1 (* title: HOL/Library/Topology_Euclidian_Space.thy
2 Author: Amine Chaieb, University of Cambridge
3 Author: Robert Himmelmann, TU Muenchen
4 Author: Brian Huffman, Portland State University
7 header {* Elementary topology in Euclidean space. *}
9 theory Topology_Euclidean_Space
10 imports SEQ Euclidean_Space "~~/src/HOL/Library/Glbs"
13 (* to be moved elsewhere *)
15 lemma euclidean_dist_l2:"dist x (y::'a::euclidean_space) = setL2 (\<lambda>i. dist(x$$i) (y$$i)) {..<DIM('a)}"
16 unfolding dist_norm norm_eq_sqrt_inner setL2_def apply(subst euclidean_inner)
17 apply(auto simp add:power2_eq_square) unfolding euclidean_component.diff ..
19 lemma dist_nth_le: "dist (x $$ i) (y $$ i) \<le> dist x (y::'a::euclidean_space)"
20 apply(subst(2) euclidean_dist_l2) apply(cases "i<DIM('a)")
21 apply(rule member_le_setL2) by auto
23 subsection{* General notion of a topology *}
25 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)"
26 typedef (open) 'a topology = "{L::('a set) set. istopology L}"
27 morphisms "openin" "topology"
28 unfolding istopology_def by blast
30 lemma istopology_open_in[intro]: "istopology(openin U)"
31 using openin[of U] by blast
33 lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
34 using topology_inverse[unfolded mem_def Collect_def] .
36 lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
37 using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
39 lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
41 {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}
43 {assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
44 hence "openin T1 = openin T2" by (metis mem_def set_eqI)
45 hence "topology (openin T1) = topology (openin T2)" by simp
46 hence "T1 = T2" unfolding openin_inverse .}
47 ultimately show ?thesis by blast
50 text{* Infer the "universe" from union of all sets in the topology. *}
52 definition "topspace T = \<Union>{S. openin T S}"
54 subsection{* Main properties of open sets *}
57 fixes U :: "'a topology"
59 "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
60 "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
61 using openin[of U] unfolding istopology_def Collect_def mem_def
62 unfolding subset_eq Ball_def mem_def by auto
64 lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
65 unfolding topspace_def by blast
66 lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
68 lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
69 using openin_clauses by simp
71 lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"
72 using openin_clauses by simp
74 lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
75 using openin_Union[of "{S,T}" U] by auto
77 lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
79 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")
81 assume ?lhs then show ?rhs by auto
84 let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
85 have "openin U ?t" by (simp add: openin_Union)
86 also have "?t = S" using H by auto
87 finally show "openin U S" .
90 subsection{* Closed sets *}
92 definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
94 lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
95 lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
96 lemma closedin_topspace[intro,simp]:
97 "closedin U (topspace U)" by (simp add: closedin_def)
98 lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
99 by (auto simp add: Diff_Un closedin_def)
101 lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
102 lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
103 shows "closedin U (\<Inter> K)" using Ke Kc unfolding closedin_def Diff_Inter by auto
105 lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
106 using closedin_Inter[of "{S,T}" U] by auto
108 lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
109 lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
110 apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
111 apply (metis openin_subset subset_eq)
114 lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
115 by (simp add: openin_closedin_eq)
117 lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
119 have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S] oS cT
120 by (auto simp add: topspace_def openin_subset)
121 then show ?thesis using oS cT by (auto simp add: closedin_def)
124 lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
126 have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S] oS cT
127 by (auto simp add: topspace_def )
128 then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
131 subsection{* Subspace topology. *}
133 definition "subtopology U V = topology {S \<inter> V |S. openin U S}"
135 lemma istopology_subtopology: "istopology {S \<inter> V |S. openin U S}" (is "istopology ?L")
137 have "{} \<in> ?L" by blast
138 {fix A B assume A: "A \<in> ?L" and B: "B \<in> ?L"
139 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
140 have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)" using Sa Sb by blast+
141 then have "A \<inter> B \<in> ?L" by blast}
143 {fix K assume K: "K \<subseteq> ?L"
144 have th0: "?L = (\<lambda>S. S \<inter> V) ` openin U "
146 apply (simp add: Ball_def image_iff)
148 from K[unfolded th0 subset_image_iff]
149 obtain Sk where Sk: "Sk \<subseteq> openin U" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
150 have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
151 moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq mem_def)
152 ultimately have "\<Union>K \<in> ?L" by blast}
153 ultimately show ?thesis unfolding istopology_def by blast
156 lemma openin_subtopology:
157 "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
158 unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
159 by (auto simp add: Collect_def)
161 lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
162 by (auto simp add: topspace_def openin_subtopology)
164 lemma closedin_subtopology:
165 "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
166 unfolding closedin_def topspace_subtopology
167 apply (simp add: openin_subtopology)
170 apply (rule_tac x="topspace U - T" in exI)
173 lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
174 unfolding openin_subtopology
175 apply (rule iffI, clarify)
176 apply (frule openin_subset[of U]) apply blast
177 apply (rule exI[where x="topspace U"])
180 lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"
181 shows "subtopology U V = U"
184 {fix T assume T: "openin U T" "S = T \<inter> V"
185 from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
186 have "openin U S" unfolding eq using T by blast}
188 {assume S: "openin U S"
189 hence "\<exists>T. openin U T \<and> S = T \<inter> V"
190 using openin_subset[OF S] UV by auto}
191 ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
192 then show ?thesis unfolding topology_eq openin_subtopology by blast
196 lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
197 by (simp add: subtopology_superset)
199 lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
200 by (simp add: subtopology_superset)
202 subsection{* The universal Euclidean versions are what we use most of the time *}
205 euclidean :: "'a::topological_space topology" where
206 "euclidean = topology open"
208 lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
209 unfolding euclidean_def
210 apply (rule cong[where x=S and y=S])
211 apply (rule topology_inverse[symmetric])
212 apply (auto simp add: istopology_def)
213 by (auto simp add: mem_def subset_eq)
215 lemma topspace_euclidean: "topspace euclidean = UNIV"
216 apply (simp add: topspace_def)
218 by (auto simp add: open_openin[symmetric])
220 lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
221 by (simp add: topspace_euclidean topspace_subtopology)
223 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
224 by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
226 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
227 by (simp add: open_openin openin_subopen[symmetric])
229 subsection{* Open and closed balls. *}
232 ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
233 "ball x e = {y. dist x y < e}"
236 cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
237 "cball x e = {y. dist x y \<le> e}"
239 lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
240 lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
242 lemma mem_ball_0 [simp]:
243 fixes x :: "'a::real_normed_vector"
244 shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
245 by (simp add: dist_norm)
247 lemma mem_cball_0 [simp]:
248 fixes x :: "'a::real_normed_vector"
249 shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
250 by (simp add: dist_norm)
252 lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e" by simp
253 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
254 lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
255 lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
256 lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
257 by (simp add: set_eq_iff) arith
259 lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
260 by (simp add: set_eq_iff)
262 lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
263 "(a::real) - b < 0 \<longleftrightarrow> a < b"
264 "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
265 lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
266 "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b" by arith+
268 lemma open_ball[intro, simp]: "open (ball x e)"
269 unfolding open_dist ball_def Collect_def Ball_def mem_def
270 unfolding dist_commute
272 apply (rule_tac x="e - dist xa x" in exI)
273 using dist_triangle_alt[where z=x]
274 apply (clarsimp simp add: diff_less_iff)
276 apply (erule_tac x="y" in allE)
277 apply (erule_tac x="xa" in allE)
280 lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
281 lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
282 unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
285 assumes "open S" "x\<in>S"
286 obtains e where "e>0" "ball x e \<subseteq> S"
287 using assms unfolding open_contains_ball by auto
289 lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
290 by (metis open_contains_ball subset_eq centre_in_ball)
292 lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
293 unfolding mem_ball set_eq_iff
294 apply (simp add: not_less)
295 by (metis zero_le_dist order_trans dist_self)
297 lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
299 subsection{* Basic "localization" results are handy for connectedness. *}
301 lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
302 by (auto simp add: openin_subtopology open_openin[symmetric])
304 lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
305 by (auto simp add: openin_open)
307 lemma open_openin_trans[trans]:
308 "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
309 by (metis Int_absorb1 openin_open_Int)
311 lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
312 by (auto simp add: openin_open)
314 lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
315 by (simp add: closedin_subtopology closed_closedin Int_ac)
317 lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
318 by (metis closedin_closed)
320 lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
321 apply (subgoal_tac "S \<inter> T = T" )
323 apply (frule closedin_closed_Int[of T S])
326 lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
327 by (auto simp add: closedin_closed)
329 lemma openin_euclidean_subtopology_iff:
330 fixes S U :: "'a::metric_space set"
331 shows "openin (subtopology euclidean U) S
332 \<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")
334 {assume ?lhs hence ?rhs unfolding openin_subtopology open_openin[symmetric]
335 by (simp add: open_dist) blast}
337 {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"
338 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)"
340 let ?T = "\<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
341 have oT: "open ?T" by auto
342 { fix x assume "x\<in>S"
343 hence "x \<in> \<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
344 apply simp apply(rule_tac x="ball x(d x)" in exI) apply auto
345 by (rule d [THEN conjunct1])
346 hence "x\<in> ?T \<inter> U" using SU and `x\<in>S` by auto }
348 { fix y assume "y\<in>?T"
349 then obtain B where "y\<in>B" "B\<in>{B. \<exists>x\<in>S. B = ball x (d x)}" by auto
350 then obtain x where "x\<in>S" and x:"y \<in> ball x (d x)" by auto
352 hence "y\<in>S" using d[OF `x\<in>S`] and x by(auto simp add: dist_commute) }
353 ultimately have "S = ?T \<inter> U" by blast
354 with oT have ?lhs unfolding openin_subtopology open_openin[symmetric] by blast}
355 ultimately show ?thesis by blast
358 text{* These "transitivity" results are handy too. *}
360 lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
361 \<Longrightarrow> openin (subtopology euclidean U) S"
362 unfolding open_openin openin_open by blast
364 lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
365 by (auto simp add: openin_open intro: openin_trans)
367 lemma closedin_trans[trans]:
368 "closedin (subtopology euclidean T) S \<Longrightarrow>
369 closedin (subtopology euclidean U) T
370 ==> closedin (subtopology euclidean U) S"
371 by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
373 lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
374 by (auto simp add: closedin_closed intro: closedin_trans)
376 subsection{* Connectedness *}
378 definition "connected S \<longleftrightarrow>
379 ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
380 \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
382 lemma connected_local:
383 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
384 openin (subtopology euclidean S) e1 \<and>
385 openin (subtopology euclidean S) e2 \<and>
386 S \<subseteq> e1 \<union> e2 \<and>
387 e1 \<inter> e2 = {} \<and>
390 unfolding connected_def openin_open by (safe, blast+)
393 fixes P :: "'a set \<Rightarrow> bool"
394 shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
396 {assume "?lhs" hence ?rhs by blast }
398 {fix S assume H: "P S"
399 have "S = - (- S)" by auto
400 with H have "P (- (- S))" by metis }
401 ultimately show ?thesis by metis
404 lemma connected_clopen: "connected S \<longleftrightarrow>
405 (\<forall>T. openin (subtopology euclidean S) T \<and>
406 closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
408 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> {})"
409 unfolding connected_def openin_open closedin_closed
410 apply (subst exists_diff) by blast
411 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> {})"
412 (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis
414 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'))"
415 (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
416 unfolding connected_def openin_open closedin_closed by auto
418 {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)"
420 then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
421 then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
422 then show ?thesis unfolding th0 th1 by simp
425 lemma connected_empty[simp, intro]: "connected {}"
426 by (simp add: connected_def)
428 subsection{* Limit points *}
431 islimpt:: "'a::topological_space \<Rightarrow> 'a set \<Rightarrow> bool"
432 (infixr "islimpt" 60) where
433 "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
436 assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
438 using assms unfolding islimpt_def by auto
441 assumes "x islimpt S" and "x \<in> T" and "open T"
442 obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
443 using assms unfolding islimpt_def by auto
445 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
447 lemma islimpt_approachable:
448 fixes x :: "'a::metric_space"
449 shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
450 unfolding islimpt_def
452 apply(erule_tac x="ball x e" in allE)
454 apply(rule_tac x=y in bexI)
455 apply (auto simp add: dist_commute)
456 apply (simp add: open_dist, drule (1) bspec)
457 apply (clarify, drule spec, drule (1) mp, auto)
460 lemma islimpt_approachable_le:
461 fixes x :: "'a::metric_space"
462 shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
463 unfolding islimpt_approachable
464 using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
467 class perfect_space =
468 assumes islimpt_UNIV [simp, intro]: "(x::'a::topological_space) islimpt UNIV"
470 lemma perfect_choose_dist:
471 fixes x :: "'a::{perfect_space, metric_space}"
472 shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
473 using islimpt_UNIV [of x]
474 by (simp add: islimpt_approachable)
476 instance real :: perfect_space
478 apply (rule islimpt_approachable [THEN iffD2])
479 apply (clarify, rule_tac x="x + e/2" in bexI)
480 apply (auto simp add: dist_norm)
483 instance euclidean_space \<subseteq> perfect_space
485 { fix e :: real assume "0 < e"
486 def a \<equiv> "x $$ 0"
487 have "a islimpt UNIV" by (rule islimpt_UNIV)
488 with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"
489 unfolding islimpt_approachable by auto
490 def y \<equiv> "\<chi>\<chi> i. if i = 0 then b else x$$i :: 'a"
491 from `b \<noteq> a` have "y \<noteq> x" unfolding a_def y_def apply(subst euclidean_eq) apply safe
492 apply(erule_tac x=0 in allE) using DIM_positive[where 'a='a] by auto
494 have *:"(\<Sum>i<DIM('a). (dist (y $$ i) (x $$ i))\<twosuperior>) = (\<Sum>i\<in>{0}. (dist (y $$ i) (x $$ i))\<twosuperior>)"
495 apply(rule setsum_mono_zero_right) unfolding y_def by auto
496 from `dist b a < e` have "dist y x < e"
497 apply(subst euclidean_dist_l2)
498 unfolding setL2_def * unfolding y_def a_def using `0 < e` by auto
499 from `y \<noteq> x` and `dist y x < e`
500 have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
502 then show "x islimpt UNIV" unfolding islimpt_approachable by blast
505 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
507 apply (subst open_subopen)
508 apply (simp add: islimpt_def subset_eq)
509 by (metis ComplE ComplI insertCI insert_absorb mem_def)
511 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
512 unfolding islimpt_def by auto
514 lemma finite_set_avoid:
515 fixes a :: "'a::metric_space"
516 assumes fS: "finite S" shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
517 proof(induct rule: finite_induct[OF fS])
518 case 1 thus ?case by (auto intro: zero_less_one)
521 from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
522 {assume "x = a" hence ?case using d by auto }
524 {assume xa: "x\<noteq>a"
525 let ?d = "min d (dist a x)"
526 have dp: "?d > 0" using xa d(1) using dist_nz by auto
527 from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
528 with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
529 ultimately show ?case by blast
532 lemma islimpt_finite:
533 fixes S :: "'a::metric_space set"
534 assumes fS: "finite S" shows "\<not> a islimpt S"
535 unfolding islimpt_approachable
536 using finite_set_avoid[OF fS, of a] by (metis dist_commute not_le)
538 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
541 apply (metis Un_upper1 Un_upper2 islimpt_subset)
542 unfolding islimpt_def
543 apply (rule ccontr, clarsimp, rename_tac A B)
544 apply (drule_tac x="A \<inter> B" in spec)
545 apply (auto simp add: open_Int)
548 lemma discrete_imp_closed:
549 fixes S :: "'a::metric_space set"
550 assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
553 {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
554 from e have e2: "e/2 > 0" by arith
555 from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
556 let ?m = "min (e/2) (dist x y) "
557 from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
558 from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
559 have th: "dist z y < e" using z y
560 by (intro dist_triangle_lt [where z=x], simp)
561 from d[rule_format, OF y(1) z(1) th] y z
562 have False by (auto simp add: dist_commute)}
563 then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
566 subsection{* Interior of a Set *}
567 definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
569 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
570 apply (simp add: set_eq_iff interior_def)
571 apply (subst (2) open_subopen) by (safe, blast+)
573 lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
575 lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
577 lemma open_interior[simp, intro]: "open(interior S)"
578 apply (simp add: interior_def)
579 apply (subst open_subopen) by blast
581 lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
582 lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
583 lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
584 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
585 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"
586 by (metis equalityI interior_maximal interior_subset open_interior)
587 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
588 apply (simp add: interior_def)
589 by (metis open_contains_ball centre_in_ball open_ball subset_trans)
591 lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
592 by (metis interior_maximal interior_subset subset_trans)
594 lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
595 apply (rule equalityI, simp)
596 apply (metis Int_lower1 Int_lower2 subset_interior)
597 by (metis Int_mono interior_subset open_Int open_interior open_subset_interior)
599 lemma interior_limit_point [intro]:
600 fixes x :: "'a::perfect_space"
601 assumes x: "x \<in> interior S" shows "x islimpt S"
602 using x islimpt_UNIV [of x]
603 unfolding interior_def islimpt_def
604 apply (clarsimp, rename_tac T T')
605 apply (drule_tac x="T \<inter> T'" in spec)
606 apply (auto simp add: open_Int)
609 lemma interior_closed_Un_empty_interior:
610 assumes cS: "closed S" and iT: "interior T = {}"
611 shows "interior(S \<union> T) = interior S"
613 show "interior S \<subseteq> interior (S\<union>T)"
614 by (rule subset_interior, blast)
616 show "interior (S \<union> T) \<subseteq> interior S"
618 fix x assume "x \<in> interior (S \<union> T)"
619 then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T"
620 unfolding interior_def by fast
621 show "x \<in> interior S"
623 assume "x \<notin> interior S"
624 with `x \<in> R` `open R` obtain y where "y \<in> R - S"
625 unfolding interior_def set_eq_iff by fast
626 from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
627 from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
628 from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
629 show "False" unfolding interior_def by fast
635 subsection{* Closure of a Set *}
637 definition "closure S = S \<union> {x | x. x islimpt S}"
639 lemma closure_interior: "closure S = - interior (- S)"
642 have "x\<in>- interior (- S) \<longleftrightarrow> x \<in> closure S" (is "?lhs = ?rhs")
644 let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> - S)"
646 hence *:"\<not> ?exT x"
647 unfolding interior_def
649 { assume "\<not> ?rhs"
651 unfolding closure_def islimpt_def
657 assume "?rhs" thus "?lhs"
658 unfolding closure_def interior_def islimpt_def
666 lemma interior_closure: "interior S = - (closure (- S))"
669 have "x \<in> interior S \<longleftrightarrow> x \<in> - (closure (- S))"
670 unfolding interior_def closure_def islimpt_def
677 lemma closed_closure[simp, intro]: "closed (closure S)"
679 have "closed (- interior (-S))" by blast
680 thus ?thesis using closure_interior[of S] by simp
683 lemma closure_hull: "closure S = closed hull S"
685 have "S \<subseteq> closure S"
686 unfolding closure_def
689 have "closed (closure S)"
690 using closed_closure[of S]
694 assume *:"S \<subseteq> t" "closed t"
697 hence "x islimpt t" using *(1)
698 using islimpt_subset[of x, of S, of t]
701 with * have "closure S \<subseteq> t"
702 unfolding closure_def
703 using closed_limpt[of t]
706 ultimately show ?thesis
707 using hull_unique[of S, of "closure S", of closed]
712 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
713 unfolding closure_hull
714 using hull_eq[of closed, unfolded mem_def, OF closed_Inter, of S]
715 by (metis mem_def subset_eq)
717 lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
718 using closure_eq[of S]
721 lemma closure_closure[simp]: "closure (closure S) = closure S"
722 unfolding closure_hull
723 using hull_hull[of closed S]
726 lemma closure_subset: "S \<subseteq> closure S"
727 unfolding closure_hull
728 using hull_subset[of S closed]
731 lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
732 unfolding closure_hull
733 using hull_mono[of S T closed]
736 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
737 using hull_minimal[of S T closed]
738 unfolding closure_hull mem_def
741 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"
742 using hull_unique[of S T closed]
743 unfolding closure_hull mem_def
746 lemma closure_empty[simp]: "closure {} = {}"
747 using closed_empty closure_closed[of "{}"]
750 lemma closure_univ[simp]: "closure UNIV = UNIV"
751 using closure_closed[of UNIV]
754 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
755 using closure_empty closure_subset[of S]
758 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
759 using closure_eq[of S] closure_subset[of S]
762 lemma open_inter_closure_eq_empty:
763 "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
764 using open_subset_interior[of S "- T"]
765 using interior_subset[of "- T"]
766 unfolding closure_interior
769 lemma open_inter_closure_subset:
770 "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
773 assume as: "open S" "x \<in> S \<inter> closure T"
774 { assume *:"x islimpt T"
775 have "x islimpt (S \<inter> T)"
776 proof (rule islimptI)
778 assume "x \<in> A" "open A"
779 with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
780 by (simp_all add: open_Int)
781 with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
783 hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
785 thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
788 then show "x \<in> closure (S \<inter> T)" using as
789 unfolding closure_def
793 lemma closure_complement: "closure(- S) = - interior(S)"
798 unfolding closure_interior
802 lemma interior_complement: "interior(- S) = - closure(S)"
803 unfolding closure_interior
806 subsection{* Frontier (aka boundary) *}
808 definition "frontier S = closure S - interior S"
810 lemma frontier_closed: "closed(frontier S)"
811 by (simp add: frontier_def closed_Diff)
813 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
814 by (auto simp add: frontier_def interior_closure)
816 lemma frontier_straddle:
817 fixes a :: "'a::metric_space"
818 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")
823 let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
825 have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
826 moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
827 unfolding frontier_closures closure_def islimpt_def using `e>0`
828 by (auto, erule_tac x="ball a e" in allE, auto)
829 ultimately have ?rhse by auto
832 { assume "a\<notin>S"
833 hence ?rhse using `?lhs`
834 unfolding frontier_closures closure_def islimpt_def
835 using open_ball[of a e] `e > 0`
836 by simp (metis centre_in_ball mem_ball open_ball)
838 ultimately have ?rhse by auto
844 { fix T assume "a\<notin>S" and
845 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"
846 from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
847 then obtain e where "e>0" "ball a e \<subseteq> T" by auto
848 then obtain y where y:"y\<in>S" "dist a y < e" using as(1) by auto
849 have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
850 using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
852 hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
854 { fix T assume "a \<in> T" "open T" "a\<in>S"
855 then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
856 obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
857 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
859 hence "a islimpt (- S) \<or> a\<notin>S" unfolding islimpt_def by auto
860 ultimately show ?lhs unfolding frontier_closures using closure_def[of "- S"] by auto
863 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
864 by (metis frontier_def closure_closed Diff_subset)
866 lemma frontier_empty[simp]: "frontier {} = {}"
867 by (simp add: frontier_def)
869 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
871 { assume "frontier S \<subseteq> S"
872 hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
873 hence "closed S" using closure_subset_eq by auto
875 thus ?thesis using frontier_subset_closed[of S] ..
878 lemma frontier_complement: "frontier(- S) = frontier S"
879 by (auto simp add: frontier_def closure_complement interior_complement)
881 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
882 using frontier_complement frontier_subset_eq[of "- S"]
883 unfolding open_closed by auto
885 subsection {* Nets and the ``eventually true'' quantifier *}
887 text {* Common nets and The "within" modifier for nets. *}
890 at_infinity :: "'a::real_normed_vector net" where
891 "at_infinity = Abs_net (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
894 indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a net" (infixr "indirection" 70) where
895 "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
897 text{* Prove That They are all nets. *}
899 lemma eventually_at_infinity:
900 "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
901 unfolding at_infinity_def
902 proof (rule eventually_Abs_net, rule is_filter.intro)
903 fix P Q :: "'a \<Rightarrow> bool"
904 assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
905 then obtain r s where
906 "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
907 then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
908 then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
911 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
913 lemma trivial_limit_within:
914 shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
916 assume "trivial_limit (at a within S)"
917 thus "\<not> a islimpt S"
918 unfolding trivial_limit_def
919 unfolding eventually_within eventually_at_topological
920 unfolding islimpt_def
921 apply (clarsimp simp add: set_eq_iff)
922 apply (rename_tac T, rule_tac x=T in exI)
923 apply (clarsimp, drule_tac x=y in bspec, simp_all)
926 assume "\<not> a islimpt S"
927 thus "trivial_limit (at a within S)"
928 unfolding trivial_limit_def
929 unfolding eventually_within eventually_at_topological
930 unfolding islimpt_def
932 apply (rule_tac x=T in exI)
937 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
938 using trivial_limit_within [of a UNIV]
939 by (simp add: within_UNIV)
941 lemma trivial_limit_at:
942 fixes a :: "'a::perfect_space"
943 shows "\<not> trivial_limit (at a)"
944 by (simp add: trivial_limit_at_iff)
946 lemma trivial_limit_at_infinity:
947 "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) net)"
948 unfolding trivial_limit_def eventually_at_infinity
950 apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
951 apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
952 apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
953 apply (drule_tac x=UNIV in spec, simp)
956 text {* Some property holds "sufficiently close" to the limit point. *}
958 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
959 "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
960 unfolding eventually_at dist_nz by auto
962 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
963 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
964 unfolding eventually_within eventually_at dist_nz by auto
966 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
967 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
968 unfolding eventually_within
969 by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl)
971 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
972 unfolding trivial_limit_def
973 by (auto elim: eventually_rev_mp)
975 lemma always_eventually: "(\<forall>x. P x) ==> eventually P net"
977 assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
978 thus "eventually P net" by simp
981 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
982 unfolding trivial_limit_def by (auto elim: eventually_rev_mp)
984 lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
985 unfolding trivial_limit_def ..
988 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
989 apply (safe elim!: trivial_limit_eventually)
990 apply (simp add: eventually_False [symmetric])
993 text{* Combining theorems for "eventually" *}
995 lemma eventually_conjI:
996 "\<lbrakk>eventually (\<lambda>x. P x) net; eventually (\<lambda>x. Q x) net\<rbrakk>
997 \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) net"
998 by (rule eventually_conj)
1000 lemma eventually_rev_mono:
1001 "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
1002 using eventually_mono [of P Q] by fast
1004 lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
1005 by (auto intro!: eventually_conjI elim: eventually_rev_mono)
1007 lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
1008 by (auto simp add: eventually_False)
1010 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
1011 by (simp add: eventually_False)
1013 subsection {* Limits *}
1015 text{* Notation Lim to avoid collition with lim defined in analysis *}
1017 Lim :: "'a net \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b" where
1018 "Lim net f = (THE l. (f ---> l) net)"
1021 "(f ---> l) net \<longleftrightarrow>
1022 trivial_limit net \<or>
1023 (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
1024 unfolding tendsto_iff trivial_limit_eq by auto
1027 text{* Show that they yield usual definitions in the various cases. *}
1029 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
1030 (\<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)"
1031 by (auto simp add: tendsto_iff eventually_within_le)
1033 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
1034 (\<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)"
1035 by (auto simp add: tendsto_iff eventually_within)
1037 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
1038 (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
1039 by (auto simp add: tendsto_iff eventually_at)
1041 lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
1042 unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
1044 lemma Lim_at_infinity:
1045 "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
1046 by (auto simp add: tendsto_iff eventually_at_infinity)
1048 lemma Lim_sequentially:
1049 "(S ---> l) sequentially \<longleftrightarrow>
1050 (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
1051 by (auto simp add: tendsto_iff eventually_sequentially)
1053 lemma Lim_sequentially_iff_LIMSEQ: "(S ---> l) sequentially \<longleftrightarrow> S ----> l"
1054 unfolding Lim_sequentially LIMSEQ_def ..
1056 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
1057 by (rule topological_tendstoI, auto elim: eventually_rev_mono)
1059 text{* The expected monotonicity property. *}
1061 lemma Lim_within_empty: "(f ---> l) (net within {})"
1062 unfolding tendsto_def Limits.eventually_within by simp
1064 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
1065 unfolding tendsto_def Limits.eventually_within
1066 by (auto elim!: eventually_elim1)
1068 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
1069 shows "(f ---> l) (net within (S \<union> T))"
1070 using assms unfolding tendsto_def Limits.eventually_within
1072 apply (drule spec, drule (1) mp, drule (1) mp)
1073 apply (drule spec, drule (1) mp, drule (1) mp)
1074 apply (auto elim: eventually_elim2)
1078 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow> S \<union> T = UNIV
1080 by (metis Lim_Un within_UNIV)
1082 text{* Interrelations between restricted and unrestricted limits. *}
1084 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
1086 unfolding tendsto_def Limits.eventually_within
1087 apply (clarify, drule spec, drule (1) mp, drule (1) mp)
1088 by (auto elim!: eventually_elim1)
1090 lemma Lim_within_open:
1091 fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
1092 assumes"a \<in> S" "open S"
1093 shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)" (is "?lhs \<longleftrightarrow> ?rhs")
1096 { fix A assume "open A" "l \<in> A"
1097 with `?lhs` have "eventually (\<lambda>x. f x \<in> A) (at a within S)"
1098 by (rule topological_tendstoD)
1099 hence "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x \<in> A) (at a)"
1100 unfolding Limits.eventually_within .
1101 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"
1102 unfolding eventually_at_topological by fast
1103 hence "open (T \<inter> S)" "a \<in> T \<inter> S" "\<forall>x\<in>(T \<inter> S). x \<noteq> a \<longrightarrow> f x \<in> A"
1105 hence "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> f x \<in> A)"
1107 hence "eventually (\<lambda>x. f x \<in> A) (at a)"
1108 unfolding eventually_at_topological .
1110 thus ?rhs by (rule topological_tendstoI)
1113 thus ?lhs by (rule Lim_at_within)
1116 lemma Lim_within_LIMSEQ:
1117 fixes a :: real and L :: "'a::metric_space"
1118 assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
1119 shows "(X ---> L) (at a within T)"
1121 assume "\<not> (X ---> L) (at a within T)"
1122 hence "\<exists>r>0. \<forall>s>0. \<exists>x\<in>T. x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> r \<le> dist (X x) L"
1123 unfolding tendsto_iff eventually_within dist_norm by (simp add: not_less[symmetric])
1124 then obtain r where r: "r > 0" "\<And>s. s > 0 \<Longrightarrow> \<exists>x\<in>T-{a}. \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r" by blast
1126 let ?F = "\<lambda>n::nat. SOME x. x \<in> T \<and> x \<noteq> a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
1127 have "\<And>n. \<exists>x. x \<in> T \<and> x \<noteq> a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
1128 using r by (simp add: Bex_def)
1129 hence F: "\<And>n. ?F n \<in> T \<and> ?F n \<noteq> a \<and> \<bar>?F n - a\<bar> < inverse (real (Suc n)) \<and> dist (X (?F n)) L \<ge> r"
1131 hence F1: "\<And>n. ?F n \<in> T \<and> ?F n \<noteq> a"
1132 and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
1133 and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
1137 proof (rule LIMSEQ_I, unfold real_norm_def)
1140 (* choose no such that inverse (real (Suc n)) < e *)
1141 then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
1142 then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
1143 show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
1144 proof (intro exI allI impI)
1146 assume mlen: "m \<le> n"
1147 have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
1149 also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
1151 also from nodef have
1152 "inverse (real (Suc m)) < e" .
1153 finally show "\<bar>?F n - a\<bar> < e" .
1156 moreover note `\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L`
1157 ultimately have "(\<lambda>n. X (?F n)) ----> L" using F1 by simp
1159 moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
1163 obtain n where "n = no + 1" by simp
1164 then have nolen: "no \<le> n" by simp
1165 (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
1166 have "dist (X (?F n)) L \<ge> r"
1168 with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
1170 then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
1171 with r have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
1172 thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
1174 ultimately show False by simp
1177 lemma Lim_right_bound:
1178 fixes f :: "real \<Rightarrow> real"
1179 assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
1180 assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
1181 shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
1183 assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
1185 assume [simp]: "{x<..} \<inter> I \<noteq> {}"
1187 proof (rule Lim_within_LIMSEQ, safe)
1188 fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"
1190 show "(\<lambda>n. f (S n)) ----> Inf (f ` ({x<..} \<inter> I))"
1191 proof (rule LIMSEQ_I, rule ccontr)
1192 fix r :: real assume "0 < r"
1193 with Inf_close[of "f ` ({x<..} \<inter> I)" r]
1194 obtain y where y: "x < y" "y \<in> I" "f y < Inf (f ` ({x <..} \<inter> I)) + r" by auto
1195 from `x < y` have "0 < y - x" by auto
1196 from S(2)[THEN LIMSEQ_D, OF this]
1197 obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto
1199 assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f ` ({x<..} \<inter> I))) < r)"
1200 moreover have "\<And>n. Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
1201 using S bnd by (intro Inf_lower[where z=K]) auto
1202 ultimately obtain n where n: "N \<le> n" "r + Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
1203 by (auto simp: not_less field_simps)
1204 with N[OF n(1)] mono[OF _ `y \<in> I`, of "S n"] S(1)[THEN spec, of n] y
1210 text{* Another limit point characterization. *}
1212 lemma islimpt_sequential:
1213 fixes x :: "'a::metric_space"
1214 shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
1218 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"
1219 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
1221 have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
1224 { fix e::real assume "e>0"
1225 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
1226 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
1227 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)
1228 moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
1229 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
1231 hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
1232 unfolding Lim_sequentially using f by auto
1233 ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
1236 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
1237 { fix e::real assume "e>0"
1238 then obtain N where "dist (f N) x < e" using f(2) by auto
1239 moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
1240 ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
1242 thus ?lhs unfolding islimpt_approachable by auto
1245 text{* Basic arithmetical combining theorems for limits. *}
1248 assumes "(f ---> l) net" "bounded_linear h"
1249 shows "((\<lambda>x. h (f x)) ---> h l) net"
1250 using `bounded_linear h` `(f ---> l) net`
1251 by (rule bounded_linear.tendsto)
1253 lemma Lim_ident_at: "((\<lambda>x. x) ---> a) (at a)"
1254 unfolding tendsto_def Limits.eventually_at_topological by fast
1256 lemma Lim_const[intro]: "((\<lambda>x. a) ---> a) net" by (rule tendsto_const)
1258 lemma Lim_cmul[intro]:
1259 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1260 shows "(f ---> l) net ==> ((\<lambda>x. c *\<^sub>R f x) ---> c *\<^sub>R l) net"
1261 by (intro tendsto_intros)
1264 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1265 shows "(f ---> l) net ==> ((\<lambda>x. -(f x)) ---> -l) net"
1266 by (rule tendsto_minus)
1268 lemma Lim_add: fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" shows
1269 "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) + g(x)) ---> l + m) net"
1270 by (rule tendsto_add)
1273 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1274 shows "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) ---> l - m) net"
1275 by (rule tendsto_diff)
1278 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1279 assumes "(c ---> d) net" "(f ---> l) net"
1280 shows "((\<lambda>x. c(x) *\<^sub>R f x) ---> (d *\<^sub>R l)) net"
1281 using assms by (rule scaleR.tendsto)
1284 fixes f :: "'a \<Rightarrow> real"
1285 assumes "(f ---> l) (net::'a net)" "l \<noteq> 0"
1286 shows "((inverse o f) ---> inverse l) net"
1287 unfolding o_def using assms by (rule tendsto_inverse)
1290 fixes c :: "'a \<Rightarrow> real" and v :: "'b::real_normed_vector"
1291 shows "(c ---> d) net ==> ((\<lambda>x. c(x) *\<^sub>R v) ---> d *\<^sub>R v) net"
1292 by (intro tendsto_intros)
1295 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1296 shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net" by (simp add: Lim dist_norm)
1298 lemma Lim_null_norm:
1299 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1300 shows "(f ---> 0) net \<longleftrightarrow> ((\<lambda>x. norm(f x)) ---> 0) net"
1301 by (simp add: Lim dist_norm)
1303 lemma Lim_null_comparison:
1304 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1305 assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
1306 shows "(f ---> 0) net"
1307 proof(simp add: tendsto_iff, rule+)
1308 fix e::real assume "0<e"
1310 assume "norm (f x) \<le> g x" "dist (g x) 0 < e"
1311 hence "dist (f x) 0 < e" by (simp add: dist_norm)
1313 thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
1314 using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
1315 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]
1316 using assms `e>0` unfolding tendsto_iff by auto
1319 lemma Lim_component:
1320 fixes f :: "'a \<Rightarrow> ('a::euclidean_space)"
1321 shows "(f ---> l) net \<Longrightarrow> ((\<lambda>a. f a $$i) ---> l$$i) net"
1322 unfolding tendsto_iff
1324 apply (drule spec, drule (1) mp)
1325 apply (erule eventually_elim1)
1326 apply (erule le_less_trans [OF dist_nth_le])
1329 lemma Lim_transform_bound:
1330 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1331 fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
1332 assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net" "(g ---> 0) net"
1333 shows "(f ---> 0) net"
1334 proof (rule tendstoI)
1335 fix e::real assume "e>0"
1337 assume "norm (f x) \<le> norm (g x)" "dist (g x) 0 < e"
1338 hence "dist (f x) 0 < e" by (simp add: dist_norm)}
1339 thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
1340 using eventually_and[of "\<lambda>x. norm (f x) \<le> norm (g x)" "\<lambda>x. dist (g x) 0 < e" net]
1341 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]
1342 using assms `e>0` unfolding tendsto_iff by blast
1345 text{* Deducing things about the limit from the elements. *}
1347 lemma Lim_in_closed_set:
1348 assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
1351 assume "l \<notin> S"
1352 with `closed S` have "open (- S)" "l \<in> - S"
1353 by (simp_all add: open_Compl)
1354 with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
1355 by (rule topological_tendstoD)
1356 with assms(2) have "eventually (\<lambda>x. False) net"
1357 by (rule eventually_elim2) simp
1358 with assms(3) show "False"
1359 by (simp add: eventually_False)
1362 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
1364 lemma Lim_dist_ubound:
1365 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
1366 shows "dist a l <= e"
1368 assume "\<not> dist a l \<le> e"
1369 then have "0 < dist a l - e" by simp
1370 with assms(2) have "eventually (\<lambda>x. dist (f x) l < dist a l - e) net"
1372 with assms(3) have "eventually (\<lambda>x. dist a (f x) \<le> e \<and> dist (f x) l < dist a l - e) net"
1373 by (rule eventually_conjI)
1374 then obtain w where "dist a (f w) \<le> e" "dist (f w) l < dist a l - e"
1375 using assms(1) eventually_happens by auto
1376 hence "dist a (f w) + dist (f w) l < e + (dist a l - e)"
1377 by (rule add_le_less_mono)
1378 hence "dist a (f w) + dist (f w) l < dist a l"
1380 also have "\<dots> \<le> dist a (f w) + dist (f w) l"
1381 by (rule dist_triangle)
1382 finally show False by simp
1385 lemma Lim_norm_ubound:
1386 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1387 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
1388 shows "norm(l) <= e"
1390 assume "\<not> norm l \<le> e"
1391 then have "0 < norm l - e" by simp
1392 with assms(2) have "eventually (\<lambda>x. dist (f x) l < norm l - e) net"
1394 with assms(3) have "eventually (\<lambda>x. norm (f x) \<le> e \<and> dist (f x) l < norm l - e) net"
1395 by (rule eventually_conjI)
1396 then obtain w where "norm (f w) \<le> e" "dist (f w) l < norm l - e"
1397 using assms(1) eventually_happens by auto
1398 hence "norm (f w - l) < norm l - e" "norm (f w) \<le> e" by (simp_all add: dist_norm)
1399 hence "norm (f w - l) + norm (f w) < norm l" by simp
1400 hence "norm (f w - l - f w) < norm l" by (rule le_less_trans [OF norm_triangle_ineq4])
1401 thus False using `\<not> norm l \<le> e` by simp
1404 lemma Lim_norm_lbound:
1405 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1406 assumes "\<not> (trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. e <= norm(f x)) net"
1407 shows "e \<le> norm l"
1409 assume "\<not> e \<le> norm l"
1410 then have "0 < e - norm l" by simp
1411 with assms(2) have "eventually (\<lambda>x. dist (f x) l < e - norm l) net"
1413 with assms(3) have "eventually (\<lambda>x. e \<le> norm (f x) \<and> dist (f x) l < e - norm l) net"
1414 by (rule eventually_conjI)
1415 then obtain w where "e \<le> norm (f w)" "dist (f w) l < e - norm l"
1416 using assms(1) eventually_happens by auto
1417 hence "norm (f w - l) + norm l < e" "e \<le> norm (f w)" by (simp_all add: dist_norm)
1418 hence "norm (f w - l) + norm l < norm (f w)" by (rule less_le_trans)
1419 hence "norm (f w - l + l) < norm (f w)" by (rule le_less_trans [OF norm_triangle_ineq])
1423 text{* Uniqueness of the limit, when nontrivial. *}
1426 fixes f :: "'a \<Rightarrow> 'b::t2_space"
1427 shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
1428 unfolding Lim_def using tendsto_unique[of net f] by auto
1430 text{* Limit under bilinear function *}
1433 assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
1434 shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
1435 using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
1436 by (rule bounded_bilinear.tendsto)
1438 text{* These are special for limits out of the same vector space. *}
1440 lemma Lim_within_id: "(id ---> a) (at a within s)"
1441 unfolding tendsto_def Limits.eventually_within eventually_at_topological
1444 lemmas Lim_intros = Lim_add Lim_const Lim_sub Lim_cmul Lim_vmul Lim_within_id
1446 lemma Lim_at_id: "(id ---> a) (at a)"
1447 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
1450 fixes a :: "'a::real_normed_vector"
1451 fixes l :: "'b::topological_space"
1452 shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
1455 { fix S assume "open S" "l \<in> S"
1456 with `?lhs` have "eventually (\<lambda>x. f x \<in> S) (at a)"
1457 by (rule topological_tendstoD)
1458 then obtain d where d: "d>0" "\<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S"
1459 unfolding Limits.eventually_at by fast
1460 { fix x::"'a" assume "x \<noteq> 0 \<and> dist x 0 < d"
1461 hence "f (a + x) \<in> S" using d
1462 apply(erule_tac x="x+a" in allE)
1463 by (auto simp add: add_commute dist_norm dist_commute)
1465 hence "\<exists>d>0. \<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
1467 hence "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
1468 unfolding Limits.eventually_at .
1470 thus "?rhs" by (rule topological_tendstoI)
1473 { fix S assume "open S" "l \<in> S"
1474 with `?rhs` have "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
1475 by (rule topological_tendstoD)
1476 then obtain d where d: "d>0" "\<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
1477 unfolding Limits.eventually_at by fast
1478 { fix x::"'a" assume "x \<noteq> a \<and> dist x a < d"
1479 hence "f x \<in> S" using d apply(erule_tac x="x-a" in allE)
1480 by(auto simp add: add_commute dist_norm dist_commute)
1482 hence "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S" using d(1) by auto
1483 hence "eventually (\<lambda>x. f x \<in> S) (at a)" unfolding Limits.eventually_at .
1485 thus "?lhs" by (rule topological_tendstoI)
1488 text{* It's also sometimes useful to extract the limit point from the net. *}
1491 netlimit :: "'a::t2_space net \<Rightarrow> 'a" where
1492 "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
1494 lemma netlimit_within:
1495 assumes "\<not> trivial_limit (at a within S)"
1496 shows "netlimit (at a within S) = a"
1497 unfolding netlimit_def
1498 apply (rule some_equality)
1499 apply (rule Lim_at_within)
1500 apply (rule Lim_ident_at)
1501 apply (erule tendsto_unique [OF assms])
1502 apply (rule Lim_at_within)
1503 apply (rule Lim_ident_at)
1507 fixes a :: "'a::{perfect_space,t2_space}"
1508 shows "netlimit (at a) = a"
1509 apply (subst within_UNIV[symmetric])
1510 using netlimit_within[of a UNIV]
1511 by (simp add: trivial_limit_at within_UNIV)
1513 text{* Transformation of limit. *}
1515 lemma Lim_transform:
1516 fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
1517 assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
1518 shows "(g ---> l) net"
1520 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
1521 thus "?thesis" using Lim_neg [of "\<lambda> x. - g x" "-l" net] by auto
1524 lemma Lim_transform_eventually:
1525 "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
1526 apply (rule topological_tendstoI)
1527 apply (drule (2) topological_tendstoD)
1528 apply (erule (1) eventually_elim2, simp)
1531 lemma Lim_transform_within:
1532 assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
1533 and "(f ---> l) (at x within S)"
1534 shows "(g ---> l) (at x within S)"
1535 proof (rule Lim_transform_eventually)
1536 show "eventually (\<lambda>x. f x = g x) (at x within S)"
1537 unfolding eventually_within
1538 using assms(1,2) by auto
1539 show "(f ---> l) (at x within S)" by fact
1542 lemma Lim_transform_at:
1543 assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
1544 and "(f ---> l) (at x)"
1545 shows "(g ---> l) (at x)"
1546 proof (rule Lim_transform_eventually)
1547 show "eventually (\<lambda>x. f x = g x) (at x)"
1548 unfolding eventually_at
1549 using assms(1,2) by auto
1550 show "(f ---> l) (at x)" by fact
1553 text{* Common case assuming being away from some crucial point like 0. *}
1555 lemma Lim_transform_away_within:
1556 fixes a b :: "'a::t1_space"
1557 assumes "a \<noteq> b" and "\<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)"
1560 proof (rule Lim_transform_eventually)
1561 show "(f ---> l) (at a within S)" by fact
1562 show "eventually (\<lambda>x. f x = g x) (at a within S)"
1563 unfolding Limits.eventually_within eventually_at_topological
1564 by (rule exI [where x="- {b}"], simp add: open_Compl assms)
1567 lemma Lim_transform_away_at:
1568 fixes a b :: "'a::t1_space"
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 assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
1579 and "(f ---> l) (at a)"
1580 shows "(g ---> l) (at a)"
1581 proof (rule Lim_transform_eventually)
1582 show "eventually (\<lambda>x. f x = g x) (at a)"
1583 unfolding eventually_at_topological
1584 using assms(1,2,3) by auto
1585 show "(f ---> l) (at a)" by fact
1588 text{* A congruence rule allowing us to transform limits assuming not at point. *}
1590 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
1592 lemma Lim_cong_within(*[cong add]*):
1593 assumes "a = b" "x = y" "S = T"
1594 assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
1595 shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
1596 unfolding tendsto_def Limits.eventually_within eventually_at_topological
1599 lemma Lim_cong_at(*[cong add]*):
1600 assumes "a = b" "x = y"
1601 assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
1602 shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
1603 unfolding tendsto_def eventually_at_topological
1606 text{* Useful lemmas on closure and set of possible sequential limits.*}
1608 lemma closure_sequential:
1609 fixes l :: "'a::metric_space"
1610 shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
1612 assume "?lhs" moreover
1613 { assume "l \<in> S"
1614 hence "?rhs" using Lim_const[of l sequentially] by auto
1616 { assume "l islimpt S"
1617 hence "?rhs" unfolding islimpt_sequential by auto
1619 show "?rhs" unfolding closure_def by auto
1622 thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
1625 lemma closed_sequential_limits:
1626 fixes S :: "'a::metric_space set"
1627 shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
1628 unfolding closed_limpt
1629 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]
1632 lemma closure_approachable:
1633 fixes S :: "'a::metric_space set"
1634 shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
1635 apply (auto simp add: closure_def islimpt_approachable)
1636 by (metis dist_self)
1638 lemma closed_approachable:
1639 fixes S :: "'a::metric_space set"
1640 shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
1641 by (metis closure_closed closure_approachable)
1643 text{* Some other lemmas about sequences. *}
1645 lemma sequentially_offset:
1646 assumes "eventually (\<lambda>i. P i) sequentially"
1647 shows "eventually (\<lambda>i. P (i + k)) sequentially"
1648 using assms unfolding eventually_sequentially by (metis trans_le_add1)
1651 assumes "(f ---> l) sequentially"
1652 shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
1653 using assms unfolding tendsto_def
1654 by clarify (rule sequentially_offset, simp)
1656 lemma seq_offset_neg:
1657 "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
1658 apply (rule topological_tendstoI)
1659 apply (drule (2) topological_tendstoD)
1660 apply (simp only: eventually_sequentially)
1661 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
1665 lemma seq_offset_rev:
1666 "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
1667 apply (rule topological_tendstoI)
1668 apply (drule (2) topological_tendstoD)
1669 apply (simp only: eventually_sequentially)
1670 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
1673 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
1675 { fix e::real assume "e>0"
1676 hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
1677 using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
1678 by (metis le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
1680 thus ?thesis unfolding Lim_sequentially dist_norm by simp
1683 subsection {* More properties of closed balls. *}
1685 lemma closed_cball: "closed (cball x e)"
1686 unfolding cball_def closed_def
1687 unfolding Collect_neg_eq [symmetric] not_le
1688 apply (clarsimp simp add: open_dist, rename_tac y)
1689 apply (rule_tac x="dist x y - e" in exI, clarsimp)
1690 apply (rename_tac x')
1691 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
1695 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
1697 { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
1698 hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
1700 { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
1701 hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
1703 show ?thesis unfolding open_contains_ball by auto
1706 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
1707 by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball mem_def)
1709 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
1710 apply (simp add: interior_def, safe)
1711 apply (force simp add: open_contains_cball)
1712 apply (rule_tac x="ball x e" in exI)
1713 apply (simp add: subset_trans [OF ball_subset_cball])
1717 fixes x y :: "'a::{real_normed_vector,perfect_space}"
1718 shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
1721 { assume "e \<le> 0"
1722 hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
1723 have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
1725 hence "e > 0" by (metis not_less)
1727 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
1728 ultimately show "?rhs" by auto
1730 assume "?rhs" hence "e>0" by auto
1731 { fix d::real assume "d>0"
1732 have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1733 proof(cases "d \<le> dist x y")
1734 case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1736 case True hence False using `d \<le> dist x y` `d>0` by auto
1737 thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
1741 have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
1742 = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1743 unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
1744 also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
1745 using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
1746 unfolding scaleR_minus_left scaleR_one
1747 by (auto simp add: norm_minus_commute)
1748 also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
1749 unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
1750 unfolding left_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
1751 also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
1752 finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
1756 have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
1757 using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
1759 have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
1760 using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
1761 unfolding dist_norm by auto
1762 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
1765 case False hence "d > dist x y" by auto
1766 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1769 obtain z where **: "z \<noteq> y" "dist z y < min e d"
1770 using perfect_choose_dist[of "min e d" y]
1771 using `d > 0` `e>0` by auto
1772 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1774 using `z \<noteq> y` **
1775 by (rule_tac x=z in bexI, auto simp add: dist_commute)
1777 case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1778 using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
1781 thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
1784 lemma closure_ball_lemma:
1785 fixes x y :: "'a::real_normed_vector"
1786 assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
1787 proof (rule islimptI)
1788 fix T assume "y \<in> T" "open T"
1789 then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
1790 unfolding open_dist by fast
1791 (* choose point between x and y, within distance r of y. *)
1792 def k \<equiv> "min 1 (r / (2 * dist x y))"
1793 def z \<equiv> "y + scaleR k (x - y)"
1794 have z_def2: "z = x + scaleR (1 - k) (y - x)"
1795 unfolding z_def by (simp add: algebra_simps)
1797 unfolding z_def k_def using `0 < r`
1798 by (simp add: dist_norm min_def)
1799 hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
1800 have "dist x z < dist x y"
1801 unfolding z_def2 dist_norm
1802 apply (simp add: norm_minus_commute)
1803 apply (simp only: dist_norm [symmetric])
1804 apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
1805 apply (rule mult_strict_right_mono)
1806 apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
1807 apply (simp add: zero_less_dist_iff `x \<noteq> y`)
1809 hence "z \<in> ball x (dist x y)" by simp
1811 unfolding z_def k_def using `x \<noteq> y` `0 < r`
1812 by (simp add: min_def)
1813 show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
1814 using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
1819 fixes x :: "'a::real_normed_vector"
1820 shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
1821 apply (rule equalityI)
1822 apply (rule closure_minimal)
1823 apply (rule ball_subset_cball)
1824 apply (rule closed_cball)
1825 apply (rule subsetI, rename_tac y)
1826 apply (simp add: le_less [where 'a=real])
1828 apply (rule subsetD [OF closure_subset], simp)
1829 apply (simp add: closure_def)
1831 apply (rule closure_ball_lemma)
1832 apply (simp add: zero_less_dist_iff)
1835 (* In a trivial vector space, this fails for e = 0. *)
1836 lemma interior_cball:
1837 fixes x :: "'a::{real_normed_vector, perfect_space}"
1838 shows "interior (cball x e) = ball x e"
1839 proof(cases "e\<ge>0")
1840 case False note cs = this
1841 from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
1842 { fix y assume "y \<in> cball x e"
1843 hence False unfolding mem_cball using dist_nz[of x y] cs by auto }
1844 hence "cball x e = {}" by auto
1845 hence "interior (cball x e) = {}" using interior_empty by auto
1846 ultimately show ?thesis by blast
1848 case True note cs = this
1849 have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
1850 { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
1851 then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
1853 then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
1854 using perfect_choose_dist [of d] by auto
1855 have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
1856 hence xa_cball:"xa \<in> cball x e" using as(1) by auto
1858 hence "y \<in> ball x e" proof(cases "x = y")
1860 hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
1861 thus "y \<in> ball x e" using `x = y ` by simp
1864 have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
1865 using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
1866 hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
1867 have "y - x \<noteq> 0" using `x \<noteq> y` by auto
1868 hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
1869 using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
1871 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)"
1872 by (auto simp add: dist_norm algebra_simps)
1873 also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1874 by (auto simp add: algebra_simps)
1875 also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
1877 also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
1878 finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
1879 thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
1881 hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
1882 ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
1885 lemma frontier_ball:
1886 fixes a :: "'a::real_normed_vector"
1887 shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
1888 apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
1889 apply (simp add: set_eq_iff)
1892 lemma frontier_cball:
1893 fixes a :: "'a::{real_normed_vector, perfect_space}"
1894 shows "frontier(cball a e) = {x. dist a x = e}"
1895 apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
1896 apply (simp add: set_eq_iff)
1899 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
1900 apply (simp add: set_eq_iff not_le)
1901 by (metis zero_le_dist dist_self order_less_le_trans)
1902 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
1904 lemma cball_eq_sing:
1905 fixes x :: "'a::{metric_space,perfect_space}"
1906 shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
1907 proof (rule linorder_cases)
1909 obtain a where "a \<noteq> x" "dist a x < e"
1910 using perfect_choose_dist [OF e] by auto
1911 hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
1912 with e show ?thesis by (auto simp add: set_eq_iff)
1916 fixes x :: "'a::metric_space"
1917 shows "e = 0 ==> cball x e = {x}"
1918 by (auto simp add: set_eq_iff)
1920 text{* For points in the interior, localization of limits makes no difference. *}
1922 lemma eventually_within_interior:
1923 assumes "x \<in> interior S"
1924 shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
1926 from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S"
1927 unfolding interior_def by fast
1929 then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
1930 unfolding Limits.eventually_within Limits.eventually_at_topological
1932 with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
1935 unfolding Limits.eventually_at_topological by auto
1937 { assume "?rhs" hence "?lhs"
1938 unfolding Limits.eventually_within
1939 by (auto elim: eventually_elim1)
1944 lemma at_within_interior:
1945 "x \<in> interior S \<Longrightarrow> at x within S = at x"
1946 by (simp add: expand_net_eq eventually_within_interior)
1948 lemma lim_within_interior:
1949 "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
1950 by (simp add: at_within_interior)
1952 lemma netlimit_within_interior:
1953 fixes x :: "'a::{t2_space,perfect_space}"
1954 assumes "x \<in> interior S"
1955 shows "netlimit (at x within S) = x"
1956 using assms by (simp add: at_within_interior netlimit_at)
1958 subsection{* Boundedness. *}
1960 (* FIXME: This has to be unified with BSEQ!! *)
1962 bounded :: "'a::metric_space set \<Rightarrow> bool" where
1963 "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
1965 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
1966 unfolding bounded_def
1968 apply (rule_tac x="dist a x + e" in exI, clarify)
1969 apply (drule (1) bspec)
1970 apply (erule order_trans [OF dist_triangle add_left_mono])
1974 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
1975 unfolding bounded_any_center [where a=0]
1976 by (simp add: dist_norm)
1978 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
1979 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
1980 by (metis bounded_def subset_eq)
1982 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
1983 by (metis bounded_subset interior_subset)
1985 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
1987 from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
1988 { fix y assume "y \<in> closure S"
1989 then obtain f where f: "\<forall>n. f n \<in> S" "(f ---> y) sequentially"
1990 unfolding closure_sequential by auto
1991 have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
1992 hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
1993 by (rule eventually_mono, simp add: f(1))
1994 have "dist x y \<le> a"
1995 apply (rule Lim_dist_ubound [of sequentially f])
1996 apply (rule trivial_limit_sequentially)
2001 thus ?thesis unfolding bounded_def by auto
2004 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
2005 apply (simp add: bounded_def)
2006 apply (rule_tac x=x in exI)
2007 apply (rule_tac x=e in exI)
2011 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
2012 by (metis ball_subset_cball bounded_cball bounded_subset)
2014 lemma finite_imp_bounded[intro]:
2015 fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"
2017 { fix a and F :: "'a set" assume as:"bounded F"
2018 then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
2019 hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
2020 hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
2022 thus ?thesis using finite_induct[of S bounded] using bounded_empty assms by auto
2025 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
2026 apply (auto simp add: bounded_def)
2027 apply (rename_tac x y r s)
2028 apply (rule_tac x=x in exI)
2029 apply (rule_tac x="max r (dist x y + s)" in exI)
2030 apply (rule ballI, rename_tac z, safe)
2031 apply (drule (1) bspec, simp)
2032 apply (drule (1) bspec)
2033 apply (rule min_max.le_supI2)
2034 apply (erule order_trans [OF dist_triangle add_left_mono])
2037 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
2038 by (induct rule: finite_induct[of F], auto)
2040 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
2041 apply (simp add: bounded_iff)
2042 apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
2045 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
2046 by (metis Int_lower1 Int_lower2 bounded_subset)
2048 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
2049 apply (metis Diff_subset bounded_subset)
2052 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
2053 by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
2055 lemma not_bounded_UNIV[simp, intro]:
2056 "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
2057 proof(auto simp add: bounded_pos not_le)
2058 obtain x :: 'a where "x \<noteq> 0"
2059 using perfect_choose_dist [OF zero_less_one] by fast
2060 fix b::real assume b: "b >0"
2061 have b1: "b +1 \<ge> 0" using b by simp
2062 with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
2063 by (simp add: norm_sgn)
2064 then show "\<exists>x::'a. b < norm x" ..
2067 lemma bounded_linear_image:
2068 assumes "bounded S" "bounded_linear f"
2069 shows "bounded(f ` S)"
2071 from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
2072 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)
2073 { fix x assume "x\<in>S"
2074 hence "norm x \<le> b" using b by auto
2075 hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
2076 by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
2078 thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
2079 using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
2082 lemma bounded_scaling:
2083 fixes S :: "'a::real_normed_vector set"
2084 shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
2085 apply (rule bounded_linear_image, assumption)
2086 apply (rule scaleR.bounded_linear_right)
2089 lemma bounded_translation:
2090 fixes S :: "'a::real_normed_vector set"
2091 assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
2093 from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
2094 { fix x assume "x\<in>S"
2095 hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
2097 thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
2098 by (auto intro!: add exI[of _ "b + norm a"])
2102 text{* Some theorems on sups and infs using the notion "bounded". *}
2105 fixes S :: "real set"
2106 shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"
2107 by (simp add: bounded_iff)
2109 lemma bounded_has_Sup:
2110 fixes S :: "real set"
2111 assumes "bounded S" "S \<noteq> {}"
2112 shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
2114 fix x assume "x\<in>S"
2115 thus "x \<le> Sup S"
2116 by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
2118 show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
2119 by (metis SupInf.Sup_least)
2123 fixes S :: "real set"
2124 shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
2125 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal)
2127 lemma Sup_insert_finite:
2128 fixes S :: "real set"
2129 shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
2130 apply (rule Sup_insert)
2131 apply (rule finite_imp_bounded)
2134 lemma bounded_has_Inf:
2135 fixes S :: "real set"
2136 assumes "bounded S" "S \<noteq> {}"
2137 shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
2139 fix x assume "x\<in>S"
2140 from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
2141 thus "x \<ge> Inf S" using `x\<in>S`
2142 by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
2144 show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
2145 by (metis SupInf.Inf_greatest)
2149 fixes S :: "real set"
2150 shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
2151 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)
2152 lemma Inf_insert_finite:
2153 fixes S :: "real set"
2154 shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
2155 by (rule Inf_insert, rule finite_imp_bounded, simp)
2158 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
2159 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
2160 apply (frule isGlb_isLb)
2161 apply (frule_tac x = y in isGlb_isLb)
2162 apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
2165 subsection {* Equivalent versions of compactness *}
2167 subsubsection{* Sequential compactness *}
2170 compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
2171 "compact S \<longleftrightarrow>
2172 (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
2173 (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
2176 assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
2178 unfolding compact_def using assms by fast
2181 assumes "compact S" "\<forall>n. f n \<in> S"
2182 obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
2183 using assms unfolding compact_def by fast
2186 A metric space (or topological vector space) is said to have the
2187 Heine-Borel property if every closed and bounded subset is compact.
2191 assumes bounded_imp_convergent_subsequence:
2192 "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
2193 \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2195 lemma bounded_closed_imp_compact:
2196 fixes s::"'a::heine_borel set"
2197 assumes "bounded s" and "closed s" shows "compact s"
2198 proof (unfold compact_def, clarify)
2199 fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
2200 obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
2201 using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
2202 from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
2203 have "l \<in> s" using `closed s` fr l
2204 unfolding closed_sequential_limits by blast
2205 show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2206 using `l \<in> s` r l by blast
2209 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
2211 show "0 \<le> r 0" by auto
2213 fix n assume "n \<le> r n"
2214 moreover have "r n < r (Suc n)"
2215 using assms [unfolded subseq_def] by auto
2216 ultimately show "Suc n \<le> r (Suc n)" by auto
2219 lemma eventually_subseq:
2220 assumes r: "subseq r"
2221 shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
2222 unfolding eventually_sequentially
2223 by (metis subseq_bigger [OF r] le_trans)
2226 "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
2227 unfolding tendsto_def eventually_sequentially o_def
2228 by (metis subseq_bigger le_trans)
2230 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
2232 apply (rule_tac x="nat_rec e f" in exI)
2234 apply (rule def_nat_rec_0, simp)
2235 apply (rule allI, rule def_nat_rec_Suc, simp)
2236 apply (rule allI, rule impI, rule ext)
2238 apply (induct_tac x)
2240 apply (erule_tac x="n" in allE)
2244 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
2245 assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
2246 shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N. abs(s n - l) < e"
2248 have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
2249 then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
2250 { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
2252 obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
2253 have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
2254 with n have "s N \<le> t - e" using `e>0` by auto
2255 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 }
2256 hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
2257 hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto }
2258 thus ?thesis by blast
2261 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
2262 assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
2263 shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
2264 using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
2265 unfolding monoseq_def incseq_def
2266 apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
2267 unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
2269 (* TODO: merge this lemma with the ones above *)
2270 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"
2271 assumes "bounded {s n| n::nat. True}" "\<forall>n. (s n) \<le>(s(Suc n))"
2272 shows "\<exists>l. (s ---> l) sequentially"
2274 obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le> a" using assms(1)[unfolded bounded_iff] by auto
2276 have "\<And> n. n\<ge>m \<longrightarrow> (s m) \<le> (s n)"
2277 apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)
2278 apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq) }
2279 hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto
2280 then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar> (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a]
2281 unfolding monoseq_def by auto
2282 thus ?thesis unfolding Lim_sequentially apply(rule_tac x="l" in exI)
2283 unfolding dist_norm by auto
2286 lemma compact_real_lemma:
2287 assumes "\<forall>n::nat. abs(s n) \<le> b"
2288 shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
2290 obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
2291 using seq_monosub[of s] by auto
2292 thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
2293 unfolding tendsto_iff dist_norm eventually_sequentially by auto
2296 instance real :: heine_borel
2298 fix s :: "real set" and f :: "nat \<Rightarrow> real"
2299 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2300 then obtain b where b: "\<forall>n. abs (f n) \<le> b"
2301 unfolding bounded_iff by auto
2302 obtain l :: real and r :: "nat \<Rightarrow> nat" where
2303 r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
2304 using compact_real_lemma [OF b] by auto
2305 thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2309 lemma bounded_component: "bounded s \<Longrightarrow>
2310 bounded ((\<lambda>x. x $$ i) ` (s::'a::euclidean_space set))"
2311 unfolding bounded_def
2313 apply (rule_tac x="x $$ i" in exI)
2314 apply (rule_tac x="e" in exI)
2316 apply (rule order_trans[OF dist_nth_le],simp)
2319 lemma compact_lemma:
2320 fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
2321 assumes "bounded s" and "\<forall>n. f n \<in> s"
2322 shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>
2323 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2325 fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"
2326 have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
2327 hence "\<exists>l::'a. \<exists>r. subseq r \<and>
2328 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2329 proof(induct d) case empty thus ?case unfolding subseq_def by auto
2330 next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto
2331 have s': "bounded ((\<lambda>x. x $$ k) ` s)" using `bounded s` by (rule bounded_component)
2332 obtain l1::"'a" and r1 where r1:"subseq r1" and
2333 lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially"
2334 using insert(3) using insert(4) by auto
2335 have f': "\<forall>n. f (r1 n) $$ k \<in> (\<lambda>x. x $$ k) ` s" using `\<forall>n. f n \<in> s` by simp
2336 obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $$ k) ---> l2) sequentially"
2337 using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
2338 def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
2339 using r1 and r2 unfolding r_def o_def subseq_def by auto
2341 def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"
2342 { fix e::real assume "e>0"
2343 from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially" by blast
2344 from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $$ k) l2 < e) sequentially" by (rule tendstoD)
2345 from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$ i) (l1 $$ i) < e) sequentially"
2346 by (rule eventually_subseq)
2347 have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$ i) (l $$ i) < e) sequentially"
2348 using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def
2349 using insert.prems by auto
2351 ultimately show ?case by auto
2353 thus "\<exists>l::'a. \<exists>r. subseq r \<and>
2354 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2355 apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe
2356 apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe
2357 apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)
2358 apply(erule_tac x=i in ballE)
2359 proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a
2360 assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"
2361 hence *:"i\<ge>DIM('a)" by auto
2362 thus "dist (f (r n) $$ i) (l $$ i) < e" using e by auto
2366 instance euclidean_space \<subseteq> heine_borel
2368 fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"
2369 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2370 then obtain l::'a and r where r: "subseq r"
2371 and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $$ i) (l $$ i) < e) sequentially"
2372 using compact_lemma [OF s f] by blast
2373 let ?d = "{..<DIM('a)}"
2374 { fix e::real assume "e>0"
2375 hence "0 < e / (real_of_nat (card ?d))"
2376 using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
2377 with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))) sequentially"
2380 { fix n assume n: "\<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))"
2381 have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $$ i) (l $$ i))"
2382 apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)
2383 also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
2384 apply(rule setsum_strict_mono) using n by auto
2385 finally have "dist (f (r n)) l < e" unfolding setsum_constant
2386 using DIM_positive[where 'a='a] by auto
2388 ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
2389 by (rule eventually_elim1)
2391 hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
2392 with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
2395 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
2396 unfolding bounded_def
2398 apply (rule_tac x="a" in exI)
2399 apply (rule_tac x="e" in exI)
2401 apply (drule (1) bspec)
2402 apply (simp add: dist_Pair_Pair)
2403 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
2406 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
2407 unfolding bounded_def
2409 apply (rule_tac x="b" in exI)
2410 apply (rule_tac x="e" in exI)
2412 apply (drule (1) bspec)
2413 apply (simp add: dist_Pair_Pair)
2414 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
2417 instance prod :: (heine_borel, heine_borel) heine_borel
2419 fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
2420 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2421 from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
2422 from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
2423 obtain l1 r1 where r1: "subseq r1"
2424 and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
2425 using bounded_imp_convergent_subsequence [OF s1 f1]
2426 unfolding o_def by fast
2427 from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
2428 from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
2429 obtain l2 r2 where r2: "subseq r2"
2430 and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
2431 using bounded_imp_convergent_subsequence [OF s2 f2]
2432 unfolding o_def by fast
2433 have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
2434 using lim_subseq [OF r2 l1] unfolding o_def .
2435 have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
2436 using tendsto_Pair [OF l1' l2] unfolding o_def by simp
2437 have r: "subseq (r1 \<circ> r2)"
2438 using r1 r2 unfolding subseq_def by simp
2439 show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2443 subsubsection{* Completeness *}
2446 "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
2447 unfolding Cauchy_def by blast
2450 complete :: "'a::metric_space set \<Rightarrow> bool" where
2451 "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
2452 --> (\<exists>l \<in> s. (f ---> l) sequentially))"
2454 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
2459 with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
2460 by (erule_tac x="e/2" in allE) auto
2462 assume nm:"N \<le> m \<and> N \<le> n"
2463 hence "dist (s m) (s n) < e" using N
2464 using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
2467 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
2471 unfolding cauchy_def
2475 unfolding cauchy_def
2476 using dist_triangle_half_l
2480 lemma convergent_imp_cauchy:
2481 "(s ---> l) sequentially ==> Cauchy s"
2482 proof(simp only: cauchy_def, rule, rule)
2483 fix e::real assume "e>0" "(s ---> l) sequentially"
2484 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
2485 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
2488 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
2490 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
2491 hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
2493 have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
2494 then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
2495 unfolding bounded_any_center [where a="s N"] by auto
2496 ultimately show "?thesis"
2497 unfolding bounded_any_center [where a="s N"]
2498 apply(rule_tac x="max a 1" in exI) apply auto
2499 apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto
2502 lemma compact_imp_complete: assumes "compact s" shows "complete s"
2504 { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
2505 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
2507 note lr' = subseq_bigger [OF lr(2)]
2509 { fix e::real assume "e>0"
2510 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
2511 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
2512 { fix n::nat assume n:"n \<ge> max N M"
2513 have "dist ((f \<circ> r) n) l < e/2" using n M by auto
2514 moreover have "r n \<ge> N" using lr'[of n] n by auto
2515 hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
2516 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) }
2517 hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast }
2518 hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding Lim_sequentially by auto }
2519 thus ?thesis unfolding complete_def by auto
2522 instance heine_borel < complete_space
2524 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2525 hence "bounded (range f)"
2526 by (rule cauchy_imp_bounded)
2527 hence "compact (closure (range f))"
2528 using bounded_closed_imp_compact [of "closure (range f)"] by auto
2529 hence "complete (closure (range f))"
2530 by (rule compact_imp_complete)
2531 moreover have "\<forall>n. f n \<in> closure (range f)"
2532 using closure_subset [of "range f"] by auto
2533 ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
2534 using `Cauchy f` unfolding complete_def by auto
2535 then show "convergent f"
2536 unfolding convergent_def by auto
2539 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
2540 proof(simp add: complete_def, rule, rule)
2541 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2542 hence "convergent f" by (rule Cauchy_convergent)
2543 thus "\<exists>l. f ----> l" unfolding convergent_def .
2546 lemma complete_imp_closed: assumes "complete s" shows "closed s"
2548 { fix x assume "x islimpt s"
2549 then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
2550 unfolding islimpt_sequential by auto
2551 then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
2552 using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
2553 hence "x \<in> s" using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
2555 thus "closed s" unfolding closed_limpt by auto
2558 lemma complete_eq_closed:
2559 fixes s :: "'a::complete_space set"
2560 shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
2562 assume ?lhs thus ?rhs by (rule complete_imp_closed)
2565 { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
2566 then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
2567 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 }
2568 thus ?lhs unfolding complete_def by auto
2571 lemma convergent_eq_cauchy:
2572 fixes s :: "nat \<Rightarrow> 'a::complete_space"
2573 shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s" (is "?lhs = ?rhs")
2575 assume ?lhs then obtain l where "(s ---> l) sequentially" by auto
2576 thus ?rhs using convergent_imp_cauchy by auto
2578 assume ?rhs thus ?lhs using complete_univ[unfolded complete_def, THEN spec[where x=s]] by auto
2581 lemma convergent_imp_bounded:
2582 fixes s :: "nat \<Rightarrow> 'a::metric_space"
2583 shows "(s ---> l) sequentially ==> bounded (s ` (UNIV::(nat set)))"
2584 using convergent_imp_cauchy[of s]
2585 using cauchy_imp_bounded[of s]
2589 subsubsection{* Total boundedness *}
2591 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
2592 "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
2593 declare helper_1.simps[simp del]
2595 lemma compact_imp_totally_bounded:
2597 shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
2598 proof(rule, rule, rule ccontr)
2599 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)"
2600 def x \<equiv> "helper_1 s e"
2602 have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
2603 proof(induct_tac rule:nat_less_induct)
2604 fix n def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
2605 assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
2606 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
2607 then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
2608 have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
2609 apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
2610 thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
2612 hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
2613 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
2614 from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
2615 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
2617 using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
2618 using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
2619 using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
2622 subsubsection{* Heine-Borel theorem *}
2624 text {* Following Burkill \& Burkill vol. 2. *}
2626 lemma heine_borel_lemma: fixes s::"'a::metric_space set"
2627 assumes "compact s" "s \<subseteq> (\<Union> t)" "\<forall>b \<in> t. open b"
2628 shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
2630 assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"
2631 hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto
2633 have "1 / real (n + 1) > 0" by auto
2634 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 }
2635 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
2636 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)"
2637 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
2639 then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
2640 using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
2642 obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
2643 then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"
2644 using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto
2646 then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"
2647 using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
2649 obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
2650 have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
2651 apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
2652 using subseq_bigger[OF r, of "N1 + N2"] by auto
2654 def x \<equiv> "(f (r (N1 + N2)))"
2655 have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
2656 using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto
2657 have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto
2658 then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto
2660 have "dist x l < e/2" using N1 unfolding x_def o_def by auto
2661 hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)
2663 thus False using e and `y\<notin>b` by auto
2666 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2667 \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"
2669 fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"
2670 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
2671 hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto
2672 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
2673 then obtain bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast
2675 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
2676 then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto
2678 have "finite (bb ` k)" using k(1) by auto
2680 { fix x assume "x\<in>s"
2681 hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3) unfolding subset_eq by auto
2682 hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast
2683 hence "x \<in> \<Union>(bb ` k)" using Union_iff[of x "bb ` k"] by auto
2685 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
2688 subsubsection {* Bolzano-Weierstrass property *}
2690 lemma heine_borel_imp_bolzano_weierstrass:
2691 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'))"
2692 "infinite t" "t \<subseteq> s"
2693 shows "\<exists>x \<in> s. x islimpt t"
2695 assume "\<not> (\<exists>x \<in> s. x islimpt t)"
2696 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
2697 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
2698 obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
2699 using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
2700 from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
2701 { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
2702 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
2703 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 }
2704 hence "inj_on f t" unfolding inj_on_def by simp
2705 hence "infinite (f ` t)" using assms(2) using finite_imageD by auto
2707 { fix x assume "x\<in>t" "f x \<notin> g"
2708 from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
2709 then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
2710 hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
2711 hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto }
2712 hence "f ` t \<subseteq> g" by auto
2713 ultimately show False using g(2) using finite_subset by auto
2716 subsubsection {* Complete the chain of compactness variants *}
2718 lemma islimpt_range_imp_convergent_subsequence:
2719 fixes f :: "nat \<Rightarrow> 'a::metric_space"
2720 assumes "l islimpt (range f)"
2721 shows "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2722 proof (intro exI conjI)
2723 have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"
2724 using assms unfolding islimpt_def
2725 by (drule_tac x="ball l e" in spec)
2726 (auto simp add: zero_less_dist_iff dist_commute)
2728 def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"
2729 have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"
2730 unfolding t_def by (rule LeastI2_ex [OF * conjunct1])
2731 have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"
2732 unfolding t_def by (rule LeastI2_ex [OF * conjunct2])
2733 have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"
2734 unfolding t_def by (simp add: Least_le)
2735 have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"
2736 unfolding t_def by (drule not_less_Least) simp
2737 have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"
2739 apply (erule f_t_neq)
2740 apply (erule (1) less_le_trans [OF f_t_closer])
2742 have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"
2743 by (drule f_t_closer) auto
2744 have t_less: "\<And>e. 0 < e \<Longrightarrow> t e < t (dist (f (t e)) l)"
2745 apply (subst less_le)
2747 apply (rule t_antimono)
2748 apply (erule f_t_neq)
2749 apply (erule f_t_closer [THEN less_imp_le])
2750 apply (rule t_dist_f_neq [symmetric])
2751 apply (erule f_t_neq)
2753 have dist_f_t_less':
2754 "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"
2755 apply (simp add: le_less)
2757 apply (rule less_trans)
2758 apply (erule f_t_closer)
2759 apply (rule le_less_trans)
2760 apply (erule less_tD)
2761 apply (erule f_t_neq)
2762 apply (erule f_t_closer)
2764 apply (erule f_t_closer)
2767 def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"
2768 have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"
2769 unfolding r_def by simp_all
2770 have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"
2771 by (induct_tac n) (simp_all add: r_simps f_t_neq)
2774 unfolding subseq_Suc_iff
2777 apply (simp_all add: r_simps)
2778 apply (rule t_less, rule zero_less_one)
2779 apply (rule t_less, rule f_r_neq)
2781 show "((f \<circ> r) ---> l) sequentially"
2782 unfolding Lim_sequentially o_def
2783 apply (clarify, rule_tac x="t e" in exI, clarify)
2784 apply (drule le_trans, rule seq_suble [OF `subseq r`])
2785 apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)
2789 lemma finite_range_imp_infinite_repeats:
2790 fixes f :: "nat \<Rightarrow> 'a"
2791 assumes "finite (range f)"
2792 shows "\<exists>k. infinite {n. f n = k}"
2794 { fix A :: "'a set" assume "finite A"
2795 hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"
2797 case empty thus ?case by simp
2801 proof (cases "finite {n. f n = x}")
2803 with `infinite {n. f n \<in> insert x A}`
2804 have "infinite {n. f n \<in> A}" by simp
2805 thus "\<exists>k. infinite {n. f n = k}" by (rule insert)
2807 case False thus "\<exists>k. infinite {n. f n = k}" ..
2811 from assms show "\<exists>k. infinite {n. f n = k}"
2815 lemma bolzano_weierstrass_imp_compact:
2816 fixes s :: "'a::metric_space set"
2817 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2820 { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
2821 have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2822 proof (cases "finite (range f)")
2824 hence "\<exists>l. infinite {n. f n = l}"
2825 by (rule finite_range_imp_infinite_repeats)
2826 then obtain l where "infinite {n. f n = l}" ..
2827 hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"
2828 by (rule infinite_enumerate)
2829 then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto
2830 hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2831 unfolding o_def by (simp add: fr Lim_const)
2832 hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2834 from f have "\<forall>n. f (r n) \<in> s" by simp
2835 hence "l \<in> s" by (simp add: fr)
2836 thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2837 by (rule rev_bexI) fact
2840 with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto
2841 then obtain l where "l \<in> s" "l islimpt (range f)" ..
2842 have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2843 using `l islimpt (range f)`
2844 by (rule islimpt_range_imp_convergent_subsequence)
2845 with `l \<in> s` show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
2848 thus ?thesis unfolding compact_def by auto
2851 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where
2852 "helper_2 beyond 0 = beyond 0" |
2853 "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"
2855 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"
2856 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2859 assume "\<not> bounded s"
2860 then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"
2861 unfolding bounded_any_center [where a=undefined]
2862 apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto
2863 hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"
2864 unfolding linorder_not_le by auto
2865 def x \<equiv> "helper_2 beyond"
2867 { fix m n ::nat assume "m<n"
2868 hence "dist undefined (x m) + 1 < dist undefined (x n)"
2870 case 0 thus ?case by auto
2873 have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"
2874 unfolding x_def and helper_2.simps
2875 using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto
2876 thus ?case proof(cases "m < n")
2877 case True thus ?thesis using Suc and * by auto
2879 case False hence "m = n" using Suc(2) by auto
2880 thus ?thesis using * by auto
2883 { fix m n ::nat assume "m\<noteq>n"
2884 have "1 < dist (x m) (x n)"
2887 hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto
2888 thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith
2890 case False hence "n<m" using `m\<noteq>n` by auto
2891 hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto
2892 thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith
2893 qed } note ** = this
2894 { fix a b assume "x a = x b" "a \<noteq> b"
2895 hence False using **[of a b] by auto }
2896 hence "inj x" unfolding inj_on_def by auto
2900 proof(cases "n = 0")
2901 case True thus ?thesis unfolding x_def using beyond by auto
2903 case False then obtain z where "n = Suc z" using not0_implies_Suc by auto
2904 thus ?thesis unfolding x_def using beyond by auto
2906 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
2908 then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto
2909 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
2910 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"]]
2911 unfolding dist_nz by auto
2912 show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto
2915 lemma sequence_infinite_lemma:
2916 fixes f :: "nat \<Rightarrow> 'a::t1_space"
2917 assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"
2918 shows "infinite (range f)"
2920 assume "finite (range f)"
2921 hence "closed (range f)" by (rule finite_imp_closed)
2922 hence "open (- range f)" by (rule open_Compl)
2923 from assms(1) have "l \<in> - range f" by auto
2924 from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
2925 using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)
2926 thus False unfolding eventually_sequentially by auto
2929 lemma closure_insert:
2930 fixes x :: "'a::t1_space"
2931 shows "closure (insert x s) = insert x (closure s)"
2932 apply (rule closure_unique)
2933 apply (rule conjI [OF insert_mono [OF closure_subset]])
2934 apply (rule conjI [OF closed_insert [OF closed_closure]])
2935 apply (simp add: closure_minimal)
2938 lemma islimpt_insert:
2939 fixes x :: "'a::t1_space"
2940 shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
2942 assume *: "x islimpt (insert a s)"
2944 proof (rule islimptI)
2945 fix t assume t: "x \<in> t" "open t"
2946 show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
2947 proof (cases "x = a")
2949 obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
2950 using * t by (rule islimptE)
2951 with `x = a` show ?thesis by auto
2954 with t have t': "x \<in> t - {a}" "open (t - {a})"
2955 by (simp_all add: open_Diff)
2956 obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
2957 using * t' by (rule islimptE)
2958 thus ?thesis by auto
2962 assume "x islimpt s" thus "x islimpt (insert a s)"
2963 by (rule islimpt_subset) auto
2966 lemma islimpt_union_finite:
2967 fixes x :: "'a::t1_space"
2968 shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
2969 by (induct set: finite, simp_all add: islimpt_insert)
2971 lemma sequence_unique_limpt:
2972 fixes f :: "nat \<Rightarrow> 'a::t2_space"
2973 assumes "(f ---> l) sequentially" and "l' islimpt (range f)"
2976 assume "l' \<noteq> l"
2977 obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
2978 using hausdorff [OF `l' \<noteq> l`] by auto
2979 have "eventually (\<lambda>n. f n \<in> t) sequentially"
2980 using assms(1) `open t` `l \<in> t` by (rule topological_tendstoD)
2981 then obtain N where "\<forall>n\<ge>N. f n \<in> t"
2982 unfolding eventually_sequentially by auto
2984 have "UNIV = {..<N} \<union> {N..}" by auto
2985 hence "l' islimpt (f ` ({..<N} \<union> {N..}))" using assms(2) by simp
2986 hence "l' islimpt (f ` {..<N} \<union> f ` {N..})" by (simp add: image_Un)
2987 hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite)
2988 then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
2989 using `l' \<in> s` `open s` by (rule islimptE)
2990 then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto
2991 with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t" by simp
2992 with `s \<inter> t = {}` show False by simp
2995 lemma bolzano_weierstrass_imp_closed:
2996 fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)
2997 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
3000 { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
3002 proof(cases "\<forall>n. x n \<noteq> l")
3003 case False thus "l\<in>s" using as(1) by auto
3005 case True note cas = this
3006 with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto
3007 then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto
3008 thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
3010 thus ?thesis unfolding closed_sequential_limits by fast
3013 text{* Hence express everything as an equivalence. *}
3015 lemma compact_eq_heine_borel:
3016 fixes s :: "'a::metric_space set"
3017 shows "compact s \<longleftrightarrow>
3018 (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
3019 --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")
3021 assume ?lhs thus ?rhs by (rule compact_imp_heine_borel)
3024 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
3025 by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])
3026 thus ?lhs by (rule bolzano_weierstrass_imp_compact)
3029 lemma compact_eq_bolzano_weierstrass:
3030 fixes s :: "'a::metric_space set"
3031 shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
3033 assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto
3035 assume ?rhs thus ?lhs by (rule bolzano_weierstrass_imp_compact)
3038 lemma compact_eq_bounded_closed:
3039 fixes s :: "'a::heine_borel set"
3040 shows "compact s \<longleftrightarrow> bounded s \<and> closed s" (is "?lhs = ?rhs")
3042 assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto
3044 assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto
3047 lemma compact_imp_bounded:
3048 fixes s :: "'a::metric_space set"
3049 shows "compact s ==> bounded s"
3052 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')"
3053 by (rule compact_imp_heine_borel)
3054 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
3055 using heine_borel_imp_bolzano_weierstrass[of s] by auto
3057 by (rule bolzano_weierstrass_imp_bounded)
3060 lemma compact_imp_closed:
3061 fixes s :: "'a::metric_space set"
3062 shows "compact s ==> closed s"
3065 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')"
3066 by (rule compact_imp_heine_borel)
3067 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
3068 using heine_borel_imp_bolzano_weierstrass[of s] by auto
3070 by (rule bolzano_weierstrass_imp_closed)
3073 text{* In particular, some common special cases. *}
3075 lemma compact_empty[simp]:
3077 unfolding compact_def
3080 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
3081 unfolding subseq_def by simp (* TODO: move somewhere else *)
3083 lemma compact_union [intro]:
3084 assumes "compact s" and "compact t"
3085 shows "compact (s \<union> t)"
3086 proof (rule compactI)
3087 fix f :: "nat \<Rightarrow> 'a"
3088 assume "\<forall>n. f n \<in> s \<union> t"
3089 hence "infinite {n. f n \<in> s \<union> t}" by simp
3090 hence "infinite {n. f n \<in> s} \<or> infinite {n. f n \<in> t}" by simp
3091 thus "\<exists>l\<in>s \<union> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
3093 assume "infinite {n. f n \<in> s}"
3094 from infinite_enumerate [OF this]
3095 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> s" by auto
3096 obtain r l where "l \<in> s" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
3097 using `compact s` `\<forall>n. (f \<circ> q) n \<in> s` by (rule compactE)
3098 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
3099 using `subseq q` by (simp_all add: subseq_o o_assoc)
3100 thus ?thesis by auto
3102 assume "infinite {n. f n \<in> t}"
3103 from infinite_enumerate [OF this]
3104 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> t" by auto
3105 obtain r l where "l \<in> t" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
3106 using `compact t` `\<forall>n. (f \<circ> q) n \<in> t` by (rule compactE)
3107 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
3108 using `subseq q` by (simp_all add: subseq_o o_assoc)
3109 thus ?thesis by auto
3113 lemma compact_inter_closed [intro]:
3114 assumes "compact s" and "closed t"
3115 shows "compact (s \<inter> t)"
3116 proof (rule compactI)
3117 fix f :: "nat \<Rightarrow> 'a"
3118 assume "\<forall>n. f n \<in> s \<inter> t"
3119 hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t" by simp_all
3120 obtain l r where "l \<in> s" "subseq r" "((f \<circ> r) ---> l) sequentially"
3121 using `compact s` `\<forall>n. f n \<in> s` by (rule compactE)
3123 from `closed t` `\<forall>n. f n \<in> t` `((f \<circ> r) ---> l) sequentially` have "l \<in> t"
3124 unfolding closed_sequential_limits o_def by fast
3125 ultimately show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
3129 lemma closed_inter_compact [intro]:
3130 assumes "closed s" and "compact t"
3131 shows "compact (s \<inter> t)"
3132 using compact_inter_closed [of t s] assms
3133 by (simp add: Int_commute)
3135 lemma compact_inter [intro]:
3136 assumes "compact s" and "compact t"
3137 shows "compact (s \<inter> t)"
3138 using assms by (intro compact_inter_closed compact_imp_closed)
3140 lemma compact_sing [simp]: "compact {a}"
3141 unfolding compact_def o_def subseq_def
3142 by (auto simp add: tendsto_const)
3144 lemma compact_insert [simp]:
3145 assumes "compact s" shows "compact (insert x s)"
3147 have "compact ({x} \<union> s)"
3148 using compact_sing assms by (rule compact_union)
3149 thus ?thesis by simp
3152 lemma finite_imp_compact:
3153 shows "finite s \<Longrightarrow> compact s"
3154 by (induct set: finite) simp_all
3156 lemma compact_cball[simp]:
3157 fixes x :: "'a::heine_borel"
3158 shows "compact(cball x e)"
3159 using compact_eq_bounded_closed bounded_cball closed_cball
3162 lemma compact_frontier_bounded[intro]:
3163 fixes s :: "'a::heine_borel set"
3164 shows "bounded s ==> compact(frontier s)"
3165 unfolding frontier_def
3166 using compact_eq_bounded_closed
3169 lemma compact_frontier[intro]:
3170 fixes s :: "'a::heine_borel set"
3171 shows "compact s ==> compact (frontier s)"
3172 using compact_eq_bounded_closed compact_frontier_bounded
3175 lemma frontier_subset_compact:
3176 fixes s :: "'a::heine_borel set"
3177 shows "compact s ==> frontier s \<subseteq> s"
3178 using frontier_subset_closed compact_eq_bounded_closed
3182 fixes s :: "'a::t1_space set"
3183 shows "open s \<Longrightarrow> open (s - {x})"
3184 by (simp add: open_Diff)
3186 text{* Finite intersection property. I could make it an equivalence in fact. *}
3188 lemma compact_imp_fip:
3189 assumes "compact s" "\<forall>t \<in> f. closed t"
3190 "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"
3191 shows "s \<inter> (\<Inter> f) \<noteq> {}"
3193 assume as:"s \<inter> (\<Inter> f) = {}"
3194 hence "s \<subseteq> \<Union> uminus ` f" by auto
3195 moreover have "Ball (uminus ` f) open" using open_Diff closed_Diff using assms(2) by auto
3196 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
3197 hence "finite (uminus ` f') \<and> uminus ` f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)
3198 hence "s \<inter> \<Inter>uminus ` f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus ` f'"]] by auto
3199 thus False using f'(3) unfolding subset_eq and Union_iff by blast
3202 subsection{* Bounded closed nest property (proof does not use Heine-Borel). *}
3204 lemma bounded_closed_nest:
3205 assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
3206 "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)" "bounded(s 0)"
3207 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
3209 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
3210 from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
3212 then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
3213 unfolding compact_def apply(erule_tac x=x in allE) using x using assms(3) by blast
3216 { fix e::real assume "e>0"
3217 with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
3218 hence "dist ((x \<circ> r) (max N n)) l < e" by auto
3220 have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
3221 hence "(x \<circ> r) (max N n) \<in> s n"
3222 using x apply(erule_tac x=n in allE)
3223 using x apply(erule_tac x="r (max N n)" in allE)
3224 using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto
3225 ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
3227 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
3229 thus ?thesis by auto
3232 text{* Decreasing case does not even need compactness, just completeness. *}
3234 lemma decreasing_closed_nest:
3235 assumes "\<forall>n. closed(s n)"
3236 "\<forall>n. (s n \<noteq> {})"
3237 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
3238 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
3239 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s n"
3241 have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
3242 hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
3243 then obtain t where t: "\<forall>n. t n \<in> s n" by auto
3244 { fix e::real assume "e>0"
3245 then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
3246 { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
3247 hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding subset_eq t by blast+
3248 hence "dist (t m) (t n) < e" using N by auto
3250 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
3252 hence "Cauchy t" unfolding cauchy_def by auto
3253 then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
3255 { fix e::real assume "e>0"
3256 then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded Lim_sequentially] by auto
3257 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
3258 hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
3260 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
3262 then show ?thesis by auto
3265 text{* Strengthen it to the intersection actually being a singleton. *}
3267 lemma decreasing_closed_nest_sing:
3268 fixes s :: "nat \<Rightarrow> 'a::heine_borel set"
3269 assumes "\<forall>n. closed(s n)"
3270 "\<forall>n. s n \<noteq> {}"
3271 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
3272 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
3273 shows "\<exists>a. \<Inter>(range s) = {a}"
3275 obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
3276 { fix b assume b:"b \<in> \<Inter>(range s)"
3277 { fix e::real assume "e>0"
3278 hence "dist a b < e" using assms(4 )using b using a by blast
3280 hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)
3282 with a have "\<Inter>(range s) = {a}" unfolding image_def by auto
3286 text{* Cauchy-type criteria for uniform convergence. *}
3288 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows
3289 "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
3290 (\<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")
3293 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
3294 { fix e::real assume "e>0"
3295 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
3296 { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
3297 hence "dist (s m x) (s n x) < e"
3298 using N[THEN spec[where x=m], THEN spec[where x=x]]
3299 using N[THEN spec[where x=n], THEN spec[where x=x]]
3300 using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto }
3301 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 }
3305 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
3306 then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
3307 using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
3308 { fix e::real assume "e>0"
3309 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"
3310 using `?rhs`[THEN spec[where x="e/2"]] by auto
3311 { fix x assume "P x"
3312 then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
3313 using l[THEN spec[where x=x], unfolded Lim_sequentially] using `e>0` by(auto elim!: allE[where x="e/2"])
3314 fix n::nat assume "n\<ge>N"
3315 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]]
3316 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) }
3317 hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
3321 lemma uniformly_cauchy_imp_uniformly_convergent:
3322 fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"
3323 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"
3324 "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
3325 shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
3327 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"
3328 using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
3330 { fix x assume "P x"
3331 hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
3332 using l and assms(2) unfolding Lim_sequentially by blast }
3333 ultimately show ?thesis by auto
3336 subsection {* Continuity *}
3338 text {* Define continuity over a net to take in restrictions of the set. *}
3341 continuous :: "'a::t2_space net \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
3342 "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
3344 lemma continuous_trivial_limit:
3345 "trivial_limit net ==> continuous net f"
3346 unfolding continuous_def tendsto_def trivial_limit_eq by auto
3348 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
3349 unfolding continuous_def
3350 unfolding tendsto_def
3351 using netlimit_within[of x s]
3352 by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
3354 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
3355 using continuous_within [of x UNIV f] by (simp add: within_UNIV)
3357 lemma continuous_at_within:
3358 assumes "continuous (at x) f" shows "continuous (at x within s) f"
3359 using assms unfolding continuous_at continuous_within
3360 by (rule Lim_at_within)
3362 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
3364 lemma continuous_within_eps_delta:
3365 "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)"
3366 unfolding continuous_within and Lim_within
3367 apply auto unfolding dist_nz[THEN sym] apply(auto elim!:allE) apply(rule_tac x=d in exI) by auto
3369 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3370 \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
3371 using continuous_within_eps_delta[of x UNIV f]
3372 unfolding within_UNIV by blast
3374 text{* Versions in terms of open balls. *}
3376 lemma continuous_within_ball:
3377 "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3378 f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
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"
3383 using `?lhs`[unfolded continuous_within Lim_within] by auto
3384 { fix y assume "y\<in>f ` (ball x d \<inter> s)"
3385 hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
3386 apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
3388 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) }
3391 assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
3392 apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
3395 lemma continuous_at_ball:
3396 "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3398 assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3399 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)
3400 unfolding dist_nz[THEN sym] by auto
3402 assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3403 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)
3406 text{* Define setwise continuity in terms of limits within the set. *}
3410 "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
3412 "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
3414 lemma continuous_on_topological:
3415 "continuous_on s f \<longleftrightarrow>
3416 (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
3417 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
3418 unfolding continuous_on_def tendsto_def
3419 unfolding Limits.eventually_within eventually_at_topological
3420 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
3422 lemma continuous_on_iff:
3423 "continuous_on s f \<longleftrightarrow>
3424 (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
3425 unfolding continuous_on_def Lim_within
3426 apply (intro ball_cong [OF refl] all_cong ex_cong)
3427 apply (rename_tac y, case_tac "y = x", simp)
3428 apply (simp add: dist_nz)
3432 uniformly_continuous_on ::
3433 "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
3435 "uniformly_continuous_on s f \<longleftrightarrow>
3436 (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
3438 text{* Some simple consequential lemmas. *}
3440 lemma uniformly_continuous_imp_continuous:
3441 " uniformly_continuous_on s f ==> continuous_on s f"
3442 unfolding uniformly_continuous_on_def continuous_on_iff by blast
3444 lemma continuous_at_imp_continuous_within:
3445 "continuous (at x) f ==> continuous (at x within s) f"
3446 unfolding continuous_within continuous_at using Lim_at_within by auto
3448 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
3449 unfolding tendsto_def by (simp add: trivial_limit_eq)
3451 lemma continuous_at_imp_continuous_on:
3452 assumes "\<forall>x\<in>s. continuous (at x) f"
3453 shows "continuous_on s f"
3454 unfolding continuous_on_def
3456 fix x assume "x \<in> s"
3457 with assms have *: "(f ---> f (netlimit (at x))) (at x)"
3458 unfolding continuous_def by simp
3459 have "(f ---> f x) (at x)"
3460 proof (cases "trivial_limit (at x)")
3461 case True thus ?thesis
3462 by (rule Lim_trivial_limit)
3465 hence 1: "netlimit (at x) = x"
3466 using netlimit_within [of x UNIV]
3467 by (simp add: within_UNIV)
3468 with * show ?thesis by simp
3470 thus "(f ---> f x) (at x within s)"
3471 by (rule Lim_at_within)
3474 lemma continuous_on_eq_continuous_within:
3475 "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"
3476 unfolding continuous_on_def continuous_def
3477 apply (rule ball_cong [OF refl])
3478 apply (case_tac "trivial_limit (at x within s)")
3479 apply (simp add: Lim_trivial_limit)
3480 apply (simp add: netlimit_within)
3483 lemmas continuous_on = continuous_on_def -- "legacy theorem name"
3485 lemma continuous_on_eq_continuous_at:
3486 shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
3487 by (auto simp add: continuous_on continuous_at Lim_within_open)
3489 lemma continuous_within_subset:
3490 "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
3491 ==> continuous (at x within t) f"
3492 unfolding continuous_within by(metis Lim_within_subset)
3494 lemma continuous_on_subset:
3495 shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
3496 unfolding continuous_on by (metis subset_eq Lim_within_subset)
3498 lemma continuous_on_interior:
3499 shows "continuous_on s f \<Longrightarrow> x \<in> interior s ==> continuous (at x) f"
3500 unfolding interior_def
3502 by (meson continuous_on_eq_continuous_at continuous_on_subset)
3504 lemma continuous_on_eq:
3505 "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"
3506 unfolding continuous_on_def tendsto_def Limits.eventually_within
3509 text{* Characterization of various kinds of continuity in terms of sequences. *}
3511 (* \<longrightarrow> could be generalized, but \<longleftarrow> requires metric space *)
3512 lemma continuous_within_sequentially:
3513 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3514 shows "continuous (at a within s) f \<longleftrightarrow>
3515 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
3516 --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
3519 { 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"
3520 fix e::real assume "e>0"
3521 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
3522 from x(2) `d>0` obtain N where N:"\<forall>n\<ge>N. dist (x n) a < d" by auto
3523 hence "\<exists>N. \<forall>n\<ge>N. dist ((f \<circ> x) n) (f a) < e"
3524 apply(rule_tac x=N in exI) using N d apply auto using x(1)
3525 apply(erule_tac x=n in allE) apply(erule_tac x=n in allE)
3526 apply(erule_tac x="x n" in ballE) apply auto unfolding dist_nz[THEN sym] apply auto using `e>0` by auto
3528 thus ?rhs unfolding continuous_within unfolding Lim_sequentially by simp
3531 { fix e::real assume "e>0"
3532 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)"
3533 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
3534 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)"
3535 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
3536 { fix d::real assume "d>0"
3537 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
3538 then obtain N::nat where N:"inverse (real (N + 1)) < d" by auto
3539 { fix n::nat assume n:"n\<ge>N"
3540 hence "dist (x (inverse (real (n + 1)))) a < inverse (real (n + 1))" using x[THEN spec[where x="inverse (real (n + 1))"]] by auto
3541 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)
3542 ultimately have "dist (x (inverse (real (n + 1)))) a < d" by auto
3544 hence "\<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < d" by auto
3546 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
3547 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
3548 hence "False" apply(erule_tac x=e in allE) using `e>0` using x by auto
3550 thus ?lhs unfolding continuous_within unfolding Lim_within unfolding Lim_sequentially by blast
3553 lemma continuous_at_sequentially:
3554 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3555 shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
3556 --> ((f o x) ---> f a) sequentially)"
3557 using continuous_within_sequentially[of a UNIV f] unfolding within_UNIV by auto
3559 lemma continuous_on_sequentially:
3560 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3561 shows "continuous_on s f \<longleftrightarrow>
3562 (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
3563 --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
3565 assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
3567 assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
3570 lemma uniformly_continuous_on_sequentially':
3571 "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
3572 ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially
3573 \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
3576 { fix x y assume x:"\<forall>n. x n \<in> s" and y:"\<forall>n. y n \<in> s" and xy:"((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"
3577 { fix e::real assume "e>0"
3578 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"
3579 using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
3580 obtain N where N:"\<forall>n\<ge>N. dist (x n) (y n) < d" using xy[unfolded Lim_sequentially dist_norm] and `d>0` by auto
3581 { fix n assume "n\<ge>N"
3582 hence "dist (f (x n)) (f (y n)) < e"
3583 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
3584 unfolding dist_commute by simp }
3585 hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" by auto }
3586 hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding Lim_sequentially and dist_real_def by auto }
3590 { assume "\<not> ?lhs"
3591 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
3592 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"
3593 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
3594 by (auto simp add: dist_commute)
3595 def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
3596 def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
3597 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"
3598 unfolding x_def and y_def using fa by auto
3599 { fix e::real assume "e>0"
3600 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
3601 { fix n::nat assume "n\<ge>N"
3602 hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
3603 also have "\<dots> < e" using N by auto
3604 finally have "inverse (real n + 1) < e" by auto
3605 hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto }
3606 hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto }
3607 hence "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" using `?rhs`[THEN spec[where x=x], THEN spec[where x=y]] and xyn unfolding Lim_sequentially dist_real_def by auto
3608 hence False using fxy and `e>0` by auto }
3609 thus ?lhs unfolding uniformly_continuous_on_def by blast
3612 lemma uniformly_continuous_on_sequentially:
3613 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
3614 shows "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
3615 ((\<lambda>n. x n - y n) ---> 0) sequentially
3616 \<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")
3617 (* BH: maybe the previous lemma should replace this one? *)
3618 unfolding uniformly_continuous_on_sequentially'
3619 unfolding dist_norm Lim_null_norm [symmetric] ..
3621 text{* The usual transformation theorems. *}
3623 lemma continuous_transform_within:
3624 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3625 assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
3626 "continuous (at x within s) f"
3627 shows "continuous (at x within s) g"
3628 unfolding continuous_within
3629 proof (rule Lim_transform_within)
3630 show "0 < d" by fact
3631 show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
3632 using assms(3) by auto
3634 using assms(1,2,3) by auto
3635 thus "(f ---> g x) (at x within s)"
3636 using assms(4) unfolding continuous_within by simp
3639 lemma continuous_transform_at:
3640 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3641 assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
3642 "continuous (at x) f"
3643 shows "continuous (at x) g"
3644 using continuous_transform_within [of d x UNIV f g] assms
3645 by (simp add: within_UNIV)
3647 text{* Combination results for pointwise continuity. *}
3649 lemma continuous_const: "continuous net (\<lambda>x. c)"
3650 by (auto simp add: continuous_def Lim_const)
3652 lemma continuous_cmul:
3653 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3654 shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"
3655 by (auto simp add: continuous_def Lim_cmul)
3657 lemma continuous_neg:
3658 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3659 shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"
3660 by (auto simp add: continuous_def Lim_neg)
3662 lemma continuous_add:
3663 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3664 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"
3665 by (auto simp add: continuous_def Lim_add)
3667 lemma continuous_sub:
3668 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3669 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"
3670 by (auto simp add: continuous_def Lim_sub)
3673 text{* Same thing for setwise continuity. *}
3675 lemma continuous_on_const:
3676 "continuous_on s (\<lambda>x. c)"
3677 unfolding continuous_on_def by auto
3679 lemma continuous_on_cmul:
3680 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3681 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c *\<^sub>R (f x))"
3682 unfolding continuous_on_def by (auto intro: tendsto_intros)
3684 lemma continuous_on_neg:
3685 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3686 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
3687 unfolding continuous_on_def by (auto intro: tendsto_intros)
3689 lemma continuous_on_add:
3690 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3691 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3692 \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
3693 unfolding continuous_on_def by (auto intro: tendsto_intros)
3695 lemma continuous_on_sub:
3696 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3697 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3698 \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
3699 unfolding continuous_on_def by (auto intro: tendsto_intros)
3701 text{* Same thing for uniform continuity, using sequential formulations. *}
3703 lemma uniformly_continuous_on_const:
3704 "uniformly_continuous_on s (\<lambda>x. c)"
3705 unfolding uniformly_continuous_on_def by simp
3707 lemma uniformly_continuous_on_cmul:
3708 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3709 assumes "uniformly_continuous_on s f"
3710 shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
3712 { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
3713 hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"
3714 using Lim_cmul[of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
3715 unfolding scaleR_zero_right scaleR_right_diff_distrib by auto
3717 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
3718 unfolding dist_norm Lim_null_norm [symmetric] by auto
3722 fixes x y :: "'a::real_normed_vector"
3723 shows "dist (- x) (- y) = dist x y"
3724 unfolding dist_norm minus_diff_minus norm_minus_cancel ..
3726 lemma uniformly_continuous_on_neg:
3727 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3728 shows "uniformly_continuous_on s f
3729 ==> uniformly_continuous_on s (\<lambda>x. -(f x))"
3730 unfolding uniformly_continuous_on_def dist_minus .
3732 lemma uniformly_continuous_on_add:
3733 fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3734 assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"
3735 shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
3737 { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
3738 "((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"
3739 hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"
3740 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
3741 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 }
3742 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
3743 unfolding dist_norm Lim_null_norm [symmetric] by auto
3746 lemma uniformly_continuous_on_sub:
3747 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3748 shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s g
3749 ==> uniformly_continuous_on s (\<lambda>x. f x - g x)"
3750 unfolding ab_diff_minus
3751 using uniformly_continuous_on_add[of s f "\<lambda>x. - g x"]
3752 using uniformly_continuous_on_neg[of s g] by auto
3754 text{* Identity function is continuous in every sense. *}
3756 lemma continuous_within_id:
3757 "continuous (at a within s) (\<lambda>x. x)"
3758 unfolding continuous_within by (rule Lim_at_within [OF Lim_ident_at])
3760 lemma continuous_at_id:
3761 "continuous (at a) (\<lambda>x. x)"
3762 unfolding continuous_at by (rule Lim_ident_at)
3764 lemma continuous_on_id:
3765 "continuous_on s (\<lambda>x. x)"
3766 unfolding continuous_on_def by (auto intro: tendsto_ident_at_within)
3768 lemma uniformly_continuous_on_id:
3769 "uniformly_continuous_on s (\<lambda>x. x)"
3770 unfolding uniformly_continuous_on_def by auto
3772 text{* Continuity of all kinds is preserved under composition. *}
3774 lemma continuous_within_topological:
3775 "continuous (at x within s) f \<longleftrightarrow>
3776 (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
3777 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
3778 unfolding continuous_within
3779 unfolding tendsto_def Limits.eventually_within eventually_at_topological
3780 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
3782 lemma continuous_within_compose:
3783 assumes "continuous (at x within s) f"
3784 assumes "continuous (at (f x) within f ` s) g"
3785 shows "continuous (at x within s) (g o f)"
3786 using assms unfolding continuous_within_topological by simp metis
3788 lemma continuous_at_compose:
3789 assumes "continuous (at x) f" "continuous (at (f x)) g"
3790 shows "continuous (at x) (g o f)"
3792 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
3793 thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g, unfolded within_UNIV] by auto
3796 lemma continuous_on_compose:
3797 "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
3798 unfolding continuous_on_topological by simp metis
3800 lemma uniformly_continuous_on_compose:
3801 assumes "uniformly_continuous_on s f" "uniformly_continuous_on (f ` s) g"
3802 shows "uniformly_continuous_on s (g o f)"
3804 { fix e::real assume "e>0"
3805 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
3806 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
3807 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 }
3808 thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
3811 text{* Continuity in terms of open preimages. *}
3813 lemma continuous_at_open:
3814 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)))"
3815 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]
3816 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto
3818 lemma continuous_on_open:
3819 shows "continuous_on s f \<longleftrightarrow>
3820 (\<forall>t. openin (subtopology euclidean (f ` s)) t
3821 --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
3824 assume 1: "continuous_on s f"
3825 assume 2: "openin (subtopology euclidean (f ` s)) t"
3826 from 2 obtain B where B: "open B" and t: "t = f ` s \<inter> B"
3827 unfolding openin_open by auto
3828 def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"
3829 have "open U" unfolding U_def by (simp add: open_Union)
3830 moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"
3831 proof (intro ballI iffI)
3832 fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"
3833 unfolding U_def t by auto
3835 fix x assume "x \<in> s" and "f x \<in> t"
3836 hence "x \<in> s" and "f x \<in> B"
3838 with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"
3839 unfolding t continuous_on_topological by metis
3840 then show "x \<in> U"
3841 unfolding U_def by auto
3843 ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto
3844 then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3845 unfolding openin_open by fast
3847 assume "?rhs" show "continuous_on s f"
3848 unfolding continuous_on_topological
3850 fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"
3851 have "openin (subtopology euclidean (f ` s)) (f ` s \<inter> B)"
3852 unfolding openin_open using `open B` by auto
3853 then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f ` s \<inter> B}"
3854 using `?rhs` by fast
3855 then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
3856 unfolding openin_open using `x \<in> s` and `f x \<in> B` by auto
3860 text {* Similarly in terms of closed sets. *}
3862 lemma continuous_on_closed:
3863 shows "continuous_on s f \<longleftrightarrow> (\<forall>t. closedin (subtopology euclidean (f ` s)) t --> closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
3867 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3868 have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
3869 assume as:"closedin (subtopology euclidean (f ` s)) t"
3870 hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
3871 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"]]
3872 unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto }
3877 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3878 assume as:"openin (subtopology euclidean (f ` s)) t"
3879 hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
3880 unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
3881 thus ?lhs unfolding continuous_on_open by auto
3884 text{* Half-global and completely global cases. *}
3886 lemma continuous_open_in_preimage:
3887 assumes "continuous_on s f" "open t"
3888 shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3890 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
3891 have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3892 using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
3893 thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3896 lemma continuous_closed_in_preimage:
3897 assumes "continuous_on s f" "closed t"
3898 shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3900 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
3901 have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3902 using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
3904 using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3907 lemma continuous_open_preimage:
3908 assumes "continuous_on s f" "open s" "open t"
3909 shows "open {x \<in> s. f x \<in> t}"
3911 obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3912 using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
3913 thus ?thesis using open_Int[of s T, OF assms(2)] by auto
3916 lemma continuous_closed_preimage:
3917 assumes "continuous_on s f" "closed s" "closed t"
3918 shows "closed {x \<in> s. f x \<in> t}"
3920 obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3921 using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
3922 thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
3925 lemma continuous_open_preimage_univ:
3926 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
3927 using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
3929 lemma continuous_closed_preimage_univ:
3930 shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
3931 using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
3933 lemma continuous_open_vimage:
3934 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
3935 unfolding vimage_def by (rule continuous_open_preimage_univ)
3937 lemma continuous_closed_vimage:
3938 shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
3939 unfolding vimage_def by (rule continuous_closed_preimage_univ)
3941 lemma interior_image_subset:
3942 assumes "\<forall>x. continuous (at x) f" "inj f"
3943 shows "interior (f ` s) \<subseteq> f ` (interior s)"
3944 apply rule unfolding interior_def mem_Collect_eq image_iff apply safe
3945 proof- fix x T assume as:"open T" "x \<in> T" "T \<subseteq> f ` s"
3946 hence "x \<in> f ` s" by auto then guess y unfolding image_iff .. note y=this
3947 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
3948 apply safe apply(rule_tac x="{x. f x \<in> T}" in exI) apply(safe,rule continuous_open_preimage_univ)
3949 proof- fix x assume "f x \<in> T" hence "f x \<in> f ` s" using as by auto
3950 thus "x \<in> s" unfolding inj_image_mem_iff[OF assms(2)] . qed auto qed
3952 text{* Equality of continuous functions on closure and related results. *}
3954 lemma continuous_closed_in_preimage_constant:
3955 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3956 shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
3957 using continuous_closed_in_preimage[of s f "{a}"] by auto
3959 lemma continuous_closed_preimage_constant:
3960 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3961 shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
3962 using continuous_closed_preimage[of s f "{a}"] by auto
3964 lemma continuous_constant_on_closure:
3965 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3966 assumes "continuous_on (closure s) f"
3967 "\<forall>x \<in> s. f x = a"
3968 shows "\<forall>x \<in> (closure s). f x = a"
3969 using continuous_closed_preimage_constant[of "closure s" f a]
3970 assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
3972 lemma image_closure_subset:
3973 assumes "continuous_on (closure s) f" "closed t" "(f ` s) \<subseteq> t"
3974 shows "f ` (closure s) \<subseteq> t"
3976 have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
3977 moreover have "closed {x \<in> closure s. f x \<in> t}"
3978 using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
3979 ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
3980 using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
3981 thus ?thesis by auto
3984 lemma continuous_on_closure_norm_le:
3985 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3986 assumes "continuous_on (closure s) f" "\<forall>y \<in> s. norm(f y) \<le> b" "x \<in> (closure s)"
3987 shows "norm(f x) \<le> b"
3989 have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
3991 using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
3992 unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
3995 text{* Making a continuous function avoid some value in a neighbourhood. *}
3997 lemma continuous_within_avoid:
3998 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3999 assumes "continuous (at x within s) f" "x \<in> s" "f x \<noteq> a"
4000 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
4002 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"
4003 using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto
4004 { fix y assume " y\<in>s" "dist x y < d"
4005 hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]
4006 apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }
4007 thus ?thesis using `d>0` by auto
4010 lemma continuous_at_avoid:
4011 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
4012 assumes "continuous (at x) f" "f x \<noteq> a"
4013 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
4014 using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto
4016 lemma continuous_on_avoid:
4017 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
4018 assumes "continuous_on s f" "x \<in> s" "f x \<noteq> a"
4019 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
4020 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
4022 lemma continuous_on_open_avoid:
4023 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
4024 assumes "continuous_on s f" "open s" "x \<in> s" "f x \<noteq> a"
4025 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
4026 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
4028 text{* Proving a function is constant by proving open-ness of level set. *}
4030 lemma continuous_levelset_open_in_cases:
4031 fixes f :: "_ \<Rightarrow> 'b::t1_space"
4032 shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
4033 openin (subtopology euclidean s) {x \<in> s. f x = a}
4034 ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
4035 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
4037 lemma continuous_levelset_open_in:
4038 fixes f :: "_ \<Rightarrow> 'b::t1_space"
4039 shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
4040 openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
4041 (\<exists>x \<in> s. f x = a) ==> (\<forall>x \<in> s. f x = a)"
4042 using continuous_levelset_open_in_cases[of s f ]
4045 lemma continuous_levelset_open:
4046 fixes f :: "_ \<Rightarrow> 'b::t1_space"
4047 assumes "connected s" "continuous_on s f" "open {x \<in> s. f x = a}" "\<exists>x \<in> s. f x = a"
4048 shows "\<forall>x \<in> s. f x = a"
4049 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
4051 text{* Some arithmetical combinations (more to prove). *}
4053 lemma open_scaling[intro]:
4054 fixes s :: "'a::real_normed_vector set"
4055 assumes "c \<noteq> 0" "open s"
4056 shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
4058 { fix x assume "x \<in> s"
4059 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
4060 have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF `e>0`] by auto
4062 { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
4063 hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
4064 using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
4065 assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
4066 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 }
4067 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 }
4068 thus ?thesis unfolding open_dist by auto
4071 lemma minus_image_eq_vimage:
4072 fixes A :: "'a::ab_group_add set"
4073 shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
4074 by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
4076 lemma open_negations:
4077 fixes s :: "'a::real_normed_vector set"
4078 shows "open s ==> open ((\<lambda> x. -x) ` s)"
4079 unfolding scaleR_minus1_left [symmetric]
4080 by (rule open_scaling, auto)
4082 lemma open_translation:
4083 fixes s :: "'a::real_normed_vector set"
4084 assumes "open s" shows "open((\<lambda>x. a + x) ` s)"
4086 { 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 }
4087 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
4088 ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
4091 lemma open_affinity:
4092 fixes s :: "'a::real_normed_vector set"
4093 assumes "open s" "c \<noteq> 0"
4094 shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
4096 have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
4097 have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
4098 thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
4101 lemma interior_translation:
4102 fixes s :: "'a::real_normed_vector set"
4103 shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
4104 proof (rule set_eqI, rule)
4105 fix x assume "x \<in> interior (op + a ` s)"
4106 then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
4107 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
4108 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
4110 fix x assume "x \<in> op + a ` interior s"
4111 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
4112 { fix z have *:"a + y - z = y + a - z" by auto
4113 assume "z\<in>ball x e"
4114 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
4115 hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"]) }
4116 hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
4117 thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
4120 text {* We can now extend limit compositions to consider the scalar multiplier. *}
4122 lemma continuous_vmul:
4123 fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
4124 shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"
4125 unfolding continuous_def using Lim_vmul[of c] by auto
4127 lemma continuous_mul:
4128 fixes c :: "'a::metric_space \<Rightarrow> real"
4129 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
4130 shows "continuous net c \<Longrightarrow> continuous net f
4131 ==> continuous net (\<lambda>x. c(x) *\<^sub>R f x) "
4132 unfolding continuous_def by (intro tendsto_intros)
4134 lemmas continuous_intros = continuous_add continuous_vmul continuous_cmul
4135 continuous_const continuous_sub continuous_at_id continuous_within_id continuous_mul
4137 lemma continuous_on_vmul:
4138 fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
4139 shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R v)"
4140 unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
4142 lemma continuous_on_mul:
4143 fixes c :: "'a::metric_space \<Rightarrow> real"
4144 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
4145 shows "continuous_on s c \<Longrightarrow> continuous_on s f
4146 ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R f x)"
4147 unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
4149 lemma continuous_on_mul_real:
4150 fixes f :: "'a::metric_space \<Rightarrow> real"
4151 fixes g :: "'a::metric_space \<Rightarrow> real"
4152 shows "continuous_on s f \<Longrightarrow> continuous_on s g
4153 ==> continuous_on s (\<lambda>x. f x * g x)"
4154 using continuous_on_mul[of s f g] unfolding real_scaleR_def .
4156 lemmas continuous_on_intros = continuous_on_add continuous_on_const
4157 continuous_on_id continuous_on_compose continuous_on_cmul continuous_on_neg
4158 continuous_on_sub continuous_on_mul continuous_on_vmul continuous_on_mul_real
4159 uniformly_continuous_on_add uniformly_continuous_on_const
4160 uniformly_continuous_on_id uniformly_continuous_on_compose
4161 uniformly_continuous_on_cmul uniformly_continuous_on_neg
4162 uniformly_continuous_on_sub
4164 text{* And so we have continuity of inverse. *}
4166 lemma continuous_inv:
4167 fixes f :: "'a::metric_space \<Rightarrow> real"
4168 shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
4169 ==> continuous net (inverse o f)"
4170 unfolding continuous_def using Lim_inv by auto
4172 lemma continuous_at_within_inv:
4173 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
4174 assumes "continuous (at a within s) f" "f a \<noteq> 0"
4175 shows "continuous (at a within s) (inverse o f)"
4176 using assms unfolding continuous_within o_def
4177 by (intro tendsto_intros)
4179 lemma continuous_at_inv:
4180 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
4181 shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
4182 ==> continuous (at a) (inverse o f) "
4183 using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
4185 text {* Topological properties of linear functions. *}
4188 assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
4190 interpret f: bounded_linear f by fact
4191 have "(f ---> f 0) (at 0)"
4192 using tendsto_ident_at by (rule f.tendsto)
4193 thus ?thesis unfolding f.zero .
4196 lemma linear_continuous_at:
4197 assumes "bounded_linear f" shows "continuous (at a) f"
4198 unfolding continuous_at using assms
4199 apply (rule bounded_linear.tendsto)
4200 apply (rule tendsto_ident_at)
4203 lemma linear_continuous_within:
4204 shows "bounded_linear f ==> continuous (at x within s) f"
4205 using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
4207 lemma linear_continuous_on:
4208 shows "bounded_linear f ==> continuous_on s f"
4209 using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
4211 text{* Also bilinear functions, in composition form. *}
4213 lemma bilinear_continuous_at_compose:
4214 shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
4215 ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
4216 unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
4218 lemma bilinear_continuous_within_compose:
4219 shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
4220 ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
4221 unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
4223 lemma bilinear_continuous_on_compose:
4224 shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
4225 ==> continuous_on s (\<lambda>x. h (f x) (g x))"
4226 unfolding continuous_on_def
4227 by (fast elim: bounded_bilinear.tendsto)
4229 text {* Preservation of compactness and connectedness under continuous function. *}
4231 lemma compact_continuous_image:
4232 assumes "continuous_on s f" "compact s"
4233 shows "compact(f ` s)"
4235 { fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
4236 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
4237 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
4238 { fix e::real assume "e>0"
4239 then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' l < d \<longrightarrow> dist (f x') (f l) < e" using assms(1)[unfolded continuous_on_iff, THEN bspec[where x=l], OF `l\<in>s`] by auto
4240 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
4241 { 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 }
4242 hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto }
4243 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 }
4244 thus ?thesis unfolding compact_def by auto
4247 lemma connected_continuous_image:
4248 assumes "continuous_on s f" "connected s"
4249 shows "connected(f ` s)"
4251 { fix T assume as: "T \<noteq> {}" "T \<noteq> f ` s" "openin (subtopology euclidean (f ` s)) T" "closedin (subtopology euclidean (f ` s)) T"
4252 have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
4253 using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
4254 using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
4255 using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
4256 hence False using as(1,2)
4257 using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
4258 thus ?thesis unfolding connected_clopen by auto
4261 text{* Continuity implies uniform continuity on a compact domain. *}
4263 lemma compact_uniformly_continuous:
4264 assumes "continuous_on s f" "compact s"
4265 shows "uniformly_continuous_on s f"
4267 { fix x assume x:"x\<in>s"
4268 hence "\<forall>xa. \<exists>y. 0 < xa \<longrightarrow> (y > 0 \<and> (\<forall>x'\<in>s. dist x' x < y \<longrightarrow> dist (f x') (f x) < xa))" using assms(1)[unfolded continuous_on_iff, THEN bspec[where x=x]] by auto
4269 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 }
4270 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
4271 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)"
4272 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
4274 { fix e::real assume "e>0"
4276 { 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 }
4277 hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto
4279 { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto }
4280 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
4282 { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"
4283 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
4284 hence "x\<in>ball z (d z (e / 2))" using `ea>0` unfolding subset_eq by auto
4285 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`
4286 by (auto simp add: dist_commute)
4287 moreover have "y\<in>ball z (d z (e / 2))" using as and `ea>0` and z[unfolded subset_eq]
4288 by (auto simp add: dist_commute)
4289 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`
4290 by (auto simp add: dist_commute)
4291 ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]
4292 by (auto simp add: dist_commute) }
4293 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 }
4294 thus ?thesis unfolding uniformly_continuous_on_def by auto
4297 text{* Continuity of inverse function on compact domain. *}
4299 lemma continuous_on_inverse:
4300 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
4301 (* TODO: can this be generalized more? *)
4302 assumes "continuous_on s f" "compact s" "\<forall>x \<in> s. g (f x) = x"
4303 shows "continuous_on (f ` s) g"
4305 have *:"g ` f ` s = s" using assms(3) by (auto simp add: image_iff)
4306 { fix t assume t:"closedin (subtopology euclidean (g ` f ` s)) t"
4307 then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto
4308 have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]
4309 unfolding T(2) and Int_left_absorb by auto
4310 moreover have "compact (s \<inter> T)"
4311 using assms(2) unfolding compact_eq_bounded_closed
4312 using bounded_subset[of s "s \<inter> T"] and T(1) by auto
4313 ultimately have "closed (f ` t)" using T(1) unfolding T(2)
4314 using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto
4315 moreover have "{x \<in> f ` s. g x \<in> t} = f ` s \<inter> f ` t" using assms(3) unfolding T(2) by auto
4316 ultimately have "closedin (subtopology euclidean (f ` s)) {x \<in> f ` s. g x \<in> t}"
4317 unfolding closedin_closed by auto }
4318 thus ?thesis unfolding continuous_on_closed by auto
4321 text {* A uniformly convergent limit of continuous functions is continuous. *}
4323 lemma norm_triangle_lt:
4324 fixes x y :: "'a::real_normed_vector"
4325 shows "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
4326 by (rule le_less_trans [OF norm_triangle_ineq])
4328 lemma continuous_uniform_limit:
4329 fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::real_normed_vector"
4330 assumes "\<not> (trivial_limit net)" "eventually (\<lambda>n. continuous_on s (f n)) net"
4331 "\<forall>e>0. eventually (\<lambda>n. \<forall>x \<in> s. norm(f n x - g x) < e) net"
4332 shows "continuous_on s g"
4334 { fix x and e::real assume "x\<in>s" "e>0"
4335 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
4336 then obtain n where n:"\<forall>xa\<in>s. norm (f n xa - g xa) < e / 3" "continuous_on s (f n)"
4337 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
4338 have "e / 3 > 0" using `e>0` by auto
4339 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"
4340 using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
4341 { fix y assume "y\<in>s" "dist y x < d"
4342 hence "dist (f n y) (f n x) < e / 3" using d[THEN bspec[where x=y]] by auto
4343 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"]
4344 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
4345 hence "dist (g y) (g x) < e" unfolding dist_norm using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
4346 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) }
4347 hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using `d>0` by auto }
4348 thus ?thesis unfolding continuous_on_iff by auto
4351 subsection{* Topological stuff lifted from and dropped to R *}
4355 fixes s :: "real set" shows
4356 "open s \<longleftrightarrow>
4357 (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
4358 unfolding open_dist dist_norm by simp
4360 lemma islimpt_approachable_real:
4361 fixes s :: "real set"
4362 shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
4363 unfolding islimpt_approachable dist_norm by simp
4366 fixes s :: "real set"
4367 shows "closed s \<longleftrightarrow>
4368 (\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
4370 unfolding closed_limpt islimpt_approachable dist_norm by simp
4372 lemma continuous_at_real_range:
4373 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4374 shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
4375 \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
4376 unfolding continuous_at unfolding Lim_at
4377 unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
4378 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
4379 apply(erule_tac x=e in allE) by auto
4381 lemma continuous_on_real_range:
4382 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4383 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))"
4384 unfolding continuous_on_iff dist_norm by simp
4386 lemma continuous_at_norm: "continuous (at x) norm"
4387 unfolding continuous_at by (intro tendsto_intros)
4389 lemma continuous_on_norm: "continuous_on s norm"
4390 unfolding continuous_on by (intro ballI tendsto_intros)
4392 lemma continuous_at_infnorm: "continuous (at x) infnorm"
4393 unfolding continuous_at Lim_at o_def unfolding dist_norm
4394 apply auto apply (rule_tac x=e in exI) apply auto
4395 using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
4397 text{* Hence some handy theorems on distance, diameter etc. of/from a set. *}
4399 lemma compact_attains_sup:
4400 fixes s :: "real set"
4401 assumes "compact s" "s \<noteq> {}"
4402 shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
4404 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4405 { 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"
4406 have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto
4407 moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
4408 ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using `e>0` by auto }
4409 thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]
4410 apply(rule_tac x="Sup s" in bexI) by auto
4414 fixes S :: "real set"
4415 shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
4416 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def)
4418 lemma compact_attains_inf:
4419 fixes s :: "real set"
4420 assumes "compact s" "s \<noteq> {}" shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
4422 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4423 { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s" "Inf s \<notin> s" "0 < e"
4424 "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"
4425 have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto
4427 { fix x assume "x \<in> s"
4428 hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto
4429 have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
4430 hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto
4431 ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using `e>0` by auto }
4432 thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]
4433 apply(rule_tac x="Inf s" in bexI) by auto
4436 lemma continuous_attains_sup:
4437 fixes f :: "'a::metric_space \<Rightarrow> real"
4438 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4439 ==> (\<exists>x \<in> s. \<forall>y \<in> s. f y \<le> f x)"
4440 using compact_attains_sup[of "f ` s"]
4441 using compact_continuous_image[of s f] by auto
4443 lemma continuous_attains_inf:
4444 fixes f :: "'a::metric_space \<Rightarrow> real"
4445 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4446 \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
4447 using compact_attains_inf[of "f ` s"]
4448 using compact_continuous_image[of s f] by auto
4450 lemma distance_attains_sup:
4451 assumes "compact s" "s \<noteq> {}"
4452 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
4453 proof (rule continuous_attains_sup [OF assms])
4454 { fix x assume "x\<in>s"
4455 have "(dist a ---> dist a x) (at x within s)"
4456 by (intro tendsto_dist tendsto_const Lim_at_within Lim_ident_at)
4458 thus "continuous_on s (dist a)"
4459 unfolding continuous_on ..
4462 text{* For *minimal* distance, we only need closure, not compactness. *}
4464 lemma distance_attains_inf:
4465 fixes a :: "'a::heine_borel"
4466 assumes "closed s" "s \<noteq> {}"
4467 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
4469 from assms(2) obtain b where "b\<in>s" by auto
4470 let ?B = "cball a (dist b a) \<inter> s"
4471 have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
4472 hence "?B \<noteq> {}" by auto
4474 { fix x assume "x\<in>?B"
4475 fix e::real assume "e>0"
4476 { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
4477 from as have "\<bar>dist a x' - dist a x\<bar> < e"
4478 unfolding abs_less_iff minus_diff_eq
4479 using dist_triangle2 [of a x' x]
4480 using dist_triangle [of a x x']
4483 hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
4486 hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
4487 unfolding continuous_on Lim_within dist_norm real_norm_def
4489 moreover have "compact ?B"
4490 using compact_cball[of a "dist b a"]
4491 unfolding compact_eq_bounded_closed
4492 using bounded_Int and closed_Int and assms(1) by auto
4493 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"
4494 using continuous_attains_inf[of ?B "dist a"] by fastsimp
4495 thus ?thesis by fastsimp
4498 subsection {* Pasted sets *}
4500 lemma bounded_Times:
4501 assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
4503 obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
4504 using assms [unfolded bounded_def] by auto
4505 then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
4506 by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
4507 thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
4510 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
4513 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
4514 unfolding compact_def
4516 apply (drule_tac x="fst \<circ> f" in spec)
4517 apply (drule mp, simp add: mem_Times_iff)
4518 apply (clarify, rename_tac l1 r1)
4519 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
4520 apply (drule mp, simp add: mem_Times_iff)
4521 apply (clarify, rename_tac l2 r2)
4522 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
4523 apply (rule_tac x="r1 \<circ> r2" in exI)
4524 apply (rule conjI, simp add: subseq_def)
4525 apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
4526 apply (drule (1) tendsto_Pair) back
4527 apply (simp add: o_def)
4530 text{* Hence some useful properties follow quite easily. *}
4532 lemma compact_scaling:
4533 fixes s :: "'a::real_normed_vector set"
4534 assumes "compact s" shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
4536 let ?f = "\<lambda>x. scaleR c x"
4537 have *:"bounded_linear ?f" by (rule scaleR.bounded_linear_right)
4538 show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
4539 using linear_continuous_at[OF *] assms by auto
4542 lemma compact_negations:
4543 fixes s :: "'a::real_normed_vector set"
4544 assumes "compact s" shows "compact ((\<lambda>x. -x) ` s)"
4545 using compact_scaling [OF assms, of "- 1"] by auto
4548 fixes s t :: "'a::real_normed_vector set"
4549 assumes "compact s" "compact t" shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
4551 have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
4552 apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
4553 have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
4554 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
4555 thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
4558 lemma compact_differences:
4559 fixes s t :: "'a::real_normed_vector set"
4560 assumes "compact s" "compact t" shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
4562 have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
4563 apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4564 thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
4567 lemma compact_translation:
4568 fixes s :: "'a::real_normed_vector set"
4569 assumes "compact s" shows "compact ((\<lambda>x. a + x) ` s)"
4571 have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
4572 thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
4575 lemma compact_affinity:
4576 fixes s :: "'a::real_normed_vector set"
4577 assumes "compact s" shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
4579 have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
4580 thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
4583 text{* Hence we get the following. *}
4585 lemma compact_sup_maxdistance:
4586 fixes s :: "'a::real_normed_vector set"
4587 assumes "compact s" "s \<noteq> {}"
4588 shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
4590 have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
4591 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"
4592 using compact_differences[OF assms(1) assms(1)]
4593 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
4594 from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
4595 thus ?thesis using x(2)[unfolded `x = a - b`] by blast
4598 text{* We can state this in terms of diameter of a set. *}
4600 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"
4601 (* TODO: generalize to class metric_space *)
4603 lemma diameter_bounded:
4605 shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4606 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"
4608 let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"
4609 obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto
4610 { fix x y assume "x \<in> s" "y \<in> s"
4611 hence "norm (x - y) \<le> 2 * a" using norm_triangle_ineq[of x "-y", unfolded norm_minus_cancel] a[THEN bspec[where x=x]] a[THEN bspec[where x=y]] by (auto simp add: field_simps) }
4613 { fix x y assume "x\<in>s" "y\<in>s" hence "s \<noteq> {}" by auto
4614 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`
4615 by simp (blast intro!: Sup_upper *) }
4617 { fix d::real assume "d>0" "d < diameter s"
4618 hence "s\<noteq>{}" unfolding diameter_def by auto
4619 have "\<exists>d' \<in> ?D. d' > d"
4621 assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"
4622 hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE)
4623 thus False using `d < diameter s` `s\<noteq>{}`
4624 apply (auto simp add: diameter_def)
4625 apply (drule Sup_real_iff [THEN [2] rev_iffD2])
4629 hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto }
4630 ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4631 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto
4634 lemma diameter_bounded_bound:
4635 "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"
4636 using diameter_bounded by blast
4638 lemma diameter_compact_attained:
4639 fixes s :: "'a::real_normed_vector set"
4640 assumes "compact s" "s \<noteq> {}"
4641 shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
4643 have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
4644 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
4645 hence "diameter s \<le> norm (x - y)"
4646 unfolding diameter_def by clarsimp (rule Sup_least, fast+)
4648 by (metis b diameter_bounded_bound order_antisym xys)
4651 text{* Related results with closure as the conclusion. *}
4653 lemma closed_scaling:
4654 fixes s :: "'a::real_normed_vector set"
4655 assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
4657 case True thus ?thesis by auto
4662 have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
4663 case True thus ?thesis apply auto unfolding * by auto
4666 { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s" "(x ---> l) sequentially"
4667 { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
4668 using as(1)[THEN spec[where x=n]]
4669 using `c\<noteq>0` by auto
4672 { fix e::real assume "e>0"
4673 hence "0 < e *\<bar>c\<bar>" using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
4674 then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
4675 using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto
4676 hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
4677 unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
4678 using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto }
4679 hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto
4680 ultimately have "l \<in> scaleR c ` s"
4681 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"]]
4682 unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto }
4683 thus ?thesis unfolding closed_sequential_limits by fast
4687 lemma closed_negations:
4688 fixes s :: "'a::real_normed_vector set"
4689 assumes "closed s" shows "closed ((\<lambda>x. -x) ` s)"
4690 using closed_scaling[OF assms, of "- 1"] by simp
4692 lemma compact_closed_sums:
4693 fixes s :: "'a::real_normed_vector set"
4694 assumes "compact s" "closed t" shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4696 let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
4697 { fix x l assume as:"\<forall>n. x n \<in> ?S" "(x ---> l) sequentially"
4698 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"
4699 using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
4700 obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
4701 using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
4702 have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
4703 using Lim_sub[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
4704 hence "l - l' \<in> t"
4705 using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
4707 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
4709 thus ?thesis unfolding closed_sequential_limits by fast
4712 lemma closed_compact_sums:
4713 fixes s t :: "'a::real_normed_vector set"
4714 assumes "closed s" "compact t"
4715 shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4717 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
4718 apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
4719 thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
4722 lemma compact_closed_differences:
4723 fixes s t :: "'a::real_normed_vector set"
4724 assumes "compact s" "closed t"
4725 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4727 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4728 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4729 thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
4732 lemma closed_compact_differences:
4733 fixes s t :: "'a::real_normed_vector set"
4734 assumes "closed s" "compact t"
4735 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4737 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4738 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4739 thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
4742 lemma closed_translation:
4743 fixes a :: "'a::real_normed_vector"
4744 assumes "closed s" shows "closed ((\<lambda>x. a + x) ` s)"
4746 have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
4747 thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
4750 lemma translation_Compl:
4751 fixes a :: "'a::ab_group_add"
4752 shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
4753 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto
4755 lemma translation_UNIV:
4756 fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
4757 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
4759 lemma translation_diff:
4760 fixes a :: "'a::ab_group_add"
4761 shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
4764 lemma closure_translation:
4765 fixes a :: "'a::real_normed_vector"
4766 shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
4768 have *:"op + a ` (- s) = - op + a ` s"
4769 apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
4770 show ?thesis unfolding closure_interior translation_Compl
4771 using interior_translation[of a "- s"] unfolding * by auto
4774 lemma frontier_translation:
4775 fixes a :: "'a::real_normed_vector"
4776 shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
4777 unfolding frontier_def translation_diff interior_translation closure_translation by auto
4779 subsection{* Separation between points and sets. *}
4781 lemma separate_point_closed:
4782 fixes s :: "'a::heine_borel set"
4783 shows "closed s \<Longrightarrow> a \<notin> s ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
4784 proof(cases "s = {}")
4786 thus ?thesis by(auto intro!: exI[where x=1])
4789 assume "closed s" "a \<notin> s"
4790 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
4791 with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
4794 lemma separate_compact_closed:
4795 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4796 (* TODO: does this generalize to heine_borel? *)
4797 assumes "compact s" and "closed t" and "s \<inter> t = {}"
4798 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4800 have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto
4801 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"
4802 using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto
4803 { fix x y assume "x\<in>s" "y\<in>t"
4804 hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto
4805 hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute
4806 by (auto simp add: dist_commute)
4807 hence "d \<le> dist x y" unfolding dist_norm by auto }
4808 thus ?thesis using `d>0` by auto
4811 lemma separate_closed_compact:
4812 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4813 assumes "closed s" and "compact t" and "s \<inter> t = {}"
4814 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4816 have *:"t \<inter> s = {}" using assms(3) by auto
4817 show ?thesis using separate_compact_closed[OF assms(2,1) *]
4818 apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
4819 by (auto simp add: dist_commute)
4822 subsection {* Intervals *}
4824 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows
4825 "{a <..< b} = {x::'a. \<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i}" and
4826 "{a .. b} = {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i}"
4827 by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
4829 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows
4830 "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i)"
4831 "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i)"
4832 using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
4834 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows
4835 "({a <..< b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i))" (is ?th1) and
4836 "({a .. b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i < a$$i))" (is ?th2)
4838 { fix i x assume i:"i<DIM('a)" and as:"b$$i \<le> a$$i" and x:"x\<in>{a <..< b}"
4839 hence "a $$ i < x $$ i \<and> x $$ i < b $$ i" unfolding mem_interval by auto
4840 hence "a$$i < b$$i" by auto
4841 hence False using as by auto }
4843 { assume as:"\<forall>i<DIM('a). \<not> (b$$i \<le> a$$i)"
4844 let ?x = "(1/2) *\<^sub>R (a + b)"
4845 { fix i assume i:"i<DIM('a)"
4846 have "a$$i < b$$i" using as[THEN spec[where x=i]] using i by auto
4847 hence "a$$i < ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i < b$$i"
4848 unfolding euclidean_simps by auto }
4849 hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto }
4850 ultimately show ?th1 by blast
4852 { fix i x assume i:"i<DIM('a)" and as:"b$$i < a$$i" and x:"x\<in>{a .. b}"
4853 hence "a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i" unfolding mem_interval by auto
4854 hence "a$$i \<le> b$$i" by auto
4855 hence False using as by auto }
4857 { assume as:"\<forall>i<DIM('a). \<not> (b$$i < a$$i)"
4858 let ?x = "(1/2) *\<^sub>R (a + b)"
4859 { fix i assume i:"i<DIM('a)"
4860 have "a$$i \<le> b$$i" using as[THEN spec[where x=i]] by auto
4861 hence "a$$i \<le> ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i \<le> b$$i"
4862 unfolding euclidean_simps by auto }
4863 hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto }
4864 ultimately show ?th2 by blast
4867 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows
4868 "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> b$$i)" and
4869 "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"
4870 unfolding interval_eq_empty[of a b] by fastsimp+
4872 lemma interval_sing: fixes a :: "'a::ordered_euclidean_space" shows
4873 "{a .. a} = {a}" "{a<..<a} = {}"
4874 apply(auto simp add: set_eq_iff euclidean_eq[where 'a='a] eucl_less[where 'a='a] eucl_le[where 'a='a])
4875 apply (simp add: order_eq_iff) apply(rule_tac x=0 in exI) by (auto simp add: not_less less_imp_le)
4877 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows
4878 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
4879 "(\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
4880 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
4881 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
4882 unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
4883 by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
4885 lemma interval_open_subset_closed: fixes a :: "'a::ordered_euclidean_space" shows
4886 "{a<..<b} \<subseteq> {a .. b}"
4887 proof(simp add: subset_eq, rule)
4889 assume x:"x \<in>{a<..<b}"
4890 { fix i assume "i<DIM('a)"
4891 hence "a $$ i \<le> x $$ i"
4892 using x order_less_imp_le[of "a$$i" "x$$i"]
4893 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4896 { fix i assume "i<DIM('a)"
4897 hence "x $$ i \<le> b $$ i"
4898 using x order_less_imp_le[of "x$$i" "b$$i"]
4899 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4902 show "a \<le> x \<and> x \<le> b"
4903 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4906 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows
4907 "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th1) and
4908 "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i)" (is ?th2) and
4909 "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th3) and
4910 "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th4)
4912 show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
4913 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)
4914 { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i<DIM('a). c$$i < d$$i"
4915 hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
4916 fix i assume i:"i<DIM('a)"
4917 (** TODO combine the following two parts as done in the HOL_light version. **)
4918 { let ?x = "(\<chi>\<chi> j. (if j=i then ((min (a$$j) (d$$j))+c$$j)/2 else (c$$j+d$$j)/2))::'a"
4919 assume as2: "a$$i > c$$i"
4920 { fix j assume j:"j<DIM('a)"
4921 hence "c $$ j < ?x $$ j \<and> ?x $$ j < d $$ j"
4922 apply(cases "j=i") using as(2)[THEN spec[where x=j]] i
4923 by (auto simp add: as2) }
4924 hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto
4926 have "?x\<notin>{a .. b}"
4927 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4928 using as(2)[THEN spec[where x=i]] and as2 i
4930 ultimately have False using as by auto }
4931 hence "a$$i \<le> c$$i" by(rule ccontr)auto
4933 { let ?x = "(\<chi>\<chi> j. (if j=i then ((max (b$$j) (c$$j))+d$$j)/2 else (c$$j+d$$j)/2))::'a"
4934 assume as2: "b$$i < d$$i"
4935 { fix j assume "j<DIM('a)"
4936 hence "d $$ j > ?x $$ j \<and> ?x $$ j > c $$ j"
4937 apply(cases "j=i") using as(2)[THEN spec[where x=j]]
4938 by (auto simp add: as2) }
4939 hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
4941 have "?x\<notin>{a .. b}"
4942 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4943 using as(2)[THEN spec[where x=i]] and as2 using i
4945 ultimately have False using as by auto }
4946 hence "b$$i \<ge> d$$i" by(rule ccontr)auto
4948 have "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" by auto
4950 show ?th3 unfolding subset_eq and Ball_def and mem_interval
4951 apply(rule,rule,rule,rule) apply(rule part1) unfolding subset_eq and Ball_def and mem_interval
4952 prefer 4 apply auto by(erule_tac x=i in allE,erule_tac x=i in allE,fastsimp)+
4953 { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i<DIM('a). c$$i < d$$i"
4954 fix i assume i:"i<DIM('a)"
4955 from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
4956 hence "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" using part1 and as(2) using i by auto } note * = this
4957 show ?th4 unfolding subset_eq and Ball_def and mem_interval
4958 apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4
4959 apply auto by(erule_tac x=i in allE, simp)+
4962 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows
4963 "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i < c$$i \<or> b$$i < c$$i \<or> d$$i < a$$i))" (is ?th1) and
4964 "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th2) and
4965 "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i < c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th3) and
4966 "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th4)
4968 let ?z = "(\<chi>\<chi> i. ((max (a$$i) (c$$i)) + (min (b$$i) (d$$i))) / 2)::'a"
4969 note * = set_eq_iff Int_iff empty_iff mem_interval all_conj_distrib[THEN sym] eq_False
4970 show ?th1 unfolding * apply safe apply(erule_tac x="?z" in allE)
4971 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4972 show ?th2 unfolding * apply safe apply(erule_tac x="?z" in allE)
4973 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4974 show ?th3 unfolding * apply safe apply(erule_tac x="?z" in allE)
4975 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4976 show ?th4 unfolding * apply safe apply(erule_tac x="?z" in allE)
4977 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4980 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows
4981 "{a .. b} \<inter> {c .. d} = {(\<chi>\<chi> i. max (a$$i) (c$$i)) .. (\<chi>\<chi> i. min (b$$i) (d$$i))}"
4982 unfolding set_eq_iff and Int_iff and mem_interval
4985 (* Moved interval_open_subset_closed a bit upwards *)
4987 lemma open_interval_lemma: fixes x :: "real" shows
4988 "a < x \<Longrightarrow> x < b ==> (\<exists>d>0. \<forall>x'. abs(x' - x) < d --> a < x' \<and> x' < b)"
4989 by(rule_tac x="min (x - a) (b - x)" in exI, auto)
4991 lemma open_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
4993 { fix x assume x:"x\<in>{a<..<b}"
4994 { fix i assume "i<DIM('a)"
4995 hence "\<exists>d>0. \<forall>x'. abs (x' - (x$$i)) < d \<longrightarrow> a$$i < x' \<and> x' < b$$i"
4996 using x[unfolded mem_interval, THEN spec[where x=i]]
4997 using open_interval_lemma[of "a$$i" "x$$i" "b$$i"] by auto }
4998 hence "\<forall>i\<in>{..<DIM('a)}. \<exists>d>0. \<forall>x'. abs (x' - (x$$i)) < d \<longrightarrow> a$$i < x' \<and> x' < b$$i" by auto
4999 from bchoice[OF this] guess d .. note d=this
5000 let ?d = "Min (d ` {..<DIM('a)})"
5001 have **:"finite (d ` {..<DIM('a)})" "d ` {..<DIM('a)} \<noteq> {}" by auto
5002 have "?d>0" using Min_gr_iff[OF **] using d by auto
5004 { fix x' assume as:"dist x' x < ?d"
5005 { fix i assume i:"i<DIM('a)"
5006 hence "\<bar>x'$$i - x $$ i\<bar> < d i"
5007 using norm_bound_component_lt[OF as[unfolded dist_norm], of i]
5008 unfolding euclidean_simps Min_gr_iff[OF **] by auto
5009 hence "a $$ i < x' $$ i" "x' $$ i < b $$ i" using i and d[THEN bspec[where x=i]] by auto }
5010 hence "a < x' \<and> x' < b" apply(subst(2) eucl_less,subst(1) eucl_less) by auto }
5011 ultimately have "\<exists>e>0. \<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a<..<b}" by auto
5013 thus ?thesis unfolding open_dist using open_interval_lemma by auto
5016 lemma closed_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
5018 { fix x i assume i:"i<DIM('a)"
5019 assume as:"\<forall>e>0. \<exists>x'\<in>{a..b}. x' \<noteq> x \<and> dist x' x < e"(* and xab:"a$$i > x$$i \<or> b$$i < x$$i"*)
5020 { assume xa:"a$$i > x$$i"
5021 with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < a$$i - x$$i" by(erule_tac x="a$$i - x$$i" in allE)auto
5022 hence False unfolding mem_interval and dist_norm
5023 using component_le_norm[of "y-x" i, unfolded euclidean_simps] and xa using i
5024 by(auto elim!: allE[where x=i])
5025 } hence "a$$i \<le> x$$i" by(rule ccontr)auto
5027 { assume xb:"b$$i < x$$i"
5028 with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < x$$i - b$$i"
5029 by(erule_tac x="x$$i - b$$i" in allE)auto
5030 hence False unfolding mem_interval and dist_norm
5031 using component_le_norm[of "y-x" i, unfolded euclidean_simps] and xb using i
5032 by(auto elim!: allE[where x=i])
5033 } hence "x$$i \<le> b$$i" by(rule ccontr)auto
5035 have "a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i" by auto }
5036 thus ?thesis unfolding closed_limpt islimpt_approachable mem_interval by auto
5039 lemma interior_closed_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows
5040 "interior {a .. b} = {a<..<b}" (is "?L = ?R")
5041 proof(rule subset_antisym)
5042 show "?R \<subseteq> ?L" using interior_maximal[OF interval_open_subset_closed open_interval] by auto
5044 { fix x assume "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> {a..b}"
5045 then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" by auto
5046 then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto
5047 { fix i assume i:"i<DIM('a)"
5048 have "dist (x - (e / 2) *\<^sub>R basis i) x < e"
5049 "dist (x + (e / 2) *\<^sub>R basis i) x < e"
5050 unfolding dist_norm apply auto
5051 unfolding norm_minus_cancel using norm_basis and `e>0` by auto
5052 hence "a $$ i \<le> (x - (e / 2) *\<^sub>R basis i) $$ i"
5053 "(x + (e / 2) *\<^sub>R basis i) $$ i \<le> b $$ i"
5054 using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]
5055 and e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]
5056 unfolding mem_interval by (auto elim!: allE[where x=i])
5057 hence "a $$ i < x $$ i" and "x $$ i < b $$ i" unfolding euclidean_simps
5058 unfolding basis_component using `e>0` i by auto }
5059 hence "x \<in> {a<..<b}" unfolding mem_interval by auto }
5060 thus "?L \<subseteq> ?R" unfolding interior_def and subset_eq by auto
5063 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"
5065 let ?b = "\<Sum>i<DIM('a). \<bar>a$$i\<bar> + \<bar>b$$i\<bar>"
5066 { fix x::"'a" assume x:"\<forall>i<DIM('a). a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i"
5067 { fix i assume "i<DIM('a)"
5068 hence "\<bar>x$$i\<bar> \<le> \<bar>a$$i\<bar> + \<bar>b$$i\<bar>" using x[THEN spec[where x=i]] by auto }
5069 hence "(\<Sum>i<DIM('a). \<bar>x $$ i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto
5070 hence "norm x \<le> ?b" using norm_le_l1[of x] by auto }
5071 thus ?thesis unfolding interval and bounded_iff by auto
5074 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows
5075 "bounded {a .. b} \<and> bounded {a<..<b}"
5076 using bounded_closed_interval[of a b]
5077 using interval_open_subset_closed[of a b]
5078 using bounded_subset[of "{a..b}" "{a<..<b}"]
5081 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows
5082 "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
5083 using bounded_interval[of a b] by auto
5085 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"
5086 using bounded_closed_imp_compact[of "{a..b}"] using bounded_interval[of a b]
5089 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"
5090 assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
5092 { fix i assume "i<DIM('a)"
5093 hence "a $$ i < ((1 / 2) *\<^sub>R (a + b)) $$ i \<and> ((1 / 2) *\<^sub>R (a + b)) $$ i < b $$ i"
5094 using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
5095 unfolding euclidean_simps by auto }
5096 thus ?thesis unfolding mem_interval by auto
5099 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"
5100 assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
5101 shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
5103 { fix i assume i:"i<DIM('a)"
5104 have "a $$ i = e * a$$i + (1 - e) * a$$i" unfolding left_diff_distrib by simp
5105 also have "\<dots> < e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
5106 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
5107 using x unfolding mem_interval using i apply simp
5108 using y unfolding mem_interval using i apply simp
5110 finally have "a $$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i" unfolding euclidean_simps by auto
5112 have "b $$ i = e * b$$i + (1 - e) * b$$i" unfolding left_diff_distrib by simp
5113 also have "\<dots> > e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
5114 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
5115 using x unfolding mem_interval using i apply simp
5116 using y unfolding mem_interval using i apply simp
5118 finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i < b $$ i" unfolding euclidean_simps by auto
5119 } ultimately have "a $$ 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 }
5120 thus ?thesis unfolding mem_interval by auto
5123 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"
5124 assumes "{a<..<b} \<noteq> {}"
5125 shows "closure {a<..<b} = {a .. b}"
5127 have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto
5128 let ?c = "(1 / 2) *\<^sub>R (a + b)"
5129 { fix x assume as:"x \<in> {a .. b}"
5130 def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
5131 { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
5132 have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
5133 have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
5134 x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
5135 by (auto simp add: algebra_simps)
5136 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
5137 hence False using fn unfolding f_def using xc by auto }
5139 { assume "\<not> (f ---> x) sequentially"
5140 { fix e::real assume "e>0"
5141 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
5142 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
5143 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)
5144 hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto }
5145 hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
5146 unfolding Lim_sequentially by(auto simp add: dist_norm)
5147 hence "(f ---> x) sequentially" unfolding f_def
5148 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]
5149 using Lim_vmul[of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto }
5150 ultimately have "x \<in> closure {a<..<b}"
5151 using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto }
5152 thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
5155 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"
5156 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a<..<a}"
5158 obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
5159 def a \<equiv> "(\<chi>\<chi> i. b+1)::'a"
5160 { fix x assume "x\<in>s"
5161 fix i assume i:"i<DIM('a)"
5162 hence "(-a)$$i < x$$i" and "x$$i < a$$i" using b[THEN bspec[where x=x], OF `x\<in>s`]
5163 and component_le_norm[of x i] unfolding euclidean_simps and a_def by auto }
5164 thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
5167 lemma bounded_subset_open_interval:
5168 fixes s :: "('a::ordered_euclidean_space) set"
5169 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
5170 by (auto dest!: bounded_subset_open_interval_symmetric)
5172 lemma bounded_subset_closed_interval_symmetric:
5173 fixes s :: "('a::ordered_euclidean_space) set"
5174 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
5176 obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
5177 thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
5180 lemma bounded_subset_closed_interval:
5181 fixes s :: "('a::ordered_euclidean_space) set"
5182 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
5183 using bounded_subset_closed_interval_symmetric[of s] by auto
5185 lemma frontier_closed_interval:
5186 fixes a b :: "'a::ordered_euclidean_space"
5187 shows "frontier {a .. b} = {a .. b} - {a<..<b}"
5188 unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
5190 lemma frontier_open_interval:
5191 fixes a b :: "'a::ordered_euclidean_space"
5192 shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
5193 proof(cases "{a<..<b} = {}")
5194 case True thus ?thesis using frontier_empty by auto
5196 case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
5199 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"
5200 assumes "{c<..<d} \<noteq> {}" shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
5201 unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
5204 (* Some stuff for half-infinite intervals too; FIXME: notation? *)
5206 lemma closed_interval_left: fixes b::"'a::euclidean_space"
5207 shows "closed {x::'a. \<forall>i<DIM('a). x$$i \<le> b$$i}"
5209 { fix i assume i:"i<DIM('a)"
5210 fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). x $$ i \<le> b $$ i}. x' \<noteq> x \<and> dist x' x < e"
5211 { assume "x$$i > b$$i"
5212 then obtain y where "y $$ i \<le> b $$ i" "y \<noteq> x" "dist y x < x$$i - b$$i"
5213 using x[THEN spec[where x="x$$i - b$$i"]] using i by auto
5214 hence False using component_le_norm[of "y - x" i] unfolding dist_norm euclidean_simps using i
5216 hence "x$$i \<le> b$$i" by(rule ccontr)auto }
5217 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
5220 lemma closed_interval_right: fixes a::"'a::euclidean_space"
5221 shows "closed {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i}"
5223 { fix i assume i:"i<DIM('a)"
5224 fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). a $$ i \<le> x $$ i}. x' \<noteq> x \<and> dist x' x < e"
5225 { assume "a$$i > x$$i"
5226 then obtain y where "a $$ i \<le> y $$ i" "y \<noteq> x" "dist y x < a$$i - x$$i"
5227 using x[THEN spec[where x="a$$i - x$$i"]] i by auto
5228 hence False using component_le_norm[of "y - x" i] unfolding dist_norm and euclidean_simps by auto }
5229 hence "a$$i \<le> x$$i" by(rule ccontr)auto }
5230 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
5233 text {* Intervals in general, including infinite and mixtures of open and closed. *}
5235 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
5236 (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i<DIM('a). ((a$$i \<le> x$$i \<and> x$$i \<le> b$$i) \<or> (b$$i \<le> x$$i \<and> x$$i \<le> a$$i))) \<longrightarrow> x \<in> s)"
5238 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)
5239 "is_interval {a<..<b}" (is ?th2) proof -
5240 have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
5241 show ?th1 ?th2 unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
5242 by(meson order_trans le_less_trans less_le_trans *)+ qed
5244 lemma is_interval_empty:
5246 unfolding is_interval_def
5249 lemma is_interval_univ:
5251 unfolding is_interval_def
5254 subsection{* Closure of halfspaces and hyperplanes. *}
5257 assumes "(f ---> l) net" shows "((\<lambda>y. inner a (f y)) ---> inner a l) net"
5258 by (intro tendsto_intros assms)
5260 lemma continuous_at_inner: "continuous (at x) (inner a)"
5261 unfolding continuous_at by (intro tendsto_intros)
5263 lemma continuous_at_euclidean_component[intro!, simp]: "continuous (at x) (\<lambda>x. x $$ i)"
5264 unfolding euclidean_component_def by (rule continuous_at_inner)
5266 lemma continuous_on_inner:
5267 fixes s :: "'a::real_inner set"
5268 shows "continuous_on s (inner a)"
5269 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
5271 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
5273 have "\<forall>x. continuous (at x) (inner a)"
5274 unfolding continuous_at by (rule allI) (intro tendsto_intros)
5275 hence "closed (inner a -` {..b})"
5276 using closed_real_atMost by (rule continuous_closed_vimage)
5277 moreover have "{x. inner a x \<le> b} = inner a -` {..b}" by auto
5278 ultimately show ?thesis by simp
5281 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
5282 using closed_halfspace_le[of "-a" "-b"] unfolding inner_minus_left by auto
5284 lemma closed_hyperplane: "closed {x. inner a x = b}"
5286 have "{x. inner a x = b} = {x. inner a x \<ge> b} \<inter> {x. inner a x \<le> b}" by auto
5287 thus ?thesis using closed_halfspace_le[of a b] and closed_halfspace_ge[of b a] using closed_Int by auto
5290 lemma closed_halfspace_component_le:
5291 shows "closed {x::'a::euclidean_space. x$$i \<le> a}"
5292 using closed_halfspace_le[of "(basis i)::'a" a] unfolding euclidean_component_def .
5294 lemma closed_halfspace_component_ge:
5295 shows "closed {x::'a::euclidean_space. x$$i \<ge> a}"
5296 using closed_halfspace_ge[of a "(basis i)::'a"] unfolding euclidean_component_def .
5298 text{* Openness of halfspaces. *}
5300 lemma open_halfspace_lt: "open {x. inner a x < b}"
5302 have "- {x. b \<le> inner a x} = {x. inner a x < b}" by auto
5303 thus ?thesis using closed_halfspace_ge[unfolded closed_def, of b a] by auto
5306 lemma open_halfspace_gt: "open {x. inner a x > b}"
5308 have "- {x. b \<ge> inner a x} = {x. inner a x > b}" by auto
5309 thus ?thesis using closed_halfspace_le[unfolded closed_def, of a b] by auto
5312 lemma open_halfspace_component_lt:
5313 shows "open {x::'a::euclidean_space. x$$i < a}"
5314 using open_halfspace_lt[of "(basis i)::'a" a] unfolding euclidean_component_def .
5316 lemma open_halfspace_component_gt:
5317 shows "open {x::'a::euclidean_space. x$$i > a}"
5318 using open_halfspace_gt[of a "(basis i)::'a"] unfolding euclidean_component_def .
5320 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}
5322 lemma eucl_lessThan_eq_halfspaces:
5323 fixes a :: "'a\<Colon>ordered_euclidean_space"
5324 shows "{..<a} = (\<Inter>i<DIM('a). {x. x $$ i < a $$ i})"
5325 by (auto simp: eucl_less[where 'a='a])
5327 lemma eucl_greaterThan_eq_halfspaces:
5328 fixes a :: "'a\<Colon>ordered_euclidean_space"
5329 shows "{a<..} = (\<Inter>i<DIM('a). {x. a $$ i < x $$ i})"
5330 by (auto simp: eucl_less[where 'a='a])
5332 lemma eucl_atMost_eq_halfspaces:
5333 fixes a :: "'a\<Colon>ordered_euclidean_space"
5334 shows "{.. a} = (\<Inter>i<DIM('a). {x. x $$ i \<le> a $$ i})"
5335 by (auto simp: eucl_le[where 'a='a])
5337 lemma eucl_atLeast_eq_halfspaces:
5338 fixes a :: "'a\<Colon>ordered_euclidean_space"
5339 shows "{a ..} = (\<Inter>i<DIM('a). {x. a $$ i \<le> x $$ i})"
5340 by (auto simp: eucl_le[where 'a='a])
5342 lemma open_eucl_lessThan[simp, intro]:
5343 fixes a :: "'a\<Colon>ordered_euclidean_space"
5344 shows "open {..< a}"
5345 by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)
5347 lemma open_eucl_greaterThan[simp, intro]:
5348 fixes a :: "'a\<Colon>ordered_euclidean_space"
5349 shows "open {a <..}"
5350 by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)
5352 lemma closed_eucl_atMost[simp, intro]:
5353 fixes a :: "'a\<Colon>ordered_euclidean_space"
5354 shows "closed {.. a}"
5355 unfolding eucl_atMost_eq_halfspaces
5356 proof (safe intro!: closed_INT)
5358 have "- {x::'a. x $$ i \<le> a $$ i} = {x. a $$ i < x $$ i}" by auto
5359 then show "closed {x::'a. x $$ i \<le> a $$ i}"
5360 by (simp add: closed_def open_halfspace_component_gt)
5363 lemma closed_eucl_atLeast[simp, intro]:
5364 fixes a :: "'a\<Colon>ordered_euclidean_space"
5365 shows "closed {a ..}"
5366 unfolding eucl_atLeast_eq_halfspaces
5367 proof (safe intro!: closed_INT)
5369 have "- {x::'a. a $$ i \<le> x $$ i} = {x. x $$ i < a $$ i}" by auto
5370 then show "closed {x::'a. a $$ i \<le> x $$ i}"
5371 by (simp add: closed_def open_halfspace_component_lt)
5374 lemma open_vimage_euclidean_component: "open S \<Longrightarrow> open ((\<lambda>x. x $$ i) -` S)"
5375 by (auto intro!: continuous_open_vimage)
5377 text{* This gives a simple derivation of limit component bounds. *}
5379 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5380 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f(x)$$i \<le> b) net"
5381 shows "l$$i \<le> b"
5383 { fix x have "x \<in> {x::'b. inner (basis i) x \<le> b} \<longleftrightarrow> x$$i \<le> b"
5384 unfolding euclidean_component_def by auto } note * = this
5385 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *
5386 using closed_halfspace_le[of "(basis i)::'b" b] and assms(1,2,3) by auto
5389 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5390 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$$i) net"
5391 shows "b \<le> l$$i"
5393 { fix x have "x \<in> {x::'b. inner (basis i) x \<ge> b} \<longleftrightarrow> x$$i \<ge> b"
5394 unfolding euclidean_component_def by auto } note * = this
5395 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *
5396 using closed_halfspace_ge[of b "(basis i)"] and assms(1,2,3) by auto
5399 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5400 assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$$i = b) net"
5402 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
5403 text{* Limits relative to a union. *}
5405 lemma eventually_within_Un:
5406 "eventually P (net within (s \<union> t)) \<longleftrightarrow>
5407 eventually P (net within s) \<and> eventually P (net within t)"
5408 unfolding Limits.eventually_within
5409 by (auto elim!: eventually_rev_mp)
5411 lemma Lim_within_union:
5412 "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>
5413 (f ---> l) (net within s) \<and> (f ---> l) (net within t)"
5414 unfolding tendsto_def
5415 by (auto simp add: eventually_within_Un)
5417 lemma Lim_topological:
5418 "(f ---> l) net \<longleftrightarrow>
5419 trivial_limit net \<or>
5420 (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
5421 unfolding tendsto_def trivial_limit_eq by auto
5423 lemma continuous_on_union:
5424 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
5425 shows "continuous_on (s \<union> t) f"
5426 using assms unfolding continuous_on Lim_within_union
5427 unfolding Lim_topological trivial_limit_within closed_limpt by auto
5429 lemma continuous_on_cases:
5430 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
5431 "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
5432 shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
5434 let ?h = "(\<lambda>x. if P x then f x else g x)"
5435 have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto
5436 hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto
5438 have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto
5439 hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto
5440 ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto
5444 text{* Some more convenient intermediate-value theorem formulations. *}
5446 lemma connected_ivt_hyperplane:
5447 assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
5448 shows "\<exists>z \<in> s. inner a z = b"
5450 assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
5451 let ?A = "{x. inner a x < b}"
5452 let ?B = "{x. inner a x > b}"
5453 have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
5454 moreover have "?A \<inter> ?B = {}" by auto
5455 moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
5456 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
5459 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows
5460 "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)"
5461 using connected_ivt_hyperplane[of s x y "(basis k)::'a" a]
5462 unfolding euclidean_component_def by auto
5464 subsection {* Homeomorphisms *}
5466 definition "homeomorphism s t f g \<equiv>
5467 (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
5468 (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
5471 homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"
5472 (infixr "homeomorphic" 60) where
5473 homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
5475 lemma homeomorphic_refl: "s homeomorphic s"
5476 unfolding homeomorphic_def
5477 unfolding homeomorphism_def
5478 using continuous_on_id
5479 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5480 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5483 lemma homeomorphic_sym:
5484 "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
5485 unfolding homeomorphic_def
5486 unfolding homeomorphism_def
5489 lemma homeomorphic_trans:
5490 assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
5492 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"
5493 using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
5494 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"
5495 using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
5497 { 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 }
5498 moreover have "(f2 \<circ> f1) ` s = u" using fg1(2) fg2(2) by auto
5499 moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
5500 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 }
5501 moreover have "(g1 \<circ> g2) ` u = s" using fg1(5) fg2(5) by auto
5502 moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6) unfolding fg2(5) by auto
5503 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
5506 lemma homeomorphic_minimal:
5507 "s homeomorphic t \<longleftrightarrow>
5508 (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
5509 (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
5510 continuous_on s f \<and> continuous_on t g)"
5511 unfolding homeomorphic_def homeomorphism_def
5512 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
5513 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
5515 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
5516 apply auto apply(rule_tac x="g x" in bexI) apply auto
5517 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
5518 apply auto apply(rule_tac x="f x" in bexI) by auto
5520 text {* Relatively weak hypotheses if a set is compact. *}
5522 lemma homeomorphism_compact:
5523 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5524 (* class constraint due to continuous_on_inverse *)
5525 assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s"
5526 shows "\<exists>g. homeomorphism s t f g"
5528 def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
5529 have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
5530 { fix y assume "y\<in>t"
5531 then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
5532 hence "g (f x) = x" using g by auto
5533 hence "f (g y) = y" unfolding x(1)[THEN sym] by auto }
5534 hence g':"\<forall>x\<in>t. f (g x) = x" by auto
5537 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"])
5539 { assume "x\<in>g ` t"
5540 then obtain y where y:"y\<in>t" "g y = x" by auto
5541 then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
5542 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 }
5543 ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" .. }
5544 hence "g ` t = s" by auto
5546 show ?thesis unfolding homeomorphism_def homeomorphic_def
5547 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
5550 lemma homeomorphic_compact:
5551 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5552 (* class constraint due to continuous_on_inverse *)
5553 shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
5554 \<Longrightarrow> s homeomorphic t"
5555 unfolding homeomorphic_def by (metis homeomorphism_compact)
5557 text{* Preservation of topological properties. *}
5559 lemma homeomorphic_compactness:
5560 "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
5561 unfolding homeomorphic_def homeomorphism_def
5562 by (metis compact_continuous_image)
5564 text{* Results on translation, scaling etc. *}
5566 lemma homeomorphic_scaling:
5567 fixes s :: "'a::real_normed_vector set"
5568 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
5569 unfolding homeomorphic_minimal
5570 apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
5571 apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
5572 using assms apply auto
5573 using continuous_on_cmul[OF continuous_on_id] by auto
5575 lemma homeomorphic_translation:
5576 fixes s :: "'a::real_normed_vector set"
5577 shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
5578 unfolding homeomorphic_minimal
5579 apply(rule_tac x="\<lambda>x. a + x" in exI)
5580 apply(rule_tac x="\<lambda>x. -a + x" in exI)
5581 using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
5583 lemma homeomorphic_affinity:
5584 fixes s :: "'a::real_normed_vector set"
5585 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
5587 have *:"op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
5589 using homeomorphic_trans
5590 using homeomorphic_scaling[OF assms, of s]
5591 using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a] unfolding * by auto
5594 lemma homeomorphic_balls:
5595 fixes a b ::"'a::real_normed_vector" (* FIXME: generalize to metric_space *)
5596 assumes "0 < d" "0 < e"
5597 shows "(ball a d) homeomorphic (ball b e)" (is ?th)
5598 "(cball a d) homeomorphic (cball b e)" (is ?cth)
5600 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5601 show ?th unfolding homeomorphic_minimal
5602 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5603 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5604 using assms apply (auto simp add: dist_commute)
5606 apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)
5607 unfolding continuous_on
5608 by (intro ballI tendsto_intros, simp)+
5610 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5611 show ?cth unfolding homeomorphic_minimal
5612 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5613 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5614 using assms apply (auto simp add: dist_commute)
5616 apply (auto simp add: pos_divide_le_eq)
5617 unfolding continuous_on
5618 by (intro ballI tendsto_intros, simp)+
5621 text{* "Isometry" (up to constant bounds) of injective linear map etc. *}
5623 lemma cauchy_isometric:
5624 fixes x :: "nat \<Rightarrow> 'a::euclidean_space"
5625 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)"
5628 interpret f: bounded_linear f by fact
5629 { fix d::real assume "d>0"
5630 then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
5631 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
5632 { fix n assume "n\<ge>N"
5633 hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding f.diff[THEN sym] by auto
5634 moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
5635 using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
5636 using normf[THEN bspec[where x="x n - x N"]] by auto
5637 ultimately have "norm (x n - x N) < d" using `e>0`
5638 using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto }
5639 hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
5640 thus ?thesis unfolding cauchy and dist_norm by auto
5643 lemma complete_isometric_image:
5644 fixes f :: "'a::euclidean_space => 'b::euclidean_space"
5645 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"
5646 shows "complete(f ` s)"
5648 { fix g assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
5649 then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
5650 using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
5651 hence x:"\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
5652 hence "f \<circ> x = g" unfolding fun_eq_iff by auto
5653 then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
5654 using cs[unfolded complete_def, THEN spec[where x="x"]]
5655 using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
5656 hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
5657 using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
5658 unfolding `f \<circ> x = g` by auto }
5659 thus ?thesis unfolding complete_def by auto
5663 fixes x :: "'a::real_normed_vector"
5664 shows "dist 0 x = norm x"
5665 unfolding dist_norm by simp
5667 lemma injective_imp_isometric: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
5668 assumes s:"closed s" "subspace s" and f:"bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
5669 shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
5670 proof(cases "s \<subseteq> {0::'a}")
5672 { fix x assume "x \<in> s"
5673 hence "x = 0" using True by auto
5674 hence "norm x \<le> norm (f x)" by auto }
5675 thus ?thesis by(auto intro!: exI[where x=1])
5677 interpret f: bounded_linear f by fact
5679 then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
5680 from False have "s \<noteq> {}" by auto
5681 let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
5682 let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
5683 let ?S'' = "{x::'a. norm x = norm a}"
5685 have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by auto
5686 hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
5687 moreover have "?S' = s \<inter> ?S''" by auto
5688 ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
5689 moreover have *:"f ` ?S' = ?S" by auto
5690 ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
5691 hence "closed ?S" using compact_imp_closed by auto
5692 moreover have "?S \<noteq> {}" using a by auto
5693 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
5694 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
5696 let ?e = "norm (f b) / norm b"
5697 have "norm b > 0" using ba and a and norm_ge_zero by auto
5698 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
5699 ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
5701 { fix x assume "x\<in>s"
5702 hence "norm (f b) / norm b * norm x \<le> norm (f x)"
5704 case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
5707 hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
5708 have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def] by auto
5709 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
5710 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"]]
5711 unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
5712 by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq)
5715 show ?thesis by auto
5718 lemma closed_injective_image_subspace:
5719 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
5720 assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
5721 shows "closed(f ` s)"
5723 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
5724 show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
5725 unfolding complete_eq_closed[THEN sym] by auto
5728 subsection{* Some properties of a canonical subspace. *}
5731 declare euclidean_component.zero[simp]
5733 lemma subspace_substandard:
5734 "subspace {x::'a::euclidean_space. (\<forall>i<DIM('a). P i \<longrightarrow> x$$i = 0)}"
5735 unfolding subspace_def by(auto simp add: euclidean_simps)
5737 lemma closed_substandard:
5738 "closed {x::'a::euclidean_space. \<forall>i<DIM('a). P i --> x$$i = 0}" (is "closed ?A")
5740 let ?D = "{i. P i} \<inter> {..<DIM('a)}"
5741 let ?Bs = "{{x::'a. inner (basis i) x = 0}| i. i \<in> ?D}"
5744 hence x:"\<forall>i\<in>?D. x $$ i = 0" by auto
5745 hence "x\<in> \<Inter> ?Bs" by(auto simp add: x euclidean_component_def) }
5747 { assume x:"x\<in>\<Inter>?Bs"
5748 { fix i assume i:"i \<in> ?D"
5749 then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::'a. inner (basis i) x = 0}" by auto
5750 hence "x $$ i = 0" unfolding B using x unfolding euclidean_component_def by auto }
5751 hence "x\<in>?A" by auto }
5752 ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" .. }
5753 hence "?A = \<Inter> ?Bs" by auto
5754 thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
5757 lemma dim_substandard: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
5758 shows "dim {x::'a::euclidean_space. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0} = card d" (is "dim ?A = _")
5760 let ?D = "{..<DIM('a)}"
5761 let ?B = "(basis::nat => 'a) ` d"
5762 let ?bas = "basis::nat \<Rightarrow> 'a"
5763 have "?B \<subseteq> ?A" by(auto simp add:basis_component)
5765 { fix x::"'a" assume "x\<in>?A"
5766 hence "finite d" "x\<in>?A" using assms by(auto intro:finite_subset)
5767 hence "x\<in> span ?B"
5768 proof(induct d arbitrary: x)
5769 case empty hence "x=0" apply(subst euclidean_eq) by auto
5770 thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
5773 hence *:"\<forall>i<DIM('a). i \<notin> insert k F \<longrightarrow> x $$ i = 0" by auto
5774 have **:"F \<subseteq> insert k F" by auto
5775 def y \<equiv> "x - x$$k *\<^sub>R basis k"
5776 have y:"x = y + (x$$k) *\<^sub>R basis k" unfolding y_def by auto
5777 { fix i assume i':"i \<notin> F"
5778 hence "y $$ i = 0" unfolding y_def
5779 using *[THEN spec[where x=i]] by(auto simp add: euclidean_simps basis_component) }
5780 hence "y \<in> span (basis ` F)" using insert(3) by auto
5781 hence "y \<in> span (basis ` (insert k F))"
5782 using span_mono[of "?bas ` F" "?bas ` (insert k F)"]
5783 using image_mono[OF **, of basis] using assms by auto
5785 have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
5786 hence "x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5787 using span_mul by auto
5789 have "y + x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5790 using span_add by auto
5791 thus ?case using y by auto
5794 hence "?A \<subseteq> span ?B" by auto
5796 { fix x assume "x \<in> ?B"
5797 hence "x\<in>{(basis i)::'a |i. i \<in> ?D}" using assms by auto }
5798 hence "independent ?B" using independent_mono[OF independent_basis, of ?B] and assms by auto
5800 have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
5801 hence *:"inj_on (basis::nat\<Rightarrow>'a) d" using subset_inj_on[OF basis_inj, of "d"] by auto
5802 have "card ?B = card d" unfolding card_image[OF *] by auto
5803 ultimately show ?thesis using dim_unique[of "basis ` d" ?A] by auto
5806 text{* Hence closure and completeness of all subspaces. *}
5808 lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"
5810 apply (rule_tac x="{}" in exI, simp)
5812 apply (subgoal_tac "\<exists>x. x \<notin> A")
5814 apply (rule_tac x="insert x A" in exI, simp)
5815 apply (subgoal_tac "A \<noteq> UNIV", auto)
5818 lemma closed_subspace: fixes s::"('a::euclidean_space) set"
5819 assumes "subspace s" shows "closed s"
5821 have *:"dim s \<le> DIM('a)" using dim_subset_UNIV by auto
5822 def d \<equiv> "{..<dim s}" have t:"card d = dim s" unfolding d_def by auto
5823 let ?t = "{x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0}"
5824 have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0} = s \<and>
5825 inj_on f {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0}"
5826 apply(rule subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]])
5827 using dim_substandard[of d,where 'a='a] and t unfolding d_def using * assms by auto
5828 then guess f apply-by(erule exE conjE)+ note f = this
5829 interpret f: bounded_linear f using f unfolding linear_conv_bounded_linear by auto
5830 have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using f.zero using f(3)[unfolded inj_on_def]
5831 by(erule_tac x=0 in ballE) auto
5832 moreover have "closed ?t" using closed_substandard .
5833 moreover have "subspace ?t" using subspace_substandard .
5834 ultimately show ?thesis using closed_injective_image_subspace[of ?t f]
5835 unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
5838 lemma complete_subspace:
5839 fixes s :: "('a::euclidean_space) set" shows "subspace s ==> complete s"
5840 using complete_eq_closed closed_subspace
5844 fixes s :: "('a::euclidean_space) set"
5845 shows "dim(closure s) = dim s" (is "?dc = ?d")
5847 have "?dc \<le> ?d" using closure_minimal[OF span_inc, of s]
5848 using closed_subspace[OF subspace_span, of s]
5849 using dim_subset[of "closure s" "span s"] unfolding dim_span by auto
5850 thus ?thesis using dim_subset[OF closure_subset, of s] by auto
5853 subsection {* Affine transformations of intervals *}
5855 lemma real_affinity_le:
5856 "0 < (m::'a::linordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"
5857 by (simp add: field_simps inverse_eq_divide)
5859 lemma real_le_affinity:
5860 "0 < (m::'a::linordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"
5861 by (simp add: field_simps inverse_eq_divide)
5863 lemma real_affinity_lt:
5864 "0 < (m::'a::linordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"
5865 by (simp add: field_simps inverse_eq_divide)
5867 lemma real_lt_affinity:
5868 "0 < (m::'a::linordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"
5869 by (simp add: field_simps inverse_eq_divide)
5871 lemma real_affinity_eq:
5872 "(m::'a::linordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"
5873 by (simp add: field_simps inverse_eq_divide)
5875 lemma real_eq_affinity:
5876 "(m::'a::linordered_field) \<noteq> 0 ==> (y = m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"
5877 by (simp add: field_simps inverse_eq_divide)
5879 lemma image_affinity_interval: fixes m::real
5880 fixes a b c :: "'a::ordered_euclidean_space"
5881 shows "(\<lambda>x. m *\<^sub>R x + c) ` {a .. b} =
5882 (if {a .. b} = {} then {}
5883 else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
5884 else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
5886 { fix x assume "x \<le> c" "c \<le> x"
5887 hence "x=c" unfolding eucl_le[where 'a='a] apply-
5888 apply(subst euclidean_eq) by (auto intro: order_antisym) }
5890 moreover have "c \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}" unfolding True by(auto simp add: eucl_le[where 'a='a])
5891 ultimately show ?thesis by auto
5894 { fix y assume "a \<le> y" "y \<le> b" "m > 0"
5895 hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
5896 unfolding eucl_le[where 'a='a] by(auto simp add: euclidean_simps)
5898 { fix y assume "a \<le> y" "y \<le> b" "m < 0"
5899 hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
5900 unfolding eucl_le[where 'a='a] by(auto simp add: mult_left_mono_neg euclidean_simps)
5902 { fix y assume "m > 0" "m *\<^sub>R a + c \<le> y" "y \<le> m *\<^sub>R b + c"
5903 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5904 unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
5905 apply(auto simp add: pth_3[symmetric]
5906 intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
5907 by(auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff euclidean_simps)
5909 { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
5910 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5911 unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
5912 apply(auto simp add: pth_3[symmetric]
5913 intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
5914 by(auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff euclidean_simps)
5916 ultimately show ?thesis using False by auto
5919 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a..b} =
5920 (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})"
5921 using image_affinity_interval[of m 0 a b] by auto
5923 subsection{* Banach fixed point theorem (not really topological...) *}
5926 assumes s:"complete s" "s \<noteq> {}" and c:"0 \<le> c" "c < 1" and f:"(f ` s) \<subseteq> s" and
5927 lipschitz:"\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
5928 shows "\<exists>! x\<in>s. (f x = x)"
5930 have "1 - c > 0" using c by auto
5932 from s(2) obtain z0 where "z0 \<in> s" by auto
5933 def z \<equiv> "\<lambda>n. (f ^^ n) z0"
5935 have "z n \<in> s" unfolding z_def
5936 proof(induct n) case 0 thus ?case using `z0 \<in>s` by auto
5937 next case Suc thus ?case using f by auto qed }
5940 def d \<equiv> "dist (z 0) (z 1)"
5942 have fzn:"\<And>n. f (z n) = z (Suc n)" unfolding z_def by auto
5944 have "dist (z n) (z (Suc n)) \<le> (c ^ n) * d"
5946 case 0 thus ?case unfolding d_def by auto
5949 hence "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
5950 using `0 \<le> c` using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by auto
5951 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]
5952 unfolding fzn and mult_le_cancel_left by auto
5957 have "(1 - c) * dist (z m) (z (m+n)) \<le> (c ^ m) * d * (1 - c ^ n)"
5959 case 0 show ?case by auto
5962 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))))"
5963 using dist_triangle and c by(auto simp add: dist_triangle)
5964 also have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
5965 using cf_z[of "m + k"] and c by auto
5966 also have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
5967 using Suc by (auto simp add: field_simps)
5968 also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
5969 unfolding power_add by (auto simp add: field_simps)
5970 also have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
5971 using c by (auto simp add: field_simps)
5972 finally show ?case by auto
5975 { fix e::real assume "e>0"
5976 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e"
5977 proof(cases "d = 0")
5979 have *: "\<And>x. ((1 - c) * x \<le> 0) = (x \<le> 0)" using `1 - c > 0`
5980 by (metis mult_zero_left real_mult_commute real_mult_le_cancel_iff1)
5981 from True have "\<And>n. z n = z0" using cf_z2[of 0] and c unfolding z_def
5983 thus ?thesis using `e>0` by auto
5985 case False hence "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]
5986 by (metis False d_def less_le)
5987 hence "0 < e * (1 - c) / d" using `e>0` and `1-c>0`
5988 using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"] by auto
5989 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
5990 { fix m n::nat assume "m>n" and as:"m\<ge>N" "n\<ge>N"
5991 have *:"c ^ n \<le> c ^ N" using `n\<ge>N` and c using power_decreasing[OF `n\<ge>N`, of c] by auto
5992 have "1 - c ^ (m - n) > 0" using c and power_strict_mono[of c 1 "m - n"] using `m>n` by auto
5993 hence **:"d * (1 - c ^ (m - n)) / (1 - c) > 0"
5994 using mult_pos_pos[OF `d>0`, of "1 - c ^ (m - n)"]
5995 using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]
5996 using `0 < 1 - c` by auto
5998 have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
5999 using cf_z2[of n "m - n"] and `m>n` unfolding pos_le_divide_eq[OF `1-c>0`]
6000 by (auto simp add: mult_commute dist_commute)
6001 also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
6002 using mult_right_mono[OF * order_less_imp_le[OF **]]
6003 unfolding mult_assoc by auto
6004 also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
6005 using mult_strict_right_mono[OF N **] unfolding mult_assoc by auto
6006 also have "\<dots> = e * (1 - c ^ (m - n))" using c and `d>0` and `1 - c > 0` by auto
6007 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
6008 finally have "dist (z m) (z n) < e" by auto
6010 { fix m n::nat assume as:"N\<le>m" "N\<le>n"
6011 hence "dist (z n) (z m) < e"
6012 proof(cases "n = m")
6013 case True thus ?thesis using `e>0` by auto
6015 case False thus ?thesis using as and *[of n m] *[of m n] unfolding nat_neq_iff by (auto simp add: dist_commute)
6017 thus ?thesis by auto
6020 hence "Cauchy z" unfolding cauchy_def by auto
6021 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
6023 def e \<equiv> "dist (f x) x"
6024 have "e = 0" proof(rule ccontr)
6025 assume "e \<noteq> 0" hence "e>0" unfolding e_def using zero_le_dist[of "f x" x]
6026 by (metis dist_eq_0_iff dist_nz e_def)
6027 then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"
6028 using x[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
6029 hence N':"dist (z N) x < e / 2" by auto
6031 have *:"c * dist (z N) x \<le> dist (z N) x" unfolding mult_le_cancel_right2
6032 using zero_le_dist[of "z N" x] and c
6033 by (metis dist_eq_0_iff dist_nz order_less_asym less_le)
6034 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]]
6035 using z_in_s[of N] `x\<in>s` using c by auto
6036 also have "\<dots> < e / 2" using N' and c using * by auto
6037 finally show False unfolding fzn
6038 using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
6039 unfolding e_def by auto
6041 hence "f x = x" unfolding e_def by auto
6043 { fix y assume "f y = y" "y\<in>s"
6044 hence "dist x y \<le> c * dist x y" using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]
6045 using `x\<in>s` and `f x = x` by auto
6046 hence "dist x y = 0" unfolding mult_le_cancel_right1
6047 using c and zero_le_dist[of x y] by auto
6048 hence "y = x" by auto
6050 ultimately show ?thesis using `x\<in>s` by blast+
6053 subsection{* Edelstein fixed point theorem. *}
6055 lemma edelstein_fix:
6056 fixes s :: "'a::real_normed_vector set"
6057 assumes s:"compact s" "s \<noteq> {}" and gs:"(g ` s) \<subseteq> s"
6058 and dist:"\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
6059 shows "\<exists>! x\<in>s. g x = x"
6060 proof(cases "\<exists>x\<in>s. g x \<noteq> x")
6061 obtain x where "x\<in>s" using s(2) by auto
6062 case False hence g:"\<forall>x\<in>s. g x = x" by auto
6063 { fix y assume "y\<in>s"
6064 hence "x = y" using `x\<in>s` and dist[THEN bspec[where x=x], THEN bspec[where x=y]]
6065 unfolding g[THEN bspec[where x=x], OF `x\<in>s`]
6066 unfolding g[THEN bspec[where x=y], OF `y\<in>s`] by auto }
6067 thus ?thesis using `x\<in>s` and g by blast+
6070 then obtain x where [simp]:"x\<in>s" and "g x \<noteq> x" by auto
6071 { fix x y assume "x \<in> s" "y \<in> s"
6072 hence "dist (g x) (g y) \<le> dist x y"
6073 using dist[THEN bspec[where x=x], THEN bspec[where x=y]] by auto } note dist' = this
6074 def y \<equiv> "g x"
6075 have [simp]:"y\<in>s" unfolding y_def using gs[unfolded image_subset_iff] and `x\<in>s` by blast
6076 def f \<equiv> "\<lambda>n. g ^^ n"
6077 have [simp]:"\<And>n z. g (f n z) = f (Suc n) z" unfolding f_def by auto
6078 have [simp]:"\<And>z. f 0 z = z" unfolding f_def by auto
6079 { fix n::nat and z assume "z\<in>s"
6080 have "f n z \<in> s" unfolding f_def
6082 case 0 thus ?case using `z\<in>s` by simp
6084 case (Suc n) thus ?case using gs[unfolded image_subset_iff] by auto
6085 qed } note fs = this
6086 { fix m n ::nat assume "m\<le>n"
6087 fix w z assume "w\<in>s" "z\<in>s"
6088 have "dist (f n w) (f n z) \<le> dist (f m w) (f m z)" using `m\<le>n`
6090 case 0 thus ?case by auto
6093 thus ?case proof(cases "m\<le>n")
6094 case True thus ?thesis using Suc(1)
6095 using dist'[OF fs fs, OF `w\<in>s` `z\<in>s`, of n n] by auto
6097 case False hence mn:"m = Suc n" using Suc(2) by simp
6098 show ?thesis unfolding mn by auto
6100 qed } note distf = this
6102 def h \<equiv> "\<lambda>n. (f n x, f n y)"
6103 let ?s2 = "s \<times> s"
6104 obtain l r where "l\<in>?s2" and r:"subseq r" and lr:"((h \<circ> r) ---> l) sequentially"
6105 using compact_Times [OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def
6106 using fs[OF `x\<in>s`] and fs[OF `y\<in>s`] by blast
6107 def a \<equiv> "fst l" def b \<equiv> "snd l"
6108 have lab:"l = (a, b)" unfolding a_def b_def by simp
6109 have [simp]:"a\<in>s" "b\<in>s" unfolding a_def b_def using `l\<in>?s2` by auto
6111 have lima:"((fst \<circ> (h \<circ> r)) ---> a) sequentially"
6112 and limb:"((snd \<circ> (h \<circ> r)) ---> b) sequentially"
6114 unfolding o_def a_def b_def by (simp_all add: tendsto_intros)
6117 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
6119 have "dist (-x) (-y) = dist x y" unfolding dist_norm
6120 using norm_minus_cancel[of "x - y"] by (auto simp add: uminus_add_conv_diff) } note ** = this
6122 { assume as:"dist a b > dist (f n x) (f n y)"
6123 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"
6124 and "\<forall>m\<ge>Nb. dist (f (r m) y) b < (dist a b - dist (f n x) (f n y)) / 2"
6125 using lima limb unfolding h_def Lim_sequentially by (fastsimp simp del: less_divide_eq_number_of1)
6126 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)"
6127 apply(erule_tac x="Na+Nb+n" in allE)
6128 apply(erule_tac x="Na+Nb+n" in allE) apply simp
6129 using dist_triangle_add_half[of a "f (r (Na + Nb + n)) x" "dist a b - dist (f n x) (f n y)"
6130 "-b" "- f (r (Na + Nb + n)) y"]
6131 unfolding ** by (auto simp add: algebra_simps dist_commute)
6133 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)"
6134 using distf[of n "r (Na+Nb+n)", OF _ `x\<in>s` `y\<in>s`]
6135 using subseq_bigger[OF r, of "Na+Nb+n"]
6136 using *[of "f (r (Na + Nb + n)) x" "f (r (Na + Nb + n)) y" "f n x" "f n y"] by auto
6137 ultimately have False by simp
6139 hence "dist a b \<le> dist (f n x) (f n y)" by(rule ccontr)auto }
6142 have [simp]:"a = b" proof(rule ccontr)
6143 def e \<equiv> "dist a b - dist (g a) (g b)"
6144 assume "a\<noteq>b" hence "e > 0" unfolding e_def using dist by fastsimp
6145 hence "\<exists>n. dist (f n x) a < e/2 \<and> dist (f n y) b < e/2"
6146 using lima limb unfolding Lim_sequentially
6147 apply (auto elim!: allE[where x="e/2"]) apply(rule_tac x="r (max N Na)" in exI) unfolding h_def by fastsimp
6148 then obtain n where n:"dist (f n x) a < e/2 \<and> dist (f n y) b < e/2" by auto
6149 have "dist (f (Suc n) x) (g a) \<le> dist (f n x) a"
6150 using dist[THEN bspec[where x="f n x"], THEN bspec[where x="a"]] and fs by auto
6151 moreover have "dist (f (Suc n) y) (g b) \<le> dist (f n y) b"
6152 using dist[THEN bspec[where x="f n y"], THEN bspec[where x="b"]] and fs by auto
6153 ultimately have "dist (f (Suc n) x) (g a) + dist (f (Suc n) y) (g b) < e" using n by auto
6154 thus False unfolding e_def using ab_fn[of "Suc n"] by norm
6157 have [simp]:"\<And>n. f (Suc n) x = f n y" unfolding f_def y_def by(induct_tac n)auto
6158 { fix x y assume "x\<in>s" "y\<in>s" moreover
6159 fix e::real assume "e>0" ultimately
6160 have "dist y x < e \<longrightarrow> dist (g y) (g x) < e" using dist by fastsimp }
6161 hence "continuous_on s g" unfolding continuous_on_iff by auto
6163 hence "((snd \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
6164 apply (rule allE[where x="\<lambda>n. (fst \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
6165 using lima unfolding h_def o_def using fs[OF `x\<in>s`] by (auto simp add: y_def)
6166 hence "g a = a" using tendsto_unique[OF trivial_limit_sequentially limb, of "g a"]
6167 unfolding `a=b` and o_assoc by auto
6169 { fix x assume "x\<in>s" "g x = x" "x\<noteq>a"
6170 hence "False" using dist[THEN bspec[where x=a], THEN bspec[where x=x]]
6171 using `g a = a` and `a\<in>s` by auto }
6172 ultimately show "\<exists>!x\<in>s. g x = x" using `a\<in>s` by blast
6176 (** TODO move this someplace else within this theory **)
6177 instance euclidean_space \<subseteq> banach ..