unfolding cauchy_def apply auto by (erule_tac x=e in allE)auto
2954 then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
2955 using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
2956 { fix e::real assume "e>0"
2957 then obtain N where N:"\<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x \<longrightarrow> dist (s m x) (s n x) < e/2"
2958 using `?rhs`[THEN spec[where x="e/2"]] by auto
2959 { fix x assume "P x"
2960 then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
2961 using l[THEN spec[where x=x], unfolded LIMSEQ_def] using `e>0` by(auto elim!: allE[where x="e/2"])
2962 fix n::nat assume "n\<ge>N"
2963 hence "dist(s n x)(l x) < e" using `P x`and N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]
2964 using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"] by (auto simp add: dist_commute) }
2965 hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
2969 lemma uniformly_cauchy_imp_uniformly_convergent:
2970 fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"
2971 assumes "\<forall>e>0.\<exists>N. \<forall>m (n::nat) x. N \<le> m \<and> N \<le> n \<and> P x --> dist(s m x)(s n x) < e"
2972 "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
2973 shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
2975 obtain l' where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l' x) < e"
2976 using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
2978 { fix x assume "P x"
2979 hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
2980 using l and assms(2) unfolding LIMSEQ_def by blast }
2981 ultimately show ?thesis by auto
2985 subsection {* Continuity *}
2987 text {* Define continuity over a net to take in restrictions of the set. *}
2990 continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
2991 where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
2993 lemma continuous_trivial_limit:
2994 "trivial_limit net ==> continuous net f"
2995 unfolding continuous_def tendsto_def trivial_limit_eq by auto
2997 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
2998 unfolding continuous_def
2999 unfolding tendsto_def
3000 using netlimit_within[of x s]
3001 by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
3003 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
3004 using continuous_within [of x UNIV f] by simp
3006 lemma continuous_at_within:
3007 assumes "continuous (at x) f" shows "continuous (at x within s) f"
3008 using assms unfolding continuous_at continuous_within
3009 by (rule Lim_at_within)
3011 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
3013 lemma continuous_within_eps_delta:
3014 "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'\<in> s. dist x' x < d --> dist (f x') (f x) < e)"
3015 unfolding continuous_within and Lim_within
3016 apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto
3018 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3019 \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
3020 using continuous_within_eps_delta [of x UNIV f] by simp
3022 text{* Versions in terms of open balls. *}
3024 lemma continuous_within_ball:
3025 "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3026 f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3029 { fix e::real assume "e>0"
3030 then obtain d where d: "d>0" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"
3031 using `?lhs`[unfolded continuous_within Lim_within] by auto
3032 { fix y assume "y\<in>f ` (ball x d \<inter> s)"
3033 hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
3034 apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
3036 hence "\<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e" using `d>0` unfolding subset_eq ball_def by (auto simp add: dist_commute) }
3039 assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
3040 apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
3043 lemma continuous_at_ball:
3044 "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3046 assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3047 apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x=xa in allE) apply (auto simp add: dist_commute dist_nz)
3048 unfolding dist_nz[THEN sym] by auto
3050 assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3051 apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x="f xa" in allE) by (auto simp add: dist_commute dist_nz)
3054 text{* Define setwise continuity in terms of limits within the set. *}
3058 "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
3060 "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
3062 lemma continuous_on_topological:
3063 "continuous_on s f \<longleftrightarrow>
3064 (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
3065 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
3066 unfolding continuous_on_def tendsto_def
3067 unfolding Limits.eventually_within eventually_at_topological
3068 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
3070 lemma continuous_on_iff:
3071 "continuous_on s f \<longleftrightarrow>
3072 (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
3073 unfolding continuous_on_def Lim_within
3074 apply (intro ball_cong [OF refl] all_cong ex_cong)
3075 apply (rename_tac y, case_tac "y = x", simp)
3076 apply (simp add: dist_nz)
3080 uniformly_continuous_on ::
3081 "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
3083 "uniformly_continuous_on s f \<longleftrightarrow>
3084 (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
3086 text{* Some simple consequential lemmas. *}
3088 lemma uniformly_continuous_imp_continuous:
3089 " uniformly_continuous_on s f ==> continuous_on s f"
3090 unfolding uniformly_continuous_on_def continuous_on_iff by blast
3092 lemma continuous_at_imp_continuous_within:
3093 "continuous (at x) f ==> continuous (at x within s) f"
3094 unfolding continuous_within continuous_at using Lim_at_within by auto
3096 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
3097 unfolding tendsto_def by (simp add: trivial_limit_eq)
3099 lemma continuous_at_imp_continuous_on:
3100 assumes "\<forall>x\<in>s. continuous (at x) f"
3101 shows "continuous_on s f"
3102 unfolding continuous_on_def
3104 fix x assume "x \<in> s"
3105 with assms have *: "(f ---> f (netlimit (at x))) (at x)"
3106 unfolding continuous_def by simp
3107 have "(f ---> f x) (at x)"
3108 proof (cases "trivial_limit (at x)")
3109 case True thus ?thesis
3110 by (rule Lim_trivial_limit)
3113 hence 1: "netlimit (at x) = x"
3114 using netlimit_within [of x UNIV] by simp
3115 with * show ?thesis by simp
3117 thus "(f ---> f x) (at x within s)"
3118 by (rule Lim_at_within)
3121 lemma continuous_on_eq_continuous_within:
3122 "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"
3123 unfolding continuous_on_def continuous_def
3124 apply (rule ball_cong [OF refl])
3125 apply (case_tac "trivial_limit (at x within s)")
3126 apply (simp add: Lim_trivial_limit)
3127 apply (simp add: netlimit_within)
3130 lemmas continuous_on = continuous_on_def -- "legacy theorem name"
3132 lemma continuous_on_eq_continuous_at:
3133 shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
3134 by (auto simp add: continuous_on continuous_at Lim_within_open)
3136 lemma continuous_within_subset:
3137 "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
3138 ==> continuous (at x within t) f"
3139 unfolding continuous_within by(metis Lim_within_subset)
3141 lemma continuous_on_subset:
3142 shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
3143 unfolding continuous_on by (metis subset_eq Lim_within_subset)
3145 lemma continuous_on_interior:
3146 shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"
3147 by (erule interiorE, drule (1) continuous_on_subset,
3148 simp add: continuous_on_eq_continuous_at)
3150 lemma continuous_on_eq:
3151 "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"
3152 unfolding continuous_on_def tendsto_def Limits.eventually_within
3155 text {* Characterization of various kinds of continuity in terms of sequences. *}
3157 lemma continuous_within_sequentially:
3158 fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3159 shows "continuous (at a within s) f \<longleftrightarrow>
3160 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
3161 --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
3164 { fix x::"nat \<Rightarrow> 'a" assume x:"\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"
3165 fix T::"'b set" assume "open T" and "f a \<in> T"
3166 with `?lhs` obtain d where "d>0" and d:"\<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> f x \<in> T"
3167 unfolding continuous_within tendsto_def eventually_within by auto
3168 have "eventually (\<lambda>n. dist (x n) a < d) sequentially"
3169 using x(2) `d>0` by simp
3170 hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"
3171 proof (rule eventually_elim1)
3172 fix n assume "dist (x n) a < d" thus "(f \<circ> x) n \<in> T"
3173 using d x(1) `f a \<in> T` unfolding dist_nz[THEN sym] by auto
3176 thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp
3178 assume ?rhs thus ?lhs
3179 unfolding continuous_within tendsto_def [where l="f a"]
3180 by (simp add: sequentially_imp_eventually_within)
3183 lemma continuous_at_sequentially:
3184 fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3185 shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
3186 --> ((f o x) ---> f a) sequentially)"
3187 using continuous_within_sequentially[of a UNIV f] by simp
3189 lemma continuous_on_sequentially:
3190 fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3191 shows "continuous_on s f \<longleftrightarrow>
3192 (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
3193 --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
3195 assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
3197 assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
3200 lemma uniformly_continuous_on_sequentially:
3201 "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
3202 ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially
3203 \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
3206 { fix x y assume x:"\<forall>n. x n \<in> s" and y:"\<forall>n. y n \<in> s" and xy:"((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"
3207 { fix e::real assume "e>0"
3208 then obtain d where "d>0" and d:"\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
3209 using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
3210 obtain N where N:"\<forall>n\<ge>N. dist (x n) (y n) < d" using xy[unfolded LIMSEQ_def dist_norm] and `d>0` by auto
3211 { fix n assume "n\<ge>N"
3212 hence "dist (f (x n)) (f (y n)) < e"
3213 using N[THEN spec[where x=n]] using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]] using x and y
3214 unfolding dist_commute by simp }
3215 hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" by auto }
3216 hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto }
3220 { assume "\<not> ?lhs"
3221 then obtain e where "e>0" "\<forall>d>0. \<exists>x\<in>s. \<exists>x'\<in>s. dist x' x < d \<and> \<not> dist (f x') (f x) < e" unfolding uniformly_continuous_on_def by auto
3222 then obtain fa where fa:"\<forall>x. 0 < x \<longrightarrow> fst (fa x) \<in> s \<and> snd (fa x) \<in> s \<and> dist (fst (fa x)) (snd (fa x)) < x \<and> \<not> dist (f (fst (fa x))) (f (snd (fa x))) < e"
3223 using choice[of "\<lambda>d x. d>0 \<longrightarrow> fst x \<in> s \<and> snd x \<in> s \<and> dist (snd x) (fst x) < d \<and> \<not> dist (f (snd x)) (f (fst x)) < e"] unfolding Bex_def
3224 by (auto simp add: dist_commute)
3225 def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
3226 def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
3227 have xyn:"\<forall>n. x n \<in> s \<and> y n \<in> s" and xy0:"\<forall>n. dist (x n) (y n) < inverse (real n + 1)" and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"
3228 unfolding x_def and y_def using fa by auto
3229 { fix e::real assume "e>0"
3230 then obtain N::nat where "N \<noteq> 0" and N:"0 < inverse (real N) \<and> inverse (real N) < e" unfolding real_arch_inv[of e] by auto
3231 { fix n::nat assume "n\<ge>N"
3232 hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
3233 also have "\<dots> < e" using N by auto
3234 finally have "inverse (real n + 1) < e" by auto
3235 hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto }
3236 hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto }
3237 hence "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" using `?rhs`[THEN spec[where x=x], THEN spec[where x=y]] and xyn unfolding LIMSEQ_def dist_real_def by auto
3238 hence False using fxy and `e>0` by auto }
3239 thus ?lhs unfolding uniformly_continuous_on_def by blast
3242 text{* The usual transformation theorems. *}
3244 lemma continuous_transform_within:
3245 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3246 assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
3247 "continuous (at x within s) f"
3248 shows "continuous (at x within s) g"
3249 unfolding continuous_within
3250 proof (rule Lim_transform_within)
3251 show "0 < d" by fact
3252 show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
3253 using assms(3) by auto
3255 using assms(1,2,3) by auto
3256 thus "(f ---> g x) (at x within s)"
3257 using assms(4) unfolding continuous_within by simp
3260 lemma continuous_transform_at:
3261 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3262 assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
3263 "continuous (at x) f"
3264 shows "continuous (at x) g"
3265 using continuous_transform_within [of d x UNIV f g] assms by simp
3267 subsubsection {* Structural rules for pointwise continuity *}
3269 lemma continuous_within_id: "continuous (at a within s) (\<lambda>x. x)"
3270 unfolding continuous_within by (rule tendsto_ident_at_within)
3272 lemma continuous_at_id: "continuous (at a) (\<lambda>x. x)"
3273 unfolding continuous_at by (rule tendsto_ident_at)
3275 lemma continuous_const: "continuous F (\<lambda>x. c)"
3276 unfolding continuous_def by (rule tendsto_const)
3278 lemma continuous_dist:
3279 assumes "continuous F f" and "continuous F g"
3280 shows "continuous F (\<lambda>x. dist (f x) (g x))"
3281 using assms unfolding continuous_def by (rule tendsto_dist)
3283 lemma continuous_norm:
3284 shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
3285 unfolding continuous_def by (rule tendsto_norm)
3287 lemma continuous_infnorm:
3288 shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"
3289 unfolding continuous_def by (rule tendsto_infnorm)
3291 lemma continuous_add:
3292 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3293 shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
3294 unfolding continuous_def by (rule tendsto_add)
3296 lemma continuous_minus:
3297 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3298 shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
3299 unfolding continuous_def by (rule tendsto_minus)
3301 lemma continuous_diff:
3302 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3303 shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
3304 unfolding continuous_def by (rule tendsto_diff)
3306 lemma continuous_scaleR:
3307 fixes g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3308 shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x *\<^sub>R g x)"
3309 unfolding continuous_def by (rule tendsto_scaleR)
3311 lemma continuous_mult:
3312 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
3313 shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x * g x)"
3314 unfolding continuous_def by (rule tendsto_mult)
3316 lemma continuous_inner:
3317 assumes "continuous F f" and "continuous F g"
3318 shows "continuous F (\<lambda>x. inner (f x) (g x))"
3319 using assms unfolding continuous_def by (rule tendsto_inner)
3321 lemma continuous_euclidean_component:
3322 shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $$ i)"
3323 unfolding continuous_def by (rule tendsto_euclidean_component)
3325 lemma continuous_inverse:
3326 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
3327 assumes "continuous F f" and "f (netlimit F) \<noteq> 0"
3328 shows "continuous F (\<lambda>x. inverse (f x))"
3329 using assms unfolding continuous_def by (rule tendsto_inverse)
3331 lemma continuous_at_within_inverse:
3332 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
3333 assumes "continuous (at a within s) f" and "f a \<noteq> 0"
3334 shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
3335 using assms unfolding continuous_within by (rule tendsto_inverse)
3337 lemma continuous_at_inverse:
3338 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
3339 assumes "continuous (at a) f" and "f a \<noteq> 0"
3340 shows "continuous (at a) (\<lambda>x. inverse (f x))"
3341 using assms unfolding continuous_at by (rule tendsto_inverse)
3343 lemmas continuous_intros = continuous_at_id continuous_within_id
3344 continuous_const continuous_dist continuous_norm continuous_infnorm
3345 continuous_add continuous_minus continuous_diff
3346 continuous_scaleR continuous_mult
3347 continuous_inner continuous_euclidean_component
3348 continuous_at_inverse continuous_at_within_inverse
3350 subsubsection {* Structural rules for setwise continuity *}
3352 lemma continuous_on_id: "continuous_on s (\<lambda>x. x)"
3353 unfolding continuous_on_def by (fast intro: tendsto_ident_at_within)
3355 lemma continuous_on_const: "continuous_on s (\<lambda>x. c)"
3356 unfolding continuous_on_def by (auto intro: tendsto_intros)
3358 lemma continuous_on_norm:
3359 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
3360 unfolding continuous_on_def by (fast intro: tendsto_norm)
3362 lemma continuous_on_infnorm:
3363 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"
3364 unfolding continuous_on by (fast intro: tendsto_infnorm)
3366 lemma continuous_on_minus:
3367 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3368 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
3369 unfolding continuous_on_def by (auto intro: tendsto_intros)
3371 lemma continuous_on_add:
3372 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3373 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3374 \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
3375 unfolding continuous_on_def by (auto intro: tendsto_intros)
3377 lemma continuous_on_diff:
3378 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3379 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3380 \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
3381 unfolding continuous_on_def by (auto intro: tendsto_intros)
3383 lemma (in bounded_linear) continuous_on:
3384 "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
3385 unfolding continuous_on_def by (fast intro: tendsto)
3387 lemma (in bounded_bilinear) continuous_on:
3388 "\<lbrakk>continuous_on s f; continuous_on s g\<rbrakk> \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
3389 unfolding continuous_on_def by (fast intro: tendsto)
3391 lemma continuous_on_scaleR:
3392 fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3393 assumes "continuous_on s f" and "continuous_on s g"
3394 shows "continuous_on s (\<lambda>x. f x *\<^sub>R g x)"
3395 using bounded_bilinear_scaleR assms
3396 by (rule bounded_bilinear.continuous_on)
3398 lemma continuous_on_mult:
3399 fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"
3400 assumes "continuous_on s f" and "continuous_on s g"
3401 shows "continuous_on s (\<lambda>x. f x * g x)"
3402 using bounded_bilinear_mult assms
3403 by (rule bounded_bilinear.continuous_on)
3405 lemma continuous_on_inner:
3406 fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"
3407 assumes "continuous_on s f" and "continuous_on s g"
3408 shows "continuous_on s (\<lambda>x. inner (f x) (g x))"
3409 using bounded_bilinear_inner assms
3410 by (rule bounded_bilinear.continuous_on)
3412 lemma continuous_on_euclidean_component:
3413 "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $$ i)"
3414 using bounded_linear_euclidean_component
3415 by (rule bounded_linear.continuous_on)
3417 lemma continuous_on_inverse:
3418 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
3419 assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
3420 shows "continuous_on s (\<lambda>x. inverse (f x))"
3421 using assms unfolding continuous_on by (fast intro: tendsto_inverse)
3423 subsubsection {* Structural rules for uniform continuity *}
3425 lemma uniformly_continuous_on_id:
3426 shows "uniformly_continuous_on s (\<lambda>x. x)"
3427 unfolding uniformly_continuous_on_def by auto
3429 lemma uniformly_continuous_on_const:
3430 shows "uniformly_continuous_on s (\<lambda>x. c)"
3431 unfolding uniformly_continuous_on_def by simp
3433 lemma uniformly_continuous_on_dist:
3434 fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3435 assumes "uniformly_continuous_on s f"
3436 assumes "uniformly_continuous_on s g"
3437 shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
3439 { fix a b c d :: 'b have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
3440 using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
3441 using dist_triangle3 [of c d a] dist_triangle [of a d b]
3445 assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"
3446 assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"
3447 have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"
3448 by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
3451 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially
3452 unfolding dist_real_def by simp
3455 lemma uniformly_continuous_on_norm:
3456 assumes "uniformly_continuous_on s f"
3457 shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"
3458 unfolding norm_conv_dist using assms
3459 by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)
3461 lemma (in bounded_linear) uniformly_continuous_on:
3462 assumes "uniformly_continuous_on s g"
3463 shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
3464 using assms unfolding uniformly_continuous_on_sequentially
3465 unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
3466 by (auto intro: tendsto_zero)
3468 lemma uniformly_continuous_on_cmul:
3469 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3470 assumes "uniformly_continuous_on s f"
3471 shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
3472 using bounded_linear_scaleR_right assms
3473 by (rule bounded_linear.uniformly_continuous_on)
3476 fixes x y :: "'a::real_normed_vector"
3477 shows "dist (- x) (- y) = dist x y"
3478 unfolding dist_norm minus_diff_minus norm_minus_cancel ..
3480 lemma uniformly_continuous_on_minus:
3481 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3482 shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"
3483 unfolding uniformly_continuous_on_def dist_minus .
3485 lemma uniformly_continuous_on_add:
3486 fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3487 assumes "uniformly_continuous_on s f"
3488 assumes "uniformly_continuous_on s g"
3489 shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
3490 using assms unfolding uniformly_continuous_on_sequentially
3491 unfolding dist_norm tendsto_norm_zero_iff add_diff_add
3492 by (auto intro: tendsto_add_zero)
3494 lemma uniformly_continuous_on_diff:
3495 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3496 assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g"
3497 shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
3498 unfolding ab_diff_minus using assms
3499 by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)
3501 text{* Continuity of all kinds is preserved under composition. *}
3503 lemma continuous_within_topological:
3504 "continuous (at x within s) f \<longleftrightarrow>
3505 (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
3506 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
3507 unfolding continuous_within
3508 unfolding tendsto_def Limits.eventually_within eventually_at_topological
3509 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
3511 lemma continuous_within_compose:
3512 assumes "continuous (at x within s) f"
3513 assumes "continuous (at (f x) within f ` s) g"
3514 shows "continuous (at x within s) (g o f)"
3515 using assms unfolding continuous_within_topological by simp metis
3517 lemma continuous_at_compose:
3518 assumes "continuous (at x) f" and "continuous (at (f x)) g"
3519 shows "continuous (at x) (g o f)"
3521 have "continuous (at (f x) within range f) g" using assms(2)
3522 using continuous_within_subset[of "f x" UNIV g "range f"] by simp
3523 thus ?thesis using assms(1)
3524 using continuous_within_compose[of x UNIV f g] by simp
3527 lemma continuous_on_compose:
3528 "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
3529 unfolding continuous_on_topological by simp metis
3531 lemma uniformly_continuous_on_compose:
3532 assumes "uniformly_continuous_on s f" "uniformly_continuous_on (f ` s) g"
3533 shows "uniformly_continuous_on s (g o f)"
3535 { fix e::real assume "e>0"
3536 then obtain d where "d>0" and d:"\<forall>x\<in>f ` s. \<forall>x'\<in>f ` s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using assms(2) unfolding uniformly_continuous_on_def by auto
3537 obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d" using `d>0` using assms(1) unfolding uniformly_continuous_on_def by auto
3538 hence "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist ((g \<circ> f) x') ((g \<circ> f) x) < e" using `d>0` using d by auto }
3539 thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
3542 lemmas continuous_on_intros = continuous_on_id continuous_on_const
3543 continuous_on_compose continuous_on_norm continuous_on_infnorm
3544 continuous_on_add continuous_on_minus continuous_on_diff
3545 continuous_on_scaleR continuous_on_mult continuous_on_inverse
3546 continuous_on_inner continuous_on_euclidean_component
3547 uniformly_continuous_on_id uniformly_continuous_on_const
3548 uniformly_continuous_on_dist uniformly_continuous_on_norm
3549 uniformly_continuous_on_compose uniformly_continuous_on_add
3550 uniformly_continuous_on_minus uniformly_continuous_on_diff
3551 uniformly_continuous_on_cmul
3553 text{* Continuity in terms of open preimages. *}
3555 lemma continuous_at_open:
3556 shows "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"
3557 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]
3558 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto
3560 lemma continuous_on_open:
3561 shows "continuous_on s f \<longleftrightarrow>
3562 (\<forall>t. openin (subtopology euclidean (f ` s)) t
3563 --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
3566 assume 1: "continuous_on s f"
3567 assume 2: "openin (subtopology euclidean (f ` s)) t"
3568 from 2 obtain B where B: "open B" and t: "t = f ` s \<inter> B"
3569 unfolding openin_open by auto
3570 def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"
3571 have "open U" unfolding U_def by (simp add: open_Union)
3572 moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"
3573 proof (intro ballI iffI)
3574 fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"
3575 unfolding U_def t by auto
3577 fix x assume "x \<in> s" and "f x \<in> t"
3578 hence "x \<in> s" and "f x \<in> B"
3580 with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"
3581 unfolding t continuous_on_topological by metis
3582 then show "x \<in> U"
3583 unfolding U_def by auto
3585 ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto
3586 then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3587 unfolding openin_open by fast
3589 assume "?rhs" show "continuous_on s f"
3590 unfolding continuous_on_topological
3592 fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"
3593 have "openin (subtopology euclidean (f ` s)) (f ` s \<inter> B)"
3594 unfolding openin_open using `open B` by auto
3595 then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f ` s \<inter> B}"
3596 using `?rhs` by fast
3597 then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
3598 unfolding openin_open using `x \<in> s` and `f x \<in> B` by auto
3602 text {* Similarly in terms of closed sets. *}
3604 lemma continuous_on_closed:
3605 shows "continuous_on s f \<longleftrightarrow> (\<forall>t. closedin (subtopology euclidean (f ` s)) t --> closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
3609 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3610 have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
3611 assume as:"closedin (subtopology euclidean (f ` s)) t"
3612 hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
3613 hence "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?lhs`[unfolded continuous_on_open, THEN spec[where x="(f ` s) - t"]]
3614 unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto }
3619 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3620 assume as:"openin (subtopology euclidean (f ` s)) t"
3621 hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
3622 unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
3623 thus ?lhs unfolding continuous_on_open by auto
3626 text {* Half-global and completely global cases. *}
3628 lemma continuous_open_in_preimage:
3629 assumes "continuous_on s f" "open t"
3630 shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3632 have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f ` s)" by auto
3633 have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3634 using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
3635 thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3638 lemma continuous_closed_in_preimage:
3639 assumes "continuous_on s f" "closed t"
3640 shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3642 have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f ` s)" by auto
3643 have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3644 using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
3646 using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3649 lemma continuous_open_preimage:
3650 assumes "continuous_on s f" "open s" "open t"
3651 shows "open {x \<in> s. f x \<in> t}"
3653 obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3654 using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
3655 thus ?thesis using open_Int[of s T, OF assms(2)] by auto
3658 lemma continuous_closed_preimage:
3659 assumes "continuous_on s f" "closed s" "closed t"
3660 shows "closed {x \<in> s. f x \<in> t}"
3662 obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3663 using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
3664 thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
3667 lemma continuous_open_preimage_univ:
3668 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
3669 using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
3671 lemma continuous_closed_preimage_univ:
3672 shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
3673 using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
3675 lemma continuous_open_vimage:
3676 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
3677 unfolding vimage_def by (rule continuous_open_preimage_univ)
3679 lemma continuous_closed_vimage:
3680 shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
3681 unfolding vimage_def by (rule continuous_closed_preimage_univ)
3683 lemma interior_image_subset:
3684 assumes "\<forall>x. continuous (at x) f" "inj f"
3685 shows "interior (f ` s) \<subseteq> f ` (interior s)"
3687 fix x assume "x \<in> interior (f ` s)"
3688 then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f ` s" ..
3689 hence "x \<in> f ` s" by auto
3690 then obtain y where y: "y \<in> s" "x = f y" by auto
3691 have "open (vimage f T)"
3692 using assms(1) `open T` by (rule continuous_open_vimage)
3693 moreover have "y \<in> vimage f T"
3694 using `x = f y` `x \<in> T` by simp
3695 moreover have "vimage f T \<subseteq> s"
3696 using `T \<subseteq> image f s` `inj f` unfolding inj_on_def subset_eq by auto
3697 ultimately have "y \<in> interior s" ..
3698 with `x = f y` show "x \<in> f ` interior s" ..
3701 text {* Equality of continuous functions on closure and related results. *}
3703 lemma continuous_closed_in_preimage_constant:
3704 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3705 shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
3706 using continuous_closed_in_preimage[of s f "{a}"] by auto
3708 lemma continuous_closed_preimage_constant:
3709 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3710 shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
3711 using continuous_closed_preimage[of s f "{a}"] by auto
3713 lemma continuous_constant_on_closure:
3714 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3715 assumes "continuous_on (closure s) f"
3716 "\<forall>x \<in> s. f x = a"
3717 shows "\<forall>x \<in> (closure s). f x = a"
3718 using continuous_closed_preimage_constant[of "closure s" f a]
3719 assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
3721 lemma image_closure_subset:
3722 assumes "continuous_on (closure s) f" "closed t" "(f ` s) \<subseteq> t"
3723 shows "f ` (closure s) \<subseteq> t"
3725 have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
3726 moreover have "closed {x \<in> closure s. f x \<in> t}"
3727 using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
3728 ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
3729 using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
3730 thus ?thesis by auto
3733 lemma continuous_on_closure_norm_le:
3734 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3735 assumes "continuous_on (closure s) f" "\<forall>y \<in> s. norm(f y) \<le> b" "x \<in> (closure s)"
3736 shows "norm(f x) \<le> b"
3738 have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
3740 using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
3741 unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
3744 text {* Making a continuous function avoid some value in a neighbourhood. *}
3746 lemma continuous_within_avoid:
3747 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3748 assumes "continuous (at x within s) f" "x \<in> s" "f x \<noteq> a"
3749 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
3751 obtain d where "d>0" and d:"\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < dist (f x) a"
3752 using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto
3753 { fix y assume " y\<in>s" "dist x y < d"
3754 hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]
3755 apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }
3756 thus ?thesis using `d>0` by auto
3759 lemma continuous_at_avoid:
3760 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3761 assumes "continuous (at x) f" and "f x \<noteq> a"
3762 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
3763 using assms continuous_within_avoid[of x UNIV f a] by simp
3765 lemma continuous_on_avoid:
3766 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
3767 assumes "continuous_on s f" "x \<in> s" "f x \<noteq> a"
3768 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
3769 using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x], OF assms(2)] continuous_within_avoid[of x s f a] assms(2,3) by auto
3771 lemma continuous_on_open_avoid:
3772 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
3773 assumes "continuous_on s f" "open s" "x \<in> s" "f x \<noteq> a"
3774 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
3775 using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)] continuous_at_avoid[of x f a] assms(3,4) by auto
3777 text {* Proving a function is constant by proving open-ness of level set. *}
3779 lemma continuous_levelset_open_in_cases:
3780 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3781 shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
3782 openin (subtopology euclidean s) {x \<in> s. f x = a}
3783 ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
3784 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
3786 lemma continuous_levelset_open_in:
3787 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3788 shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
3789 openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
3790 (\<exists>x \<in> s. f x = a) ==> (\<forall>x \<in> s. f x = a)"
3791 using continuous_levelset_open_in_cases[of s f ]
3794 lemma continuous_levelset_open:
3795 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3796 assumes "connected s" "continuous_on s f" "open {x \<in> s. f x = a}" "\<exists>x \<in> s. f x = a"
3797 shows "\<forall>x \<in> s. f x = a"
3798 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
3800 text {* Some arithmetical combinations (more to prove). *}
3802 lemma open_scaling[intro]:
3803 fixes s :: "'a::real_normed_vector set"
3804 assumes "c \<noteq> 0" "open s"
3805 shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
3807 { fix x assume "x \<in> s"
3808 then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]] by auto
3809 have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF `e>0`] by auto
3811 { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
3812 hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
3813 using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
3814 assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
3815 hence "y \<in> op *\<^sub>R c ` s" using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"] e[THEN spec[where x="(1 / c) *\<^sub>R y"]] assms(1) unfolding dist_norm scaleR_scaleR by auto }
3816 ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c ` s" apply(rule_tac x="e * abs c" in exI) by auto }
3817 thus ?thesis unfolding open_dist by auto
3820 lemma minus_image_eq_vimage:
3821 fixes A :: "'a::ab_group_add set"
3822 shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
3823 by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
3825 lemma open_negations:
3826 fixes s :: "'a::real_normed_vector set"
3827 shows "open s ==> open ((\<lambda> x. -x) ` s)"
3828 unfolding scaleR_minus1_left [symmetric]
3829 by (rule open_scaling, auto)
3831 lemma open_translation:
3832 fixes s :: "'a::real_normed_vector set"
3833 assumes "open s" shows "open((\<lambda>x. a + x) ` s)"
3835 { fix x have "continuous (at x) (\<lambda>x. x - a)"
3836 by (intro continuous_diff continuous_at_id continuous_const) }
3837 moreover have "{x. x - a \<in> s} = op + a ` s" by force
3838 ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
3841 lemma open_affinity:
3842 fixes s :: "'a::real_normed_vector set"
3843 assumes "open s" "c \<noteq> 0"
3844 shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
3846 have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
3847 have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
3848 thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
3851 lemma interior_translation:
3852 fixes s :: "'a::real_normed_vector set"
3853 shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
3854 proof (rule set_eqI, rule)
3855 fix x assume "x \<in> interior (op + a ` s)"
3856 then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
3857 hence "ball (x - a) e \<subseteq> s" unfolding subset_eq Ball_def mem_ball dist_norm apply auto apply(erule_tac x="a + xa" in allE) unfolding ab_group_add_class.diff_diff_eq[THEN sym] by auto
3858 thus "x \<in> op + a ` interior s" unfolding image_iff apply(rule_tac x="x - a" in bexI) unfolding mem_interior using `e > 0` by auto
3860 fix x assume "x \<in> op + a ` interior s"
3861 then obtain y e where "e>0" and e:"ball y e \<subseteq> s" and y:"x = a + y" unfolding image_iff Bex_def mem_interior by auto
3862 { fix z have *:"a + y - z = y + a - z" by auto
3863 assume "z\<in>ball x e"
3864 hence "z - a \<in> s" using e[unfolded subset_eq, THEN bspec[where x="z - a"]] unfolding mem_ball dist_norm y ab_group_add_class.diff_diff_eq2 * by auto
3865 hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"]) }
3866 hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
3867 thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
3870 text {* Topological properties of linear functions. *}
3873 assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
3875 interpret f: bounded_linear f by fact
3876 have "(f ---> f 0) (at 0)"
3877 using tendsto_ident_at by (rule f.tendsto)
3878 thus ?thesis unfolding f.zero .
3881 lemma linear_continuous_at:
3882 assumes "bounded_linear f" shows "continuous (at a) f"
3883 unfolding continuous_at using assms
3884 apply (rule bounded_linear.tendsto)
3885 apply (rule tendsto_ident_at)
3888 lemma linear_continuous_within:
3889 shows "bounded_linear f ==> continuous (at x within s) f"
3890 using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
3892 lemma linear_continuous_on:
3893 shows "bounded_linear f ==> continuous_on s f"
3894 using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
3896 text {* Also bilinear functions, in composition form. *}
3898 lemma bilinear_continuous_at_compose:
3899 shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
3900 ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
3901 unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
3903 lemma bilinear_continuous_within_compose:
3904 shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
3905 ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
3906 unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
3908 lemma bilinear_continuous_on_compose:
3909 shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
3910 ==> continuous_on s (\<lambda>x. h (f x) (g x))"
3911 unfolding continuous_on_def
3912 by (fast elim: bounded_bilinear.tendsto)
3914 text {* Preservation of compactness and connectedness under continuous function. *}
3916 lemma compact_continuous_image:
3917 assumes "continuous_on s f" "compact s"
3918 shows "compact(f ` s)"
3920 { fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
3921 then obtain y where y:"\<forall>n. y n \<in> s \<and> x n = f (y n)" unfolding image_iff Bex_def using choice[of "\<lambda>n xa. xa \<in> s \<and> x n = f xa"] by auto
3922 then obtain l r where "l\<in>s" and r:"subseq r" and lr:"((y \<circ> r) ---> l) sequentially" using assms(2)[unfolded compact_def, THEN spec[where x=y]] by auto
3923 { fix e::real assume "e>0"
3924 then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' l < d \<longrightarrow> dist (f x') (f l) < e" using assms(1)[unfolded continuous_on_iff, THEN bspec[where x=l], OF `l\<in>s`] by auto
3925 then obtain N::nat where N:"\<forall>n\<ge>N. dist ((y \<circ> r) n) l < d" using lr[unfolded LIMSEQ_def, THEN spec[where x=d]] by auto
3926 { fix n::nat assume "n\<ge>N" hence "dist ((x \<circ> r) n) (f l) < e" using N[THEN spec[where x=n]] d[THEN bspec[where x="y (r n)"]] y[THEN spec[where x="r n"]] by auto }
3927 hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto }
3928 hence "\<exists>l\<in>f ` s. \<exists>r. subseq r \<and> ((x \<circ> r) ---> l) sequentially" unfolding LIMSEQ_def using r lr `l\<in>s` by auto }
3929 thus ?thesis unfolding compact_def by auto
3932 lemma connected_continuous_image:
3933 assumes "continuous_on s f" "connected s"
3934 shows "connected(f ` s)"
3936 { fix T assume as: "T \<noteq> {}" "T \<noteq> f ` s" "openin (subtopology euclidean (f ` s)) T" "closedin (subtopology euclidean (f ` s)) T"
3937 have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
3938 using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
3939 using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
3940 using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
3941 hence False using as(1,2)
3942 using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
3943 thus ?thesis unfolding connected_clopen by auto
3946 text {* Continuity implies uniform continuity on a compact domain. *}
3948 lemma compact_uniformly_continuous:
3949 assumes "continuous_on s f" "compact s"
3950 shows "uniformly_continuous_on s f"
3952 { fix x assume x:"x\<in>s"
3953 hence "\<forall>xa. \<exists>y. 0 < xa \<longrightarrow> (y > 0 \<and> (\<forall>x'\<in>s. dist x' x < y \<longrightarrow> dist (f x') (f x) < xa))" using assms(1)[unfolded continuous_on_iff, THEN bspec[where x=x]] by auto
3954 hence "\<exists>fa. \<forall>xa>0. \<forall>x'\<in>s. fa xa > 0 \<and> (dist x' x < fa xa \<longrightarrow> dist (f x') (f x) < xa)" using choice[of "\<lambda>e d. e>0 \<longrightarrow> d>0 \<and>(\<forall>x'\<in>s. (dist x' x < d \<longrightarrow> dist (f x') (f x) < e))"] by auto }
3955 then have "\<forall>x\<in>s. \<exists>y. \<forall>xa. 0 < xa \<longrightarrow> (\<forall>x'\<in>s. y xa > 0 \<and> (dist x' x < y xa \<longrightarrow> dist (f x') (f x) < xa))" by auto
3956 then obtain d where d:"\<forall>e>0. \<forall>x\<in>s. \<forall>x'\<in>s. d x e > 0 \<and> (dist x' x < d x e \<longrightarrow> dist (f x') (f x) < e)"
3957 using bchoice[of s "\<lambda>x fa. \<forall>xa>0. \<forall>x'\<in>s. fa xa > 0 \<and> (dist x' x < fa xa \<longrightarrow> dist (f x') (f x) < xa)"] by blast
3959 { fix e::real assume "e>0"
3961 { fix x assume "x\<in>s" hence "x \<in> ball x (d x (e / 2))" unfolding centre_in_ball using d[THEN spec[where x="e/2"]] using `e>0` by auto }
3962 hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto
3964 { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto }
3965 ultimately obtain ea where "ea>0" and ea:"\<forall>x\<in>s. \<exists>b\<in>{ball x (d x (e / 2)) |x. x \<in> s}. ball x ea \<subseteq> b" using heine_borel_lemma[OF assms(2), of "{ball x (d x (e / 2)) | x. x\<in>s }"] by auto
3967 { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"
3968 obtain z where "z\<in>s" and z:"ball x ea \<subseteq> ball z (d z (e / 2))" using ea[THEN bspec[where x=x]] and `x\<in>s` by auto
3969 hence "x\<in>ball z (d z (e / 2))" using `ea>0` unfolding subset_eq by auto
3970 hence "dist (f z) (f x) < e / 2" using d[THEN spec[where x="e/2"]] and `e>0` and `x\<in>s` and `z\<in>s`
3971 by (auto simp add: dist_commute)
3972 moreover have "y\<in>ball z (d z (e / 2))" using as and `ea>0` and z[unfolded subset_eq]
3973 by (auto simp add: dist_commute)
3974 hence "dist (f z) (f y) < e / 2" using d[THEN spec[where x="e/2"]] and `e>0` and `y\<in>s` and `z\<in>s`
3975 by (auto simp add: dist_commute)
3976 ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]
3977 by (auto simp add: dist_commute) }
3978 then have "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e" using `ea>0` by auto }
3979 thus ?thesis unfolding uniformly_continuous_on_def by auto
3982 text{* Continuity of inverse function on compact domain. *}
3984 lemma continuous_on_inv:
3985 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
3986 (* TODO: can this be generalized more? *)
3987 assumes "continuous_on s f" "compact s" "\<forall>x \<in> s. g (f x) = x"
3988 shows "continuous_on (f ` s) g"
3990 have *:"g ` f ` s = s" using assms(3) by (auto simp add: image_iff)
3991 { fix t assume t:"closedin (subtopology euclidean (g ` f ` s)) t"
3992 then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto
3993 have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]
3994 unfolding T(2) and Int_left_absorb by auto
3995 moreover have "compact (s \<inter> T)"
3996 using assms(2) unfolding compact_eq_bounded_closed
3997 using bounded_subset[of s "s \<inter> T"] and T(1) by auto
3998 ultimately have "closed (f ` t)" using T(1) unfolding T(2)
3999 using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto
4000 moreover have "{x \<in> f ` s. g x \<in> t} = f ` s \<inter> f ` t" using assms(3) unfolding T(2) by auto
4001 ultimately have "closedin (subtopology euclidean (f ` s)) {x \<in> f ` s. g x \<in> t}"
4002 unfolding closedin_closed by auto }
4003 thus ?thesis unfolding continuous_on_closed by auto
4006 text {* A uniformly convergent limit of continuous functions is continuous. *}
4008 lemma continuous_uniform_limit:
4009 fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"
4010 assumes "\<not> trivial_limit F"
4011 assumes "eventually (\<lambda>n. continuous_on s (f n)) F"
4012 assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"
4013 shows "continuous_on s g"
4015 { fix x and e::real assume "x\<in>s" "e>0"
4016 have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"
4017 using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto
4018 from eventually_happens [OF eventually_conj [OF this assms(2)]]
4019 obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3" "continuous_on s (f n)"
4020 using assms(1) by blast
4021 have "e / 3 > 0" using `e>0` by auto
4022 then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f n x') (f n x) < e / 3"
4023 using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
4024 { fix y assume "y \<in> s" and "dist y x < d"
4025 hence "dist (f n y) (f n x) < e / 3"
4026 by (rule d [rule_format])
4027 hence "dist (f n y) (g x) < 2 * e / 3"
4028 using dist_triangle [of "f n y" "g x" "f n x"]
4029 using n(1)[THEN bspec[where x=x], OF `x\<in>s`]
4031 hence "dist (g y) (g x) < e"
4032 using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
4033 using dist_triangle3 [of "g y" "g x" "f n y"]
4035 hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
4036 using `d>0` by auto }
4037 thus ?thesis unfolding continuous_on_iff by auto
4041 subsection {* Topological stuff lifted from and dropped to R *}
4044 fixes s :: "real set" shows
4045 "open s \<longleftrightarrow>
4046 (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
4047 unfolding open_dist dist_norm by simp
4049 lemma islimpt_approachable_real:
4050 fixes s :: "real set"
4051 shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
4052 unfolding islimpt_approachable dist_norm by simp
4055 fixes s :: "real set"
4056 shows "closed s \<longleftrightarrow>
4057 (\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
4059 unfolding closed_limpt islimpt_approachable dist_norm by simp
4061 lemma continuous_at_real_range:
4062 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4063 shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
4064 \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
4065 unfolding continuous_at unfolding Lim_at
4066 unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
4067 apply(erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=x' in allE) apply auto
4068 apply(erule_tac x=e in allE) by auto
4070 lemma continuous_on_real_range:
4071 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4072 shows "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d --> abs(f x' - f x) < e))"
4073 unfolding continuous_on_iff dist_norm by simp
4075 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}
4077 lemma compact_attains_sup:
4078 fixes s :: "real set"
4079 assumes "compact s" "s \<noteq> {}"
4080 shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
4082 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4083 { fix e::real assume as: "\<forall>x\<in>s. x \<le> Sup s" "Sup s \<notin> s" "0 < e" "\<forall>x'\<in>s. x' = Sup s \<or> \<not> Sup s - x' < e"
4084 have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto
4085 moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
4086 ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using `e>0` by auto }
4087 thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]
4088 apply(rule_tac x="Sup s" in bexI) by auto
4092 fixes S :: "real set"
4093 shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
4094 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def)
4096 lemma compact_attains_inf:
4097 fixes s :: "real set"
4098 assumes "compact s" "s \<noteq> {}" shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
4100 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4101 { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s" "Inf s \<notin> s" "0 < e"
4102 "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"
4103 have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto
4105 { fix x assume "x \<in> s"
4106 hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto
4107 have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
4108 hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto
4109 ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using `e>0` by auto }
4110 thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]
4111 apply(rule_tac x="Inf s" in bexI) by auto
4114 lemma continuous_attains_sup:
4115 fixes f :: "'a::metric_space \<Rightarrow> real"
4116 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4117 ==> (\<exists>x \<in> s. \<forall>y \<in> s. f y \<le> f x)"
4118 using compact_attains_sup[of "f ` s"]
4119 using compact_continuous_image[of s f] by auto
4121 lemma continuous_attains_inf:
4122 fixes f :: "'a::metric_space \<Rightarrow> real"
4123 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4124 \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
4125 using compact_attains_inf[of "f ` s"]
4126 using compact_continuous_image[of s f] by auto
4128 lemma distance_attains_sup:
4129 assumes "compact s" "s \<noteq> {}"
4130 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
4131 proof (rule continuous_attains_sup [OF assms])
4132 { fix x assume "x\<in>s"
4133 have "(dist a ---> dist a x) (at x within s)"
4134 by (intro tendsto_dist tendsto_const Lim_at_within tendsto_ident_at)
4136 thus "continuous_on s (dist a)"
4137 unfolding continuous_on ..
4140 text {* For \emph{minimal} distance, we only need closure, not compactness. *}
4142 lemma distance_attains_inf:
4143 fixes a :: "'a::heine_borel"
4144 assumes "closed s" "s \<noteq> {}"
4145 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
4147 from assms(2) obtain b where "b\<in>s" by auto
4148 let ?B = "cball a (dist b a) \<inter> s"
4149 have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
4150 hence "?B \<noteq> {}" by auto
4152 { fix x assume "x\<in>?B"
4153 fix e::real assume "e>0"
4154 { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
4155 from as have "\<bar>dist a x' - dist a x\<bar> < e"
4156 unfolding abs_less_iff minus_diff_eq
4157 using dist_triangle2 [of a x' x]
4158 using dist_triangle [of a x x']
4161 hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
4164 hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
4165 unfolding continuous_on Lim_within dist_norm real_norm_def
4167 moreover have "compact ?B"
4168 using compact_cball[of a "dist b a"]
4169 unfolding compact_eq_bounded_closed
4170 using bounded_Int and closed_Int and assms(1) by auto
4171 ultimately obtain x where "x\<in>cball a (dist b a) \<inter> s" "\<forall>y\<in>cball a (dist b a) \<inter> s. dist a x \<le> dist a y"
4172 using continuous_attains_inf[of ?B "dist a"] by fastforce
4173 thus ?thesis by fastforce
4177 subsection {* Pasted sets *}
4179 lemma bounded_Times:
4180 assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
4182 obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
4183 using assms [unfolded bounded_def] by auto
4184 then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
4185 by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
4186 thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
4189 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
4192 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
4193 unfolding compact_def
4195 apply (drule_tac x="fst \<circ> f" in spec)
4196 apply (drule mp, simp add: mem_Times_iff)
4197 apply (clarify, rename_tac l1 r1)
4198 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
4199 apply (drule mp, simp add: mem_Times_iff)
4200 apply (clarify, rename_tac l2 r2)
4201 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
4202 apply (rule_tac x="r1 \<circ> r2" in exI)
4203 apply (rule conjI, simp add: subseq_def)
4204 apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
4205 apply (drule (1) tendsto_Pair) back
4206 apply (simp add: o_def)
4209 text{* Hence some useful properties follow quite easily. *}
4211 lemma compact_scaling:
4212 fixes s :: "'a::real_normed_vector set"
4213 assumes "compact s" shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
4215 let ?f = "\<lambda>x. scaleR c x"
4216 have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)
4217 show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
4218 using linear_continuous_at[OF *] assms by auto
4221 lemma compact_negations:
4222 fixes s :: "'a::real_normed_vector set"
4223 assumes "compact s" shows "compact ((\<lambda>x. -x) ` s)"
4224 using compact_scaling [OF assms, of "- 1"] by auto
4227 fixes s t :: "'a::real_normed_vector set"
4228 assumes "compact s" "compact t" shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
4230 have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
4231 apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
4232 have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
4233 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
4234 thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
4237 lemma compact_differences:
4238 fixes s t :: "'a::real_normed_vector set"
4239 assumes "compact s" "compact t" shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
4241 have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
4242 apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4243 thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
4246 lemma compact_translation:
4247 fixes s :: "'a::real_normed_vector set"
4248 assumes "compact s" shows "compact ((\<lambda>x. a + x) ` s)"
4250 have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
4251 thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
4254 lemma compact_affinity:
4255 fixes s :: "'a::real_normed_vector set"
4256 assumes "compact s" shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
4258 have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
4259 thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
4262 text {* Hence we get the following. *}
4264 lemma compact_sup_maxdistance:
4265 fixes s :: "'a::real_normed_vector set"
4266 assumes "compact s" "s \<noteq> {}"
4267 shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
4269 have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
4270 then obtain x where x:"x\<in>{x - y |x y. x \<in> s \<and> y \<in> s}" "\<forall>y\<in>{x - y |x y. x \<in> s \<and> y \<in> s}. norm y \<le> norm x"
4271 using compact_differences[OF assms(1) assms(1)]
4272 using distance_attains_sup[where 'a="'a", unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by auto
4273 from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
4274 thus ?thesis using x(2)[unfolded `x = a - b`] by blast
4277 text {* We can state this in terms of diameter of a set. *}
4279 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"
4280 (* TODO: generalize to class metric_space *)
4282 lemma diameter_bounded:
4284 shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4285 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"
4287 let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"
4288 obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto
4289 { fix x y assume "x \<in> s" "y \<in> s"
4290 hence "norm (x - y) \<le> 2 * a" using norm_triangle_ineq[of x "-y", unfolded norm_minus_cancel] a[THEN bspec[where x=x]] a[THEN bspec[where x=y]] by (auto simp add: field_simps) }
4292 { fix x y assume "x\<in>s" "y\<in>s" hence "s \<noteq> {}" by auto
4293 have "norm(x - y) \<le> diameter s" unfolding diameter_def using `s\<noteq>{}` *[OF `x\<in>s` `y\<in>s`] `x\<in>s` `y\<in>s`
4294 by simp (blast del: Sup_upper intro!: * Sup_upper) }
4296 { fix d::real assume "d>0" "d < diameter s"
4297 hence "s\<noteq>{}" unfolding diameter_def by auto
4298 have "\<exists>d' \<in> ?D. d' > d"
4300 assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"
4301 hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE)
4302 thus False using `d < diameter s` `s\<noteq>{}`
4303 apply (auto simp add: diameter_def)
4304 apply (drule Sup_real_iff [THEN [2] rev_iffD2])
4308 hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto }
4309 ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4310 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto
4313 lemma diameter_bounded_bound:
4314 "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"
4315 using diameter_bounded by blast
4317 lemma diameter_compact_attained:
4318 fixes s :: "'a::real_normed_vector set"
4319 assumes "compact s" "s \<noteq> {}"
4320 shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
4322 have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
4323 then obtain x y where xys:"x\<in>s" "y\<in>s" and xy:"\<forall>u\<in>s. \<forall>v\<in>s. norm (u - v) \<le> norm (x - y)" using compact_sup_maxdistance[OF assms] by auto
4324 hence "diameter s \<le> norm (x - y)"
4325 unfolding diameter_def by clarsimp (rule Sup_least, fast+)
4327 by (metis b diameter_bounded_bound order_antisym xys)
4330 text {* Related results with closure as the conclusion. *}
4332 lemma closed_scaling:
4333 fixes s :: "'a::real_normed_vector set"
4334 assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
4336 case True thus ?thesis by auto
4341 have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
4342 case True thus ?thesis apply auto unfolding * by auto
4345 { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s" "(x ---> l) sequentially"
4346 { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
4347 using as(1)[THEN spec[where x=n]]
4348 using `c\<noteq>0` by auto
4351 { fix e::real assume "e>0"
4352 hence "0 < e *\<bar>c\<bar>" using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
4353 then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
4354 using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto
4355 hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
4356 unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
4357 using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto }
4358 hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto
4359 ultimately have "l \<in> scaleR c ` s"
4360 using assms[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. scaleR (1/c) (x n)"], THEN spec[where x="scaleR (1/c) l"]]
4361 unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto }
4362 thus ?thesis unfolding closed_sequential_limits by fast
4366 lemma closed_negations:
4367 fixes s :: "'a::real_normed_vector set"
4368 assumes "closed s" shows "closed ((\<lambda>x. -x) ` s)"
4369 using closed_scaling[OF assms, of "- 1"] by simp
4371 lemma compact_closed_sums:
4372 fixes s :: "'a::real_normed_vector set"
4373 assumes "compact s" "closed t" shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4375 let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
4376 { fix x l assume as:"\<forall>n. x n \<in> ?S" "(x ---> l) sequentially"
4377 from as(1) obtain f where f:"\<forall>n. x n = fst (f n) + snd (f n)" "\<forall>n. fst (f n) \<in> s" "\<forall>n. snd (f n) \<in> t"
4378 using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
4379 obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
4380 using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
4381 have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
4382 using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
4383 hence "l - l' \<in> t"
4384 using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
4386 hence "l \<in> ?S" using `l' \<in> s` apply auto apply(rule_tac x=l' in exI) apply(rule_tac x="l - l'" in exI) by auto
4388 thus ?thesis unfolding closed_sequential_limits by fast
4391 lemma closed_compact_sums:
4392 fixes s t :: "'a::real_normed_vector set"
4393 assumes "closed s" "compact t"
4394 shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4396 have "{x + y |x y. x \<in> t \<and> y \<in> s} = {x + y |x y. x \<in> s \<and> y \<in> t}" apply auto
4397 apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
4398 thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
4401 lemma compact_closed_differences:
4402 fixes s t :: "'a::real_normed_vector set"
4403 assumes "compact s" "closed t"
4404 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4406 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4407 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4408 thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
4411 lemma closed_compact_differences:
4412 fixes s t :: "'a::real_normed_vector set"
4413 assumes "closed s" "compact t"
4414 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4416 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4417 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4418 thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
4421 lemma closed_translation:
4422 fixes a :: "'a::real_normed_vector"
4423 assumes "closed s" shows "closed ((\<lambda>x. a + x) ` s)"
4425 have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
4426 thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
4429 lemma translation_Compl:
4430 fixes a :: "'a::ab_group_add"
4431 shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
4432 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto
4434 lemma translation_UNIV:
4435 fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
4436 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
4438 lemma translation_diff:
4439 fixes a :: "'a::ab_group_add"
4440 shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
4443 lemma closure_translation:
4444 fixes a :: "'a::real_normed_vector"
4445 shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
4447 have *:"op + a ` (- s) = - op + a ` s"
4448 apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
4449 show ?thesis unfolding closure_interior translation_Compl
4450 using interior_translation[of a "- s"] unfolding * by auto
4453 lemma frontier_translation:
4454 fixes a :: "'a::real_normed_vector"
4455 shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
4456 unfolding frontier_def translation_diff interior_translation closure_translation by auto
4459 subsection {* Separation between points and sets *}
4461 lemma separate_point_closed:
4462 fixes s :: "'a::heine_borel set"
4463 shows "closed s \<Longrightarrow> a \<notin> s ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
4464 proof(cases "s = {}")
4466 thus ?thesis by(auto intro!: exI[where x=1])
4469 assume "closed s" "a \<notin> s"
4470 then obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y" using `s \<noteq> {}` distance_attains_inf [of s a] by blast
4471 with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
4474 lemma separate_compact_closed:
4475 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4476 (* TODO: does this generalize to heine_borel? *)
4477 assumes "compact s" and "closed t" and "s \<inter> t = {}"
4478 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4480 have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto
4481 then obtain d where "d>0" and d:"\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. d \<le> dist 0 x"
4482 using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto
4483 { fix x y assume "x\<in>s" "y\<in>t"
4484 hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto
4485 hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute
4486 by (auto simp add: dist_commute)
4487 hence "d \<le> dist x y" unfolding dist_norm by auto }
4488 thus ?thesis using `d>0` by auto
4491 lemma separate_closed_compact:
4492 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4493 assumes "closed s" and "compact t" and "s \<inter> t = {}"
4494 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4496 have *:"t \<inter> s = {}" using assms(3) by auto
4497 show ?thesis using separate_compact_closed[OF assms(2,1) *]
4498 apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
4499 by (auto simp add: dist_commute)
4503 subsection {* Intervals *}
4505 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows
4506 "{a <..< b} = {x::'a. \<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i}" and
4507 "{a .. b} = {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i}"
4508 by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
4510 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows
4511 "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i)"
4512 "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i)"
4513 using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
4515 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows
4516 "({a <..< b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i))" (is ?th1) and
4517 "({a .. b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i < a$$i))" (is ?th2)
4519 { fix i x assume i:"i<DIM('a)" and as:"b$$i \<le> a$$i" and x:"x\<in>{a <..< b}"
4520 hence "a $$ i < x $$ i \<and> x $$ i < b $$ i" unfolding mem_interval by auto
4521 hence "a$$i < b$$i" by auto
4522 hence False using as by auto }
4524 { assume as:"\<forall>i<DIM('a). \<not> (b$$i \<le> a$$i)"
4525 let ?x = "(1/2) *\<^sub>R (a + b)"
4526 { fix i assume i:"i<DIM('a)"
4527 have "a$$i < b$$i" using as[THEN spec[where x=i]] using i by auto
4528 hence "a$$i < ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i < b$$i"
4529 unfolding euclidean_simps by auto }
4530 hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto }
4531 ultimately show ?th1 by blast
4533 { fix i x assume i:"i<DIM('a)" and as:"b$$i < a$$i" and x:"x\<in>{a .. b}"
4534 hence "a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i" unfolding mem_interval by auto
4535 hence "a$$i \<le> b$$i" by auto
4536 hence False using as by auto }
4538 { assume as:"\<forall>i<DIM('a). \<not> (b$$i < a$$i)"
4539 let ?x = "(1/2) *\<^sub>R (a + b)"
4540 { fix i assume i:"i<DIM('a)"
4541 have "a$$i \<le> b$$i" using as[THEN spec[where x=i]] by auto
4542 hence "a$$i \<le> ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i \<le> b$$i"
4543 unfolding euclidean_simps by auto }
4544 hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto }
4545 ultimately show ?th2 by blast
4548 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows
4549 "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> b$$i)" and
4550 "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"
4551 unfolding interval_eq_empty[of a b] by fastforce+
4553 lemma interval_sing:
4554 fixes a :: "'a::ordered_euclidean_space"
4555 shows "{a .. a} = {a}" and "{a<..<a} = {}"
4556 unfolding set_eq_iff mem_interval eq_iff [symmetric]
4557 by (auto simp add: euclidean_eq[where 'a='a] eq_commute
4558 eucl_less[where 'a='a] eucl_le[where 'a='a])
4560 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows
4561 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
4562 "(\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
4563 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
4564 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
4565 unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
4566 by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+
4568 lemma interval_open_subset_closed:
4569 fixes a :: "'a::ordered_euclidean_space"
4570 shows "{a<..<b} \<subseteq> {a .. b}"
4571 unfolding subset_eq [unfolded Ball_def] mem_interval
4572 by (fast intro: less_imp_le)
4574 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows
4575 "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th1) and
4576 "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i)" (is ?th2) and
4577 "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th3) and
4578 "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th4)
4580 show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
4581 show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
4582 { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i<DIM('a). c$$i < d$$i"
4583 hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
4584 fix i assume i:"i<DIM('a)"
4585 (** TODO combine the following two parts as done in the HOL_light version. **)
4586 { let ?x = "(\<chi>\<chi> j. (if j=i then ((min (a$$j) (d$$j))+c$$j)/2 else (c$$j+d$$j)/2))::'a"
4587 assume as2: "a$$i > c$$i"
4588 { fix j assume j:"j<DIM('a)"
4589 hence "c $$ j < ?x $$ j \<and> ?x $$ j < d $$ j"
4590 apply(cases "j=i") using as(2)[THEN spec[where x=j]] i
4591 by (auto simp add: as2) }
4592 hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto
4594 have "?x\<notin>{a .. b}"
4595 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4596 using as(2)[THEN spec[where x=i]] and as2 i
4598 ultimately have False using as by auto }
4599 hence "a$$i \<le> c$$i" by(rule ccontr)auto
4601 { let ?x = "(\<chi>\<chi> j. (if j=i then ((max (b$$j) (c$$j))+d$$j)/2 else (c$$j+d$$j)/2))::'a"
4602 assume as2: "b$$i < d$$i"
4603 { fix j assume "j<DIM('a)"
4604 hence "d $$ j > ?x $$ j \<and> ?x $$ j > c $$ j"
4605 apply(cases "j=i") using as(2)[THEN spec[where x=j]]
4606 by (auto simp add: as2) }
4607 hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
4609 have "?x\<notin>{a .. b}"
4610 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4611 using as(2)[THEN spec[where x=i]] and as2 using i
4613 ultimately have False using as by auto }
4614 hence "b$$i \<ge> d$$i" by(rule ccontr)auto
4616 have "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" by auto
4618 show ?th3 unfolding subset_eq and Ball_def and mem_interval
4619 apply(rule,rule,rule,rule) apply(rule part1) unfolding subset_eq and Ball_def and mem_interval
4620 prefer 4 apply auto by(erule_tac x=i in allE,erule_tac x=i in allE,fastforce)+
4621 { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i<DIM('a). c$$i < d$$i"
4622 fix i assume i:"i<DIM('a)"
4623 from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
4624 hence "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" using part1 and as(2) using i by auto } note * = this
4625 show ?th4 unfolding subset_eq and Ball_def and mem_interval
4626 apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4
4627 apply auto by(erule_tac x=i in allE, simp)+
4630 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows
4631 "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i < c$$i \<or> b$$i < c$$i \<or> d$$i < a$$i))" (is ?th1) and
4632 "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th2) and
4633 "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i < c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th3) and
4634 "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th4)
4636 let ?z = "(\<chi>\<chi> i. ((max (a$$i) (c$$i)) + (min (b$$i) (d$$i))) / 2)::'a"
4637 note * = set_eq_iff Int_iff empty_iff mem_interval all_conj_distrib[THEN sym] eq_False
4638 show ?th1 unfolding * apply safe apply(erule_tac x="?z" in allE)
4639 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4640 show ?th2 unfolding * apply safe apply(erule_tac x="?z" in allE)
4641 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4642 show ?th3 unfolding * apply safe apply(erule_tac x="?z" in allE)
4643 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4644 show ?th4 unfolding * apply safe apply(erule_tac x="?z" in allE)
4645 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4648 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows
4649 "{a .. b} \<inter> {c .. d} = {(\<chi>\<chi> i. max (a$$i) (c$$i)) .. (\<chi>\<chi> i. min (b$$i) (d$$i))}"
4650 unfolding set_eq_iff and Int_iff and mem_interval
4653 (* Moved interval_open_subset_closed a bit upwards *)
4655 lemma open_interval[intro]:
4656 fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
4658 have "open (\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i<..<b$$i})"
4659 by (intro open_INT finite_lessThan ballI continuous_open_vimage allI
4660 linear_continuous_at bounded_linear_euclidean_component
4661 open_real_greaterThanLessThan)
4662 also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i<..<b$$i}) = {a<..<b}"
4663 by (auto simp add: eucl_less [where 'a='a])
4664 finally show "open {a<..<b}" .
4667 lemma closed_interval[intro]:
4668 fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
4670 have "closed (\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i .. b$$i})"
4671 by (intro closed_INT ballI continuous_closed_vimage allI
4672 linear_continuous_at bounded_linear_euclidean_component
4673 closed_real_atLeastAtMost)
4674 also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i .. b$$i}) = {a .. b}"
4675 by (auto simp add: eucl_le [where 'a='a])
4676 finally show "closed {a .. b}" .
4679 lemma interior_closed_interval [intro]:
4680 fixes a b :: "'a::ordered_euclidean_space"
4681 shows "interior {a..b} = {a<..<b}" (is "?L = ?R")
4682 proof(rule subset_antisym)
4683 show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval
4684 by (rule interior_maximal)
4686 { fix x assume "x \<in> interior {a..b}"
4687 then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..
4688 then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto
4689 { fix i assume i:"i<DIM('a)"
4690 have "dist (x - (e / 2) *\<^sub>R basis i) x < e"
4691 "dist (x + (e / 2) *\<^sub>R basis i) x < e"
4692 unfolding dist_norm apply auto
4693 unfolding norm_minus_cancel using norm_basis and `e>0` by auto
4694 hence "a $$ i \<le> (x - (e / 2) *\<^sub>R basis i) $$ i"
4695 "(x + (e / 2) *\<^sub>R basis i) $$ i \<le> b $$ i"
4696 using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]
4697 and e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]
4698 unfolding mem_interval using i by blast+
4699 hence "a $$ i < x $$ i" and "x $$ i < b $$ i" unfolding euclidean_simps
4700 unfolding basis_component using `e>0` i by auto }
4701 hence "x \<in> {a<..<b}" unfolding mem_interval by auto }
4702 thus "?L \<subseteq> ?R" ..
4705 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"
4707 let ?b = "\<Sum>i<DIM('a). \<bar>a$$i\<bar> + \<bar>b$$i\<bar>"
4708 { fix x::"'a" assume x:"\<forall>i<DIM('a). a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i"
4709 { fix i assume "i<DIM('a)"
4710 hence "\<bar>x$$i\<bar> \<le> \<bar>a$$i\<bar> + \<bar>b$$i\<bar>" using x[THEN spec[where x=i]] by auto }
4711 hence "(\<Sum>i<DIM('a). \<bar>x $$ i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto
4712 hence "norm x \<le> ?b" using norm_le_l1[of x] by auto }
4713 thus ?thesis unfolding interval and bounded_iff by auto
4716 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows
4717 "bounded {a .. b} \<and> bounded {a<..<b}"
4718 using bounded_closed_interval[of a b]
4719 using interval_open_subset_closed[of a b]
4720 using bounded_subset[of "{a..b}" "{a<..<b}"]
4723 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows
4724 "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
4725 using bounded_interval[of a b] by auto
4727 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"
4728 using bounded_closed_imp_compact[of "{a..b}"] using bounded_interval[of a b]
4731 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"
4732 assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
4734 { fix i assume "i<DIM('a)"
4735 hence "a $$ i < ((1 / 2) *\<^sub>R (a + b)) $$ i \<and> ((1 / 2) *\<^sub>R (a + b)) $$ i < b $$ i"
4736 using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
4737 unfolding euclidean_simps by auto }
4738 thus ?thesis unfolding mem_interval by auto
4741 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"
4742 assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
4743 shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
4745 { fix i assume i:"i<DIM('a)"
4746 have "a $$ i = e * a$$i + (1 - e) * a$$i" unfolding left_diff_distrib by simp
4747 also have "\<dots> < e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
4748 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
4749 using x unfolding mem_interval using i apply simp
4750 using y unfolding mem_interval using i apply simp
4752 finally have "a $$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i" unfolding euclidean_simps by auto
4754 have "b $$ i = e * b$$i + (1 - e) * b$$i" unfolding left_diff_distrib by simp
4755 also have "\<dots> > e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
4756 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
4757 using x unfolding mem_interval using i apply simp
4758 using y unfolding mem_interval using i apply simp
4760 finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i < b $$ i" unfolding euclidean_simps by auto
4761 } ultimately have "a $$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i < b $$ i" by auto }
4762 thus ?thesis unfolding mem_interval by auto
4765 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"
4766 assumes "{a<..<b} \<noteq> {}"
4767 shows "closure {a<..<b} = {a .. b}"
4769 have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto
4770 let ?c = "(1 / 2) *\<^sub>R (a + b)"
4771 { fix x assume as:"x \<in> {a .. b}"
4772 def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
4773 { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
4774 have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
4775 have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
4776 x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
4777 by (auto simp add: algebra_simps)
4778 hence "f n < b" and "a < f n" using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *] unfolding f_def by auto
4779 hence False using fn unfolding f_def using xc by auto }
4781 { assume "\<not> (f ---> x) sequentially"
4782 { fix e::real assume "e>0"
4783 hence "\<exists>N::nat. inverse (real (N + 1)) < e" using real_arch_inv[of e] apply (auto simp add: Suc_pred') apply(rule_tac x="n - 1" in exI) by auto
4784 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
4785 hence "\<forall>n\<ge>N. inverse (real n + 1) < e" by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
4786 hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto }
4787 hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
4788 unfolding LIMSEQ_def by(auto simp add: dist_norm)
4789 hence "(f ---> x) sequentially" unfolding f_def
4790 using tendsto_add[OF tendsto_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
4791 using tendsto_scaleR [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto }
4792 ultimately have "x \<in> closure {a<..<b}"
4793 using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto }
4794 thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
4797 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"
4798 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a<..<a}"
4800 obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
4801 def a \<equiv> "(\<chi>\<chi> i. b+1)::'a"
4802 { fix x assume "x\<in>s"
4803 fix i assume i:"i<DIM('a)"
4804 hence "(-a)$$i < x$$i" and "x$$i < a$$i" using b[THEN bspec[where x=x], OF `x\<in>s`]
4805 and component_le_norm[of x i] unfolding euclidean_simps and a_def by auto }
4806 thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
4809 lemma bounded_subset_open_interval:
4810 fixes s :: "('a::ordered_euclidean_space) set"
4811 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
4812 by (auto dest!: bounded_subset_open_interval_symmetric)
4814 lemma bounded_subset_closed_interval_symmetric:
4815 fixes s :: "('a::ordered_euclidean_space) set"
4816 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
4818 obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
4819 thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
4822 lemma bounded_subset_closed_interval:
4823 fixes s :: "('a::ordered_euclidean_space) set"
4824 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
4825 using bounded_subset_closed_interval_symmetric[of s] by auto
4827 lemma frontier_closed_interval:
4828 fixes a b :: "'a::ordered_euclidean_space"
4829 shows "frontier {a .. b} = {a .. b} - {a<..<b}"
4830 unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
4832 lemma frontier_open_interval:
4833 fixes a b :: "'a::ordered_euclidean_space"
4834 shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
4835 proof(cases "{a<..<b} = {}")
4836 case True thus ?thesis using frontier_empty by auto
4838 case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
4841 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"
4842 assumes "{c<..<d} \<noteq> {}" shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
4843 unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
4846 (* Some stuff for half-infinite intervals too; FIXME: notation? *)
4848 lemma closed_interval_left: fixes b::"'a::euclidean_space"
4849 shows "closed {x::'a. \<forall>i<DIM('a). x$$i \<le> b$$i}"
4851 { fix i assume i:"i<DIM('a)"
4852 fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). x $$ i \<le> b $$ i}. x' \<noteq> x \<and> dist x' x < e"
4853 { assume "x$$i > b$$i"
4854 then obtain y where "y $$ i \<le> b $$ i" "y \<noteq> x" "dist y x < x$$i - b$$i"
4855 using x[THEN spec[where x="x$$i - b$$i"]] using i by auto
4856 hence False using component_le_norm[of "y - x" i] unfolding dist_norm euclidean_simps using i
4858 hence "x$$i \<le> b$$i" by(rule ccontr)auto }
4859 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
4862 lemma closed_interval_right: fixes a::"'a::euclidean_space"
4863 shows "closed {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i}"
4865 { fix i assume i:"i<DIM('a)"
4866 fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). a $$ i \<le> x $$ i}. x' \<noteq> x \<and> dist x' x < e"
4867 { assume "a$$i > x$$i"
4868 then obtain y where "a $$ i \<le> y $$ i" "y \<noteq> x" "dist y x < a$$i - x$$i"
4869 using x[THEN spec[where x="a$$i - x$$i"]] i by auto
4870 hence False using component_le_norm[of "y - x" i] unfolding dist_norm and euclidean_simps by auto }
4871 hence "a$$i \<le> x$$i" by(rule ccontr)auto }
4872 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
4875 text {* Intervals in general, including infinite and mixtures of open and closed. *}
4877 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
4878 (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i<DIM('a). ((a$$i \<le> x$$i \<and> x$$i \<le> b$$i) \<or> (b$$i \<le> x$$i \<and> x$$i \<le> a$$i))) \<longrightarrow> x \<in> s)"
4880 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)
4881 "is_interval {a<..<b}" (is ?th2) proof -
4882 show ?th1 ?th2 unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
4883 by(meson order_trans le_less_trans less_le_trans less_trans)+ qed
4885 lemma is_interval_empty:
4887 unfolding is_interval_def
4890 lemma is_interval_univ:
4892 unfolding is_interval_def
4896 subsection {* Closure of halfspaces and hyperplanes *}
4898 lemma isCont_open_vimage:
4899 assumes "\<And>x. isCont f x" and "open s" shows "open (f -` s)"
4901 from assms(1) have "continuous_on UNIV f"
4902 unfolding isCont_def continuous_on_def within_UNIV by simp
4903 hence "open {x \<in> UNIV. f x \<in> s}"
4904 using open_UNIV `open s` by (rule continuous_open_preimage)
4905 thus "open (f -` s)"
4906 by (simp add: vimage_def)
4909 lemma isCont_closed_vimage:
4910 assumes "\<And>x. isCont f x" and "closed s" shows "closed (f -` s)"
4911 using assms unfolding closed_def vimage_Compl [symmetric]
4912 by (rule isCont_open_vimage)
4914 lemma open_Collect_less:
4915 fixes f g :: "'a::topological_space \<Rightarrow> real"
4916 assumes f: "\<And>x. isCont f x"
4917 assumes g: "\<And>x. isCont g x"
4918 shows "open {x. f x < g x}"
4920 have "open ((\<lambda>x. g x - f x) -` {0<..})"
4921 using isCont_diff [OF g f] open_real_greaterThan
4922 by (rule isCont_open_vimage)
4923 also have "((\<lambda>x. g x - f x) -` {0<..}) = {x. f x < g x}"
4925 finally show ?thesis .
4928 lemma closed_Collect_le:
4929 fixes f g :: "'a::topological_space \<Rightarrow> real"
4930 assumes f: "\<And>x. isCont f x"
4931 assumes g: "\<And>x. isCont g x"
4932 shows "closed {x. f x \<le> g x}"
4934 have "closed ((\<lambda>x. g x - f x) -` {0..})"
4935 using isCont_diff [OF g f] closed_real_atLeast
4936 by (rule isCont_closed_vimage)
4937 also have "((\<lambda>x. g x - f x) -` {0..}) = {x. f x \<le> g x}"
4939 finally show ?thesis .
4942 lemma closed_Collect_eq:
4943 fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
4944 assumes f: "\<And>x. isCont f x"
4945 assumes g: "\<And>x. isCont g x"
4946 shows "closed {x. f x = g x}"
4948 have "open {(x::'b, y::'b). x \<noteq> y}"
4949 unfolding open_prod_def by (auto dest!: hausdorff)
4950 hence "closed {(x::'b, y::'b). x = y}"
4951 unfolding closed_def split_def Collect_neg_eq .
4952 with isCont_Pair [OF f g]
4953 have "closed ((\<lambda>x. (f x, g x)) -` {(x, y). x = y})"
4954 by (rule isCont_closed_vimage)
4955 also have "\<dots> = {x. f x = g x}" by auto
4956 finally show ?thesis .
4959 lemma continuous_at_inner: "continuous (at x) (inner a)"
4960 unfolding continuous_at by (intro tendsto_intros)
4962 lemma continuous_at_euclidean_component[intro!, simp]: "continuous (at x) (\<lambda>x. x $$ i)"
4963 unfolding euclidean_component_def by (rule continuous_at_inner)
4965 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
4966 by (simp add: closed_Collect_le)
4968 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
4969 by (simp add: closed_Collect_le)
4971 lemma closed_hyperplane: "closed {x. inner a x = b}"
4972 by (simp add: closed_Collect_eq)
4974 lemma closed_halfspace_component_le:
4975 shows "closed {x::'a::euclidean_space. x$$i \<le> a}"
4976 by (simp add: closed_Collect_le)
4978 lemma closed_halfspace_component_ge:
4979 shows "closed {x::'a::euclidean_space. x$$i \<ge> a}"
4980 by (simp add: closed_Collect_le)
4982 text {* Openness of halfspaces. *}
4984 lemma open_halfspace_lt: "open {x. inner a x < b}"
4985 by (simp add: open_Collect_less)
4987 lemma open_halfspace_gt: "open {x. inner a x > b}"
4988 by (simp add: open_Collect_less)
4990 lemma open_halfspace_component_lt:
4991 shows "open {x::'a::euclidean_space. x$$i < a}"
4992 by (simp add: open_Collect_less)
4994 lemma open_halfspace_component_gt:
4995 shows "open {x::'a::euclidean_space. x$$i > a}"
4996 by (simp add: open_Collect_less)
4998 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}
5000 lemma eucl_lessThan_eq_halfspaces:
5001 fixes a :: "'a\<Colon>ordered_euclidean_space"
5002 shows "{..<a} = (\<Inter>i<DIM('a). {x. x $$ i < a $$ i})"
5003 by (auto simp: eucl_less[where 'a='a])
5005 lemma eucl_greaterThan_eq_halfspaces:
5006 fixes a :: "'a\<Colon>ordered_euclidean_space"
5007 shows "{a<..} = (\<Inter>i<DIM('a). {x. a $$ i < x $$ i})"
5008 by (auto simp: eucl_less[where 'a='a])
5010 lemma eucl_atMost_eq_halfspaces:
5011 fixes a :: "'a\<Colon>ordered_euclidean_space"
5012 shows "{.. a} = (\<Inter>i<DIM('a). {x. x $$ i \<le> a $$ i})"
5013 by (auto simp: eucl_le[where 'a='a])
5015 lemma eucl_atLeast_eq_halfspaces:
5016 fixes a :: "'a\<Colon>ordered_euclidean_space"
5017 shows "{a ..} = (\<Inter>i<DIM('a). {x. a $$ i \<le> x $$ i})"
5018 by (auto simp: eucl_le[where 'a='a])
5020 lemma open_eucl_lessThan[simp, intro]:
5021 fixes a :: "'a\<Colon>ordered_euclidean_space"
5022 shows "open {..< a}"
5023 by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)
5025 lemma open_eucl_greaterThan[simp, intro]:
5026 fixes a :: "'a\<Colon>ordered_euclidean_space"
5027 shows "open {a <..}"
5028 by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)
5030 lemma closed_eucl_atMost[simp, intro]:
5031 fixes a :: "'a\<Colon>ordered_euclidean_space"
5032 shows "closed {.. a}"
5033 unfolding eucl_atMost_eq_halfspaces
5034 by (simp add: closed_INT closed_Collect_le)
5036 lemma closed_eucl_atLeast[simp, intro]:
5037 fixes a :: "'a\<Colon>ordered_euclidean_space"
5038 shows "closed {a ..}"
5039 unfolding eucl_atLeast_eq_halfspaces
5040 by (simp add: closed_INT closed_Collect_le)
5042 lemma open_vimage_euclidean_component: "open S \<Longrightarrow> open ((\<lambda>x. x $$ i) -` S)"
5043 by (auto intro!: continuous_open_vimage)
5045 text {* This gives a simple derivation of limit component bounds. *}
5047 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5048 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f(x)$$i \<le> b) net"
5049 shows "l$$i \<le> b"
5051 { fix x have "x \<in> {x::'b. inner (basis i) x \<le> b} \<longleftrightarrow> x$$i \<le> b"
5052 unfolding euclidean_component_def by auto } note * = this
5053 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *
5054 using closed_halfspace_le[of "(basis i)::'b" b] and assms(1,2,3) by auto
5057 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5058 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$$i) net"
5059 shows "b \<le> l$$i"
5061 { fix x have "x \<in> {x::'b. inner (basis i) x \<ge> b} \<longleftrightarrow> x$$i \<ge> b"
5062 unfolding euclidean_component_def by auto } note * = this
5063 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *
5064 using closed_halfspace_ge[of b "(basis i)"] and assms(1,2,3) by auto
5067 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5068 assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$$i = b) net"
5070 using ev[unfolded order_eq_iff eventually_conj_iff] using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto
5071 text{* Limits relative to a union. *}
5073 lemma eventually_within_Un:
5074 "eventually P (net within (s \<union> t)) \<longleftrightarrow>
5075 eventually P (net within s) \<and> eventually P (net within t)"
5076 unfolding Limits.eventually_within
5077 by (auto elim!: eventually_rev_mp)
5079 lemma Lim_within_union:
5080 "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>
5081 (f ---> l) (net within s) \<and> (f ---> l) (net within t)"
5082 unfolding tendsto_def
5083 by (auto simp add: eventually_within_Un)
5085 lemma Lim_topological:
5086 "(f ---> l) net \<longleftrightarrow>
5087 trivial_limit net \<or>
5088 (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
5089 unfolding tendsto_def trivial_limit_eq by auto
5091 lemma continuous_on_union:
5092 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
5093 shows "continuous_on (s \<union> t) f"
5094 using assms unfolding continuous_on Lim_within_union
5095 unfolding Lim_topological trivial_limit_within closed_limpt by auto
5097 lemma continuous_on_cases:
5098 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
5099 "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
5100 shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
5102 let ?h = "(\<lambda>x. if P x then f x else g x)"
5103 have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto
5104 hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto
5106 have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto
5107 hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto
5108 ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto
5112 text{* Some more convenient intermediate-value theorem formulations. *}
5114 lemma connected_ivt_hyperplane:
5115 assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
5116 shows "\<exists>z \<in> s. inner a z = b"
5118 assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
5119 let ?A = "{x. inner a x < b}"
5120 let ?B = "{x. inner a x > b}"
5121 have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
5122 moreover have "?A \<inter> ?B = {}" by auto
5123 moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
5124 ultimately show False using assms(1)[unfolded connected_def not_ex, THEN spec[where x="?A"], THEN spec[where x="?B"]] and assms(2-5) by auto
5127 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows
5128 "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$$k \<le> a \<Longrightarrow> a \<le> y$$k \<Longrightarrow> (\<exists>z\<in>s. z$$k = a)"
5129 using connected_ivt_hyperplane[of s x y "(basis k)::'a" a]
5130 unfolding euclidean_component_def by auto
5133 subsection {* Homeomorphisms *}
5135 definition "homeomorphism s t f g \<equiv>
5136 (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
5137 (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
5140 homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"
5141 (infixr "homeomorphic" 60) where
5142 homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
5144 lemma homeomorphic_refl: "s homeomorphic s"
5145 unfolding homeomorphic_def
5146 unfolding homeomorphism_def
5147 using continuous_on_id
5148 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5149 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5152 lemma homeomorphic_sym:
5153 "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
5154 unfolding homeomorphic_def
5155 unfolding homeomorphism_def
5158 lemma homeomorphic_trans:
5159 assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
5161 obtain f1 g1 where fg1:"\<forall>x\<in>s. g1 (f1 x) = x" "f1 ` s = t" "continuous_on s f1" "\<forall>y\<in>t. f1 (g1 y) = y" "g1 ` t = s" "continuous_on t g1"
5162 using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
5163 obtain f2 g2 where fg2:"\<forall>x\<in>t. g2 (f2 x) = x" "f2 ` t = u" "continuous_on t f2" "\<forall>y\<in>u. f2 (g2 y) = y" "g2 ` u = t" "continuous_on u g2"
5164 using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
5166 { fix x assume "x\<in>s" hence "(g1 \<circ> g2) ((f2 \<circ> f1) x) = x" using fg1(1)[THEN bspec[where x=x]] and fg2(1)[THEN bspec[where x="f1 x"]] and fg1(2) by auto }
5167 moreover have "(f2 \<circ> f1) ` s = u" using fg1(2) fg2(2) by auto
5168 moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
5169 moreover { fix y assume "y\<in>u" hence "(f2 \<circ> f1) ((g1 \<circ> g2) y) = y" using fg2(4)[THEN bspec[where x=y]] and fg1(4)[THEN bspec[where x="g2 y"]] and fg2(5) by auto }
5170 moreover have "(g1 \<circ> g2) ` u = s" using fg1(5) fg2(5) by auto
5171 moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6) unfolding fg2(5) by auto
5172 ultimately show ?thesis unfolding homeomorphic_def homeomorphism_def apply(rule_tac x="f2 \<circ> f1" in exI) apply(rule_tac x="g1 \<circ> g2" in exI) by auto
5175 lemma homeomorphic_minimal:
5176 "s homeomorphic t \<longleftrightarrow>
5177 (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
5178 (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
5179 continuous_on s f \<and> continuous_on t g)"
5180 unfolding homeomorphic_def homeomorphism_def
5181 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
5182 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
5184 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
5185 apply auto apply(rule_tac x="g x" in bexI) apply auto
5186 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
5187 apply auto apply(rule_tac x="f x" in bexI) by auto
5189 text {* Relatively weak hypotheses if a set is compact. *}
5191 lemma homeomorphism_compact:
5192 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5193 (* class constraint due to continuous_on_inv *)
5194 assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s"
5195 shows "\<exists>g. homeomorphism s t f g"
5197 def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
5198 have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
5199 { fix y assume "y\<in>t"
5200 then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
5201 hence "g (f x) = x" using g by auto
5202 hence "f (g y) = y" unfolding x(1)[THEN sym] by auto }
5203 hence g':"\<forall>x\<in>t. f (g x) = x" by auto
5206 have "x\<in>s \<Longrightarrow> x \<in> g ` t" using g[THEN bspec[where x=x]] unfolding image_iff using assms(3) by(auto intro!: bexI[where x="f x"])
5208 { assume "x\<in>g ` t"
5209 then obtain y where y:"y\<in>t" "g y = x" by auto
5210 then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
5211 hence "x \<in> s" unfolding g_def using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"] unfolding y(2)[THEN sym] and g_def by auto }
5212 ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" .. }
5213 hence "g ` t = s" by auto
5215 show ?thesis unfolding homeomorphism_def homeomorphic_def
5216 apply(rule_tac x=g in exI) using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2) by auto
5219 lemma homeomorphic_compact:
5220 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5221 (* class constraint due to continuous_on_inv *)
5222 shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
5223 \<Longrightarrow> s homeomorphic t"
5224 unfolding homeomorphic_def by (metis homeomorphism_compact)
5226 text{* Preservation of topological properties. *}
5228 lemma homeomorphic_compactness:
5229 "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
5230 unfolding homeomorphic_def homeomorphism_def
5231 by (metis compact_continuous_image)
5233 text{* Results on translation, scaling etc. *}
5235 lemma homeomorphic_scaling:
5236 fixes s :: "'a::real_normed_vector set"
5237 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
5238 unfolding homeomorphic_minimal
5239 apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
5240 apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
5241 using assms by (auto simp add: continuous_on_intros)
5243 lemma homeomorphic_translation:
5244 fixes s :: "'a::real_normed_vector set"
5245 shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
5246 unfolding homeomorphic_minimal
5247 apply(rule_tac x="\<lambda>x. a + x" in exI)
5248 apply(rule_tac x="\<lambda>x. -a + x" in exI)
5249 using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
5251 lemma homeomorphic_affinity:
5252 fixes s :: "'a::real_normed_vector set"
5253 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
5255 have *:"op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
5257 using homeomorphic_trans
5258 using homeomorphic_scaling[OF assms, of s]
5259 using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a] unfolding * by auto
5262 lemma homeomorphic_balls:
5263 fixes a b ::"'a::real_normed_vector" (* FIXME: generalize to metric_space *)
5264 assumes "0 < d" "0 < e"
5265 shows "(ball a d) homeomorphic (ball b e)" (is ?th)
5266 "(cball a d) homeomorphic (cball b e)" (is ?cth)
5268 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5269 show ?th unfolding homeomorphic_minimal
5270 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5271 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5272 using assms apply (auto simp add: dist_commute)
5274 apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)
5275 unfolding continuous_on
5276 by (intro ballI tendsto_intros, simp)+
5278 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5279 show ?cth unfolding homeomorphic_minimal
5280 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5281 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5282 using assms apply (auto simp add: dist_commute)
5284 apply (auto simp add: pos_divide_le_eq)
5285 unfolding continuous_on
5286 by (intro ballI tendsto_intros, simp)+
5289 text{* "Isometry" (up to constant bounds) of injective linear map etc. *}
5291 lemma cauchy_isometric:
5292 fixes x :: "nat \<Rightarrow> 'a::euclidean_space"
5293 assumes e:"0 < e" and s:"subspace s" and f:"bounded_linear f" and normf:"\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)" and xs:"\<forall>n::nat. x n \<in> s" and cf:"Cauchy(f o x)"
5296 interpret f: bounded_linear f by fact
5297 { fix d::real assume "d>0"
5298 then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
5299 using cf[unfolded cauchy o_def dist_norm, THEN spec[where x="e*d"]] and e and mult_pos_pos[of e d] by auto
5300 { fix n assume "n\<ge>N"
5301 hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding f.diff[THEN sym] by auto
5302 moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
5303 using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
5304 using normf[THEN bspec[where x="x n - x N"]] by auto
5305 ultimately have "norm (x n - x N) < d" using `e>0`
5306 using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto }
5307 hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
5308 thus ?thesis unfolding cauchy and dist_norm by auto
5311 lemma complete_isometric_image:
5312 fixes f :: "'a::euclidean_space => 'b::euclidean_space"
5313 assumes "0 < e" and s:"subspace s" and f:"bounded_linear f" and normf:"\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)" and cs:"complete s"
5314 shows "complete(f ` s)"
5316 { fix g assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
5317 then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
5318 using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
5319 hence x:"\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
5320 hence "f \<circ> x = g" unfolding fun_eq_iff by auto
5321 then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
5322 using cs[unfolded complete_def, THEN spec[where x="x"]]
5323 using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
5324 hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
5325 using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
5326 unfolding `f \<circ> x = g` by auto }
5327 thus ?thesis unfolding complete_def by auto
5331 fixes x :: "'a::real_normed_vector"
5332 shows "dist 0 x = norm x"
5333 unfolding dist_norm by simp
5335 lemma injective_imp_isometric: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
5336 assumes s:"closed s" "subspace s" and f:"bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
5337 shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
5338 proof(cases "s \<subseteq> {0::'a}")
5340 { fix x assume "x \<in> s"
5341 hence "x = 0" using True by auto
5342 hence "norm x \<le> norm (f x)" by auto }
5343 thus ?thesis by(auto intro!: exI[where x=1])
5345 interpret f: bounded_linear f by fact
5347 then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
5348 from False have "s \<noteq> {}" by auto
5349 let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
5350 let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
5351 let ?S'' = "{x::'a. norm x = norm a}"
5353 have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by auto
5354 hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
5355 moreover have "?S' = s \<inter> ?S''" by auto
5356 ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
5357 moreover have *:"f ` ?S' = ?S" by auto
5358 ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
5359 hence "closed ?S" using compact_imp_closed by auto
5360 moreover have "?S \<noteq> {}" using a by auto
5361 ultimately obtain b' where "b'\<in>?S" "\<forall>y\<in>?S. norm b' \<le> norm y" using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto
5362 then obtain b where "b\<in>s" and ba:"norm b = norm a" and b:"\<forall>x\<in>{x \<in> s. norm x = norm a}. norm (f b) \<le> norm (f x)" unfolding *[THEN sym] unfolding image_iff by auto
5364 let ?e = "norm (f b) / norm b"
5365 have "norm b > 0" using ba and a and norm_ge_zero by auto
5366 moreover have "norm (f b) > 0" using f(2)[THEN bspec[where x=b], OF `b\<in>s`] using `norm b >0` unfolding zero_less_norm_iff by auto
5367 ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
5369 { fix x assume "x\<in>s"
5370 hence "norm (f b) / norm b * norm x \<le> norm (f x)"
5372 case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
5375 hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
5376 have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def] by auto
5377 hence "(norm a / norm x) *\<^sub>R x \<in> {x \<in> s. norm x = norm a}" using `x\<in>s` and `x\<noteq>0` by auto
5378 thus "norm (f b) / norm b * norm x \<le> norm (f x)" using b[THEN bspec[where x="(norm a / norm x) *\<^sub>R x"]]
5379 unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
5380 by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq)
5383 show ?thesis by auto
5386 lemma closed_injective_image_subspace:
5387 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
5388 assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
5389 shows "closed(f ` s)"
5391 obtain e where "e>0" and e:"\<forall>x\<in>s. e * norm x \<le> norm (f x)" using injective_imp_isometric[OF assms(4,1,2,3)] by auto
5392 show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
5393 unfolding complete_eq_closed[THEN sym] by auto
5397 subsection {* Some properties of a canonical subspace *}
5399 lemma subspace_substandard:
5400 "subspace {x::'a::euclidean_space. (\<forall>i<DIM('a). P i \<longrightarrow> x$$i = 0)}"
5401 unfolding subspace_def by auto
5403 lemma closed_substandard:
5404 "closed {x::'a::euclidean_space. \<forall>i<DIM('a). P i --> x$$i = 0}" (is "closed ?A")
5406 let ?D = "{i. P i} \<inter> {..<DIM('a)}"
5407 have "closed (\<Inter>i\<in>?D. {x::'a. x$$i = 0})"
5408 by (simp add: closed_INT closed_Collect_eq)
5409 also have "(\<Inter>i\<in>?D. {x::'a. x$$i = 0}) = ?A"
5411 finally show "closed ?A" .
5414 lemma dim_substandard: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
5415 shows "dim {x::'a::euclidean_space. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0} = card d" (is "dim ?A = _")
5417 let ?D = "{..<DIM('a)}"
5418 let ?B = "(basis::nat => 'a) ` d"
5419 let ?bas = "basis::nat \<Rightarrow> 'a"
5420 have "?B \<subseteq> ?A" by auto
5422 { fix x::"'a" assume "x\<in>?A"
5423 hence "finite d" "x\<in>?A" using assms by(auto intro:finite_subset)
5424 hence "x\<in> span ?B"
5425 proof(induct d arbitrary: x)
5426 case empty hence "x=0" apply(subst euclidean_eq) by auto
5427 thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
5430 hence *:"\<forall>i<DIM('a). i \<notin> insert k F \<longrightarrow> x $$ i = 0" by auto
5431 have **:"F \<subseteq> insert k F" by auto
5432 def y \<equiv> "x - x$$k *\<^sub>R basis k"
5433 have y:"x = y + (x$$k) *\<^sub>R basis k" unfolding y_def by auto
5434 { fix i assume i':"i \<notin> F"
5435 hence "y $$ i = 0" unfolding y_def
5436 using *[THEN spec[where x=i]] by auto }
5437 hence "y \<in> span (basis ` F)" using insert(3) by auto
5438 hence "y \<in> span (basis ` (insert k F))"
5439 using span_mono[of "?bas ` F" "?bas ` (insert k F)"]
5440 using image_mono[OF **, of basis] using assms by auto
5442 have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
5443 hence "x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5444 using span_mul by auto
5446 have "y + x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5447 using span_add by auto
5448 thus ?case using y by auto
5451 hence "?A \<subseteq> span ?B" by auto
5453 { fix x assume "x \<in> ?B"
5454 hence "x\<in>{(basis i)::'a |i. i \<in> ?D}" using assms by auto }
5455 hence "independent ?B" using independent_mono[OF independent_basis, of ?B] and assms by auto
5457 have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
5458 hence *:"inj_on (basis::nat\<Rightarrow>'a) d" using subset_inj_on[OF basis_inj, of "d"] by auto
5459 have "card ?B = card d" unfolding card_image[OF *] by auto
5460 ultimately show ?thesis using dim_unique[of "basis ` d" ?A] by auto
5463 text{* Hence closure and completeness of all subspaces. *}
5465 lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"
5467 apply (rule_tac x="{}" in exI, simp)
5469 apply (subgoal_tac "\<exists>x. x \<notin> A")
5471 apply (rule_tac x="insert x A" in exI, simp)
5472 apply (subgoal_tac "A \<noteq> UNIV", auto)
5475 lemma closed_subspace: fixes s::"('a::euclidean_space) set"
5476 assumes "subspace s" shows "closed s"
5478 have *:"dim s \<le> DIM('a)" using dim_subset_UNIV by auto
5479 def d \<equiv> "{..<dim s}" have t:"card d = dim s" unfolding d_def by auto
5480 let ?t = "{x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0}"
5481 have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0} = s \<and>
5482 inj_on f {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0}"
5483 apply(rule subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]])
5484 using dim_substandard[of d,where 'a='a] and t unfolding d_def using * assms by auto
5485 then guess f apply-by(erule exE conjE)+ note f = this
5486 interpret f: bounded_linear f using f unfolding linear_conv_bounded_linear by auto
5487 have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using f.zero using f(3)[unfolded inj_on_def]
5488 by(erule_tac x=0 in ballE) auto
5489 moreover have "closed ?t" using closed_substandard .
5490 moreover have "subspace ?t" using subspace_substandard .
5491 ultimately show ?thesis using closed_injective_image_subspace[of ?t f]
5492 unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
5495 lemma complete_subspace:
5496 fixes s :: "('a::euclidean_space) set" shows "subspace s ==> complete s"
5497 using complete_eq_closed closed_subspace
5501 fixes s :: "('a::euclidean_space) set"
5502 shows "dim(closure s) = dim s" (is "?dc = ?d")
5504 have "?dc \<le> ?d" using closure_minimal[OF span_inc, of s]
5505 using closed_subspace[OF subspace_span, of s]
5506 using dim_subset[of "closure s" "span s"] unfolding dim_span by auto
5507 thus ?thesis using dim_subset[OF closure_subset, of s] by auto
5511 subsection {* Affine transformations of intervals *}
5513 lemma real_affinity_le:
5514 "0 < (m::'a::linordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"
5515 by (simp add: field_simps inverse_eq_divide)
5517 lemma real_le_affinity:
5518 "0 < (m::'a::linordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"
5519 by (simp add: field_simps inverse_eq_divide)
5521 lemma real_affinity_lt:
5522 "0 < (m::'a::linordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"
5523 by (simp add: field_simps inverse_eq_divide)
5525 lemma real_lt_affinity:
5526 "0 < (m::'a::linordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"
5527 by (simp add: field_simps inverse_eq_divide)
5529 lemma real_affinity_eq:
5530 "(m::'a::linordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"
5531 by (simp add: field_simps inverse_eq_divide)
5533 lemma real_eq_affinity:
5534 "(m::'a::linordered_field) \<noteq> 0 ==> (y = m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"
5535 by (simp add: field_simps inverse_eq_divide)
5537 lemma image_affinity_interval: fixes m::real
5538 fixes a b c :: "'a::ordered_euclidean_space"
5539 shows "(\<lambda>x. m *\<^sub>R x + c) ` {a .. b} =
5540 (if {a .. b} = {} then {}
5541 else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
5542 else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
5544 { fix x assume "x \<le> c" "c \<le> x"
5545 hence "x=c" unfolding eucl_le[where 'a='a] apply-
5546 apply(subst euclidean_eq) by (auto intro: order_antisym) }
5548 moreover have "c \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}" unfolding True by(auto simp add: eucl_le[where 'a='a])
5549 ultimately show ?thesis by auto
5552 { fix y assume "a \<le> y" "y \<le> b" "m > 0"
5553 hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
5554 unfolding eucl_le[where 'a='a] by auto
5556 { fix y assume "a \<le> y" "y \<le> b" "m < 0"
5557 hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
5558 unfolding eucl_le[where 'a='a] by(auto simp add: mult_left_mono_neg)
5560 { fix y assume "m > 0" "m *\<^sub>R a + c \<le> y" "y \<le> m *\<^sub>R b + c"
5561 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5562 unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
5563 apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
5564 by(auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff)
5566 { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
5567 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5568 unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
5569 apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
5570 by(auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff)
5572 ultimately show ?thesis using False by auto
5575 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a..b} =
5576 (if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
5577 using image_affinity_interval[of m 0 a b] by auto
5580 subsection {* Banach fixed point theorem (not really topological...) *}
5583 assumes s:"complete s" "s \<noteq> {}" and c:"0 \<le> c" "c < 1" and f:"(f ` s) \<subseteq> s" and
5584 lipschitz:"\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
5585 shows "\<exists>! x\<in>s. (f x = x)"
5587 have "1 - c > 0" using c by auto
5589 from s(2) obtain z0 where "z0 \<in> s" by auto
5590 def z \<equiv> "\<lambda>n. (f ^^ n) z0"
5592 have "z n \<in> s" unfolding z_def
5593 proof(induct n) case 0 thus ?case using `z0 \<in>s` by auto
5594 next case Suc thus ?case using f by auto qed }
5597 def d \<equiv> "dist (z 0) (z 1)"
5599 have fzn:"\<And>n. f (z n) = z (Suc n)" unfolding z_def by auto
5601 have "dist (z n) (z (Suc n)) \<le> (c ^ n) * d"
5603 case 0 thus ?case unfolding d_def by auto
5606 hence "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
5607 using `0 \<le> c` using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by auto
5608 thus ?case using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]
5609 unfolding fzn and mult_le_cancel_left by auto
5614 have "(1 - c) * dist (z m) (z (m+n)) \<le> (c ^ m) * d * (1 - c ^ n)"
5616 case 0 show ?case by auto
5619 have "(1 - c) * dist (z m) (z (m + Suc k)) \<le> (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))"
5620 using dist_triangle and c by(auto simp add: dist_triangle)
5621 also have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
5622 using cf_z[of "m + k"] and c by auto
5623 also have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
5624 using Suc by (auto simp add: field_simps)
5625 also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
5626 unfolding power_add by (auto simp add: field_simps)
5627 also have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
5628 using c by (auto simp add: field_simps)
5629 finally show ?case by auto
5632 { fix e::real assume "e>0"
5633 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e"
5634 proof(cases "d = 0")
5636 have *: "\<And>x. ((1 - c) * x \<le> 0) = (x \<le> 0)" using `1 - c > 0`
5637 by (metis mult_zero_left real_mult_commute real_mult_le_cancel_iff1)
5638 from True have "\<And>n. z n = z0" using cf_z2[of 0] and c unfolding z_def
5640 thus ?thesis using `e>0` by auto
5642 case False hence "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]
5643 by (metis False d_def less_le)
5644 hence "0 < e * (1 - c) / d" using `e>0` and `1-c>0`
5645 using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"] by auto
5646 then obtain N where N:"c ^ N < e * (1 - c) / d" using real_arch_pow_inv[of "e * (1 - c) / d" c] and c by auto
5647 { fix m n::nat assume "m>n" and as:"m\<ge>N" "n\<ge>N"
5648 have *:"c ^ n \<le> c ^ N" using `n\<ge>N` and c using power_decreasing[OF `n\<ge>N`, of c] by auto
5649 have "1 - c ^ (m - n) > 0" using c and power_strict_mono[of c 1 "m - n"] using `m>n` by auto
5650 hence **:"d * (1 - c ^ (m - n)) / (1 - c) > 0"
5651 using mult_pos_pos[OF `d>0`, of "1 - c ^ (m - n)"]
5652 using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]
5653 using `0 < 1 - c` by auto
5655 have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
5656 using cf_z2[of n "m - n"] and `m>n` unfolding pos_le_divide_eq[OF `1-c>0`]
5657 by (auto simp add: mult_commute dist_commute)
5658 also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
5659 using mult_right_mono[OF * order_less_imp_le[OF **]]
5660 unfolding mult_assoc by auto
5661 also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
5662 using mult_strict_right_mono[OF N **] unfolding mult_assoc by auto
5663 also have "\<dots> = e * (1 - c ^ (m - n))" using c and `d>0` and `1 - c > 0` by auto
5664 also have "\<dots> \<le> e" using c and `1 - c ^ (m - n) > 0` and `e>0` using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto
5665 finally have "dist (z m) (z n) < e" by auto
5667 { fix m n::nat assume as:"N\<le>m" "N\<le>n"
5668 hence "dist (z n) (z m) < e"
5669 proof(cases "n = m")
5670 case True thus ?thesis using `e>0` by auto
5672 case False thus ?thesis using as and *[of n m] *[of m n] unfolding nat_neq_iff by (auto simp add: dist_commute)
5674 thus ?thesis by auto
5677 hence "Cauchy z" unfolding cauchy_def by auto
5678 then obtain x where "x\<in>s" and x:"(z ---> x) sequentially" using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto
5680 def e \<equiv> "dist (f x) x"
5681 have "e = 0" proof(rule ccontr)
5682 assume "e \<noteq> 0" hence "e>0" unfolding e_def using zero_le_dist[of "f x" x]
5683 by (metis dist_eq_0_iff dist_nz e_def)
5684 then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"
5685 using x[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] by auto
5686 hence N':"dist (z N) x < e / 2" by auto
5688 have *:"c * dist (z N) x \<le> dist (z N) x" unfolding mult_le_cancel_right2
5689 using zero_le_dist[of "z N" x] and c
5690 by (metis dist_eq_0_iff dist_nz order_less_asym less_le)
5691 have "dist (f (z N)) (f x) \<le> c * dist (z N) x" using lipschitz[THEN bspec[where x="z N"], THEN bspec[where x=x]]
5692 using z_in_s[of N] `x\<in>s` using c by auto
5693 also have "\<dots> < e / 2" using N' and c using * by auto
5694 finally show False unfolding fzn
5695 using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
5696 unfolding e_def by auto
5698 hence "f x = x" unfolding e_def by auto
5700 { fix y assume "f y = y" "y\<in>s"
5701 hence "dist x y \<le> c * dist x y" using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]
5702 using `x\<in>s` and `f x = x` by auto
5703 hence "dist x y = 0" unfolding mult_le_cancel_right1
5704 using c and zero_le_dist[of x y] by auto
5705 hence "y = x" by auto
5707 ultimately show ?thesis using `x\<in>s` by blast+
5710 subsection {* Edelstein fixed point theorem *}
5712 lemma edelstein_fix:
5713 fixes s :: "'a::real_normed_vector set"
5714 assumes s:"compact s" "s \<noteq> {}" and gs:"(g ` s) \<subseteq> s"
5715 and dist:"\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
5716 shows "\<exists>! x\<in>s. g x = x"
5717 proof(cases "\<exists>x\<in>s. g x \<noteq> x")
5718 obtain x where "x\<in>s" using s(2) by auto
5719 case False hence g:"\<forall>x\<in>s. g x = x" by auto
5720 { fix y assume "y\<in>s"
5721 hence "x = y" using `x\<in>s` and dist[THEN bspec[where x=x], THEN bspec[where x=y]]
5722 unfolding g[THEN bspec[where x=x], OF `x\<in>s`]
5723 unfolding g[THEN bspec[where x=y], OF `y\<in>s`] by auto }
5724 thus ?thesis using `x\<in>s` and g by blast+
5727 then obtain x where [simp]:"x\<in>s" and "g x \<noteq> x" by auto
5728 { fix x y assume "x \<in> s" "y \<in> s"
5729 hence "dist (g x) (g y) \<le> dist x y"
5730 using dist[THEN bspec[where x=x], THEN bspec[where x=y]] by auto } note dist' = this
5731 def y \<equiv> "g x"
5732 have [simp]:"y\<in>s" unfolding y_def using gs[unfolded image_subset_iff] and `x\<in>s` by blast
5733 def f \<equiv> "\<lambda>n. g ^^ n"
5734 have [simp]:"\<And>n z. g (f n z) = f (Suc n) z" unfolding f_def by auto
5735 have [simp]:"\<And>z. f 0 z = z" unfolding f_def by auto
5736 { fix n::nat and z assume "z\<in>s"
5737 have "f n z \<in> s" unfolding f_def
5739 case 0 thus ?case using `z\<in>s` by simp
5741 case (Suc n) thus ?case using gs[unfolded image_subset_iff] by auto
5742 qed } note fs = this
5743 { fix m n ::nat assume "m\<le>n"
5744 fix w z assume "w\<in>s" "z\<in>s"
5745 have "dist (f n w) (f n z) \<le> dist (f m w) (f m z)" using `m\<le>n`
5747 case 0 thus ?case by auto
5750 thus ?case proof(cases "m\<le>n")
5751 case True thus ?thesis using Suc(1)
5752 using dist'[OF fs fs, OF `w\<in>s` `z\<in>s`, of n n] by auto
5754 case False hence mn:"m = Suc n" using Suc(2) by simp
5755 show ?thesis unfolding mn by auto
5757 qed } note distf = this
5759 def h \<equiv> "\<lambda>n. (f n x, f n y)"
5760 let ?s2 = "s \<times> s"
5761 obtain l r where "l\<in>?s2" and r:"subseq r" and lr:"((h \<circ> r) ---> l) sequentially"
5762 using compact_Times [OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def
5763 using fs[OF `x\<in>s`] and fs[OF `y\<in>s`] by blast
5764 def a \<equiv> "fst l" def b \<equiv> "snd l"
5765 have lab:"l = (a, b)" unfolding a_def b_def by simp
5766 have [simp]:"a\<in>s" "b\<in>s" unfolding a_def b_def using `l\<in>?s2` by auto
5768 have lima:"((fst \<circ> (h \<circ> r)) ---> a) sequentially"
5769 and limb:"((snd \<circ> (h \<circ> r)) ---> b) sequentially"
5771 unfolding o_def a_def b_def by (rule tendsto_intros)+
5774 have *:"\<And>fx fy (x::'a) y. dist fx fy \<le> dist x y \<Longrightarrow> \<not> (dist (fx - fy) (a - b) < dist a b - dist x y)" unfolding dist_norm by norm
5776 have "dist (-x) (-y) = dist x y" unfolding dist_norm
5777 using norm_minus_cancel[of "x - y"] by (auto simp add: uminus_add_conv_diff) } note ** = this
5779 { assume as:"dist a b > dist (f n x) (f n y)"
5780 then obtain Na Nb where "\<forall>m\<ge>Na. dist (f (r m) x) a < (dist a b - dist (f n x) (f n y)) / 2"
5781 and "\<forall>m\<ge>Nb. dist (f (r m) y) b < (dist a b - dist (f n x) (f n y)) / 2"
5782 using lima limb unfolding h_def LIMSEQ_def by (fastforce simp del: less_divide_eq_number_of1)
5783 hence "dist (f (r (Na + Nb + n)) x - f (r (Na + Nb + n)) y) (a - b) < dist a b - dist (f n x) (f n y)"
5784 apply(erule_tac x="Na+Nb+n" in allE)
5785 apply(erule_tac x="Na+Nb+n" in allE) apply simp
5786 using dist_triangle_add_half[of a "f (r (Na + Nb + n)) x" "dist a b - dist (f n x) (f n y)"
5787 "-b" "- f (r (Na + Nb + n)) y"]
5788 unfolding ** by (auto simp add: algebra_simps dist_commute)
5790 have "dist (f (r (Na + Nb + n)) x - f (r (Na + Nb + n)) y) (a - b) \<ge> dist a b - dist (f n x) (f n y)"
5791 using distf[of n "r (Na+Nb+n)", OF _ `x\<in>s` `y\<in>s`]
5792 using subseq_bigger[OF r, of "Na+Nb+n"]
5793 using *[of "f (r (Na + Nb + n)) x" "f (r (Na + Nb + n)) y" "f n x" "f n y"] by auto
5794 ultimately have False by simp
5796 hence "dist a b \<le> dist (f n x) (f n y)" by(rule ccontr)auto }
5799 have [simp]:"a = b" proof(rule ccontr)
5800 def e \<equiv> "dist a b - dist (g a) (g b)"
5801 assume "a\<noteq>b" hence "e > 0" unfolding e_def using dist by fastforce
5802 hence "\<exists>n. dist (f n x) a < e/2 \<and> dist (f n y) b < e/2"
5803 using lima limb unfolding LIMSEQ_def
5804 apply (auto elim!: allE[where x="e/2"]) apply(rename_tac N N', rule_tac x="r (max N N')" in exI) unfolding h_def by fastforce
5805 then obtain n where n:"dist (f n x) a < e/2 \<and> dist (f n y) b < e/2" by auto
5806 have "dist (f (Suc n) x) (g a) \<le> dist (f n x) a"
5807 using dist[THEN bspec[where x="f n x"], THEN bspec[where x="a"]] and fs by auto
5808 moreover have "dist (f (Suc n) y) (g b) \<le> dist (f n y) b"
5809 using dist[THEN bspec[where x="f n y"], THEN bspec[where x="b"]] and fs by auto
5810 ultimately have "dist (f (Suc n) x) (g a) + dist (f (Suc n) y) (g b) < e" using n by auto
5811 thus False unfolding e_def using ab_fn[of "Suc n"] by norm
5814 have [simp]:"\<And>n. f (Suc n) x = f n y" unfolding f_def y_def by(induct_tac n)auto
5815 { fix x y assume "x\<in>s" "y\<in>s" moreover
5816 fix e::real assume "e>0" ultimately
5817 have "dist y x < e \<longrightarrow> dist (g y) (g x) < e" using dist by fastforce }
5818 hence "continuous_on s g" unfolding continuous_on_iff by auto
5820 hence "((snd \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
5821 apply (rule allE[where x="\<lambda>n. (fst \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
5822 using lima unfolding h_def o_def using fs[OF `x\<in>s`] by (auto simp add: y_def)
5823 hence "g a = a" using tendsto_unique[OF trivial_limit_sequentially limb, of "g a"]
5824 unfolding `a=b` and o_assoc by auto
5826 { fix x assume "x\<in>s" "g x = x" "x\<noteq>a"
5827 hence "False" using dist[THEN bspec[where x=a], THEN bspec[where x=x]]
5828 using `g a = a` and `a\<in>s` by auto }
5829 ultimately show "\<exists>!x\<in>s. g x = x" using `a\<in>s` by blast
5832 declare tendsto_const [intro] (* FIXME: move *)