1 (* title: HOL/Library/Topology_Euclidian_Space.thy
2 Author: Amine Chaieb, University of Cambridge
3 Author: Robert Himmelmann, TU Muenchen
6 header {* Elementary topology in Euclidean space. *}
8 theory Topology_Euclidean_Space
9 imports SEQ Euclidean_Space Product_Vector
12 subsection{* General notion of a topology *}
14 definition "istopology L \<longleftrightarrow> {} \<in> L \<and> (\<forall>S \<in>L. \<forall>T \<in>L. S \<inter> T \<in> L) \<and> (\<forall>K. K \<subseteq>L \<longrightarrow> \<Union> K \<in> L)"
15 typedef (open) 'a topology = "{L::('a set) set. istopology L}"
16 morphisms "openin" "topology"
17 unfolding istopology_def by blast
19 lemma istopology_open_in[intro]: "istopology(openin U)"
20 using openin[of U] by blast
22 lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
23 using topology_inverse[unfolded mem_def Collect_def] .
25 lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
26 using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
28 lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
30 {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}
32 {assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
33 hence "openin T1 = openin T2" by (metis mem_def set_ext)
34 hence "topology (openin T1) = topology (openin T2)" by simp
35 hence "T1 = T2" unfolding openin_inverse .}
36 ultimately show ?thesis by blast
39 text{* Infer the "universe" from union of all sets in the topology. *}
41 definition "topspace T = \<Union>{S. openin T S}"
43 subsection{* Main properties of open sets *}
46 fixes U :: "'a topology"
48 "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
49 "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
50 using openin[of U] unfolding istopology_def Collect_def mem_def
51 by (metis mem_def subset_eq)+
53 lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
54 unfolding topspace_def by blast
55 lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
57 lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
58 by (simp add: openin_clauses)
60 lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)" by (simp add: openin_clauses)
62 lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
63 using openin_Union[of "{S,T}" U] by auto
65 lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
67 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")
69 {assume ?lhs then have ?rhs by auto }
72 then obtain t where t: "\<forall>x\<in>S. openin U (t x) \<and> x \<in> t x \<and> t x \<subseteq> S"
73 unfolding Ball_def ex_simps(6)[symmetric] choice_iff by blast
74 from t have th0: "\<forall>x\<in> t`S. openin U x" by auto
75 have "\<Union> t`S = S" using t by auto
76 with openin_Union[OF th0] have "openin U S" by simp }
77 ultimately show ?thesis by blast
80 subsection{* 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 subsection{* Subspace topology. *}
123 definition "subtopology U V = topology {S \<inter> V |S. openin U S}"
125 lemma istopology_subtopology: "istopology {S \<inter> V |S. openin U S}" (is "istopology ?L")
127 have "{} \<in> ?L" by blast
128 {fix A B assume A: "A \<in> ?L" and B: "B \<in> ?L"
129 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
130 have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)" using Sa Sb by blast+
131 then have "A \<inter> B \<in> ?L" by blast}
133 {fix K assume K: "K \<subseteq> ?L"
134 have th0: "?L = (\<lambda>S. S \<inter> V) ` openin U "
136 apply (simp add: Ball_def image_iff)
138 from K[unfolded th0 subset_image_iff]
139 obtain Sk where Sk: "Sk \<subseteq> openin U" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
140 have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
141 moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq mem_def)
142 ultimately have "\<Union>K \<in> ?L" by blast}
143 ultimately show ?thesis unfolding istopology_def by blast
146 lemma openin_subtopology:
147 "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
148 unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
149 by (auto simp add: Collect_def)
151 lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
152 by (auto simp add: topspace_def openin_subtopology)
154 lemma closedin_subtopology:
155 "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
156 unfolding closedin_def topspace_subtopology
157 apply (simp add: openin_subtopology)
160 apply (rule_tac x="topspace U - T" in exI)
163 lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
164 unfolding openin_subtopology
165 apply (rule iffI, clarify)
166 apply (frule openin_subset[of U]) apply blast
167 apply (rule exI[where x="topspace U"])
170 lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"
171 shows "subtopology U V = U"
174 {fix T assume T: "openin U T" "S = T \<inter> V"
175 from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
176 have "openin U S" unfolding eq using T by blast}
178 {assume S: "openin U S"
179 hence "\<exists>T. openin U T \<and> S = T \<inter> V"
180 using openin_subset[OF S] UV by auto}
181 ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
182 then show ?thesis unfolding topology_eq openin_subtopology by blast
186 lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
187 by (simp add: subtopology_superset)
189 lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
190 by (simp add: subtopology_superset)
192 subsection{* The universal Euclidean versions are what we use most of the time *}
195 euclidean :: "'a::topological_space topology" where
196 "euclidean = topology open"
198 lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
199 unfolding euclidean_def
200 apply (rule cong[where x=S and y=S])
201 apply (rule topology_inverse[symmetric])
202 apply (auto simp add: istopology_def)
203 by (auto simp add: mem_def subset_eq)
205 lemma topspace_euclidean: "topspace euclidean = UNIV"
206 apply (simp add: topspace_def)
208 by (auto simp add: open_openin[symmetric])
210 lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
211 by (simp add: topspace_euclidean topspace_subtopology)
213 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
214 by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
216 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
217 by (simp add: open_openin openin_subopen[symmetric])
219 subsection{* Open and closed balls. *}
222 ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
223 "ball x e = {y. dist x y < e}"
226 cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
227 "cball x e = {y. dist x y \<le> e}"
229 lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
230 lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
232 lemma mem_ball_0 [simp]:
233 fixes x :: "'a::real_normed_vector"
234 shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
235 by (simp add: dist_norm)
237 lemma mem_cball_0 [simp]:
238 fixes x :: "'a::real_normed_vector"
239 shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
240 by (simp add: dist_norm)
242 lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e" by simp
243 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
244 lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
245 lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
246 lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
247 by (simp add: expand_set_eq) arith
249 lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
250 by (simp add: expand_set_eq)
252 subsection{* Topological properties of open balls *}
254 lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
255 "(a::real) - b < 0 \<longleftrightarrow> a < b"
256 "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
257 lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
258 "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b" by arith+
260 lemma open_ball[intro, simp]: "open (ball x e)"
261 unfolding open_dist ball_def Collect_def Ball_def mem_def
262 unfolding dist_commute
264 apply (rule_tac x="e - dist xa x" in exI)
265 using dist_triangle_alt[where z=x]
266 apply (clarsimp simp add: diff_less_iff)
268 apply (erule_tac x="y" in allE)
269 apply (erule_tac x="xa" in allE)
272 lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
273 lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
274 unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
277 assumes "open S" "x\<in>S"
278 obtains e where "e>0" "ball x e \<subseteq> S"
279 using assms unfolding open_contains_ball by auto
281 lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
282 by (metis open_contains_ball subset_eq centre_in_ball)
284 lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
285 unfolding mem_ball expand_set_eq
286 apply (simp add: not_less)
287 by (metis zero_le_dist order_trans dist_self)
289 lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
291 subsection{* Basic "localization" results are handy for connectedness. *}
293 lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
294 by (auto simp add: openin_subtopology open_openin[symmetric])
296 lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
297 by (auto simp add: openin_open)
299 lemma open_openin_trans[trans]:
300 "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
301 by (metis Int_absorb1 openin_open_Int)
303 lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
304 by (auto simp add: openin_open)
306 lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
307 by (simp add: closedin_subtopology closed_closedin Int_ac)
309 lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
310 by (metis closedin_closed)
312 lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
313 apply (subgoal_tac "S \<inter> T = T" )
315 apply (frule closedin_closed_Int[of T S])
318 lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
319 by (auto simp add: closedin_closed)
321 lemma openin_euclidean_subtopology_iff:
322 fixes S U :: "'a::metric_space set"
323 shows "openin (subtopology euclidean U) S
324 \<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")
326 {assume ?lhs hence ?rhs unfolding openin_subtopology open_openin[symmetric]
327 by (simp add: open_dist) blast}
329 {assume SU: "S \<subseteq> U" and H: "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x' \<in> S"
330 from H obtain d where d: "\<And>x . x\<in> S \<Longrightarrow> d x > 0 \<and> (\<forall>x' \<in> U. dist x' x < d x \<longrightarrow> x' \<in> S)"
332 let ?T = "\<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
333 have oT: "open ?T" by auto
334 { fix x assume "x\<in>S"
335 hence "x \<in> \<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
336 apply simp apply(rule_tac x="ball x(d x)" in exI) apply auto
337 by (rule d [THEN conjunct1])
338 hence "x\<in> ?T \<inter> U" using SU and `x\<in>S` by auto }
340 { fix y assume "y\<in>?T"
341 then obtain B where "y\<in>B" "B\<in>{B. \<exists>x\<in>S. B = ball x (d x)}" by auto
342 then obtain x where "x\<in>S" and x:"y \<in> ball x (d x)" by auto
344 hence "y\<in>S" using d[OF `x\<in>S`] and x by(auto simp add: dist_commute) }
345 ultimately have "S = ?T \<inter> U" by blast
346 with oT have ?lhs unfolding openin_subtopology open_openin[symmetric] by blast}
347 ultimately show ?thesis by blast
350 text{* These "transitivity" results are handy too. *}
352 lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
353 \<Longrightarrow> openin (subtopology euclidean U) S"
354 unfolding open_openin openin_open by blast
356 lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
357 by (auto simp add: openin_open intro: openin_trans)
359 lemma closedin_trans[trans]:
360 "closedin (subtopology euclidean T) S \<Longrightarrow>
361 closedin (subtopology euclidean U) T
362 ==> closedin (subtopology euclidean U) S"
363 by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
365 lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
366 by (auto simp add: closedin_closed intro: closedin_trans)
368 subsection{* Connectedness *}
370 definition "connected S \<longleftrightarrow>
371 ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
372 \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
374 lemma connected_local:
375 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
376 openin (subtopology euclidean S) e1 \<and>
377 openin (subtopology euclidean S) e2 \<and>
378 S \<subseteq> e1 \<union> e2 \<and>
379 e1 \<inter> e2 = {} \<and>
382 unfolding connected_def openin_open by (safe, blast+)
385 fixes P :: "'a set \<Rightarrow> bool"
386 shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
388 {assume "?lhs" hence ?rhs by blast }
390 {fix S assume H: "P S"
391 have "S = - (- S)" by auto
392 with H have "P (- (- S))" by metis }
393 ultimately show ?thesis by metis
396 lemma connected_clopen: "connected S \<longleftrightarrow>
397 (\<forall>T. openin (subtopology euclidean S) T \<and>
398 closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
400 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> {})"
401 unfolding connected_def openin_open closedin_closed
402 apply (subst exists_diff) by blast
403 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> {})"
404 (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis
406 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'))"
407 (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
408 unfolding connected_def openin_open closedin_closed by auto
410 {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)"
412 then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
413 then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
414 then show ?thesis unfolding th0 th1 by simp
417 lemma connected_empty[simp, intro]: "connected {}"
418 by (simp add: connected_def)
420 subsection{* Hausdorff and other separation properties *}
423 assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
426 assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<notin> U \<and> x \<notin> V \<and> y \<in> V"
431 qed (fast dest: t1_space)
435 text {* T2 spaces are also known as Hausdorff spaces. *}
438 assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
443 qed (fast dest: hausdorff)
447 instance metric_space \<subseteq> t2_space
449 fix x y :: "'a::metric_space"
450 assume xy: "x \<noteq> y"
451 let ?U = "ball x (dist x y / 2)"
452 let ?V = "ball y (dist x y / 2)"
453 have th0: "\<And>d x y z. (d x z :: real) <= d x y + d y z \<Longrightarrow> d y z = d z y
454 ==> ~(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
455 have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
456 using dist_pos_lt[OF xy] th0[of dist,OF dist_triangle dist_commute]
457 by (auto simp add: expand_set_eq)
458 then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
463 fixes x y :: "'a::t2_space"
464 shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
465 using hausdorff[of x y] by blast
468 fixes x y :: "'a::t1_space"
469 shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in>U \<and> y\<notin> U \<and> x\<notin>V \<and> y\<in>V)"
470 using t1_space[of x y] by blast
473 fixes x y :: "'a::t0_space"
474 shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
475 using t0_space[of x y] by blast
477 subsection{* Limit points *}
480 islimpt:: "'a::topological_space \<Rightarrow> 'a set \<Rightarrow> bool"
481 (infixr "islimpt" 60) where
482 "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
485 assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
487 using assms unfolding islimpt_def by auto
490 assumes "x islimpt S" and "x \<in> T" and "open T"
491 obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
492 using assms unfolding islimpt_def by auto
494 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
496 lemma islimpt_approachable:
497 fixes x :: "'a::metric_space"
498 shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
499 unfolding islimpt_def
501 apply(erule_tac x="ball x e" in allE)
503 apply(rule_tac x=y in bexI)
504 apply (auto simp add: dist_commute)
505 apply (simp add: open_dist, drule (1) bspec)
506 apply (clarify, drule spec, drule (1) mp, auto)
509 lemma islimpt_approachable_le:
510 fixes x :: "'a::metric_space"
511 shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
512 unfolding islimpt_approachable
513 using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
516 class perfect_space =
517 (* FIXME: perfect_space should inherit from topological_space *)
518 assumes islimpt_UNIV [simp, intro]: "(x::'a::metric_space) islimpt UNIV"
520 lemma perfect_choose_dist:
521 fixes x :: "'a::perfect_space"
522 shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
523 using islimpt_UNIV [of x]
524 by (simp add: islimpt_approachable)
526 instance real :: perfect_space
528 apply (rule islimpt_approachable [THEN iffD2])
529 apply (clarify, rule_tac x="x + e/2" in bexI)
530 apply (auto simp add: dist_norm)
533 instance cart :: (perfect_space, finite) perfect_space
537 fix e :: real assume "0 < e"
538 def a \<equiv> "x $ undefined"
539 have "a islimpt UNIV" by (rule islimpt_UNIV)
540 with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"
541 unfolding islimpt_approachable by auto
542 def y \<equiv> "Cart_lambda ((Cart_nth x)(undefined := b))"
543 from `b \<noteq> a` have "y \<noteq> x"
544 unfolding a_def y_def by (simp add: Cart_eq)
545 from `dist b a < e` have "dist y x < e"
546 unfolding dist_vector_def a_def y_def
548 apply (rule le_less_trans [OF setL2_le_setsum [OF zero_le_dist]])
549 apply (subst setsum_diff1' [where a=undefined], simp, simp, simp)
551 from `y \<noteq> x` and `dist y x < e`
552 have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
554 then show "x islimpt UNIV" unfolding islimpt_approachable by blast
557 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
559 apply (subst open_subopen)
560 apply (simp add: islimpt_def subset_eq)
561 by (metis ComplE ComplI insertCI insert_absorb mem_def)
563 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
564 unfolding islimpt_def by auto
566 lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
568 let ?U = "UNIV :: 'n set"
569 let ?O = "{x::real^'n. \<forall>i. x$i\<ge>0}"
570 {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"
572 from xi have th0: "-x$i > 0" by arith
573 from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x $ i" by blast
574 have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith
575 have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith
576 have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using x'(1) xi
577 apply (simp only: vector_component)
579 have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using component_le_norm[of "x'-x" i]
580 apply (simp add: dist_norm) by norm
581 from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) }
582 then show ?thesis unfolding closed_limpt islimpt_approachable
583 unfolding not_le[symmetric] by blast
586 lemma finite_set_avoid:
587 fixes a :: "'a::metric_space"
588 assumes fS: "finite S" shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
589 proof(induct rule: finite_induct[OF fS])
590 case 1 thus ?case apply auto by ferrack
593 from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
594 {assume "x = a" hence ?case using d by auto }
596 {assume xa: "x\<noteq>a"
597 let ?d = "min d (dist a x)"
598 have dp: "?d > 0" using xa d(1) using dist_nz by auto
599 from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
600 with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
601 ultimately show ?case by blast
604 lemma islimpt_finite:
605 fixes S :: "'a::metric_space set"
606 assumes fS: "finite S" shows "\<not> a islimpt S"
607 unfolding islimpt_approachable
608 using finite_set_avoid[OF fS, of a] by (metis dist_commute not_le)
610 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
613 apply (metis Un_upper1 Un_upper2 islimpt_subset)
614 unfolding islimpt_def
615 apply (rule ccontr, clarsimp, rename_tac A B)
616 apply (drule_tac x="A \<inter> B" in spec)
617 apply (auto simp add: open_Int)
620 lemma discrete_imp_closed:
621 fixes S :: "'a::metric_space set"
622 assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
625 {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
626 from e have e2: "e/2 > 0" by arith
627 from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
628 let ?m = "min (e/2) (dist x y) "
629 from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
630 from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
631 have th: "dist z y < e" using z y
632 by (intro dist_triangle_lt [where z=x], simp)
633 from d[rule_format, OF y(1) z(1) th] y z
634 have False by (auto simp add: dist_commute)}
635 then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
638 subsection{* Interior of a Set *}
639 definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
641 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
642 apply (simp add: expand_set_eq interior_def)
643 apply (subst (2) open_subopen) by (safe, blast+)
645 lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
647 lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
649 lemma open_interior[simp, intro]: "open(interior S)"
650 apply (simp add: interior_def)
651 apply (subst open_subopen) by blast
653 lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
654 lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
655 lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
656 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
657 lemma interior_unique: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> (\<forall>T'. T' \<subseteq> S \<and> open T' \<longrightarrow> T' \<subseteq> T) \<Longrightarrow> interior S = T"
658 by (metis equalityI interior_maximal interior_subset open_interior)
659 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
660 apply (simp add: interior_def)
661 by (metis open_contains_ball centre_in_ball open_ball subset_trans)
663 lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
664 by (metis interior_maximal interior_subset subset_trans)
666 lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
667 apply (rule equalityI, simp)
668 apply (metis Int_lower1 Int_lower2 subset_interior)
669 by (metis Int_mono interior_subset open_Int open_interior open_subset_interior)
671 lemma interior_limit_point [intro]:
672 fixes x :: "'a::perfect_space"
673 assumes x: "x \<in> interior S" shows "x islimpt S"
675 from x obtain e where e: "e>0" "\<forall>x'. dist x x' < e \<longrightarrow> x' \<in> S"
676 unfolding mem_interior subset_eq Ball_def mem_ball by blast
678 fix d::real assume d: "d>0"
680 have mde2: "0 < ?m" using e(1) d(1) by simp
681 from perfect_choose_dist [OF mde2, of x]
682 obtain y where "y \<noteq> x" and "dist y x < ?m" by blast
683 then have "dist y x < e" "dist y x < d" by simp_all
684 from `dist y x < e` e(2) have "y \<in> S" by (simp add: dist_commute)
685 have "\<exists>x'\<in>S. x'\<noteq> x \<and> dist x' x < d"
686 using `y \<in> S` `y \<noteq> x` `dist y x < d` by fast
688 then show ?thesis unfolding islimpt_approachable by blast
691 lemma interior_closed_Un_empty_interior:
692 assumes cS: "closed S" and iT: "interior T = {}"
693 shows "interior(S \<union> T) = interior S"
695 show "interior S \<subseteq> interior (S\<union>T)"
696 by (rule subset_interior, blast)
698 show "interior (S \<union> T) \<subseteq> interior S"
700 fix x assume "x \<in> interior (S \<union> T)"
701 then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T"
702 unfolding interior_def by fast
703 show "x \<in> interior S"
705 assume "x \<notin> interior S"
706 with `x \<in> R` `open R` obtain y where "y \<in> R - S"
707 unfolding interior_def expand_set_eq by fast
708 from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
709 from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
710 from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
711 show "False" unfolding interior_def by fast
717 subsection{* Closure of a Set *}
719 definition "closure S = S \<union> {x | x. x islimpt S}"
721 lemma closure_interior: "closure S = - interior (- S)"
724 have "x\<in>- interior (- S) \<longleftrightarrow> x \<in> closure S" (is "?lhs = ?rhs")
726 let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> - S)"
728 hence *:"\<not> ?exT x"
729 unfolding interior_def
731 { assume "\<not> ?rhs"
733 unfolding closure_def islimpt_def
739 assume "?rhs" thus "?lhs"
740 unfolding closure_def interior_def islimpt_def
748 lemma interior_closure: "interior S = - (closure (- S))"
751 have "x \<in> interior S \<longleftrightarrow> x \<in> - (closure (- S))"
752 unfolding interior_def closure_def islimpt_def
759 lemma closed_closure[simp, intro]: "closed (closure S)"
761 have "closed (- interior (-S))" by blast
762 thus ?thesis using closure_interior[of S] by simp
765 lemma closure_hull: "closure S = closed hull S"
767 have "S \<subseteq> closure S"
768 unfolding closure_def
771 have "closed (closure S)"
772 using closed_closure[of S]
776 assume *:"S \<subseteq> t" "closed t"
779 hence "x islimpt t" using *(1)
780 using islimpt_subset[of x, of S, of t]
783 with * have "closure S \<subseteq> t"
784 unfolding closure_def
785 using closed_limpt[of t]
788 ultimately show ?thesis
789 using hull_unique[of S, of "closure S", of closed]
794 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
795 unfolding closure_hull
796 using hull_eq[of closed, unfolded mem_def, OF closed_Inter, of S]
797 by (metis mem_def subset_eq)
799 lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
800 using closure_eq[of S]
803 lemma closure_closure[simp]: "closure (closure S) = closure S"
804 unfolding closure_hull
805 using hull_hull[of closed S]
808 lemma closure_subset: "S \<subseteq> closure S"
809 unfolding closure_hull
810 using hull_subset[of S closed]
813 lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
814 unfolding closure_hull
815 using hull_mono[of S T closed]
818 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
819 using hull_minimal[of S T closed]
820 unfolding closure_hull mem_def
823 lemma closure_unique: "S \<subseteq> T \<and> closed T \<and> (\<forall> T'. S \<subseteq> T' \<and> closed T' \<longrightarrow> T \<subseteq> T') \<Longrightarrow> closure S = T"
824 using hull_unique[of S T closed]
825 unfolding closure_hull mem_def
828 lemma closure_empty[simp]: "closure {} = {}"
829 using closed_empty closure_closed[of "{}"]
832 lemma closure_univ[simp]: "closure UNIV = UNIV"
833 using closure_closed[of UNIV]
836 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
837 using closure_empty closure_subset[of S]
840 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
841 using closure_eq[of S] closure_subset[of S]
844 lemma open_inter_closure_eq_empty:
845 "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
846 using open_subset_interior[of S "- T"]
847 using interior_subset[of "- T"]
848 unfolding closure_interior
851 lemma open_inter_closure_subset:
852 "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
855 assume as: "open S" "x \<in> S \<inter> closure T"
856 { assume *:"x islimpt T"
857 have "x islimpt (S \<inter> T)"
858 proof (rule islimptI)
860 assume "x \<in> A" "open A"
861 with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
862 by (simp_all add: open_Int)
863 with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
865 hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
867 thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
870 then show "x \<in> closure (S \<inter> T)" using as
871 unfolding closure_def
875 lemma closure_complement: "closure(- S) = - interior(S)"
880 unfolding closure_interior
884 lemma interior_complement: "interior(- S) = - closure(S)"
885 unfolding closure_interior
888 subsection{* Frontier (aka boundary) *}
890 definition "frontier S = closure S - interior S"
892 lemma frontier_closed: "closed(frontier S)"
893 by (simp add: frontier_def closed_Diff)
895 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
896 by (auto simp add: frontier_def interior_closure)
898 lemma frontier_straddle:
899 fixes a :: "'a::metric_space"
900 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))" (is "?lhs \<longleftrightarrow> ?rhs")
905 let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
907 have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
908 moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
909 unfolding frontier_closures closure_def islimpt_def using `e>0`
910 by (auto, erule_tac x="ball a e" in allE, auto)
911 ultimately have ?rhse by auto
914 { assume "a\<notin>S"
915 hence ?rhse using `?lhs`
916 unfolding frontier_closures closure_def islimpt_def
917 using open_ball[of a e] `e > 0`
918 by simp (metis centre_in_ball mem_ball open_ball)
920 ultimately have ?rhse by auto
926 { fix T assume "a\<notin>S" and
927 as:"\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)" "a \<notin> S" "a \<in> T" "open T"
928 from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
929 then obtain e where "e>0" "ball a e \<subseteq> T" by auto
930 then obtain y where y:"y\<in>S" "dist a y < e" using as(1) by auto
931 have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
932 using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
934 hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
936 { fix T assume "a \<in> T" "open T" "a\<in>S"
937 then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
938 obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
939 hence "\<exists>y\<in>- S. y \<in> T \<and> y \<noteq> a" using balle `a\<in>S` unfolding ball_def by (rule_tac x=x in bexI)auto
941 hence "a islimpt (- S) \<or> a\<notin>S" unfolding islimpt_def by auto
942 ultimately show ?lhs unfolding frontier_closures using closure_def[of "- S"] by auto
945 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
946 by (metis frontier_def closure_closed Diff_subset)
948 lemma frontier_empty[simp]: "frontier {} = {}"
949 by (simp add: frontier_def closure_empty)
951 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
953 { assume "frontier S \<subseteq> S"
954 hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
955 hence "closed S" using closure_subset_eq by auto
957 thus ?thesis using frontier_subset_closed[of S] by auto
960 lemma frontier_complement: "frontier(- S) = frontier S"
961 by (auto simp add: frontier_def closure_complement interior_complement)
963 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
964 using frontier_complement frontier_subset_eq[of "- S"]
965 unfolding open_closed by auto
967 subsection{* Common nets and The "within" modifier for nets. *}
970 at_infinity :: "'a::real_normed_vector net" where
971 "at_infinity = Abs_net (range (\<lambda>r. {x. r \<le> norm x}))"
974 indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a net" (infixr "indirection" 70) where
975 "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
977 text{* Prove That They are all nets. *}
979 lemma Rep_net_at_infinity:
980 "Rep_net at_infinity = range (\<lambda>r. {x. r \<le> norm x})"
981 unfolding at_infinity_def
982 apply (rule Abs_net_inverse')
983 apply (rule image_nonempty, simp)
984 apply (clarsimp, rename_tac r s)
985 apply (rule_tac x="max r s" in exI, auto)
988 lemma within_UNIV: "net within UNIV = net"
989 by (simp add: Rep_net_inject [symmetric] Rep_net_within)
991 subsection{* Identify Trivial limits, where we can't approach arbitrarily closely. *}
994 trivial_limit :: "'a net \<Rightarrow> bool" where
995 "trivial_limit net \<longleftrightarrow> {} \<in> Rep_net net"
997 lemma trivial_limit_within:
998 shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
1000 assume "trivial_limit (at a within S)"
1001 thus "\<not> a islimpt S"
1002 unfolding trivial_limit_def
1003 unfolding Rep_net_within Rep_net_at
1004 unfolding islimpt_def
1005 apply (clarsimp simp add: expand_set_eq)
1006 apply (rename_tac T, rule_tac x=T in exI)
1007 apply (clarsimp, drule_tac x=y in spec, simp)
1010 assume "\<not> a islimpt S"
1011 thus "trivial_limit (at a within S)"
1012 unfolding trivial_limit_def
1013 unfolding Rep_net_within Rep_net_at
1014 unfolding islimpt_def
1015 apply (clarsimp simp add: image_image)
1016 apply (rule_tac x=T in image_eqI)
1017 apply (auto simp add: expand_set_eq)
1021 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
1022 using trivial_limit_within [of a UNIV]
1023 by (simp add: within_UNIV)
1025 lemma trivial_limit_at:
1026 fixes a :: "'a::perfect_space"
1027 shows "\<not> trivial_limit (at a)"
1028 by (simp add: trivial_limit_at_iff)
1030 lemma trivial_limit_at_infinity:
1031 "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,zero_neq_one}) net)"
1032 (* FIXME: find a more appropriate type class *)
1033 unfolding trivial_limit_def Rep_net_at_infinity
1034 apply (clarsimp simp add: expand_set_eq)
1035 apply (drule_tac x="scaleR r (sgn 1)" in spec)
1036 apply (simp add: norm_sgn)
1039 lemma trivial_limit_sequentially[intro]: "\<not> trivial_limit sequentially"
1040 by (auto simp add: trivial_limit_def Rep_net_sequentially)
1042 subsection{* Some property holds "sufficiently close" to the limit point. *}
1044 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
1045 "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
1046 unfolding eventually_at dist_nz by auto
1048 lemma eventually_at_infinity:
1049 "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
1050 unfolding eventually_def Rep_net_at_infinity by auto
1052 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
1053 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
1054 unfolding eventually_within eventually_at dist_nz by auto
1056 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
1057 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
1058 unfolding eventually_within
1059 by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl)
1061 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
1062 unfolding eventually_def trivial_limit_def
1063 using Rep_net_nonempty [of net] by auto
1065 lemma always_eventually: "(\<forall>x. P x) ==> eventually P net"
1066 unfolding eventually_def trivial_limit_def
1067 using Rep_net_nonempty [of net] by auto
1069 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
1070 unfolding trivial_limit_def eventually_def by auto
1072 lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
1073 unfolding trivial_limit_def eventually_def by auto
1075 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
1076 apply (safe elim!: trivial_limit_eventually)
1077 apply (simp add: eventually_False [symmetric])
1080 text{* Combining theorems for "eventually" *}
1082 lemma eventually_conjI:
1083 "\<lbrakk>eventually (\<lambda>x. P x) net; eventually (\<lambda>x. Q x) net\<rbrakk>
1084 \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) net"
1085 by (rule eventually_conj)
1087 lemma eventually_rev_mono:
1088 "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
1089 using eventually_mono [of P Q] by fast
1091 lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
1092 by (auto intro!: eventually_conjI elim: eventually_rev_mono)
1094 lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
1095 by (auto simp add: eventually_False)
1097 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
1098 by (simp add: eventually_False)
1100 subsection{* Limits, defined as vacuously true when the limit is trivial. *}
1102 text{* Notation Lim to avoid collition with lim defined in analysis *}
1104 Lim :: "'a net \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b" where
1105 "Lim net f = (THE l. (f ---> l) net)"
1108 "(f ---> l) net \<longleftrightarrow>
1109 trivial_limit net \<or>
1110 (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
1111 unfolding tendsto_iff trivial_limit_eq by auto
1114 text{* Show that they yield usual definitions in the various cases. *}
1116 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
1117 (\<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)"
1118 by (auto simp add: tendsto_iff eventually_within_le)
1120 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
1121 (\<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)"
1122 by (auto simp add: tendsto_iff eventually_within)
1124 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
1125 (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
1126 by (auto simp add: tendsto_iff eventually_at)
1128 lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
1129 unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
1131 lemma Lim_at_infinity:
1132 "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
1133 by (auto simp add: tendsto_iff eventually_at_infinity)
1135 lemma Lim_sequentially:
1136 "(S ---> l) sequentially \<longleftrightarrow>
1137 (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
1138 by (auto simp add: tendsto_iff eventually_sequentially)
1140 lemma Lim_sequentially_iff_LIMSEQ: "(S ---> l) sequentially \<longleftrightarrow> S ----> l"
1141 unfolding Lim_sequentially LIMSEQ_def ..
1143 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
1144 by (rule topological_tendstoI, auto elim: eventually_rev_mono)
1146 text{* The expected monotonicity property. *}
1148 lemma Lim_within_empty: "(f ---> l) (net within {})"
1149 unfolding tendsto_def Limits.eventually_within by simp
1151 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
1152 unfolding tendsto_def Limits.eventually_within
1153 by (auto elim!: eventually_elim1)
1155 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
1156 shows "(f ---> l) (net within (S \<union> T))"
1157 using assms unfolding tendsto_def Limits.eventually_within
1159 apply (drule spec, drule (1) mp, drule (1) mp)
1160 apply (drule spec, drule (1) mp, drule (1) mp)
1161 apply (auto elim: eventually_elim2)
1165 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow> S \<union> T = UNIV
1167 by (metis Lim_Un within_UNIV)
1169 text{* Interrelations between restricted and unrestricted limits. *}
1171 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
1173 unfolding tendsto_def Limits.eventually_within
1174 apply (clarify, drule spec, drule (1) mp, drule (1) mp)
1175 by (auto elim!: eventually_elim1)
1177 lemma Lim_within_open:
1178 fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
1179 assumes"a \<in> S" "open S"
1180 shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)" (is "?lhs \<longleftrightarrow> ?rhs")
1183 { fix A assume "open A" "l \<in> A"
1184 with `?lhs` have "eventually (\<lambda>x. f x \<in> A) (at a within S)"
1185 by (rule topological_tendstoD)
1186 hence "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x \<in> A) (at a)"
1187 unfolding Limits.eventually_within .
1188 then obtain T where "open T" "a \<in> T" "\<forall>x\<in>T. x \<noteq> a \<longrightarrow> x \<in> S \<longrightarrow> f x \<in> A"
1189 unfolding eventually_at_topological by fast
1190 hence "open (T \<inter> S)" "a \<in> T \<inter> S" "\<forall>x\<in>(T \<inter> S). x \<noteq> a \<longrightarrow> f x \<in> A"
1192 hence "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> f x \<in> A)"
1194 hence "eventually (\<lambda>x. f x \<in> A) (at a)"
1195 unfolding eventually_at_topological .
1197 thus ?rhs by (rule topological_tendstoI)
1200 thus ?lhs by (rule Lim_at_within)
1203 text{* Another limit point characterization. *}
1205 lemma islimpt_sequential:
1206 fixes x :: "'a::metric_space" (* FIXME: generalize to topological_space *)
1207 shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
1211 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"
1212 unfolding islimpt_approachable using choice[of "\<lambda>e y. e>0 \<longrightarrow> y\<in>S \<and> y\<noteq>x \<and> dist y x < e"] by auto
1214 have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
1217 { fix e::real assume "e>0"
1218 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
1219 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
1220 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)
1221 moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
1222 ultimately have "\<exists>N::nat. \<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < e" apply(rule_tac x=N in exI) apply auto apply(erule_tac x=n in allE)+ by auto
1224 hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
1225 unfolding Lim_sequentially using f by auto
1226 ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
1229 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 Lim_sequentially by auto
1230 { fix e::real assume "e>0"
1231 then obtain N where "dist (f N) x < e" using f(2) by auto
1232 moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
1233 ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
1235 thus ?lhs unfolding islimpt_approachable by auto
1238 text{* Basic arithmetical combining theorems for limits. *}
1241 assumes "(f ---> l) net" "bounded_linear h"
1242 shows "((\<lambda>x. h (f x)) ---> h l) net"
1243 using `bounded_linear h` `(f ---> l) net`
1244 by (rule bounded_linear.tendsto)
1246 lemma Lim_ident_at: "((\<lambda>x. x) ---> a) (at a)"
1247 unfolding tendsto_def Limits.eventually_at_topological by fast
1249 lemma Lim_const[intro]: "((\<lambda>x. a) ---> a) net" by (rule tendsto_const)
1251 lemma Lim_cmul[intro]:
1252 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1253 shows "(f ---> l) net ==> ((\<lambda>x. c *\<^sub>R f x) ---> c *\<^sub>R l) net"
1254 by (intro tendsto_intros)
1257 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1258 shows "(f ---> l) net ==> ((\<lambda>x. -(f x)) ---> -l) net"
1259 by (rule tendsto_minus)
1261 lemma Lim_add: fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" shows
1262 "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) + g(x)) ---> l + m) net"
1263 by (rule tendsto_add)
1266 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1267 shows "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) ---> l - m) net"
1268 by (rule tendsto_diff)
1271 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1272 shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net" by (simp add: Lim dist_norm)
1274 lemma Lim_null_norm:
1275 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1276 shows "(f ---> 0) net \<longleftrightarrow> ((\<lambda>x. norm(f x)) ---> 0) net"
1277 by (simp add: Lim dist_norm)
1279 lemma Lim_null_comparison:
1280 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1281 assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
1282 shows "(f ---> 0) net"
1283 proof(simp add: tendsto_iff, rule+)
1284 fix e::real assume "0<e"
1286 assume "norm (f x) \<le> g x" "dist (g x) 0 < e"
1287 hence "dist (f x) 0 < e" by (simp add: dist_norm)
1289 thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
1290 using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
1291 using eventually_mono[of "(\<lambda>x. norm (f x) \<le> g x \<and> dist (g x) 0 < e)" "(\<lambda>x. dist (f x) 0 < e)" net]
1292 using assms `e>0` unfolding tendsto_iff by auto
1295 lemma Lim_component:
1296 fixes f :: "'a \<Rightarrow> 'b::metric_space ^ 'n"
1297 shows "(f ---> l) net \<Longrightarrow> ((\<lambda>a. f a $i) ---> l$i) net"
1298 unfolding tendsto_iff
1300 apply (drule spec, drule (1) mp)
1301 apply (erule eventually_elim1)
1302 apply (erule le_less_trans [OF dist_nth_le])
1305 lemma Lim_transform_bound:
1306 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1307 fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
1308 assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net" "(g ---> 0) net"
1309 shows "(f ---> 0) net"
1310 proof (rule tendstoI)
1311 fix e::real assume "e>0"
1313 assume "norm (f x) \<le> norm (g x)" "dist (g x) 0 < e"
1314 hence "dist (f x) 0 < e" by (simp add: dist_norm)}
1315 thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
1316 using eventually_and[of "\<lambda>x. norm (f x) \<le> norm (g x)" "\<lambda>x. dist (g x) 0 < e" net]
1317 using eventually_mono[of "\<lambda>x. norm (f x) \<le> norm (g x) \<and> dist (g x) 0 < e" "\<lambda>x. dist (f x) 0 < e" net]
1318 using assms `e>0` unfolding tendsto_iff by blast
1321 text{* Deducing things about the limit from the elements. *}
1323 lemma Lim_in_closed_set:
1324 assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
1327 assume "l \<notin> S"
1328 with `closed S` have "open (- S)" "l \<in> - S"
1329 by (simp_all add: open_Compl)
1330 with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
1331 by (rule topological_tendstoD)
1332 with assms(2) have "eventually (\<lambda>x. False) net"
1333 by (rule eventually_elim2) simp
1334 with assms(3) show "False"
1335 by (simp add: eventually_False)
1338 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
1340 lemma Lim_dist_ubound:
1341 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
1342 shows "dist a l <= e"
1344 assume "\<not> dist a l \<le> e"
1345 then have "0 < dist a l - e" by simp
1346 with assms(2) have "eventually (\<lambda>x. dist (f x) l < dist a l - e) net"
1348 with assms(3) have "eventually (\<lambda>x. dist a (f x) \<le> e \<and> dist (f x) l < dist a l - e) net"
1349 by (rule eventually_conjI)
1350 then obtain w where "dist a (f w) \<le> e" "dist (f w) l < dist a l - e"
1351 using assms(1) eventually_happens by auto
1352 hence "dist a (f w) + dist (f w) l < e + (dist a l - e)"
1353 by (rule add_le_less_mono)
1354 hence "dist a (f w) + dist (f w) l < dist a l"
1356 also have "\<dots> \<le> dist a (f w) + dist (f w) l"
1357 by (rule dist_triangle)
1358 finally show False by simp
1361 lemma Lim_norm_ubound:
1362 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1363 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
1364 shows "norm(l) <= e"
1366 assume "\<not> norm l \<le> e"
1367 then have "0 < norm l - e" by simp
1368 with assms(2) have "eventually (\<lambda>x. dist (f x) l < norm l - e) net"
1370 with assms(3) have "eventually (\<lambda>x. norm (f x) \<le> e \<and> dist (f x) l < norm l - e) net"
1371 by (rule eventually_conjI)
1372 then obtain w where "norm (f w) \<le> e" "dist (f w) l < norm l - e"
1373 using assms(1) eventually_happens by auto
1374 hence "norm (f w - l) < norm l - e" "norm (f w) \<le> e" by (simp_all add: dist_norm)
1375 hence "norm (f w - l) + norm (f w) < norm l" by simp
1376 hence "norm (f w - l - f w) < norm l" by (rule le_less_trans [OF norm_triangle_ineq4])
1377 thus False using `\<not> norm l \<le> e` by simp
1380 lemma Lim_norm_lbound:
1381 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1382 assumes "\<not> (trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. e <= norm(f x)) net"
1383 shows "e \<le> norm l"
1385 assume "\<not> e \<le> norm l"
1386 then have "0 < e - norm l" by simp
1387 with assms(2) have "eventually (\<lambda>x. dist (f x) l < e - norm l) net"
1389 with assms(3) have "eventually (\<lambda>x. e \<le> norm (f x) \<and> dist (f x) l < e - norm l) net"
1390 by (rule eventually_conjI)
1391 then obtain w where "e \<le> norm (f w)" "dist (f w) l < e - norm l"
1392 using assms(1) eventually_happens by auto
1393 hence "norm (f w - l) + norm l < e" "e \<le> norm (f w)" by (simp_all add: dist_norm)
1394 hence "norm (f w - l) + norm l < norm (f w)" by (rule less_le_trans)
1395 hence "norm (f w - l + l) < norm (f w)" by (rule le_less_trans [OF norm_triangle_ineq])
1399 text{* Uniqueness of the limit, when nontrivial. *}
1402 fixes f :: "'a \<Rightarrow> 'b::t2_space"
1403 assumes "\<not> trivial_limit net" "(f ---> l) net" "(f ---> l') net"
1406 assume "l \<noteq> l'"
1407 obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
1408 using hausdorff [OF `l \<noteq> l'`] by fast
1409 have "eventually (\<lambda>x. f x \<in> U) net"
1410 using `(f ---> l) net` `open U` `l \<in> U` by (rule topological_tendstoD)
1412 have "eventually (\<lambda>x. f x \<in> V) net"
1413 using `(f ---> l') net` `open V` `l' \<in> V` by (rule topological_tendstoD)
1415 have "eventually (\<lambda>x. False) net"
1416 proof (rule eventually_elim2)
1418 assume "f x \<in> U" "f x \<in> V"
1419 hence "f x \<in> U \<inter> V" by simp
1420 with `U \<inter> V = {}` show "False" by simp
1422 with `\<not> trivial_limit net` show "False"
1423 by (simp add: eventually_False)
1427 fixes f :: "'a \<Rightarrow> 'b::t2_space"
1428 shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
1429 unfolding Lim_def using Lim_unique[of net f] by auto
1431 text{* Limit under bilinear function *}
1434 assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
1435 shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
1436 using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
1437 by (rule bounded_bilinear.tendsto)
1439 text{* These are special for limits out of the same vector space. *}
1441 lemma Lim_within_id: "(id ---> a) (at a within s)"
1442 unfolding tendsto_def Limits.eventually_within eventually_at_topological
1445 lemma Lim_at_id: "(id ---> a) (at a)"
1446 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
1449 fixes a :: "'a::real_normed_vector"
1450 fixes l :: "'b::topological_space"
1451 shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
1454 { fix S assume "open S" "l \<in> S"
1455 with `?lhs` have "eventually (\<lambda>x. f x \<in> S) (at a)"
1456 by (rule topological_tendstoD)
1457 then obtain d where d: "d>0" "\<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S"
1458 unfolding Limits.eventually_at by fast
1459 { fix x::"'a" assume "x \<noteq> 0 \<and> dist x 0 < d"
1460 hence "f (a + x) \<in> S" using d
1461 apply(erule_tac x="x+a" in allE)
1462 by(auto simp add: comm_monoid_add.mult_commute dist_norm dist_commute)
1464 hence "\<exists>d>0. \<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
1466 hence "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
1467 unfolding Limits.eventually_at .
1469 thus "?rhs" by (rule topological_tendstoI)
1472 { fix S assume "open S" "l \<in> S"
1473 with `?rhs` have "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
1474 by (rule topological_tendstoD)
1475 then obtain d where d: "d>0" "\<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
1476 unfolding Limits.eventually_at by fast
1477 { fix x::"'a" assume "x \<noteq> a \<and> dist x a < d"
1478 hence "f x \<in> S" using d apply(erule_tac x="x-a" in allE)
1479 by(auto simp add: comm_monoid_add.mult_commute dist_norm dist_commute)
1481 hence "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S" using d(1) by auto
1482 hence "eventually (\<lambda>x. f x \<in> S) (at a)" unfolding Limits.eventually_at .
1484 thus "?lhs" by (rule topological_tendstoI)
1487 text{* It's also sometimes useful to extract the limit point from the net. *}
1490 netlimit :: "'a::t2_space net \<Rightarrow> 'a" where
1491 "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
1493 lemma netlimit_within:
1494 assumes "\<not> trivial_limit (at a within S)"
1495 shows "netlimit (at a within S) = a"
1496 unfolding netlimit_def
1497 apply (rule some_equality)
1498 apply (rule Lim_at_within)
1499 apply (rule Lim_ident_at)
1500 apply (erule Lim_unique [OF assms])
1501 apply (rule Lim_at_within)
1502 apply (rule Lim_ident_at)
1506 fixes a :: "'a::perfect_space"
1507 shows "netlimit (at a) = a"
1508 apply (subst within_UNIV[symmetric])
1509 using netlimit_within[of a UNIV]
1510 by (simp add: trivial_limit_at within_UNIV)
1512 text{* Transformation of limit. *}
1514 lemma Lim_transform:
1515 fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
1516 assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
1517 shows "(g ---> l) net"
1519 from assms have "((\<lambda>x. f x - g x - f x) ---> 0 - l) net" using Lim_sub[of "\<lambda>x. f x - g x" 0 net f l] by auto
1520 thus "?thesis" using Lim_neg [of "\<lambda> x. - g x" "-l" net] by auto
1523 lemma Lim_transform_eventually:
1524 "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net ==> (g ---> l) net"
1525 apply (rule topological_tendstoI)
1526 apply (drule (2) topological_tendstoD)
1527 apply (erule (1) eventually_elim2, simp)
1530 lemma Lim_transform_within:
1531 fixes l :: "'b::metric_space" (* TODO: generalize *)
1532 assumes "0 < d" "(\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x')"
1533 "(f ---> l) (at x within S)"
1534 shows "(g ---> l) (at x within S)"
1535 using assms(1,3) unfolding Lim_within
1537 apply (clarify, rename_tac e)
1538 apply (drule_tac x=e in spec, clarsimp, rename_tac r)
1539 apply (rule_tac x="min d r" in exI, clarsimp, rename_tac y)
1540 apply (drule_tac x=y in bspec, assumption, clarsimp)
1541 apply (simp add: assms(2))
1544 lemma Lim_transform_at:
1545 fixes l :: "'b::metric_space" (* TODO: generalize *)
1546 shows "0 < d \<Longrightarrow> (\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x') \<Longrightarrow>
1547 (f ---> l) (at x) ==> (g ---> l) (at x)"
1548 apply (subst within_UNIV[symmetric])
1549 using Lim_transform_within[of d UNIV x f g l]
1550 by (auto simp add: within_UNIV)
1552 text{* Common case assuming being away from some crucial point like 0. *}
1554 lemma Lim_transform_away_within:
1555 fixes a b :: "'a::metric_space"
1556 fixes l :: "'b::metric_space" (* TODO: generalize *)
1557 assumes "a\<noteq>b" "\<forall>x\<in> S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1558 and "(f ---> l) (at a within S)"
1559 shows "(g ---> l) (at a within S)"
1561 have "\<forall>x'\<in>S. 0 < dist x' a \<and> dist x' a < dist a b \<longrightarrow> f x' = g x'" using assms(2)
1562 apply auto apply(erule_tac x=x' in ballE) by (auto simp add: dist_commute)
1563 thus ?thesis using Lim_transform_within[of "dist a b" S a f g l] using assms(1,3) unfolding dist_nz by auto
1566 lemma Lim_transform_away_at:
1567 fixes a b :: "'a::metric_space"
1568 fixes l :: "'b::metric_space" (* TODO: generalize *)
1569 assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1570 and fl: "(f ---> l) (at a)"
1571 shows "(g ---> l) (at a)"
1572 using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
1573 by (auto simp add: within_UNIV)
1575 text{* Alternatively, within an open set. *}
1577 lemma Lim_transform_within_open:
1578 fixes a :: "'a::metric_space"
1579 fixes l :: "'b::metric_space" (* TODO: generalize *)
1580 assumes "open S" "a \<in> S" "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x" "(f ---> l) (at a)"
1581 shows "(g ---> l) (at a)"
1583 from assms(1,2) obtain e::real where "e>0" and e:"ball a e \<subseteq> S" unfolding open_contains_ball by auto
1584 hence "\<forall>x'. 0 < dist x' a \<and> dist x' a < e \<longrightarrow> f x' = g x'" using assms(3)
1585 unfolding ball_def subset_eq apply auto apply(erule_tac x=x' in allE) apply(erule_tac x=x' in ballE) by(auto simp add: dist_commute)
1586 thus ?thesis using Lim_transform_at[of e a f g l] `e>0` assms(4) by auto
1589 text{* A congruence rule allowing us to transform limits assuming not at point. *}
1591 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
1593 lemma Lim_cong_within[cong add]:
1594 fixes a :: "'a::metric_space"
1595 fixes l :: "'b::metric_space" (* TODO: generalize *)
1596 shows "(\<And>x. x \<noteq> a \<Longrightarrow> f x = g x) ==> ((\<lambda>x. f x) ---> l) (at a within S) \<longleftrightarrow> ((g ---> l) (at a within S))"
1597 by (simp add: Lim_within dist_nz[symmetric])
1599 lemma Lim_cong_at[cong add]:
1600 fixes a :: "'a::metric_space"
1601 fixes l :: "'b::metric_space" (* TODO: generalize *)
1602 shows "(\<And>x. x \<noteq> a ==> f x = g x) ==> (((\<lambda>x. f x) ---> l) (at a) \<longleftrightarrow> ((g ---> l) (at a)))"
1603 by (simp add: Lim_at dist_nz[symmetric])
1605 text{* Useful lemmas on closure and set of possible sequential limits.*}
1607 lemma closure_sequential:
1608 fixes l :: "'a::metric_space" (* TODO: generalize *)
1609 shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
1611 assume "?lhs" moreover
1612 { assume "l \<in> S"
1613 hence "?rhs" using Lim_const[of l sequentially] by auto
1615 { assume "l islimpt S"
1616 hence "?rhs" unfolding islimpt_sequential by auto
1618 show "?rhs" unfolding closure_def by auto
1621 thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
1624 lemma closed_sequential_limits:
1625 fixes S :: "'a::metric_space set"
1626 shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
1627 unfolding closed_limpt
1628 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]
1631 lemma closure_approachable:
1632 fixes S :: "'a::metric_space set"
1633 shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
1634 apply (auto simp add: closure_def islimpt_approachable)
1635 by (metis dist_self)
1637 lemma closed_approachable:
1638 fixes S :: "'a::metric_space set"
1639 shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
1640 by (metis closure_closed closure_approachable)
1642 text{* Some other lemmas about sequences. *}
1645 fixes l :: "'a::metric_space" (* TODO: generalize *)
1646 shows "(f ---> l) sequentially ==> ((\<lambda>i. f( i + k)) ---> l) sequentially"
1647 apply (auto simp add: Lim_sequentially)
1648 by (metis trans_le_add1 )
1650 lemma seq_offset_neg:
1651 "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
1652 apply (rule topological_tendstoI)
1653 apply (drule (2) topological_tendstoD)
1654 apply (simp only: eventually_sequentially)
1655 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
1659 lemma seq_offset_rev:
1660 "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
1661 apply (rule topological_tendstoI)
1662 apply (drule (2) topological_tendstoD)
1663 apply (simp only: eventually_sequentially)
1664 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
1667 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
1669 { fix e::real assume "e>0"
1670 hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
1671 using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
1672 by (metis not_le le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
1674 thus ?thesis unfolding Lim_sequentially dist_norm by simp
1677 text{* More properties of closed balls. *}
1679 lemma closed_cball: "closed (cball x e)"
1680 unfolding cball_def closed_def
1681 unfolding Collect_neg_eq [symmetric] not_le
1682 apply (clarsimp simp add: open_dist, rename_tac y)
1683 apply (rule_tac x="dist x y - e" in exI, clarsimp)
1684 apply (rename_tac x')
1685 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
1689 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
1691 { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
1692 hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
1694 { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
1695 hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
1697 show ?thesis unfolding open_contains_ball by auto
1700 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
1701 by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball mem_def)
1703 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
1704 apply (simp add: interior_def, safe)
1705 apply (force simp add: open_contains_cball)
1706 apply (rule_tac x="ball x e" in exI)
1707 apply (simp add: open_ball centre_in_ball subset_trans [OF ball_subset_cball])
1711 fixes x y :: "'a::{real_normed_vector,perfect_space}"
1712 shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
1715 { assume "e \<le> 0"
1716 hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
1717 have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
1719 hence "e > 0" by (metis not_less)
1721 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
1722 ultimately show "?rhs" by auto
1724 assume "?rhs" hence "e>0" by auto
1725 { fix d::real assume "d>0"
1726 have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1727 proof(cases "d \<le> dist x y")
1728 case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1730 case True hence False using `d \<le> dist x y` `d>0` by auto
1731 thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
1735 have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
1736 = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1737 unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
1738 also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
1739 using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
1740 unfolding scaleR_minus_left scaleR_one
1741 by (auto simp add: norm_minus_commute)
1742 also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
1743 unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
1744 unfolding real_add_mult_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
1745 also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
1746 finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
1750 have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
1751 using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
1753 have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
1754 using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
1755 unfolding dist_norm by auto
1756 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
1759 case False hence "d > dist x y" by auto
1760 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1763 obtain z where **: "z \<noteq> y" "dist z y < min e d"
1764 using perfect_choose_dist[of "min e d" y]
1765 using `d > 0` `e>0` by auto
1766 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1768 using `z \<noteq> y` **
1769 by (rule_tac x=z in bexI, auto simp add: dist_commute)
1771 case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1772 using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
1775 thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
1778 lemma closure_ball_lemma:
1779 fixes x y :: "'a::real_normed_vector"
1780 assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
1781 proof (rule islimptI)
1782 fix T assume "y \<in> T" "open T"
1783 then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
1784 unfolding open_dist by fast
1785 (* choose point between x and y, within distance r of y. *)
1786 def k \<equiv> "min 1 (r / (2 * dist x y))"
1787 def z \<equiv> "y + scaleR k (x - y)"
1788 have z_def2: "z = x + scaleR (1 - k) (y - x)"
1789 unfolding z_def by (simp add: algebra_simps)
1791 unfolding z_def k_def using `0 < r`
1792 by (simp add: dist_norm min_def)
1793 hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
1794 have "dist x z < dist x y"
1795 unfolding z_def2 dist_norm
1796 apply (simp add: norm_minus_commute)
1797 apply (simp only: dist_norm [symmetric])
1798 apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
1799 apply (rule mult_strict_right_mono)
1800 apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
1801 apply (simp add: zero_less_dist_iff `x \<noteq> y`)
1803 hence "z \<in> ball x (dist x y)" by simp
1805 unfolding z_def k_def using `x \<noteq> y` `0 < r`
1806 by (simp add: min_def)
1807 show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
1808 using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
1813 fixes x :: "'a::real_normed_vector"
1814 shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
1815 apply (rule equalityI)
1816 apply (rule closure_minimal)
1817 apply (rule ball_subset_cball)
1818 apply (rule closed_cball)
1819 apply (rule subsetI, rename_tac y)
1820 apply (simp add: le_less [where 'a=real])
1822 apply (rule subsetD [OF closure_subset], simp)
1823 apply (simp add: closure_def)
1825 apply (rule closure_ball_lemma)
1826 apply (simp add: zero_less_dist_iff)
1829 (* In a trivial vector space, this fails for e = 0. *)
1830 lemma interior_cball:
1831 fixes x :: "'a::{real_normed_vector, perfect_space}"
1832 shows "interior (cball x e) = ball x e"
1833 proof(cases "e\<ge>0")
1834 case False note cs = this
1835 from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
1836 { fix y assume "y \<in> cball x e"
1837 hence False unfolding mem_cball using dist_nz[of x y] cs by auto }
1838 hence "cball x e = {}" by auto
1839 hence "interior (cball x e) = {}" using interior_empty by auto
1840 ultimately show ?thesis by blast
1842 case True note cs = this
1843 have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
1844 { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
1845 then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
1847 then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
1848 using perfect_choose_dist [of d] by auto
1849 have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
1850 hence xa_cball:"xa \<in> cball x e" using as(1) by auto
1852 hence "y \<in> ball x e" proof(cases "x = y")
1854 hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
1855 thus "y \<in> ball x e" using `x = y ` by simp
1858 have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
1859 using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
1860 hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
1861 have "y - x \<noteq> 0" using `x \<noteq> y` by auto
1862 hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
1863 using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
1865 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)"
1866 by (auto simp add: dist_norm algebra_simps)
1867 also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1868 by (auto simp add: algebra_simps)
1869 also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
1871 also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
1872 finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
1873 thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
1875 hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
1876 ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
1879 lemma frontier_ball:
1880 fixes a :: "'a::real_normed_vector"
1881 shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
1882 apply (simp add: frontier_def closure_ball interior_open open_ball order_less_imp_le)
1883 apply (simp add: expand_set_eq)
1886 lemma frontier_cball:
1887 fixes a :: "'a::{real_normed_vector, perfect_space}"
1888 shows "frontier(cball a e) = {x. dist a x = e}"
1889 apply (simp add: frontier_def interior_cball closed_cball closure_closed order_less_imp_le)
1890 apply (simp add: expand_set_eq)
1893 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
1894 apply (simp add: expand_set_eq not_le)
1895 by (metis zero_le_dist dist_self order_less_le_trans)
1896 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
1898 lemma cball_eq_sing:
1899 fixes x :: "'a::perfect_space"
1900 shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
1901 proof (rule linorder_cases)
1903 obtain a where "a \<noteq> x" "dist a x < e"
1904 using perfect_choose_dist [OF e] by auto
1905 hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
1906 with e show ?thesis by (auto simp add: expand_set_eq)
1910 fixes x :: "'a::metric_space"
1911 shows "e = 0 ==> cball x e = {x}"
1912 by (auto simp add: expand_set_eq)
1914 text{* For points in the interior, localization of limits makes no difference. *}
1916 lemma eventually_within_interior:
1917 assumes "x \<in> interior S"
1918 shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
1920 from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S"
1921 unfolding interior_def by fast
1923 then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
1924 unfolding Limits.eventually_within Limits.eventually_at_topological
1926 with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
1929 unfolding Limits.eventually_at_topological by auto
1931 { assume "?rhs" hence "?lhs"
1932 unfolding Limits.eventually_within
1933 by (auto elim: eventually_elim1)
1938 lemma lim_within_interior:
1939 "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
1940 unfolding tendsto_def by (simp add: eventually_within_interior)
1942 lemma netlimit_within_interior:
1943 fixes x :: "'a::{perfect_space, real_normed_vector}"
1944 (* FIXME: generalize to perfect_space *)
1945 assumes "x \<in> interior S"
1946 shows "netlimit(at x within S) = x" (is "?lhs = ?rhs")
1948 from assms obtain e::real where e:"e>0" "ball x e \<subseteq> S" using open_interior[of S] unfolding open_contains_ball using interior_subset[of S] by auto
1949 hence "\<not> trivial_limit (at x within S)" using islimpt_subset[of x "ball x e" S] unfolding trivial_limit_within islimpt_ball centre_in_cball by auto
1950 thus ?thesis using netlimit_within by auto
1953 subsection{* Boundedness. *}
1955 (* FIXME: This has to be unified with BSEQ!! *)
1957 bounded :: "'a::metric_space set \<Rightarrow> bool" where
1958 "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
1960 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
1961 unfolding bounded_def
1963 apply (rule_tac x="dist a x + e" in exI, clarify)
1964 apply (drule (1) bspec)
1965 apply (erule order_trans [OF dist_triangle add_left_mono])
1969 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
1970 unfolding bounded_any_center [where a=0]
1971 by (simp add: dist_norm)
1973 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
1974 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
1975 by (metis bounded_def subset_eq)
1977 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
1978 by (metis bounded_subset interior_subset)
1980 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
1982 from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
1983 { fix y assume "y \<in> closure S"
1984 then obtain f where f: "\<forall>n. f n \<in> S" "(f ---> y) sequentially"
1985 unfolding closure_sequential by auto
1986 have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
1987 hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
1988 by (rule eventually_mono, simp add: f(1))
1989 have "dist x y \<le> a"
1990 apply (rule Lim_dist_ubound [of sequentially f])
1991 apply (rule trivial_limit_sequentially)
1996 thus ?thesis unfolding bounded_def by auto
1999 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
2000 apply (simp add: bounded_def)
2001 apply (rule_tac x=x in exI)
2002 apply (rule_tac x=e in exI)
2006 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
2007 by (metis ball_subset_cball bounded_cball bounded_subset)
2009 lemma finite_imp_bounded[intro]: assumes "finite S" shows "bounded S"
2011 { fix a F assume as:"bounded F"
2012 then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
2013 hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
2014 hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
2016 thus ?thesis using finite_induct[of S bounded] using bounded_empty assms by auto
2019 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
2020 apply (auto simp add: bounded_def)
2021 apply (rename_tac x y r s)
2022 apply (rule_tac x=x in exI)
2023 apply (rule_tac x="max r (dist x y + s)" in exI)
2024 apply (rule ballI, rename_tac z, safe)
2025 apply (drule (1) bspec, simp)
2026 apply (drule (1) bspec)
2027 apply (rule min_max.le_supI2)
2028 apply (erule order_trans [OF dist_triangle add_left_mono])
2031 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
2032 by (induct rule: finite_induct[of F], auto)
2034 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
2035 apply (simp add: bounded_iff)
2036 apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
2039 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
2040 by (metis Int_lower1 Int_lower2 bounded_subset)
2042 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
2043 apply (metis Diff_subset bounded_subset)
2046 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
2047 by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
2049 lemma not_bounded_UNIV[simp, intro]:
2050 "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
2051 proof(auto simp add: bounded_pos not_le)
2052 obtain x :: 'a where "x \<noteq> 0"
2053 using perfect_choose_dist [OF zero_less_one] by fast
2054 fix b::real assume b: "b >0"
2055 have b1: "b +1 \<ge> 0" using b by simp
2056 with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
2057 by (simp add: norm_sgn)
2058 then show "\<exists>x::'a. b < norm x" ..
2061 lemma bounded_linear_image:
2062 assumes "bounded S" "bounded_linear f"
2063 shows "bounded(f ` S)"
2065 from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
2066 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)
2067 { fix x assume "x\<in>S"
2068 hence "norm x \<le> b" using b by auto
2069 hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
2070 by (metis B(1) B(2) real_le_trans real_mult_le_cancel_iff2)
2072 thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
2073 using b B real_mult_order[of b B] by (auto simp add: real_mult_commute)
2076 lemma bounded_scaling:
2077 fixes S :: "'a::real_normed_vector set"
2078 shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
2079 apply (rule bounded_linear_image, assumption)
2080 apply (rule scaleR.bounded_linear_right)
2083 lemma bounded_translation:
2084 fixes S :: "'a::real_normed_vector set"
2085 assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
2087 from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
2088 { fix x assume "x\<in>S"
2089 hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
2091 thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
2092 by (auto intro!: add exI[of _ "b + norm a"])
2096 text{* Some theorems on sups and infs using the notion "bounded". *}
2099 fixes S :: "real set"
2100 shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"
2101 by (simp add: bounded_iff)
2103 lemma bounded_has_Sup:
2104 fixes S :: "real set"
2105 assumes "bounded S" "S \<noteq> {}"
2106 shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
2108 fix x assume "x\<in>S"
2109 thus "x \<le> Sup S"
2110 by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
2112 show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
2113 by (metis SupInf.Sup_least)
2117 fixes S :: "real set"
2118 shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
2119 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal)
2121 lemma Sup_insert_finite:
2122 fixes S :: "real set"
2123 shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
2124 apply (rule Sup_insert)
2125 apply (rule finite_imp_bounded)
2128 lemma bounded_has_Inf:
2129 fixes S :: "real set"
2130 assumes "bounded S" "S \<noteq> {}"
2131 shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
2133 fix x assume "x\<in>S"
2134 from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
2135 thus "x \<ge> Inf S" using `x\<in>S`
2136 by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
2138 show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
2139 by (metis SupInf.Inf_greatest)
2143 fixes S :: "real set"
2144 shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
2145 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)
2146 lemma Inf_insert_finite:
2147 fixes S :: "real set"
2148 shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
2149 by (rule Inf_insert, rule finite_imp_bounded, simp)
2152 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
2153 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
2154 apply (frule isGlb_isLb)
2155 apply (frule_tac x = y in isGlb_isLb)
2156 apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
2159 subsection{* Compactness (the definition is the one based on convegent subsequences). *}
2162 compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
2163 "compact S \<longleftrightarrow>
2164 (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
2165 (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
2168 A metric space (or topological vector space) is said to have the
2169 Heine-Borel property if every closed and bounded subset is compact.
2173 assumes bounded_imp_convergent_subsequence:
2174 "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
2175 \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2177 lemma bounded_closed_imp_compact:
2178 fixes s::"'a::heine_borel set"
2179 assumes "bounded s" and "closed s" shows "compact s"
2180 proof (unfold compact_def, clarify)
2181 fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
2182 obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
2183 using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
2184 from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
2185 have "l \<in> s" using `closed s` fr l
2186 unfolding closed_sequential_limits by blast
2187 show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2188 using `l \<in> s` r l by blast
2191 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
2193 show "0 \<le> r 0" by auto
2195 fix n assume "n \<le> r n"
2196 moreover have "r n < r (Suc n)"
2197 using assms [unfolded subseq_def] by auto
2198 ultimately show "Suc n \<le> r (Suc n)" by auto
2201 lemma eventually_subseq:
2202 assumes r: "subseq r"
2203 shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
2204 unfolding eventually_sequentially
2205 by (metis subseq_bigger [OF r] le_trans)
2208 "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
2209 unfolding tendsto_def eventually_sequentially o_def
2210 by (metis subseq_bigger le_trans)
2212 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
2214 apply (rule_tac x="nat_rec e f" in exI)
2216 apply (rule def_nat_rec_0, simp)
2217 apply (rule allI, rule def_nat_rec_Suc, simp)
2218 apply (rule allI, rule impI, rule ext)
2220 apply (induct_tac x)
2221 apply (simp add: nat_rec_0)
2222 apply (erule_tac x="n" in allE)
2226 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
2227 assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
2228 shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N. abs(s n - l) < e"
2230 have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
2231 then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
2232 { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
2234 obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
2235 have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
2236 with n have "s N \<le> t - e" using `e>0` by auto
2237 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 }
2238 hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
2239 hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto }
2240 thus ?thesis by blast
2243 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
2244 assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
2245 shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
2246 using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
2247 unfolding monoseq_def incseq_def
2248 apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
2249 unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
2251 lemma compact_real_lemma:
2252 assumes "\<forall>n::nat. abs(s n) \<le> b"
2253 shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
2255 obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
2256 using seq_monosub[of s] by auto
2257 thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
2258 unfolding tendsto_iff dist_norm eventually_sequentially by auto
2261 instance real :: heine_borel
2263 fix s :: "real set" and f :: "nat \<Rightarrow> real"
2264 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2265 then obtain b where b: "\<forall>n. abs (f n) \<le> b"
2266 unfolding bounded_iff by auto
2267 obtain l :: real and r :: "nat \<Rightarrow> nat" where
2268 r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
2269 using compact_real_lemma [OF b] by auto
2270 thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2274 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
2275 unfolding bounded_def
2277 apply (rule_tac x="x $ i" in exI)
2278 apply (rule_tac x="e" in exI)
2280 apply (rule order_trans [OF dist_nth_le], simp)
2283 lemma compact_lemma:
2284 fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
2285 assumes "bounded s" and "\<forall>n. f n \<in> s"
2287 \<exists>l r. subseq r \<and>
2288 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
2290 fix d::"'n set" have "finite d" by simp
2291 thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
2292 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
2293 proof(induct d) case empty thus ?case unfolding subseq_def by auto
2294 next case (insert k d)
2295 have s': "bounded ((\<lambda>x. x $ k) ` s)" using `bounded s` by (rule bounded_component)
2296 obtain l1::"'a^'n" and r1 where r1:"subseq r1" and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
2297 using insert(3) by auto
2298 have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` s" using `\<forall>n. f n \<in> s` by simp
2299 obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
2300 using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
2301 def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
2302 using r1 and r2 unfolding r_def o_def subseq_def by auto
2304 def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
2305 { fix e::real assume "e>0"
2306 from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially" by blast
2307 from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially" by (rule tendstoD)
2308 from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
2309 by (rule eventually_subseq)
2310 have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
2311 using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
2313 ultimately show ?case by auto
2317 instance cart :: (heine_borel, finite) heine_borel
2319 fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"
2320 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2321 then obtain l r where r: "subseq r"
2322 and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
2323 using compact_lemma [OF s f] by blast
2324 let ?d = "UNIV::'b set"
2325 { fix e::real assume "e>0"
2326 hence "0 < e / (real_of_nat (card ?d))"
2327 using zero_less_card_finite using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
2328 with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
2331 { fix n assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
2332 have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
2333 unfolding dist_vector_def using zero_le_dist by (rule setL2_le_setsum)
2334 also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
2335 by (rule setsum_strict_mono) (simp_all add: n)
2336 finally have "dist (f (r n)) l < e" by simp
2338 ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
2339 by (rule eventually_elim1)
2341 hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
2342 with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
2345 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
2346 unfolding bounded_def
2348 apply (rule_tac x="a" in exI)
2349 apply (rule_tac x="e" in exI)
2351 apply (drule (1) bspec)
2352 apply (simp add: dist_Pair_Pair)
2353 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
2356 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
2357 unfolding bounded_def
2359 apply (rule_tac x="b" in exI)
2360 apply (rule_tac x="e" in exI)
2362 apply (drule (1) bspec)
2363 apply (simp add: dist_Pair_Pair)
2364 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
2367 instance "*" :: (heine_borel, heine_borel) heine_borel
2369 fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
2370 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2371 from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
2372 from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
2373 obtain l1 r1 where r1: "subseq r1"
2374 and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
2375 using bounded_imp_convergent_subsequence [OF s1 f1]
2376 unfolding o_def by fast
2377 from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
2378 from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
2379 obtain l2 r2 where r2: "subseq r2"
2380 and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
2381 using bounded_imp_convergent_subsequence [OF s2 f2]
2382 unfolding o_def by fast
2383 have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
2384 using lim_subseq [OF r2 l1] unfolding o_def .
2385 have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
2386 using tendsto_Pair [OF l1' l2] unfolding o_def by simp
2387 have r: "subseq (r1 \<circ> r2)"
2388 using r1 r2 unfolding subseq_def by simp
2389 show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2393 subsection{* Completeness. *}
2396 "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
2397 unfolding Cauchy_def by blast
2400 complete :: "'a::metric_space set \<Rightarrow> bool" where
2401 "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
2402 --> (\<exists>l \<in> s. (f ---> l) sequentially))"
2404 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
2409 with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
2410 by (erule_tac x="e/2" in allE) auto
2412 assume nm:"N \<le> m \<and> N \<le> n"
2413 hence "dist (s m) (s n) < e" using N
2414 using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
2417 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
2421 unfolding cauchy_def
2425 unfolding cauchy_def
2426 using dist_triangle_half_l
2430 lemma convergent_imp_cauchy:
2431 "(s ---> l) sequentially ==> Cauchy s"
2432 proof(simp only: cauchy_def, rule, rule)
2433 fix e::real assume "e>0" "(s ---> l) sequentially"
2434 then obtain N::nat where N:"\<forall>n\<ge>N. dist (s n) l < e/2" unfolding Lim_sequentially by(erule_tac x="e/2" in allE) auto
2435 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
2438 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
2440 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
2441 hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
2443 have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
2444 then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
2445 unfolding bounded_any_center [where a="s N"] by auto
2446 ultimately show "?thesis"
2447 unfolding bounded_any_center [where a="s N"]
2448 apply(rule_tac x="max a 1" in exI) apply auto
2449 apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto
2452 lemma compact_imp_complete: assumes "compact s" shows "complete s"
2454 { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
2455 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
2457 note lr' = subseq_bigger [OF lr(2)]
2459 { fix e::real assume "e>0"
2460 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
2461 from lr(3)[unfolded Lim_sequentially, 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
2462 { fix n::nat assume n:"n \<ge> max N M"
2463 have "dist ((f \<circ> r) n) l < e/2" using n M by auto
2464 moreover have "r n \<ge> N" using lr'[of n] n by auto
2465 hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
2466 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) }
2467 hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast }
2468 hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding Lim_sequentially by auto }
2469 thus ?thesis unfolding complete_def by auto
2472 instance heine_borel < complete_space
2474 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2475 hence "bounded (range f)"
2476 by (rule cauchy_imp_bounded)
2477 hence "compact (closure (range f))"
2478 using bounded_closed_imp_compact [of "closure (range f)"] by auto
2479 hence "complete (closure (range f))"
2480 by (rule compact_imp_complete)
2481 moreover have "\<forall>n. f n \<in> closure (range f)"
2482 using closure_subset [of "range f"] by auto
2483 ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
2484 using `Cauchy f` unfolding complete_def by auto
2485 then show "convergent f"
2486 unfolding convergent_def LIMSEQ_conv_tendsto [symmetric] by auto
2489 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
2490 proof(simp add: complete_def, rule, rule)
2491 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2492 hence "convergent f" by (rule Cauchy_convergent)
2493 hence "\<exists>l. f ----> l" unfolding convergent_def .
2494 thus "\<exists>l. (f ---> l) sequentially" unfolding LIMSEQ_conv_tendsto .
2497 lemma complete_imp_closed: assumes "complete s" shows "closed s"
2499 { fix x assume "x islimpt s"
2500 then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
2501 unfolding islimpt_sequential by auto
2502 then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
2503 using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
2504 hence "x \<in> s" using Lim_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
2506 thus "closed s" unfolding closed_limpt by auto
2509 lemma complete_eq_closed:
2510 fixes s :: "'a::complete_space set"
2511 shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
2513 assume ?lhs thus ?rhs by (rule complete_imp_closed)
2516 { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
2517 then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
2518 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 }
2519 thus ?lhs unfolding complete_def by auto
2522 lemma convergent_eq_cauchy:
2523 fixes s :: "nat \<Rightarrow> 'a::complete_space"
2524 shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s" (is "?lhs = ?rhs")
2526 assume ?lhs then obtain l where "(s ---> l) sequentially" by auto
2527 thus ?rhs using convergent_imp_cauchy by auto
2529 assume ?rhs thus ?lhs using complete_univ[unfolded complete_def, THEN spec[where x=s]] by auto
2532 lemma convergent_imp_bounded:
2533 fixes s :: "nat \<Rightarrow> 'a::metric_space"
2534 shows "(s ---> l) sequentially ==> bounded (s ` (UNIV::(nat set)))"
2535 using convergent_imp_cauchy[of s]
2536 using cauchy_imp_bounded[of s]
2540 subsection{* Total boundedness. *}
2542 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
2543 "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
2544 declare helper_1.simps[simp del]
2546 lemma compact_imp_totally_bounded:
2548 shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
2549 proof(rule, rule, rule ccontr)
2550 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)"
2551 def x \<equiv> "helper_1 s e"
2553 have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
2554 proof(induct_tac rule:nat_less_induct)
2555 fix n def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
2556 assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
2557 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
2558 then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
2559 have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
2560 apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
2561 thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
2563 hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
2564 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
2565 from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
2566 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
2568 using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
2569 using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
2570 using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
2573 subsection{* Heine-Borel theorem (following Burkill \& Burkill vol. 2) *}
2575 lemma heine_borel_lemma: fixes s::"'a::metric_space set"
2576 assumes "compact s" "s \<subseteq> (\<Union> t)" "\<forall>b \<in> t. open b"
2577 shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
2579 assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"
2580 hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto
2582 have "1 / real (n + 1) > 0" by auto
2583 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 }
2584 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
2585 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)"
2586 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
2588 then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
2589 using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
2591 obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
2592 then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"
2593 using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto
2595 then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"
2596 using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
2598 obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
2599 have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
2600 apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
2601 using subseq_bigger[OF r, of "N1 + N2"] by auto
2603 def x \<equiv> "(f (r (N1 + N2)))"
2604 have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
2605 using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto
2606 have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto
2607 then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto
2609 have "dist x l < e/2" using N1 unfolding x_def o_def by auto
2610 hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)
2612 thus False using e and `y\<notin>b` by auto
2615 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2616 \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"
2618 fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"
2619 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
2620 hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto
2621 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
2622 then obtain bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast
2624 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
2625 then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto
2627 have "finite (bb ` k)" using k(1) by auto
2629 { fix x assume "x\<in>s"
2630 hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3) unfolding subset_eq by auto
2631 hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast
2632 hence "x \<in> \<Union>(bb ` k)" using Union_iff[of x "bb ` k"] by auto
2634 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
2637 subsection{* Bolzano-Weierstrass property. *}
2639 lemma heine_borel_imp_bolzano_weierstrass:
2640 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'))"
2641 "infinite t" "t \<subseteq> s"
2642 shows "\<exists>x \<in> s. x islimpt t"
2644 assume "\<not> (\<exists>x \<in> s. x islimpt t)"
2645 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
2646 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
2647 obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
2648 using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
2649 from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
2650 { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
2651 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
2652 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 }
2653 hence "infinite (f ` t)" using assms(2) using finite_imageD[unfolded inj_on_def, of f t] by auto
2655 { fix x assume "x\<in>t" "f x \<notin> g"
2656 from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
2657 then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
2658 hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
2659 hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto }
2660 hence "f ` t \<subseteq> g" by auto
2661 ultimately show False using g(2) using finite_subset by auto
2664 subsection{* Complete the chain of compactness variants. *}
2666 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where
2667 "helper_2 beyond 0 = beyond 0" |
2668 "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"
2670 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"
2671 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2674 assume "\<not> bounded s"
2675 then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"
2676 unfolding bounded_any_center [where a=undefined]
2677 apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto
2678 hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"
2679 unfolding linorder_not_le by auto
2680 def x \<equiv> "helper_2 beyond"
2682 { fix m n ::nat assume "m<n"
2683 hence "dist undefined (x m) + 1 < dist undefined (x n)"
2685 case 0 thus ?case by auto
2688 have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"
2689 unfolding x_def and helper_2.simps
2690 using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto
2691 thus ?case proof(cases "m < n")
2692 case True thus ?thesis using Suc and * by auto
2694 case False hence "m = n" using Suc(2) by auto
2695 thus ?thesis using * by auto
2698 { fix m n ::nat assume "m\<noteq>n"
2699 have "1 < dist (x m) (x n)"
2702 hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto
2703 thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith
2705 case False hence "n<m" using `m\<noteq>n` by auto
2706 hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto
2707 thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith
2708 qed } note ** = this
2709 { fix a b assume "x a = x b" "a \<noteq> b"
2710 hence False using **[of a b] by auto }
2711 hence "inj x" unfolding inj_on_def by auto
2715 proof(cases "n = 0")
2716 case True thus ?thesis unfolding x_def using beyond by auto
2718 case False then obtain z where "n = Suc z" using not0_implies_Suc by auto
2719 thus ?thesis unfolding x_def using beyond by auto
2721 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
2723 then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto
2724 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
2725 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"]]
2726 unfolding dist_nz by auto
2727 show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto
2730 lemma sequence_infinite_lemma:
2731 fixes l :: "'a::metric_space" (* TODO: generalize *)
2732 assumes "\<forall>n::nat. (f n \<noteq> l)" "(f ---> l) sequentially"
2733 shows "infinite (range f)"
2735 let ?A = "(\<lambda>x. dist x l) ` range f"
2736 assume "finite (range f)"
2737 hence **:"finite ?A" "?A \<noteq> {}" by auto
2738 obtain k where k:"dist (f k) l = Min ?A" using Min_in[OF **] by auto
2739 have "0 < Min ?A" using assms(1) unfolding dist_nz unfolding Min_gr_iff[OF **] by auto
2740 then obtain N where "dist (f N) l < Min ?A" using assms(2)[unfolded Lim_sequentially, THEN spec[where x="Min ?A"]] by auto
2741 moreover have "dist (f N) l \<in> ?A" by auto
2742 ultimately show False using Min_le[OF **(1), of "dist (f N) l"] by auto
2745 lemma sequence_unique_limpt:
2746 fixes l :: "'a::metric_space" (* TODO: generalize *)
2747 assumes "\<forall>n::nat. (f n \<noteq> l)" "(f ---> l) sequentially" "l' islimpt (range f)"
2750 def e \<equiv> "dist l' l"
2751 assume "l' \<noteq> l" hence "e>0" unfolding dist_nz e_def by auto
2752 then obtain N::nat where N:"\<forall>n\<ge>N. dist (f n) l < e / 2"
2753 using assms(2)[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
2754 def d \<equiv> "Min (insert (e/2) ((\<lambda>n. if dist (f n) l' = 0 then e/2 else dist (f n) l') ` {0 .. N}))"
2755 have "d>0" using `e>0` unfolding d_def e_def using zero_le_dist[of _ l', unfolded order_le_less] by auto
2756 obtain k where k:"f k \<noteq> l'" "dist (f k) l' < d" using `d>0` and assms(3)[unfolded islimpt_approachable, THEN spec[where x="d"]] by auto
2757 have "k\<ge>N" using k(1)[unfolded dist_nz] using k(2)[unfolded d_def]
2759 hence "dist l' l < e" using N[THEN spec[where x=k]] using k(2)[unfolded d_def] and dist_triangle_half_r[of "f k" l' e l] by auto
2760 thus False unfolding e_def by auto
2763 lemma bolzano_weierstrass_imp_closed:
2764 fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)
2765 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2768 { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
2770 proof(cases "\<forall>n. x n \<noteq> l")
2771 case False thus "l\<in>s" using as(1) by auto
2773 case True note cas = this
2774 with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto
2775 then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto
2776 thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
2778 thus ?thesis unfolding closed_sequential_limits by fast
2781 text{* Hence express everything as an equivalence. *}
2783 lemma compact_eq_heine_borel:
2784 fixes s :: "'a::heine_borel set"
2785 shows "compact s \<longleftrightarrow>
2786 (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2787 --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")
2789 assume ?lhs thus ?rhs using compact_imp_heine_borel[of s] by blast
2792 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
2793 by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])
2794 thus ?lhs using bolzano_weierstrass_imp_bounded[of s] bolzano_weierstrass_imp_closed[of s] bounded_closed_imp_compact[of s] by blast
2797 lemma compact_eq_bolzano_weierstrass:
2798 fixes s :: "'a::heine_borel set"
2799 shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
2801 assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto
2803 assume ?rhs thus ?lhs using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed bounded_closed_imp_compact by auto
2806 lemma compact_eq_bounded_closed:
2807 fixes s :: "'a::heine_borel set"
2808 shows "compact s \<longleftrightarrow> bounded s \<and> closed s" (is "?lhs = ?rhs")
2810 assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto
2812 assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto
2815 lemma compact_imp_bounded:
2816 fixes s :: "'a::metric_space set"
2817 shows "compact s ==> bounded s"
2820 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')"
2821 by (rule compact_imp_heine_borel)
2822 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
2823 using heine_borel_imp_bolzano_weierstrass[of s] by auto
2825 by (rule bolzano_weierstrass_imp_bounded)
2828 lemma compact_imp_closed:
2829 fixes s :: "'a::metric_space set"
2830 shows "compact s ==> closed s"
2833 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')"
2834 by (rule compact_imp_heine_borel)
2835 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
2836 using heine_borel_imp_bolzano_weierstrass[of s] by auto
2838 by (rule bolzano_weierstrass_imp_closed)
2841 text{* In particular, some common special cases. *}
2843 lemma compact_empty[simp]:
2845 unfolding compact_def
2848 (* TODO: can any of the next 3 lemmas be generalized to metric spaces? *)
2850 (* FIXME : Rename *)
2851 lemma compact_union[intro]:
2852 fixes s t :: "'a::heine_borel set"
2853 shows "compact s \<Longrightarrow> compact t ==> compact (s \<union> t)"
2854 unfolding compact_eq_bounded_closed
2855 using bounded_Un[of s t]
2856 using closed_Un[of s t]
2859 lemma compact_inter[intro]:
2860 fixes s t :: "'a::heine_borel set"
2861 shows "compact s \<Longrightarrow> compact t ==> compact (s \<inter> t)"
2862 unfolding compact_eq_bounded_closed
2863 using bounded_Int[of s t]
2864 using closed_Int[of s t]
2867 lemma compact_inter_closed[intro]:
2868 fixes s t :: "'a::heine_borel set"
2869 shows "compact s \<Longrightarrow> closed t ==> compact (s \<inter> t)"
2870 unfolding compact_eq_bounded_closed
2871 using closed_Int[of s t]
2872 using bounded_subset[of "s \<inter> t" s]
2875 lemma closed_inter_compact[intro]:
2876 fixes s t :: "'a::heine_borel set"
2877 shows "closed s \<Longrightarrow> compact t ==> compact (s \<inter> t)"
2879 assume "closed s" "compact t"
2881 have "s \<inter> t = t \<inter> s" by auto ultimately
2883 using compact_inter_closed[of t s]
2887 lemma closed_sing [simp]:
2888 fixes a :: "'a::metric_space"
2890 apply (clarsimp simp add: closed_def open_dist)
2892 apply (drule_tac x="dist x a" in spec)
2893 apply (simp add: dist_nz dist_commute)
2896 lemma finite_imp_closed:
2897 fixes s :: "'a::metric_space set"
2898 shows "finite s ==> closed s"
2899 proof (induct set: finite)
2900 case empty show "closed {}" by simp
2903 hence "closed ({x} \<union> F)" by (simp only: closed_Un closed_sing)
2904 thus "closed (insert x F)" by simp
2907 lemma finite_imp_compact:
2908 fixes s :: "'a::heine_borel set"
2909 shows "finite s ==> compact s"
2910 unfolding compact_eq_bounded_closed
2911 using finite_imp_closed finite_imp_bounded
2914 lemma compact_sing [simp]: "compact {a}"
2915 unfolding compact_def o_def subseq_def
2916 by (auto simp add: tendsto_const)
2918 lemma compact_cball[simp]:
2919 fixes x :: "'a::heine_borel"
2920 shows "compact(cball x e)"
2921 using compact_eq_bounded_closed bounded_cball closed_cball
2924 lemma compact_frontier_bounded[intro]:
2925 fixes s :: "'a::heine_borel set"
2926 shows "bounded s ==> compact(frontier s)"
2927 unfolding frontier_def
2928 using compact_eq_bounded_closed
2931 lemma compact_frontier[intro]:
2932 fixes s :: "'a::heine_borel set"
2933 shows "compact s ==> compact (frontier s)"
2934 using compact_eq_bounded_closed compact_frontier_bounded
2937 lemma frontier_subset_compact:
2938 fixes s :: "'a::heine_borel set"
2939 shows "compact s ==> frontier s \<subseteq> s"
2940 using frontier_subset_closed compact_eq_bounded_closed
2944 fixes s :: "'a::metric_space set"
2945 shows "open s ==> open(s - {x})"
2946 using open_Diff[of s "{x}"] closed_sing
2949 text{* Finite intersection property. I could make it an equivalence in fact. *}
2951 lemma compact_imp_fip:
2952 fixes s :: "'a::heine_borel set"
2953 assumes "compact s" "\<forall>t \<in> f. closed t"
2954 "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"
2955 shows "s \<inter> (\<Inter> f) \<noteq> {}"
2957 assume as:"s \<inter> (\<Inter> f) = {}"
2958 hence "s \<subseteq> \<Union> uminus ` f" by auto
2959 moreover have "Ball (uminus ` f) open" using open_Diff closed_Diff using assms(2) by auto
2960 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
2961 hence "finite (uminus ` f') \<and> uminus ` f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)
2962 hence "s \<inter> \<Inter>uminus ` f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus ` f'"]] by auto
2963 thus False using f'(3) unfolding subset_eq and Union_iff by blast
2966 subsection{* Bounded closed nest property (proof does not use Heine-Borel). *}
2968 lemma bounded_closed_nest:
2969 assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
2970 "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)" "bounded(s 0)"
2971 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
2973 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
2974 from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
2976 then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
2977 unfolding compact_def apply(erule_tac x=x in allE) using x using assms(3) by blast
2980 { fix e::real assume "e>0"
2981 with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
2982 hence "dist ((x \<circ> r) (max N n)) l < e" by auto
2984 have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
2985 hence "(x \<circ> r) (max N n) \<in> s n"
2986 using x apply(erule_tac x=n in allE)
2987 using x apply(erule_tac x="r (max N n)" in allE)
2988 using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto
2989 ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
2991 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
2993 thus ?thesis by auto
2996 text{* Decreasing case does not even need compactness, just completeness. *}
2998 lemma decreasing_closed_nest:
2999 assumes "\<forall>n. closed(s n)"
3000 "\<forall>n. (s n \<noteq> {})"
3001 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
3002 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
3003 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s n"
3005 have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
3006 hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
3007 then obtain t where t: "\<forall>n. t n \<in> s n" by auto
3008 { fix e::real assume "e>0"
3009 then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
3010 { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
3011 hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding subset_eq t by blast+
3012 hence "dist (t m) (t n) < e" using N by auto
3014 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
3016 hence "Cauchy t" unfolding cauchy_def by auto
3017 then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
3019 { fix e::real assume "e>0"
3020 then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded Lim_sequentially] by auto
3021 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
3022 hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
3024 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
3026 then show ?thesis by auto
3029 text{* Strengthen it to the intersection actually being a singleton. *}
3031 lemma decreasing_closed_nest_sing:
3032 fixes s :: "nat \<Rightarrow> 'a::heine_borel set"
3033 assumes "\<forall>n. closed(s n)"
3034 "\<forall>n. s n \<noteq> {}"
3035 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
3036 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
3037 shows "\<exists>a. \<Inter>(range s) = {a}"
3039 obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
3040 { fix b assume b:"b \<in> \<Inter>(range s)"
3041 { fix e::real assume "e>0"
3042 hence "dist a b < e" using assms(4 )using b using a by blast
3044 hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz real_less_def)
3046 with a have "\<Inter>(range s) = {a}" unfolding image_def by auto
3050 text{* Cauchy-type criteria for uniform convergence. *}
3052 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows
3053 "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
3054 (\<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")
3057 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
3058 { fix e::real assume "e>0"
3059 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
3060 { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
3061 hence "dist (s m x) (s n x) < e"
3062 using N[THEN spec[where x=m], THEN spec[where x=x]]
3063 using N[THEN spec[where x=n], THEN spec[where x=x]]
3064 using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto }
3065 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 }
3069 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
3070 then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
3071 using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
3072 { fix e::real assume "e>0"
3073 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"
3074 using `?rhs`[THEN spec[where x="e/2"]] by auto
3075 { fix x assume "P x"
3076 then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
3077 using l[THEN spec[where x=x], unfolded Lim_sequentially] using `e>0` by(auto elim!: allE[where x="e/2"])
3078 fix n::nat assume "n\<ge>N"
3079 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]]
3080 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) }
3081 hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
3085 lemma uniformly_cauchy_imp_uniformly_convergent:
3086 fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"
3087 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"
3088 "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
3089 shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
3091 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"
3092 using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
3094 { fix x assume "P x"
3095 hence "l x = l' x" using Lim_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
3096 using l and assms(2) unfolding Lim_sequentially by blast }
3097 ultimately show ?thesis by auto
3100 subsection{* Define continuity over a net to take in restrictions of the set. *}
3103 continuous :: "'a::t2_space net \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
3104 "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
3106 lemma continuous_trivial_limit:
3107 "trivial_limit net ==> continuous net f"
3108 unfolding continuous_def tendsto_def trivial_limit_eq by auto
3110 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
3111 unfolding continuous_def
3112 unfolding tendsto_def
3113 using netlimit_within[of x s]
3114 by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
3116 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
3117 using continuous_within [of x UNIV f] by (simp add: within_UNIV)
3119 lemma continuous_at_within:
3120 assumes "continuous (at x) f" shows "continuous (at x within s) f"
3121 using assms unfolding continuous_at continuous_within
3122 by (rule Lim_at_within)
3124 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
3126 lemma continuous_within_eps_delta:
3127 "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)"
3128 unfolding continuous_within and Lim_within
3129 apply auto unfolding dist_nz[THEN sym] apply(auto elim!:allE) apply(rule_tac x=d in exI) by auto
3131 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3132 \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
3133 using continuous_within_eps_delta[of x UNIV f]
3134 unfolding within_UNIV by blast
3136 text{* Versions in terms of open balls. *}
3138 lemma continuous_within_ball:
3139 "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3140 f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3143 { fix e::real assume "e>0"
3144 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"
3145 using `?lhs`[unfolded continuous_within Lim_within] by auto
3146 { fix y assume "y\<in>f ` (ball x d \<inter> s)"
3147 hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
3148 apply (auto simp add: dist_commute mem_ball) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
3150 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) }
3153 assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
3154 apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
3157 lemma continuous_at_ball:
3158 "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3160 assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3161 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)
3162 unfolding dist_nz[THEN sym] by auto
3164 assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3165 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)
3168 text{* For setwise continuity, just start from the epsilon-delta definitions. *}
3171 continuous_on :: "'a::metric_space set \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> bool" where
3172 "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d::real>0. \<forall>x' \<in> s. dist x' x < d --> dist (f x') (f x) < e)"
3176 uniformly_continuous_on ::
3177 "'a::metric_space set \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> bool" where
3178 "uniformly_continuous_on s f \<longleftrightarrow>
3179 (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall> x'\<in>s. dist x' x < d
3180 --> dist (f x') (f x) < e)"
3183 text{* Lifting and dropping *}
3185 lemma continuous_on_o_dest_vec1: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
3186 assumes "continuous_on {a..b::real} f" shows "continuous_on {vec1 a..vec1 b} (f o dest_vec1)"
3187 using assms unfolding continuous_on_def apply safe
3188 apply(erule_tac x="x$1" in ballE,erule_tac x=e in allE) apply safe
3189 apply(rule_tac x=d in exI) apply safe unfolding o_def dist_real_def dist_real
3190 apply(erule_tac x="dest_vec1 x'" in ballE) by(auto simp add:vector_le_def)
3192 lemma continuous_on_o_vec1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
3193 assumes "continuous_on {a..b} f" shows "continuous_on {dest_vec1 a..dest_vec1 b} (f o vec1)"
3194 using assms unfolding continuous_on_def apply safe
3195 apply(erule_tac x="vec x" in ballE,erule_tac x=e in allE) apply safe
3196 apply(rule_tac x=d in exI) apply safe unfolding o_def dist_real_def dist_real
3197 apply(erule_tac x="vec1 x'" in ballE) by(auto simp add:vector_le_def)
3199 text{* Some simple consequential lemmas. *}
3201 lemma uniformly_continuous_imp_continuous:
3202 " uniformly_continuous_on s f ==> continuous_on s f"
3203 unfolding uniformly_continuous_on_def continuous_on_def by blast
3205 lemma continuous_at_imp_continuous_within:
3206 "continuous (at x) f ==> continuous (at x within s) f"
3207 unfolding continuous_within continuous_at using Lim_at_within by auto
3209 lemma continuous_at_imp_continuous_on: assumes "(\<forall>x \<in> s. continuous (at x) f)"
3210 shows "continuous_on s f"
3211 proof(simp add: continuous_at continuous_on_def, rule, rule, rule)
3212 fix x and e::real assume "x\<in>s" "e>0"
3213 hence "eventually (\<lambda>xa. dist (f xa) (f x) < e) (at x)" using assms unfolding continuous_at tendsto_iff by auto
3214 then obtain d where d:"d>0" "\<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e" unfolding eventually_at by auto
3215 { fix x' assume "\<not> 0 < dist x' x"
3217 using dist_nz[of x' x] by auto
3218 hence "dist (f x') (f x) < e" using `e>0` by auto
3220 thus "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e" using d by auto
3223 lemma continuous_on_eq_continuous_within:
3224 "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)" (is "?lhs = ?rhs")
3227 { fix x assume "x\<in>s"
3228 fix e::real assume "e>0"
3229 assume "\<exists>d>0. \<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"
3230 then 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) < e" by auto
3231 { fix x' assume as:"x'\<in>s" "dist x' x < d"
3232 hence "dist (f x') (f x) < e" using `e>0` d `x'\<in>s` dist_eq_0_iff[of x' x] zero_le_dist[of x' x] as(2) by (metis dist_eq_0_iff dist_nz) }
3233 hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e" using `d>0` by auto
3235 thus ?lhs using `?rhs` unfolding continuous_on_def continuous_within Lim_within by auto
3238 thus ?rhs unfolding continuous_on_def continuous_within Lim_within by blast
3241 lemma continuous_on:
3242 "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. (f ---> f(x)) (at x within s))"
3243 by (auto simp add: continuous_on_eq_continuous_within continuous_within)
3245 lemma continuous_on_eq_continuous_at:
3246 "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
3247 by (auto simp add: continuous_on continuous_at Lim_within_open)
3249 lemma continuous_within_subset:
3250 "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
3251 ==> continuous (at x within t) f"
3252 unfolding continuous_within by(metis Lim_within_subset)
3254 lemma continuous_on_subset:
3255 "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
3256 unfolding continuous_on by (metis subset_eq Lim_within_subset)
3258 lemma continuous_on_interior:
3259 "continuous_on s f \<Longrightarrow> x \<in> interior s ==> continuous (at x) f"
3260 unfolding interior_def
3262 by (meson continuous_on_eq_continuous_at continuous_on_subset)
3264 lemma continuous_on_eq:
3265 "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f
3266 ==> continuous_on s g"
3267 by (simp add: continuous_on_def)
3269 text{* Characterization of various kinds of continuity in terms of sequences. *}
3271 (* \<longrightarrow> could be generalized, but \<longleftarrow> requires metric space *)
3272 lemma continuous_within_sequentially:
3273 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3274 shows "continuous (at a within s) f \<longleftrightarrow>
3275 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
3276 --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
3279 { fix x::"nat \<Rightarrow> 'a" assume x:"\<forall>n. x n \<in> s" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (x n) a < e"
3280 fix e::real assume "e>0"
3281 from `?lhs` obtain d where "d>0" and d:"\<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) (f a) < e" unfolding continuous_within Lim_within using `e>0` by auto
3282 from x(2) `d>0` obtain N where N:"\<forall>n\<ge>N. dist (x n) a < d" by auto
3283 hence "\<exists>N. \<forall>n\<ge>N. dist ((f \<circ> x) n) (f a) < e"
3284 apply(rule_tac x=N in exI) using N d apply auto using x(1)
3285 apply(erule_tac x=n in allE) apply(erule_tac x=n in allE)
3286 apply(erule_tac x="x n" in ballE) apply auto unfolding dist_nz[THEN sym] apply auto using `e>0` by auto
3288 thus ?rhs unfolding continuous_within unfolding Lim_sequentially by simp
3291 { fix e::real assume "e>0"
3292 assume "\<not> (\<exists>d>0. \<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) (f a) < e)"
3293 hence "\<forall>d. \<exists>x. d>0 \<longrightarrow> x\<in>s \<and> (0 < dist x a \<and> dist x a < d \<and> \<not> dist (f x) (f a) < e)" by blast
3294 then obtain x where x:"\<forall>d>0. x d \<in> s \<and> (0 < dist (x d) a \<and> dist (x d) a < d \<and> \<not> dist (f (x d)) (f a) < e)"
3295 using choice[of "\<lambda>d x.0<d \<longrightarrow> x\<in>s \<and> (0 < dist x a \<and> dist x a < d \<and> \<not> dist (f x) (f a) < e)"] by auto
3296 { fix d::real assume "d>0"
3297 hence "\<exists>N::nat. inverse (real (N + 1)) < d" using real_arch_inv[of d] by (auto, rule_tac x="n - 1" in exI)auto
3298 then obtain N::nat where N:"inverse (real (N + 1)) < d" by auto
3299 { fix n::nat assume n:"n\<ge>N"
3300 hence "dist (x (inverse (real (n + 1)))) a < inverse (real (n + 1))" using x[THEN spec[where x="inverse (real (n + 1))"]] by auto
3301 moreover have "inverse (real (n + 1)) < d" using N n 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)
3302 ultimately have "dist (x (inverse (real (n + 1)))) a < d" by auto
3304 hence "\<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < d" by auto
3306 hence "(\<forall>n::nat. x (inverse (real (n + 1))) \<in> s) \<and> (\<forall>e>0. \<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < e)" using x by auto
3307 hence "\<forall>e>0. \<exists>N::nat. \<forall>n\<ge>N. dist (f (x (inverse (real (n + 1))))) (f a) < e" using `?rhs`[THEN spec[where x="\<lambda>n::nat. x (inverse (real (n+1)))"], unfolded Lim_sequentially] by auto
3308 hence "False" apply(erule_tac x=e in allE) using `e>0` using x by auto
3310 thus ?lhs unfolding continuous_within unfolding Lim_within unfolding Lim_sequentially by blast
3313 lemma continuous_at_sequentially:
3314 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3315 shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
3316 --> ((f o x) ---> f a) sequentially)"
3317 using continuous_within_sequentially[of a UNIV f] unfolding within_UNIV by auto
3319 lemma continuous_on_sequentially:
3320 "continuous_on s f \<longleftrightarrow> (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
3321 --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
3323 assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
3325 assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
3328 lemma uniformly_continuous_on_sequentially:
3329 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
3330 shows "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
3331 ((\<lambda>n. x n - y n) ---> 0) sequentially
3332 \<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")
3335 { fix x y assume x:"\<forall>n. x n \<in> s" and y:"\<forall>n. y n \<in> s" and xy:"((\<lambda>n. x n - y n) ---> 0) sequentially"
3336 { fix e::real assume "e>0"
3337 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"
3338 using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
3339 obtain N where N:"\<forall>n\<ge>N. norm (x n - y n - 0) < d" using xy[unfolded Lim_sequentially dist_norm] and `d>0` by auto
3340 { fix n assume "n\<ge>N"
3341 hence "norm (f (x n) - f (y n) - 0) < e"
3342 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
3343 unfolding dist_commute and dist_norm by simp }
3344 hence "\<exists>N. \<forall>n\<ge>N. norm (f (x n) - f (y n) - 0) < e" by auto }
3345 hence "((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially" unfolding Lim_sequentially and dist_norm by auto }
3349 { assume "\<not> ?lhs"
3350 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
3351 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"
3352 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
3353 by (auto simp add: dist_commute)
3354 def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
3355 def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
3356 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"
3357 unfolding x_def and y_def using fa by auto
3358 have 1:"\<And>(x::'a) y. dist (x - y) 0 = dist x y" unfolding dist_norm by auto
3359 have 2:"\<And>(x::'b) y. dist (x - y) 0 = dist x y" unfolding dist_norm by auto
3360 { fix e::real assume "e>0"
3361 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
3362 { fix n::nat assume "n\<ge>N"
3363 hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
3364 also have "\<dots> < e" using N by auto
3365 finally have "inverse (real n + 1) < e" by auto
3366 hence "dist (x n - y n) 0 < e" unfolding 1 using xy0[THEN spec[where x=n]] by auto }
3367 hence "\<exists>N. \<forall>n\<ge>N. dist (x n - y n) 0 < e" by auto }
3368 hence "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n) - f (y n)) 0 < e" using `?rhs`[THEN spec[where x=x], THEN spec[where x=y]] and xyn unfolding Lim_sequentially by auto
3369 hence False unfolding 2 using fxy and `e>0` by auto }
3370 thus ?lhs unfolding uniformly_continuous_on_def by blast
3373 text{* The usual transformation theorems. *}
3375 lemma continuous_transform_within:
3376 fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3377 assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
3378 "continuous (at x within s) f"
3379 shows "continuous (at x within s) g"
3381 { fix e::real assume "e>0"
3382 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" using assms(4) unfolding continuous_within Lim_within by auto
3383 { fix x' assume "x'\<in>s" "0 < dist x' x" "dist x' x < (min d d')"
3384 hence "dist (f x') (g x) < e" using assms(2,3) apply(erule_tac x=x in ballE) using d' by auto }
3385 hence "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < (min d d') \<longrightarrow> dist (f xa) (g x) < e" by blast
3386 hence "\<exists>d>0. \<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (g x) < e" using `d>0` `d'>0` by(rule_tac x="min d d'" in exI)auto }
3387 hence "(f ---> g x) (at x within s)" unfolding Lim_within using assms(1) by auto
3388 thus ?thesis unfolding continuous_within using Lim_transform_within[of d s x f g "g x"] using assms by blast
3391 lemma continuous_transform_at:
3392 fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3393 assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
3394 "continuous (at x) f"
3395 shows "continuous (at x) g"
3397 { fix e::real assume "e>0"
3398 then obtain d' where d':"d'>0" "\<forall>xa. 0 < dist xa x \<and> dist xa x < d' \<longrightarrow> dist (f xa) (f x) < e" using assms(3) unfolding continuous_at Lim_at by auto
3399 { fix x' assume "0 < dist x' x" "dist x' x < (min d d')"
3400 hence "dist (f x') (g x) < e" using assms(2) apply(erule_tac x=x in allE) using d' by auto
3402 hence "\<forall>xa. 0 < dist xa x \<and> dist xa x < (min d d') \<longrightarrow> dist (f xa) (g x) < e" by blast
3403 hence "\<exists>d>0. \<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (g x) < e" using `d>0` `d'>0` by(rule_tac x="min d d'" in exI)auto
3405 hence "(f ---> g x) (at x)" unfolding Lim_at using assms(1) by auto
3406 thus ?thesis unfolding continuous_at using Lim_transform_at[of d x f g "g x"] using assms by blast
3409 text{* Combination results for pointwise continuity. *}
3411 lemma continuous_const: "continuous net (\<lambda>x. c)"
3412 by (auto simp add: continuous_def Lim_const)
3414 lemma continuous_cmul:
3415 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3416 shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"
3417 by (auto simp add: continuous_def Lim_cmul)
3419 lemma continuous_neg:
3420 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3421 shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"
3422 by (auto simp add: continuous_def Lim_neg)
3424 lemma continuous_add:
3425 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3426 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"
3427 by (auto simp add: continuous_def Lim_add)
3429 lemma continuous_sub:
3430 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3431 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"
3432 by (auto simp add: continuous_def Lim_sub)
3435 text{* Same thing for setwise continuity. *}
3437 lemma continuous_on_const:
3438 "continuous_on s (\<lambda>x. c)"
3439 unfolding continuous_on_eq_continuous_within using continuous_const by blast
3441 lemma continuous_on_cmul:
3442 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3443 shows "continuous_on s f ==> continuous_on s (\<lambda>x. c *\<^sub>R (f x))"
3444 unfolding continuous_on_eq_continuous_within using continuous_cmul by blast
3446 lemma continuous_on_neg:
3447 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3448 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
3449 unfolding continuous_on_eq_continuous_within using continuous_neg by blast
3451 lemma continuous_on_add:
3452 fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3453 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3454 \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
3455 unfolding continuous_on_eq_continuous_within using continuous_add by blast
3457 lemma continuous_on_sub:
3458 fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3459 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3460 \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
3461 unfolding continuous_on_eq_continuous_within using continuous_sub by blast
3463 text{* Same thing for uniform continuity, using sequential formulations. *}
3465 lemma uniformly_continuous_on_const:
3466 "uniformly_continuous_on s (\<lambda>x. c)"
3467 unfolding uniformly_continuous_on_def by simp
3469 lemma uniformly_continuous_on_cmul:
3470 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
3471 (* FIXME: generalize 'a to metric_space *)
3472 assumes "uniformly_continuous_on s f"
3473 shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
3475 { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
3476 hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"
3477 using Lim_cmul[of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
3478 unfolding scaleR_zero_right scaleR_right_diff_distrib by auto
3480 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially by auto
3484 fixes x y :: "'a::real_normed_vector"
3485 shows "dist (- x) (- y) = dist x y"
3486 unfolding dist_norm minus_diff_minus norm_minus_cancel ..
3488 lemma uniformly_continuous_on_neg:
3489 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3490 shows "uniformly_continuous_on s f
3491 ==> uniformly_continuous_on s (\<lambda>x. -(f x))"
3492 unfolding uniformly_continuous_on_def dist_minus .
3494 lemma uniformly_continuous_on_add:
3495 fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" (* FIXME: generalize 'a *)
3496 assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"
3497 shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
3499 { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
3500 "((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"
3501 hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"
3502 using Lim_add[of "\<lambda> n. f (x n) - f (y n)" 0 sequentially "\<lambda> n. g (x n) - g (y n)" 0] by auto
3503 hence "((\<lambda>n. f (x n) + g (x n) - (f (y n) + g (y n))) ---> 0) sequentially" unfolding Lim_sequentially and add_diff_add [symmetric] by auto }
3504 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially by auto
3507 lemma uniformly_continuous_on_sub:
3508 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" (* FIXME: generalize 'a *)
3509 shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s g
3510 ==> uniformly_continuous_on s (\<lambda>x. f x - g x)"
3511 unfolding ab_diff_minus
3512 using uniformly_continuous_on_add[of s f "\<lambda>x. - g x"]
3513 using uniformly_continuous_on_neg[of s g] by auto
3515 text{* Identity function is continuous in every sense. *}
3517 lemma continuous_within_id:
3518 "continuous (at a within s) (\<lambda>x. x)"
3519 unfolding continuous_within by (rule Lim_at_within [OF Lim_ident_at])
3521 lemma continuous_at_id:
3522 "continuous (at a) (\<lambda>x. x)"
3523 unfolding continuous_at by (rule Lim_ident_at)
3525 lemma continuous_on_id:
3526 "continuous_on s (\<lambda>x. x)"
3527 unfolding continuous_on Lim_within by auto
3529 lemma uniformly_continuous_on_id:
3530 "uniformly_continuous_on s (\<lambda>x. x)"
3531 unfolding uniformly_continuous_on_def by auto
3533 text{* Continuity of all kinds is preserved under composition. *}
3535 lemma continuous_within_compose:
3536 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3537 fixes g :: "'b::metric_space \<Rightarrow> 'c::metric_space"
3538 assumes "continuous (at x within s) f" "continuous (at (f x) within f ` s) g"
3539 shows "continuous (at x within s) (g o f)"
3541 { fix e::real assume "e>0"
3542 with assms(2)[unfolded continuous_within Lim_within] obtain d where "d>0" and d:"\<forall>xa\<in>f ` s. 0 < dist xa (f x) \<and> dist xa (f x) < d \<longrightarrow> dist (g xa) (g (f x)) < e" by auto
3543 from assms(1)[unfolded continuous_within Lim_within] 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) < d" using `d>0` by auto
3544 { fix y assume as:"y\<in>s" "0 < dist y x" "dist y x < d'"
3545 hence "dist (f y) (f x) < d" using d'[THEN bspec[where x=y]] by (auto simp add:dist_commute)
3546 hence "dist (g (f y)) (g (f x)) < e" using as(1) d[THEN bspec[where x="f y"]] unfolding dist_nz[THEN sym] using `e>0` by auto }
3547 hence "\<exists>d>0. \<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (g (f xa)) (g (f x)) < e" using `d'>0` by auto }
3548 thus ?thesis unfolding continuous_within Lim_within by auto
3551 lemma continuous_at_compose:
3552 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3553 fixes g :: "'b::metric_space \<Rightarrow> 'c::metric_space"
3554 assumes "continuous (at x) f" "continuous (at (f x)) g"
3555 shows "continuous (at x) (g o f)"
3557 have " continuous (at (f x) within range f) g" using assms(2) using continuous_within_subset[of "f x" UNIV g "range f", unfolded within_UNIV] by auto
3558 thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g, unfolded within_UNIV] by auto
3561 lemma continuous_on_compose:
3562 "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
3563 unfolding continuous_on_eq_continuous_within using continuous_within_compose[of _ s f g] by auto
3565 lemma uniformly_continuous_on_compose:
3566 assumes "uniformly_continuous_on s f" "uniformly_continuous_on (f ` s) g"
3567 shows "uniformly_continuous_on s (g o f)"
3569 { fix e::real assume "e>0"
3570 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
3571 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
3572 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 }
3573 thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
3576 text{* Continuity in terms of open preimages. *}
3578 lemma continuous_at_open:
3579 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3580 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)))" (is "?lhs = ?rhs")
3583 { fix t assume as: "open t" "f x \<in> t"
3584 then obtain e where "e>0" and e:"ball (f x) e \<subseteq> t" unfolding open_contains_ball by auto
3586 obtain d where "d>0" and d:"\<forall>y. 0 < dist y x \<and> dist y x < d \<longrightarrow> dist (f y) (f x) < e" using `e>0` using `?lhs`[unfolded continuous_at Lim_at open_dist] by auto
3588 have "open (ball x d)" using open_ball by auto
3589 moreover have "x \<in> ball x d" unfolding centre_in_ball using `d>0` by simp
3591 { fix x' assume "x'\<in>ball x d" hence "f x' \<in> t"
3592 using e[unfolded subset_eq Ball_def mem_ball, THEN spec[where x="f x'"]] d[THEN spec[where x=x']]
3593 unfolding mem_ball apply (auto simp add: dist_commute)
3594 unfolding dist_nz[THEN sym] using as(2) by auto }
3595 hence "\<forall>x'\<in>ball x d. f x' \<in> t" by auto
3596 ultimately have "\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x'\<in>s. f x' \<in> t)"
3597 apply(rule_tac x="ball x d" in exI) by simp }
3601 { fix e::real assume "e>0"
3602 then obtain s where s: "open s" "x \<in> s" "\<forall>x'\<in>s. f x' \<in> ball (f x) e" using `?rhs`[unfolded continuous_at Lim_at, THEN spec[where x="ball (f x) e"]]
3603 unfolding centre_in_ball[of "f x" e, THEN sym] by auto
3604 then obtain d where "d>0" and d:"ball x d \<subseteq> s" unfolding open_contains_ball by auto
3605 { fix y assume "0 < dist y x \<and> dist y x < d"
3606 hence "dist (f y) (f x) < e" using d[unfolded subset_eq Ball_def mem_ball, THEN spec[where x=y]]
3607 using s(3)[THEN bspec[where x=y], unfolded mem_ball] by (auto simp add: dist_commute) }
3608 hence "\<exists>d>0. \<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e" using `d>0` by auto }
3609 thus ?lhs unfolding continuous_at Lim_at by auto
3612 lemma continuous_on_open:
3613 "continuous_on s f \<longleftrightarrow>
3614 (\<forall>t. openin (subtopology euclidean (f ` s)) t
3615 --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
3618 { fix t assume as:"openin (subtopology euclidean (f ` s)) t"
3619 have "{x \<in> s. f x \<in> t} \<subseteq> s" using as[unfolded openin_euclidean_subtopology_iff] by auto
3621 { fix x assume as':"x\<in>{x \<in> s. f x \<in> t}"
3622 then obtain e where e: "e>0" "\<forall>x'\<in>f ` s. dist x' (f x) < e \<longrightarrow> x' \<in> t" using as[unfolded openin_euclidean_subtopology_iff, THEN conjunct2, THEN bspec[where x="f x"]] by auto
3623 from this(1) 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" using `?lhs`[unfolded continuous_on Lim_within, THEN bspec[where x=x]] using as' by auto
3624 have "\<exists>e>0. \<forall>x'\<in>s. dist x' x < e \<longrightarrow> x' \<in> {x \<in> s. f x \<in> t}" using d e unfolding dist_nz[THEN sym] by (rule_tac x=d in exI, auto) }
3625 ultimately have "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" unfolding openin_euclidean_subtopology_iff by auto }
3626 thus ?rhs unfolding continuous_on Lim_within using openin by auto
3629 { fix e::real and x assume "x\<in>s" "e>0"
3630 { fix xa x' assume "dist (f xa) (f x) < e" "xa \<in> s" "x' \<in> s" "dist (f xa) (f x') < e - dist (f xa) (f x)"
3631 hence "dist (f x') (f x) < e" using dist_triangle[of "f x'" "f x" "f xa"]
3632 by (auto simp add: dist_commute) }
3633 hence "ball (f x) e \<inter> f ` s \<subseteq> f ` s \<and> (\<forall>xa\<in>ball (f x) e \<inter> f ` s. \<exists>ea>0. \<forall>x'\<in>f ` s. dist x' xa < ea \<longrightarrow> x' \<in> ball (f x) e \<inter> f ` s)" apply auto
3634 apply(rule_tac x="e - dist (f xa) (f x)" in exI) using `e>0` by (auto simp add: dist_commute)
3635 hence "\<forall>xa\<in>{xa \<in> s. f xa \<in> ball (f x) e \<inter> f ` s}. \<exists>ea>0. \<forall>x'\<in>s. dist x' xa < ea \<longrightarrow> x' \<in> {xa \<in> s. f xa \<in> ball (f x) e \<inter> f ` s}"
3636 using `?rhs`[unfolded openin_euclidean_subtopology_iff, THEN spec[where x="ball (f x) e \<inter> f ` s"]] by auto
3637 hence "\<exists>d>0. \<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e" apply(erule_tac x=x in ballE) apply auto using `e>0` `x\<in>s` by (auto simp add: dist_commute) }
3638 thus ?lhs unfolding continuous_on Lim_within by auto
3641 (* ------------------------------------------------------------------------- *)
3642 (* Similarly in terms of closed sets. *)
3643 (* ------------------------------------------------------------------------- *)
3645 lemma continuous_on_closed:
3646 "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")
3650 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3651 have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
3652 assume as:"closedin (subtopology euclidean (f ` s)) t"
3653 hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
3654 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"]]
3655 unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto }
3660 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3661 assume as:"openin (subtopology euclidean (f ` s)) t"
3662 hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
3663 unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
3664 thus ?lhs unfolding continuous_on_open by auto
3667 text{* Half-global and completely global cases. *}
3669 lemma continuous_open_in_preimage:
3670 assumes "continuous_on s f" "open t"
3671 shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3673 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
3674 have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3675 using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
3676 thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3679 lemma continuous_closed_in_preimage:
3680 assumes "continuous_on s f" "closed t"
3681 shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3683 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
3684 have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3685 using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
3687 using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3690 lemma continuous_open_preimage:
3691 assumes "continuous_on s f" "open s" "open t"
3692 shows "open {x \<in> s. f x \<in> t}"
3694 obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3695 using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
3696 thus ?thesis using open_Int[of s T, OF assms(2)] by auto
3699 lemma continuous_closed_preimage:
3700 assumes "continuous_on s f" "closed s" "closed t"
3701 shows "closed {x \<in> s. f x \<in> t}"
3703 obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3704 using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
3705 thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
3708 lemma continuous_open_preimage_univ:
3709 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3710 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
3711 using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
3713 lemma continuous_closed_preimage_univ:
3714 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3715 shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
3716 using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
3718 lemma continuous_open_vimage:
3719 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3720 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
3721 unfolding vimage_def by (rule continuous_open_preimage_univ)
3723 lemma continuous_closed_vimage:
3724 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3725 shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
3726 unfolding vimage_def by (rule continuous_closed_preimage_univ)
3728 lemma interior_image_subset: fixes f::"_::metric_space \<Rightarrow> _::metric_space"
3729 assumes "\<forall>x. continuous (at x) f" "inj f"
3730 shows "interior (f ` s) \<subseteq> f ` (interior s)"
3731 apply rule unfolding interior_def mem_Collect_eq image_iff apply safe
3732 proof- fix x T assume as:"open T" "x \<in> T" "T \<subseteq> f ` s"
3733 hence "x \<in> f ` s" by auto then guess y unfolding image_iff .. note y=this
3734 thus "\<exists>xa\<in>{x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> s}. x = f xa" apply(rule_tac x=y in bexI) using assms as
3735 apply safe apply(rule_tac x="{x. f x \<in> T}" in exI) apply(safe,rule continuous_open_preimage_univ)
3736 proof- fix x assume "f x \<in> T" hence "f x \<in> f ` s" using as by auto
3737 thus "x \<in> s" unfolding inj_image_mem_iff[OF assms(2)] . qed auto qed
3739 text{* Equality of continuous functions on closure and related results. *}
3741 lemma continuous_closed_in_preimage_constant:
3742 "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
3743 using continuous_closed_in_preimage[of s f "{a}"] closed_sing by auto
3745 lemma continuous_closed_preimage_constant:
3746 "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
3747 using continuous_closed_preimage[of s f "{a}"] closed_sing by auto
3749 lemma continuous_constant_on_closure:
3750 assumes "continuous_on (closure s) f"
3751 "\<forall>x \<in> s. f x = a"
3752 shows "\<forall>x \<in> (closure s). f x = a"
3753 using continuous_closed_preimage_constant[of "closure s" f a]
3754 assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
3756 lemma image_closure_subset:
3757 assumes "continuous_on (closure s) f" "closed t" "(f ` s) \<subseteq> t"
3758 shows "f ` (closure s) \<subseteq> t"
3760 have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
3761 moreover have "closed {x \<in> closure s. f x \<in> t}"
3762 using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
3763 ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
3764 using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
3765 thus ?thesis by auto
3768 lemma continuous_on_closure_norm_le:
3769 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3770 assumes "continuous_on (closure s) f" "\<forall>y \<in> s. norm(f y) \<le> b" "x \<in> (closure s)"
3771 shows "norm(f x) \<le> b"
3773 have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
3775 using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
3776 unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
3779 text{* Making a continuous function avoid some value in a neighbourhood. *}
3781 lemma continuous_within_avoid:
3782 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3783 assumes "continuous (at x within s) f" "x \<in> s" "f x \<noteq> a"
3784 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
3786 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"
3787 using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto
3788 { fix y assume " y\<in>s" "dist x y < d"
3789 hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]
3790 apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }
3791 thus ?thesis using `d>0` by auto
3794 lemma continuous_at_avoid:
3795 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3796 assumes "continuous (at x) f" "f x \<noteq> a"
3797 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
3798 using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto
3800 lemma continuous_on_avoid:
3801 assumes "continuous_on s f" "x \<in> s" "f x \<noteq> a"
3802 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
3803 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
3805 lemma continuous_on_open_avoid:
3806 assumes "continuous_on s f" "open s" "x \<in> s" "f x \<noteq> a"
3807 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
3808 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
3810 text{* Proving a function is constant by proving open-ness of level set. *}
3812 lemma continuous_levelset_open_in_cases:
3813 "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
3814 openin (subtopology euclidean s) {x \<in> s. f x = a}
3815 ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
3816 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
3818 lemma continuous_levelset_open_in:
3819 "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
3820 openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
3821 (\<exists>x \<in> s. f x = a) ==> (\<forall>x \<in> s. f x = a)"
3822 using continuous_levelset_open_in_cases[of s f ]
3825 lemma continuous_levelset_open:
3826 assumes "connected s" "continuous_on s f" "open {x \<in> s. f x = a}" "\<exists>x \<in> s. f x = a"
3827 shows "\<forall>x \<in> s. f x = a"
3828 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by auto
3830 text{* Some arithmetical combinations (more to prove). *}
3832 lemma open_scaling[intro]:
3833 fixes s :: "'a::real_normed_vector set"
3834 assumes "c \<noteq> 0" "open s"
3835 shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
3837 { fix x assume "x \<in> s"
3838 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
3839 have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using real_mult_order[OF `e>0`] by auto
3841 { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
3842 hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
3843 using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
3844 assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
3845 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 }
3846 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 }
3847 thus ?thesis unfolding open_dist by auto
3850 lemma minus_image_eq_vimage:
3851 fixes A :: "'a::ab_group_add set"
3852 shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
3853 by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
3855 lemma open_negations:
3856 fixes s :: "'a::real_normed_vector set"
3857 shows "open s ==> open ((\<lambda> x. -x) ` s)"
3858 unfolding scaleR_minus1_left [symmetric]
3859 by (rule open_scaling, auto)
3861 lemma open_translation:
3862 fixes s :: "'a::real_normed_vector set"
3863 assumes "open s" shows "open((\<lambda>x. a + x) ` s)"
3865 { fix x have "continuous (at x) (\<lambda>x. x - a)" using continuous_sub[of "at x" "\<lambda>x. x" "\<lambda>x. a"] continuous_at_id[of x] continuous_const[of "at x" a] by auto }
3866 moreover have "{x. x - a \<in> s} = op + a ` s" apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
3867 ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
3870 lemma open_affinity:
3871 fixes s :: "'a::real_normed_vector set"
3872 assumes "open s" "c \<noteq> 0"
3873 shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
3875 have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
3876 have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
3877 thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
3880 lemma interior_translation:
3881 fixes s :: "'a::real_normed_vector set"
3882 shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
3883 proof (rule set_ext, rule)
3884 fix x assume "x \<in> interior (op + a ` s)"
3885 then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
3886 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
3887 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
3889 fix x assume "x \<in> op + a ` interior s"
3890 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
3891 { fix z have *:"a + y - z = y + a - z" by auto
3892 assume "z\<in>ball x e"
3893 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
3894 hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"]) }
3895 hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
3896 thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
3899 subsection {* Preservation of compactness and connectedness under continuous function. *}
3901 lemma compact_continuous_image:
3902 assumes "continuous_on s f" "compact s"
3903 shows "compact(f ` s)"
3905 { fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
3906 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
3907 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
3908 { fix e::real assume "e>0"
3909 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_def, THEN bspec[where x=l], OF `l\<in>s`] by auto
3910 then obtain N::nat where N:"\<forall>n\<ge>N. dist ((y \<circ> r) n) l < d" using lr[unfolded Lim_sequentially, THEN spec[where x=d]] by auto
3911 { 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 }
3912 hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto }
3913 hence "\<exists>l\<in>f ` s. \<exists>r. subseq r \<and> ((x \<circ> r) ---> l) sequentially" unfolding Lim_sequentially using r lr `l\<in>s` by auto }
3914 thus ?thesis unfolding compact_def by auto
3917 lemma connected_continuous_image:
3918 assumes "continuous_on s f" "connected s"
3919 shows "connected(f ` s)"
3921 { fix T assume as: "T \<noteq> {}" "T \<noteq> f ` s" "openin (subtopology euclidean (f ` s)) T" "closedin (subtopology euclidean (f ` s)) T"
3922 have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
3923 using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
3924 using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
3925 using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
3926 hence False using as(1,2)
3927 using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
3928 thus ?thesis unfolding connected_clopen by auto
3931 text{* Continuity implies uniform continuity on a compact domain. *}
3933 lemma compact_uniformly_continuous:
3934 assumes "continuous_on s f" "compact s"
3935 shows "uniformly_continuous_on s f"
3937 { fix x assume x:"x\<in>s"
3938 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_def, THEN bspec[where x=x]] by auto
3939 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 }
3940 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
3941 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)"
3942 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
3944 { fix e::real assume "e>0"
3946 { 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 }
3947 hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto
3949 { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto }
3950 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
3952 { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"
3953 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
3954 hence "x\<in>ball z (d z (e / 2))" using `ea>0` unfolding subset_eq by auto
3955 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`
3956 by (auto simp add: dist_commute)
3957 moreover have "y\<in>ball z (d z (e / 2))" using as and `ea>0` and z[unfolded subset_eq]
3958 by (auto simp add: dist_commute)
3959 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`
3960 by (auto simp add: dist_commute)
3961 ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]
3962 by (auto simp add: dist_commute) }
3963 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 }
3964 thus ?thesis unfolding uniformly_continuous_on_def by auto
3967 text{* Continuity of inverse function on compact domain. *}
3969 lemma continuous_on_inverse:
3970 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
3971 (* TODO: can this be generalized more? *)
3972 assumes "continuous_on s f" "compact s" "\<forall>x \<in> s. g (f x) = x"
3973 shows "continuous_on (f ` s) g"
3975 have *:"g ` f ` s = s" using assms(3) by (auto simp add: image_iff)
3976 { fix t assume t:"closedin (subtopology euclidean (g ` f ` s)) t"
3977 then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto
3978 have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]
3979 unfolding T(2) and Int_left_absorb by auto
3980 moreover have "compact (s \<inter> T)"
3981 using assms(2) unfolding compact_eq_bounded_closed
3982 using bounded_subset[of s "s \<inter> T"] and T(1) by auto
3983 ultimately have "closed (f ` t)" using T(1) unfolding T(2)
3984 using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto
3985 moreover have "{x \<in> f ` s. g x \<in> t} = f ` s \<inter> f ` t" using assms(3) unfolding T(2) by auto
3986 ultimately have "closedin (subtopology euclidean (f ` s)) {x \<in> f ` s. g x \<in> t}"
3987 unfolding closedin_closed by auto }
3988 thus ?thesis unfolding continuous_on_closed by auto
3991 subsection{* A uniformly convergent limit of continuous functions is continuous. *}
3993 lemma norm_triangle_lt:
3994 fixes x y :: "'a::real_normed_vector"
3995 shows "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
3996 by (rule le_less_trans [OF norm_triangle_ineq])
3998 lemma continuous_uniform_limit:
3999 fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::real_normed_vector"
4000 assumes "\<not> (trivial_limit net)" "eventually (\<lambda>n. continuous_on s (f n)) net"
4001 "\<forall>e>0. eventually (\<lambda>n. \<forall>x \<in> s. norm(f n x - g x) < e) net"
4002 shows "continuous_on s g"
4004 { fix x and e::real assume "x\<in>s" "e>0"
4005 have "eventually (\<lambda>n. \<forall>x\<in>s. norm (f n x - g x) < e / 3) net" using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto
4006 then obtain n where n:"\<forall>xa\<in>s. norm (f n xa - g xa) < e / 3" "continuous_on s (f n)"
4007 using eventually_and[of "(\<lambda>n. \<forall>x\<in>s. norm (f n x - g x) < e / 3)" "(\<lambda>n. continuous_on s (f n))" net] assms(1,2) eventually_happens by blast
4008 have "e / 3 > 0" using `e>0` by auto
4009 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"
4010 using n(2)[unfolded continuous_on_def, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
4011 { fix y assume "y\<in>s" "dist y x < d"
4012 hence "dist (f n y) (f n x) < e / 3" using d[THEN bspec[where x=y]] by auto
4013 hence "norm (f n y - g x) < 2 * e / 3" using norm_triangle_lt[of "f n y - f n x" "f n x - g x" "2*e/3"]
4014 using n(1)[THEN bspec[where x=x], OF `x\<in>s`] unfolding dist_norm unfolding ab_group_add_class.ab_diff_minus by auto
4015 hence "dist (g y) (g x) < e" unfolding dist_norm using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
4016 unfolding norm_minus_cancel[of "f n y - g y", THEN sym] using norm_triangle_lt[of "f n y - g x" "g y - f n y" e] by (auto simp add: uminus_add_conv_diff) }
4017 hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using `d>0` by auto }
4018 thus ?thesis unfolding continuous_on_def by auto
4021 subsection{* Topological properties of linear functions. *}
4024 assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
4026 interpret f: bounded_linear f by fact
4027 have "(f ---> f 0) (at 0)"
4028 using tendsto_ident_at by (rule f.tendsto)
4029 thus ?thesis unfolding f.zero .
4032 lemma linear_continuous_at:
4033 assumes "bounded_linear f" shows "continuous (at a) f"
4034 unfolding continuous_at using assms
4035 apply (rule bounded_linear.tendsto)
4036 apply (rule tendsto_ident_at)
4039 lemma linear_continuous_within:
4040 shows "bounded_linear f ==> continuous (at x within s) f"
4041 using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
4043 lemma linear_continuous_on:
4044 shows "bounded_linear f ==> continuous_on s f"
4045 using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
4047 lemma continuous_on_vec1:"continuous_on A (vec1::real\<Rightarrow>real^1)"
4048 by(rule linear_continuous_on[OF bounded_linear_vec1])
4050 text{* Also bilinear functions, in composition form. *}
4052 lemma bilinear_continuous_at_compose:
4053 shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
4054 ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
4055 unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
4057 lemma bilinear_continuous_within_compose:
4058 shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
4059 ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
4060 unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
4062 lemma bilinear_continuous_on_compose:
4063 shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
4064 ==> continuous_on s (\<lambda>x. h (f x) (g x))"
4065 unfolding continuous_on_eq_continuous_within apply auto apply(erule_tac x=x in ballE) apply auto apply(erule_tac x=x in ballE) apply auto
4066 using bilinear_continuous_within_compose[of _ s f g h] by auto
4068 subsection{* Topological stuff lifted from and dropped to R *}
4072 fixes s :: "real set" shows
4073 "open s \<longleftrightarrow>
4074 (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
4075 unfolding open_dist dist_norm by simp
4077 lemma islimpt_approachable_real:
4078 fixes s :: "real set"
4079 shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
4080 unfolding islimpt_approachable dist_norm by simp
4083 fixes s :: "real set"
4084 shows "closed s \<longleftrightarrow>
4085 (\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
4087 unfolding closed_limpt islimpt_approachable dist_norm by simp
4089 lemma continuous_at_real_range:
4090 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4091 shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
4092 \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
4093 unfolding continuous_at unfolding Lim_at
4094 unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
4095 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
4096 apply(erule_tac x=e in allE) by auto
4098 lemma continuous_on_real_range:
4099 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4100 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))"
4101 unfolding continuous_on_def dist_norm by simp
4103 lemma continuous_at_norm: "continuous (at x) norm"
4104 unfolding continuous_at by (intro tendsto_intros)
4106 lemma continuous_on_norm: "continuous_on s norm"
4107 unfolding continuous_on by (intro ballI tendsto_intros)
4109 lemma continuous_at_component: "continuous (at a) (\<lambda>x. x $ i)"
4110 unfolding continuous_at by (intro tendsto_intros)
4112 lemma continuous_on_component: "continuous_on s (\<lambda>x. x $ i)"
4113 unfolding continuous_on by (intro ballI tendsto_intros)
4115 lemma continuous_at_infnorm: "continuous (at x) infnorm"
4116 unfolding continuous_at Lim_at o_def unfolding dist_norm
4117 apply auto apply (rule_tac x=e in exI) apply auto
4118 using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
4120 text{* Hence some handy theorems on distance, diameter etc. of/from a set. *}
4122 lemma compact_attains_sup:
4123 fixes s :: "real set"
4124 assumes "compact s" "s \<noteq> {}"
4125 shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
4127 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4128 { 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"
4129 have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto
4130 moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
4131 ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using `e>0` by auto }
4132 thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]
4133 apply(rule_tac x="Sup s" in bexI) by auto
4137 fixes S :: "real set"
4138 shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
4139 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def)
4141 lemma compact_attains_inf:
4142 fixes s :: "real set"
4143 assumes "compact s" "s \<noteq> {}" shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
4145 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4146 { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s" "Inf s \<notin> s" "0 < e"
4147 "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"
4148 have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto
4150 { fix x assume "x \<in> s"
4151 hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto
4152 have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
4153 hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto
4154 ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using `e>0` by auto }
4155 thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]
4156 apply(rule_tac x="Inf s" in bexI) by auto
4159 lemma continuous_attains_sup:
4160 fixes f :: "'a::metric_space \<Rightarrow> real"
4161 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4162 ==> (\<exists>x \<in> s. \<forall>y \<in> s. f y \<le> f x)"
4163 using compact_attains_sup[of "f ` s"]
4164 using compact_continuous_image[of s f] by auto
4166 lemma continuous_attains_inf:
4167 fixes f :: "'a::metric_space \<Rightarrow> real"
4168 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4169 \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
4170 using compact_attains_inf[of "f ` s"]
4171 using compact_continuous_image[of s f] by auto
4173 lemma distance_attains_sup:
4174 assumes "compact s" "s \<noteq> {}"
4175 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
4176 proof (rule continuous_attains_sup [OF assms])
4177 { fix x assume "x\<in>s"
4178 have "(dist a ---> dist a x) (at x within s)"
4179 by (intro tendsto_dist tendsto_const Lim_at_within Lim_ident_at)
4181 thus "continuous_on s (dist a)"
4182 unfolding continuous_on ..
4185 text{* For *minimal* distance, we only need closure, not compactness. *}
4187 lemma distance_attains_inf:
4188 fixes a :: "'a::heine_borel"
4189 assumes "closed s" "s \<noteq> {}"
4190 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
4192 from assms(2) obtain b where "b\<in>s" by auto
4193 let ?B = "cball a (dist b a) \<inter> s"
4194 have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
4195 hence "?B \<noteq> {}" by auto
4197 { fix x assume "x\<in>?B"
4198 fix e::real assume "e>0"
4199 { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
4200 from as have "\<bar>dist a x' - dist a x\<bar> < e"
4201 unfolding abs_less_iff minus_diff_eq
4202 using dist_triangle2 [of a x' x]
4203 using dist_triangle [of a x x']
4206 hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
4209 hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
4210 unfolding continuous_on Lim_within dist_norm real_norm_def
4212 moreover have "compact ?B"
4213 using compact_cball[of a "dist b a"]
4214 unfolding compact_eq_bounded_closed
4215 using bounded_Int and closed_Int and assms(1) by auto
4216 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"
4217 using continuous_attains_inf[of ?B "dist a"] by fastsimp
4218 thus ?thesis by fastsimp
4221 subsection{* We can now extend limit compositions to consider the scalar multiplier. *}
4224 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
4225 assumes "(c ---> d) net" "(f ---> l) net"
4226 shows "((\<lambda>x. c(x) *\<^sub>R f x) ---> (d *\<^sub>R l)) net"
4227 using assms by (rule scaleR.tendsto)
4230 fixes c :: "'a \<Rightarrow> real" and v :: "'b::real_normed_vector"
4231 shows "(c ---> d) net ==> ((\<lambda>x. c(x) *\<^sub>R v) ---> d *\<^sub>R v) net"
4232 by (intro tendsto_intros)
4234 lemmas Lim_intros = Lim_add Lim_const Lim_sub Lim_cmul Lim_vmul Lim_within_id
4236 lemma continuous_vmul:
4237 fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
4238 shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"
4239 unfolding continuous_def using Lim_vmul[of c] by auto
4241 lemma continuous_mul:
4242 fixes c :: "'a::metric_space \<Rightarrow> real"
4243 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
4244 shows "continuous net c \<Longrightarrow> continuous net f
4245 ==> continuous net (\<lambda>x. c(x) *\<^sub>R f x) "
4246 unfolding continuous_def by (intro tendsto_intros)
4248 lemmas continuous_intros = continuous_add continuous_vmul continuous_cmul continuous_const continuous_sub continuous_at_id continuous_within_id continuous_mul
4250 lemma continuous_on_vmul:
4251 fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
4252 shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R v)"
4253 unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
4255 lemma continuous_on_mul:
4256 fixes c :: "'a::metric_space \<Rightarrow> real"
4257 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
4258 shows "continuous_on s c \<Longrightarrow> continuous_on s f
4259 ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R f x)"
4260 unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
4262 lemmas continuous_on_intros = continuous_on_add continuous_on_const continuous_on_id continuous_on_compose continuous_on_cmul continuous_on_neg continuous_on_sub
4263 uniformly_continuous_on_add uniformly_continuous_on_const uniformly_continuous_on_id uniformly_continuous_on_compose uniformly_continuous_on_cmul uniformly_continuous_on_neg uniformly_continuous_on_sub
4264 continuous_on_mul continuous_on_vmul
4266 text{* And so we have continuity of inverse. *}
4269 fixes f :: "'a \<Rightarrow> real"
4270 assumes "(f ---> l) (net::'a net)" "l \<noteq> 0"
4271 shows "((inverse o f) ---> inverse l) net"
4272 unfolding o_def using assms by (rule tendsto_inverse)
4274 lemma continuous_inv:
4275 fixes f :: "'a::metric_space \<Rightarrow> real"
4276 shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
4277 ==> continuous net (inverse o f)"
4278 unfolding continuous_def using Lim_inv by auto
4280 lemma continuous_at_within_inv:
4281 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
4282 assumes "continuous (at a within s) f" "f a \<noteq> 0"
4283 shows "continuous (at a within s) (inverse o f)"
4284 using assms unfolding continuous_within o_def
4285 by (intro tendsto_intros)
4287 lemma continuous_at_inv:
4288 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
4289 shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
4290 ==> continuous (at a) (inverse o f) "
4291 using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
4293 subsection{* Preservation properties for pasted sets. *}
4295 lemma bounded_pastecart:
4296 fixes s :: "('a::real_normed_vector ^ _) set" (* FIXME: generalize to metric_space *)
4297 assumes "bounded s" "bounded t"
4298 shows "bounded { pastecart x y | x y . (x \<in> s \<and> y \<in> t)}"
4300 obtain a b where ab:"\<forall>x\<in>s. norm x \<le> a" "\<forall>x\<in>t. norm x \<le> b" using assms[unfolded bounded_iff] by auto
4301 { fix x y assume "x\<in>s" "y\<in>t"
4302 hence "norm x \<le> a" "norm y \<le> b" using ab by auto
4303 hence "norm (pastecart x y) \<le> a + b" using norm_pastecart[of x y] by auto }
4304 thus ?thesis unfolding bounded_iff by auto
4307 lemma bounded_Times:
4308 assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
4310 obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
4311 using assms [unfolded bounded_def] by auto
4312 then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
4313 by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
4314 thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
4317 lemma closed_pastecart:
4318 fixes s :: "(real ^ 'a) set" (* FIXME: generalize *)
4319 assumes "closed s" "closed t"
4320 shows "closed {pastecart x y | x y . x \<in> s \<and> y \<in> t}"
4322 { fix x l assume as:"\<forall>n::nat. x n \<in> {pastecart x y |x y. x \<in> s \<and> y \<in> t}" "(x ---> l) sequentially"
4323 { fix n::nat have "fstcart (x n) \<in> s" "sndcart (x n) \<in> t" using as(1)[THEN spec[where x=n]] by auto } note * = this
4325 { fix e::real assume "e>0"
4326 then obtain N::nat where N:"\<forall>n\<ge>N. dist (x n) l < e" using as(2)[unfolded Lim_sequentially, THEN spec[where x=e]] by auto
4327 { fix n::nat assume "n\<ge>N"
4328 hence "dist (fstcart (x n)) (fstcart l) < e" "dist (sndcart (x n)) (sndcart l) < e"
4329 using N[THEN spec[where x=n]] dist_fstcart[of "x n" l] dist_sndcart[of "x n" l] by auto }
4330 hence "\<exists>N. \<forall>n\<ge>N. dist (fstcart (x n)) (fstcart l) < e" "\<exists>N. \<forall>n\<ge>N. dist (sndcart (x n)) (sndcart l) < e" by auto }
4331 ultimately have "fstcart l \<in> s" "sndcart l \<in> t"
4332 using assms(1)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. fstcart (x n)"], THEN spec[where x="fstcart l"]]
4333 using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. sndcart (x n)"], THEN spec[where x="sndcart l"]]
4334 unfolding Lim_sequentially by auto
4335 hence "l \<in> {pastecart x y |x y. x \<in> s \<and> y \<in> t}" apply- unfolding mem_Collect_eq apply(rule_tac x="fstcart l" in exI,rule_tac x="sndcart l" in exI) by auto }
4336 thus ?thesis unfolding closed_sequential_limits by auto
4339 lemma compact_pastecart:
4340 fixes s t :: "(real ^ _) set"
4341 shows "compact s \<Longrightarrow> compact t ==> compact {pastecart x y | x y . x \<in> s \<and> y \<in> t}"
4342 unfolding compact_eq_bounded_closed using bounded_pastecart[of s t] closed_pastecart[of s t] by auto
4344 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
4347 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
4348 unfolding compact_def
4350 apply (drule_tac x="fst \<circ> f" in spec)
4351 apply (drule mp, simp add: mem_Times_iff)
4352 apply (clarify, rename_tac l1 r1)
4353 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
4354 apply (drule mp, simp add: mem_Times_iff)
4355 apply (clarify, rename_tac l2 r2)
4356 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
4357 apply (rule_tac x="r1 \<circ> r2" in exI)
4358 apply (rule conjI, simp add: subseq_def)
4359 apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
4360 apply (drule (1) tendsto_Pair) back
4361 apply (simp add: o_def)
4364 text{* Hence some useful properties follow quite easily. *}
4366 lemma compact_scaling:
4367 fixes s :: "'a::real_normed_vector set"
4368 assumes "compact s" shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
4370 let ?f = "\<lambda>x. scaleR c x"
4371 have *:"bounded_linear ?f" by (rule scaleR.bounded_linear_right)
4372 show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
4373 using linear_continuous_at[OF *] assms by auto
4376 lemma compact_negations:
4377 fixes s :: "'a::real_normed_vector set"
4378 assumes "compact s" shows "compact ((\<lambda>x. -x) ` s)"
4379 using compact_scaling [OF assms, of "- 1"] by auto
4382 fixes s t :: "'a::real_normed_vector set"
4383 assumes "compact s" "compact t" shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
4385 have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
4386 apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
4387 have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
4388 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
4389 thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
4392 lemma compact_differences:
4393 fixes s t :: "'a::real_normed_vector set"
4394 assumes "compact s" "compact t" shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
4396 have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
4397 apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4398 thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
4401 lemma compact_translation:
4402 fixes s :: "'a::real_normed_vector set"
4403 assumes "compact s" shows "compact ((\<lambda>x. a + x) ` s)"
4405 have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
4406 thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
4409 lemma compact_affinity:
4410 fixes s :: "'a::real_normed_vector set"
4411 assumes "compact s" shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
4413 have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
4414 thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
4417 text{* Hence we get the following. *}
4419 lemma compact_sup_maxdistance:
4420 fixes s :: "'a::real_normed_vector set"
4421 assumes "compact s" "s \<noteq> {}"
4422 shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
4424 have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
4425 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"
4426 using compact_differences[OF assms(1) assms(1)]
4427 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 simp add: norm_minus_cancel)
4428 from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
4429 thus ?thesis using x(2)[unfolded `x = a - b`] by blast
4432 text{* We can state this in terms of diameter of a set. *}
4434 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"
4435 (* TODO: generalize to class metric_space *)
4437 lemma diameter_bounded:
4439 shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4440 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"
4442 let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"
4443 obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto
4444 { fix x y assume "x \<in> s" "y \<in> s"
4445 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: ring_simps) }
4447 { fix x y assume "x\<in>s" "y\<in>s" hence "s \<noteq> {}" by auto
4448 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`
4449 by simp (blast intro!: Sup_upper *) }
4451 { fix d::real assume "d>0" "d < diameter s"
4452 hence "s\<noteq>{}" unfolding diameter_def by auto
4453 have "\<exists>d' \<in> ?D. d' > d"
4455 assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"
4456 hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE)
4457 thus False using `d < diameter s` `s\<noteq>{}`
4458 apply (auto simp add: diameter_def)
4459 apply (drule Sup_real_iff [THEN [2] rev_iffD2])
4463 hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto }
4464 ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4465 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto
4468 lemma diameter_bounded_bound:
4469 "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"
4470 using diameter_bounded by blast
4472 lemma diameter_compact_attained:
4473 fixes s :: "'a::real_normed_vector set"
4474 assumes "compact s" "s \<noteq> {}"
4475 shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
4477 have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
4478 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
4479 hence "diameter s \<le> norm (x - y)"
4480 by (force simp add: diameter_def intro!: Sup_least)
4482 by (metis b diameter_bounded_bound order_antisym xys)
4485 text{* Related results with closure as the conclusion. *}
4487 lemma closed_scaling:
4488 fixes s :: "'a::real_normed_vector set"
4489 assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
4491 case True thus ?thesis by auto
4496 have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
4497 case True thus ?thesis apply auto unfolding * using closed_sing by auto
4500 { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s" "(x ---> l) sequentially"
4501 { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
4502 using as(1)[THEN spec[where x=n]]
4503 using `c\<noteq>0` by (auto simp add: vector_smult_assoc)
4506 { fix e::real assume "e>0"
4507 hence "0 < e *\<bar>c\<bar>" using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
4508 then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
4509 using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto
4510 hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
4511 unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
4512 using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto }
4513 hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto
4514 ultimately have "l \<in> scaleR c ` s"
4515 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"]]
4516 unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto }
4517 thus ?thesis unfolding closed_sequential_limits by fast
4521 lemma closed_negations:
4522 fixes s :: "'a::real_normed_vector set"
4523 assumes "closed s" shows "closed ((\<lambda>x. -x) ` s)"
4524 using closed_scaling[OF assms, of "- 1"] by simp
4526 lemma compact_closed_sums:
4527 fixes s :: "'a::real_normed_vector set"
4528 assumes "compact s" "closed t" shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4530 let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
4531 { fix x l assume as:"\<forall>n. x n \<in> ?S" "(x ---> l) sequentially"
4532 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"
4533 using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
4534 obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
4535 using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
4536 have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
4537 using Lim_sub[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
4538 hence "l - l' \<in> t"
4539 using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
4541 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
4543 thus ?thesis unfolding closed_sequential_limits by fast
4546 lemma closed_compact_sums:
4547 fixes s t :: "'a::real_normed_vector set"
4548 assumes "closed s" "compact t"
4549 shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4551 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
4552 apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
4553 thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
4556 lemma compact_closed_differences:
4557 fixes s t :: "'a::real_normed_vector set"
4558 assumes "compact s" "closed t"
4559 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4561 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4562 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4563 thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
4566 lemma closed_compact_differences:
4567 fixes s t :: "'a::real_normed_vector set"
4568 assumes "closed s" "compact t"
4569 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4571 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4572 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4573 thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
4576 lemma closed_translation:
4577 fixes a :: "'a::real_normed_vector"
4578 assumes "closed s" shows "closed ((\<lambda>x. a + x) ` s)"
4580 have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
4581 thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
4584 lemma translation_Compl:
4585 fixes a :: "'a::ab_group_add"
4586 shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
4587 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto
4589 lemma translation_UNIV:
4590 fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
4591 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
4593 lemma translation_diff:
4594 fixes a :: "'a::ab_group_add"
4595 shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
4598 lemma closure_translation:
4599 fixes a :: "'a::real_normed_vector"
4600 shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
4602 have *:"op + a ` (- s) = - op + a ` s"
4603 apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
4604 show ?thesis unfolding closure_interior translation_Compl
4605 using interior_translation[of a "- s"] unfolding * by auto
4608 lemma frontier_translation:
4609 fixes a :: "'a::real_normed_vector"
4610 shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
4611 unfolding frontier_def translation_diff interior_translation closure_translation by auto
4613 subsection{* Separation between points and sets. *}
4615 lemma separate_point_closed:
4616 fixes s :: "'a::heine_borel set"
4617 shows "closed s \<Longrightarrow> a \<notin> s ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
4618 proof(cases "s = {}")
4620 thus ?thesis by(auto intro!: exI[where x=1])
4623 assume "closed s" "a \<notin> s"
4624 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
4625 with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
4628 lemma separate_compact_closed:
4629 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4630 (* TODO: does this generalize to heine_borel? *)
4631 assumes "compact s" and "closed t" and "s \<inter> t = {}"
4632 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4634 have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto
4635 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"
4636 using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto
4637 { fix x y assume "x\<in>s" "y\<in>t"
4638 hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto
4639 hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute
4640 by (auto simp add: dist_commute)
4641 hence "d \<le> dist x y" unfolding dist_norm by auto }
4642 thus ?thesis using `d>0` by auto
4645 lemma separate_closed_compact:
4646 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4647 assumes "closed s" and "compact t" and "s \<inter> t = {}"
4648 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4650 have *:"t \<inter> s = {}" using assms(3) by auto
4651 show ?thesis using separate_compact_closed[OF assms(2,1) *]
4652 apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
4653 by (auto simp add: dist_commute)
4656 (* A cute way of denoting open and closed intervals using overloading. *)
4658 lemma interval: fixes a :: "'a::ord^'n" shows
4659 "{a <..< b} = {x::'a^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" and
4660 "{a .. b} = {x::'a^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
4661 by (auto simp add: expand_set_eq vector_less_def vector_le_def)
4663 lemma mem_interval: fixes a :: "'a::ord^'n" shows
4664 "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
4665 "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
4666 using interval[of a b] by(auto simp add: expand_set_eq vector_less_def vector_le_def)
4668 lemma mem_interval_1: fixes x :: "real^1" shows
4669 "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
4670 "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
4671 by(simp_all add: Cart_eq vector_less_def vector_le_def forall_1)
4673 lemma vec1_interval:fixes a::"real" shows
4674 "vec1 ` {a .. b} = {vec1 a .. vec1 b}"
4675 "vec1 ` {a<..<b} = {vec1 a<..<vec1 b}"
4676 apply(rule_tac[!] set_ext) unfolding image_iff vector_less_def unfolding mem_interval
4677 unfolding forall_1 unfolding vec1_dest_vec1_simps
4678 apply rule defer apply(rule_tac x="dest_vec1 x" in bexI) prefer 3 apply rule defer
4679 apply(rule_tac x="dest_vec1 x" in bexI) by auto
4682 lemma interval_eq_empty: fixes a :: "real^'n" shows
4683 "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) and
4684 "({a .. b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
4686 { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>{a <..< b}"
4687 hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval by auto
4688 hence "a$i < b$i" by auto
4689 hence False using as by auto }
4691 { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
4692 let ?x = "(1/2) *\<^sub>R (a + b)"
4694 have "a$i < b$i" using as[THEN spec[where x=i]] by auto
4695 hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
4696 unfolding vector_smult_component and vector_add_component
4697 by (auto simp add: less_divide_eq_number_of1) }
4698 hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto }
4699 ultimately show ?th1 by blast
4701 { fix i x assume as:"b$i < a$i" and x:"x\<in>{a .. b}"
4702 hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval by auto
4703 hence "a$i \<le> b$i" by auto
4704 hence False using as by auto }
4706 { assume as:"\<forall>i. \<not> (b$i < a$i)"
4707 let ?x = "(1/2) *\<^sub>R (a + b)"
4709 have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
4710 hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
4711 unfolding vector_smult_component and vector_add_component
4712 by (auto simp add: less_divide_eq_number_of1) }
4713 hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto }
4714 ultimately show ?th2 by blast
4717 lemma interval_ne_empty: fixes a :: "real^'n" shows
4718 "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)" and
4719 "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
4720 unfolding interval_eq_empty[of a b] by (auto simp add: not_less not_le) (* BH: Why doesn't just "auto" work here? *)
4722 lemma subset_interval_imp: fixes a :: "real^'n" shows
4723 "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
4724 "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
4725 "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
4726 "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
4727 unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
4728 by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
4730 lemma interval_sing: fixes a :: "'a::linorder^'n" shows
4731 "{a .. a} = {a} \<and> {a<..<a} = {}"
4732 apply(auto simp add: expand_set_eq vector_less_def vector_le_def Cart_eq)
4733 apply (simp add: order_eq_iff)
4734 apply (auto simp add: not_less less_imp_le)
4737 lemma interval_open_subset_closed: fixes a :: "'a::preorder^'n" shows
4738 "{a<..<b} \<subseteq> {a .. b}"
4739 proof(simp add: subset_eq, rule)
4741 assume x:"x \<in>{a<..<b}"
4743 have "a $ i \<le> x $ i"
4744 using x order_less_imp_le[of "a$i" "x$i"]
4745 by(simp add: expand_set_eq vector_less_def vector_le_def Cart_eq)
4749 have "x $ i \<le> b $ i"
4750 using x order_less_imp_le[of "x$i" "b$i"]
4751 by(simp add: expand_set_eq vector_less_def vector_le_def Cart_eq)
4754 show "a \<le> x \<and> x \<le> b"
4755 by(simp add: expand_set_eq vector_less_def vector_le_def Cart_eq)
4758 lemma subset_interval: fixes a :: "real^'n" shows
4759 "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1) and
4760 "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2) and
4761 "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3) and
4762 "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
4764 show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
4765 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)
4766 { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i. c$i < d$i"
4767 hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by (auto, drule_tac x=i in spec, simp) (* BH: Why doesn't just "auto" work? *)
4769 (** TODO combine the following two parts as done in the HOL_light version. **)
4770 { let ?x = "(\<chi> j. (if j=i then ((min (a$j) (d$j))+c$j)/2 else (c$j+d$j)/2))::real^'n"
4771 assume as2: "a$i > c$i"
4773 have "c $ j < ?x $ j \<and> ?x $ j < d $ j" unfolding Cart_lambda_beta
4774 apply(cases "j=i") using as(2)[THEN spec[where x=j]]
4775 by (auto simp add: less_divide_eq_number_of1 as2) }
4776 hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
4778 have "?x\<notin>{a .. b}"
4779 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4780 using as(2)[THEN spec[where x=i]] and as2
4781 by (auto simp add: less_divide_eq_number_of1)
4782 ultimately have False using as by auto }
4783 hence "a$i \<le> c$i" by(rule ccontr)auto
4785 { let ?x = "(\<chi> j. (if j=i then ((max (b$j) (c$j))+d$j)/2 else (c$j+d$j)/2))::real^'n"
4786 assume as2: "b$i < d$i"
4788 have "d $ j > ?x $ j \<and> ?x $ j > c $ j" unfolding Cart_lambda_beta
4789 apply(cases "j=i") using as(2)[THEN spec[where x=j]]
4790 by (auto simp add: less_divide_eq_number_of1 as2) }
4791 hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
4793 have "?x\<notin>{a .. b}"
4794 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4795 using as(2)[THEN spec[where x=i]] and as2
4796 by (auto simp add: less_divide_eq_number_of1)
4797 ultimately have False using as by auto }
4798 hence "b$i \<ge> d$i" by(rule ccontr)auto
4800 have "a$i \<le> c$i \<and> d$i \<le> b$i" by auto
4802 thus ?th3 unfolding subset_eq and Ball_def and mem_interval apply auto apply (erule_tac x=ia in allE, simp)+ by (erule_tac x=i in allE, erule_tac x=i in allE, simp)+
4803 { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i. c$i < d$i"
4805 from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
4806 hence "a$i \<le> c$i \<and> d$i \<le> b$i" using part1 and as(2) by auto } note * = this
4807 thus ?th4 unfolding subset_eq and Ball_def and mem_interval apply auto apply (erule_tac x=ia in allE, simp)+ by (erule_tac x=i in allE, erule_tac x=i in allE, simp)+
4810 lemma disjoint_interval: fixes a::"real^'n" shows
4811 "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1) and
4812 "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (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
4813 "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (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
4814 "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (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)
4816 let ?z = "(\<chi> i. ((max (a$i) (c$i)) + (min (b$i) (d$i))) / 2)::real^'n"
4817 show ?th1 ?th2 ?th3 ?th4
4818 unfolding expand_set_eq and Int_iff and empty_iff and mem_interval and all_conj_distrib[THEN sym] and eq_False
4819 apply (auto elim!: allE[where x="?z"])
4820 apply ((rule_tac x=x in exI, force) | (rule_tac x=i in exI, force))+
4824 lemma inter_interval: fixes a :: "'a::linorder^'n" shows
4825 "{a .. b} \<inter> {c .. d} = {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
4826 unfolding expand_set_eq and Int_iff and mem_interval
4827 by (auto simp add: less_divide_eq_number_of1 intro!: bexI)
4829 (* Moved interval_open_subset_closed a bit upwards *)
4831 lemma open_interval_lemma: fixes x :: "real" shows
4832 "a < x \<Longrightarrow> x < b ==> (\<exists>d>0. \<forall>x'. abs(x' - x) < d --> a < x' \<and> x' < b)"
4833 by(rule_tac x="min (x - a) (b - x)" in exI, auto)
4835 lemma open_interval[intro]: fixes a :: "real^'n" shows "open {a<..<b}"
4837 { fix x assume x:"x\<in>{a<..<b}"
4839 have "\<exists>d>0. \<forall>x'. abs (x' - (x$i)) < d \<longrightarrow> a$i < x' \<and> x' < b$i"
4840 using x[unfolded mem_interval, THEN spec[where x=i]]
4841 using open_interval_lemma[of "a$i" "x$i" "b$i"] by auto }
4843 hence "\<forall>i. \<exists>d>0. \<forall>x'. abs (x' - (x$i)) < d \<longrightarrow> a$i < x' \<and> x' < b$i" by auto
4844 then obtain d where d:"\<forall>i. 0 < d i \<and> (\<forall>x'. \<bar>x' - x $ i\<bar> < d i \<longrightarrow> a $ i < x' \<and> x' < b $ i)"
4845 using bchoice[of "UNIV" "\<lambda>i d. d>0 \<and> (\<forall>x'. \<bar>x' - x $ i\<bar> < d \<longrightarrow> a $ i < x' \<and> x' < b $ i)"] by auto
4847 let ?d = "Min (range d)"
4848 have **:"finite (range d)" "range d \<noteq> {}" by auto
4849 have "?d>0" unfolding Min_gr_iff[OF **] using d by auto
4851 { fix x' assume as:"dist x' x < ?d"
4853 have "\<bar>x'$i - x $ i\<bar> < d i"
4854 using norm_bound_component_lt[OF as[unfolded dist_norm], of i]
4855 unfolding vector_minus_component and Min_gr_iff[OF **] by auto
4856 hence "a $ i < x' $ i" "x' $ i < b $ i" using d[THEN spec[where x=i]] by auto }
4857 hence "a < x' \<and> x' < b" unfolding vector_less_def by auto }
4858 ultimately have "\<exists>e>0. \<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a<..<b}" by (auto, rule_tac x="?d" in exI, simp)
4860 thus ?thesis unfolding open_dist using open_interval_lemma by auto
4863 lemma open_interval_real[intro]: fixes a :: "real" shows "open {a<..<b}"
4864 using open_interval[of "vec1 a" "vec1 b"] unfolding open_contains_ball
4865 apply-apply(rule,erule_tac x="vec1 x" in ballE) apply(erule exE,rule_tac x=e in exI)
4866 unfolding subset_eq mem_ball apply(rule) defer apply(rule,erule conjE,erule_tac x="vec1 xa" in ballE)
4867 by(auto simp add: vec1_dest_vec1_simps vector_less_def forall_1)
4869 lemma closed_interval[intro]: fixes a :: "real^'n" shows "closed {a .. b}"
4871 { fix x i assume as:"\<forall>e>0. \<exists>x'\<in>{a..b}. x' \<noteq> x \<and> dist x' x < e"(* and xab:"a$i > x$i \<or> b$i < x$i"*)
4872 { assume xa:"a$i > x$i"
4873 with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < a$i - x$i" by(erule_tac x="a$i - x$i" in allE)auto
4874 hence False unfolding mem_interval and dist_norm
4875 using component_le_norm[of "y-x" i, unfolded vector_minus_component] and xa by(auto elim!: allE[where x=i])
4876 } hence "a$i \<le> x$i" by(rule ccontr)auto
4878 { assume xb:"b$i < x$i"
4879 with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < x$i - b$i" by(erule_tac x="x$i - b$i" in allE)auto
4880 hence False unfolding mem_interval and dist_norm
4881 using component_le_norm[of "y-x" i, unfolded vector_minus_component] and xb by(auto elim!: allE[where x=i])
4882 } hence "x$i \<le> b$i" by(rule ccontr)auto
4884 have "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" by auto }
4885 thus ?thesis unfolding closed_limpt islimpt_approachable mem_interval by auto
4888 lemma interior_closed_interval[intro]: fixes a :: "real^'n" shows
4889 "interior {a .. b} = {a<..<b}" (is "?L = ?R")
4890 proof(rule subset_antisym)
4891 show "?R \<subseteq> ?L" using interior_maximal[OF interval_open_subset_closed open_interval] by auto
4893 { fix x assume "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> {a..b}"
4894 then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" by auto
4895 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
4897 have "dist (x - (e / 2) *\<^sub>R basis i) x < e"
4898 "dist (x + (e / 2) *\<^sub>R basis i) x < e"
4899 unfolding dist_norm apply auto
4900 unfolding norm_minus_cancel using norm_basis[of i] and `e>0` by auto
4901 hence "a $ i \<le> (x - (e / 2) *\<^sub>R basis i) $ i"
4902 "(x + (e / 2) *\<^sub>R basis i) $ i \<le> b $ i"
4903 using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]
4904 and e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]
4905 unfolding mem_interval by (auto elim!: allE[where x=i])
4906 hence "a $ i < x $ i" and "x $ i < b $ i"
4907 unfolding vector_minus_component and vector_add_component
4908 unfolding vector_smult_component and basis_component using `e>0` by auto }
4909 hence "x \<in> {a<..<b}" unfolding mem_interval by auto }
4910 thus "?L \<subseteq> ?R" unfolding interior_def and subset_eq by auto
4913 lemma bounded_closed_interval: fixes a :: "real^'n" shows
4916 let ?b = "\<Sum>i\<in>UNIV. \<bar>a$i\<bar> + \<bar>b$i\<bar>"
4917 { fix x::"real^'n" assume x:"\<forall>i. a $ i \<le> x $ i \<and> x $ i \<le> b $ i"
4919 have "\<bar>x$i\<bar> \<le> \<bar>a$i\<bar> + \<bar>b$i\<bar>" using x[THEN spec[where x=i]] by auto }
4920 hence "(\<Sum>i\<in>UNIV. \<bar>x $ i\<bar>) \<le> ?b" by(rule setsum_mono)
4921 hence "norm x \<le> ?b" using norm_le_l1[of x] by auto }
4922 thus ?thesis unfolding interval and bounded_iff by auto
4925 lemma bounded_interval: fixes a :: "real^'n" shows
4926 "bounded {a .. b} \<and> bounded {a<..<b}"
4927 using bounded_closed_interval[of a b]
4928 using interval_open_subset_closed[of a b]
4929 using bounded_subset[of "{a..b}" "{a<..<b}"]
4932 lemma not_interval_univ: fixes a :: "real^'n" shows
4933 "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
4934 using bounded_interval[of a b]
4937 lemma compact_interval: fixes a :: "real^'n" shows
4939 using bounded_closed_imp_compact using bounded_interval[of a b] using closed_interval[of a b] by auto
4941 lemma open_interval_midpoint: fixes a :: "real^'n"
4942 assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
4945 have "a $ i < ((1 / 2) *\<^sub>R (a + b)) $ i \<and> ((1 / 2) *\<^sub>R (a + b)) $ i < b $ i"
4946 using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
4947 unfolding vector_smult_component and vector_add_component
4948 by(auto simp add: less_divide_eq_number_of1) }
4949 thus ?thesis unfolding mem_interval by auto
4952 lemma open_closed_interval_convex: fixes x :: "real^'n"
4953 assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
4954 shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
4957 have "a $ i = e * a$i + (1 - e) * a$i" unfolding left_diff_distrib by simp
4958 also have "\<dots> < e * x $ i + (1 - e) * y $ i" apply(rule add_less_le_mono)
4959 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
4960 using x unfolding mem_interval apply simp
4961 using y unfolding mem_interval apply simp
4963 finally have "a $ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $ i" by auto
4965 have "b $ i = e * b$i + (1 - e) * b$i" unfolding left_diff_distrib by simp
4966 also have "\<dots> > e * x $ i + (1 - e) * y $ i" apply(rule add_less_le_mono)
4967 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
4968 using x unfolding mem_interval apply simp
4969 using y unfolding mem_interval apply simp
4971 finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $ i < b $ i" by auto
4972 } 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 }
4973 thus ?thesis unfolding mem_interval by auto
4976 lemma closure_open_interval: fixes a :: "real^'n"
4977 assumes "{a<..<b} \<noteq> {}"
4978 shows "closure {a<..<b} = {a .. b}"
4980 have ab:"a < b" using assms[unfolded interval_ne_empty] unfolding vector_less_def by auto
4981 let ?c = "(1 / 2) *\<^sub>R (a + b)"
4982 { fix x assume as:"x \<in> {a .. b}"
4983 def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
4984 { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
4985 have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
4986 have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
4987 x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
4988 by (auto simp add: algebra_simps)
4989 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
4990 hence False using fn unfolding f_def using xc by(auto simp add: vector_mul_lcancel vector_ssub_ldistrib) }
4992 { assume "\<not> (f ---> x) sequentially"
4993 { fix e::real assume "e>0"
4994 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
4995 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
4996 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)
4997 hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto }
4998 hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
4999 unfolding Lim_sequentially by(auto simp add: dist_norm)
5000 hence "(f ---> x) sequentially" unfolding f_def
5001 using Lim_add[OF Lim_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
5002 using Lim_vmul[of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto }
5003 ultimately have "x \<in> closure {a<..<b}"
5004 using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto }
5005 thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
5008 lemma bounded_subset_open_interval_symmetric: fixes s::"(real^'n) set"
5009 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a<..<a}"
5011 obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
5012 def a \<equiv> "(\<chi> i. b+1)::real^'n"
5013 { fix x assume "x\<in>s"
5015 have "(-a)$i < x$i" and "x$i < a$i" using b[THEN bspec[where x=x], OF `x\<in>s`] and component_le_norm[of x i]
5016 unfolding vector_uminus_component and a_def and Cart_lambda_beta by auto
5018 thus ?thesis by(auto intro: exI[where x=a] simp add: vector_less_def)
5021 lemma bounded_subset_open_interval:
5022 fixes s :: "(real ^ 'n) set"
5023 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
5024 by (auto dest!: bounded_subset_open_interval_symmetric)
5026 lemma bounded_subset_closed_interval_symmetric:
5027 fixes s :: "(real ^ 'n) set"
5028 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
5030 obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
5031 thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
5034 lemma bounded_subset_closed_interval:
5035 fixes s :: "(real ^ 'n) set"
5036 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
5037 using bounded_subset_closed_interval_symmetric[of s] by auto
5039 lemma frontier_closed_interval:
5040 fixes a b :: "real ^ _"
5041 shows "frontier {a .. b} = {a .. b} - {a<..<b}"
5042 unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
5044 lemma frontier_open_interval:
5045 fixes a b :: "real ^ _"
5046 shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
5047 proof(cases "{a<..<b} = {}")
5048 case True thus ?thesis using frontier_empty by auto
5050 case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
5053 lemma inter_interval_mixed_eq_empty: fixes a :: "real^'n"
5054 assumes "{c<..<d} \<noteq> {}" shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
5055 unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
5058 (* Some special cases for intervals in R^1. *)
5060 lemma interval_cases_1: fixes x :: "real^1" shows
5061 "x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"
5062 unfolding Cart_eq vector_less_def vector_le_def mem_interval by(auto simp del:dest_vec1_eq)
5064 lemma in_interval_1: fixes x :: "real^1" shows
5065 "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b) \<and>
5066 (x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
5067 unfolding Cart_eq vector_less_def vector_le_def mem_interval by(auto simp del:dest_vec1_eq)
5069 lemma interval_eq_empty_1: fixes a :: "real^1" shows
5070 "{a .. b} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a"
5071 "{a<..<b} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
5072 unfolding interval_eq_empty and ex_1 by auto
5074 lemma subset_interval_1: fixes a :: "real^1" shows
5075 "({a .. b} \<subseteq> {c .. d} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or>
5076 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
5077 "({a .. b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or>
5078 dest_vec1 c < dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b < dest_vec1 d)"
5079 "({a<..<b} \<subseteq> {c .. d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
5080 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
5081 "({a<..<b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
5082 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
5083 unfolding subset_interval[of a b c d] unfolding forall_1 by auto
5085 lemma eq_interval_1: fixes a :: "real^1" shows
5086 "{a .. b} = {c .. d} \<longleftrightarrow>
5087 dest_vec1 b < dest_vec1 a \<and> dest_vec1 d < dest_vec1 c \<or>
5088 dest_vec1 a = dest_vec1 c \<and> dest_vec1 b = dest_vec1 d"
5089 unfolding set_eq_subset[of "{a .. b}" "{c .. d}"]
5090 unfolding subset_interval_1(1)[of a b c d]
5091 unfolding subset_interval_1(1)[of c d a b]
5094 lemma disjoint_interval_1: fixes a :: "real^1" shows
5095 "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or> dest_vec1 d < dest_vec1 c \<or> dest_vec1 b < dest_vec1 c \<or> dest_vec1 d < dest_vec1 a"
5096 "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c \<or> dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
5097 "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d < dest_vec1 c \<or> dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
5098 "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c \<or> dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
5099 unfolding disjoint_interval and ex_1 by auto
5101 lemma open_closed_interval_1: fixes a :: "real^1" shows
5102 "{a<..<b} = {a .. b} - {a, b}"
5103 unfolding expand_set_eq apply simp unfolding vector_less_def and vector_le_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)
5105 lemma closed_open_interval_1: "dest_vec1 (a::real^1) \<le> dest_vec1 b ==> {a .. b} = {a<..<b} \<union> {a,b}"
5106 unfolding expand_set_eq apply simp unfolding vector_less_def and vector_le_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)
5108 (* Some stuff for half-infinite intervals too; FIXME: notation? *)
5110 lemma closed_interval_left: fixes b::"real^'n"
5111 shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
5114 fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i. x $ i \<le> b $ i}. x' \<noteq> x \<and> dist x' x < e"
5115 { assume "x$i > b$i"
5116 then obtain y where "y $ i \<le> b $ i" "y \<noteq> x" "dist y x < x$i - b$i" using x[THEN spec[where x="x$i - b$i"]] by auto
5117 hence False using component_le_norm[of "y - x" i] unfolding dist_norm and vector_minus_component by auto }
5118 hence "x$i \<le> b$i" by(rule ccontr)auto }
5119 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
5122 lemma closed_interval_right: fixes a::"real^'n"
5123 shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
5126 fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i. a $ i \<le> x $ i}. x' \<noteq> x \<and> dist x' x < e"
5127 { assume "a$i > x$i"
5128 then obtain y where "a $ i \<le> y $ i" "y \<noteq> x" "dist y x < a$i - x$i" using x[THEN spec[where x="a$i - x$i"]] by auto
5129 hence False using component_le_norm[of "y - x" i] unfolding dist_norm and vector_minus_component by auto }
5130 hence "a$i \<le> x$i" by(rule ccontr)auto }
5131 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
5134 subsection{* Intervals in general, including infinite and mixtures of open and closed. *}
5136 definition "is_interval s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((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)"
5138 lemma is_interval_interval: "is_interval {a .. b::real^'n}" (is ?th1) "is_interval {a<..<b}" (is ?th2) proof -
5139 have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
5140 show ?th1 ?th2 unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
5141 by(meson real_le_trans le_less_trans less_le_trans *)+ qed
5143 lemma is_interval_empty:
5145 unfolding is_interval_def
5148 lemma is_interval_univ:
5150 unfolding is_interval_def
5153 subsection{* Closure of halfspaces and hyperplanes. *}
5156 assumes "(f ---> l) net" shows "((\<lambda>y. inner a (f y)) ---> inner a l) net"
5157 by (intro tendsto_intros assms)
5159 lemma continuous_at_inner: "continuous (at x) (inner a)"
5160 unfolding continuous_at by (intro tendsto_intros)
5162 lemma continuous_on_inner:
5163 fixes s :: "'a::real_inner set"
5164 shows "continuous_on s (inner a)"
5165 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
5167 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
5169 have "\<forall>x. continuous (at x) (inner a)"
5170 unfolding continuous_at by (rule allI) (intro tendsto_intros)
5171 hence "closed (inner a -` {..b})"
5172 using closed_real_atMost by (rule continuous_closed_vimage)
5173 moreover have "{x. inner a x \<le> b} = inner a -` {..b}" by auto
5174 ultimately show ?thesis by simp
5177 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
5178 using closed_halfspace_le[of "-a" "-b"] unfolding inner_minus_left by auto
5180 lemma closed_hyperplane: "closed {x. inner a x = b}"
5182 have "{x. inner a x = b} = {x. inner a x \<ge> b} \<inter> {x. inner a x \<le> b}" by auto
5183 thus ?thesis using closed_halfspace_le[of a b] and closed_halfspace_ge[of b a] using closed_Int by auto
5186 lemma closed_halfspace_component_le:
5187 shows "closed {x::real^'n. x$i \<le> a}"
5188 using closed_halfspace_le[of "(basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto
5190 lemma closed_halfspace_component_ge:
5191 shows "closed {x::real^'n. x$i \<ge> a}"
5192 using closed_halfspace_ge[of a "(basis i)::real^'n"] unfolding inner_basis[OF assms] by auto
5194 text{* Openness of halfspaces. *}
5196 lemma open_halfspace_lt: "open {x. inner a x < b}"
5198 have "- {x. b \<le> inner a x} = {x. inner a x < b}" by auto
5199 thus ?thesis using closed_halfspace_ge[unfolded closed_def, of b a] by auto
5202 lemma open_halfspace_gt: "open {x. inner a x > b}"
5204 have "- {x. b \<ge> inner a x} = {x. inner a x > b}" by auto
5205 thus ?thesis using closed_halfspace_le[unfolded closed_def, of a b] by auto
5208 lemma open_halfspace_component_lt:
5209 shows "open {x::real^'n. x$i < a}"
5210 using open_halfspace_lt[of "(basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto
5212 lemma open_halfspace_component_gt:
5213 shows "open {x::real^'n. x$i > a}"
5214 using open_halfspace_gt[of a "(basis i)::real^'n"] unfolding inner_basis[OF assms] by auto
5216 text{* This gives a simple derivation of limit component bounds. *}
5218 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> real^'n"
5219 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f(x)$i \<le> b) net"
5222 { fix x have "x \<in> {x::real^'n. inner (basis i) x \<le> b} \<longleftrightarrow> x$i \<le> b" unfolding inner_basis by auto } note * = this
5223 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *
5224 using closed_halfspace_le[of "(basis i)::real^'n" b] and assms(1,2,3) by auto
5227 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> real^'n"
5228 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$i) net"
5231 { fix x have "x \<in> {x::real^'n. inner (basis i) x \<ge> b} \<longleftrightarrow> x$i \<ge> b" unfolding inner_basis by auto } note * = this
5232 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *
5233 using closed_halfspace_ge[of b "(basis i)::real^'n"] and assms(1,2,3) by auto
5236 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> real^'n"
5237 assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
5239 using ev[unfolded order_eq_iff eventually_and] using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto
5241 lemma Lim_drop_le: fixes f :: "'a \<Rightarrow> real^1" shows
5242 "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. dest_vec1 (f x) \<le> b) net ==> dest_vec1 l \<le> b"
5243 using Lim_component_le[of f l net 1 b] by auto
5245 lemma Lim_drop_ge: fixes f :: "'a \<Rightarrow> real^1" shows
5246 "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. b \<le> dest_vec1 (f x)) net ==> b \<le> dest_vec1 l"
5247 using Lim_component_ge[of f l net b 1] by auto
5249 text{* Limits relative to a union. *}
5251 lemma eventually_within_Un:
5252 "eventually P (net within (s \<union> t)) \<longleftrightarrow>
5253 eventually P (net within s) \<and> eventually P (net within t)"
5254 unfolding Limits.eventually_within
5255 by (auto elim!: eventually_rev_mp)
5257 lemma Lim_within_union:
5258 "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>
5259 (f ---> l) (net within s) \<and> (f ---> l) (net within t)"
5260 unfolding tendsto_def
5261 by (auto simp add: eventually_within_Un)
5263 lemma continuous_on_union:
5264 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
5265 shows "continuous_on (s \<union> t) f"
5266 using assms unfolding continuous_on unfolding Lim_within_union
5267 unfolding Lim unfolding trivial_limit_within unfolding closed_limpt by auto
5269 lemma continuous_on_cases:
5270 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
5271 "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
5272 shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
5274 let ?h = "(\<lambda>x. if P x then f x else g x)"
5275 have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto
5276 hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto
5278 have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto
5279 hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto
5280 ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto
5284 text{* Some more convenient intermediate-value theorem formulations. *}
5286 lemma connected_ivt_hyperplane:
5287 assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
5288 shows "\<exists>z \<in> s. inner a z = b"
5290 assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
5291 let ?A = "{x. inner a x < b}"
5292 let ?B = "{x. inner a x > b}"
5293 have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
5294 moreover have "?A \<inter> ?B = {}" by auto
5295 moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
5296 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
5299 lemma connected_ivt_component: fixes x::"real^'n" shows
5300 "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)"
5301 using connected_ivt_hyperplane[of s x y "(basis k)::real^'n" a] by (auto simp add: inner_basis)
5303 text{* Also more convenient formulations of monotone convergence. *}
5305 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real^1"
5306 assumes "bounded {s n| n::nat. True}" "\<forall>n. dest_vec1(s n) \<le> dest_vec1(s(Suc n))"
5307 shows "\<exists>l. (s ---> l) sequentially"
5309 obtain a where a:"\<forall>n. \<bar>dest_vec1 (s n)\<bar> \<le> a" using assms(1)[unfolded bounded_iff abs_dest_vec1] by auto
5311 have "\<And> n. n\<ge>m \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
5312 apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE) by(auto simp add: not_less_eq_eq) }
5313 hence "\<forall>m n. m \<le> n \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)" by auto
5314 then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>dest_vec1 (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a] unfolding monoseq_def by auto
5315 thus ?thesis unfolding Lim_sequentially apply(rule_tac x="vec1 l" in exI)
5316 unfolding dist_norm unfolding abs_dest_vec1 by auto
5319 subsection{* Basic homeomorphism definitions. *}
5321 definition "homeomorphism s t f g \<equiv>
5322 (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
5323 (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
5326 homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"
5327 (infixr "homeomorphic" 60) where
5328 homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
5330 lemma homeomorphic_refl: "s homeomorphic s"
5331 unfolding homeomorphic_def
5332 unfolding homeomorphism_def
5333 using continuous_on_id
5334 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5335 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5338 lemma homeomorphic_sym:
5339 "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
5340 unfolding homeomorphic_def
5341 unfolding homeomorphism_def
5344 lemma homeomorphic_trans:
5345 assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
5347 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"
5348 using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
5349 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"
5350 using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
5352 { 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 }
5353 moreover have "(f2 \<circ> f1) ` s = u" using fg1(2) fg2(2) by auto
5354 moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
5355 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 }
5356 moreover have "(g1 \<circ> g2) ` u = s" using fg1(5) fg2(5) by auto
5357 moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6) unfolding fg2(5) by auto
5358 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
5361 lemma homeomorphic_minimal:
5362 "s homeomorphic t \<longleftrightarrow>
5363 (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
5364 (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
5365 continuous_on s f \<and> continuous_on t g)"
5366 unfolding homeomorphic_def homeomorphism_def
5367 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
5368 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
5370 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
5371 apply auto apply(rule_tac x="g x" in bexI) apply auto
5372 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
5373 apply auto apply(rule_tac x="f x" in bexI) by auto
5375 subsection{* Relatively weak hypotheses if a set is compact. *}
5377 definition "inv_on f s = (\<lambda>x. SOME y. y\<in>s \<and> f y = x)"
5379 lemma assumes "inj_on f s" "x\<in>s"
5380 shows "inv_on f s (f x) = x"
5381 using assms unfolding inj_on_def inv_on_def by auto
5383 lemma homeomorphism_compact:
5384 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5385 (* class constraint due to continuous_on_inverse *)
5386 assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s"
5387 shows "\<exists>g. homeomorphism s t f g"
5389 def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
5390 have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
5391 { fix y assume "y\<in>t"
5392 then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
5393 hence "g (f x) = x" using g by auto
5394 hence "f (g y) = y" unfolding x(1)[THEN sym] by auto }
5395 hence g':"\<forall>x\<in>t. f (g x) = x" by auto
5398 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"])
5400 { assume "x\<in>g ` t"
5401 then obtain y where y:"y\<in>t" "g y = x" by auto
5402 then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
5403 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 }
5404 ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" by auto }
5405 hence "g ` t = s" by auto
5407 show ?thesis unfolding homeomorphism_def homeomorphic_def
5408 apply(rule_tac x=g in exI) using g and assms(3) and continuous_on_inverse[OF assms(2,1), of g, unfolded assms(3)] and assms(2) by auto
5411 lemma homeomorphic_compact:
5412 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5413 (* class constraint due to continuous_on_inverse *)
5414 shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
5415 \<Longrightarrow> s homeomorphic t"
5416 unfolding homeomorphic_def by(metis homeomorphism_compact)
5418 text{* Preservation of topological properties. *}
5420 lemma homeomorphic_compactness:
5421 "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
5422 unfolding homeomorphic_def homeomorphism_def
5423 by (metis compact_continuous_image)
5425 text{* Results on translation, scaling etc. *}
5427 lemma homeomorphic_scaling:
5428 fixes s :: "'a::real_normed_vector set"
5429 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
5430 unfolding homeomorphic_minimal
5431 apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
5432 apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
5433 using assms apply auto
5434 using continuous_on_cmul[OF continuous_on_id] by auto
5436 lemma homeomorphic_translation:
5437 fixes s :: "'a::real_normed_vector set"
5438 shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
5439 unfolding homeomorphic_minimal
5440 apply(rule_tac x="\<lambda>x. a + x" in exI)
5441 apply(rule_tac x="\<lambda>x. -a + x" in exI)
5442 using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
5444 lemma homeomorphic_affinity:
5445 fixes s :: "'a::real_normed_vector set"
5446 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
5448 have *:"op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
5450 using homeomorphic_trans
5451 using homeomorphic_scaling[OF assms, of s]
5452 using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a] unfolding * by auto
5455 lemma homeomorphic_balls:
5456 fixes a b ::"'a::real_normed_vector" (* FIXME: generalize to metric_space *)
5457 assumes "0 < d" "0 < e"
5458 shows "(ball a d) homeomorphic (ball b e)" (is ?th)
5459 "(cball a d) homeomorphic (cball b e)" (is ?cth)
5461 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5462 show ?th unfolding homeomorphic_minimal
5463 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5464 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5465 using assms apply (auto simp add: dist_commute)
5467 apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)
5468 unfolding continuous_on
5469 by (intro ballI tendsto_intros, simp, assumption)+
5471 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5472 show ?cth unfolding homeomorphic_minimal
5473 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5474 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5475 using assms apply (auto simp add: dist_commute)
5477 apply (auto simp add: pos_divide_le_eq)
5478 unfolding continuous_on
5479 by (intro ballI tendsto_intros, simp, assumption)+
5482 text{* "Isometry" (up to constant bounds) of injective linear map etc. *}
5484 lemma cauchy_isometric:
5485 fixes x :: "nat \<Rightarrow> real ^ 'n"
5486 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)"
5489 interpret f: bounded_linear f by fact
5490 { fix d::real assume "d>0"
5491 then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
5492 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
5493 { fix n assume "n\<ge>N"
5494 hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding f.diff[THEN sym] by auto
5495 moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
5496 using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
5497 using normf[THEN bspec[where x="x n - x N"]] by auto
5498 ultimately have "norm (x n - x N) < d" using `e>0`
5499 using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto }
5500 hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
5501 thus ?thesis unfolding cauchy and dist_norm by auto
5504 lemma complete_isometric_image:
5505 fixes f :: "real ^ _ \<Rightarrow> real ^ _"
5506 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"
5507 shows "complete(f ` s)"
5509 { fix g assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
5510 then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
5511 using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
5512 hence x:"\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
5513 hence "f \<circ> x = g" unfolding expand_fun_eq by auto
5514 then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
5515 using cs[unfolded complete_def, THEN spec[where x="x"]]
5516 using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
5517 hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
5518 using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
5519 unfolding `f \<circ> x = g` by auto }
5520 thus ?thesis unfolding complete_def by auto
5524 fixes x :: "'a::real_normed_vector"
5525 shows "dist 0 x = norm x"
5526 unfolding dist_norm by simp
5528 lemma injective_imp_isometric: fixes f::"real^'m \<Rightarrow> real^'n"
5529 assumes s:"closed s" "subspace s" and f:"bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
5530 shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
5531 proof(cases "s \<subseteq> {0::real^'m}")
5533 { fix x assume "x \<in> s"
5534 hence "x = 0" using True by auto
5535 hence "norm x \<le> norm (f x)" by auto }
5536 thus ?thesis by(auto intro!: exI[where x=1])
5538 interpret f: bounded_linear f by fact
5540 then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
5541 from False have "s \<noteq> {}" by auto
5542 let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
5543 let ?S' = "{x::real^'m. x\<in>s \<and> norm x = norm a}"
5544 let ?S'' = "{x::real^'m. norm x = norm a}"
5546 have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by (auto simp add: norm_minus_cancel)
5547 hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
5548 moreover have "?S' = s \<inter> ?S''" by auto
5549 ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
5550 moreover have *:"f ` ?S' = ?S" by auto
5551 ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
5552 hence "closed ?S" using compact_imp_closed by auto
5553 moreover have "?S \<noteq> {}" using a by auto
5554 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
5555 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
5557 let ?e = "norm (f b) / norm b"
5558 have "norm b > 0" using ba and a and norm_ge_zero by auto
5559 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
5560 ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
5562 { fix x assume "x\<in>s"
5563 hence "norm (f b) / norm b * norm x \<le> norm (f x)"
5565 case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
5568 hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
5569 have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def smult_conv_scaleR] by auto
5570 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
5571 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"]]
5572 unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
5573 by (auto simp add: real_mult_commute pos_le_divide_eq pos_divide_le_eq)
5576 show ?thesis by auto
5579 lemma closed_injective_image_subspace:
5580 fixes f :: "real ^ _ \<Rightarrow> real ^ _"
5581 assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
5582 shows "closed(f ` s)"
5584 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
5585 show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
5586 unfolding complete_eq_closed[THEN sym] by auto
5589 subsection{* Some properties of a canonical subspace. *}
5591 lemma subspace_substandard:
5592 "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
5593 unfolding subspace_def by(auto simp add: vector_add_component vector_smult_component elim!: ballE)
5595 lemma closed_substandard:
5596 "closed {x::real^'n. \<forall>i. P i --> x$i = 0}" (is "closed ?A")
5599 let ?Bs = "{{x::real^'n. inner (basis i) x = 0}| i. i \<in> ?D}"
5602 hence x:"\<forall>i\<in>?D. x $ i = 0" by auto
5603 hence "x\<in> \<Inter> ?Bs" by(auto simp add: inner_basis x) }
5605 { assume x:"x\<in>\<Inter>?Bs"
5606 { fix i assume i:"i \<in> ?D"
5607 then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::real^'n. inner (basis i) x = 0}" by auto
5608 hence "x $ i = 0" unfolding B using x unfolding inner_basis by auto }
5609 hence "x\<in>?A" by auto }
5610 ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" by auto }
5611 hence "?A = \<Inter> ?Bs" by auto
5612 thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
5615 lemma dim_substandard:
5616 shows "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d" (is "dim ?A = _")
5618 let ?D = "UNIV::'n set"
5619 let ?B = "(basis::'n\<Rightarrow>real^'n) ` d"
5621 let ?bas = "basis::'n \<Rightarrow> real^'n"
5623 have "?B \<subseteq> ?A" by auto
5626 { fix x::"real^'n" assume "x\<in>?A"
5628 have "x\<in> span ?B"
5629 proof(induct d arbitrary: x)
5630 case empty hence "x=0" unfolding Cart_eq by auto
5631 thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
5634 hence *:"\<forall>i. i \<notin> insert k F \<longrightarrow> x $ i = 0" by auto
5635 have **:"F \<subseteq> insert k F" by auto
5636 def y \<equiv> "x - x$k *\<^sub>R basis k"
5637 have y:"x = y + (x$k) *\<^sub>R basis k" unfolding y_def by auto
5638 { fix i assume i':"i \<notin> F"
5639 hence "y $ i = 0" unfolding y_def unfolding vector_minus_component
5640 and vector_smult_component and basis_component
5641 using *[THEN spec[where x=i]] by auto }
5642 hence "y \<in> span (basis ` (insert k F))" using insert(3)
5643 using span_mono[of "?bas ` F" "?bas ` (insert k F)"]
5644 using image_mono[OF **, of basis] by auto
5646 have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
5647 hence "x$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5648 using span_mul [where 'a=real, unfolded smult_conv_scaleR] by auto
5650 have "y + x$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5651 using span_add by auto
5652 thus ?case using y by auto
5655 hence "?A \<subseteq> span ?B" by auto
5658 { fix x assume "x \<in> ?B"
5659 hence "x\<in>{(basis i)::real^'n |i. i \<in> ?D}" using assms by auto }
5660 hence "independent ?B" using independent_mono[OF independent_stdbasis, of ?B] and assms by auto
5663 have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
5664 hence *:"inj_on (basis::'n\<Rightarrow>real^'n) d" using subset_inj_on[OF basis_inj, of "d"] by auto
5665 have "card ?B = card d" unfolding card_image[OF *] by auto
5667 ultimately show ?thesis using dim_unique[of "basis ` d" ?A] by auto
5670 text{* Hence closure and completeness of all subspaces. *}
5672 lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"
5674 apply (rule_tac x="{}" in exI, simp)
5676 apply (subgoal_tac "\<exists>x. x \<notin> A")
5678 apply (rule_tac x="insert x A" in exI, simp)
5679 apply (subgoal_tac "A \<noteq> UNIV", auto)
5682 lemma closed_subspace: fixes s::"(real^'n) set"
5683 assumes "subspace s" shows "closed s"
5685 have "dim s \<le> card (UNIV :: 'n set)" using dim_subset_univ by auto
5686 then obtain d::"'n set" where t: "card d = dim s"
5687 using closed_subspace_lemma by auto
5688 let ?t = "{x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0}"
5689 obtain f where f:"bounded_linear f" "f ` ?t = s" "inj_on f ?t"
5690 using subspace_isomorphism[unfolded linear_conv_bounded_linear, OF subspace_substandard[of "\<lambda>i. i \<notin> d"] assms]
5691 using dim_substandard[of d] and t by auto
5692 interpret f: bounded_linear f by fact
5693 have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using f.zero using f(3)[unfolded inj_on_def]
5694 by(erule_tac x=0 in ballE) auto
5695 moreover have "closed ?t" using closed_substandard .
5696 moreover have "subspace ?t" using subspace_substandard .
5697 ultimately show ?thesis using closed_injective_image_subspace[of ?t f]
5698 unfolding f(2) using f(1) by auto
5701 lemma complete_subspace:
5702 fixes s :: "(real ^ _) set" shows "subspace s ==> complete s"
5703 using complete_eq_closed closed_subspace
5707 fixes s :: "(real ^ _) set"
5708 shows "dim(closure s) = dim s" (is "?dc = ?d")
5710 have "?dc \<le> ?d" using closure_minimal[OF span_inc, of s]
5711 using closed_subspace[OF subspace_span, of s]
5712 using dim_subset[of "closure s" "span s"] unfolding dim_span by auto
5713 thus ?thesis using dim_subset[OF closure_subset, of s] by auto
5716 text{* Affine transformations of intervals. *}
5718 lemma affinity_inverses:
5719 assumes m0: "m \<noteq> (0::'a::field)"
5720 shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
5721 "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
5723 apply (auto simp add: expand_fun_eq vector_add_ldistrib vector_smult_assoc)
5724 by (simp add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1[symmetric])
5726 lemma real_affinity_le:
5727 "0 < (m::'a::linordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"
5728 by (simp add: field_simps inverse_eq_divide)
5730 lemma real_le_affinity:
5731 "0 < (m::'a::linordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"
5732 by (simp add: field_simps inverse_eq_divide)
5734 lemma real_affinity_lt:
5735 "0 < (m::'a::linordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"
5736 by (simp add: field_simps inverse_eq_divide)
5738 lemma real_lt_affinity:
5739 "0 < (m::'a::linordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"
5740 by (simp add: field_simps inverse_eq_divide)
5742 lemma real_affinity_eq:
5743 "(m::'a::linordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"
5744 by (simp add: field_simps inverse_eq_divide)
5746 lemma real_eq_affinity:
5747 "(m::'a::linordered_field) \<noteq> 0 ==> (y = m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"
5748 by (simp add: field_simps inverse_eq_divide)
5750 lemma vector_affinity_eq:
5751 assumes m0: "(m::'a::field) \<noteq> 0"
5752 shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
5754 assume h: "m *s x + c = y"
5755 hence "m *s x = y - c" by (simp add: ring_simps)
5756 hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
5757 then show "x = inverse m *s y + - (inverse m *s c)"
5758 using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
5760 assume h: "x = inverse m *s y + - (inverse m *s c)"
5761 show "m *s x + c = y" unfolding h diff_minus[symmetric]
5762 using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
5765 lemma vector_eq_affinity:
5766 "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
5767 using vector_affinity_eq[where m=m and x=x and y=y and c=c]
5770 lemma image_affinity_interval: fixes m::real
5771 fixes a b c :: "real^'n"
5772 shows "(\<lambda>x. m *\<^sub>R x + c) ` {a .. b} =
5773 (if {a .. b} = {} then {}
5774 else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
5775 else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
5777 { fix x assume "x \<le> c" "c \<le> x"
5778 hence "x=c" unfolding vector_le_def and Cart_eq by (auto intro: order_antisym) }
5780 moreover have "c \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}" unfolding True by(auto simp add: vector_le_def)
5781 ultimately show ?thesis by auto
5784 { fix y assume "a \<le> y" "y \<le> b" "m > 0"
5785 hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
5786 unfolding vector_le_def by(auto simp add: vector_smult_component vector_add_component)
5788 { fix y assume "a \<le> y" "y \<le> b" "m < 0"
5789 hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
5790 unfolding vector_le_def by(auto simp add: vector_smult_component vector_add_component mult_left_mono_neg elim!:ballE)
5792 { fix y assume "m > 0" "m *\<^sub>R a + c \<le> y" "y \<le> m *\<^sub>R b + c"
5793 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5794 unfolding image_iff Bex_def mem_interval vector_le_def
5795 apply(auto simp add: vector_smult_component vector_add_component vector_minus_component vector_smult_assoc pth_3[symmetric]
5796 intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
5797 by(auto simp add: pos_le_divide_eq pos_divide_le_eq real_mult_commute diff_le_iff)
5799 { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
5800 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5801 unfolding image_iff Bex_def mem_interval vector_le_def
5802 apply(auto simp add: vector_smult_component vector_add_component vector_minus_component vector_smult_assoc pth_3[symmetric]
5803 intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
5804 by(auto simp add: neg_le_divide_eq neg_divide_le_eq real_mult_commute diff_le_iff)
5806 ultimately show ?thesis using False by auto
5809 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::real^'n)) ` {a..b} =
5810 (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})"
5811 using image_affinity_interval[of m 0 a b] by auto
5813 subsection{* Banach fixed point theorem (not really topological...) *}
5816 assumes s:"complete s" "s \<noteq> {}" and c:"0 \<le> c" "c < 1" and f:"(f ` s) \<subseteq> s" and
5817 lipschitz:"\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
5818 shows "\<exists>! x\<in>s. (f x = x)"
5820 have "1 - c > 0" using c by auto
5822 from s(2) obtain z0 where "z0 \<in> s" by auto
5823 def z \<equiv> "\<lambda>n. (f ^^ n) z0"
5825 have "z n \<in> s" unfolding z_def
5826 proof(induct n) case 0 thus ?case using `z0 \<in>s` by auto
5827 next case Suc thus ?case using f by auto qed }
5830 def d \<equiv> "dist (z 0) (z 1)"
5832 have fzn:"\<And>n. f (z n) = z (Suc n)" unfolding z_def by auto
5834 have "dist (z n) (z (Suc n)) \<le> (c ^ n) * d"
5836 case 0 thus ?case unfolding d_def by auto
5839 hence "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
5840 using `0 \<le> c` using mult_mono1_class.mult_mono1[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by auto
5841 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]
5842 unfolding fzn and mult_le_cancel_left by auto
5847 have "(1 - c) * dist (z m) (z (m+n)) \<le> (c ^ m) * d * (1 - c ^ n)"
5849 case 0 show ?case by auto
5852 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))))"
5853 using dist_triangle and c by(auto simp add: dist_triangle)
5854 also have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
5855 using cf_z[of "m + k"] and c by auto
5856 also have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
5857 using Suc by (auto simp add: ring_simps)
5858 also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
5859 unfolding power_add by (auto simp add: ring_simps)
5860 also have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
5861 using c by (auto simp add: ring_simps)
5862 finally show ?case by auto
5865 { fix e::real assume "e>0"
5866 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e"
5867 proof(cases "d = 0")
5869 hence "\<And>n. z n = z0" using cf_z2[of 0] and c unfolding z_def by (auto simp add: pos_prod_le[OF `1 - c > 0`])
5870 thus ?thesis using `e>0` by auto
5872 case False hence "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]
5873 by (metis False d_def real_less_def)
5874 hence "0 < e * (1 - c) / d" using `e>0` and `1-c>0`
5875 using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"] by auto
5876 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
5877 { fix m n::nat assume "m>n" and as:"m\<ge>N" "n\<ge>N"
5878 have *:"c ^ n \<le> c ^ N" using `n\<ge>N` and c using power_decreasing[OF `n\<ge>N`, of c] by auto
5879 have "1 - c ^ (m - n) > 0" using c and power_strict_mono[of c 1 "m - n"] using `m>n` by auto
5880 hence **:"d * (1 - c ^ (m - n)) / (1 - c) > 0"
5881 using real_mult_order[OF `d>0`, of "1 - c ^ (m - n)"]
5882 using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]
5883 using `0 < 1 - c` by auto
5885 have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
5886 using cf_z2[of n "m - n"] and `m>n` unfolding pos_le_divide_eq[OF `1-c>0`]
5887 by (auto simp add: real_mult_commute dist_commute)
5888 also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
5889 using mult_right_mono[OF * order_less_imp_le[OF **]]
5890 unfolding real_mult_assoc by auto
5891 also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
5892 using mult_strict_right_mono[OF N **] unfolding real_mult_assoc by auto
5893 also have "\<dots> = e * (1 - c ^ (m - n))" using c and `d>0` and `1 - c > 0` by auto
5894 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
5895 finally have "dist (z m) (z n) < e" by auto
5897 { fix m n::nat assume as:"N\<le>m" "N\<le>n"
5898 hence "dist (z n) (z m) < e"
5899 proof(cases "n = m")
5900 case True thus ?thesis using `e>0` by auto
5902 case False thus ?thesis using as and *[of n m] *[of m n] unfolding nat_neq_iff by (auto simp add: dist_commute)
5904 thus ?thesis by auto
5907 hence "Cauchy z" unfolding cauchy_def by auto
5908 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
5910 def e \<equiv> "dist (f x) x"
5911 have "e = 0" proof(rule ccontr)
5912 assume "e \<noteq> 0" hence "e>0" unfolding e_def using zero_le_dist[of "f x" x]
5913 by (metis dist_eq_0_iff dist_nz e_def)
5914 then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"
5915 using x[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
5916 hence N':"dist (z N) x < e / 2" by auto
5918 have *:"c * dist (z N) x \<le> dist (z N) x" unfolding mult_le_cancel_right2
5919 using zero_le_dist[of "z N" x] and c
5920 by (metis dist_eq_0_iff dist_nz order_less_asym real_less_def)
5921 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]]
5922 using z_in_s[of N] `x\<in>s` using c by auto
5923 also have "\<dots> < e / 2" using N' and c using * by auto
5924 finally show False unfolding fzn
5925 using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
5926 unfolding e_def by auto
5928 hence "f x = x" unfolding e_def by auto
5930 { fix y assume "f y = y" "y\<in>s"
5931 hence "dist x y \<le> c * dist x y" using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]
5932 using `x\<in>s` and `f x = x` by auto
5933 hence "dist x y = 0" unfolding mult_le_cancel_right1
5934 using c and zero_le_dist[of x y] by auto
5935 hence "y = x" by auto
5937 ultimately show ?thesis using `x\<in>s` by blast+
5940 subsection{* Edelstein fixed point theorem. *}
5942 lemma edelstein_fix:
5943 fixes s :: "'a::real_normed_vector set"
5944 assumes s:"compact s" "s \<noteq> {}" and gs:"(g ` s) \<subseteq> s"
5945 and dist:"\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
5946 shows "\<exists>! x\<in>s. g x = x"
5947 proof(cases "\<exists>x\<in>s. g x \<noteq> x")
5948 obtain x where "x\<in>s" using s(2) by auto
5949 case False hence g:"\<forall>x\<in>s. g x = x" by auto
5950 { fix y assume "y\<in>s"
5951 hence "x = y" using `x\<in>s` and dist[THEN bspec[where x=x], THEN bspec[where x=y]]
5952 unfolding g[THEN bspec[where x=x], OF `x\<in>s`]
5953 unfolding g[THEN bspec[where x=y], OF `y\<in>s`] by auto }
5954 thus ?thesis using `x\<in>s` and g by blast+
5957 then obtain x where [simp]:"x\<in>s" and "g x \<noteq> x" by auto
5958 { fix x y assume "x \<in> s" "y \<in> s"
5959 hence "dist (g x) (g y) \<le> dist x y"
5960 using dist[THEN bspec[where x=x], THEN bspec[where x=y]] by auto } note dist' = this
5961 def y \<equiv> "g x"
5962 have [simp]:"y\<in>s" unfolding y_def using gs[unfolded image_subset_iff] and `x\<in>s` by blast
5963 def f \<equiv> "\<lambda>n. g ^^ n"
5964 have [simp]:"\<And>n z. g (f n z) = f (Suc n) z" unfolding f_def by auto
5965 have [simp]:"\<And>z. f 0 z = z" unfolding f_def by auto
5966 { fix n::nat and z assume "z\<in>s"
5967 have "f n z \<in> s" unfolding f_def
5969 case 0 thus ?case using `z\<in>s` by simp
5971 case (Suc n) thus ?case using gs[unfolded image_subset_iff] by auto
5972 qed } note fs = this
5973 { fix m n ::nat assume "m\<le>n"
5974 fix w z assume "w\<in>s" "z\<in>s"
5975 have "dist (f n w) (f n z) \<le> dist (f m w) (f m z)" using `m\<le>n`
5977 case 0 thus ?case by auto
5980 thus ?case proof(cases "m\<le>n")
5981 case True thus ?thesis using Suc(1)
5982 using dist'[OF fs fs, OF `w\<in>s` `z\<in>s`, of n n] by auto
5984 case False hence mn:"m = Suc n" using Suc(2) by simp
5985 show ?thesis unfolding mn by auto
5987 qed } note distf = this
5989 def h \<equiv> "\<lambda>n. (f n x, f n y)"
5990 let ?s2 = "s \<times> s"
5991 obtain l r where "l\<in>?s2" and r:"subseq r" and lr:"((h \<circ> r) ---> l) sequentially"
5992 using compact_Times [OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def
5993 using fs[OF `x\<in>s`] and fs[OF `y\<in>s`] by blast
5994 def a \<equiv> "fst l" def b \<equiv> "snd l"
5995 have lab:"l = (a, b)" unfolding a_def b_def by simp
5996 have [simp]:"a\<in>s" "b\<in>s" unfolding a_def b_def using `l\<in>?s2` by auto
5998 have lima:"((fst \<circ> (h \<circ> r)) ---> a) sequentially"
5999 and limb:"((snd \<circ> (h \<circ> r)) ---> b) sequentially"
6001 unfolding o_def a_def b_def by (simp_all add: tendsto_intros)
6004 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
6006 have "dist (-x) (-y) = dist x y" unfolding dist_norm
6007 using norm_minus_cancel[of "x - y"] by (auto simp add: uminus_add_conv_diff) } note ** = this
6009 { assume as:"dist a b > dist (f n x) (f n y)"
6010 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"
6011 and "\<forall>m\<ge>Nb. dist (f (r m) y) b < (dist a b - dist (f n x) (f n y)) / 2"
6012 using lima limb unfolding h_def Lim_sequentially by (fastsimp simp del: less_divide_eq_number_of1)
6013 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)"
6014 apply(erule_tac x="Na+Nb+n" in allE)
6015 apply(erule_tac x="Na+Nb+n" in allE) apply simp
6016 using dist_triangle_add_half[of a "f (r (Na + Nb + n)) x" "dist a b - dist (f n x) (f n y)"
6017 "-b" "- f (r (Na + Nb + n)) y"]
6018 unfolding ** unfolding group_simps(12) by (auto simp add: dist_commute)
6020 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)"
6021 using distf[of n "r (Na+Nb+n)", OF _ `x\<in>s` `y\<in>s`]
6022 using subseq_bigger[OF r, of "Na+Nb+n"]
6023 using *[of "f (r (Na + Nb + n)) x" "f (r (Na + Nb + n)) y" "f n x" "f n y"] by auto
6024 ultimately have False by simp
6026 hence "dist a b \<le> dist (f n x) (f n y)" by(rule ccontr)auto }
6029 have [simp]:"a = b" proof(rule ccontr)
6030 def e \<equiv> "dist a b - dist (g a) (g b)"
6031 assume "a\<noteq>b" hence "e > 0" unfolding e_def using dist by fastsimp
6032 hence "\<exists>n. dist (f n x) a < e/2 \<and> dist (f n y) b < e/2"
6033 using lima limb unfolding Lim_sequentially
6034 apply (auto elim!: allE[where x="e/2"]) apply(rule_tac x="r (max N Na)" in exI) unfolding h_def by fastsimp
6035 then obtain n where n:"dist (f n x) a < e/2 \<and> dist (f n y) b < e/2" by auto
6036 have "dist (f (Suc n) x) (g a) \<le> dist (f n x) a"
6037 using dist[THEN bspec[where x="f n x"], THEN bspec[where x="a"]] and fs by auto
6038 moreover have "dist (f (Suc n) y) (g b) \<le> dist (f n y) b"
6039 using dist[THEN bspec[where x="f n y"], THEN bspec[where x="b"]] and fs by auto
6040 ultimately have "dist (f (Suc n) x) (g a) + dist (f (Suc n) y) (g b) < e" using n by auto
6041 thus False unfolding e_def using ab_fn[of "Suc n"] by norm
6044 have [simp]:"\<And>n. f (Suc n) x = f n y" unfolding f_def y_def by(induct_tac n)auto
6045 { fix x y assume "x\<in>s" "y\<in>s" moreover
6046 fix e::real assume "e>0" ultimately
6047 have "dist y x < e \<longrightarrow> dist (g y) (g x) < e" using dist by fastsimp }
6048 hence "continuous_on s g" unfolding continuous_on_def by auto
6050 hence "((snd \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
6051 apply (rule allE[where x="\<lambda>n. (fst \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
6052 using lima unfolding h_def o_def using fs[OF `x\<in>s`] by (auto simp add: y_def)
6053 hence "g a = a" using Lim_unique[OF trivial_limit_sequentially limb, of "g a"]
6054 unfolding `a=b` and o_assoc by auto
6056 { fix x assume "x\<in>s" "g x = x" "x\<noteq>a"
6057 hence "False" using dist[THEN bspec[where x=a], THEN bspec[where x=x]]
6058 using `g a = a` and `a\<in>s` by auto }
6059 ultimately show "\<exists>!x\<in>s. g x = x" using `a\<in>s` by blast