1 (* title: HOL/Library/Topology_Euclidian_Space.thy
2 Author: Amine Chaieb, University of Cambridge
3 Author: Robert Himmelmann, TU Muenchen
4 Author: Brian Huffman, Portland State University
7 header {* Elementary topology in Euclidean space. *}
9 theory Topology_Euclidean_Space
10 imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs"
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 euclidean_space \<subseteq> perfect_space
479 { fix e :: real assume "0 < e"
480 def y \<equiv> "x + scaleR (e/2) (sgn (basis 0))"
481 from `0 < e` have "y \<noteq> x"
482 unfolding y_def by (simp add: sgn_zero_iff basis_eq_0_iff DIM_positive)
483 from `0 < e` have "dist y x < e"
484 unfolding y_def by (simp add: dist_norm norm_sgn)
485 from `y \<noteq> x` and `dist y x < e`
486 have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
488 then show "x islimpt UNIV" unfolding islimpt_approachable by blast
491 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
493 apply (subst open_subopen)
494 apply (simp add: islimpt_def subset_eq)
495 by (metis ComplE ComplI insertCI insert_absorb mem_def)
497 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
498 unfolding islimpt_def by auto
500 lemma finite_set_avoid:
501 fixes a :: "'a::metric_space"
502 assumes fS: "finite S" shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
503 proof(induct rule: finite_induct[OF fS])
504 case 1 thus ?case by (auto intro: zero_less_one)
507 from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
508 {assume "x = a" hence ?case using d by auto }
510 {assume xa: "x\<noteq>a"
511 let ?d = "min d (dist a x)"
512 have dp: "?d > 0" using xa d(1) using dist_nz by auto
513 from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
514 with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
515 ultimately show ?case by blast
518 lemma islimpt_finite:
519 fixes S :: "'a::metric_space set"
520 assumes fS: "finite S" shows "\<not> a islimpt S"
521 unfolding islimpt_approachable
522 using finite_set_avoid[OF fS, of a] by (metis dist_commute not_le)
524 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
527 apply (metis Un_upper1 Un_upper2 islimpt_subset)
528 unfolding islimpt_def
529 apply (rule ccontr, clarsimp, rename_tac A B)
530 apply (drule_tac x="A \<inter> B" in spec)
531 apply (auto simp add: open_Int)
534 lemma discrete_imp_closed:
535 fixes S :: "'a::metric_space set"
536 assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
539 {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
540 from e have e2: "e/2 > 0" by arith
541 from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
542 let ?m = "min (e/2) (dist x y) "
543 from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
544 from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
545 have th: "dist z y < e" using z y
546 by (intro dist_triangle_lt [where z=x], simp)
547 from d[rule_format, OF y(1) z(1) th] y z
548 have False by (auto simp add: dist_commute)}
549 then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
552 subsection{* Interior of a Set *}
553 definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
555 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
556 apply (simp add: set_eq_iff interior_def)
557 apply (subst (2) open_subopen) by (safe, blast+)
559 lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
561 lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
563 lemma open_interior[simp, intro]: "open(interior S)"
564 apply (simp add: interior_def)
565 apply (subst open_subopen) by blast
567 lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
568 lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
569 lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
570 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
571 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"
572 by (metis equalityI interior_maximal interior_subset open_interior)
573 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
574 apply (simp add: interior_def)
575 by (metis open_contains_ball centre_in_ball open_ball subset_trans)
577 lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
578 by (metis interior_maximal interior_subset subset_trans)
580 lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
581 apply (rule equalityI, simp)
582 apply (metis Int_lower1 Int_lower2 subset_interior)
583 by (metis Int_mono interior_subset open_Int open_interior open_subset_interior)
585 lemma interior_limit_point [intro]:
586 fixes x :: "'a::perfect_space"
587 assumes x: "x \<in> interior S" shows "x islimpt S"
588 using x islimpt_UNIV [of x]
589 unfolding interior_def islimpt_def
590 apply (clarsimp, rename_tac T T')
591 apply (drule_tac x="T \<inter> T'" in spec)
592 apply (auto simp add: open_Int)
595 lemma interior_closed_Un_empty_interior:
596 assumes cS: "closed S" and iT: "interior T = {}"
597 shows "interior(S \<union> T) = interior S"
599 show "interior S \<subseteq> interior (S\<union>T)"
600 by (rule subset_interior, blast)
602 show "interior (S \<union> T) \<subseteq> interior S"
604 fix x assume "x \<in> interior (S \<union> T)"
605 then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T"
606 unfolding interior_def by fast
607 show "x \<in> interior S"
609 assume "x \<notin> interior S"
610 with `x \<in> R` `open R` obtain y where "y \<in> R - S"
611 unfolding interior_def set_eq_iff by fast
612 from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
613 from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
614 from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
615 show "False" unfolding interior_def by fast
621 subsection{* Closure of a Set *}
623 definition "closure S = S \<union> {x | x. x islimpt S}"
625 lemma closure_interior: "closure S = - interior (- S)"
628 have "x\<in>- interior (- S) \<longleftrightarrow> x \<in> closure S" (is "?lhs = ?rhs")
630 let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> - S)"
632 hence *:"\<not> ?exT x"
633 unfolding interior_def
635 { assume "\<not> ?rhs"
637 unfolding closure_def islimpt_def
643 assume "?rhs" thus "?lhs"
644 unfolding closure_def interior_def islimpt_def
652 lemma interior_closure: "interior S = - (closure (- S))"
655 have "x \<in> interior S \<longleftrightarrow> x \<in> - (closure (- S))"
656 unfolding interior_def closure_def islimpt_def
663 lemma closed_closure[simp, intro]: "closed (closure S)"
665 have "closed (- interior (-S))" by blast
666 thus ?thesis using closure_interior[of S] by simp
669 lemma closure_hull: "closure S = closed hull S"
671 have "S \<subseteq> closure S"
672 unfolding closure_def
675 have "closed (closure S)"
676 using closed_closure[of S]
680 assume *:"S \<subseteq> t" "closed t"
683 hence "x islimpt t" using *(1)
684 using islimpt_subset[of x, of S, of t]
687 with * have "closure S \<subseteq> t"
688 unfolding closure_def
689 using closed_limpt[of t]
692 ultimately show ?thesis
693 using hull_unique[of S, of "closure S", of closed]
698 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
699 unfolding closure_hull
700 using hull_eq[of closed, unfolded mem_def, OF closed_Inter, of S]
701 by (metis mem_def subset_eq)
703 lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
704 using closure_eq[of S]
707 lemma closure_closure[simp]: "closure (closure S) = closure S"
708 unfolding closure_hull
709 using hull_hull[of closed S]
712 lemma closure_subset: "S \<subseteq> closure S"
713 unfolding closure_hull
714 using hull_subset[of S closed]
717 lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
718 unfolding closure_hull
719 using hull_mono[of S T closed]
722 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
723 using hull_minimal[of S T closed]
724 unfolding closure_hull mem_def
727 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"
728 using hull_unique[of S T closed]
729 unfolding closure_hull mem_def
732 lemma closure_empty[simp]: "closure {} = {}"
733 using closed_empty closure_closed[of "{}"]
736 lemma closure_univ[simp]: "closure UNIV = UNIV"
737 using closure_closed[of UNIV]
740 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
741 using closure_empty closure_subset[of S]
744 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
745 using closure_eq[of S] closure_subset[of S]
748 lemma open_inter_closure_eq_empty:
749 "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
750 using open_subset_interior[of S "- T"]
751 using interior_subset[of "- T"]
752 unfolding closure_interior
755 lemma open_inter_closure_subset:
756 "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
759 assume as: "open S" "x \<in> S \<inter> closure T"
760 { assume *:"x islimpt T"
761 have "x islimpt (S \<inter> T)"
762 proof (rule islimptI)
764 assume "x \<in> A" "open A"
765 with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
766 by (simp_all add: open_Int)
767 with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
769 hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
771 thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
774 then show "x \<in> closure (S \<inter> T)" using as
775 unfolding closure_def
779 lemma closure_complement: "closure(- S) = - interior(S)"
784 unfolding closure_interior
788 lemma interior_complement: "interior(- S) = - closure(S)"
789 unfolding closure_interior
792 subsection{* Frontier (aka boundary) *}
794 definition "frontier S = closure S - interior S"
796 lemma frontier_closed: "closed(frontier S)"
797 by (simp add: frontier_def closed_Diff)
799 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
800 by (auto simp add: frontier_def interior_closure)
802 lemma frontier_straddle:
803 fixes a :: "'a::metric_space"
804 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")
809 let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
811 have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
812 moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
813 unfolding frontier_closures closure_def islimpt_def using `e>0`
814 by (auto, erule_tac x="ball a e" in allE, auto)
815 ultimately have ?rhse by auto
818 { assume "a\<notin>S"
819 hence ?rhse using `?lhs`
820 unfolding frontier_closures closure_def islimpt_def
821 using open_ball[of a e] `e > 0`
822 by simp (metis centre_in_ball mem_ball open_ball)
824 ultimately have ?rhse by auto
830 { fix T assume "a\<notin>S" and
831 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"
832 from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
833 then obtain e where "e>0" "ball a e \<subseteq> T" by auto
834 then obtain y where y:"y\<in>S" "dist a y < e" using as(1) by auto
835 have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
836 using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
838 hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
840 { fix T assume "a \<in> T" "open T" "a\<in>S"
841 then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
842 obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
843 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
845 hence "a islimpt (- S) \<or> a\<notin>S" unfolding islimpt_def by auto
846 ultimately show ?lhs unfolding frontier_closures using closure_def[of "- S"] by auto
849 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
850 by (metis frontier_def closure_closed Diff_subset)
852 lemma frontier_empty[simp]: "frontier {} = {}"
853 by (simp add: frontier_def)
855 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
857 { assume "frontier S \<subseteq> S"
858 hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
859 hence "closed S" using closure_subset_eq by auto
861 thus ?thesis using frontier_subset_closed[of S] ..
864 lemma frontier_complement: "frontier(- S) = frontier S"
865 by (auto simp add: frontier_def closure_complement interior_complement)
867 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
868 using frontier_complement frontier_subset_eq[of "- S"]
869 unfolding open_closed by auto
871 subsection {* Filters and the ``eventually true'' quantifier *}
873 text {* Common filters and The "within" modifier for filters. *}
876 at_infinity :: "'a::real_normed_vector filter" where
877 "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
880 indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
881 (infixr "indirection" 70) where
882 "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
884 text{* Prove That They are all filters. *}
886 lemma eventually_at_infinity:
887 "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
888 unfolding at_infinity_def
889 proof (rule eventually_Abs_filter, rule is_filter.intro)
890 fix P Q :: "'a \<Rightarrow> bool"
891 assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
892 then obtain r s where
893 "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
894 then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
895 then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
898 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
900 lemma trivial_limit_within:
901 shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
903 assume "trivial_limit (at a within S)"
904 thus "\<not> a islimpt S"
905 unfolding trivial_limit_def
906 unfolding eventually_within eventually_at_topological
907 unfolding islimpt_def
908 apply (clarsimp simp add: set_eq_iff)
909 apply (rename_tac T, rule_tac x=T in exI)
910 apply (clarsimp, drule_tac x=y in bspec, simp_all)
913 assume "\<not> a islimpt S"
914 thus "trivial_limit (at a within S)"
915 unfolding trivial_limit_def
916 unfolding eventually_within eventually_at_topological
917 unfolding islimpt_def
919 apply (rule_tac x=T in exI)
924 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
925 using trivial_limit_within [of a UNIV]
926 by (simp add: within_UNIV)
928 lemma trivial_limit_at:
929 fixes a :: "'a::perfect_space"
930 shows "\<not> trivial_limit (at a)"
931 by (simp add: trivial_limit_at_iff)
933 lemma trivial_limit_at_infinity:
934 "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
935 unfolding trivial_limit_def eventually_at_infinity
937 apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
938 apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
939 apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
940 apply (drule_tac x=UNIV in spec, simp)
943 text {* Some property holds "sufficiently close" to the limit point. *}
945 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
946 "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
947 unfolding eventually_at dist_nz by auto
949 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
950 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
951 unfolding eventually_within eventually_at dist_nz by auto
953 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
954 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
955 unfolding eventually_within
956 by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl)
958 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
959 unfolding trivial_limit_def
960 by (auto elim: eventually_rev_mp)
962 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
963 unfolding trivial_limit_def by (auto elim: eventually_rev_mp)
965 lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
966 unfolding trivial_limit_def ..
969 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
970 apply (safe elim!: trivial_limit_eventually)
971 apply (simp add: eventually_False [symmetric])
974 text{* Combining theorems for "eventually" *}
976 lemma eventually_conjI:
977 "\<lbrakk>eventually (\<lambda>x. P x) net; eventually (\<lambda>x. Q x) net\<rbrakk>
978 \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) net"
979 by (rule eventually_conj)
981 lemma eventually_rev_mono:
982 "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
983 using eventually_mono [of P Q] by fast
985 lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
986 by (auto intro!: eventually_conjI elim: eventually_rev_mono)
988 lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
989 by (auto simp add: eventually_False)
991 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
992 by (simp add: eventually_False)
994 subsection {* Limits *}
996 text{* Notation Lim to avoid collition with lim defined in analysis *}
998 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"
999 where "Lim A f = (THE l. (f ---> l) A)"
1002 "(f ---> l) net \<longleftrightarrow>
1003 trivial_limit net \<or>
1004 (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
1005 unfolding tendsto_iff trivial_limit_eq by auto
1008 text{* Show that they yield usual definitions in the various cases. *}
1010 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
1011 (\<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)"
1012 by (auto simp add: tendsto_iff eventually_within_le)
1014 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
1015 (\<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)"
1016 by (auto simp add: tendsto_iff eventually_within)
1018 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
1019 (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
1020 by (auto simp add: tendsto_iff eventually_at)
1022 lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
1023 unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
1025 lemma Lim_at_infinity:
1026 "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
1027 by (auto simp add: tendsto_iff eventually_at_infinity)
1029 lemma Lim_sequentially:
1030 "(S ---> l) sequentially \<longleftrightarrow>
1031 (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
1032 by (auto simp add: tendsto_iff eventually_sequentially)
1034 lemma Lim_sequentially_iff_LIMSEQ: "(S ---> l) sequentially \<longleftrightarrow> S ----> l"
1035 unfolding Lim_sequentially LIMSEQ_def ..
1037 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
1038 by (rule topological_tendstoI, auto elim: eventually_rev_mono)
1040 text{* The expected monotonicity property. *}
1042 lemma Lim_within_empty: "(f ---> l) (net within {})"
1043 unfolding tendsto_def Limits.eventually_within by simp
1045 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
1046 unfolding tendsto_def Limits.eventually_within
1047 by (auto elim!: eventually_elim1)
1049 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
1050 shows "(f ---> l) (net within (S \<union> T))"
1051 using assms unfolding tendsto_def Limits.eventually_within
1053 apply (drule spec, drule (1) mp, drule (1) mp)
1054 apply (drule spec, drule (1) mp, drule (1) mp)
1055 apply (auto elim: eventually_elim2)
1059 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow> S \<union> T = UNIV
1061 by (metis Lim_Un within_UNIV)
1063 text{* Interrelations between restricted and unrestricted limits. *}
1065 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
1067 unfolding tendsto_def Limits.eventually_within
1068 apply (clarify, drule spec, drule (1) mp, drule (1) mp)
1069 by (auto elim!: eventually_elim1)
1071 lemma Lim_within_open:
1072 fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
1073 assumes"a \<in> S" "open S"
1074 shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)" (is "?lhs \<longleftrightarrow> ?rhs")
1077 { fix A assume "open A" "l \<in> A"
1078 with `?lhs` have "eventually (\<lambda>x. f x \<in> A) (at a within S)"
1079 by (rule topological_tendstoD)
1080 hence "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x \<in> A) (at a)"
1081 unfolding Limits.eventually_within .
1082 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"
1083 unfolding eventually_at_topological by fast
1084 hence "open (T \<inter> S)" "a \<in> T \<inter> S" "\<forall>x\<in>(T \<inter> S). x \<noteq> a \<longrightarrow> f x \<in> A"
1086 hence "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> f x \<in> A)"
1088 hence "eventually (\<lambda>x. f x \<in> A) (at a)"
1089 unfolding eventually_at_topological .
1091 thus ?rhs by (rule topological_tendstoI)
1094 thus ?lhs by (rule Lim_at_within)
1097 lemma Lim_within_LIMSEQ:
1098 fixes a :: real and L :: "'a::metric_space"
1099 assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
1100 shows "(X ---> L) (at a within T)"
1102 assume "\<not> (X ---> L) (at a within T)"
1103 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"
1104 unfolding tendsto_iff eventually_within dist_norm by (simp add: not_less[symmetric])
1105 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
1107 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"
1108 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"
1109 using r by (simp add: Bex_def)
1110 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"
1112 hence F1: "\<And>n. ?F n \<in> T \<and> ?F n \<noteq> a"
1113 and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
1114 and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
1118 proof (rule LIMSEQ_I, unfold real_norm_def)
1121 (* choose no such that inverse (real (Suc n)) < e *)
1122 then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
1123 then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
1124 show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
1125 proof (intro exI allI impI)
1127 assume mlen: "m \<le> n"
1128 have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
1130 also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
1132 also from nodef have
1133 "inverse (real (Suc m)) < e" .
1134 finally show "\<bar>?F n - a\<bar> < e" .
1137 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`
1138 ultimately have "(\<lambda>n. X (?F n)) ----> L" using F1 by simp
1140 moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
1144 obtain n where "n = no + 1" by simp
1145 then have nolen: "no \<le> n" by simp
1146 (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
1147 have "dist (X (?F n)) L \<ge> r"
1149 with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
1151 then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
1152 with r have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
1153 thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
1155 ultimately show False by simp
1158 lemma Lim_right_bound:
1159 fixes f :: "real \<Rightarrow> real"
1160 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"
1161 assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
1162 shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
1164 assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
1166 assume [simp]: "{x<..} \<inter> I \<noteq> {}"
1168 proof (rule Lim_within_LIMSEQ, safe)
1169 fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"
1171 show "(\<lambda>n. f (S n)) ----> Inf (f ` ({x<..} \<inter> I))"
1172 proof (rule LIMSEQ_I, rule ccontr)
1173 fix r :: real assume "0 < r"
1174 with Inf_close[of "f ` ({x<..} \<inter> I)" r]
1175 obtain y where y: "x < y" "y \<in> I" "f y < Inf (f ` ({x <..} \<inter> I)) + r" by auto
1176 from `x < y` have "0 < y - x" by auto
1177 from S(2)[THEN LIMSEQ_D, OF this]
1178 obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto
1180 assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f ` ({x<..} \<inter> I))) < r)"
1181 moreover have "\<And>n. Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
1182 using S bnd by (intro Inf_lower[where z=K]) auto
1183 ultimately obtain n where n: "N \<le> n" "r + Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
1184 by (auto simp: not_less field_simps)
1185 with N[OF n(1)] mono[OF _ `y \<in> I`, of "S n"] S(1)[THEN spec, of n] y
1191 text{* Another limit point characterization. *}
1193 lemma islimpt_sequential:
1194 fixes x :: "'a::metric_space"
1195 shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
1199 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"
1200 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
1202 have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
1205 { fix e::real assume "e>0"
1206 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
1207 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
1208 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)
1209 moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
1210 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
1212 hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
1213 unfolding Lim_sequentially using f by auto
1214 ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
1217 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
1218 { fix e::real assume "e>0"
1219 then obtain N where "dist (f N) x < e" using f(2) by auto
1220 moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
1221 ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
1223 thus ?lhs unfolding islimpt_approachable by auto
1226 lemma Lim_inv: (* TODO: delete *)
1227 fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
1228 assumes "(f ---> l) A" and "l \<noteq> 0"
1229 shows "((inverse o f) ---> inverse l) A"
1230 unfolding o_def using assms by (rule tendsto_inverse)
1233 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1234 shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
1235 by (simp add: Lim dist_norm)
1237 lemma Lim_null_comparison:
1238 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1239 assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
1240 shows "(f ---> 0) net"
1241 proof(simp add: tendsto_iff, rule+)
1242 fix e::real assume "0<e"
1244 assume "norm (f x) \<le> g x" "dist (g x) 0 < e"
1245 hence "dist (f x) 0 < e" by (simp add: dist_norm)
1247 thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
1248 using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
1249 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]
1250 using assms `e>0` unfolding tendsto_iff by auto
1253 lemma Lim_transform_bound:
1254 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1255 fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
1256 assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net" "(g ---> 0) net"
1257 shows "(f ---> 0) net"
1258 proof (rule tendstoI)
1259 fix e::real assume "e>0"
1261 assume "norm (f x) \<le> norm (g x)" "dist (g x) 0 < e"
1262 hence "dist (f x) 0 < e" by (simp add: dist_norm)}
1263 thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
1264 using eventually_and[of "\<lambda>x. norm (f x) \<le> norm (g x)" "\<lambda>x. dist (g x) 0 < e" net]
1265 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]
1266 using assms `e>0` unfolding tendsto_iff by blast
1269 text{* Deducing things about the limit from the elements. *}
1271 lemma Lim_in_closed_set:
1272 assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
1275 assume "l \<notin> S"
1276 with `closed S` have "open (- S)" "l \<in> - S"
1277 by (simp_all add: open_Compl)
1278 with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
1279 by (rule topological_tendstoD)
1280 with assms(2) have "eventually (\<lambda>x. False) net"
1281 by (rule eventually_elim2) simp
1282 with assms(3) show "False"
1283 by (simp add: eventually_False)
1286 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
1288 lemma Lim_dist_ubound:
1289 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
1290 shows "dist a l <= e"
1292 assume "\<not> dist a l \<le> e"
1293 then have "0 < dist a l - e" by simp
1294 with assms(2) have "eventually (\<lambda>x. dist (f x) l < dist a l - e) net"
1296 with assms(3) have "eventually (\<lambda>x. dist a (f x) \<le> e \<and> dist (f x) l < dist a l - e) net"
1297 by (rule eventually_conjI)
1298 then obtain w where "dist a (f w) \<le> e" "dist (f w) l < dist a l - e"
1299 using assms(1) eventually_happens by auto
1300 hence "dist a (f w) + dist (f w) l < e + (dist a l - e)"
1301 by (rule add_le_less_mono)
1302 hence "dist a (f w) + dist (f w) l < dist a l"
1304 also have "\<dots> \<le> dist a (f w) + dist (f w) l"
1305 by (rule dist_triangle)
1306 finally show False by simp
1309 lemma Lim_norm_ubound:
1310 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1311 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
1312 shows "norm(l) <= e"
1314 assume "\<not> norm l \<le> e"
1315 then have "0 < norm l - e" by simp
1316 with assms(2) have "eventually (\<lambda>x. dist (f x) l < norm l - e) net"
1318 with assms(3) have "eventually (\<lambda>x. norm (f x) \<le> e \<and> dist (f x) l < norm l - e) net"
1319 by (rule eventually_conjI)
1320 then obtain w where "norm (f w) \<le> e" "dist (f w) l < norm l - e"
1321 using assms(1) eventually_happens by auto
1322 hence "norm (f w - l) < norm l - e" "norm (f w) \<le> e" by (simp_all add: dist_norm)
1323 hence "norm (f w - l) + norm (f w) < norm l" by simp
1324 hence "norm (f w - l - f w) < norm l" by (rule le_less_trans [OF norm_triangle_ineq4])
1325 thus False using `\<not> norm l \<le> e` by simp
1328 lemma Lim_norm_lbound:
1329 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1330 assumes "\<not> (trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. e <= norm(f x)) net"
1331 shows "e \<le> norm l"
1333 assume "\<not> e \<le> norm l"
1334 then have "0 < e - norm l" by simp
1335 with assms(2) have "eventually (\<lambda>x. dist (f x) l < e - norm l) net"
1337 with assms(3) have "eventually (\<lambda>x. e \<le> norm (f x) \<and> dist (f x) l < e - norm l) net"
1338 by (rule eventually_conjI)
1339 then obtain w where "e \<le> norm (f w)" "dist (f w) l < e - norm l"
1340 using assms(1) eventually_happens by auto
1341 hence "norm (f w - l) + norm l < e" "e \<le> norm (f w)" by (simp_all add: dist_norm)
1342 hence "norm (f w - l) + norm l < norm (f w)" by (rule less_le_trans)
1343 hence "norm (f w - l + l) < norm (f w)" by (rule le_less_trans [OF norm_triangle_ineq])
1347 text{* Uniqueness of the limit, when nontrivial. *}
1350 fixes f :: "'a \<Rightarrow> 'b::t2_space"
1351 shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
1352 unfolding Lim_def using tendsto_unique[of net f] by auto
1354 text{* Limit under bilinear function *}
1357 assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
1358 shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
1359 using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
1360 by (rule bounded_bilinear.tendsto)
1362 text{* These are special for limits out of the same vector space. *}
1364 lemma Lim_within_id: "(id ---> a) (at a within s)"
1365 unfolding tendsto_def Limits.eventually_within eventually_at_topological
1368 lemma Lim_at_id: "(id ---> a) (at a)"
1369 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
1372 fixes a :: "'a::real_normed_vector"
1373 fixes l :: "'b::topological_space"
1374 shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
1377 { fix S assume "open S" "l \<in> S"
1378 with `?lhs` have "eventually (\<lambda>x. f x \<in> S) (at a)"
1379 by (rule topological_tendstoD)
1380 then obtain d where d: "d>0" "\<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S"
1381 unfolding Limits.eventually_at by fast
1382 { fix x::"'a" assume "x \<noteq> 0 \<and> dist x 0 < d"
1383 hence "f (a + x) \<in> S" using d
1384 apply(erule_tac x="x+a" in allE)
1385 by (auto simp add: add_commute dist_norm dist_commute)
1387 hence "\<exists>d>0. \<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
1389 hence "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
1390 unfolding Limits.eventually_at .
1392 thus "?rhs" by (rule topological_tendstoI)
1395 { fix S assume "open S" "l \<in> S"
1396 with `?rhs` have "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
1397 by (rule topological_tendstoD)
1398 then obtain d where d: "d>0" "\<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
1399 unfolding Limits.eventually_at by fast
1400 { fix x::"'a" assume "x \<noteq> a \<and> dist x a < d"
1401 hence "f x \<in> S" using d apply(erule_tac x="x-a" in allE)
1402 by(auto simp add: add_commute dist_norm dist_commute)
1404 hence "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S" using d(1) by auto
1405 hence "eventually (\<lambda>x. f x \<in> S) (at a)" unfolding Limits.eventually_at .
1407 thus "?lhs" by (rule topological_tendstoI)
1410 text{* It's also sometimes useful to extract the limit point from the filter. *}
1413 netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
1414 "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
1416 lemma netlimit_within:
1417 assumes "\<not> trivial_limit (at a within S)"
1418 shows "netlimit (at a within S) = a"
1419 unfolding netlimit_def
1420 apply (rule some_equality)
1421 apply (rule Lim_at_within)
1422 apply (rule LIM_ident)
1423 apply (erule tendsto_unique [OF assms])
1424 apply (rule Lim_at_within)
1425 apply (rule LIM_ident)
1429 fixes a :: "'a::{perfect_space,t2_space}"
1430 shows "netlimit (at a) = a"
1431 apply (subst within_UNIV[symmetric])
1432 using netlimit_within[of a UNIV]
1433 by (simp add: trivial_limit_at within_UNIV)
1435 text{* Transformation of limit. *}
1437 lemma Lim_transform:
1438 fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
1439 assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
1440 shows "(g ---> l) net"
1442 from assms have "((\<lambda>x. f x - g x - f x) ---> 0 - l) net" using tendsto_diff[of "\<lambda>x. f x - g x" 0 net f l] by auto
1443 thus "?thesis" using tendsto_minus [of "\<lambda> x. - g x" "-l" net] by auto
1446 lemma Lim_transform_eventually:
1447 "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
1448 apply (rule topological_tendstoI)
1449 apply (drule (2) topological_tendstoD)
1450 apply (erule (1) eventually_elim2, simp)
1453 lemma Lim_transform_within:
1454 assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
1455 and "(f ---> l) (at x within S)"
1456 shows "(g ---> l) (at x within S)"
1457 proof (rule Lim_transform_eventually)
1458 show "eventually (\<lambda>x. f x = g x) (at x within S)"
1459 unfolding eventually_within
1460 using assms(1,2) by auto
1461 show "(f ---> l) (at x within S)" by fact
1464 lemma Lim_transform_at:
1465 assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
1466 and "(f ---> l) (at x)"
1467 shows "(g ---> l) (at x)"
1468 proof (rule Lim_transform_eventually)
1469 show "eventually (\<lambda>x. f x = g x) (at x)"
1470 unfolding eventually_at
1471 using assms(1,2) by auto
1472 show "(f ---> l) (at x)" by fact
1475 text{* Common case assuming being away from some crucial point like 0. *}
1477 lemma Lim_transform_away_within:
1478 fixes a b :: "'a::t1_space"
1479 assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1480 and "(f ---> l) (at a within S)"
1481 shows "(g ---> l) (at a within S)"
1482 proof (rule Lim_transform_eventually)
1483 show "(f ---> l) (at a within S)" by fact
1484 show "eventually (\<lambda>x. f x = g x) (at a within S)"
1485 unfolding Limits.eventually_within eventually_at_topological
1486 by (rule exI [where x="- {b}"], simp add: open_Compl assms)
1489 lemma Lim_transform_away_at:
1490 fixes a b :: "'a::t1_space"
1491 assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1492 and fl: "(f ---> l) (at a)"
1493 shows "(g ---> l) (at a)"
1494 using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
1495 by (auto simp add: within_UNIV)
1497 text{* Alternatively, within an open set. *}
1499 lemma Lim_transform_within_open:
1500 assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
1501 and "(f ---> l) (at a)"
1502 shows "(g ---> l) (at a)"
1503 proof (rule Lim_transform_eventually)
1504 show "eventually (\<lambda>x. f x = g x) (at a)"
1505 unfolding eventually_at_topological
1506 using assms(1,2,3) by auto
1507 show "(f ---> l) (at a)" by fact
1510 text{* A congruence rule allowing us to transform limits assuming not at point. *}
1512 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
1514 lemma Lim_cong_within(*[cong add]*):
1515 assumes "a = b" "x = y" "S = T"
1516 assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
1517 shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
1518 unfolding tendsto_def Limits.eventually_within eventually_at_topological
1521 lemma Lim_cong_at(*[cong add]*):
1522 assumes "a = b" "x = y"
1523 assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
1524 shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
1525 unfolding tendsto_def eventually_at_topological
1528 text{* Useful lemmas on closure and set of possible sequential limits.*}
1530 lemma closure_sequential:
1531 fixes l :: "'a::metric_space"
1532 shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
1534 assume "?lhs" moreover
1535 { assume "l \<in> S"
1536 hence "?rhs" using tendsto_const[of l sequentially] by auto
1538 { assume "l islimpt S"
1539 hence "?rhs" unfolding islimpt_sequential by auto
1541 show "?rhs" unfolding closure_def by auto
1544 thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
1547 lemma closed_sequential_limits:
1548 fixes S :: "'a::metric_space set"
1549 shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
1550 unfolding closed_limpt
1551 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]
1554 lemma closure_approachable:
1555 fixes S :: "'a::metric_space set"
1556 shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
1557 apply (auto simp add: closure_def islimpt_approachable)
1558 by (metis dist_self)
1560 lemma closed_approachable:
1561 fixes S :: "'a::metric_space set"
1562 shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
1563 by (metis closure_closed closure_approachable)
1565 text{* Some other lemmas about sequences. *}
1567 lemma sequentially_offset:
1568 assumes "eventually (\<lambda>i. P i) sequentially"
1569 shows "eventually (\<lambda>i. P (i + k)) sequentially"
1570 using assms unfolding eventually_sequentially by (metis trans_le_add1)
1573 assumes "(f ---> l) sequentially"
1574 shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
1575 using assms unfolding tendsto_def
1576 by clarify (rule sequentially_offset, simp)
1578 lemma seq_offset_neg:
1579 "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
1580 apply (rule topological_tendstoI)
1581 apply (drule (2) topological_tendstoD)
1582 apply (simp only: eventually_sequentially)
1583 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
1587 lemma seq_offset_rev:
1588 "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
1589 apply (rule topological_tendstoI)
1590 apply (drule (2) topological_tendstoD)
1591 apply (simp only: eventually_sequentially)
1592 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
1595 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
1597 { fix e::real assume "e>0"
1598 hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
1599 using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
1600 by (metis le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
1602 thus ?thesis unfolding Lim_sequentially dist_norm by simp
1605 subsection {* More properties of closed balls. *}
1607 lemma closed_cball: "closed (cball x e)"
1608 unfolding cball_def closed_def
1609 unfolding Collect_neg_eq [symmetric] not_le
1610 apply (clarsimp simp add: open_dist, rename_tac y)
1611 apply (rule_tac x="dist x y - e" in exI, clarsimp)
1612 apply (rename_tac x')
1613 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
1617 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
1619 { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
1620 hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
1622 { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
1623 hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
1625 show ?thesis unfolding open_contains_ball by auto
1628 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
1629 by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball mem_def)
1631 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
1632 apply (simp add: interior_def, safe)
1633 apply (force simp add: open_contains_cball)
1634 apply (rule_tac x="ball x e" in exI)
1635 apply (simp add: subset_trans [OF ball_subset_cball])
1639 fixes x y :: "'a::{real_normed_vector,perfect_space}"
1640 shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
1643 { assume "e \<le> 0"
1644 hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
1645 have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
1647 hence "e > 0" by (metis not_less)
1649 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
1650 ultimately show "?rhs" by auto
1652 assume "?rhs" hence "e>0" by auto
1653 { fix d::real assume "d>0"
1654 have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1655 proof(cases "d \<le> dist x y")
1656 case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1658 case True hence False using `d \<le> dist x y` `d>0` by auto
1659 thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
1663 have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
1664 = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1665 unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
1666 also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
1667 using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
1668 unfolding scaleR_minus_left scaleR_one
1669 by (auto simp add: norm_minus_commute)
1670 also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
1671 unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
1672 unfolding left_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
1673 also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
1674 finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
1678 have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
1679 using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
1681 have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
1682 using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
1683 unfolding dist_norm by auto
1684 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
1687 case False hence "d > dist x y" by auto
1688 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1691 obtain z where **: "z \<noteq> y" "dist z y < min e d"
1692 using perfect_choose_dist[of "min e d" y]
1693 using `d > 0` `e>0` by auto
1694 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1696 using `z \<noteq> y` **
1697 by (rule_tac x=z in bexI, auto simp add: dist_commute)
1699 case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1700 using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
1703 thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
1706 lemma closure_ball_lemma:
1707 fixes x y :: "'a::real_normed_vector"
1708 assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
1709 proof (rule islimptI)
1710 fix T assume "y \<in> T" "open T"
1711 then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
1712 unfolding open_dist by fast
1713 (* choose point between x and y, within distance r of y. *)
1714 def k \<equiv> "min 1 (r / (2 * dist x y))"
1715 def z \<equiv> "y + scaleR k (x - y)"
1716 have z_def2: "z = x + scaleR (1 - k) (y - x)"
1717 unfolding z_def by (simp add: algebra_simps)
1719 unfolding z_def k_def using `0 < r`
1720 by (simp add: dist_norm min_def)
1721 hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
1722 have "dist x z < dist x y"
1723 unfolding z_def2 dist_norm
1724 apply (simp add: norm_minus_commute)
1725 apply (simp only: dist_norm [symmetric])
1726 apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
1727 apply (rule mult_strict_right_mono)
1728 apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
1729 apply (simp add: zero_less_dist_iff `x \<noteq> y`)
1731 hence "z \<in> ball x (dist x y)" by simp
1733 unfolding z_def k_def using `x \<noteq> y` `0 < r`
1734 by (simp add: min_def)
1735 show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
1736 using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
1741 fixes x :: "'a::real_normed_vector"
1742 shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
1743 apply (rule equalityI)
1744 apply (rule closure_minimal)
1745 apply (rule ball_subset_cball)
1746 apply (rule closed_cball)
1747 apply (rule subsetI, rename_tac y)
1748 apply (simp add: le_less [where 'a=real])
1750 apply (rule subsetD [OF closure_subset], simp)
1751 apply (simp add: closure_def)
1753 apply (rule closure_ball_lemma)
1754 apply (simp add: zero_less_dist_iff)
1757 (* In a trivial vector space, this fails for e = 0. *)
1758 lemma interior_cball:
1759 fixes x :: "'a::{real_normed_vector, perfect_space}"
1760 shows "interior (cball x e) = ball x e"
1761 proof(cases "e\<ge>0")
1762 case False note cs = this
1763 from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
1764 { fix y assume "y \<in> cball x e"
1765 hence False unfolding mem_cball using dist_nz[of x y] cs by auto }
1766 hence "cball x e = {}" by auto
1767 hence "interior (cball x e) = {}" using interior_empty by auto
1768 ultimately show ?thesis by blast
1770 case True note cs = this
1771 have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
1772 { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
1773 then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
1775 then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
1776 using perfect_choose_dist [of d] by auto
1777 have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
1778 hence xa_cball:"xa \<in> cball x e" using as(1) by auto
1780 hence "y \<in> ball x e" proof(cases "x = y")
1782 hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
1783 thus "y \<in> ball x e" using `x = y ` by simp
1786 have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
1787 using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
1788 hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
1789 have "y - x \<noteq> 0" using `x \<noteq> y` by auto
1790 hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
1791 using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
1793 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)"
1794 by (auto simp add: dist_norm algebra_simps)
1795 also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1796 by (auto simp add: algebra_simps)
1797 also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
1799 also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
1800 finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
1801 thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
1803 hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
1804 ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
1807 lemma frontier_ball:
1808 fixes a :: "'a::real_normed_vector"
1809 shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
1810 apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
1811 apply (simp add: set_eq_iff)
1814 lemma frontier_cball:
1815 fixes a :: "'a::{real_normed_vector, perfect_space}"
1816 shows "frontier(cball a e) = {x. dist a x = e}"
1817 apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
1818 apply (simp add: set_eq_iff)
1821 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
1822 apply (simp add: set_eq_iff not_le)
1823 by (metis zero_le_dist dist_self order_less_le_trans)
1824 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
1826 lemma cball_eq_sing:
1827 fixes x :: "'a::{metric_space,perfect_space}"
1828 shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
1829 proof (rule linorder_cases)
1831 obtain a where "a \<noteq> x" "dist a x < e"
1832 using perfect_choose_dist [OF e] by auto
1833 hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
1834 with e show ?thesis by (auto simp add: set_eq_iff)
1838 fixes x :: "'a::metric_space"
1839 shows "e = 0 ==> cball x e = {x}"
1840 by (auto simp add: set_eq_iff)
1842 text{* For points in the interior, localization of limits makes no difference. *}
1844 lemma eventually_within_interior:
1845 assumes "x \<in> interior S"
1846 shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
1848 from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S"
1849 unfolding interior_def by fast
1851 then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
1852 unfolding Limits.eventually_within Limits.eventually_at_topological
1854 with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
1857 unfolding Limits.eventually_at_topological by auto
1859 { assume "?rhs" hence "?lhs"
1860 unfolding Limits.eventually_within
1861 by (auto elim: eventually_elim1)
1866 lemma at_within_interior:
1867 "x \<in> interior S \<Longrightarrow> at x within S = at x"
1868 by (simp add: filter_eq_iff eventually_within_interior)
1870 lemma lim_within_interior:
1871 "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
1872 by (simp add: at_within_interior)
1874 lemma netlimit_within_interior:
1875 fixes x :: "'a::{t2_space,perfect_space}"
1876 assumes "x \<in> interior S"
1877 shows "netlimit (at x within S) = x"
1878 using assms by (simp add: at_within_interior netlimit_at)
1880 subsection{* Boundedness. *}
1882 (* FIXME: This has to be unified with BSEQ!! *)
1884 bounded :: "'a::metric_space set \<Rightarrow> bool" where
1885 "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
1887 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
1888 unfolding bounded_def
1890 apply (rule_tac x="dist a x + e" in exI, clarify)
1891 apply (drule (1) bspec)
1892 apply (erule order_trans [OF dist_triangle add_left_mono])
1896 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
1897 unfolding bounded_any_center [where a=0]
1898 by (simp add: dist_norm)
1900 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
1901 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
1902 by (metis bounded_def subset_eq)
1904 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
1905 by (metis bounded_subset interior_subset)
1907 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
1909 from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
1910 { fix y assume "y \<in> closure S"
1911 then obtain f where f: "\<forall>n. f n \<in> S" "(f ---> y) sequentially"
1912 unfolding closure_sequential by auto
1913 have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
1914 hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
1915 by (rule eventually_mono, simp add: f(1))
1916 have "dist x y \<le> a"
1917 apply (rule Lim_dist_ubound [of sequentially f])
1918 apply (rule trivial_limit_sequentially)
1923 thus ?thesis unfolding bounded_def by auto
1926 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
1927 apply (simp add: bounded_def)
1928 apply (rule_tac x=x in exI)
1929 apply (rule_tac x=e in exI)
1933 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
1934 by (metis ball_subset_cball bounded_cball bounded_subset)
1936 lemma finite_imp_bounded[intro]:
1937 fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"
1939 { fix a and F :: "'a set" assume as:"bounded F"
1940 then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
1941 hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
1942 hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
1944 thus ?thesis using finite_induct[of S bounded] using bounded_empty assms by auto
1947 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
1948 apply (auto simp add: bounded_def)
1949 apply (rename_tac x y r s)
1950 apply (rule_tac x=x in exI)
1951 apply (rule_tac x="max r (dist x y + s)" in exI)
1952 apply (rule ballI, rename_tac z, safe)
1953 apply (drule (1) bspec, simp)
1954 apply (drule (1) bspec)
1955 apply (rule min_max.le_supI2)
1956 apply (erule order_trans [OF dist_triangle add_left_mono])
1959 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
1960 by (induct rule: finite_induct[of F], auto)
1962 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
1963 apply (simp add: bounded_iff)
1964 apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
1967 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
1968 by (metis Int_lower1 Int_lower2 bounded_subset)
1970 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
1971 apply (metis Diff_subset bounded_subset)
1974 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
1975 by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
1977 lemma not_bounded_UNIV[simp, intro]:
1978 "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
1979 proof(auto simp add: bounded_pos not_le)
1980 obtain x :: 'a where "x \<noteq> 0"
1981 using perfect_choose_dist [OF zero_less_one] by fast
1982 fix b::real assume b: "b >0"
1983 have b1: "b +1 \<ge> 0" using b by simp
1984 with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
1985 by (simp add: norm_sgn)
1986 then show "\<exists>x::'a. b < norm x" ..
1989 lemma bounded_linear_image:
1990 assumes "bounded S" "bounded_linear f"
1991 shows "bounded(f ` S)"
1993 from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
1994 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)
1995 { fix x assume "x\<in>S"
1996 hence "norm x \<le> b" using b by auto
1997 hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
1998 by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
2000 thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
2001 using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
2004 lemma bounded_scaling:
2005 fixes S :: "'a::real_normed_vector set"
2006 shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
2007 apply (rule bounded_linear_image, assumption)
2008 apply (rule scaleR.bounded_linear_right)
2011 lemma bounded_translation:
2012 fixes S :: "'a::real_normed_vector set"
2013 assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
2015 from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
2016 { fix x assume "x\<in>S"
2017 hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
2019 thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
2020 by (auto intro!: add exI[of _ "b + norm a"])
2024 text{* Some theorems on sups and infs using the notion "bounded". *}
2027 fixes S :: "real set"
2028 shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"
2029 by (simp add: bounded_iff)
2031 lemma bounded_has_Sup:
2032 fixes S :: "real set"
2033 assumes "bounded S" "S \<noteq> {}"
2034 shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
2036 fix x assume "x\<in>S"
2037 thus "x \<le> Sup S"
2038 by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
2040 show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
2041 by (metis SupInf.Sup_least)
2045 fixes S :: "real set"
2046 shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
2047 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal)
2049 lemma Sup_insert_finite:
2050 fixes S :: "real set"
2051 shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
2052 apply (rule Sup_insert)
2053 apply (rule finite_imp_bounded)
2056 lemma bounded_has_Inf:
2057 fixes S :: "real set"
2058 assumes "bounded S" "S \<noteq> {}"
2059 shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
2061 fix x assume "x\<in>S"
2062 from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
2063 thus "x \<ge> Inf S" using `x\<in>S`
2064 by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
2066 show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
2067 by (metis SupInf.Inf_greatest)
2071 fixes S :: "real set"
2072 shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
2073 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)
2074 lemma Inf_insert_finite:
2075 fixes S :: "real set"
2076 shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
2077 by (rule Inf_insert, rule finite_imp_bounded, simp)
2080 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
2081 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
2082 apply (frule isGlb_isLb)
2083 apply (frule_tac x = y in isGlb_isLb)
2084 apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
2087 subsection {* Equivalent versions of compactness *}
2089 subsubsection{* Sequential compactness *}
2092 compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
2093 "compact S \<longleftrightarrow>
2094 (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
2095 (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
2098 assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
2100 unfolding compact_def using assms by fast
2103 assumes "compact S" "\<forall>n. f n \<in> S"
2104 obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
2105 using assms unfolding compact_def by fast
2108 A metric space (or topological vector space) is said to have the
2109 Heine-Borel property if every closed and bounded subset is compact.
2113 assumes bounded_imp_convergent_subsequence:
2114 "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
2115 \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2117 lemma bounded_closed_imp_compact:
2118 fixes s::"'a::heine_borel set"
2119 assumes "bounded s" and "closed s" shows "compact s"
2120 proof (unfold compact_def, clarify)
2121 fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
2122 obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
2123 using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
2124 from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
2125 have "l \<in> s" using `closed s` fr l
2126 unfolding closed_sequential_limits by blast
2127 show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2128 using `l \<in> s` r l by blast
2131 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
2133 show "0 \<le> r 0" by auto
2135 fix n assume "n \<le> r n"
2136 moreover have "r n < r (Suc n)"
2137 using assms [unfolded subseq_def] by auto
2138 ultimately show "Suc n \<le> r (Suc n)" by auto
2141 lemma eventually_subseq:
2142 assumes r: "subseq r"
2143 shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
2144 unfolding eventually_sequentially
2145 by (metis subseq_bigger [OF r] le_trans)
2148 "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
2149 unfolding tendsto_def eventually_sequentially o_def
2150 by (metis subseq_bigger le_trans)
2152 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
2154 apply (rule_tac x="nat_rec e f" in exI)
2156 apply (rule def_nat_rec_0, simp)
2157 apply (rule allI, rule def_nat_rec_Suc, simp)
2158 apply (rule allI, rule impI, rule ext)
2160 apply (induct_tac x)
2162 apply (erule_tac x="n" in allE)
2166 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
2167 assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
2168 shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N. abs(s n - l) < e"
2170 have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
2171 then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
2172 { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
2174 obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
2175 have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
2176 with n have "s N \<le> t - e" using `e>0` by auto
2177 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 }
2178 hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
2179 hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto }
2180 thus ?thesis by blast
2183 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
2184 assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
2185 shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
2186 using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
2187 unfolding monoseq_def incseq_def
2188 apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
2189 unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
2191 (* TODO: merge this lemma with the ones above *)
2192 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"
2193 assumes "bounded {s n| n::nat. True}" "\<forall>n. (s n) \<le>(s(Suc n))"
2194 shows "\<exists>l. (s ---> l) sequentially"
2196 obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le> a" using assms(1)[unfolded bounded_iff] by auto
2198 have "\<And> n. n\<ge>m \<longrightarrow> (s m) \<le> (s n)"
2199 apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)
2200 apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq) }
2201 hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto
2202 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]
2203 unfolding monoseq_def by auto
2204 thus ?thesis unfolding Lim_sequentially apply(rule_tac x="l" in exI)
2205 unfolding dist_norm by auto
2208 lemma compact_real_lemma:
2209 assumes "\<forall>n::nat. abs(s n) \<le> b"
2210 shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
2212 obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
2213 using seq_monosub[of s] by auto
2214 thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
2215 unfolding tendsto_iff dist_norm eventually_sequentially by auto
2218 instance real :: heine_borel
2220 fix s :: "real set" and f :: "nat \<Rightarrow> real"
2221 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2222 then obtain b where b: "\<forall>n. abs (f n) \<le> b"
2223 unfolding bounded_iff by auto
2224 obtain l :: real and r :: "nat \<Rightarrow> nat" where
2225 r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
2226 using compact_real_lemma [OF b] by auto
2227 thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2231 lemma bounded_component: "bounded s \<Longrightarrow>
2232 bounded ((\<lambda>x. x $$ i) ` (s::'a::euclidean_space set))"
2233 unfolding bounded_def
2235 apply (rule_tac x="x $$ i" in exI)
2236 apply (rule_tac x="e" in exI)
2238 apply (rule order_trans[OF dist_nth_le],simp)
2241 lemma compact_lemma:
2242 fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
2243 assumes "bounded s" and "\<forall>n. f n \<in> s"
2244 shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>
2245 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2247 fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"
2248 have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
2249 hence "\<exists>l::'a. \<exists>r. subseq r \<and>
2250 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2251 proof(induct d) case empty thus ?case unfolding subseq_def by auto
2252 next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto
2253 have s': "bounded ((\<lambda>x. x $$ k) ` s)" using `bounded s` by (rule bounded_component)
2254 obtain l1::"'a" and r1 where r1:"subseq r1" and
2255 lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially"
2256 using insert(3) using insert(4) by auto
2257 have f': "\<forall>n. f (r1 n) $$ k \<in> (\<lambda>x. x $$ k) ` s" using `\<forall>n. f n \<in> s` by simp
2258 obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $$ k) ---> l2) sequentially"
2259 using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
2260 def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
2261 using r1 and r2 unfolding r_def o_def subseq_def by auto
2263 def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"
2264 { fix e::real assume "e>0"
2265 from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially" by blast
2266 from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $$ k) l2 < e) sequentially" by (rule tendstoD)
2267 from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$ i) (l1 $$ i) < e) sequentially"
2268 by (rule eventually_subseq)
2269 have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$ i) (l $$ i) < e) sequentially"
2270 using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def
2271 using insert.prems by auto
2273 ultimately show ?case by auto
2275 thus "\<exists>l::'a. \<exists>r. subseq r \<and>
2276 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2277 apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe
2278 apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe
2279 apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)
2280 apply(erule_tac x=i in ballE)
2281 proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a
2282 assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"
2283 hence *:"i\<ge>DIM('a)" by auto
2284 thus "dist (f (r n) $$ i) (l $$ i) < e" using e by auto
2288 instance euclidean_space \<subseteq> heine_borel
2290 fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"
2291 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2292 then obtain l::'a and r where r: "subseq r"
2293 and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $$ i) (l $$ i) < e) sequentially"
2294 using compact_lemma [OF s f] by blast
2295 let ?d = "{..<DIM('a)}"
2296 { fix e::real assume "e>0"
2297 hence "0 < e / (real_of_nat (card ?d))"
2298 using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
2299 with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))) sequentially"
2302 { fix n assume n: "\<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))"
2303 have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $$ i) (l $$ i))"
2304 apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)
2305 also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
2306 apply(rule setsum_strict_mono) using n by auto
2307 finally have "dist (f (r n)) l < e" unfolding setsum_constant
2308 using DIM_positive[where 'a='a] by auto
2310 ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
2311 by (rule eventually_elim1)
2313 hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
2314 with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
2317 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
2318 unfolding bounded_def
2320 apply (rule_tac x="a" in exI)
2321 apply (rule_tac x="e" in exI)
2323 apply (drule (1) bspec)
2324 apply (simp add: dist_Pair_Pair)
2325 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
2328 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
2329 unfolding bounded_def
2331 apply (rule_tac x="b" in exI)
2332 apply (rule_tac x="e" in exI)
2334 apply (drule (1) bspec)
2335 apply (simp add: dist_Pair_Pair)
2336 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
2339 instance prod :: (heine_borel, heine_borel) heine_borel
2341 fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
2342 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2343 from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
2344 from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
2345 obtain l1 r1 where r1: "subseq r1"
2346 and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
2347 using bounded_imp_convergent_subsequence [OF s1 f1]
2348 unfolding o_def by fast
2349 from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
2350 from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
2351 obtain l2 r2 where r2: "subseq r2"
2352 and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
2353 using bounded_imp_convergent_subsequence [OF s2 f2]
2354 unfolding o_def by fast
2355 have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
2356 using lim_subseq [OF r2 l1] unfolding o_def .
2357 have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
2358 using tendsto_Pair [OF l1' l2] unfolding o_def by simp
2359 have r: "subseq (r1 \<circ> r2)"
2360 using r1 r2 unfolding subseq_def by simp
2361 show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2365 subsubsection{* Completeness *}
2368 "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
2369 unfolding Cauchy_def by blast
2372 complete :: "'a::metric_space set \<Rightarrow> bool" where
2373 "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
2374 --> (\<exists>l \<in> s. (f ---> l) sequentially))"
2376 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
2381 with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
2382 by (erule_tac x="e/2" in allE) auto
2384 assume nm:"N \<le> m \<and> N \<le> n"
2385 hence "dist (s m) (s n) < e" using N
2386 using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
2389 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
2393 unfolding cauchy_def
2397 unfolding cauchy_def
2398 using dist_triangle_half_l
2402 lemma convergent_imp_cauchy:
2403 "(s ---> l) sequentially ==> Cauchy s"
2404 proof(simp only: cauchy_def, rule, rule)
2405 fix e::real assume "e>0" "(s ---> l) sequentially"
2406 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
2407 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
2410 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
2412 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
2413 hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
2415 have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
2416 then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
2417 unfolding bounded_any_center [where a="s N"] by auto
2418 ultimately show "?thesis"
2419 unfolding bounded_any_center [where a="s N"]
2420 apply(rule_tac x="max a 1" in exI) apply auto
2421 apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto
2424 lemma compact_imp_complete: assumes "compact s" shows "complete s"
2426 { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
2427 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
2429 note lr' = subseq_bigger [OF lr(2)]
2431 { fix e::real assume "e>0"
2432 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
2433 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
2434 { fix n::nat assume n:"n \<ge> max N M"
2435 have "dist ((f \<circ> r) n) l < e/2" using n M by auto
2436 moreover have "r n \<ge> N" using lr'[of n] n by auto
2437 hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
2438 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) }
2439 hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast }
2440 hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding Lim_sequentially by auto }
2441 thus ?thesis unfolding complete_def by auto
2444 instance heine_borel < complete_space
2446 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2447 hence "bounded (range f)"
2448 by (rule cauchy_imp_bounded)
2449 hence "compact (closure (range f))"
2450 using bounded_closed_imp_compact [of "closure (range f)"] by auto
2451 hence "complete (closure (range f))"
2452 by (rule compact_imp_complete)
2453 moreover have "\<forall>n. f n \<in> closure (range f)"
2454 using closure_subset [of "range f"] by auto
2455 ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
2456 using `Cauchy f` unfolding complete_def by auto
2457 then show "convergent f"
2458 unfolding convergent_def by auto
2461 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
2462 proof(simp add: complete_def, rule, rule)
2463 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2464 hence "convergent f" by (rule Cauchy_convergent)
2465 thus "\<exists>l. f ----> l" unfolding convergent_def .
2468 lemma complete_imp_closed: assumes "complete s" shows "closed s"
2470 { fix x assume "x islimpt s"
2471 then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
2472 unfolding islimpt_sequential by auto
2473 then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
2474 using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
2475 hence "x \<in> s" using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
2477 thus "closed s" unfolding closed_limpt by auto
2480 lemma complete_eq_closed:
2481 fixes s :: "'a::complete_space set"
2482 shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
2484 assume ?lhs thus ?rhs by (rule complete_imp_closed)
2487 { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
2488 then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
2489 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 }
2490 thus ?lhs unfolding complete_def by auto
2493 lemma convergent_eq_cauchy:
2494 fixes s :: "nat \<Rightarrow> 'a::complete_space"
2495 shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s" (is "?lhs = ?rhs")
2497 assume ?lhs then obtain l where "(s ---> l) sequentially" by auto
2498 thus ?rhs using convergent_imp_cauchy by auto
2500 assume ?rhs thus ?lhs using complete_univ[unfolded complete_def, THEN spec[where x=s]] by auto
2503 lemma convergent_imp_bounded:
2504 fixes s :: "nat \<Rightarrow> 'a::metric_space"
2505 shows "(s ---> l) sequentially ==> bounded (s ` (UNIV::(nat set)))"
2506 using convergent_imp_cauchy[of s]
2507 using cauchy_imp_bounded[of s]
2511 subsubsection{* Total boundedness *}
2513 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
2514 "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
2515 declare helper_1.simps[simp del]
2517 lemma compact_imp_totally_bounded:
2519 shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
2520 proof(rule, rule, rule ccontr)
2521 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)"
2522 def x \<equiv> "helper_1 s e"
2524 have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
2525 proof(induct_tac rule:nat_less_induct)
2526 fix n def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
2527 assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
2528 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
2529 then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
2530 have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
2531 apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
2532 thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
2534 hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
2535 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
2536 from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
2537 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
2539 using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
2540 using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
2541 using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
2544 subsubsection{* Heine-Borel theorem *}
2546 text {* Following Burkill \& Burkill vol. 2. *}
2548 lemma heine_borel_lemma: fixes s::"'a::metric_space set"
2549 assumes "compact s" "s \<subseteq> (\<Union> t)" "\<forall>b \<in> t. open b"
2550 shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
2552 assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"
2553 hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto
2555 have "1 / real (n + 1) > 0" by auto
2556 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 }
2557 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
2558 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)"
2559 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
2561 then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
2562 using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
2564 obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
2565 then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"
2566 using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto
2568 then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"
2569 using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
2571 obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
2572 have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
2573 apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
2574 using subseq_bigger[OF r, of "N1 + N2"] by auto
2576 def x \<equiv> "(f (r (N1 + N2)))"
2577 have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
2578 using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto
2579 have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto
2580 then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto
2582 have "dist x l < e/2" using N1 unfolding x_def o_def by auto
2583 hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)
2585 thus False using e and `y\<notin>b` by auto
2588 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2589 \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"
2591 fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"
2592 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
2593 hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto
2594 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
2595 then obtain bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast
2597 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
2598 then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto
2600 have "finite (bb ` k)" using k(1) by auto
2602 { fix x assume "x\<in>s"
2603 hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3) unfolding subset_eq by auto
2604 hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast
2605 hence "x \<in> \<Union>(bb ` k)" using Union_iff[of x "bb ` k"] by auto
2607 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
2610 subsubsection {* Bolzano-Weierstrass property *}
2612 lemma heine_borel_imp_bolzano_weierstrass:
2613 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'))"
2614 "infinite t" "t \<subseteq> s"
2615 shows "\<exists>x \<in> s. x islimpt t"
2617 assume "\<not> (\<exists>x \<in> s. x islimpt t)"
2618 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
2619 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
2620 obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
2621 using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
2622 from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
2623 { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
2624 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
2625 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 }
2626 hence "inj_on f t" unfolding inj_on_def by simp
2627 hence "infinite (f ` t)" using assms(2) using finite_imageD by auto
2629 { fix x assume "x\<in>t" "f x \<notin> g"
2630 from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
2631 then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
2632 hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
2633 hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto }
2634 hence "f ` t \<subseteq> g" by auto
2635 ultimately show False using g(2) using finite_subset by auto
2638 subsubsection {* Complete the chain of compactness variants *}
2640 lemma islimpt_range_imp_convergent_subsequence:
2641 fixes f :: "nat \<Rightarrow> 'a::metric_space"
2642 assumes "l islimpt (range f)"
2643 shows "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2644 proof (intro exI conjI)
2645 have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"
2646 using assms unfolding islimpt_def
2647 by (drule_tac x="ball l e" in spec)
2648 (auto simp add: zero_less_dist_iff dist_commute)
2650 def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"
2651 have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"
2652 unfolding t_def by (rule LeastI2_ex [OF * conjunct1])
2653 have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"
2654 unfolding t_def by (rule LeastI2_ex [OF * conjunct2])
2655 have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"
2656 unfolding t_def by (simp add: Least_le)
2657 have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"
2658 unfolding t_def by (drule not_less_Least) simp
2659 have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"
2661 apply (erule f_t_neq)
2662 apply (erule (1) less_le_trans [OF f_t_closer])
2664 have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"
2665 by (drule f_t_closer) auto
2666 have t_less: "\<And>e. 0 < e \<Longrightarrow> t e < t (dist (f (t e)) l)"
2667 apply (subst less_le)
2669 apply (rule t_antimono)
2670 apply (erule f_t_neq)
2671 apply (erule f_t_closer [THEN less_imp_le])
2672 apply (rule t_dist_f_neq [symmetric])
2673 apply (erule f_t_neq)
2675 have dist_f_t_less':
2676 "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"
2677 apply (simp add: le_less)
2679 apply (rule less_trans)
2680 apply (erule f_t_closer)
2681 apply (rule le_less_trans)
2682 apply (erule less_tD)
2683 apply (erule f_t_neq)
2684 apply (erule f_t_closer)
2686 apply (erule f_t_closer)
2689 def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"
2690 have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"
2691 unfolding r_def by simp_all
2692 have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"
2693 by (induct_tac n) (simp_all add: r_simps f_t_neq)
2696 unfolding subseq_Suc_iff
2699 apply (simp_all add: r_simps)
2700 apply (rule t_less, rule zero_less_one)
2701 apply (rule t_less, rule f_r_neq)
2703 show "((f \<circ> r) ---> l) sequentially"
2704 unfolding Lim_sequentially o_def
2705 apply (clarify, rule_tac x="t e" in exI, clarify)
2706 apply (drule le_trans, rule seq_suble [OF `subseq r`])
2707 apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)
2711 lemma finite_range_imp_infinite_repeats:
2712 fixes f :: "nat \<Rightarrow> 'a"
2713 assumes "finite (range f)"
2714 shows "\<exists>k. infinite {n. f n = k}"
2716 { fix A :: "'a set" assume "finite A"
2717 hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"
2719 case empty thus ?case by simp
2723 proof (cases "finite {n. f n = x}")
2725 with `infinite {n. f n \<in> insert x A}`
2726 have "infinite {n. f n \<in> A}" by simp
2727 thus "\<exists>k. infinite {n. f n = k}" by (rule insert)
2729 case False thus "\<exists>k. infinite {n. f n = k}" ..
2733 from assms show "\<exists>k. infinite {n. f n = k}"
2737 lemma bolzano_weierstrass_imp_compact:
2738 fixes s :: "'a::metric_space set"
2739 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2742 { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
2743 have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2744 proof (cases "finite (range f)")
2746 hence "\<exists>l. infinite {n. f n = l}"
2747 by (rule finite_range_imp_infinite_repeats)
2748 then obtain l where "infinite {n. f n = l}" ..
2749 hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"
2750 by (rule infinite_enumerate)
2751 then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto
2752 hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2753 unfolding o_def by (simp add: fr tendsto_const)
2754 hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2756 from f have "\<forall>n. f (r n) \<in> s" by simp
2757 hence "l \<in> s" by (simp add: fr)
2758 thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2759 by (rule rev_bexI) fact
2762 with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto
2763 then obtain l where "l \<in> s" "l islimpt (range f)" ..
2764 have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2765 using `l islimpt (range f)`
2766 by (rule islimpt_range_imp_convergent_subsequence)
2767 with `l \<in> s` show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
2770 thus ?thesis unfolding compact_def by auto
2773 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where
2774 "helper_2 beyond 0 = beyond 0" |
2775 "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"
2777 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"
2778 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2781 assume "\<not> bounded s"
2782 then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"
2783 unfolding bounded_any_center [where a=undefined]
2784 apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto
2785 hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"
2786 unfolding linorder_not_le by auto
2787 def x \<equiv> "helper_2 beyond"
2789 { fix m n ::nat assume "m<n"
2790 hence "dist undefined (x m) + 1 < dist undefined (x n)"
2792 case 0 thus ?case by auto
2795 have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"
2796 unfolding x_def and helper_2.simps
2797 using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto
2798 thus ?case proof(cases "m < n")
2799 case True thus ?thesis using Suc and * by auto
2801 case False hence "m = n" using Suc(2) by auto
2802 thus ?thesis using * by auto
2805 { fix m n ::nat assume "m\<noteq>n"
2806 have "1 < dist (x m) (x n)"
2809 hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto
2810 thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith
2812 case False hence "n<m" using `m\<noteq>n` by auto
2813 hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto
2814 thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith
2815 qed } note ** = this
2816 { fix a b assume "x a = x b" "a \<noteq> b"
2817 hence False using **[of a b] by auto }
2818 hence "inj x" unfolding inj_on_def by auto
2822 proof(cases "n = 0")
2823 case True thus ?thesis unfolding x_def using beyond by auto
2825 case False then obtain z where "n = Suc z" using not0_implies_Suc by auto
2826 thus ?thesis unfolding x_def using beyond by auto
2828 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
2830 then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto
2831 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
2832 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"]]
2833 unfolding dist_nz by auto
2834 show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto
2837 lemma sequence_infinite_lemma:
2838 fixes f :: "nat \<Rightarrow> 'a::t1_space"
2839 assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"
2840 shows "infinite (range f)"
2842 assume "finite (range f)"
2843 hence "closed (range f)" by (rule finite_imp_closed)
2844 hence "open (- range f)" by (rule open_Compl)
2845 from assms(1) have "l \<in> - range f" by auto
2846 from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
2847 using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)
2848 thus False unfolding eventually_sequentially by auto
2851 lemma closure_insert:
2852 fixes x :: "'a::t1_space"
2853 shows "closure (insert x s) = insert x (closure s)"
2854 apply (rule closure_unique)
2855 apply (rule conjI [OF insert_mono [OF closure_subset]])
2856 apply (rule conjI [OF closed_insert [OF closed_closure]])
2857 apply (simp add: closure_minimal)
2860 lemma islimpt_insert:
2861 fixes x :: "'a::t1_space"
2862 shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
2864 assume *: "x islimpt (insert a s)"
2866 proof (rule islimptI)
2867 fix t assume t: "x \<in> t" "open t"
2868 show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
2869 proof (cases "x = a")
2871 obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
2872 using * t by (rule islimptE)
2873 with `x = a` show ?thesis by auto
2876 with t have t': "x \<in> t - {a}" "open (t - {a})"
2877 by (simp_all add: open_Diff)
2878 obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
2879 using * t' by (rule islimptE)
2880 thus ?thesis by auto
2884 assume "x islimpt s" thus "x islimpt (insert a s)"
2885 by (rule islimpt_subset) auto
2888 lemma islimpt_union_finite:
2889 fixes x :: "'a::t1_space"
2890 shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
2891 by (induct set: finite, simp_all add: islimpt_insert)
2893 lemma sequence_unique_limpt:
2894 fixes f :: "nat \<Rightarrow> 'a::t2_space"
2895 assumes "(f ---> l) sequentially" and "l' islimpt (range f)"
2898 assume "l' \<noteq> l"
2899 obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
2900 using hausdorff [OF `l' \<noteq> l`] by auto
2901 have "eventually (\<lambda>n. f n \<in> t) sequentially"
2902 using assms(1) `open t` `l \<in> t` by (rule topological_tendstoD)
2903 then obtain N where "\<forall>n\<ge>N. f n \<in> t"
2904 unfolding eventually_sequentially by auto
2906 have "UNIV = {..<N} \<union> {N..}" by auto
2907 hence "l' islimpt (f ` ({..<N} \<union> {N..}))" using assms(2) by simp
2908 hence "l' islimpt (f ` {..<N} \<union> f ` {N..})" by (simp add: image_Un)
2909 hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite)
2910 then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
2911 using `l' \<in> s` `open s` by (rule islimptE)
2912 then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto
2913 with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t" by simp
2914 with `s \<inter> t = {}` show False by simp
2917 lemma bolzano_weierstrass_imp_closed:
2918 fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)
2919 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2922 { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
2924 proof(cases "\<forall>n. x n \<noteq> l")
2925 case False thus "l\<in>s" using as(1) by auto
2927 case True note cas = this
2928 with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto
2929 then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto
2930 thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
2932 thus ?thesis unfolding closed_sequential_limits by fast
2935 text{* Hence express everything as an equivalence. *}
2937 lemma compact_eq_heine_borel:
2938 fixes s :: "'a::metric_space set"
2939 shows "compact s \<longleftrightarrow>
2940 (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2941 --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")
2943 assume ?lhs thus ?rhs by (rule compact_imp_heine_borel)
2946 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
2947 by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])
2948 thus ?lhs by (rule bolzano_weierstrass_imp_compact)
2951 lemma compact_eq_bolzano_weierstrass:
2952 fixes s :: "'a::metric_space set"
2953 shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
2955 assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto
2957 assume ?rhs thus ?lhs by (rule bolzano_weierstrass_imp_compact)
2960 lemma compact_eq_bounded_closed:
2961 fixes s :: "'a::heine_borel set"
2962 shows "compact s \<longleftrightarrow> bounded s \<and> closed s" (is "?lhs = ?rhs")
2964 assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto
2966 assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto
2969 lemma compact_imp_bounded:
2970 fixes s :: "'a::metric_space set"
2971 shows "compact s ==> bounded s"
2974 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')"
2975 by (rule compact_imp_heine_borel)
2976 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
2977 using heine_borel_imp_bolzano_weierstrass[of s] by auto
2979 by (rule bolzano_weierstrass_imp_bounded)
2982 lemma compact_imp_closed:
2983 fixes s :: "'a::metric_space set"
2984 shows "compact s ==> closed s"
2987 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')"
2988 by (rule compact_imp_heine_borel)
2989 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
2990 using heine_borel_imp_bolzano_weierstrass[of s] by auto
2992 by (rule bolzano_weierstrass_imp_closed)
2995 text{* In particular, some common special cases. *}
2997 lemma compact_empty[simp]:
2999 unfolding compact_def
3002 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
3003 unfolding subseq_def by simp (* TODO: move somewhere else *)
3005 lemma compact_union [intro]:
3006 assumes "compact s" and "compact t"
3007 shows "compact (s \<union> t)"
3008 proof (rule compactI)
3009 fix f :: "nat \<Rightarrow> 'a"
3010 assume "\<forall>n. f n \<in> s \<union> t"
3011 hence "infinite {n. f n \<in> s \<union> t}" by simp
3012 hence "infinite {n. f n \<in> s} \<or> infinite {n. f n \<in> t}" by simp
3013 thus "\<exists>l\<in>s \<union> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
3015 assume "infinite {n. f n \<in> s}"
3016 from infinite_enumerate [OF this]
3017 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> s" by auto
3018 obtain r l where "l \<in> s" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
3019 using `compact s` `\<forall>n. (f \<circ> q) n \<in> s` by (rule compactE)
3020 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
3021 using `subseq q` by (simp_all add: subseq_o o_assoc)
3022 thus ?thesis by auto
3024 assume "infinite {n. f n \<in> t}"
3025 from infinite_enumerate [OF this]
3026 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> t" by auto
3027 obtain r l where "l \<in> t" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
3028 using `compact t` `\<forall>n. (f \<circ> q) n \<in> t` by (rule compactE)
3029 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
3030 using `subseq q` by (simp_all add: subseq_o o_assoc)
3031 thus ?thesis by auto
3035 lemma compact_inter_closed [intro]:
3036 assumes "compact s" and "closed t"
3037 shows "compact (s \<inter> t)"
3038 proof (rule compactI)
3039 fix f :: "nat \<Rightarrow> 'a"
3040 assume "\<forall>n. f n \<in> s \<inter> t"
3041 hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t" by simp_all
3042 obtain l r where "l \<in> s" "subseq r" "((f \<circ> r) ---> l) sequentially"
3043 using `compact s` `\<forall>n. f n \<in> s` by (rule compactE)
3045 from `closed t` `\<forall>n. f n \<in> t` `((f \<circ> r) ---> l) sequentially` have "l \<in> t"
3046 unfolding closed_sequential_limits o_def by fast
3047 ultimately show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
3051 lemma closed_inter_compact [intro]:
3052 assumes "closed s" and "compact t"
3053 shows "compact (s \<inter> t)"
3054 using compact_inter_closed [of t s] assms
3055 by (simp add: Int_commute)
3057 lemma compact_inter [intro]:
3058 assumes "compact s" and "compact t"
3059 shows "compact (s \<inter> t)"
3060 using assms by (intro compact_inter_closed compact_imp_closed)
3062 lemma compact_sing [simp]: "compact {a}"
3063 unfolding compact_def o_def subseq_def
3064 by (auto simp add: tendsto_const)
3066 lemma compact_insert [simp]:
3067 assumes "compact s" shows "compact (insert x s)"
3069 have "compact ({x} \<union> s)"
3070 using compact_sing assms by (rule compact_union)
3071 thus ?thesis by simp
3074 lemma finite_imp_compact:
3075 shows "finite s \<Longrightarrow> compact s"
3076 by (induct set: finite) simp_all
3078 lemma compact_cball[simp]:
3079 fixes x :: "'a::heine_borel"
3080 shows "compact(cball x e)"
3081 using compact_eq_bounded_closed bounded_cball closed_cball
3084 lemma compact_frontier_bounded[intro]:
3085 fixes s :: "'a::heine_borel set"
3086 shows "bounded s ==> compact(frontier s)"
3087 unfolding frontier_def
3088 using compact_eq_bounded_closed
3091 lemma compact_frontier[intro]:
3092 fixes s :: "'a::heine_borel set"
3093 shows "compact s ==> compact (frontier s)"
3094 using compact_eq_bounded_closed compact_frontier_bounded
3097 lemma frontier_subset_compact:
3098 fixes s :: "'a::heine_borel set"
3099 shows "compact s ==> frontier s \<subseteq> s"
3100 using frontier_subset_closed compact_eq_bounded_closed
3104 fixes s :: "'a::t1_space set"
3105 shows "open s \<Longrightarrow> open (s - {x})"
3106 by (simp add: open_Diff)
3108 text{* Finite intersection property. I could make it an equivalence in fact. *}
3110 lemma compact_imp_fip:
3111 assumes "compact s" "\<forall>t \<in> f. closed t"
3112 "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"
3113 shows "s \<inter> (\<Inter> f) \<noteq> {}"
3115 assume as:"s \<inter> (\<Inter> f) = {}"
3116 hence "s \<subseteq> \<Union> uminus ` f" by auto
3117 moreover have "Ball (uminus ` f) open" using open_Diff closed_Diff using assms(2) by auto
3118 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
3119 hence "finite (uminus ` f') \<and> uminus ` f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)
3120 hence "s \<inter> \<Inter>uminus ` f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus ` f'"]] by auto
3121 thus False using f'(3) unfolding subset_eq and Union_iff by blast
3124 subsection{* Bounded closed nest property (proof does not use Heine-Borel). *}
3126 lemma bounded_closed_nest:
3127 assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
3128 "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)" "bounded(s 0)"
3129 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
3131 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
3132 from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
3134 then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
3135 unfolding compact_def apply(erule_tac x=x in allE) using x using assms(3) by blast
3138 { fix e::real assume "e>0"
3139 with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
3140 hence "dist ((x \<circ> r) (max N n)) l < e" by auto
3142 have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
3143 hence "(x \<circ> r) (max N n) \<in> s n"
3144 using x apply(erule_tac x=n in allE)
3145 using x apply(erule_tac x="r (max N n)" in allE)
3146 using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto
3147 ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
3149 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
3151 thus ?thesis by auto
3154 text{* Decreasing case does not even need compactness, just completeness. *}
3156 lemma decreasing_closed_nest:
3157 assumes "\<forall>n. closed(s n)"
3158 "\<forall>n. (s n \<noteq> {})"
3159 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
3160 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
3161 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s n"
3163 have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
3164 hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
3165 then obtain t where t: "\<forall>n. t n \<in> s n" by auto
3166 { fix e::real assume "e>0"
3167 then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
3168 { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
3169 hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding subset_eq t by blast+
3170 hence "dist (t m) (t n) < e" using N by auto
3172 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
3174 hence "Cauchy t" unfolding cauchy_def by auto
3175 then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
3177 { fix e::real assume "e>0"
3178 then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded Lim_sequentially] by auto
3179 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
3180 hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
3182 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
3184 then show ?thesis by auto
3187 text{* Strengthen it to the intersection actually being a singleton. *}
3189 lemma decreasing_closed_nest_sing:
3190 fixes s :: "nat \<Rightarrow> 'a::heine_borel set"
3191 assumes "\<forall>n. closed(s n)"
3192 "\<forall>n. s n \<noteq> {}"
3193 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
3194 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
3195 shows "\<exists>a. \<Inter>(range s) = {a}"
3197 obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
3198 { fix b assume b:"b \<in> \<Inter>(range s)"
3199 { fix e::real assume "e>0"
3200 hence "dist a b < e" using assms(4 )using b using a by blast
3202 hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)
3204 with a have "\<Inter>(range s) = {a}" unfolding image_def by auto
3208 text{* Cauchy-type criteria for uniform convergence. *}
3210 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows
3211 "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
3212 (\<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")
3215 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
3216 { fix e::real assume "e>0"
3217 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
3218 { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
3219 hence "dist (s m x) (s n x) < e"
3220 using N[THEN spec[where x=m], THEN spec[where x=x]]
3221 using N[THEN spec[where x=n], THEN spec[where x=x]]
3222 using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto }
3223 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 }
3227 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
3228 then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
3229 using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
3230 { fix e::real assume "e>0"
3231 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"
3232 using `?rhs`[THEN spec[where x="e/2"]] by auto
3233 { fix x assume "P x"
3234 then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
3235 using l[THEN spec[where x=x], unfolded Lim_sequentially] using `e>0` by(auto elim!: allE[where x="e/2"])
3236 fix n::nat assume "n\<ge>N"
3237 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]]
3238 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) }
3239 hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
3243 lemma uniformly_cauchy_imp_uniformly_convergent:
3244 fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"
3245 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"
3246 "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
3247 shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
3249 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"
3250 using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
3252 { fix x assume "P x"
3253 hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
3254 using l and assms(2) unfolding Lim_sequentially by blast }
3255 ultimately show ?thesis by auto
3258 subsection {* Continuity *}
3260 text {* Define continuity over a net to take in restrictions of the set. *}
3263 continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
3264 where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
3266 lemma continuous_trivial_limit:
3267 "trivial_limit net ==> continuous net f"
3268 unfolding continuous_def tendsto_def trivial_limit_eq by auto
3270 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
3271 unfolding continuous_def
3272 unfolding tendsto_def
3273 using netlimit_within[of x s]
3274 by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
3276 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
3277 using continuous_within [of x UNIV f] by (simp add: within_UNIV)
3279 lemma continuous_at_within:
3280 assumes "continuous (at x) f" shows "continuous (at x within s) f"
3281 using assms unfolding continuous_at continuous_within
3282 by (rule Lim_at_within)
3284 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
3286 lemma continuous_within_eps_delta:
3287 "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)"
3288 unfolding continuous_within and Lim_within
3289 apply auto unfolding dist_nz[THEN sym] apply(auto elim!:allE) apply(rule_tac x=d in exI) by auto
3291 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3292 \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
3293 using continuous_within_eps_delta[of x UNIV f]
3294 unfolding within_UNIV by blast
3296 text{* Versions in terms of open balls. *}
3298 lemma continuous_within_ball:
3299 "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3300 f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3303 { fix e::real assume "e>0"
3304 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"
3305 using `?lhs`[unfolded continuous_within Lim_within] by auto
3306 { fix y assume "y\<in>f ` (ball x d \<inter> s)"
3307 hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
3308 apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
3310 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) }
3313 assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
3314 apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
3317 lemma continuous_at_ball:
3318 "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3320 assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3321 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)
3322 unfolding dist_nz[THEN sym] by auto
3324 assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3325 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)
3328 text{* Define setwise continuity in terms of limits within the set. *}
3332 "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
3334 "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
3336 lemma continuous_on_topological:
3337 "continuous_on s f \<longleftrightarrow>
3338 (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
3339 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
3340 unfolding continuous_on_def tendsto_def
3341 unfolding Limits.eventually_within eventually_at_topological
3342 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
3344 lemma continuous_on_iff:
3345 "continuous_on s f \<longleftrightarrow>
3346 (\<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)"
3347 unfolding continuous_on_def Lim_within
3348 apply (intro ball_cong [OF refl] all_cong ex_cong)
3349 apply (rename_tac y, case_tac "y = x", simp)
3350 apply (simp add: dist_nz)
3354 uniformly_continuous_on ::
3355 "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
3357 "uniformly_continuous_on s f \<longleftrightarrow>
3358 (\<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)"
3360 text{* Some simple consequential lemmas. *}
3362 lemma uniformly_continuous_imp_continuous:
3363 " uniformly_continuous_on s f ==> continuous_on s f"
3364 unfolding uniformly_continuous_on_def continuous_on_iff by blast
3366 lemma continuous_at_imp_continuous_within:
3367 "continuous (at x) f ==> continuous (at x within s) f"
3368 unfolding continuous_within continuous_at using Lim_at_within by auto
3370 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
3371 unfolding tendsto_def by (simp add: trivial_limit_eq)
3373 lemma continuous_at_imp_continuous_on:
3374 assumes "\<forall>x\<in>s. continuous (at x) f"
3375 shows "continuous_on s f"
3376 unfolding continuous_on_def
3378 fix x assume "x \<in> s"
3379 with assms have *: "(f ---> f (netlimit (at x))) (at x)"
3380 unfolding continuous_def by simp
3381 have "(f ---> f x) (at x)"
3382 proof (cases "trivial_limit (at x)")
3383 case True thus ?thesis
3384 by (rule Lim_trivial_limit)
3387 hence 1: "netlimit (at x) = x"
3388 using netlimit_within [of x UNIV]
3389 by (simp add: within_UNIV)
3390 with * show ?thesis by simp
3392 thus "(f ---> f x) (at x within s)"
3393 by (rule Lim_at_within)
3396 lemma continuous_on_eq_continuous_within:
3397 "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"
3398 unfolding continuous_on_def continuous_def
3399 apply (rule ball_cong [OF refl])
3400 apply (case_tac "trivial_limit (at x within s)")
3401 apply (simp add: Lim_trivial_limit)
3402 apply (simp add: netlimit_within)
3405 lemmas continuous_on = continuous_on_def -- "legacy theorem name"
3407 lemma continuous_on_eq_continuous_at:
3408 shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
3409 by (auto simp add: continuous_on continuous_at Lim_within_open)
3411 lemma continuous_within_subset:
3412 "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
3413 ==> continuous (at x within t) f"
3414 unfolding continuous_within by(metis Lim_within_subset)
3416 lemma continuous_on_subset:
3417 shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
3418 unfolding continuous_on by (metis subset_eq Lim_within_subset)
3420 lemma continuous_on_interior:
3421 shows "continuous_on s f \<Longrightarrow> x \<in> interior s ==> continuous (at x) f"
3422 unfolding interior_def
3424 by (meson continuous_on_eq_continuous_at continuous_on_subset)
3426 lemma continuous_on_eq:
3427 "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"
3428 unfolding continuous_on_def tendsto_def Limits.eventually_within
3431 text{* Characterization of various kinds of continuity in terms of sequences. *}
3433 (* \<longrightarrow> could be generalized, but \<longleftarrow> requires metric space *)
3434 lemma continuous_within_sequentially:
3435 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3436 shows "continuous (at a within s) f \<longleftrightarrow>
3437 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
3438 --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
3441 { 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"
3442 fix e::real assume "e>0"
3443 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
3444 from x(2) `d>0` obtain N where N:"\<forall>n\<ge>N. dist (x n) a < d" by auto
3445 hence "\<exists>N. \<forall>n\<ge>N. dist ((f \<circ> x) n) (f a) < e"
3446 apply(rule_tac x=N in exI) using N d apply auto using x(1)
3447 apply(erule_tac x=n in allE) apply(erule_tac x=n in allE)
3448 apply(erule_tac x="x n" in ballE) apply auto unfolding dist_nz[THEN sym] apply auto using `e>0` by auto
3450 thus ?rhs unfolding continuous_within unfolding Lim_sequentially by simp
3453 { fix e::real assume "e>0"
3454 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)"
3455 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
3456 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)"
3457 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
3458 { fix d::real assume "d>0"
3459 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
3460 then obtain N::nat where N:"inverse (real (N + 1)) < d" by auto
3461 { fix n::nat assume n:"n\<ge>N"
3462 hence "dist (x (inverse (real (n + 1)))) a < inverse (real (n + 1))" using x[THEN spec[where x="inverse (real (n + 1))"]] by auto
3463 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)
3464 ultimately have "dist (x (inverse (real (n + 1)))) a < d" by auto
3466 hence "\<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < d" by auto
3468 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
3469 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
3470 hence "False" apply(erule_tac x=e in allE) using `e>0` using x by auto
3472 thus ?lhs unfolding continuous_within unfolding Lim_within unfolding Lim_sequentially by blast
3475 lemma continuous_at_sequentially:
3476 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3477 shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
3478 --> ((f o x) ---> f a) sequentially)"
3479 using continuous_within_sequentially[of a UNIV f] unfolding within_UNIV by auto
3481 lemma continuous_on_sequentially:
3482 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3483 shows "continuous_on s f \<longleftrightarrow>
3484 (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
3485 --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
3487 assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
3489 assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
3492 lemma uniformly_continuous_on_sequentially':
3493 "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
3494 ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially
3495 \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
3498 { 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"
3499 { fix e::real assume "e>0"
3500 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"
3501 using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
3502 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
3503 { fix n assume "n\<ge>N"
3504 hence "dist (f (x n)) (f (y n)) < e"
3505 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
3506 unfolding dist_commute by simp }
3507 hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" by auto }
3508 hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding Lim_sequentially and dist_real_def by auto }
3512 { assume "\<not> ?lhs"
3513 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
3514 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"
3515 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
3516 by (auto simp add: dist_commute)
3517 def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
3518 def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
3519 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"
3520 unfolding x_def and y_def using fa by auto
3521 { fix e::real assume "e>0"
3522 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
3523 { fix n::nat assume "n\<ge>N"
3524 hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
3525 also have "\<dots> < e" using N by auto
3526 finally have "inverse (real n + 1) < e" by auto
3527 hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto }
3528 hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto }
3529 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
3530 hence False using fxy and `e>0` by auto }
3531 thus ?lhs unfolding uniformly_continuous_on_def by blast
3534 lemma uniformly_continuous_on_sequentially:
3535 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
3536 shows "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
3537 ((\<lambda>n. x n - y n) ---> 0) sequentially
3538 \<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")
3539 (* BH: maybe the previous lemma should replace this one? *)
3540 unfolding uniformly_continuous_on_sequentially'
3541 unfolding dist_norm tendsto_norm_zero_iff ..
3543 text{* The usual transformation theorems. *}
3545 lemma continuous_transform_within:
3546 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3547 assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
3548 "continuous (at x within s) f"
3549 shows "continuous (at x within s) g"
3550 unfolding continuous_within
3551 proof (rule Lim_transform_within)
3552 show "0 < d" by fact
3553 show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
3554 using assms(3) by auto
3556 using assms(1,2,3) by auto
3557 thus "(f ---> g x) (at x within s)"
3558 using assms(4) unfolding continuous_within by simp
3561 lemma continuous_transform_at:
3562 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3563 assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
3564 "continuous (at x) f"
3565 shows "continuous (at x) g"
3566 using continuous_transform_within [of d x UNIV f g] assms
3567 by (simp add: within_UNIV)
3569 text{* Combination results for pointwise continuity. *}
3571 lemma continuous_const: "continuous net (\<lambda>x. c)"
3572 by (auto simp add: continuous_def tendsto_const)
3574 lemma continuous_cmul:
3575 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3576 shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"
3577 by (auto simp add: continuous_def intro: tendsto_intros)
3579 lemma continuous_neg:
3580 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3581 shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"
3582 by (auto simp add: continuous_def tendsto_minus)
3584 lemma continuous_add:
3585 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3586 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"
3587 by (auto simp add: continuous_def tendsto_add)
3589 lemma continuous_sub:
3590 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3591 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"
3592 by (auto simp add: continuous_def tendsto_diff)
3595 text{* Same thing for setwise continuity. *}
3597 lemma continuous_on_const:
3598 "continuous_on s (\<lambda>x. c)"
3599 unfolding continuous_on_def by (auto intro: tendsto_intros)
3601 lemma continuous_on_cmul:
3602 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3603 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c *\<^sub>R (f x))"
3604 unfolding continuous_on_def by (auto intro: tendsto_intros)
3606 lemma continuous_on_neg:
3607 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3608 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
3609 unfolding continuous_on_def by (auto intro: tendsto_intros)
3611 lemma continuous_on_add:
3612 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3613 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3614 \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
3615 unfolding continuous_on_def by (auto intro: tendsto_intros)
3617 lemma continuous_on_sub:
3618 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3619 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3620 \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
3621 unfolding continuous_on_def by (auto intro: tendsto_intros)
3623 text{* Same thing for uniform continuity, using sequential formulations. *}
3625 lemma uniformly_continuous_on_const:
3626 "uniformly_continuous_on s (\<lambda>x. c)"
3627 unfolding uniformly_continuous_on_def by simp
3629 lemma uniformly_continuous_on_cmul:
3630 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3631 assumes "uniformly_continuous_on s f"
3632 shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
3634 { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
3635 hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"
3636 using scaleR.tendsto [OF tendsto_const, of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
3637 unfolding scaleR_zero_right scaleR_right_diff_distrib by auto
3639 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
3640 unfolding dist_norm tendsto_norm_zero_iff by auto
3644 fixes x y :: "'a::real_normed_vector"
3645 shows "dist (- x) (- y) = dist x y"
3646 unfolding dist_norm minus_diff_minus norm_minus_cancel ..
3648 lemma uniformly_continuous_on_neg:
3649 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3650 shows "uniformly_continuous_on s f
3651 ==> uniformly_continuous_on s (\<lambda>x. -(f x))"
3652 unfolding uniformly_continuous_on_def dist_minus .
3654 lemma uniformly_continuous_on_add:
3655 fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3656 assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"
3657 shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
3659 { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
3660 "((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"
3661 hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"
3662 using tendsto_add[of "\<lambda> n. f (x n) - f (y n)" 0 sequentially "\<lambda> n. g (x n) - g (y n)" 0] by auto
3663 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 }
3664 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
3665 unfolding dist_norm tendsto_norm_zero_iff by auto
3668 lemma uniformly_continuous_on_sub:
3669 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3670 shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s g
3671 ==> uniformly_continuous_on s (\<lambda>x. f x - g x)"
3672 unfolding ab_diff_minus
3673 using uniformly_continuous_on_add[of s f "\<lambda>x. - g x"]
3674 using uniformly_continuous_on_neg[of s g] by auto
3676 text{* Identity function is continuous in every sense. *}
3678 lemma continuous_within_id:
3679 "continuous (at a within s) (\<lambda>x. x)"
3680 unfolding continuous_within by (rule Lim_at_within [OF LIM_ident])
3682 lemma continuous_at_id:
3683 "continuous (at a) (\<lambda>x. x)"
3684 unfolding continuous_at by (rule LIM_ident)
3686 lemma continuous_on_id:
3687 "continuous_on s (\<lambda>x. x)"
3688 unfolding continuous_on_def by (auto intro: tendsto_ident_at_within)
3690 lemma uniformly_continuous_on_id:
3691 "uniformly_continuous_on s (\<lambda>x. x)"
3692 unfolding uniformly_continuous_on_def by auto
3694 text{* Continuity of all kinds is preserved under composition. *}
3696 lemma continuous_within_topological:
3697 "continuous (at x within s) f \<longleftrightarrow>
3698 (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
3699 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
3700 unfolding continuous_within
3701 unfolding tendsto_def Limits.eventually_within eventually_at_topological
3702 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
3704 lemma continuous_within_compose:
3705 assumes "continuous (at x within s) f"
3706 assumes "continuous (at (f x) within f ` s) g"
3707 shows "continuous (at x within s) (g o f)"
3708 using assms unfolding continuous_within_topological by simp metis
3710 lemma continuous_at_compose:
3711 assumes "continuous (at x) f" "continuous (at (f x)) g"
3712 shows "continuous (at x) (g o f)"
3714 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
3715 thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g, unfolded within_UNIV] by auto
3718 lemma continuous_on_compose:
3719 "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
3720 unfolding continuous_on_topological by simp metis
3722 lemma uniformly_continuous_on_compose:
3723 assumes "uniformly_continuous_on s f" "uniformly_continuous_on (f ` s) g"
3724 shows "uniformly_continuous_on s (g o f)"
3726 { fix e::real assume "e>0"
3727 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
3728 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
3729 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 }
3730 thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
3733 text{* Continuity in terms of open preimages. *}
3735 lemma continuous_at_open:
3736 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)))"
3737 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]
3738 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto
3740 lemma continuous_on_open:
3741 shows "continuous_on s f \<longleftrightarrow>
3742 (\<forall>t. openin (subtopology euclidean (f ` s)) t
3743 --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
3746 assume 1: "continuous_on s f"
3747 assume 2: "openin (subtopology euclidean (f ` s)) t"
3748 from 2 obtain B where B: "open B" and t: "t = f ` s \<inter> B"
3749 unfolding openin_open by auto
3750 def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"
3751 have "open U" unfolding U_def by (simp add: open_Union)
3752 moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"
3753 proof (intro ballI iffI)
3754 fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"
3755 unfolding U_def t by auto
3757 fix x assume "x \<in> s" and "f x \<in> t"
3758 hence "x \<in> s" and "f x \<in> B"
3760 with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"
3761 unfolding t continuous_on_topological by metis
3762 then show "x \<in> U"
3763 unfolding U_def by auto
3765 ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto
3766 then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3767 unfolding openin_open by fast
3769 assume "?rhs" show "continuous_on s f"
3770 unfolding continuous_on_topological
3772 fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"
3773 have "openin (subtopology euclidean (f ` s)) (f ` s \<inter> B)"
3774 unfolding openin_open using `open B` by auto
3775 then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f ` s \<inter> B}"
3776 using `?rhs` by fast
3777 then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
3778 unfolding openin_open using `x \<in> s` and `f x \<in> B` by auto
3782 text {* Similarly in terms of closed sets. *}
3784 lemma continuous_on_closed:
3785 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")
3789 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3790 have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
3791 assume as:"closedin (subtopology euclidean (f ` s)) t"
3792 hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
3793 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"]]
3794 unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto }
3799 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3800 assume as:"openin (subtopology euclidean (f ` s)) t"
3801 hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
3802 unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
3803 thus ?lhs unfolding continuous_on_open by auto
3806 text{* Half-global and completely global cases. *}
3808 lemma continuous_open_in_preimage:
3809 assumes "continuous_on s f" "open t"
3810 shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3812 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
3813 have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3814 using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
3815 thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3818 lemma continuous_closed_in_preimage:
3819 assumes "continuous_on s f" "closed t"
3820 shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3822 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
3823 have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3824 using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
3826 using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3829 lemma continuous_open_preimage:
3830 assumes "continuous_on s f" "open s" "open t"
3831 shows "open {x \<in> s. f x \<in> t}"
3833 obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3834 using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
3835 thus ?thesis using open_Int[of s T, OF assms(2)] by auto
3838 lemma continuous_closed_preimage:
3839 assumes "continuous_on s f" "closed s" "closed t"
3840 shows "closed {x \<in> s. f x \<in> t}"
3842 obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3843 using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
3844 thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
3847 lemma continuous_open_preimage_univ:
3848 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
3849 using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
3851 lemma continuous_closed_preimage_univ:
3852 shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
3853 using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
3855 lemma continuous_open_vimage:
3856 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
3857 unfolding vimage_def by (rule continuous_open_preimage_univ)
3859 lemma continuous_closed_vimage:
3860 shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
3861 unfolding vimage_def by (rule continuous_closed_preimage_univ)
3863 lemma interior_image_subset:
3864 assumes "\<forall>x. continuous (at x) f" "inj f"
3865 shows "interior (f ` s) \<subseteq> f ` (interior s)"
3866 apply rule unfolding interior_def mem_Collect_eq image_iff apply safe
3867 proof- fix x T assume as:"open T" "x \<in> T" "T \<subseteq> f ` s"
3868 hence "x \<in> f ` s" by auto then guess y unfolding image_iff .. note y=this
3869 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
3870 apply safe apply(rule_tac x="{x. f x \<in> T}" in exI) apply(safe,rule continuous_open_preimage_univ)
3871 proof- fix x assume "f x \<in> T" hence "f x \<in> f ` s" using as by auto
3872 thus "x \<in> s" unfolding inj_image_mem_iff[OF assms(2)] . qed auto qed
3874 text{* Equality of continuous functions on closure and related results. *}
3876 lemma continuous_closed_in_preimage_constant:
3877 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3878 shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
3879 using continuous_closed_in_preimage[of s f "{a}"] by auto
3881 lemma continuous_closed_preimage_constant:
3882 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3883 shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
3884 using continuous_closed_preimage[of s f "{a}"] by auto
3886 lemma continuous_constant_on_closure:
3887 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3888 assumes "continuous_on (closure s) f"
3889 "\<forall>x \<in> s. f x = a"
3890 shows "\<forall>x \<in> (closure s). f x = a"
3891 using continuous_closed_preimage_constant[of "closure s" f a]
3892 assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
3894 lemma image_closure_subset:
3895 assumes "continuous_on (closure s) f" "closed t" "(f ` s) \<subseteq> t"
3896 shows "f ` (closure s) \<subseteq> t"
3898 have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
3899 moreover have "closed {x \<in> closure s. f x \<in> t}"
3900 using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
3901 ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
3902 using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
3903 thus ?thesis by auto
3906 lemma continuous_on_closure_norm_le:
3907 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3908 assumes "continuous_on (closure s) f" "\<forall>y \<in> s. norm(f y) \<le> b" "x \<in> (closure s)"
3909 shows "norm(f x) \<le> b"
3911 have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
3913 using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
3914 unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
3917 text{* Making a continuous function avoid some value in a neighbourhood. *}
3919 lemma continuous_within_avoid:
3920 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3921 assumes "continuous (at x within s) f" "x \<in> s" "f x \<noteq> a"
3922 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
3924 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"
3925 using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto
3926 { fix y assume " y\<in>s" "dist x y < d"
3927 hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]
3928 apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }
3929 thus ?thesis using `d>0` by auto
3932 lemma continuous_at_avoid:
3933 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3934 assumes "continuous (at x) f" "f x \<noteq> a"
3935 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
3936 using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto
3938 lemma continuous_on_avoid:
3939 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
3940 assumes "continuous_on s f" "x \<in> s" "f x \<noteq> a"
3941 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
3942 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
3944 lemma continuous_on_open_avoid:
3945 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
3946 assumes "continuous_on s f" "open s" "x \<in> s" "f x \<noteq> a"
3947 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
3948 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
3950 text{* Proving a function is constant by proving open-ness of level set. *}
3952 lemma continuous_levelset_open_in_cases:
3953 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3954 shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
3955 openin (subtopology euclidean s) {x \<in> s. f x = a}
3956 ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
3957 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
3959 lemma continuous_levelset_open_in:
3960 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3961 shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
3962 openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
3963 (\<exists>x \<in> s. f x = a) ==> (\<forall>x \<in> s. f x = a)"
3964 using continuous_levelset_open_in_cases[of s f ]
3967 lemma continuous_levelset_open:
3968 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3969 assumes "connected s" "continuous_on s f" "open {x \<in> s. f x = a}" "\<exists>x \<in> s. f x = a"
3970 shows "\<forall>x \<in> s. f x = a"
3971 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
3973 text{* Some arithmetical combinations (more to prove). *}
3975 lemma open_scaling[intro]:
3976 fixes s :: "'a::real_normed_vector set"
3977 assumes "c \<noteq> 0" "open s"
3978 shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
3980 { fix x assume "x \<in> s"
3981 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
3982 have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF `e>0`] by auto
3984 { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
3985 hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
3986 using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
3987 assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
3988 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 }
3989 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 }
3990 thus ?thesis unfolding open_dist by auto
3993 lemma minus_image_eq_vimage:
3994 fixes A :: "'a::ab_group_add set"
3995 shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
3996 by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
3998 lemma open_negations:
3999 fixes s :: "'a::real_normed_vector set"
4000 shows "open s ==> open ((\<lambda> x. -x) ` s)"
4001 unfolding scaleR_minus1_left [symmetric]
4002 by (rule open_scaling, auto)
4004 lemma open_translation:
4005 fixes s :: "'a::real_normed_vector set"
4006 assumes "open s" shows "open((\<lambda>x. a + x) ` s)"
4008 { 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 }
4009 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
4010 ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
4013 lemma open_affinity:
4014 fixes s :: "'a::real_normed_vector set"
4015 assumes "open s" "c \<noteq> 0"
4016 shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
4018 have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
4019 have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
4020 thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
4023 lemma interior_translation:
4024 fixes s :: "'a::real_normed_vector set"
4025 shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
4026 proof (rule set_eqI, rule)
4027 fix x assume "x \<in> interior (op + a ` s)"
4028 then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
4029 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
4030 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
4032 fix x assume "x \<in> op + a ` interior s"
4033 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
4034 { fix z have *:"a + y - z = y + a - z" by auto
4035 assume "z\<in>ball x e"
4036 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
4037 hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"]) }
4038 hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
4039 thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
4042 text {* We can now extend limit compositions to consider the scalar multiplier. *}
4044 lemma continuous_vmul:
4045 fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
4046 shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"
4047 unfolding continuous_def by (intro tendsto_intros)
4049 lemma continuous_mul:
4050 fixes c :: "'a::metric_space \<Rightarrow> real"
4051 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
4052 shows "continuous net c \<Longrightarrow> continuous net f
4053 ==> continuous net (\<lambda>x. c(x) *\<^sub>R f x) "
4054 unfolding continuous_def by (intro tendsto_intros)
4056 lemmas continuous_intros = continuous_add continuous_vmul continuous_cmul
4057 continuous_const continuous_sub continuous_at_id continuous_within_id continuous_mul
4059 lemma continuous_on_vmul:
4060 fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
4061 shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R v)"
4062 unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
4064 lemma continuous_on_mul:
4065 fixes c :: "'a::metric_space \<Rightarrow> real"
4066 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
4067 shows "continuous_on s c \<Longrightarrow> continuous_on s f
4068 ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R f x)"
4069 unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
4071 lemma continuous_on_mul_real:
4072 fixes f :: "'a::metric_space \<Rightarrow> real"
4073 fixes g :: "'a::metric_space \<Rightarrow> real"
4074 shows "continuous_on s f \<Longrightarrow> continuous_on s g
4075 ==> continuous_on s (\<lambda>x. f x * g x)"
4076 using continuous_on_mul[of s f g] unfolding real_scaleR_def .
4078 lemmas continuous_on_intros = continuous_on_add continuous_on_const
4079 continuous_on_id continuous_on_compose continuous_on_cmul continuous_on_neg
4080 continuous_on_sub continuous_on_mul continuous_on_vmul continuous_on_mul_real
4081 uniformly_continuous_on_add uniformly_continuous_on_const
4082 uniformly_continuous_on_id uniformly_continuous_on_compose
4083 uniformly_continuous_on_cmul uniformly_continuous_on_neg
4084 uniformly_continuous_on_sub
4086 text{* And so we have continuity of inverse. *}
4088 lemma continuous_inv:
4089 fixes f :: "'a::metric_space \<Rightarrow> real"
4090 shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
4091 ==> continuous net (inverse o f)"
4092 unfolding continuous_def using Lim_inv by auto
4094 lemma continuous_at_within_inv:
4095 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
4096 assumes "continuous (at a within s) f" "f a \<noteq> 0"
4097 shows "continuous (at a within s) (inverse o f)"
4098 using assms unfolding continuous_within o_def
4099 by (intro tendsto_intros)
4101 lemma continuous_at_inv:
4102 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
4103 shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
4104 ==> continuous (at a) (inverse o f) "
4105 using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
4107 text {* Topological properties of linear functions. *}
4110 assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
4112 interpret f: bounded_linear f by fact
4113 have "(f ---> f 0) (at 0)"
4114 using tendsto_ident_at by (rule f.tendsto)
4115 thus ?thesis unfolding f.zero .
4118 lemma linear_continuous_at:
4119 assumes "bounded_linear f" shows "continuous (at a) f"
4120 unfolding continuous_at using assms
4121 apply (rule bounded_linear.tendsto)
4122 apply (rule tendsto_ident_at)
4125 lemma linear_continuous_within:
4126 shows "bounded_linear f ==> continuous (at x within s) f"
4127 using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
4129 lemma linear_continuous_on:
4130 shows "bounded_linear f ==> continuous_on s f"
4131 using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
4133 text{* Also bilinear functions, in composition form. *}
4135 lemma bilinear_continuous_at_compose:
4136 shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
4137 ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
4138 unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
4140 lemma bilinear_continuous_within_compose:
4141 shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
4142 ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
4143 unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
4145 lemma bilinear_continuous_on_compose:
4146 shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
4147 ==> continuous_on s (\<lambda>x. h (f x) (g x))"
4148 unfolding continuous_on_def
4149 by (fast elim: bounded_bilinear.tendsto)
4151 text {* Preservation of compactness and connectedness under continuous function. *}
4153 lemma compact_continuous_image:
4154 assumes "continuous_on s f" "compact s"
4155 shows "compact(f ` s)"
4157 { fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
4158 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
4159 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
4160 { fix e::real assume "e>0"
4161 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
4162 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
4163 { 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 }
4164 hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto }
4165 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 }
4166 thus ?thesis unfolding compact_def by auto
4169 lemma connected_continuous_image:
4170 assumes "continuous_on s f" "connected s"
4171 shows "connected(f ` s)"
4173 { fix T assume as: "T \<noteq> {}" "T \<noteq> f ` s" "openin (subtopology euclidean (f ` s)) T" "closedin (subtopology euclidean (f ` s)) T"
4174 have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
4175 using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
4176 using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
4177 using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
4178 hence False using as(1,2)
4179 using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
4180 thus ?thesis unfolding connected_clopen by auto
4183 text{* Continuity implies uniform continuity on a compact domain. *}
4185 lemma compact_uniformly_continuous:
4186 assumes "continuous_on s f" "compact s"
4187 shows "uniformly_continuous_on s f"
4189 { fix x assume x:"x\<in>s"
4190 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
4191 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 }
4192 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
4193 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)"
4194 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
4196 { fix e::real assume "e>0"
4198 { 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 }
4199 hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto
4201 { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto }
4202 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
4204 { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"
4205 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
4206 hence "x\<in>ball z (d z (e / 2))" using `ea>0` unfolding subset_eq by auto
4207 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`
4208 by (auto simp add: dist_commute)
4209 moreover have "y\<in>ball z (d z (e / 2))" using as and `ea>0` and z[unfolded subset_eq]
4210 by (auto simp add: dist_commute)
4211 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`
4212 by (auto simp add: dist_commute)
4213 ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]
4214 by (auto simp add: dist_commute) }
4215 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 }
4216 thus ?thesis unfolding uniformly_continuous_on_def by auto
4219 text{* Continuity of inverse function on compact domain. *}
4221 lemma continuous_on_inverse:
4222 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
4223 (* TODO: can this be generalized more? *)
4224 assumes "continuous_on s f" "compact s" "\<forall>x \<in> s. g (f x) = x"
4225 shows "continuous_on (f ` s) g"
4227 have *:"g ` f ` s = s" using assms(3) by (auto simp add: image_iff)
4228 { fix t assume t:"closedin (subtopology euclidean (g ` f ` s)) t"
4229 then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto
4230 have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]
4231 unfolding T(2) and Int_left_absorb by auto
4232 moreover have "compact (s \<inter> T)"
4233 using assms(2) unfolding compact_eq_bounded_closed
4234 using bounded_subset[of s "s \<inter> T"] and T(1) by auto
4235 ultimately have "closed (f ` t)" using T(1) unfolding T(2)
4236 using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto
4237 moreover have "{x \<in> f ` s. g x \<in> t} = f ` s \<inter> f ` t" using assms(3) unfolding T(2) by auto
4238 ultimately have "closedin (subtopology euclidean (f ` s)) {x \<in> f ` s. g x \<in> t}"
4239 unfolding closedin_closed by auto }
4240 thus ?thesis unfolding continuous_on_closed by auto
4243 text {* A uniformly convergent limit of continuous functions is continuous. *}
4245 lemma norm_triangle_lt:
4246 fixes x y :: "'a::real_normed_vector"
4247 shows "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
4248 by (rule le_less_trans [OF norm_triangle_ineq])
4250 lemma continuous_uniform_limit:
4251 fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::real_normed_vector"
4252 assumes "\<not> (trivial_limit net)" "eventually (\<lambda>n. continuous_on s (f n)) net"
4253 "\<forall>e>0. eventually (\<lambda>n. \<forall>x \<in> s. norm(f n x - g x) < e) net"
4254 shows "continuous_on s g"
4256 { fix x and e::real assume "x\<in>s" "e>0"
4257 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
4258 then obtain n where n:"\<forall>xa\<in>s. norm (f n xa - g xa) < e / 3" "continuous_on s (f n)"
4259 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
4260 have "e / 3 > 0" using `e>0` by auto
4261 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"
4262 using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
4263 { fix y assume "y\<in>s" "dist y x < d"
4264 hence "dist (f n y) (f n x) < e / 3" using d[THEN bspec[where x=y]] by auto
4265 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"]
4266 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
4267 hence "dist (g y) (g x) < e" unfolding dist_norm using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
4268 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) }
4269 hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using `d>0` by auto }
4270 thus ?thesis unfolding continuous_on_iff by auto
4273 subsection{* Topological stuff lifted from and dropped to R *}
4277 fixes s :: "real set" shows
4278 "open s \<longleftrightarrow>
4279 (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
4280 unfolding open_dist dist_norm by simp
4282 lemma islimpt_approachable_real:
4283 fixes s :: "real set"
4284 shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
4285 unfolding islimpt_approachable dist_norm by simp
4288 fixes s :: "real set"
4289 shows "closed s \<longleftrightarrow>
4290 (\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
4292 unfolding closed_limpt islimpt_approachable dist_norm by simp
4294 lemma continuous_at_real_range:
4295 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4296 shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
4297 \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
4298 unfolding continuous_at unfolding Lim_at
4299 unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
4300 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
4301 apply(erule_tac x=e in allE) by auto
4303 lemma continuous_on_real_range:
4304 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4305 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))"
4306 unfolding continuous_on_iff dist_norm by simp
4308 lemma continuous_at_norm: "continuous (at x) norm"
4309 unfolding continuous_at by (intro tendsto_intros)
4311 lemma continuous_on_norm: "continuous_on s norm"
4312 unfolding continuous_on by (intro ballI tendsto_intros)
4314 lemma continuous_at_infnorm: "continuous (at x) infnorm"
4315 unfolding continuous_at Lim_at o_def unfolding dist_norm
4316 apply auto apply (rule_tac x=e in exI) apply auto
4317 using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
4319 text{* Hence some handy theorems on distance, diameter etc. of/from a set. *}
4321 lemma compact_attains_sup:
4322 fixes s :: "real set"
4323 assumes "compact s" "s \<noteq> {}"
4324 shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
4326 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4327 { 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"
4328 have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto
4329 moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
4330 ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using `e>0` by auto }
4331 thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]
4332 apply(rule_tac x="Sup s" in bexI) by auto
4336 fixes S :: "real set"
4337 shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
4338 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def)
4340 lemma compact_attains_inf:
4341 fixes s :: "real set"
4342 assumes "compact s" "s \<noteq> {}" shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
4344 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4345 { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s" "Inf s \<notin> s" "0 < e"
4346 "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"
4347 have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto
4349 { fix x assume "x \<in> s"
4350 hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto
4351 have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
4352 hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto
4353 ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using `e>0` by auto }
4354 thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]
4355 apply(rule_tac x="Inf s" in bexI) by auto
4358 lemma continuous_attains_sup:
4359 fixes f :: "'a::metric_space \<Rightarrow> real"
4360 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4361 ==> (\<exists>x \<in> s. \<forall>y \<in> s. f y \<le> f x)"
4362 using compact_attains_sup[of "f ` s"]
4363 using compact_continuous_image[of s f] by auto
4365 lemma continuous_attains_inf:
4366 fixes f :: "'a::metric_space \<Rightarrow> real"
4367 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4368 \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
4369 using compact_attains_inf[of "f ` s"]
4370 using compact_continuous_image[of s f] by auto
4372 lemma distance_attains_sup:
4373 assumes "compact s" "s \<noteq> {}"
4374 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
4375 proof (rule continuous_attains_sup [OF assms])
4376 { fix x assume "x\<in>s"
4377 have "(dist a ---> dist a x) (at x within s)"
4378 by (intro tendsto_dist tendsto_const Lim_at_within LIM_ident)
4380 thus "continuous_on s (dist a)"
4381 unfolding continuous_on ..
4384 text{* For *minimal* distance, we only need closure, not compactness. *}
4386 lemma distance_attains_inf:
4387 fixes a :: "'a::heine_borel"
4388 assumes "closed s" "s \<noteq> {}"
4389 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
4391 from assms(2) obtain b where "b\<in>s" by auto
4392 let ?B = "cball a (dist b a) \<inter> s"
4393 have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
4394 hence "?B \<noteq> {}" by auto
4396 { fix x assume "x\<in>?B"
4397 fix e::real assume "e>0"
4398 { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
4399 from as have "\<bar>dist a x' - dist a x\<bar> < e"
4400 unfolding abs_less_iff minus_diff_eq
4401 using dist_triangle2 [of a x' x]
4402 using dist_triangle [of a x x']
4405 hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
4408 hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
4409 unfolding continuous_on Lim_within dist_norm real_norm_def
4411 moreover have "compact ?B"
4412 using compact_cball[of a "dist b a"]
4413 unfolding compact_eq_bounded_closed
4414 using bounded_Int and closed_Int and assms(1) by auto
4415 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"
4416 using continuous_attains_inf[of ?B "dist a"] by fastsimp
4417 thus ?thesis by fastsimp
4420 subsection {* Pasted sets *}
4422 lemma bounded_Times:
4423 assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
4425 obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
4426 using assms [unfolded bounded_def] by auto
4427 then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
4428 by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
4429 thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
4432 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
4435 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
4436 unfolding compact_def
4438 apply (drule_tac x="fst \<circ> f" in spec)
4439 apply (drule mp, simp add: mem_Times_iff)
4440 apply (clarify, rename_tac l1 r1)
4441 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
4442 apply (drule mp, simp add: mem_Times_iff)
4443 apply (clarify, rename_tac l2 r2)
4444 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
4445 apply (rule_tac x="r1 \<circ> r2" in exI)
4446 apply (rule conjI, simp add: subseq_def)
4447 apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
4448 apply (drule (1) tendsto_Pair) back
4449 apply (simp add: o_def)
4452 text{* Hence some useful properties follow quite easily. *}
4454 lemma compact_scaling:
4455 fixes s :: "'a::real_normed_vector set"
4456 assumes "compact s" shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
4458 let ?f = "\<lambda>x. scaleR c x"
4459 have *:"bounded_linear ?f" by (rule scaleR.bounded_linear_right)
4460 show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
4461 using linear_continuous_at[OF *] assms by auto
4464 lemma compact_negations:
4465 fixes s :: "'a::real_normed_vector set"
4466 assumes "compact s" shows "compact ((\<lambda>x. -x) ` s)"
4467 using compact_scaling [OF assms, of "- 1"] by auto
4470 fixes s t :: "'a::real_normed_vector set"
4471 assumes "compact s" "compact t" shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
4473 have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
4474 apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
4475 have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
4476 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
4477 thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
4480 lemma compact_differences:
4481 fixes s t :: "'a::real_normed_vector set"
4482 assumes "compact s" "compact t" shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
4484 have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
4485 apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4486 thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
4489 lemma compact_translation:
4490 fixes s :: "'a::real_normed_vector set"
4491 assumes "compact s" shows "compact ((\<lambda>x. a + x) ` s)"
4493 have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
4494 thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
4497 lemma compact_affinity:
4498 fixes s :: "'a::real_normed_vector set"
4499 assumes "compact s" shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
4501 have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
4502 thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
4505 text{* Hence we get the following. *}
4507 lemma compact_sup_maxdistance:
4508 fixes s :: "'a::real_normed_vector set"
4509 assumes "compact s" "s \<noteq> {}"
4510 shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
4512 have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
4513 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"
4514 using compact_differences[OF assms(1) assms(1)]
4515 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
4516 from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
4517 thus ?thesis using x(2)[unfolded `x = a - b`] by blast
4520 text{* We can state this in terms of diameter of a set. *}
4522 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"
4523 (* TODO: generalize to class metric_space *)
4525 lemma diameter_bounded:
4527 shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4528 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"
4530 let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"
4531 obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto
4532 { fix x y assume "x \<in> s" "y \<in> s"
4533 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) }
4535 { fix x y assume "x\<in>s" "y\<in>s" hence "s \<noteq> {}" by auto
4536 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`
4537 by simp (blast intro!: Sup_upper *) }
4539 { fix d::real assume "d>0" "d < diameter s"
4540 hence "s\<noteq>{}" unfolding diameter_def by auto
4541 have "\<exists>d' \<in> ?D. d' > d"
4543 assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"
4544 hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE)
4545 thus False using `d < diameter s` `s\<noteq>{}`
4546 apply (auto simp add: diameter_def)
4547 apply (drule Sup_real_iff [THEN [2] rev_iffD2])
4551 hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto }
4552 ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4553 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto
4556 lemma diameter_bounded_bound:
4557 "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"
4558 using diameter_bounded by blast
4560 lemma diameter_compact_attained:
4561 fixes s :: "'a::real_normed_vector set"
4562 assumes "compact s" "s \<noteq> {}"
4563 shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
4565 have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
4566 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
4567 hence "diameter s \<le> norm (x - y)"
4568 unfolding diameter_def by clarsimp (rule Sup_least, fast+)
4570 by (metis b diameter_bounded_bound order_antisym xys)
4573 text{* Related results with closure as the conclusion. *}
4575 lemma closed_scaling:
4576 fixes s :: "'a::real_normed_vector set"
4577 assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
4579 case True thus ?thesis by auto
4584 have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
4585 case True thus ?thesis apply auto unfolding * by auto
4588 { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s" "(x ---> l) sequentially"
4589 { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
4590 using as(1)[THEN spec[where x=n]]
4591 using `c\<noteq>0` by auto
4594 { fix e::real assume "e>0"
4595 hence "0 < e *\<bar>c\<bar>" using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
4596 then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
4597 using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto
4598 hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
4599 unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
4600 using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto }
4601 hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto
4602 ultimately have "l \<in> scaleR c ` s"
4603 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"]]
4604 unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto }
4605 thus ?thesis unfolding closed_sequential_limits by fast
4609 lemma closed_negations:
4610 fixes s :: "'a::real_normed_vector set"
4611 assumes "closed s" shows "closed ((\<lambda>x. -x) ` s)"
4612 using closed_scaling[OF assms, of "- 1"] by simp
4614 lemma compact_closed_sums:
4615 fixes s :: "'a::real_normed_vector set"
4616 assumes "compact s" "closed t" shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4618 let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
4619 { fix x l assume as:"\<forall>n. x n \<in> ?S" "(x ---> l) sequentially"
4620 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"
4621 using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
4622 obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
4623 using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
4624 have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
4625 using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
4626 hence "l - l' \<in> t"
4627 using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
4629 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
4631 thus ?thesis unfolding closed_sequential_limits by fast
4634 lemma closed_compact_sums:
4635 fixes s t :: "'a::real_normed_vector set"
4636 assumes "closed s" "compact t"
4637 shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4639 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
4640 apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
4641 thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
4644 lemma compact_closed_differences:
4645 fixes s t :: "'a::real_normed_vector set"
4646 assumes "compact s" "closed t"
4647 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4649 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4650 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4651 thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
4654 lemma closed_compact_differences:
4655 fixes s t :: "'a::real_normed_vector set"
4656 assumes "closed s" "compact t"
4657 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4659 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4660 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4661 thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
4664 lemma closed_translation:
4665 fixes a :: "'a::real_normed_vector"
4666 assumes "closed s" shows "closed ((\<lambda>x. a + x) ` s)"
4668 have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
4669 thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
4672 lemma translation_Compl:
4673 fixes a :: "'a::ab_group_add"
4674 shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
4675 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto
4677 lemma translation_UNIV:
4678 fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
4679 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
4681 lemma translation_diff:
4682 fixes a :: "'a::ab_group_add"
4683 shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
4686 lemma closure_translation:
4687 fixes a :: "'a::real_normed_vector"
4688 shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
4690 have *:"op + a ` (- s) = - op + a ` s"
4691 apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
4692 show ?thesis unfolding closure_interior translation_Compl
4693 using interior_translation[of a "- s"] unfolding * by auto
4696 lemma frontier_translation:
4697 fixes a :: "'a::real_normed_vector"
4698 shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
4699 unfolding frontier_def translation_diff interior_translation closure_translation by auto
4701 subsection{* Separation between points and sets. *}
4703 lemma separate_point_closed:
4704 fixes s :: "'a::heine_borel set"
4705 shows "closed s \<Longrightarrow> a \<notin> s ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
4706 proof(cases "s = {}")
4708 thus ?thesis by(auto intro!: exI[where x=1])
4711 assume "closed s" "a \<notin> s"
4712 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
4713 with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
4716 lemma separate_compact_closed:
4717 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4718 (* TODO: does this generalize to heine_borel? *)
4719 assumes "compact s" and "closed t" and "s \<inter> t = {}"
4720 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4722 have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto
4723 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"
4724 using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto
4725 { fix x y assume "x\<in>s" "y\<in>t"
4726 hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto
4727 hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute
4728 by (auto simp add: dist_commute)
4729 hence "d \<le> dist x y" unfolding dist_norm by auto }
4730 thus ?thesis using `d>0` by auto
4733 lemma separate_closed_compact:
4734 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4735 assumes "closed s" and "compact t" and "s \<inter> t = {}"
4736 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4738 have *:"t \<inter> s = {}" using assms(3) by auto
4739 show ?thesis using separate_compact_closed[OF assms(2,1) *]
4740 apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
4741 by (auto simp add: dist_commute)
4744 subsection {* Intervals *}
4746 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows
4747 "{a <..< b} = {x::'a. \<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i}" and
4748 "{a .. b} = {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i}"
4749 by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
4751 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows
4752 "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i)"
4753 "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i)"
4754 using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
4756 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows
4757 "({a <..< b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i))" (is ?th1) and
4758 "({a .. b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i < a$$i))" (is ?th2)
4760 { fix i x assume i:"i<DIM('a)" and as:"b$$i \<le> a$$i" and x:"x\<in>{a <..< b}"
4761 hence "a $$ i < x $$ i \<and> x $$ i < b $$ i" unfolding mem_interval by auto
4762 hence "a$$i < b$$i" by auto
4763 hence False using as by auto }
4765 { assume as:"\<forall>i<DIM('a). \<not> (b$$i \<le> a$$i)"
4766 let ?x = "(1/2) *\<^sub>R (a + b)"
4767 { fix i assume i:"i<DIM('a)"
4768 have "a$$i < b$$i" using as[THEN spec[where x=i]] using i by auto
4769 hence "a$$i < ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i < b$$i"
4770 unfolding euclidean_simps by auto }
4771 hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto }
4772 ultimately show ?th1 by blast
4774 { fix i x assume i:"i<DIM('a)" and as:"b$$i < a$$i" and x:"x\<in>{a .. b}"
4775 hence "a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i" unfolding mem_interval by auto
4776 hence "a$$i \<le> b$$i" by auto
4777 hence False using as by auto }
4779 { assume as:"\<forall>i<DIM('a). \<not> (b$$i < a$$i)"
4780 let ?x = "(1/2) *\<^sub>R (a + b)"
4781 { fix i assume i:"i<DIM('a)"
4782 have "a$$i \<le> b$$i" using as[THEN spec[where x=i]] by auto
4783 hence "a$$i \<le> ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i \<le> b$$i"
4784 unfolding euclidean_simps by auto }
4785 hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto }
4786 ultimately show ?th2 by blast
4789 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows
4790 "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> b$$i)" and
4791 "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"
4792 unfolding interval_eq_empty[of a b] by fastsimp+
4794 lemma interval_sing: fixes a :: "'a::ordered_euclidean_space" shows
4795 "{a .. a} = {a}" "{a<..<a} = {}"
4796 apply(auto simp add: set_eq_iff euclidean_eq[where 'a='a] eucl_less[where 'a='a] eucl_le[where 'a='a])
4797 apply (simp add: order_eq_iff) apply(rule_tac x=0 in exI) by (auto simp add: not_less less_imp_le)
4799 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows
4800 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
4801 "(\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
4802 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
4803 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
4804 unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
4805 by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
4807 lemma interval_open_subset_closed: fixes a :: "'a::ordered_euclidean_space" shows
4808 "{a<..<b} \<subseteq> {a .. b}"
4809 proof(simp add: subset_eq, rule)
4811 assume x:"x \<in>{a<..<b}"
4812 { fix i assume "i<DIM('a)"
4813 hence "a $$ i \<le> x $$ i"
4814 using x order_less_imp_le[of "a$$i" "x$$i"]
4815 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4818 { fix i assume "i<DIM('a)"
4819 hence "x $$ i \<le> b $$ i"
4820 using x order_less_imp_le[of "x$$i" "b$$i"]
4821 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4824 show "a \<le> x \<and> x \<le> b"
4825 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4828 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows
4829 "{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
4830 "{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
4831 "{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
4832 "{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)
4834 show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
4835 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)
4836 { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i<DIM('a). c$$i < d$$i"
4837 hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
4838 fix i assume i:"i<DIM('a)"
4839 (** TODO combine the following two parts as done in the HOL_light version. **)
4840 { let ?x = "(\<chi>\<chi> j. (if j=i then ((min (a$$j) (d$$j))+c$$j)/2 else (c$$j+d$$j)/2))::'a"
4841 assume as2: "a$$i > c$$i"
4842 { fix j assume j:"j<DIM('a)"
4843 hence "c $$ j < ?x $$ j \<and> ?x $$ j < d $$ j"
4844 apply(cases "j=i") using as(2)[THEN spec[where x=j]] i
4845 by (auto simp add: as2) }
4846 hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto
4848 have "?x\<notin>{a .. b}"
4849 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4850 using as(2)[THEN spec[where x=i]] and as2 i
4852 ultimately have False using as by auto }
4853 hence "a$$i \<le> c$$i" by(rule ccontr)auto
4855 { let ?x = "(\<chi>\<chi> j. (if j=i then ((max (b$$j) (c$$j))+d$$j)/2 else (c$$j+d$$j)/2))::'a"
4856 assume as2: "b$$i < d$$i"
4857 { fix j assume "j<DIM('a)"
4858 hence "d $$ j > ?x $$ j \<and> ?x $$ j > c $$ j"
4859 apply(cases "j=i") using as(2)[THEN spec[where x=j]]
4860 by (auto simp add: as2) }
4861 hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
4863 have "?x\<notin>{a .. b}"
4864 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4865 using as(2)[THEN spec[where x=i]] and as2 using i
4867 ultimately have False using as by auto }
4868 hence "b$$i \<ge> d$$i" by(rule ccontr)auto
4870 have "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" by auto
4872 show ?th3 unfolding subset_eq and Ball_def and mem_interval
4873 apply(rule,rule,rule,rule) apply(rule part1) unfolding subset_eq and Ball_def and mem_interval
4874 prefer 4 apply auto by(erule_tac x=i in allE,erule_tac x=i in allE,fastsimp)+
4875 { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i<DIM('a). c$$i < d$$i"
4876 fix i assume i:"i<DIM('a)"
4877 from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
4878 hence "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" using part1 and as(2) using i by auto } note * = this
4879 show ?th4 unfolding subset_eq and Ball_def and mem_interval
4880 apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4
4881 apply auto by(erule_tac x=i in allE, simp)+
4884 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows
4885 "{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
4886 "{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
4887 "{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
4888 "{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)
4890 let ?z = "(\<chi>\<chi> i. ((max (a$$i) (c$$i)) + (min (b$$i) (d$$i))) / 2)::'a"
4891 note * = set_eq_iff Int_iff empty_iff mem_interval all_conj_distrib[THEN sym] eq_False
4892 show ?th1 unfolding * apply safe apply(erule_tac x="?z" in allE)
4893 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4894 show ?th2 unfolding * apply safe apply(erule_tac x="?z" in allE)
4895 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4896 show ?th3 unfolding * apply safe apply(erule_tac x="?z" in allE)
4897 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4898 show ?th4 unfolding * apply safe apply(erule_tac x="?z" in allE)
4899 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4902 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows
4903 "{a .. b} \<inter> {c .. d} = {(\<chi>\<chi> i. max (a$$i) (c$$i)) .. (\<chi>\<chi> i. min (b$$i) (d$$i))}"
4904 unfolding set_eq_iff and Int_iff and mem_interval
4907 (* Moved interval_open_subset_closed a bit upwards *)
4909 lemma open_interval_lemma: fixes x :: "real" shows
4910 "a < x \<Longrightarrow> x < b ==> (\<exists>d>0. \<forall>x'. abs(x' - x) < d --> a < x' \<and> x' < b)"
4911 by(rule_tac x="min (x - a) (b - x)" in exI, auto)
4913 lemma open_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
4915 { fix x assume x:"x\<in>{a<..<b}"
4916 { fix i assume "i<DIM('a)"
4917 hence "\<exists>d>0. \<forall>x'. abs (x' - (x$$i)) < d \<longrightarrow> a$$i < x' \<and> x' < b$$i"
4918 using x[unfolded mem_interval, THEN spec[where x=i]]
4919 using open_interval_lemma[of "a$$i" "x$$i" "b$$i"] by auto }
4920 hence "\<forall>i\<in>{..<DIM('a)}. \<exists>d>0. \<forall>x'. abs (x' - (x$$i)) < d \<longrightarrow> a$$i < x' \<and> x' < b$$i" by auto
4921 from bchoice[OF this] guess d .. note d=this
4922 let ?d = "Min (d ` {..<DIM('a)})"
4923 have **:"finite (d ` {..<DIM('a)})" "d ` {..<DIM('a)} \<noteq> {}" by auto
4924 have "?d>0" using Min_gr_iff[OF **] using d by auto
4926 { fix x' assume as:"dist x' x < ?d"
4927 { fix i assume i:"i<DIM('a)"
4928 hence "\<bar>x'$$i - x $$ i\<bar> < d i"
4929 using norm_bound_component_lt[OF as[unfolded dist_norm], of i]
4930 unfolding euclidean_simps Min_gr_iff[OF **] by auto
4931 hence "a $$ i < x' $$ i" "x' $$ i < b $$ i" using i and d[THEN bspec[where x=i]] by auto }
4932 hence "a < x' \<and> x' < b" apply(subst(2) eucl_less,subst(1) eucl_less) by auto }
4933 ultimately have "\<exists>e>0. \<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a<..<b}" by auto
4935 thus ?thesis unfolding open_dist using open_interval_lemma by auto
4938 lemma closed_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
4940 { fix x i assume i:"i<DIM('a)"
4941 assume as:"\<forall>e>0. \<exists>x'\<in>{a..b}. x' \<noteq> x \<and> dist x' x < e"(* and xab:"a$$i > x$$i \<or> b$$i < x$$i"*)
4942 { assume xa:"a$$i > x$$i"
4943 with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < a$$i - x$$i" by(erule_tac x="a$$i - x$$i" in allE)auto
4944 hence False unfolding mem_interval and dist_norm
4945 using component_le_norm[of "y-x" i, unfolded euclidean_simps] and xa using i
4946 by(auto elim!: allE[where x=i])
4947 } hence "a$$i \<le> x$$i" by(rule ccontr)auto
4949 { assume xb:"b$$i < x$$i"
4950 with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < x$$i - b$$i"
4951 by(erule_tac x="x$$i - b$$i" in allE)auto
4952 hence False unfolding mem_interval and dist_norm
4953 using component_le_norm[of "y-x" i, unfolded euclidean_simps] and xb using i
4954 by(auto elim!: allE[where x=i])
4955 } hence "x$$i \<le> b$$i" by(rule ccontr)auto
4957 have "a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i" by auto }
4958 thus ?thesis unfolding closed_limpt islimpt_approachable mem_interval by auto
4961 lemma interior_closed_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows
4962 "interior {a .. b} = {a<..<b}" (is "?L = ?R")
4963 proof(rule subset_antisym)
4964 show "?R \<subseteq> ?L" using interior_maximal[OF interval_open_subset_closed open_interval] by auto
4966 { fix x assume "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> {a..b}"
4967 then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" by auto
4968 then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto
4969 { fix i assume i:"i<DIM('a)"
4970 have "dist (x - (e / 2) *\<^sub>R basis i) x < e"
4971 "dist (x + (e / 2) *\<^sub>R basis i) x < e"
4972 unfolding dist_norm apply auto
4973 unfolding norm_minus_cancel using norm_basis and `e>0` by auto
4974 hence "a $$ i \<le> (x - (e / 2) *\<^sub>R basis i) $$ i"
4975 "(x + (e / 2) *\<^sub>R basis i) $$ i \<le> b $$ i"
4976 using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]
4977 and e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]
4978 unfolding mem_interval by (auto elim!: allE[where x=i])
4979 hence "a $$ i < x $$ i" and "x $$ i < b $$ i" unfolding euclidean_simps
4980 unfolding basis_component using `e>0` i by auto }
4981 hence "x \<in> {a<..<b}" unfolding mem_interval by auto }
4982 thus "?L \<subseteq> ?R" unfolding interior_def and subset_eq by auto
4985 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"
4987 let ?b = "\<Sum>i<DIM('a). \<bar>a$$i\<bar> + \<bar>b$$i\<bar>"
4988 { fix x::"'a" assume x:"\<forall>i<DIM('a). a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i"
4989 { fix i assume "i<DIM('a)"
4990 hence "\<bar>x$$i\<bar> \<le> \<bar>a$$i\<bar> + \<bar>b$$i\<bar>" using x[THEN spec[where x=i]] by auto }
4991 hence "(\<Sum>i<DIM('a). \<bar>x $$ i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto
4992 hence "norm x \<le> ?b" using norm_le_l1[of x] by auto }
4993 thus ?thesis unfolding interval and bounded_iff by auto
4996 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows
4997 "bounded {a .. b} \<and> bounded {a<..<b}"
4998 using bounded_closed_interval[of a b]
4999 using interval_open_subset_closed[of a b]
5000 using bounded_subset[of "{a..b}" "{a<..<b}"]
5003 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows
5004 "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
5005 using bounded_interval[of a b] by auto
5007 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"
5008 using bounded_closed_imp_compact[of "{a..b}"] using bounded_interval[of a b]
5011 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"
5012 assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
5014 { fix i assume "i<DIM('a)"
5015 hence "a $$ i < ((1 / 2) *\<^sub>R (a + b)) $$ i \<and> ((1 / 2) *\<^sub>R (a + b)) $$ i < b $$ i"
5016 using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
5017 unfolding euclidean_simps by auto }
5018 thus ?thesis unfolding mem_interval by auto
5021 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"
5022 assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
5023 shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
5025 { fix i assume i:"i<DIM('a)"
5026 have "a $$ i = e * a$$i + (1 - e) * a$$i" unfolding left_diff_distrib by simp
5027 also have "\<dots> < e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
5028 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
5029 using x unfolding mem_interval using i apply simp
5030 using y unfolding mem_interval using i apply simp
5032 finally have "a $$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i" unfolding euclidean_simps by auto
5034 have "b $$ i = e * b$$i + (1 - e) * b$$i" unfolding left_diff_distrib by simp
5035 also have "\<dots> > e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
5036 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
5037 using x unfolding mem_interval using i apply simp
5038 using y unfolding mem_interval using i apply simp
5040 finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i < b $$ i" unfolding euclidean_simps by auto
5041 } ultimately have "a $$ 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 }
5042 thus ?thesis unfolding mem_interval by auto
5045 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"
5046 assumes "{a<..<b} \<noteq> {}"
5047 shows "closure {a<..<b} = {a .. b}"
5049 have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto
5050 let ?c = "(1 / 2) *\<^sub>R (a + b)"
5051 { fix x assume as:"x \<in> {a .. b}"
5052 def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
5053 { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
5054 have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
5055 have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
5056 x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
5057 by (auto simp add: algebra_simps)
5058 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
5059 hence False using fn unfolding f_def using xc by auto }
5061 { assume "\<not> (f ---> x) sequentially"
5062 { fix e::real assume "e>0"
5063 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
5064 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
5065 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)
5066 hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto }
5067 hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
5068 unfolding Lim_sequentially by(auto simp add: dist_norm)
5069 hence "(f ---> x) sequentially" unfolding f_def
5070 using tendsto_add[OF tendsto_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
5071 using scaleR.tendsto [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto }
5072 ultimately have "x \<in> closure {a<..<b}"
5073 using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto }
5074 thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
5077 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"
5078 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a<..<a}"
5080 obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
5081 def a \<equiv> "(\<chi>\<chi> i. b+1)::'a"
5082 { fix x assume "x\<in>s"
5083 fix i assume i:"i<DIM('a)"
5084 hence "(-a)$$i < x$$i" and "x$$i < a$$i" using b[THEN bspec[where x=x], OF `x\<in>s`]
5085 and component_le_norm[of x i] unfolding euclidean_simps and a_def by auto }
5086 thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
5089 lemma bounded_subset_open_interval:
5090 fixes s :: "('a::ordered_euclidean_space) set"
5091 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
5092 by (auto dest!: bounded_subset_open_interval_symmetric)
5094 lemma bounded_subset_closed_interval_symmetric:
5095 fixes s :: "('a::ordered_euclidean_space) set"
5096 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
5098 obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
5099 thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
5102 lemma bounded_subset_closed_interval:
5103 fixes s :: "('a::ordered_euclidean_space) set"
5104 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
5105 using bounded_subset_closed_interval_symmetric[of s] by auto
5107 lemma frontier_closed_interval:
5108 fixes a b :: "'a::ordered_euclidean_space"
5109 shows "frontier {a .. b} = {a .. b} - {a<..<b}"
5110 unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
5112 lemma frontier_open_interval:
5113 fixes a b :: "'a::ordered_euclidean_space"
5114 shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
5115 proof(cases "{a<..<b} = {}")
5116 case True thus ?thesis using frontier_empty by auto
5118 case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
5121 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"
5122 assumes "{c<..<d} \<noteq> {}" shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
5123 unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
5126 (* Some stuff for half-infinite intervals too; FIXME: notation? *)
5128 lemma closed_interval_left: fixes b::"'a::euclidean_space"
5129 shows "closed {x::'a. \<forall>i<DIM('a). x$$i \<le> b$$i}"
5131 { fix i assume i:"i<DIM('a)"
5132 fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). x $$ i \<le> b $$ i}. x' \<noteq> x \<and> dist x' x < e"
5133 { assume "x$$i > b$$i"
5134 then obtain y where "y $$ i \<le> b $$ i" "y \<noteq> x" "dist y x < x$$i - b$$i"
5135 using x[THEN spec[where x="x$$i - b$$i"]] using i by auto
5136 hence False using component_le_norm[of "y - x" i] unfolding dist_norm euclidean_simps using i
5138 hence "x$$i \<le> b$$i" by(rule ccontr)auto }
5139 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
5142 lemma closed_interval_right: fixes a::"'a::euclidean_space"
5143 shows "closed {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i}"
5145 { fix i assume i:"i<DIM('a)"
5146 fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). a $$ i \<le> x $$ i}. x' \<noteq> x \<and> dist x' x < e"
5147 { assume "a$$i > x$$i"
5148 then obtain y where "a $$ i \<le> y $$ i" "y \<noteq> x" "dist y x < a$$i - x$$i"
5149 using x[THEN spec[where x="a$$i - x$$i"]] i by auto
5150 hence False using component_le_norm[of "y - x" i] unfolding dist_norm and euclidean_simps by auto }
5151 hence "a$$i \<le> x$$i" by(rule ccontr)auto }
5152 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
5155 text {* Intervals in general, including infinite and mixtures of open and closed. *}
5157 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
5158 (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i<DIM('a). ((a$$i \<le> x$$i \<and> x$$i \<le> b$$i) \<or> (b$$i \<le> x$$i \<and> x$$i \<le> a$$i))) \<longrightarrow> x \<in> s)"
5160 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)
5161 "is_interval {a<..<b}" (is ?th2) proof -
5162 have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
5163 show ?th1 ?th2 unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
5164 by(meson order_trans le_less_trans less_le_trans *)+ qed
5166 lemma is_interval_empty:
5168 unfolding is_interval_def
5171 lemma is_interval_univ:
5173 unfolding is_interval_def
5176 subsection{* Closure of halfspaces and hyperplanes. *}
5179 assumes "(f ---> l) net" shows "((\<lambda>y. inner a (f y)) ---> inner a l) net"
5180 by (intro tendsto_intros assms)
5182 lemma continuous_at_inner: "continuous (at x) (inner a)"
5183 unfolding continuous_at by (intro tendsto_intros)
5185 lemma continuous_at_euclidean_component[intro!, simp]: "continuous (at x) (\<lambda>x. x $$ i)"
5186 unfolding euclidean_component_def by (rule continuous_at_inner)
5188 lemma continuous_on_inner:
5189 fixes s :: "'a::real_inner set"
5190 shows "continuous_on s (inner a)"
5191 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
5193 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
5195 have "\<forall>x. continuous (at x) (inner a)"
5196 unfolding continuous_at by (rule allI) (intro tendsto_intros)
5197 hence "closed (inner a -` {..b})"
5198 using closed_real_atMost by (rule continuous_closed_vimage)
5199 moreover have "{x. inner a x \<le> b} = inner a -` {..b}" by auto
5200 ultimately show ?thesis by simp
5203 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
5204 using closed_halfspace_le[of "-a" "-b"] unfolding inner_minus_left by auto
5206 lemma closed_hyperplane: "closed {x. inner a x = b}"
5208 have "{x. inner a x = b} = {x. inner a x \<ge> b} \<inter> {x. inner a x \<le> b}" by auto
5209 thus ?thesis using closed_halfspace_le[of a b] and closed_halfspace_ge[of b a] using closed_Int by auto
5212 lemma closed_halfspace_component_le:
5213 shows "closed {x::'a::euclidean_space. x$$i \<le> a}"
5214 using closed_halfspace_le[of "(basis i)::'a" a] unfolding euclidean_component_def .
5216 lemma closed_halfspace_component_ge:
5217 shows "closed {x::'a::euclidean_space. x$$i \<ge> a}"
5218 using closed_halfspace_ge[of a "(basis i)::'a"] unfolding euclidean_component_def .
5220 text{* Openness of halfspaces. *}
5222 lemma open_halfspace_lt: "open {x. inner a x < b}"
5224 have "- {x. b \<le> inner a x} = {x. inner a x < b}" by auto
5225 thus ?thesis using closed_halfspace_ge[unfolded closed_def, of b a] by auto
5228 lemma open_halfspace_gt: "open {x. inner a x > b}"
5230 have "- {x. b \<ge> inner a x} = {x. inner a x > b}" by auto
5231 thus ?thesis using closed_halfspace_le[unfolded closed_def, of a b] by auto
5234 lemma open_halfspace_component_lt:
5235 shows "open {x::'a::euclidean_space. x$$i < a}"
5236 using open_halfspace_lt[of "(basis i)::'a" a] unfolding euclidean_component_def .
5238 lemma open_halfspace_component_gt:
5239 shows "open {x::'a::euclidean_space. x$$i > a}"
5240 using open_halfspace_gt[of a "(basis i)::'a"] unfolding euclidean_component_def .
5242 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}
5244 lemma eucl_lessThan_eq_halfspaces:
5245 fixes a :: "'a\<Colon>ordered_euclidean_space"
5246 shows "{..<a} = (\<Inter>i<DIM('a). {x. x $$ i < a $$ i})"
5247 by (auto simp: eucl_less[where 'a='a])
5249 lemma eucl_greaterThan_eq_halfspaces:
5250 fixes a :: "'a\<Colon>ordered_euclidean_space"
5251 shows "{a<..} = (\<Inter>i<DIM('a). {x. a $$ i < x $$ i})"
5252 by (auto simp: eucl_less[where 'a='a])
5254 lemma eucl_atMost_eq_halfspaces:
5255 fixes a :: "'a\<Colon>ordered_euclidean_space"
5256 shows "{.. a} = (\<Inter>i<DIM('a). {x. x $$ i \<le> a $$ i})"
5257 by (auto simp: eucl_le[where 'a='a])
5259 lemma eucl_atLeast_eq_halfspaces:
5260 fixes a :: "'a\<Colon>ordered_euclidean_space"
5261 shows "{a ..} = (\<Inter>i<DIM('a). {x. a $$ i \<le> x $$ i})"
5262 by (auto simp: eucl_le[where 'a='a])
5264 lemma open_eucl_lessThan[simp, intro]:
5265 fixes a :: "'a\<Colon>ordered_euclidean_space"
5266 shows "open {..< a}"
5267 by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)
5269 lemma open_eucl_greaterThan[simp, intro]:
5270 fixes a :: "'a\<Colon>ordered_euclidean_space"
5271 shows "open {a <..}"
5272 by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)
5274 lemma closed_eucl_atMost[simp, intro]:
5275 fixes a :: "'a\<Colon>ordered_euclidean_space"
5276 shows "closed {.. a}"
5277 unfolding eucl_atMost_eq_halfspaces
5278 proof (safe intro!: closed_INT)
5280 have "- {x::'a. x $$ i \<le> a $$ i} = {x. a $$ i < x $$ i}" by auto
5281 then show "closed {x::'a. x $$ i \<le> a $$ i}"
5282 by (simp add: closed_def open_halfspace_component_gt)
5285 lemma closed_eucl_atLeast[simp, intro]:
5286 fixes a :: "'a\<Colon>ordered_euclidean_space"
5287 shows "closed {a ..}"
5288 unfolding eucl_atLeast_eq_halfspaces
5289 proof (safe intro!: closed_INT)
5291 have "- {x::'a. a $$ i \<le> x $$ i} = {x. x $$ i < a $$ i}" by auto
5292 then show "closed {x::'a. a $$ i \<le> x $$ i}"
5293 by (simp add: closed_def open_halfspace_component_lt)
5296 lemma open_vimage_euclidean_component: "open S \<Longrightarrow> open ((\<lambda>x. x $$ i) -` S)"
5297 by (auto intro!: continuous_open_vimage)
5299 text{* This gives a simple derivation of limit component bounds. *}
5301 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5302 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f(x)$$i \<le> b) net"
5303 shows "l$$i \<le> b"
5305 { fix x have "x \<in> {x::'b. inner (basis i) x \<le> b} \<longleftrightarrow> x$$i \<le> b"
5306 unfolding euclidean_component_def by auto } note * = this
5307 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *
5308 using closed_halfspace_le[of "(basis i)::'b" b] and assms(1,2,3) by auto
5311 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5312 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$$i) net"
5313 shows "b \<le> l$$i"
5315 { fix x have "x \<in> {x::'b. inner (basis i) x \<ge> b} \<longleftrightarrow> x$$i \<ge> b"
5316 unfolding euclidean_component_def by auto } note * = this
5317 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *
5318 using closed_halfspace_ge[of b "(basis i)"] and assms(1,2,3) by auto
5321 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5322 assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$$i = b) net"
5324 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
5325 text{* Limits relative to a union. *}
5327 lemma eventually_within_Un:
5328 "eventually P (net within (s \<union> t)) \<longleftrightarrow>
5329 eventually P (net within s) \<and> eventually P (net within t)"
5330 unfolding Limits.eventually_within
5331 by (auto elim!: eventually_rev_mp)
5333 lemma Lim_within_union:
5334 "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>
5335 (f ---> l) (net within s) \<and> (f ---> l) (net within t)"
5336 unfolding tendsto_def
5337 by (auto simp add: eventually_within_Un)
5339 lemma Lim_topological:
5340 "(f ---> l) net \<longleftrightarrow>
5341 trivial_limit net \<or>
5342 (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
5343 unfolding tendsto_def trivial_limit_eq by auto
5345 lemma continuous_on_union:
5346 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
5347 shows "continuous_on (s \<union> t) f"
5348 using assms unfolding continuous_on Lim_within_union
5349 unfolding Lim_topological trivial_limit_within closed_limpt by auto
5351 lemma continuous_on_cases:
5352 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
5353 "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
5354 shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
5356 let ?h = "(\<lambda>x. if P x then f x else g x)"
5357 have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto
5358 hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto
5360 have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto
5361 hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto
5362 ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto
5366 text{* Some more convenient intermediate-value theorem formulations. *}
5368 lemma connected_ivt_hyperplane:
5369 assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
5370 shows "\<exists>z \<in> s. inner a z = b"
5372 assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
5373 let ?A = "{x. inner a x < b}"
5374 let ?B = "{x. inner a x > b}"
5375 have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
5376 moreover have "?A \<inter> ?B = {}" by auto
5377 moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
5378 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
5381 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows
5382 "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)"
5383 using connected_ivt_hyperplane[of s x y "(basis k)::'a" a]
5384 unfolding euclidean_component_def by auto
5386 subsection {* Homeomorphisms *}
5388 definition "homeomorphism s t f g \<equiv>
5389 (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
5390 (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
5393 homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"
5394 (infixr "homeomorphic" 60) where
5395 homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
5397 lemma homeomorphic_refl: "s homeomorphic s"
5398 unfolding homeomorphic_def
5399 unfolding homeomorphism_def
5400 using continuous_on_id
5401 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5402 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5405 lemma homeomorphic_sym:
5406 "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
5407 unfolding homeomorphic_def
5408 unfolding homeomorphism_def
5411 lemma homeomorphic_trans:
5412 assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
5414 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"
5415 using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
5416 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"
5417 using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
5419 { 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 }
5420 moreover have "(f2 \<circ> f1) ` s = u" using fg1(2) fg2(2) by auto
5421 moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
5422 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 }
5423 moreover have "(g1 \<circ> g2) ` u = s" using fg1(5) fg2(5) by auto
5424 moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6) unfolding fg2(5) by auto
5425 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
5428 lemma homeomorphic_minimal:
5429 "s homeomorphic t \<longleftrightarrow>
5430 (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
5431 (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
5432 continuous_on s f \<and> continuous_on t g)"
5433 unfolding homeomorphic_def homeomorphism_def
5434 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
5435 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
5437 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
5438 apply auto apply(rule_tac x="g x" in bexI) apply auto
5439 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
5440 apply auto apply(rule_tac x="f x" in bexI) by auto
5442 text {* Relatively weak hypotheses if a set is compact. *}
5444 lemma homeomorphism_compact:
5445 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5446 (* class constraint due to continuous_on_inverse *)
5447 assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s"
5448 shows "\<exists>g. homeomorphism s t f g"
5450 def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
5451 have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
5452 { fix y assume "y\<in>t"
5453 then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
5454 hence "g (f x) = x" using g by auto
5455 hence "f (g y) = y" unfolding x(1)[THEN sym] by auto }
5456 hence g':"\<forall>x\<in>t. f (g x) = x" by auto
5459 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"])
5461 { assume "x\<in>g ` t"
5462 then obtain y where y:"y\<in>t" "g y = x" by auto
5463 then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
5464 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 }
5465 ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" .. }
5466 hence "g ` t = s" by auto
5468 show ?thesis unfolding homeomorphism_def homeomorphic_def
5469 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
5472 lemma homeomorphic_compact:
5473 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5474 (* class constraint due to continuous_on_inverse *)
5475 shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
5476 \<Longrightarrow> s homeomorphic t"
5477 unfolding homeomorphic_def by (metis homeomorphism_compact)
5479 text{* Preservation of topological properties. *}
5481 lemma homeomorphic_compactness:
5482 "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
5483 unfolding homeomorphic_def homeomorphism_def
5484 by (metis compact_continuous_image)
5486 text{* Results on translation, scaling etc. *}
5488 lemma homeomorphic_scaling:
5489 fixes s :: "'a::real_normed_vector set"
5490 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
5491 unfolding homeomorphic_minimal
5492 apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
5493 apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
5494 using assms apply auto
5495 using continuous_on_cmul[OF continuous_on_id] by auto
5497 lemma homeomorphic_translation:
5498 fixes s :: "'a::real_normed_vector set"
5499 shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
5500 unfolding homeomorphic_minimal
5501 apply(rule_tac x="\<lambda>x. a + x" in exI)
5502 apply(rule_tac x="\<lambda>x. -a + x" in exI)
5503 using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
5505 lemma homeomorphic_affinity:
5506 fixes s :: "'a::real_normed_vector set"
5507 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
5509 have *:"op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
5511 using homeomorphic_trans
5512 using homeomorphic_scaling[OF assms, of s]
5513 using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a] unfolding * by auto
5516 lemma homeomorphic_balls:
5517 fixes a b ::"'a::real_normed_vector" (* FIXME: generalize to metric_space *)
5518 assumes "0 < d" "0 < e"
5519 shows "(ball a d) homeomorphic (ball b e)" (is ?th)
5520 "(cball a d) homeomorphic (cball b e)" (is ?cth)
5522 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5523 show ?th unfolding homeomorphic_minimal
5524 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5525 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5526 using assms apply (auto simp add: dist_commute)
5528 apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)
5529 unfolding continuous_on
5530 by (intro ballI tendsto_intros, simp)+
5532 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5533 show ?cth unfolding homeomorphic_minimal
5534 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5535 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5536 using assms apply (auto simp add: dist_commute)
5538 apply (auto simp add: pos_divide_le_eq)
5539 unfolding continuous_on
5540 by (intro ballI tendsto_intros, simp)+
5543 text{* "Isometry" (up to constant bounds) of injective linear map etc. *}
5545 lemma cauchy_isometric:
5546 fixes x :: "nat \<Rightarrow> 'a::euclidean_space"
5547 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)"
5550 interpret f: bounded_linear f by fact
5551 { fix d::real assume "d>0"
5552 then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
5553 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
5554 { fix n assume "n\<ge>N"
5555 hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding f.diff[THEN sym] by auto
5556 moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
5557 using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
5558 using normf[THEN bspec[where x="x n - x N"]] by auto
5559 ultimately have "norm (x n - x N) < d" using `e>0`
5560 using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto }
5561 hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
5562 thus ?thesis unfolding cauchy and dist_norm by auto
5565 lemma complete_isometric_image:
5566 fixes f :: "'a::euclidean_space => 'b::euclidean_space"
5567 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"
5568 shows "complete(f ` s)"
5570 { fix g assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
5571 then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
5572 using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
5573 hence x:"\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
5574 hence "f \<circ> x = g" unfolding fun_eq_iff by auto
5575 then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
5576 using cs[unfolded complete_def, THEN spec[where x="x"]]
5577 using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
5578 hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
5579 using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
5580 unfolding `f \<circ> x = g` by auto }
5581 thus ?thesis unfolding complete_def by auto
5585 fixes x :: "'a::real_normed_vector"
5586 shows "dist 0 x = norm x"
5587 unfolding dist_norm by simp
5589 lemma injective_imp_isometric: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
5590 assumes s:"closed s" "subspace s" and f:"bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
5591 shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
5592 proof(cases "s \<subseteq> {0::'a}")
5594 { fix x assume "x \<in> s"
5595 hence "x = 0" using True by auto
5596 hence "norm x \<le> norm (f x)" by auto }
5597 thus ?thesis by(auto intro!: exI[where x=1])
5599 interpret f: bounded_linear f by fact
5601 then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
5602 from False have "s \<noteq> {}" by auto
5603 let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
5604 let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
5605 let ?S'' = "{x::'a. norm x = norm a}"
5607 have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by auto
5608 hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
5609 moreover have "?S' = s \<inter> ?S''" by auto
5610 ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
5611 moreover have *:"f ` ?S' = ?S" by auto
5612 ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
5613 hence "closed ?S" using compact_imp_closed by auto
5614 moreover have "?S \<noteq> {}" using a by auto
5615 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
5616 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
5618 let ?e = "norm (f b) / norm b"
5619 have "norm b > 0" using ba and a and norm_ge_zero by auto
5620 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
5621 ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
5623 { fix x assume "x\<in>s"
5624 hence "norm (f b) / norm b * norm x \<le> norm (f x)"
5626 case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
5629 hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
5630 have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def] by auto
5631 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
5632 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"]]
5633 unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
5634 by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq)
5637 show ?thesis by auto
5640 lemma closed_injective_image_subspace:
5641 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
5642 assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
5643 shows "closed(f ` s)"
5645 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
5646 show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
5647 unfolding complete_eq_closed[THEN sym] by auto
5650 subsection{* Some properties of a canonical subspace. *}
5653 declare euclidean_component.zero[simp]
5655 lemma subspace_substandard:
5656 "subspace {x::'a::euclidean_space. (\<forall>i<DIM('a). P i \<longrightarrow> x$$i = 0)}"
5657 unfolding subspace_def by(auto simp add: euclidean_simps)
5659 lemma closed_substandard:
5660 "closed {x::'a::euclidean_space. \<forall>i<DIM('a). P i --> x$$i = 0}" (is "closed ?A")
5662 let ?D = "{i. P i} \<inter> {..<DIM('a)}"
5663 let ?Bs = "{{x::'a. inner (basis i) x = 0}| i. i \<in> ?D}"
5666 hence x:"\<forall>i\<in>?D. x $$ i = 0" by auto
5667 hence "x\<in> \<Inter> ?Bs" by(auto simp add: x euclidean_component_def) }
5669 { assume x:"x\<in>\<Inter>?Bs"
5670 { fix i assume i:"i \<in> ?D"
5671 then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::'a. inner (basis i) x = 0}" by auto
5672 hence "x $$ i = 0" unfolding B using x unfolding euclidean_component_def by auto }
5673 hence "x\<in>?A" by auto }
5674 ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" .. }
5675 hence "?A = \<Inter> ?Bs" by auto
5676 thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
5679 lemma dim_substandard: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
5680 shows "dim {x::'a::euclidean_space. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0} = card d" (is "dim ?A = _")
5682 let ?D = "{..<DIM('a)}"
5683 let ?B = "(basis::nat => 'a) ` d"
5684 let ?bas = "basis::nat \<Rightarrow> 'a"
5685 have "?B \<subseteq> ?A" by(auto simp add:basis_component)
5687 { fix x::"'a" assume "x\<in>?A"
5688 hence "finite d" "x\<in>?A" using assms by(auto intro:finite_subset)
5689 hence "x\<in> span ?B"
5690 proof(induct d arbitrary: x)
5691 case empty hence "x=0" apply(subst euclidean_eq) by auto
5692 thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
5695 hence *:"\<forall>i<DIM('a). i \<notin> insert k F \<longrightarrow> x $$ i = 0" by auto
5696 have **:"F \<subseteq> insert k F" by auto
5697 def y \<equiv> "x - x$$k *\<^sub>R basis k"
5698 have y:"x = y + (x$$k) *\<^sub>R basis k" unfolding y_def by auto
5699 { fix i assume i':"i \<notin> F"
5700 hence "y $$ i = 0" unfolding y_def
5701 using *[THEN spec[where x=i]] by(auto simp add: euclidean_simps basis_component) }
5702 hence "y \<in> span (basis ` F)" using insert(3) by auto
5703 hence "y \<in> span (basis ` (insert k F))"
5704 using span_mono[of "?bas ` F" "?bas ` (insert k F)"]
5705 using image_mono[OF **, of basis] using assms by auto
5707 have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
5708 hence "x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5709 using span_mul by auto
5711 have "y + x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5712 using span_add by auto
5713 thus ?case using y by auto
5716 hence "?A \<subseteq> span ?B" by auto
5718 { fix x assume "x \<in> ?B"
5719 hence "x\<in>{(basis i)::'a |i. i \<in> ?D}" using assms by auto }
5720 hence "independent ?B" using independent_mono[OF independent_basis, of ?B] and assms by auto
5722 have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
5723 hence *:"inj_on (basis::nat\<Rightarrow>'a) d" using subset_inj_on[OF basis_inj, of "d"] by auto
5724 have "card ?B = card d" unfolding card_image[OF *] by auto
5725 ultimately show ?thesis using dim_unique[of "basis ` d" ?A] by auto
5728 text{* Hence closure and completeness of all subspaces. *}
5730 lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"
5732 apply (rule_tac x="{}" in exI, simp)
5734 apply (subgoal_tac "\<exists>x. x \<notin> A")
5736 apply (rule_tac x="insert x A" in exI, simp)
5737 apply (subgoal_tac "A \<noteq> UNIV", auto)
5740 lemma closed_subspace: fixes s::"('a::euclidean_space) set"
5741 assumes "subspace s" shows "closed s"
5743 have *:"dim s \<le> DIM('a)" using dim_subset_UNIV by auto
5744 def d \<equiv> "{..<dim s}" have t:"card d = dim s" unfolding d_def by auto
5745 let ?t = "{x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0}"
5746 have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0} = s \<and>
5747 inj_on f {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0}"
5748 apply(rule subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]])
5749 using dim_substandard[of d,where 'a='a] and t unfolding d_def using * assms by auto
5750 then guess f apply-by(erule exE conjE)+ note f = this
5751 interpret f: bounded_linear f using f unfolding linear_conv_bounded_linear by auto
5752 have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using f.zero using f(3)[unfolded inj_on_def]
5753 by(erule_tac x=0 in ballE) auto
5754 moreover have "closed ?t" using closed_substandard .
5755 moreover have "subspace ?t" using subspace_substandard .
5756 ultimately show ?thesis using closed_injective_image_subspace[of ?t f]
5757 unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
5760 lemma complete_subspace:
5761 fixes s :: "('a::euclidean_space) set" shows "subspace s ==> complete s"
5762 using complete_eq_closed closed_subspace
5766 fixes s :: "('a::euclidean_space) set"
5767 shows "dim(closure s) = dim s" (is "?dc = ?d")
5769 have "?dc \<le> ?d" using closure_minimal[OF span_inc, of s]
5770 using closed_subspace[OF subspace_span, of s]
5771 using dim_subset[of "closure s" "span s"] unfolding dim_span by auto
5772 thus ?thesis using dim_subset[OF closure_subset, of s] by auto
5775 subsection {* Affine transformations of intervals *}
5777 lemma real_affinity_le:
5778 "0 < (m::'a::linordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"
5779 by (simp add: field_simps inverse_eq_divide)
5781 lemma real_le_affinity:
5782 "0 < (m::'a::linordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"
5783 by (simp add: field_simps inverse_eq_divide)
5785 lemma real_affinity_lt:
5786 "0 < (m::'a::linordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"
5787 by (simp add: field_simps inverse_eq_divide)
5789 lemma real_lt_affinity:
5790 "0 < (m::'a::linordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"
5791 by (simp add: field_simps inverse_eq_divide)
5793 lemma real_affinity_eq:
5794 "(m::'a::linordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"
5795 by (simp add: field_simps inverse_eq_divide)
5797 lemma real_eq_affinity:
5798 "(m::'a::linordered_field) \<noteq> 0 ==> (y = m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"
5799 by (simp add: field_simps inverse_eq_divide)
5801 lemma image_affinity_interval: fixes m::real
5802 fixes a b c :: "'a::ordered_euclidean_space"
5803 shows "(\<lambda>x. m *\<^sub>R x + c) ` {a .. b} =
5804 (if {a .. b} = {} then {}
5805 else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
5806 else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
5808 { fix x assume "x \<le> c" "c \<le> x"
5809 hence "x=c" unfolding eucl_le[where 'a='a] apply-
5810 apply(subst euclidean_eq) by (auto intro: order_antisym) }
5812 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])
5813 ultimately show ?thesis by auto
5816 { fix y assume "a \<le> y" "y \<le> b" "m > 0"
5817 hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
5818 unfolding eucl_le[where 'a='a] by(auto simp add: euclidean_simps)
5820 { fix y assume "a \<le> y" "y \<le> b" "m < 0"
5821 hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
5822 unfolding eucl_le[where 'a='a] by(auto simp add: mult_left_mono_neg euclidean_simps)
5824 { fix y assume "m > 0" "m *\<^sub>R a + c \<le> y" "y \<le> m *\<^sub>R b + c"
5825 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5826 unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
5827 apply(auto simp add: pth_3[symmetric]
5828 intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
5829 by(auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff euclidean_simps)
5831 { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
5832 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5833 unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
5834 apply(auto simp add: pth_3[symmetric]
5835 intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
5836 by(auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff euclidean_simps)
5838 ultimately show ?thesis using False by auto
5841 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a..b} =
5842 (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})"
5843 using image_affinity_interval[of m 0 a b] by auto
5845 subsection{* Banach fixed point theorem (not really topological...) *}
5848 assumes s:"complete s" "s \<noteq> {}" and c:"0 \<le> c" "c < 1" and f:"(f ` s) \<subseteq> s" and
5849 lipschitz:"\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
5850 shows "\<exists>! x\<in>s. (f x = x)"
5852 have "1 - c > 0" using c by auto
5854 from s(2) obtain z0 where "z0 \<in> s" by auto
5855 def z \<equiv> "\<lambda>n. (f ^^ n) z0"
5857 have "z n \<in> s" unfolding z_def
5858 proof(induct n) case 0 thus ?case using `z0 \<in>s` by auto
5859 next case Suc thus ?case using f by auto qed }
5862 def d \<equiv> "dist (z 0) (z 1)"
5864 have fzn:"\<And>n. f (z n) = z (Suc n)" unfolding z_def by auto
5866 have "dist (z n) (z (Suc n)) \<le> (c ^ n) * d"
5868 case 0 thus ?case unfolding d_def by auto
5871 hence "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
5872 using `0 \<le> c` using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by auto
5873 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]
5874 unfolding fzn and mult_le_cancel_left by auto
5879 have "(1 - c) * dist (z m) (z (m+n)) \<le> (c ^ m) * d * (1 - c ^ n)"
5881 case 0 show ?case by auto
5884 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))))"
5885 using dist_triangle and c by(auto simp add: dist_triangle)
5886 also have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
5887 using cf_z[of "m + k"] and c by auto
5888 also have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
5889 using Suc by (auto simp add: field_simps)
5890 also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
5891 unfolding power_add by (auto simp add: field_simps)
5892 also have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
5893 using c by (auto simp add: field_simps)
5894 finally show ?case by auto
5897 { fix e::real assume "e>0"
5898 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e"
5899 proof(cases "d = 0")
5901 have *: "\<And>x. ((1 - c) * x \<le> 0) = (x \<le> 0)" using `1 - c > 0`
5902 by (metis mult_zero_left real_mult_commute real_mult_le_cancel_iff1)
5903 from True have "\<And>n. z n = z0" using cf_z2[of 0] and c unfolding z_def
5905 thus ?thesis using `e>0` by auto
5907 case False hence "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]
5908 by (metis False d_def less_le)
5909 hence "0 < e * (1 - c) / d" using `e>0` and `1-c>0`
5910 using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"] by auto
5911 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
5912 { fix m n::nat assume "m>n" and as:"m\<ge>N" "n\<ge>N"
5913 have *:"c ^ n \<le> c ^ N" using `n\<ge>N` and c using power_decreasing[OF `n\<ge>N`, of c] by auto
5914 have "1 - c ^ (m - n) > 0" using c and power_strict_mono[of c 1 "m - n"] using `m>n` by auto
5915 hence **:"d * (1 - c ^ (m - n)) / (1 - c) > 0"
5916 using mult_pos_pos[OF `d>0`, of "1 - c ^ (m - n)"]
5917 using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]
5918 using `0 < 1 - c` by auto
5920 have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
5921 using cf_z2[of n "m - n"] and `m>n` unfolding pos_le_divide_eq[OF `1-c>0`]
5922 by (auto simp add: mult_commute dist_commute)
5923 also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
5924 using mult_right_mono[OF * order_less_imp_le[OF **]]
5925 unfolding mult_assoc by auto
5926 also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
5927 using mult_strict_right_mono[OF N **] unfolding mult_assoc by auto
5928 also have "\<dots> = e * (1 - c ^ (m - n))" using c and `d>0` and `1 - c > 0` by auto
5929 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
5930 finally have "dist (z m) (z n) < e" by auto
5932 { fix m n::nat assume as:"N\<le>m" "N\<le>n"
5933 hence "dist (z n) (z m) < e"
5934 proof(cases "n = m")
5935 case True thus ?thesis using `e>0` by auto
5937 case False thus ?thesis using as and *[of n m] *[of m n] unfolding nat_neq_iff by (auto simp add: dist_commute)
5939 thus ?thesis by auto
5942 hence "Cauchy z" unfolding cauchy_def by auto
5943 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
5945 def e \<equiv> "dist (f x) x"
5946 have "e = 0" proof(rule ccontr)
5947 assume "e \<noteq> 0" hence "e>0" unfolding e_def using zero_le_dist[of "f x" x]
5948 by (metis dist_eq_0_iff dist_nz e_def)
5949 then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"
5950 using x[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
5951 hence N':"dist (z N) x < e / 2" by auto
5953 have *:"c * dist (z N) x \<le> dist (z N) x" unfolding mult_le_cancel_right2
5954 using zero_le_dist[of "z N" x] and c
5955 by (metis dist_eq_0_iff dist_nz order_less_asym less_le)
5956 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]]
5957 using z_in_s[of N] `x\<in>s` using c by auto
5958 also have "\<dots> < e / 2" using N' and c using * by auto
5959 finally show False unfolding fzn
5960 using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
5961 unfolding e_def by auto
5963 hence "f x = x" unfolding e_def by auto
5965 { fix y assume "f y = y" "y\<in>s"
5966 hence "dist x y \<le> c * dist x y" using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]
5967 using `x\<in>s` and `f x = x` by auto
5968 hence "dist x y = 0" unfolding mult_le_cancel_right1
5969 using c and zero_le_dist[of x y] by auto
5970 hence "y = x" by auto
5972 ultimately show ?thesis using `x\<in>s` by blast+
5975 subsection{* Edelstein fixed point theorem. *}
5977 lemma edelstein_fix:
5978 fixes s :: "'a::real_normed_vector set"
5979 assumes s:"compact s" "s \<noteq> {}" and gs:"(g ` s) \<subseteq> s"
5980 and dist:"\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
5981 shows "\<exists>! x\<in>s. g x = x"
5982 proof(cases "\<exists>x\<in>s. g x \<noteq> x")
5983 obtain x where "x\<in>s" using s(2) by auto
5984 case False hence g:"\<forall>x\<in>s. g x = x" by auto
5985 { fix y assume "y\<in>s"
5986 hence "x = y" using `x\<in>s` and dist[THEN bspec[where x=x], THEN bspec[where x=y]]
5987 unfolding g[THEN bspec[where x=x], OF `x\<in>s`]
5988 unfolding g[THEN bspec[where x=y], OF `y\<in>s`] by auto }
5989 thus ?thesis using `x\<in>s` and g by blast+
5992 then obtain x where [simp]:"x\<in>s" and "g x \<noteq> x" by auto
5993 { fix x y assume "x \<in> s" "y \<in> s"
5994 hence "dist (g x) (g y) \<le> dist x y"
5995 using dist[THEN bspec[where x=x], THEN bspec[where x=y]] by auto } note dist' = this
5996 def y \<equiv> "g x"
5997 have [simp]:"y\<in>s" unfolding y_def using gs[unfolded image_subset_iff] and `x\<in>s` by blast
5998 def f \<equiv> "\<lambda>n. g ^^ n"
5999 have [simp]:"\<And>n z. g (f n z) = f (Suc n) z" unfolding f_def by auto
6000 have [simp]:"\<And>z. f 0 z = z" unfolding f_def by auto
6001 { fix n::nat and z assume "z\<in>s"
6002 have "f n z \<in> s" unfolding f_def
6004 case 0 thus ?case using `z\<in>s` by simp
6006 case (Suc n) thus ?case using gs[unfolded image_subset_iff] by auto
6007 qed } note fs = this
6008 { fix m n ::nat assume "m\<le>n"
6009 fix w z assume "w\<in>s" "z\<in>s"
6010 have "dist (f n w) (f n z) \<le> dist (f m w) (f m z)" using `m\<le>n`
6012 case 0 thus ?case by auto
6015 thus ?case proof(cases "m\<le>n")
6016 case True thus ?thesis using Suc(1)
6017 using dist'[OF fs fs, OF `w\<in>s` `z\<in>s`, of n n] by auto
6019 case False hence mn:"m = Suc n" using Suc(2) by simp
6020 show ?thesis unfolding mn by auto
6022 qed } note distf = this
6024 def h \<equiv> "\<lambda>n. (f n x, f n y)"
6025 let ?s2 = "s \<times> s"
6026 obtain l r where "l\<in>?s2" and r:"subseq r" and lr:"((h \<circ> r) ---> l) sequentially"
6027 using compact_Times [OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def
6028 using fs[OF `x\<in>s`] and fs[OF `y\<in>s`] by blast
6029 def a \<equiv> "fst l" def b \<equiv> "snd l"
6030 have lab:"l = (a, b)" unfolding a_def b_def by simp
6031 have [simp]:"a\<in>s" "b\<in>s" unfolding a_def b_def using `l\<in>?s2` by auto
6033 have lima:"((fst \<circ> (h \<circ> r)) ---> a) sequentially"
6034 and limb:"((snd \<circ> (h \<circ> r)) ---> b) sequentially"
6036 unfolding o_def a_def b_def by (simp_all add: tendsto_intros)
6039 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
6041 have "dist (-x) (-y) = dist x y" unfolding dist_norm
6042 using norm_minus_cancel[of "x - y"] by (auto simp add: uminus_add_conv_diff) } note ** = this
6044 { assume as:"dist a b > dist (f n x) (f n y)"
6045 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"
6046 and "\<forall>m\<ge>Nb. dist (f (r m) y) b < (dist a b - dist (f n x) (f n y)) / 2"
6047 using lima limb unfolding h_def Lim_sequentially by (fastsimp simp del: less_divide_eq_number_of1)
6048 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)"
6049 apply(erule_tac x="Na+Nb+n" in allE)
6050 apply(erule_tac x="Na+Nb+n" in allE) apply simp
6051 using dist_triangle_add_half[of a "f (r (Na + Nb + n)) x" "dist a b - dist (f n x) (f n y)"
6052 "-b" "- f (r (Na + Nb + n)) y"]
6053 unfolding ** by (auto simp add: algebra_simps dist_commute)
6055 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)"
6056 using distf[of n "r (Na+Nb+n)", OF _ `x\<in>s` `y\<in>s`]
6057 using subseq_bigger[OF r, of "Na+Nb+n"]
6058 using *[of "f (r (Na + Nb + n)) x" "f (r (Na + Nb + n)) y" "f n x" "f n y"] by auto
6059 ultimately have False by simp
6061 hence "dist a b \<le> dist (f n x) (f n y)" by(rule ccontr)auto }
6064 have [simp]:"a = b" proof(rule ccontr)
6065 def e \<equiv> "dist a b - dist (g a) (g b)"
6066 assume "a\<noteq>b" hence "e > 0" unfolding e_def using dist by fastsimp
6067 hence "\<exists>n. dist (f n x) a < e/2 \<and> dist (f n y) b < e/2"
6068 using lima limb unfolding Lim_sequentially
6069 apply (auto elim!: allE[where x="e/2"]) apply(rule_tac x="r (max N Na)" in exI) unfolding h_def by fastsimp
6070 then obtain n where n:"dist (f n x) a < e/2 \<and> dist (f n y) b < e/2" by auto
6071 have "dist (f (Suc n) x) (g a) \<le> dist (f n x) a"
6072 using dist[THEN bspec[where x="f n x"], THEN bspec[where x="a"]] and fs by auto
6073 moreover have "dist (f (Suc n) y) (g b) \<le> dist (f n y) b"
6074 using dist[THEN bspec[where x="f n y"], THEN bspec[where x="b"]] and fs by auto
6075 ultimately have "dist (f (Suc n) x) (g a) + dist (f (Suc n) y) (g b) < e" using n by auto
6076 thus False unfolding e_def using ab_fn[of "Suc n"] by norm
6079 have [simp]:"\<And>n. f (Suc n) x = f n y" unfolding f_def y_def by(induct_tac n)auto
6080 { fix x y assume "x\<in>s" "y\<in>s" moreover
6081 fix e::real assume "e>0" ultimately
6082 have "dist y x < e \<longrightarrow> dist (g y) (g x) < e" using dist by fastsimp }
6083 hence "continuous_on s g" unfolding continuous_on_iff by auto
6085 hence "((snd \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
6086 apply (rule allE[where x="\<lambda>n. (fst \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
6087 using lima unfolding h_def o_def using fs[OF `x\<in>s`] by (auto simp add: y_def)
6088 hence "g a = a" using tendsto_unique[OF trivial_limit_sequentially limb, of "g a"]
6089 unfolding `a=b` and o_assoc by auto
6091 { fix x assume "x\<in>s" "g x = x" "x\<noteq>a"
6092 hence "False" using dist[THEN bspec[where x=a], THEN bspec[where x=x]]
6093 using `g a = a` and `a\<in>s` by auto }
6094 ultimately show "\<exists>!x\<in>s. g x = x" using `a\<in>s` by blast
6098 (** TODO move this someplace else within this theory **)
6099 instance euclidean_space \<subseteq> banach ..
6101 declare tendsto_const [intro] (* FIXME: move *)
6103 text {* Legacy theorem names *}
6105 lemmas Lim_ident_at = LIM_ident
6106 lemmas Lim_const = tendsto_const
6107 lemmas Lim_cmul = scaleR.tendsto [OF tendsto_const]
6108 lemmas Lim_neg = tendsto_minus
6109 lemmas Lim_add = tendsto_add
6110 lemmas Lim_sub = tendsto_diff
6111 lemmas Lim_mul = scaleR.tendsto
6112 lemmas Lim_vmul = scaleR.tendsto [OF _ tendsto_const]
6113 lemmas Lim_null_norm = tendsto_norm_zero_iff [symmetric]
6114 lemmas Lim_linear = bounded_linear.tendsto [COMP swap_prems_rl]
6115 lemmas Lim_component = euclidean_component.tendsto
6116 lemmas Lim_intros = Lim_add Lim_const Lim_sub Lim_cmul Lim_vmul Lim_within_id