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 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_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
1035 by (rule topological_tendstoI, auto elim: eventually_rev_mono)
1037 text{* The expected monotonicity property. *}
1039 lemma Lim_within_empty: "(f ---> l) (net within {})"
1040 unfolding tendsto_def Limits.eventually_within by simp
1042 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
1043 unfolding tendsto_def Limits.eventually_within
1044 by (auto elim!: eventually_elim1)
1046 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
1047 shows "(f ---> l) (net within (S \<union> T))"
1048 using assms unfolding tendsto_def Limits.eventually_within
1050 apply (drule spec, drule (1) mp, drule (1) mp)
1051 apply (drule spec, drule (1) mp, drule (1) mp)
1052 apply (auto elim: eventually_elim2)
1056 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow> S \<union> T = UNIV
1058 by (metis Lim_Un within_UNIV)
1060 text{* Interrelations between restricted and unrestricted limits. *}
1062 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
1064 unfolding tendsto_def Limits.eventually_within
1065 apply (clarify, drule spec, drule (1) mp, drule (1) mp)
1066 by (auto elim!: eventually_elim1)
1068 lemma Lim_within_open:
1069 fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
1070 assumes"a \<in> S" "open S"
1071 shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)" (is "?lhs \<longleftrightarrow> ?rhs")
1074 { fix A assume "open A" "l \<in> A"
1075 with `?lhs` have "eventually (\<lambda>x. f x \<in> A) (at a within S)"
1076 by (rule topological_tendstoD)
1077 hence "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x \<in> A) (at a)"
1078 unfolding Limits.eventually_within .
1079 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"
1080 unfolding eventually_at_topological by fast
1081 hence "open (T \<inter> S)" "a \<in> T \<inter> S" "\<forall>x\<in>(T \<inter> S). x \<noteq> a \<longrightarrow> f x \<in> A"
1083 hence "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> f x \<in> A)"
1085 hence "eventually (\<lambda>x. f x \<in> A) (at a)"
1086 unfolding eventually_at_topological .
1088 thus ?rhs by (rule topological_tendstoI)
1091 thus ?lhs by (rule Lim_at_within)
1094 lemma Lim_within_LIMSEQ:
1095 fixes a :: real and L :: "'a::metric_space"
1096 assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
1097 shows "(X ---> L) (at a within T)"
1099 assume "\<not> (X ---> L) (at a within T)"
1100 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"
1101 unfolding tendsto_iff eventually_within dist_norm by (simp add: not_less[symmetric])
1102 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
1104 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"
1105 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"
1106 using r by (simp add: Bex_def)
1107 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"
1109 hence F1: "\<And>n. ?F n \<in> T \<and> ?F n \<noteq> a"
1110 and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
1111 and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
1115 proof (rule LIMSEQ_I, unfold real_norm_def)
1118 (* choose no such that inverse (real (Suc n)) < e *)
1119 then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
1120 then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
1121 show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
1122 proof (intro exI allI impI)
1124 assume mlen: "m \<le> n"
1125 have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
1127 also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
1129 also from nodef have
1130 "inverse (real (Suc m)) < e" .
1131 finally show "\<bar>?F n - a\<bar> < e" .
1134 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`
1135 ultimately have "(\<lambda>n. X (?F n)) ----> L" using F1 by simp
1137 moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
1141 obtain n where "n = no + 1" by simp
1142 then have nolen: "no \<le> n" by simp
1143 (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
1144 have "dist (X (?F n)) L \<ge> r"
1146 with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
1148 then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
1149 with r have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
1150 thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
1152 ultimately show False by simp
1155 lemma Lim_right_bound:
1156 fixes f :: "real \<Rightarrow> real"
1157 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"
1158 assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
1159 shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
1161 assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
1163 assume [simp]: "{x<..} \<inter> I \<noteq> {}"
1165 proof (rule Lim_within_LIMSEQ, safe)
1166 fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"
1168 show "(\<lambda>n. f (S n)) ----> Inf (f ` ({x<..} \<inter> I))"
1169 proof (rule LIMSEQ_I, rule ccontr)
1170 fix r :: real assume "0 < r"
1171 with Inf_close[of "f ` ({x<..} \<inter> I)" r]
1172 obtain y where y: "x < y" "y \<in> I" "f y < Inf (f ` ({x <..} \<inter> I)) + r" by auto
1173 from `x < y` have "0 < y - x" by auto
1174 from S(2)[THEN LIMSEQ_D, OF this]
1175 obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto
1177 assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f ` ({x<..} \<inter> I))) < r)"
1178 moreover have "\<And>n. Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
1179 using S bnd by (intro Inf_lower[where z=K]) auto
1180 ultimately obtain n where n: "N \<le> n" "r + Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
1181 by (auto simp: not_less field_simps)
1182 with N[OF n(1)] mono[OF _ `y \<in> I`, of "S n"] S(1)[THEN spec, of n] y
1188 text{* Another limit point characterization. *}
1190 lemma islimpt_sequential:
1191 fixes x :: "'a::metric_space"
1192 shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
1196 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"
1197 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
1199 have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
1202 { fix e::real assume "e>0"
1203 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
1204 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
1205 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)
1206 moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
1207 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
1209 hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
1210 unfolding Lim_sequentially using f by auto
1211 ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
1214 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
1215 { fix e::real assume "e>0"
1216 then obtain N where "dist (f N) x < e" using f(2) by auto
1217 moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
1218 ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
1220 thus ?lhs unfolding islimpt_approachable by auto
1223 lemma Lim_inv: (* TODO: delete *)
1224 fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
1225 assumes "(f ---> l) A" and "l \<noteq> 0"
1226 shows "((inverse o f) ---> inverse l) A"
1227 unfolding o_def using assms by (rule tendsto_inverse)
1230 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1231 shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
1232 by (simp add: Lim dist_norm)
1234 lemma Lim_null_comparison:
1235 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1236 assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
1237 shows "(f ---> 0) net"
1238 proof(simp add: tendsto_iff, rule+)
1239 fix e::real assume "0<e"
1241 assume "norm (f x) \<le> g x" "dist (g x) 0 < e"
1242 hence "dist (f x) 0 < e" by (simp add: dist_norm)
1244 thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
1245 using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
1246 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]
1247 using assms `e>0` unfolding tendsto_iff by auto
1250 lemma Lim_transform_bound:
1251 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1252 fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
1253 assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net" "(g ---> 0) net"
1254 shows "(f ---> 0) net"
1255 proof (rule tendstoI)
1256 fix e::real assume "e>0"
1258 assume "norm (f x) \<le> norm (g x)" "dist (g x) 0 < e"
1259 hence "dist (f x) 0 < e" by (simp add: dist_norm)}
1260 thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
1261 using eventually_and[of "\<lambda>x. norm (f x) \<le> norm (g x)" "\<lambda>x. dist (g x) 0 < e" net]
1262 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]
1263 using assms `e>0` unfolding tendsto_iff by blast
1266 text{* Deducing things about the limit from the elements. *}
1268 lemma Lim_in_closed_set:
1269 assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
1272 assume "l \<notin> S"
1273 with `closed S` have "open (- S)" "l \<in> - S"
1274 by (simp_all add: open_Compl)
1275 with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
1276 by (rule topological_tendstoD)
1277 with assms(2) have "eventually (\<lambda>x. False) net"
1278 by (rule eventually_elim2) simp
1279 with assms(3) show "False"
1280 by (simp add: eventually_False)
1283 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
1285 lemma Lim_dist_ubound:
1286 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
1287 shows "dist a l <= e"
1289 assume "\<not> dist a l \<le> e"
1290 then have "0 < dist a l - e" by simp
1291 with assms(2) have "eventually (\<lambda>x. dist (f x) l < dist a l - e) net"
1293 with assms(3) have "eventually (\<lambda>x. dist a (f x) \<le> e \<and> dist (f x) l < dist a l - e) net"
1294 by (rule eventually_conjI)
1295 then obtain w where "dist a (f w) \<le> e" "dist (f w) l < dist a l - e"
1296 using assms(1) eventually_happens by auto
1297 hence "dist a (f w) + dist (f w) l < e + (dist a l - e)"
1298 by (rule add_le_less_mono)
1299 hence "dist a (f w) + dist (f w) l < dist a l"
1301 also have "\<dots> \<le> dist a (f w) + dist (f w) l"
1302 by (rule dist_triangle)
1303 finally show False by simp
1306 lemma Lim_norm_ubound:
1307 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1308 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
1309 shows "norm(l) <= e"
1311 assume "\<not> norm l \<le> e"
1312 then have "0 < norm l - e" by simp
1313 with assms(2) have "eventually (\<lambda>x. dist (f x) l < norm l - e) net"
1315 with assms(3) have "eventually (\<lambda>x. norm (f x) \<le> e \<and> dist (f x) l < norm l - e) net"
1316 by (rule eventually_conjI)
1317 then obtain w where "norm (f w) \<le> e" "dist (f w) l < norm l - e"
1318 using assms(1) eventually_happens by auto
1319 hence "norm (f w - l) < norm l - e" "norm (f w) \<le> e" by (simp_all add: dist_norm)
1320 hence "norm (f w - l) + norm (f w) < norm l" by simp
1321 hence "norm (f w - l - f w) < norm l" by (rule le_less_trans [OF norm_triangle_ineq4])
1322 thus False using `\<not> norm l \<le> e` by simp
1325 lemma Lim_norm_lbound:
1326 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1327 assumes "\<not> (trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. e <= norm(f x)) net"
1328 shows "e \<le> norm l"
1330 assume "\<not> e \<le> norm l"
1331 then have "0 < e - norm l" by simp
1332 with assms(2) have "eventually (\<lambda>x. dist (f x) l < e - norm l) net"
1334 with assms(3) have "eventually (\<lambda>x. e \<le> norm (f x) \<and> dist (f x) l < e - norm l) net"
1335 by (rule eventually_conjI)
1336 then obtain w where "e \<le> norm (f w)" "dist (f w) l < e - norm l"
1337 using assms(1) eventually_happens by auto
1338 hence "norm (f w - l) + norm l < e" "e \<le> norm (f w)" by (simp_all add: dist_norm)
1339 hence "norm (f w - l) + norm l < norm (f w)" by (rule less_le_trans)
1340 hence "norm (f w - l + l) < norm (f w)" by (rule le_less_trans [OF norm_triangle_ineq])
1344 text{* Uniqueness of the limit, when nontrivial. *}
1347 fixes f :: "'a \<Rightarrow> 'b::t2_space"
1348 shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
1349 unfolding Lim_def using tendsto_unique[of net f] by auto
1351 text{* Limit under bilinear function *}
1354 assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
1355 shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
1356 using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
1357 by (rule bounded_bilinear.tendsto)
1359 text{* These are special for limits out of the same vector space. *}
1361 lemma Lim_within_id: "(id ---> a) (at a within s)"
1362 unfolding tendsto_def Limits.eventually_within eventually_at_topological
1365 lemma Lim_at_id: "(id ---> a) (at a)"
1366 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
1369 fixes a :: "'a::real_normed_vector"
1370 fixes l :: "'b::topological_space"
1371 shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
1374 { fix S assume "open S" "l \<in> S"
1375 with `?lhs` have "eventually (\<lambda>x. f x \<in> S) (at a)"
1376 by (rule topological_tendstoD)
1377 then obtain d where d: "d>0" "\<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S"
1378 unfolding Limits.eventually_at by fast
1379 { fix x::"'a" assume "x \<noteq> 0 \<and> dist x 0 < d"
1380 hence "f (a + x) \<in> S" using d
1381 apply(erule_tac x="x+a" in allE)
1382 by (auto simp add: add_commute dist_norm dist_commute)
1384 hence "\<exists>d>0. \<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
1386 hence "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
1387 unfolding Limits.eventually_at .
1389 thus "?rhs" by (rule topological_tendstoI)
1392 { fix S assume "open S" "l \<in> S"
1393 with `?rhs` have "eventually (\<lambda>x. f (a + x) \<in> S) (at 0)"
1394 by (rule topological_tendstoD)
1395 then obtain d where d: "d>0" "\<forall>x. x \<noteq> 0 \<and> dist x 0 < d \<longrightarrow> f (a + x) \<in> S"
1396 unfolding Limits.eventually_at by fast
1397 { fix x::"'a" assume "x \<noteq> a \<and> dist x a < d"
1398 hence "f x \<in> S" using d apply(erule_tac x="x-a" in allE)
1399 by(auto simp add: add_commute dist_norm dist_commute)
1401 hence "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<in> S" using d(1) by auto
1402 hence "eventually (\<lambda>x. f x \<in> S) (at a)" unfolding Limits.eventually_at .
1404 thus "?lhs" by (rule topological_tendstoI)
1407 text{* It's also sometimes useful to extract the limit point from the filter. *}
1410 netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
1411 "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
1413 lemma netlimit_within:
1414 assumes "\<not> trivial_limit (at a within S)"
1415 shows "netlimit (at a within S) = a"
1416 unfolding netlimit_def
1417 apply (rule some_equality)
1418 apply (rule Lim_at_within)
1419 apply (rule LIM_ident)
1420 apply (erule tendsto_unique [OF assms])
1421 apply (rule Lim_at_within)
1422 apply (rule LIM_ident)
1426 fixes a :: "'a::{perfect_space,t2_space}"
1427 shows "netlimit (at a) = a"
1428 apply (subst within_UNIV[symmetric])
1429 using netlimit_within[of a UNIV]
1430 by (simp add: trivial_limit_at within_UNIV)
1432 text{* Transformation of limit. *}
1434 lemma Lim_transform:
1435 fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
1436 assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
1437 shows "(g ---> l) net"
1439 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
1440 thus "?thesis" using tendsto_minus [of "\<lambda> x. - g x" "-l" net] by auto
1443 lemma Lim_transform_eventually:
1444 "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
1445 apply (rule topological_tendstoI)
1446 apply (drule (2) topological_tendstoD)
1447 apply (erule (1) eventually_elim2, simp)
1450 lemma Lim_transform_within:
1451 assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
1452 and "(f ---> l) (at x within S)"
1453 shows "(g ---> l) (at x within S)"
1454 proof (rule Lim_transform_eventually)
1455 show "eventually (\<lambda>x. f x = g x) (at x within S)"
1456 unfolding eventually_within
1457 using assms(1,2) by auto
1458 show "(f ---> l) (at x within S)" by fact
1461 lemma Lim_transform_at:
1462 assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
1463 and "(f ---> l) (at x)"
1464 shows "(g ---> l) (at x)"
1465 proof (rule Lim_transform_eventually)
1466 show "eventually (\<lambda>x. f x = g x) (at x)"
1467 unfolding eventually_at
1468 using assms(1,2) by auto
1469 show "(f ---> l) (at x)" by fact
1472 text{* Common case assuming being away from some crucial point like 0. *}
1474 lemma Lim_transform_away_within:
1475 fixes a b :: "'a::t1_space"
1476 assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1477 and "(f ---> l) (at a within S)"
1478 shows "(g ---> l) (at a within S)"
1479 proof (rule Lim_transform_eventually)
1480 show "(f ---> l) (at a within S)" by fact
1481 show "eventually (\<lambda>x. f x = g x) (at a within S)"
1482 unfolding Limits.eventually_within eventually_at_topological
1483 by (rule exI [where x="- {b}"], simp add: open_Compl assms)
1486 lemma Lim_transform_away_at:
1487 fixes a b :: "'a::t1_space"
1488 assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1489 and fl: "(f ---> l) (at a)"
1490 shows "(g ---> l) (at a)"
1491 using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
1492 by (auto simp add: within_UNIV)
1494 text{* Alternatively, within an open set. *}
1496 lemma Lim_transform_within_open:
1497 assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
1498 and "(f ---> l) (at a)"
1499 shows "(g ---> l) (at a)"
1500 proof (rule Lim_transform_eventually)
1501 show "eventually (\<lambda>x. f x = g x) (at a)"
1502 unfolding eventually_at_topological
1503 using assms(1,2,3) by auto
1504 show "(f ---> l) (at a)" by fact
1507 text{* A congruence rule allowing us to transform limits assuming not at point. *}
1509 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
1511 lemma Lim_cong_within(*[cong add]*):
1512 assumes "a = b" "x = y" "S = T"
1513 assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
1514 shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
1515 unfolding tendsto_def Limits.eventually_within eventually_at_topological
1518 lemma Lim_cong_at(*[cong add]*):
1519 assumes "a = b" "x = y"
1520 assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
1521 shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
1522 unfolding tendsto_def eventually_at_topological
1525 text{* Useful lemmas on closure and set of possible sequential limits.*}
1527 lemma closure_sequential:
1528 fixes l :: "'a::metric_space"
1529 shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
1531 assume "?lhs" moreover
1532 { assume "l \<in> S"
1533 hence "?rhs" using tendsto_const[of l sequentially] by auto
1535 { assume "l islimpt S"
1536 hence "?rhs" unfolding islimpt_sequential by auto
1538 show "?rhs" unfolding closure_def by auto
1541 thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
1544 lemma closed_sequential_limits:
1545 fixes S :: "'a::metric_space set"
1546 shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
1547 unfolding closed_limpt
1548 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]
1551 lemma closure_approachable:
1552 fixes S :: "'a::metric_space set"
1553 shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
1554 apply (auto simp add: closure_def islimpt_approachable)
1555 by (metis dist_self)
1557 lemma closed_approachable:
1558 fixes S :: "'a::metric_space set"
1559 shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
1560 by (metis closure_closed closure_approachable)
1562 text{* Some other lemmas about sequences. *}
1564 lemma sequentially_offset:
1565 assumes "eventually (\<lambda>i. P i) sequentially"
1566 shows "eventually (\<lambda>i. P (i + k)) sequentially"
1567 using assms unfolding eventually_sequentially by (metis trans_le_add1)
1570 assumes "(f ---> l) sequentially"
1571 shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
1572 using assms unfolding tendsto_def
1573 by clarify (rule sequentially_offset, simp)
1575 lemma seq_offset_neg:
1576 "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
1577 apply (rule topological_tendstoI)
1578 apply (drule (2) topological_tendstoD)
1579 apply (simp only: eventually_sequentially)
1580 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
1584 lemma seq_offset_rev:
1585 "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
1586 apply (rule topological_tendstoI)
1587 apply (drule (2) topological_tendstoD)
1588 apply (simp only: eventually_sequentially)
1589 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
1592 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
1594 { fix e::real assume "e>0"
1595 hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
1596 using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
1597 by (metis le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
1599 thus ?thesis unfolding Lim_sequentially dist_norm by simp
1602 subsection {* More properties of closed balls. *}
1604 lemma closed_cball: "closed (cball x e)"
1605 unfolding cball_def closed_def
1606 unfolding Collect_neg_eq [symmetric] not_le
1607 apply (clarsimp simp add: open_dist, rename_tac y)
1608 apply (rule_tac x="dist x y - e" in exI, clarsimp)
1609 apply (rename_tac x')
1610 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
1614 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
1616 { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
1617 hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
1619 { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
1620 hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
1622 show ?thesis unfolding open_contains_ball by auto
1625 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
1626 by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball mem_def)
1628 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
1629 apply (simp add: interior_def, safe)
1630 apply (force simp add: open_contains_cball)
1631 apply (rule_tac x="ball x e" in exI)
1632 apply (simp add: subset_trans [OF ball_subset_cball])
1636 fixes x y :: "'a::{real_normed_vector,perfect_space}"
1637 shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
1640 { assume "e \<le> 0"
1641 hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
1642 have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
1644 hence "e > 0" by (metis not_less)
1646 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
1647 ultimately show "?rhs" by auto
1649 assume "?rhs" hence "e>0" by auto
1650 { fix d::real assume "d>0"
1651 have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1652 proof(cases "d \<le> dist x y")
1653 case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1655 case True hence False using `d \<le> dist x y` `d>0` by auto
1656 thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
1660 have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
1661 = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1662 unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
1663 also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
1664 using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
1665 unfolding scaleR_minus_left scaleR_one
1666 by (auto simp add: norm_minus_commute)
1667 also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
1668 unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
1669 unfolding left_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
1670 also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
1671 finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
1675 have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
1676 using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
1678 have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
1679 using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
1680 unfolding dist_norm by auto
1681 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
1684 case False hence "d > dist x y" by auto
1685 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1688 obtain z where **: "z \<noteq> y" "dist z y < min e d"
1689 using perfect_choose_dist[of "min e d" y]
1690 using `d > 0` `e>0` by auto
1691 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1693 using `z \<noteq> y` **
1694 by (rule_tac x=z in bexI, auto simp add: dist_commute)
1696 case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1697 using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
1700 thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
1703 lemma closure_ball_lemma:
1704 fixes x y :: "'a::real_normed_vector"
1705 assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
1706 proof (rule islimptI)
1707 fix T assume "y \<in> T" "open T"
1708 then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
1709 unfolding open_dist by fast
1710 (* choose point between x and y, within distance r of y. *)
1711 def k \<equiv> "min 1 (r / (2 * dist x y))"
1712 def z \<equiv> "y + scaleR k (x - y)"
1713 have z_def2: "z = x + scaleR (1 - k) (y - x)"
1714 unfolding z_def by (simp add: algebra_simps)
1716 unfolding z_def k_def using `0 < r`
1717 by (simp add: dist_norm min_def)
1718 hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
1719 have "dist x z < dist x y"
1720 unfolding z_def2 dist_norm
1721 apply (simp add: norm_minus_commute)
1722 apply (simp only: dist_norm [symmetric])
1723 apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
1724 apply (rule mult_strict_right_mono)
1725 apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
1726 apply (simp add: zero_less_dist_iff `x \<noteq> y`)
1728 hence "z \<in> ball x (dist x y)" by simp
1730 unfolding z_def k_def using `x \<noteq> y` `0 < r`
1731 by (simp add: min_def)
1732 show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
1733 using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
1738 fixes x :: "'a::real_normed_vector"
1739 shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
1740 apply (rule equalityI)
1741 apply (rule closure_minimal)
1742 apply (rule ball_subset_cball)
1743 apply (rule closed_cball)
1744 apply (rule subsetI, rename_tac y)
1745 apply (simp add: le_less [where 'a=real])
1747 apply (rule subsetD [OF closure_subset], simp)
1748 apply (simp add: closure_def)
1750 apply (rule closure_ball_lemma)
1751 apply (simp add: zero_less_dist_iff)
1754 (* In a trivial vector space, this fails for e = 0. *)
1755 lemma interior_cball:
1756 fixes x :: "'a::{real_normed_vector, perfect_space}"
1757 shows "interior (cball x e) = ball x e"
1758 proof(cases "e\<ge>0")
1759 case False note cs = this
1760 from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
1761 { fix y assume "y \<in> cball x e"
1762 hence False unfolding mem_cball using dist_nz[of x y] cs by auto }
1763 hence "cball x e = {}" by auto
1764 hence "interior (cball x e) = {}" using interior_empty by auto
1765 ultimately show ?thesis by blast
1767 case True note cs = this
1768 have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
1769 { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
1770 then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
1772 then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
1773 using perfect_choose_dist [of d] by auto
1774 have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
1775 hence xa_cball:"xa \<in> cball x e" using as(1) by auto
1777 hence "y \<in> ball x e" proof(cases "x = y")
1779 hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
1780 thus "y \<in> ball x e" using `x = y ` by simp
1783 have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
1784 using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
1785 hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
1786 have "y - x \<noteq> 0" using `x \<noteq> y` by auto
1787 hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
1788 using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
1790 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)"
1791 by (auto simp add: dist_norm algebra_simps)
1792 also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1793 by (auto simp add: algebra_simps)
1794 also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
1796 also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
1797 finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
1798 thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
1800 hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
1801 ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
1804 lemma frontier_ball:
1805 fixes a :: "'a::real_normed_vector"
1806 shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
1807 apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
1808 apply (simp add: set_eq_iff)
1811 lemma frontier_cball:
1812 fixes a :: "'a::{real_normed_vector, perfect_space}"
1813 shows "frontier(cball a e) = {x. dist a x = e}"
1814 apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
1815 apply (simp add: set_eq_iff)
1818 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
1819 apply (simp add: set_eq_iff not_le)
1820 by (metis zero_le_dist dist_self order_less_le_trans)
1821 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
1823 lemma cball_eq_sing:
1824 fixes x :: "'a::{metric_space,perfect_space}"
1825 shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
1826 proof (rule linorder_cases)
1828 obtain a where "a \<noteq> x" "dist a x < e"
1829 using perfect_choose_dist [OF e] by auto
1830 hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
1831 with e show ?thesis by (auto simp add: set_eq_iff)
1835 fixes x :: "'a::metric_space"
1836 shows "e = 0 ==> cball x e = {x}"
1837 by (auto simp add: set_eq_iff)
1839 text{* For points in the interior, localization of limits makes no difference. *}
1841 lemma eventually_within_interior:
1842 assumes "x \<in> interior S"
1843 shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
1845 from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S"
1846 unfolding interior_def by fast
1848 then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
1849 unfolding Limits.eventually_within Limits.eventually_at_topological
1851 with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
1854 unfolding Limits.eventually_at_topological by auto
1856 { assume "?rhs" hence "?lhs"
1857 unfolding Limits.eventually_within
1858 by (auto elim: eventually_elim1)
1863 lemma at_within_interior:
1864 "x \<in> interior S \<Longrightarrow> at x within S = at x"
1865 by (simp add: filter_eq_iff eventually_within_interior)
1867 lemma lim_within_interior:
1868 "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
1869 by (simp add: at_within_interior)
1871 lemma netlimit_within_interior:
1872 fixes x :: "'a::{t2_space,perfect_space}"
1873 assumes "x \<in> interior S"
1874 shows "netlimit (at x within S) = x"
1875 using assms by (simp add: at_within_interior netlimit_at)
1877 subsection{* Boundedness. *}
1879 (* FIXME: This has to be unified with BSEQ!! *)
1881 bounded :: "'a::metric_space set \<Rightarrow> bool" where
1882 "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
1884 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
1885 unfolding bounded_def
1887 apply (rule_tac x="dist a x + e" in exI, clarify)
1888 apply (drule (1) bspec)
1889 apply (erule order_trans [OF dist_triangle add_left_mono])
1893 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
1894 unfolding bounded_any_center [where a=0]
1895 by (simp add: dist_norm)
1897 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
1898 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
1899 by (metis bounded_def subset_eq)
1901 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
1902 by (metis bounded_subset interior_subset)
1904 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
1906 from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
1907 { fix y assume "y \<in> closure S"
1908 then obtain f where f: "\<forall>n. f n \<in> S" "(f ---> y) sequentially"
1909 unfolding closure_sequential by auto
1910 have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
1911 hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
1912 by (rule eventually_mono, simp add: f(1))
1913 have "dist x y \<le> a"
1914 apply (rule Lim_dist_ubound [of sequentially f])
1915 apply (rule trivial_limit_sequentially)
1920 thus ?thesis unfolding bounded_def by auto
1923 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
1924 apply (simp add: bounded_def)
1925 apply (rule_tac x=x in exI)
1926 apply (rule_tac x=e in exI)
1930 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
1931 by (metis ball_subset_cball bounded_cball bounded_subset)
1933 lemma finite_imp_bounded[intro]:
1934 fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"
1936 { fix a and F :: "'a set" assume as:"bounded F"
1937 then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
1938 hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
1939 hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
1941 thus ?thesis using finite_induct[of S bounded] using bounded_empty assms by auto
1944 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
1945 apply (auto simp add: bounded_def)
1946 apply (rename_tac x y r s)
1947 apply (rule_tac x=x in exI)
1948 apply (rule_tac x="max r (dist x y + s)" in exI)
1949 apply (rule ballI, rename_tac z, safe)
1950 apply (drule (1) bspec, simp)
1951 apply (drule (1) bspec)
1952 apply (rule min_max.le_supI2)
1953 apply (erule order_trans [OF dist_triangle add_left_mono])
1956 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
1957 by (induct rule: finite_induct[of F], auto)
1959 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
1960 apply (simp add: bounded_iff)
1961 apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
1964 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
1965 by (metis Int_lower1 Int_lower2 bounded_subset)
1967 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
1968 apply (metis Diff_subset bounded_subset)
1971 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
1972 by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
1974 lemma not_bounded_UNIV[simp, intro]:
1975 "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
1976 proof(auto simp add: bounded_pos not_le)
1977 obtain x :: 'a where "x \<noteq> 0"
1978 using perfect_choose_dist [OF zero_less_one] by fast
1979 fix b::real assume b: "b >0"
1980 have b1: "b +1 \<ge> 0" using b by simp
1981 with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
1982 by (simp add: norm_sgn)
1983 then show "\<exists>x::'a. b < norm x" ..
1986 lemma bounded_linear_image:
1987 assumes "bounded S" "bounded_linear f"
1988 shows "bounded(f ` S)"
1990 from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
1991 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)
1992 { fix x assume "x\<in>S"
1993 hence "norm x \<le> b" using b by auto
1994 hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
1995 by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
1997 thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
1998 using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
2001 lemma bounded_scaling:
2002 fixes S :: "'a::real_normed_vector set"
2003 shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
2004 apply (rule bounded_linear_image, assumption)
2005 apply (rule scaleR.bounded_linear_right)
2008 lemma bounded_translation:
2009 fixes S :: "'a::real_normed_vector set"
2010 assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
2012 from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
2013 { fix x assume "x\<in>S"
2014 hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
2016 thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
2017 by (auto intro!: add exI[of _ "b + norm a"])
2021 text{* Some theorems on sups and infs using the notion "bounded". *}
2024 fixes S :: "real set"
2025 shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"
2026 by (simp add: bounded_iff)
2028 lemma bounded_has_Sup:
2029 fixes S :: "real set"
2030 assumes "bounded S" "S \<noteq> {}"
2031 shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
2033 fix x assume "x\<in>S"
2034 thus "x \<le> Sup S"
2035 by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
2037 show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
2038 by (metis SupInf.Sup_least)
2042 fixes S :: "real set"
2043 shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
2044 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal)
2046 lemma Sup_insert_finite:
2047 fixes S :: "real set"
2048 shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
2049 apply (rule Sup_insert)
2050 apply (rule finite_imp_bounded)
2053 lemma bounded_has_Inf:
2054 fixes S :: "real set"
2055 assumes "bounded S" "S \<noteq> {}"
2056 shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
2058 fix x assume "x\<in>S"
2059 from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
2060 thus "x \<ge> Inf S" using `x\<in>S`
2061 by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
2063 show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
2064 by (metis SupInf.Inf_greatest)
2068 fixes S :: "real set"
2069 shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
2070 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)
2071 lemma Inf_insert_finite:
2072 fixes S :: "real set"
2073 shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
2074 by (rule Inf_insert, rule finite_imp_bounded, simp)
2077 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
2078 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
2079 apply (frule isGlb_isLb)
2080 apply (frule_tac x = y in isGlb_isLb)
2081 apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
2084 subsection {* Equivalent versions of compactness *}
2086 subsubsection{* Sequential compactness *}
2089 compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
2090 "compact S \<longleftrightarrow>
2091 (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
2092 (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
2095 assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
2097 unfolding compact_def using assms by fast
2100 assumes "compact S" "\<forall>n. f n \<in> S"
2101 obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
2102 using assms unfolding compact_def by fast
2105 A metric space (or topological vector space) is said to have the
2106 Heine-Borel property if every closed and bounded subset is compact.
2110 assumes bounded_imp_convergent_subsequence:
2111 "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
2112 \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2114 lemma bounded_closed_imp_compact:
2115 fixes s::"'a::heine_borel set"
2116 assumes "bounded s" and "closed s" shows "compact s"
2117 proof (unfold compact_def, clarify)
2118 fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
2119 obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
2120 using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
2121 from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
2122 have "l \<in> s" using `closed s` fr l
2123 unfolding closed_sequential_limits by blast
2124 show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2125 using `l \<in> s` r l by blast
2128 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
2130 show "0 \<le> r 0" by auto
2132 fix n assume "n \<le> r n"
2133 moreover have "r n < r (Suc n)"
2134 using assms [unfolded subseq_def] by auto
2135 ultimately show "Suc n \<le> r (Suc n)" by auto
2138 lemma eventually_subseq:
2139 assumes r: "subseq r"
2140 shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
2141 unfolding eventually_sequentially
2142 by (metis subseq_bigger [OF r] le_trans)
2145 "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
2146 unfolding tendsto_def eventually_sequentially o_def
2147 by (metis subseq_bigger le_trans)
2149 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
2151 apply (rule_tac x="nat_rec e f" in exI)
2153 apply (rule def_nat_rec_0, simp)
2154 apply (rule allI, rule def_nat_rec_Suc, simp)
2155 apply (rule allI, rule impI, rule ext)
2157 apply (induct_tac x)
2159 apply (erule_tac x="n" in allE)
2163 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
2164 assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
2165 shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N. abs(s n - l) < e"
2167 have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
2168 then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
2169 { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
2171 obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
2172 have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
2173 with n have "s N \<le> t - e" using `e>0` by auto
2174 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 }
2175 hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
2176 hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto }
2177 thus ?thesis by blast
2180 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
2181 assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
2182 shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
2183 using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
2184 unfolding monoseq_def incseq_def
2185 apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
2186 unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
2188 (* TODO: merge this lemma with the ones above *)
2189 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"
2190 assumes "bounded {s n| n::nat. True}" "\<forall>n. (s n) \<le>(s(Suc n))"
2191 shows "\<exists>l. (s ---> l) sequentially"
2193 obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le> a" using assms(1)[unfolded bounded_iff] by auto
2195 have "\<And> n. n\<ge>m \<longrightarrow> (s m) \<le> (s n)"
2196 apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)
2197 apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq) }
2198 hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto
2199 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]
2200 unfolding monoseq_def by auto
2201 thus ?thesis unfolding Lim_sequentially apply(rule_tac x="l" in exI)
2202 unfolding dist_norm by auto
2205 lemma compact_real_lemma:
2206 assumes "\<forall>n::nat. abs(s n) \<le> b"
2207 shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
2209 obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
2210 using seq_monosub[of s] by auto
2211 thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
2212 unfolding tendsto_iff dist_norm eventually_sequentially by auto
2215 instance real :: heine_borel
2217 fix s :: "real set" and f :: "nat \<Rightarrow> real"
2218 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2219 then obtain b where b: "\<forall>n. abs (f n) \<le> b"
2220 unfolding bounded_iff by auto
2221 obtain l :: real and r :: "nat \<Rightarrow> nat" where
2222 r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
2223 using compact_real_lemma [OF b] by auto
2224 thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2228 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $$ i) ` s)"
2229 apply (erule bounded_linear_image)
2230 apply (rule bounded_linear_euclidean_component)
2233 lemma compact_lemma:
2234 fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
2235 assumes "bounded s" and "\<forall>n. f n \<in> s"
2236 shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>
2237 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2239 fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"
2240 have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
2241 hence "\<exists>l::'a. \<exists>r. subseq r \<and>
2242 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2243 proof(induct d) case empty thus ?case unfolding subseq_def by auto
2244 next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto
2245 have s': "bounded ((\<lambda>x. x $$ k) ` s)" using `bounded s` by (rule bounded_component)
2246 obtain l1::"'a" and r1 where r1:"subseq r1" and
2247 lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially"
2248 using insert(3) using insert(4) by auto
2249 have f': "\<forall>n. f (r1 n) $$ k \<in> (\<lambda>x. x $$ k) ` s" using `\<forall>n. f n \<in> s` by simp
2250 obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $$ k) ---> l2) sequentially"
2251 using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
2252 def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
2253 using r1 and r2 unfolding r_def o_def subseq_def by auto
2255 def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"
2256 { fix e::real assume "e>0"
2257 from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially" by blast
2258 from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $$ k) l2 < e) sequentially" by (rule tendstoD)
2259 from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$ i) (l1 $$ i) < e) sequentially"
2260 by (rule eventually_subseq)
2261 have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$ i) (l $$ i) < e) sequentially"
2262 using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def
2263 using insert.prems by auto
2265 ultimately show ?case by auto
2267 thus "\<exists>l::'a. \<exists>r. subseq r \<and>
2268 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2269 apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe
2270 apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe
2271 apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)
2272 apply(erule_tac x=i in ballE)
2273 proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a
2274 assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"
2275 hence *:"i\<ge>DIM('a)" by auto
2276 thus "dist (f (r n) $$ i) (l $$ i) < e" using e by auto
2280 instance euclidean_space \<subseteq> heine_borel
2282 fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"
2283 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2284 then obtain l::'a and r where r: "subseq r"
2285 and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $$ i) (l $$ i) < e) sequentially"
2286 using compact_lemma [OF s f] by blast
2287 let ?d = "{..<DIM('a)}"
2288 { fix e::real assume "e>0"
2289 hence "0 < e / (real_of_nat (card ?d))"
2290 using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
2291 with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))) sequentially"
2294 { fix n assume n: "\<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))"
2295 have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $$ i) (l $$ i))"
2296 apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)
2297 also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
2298 apply(rule setsum_strict_mono) using n by auto
2299 finally have "dist (f (r n)) l < e" unfolding setsum_constant
2300 using DIM_positive[where 'a='a] by auto
2302 ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
2303 by (rule eventually_elim1)
2305 hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
2306 with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
2309 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
2310 unfolding bounded_def
2312 apply (rule_tac x="a" in exI)
2313 apply (rule_tac x="e" in exI)
2315 apply (drule (1) bspec)
2316 apply (simp add: dist_Pair_Pair)
2317 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
2320 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
2321 unfolding bounded_def
2323 apply (rule_tac x="b" in exI)
2324 apply (rule_tac x="e" in exI)
2326 apply (drule (1) bspec)
2327 apply (simp add: dist_Pair_Pair)
2328 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
2331 instance prod :: (heine_borel, heine_borel) heine_borel
2333 fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
2334 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2335 from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
2336 from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
2337 obtain l1 r1 where r1: "subseq r1"
2338 and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
2339 using bounded_imp_convergent_subsequence [OF s1 f1]
2340 unfolding o_def by fast
2341 from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
2342 from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
2343 obtain l2 r2 where r2: "subseq r2"
2344 and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
2345 using bounded_imp_convergent_subsequence [OF s2 f2]
2346 unfolding o_def by fast
2347 have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
2348 using lim_subseq [OF r2 l1] unfolding o_def .
2349 have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
2350 using tendsto_Pair [OF l1' l2] unfolding o_def by simp
2351 have r: "subseq (r1 \<circ> r2)"
2352 using r1 r2 unfolding subseq_def by simp
2353 show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2357 subsubsection{* Completeness *}
2360 "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
2361 unfolding Cauchy_def by blast
2364 complete :: "'a::metric_space set \<Rightarrow> bool" where
2365 "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
2366 --> (\<exists>l \<in> s. (f ---> l) sequentially))"
2368 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
2373 with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
2374 by (erule_tac x="e/2" in allE) auto
2376 assume nm:"N \<le> m \<and> N \<le> n"
2377 hence "dist (s m) (s n) < e" using N
2378 using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
2381 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
2385 unfolding cauchy_def
2389 unfolding cauchy_def
2390 using dist_triangle_half_l
2394 lemma convergent_imp_cauchy:
2395 "(s ---> l) sequentially ==> Cauchy s"
2396 proof(simp only: cauchy_def, rule, rule)
2397 fix e::real assume "e>0" "(s ---> l) sequentially"
2398 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
2399 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
2402 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
2404 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
2405 hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
2407 have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
2408 then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
2409 unfolding bounded_any_center [where a="s N"] by auto
2410 ultimately show "?thesis"
2411 unfolding bounded_any_center [where a="s N"]
2412 apply(rule_tac x="max a 1" in exI) apply auto
2413 apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto
2416 lemma compact_imp_complete: assumes "compact s" shows "complete s"
2418 { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
2419 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
2421 note lr' = subseq_bigger [OF lr(2)]
2423 { fix e::real assume "e>0"
2424 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
2425 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
2426 { fix n::nat assume n:"n \<ge> max N M"
2427 have "dist ((f \<circ> r) n) l < e/2" using n M by auto
2428 moreover have "r n \<ge> N" using lr'[of n] n by auto
2429 hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
2430 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) }
2431 hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast }
2432 hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding Lim_sequentially by auto }
2433 thus ?thesis unfolding complete_def by auto
2436 instance heine_borel < complete_space
2438 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2439 hence "bounded (range f)"
2440 by (rule cauchy_imp_bounded)
2441 hence "compact (closure (range f))"
2442 using bounded_closed_imp_compact [of "closure (range f)"] by auto
2443 hence "complete (closure (range f))"
2444 by (rule compact_imp_complete)
2445 moreover have "\<forall>n. f n \<in> closure (range f)"
2446 using closure_subset [of "range f"] by auto
2447 ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
2448 using `Cauchy f` unfolding complete_def by auto
2449 then show "convergent f"
2450 unfolding convergent_def by auto
2453 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
2454 proof(simp add: complete_def, rule, rule)
2455 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2456 hence "convergent f" by (rule Cauchy_convergent)
2457 thus "\<exists>l. f ----> l" unfolding convergent_def .
2460 lemma complete_imp_closed: assumes "complete s" shows "closed s"
2462 { fix x assume "x islimpt s"
2463 then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
2464 unfolding islimpt_sequential by auto
2465 then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
2466 using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
2467 hence "x \<in> s" using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
2469 thus "closed s" unfolding closed_limpt by auto
2472 lemma complete_eq_closed:
2473 fixes s :: "'a::complete_space set"
2474 shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
2476 assume ?lhs thus ?rhs by (rule complete_imp_closed)
2479 { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
2480 then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
2481 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 }
2482 thus ?lhs unfolding complete_def by auto
2485 lemma convergent_eq_cauchy:
2486 fixes s :: "nat \<Rightarrow> 'a::complete_space"
2487 shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s" (is "?lhs = ?rhs")
2489 assume ?lhs then obtain l where "(s ---> l) sequentially" by auto
2490 thus ?rhs using convergent_imp_cauchy by auto
2492 assume ?rhs thus ?lhs using complete_univ[unfolded complete_def, THEN spec[where x=s]] by auto
2495 lemma convergent_imp_bounded:
2496 fixes s :: "nat \<Rightarrow> 'a::metric_space"
2497 shows "(s ---> l) sequentially ==> bounded (s ` (UNIV::(nat set)))"
2498 using convergent_imp_cauchy[of s]
2499 using cauchy_imp_bounded[of s]
2503 subsubsection{* Total boundedness *}
2505 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
2506 "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
2507 declare helper_1.simps[simp del]
2509 lemma compact_imp_totally_bounded:
2511 shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
2512 proof(rule, rule, rule ccontr)
2513 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)"
2514 def x \<equiv> "helper_1 s e"
2516 have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
2517 proof(induct_tac rule:nat_less_induct)
2518 fix n def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
2519 assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
2520 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
2521 then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
2522 have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
2523 apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
2524 thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
2526 hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
2527 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
2528 from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
2529 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
2531 using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
2532 using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
2533 using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
2536 subsubsection{* Heine-Borel theorem *}
2538 text {* Following Burkill \& Burkill vol. 2. *}
2540 lemma heine_borel_lemma: fixes s::"'a::metric_space set"
2541 assumes "compact s" "s \<subseteq> (\<Union> t)" "\<forall>b \<in> t. open b"
2542 shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
2544 assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"
2545 hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto
2547 have "1 / real (n + 1) > 0" by auto
2548 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 }
2549 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
2550 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)"
2551 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
2553 then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
2554 using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
2556 obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
2557 then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"
2558 using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto
2560 then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"
2561 using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
2563 obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
2564 have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
2565 apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
2566 using subseq_bigger[OF r, of "N1 + N2"] by auto
2568 def x \<equiv> "(f (r (N1 + N2)))"
2569 have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
2570 using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto
2571 have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto
2572 then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto
2574 have "dist x l < e/2" using N1 unfolding x_def o_def by auto
2575 hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)
2577 thus False using e and `y\<notin>b` by auto
2580 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2581 \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"
2583 fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"
2584 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
2585 hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto
2586 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
2587 then obtain bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast
2589 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
2590 then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto
2592 have "finite (bb ` k)" using k(1) by auto
2594 { fix x assume "x\<in>s"
2595 hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3) unfolding subset_eq by auto
2596 hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast
2597 hence "x \<in> \<Union>(bb ` k)" using Union_iff[of x "bb ` k"] by auto
2599 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
2602 subsubsection {* Bolzano-Weierstrass property *}
2604 lemma heine_borel_imp_bolzano_weierstrass:
2605 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'))"
2606 "infinite t" "t \<subseteq> s"
2607 shows "\<exists>x \<in> s. x islimpt t"
2609 assume "\<not> (\<exists>x \<in> s. x islimpt t)"
2610 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
2611 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
2612 obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
2613 using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
2614 from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
2615 { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
2616 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
2617 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 }
2618 hence "inj_on f t" unfolding inj_on_def by simp
2619 hence "infinite (f ` t)" using assms(2) using finite_imageD by auto
2621 { fix x assume "x\<in>t" "f x \<notin> g"
2622 from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
2623 then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
2624 hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
2625 hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto }
2626 hence "f ` t \<subseteq> g" by auto
2627 ultimately show False using g(2) using finite_subset by auto
2630 subsubsection {* Complete the chain of compactness variants *}
2632 lemma islimpt_range_imp_convergent_subsequence:
2633 fixes f :: "nat \<Rightarrow> 'a::metric_space"
2634 assumes "l islimpt (range f)"
2635 shows "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2636 proof (intro exI conjI)
2637 have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"
2638 using assms unfolding islimpt_def
2639 by (drule_tac x="ball l e" in spec)
2640 (auto simp add: zero_less_dist_iff dist_commute)
2642 def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"
2643 have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"
2644 unfolding t_def by (rule LeastI2_ex [OF * conjunct1])
2645 have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"
2646 unfolding t_def by (rule LeastI2_ex [OF * conjunct2])
2647 have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"
2648 unfolding t_def by (simp add: Least_le)
2649 have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"
2650 unfolding t_def by (drule not_less_Least) simp
2651 have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"
2653 apply (erule f_t_neq)
2654 apply (erule (1) less_le_trans [OF f_t_closer])
2656 have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"
2657 by (drule f_t_closer) auto
2658 have t_less: "\<And>e. 0 < e \<Longrightarrow> t e < t (dist (f (t e)) l)"
2659 apply (subst less_le)
2661 apply (rule t_antimono)
2662 apply (erule f_t_neq)
2663 apply (erule f_t_closer [THEN less_imp_le])
2664 apply (rule t_dist_f_neq [symmetric])
2665 apply (erule f_t_neq)
2667 have dist_f_t_less':
2668 "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"
2669 apply (simp add: le_less)
2671 apply (rule less_trans)
2672 apply (erule f_t_closer)
2673 apply (rule le_less_trans)
2674 apply (erule less_tD)
2675 apply (erule f_t_neq)
2676 apply (erule f_t_closer)
2678 apply (erule f_t_closer)
2681 def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"
2682 have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"
2683 unfolding r_def by simp_all
2684 have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"
2685 by (induct_tac n) (simp_all add: r_simps f_t_neq)
2688 unfolding subseq_Suc_iff
2691 apply (simp_all add: r_simps)
2692 apply (rule t_less, rule zero_less_one)
2693 apply (rule t_less, rule f_r_neq)
2695 show "((f \<circ> r) ---> l) sequentially"
2696 unfolding Lim_sequentially o_def
2697 apply (clarify, rule_tac x="t e" in exI, clarify)
2698 apply (drule le_trans, rule seq_suble [OF `subseq r`])
2699 apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)
2703 lemma finite_range_imp_infinite_repeats:
2704 fixes f :: "nat \<Rightarrow> 'a"
2705 assumes "finite (range f)"
2706 shows "\<exists>k. infinite {n. f n = k}"
2708 { fix A :: "'a set" assume "finite A"
2709 hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"
2711 case empty thus ?case by simp
2715 proof (cases "finite {n. f n = x}")
2717 with `infinite {n. f n \<in> insert x A}`
2718 have "infinite {n. f n \<in> A}" by simp
2719 thus "\<exists>k. infinite {n. f n = k}" by (rule insert)
2721 case False thus "\<exists>k. infinite {n. f n = k}" ..
2725 from assms show "\<exists>k. infinite {n. f n = k}"
2729 lemma bolzano_weierstrass_imp_compact:
2730 fixes s :: "'a::metric_space set"
2731 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2734 { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
2735 have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2736 proof (cases "finite (range f)")
2738 hence "\<exists>l. infinite {n. f n = l}"
2739 by (rule finite_range_imp_infinite_repeats)
2740 then obtain l where "infinite {n. f n = l}" ..
2741 hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"
2742 by (rule infinite_enumerate)
2743 then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto
2744 hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2745 unfolding o_def by (simp add: fr tendsto_const)
2746 hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2748 from f have "\<forall>n. f (r n) \<in> s" by simp
2749 hence "l \<in> s" by (simp add: fr)
2750 thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2751 by (rule rev_bexI) fact
2754 with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto
2755 then obtain l where "l \<in> s" "l islimpt (range f)" ..
2756 have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2757 using `l islimpt (range f)`
2758 by (rule islimpt_range_imp_convergent_subsequence)
2759 with `l \<in> s` show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
2762 thus ?thesis unfolding compact_def by auto
2765 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where
2766 "helper_2 beyond 0 = beyond 0" |
2767 "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"
2769 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"
2770 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2773 assume "\<not> bounded s"
2774 then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"
2775 unfolding bounded_any_center [where a=undefined]
2776 apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto
2777 hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"
2778 unfolding linorder_not_le by auto
2779 def x \<equiv> "helper_2 beyond"
2781 { fix m n ::nat assume "m<n"
2782 hence "dist undefined (x m) + 1 < dist undefined (x n)"
2784 case 0 thus ?case by auto
2787 have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"
2788 unfolding x_def and helper_2.simps
2789 using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto
2790 thus ?case proof(cases "m < n")
2791 case True thus ?thesis using Suc and * by auto
2793 case False hence "m = n" using Suc(2) by auto
2794 thus ?thesis using * by auto
2797 { fix m n ::nat assume "m\<noteq>n"
2798 have "1 < dist (x m) (x n)"
2801 hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto
2802 thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith
2804 case False hence "n<m" using `m\<noteq>n` by auto
2805 hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto
2806 thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith
2807 qed } note ** = this
2808 { fix a b assume "x a = x b" "a \<noteq> b"
2809 hence False using **[of a b] by auto }
2810 hence "inj x" unfolding inj_on_def by auto
2814 proof(cases "n = 0")
2815 case True thus ?thesis unfolding x_def using beyond by auto
2817 case False then obtain z where "n = Suc z" using not0_implies_Suc by auto
2818 thus ?thesis unfolding x_def using beyond by auto
2820 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
2822 then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto
2823 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
2824 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"]]
2825 unfolding dist_nz by auto
2826 show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto
2829 lemma sequence_infinite_lemma:
2830 fixes f :: "nat \<Rightarrow> 'a::t1_space"
2831 assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"
2832 shows "infinite (range f)"
2834 assume "finite (range f)"
2835 hence "closed (range f)" by (rule finite_imp_closed)
2836 hence "open (- range f)" by (rule open_Compl)
2837 from assms(1) have "l \<in> - range f" by auto
2838 from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
2839 using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)
2840 thus False unfolding eventually_sequentially by auto
2843 lemma closure_insert:
2844 fixes x :: "'a::t1_space"
2845 shows "closure (insert x s) = insert x (closure s)"
2846 apply (rule closure_unique)
2847 apply (rule conjI [OF insert_mono [OF closure_subset]])
2848 apply (rule conjI [OF closed_insert [OF closed_closure]])
2849 apply (simp add: closure_minimal)
2852 lemma islimpt_insert:
2853 fixes x :: "'a::t1_space"
2854 shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
2856 assume *: "x islimpt (insert a s)"
2858 proof (rule islimptI)
2859 fix t assume t: "x \<in> t" "open t"
2860 show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
2861 proof (cases "x = a")
2863 obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
2864 using * t by (rule islimptE)
2865 with `x = a` show ?thesis by auto
2868 with t have t': "x \<in> t - {a}" "open (t - {a})"
2869 by (simp_all add: open_Diff)
2870 obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
2871 using * t' by (rule islimptE)
2872 thus ?thesis by auto
2876 assume "x islimpt s" thus "x islimpt (insert a s)"
2877 by (rule islimpt_subset) auto
2880 lemma islimpt_union_finite:
2881 fixes x :: "'a::t1_space"
2882 shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
2883 by (induct set: finite, simp_all add: islimpt_insert)
2885 lemma sequence_unique_limpt:
2886 fixes f :: "nat \<Rightarrow> 'a::t2_space"
2887 assumes "(f ---> l) sequentially" and "l' islimpt (range f)"
2890 assume "l' \<noteq> l"
2891 obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
2892 using hausdorff [OF `l' \<noteq> l`] by auto
2893 have "eventually (\<lambda>n. f n \<in> t) sequentially"
2894 using assms(1) `open t` `l \<in> t` by (rule topological_tendstoD)
2895 then obtain N where "\<forall>n\<ge>N. f n \<in> t"
2896 unfolding eventually_sequentially by auto
2898 have "UNIV = {..<N} \<union> {N..}" by auto
2899 hence "l' islimpt (f ` ({..<N} \<union> {N..}))" using assms(2) by simp
2900 hence "l' islimpt (f ` {..<N} \<union> f ` {N..})" by (simp add: image_Un)
2901 hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite)
2902 then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
2903 using `l' \<in> s` `open s` by (rule islimptE)
2904 then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto
2905 with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t" by simp
2906 with `s \<inter> t = {}` show False by simp
2909 lemma bolzano_weierstrass_imp_closed:
2910 fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)
2911 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2914 { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
2916 proof(cases "\<forall>n. x n \<noteq> l")
2917 case False thus "l\<in>s" using as(1) by auto
2919 case True note cas = this
2920 with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto
2921 then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto
2922 thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
2924 thus ?thesis unfolding closed_sequential_limits by fast
2927 text{* Hence express everything as an equivalence. *}
2929 lemma compact_eq_heine_borel:
2930 fixes s :: "'a::metric_space set"
2931 shows "compact s \<longleftrightarrow>
2932 (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2933 --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")
2935 assume ?lhs thus ?rhs by (rule compact_imp_heine_borel)
2938 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
2939 by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])
2940 thus ?lhs by (rule bolzano_weierstrass_imp_compact)
2943 lemma compact_eq_bolzano_weierstrass:
2944 fixes s :: "'a::metric_space set"
2945 shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
2947 assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto
2949 assume ?rhs thus ?lhs by (rule bolzano_weierstrass_imp_compact)
2952 lemma compact_eq_bounded_closed:
2953 fixes s :: "'a::heine_borel set"
2954 shows "compact s \<longleftrightarrow> bounded s \<and> closed s" (is "?lhs = ?rhs")
2956 assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto
2958 assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto
2961 lemma compact_imp_bounded:
2962 fixes s :: "'a::metric_space set"
2963 shows "compact s ==> bounded s"
2966 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')"
2967 by (rule compact_imp_heine_borel)
2968 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
2969 using heine_borel_imp_bolzano_weierstrass[of s] by auto
2971 by (rule bolzano_weierstrass_imp_bounded)
2974 lemma compact_imp_closed:
2975 fixes s :: "'a::metric_space set"
2976 shows "compact s ==> closed s"
2979 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')"
2980 by (rule compact_imp_heine_borel)
2981 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
2982 using heine_borel_imp_bolzano_weierstrass[of s] by auto
2984 by (rule bolzano_weierstrass_imp_closed)
2987 text{* In particular, some common special cases. *}
2989 lemma compact_empty[simp]:
2991 unfolding compact_def
2994 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
2995 unfolding subseq_def by simp (* TODO: move somewhere else *)
2997 lemma compact_union [intro]:
2998 assumes "compact s" and "compact t"
2999 shows "compact (s \<union> t)"
3000 proof (rule compactI)
3001 fix f :: "nat \<Rightarrow> 'a"
3002 assume "\<forall>n. f n \<in> s \<union> t"
3003 hence "infinite {n. f n \<in> s \<union> t}" by simp
3004 hence "infinite {n. f n \<in> s} \<or> infinite {n. f n \<in> t}" by simp
3005 thus "\<exists>l\<in>s \<union> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
3007 assume "infinite {n. f n \<in> s}"
3008 from infinite_enumerate [OF this]
3009 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> s" by auto
3010 obtain r l where "l \<in> s" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
3011 using `compact s` `\<forall>n. (f \<circ> q) n \<in> s` by (rule compactE)
3012 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
3013 using `subseq q` by (simp_all add: subseq_o o_assoc)
3014 thus ?thesis by auto
3016 assume "infinite {n. f n \<in> t}"
3017 from infinite_enumerate [OF this]
3018 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> t" by auto
3019 obtain r l where "l \<in> t" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
3020 using `compact t` `\<forall>n. (f \<circ> q) n \<in> t` by (rule compactE)
3021 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
3022 using `subseq q` by (simp_all add: subseq_o o_assoc)
3023 thus ?thesis by auto
3027 lemma compact_inter_closed [intro]:
3028 assumes "compact s" and "closed t"
3029 shows "compact (s \<inter> t)"
3030 proof (rule compactI)
3031 fix f :: "nat \<Rightarrow> 'a"
3032 assume "\<forall>n. f n \<in> s \<inter> t"
3033 hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t" by simp_all
3034 obtain l r where "l \<in> s" "subseq r" "((f \<circ> r) ---> l) sequentially"
3035 using `compact s` `\<forall>n. f n \<in> s` by (rule compactE)
3037 from `closed t` `\<forall>n. f n \<in> t` `((f \<circ> r) ---> l) sequentially` have "l \<in> t"
3038 unfolding closed_sequential_limits o_def by fast
3039 ultimately show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
3043 lemma closed_inter_compact [intro]:
3044 assumes "closed s" and "compact t"
3045 shows "compact (s \<inter> t)"
3046 using compact_inter_closed [of t s] assms
3047 by (simp add: Int_commute)
3049 lemma compact_inter [intro]:
3050 assumes "compact s" and "compact t"
3051 shows "compact (s \<inter> t)"
3052 using assms by (intro compact_inter_closed compact_imp_closed)
3054 lemma compact_sing [simp]: "compact {a}"
3055 unfolding compact_def o_def subseq_def
3056 by (auto simp add: tendsto_const)
3058 lemma compact_insert [simp]:
3059 assumes "compact s" shows "compact (insert x s)"
3061 have "compact ({x} \<union> s)"
3062 using compact_sing assms by (rule compact_union)
3063 thus ?thesis by simp
3066 lemma finite_imp_compact:
3067 shows "finite s \<Longrightarrow> compact s"
3068 by (induct set: finite) simp_all
3070 lemma compact_cball[simp]:
3071 fixes x :: "'a::heine_borel"
3072 shows "compact(cball x e)"
3073 using compact_eq_bounded_closed bounded_cball closed_cball
3076 lemma compact_frontier_bounded[intro]:
3077 fixes s :: "'a::heine_borel set"
3078 shows "bounded s ==> compact(frontier s)"
3079 unfolding frontier_def
3080 using compact_eq_bounded_closed
3083 lemma compact_frontier[intro]:
3084 fixes s :: "'a::heine_borel set"
3085 shows "compact s ==> compact (frontier s)"
3086 using compact_eq_bounded_closed compact_frontier_bounded
3089 lemma frontier_subset_compact:
3090 fixes s :: "'a::heine_borel set"
3091 shows "compact s ==> frontier s \<subseteq> s"
3092 using frontier_subset_closed compact_eq_bounded_closed
3096 fixes s :: "'a::t1_space set"
3097 shows "open s \<Longrightarrow> open (s - {x})"
3098 by (simp add: open_Diff)
3100 text{* Finite intersection property. I could make it an equivalence in fact. *}
3102 lemma compact_imp_fip:
3103 assumes "compact s" "\<forall>t \<in> f. closed t"
3104 "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"
3105 shows "s \<inter> (\<Inter> f) \<noteq> {}"
3107 assume as:"s \<inter> (\<Inter> f) = {}"
3108 hence "s \<subseteq> \<Union> uminus ` f" by auto
3109 moreover have "Ball (uminus ` f) open" using open_Diff closed_Diff using assms(2) by auto
3110 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
3111 hence "finite (uminus ` f') \<and> uminus ` f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)
3112 hence "s \<inter> \<Inter>uminus ` f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus ` f'"]] by auto
3113 thus False using f'(3) unfolding subset_eq and Union_iff by blast
3116 subsection{* Bounded closed nest property (proof does not use Heine-Borel). *}
3118 lemma bounded_closed_nest:
3119 assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
3120 "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)" "bounded(s 0)"
3121 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
3123 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
3124 from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
3126 then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
3127 unfolding compact_def apply(erule_tac x=x in allE) using x using assms(3) by blast
3130 { fix e::real assume "e>0"
3131 with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
3132 hence "dist ((x \<circ> r) (max N n)) l < e" by auto
3134 have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
3135 hence "(x \<circ> r) (max N n) \<in> s n"
3136 using x apply(erule_tac x=n in allE)
3137 using x apply(erule_tac x="r (max N n)" in allE)
3138 using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto
3139 ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
3141 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
3143 thus ?thesis by auto
3146 text{* Decreasing case does not even need compactness, just completeness. *}
3148 lemma decreasing_closed_nest:
3149 assumes "\<forall>n. closed(s n)"
3150 "\<forall>n. (s n \<noteq> {})"
3151 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
3152 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
3153 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s n"
3155 have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
3156 hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
3157 then obtain t where t: "\<forall>n. t n \<in> s n" by auto
3158 { fix e::real assume "e>0"
3159 then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
3160 { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
3161 hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding subset_eq t by blast+
3162 hence "dist (t m) (t n) < e" using N by auto
3164 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
3166 hence "Cauchy t" unfolding cauchy_def by auto
3167 then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
3169 { fix e::real assume "e>0"
3170 then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded Lim_sequentially] by auto
3171 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
3172 hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
3174 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
3176 then show ?thesis by auto
3179 text{* Strengthen it to the intersection actually being a singleton. *}
3181 lemma decreasing_closed_nest_sing:
3182 fixes s :: "nat \<Rightarrow> 'a::heine_borel set"
3183 assumes "\<forall>n. closed(s n)"
3184 "\<forall>n. s n \<noteq> {}"
3185 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
3186 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
3187 shows "\<exists>a. \<Inter>(range s) = {a}"
3189 obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
3190 { fix b assume b:"b \<in> \<Inter>(range s)"
3191 { fix e::real assume "e>0"
3192 hence "dist a b < e" using assms(4 )using b using a by blast
3194 hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)
3196 with a have "\<Inter>(range s) = {a}" unfolding image_def by auto
3200 text{* Cauchy-type criteria for uniform convergence. *}
3202 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows
3203 "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
3204 (\<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")
3207 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
3208 { fix e::real assume "e>0"
3209 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
3210 { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
3211 hence "dist (s m x) (s n x) < e"
3212 using N[THEN spec[where x=m], THEN spec[where x=x]]
3213 using N[THEN spec[where x=n], THEN spec[where x=x]]
3214 using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto }
3215 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 }
3219 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
3220 then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
3221 using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
3222 { fix e::real assume "e>0"
3223 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"
3224 using `?rhs`[THEN spec[where x="e/2"]] by auto
3225 { fix x assume "P x"
3226 then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
3227 using l[THEN spec[where x=x], unfolded Lim_sequentially] using `e>0` by(auto elim!: allE[where x="e/2"])
3228 fix n::nat assume "n\<ge>N"
3229 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]]
3230 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) }
3231 hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
3235 lemma uniformly_cauchy_imp_uniformly_convergent:
3236 fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"
3237 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"
3238 "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
3239 shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
3241 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"
3242 using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
3244 { fix x assume "P x"
3245 hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
3246 using l and assms(2) unfolding Lim_sequentially by blast }
3247 ultimately show ?thesis by auto
3250 subsection {* Continuity *}
3252 text {* Define continuity over a net to take in restrictions of the set. *}
3255 continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
3256 where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
3258 lemma continuous_trivial_limit:
3259 "trivial_limit net ==> continuous net f"
3260 unfolding continuous_def tendsto_def trivial_limit_eq by auto
3262 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
3263 unfolding continuous_def
3264 unfolding tendsto_def
3265 using netlimit_within[of x s]
3266 by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
3268 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
3269 using continuous_within [of x UNIV f] by (simp add: within_UNIV)
3271 lemma continuous_at_within:
3272 assumes "continuous (at x) f" shows "continuous (at x within s) f"
3273 using assms unfolding continuous_at continuous_within
3274 by (rule Lim_at_within)
3276 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
3278 lemma continuous_within_eps_delta:
3279 "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)"
3280 unfolding continuous_within and Lim_within
3281 apply auto unfolding dist_nz[THEN sym] apply(auto elim!:allE) apply(rule_tac x=d in exI) by auto
3283 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3284 \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
3285 using continuous_within_eps_delta[of x UNIV f]
3286 unfolding within_UNIV by blast
3288 text{* Versions in terms of open balls. *}
3290 lemma continuous_within_ball:
3291 "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3292 f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3295 { fix e::real assume "e>0"
3296 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"
3297 using `?lhs`[unfolded continuous_within Lim_within] by auto
3298 { fix y assume "y\<in>f ` (ball x d \<inter> s)"
3299 hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
3300 apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
3302 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) }
3305 assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
3306 apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
3309 lemma continuous_at_ball:
3310 "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3312 assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3313 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)
3314 unfolding dist_nz[THEN sym] by auto
3316 assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3317 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)
3320 text{* Define setwise continuity in terms of limits within the set. *}
3324 "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
3326 "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
3328 lemma continuous_on_topological:
3329 "continuous_on s f \<longleftrightarrow>
3330 (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
3331 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
3332 unfolding continuous_on_def tendsto_def
3333 unfolding Limits.eventually_within eventually_at_topological
3334 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
3336 lemma continuous_on_iff:
3337 "continuous_on s f \<longleftrightarrow>
3338 (\<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)"
3339 unfolding continuous_on_def Lim_within
3340 apply (intro ball_cong [OF refl] all_cong ex_cong)
3341 apply (rename_tac y, case_tac "y = x", simp)
3342 apply (simp add: dist_nz)
3346 uniformly_continuous_on ::
3347 "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
3349 "uniformly_continuous_on s f \<longleftrightarrow>
3350 (\<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)"
3352 text{* Some simple consequential lemmas. *}
3354 lemma uniformly_continuous_imp_continuous:
3355 " uniformly_continuous_on s f ==> continuous_on s f"
3356 unfolding uniformly_continuous_on_def continuous_on_iff by blast
3358 lemma continuous_at_imp_continuous_within:
3359 "continuous (at x) f ==> continuous (at x within s) f"
3360 unfolding continuous_within continuous_at using Lim_at_within by auto
3362 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
3363 unfolding tendsto_def by (simp add: trivial_limit_eq)
3365 lemma continuous_at_imp_continuous_on:
3366 assumes "\<forall>x\<in>s. continuous (at x) f"
3367 shows "continuous_on s f"
3368 unfolding continuous_on_def
3370 fix x assume "x \<in> s"
3371 with assms have *: "(f ---> f (netlimit (at x))) (at x)"
3372 unfolding continuous_def by simp
3373 have "(f ---> f x) (at x)"
3374 proof (cases "trivial_limit (at x)")
3375 case True thus ?thesis
3376 by (rule Lim_trivial_limit)
3379 hence 1: "netlimit (at x) = x"
3380 using netlimit_within [of x UNIV]
3381 by (simp add: within_UNIV)
3382 with * show ?thesis by simp
3384 thus "(f ---> f x) (at x within s)"
3385 by (rule Lim_at_within)
3388 lemma continuous_on_eq_continuous_within:
3389 "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"
3390 unfolding continuous_on_def continuous_def
3391 apply (rule ball_cong [OF refl])
3392 apply (case_tac "trivial_limit (at x within s)")
3393 apply (simp add: Lim_trivial_limit)
3394 apply (simp add: netlimit_within)
3397 lemmas continuous_on = continuous_on_def -- "legacy theorem name"
3399 lemma continuous_on_eq_continuous_at:
3400 shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
3401 by (auto simp add: continuous_on continuous_at Lim_within_open)
3403 lemma continuous_within_subset:
3404 "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
3405 ==> continuous (at x within t) f"
3406 unfolding continuous_within by(metis Lim_within_subset)
3408 lemma continuous_on_subset:
3409 shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
3410 unfolding continuous_on by (metis subset_eq Lim_within_subset)
3412 lemma continuous_on_interior:
3413 shows "continuous_on s f \<Longrightarrow> x \<in> interior s ==> continuous (at x) f"
3414 unfolding interior_def
3416 by (meson continuous_on_eq_continuous_at continuous_on_subset)
3418 lemma continuous_on_eq:
3419 "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"
3420 unfolding continuous_on_def tendsto_def Limits.eventually_within
3423 text{* Characterization of various kinds of continuity in terms of sequences. *}
3425 (* \<longrightarrow> could be generalized, but \<longleftarrow> requires metric space *)
3426 lemma continuous_within_sequentially:
3427 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3428 shows "continuous (at a within s) f \<longleftrightarrow>
3429 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
3430 --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
3433 { 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"
3434 fix e::real assume "e>0"
3435 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
3436 from x(2) `d>0` obtain N where N:"\<forall>n\<ge>N. dist (x n) a < d" by auto
3437 hence "\<exists>N. \<forall>n\<ge>N. dist ((f \<circ> x) n) (f a) < e"
3438 apply(rule_tac x=N in exI) using N d apply auto using x(1)
3439 apply(erule_tac x=n in allE) apply(erule_tac x=n in allE)
3440 apply(erule_tac x="x n" in ballE) apply auto unfolding dist_nz[THEN sym] apply auto using `e>0` by auto
3442 thus ?rhs unfolding continuous_within unfolding Lim_sequentially by simp
3445 { fix e::real assume "e>0"
3446 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)"
3447 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
3448 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)"
3449 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
3450 { fix d::real assume "d>0"
3451 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
3452 then obtain N::nat where N:"inverse (real (N + 1)) < d" by auto
3453 { fix n::nat assume n:"n\<ge>N"
3454 hence "dist (x (inverse (real (n + 1)))) a < inverse (real (n + 1))" using x[THEN spec[where x="inverse (real (n + 1))"]] by auto
3455 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)
3456 ultimately have "dist (x (inverse (real (n + 1)))) a < d" by auto
3458 hence "\<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < d" by auto
3460 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
3461 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
3462 hence "False" apply(erule_tac x=e in allE) using `e>0` using x by auto
3464 thus ?lhs unfolding continuous_within unfolding Lim_within unfolding Lim_sequentially by blast
3467 lemma continuous_at_sequentially:
3468 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3469 shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
3470 --> ((f o x) ---> f a) sequentially)"
3471 using continuous_within_sequentially[of a UNIV f] unfolding within_UNIV by auto
3473 lemma continuous_on_sequentially:
3474 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3475 shows "continuous_on s f \<longleftrightarrow>
3476 (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
3477 --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
3479 assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
3481 assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
3484 lemma uniformly_continuous_on_sequentially':
3485 "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
3486 ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially
3487 \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
3490 { 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"
3491 { fix e::real assume "e>0"
3492 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"
3493 using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
3494 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
3495 { fix n assume "n\<ge>N"
3496 hence "dist (f (x n)) (f (y n)) < e"
3497 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
3498 unfolding dist_commute by simp }
3499 hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" by auto }
3500 hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding Lim_sequentially and dist_real_def by auto }
3504 { assume "\<not> ?lhs"
3505 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
3506 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"
3507 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
3508 by (auto simp add: dist_commute)
3509 def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
3510 def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
3511 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"
3512 unfolding x_def and y_def using fa by auto
3513 { fix e::real assume "e>0"
3514 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
3515 { fix n::nat assume "n\<ge>N"
3516 hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
3517 also have "\<dots> < e" using N by auto
3518 finally have "inverse (real n + 1) < e" by auto
3519 hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto }
3520 hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto }
3521 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
3522 hence False using fxy and `e>0` by auto }
3523 thus ?lhs unfolding uniformly_continuous_on_def by blast
3526 lemma uniformly_continuous_on_sequentially:
3527 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
3528 shows "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
3529 ((\<lambda>n. x n - y n) ---> 0) sequentially
3530 \<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")
3531 (* BH: maybe the previous lemma should replace this one? *)
3532 unfolding uniformly_continuous_on_sequentially'
3533 unfolding dist_norm tendsto_norm_zero_iff ..
3535 text{* The usual transformation theorems. *}
3537 lemma continuous_transform_within:
3538 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3539 assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
3540 "continuous (at x within s) f"
3541 shows "continuous (at x within s) g"
3542 unfolding continuous_within
3543 proof (rule Lim_transform_within)
3544 show "0 < d" by fact
3545 show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
3546 using assms(3) by auto
3548 using assms(1,2,3) by auto
3549 thus "(f ---> g x) (at x within s)"
3550 using assms(4) unfolding continuous_within by simp
3553 lemma continuous_transform_at:
3554 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3555 assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
3556 "continuous (at x) f"
3557 shows "continuous (at x) g"
3558 using continuous_transform_within [of d x UNIV f g] assms
3559 by (simp add: within_UNIV)
3561 text{* Combination results for pointwise continuity. *}
3563 lemma continuous_const: "continuous net (\<lambda>x. c)"
3564 by (auto simp add: continuous_def tendsto_const)
3566 lemma continuous_cmul:
3567 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3568 shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"
3569 by (auto simp add: continuous_def intro: tendsto_intros)
3571 lemma continuous_neg:
3572 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3573 shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"
3574 by (auto simp add: continuous_def tendsto_minus)
3576 lemma continuous_add:
3577 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3578 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"
3579 by (auto simp add: continuous_def tendsto_add)
3581 lemma continuous_sub:
3582 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3583 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"
3584 by (auto simp add: continuous_def tendsto_diff)
3587 text{* Same thing for setwise continuity. *}
3589 lemma continuous_on_const:
3590 "continuous_on s (\<lambda>x. c)"
3591 unfolding continuous_on_def by (auto intro: tendsto_intros)
3593 lemma continuous_on_cmul:
3594 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3595 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c *\<^sub>R (f x))"
3596 unfolding continuous_on_def by (auto intro: tendsto_intros)
3598 lemma continuous_on_neg:
3599 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3600 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
3601 unfolding continuous_on_def by (auto intro: tendsto_intros)
3603 lemma continuous_on_add:
3604 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3605 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3606 \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
3607 unfolding continuous_on_def by (auto intro: tendsto_intros)
3609 lemma continuous_on_sub:
3610 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3611 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3612 \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
3613 unfolding continuous_on_def by (auto intro: tendsto_intros)
3615 text{* Same thing for uniform continuity, using sequential formulations. *}
3617 lemma uniformly_continuous_on_const:
3618 "uniformly_continuous_on s (\<lambda>x. c)"
3619 unfolding uniformly_continuous_on_def by simp
3621 lemma uniformly_continuous_on_cmul:
3622 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3623 assumes "uniformly_continuous_on s f"
3624 shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
3626 { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
3627 hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"
3628 using scaleR.tendsto [OF tendsto_const, of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
3629 unfolding scaleR_zero_right scaleR_right_diff_distrib by auto
3631 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
3632 unfolding dist_norm tendsto_norm_zero_iff by auto
3636 fixes x y :: "'a::real_normed_vector"
3637 shows "dist (- x) (- y) = dist x y"
3638 unfolding dist_norm minus_diff_minus norm_minus_cancel ..
3640 lemma uniformly_continuous_on_neg:
3641 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3642 shows "uniformly_continuous_on s f
3643 ==> uniformly_continuous_on s (\<lambda>x. -(f x))"
3644 unfolding uniformly_continuous_on_def dist_minus .
3646 lemma uniformly_continuous_on_add:
3647 fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3648 assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"
3649 shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
3651 { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
3652 "((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"
3653 hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"
3654 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
3655 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 }
3656 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
3657 unfolding dist_norm tendsto_norm_zero_iff by auto
3660 lemma uniformly_continuous_on_sub:
3661 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3662 shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s g
3663 ==> uniformly_continuous_on s (\<lambda>x. f x - g x)"
3664 unfolding ab_diff_minus
3665 using uniformly_continuous_on_add[of s f "\<lambda>x. - g x"]
3666 using uniformly_continuous_on_neg[of s g] by auto
3668 text{* Identity function is continuous in every sense. *}
3670 lemma continuous_within_id:
3671 "continuous (at a within s) (\<lambda>x. x)"
3672 unfolding continuous_within by (rule Lim_at_within [OF LIM_ident])
3674 lemma continuous_at_id:
3675 "continuous (at a) (\<lambda>x. x)"
3676 unfolding continuous_at by (rule LIM_ident)
3678 lemma continuous_on_id:
3679 "continuous_on s (\<lambda>x. x)"
3680 unfolding continuous_on_def by (auto intro: tendsto_ident_at_within)
3682 lemma uniformly_continuous_on_id:
3683 "uniformly_continuous_on s (\<lambda>x. x)"
3684 unfolding uniformly_continuous_on_def by auto
3686 text{* Continuity of all kinds is preserved under composition. *}
3688 lemma continuous_within_topological:
3689 "continuous (at x within s) f \<longleftrightarrow>
3690 (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
3691 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
3692 unfolding continuous_within
3693 unfolding tendsto_def Limits.eventually_within eventually_at_topological
3694 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
3696 lemma continuous_within_compose:
3697 assumes "continuous (at x within s) f"
3698 assumes "continuous (at (f x) within f ` s) g"
3699 shows "continuous (at x within s) (g o f)"
3700 using assms unfolding continuous_within_topological by simp metis
3702 lemma continuous_at_compose:
3703 assumes "continuous (at x) f" "continuous (at (f x)) g"
3704 shows "continuous (at x) (g o f)"
3706 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
3707 thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g, unfolded within_UNIV] by auto
3710 lemma continuous_on_compose:
3711 "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
3712 unfolding continuous_on_topological by simp metis
3714 lemma uniformly_continuous_on_compose:
3715 assumes "uniformly_continuous_on s f" "uniformly_continuous_on (f ` s) g"
3716 shows "uniformly_continuous_on s (g o f)"
3718 { fix e::real assume "e>0"
3719 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
3720 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
3721 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 }
3722 thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
3725 text{* Continuity in terms of open preimages. *}
3727 lemma continuous_at_open:
3728 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)))"
3729 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]
3730 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto
3732 lemma continuous_on_open:
3733 shows "continuous_on s f \<longleftrightarrow>
3734 (\<forall>t. openin (subtopology euclidean (f ` s)) t
3735 --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
3738 assume 1: "continuous_on s f"
3739 assume 2: "openin (subtopology euclidean (f ` s)) t"
3740 from 2 obtain B where B: "open B" and t: "t = f ` s \<inter> B"
3741 unfolding openin_open by auto
3742 def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"
3743 have "open U" unfolding U_def by (simp add: open_Union)
3744 moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"
3745 proof (intro ballI iffI)
3746 fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"
3747 unfolding U_def t by auto
3749 fix x assume "x \<in> s" and "f x \<in> t"
3750 hence "x \<in> s" and "f x \<in> B"
3752 with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"
3753 unfolding t continuous_on_topological by metis
3754 then show "x \<in> U"
3755 unfolding U_def by auto
3757 ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto
3758 then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3759 unfolding openin_open by fast
3761 assume "?rhs" show "continuous_on s f"
3762 unfolding continuous_on_topological
3764 fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"
3765 have "openin (subtopology euclidean (f ` s)) (f ` s \<inter> B)"
3766 unfolding openin_open using `open B` by auto
3767 then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f ` s \<inter> B}"
3768 using `?rhs` by fast
3769 then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
3770 unfolding openin_open using `x \<in> s` and `f x \<in> B` by auto
3774 text {* Similarly in terms of closed sets. *}
3776 lemma continuous_on_closed:
3777 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")
3781 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3782 have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
3783 assume as:"closedin (subtopology euclidean (f ` s)) t"
3784 hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
3785 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"]]
3786 unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto }
3791 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3792 assume as:"openin (subtopology euclidean (f ` s)) t"
3793 hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
3794 unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
3795 thus ?lhs unfolding continuous_on_open by auto
3798 text{* Half-global and completely global cases. *}
3800 lemma continuous_open_in_preimage:
3801 assumes "continuous_on s f" "open t"
3802 shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3804 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
3805 have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3806 using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
3807 thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3810 lemma continuous_closed_in_preimage:
3811 assumes "continuous_on s f" "closed t"
3812 shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3814 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
3815 have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3816 using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
3818 using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3821 lemma continuous_open_preimage:
3822 assumes "continuous_on s f" "open s" "open t"
3823 shows "open {x \<in> s. f x \<in> t}"
3825 obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3826 using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
3827 thus ?thesis using open_Int[of s T, OF assms(2)] by auto
3830 lemma continuous_closed_preimage:
3831 assumes "continuous_on s f" "closed s" "closed t"
3832 shows "closed {x \<in> s. f x \<in> t}"
3834 obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3835 using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
3836 thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
3839 lemma continuous_open_preimage_univ:
3840 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
3841 using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
3843 lemma continuous_closed_preimage_univ:
3844 shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
3845 using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
3847 lemma continuous_open_vimage:
3848 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
3849 unfolding vimage_def by (rule continuous_open_preimage_univ)
3851 lemma continuous_closed_vimage:
3852 shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
3853 unfolding vimage_def by (rule continuous_closed_preimage_univ)
3855 lemma interior_image_subset:
3856 assumes "\<forall>x. continuous (at x) f" "inj f"
3857 shows "interior (f ` s) \<subseteq> f ` (interior s)"
3858 apply rule unfolding interior_def mem_Collect_eq image_iff apply safe
3859 proof- fix x T assume as:"open T" "x \<in> T" "T \<subseteq> f ` s"
3860 hence "x \<in> f ` s" by auto then guess y unfolding image_iff .. note y=this
3861 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
3862 apply safe apply(rule_tac x="{x. f x \<in> T}" in exI) apply(safe,rule continuous_open_preimage_univ)
3863 proof- fix x assume "f x \<in> T" hence "f x \<in> f ` s" using as by auto
3864 thus "x \<in> s" unfolding inj_image_mem_iff[OF assms(2)] . qed auto qed
3866 text{* Equality of continuous functions on closure and related results. *}
3868 lemma continuous_closed_in_preimage_constant:
3869 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3870 shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
3871 using continuous_closed_in_preimage[of s f "{a}"] by auto
3873 lemma continuous_closed_preimage_constant:
3874 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3875 shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
3876 using continuous_closed_preimage[of s f "{a}"] by auto
3878 lemma continuous_constant_on_closure:
3879 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3880 assumes "continuous_on (closure s) f"
3881 "\<forall>x \<in> s. f x = a"
3882 shows "\<forall>x \<in> (closure s). f x = a"
3883 using continuous_closed_preimage_constant[of "closure s" f a]
3884 assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
3886 lemma image_closure_subset:
3887 assumes "continuous_on (closure s) f" "closed t" "(f ` s) \<subseteq> t"
3888 shows "f ` (closure s) \<subseteq> t"
3890 have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
3891 moreover have "closed {x \<in> closure s. f x \<in> t}"
3892 using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
3893 ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
3894 using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
3895 thus ?thesis by auto
3898 lemma continuous_on_closure_norm_le:
3899 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3900 assumes "continuous_on (closure s) f" "\<forall>y \<in> s. norm(f y) \<le> b" "x \<in> (closure s)"
3901 shows "norm(f x) \<le> b"
3903 have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
3905 using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
3906 unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
3909 text{* Making a continuous function avoid some value in a neighbourhood. *}
3911 lemma continuous_within_avoid:
3912 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3913 assumes "continuous (at x within s) f" "x \<in> s" "f x \<noteq> a"
3914 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
3916 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"
3917 using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto
3918 { fix y assume " y\<in>s" "dist x y < d"
3919 hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]
3920 apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }
3921 thus ?thesis using `d>0` by auto
3924 lemma continuous_at_avoid:
3925 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3926 assumes "continuous (at x) f" "f x \<noteq> a"
3927 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
3928 using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto
3930 lemma continuous_on_avoid:
3931 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
3932 assumes "continuous_on s f" "x \<in> s" "f x \<noteq> a"
3933 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
3934 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
3936 lemma continuous_on_open_avoid:
3937 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
3938 assumes "continuous_on s f" "open s" "x \<in> s" "f x \<noteq> a"
3939 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
3940 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
3942 text{* Proving a function is constant by proving open-ness of level set. *}
3944 lemma continuous_levelset_open_in_cases:
3945 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3946 shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
3947 openin (subtopology euclidean s) {x \<in> s. f x = a}
3948 ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
3949 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
3951 lemma continuous_levelset_open_in:
3952 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3953 shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
3954 openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
3955 (\<exists>x \<in> s. f x = a) ==> (\<forall>x \<in> s. f x = a)"
3956 using continuous_levelset_open_in_cases[of s f ]
3959 lemma continuous_levelset_open:
3960 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3961 assumes "connected s" "continuous_on s f" "open {x \<in> s. f x = a}" "\<exists>x \<in> s. f x = a"
3962 shows "\<forall>x \<in> s. f x = a"
3963 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
3965 text{* Some arithmetical combinations (more to prove). *}
3967 lemma open_scaling[intro]:
3968 fixes s :: "'a::real_normed_vector set"
3969 assumes "c \<noteq> 0" "open s"
3970 shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
3972 { fix x assume "x \<in> s"
3973 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
3974 have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF `e>0`] by auto
3976 { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
3977 hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
3978 using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
3979 assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
3980 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 }
3981 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 }
3982 thus ?thesis unfolding open_dist by auto
3985 lemma minus_image_eq_vimage:
3986 fixes A :: "'a::ab_group_add set"
3987 shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
3988 by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
3990 lemma open_negations:
3991 fixes s :: "'a::real_normed_vector set"
3992 shows "open s ==> open ((\<lambda> x. -x) ` s)"
3993 unfolding scaleR_minus1_left [symmetric]
3994 by (rule open_scaling, auto)
3996 lemma open_translation:
3997 fixes s :: "'a::real_normed_vector set"
3998 assumes "open s" shows "open((\<lambda>x. a + x) ` s)"
4000 { 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 }
4001 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
4002 ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
4005 lemma open_affinity:
4006 fixes s :: "'a::real_normed_vector set"
4007 assumes "open s" "c \<noteq> 0"
4008 shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
4010 have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
4011 have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
4012 thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
4015 lemma interior_translation:
4016 fixes s :: "'a::real_normed_vector set"
4017 shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
4018 proof (rule set_eqI, rule)
4019 fix x assume "x \<in> interior (op + a ` s)"
4020 then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
4021 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
4022 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
4024 fix x assume "x \<in> op + a ` interior s"
4025 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
4026 { fix z have *:"a + y - z = y + a - z" by auto
4027 assume "z\<in>ball x e"
4028 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
4029 hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"]) }
4030 hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
4031 thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
4034 text {* We can now extend limit compositions to consider the scalar multiplier. *}
4036 lemma continuous_vmul:
4037 fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
4038 shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"
4039 unfolding continuous_def by (intro tendsto_intros)
4041 lemma continuous_mul:
4042 fixes c :: "'a::metric_space \<Rightarrow> real"
4043 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
4044 shows "continuous net c \<Longrightarrow> continuous net f
4045 ==> continuous net (\<lambda>x. c(x) *\<^sub>R f x) "
4046 unfolding continuous_def by (intro tendsto_intros)
4048 lemmas continuous_intros = continuous_add continuous_vmul continuous_cmul
4049 continuous_const continuous_sub continuous_at_id continuous_within_id continuous_mul
4051 lemma continuous_on_vmul:
4052 fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
4053 shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R v)"
4054 unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
4056 lemma continuous_on_mul:
4057 fixes c :: "'a::metric_space \<Rightarrow> real"
4058 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
4059 shows "continuous_on s c \<Longrightarrow> continuous_on s f
4060 ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R f x)"
4061 unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
4063 lemma continuous_on_mul_real:
4064 fixes f :: "'a::metric_space \<Rightarrow> real"
4065 fixes g :: "'a::metric_space \<Rightarrow> real"
4066 shows "continuous_on s f \<Longrightarrow> continuous_on s g
4067 ==> continuous_on s (\<lambda>x. f x * g x)"
4068 using continuous_on_mul[of s f g] unfolding real_scaleR_def .
4070 lemmas continuous_on_intros = continuous_on_add continuous_on_const
4071 continuous_on_id continuous_on_compose continuous_on_cmul continuous_on_neg
4072 continuous_on_sub continuous_on_mul continuous_on_vmul continuous_on_mul_real
4073 uniformly_continuous_on_add uniformly_continuous_on_const
4074 uniformly_continuous_on_id uniformly_continuous_on_compose
4075 uniformly_continuous_on_cmul uniformly_continuous_on_neg
4076 uniformly_continuous_on_sub
4078 text{* And so we have continuity of inverse. *}
4080 lemma continuous_inv:
4081 fixes f :: "'a::metric_space \<Rightarrow> real"
4082 shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
4083 ==> continuous net (inverse o f)"
4084 unfolding continuous_def using Lim_inv by auto
4086 lemma continuous_at_within_inv:
4087 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
4088 assumes "continuous (at a within s) f" "f a \<noteq> 0"
4089 shows "continuous (at a within s) (inverse o f)"
4090 using assms unfolding continuous_within o_def
4091 by (intro tendsto_intros)
4093 lemma continuous_at_inv:
4094 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
4095 shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
4096 ==> continuous (at a) (inverse o f) "
4097 using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
4099 text {* Topological properties of linear functions. *}
4102 assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
4104 interpret f: bounded_linear f by fact
4105 have "(f ---> f 0) (at 0)"
4106 using tendsto_ident_at by (rule f.tendsto)
4107 thus ?thesis unfolding f.zero .
4110 lemma linear_continuous_at:
4111 assumes "bounded_linear f" shows "continuous (at a) f"
4112 unfolding continuous_at using assms
4113 apply (rule bounded_linear.tendsto)
4114 apply (rule tendsto_ident_at)
4117 lemma linear_continuous_within:
4118 shows "bounded_linear f ==> continuous (at x within s) f"
4119 using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
4121 lemma linear_continuous_on:
4122 shows "bounded_linear f ==> continuous_on s f"
4123 using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
4125 text{* Also bilinear functions, in composition form. *}
4127 lemma bilinear_continuous_at_compose:
4128 shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
4129 ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
4130 unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
4132 lemma bilinear_continuous_within_compose:
4133 shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
4134 ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
4135 unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
4137 lemma bilinear_continuous_on_compose:
4138 shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
4139 ==> continuous_on s (\<lambda>x. h (f x) (g x))"
4140 unfolding continuous_on_def
4141 by (fast elim: bounded_bilinear.tendsto)
4143 text {* Preservation of compactness and connectedness under continuous function. *}
4145 lemma compact_continuous_image:
4146 assumes "continuous_on s f" "compact s"
4147 shows "compact(f ` s)"
4149 { fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
4150 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
4151 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
4152 { fix e::real assume "e>0"
4153 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
4154 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
4155 { 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 }
4156 hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto }
4157 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 }
4158 thus ?thesis unfolding compact_def by auto
4161 lemma connected_continuous_image:
4162 assumes "continuous_on s f" "connected s"
4163 shows "connected(f ` s)"
4165 { fix T assume as: "T \<noteq> {}" "T \<noteq> f ` s" "openin (subtopology euclidean (f ` s)) T" "closedin (subtopology euclidean (f ` s)) T"
4166 have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
4167 using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
4168 using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
4169 using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
4170 hence False using as(1,2)
4171 using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
4172 thus ?thesis unfolding connected_clopen by auto
4175 text{* Continuity implies uniform continuity on a compact domain. *}
4177 lemma compact_uniformly_continuous:
4178 assumes "continuous_on s f" "compact s"
4179 shows "uniformly_continuous_on s f"
4181 { fix x assume x:"x\<in>s"
4182 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
4183 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 }
4184 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
4185 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)"
4186 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
4188 { fix e::real assume "e>0"
4190 { 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 }
4191 hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto
4193 { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto }
4194 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
4196 { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"
4197 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
4198 hence "x\<in>ball z (d z (e / 2))" using `ea>0` unfolding subset_eq by auto
4199 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`
4200 by (auto simp add: dist_commute)
4201 moreover have "y\<in>ball z (d z (e / 2))" using as and `ea>0` and z[unfolded subset_eq]
4202 by (auto simp add: dist_commute)
4203 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`
4204 by (auto simp add: dist_commute)
4205 ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]
4206 by (auto simp add: dist_commute) }
4207 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 }
4208 thus ?thesis unfolding uniformly_continuous_on_def by auto
4211 text{* Continuity of inverse function on compact domain. *}
4213 lemma continuous_on_inverse:
4214 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
4215 (* TODO: can this be generalized more? *)
4216 assumes "continuous_on s f" "compact s" "\<forall>x \<in> s. g (f x) = x"
4217 shows "continuous_on (f ` s) g"
4219 have *:"g ` f ` s = s" using assms(3) by (auto simp add: image_iff)
4220 { fix t assume t:"closedin (subtopology euclidean (g ` f ` s)) t"
4221 then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto
4222 have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]
4223 unfolding T(2) and Int_left_absorb by auto
4224 moreover have "compact (s \<inter> T)"
4225 using assms(2) unfolding compact_eq_bounded_closed
4226 using bounded_subset[of s "s \<inter> T"] and T(1) by auto
4227 ultimately have "closed (f ` t)" using T(1) unfolding T(2)
4228 using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto
4229 moreover have "{x \<in> f ` s. g x \<in> t} = f ` s \<inter> f ` t" using assms(3) unfolding T(2) by auto
4230 ultimately have "closedin (subtopology euclidean (f ` s)) {x \<in> f ` s. g x \<in> t}"
4231 unfolding closedin_closed by auto }
4232 thus ?thesis unfolding continuous_on_closed by auto
4235 text {* A uniformly convergent limit of continuous functions is continuous. *}
4237 lemma norm_triangle_lt:
4238 fixes x y :: "'a::real_normed_vector"
4239 shows "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
4240 by (rule le_less_trans [OF norm_triangle_ineq])
4242 lemma continuous_uniform_limit:
4243 fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::real_normed_vector"
4244 assumes "\<not> (trivial_limit net)" "eventually (\<lambda>n. continuous_on s (f n)) net"
4245 "\<forall>e>0. eventually (\<lambda>n. \<forall>x \<in> s. norm(f n x - g x) < e) net"
4246 shows "continuous_on s g"
4248 { fix x and e::real assume "x\<in>s" "e>0"
4249 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
4250 then obtain n where n:"\<forall>xa\<in>s. norm (f n xa - g xa) < e / 3" "continuous_on s (f n)"
4251 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
4252 have "e / 3 > 0" using `e>0` by auto
4253 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"
4254 using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
4255 { fix y assume "y\<in>s" "dist y x < d"
4256 hence "dist (f n y) (f n x) < e / 3" using d[THEN bspec[where x=y]] by auto
4257 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"]
4258 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
4259 hence "dist (g y) (g x) < e" unfolding dist_norm using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
4260 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) }
4261 hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using `d>0` by auto }
4262 thus ?thesis unfolding continuous_on_iff by auto
4265 subsection{* Topological stuff lifted from and dropped to R *}
4269 fixes s :: "real set" shows
4270 "open s \<longleftrightarrow>
4271 (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
4272 unfolding open_dist dist_norm by simp
4274 lemma islimpt_approachable_real:
4275 fixes s :: "real set"
4276 shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
4277 unfolding islimpt_approachable dist_norm by simp
4280 fixes s :: "real set"
4281 shows "closed s \<longleftrightarrow>
4282 (\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
4284 unfolding closed_limpt islimpt_approachable dist_norm by simp
4286 lemma continuous_at_real_range:
4287 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4288 shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
4289 \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
4290 unfolding continuous_at unfolding Lim_at
4291 unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
4292 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
4293 apply(erule_tac x=e in allE) by auto
4295 lemma continuous_on_real_range:
4296 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4297 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))"
4298 unfolding continuous_on_iff dist_norm by simp
4300 lemma continuous_at_norm: "continuous (at x) norm"
4301 unfolding continuous_at by (intro tendsto_intros)
4303 lemma continuous_on_norm: "continuous_on s norm"
4304 unfolding continuous_on by (intro ballI tendsto_intros)
4306 lemma continuous_at_infnorm: "continuous (at x) infnorm"
4307 unfolding continuous_at Lim_at o_def unfolding dist_norm
4308 apply auto apply (rule_tac x=e in exI) apply auto
4309 using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
4311 text{* Hence some handy theorems on distance, diameter etc. of/from a set. *}
4313 lemma compact_attains_sup:
4314 fixes s :: "real set"
4315 assumes "compact s" "s \<noteq> {}"
4316 shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
4318 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4319 { 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"
4320 have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto
4321 moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
4322 ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using `e>0` by auto }
4323 thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]
4324 apply(rule_tac x="Sup s" in bexI) by auto
4328 fixes S :: "real set"
4329 shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
4330 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def)
4332 lemma compact_attains_inf:
4333 fixes s :: "real set"
4334 assumes "compact s" "s \<noteq> {}" shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
4336 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4337 { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s" "Inf s \<notin> s" "0 < e"
4338 "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"
4339 have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto
4341 { fix x assume "x \<in> s"
4342 hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto
4343 have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
4344 hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto
4345 ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using `e>0` by auto }
4346 thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]
4347 apply(rule_tac x="Inf s" in bexI) by auto
4350 lemma continuous_attains_sup:
4351 fixes f :: "'a::metric_space \<Rightarrow> real"
4352 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4353 ==> (\<exists>x \<in> s. \<forall>y \<in> s. f y \<le> f x)"
4354 using compact_attains_sup[of "f ` s"]
4355 using compact_continuous_image[of s f] by auto
4357 lemma continuous_attains_inf:
4358 fixes f :: "'a::metric_space \<Rightarrow> real"
4359 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4360 \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
4361 using compact_attains_inf[of "f ` s"]
4362 using compact_continuous_image[of s f] by auto
4364 lemma distance_attains_sup:
4365 assumes "compact s" "s \<noteq> {}"
4366 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
4367 proof (rule continuous_attains_sup [OF assms])
4368 { fix x assume "x\<in>s"
4369 have "(dist a ---> dist a x) (at x within s)"
4370 by (intro tendsto_dist tendsto_const Lim_at_within LIM_ident)
4372 thus "continuous_on s (dist a)"
4373 unfolding continuous_on ..
4376 text{* For *minimal* distance, we only need closure, not compactness. *}
4378 lemma distance_attains_inf:
4379 fixes a :: "'a::heine_borel"
4380 assumes "closed s" "s \<noteq> {}"
4381 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
4383 from assms(2) obtain b where "b\<in>s" by auto
4384 let ?B = "cball a (dist b a) \<inter> s"
4385 have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
4386 hence "?B \<noteq> {}" by auto
4388 { fix x assume "x\<in>?B"
4389 fix e::real assume "e>0"
4390 { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
4391 from as have "\<bar>dist a x' - dist a x\<bar> < e"
4392 unfolding abs_less_iff minus_diff_eq
4393 using dist_triangle2 [of a x' x]
4394 using dist_triangle [of a x x']
4397 hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
4400 hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
4401 unfolding continuous_on Lim_within dist_norm real_norm_def
4403 moreover have "compact ?B"
4404 using compact_cball[of a "dist b a"]
4405 unfolding compact_eq_bounded_closed
4406 using bounded_Int and closed_Int and assms(1) by auto
4407 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"
4408 using continuous_attains_inf[of ?B "dist a"] by fastsimp
4409 thus ?thesis by fastsimp
4412 subsection {* Pasted sets *}
4414 lemma bounded_Times:
4415 assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
4417 obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
4418 using assms [unfolded bounded_def] by auto
4419 then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
4420 by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
4421 thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
4424 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
4427 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
4428 unfolding compact_def
4430 apply (drule_tac x="fst \<circ> f" in spec)
4431 apply (drule mp, simp add: mem_Times_iff)
4432 apply (clarify, rename_tac l1 r1)
4433 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
4434 apply (drule mp, simp add: mem_Times_iff)
4435 apply (clarify, rename_tac l2 r2)
4436 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
4437 apply (rule_tac x="r1 \<circ> r2" in exI)
4438 apply (rule conjI, simp add: subseq_def)
4439 apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
4440 apply (drule (1) tendsto_Pair) back
4441 apply (simp add: o_def)
4444 text{* Hence some useful properties follow quite easily. *}
4446 lemma compact_scaling:
4447 fixes s :: "'a::real_normed_vector set"
4448 assumes "compact s" shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
4450 let ?f = "\<lambda>x. scaleR c x"
4451 have *:"bounded_linear ?f" by (rule scaleR.bounded_linear_right)
4452 show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
4453 using linear_continuous_at[OF *] assms by auto
4456 lemma compact_negations:
4457 fixes s :: "'a::real_normed_vector set"
4458 assumes "compact s" shows "compact ((\<lambda>x. -x) ` s)"
4459 using compact_scaling [OF assms, of "- 1"] by auto
4462 fixes s t :: "'a::real_normed_vector set"
4463 assumes "compact s" "compact t" shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
4465 have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
4466 apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
4467 have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
4468 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
4469 thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
4472 lemma compact_differences:
4473 fixes s t :: "'a::real_normed_vector set"
4474 assumes "compact s" "compact t" shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
4476 have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
4477 apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4478 thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
4481 lemma compact_translation:
4482 fixes s :: "'a::real_normed_vector set"
4483 assumes "compact s" shows "compact ((\<lambda>x. a + x) ` s)"
4485 have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
4486 thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
4489 lemma compact_affinity:
4490 fixes s :: "'a::real_normed_vector set"
4491 assumes "compact s" shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
4493 have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
4494 thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
4497 text{* Hence we get the following. *}
4499 lemma compact_sup_maxdistance:
4500 fixes s :: "'a::real_normed_vector set"
4501 assumes "compact s" "s \<noteq> {}"
4502 shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
4504 have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
4505 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"
4506 using compact_differences[OF assms(1) assms(1)]
4507 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
4508 from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
4509 thus ?thesis using x(2)[unfolded `x = a - b`] by blast
4512 text{* We can state this in terms of diameter of a set. *}
4514 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"
4515 (* TODO: generalize to class metric_space *)
4517 lemma diameter_bounded:
4519 shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4520 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"
4522 let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"
4523 obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto
4524 { fix x y assume "x \<in> s" "y \<in> s"
4525 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) }
4527 { fix x y assume "x\<in>s" "y\<in>s" hence "s \<noteq> {}" by auto
4528 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`
4529 by simp (blast intro!: Sup_upper *) }
4531 { fix d::real assume "d>0" "d < diameter s"
4532 hence "s\<noteq>{}" unfolding diameter_def by auto
4533 have "\<exists>d' \<in> ?D. d' > d"
4535 assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"
4536 hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE)
4537 thus False using `d < diameter s` `s\<noteq>{}`
4538 apply (auto simp add: diameter_def)
4539 apply (drule Sup_real_iff [THEN [2] rev_iffD2])
4543 hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto }
4544 ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4545 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto
4548 lemma diameter_bounded_bound:
4549 "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"
4550 using diameter_bounded by blast
4552 lemma diameter_compact_attained:
4553 fixes s :: "'a::real_normed_vector set"
4554 assumes "compact s" "s \<noteq> {}"
4555 shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
4557 have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
4558 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
4559 hence "diameter s \<le> norm (x - y)"
4560 unfolding diameter_def by clarsimp (rule Sup_least, fast+)
4562 by (metis b diameter_bounded_bound order_antisym xys)
4565 text{* Related results with closure as the conclusion. *}
4567 lemma closed_scaling:
4568 fixes s :: "'a::real_normed_vector set"
4569 assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
4571 case True thus ?thesis by auto
4576 have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
4577 case True thus ?thesis apply auto unfolding * by auto
4580 { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s" "(x ---> l) sequentially"
4581 { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
4582 using as(1)[THEN spec[where x=n]]
4583 using `c\<noteq>0` by auto
4586 { fix e::real assume "e>0"
4587 hence "0 < e *\<bar>c\<bar>" using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
4588 then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
4589 using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto
4590 hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
4591 unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
4592 using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto }
4593 hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto
4594 ultimately have "l \<in> scaleR c ` s"
4595 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"]]
4596 unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto }
4597 thus ?thesis unfolding closed_sequential_limits by fast
4601 lemma closed_negations:
4602 fixes s :: "'a::real_normed_vector set"
4603 assumes "closed s" shows "closed ((\<lambda>x. -x) ` s)"
4604 using closed_scaling[OF assms, of "- 1"] by simp
4606 lemma compact_closed_sums:
4607 fixes s :: "'a::real_normed_vector set"
4608 assumes "compact s" "closed t" shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4610 let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
4611 { fix x l assume as:"\<forall>n. x n \<in> ?S" "(x ---> l) sequentially"
4612 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"
4613 using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
4614 obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
4615 using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
4616 have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
4617 using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
4618 hence "l - l' \<in> t"
4619 using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
4621 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
4623 thus ?thesis unfolding closed_sequential_limits by fast
4626 lemma closed_compact_sums:
4627 fixes s t :: "'a::real_normed_vector set"
4628 assumes "closed s" "compact t"
4629 shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4631 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
4632 apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
4633 thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
4636 lemma compact_closed_differences:
4637 fixes s t :: "'a::real_normed_vector set"
4638 assumes "compact s" "closed t"
4639 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4641 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4642 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4643 thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
4646 lemma closed_compact_differences:
4647 fixes s t :: "'a::real_normed_vector set"
4648 assumes "closed s" "compact t"
4649 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4651 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4652 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4653 thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
4656 lemma closed_translation:
4657 fixes a :: "'a::real_normed_vector"
4658 assumes "closed s" shows "closed ((\<lambda>x. a + x) ` s)"
4660 have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
4661 thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
4664 lemma translation_Compl:
4665 fixes a :: "'a::ab_group_add"
4666 shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
4667 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto
4669 lemma translation_UNIV:
4670 fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
4671 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
4673 lemma translation_diff:
4674 fixes a :: "'a::ab_group_add"
4675 shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
4678 lemma closure_translation:
4679 fixes a :: "'a::real_normed_vector"
4680 shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
4682 have *:"op + a ` (- s) = - op + a ` s"
4683 apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
4684 show ?thesis unfolding closure_interior translation_Compl
4685 using interior_translation[of a "- s"] unfolding * by auto
4688 lemma frontier_translation:
4689 fixes a :: "'a::real_normed_vector"
4690 shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
4691 unfolding frontier_def translation_diff interior_translation closure_translation by auto
4693 subsection{* Separation between points and sets. *}
4695 lemma separate_point_closed:
4696 fixes s :: "'a::heine_borel set"
4697 shows "closed s \<Longrightarrow> a \<notin> s ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
4698 proof(cases "s = {}")
4700 thus ?thesis by(auto intro!: exI[where x=1])
4703 assume "closed s" "a \<notin> s"
4704 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
4705 with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
4708 lemma separate_compact_closed:
4709 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4710 (* TODO: does this generalize to heine_borel? *)
4711 assumes "compact s" and "closed t" and "s \<inter> t = {}"
4712 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4714 have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto
4715 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"
4716 using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto
4717 { fix x y assume "x\<in>s" "y\<in>t"
4718 hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto
4719 hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute
4720 by (auto simp add: dist_commute)
4721 hence "d \<le> dist x y" unfolding dist_norm by auto }
4722 thus ?thesis using `d>0` by auto
4725 lemma separate_closed_compact:
4726 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4727 assumes "closed s" and "compact t" and "s \<inter> t = {}"
4728 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4730 have *:"t \<inter> s = {}" using assms(3) by auto
4731 show ?thesis using separate_compact_closed[OF assms(2,1) *]
4732 apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
4733 by (auto simp add: dist_commute)
4736 subsection {* Intervals *}
4738 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows
4739 "{a <..< b} = {x::'a. \<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i}" and
4740 "{a .. b} = {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i}"
4741 by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
4743 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows
4744 "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i)"
4745 "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i)"
4746 using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
4748 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows
4749 "({a <..< b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i))" (is ?th1) and
4750 "({a .. b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i < a$$i))" (is ?th2)
4752 { fix i x assume i:"i<DIM('a)" and as:"b$$i \<le> a$$i" and x:"x\<in>{a <..< b}"
4753 hence "a $$ i < x $$ i \<and> x $$ i < b $$ i" unfolding mem_interval by auto
4754 hence "a$$i < b$$i" by auto
4755 hence False using as by auto }
4757 { assume as:"\<forall>i<DIM('a). \<not> (b$$i \<le> a$$i)"
4758 let ?x = "(1/2) *\<^sub>R (a + b)"
4759 { fix i assume i:"i<DIM('a)"
4760 have "a$$i < b$$i" using as[THEN spec[where x=i]] using i by auto
4761 hence "a$$i < ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i < b$$i"
4762 unfolding euclidean_simps by auto }
4763 hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto }
4764 ultimately show ?th1 by blast
4766 { fix i x assume i:"i<DIM('a)" and as:"b$$i < a$$i" and x:"x\<in>{a .. b}"
4767 hence "a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i" unfolding mem_interval by auto
4768 hence "a$$i \<le> b$$i" by auto
4769 hence False using as by auto }
4771 { assume as:"\<forall>i<DIM('a). \<not> (b$$i < a$$i)"
4772 let ?x = "(1/2) *\<^sub>R (a + b)"
4773 { fix i assume i:"i<DIM('a)"
4774 have "a$$i \<le> b$$i" using as[THEN spec[where x=i]] by auto
4775 hence "a$$i \<le> ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i \<le> b$$i"
4776 unfolding euclidean_simps by auto }
4777 hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto }
4778 ultimately show ?th2 by blast
4781 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows
4782 "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> b$$i)" and
4783 "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"
4784 unfolding interval_eq_empty[of a b] by fastsimp+
4786 lemma interval_sing: fixes a :: "'a::ordered_euclidean_space" shows
4787 "{a .. a} = {a}" "{a<..<a} = {}"
4788 apply(auto simp add: set_eq_iff euclidean_eq[where 'a='a] eucl_less[where 'a='a] eucl_le[where 'a='a])
4789 apply (simp add: order_eq_iff) apply(rule_tac x=0 in exI) by (auto simp add: not_less less_imp_le)
4791 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows
4792 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
4793 "(\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
4794 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
4795 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
4796 unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
4797 by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
4799 lemma interval_open_subset_closed: fixes a :: "'a::ordered_euclidean_space" shows
4800 "{a<..<b} \<subseteq> {a .. b}"
4801 proof(simp add: subset_eq, rule)
4803 assume x:"x \<in>{a<..<b}"
4804 { fix i assume "i<DIM('a)"
4805 hence "a $$ i \<le> x $$ i"
4806 using x order_less_imp_le[of "a$$i" "x$$i"]
4807 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4810 { fix i assume "i<DIM('a)"
4811 hence "x $$ i \<le> b $$ i"
4812 using x order_less_imp_le[of "x$$i" "b$$i"]
4813 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4816 show "a \<le> x \<and> x \<le> b"
4817 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4820 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows
4821 "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th1) and
4822 "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i)" (is ?th2) and
4823 "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th3) and
4824 "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th4)
4826 show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
4827 show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
4828 { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i<DIM('a). c$$i < d$$i"
4829 hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
4830 fix i assume i:"i<DIM('a)"
4831 (** TODO combine the following two parts as done in the HOL_light version. **)
4832 { let ?x = "(\<chi>\<chi> j. (if j=i then ((min (a$$j) (d$$j))+c$$j)/2 else (c$$j+d$$j)/2))::'a"
4833 assume as2: "a$$i > c$$i"
4834 { fix j assume j:"j<DIM('a)"
4835 hence "c $$ j < ?x $$ j \<and> ?x $$ j < d $$ j"
4836 apply(cases "j=i") using as(2)[THEN spec[where x=j]] i
4837 by (auto simp add: as2) }
4838 hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto
4840 have "?x\<notin>{a .. b}"
4841 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4842 using as(2)[THEN spec[where x=i]] and as2 i
4844 ultimately have False using as by auto }
4845 hence "a$$i \<le> c$$i" by(rule ccontr)auto
4847 { let ?x = "(\<chi>\<chi> j. (if j=i then ((max (b$$j) (c$$j))+d$$j)/2 else (c$$j+d$$j)/2))::'a"
4848 assume as2: "b$$i < d$$i"
4849 { fix j assume "j<DIM('a)"
4850 hence "d $$ j > ?x $$ j \<and> ?x $$ j > c $$ j"
4851 apply(cases "j=i") using as(2)[THEN spec[where x=j]]
4852 by (auto simp add: as2) }
4853 hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
4855 have "?x\<notin>{a .. b}"
4856 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4857 using as(2)[THEN spec[where x=i]] and as2 using i
4859 ultimately have False using as by auto }
4860 hence "b$$i \<ge> d$$i" by(rule ccontr)auto
4862 have "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" by auto
4864 show ?th3 unfolding subset_eq and Ball_def and mem_interval
4865 apply(rule,rule,rule,rule) apply(rule part1) unfolding subset_eq and Ball_def and mem_interval
4866 prefer 4 apply auto by(erule_tac x=i in allE,erule_tac x=i in allE,fastsimp)+
4867 { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i<DIM('a). c$$i < d$$i"
4868 fix i assume i:"i<DIM('a)"
4869 from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
4870 hence "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" using part1 and as(2) using i by auto } note * = this
4871 show ?th4 unfolding subset_eq and Ball_def and mem_interval
4872 apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4
4873 apply auto by(erule_tac x=i in allE, simp)+
4876 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows
4877 "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i < c$$i \<or> b$$i < c$$i \<or> d$$i < a$$i))" (is ?th1) and
4878 "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th2) and
4879 "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i < c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th3) and
4880 "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th4)
4882 let ?z = "(\<chi>\<chi> i. ((max (a$$i) (c$$i)) + (min (b$$i) (d$$i))) / 2)::'a"
4883 note * = set_eq_iff Int_iff empty_iff mem_interval all_conj_distrib[THEN sym] eq_False
4884 show ?th1 unfolding * apply safe apply(erule_tac x="?z" in allE)
4885 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4886 show ?th2 unfolding * apply safe apply(erule_tac x="?z" in allE)
4887 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4888 show ?th3 unfolding * apply safe apply(erule_tac x="?z" in allE)
4889 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4890 show ?th4 unfolding * apply safe apply(erule_tac x="?z" in allE)
4891 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4894 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows
4895 "{a .. b} \<inter> {c .. d} = {(\<chi>\<chi> i. max (a$$i) (c$$i)) .. (\<chi>\<chi> i. min (b$$i) (d$$i))}"
4896 unfolding set_eq_iff and Int_iff and mem_interval
4899 (* Moved interval_open_subset_closed a bit upwards *)
4901 lemma open_interval_lemma: fixes x :: "real" shows
4902 "a < x \<Longrightarrow> x < b ==> (\<exists>d>0. \<forall>x'. abs(x' - x) < d --> a < x' \<and> x' < b)"
4903 by(rule_tac x="min (x - a) (b - x)" in exI, auto)
4905 lemma open_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
4907 { fix x assume x:"x\<in>{a<..<b}"
4908 { fix i assume "i<DIM('a)"
4909 hence "\<exists>d>0. \<forall>x'. abs (x' - (x$$i)) < d \<longrightarrow> a$$i < x' \<and> x' < b$$i"
4910 using x[unfolded mem_interval, THEN spec[where x=i]]
4911 using open_interval_lemma[of "a$$i" "x$$i" "b$$i"] by auto }
4912 hence "\<forall>i\<in>{..<DIM('a)}. \<exists>d>0. \<forall>x'. abs (x' - (x$$i)) < d \<longrightarrow> a$$i < x' \<and> x' < b$$i" by auto
4913 from bchoice[OF this] guess d .. note d=this
4914 let ?d = "Min (d ` {..<DIM('a)})"
4915 have **:"finite (d ` {..<DIM('a)})" "d ` {..<DIM('a)} \<noteq> {}" by auto
4916 have "?d>0" using Min_gr_iff[OF **] using d by auto
4918 { fix x' assume as:"dist x' x < ?d"
4919 { fix i assume i:"i<DIM('a)"
4920 hence "\<bar>x'$$i - x $$ i\<bar> < d i"
4921 using norm_bound_component_lt[OF as[unfolded dist_norm], of i]
4922 unfolding euclidean_simps Min_gr_iff[OF **] by auto
4923 hence "a $$ i < x' $$ i" "x' $$ i < b $$ i" using i and d[THEN bspec[where x=i]] by auto }
4924 hence "a < x' \<and> x' < b" apply(subst(2) eucl_less,subst(1) eucl_less) by auto }
4925 ultimately have "\<exists>e>0. \<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a<..<b}" by auto
4927 thus ?thesis unfolding open_dist using open_interval_lemma by auto
4930 lemma closed_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
4932 { fix x i assume i:"i<DIM('a)"
4933 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"*)
4934 { assume xa:"a$$i > x$$i"
4935 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
4936 hence False unfolding mem_interval and dist_norm
4937 using component_le_norm[of "y-x" i, unfolded euclidean_simps] and xa using i
4938 by(auto elim!: allE[where x=i])
4939 } hence "a$$i \<le> x$$i" by(rule ccontr)auto
4941 { assume xb:"b$$i < x$$i"
4942 with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < x$$i - b$$i"
4943 by(erule_tac x="x$$i - b$$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 xb using i
4946 by(auto elim!: allE[where x=i])
4947 } hence "x$$i \<le> b$$i" by(rule ccontr)auto
4949 have "a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i" by auto }
4950 thus ?thesis unfolding closed_limpt islimpt_approachable mem_interval by auto
4953 lemma interior_closed_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows
4954 "interior {a .. b} = {a<..<b}" (is "?L = ?R")
4955 proof(rule subset_antisym)
4956 show "?R \<subseteq> ?L" using interior_maximal[OF interval_open_subset_closed open_interval] by auto
4958 { fix x assume "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> {a..b}"
4959 then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" by auto
4960 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
4961 { fix i assume i:"i<DIM('a)"
4962 have "dist (x - (e / 2) *\<^sub>R basis i) x < e"
4963 "dist (x + (e / 2) *\<^sub>R basis i) x < e"
4964 unfolding dist_norm apply auto
4965 unfolding norm_minus_cancel using norm_basis and `e>0` by auto
4966 hence "a $$ i \<le> (x - (e / 2) *\<^sub>R basis i) $$ i"
4967 "(x + (e / 2) *\<^sub>R basis i) $$ i \<le> b $$ i"
4968 using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]
4969 and e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]
4970 unfolding mem_interval by (auto elim!: allE[where x=i])
4971 hence "a $$ i < x $$ i" and "x $$ i < b $$ i" unfolding euclidean_simps
4972 unfolding basis_component using `e>0` i by auto }
4973 hence "x \<in> {a<..<b}" unfolding mem_interval by auto }
4974 thus "?L \<subseteq> ?R" unfolding interior_def and subset_eq by auto
4977 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"
4979 let ?b = "\<Sum>i<DIM('a). \<bar>a$$i\<bar> + \<bar>b$$i\<bar>"
4980 { fix x::"'a" assume x:"\<forall>i<DIM('a). a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i"
4981 { fix i assume "i<DIM('a)"
4982 hence "\<bar>x$$i\<bar> \<le> \<bar>a$$i\<bar> + \<bar>b$$i\<bar>" using x[THEN spec[where x=i]] by auto }
4983 hence "(\<Sum>i<DIM('a). \<bar>x $$ i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto
4984 hence "norm x \<le> ?b" using norm_le_l1[of x] by auto }
4985 thus ?thesis unfolding interval and bounded_iff by auto
4988 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows
4989 "bounded {a .. b} \<and> bounded {a<..<b}"
4990 using bounded_closed_interval[of a b]
4991 using interval_open_subset_closed[of a b]
4992 using bounded_subset[of "{a..b}" "{a<..<b}"]
4995 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows
4996 "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
4997 using bounded_interval[of a b] by auto
4999 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"
5000 using bounded_closed_imp_compact[of "{a..b}"] using bounded_interval[of a b]
5003 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"
5004 assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
5006 { fix i assume "i<DIM('a)"
5007 hence "a $$ i < ((1 / 2) *\<^sub>R (a + b)) $$ i \<and> ((1 / 2) *\<^sub>R (a + b)) $$ i < b $$ i"
5008 using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
5009 unfolding euclidean_simps by auto }
5010 thus ?thesis unfolding mem_interval by auto
5013 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"
5014 assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
5015 shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
5017 { fix i assume i:"i<DIM('a)"
5018 have "a $$ i = e * a$$i + (1 - e) * a$$i" unfolding left_diff_distrib by simp
5019 also have "\<dots> < e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
5020 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
5021 using x unfolding mem_interval using i apply simp
5022 using y unfolding mem_interval using i apply simp
5024 finally have "a $$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i" unfolding euclidean_simps by auto
5026 have "b $$ i = e * b$$i + (1 - e) * b$$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 "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i < b $$ i" unfolding euclidean_simps by auto
5033 } 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 }
5034 thus ?thesis unfolding mem_interval by auto
5037 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"
5038 assumes "{a<..<b} \<noteq> {}"
5039 shows "closure {a<..<b} = {a .. b}"
5041 have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto
5042 let ?c = "(1 / 2) *\<^sub>R (a + b)"
5043 { fix x assume as:"x \<in> {a .. b}"
5044 def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
5045 { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
5046 have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
5047 have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
5048 x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
5049 by (auto simp add: algebra_simps)
5050 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
5051 hence False using fn unfolding f_def using xc by auto }
5053 { assume "\<not> (f ---> x) sequentially"
5054 { fix e::real assume "e>0"
5055 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
5056 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
5057 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)
5058 hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto }
5059 hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
5060 unfolding Lim_sequentially by(auto simp add: dist_norm)
5061 hence "(f ---> x) sequentially" unfolding f_def
5062 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]
5063 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 }
5064 ultimately have "x \<in> closure {a<..<b}"
5065 using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto }
5066 thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
5069 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"
5070 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a<..<a}"
5072 obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
5073 def a \<equiv> "(\<chi>\<chi> i. b+1)::'a"
5074 { fix x assume "x\<in>s"
5075 fix i assume i:"i<DIM('a)"
5076 hence "(-a)$$i < x$$i" and "x$$i < a$$i" using b[THEN bspec[where x=x], OF `x\<in>s`]
5077 and component_le_norm[of x i] unfolding euclidean_simps and a_def by auto }
5078 thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
5081 lemma bounded_subset_open_interval:
5082 fixes s :: "('a::ordered_euclidean_space) set"
5083 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
5084 by (auto dest!: bounded_subset_open_interval_symmetric)
5086 lemma bounded_subset_closed_interval_symmetric:
5087 fixes s :: "('a::ordered_euclidean_space) set"
5088 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
5090 obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
5091 thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
5094 lemma bounded_subset_closed_interval:
5095 fixes s :: "('a::ordered_euclidean_space) set"
5096 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
5097 using bounded_subset_closed_interval_symmetric[of s] by auto
5099 lemma frontier_closed_interval:
5100 fixes a b :: "'a::ordered_euclidean_space"
5101 shows "frontier {a .. b} = {a .. b} - {a<..<b}"
5102 unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
5104 lemma frontier_open_interval:
5105 fixes a b :: "'a::ordered_euclidean_space"
5106 shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
5107 proof(cases "{a<..<b} = {}")
5108 case True thus ?thesis using frontier_empty by auto
5110 case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
5113 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"
5114 assumes "{c<..<d} \<noteq> {}" shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
5115 unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
5118 (* Some stuff for half-infinite intervals too; FIXME: notation? *)
5120 lemma closed_interval_left: fixes b::"'a::euclidean_space"
5121 shows "closed {x::'a. \<forall>i<DIM('a). x$$i \<le> b$$i}"
5123 { fix i assume i:"i<DIM('a)"
5124 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"
5125 { assume "x$$i > b$$i"
5126 then obtain y where "y $$ i \<le> b $$ i" "y \<noteq> x" "dist y x < x$$i - b$$i"
5127 using x[THEN spec[where x="x$$i - b$$i"]] using i by auto
5128 hence False using component_le_norm[of "y - x" i] unfolding dist_norm euclidean_simps using i
5130 hence "x$$i \<le> b$$i" by(rule ccontr)auto }
5131 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
5134 lemma closed_interval_right: fixes a::"'a::euclidean_space"
5135 shows "closed {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i}"
5137 { fix i assume i:"i<DIM('a)"
5138 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"
5139 { assume "a$$i > x$$i"
5140 then obtain y where "a $$ i \<le> y $$ i" "y \<noteq> x" "dist y x < a$$i - x$$i"
5141 using x[THEN spec[where x="a$$i - x$$i"]] i by auto
5142 hence False using component_le_norm[of "y - x" i] unfolding dist_norm and euclidean_simps by auto }
5143 hence "a$$i \<le> x$$i" by(rule ccontr)auto }
5144 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
5147 text {* Intervals in general, including infinite and mixtures of open and closed. *}
5149 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
5150 (\<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)"
5152 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)
5153 "is_interval {a<..<b}" (is ?th2) proof -
5154 have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
5155 show ?th1 ?th2 unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
5156 by(meson order_trans le_less_trans less_le_trans *)+ qed
5158 lemma is_interval_empty:
5160 unfolding is_interval_def
5163 lemma is_interval_univ:
5165 unfolding is_interval_def
5168 subsection{* Closure of halfspaces and hyperplanes. *}
5171 assumes "(f ---> l) net" shows "((\<lambda>y. inner a (f y)) ---> inner a l) net"
5172 by (intro tendsto_intros assms)
5174 lemma continuous_at_inner: "continuous (at x) (inner a)"
5175 unfolding continuous_at by (intro tendsto_intros)
5177 lemma continuous_at_euclidean_component[intro!, simp]: "continuous (at x) (\<lambda>x. x $$ i)"
5178 unfolding euclidean_component_def by (rule continuous_at_inner)
5180 lemma continuous_on_inner:
5181 fixes s :: "'a::real_inner set"
5182 shows "continuous_on s (inner a)"
5183 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
5185 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
5187 have "\<forall>x. continuous (at x) (inner a)"
5188 unfolding continuous_at by (rule allI) (intro tendsto_intros)
5189 hence "closed (inner a -` {..b})"
5190 using closed_real_atMost by (rule continuous_closed_vimage)
5191 moreover have "{x. inner a x \<le> b} = inner a -` {..b}" by auto
5192 ultimately show ?thesis by simp
5195 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
5196 using closed_halfspace_le[of "-a" "-b"] unfolding inner_minus_left by auto
5198 lemma closed_hyperplane: "closed {x. inner a x = b}"
5200 have "{x. inner a x = b} = {x. inner a x \<ge> b} \<inter> {x. inner a x \<le> b}" by auto
5201 thus ?thesis using closed_halfspace_le[of a b] and closed_halfspace_ge[of b a] using closed_Int by auto
5204 lemma closed_halfspace_component_le:
5205 shows "closed {x::'a::euclidean_space. x$$i \<le> a}"
5206 using closed_halfspace_le[of "(basis i)::'a" a] unfolding euclidean_component_def .
5208 lemma closed_halfspace_component_ge:
5209 shows "closed {x::'a::euclidean_space. x$$i \<ge> a}"
5210 using closed_halfspace_ge[of a "(basis i)::'a"] unfolding euclidean_component_def .
5212 text{* Openness of halfspaces. *}
5214 lemma open_halfspace_lt: "open {x. inner a x < b}"
5216 have "- {x. b \<le> inner a x} = {x. inner a x < b}" by auto
5217 thus ?thesis using closed_halfspace_ge[unfolded closed_def, of b a] by auto
5220 lemma open_halfspace_gt: "open {x. inner a x > b}"
5222 have "- {x. b \<ge> inner a x} = {x. inner a x > b}" by auto
5223 thus ?thesis using closed_halfspace_le[unfolded closed_def, of a b] by auto
5226 lemma open_halfspace_component_lt:
5227 shows "open {x::'a::euclidean_space. x$$i < a}"
5228 using open_halfspace_lt[of "(basis i)::'a" a] unfolding euclidean_component_def .
5230 lemma open_halfspace_component_gt:
5231 shows "open {x::'a::euclidean_space. x$$i > a}"
5232 using open_halfspace_gt[of a "(basis i)::'a"] unfolding euclidean_component_def .
5234 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}
5236 lemma eucl_lessThan_eq_halfspaces:
5237 fixes a :: "'a\<Colon>ordered_euclidean_space"
5238 shows "{..<a} = (\<Inter>i<DIM('a). {x. x $$ i < a $$ i})"
5239 by (auto simp: eucl_less[where 'a='a])
5241 lemma eucl_greaterThan_eq_halfspaces:
5242 fixes a :: "'a\<Colon>ordered_euclidean_space"
5243 shows "{a<..} = (\<Inter>i<DIM('a). {x. a $$ i < x $$ i})"
5244 by (auto simp: eucl_less[where 'a='a])
5246 lemma eucl_atMost_eq_halfspaces:
5247 fixes a :: "'a\<Colon>ordered_euclidean_space"
5248 shows "{.. a} = (\<Inter>i<DIM('a). {x. x $$ i \<le> a $$ i})"
5249 by (auto simp: eucl_le[where 'a='a])
5251 lemma eucl_atLeast_eq_halfspaces:
5252 fixes a :: "'a\<Colon>ordered_euclidean_space"
5253 shows "{a ..} = (\<Inter>i<DIM('a). {x. a $$ i \<le> x $$ i})"
5254 by (auto simp: eucl_le[where 'a='a])
5256 lemma open_eucl_lessThan[simp, intro]:
5257 fixes a :: "'a\<Colon>ordered_euclidean_space"
5258 shows "open {..< a}"
5259 by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)
5261 lemma open_eucl_greaterThan[simp, intro]:
5262 fixes a :: "'a\<Colon>ordered_euclidean_space"
5263 shows "open {a <..}"
5264 by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)
5266 lemma closed_eucl_atMost[simp, intro]:
5267 fixes a :: "'a\<Colon>ordered_euclidean_space"
5268 shows "closed {.. a}"
5269 unfolding eucl_atMost_eq_halfspaces
5270 proof (safe intro!: closed_INT)
5272 have "- {x::'a. x $$ i \<le> a $$ i} = {x. a $$ i < x $$ i}" by auto
5273 then show "closed {x::'a. x $$ i \<le> a $$ i}"
5274 by (simp add: closed_def open_halfspace_component_gt)
5277 lemma closed_eucl_atLeast[simp, intro]:
5278 fixes a :: "'a\<Colon>ordered_euclidean_space"
5279 shows "closed {a ..}"
5280 unfolding eucl_atLeast_eq_halfspaces
5281 proof (safe intro!: closed_INT)
5283 have "- {x::'a. a $$ i \<le> x $$ i} = {x. x $$ i < a $$ i}" by auto
5284 then show "closed {x::'a. a $$ i \<le> x $$ i}"
5285 by (simp add: closed_def open_halfspace_component_lt)
5288 lemma open_vimage_euclidean_component: "open S \<Longrightarrow> open ((\<lambda>x. x $$ i) -` S)"
5289 by (auto intro!: continuous_open_vimage)
5291 text{* This gives a simple derivation of limit component bounds. *}
5293 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5294 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f(x)$$i \<le> b) net"
5295 shows "l$$i \<le> b"
5297 { fix x have "x \<in> {x::'b. inner (basis i) x \<le> b} \<longleftrightarrow> x$$i \<le> b"
5298 unfolding euclidean_component_def by auto } note * = this
5299 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *
5300 using closed_halfspace_le[of "(basis i)::'b" b] and assms(1,2,3) by auto
5303 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5304 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$$i) net"
5305 shows "b \<le> l$$i"
5307 { fix x have "x \<in> {x::'b. inner (basis i) x \<ge> b} \<longleftrightarrow> x$$i \<ge> b"
5308 unfolding euclidean_component_def by auto } note * = this
5309 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *
5310 using closed_halfspace_ge[of b "(basis i)"] and assms(1,2,3) by auto
5313 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5314 assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$$i = b) net"
5316 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
5317 text{* Limits relative to a union. *}
5319 lemma eventually_within_Un:
5320 "eventually P (net within (s \<union> t)) \<longleftrightarrow>
5321 eventually P (net within s) \<and> eventually P (net within t)"
5322 unfolding Limits.eventually_within
5323 by (auto elim!: eventually_rev_mp)
5325 lemma Lim_within_union:
5326 "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>
5327 (f ---> l) (net within s) \<and> (f ---> l) (net within t)"
5328 unfolding tendsto_def
5329 by (auto simp add: eventually_within_Un)
5331 lemma Lim_topological:
5332 "(f ---> l) net \<longleftrightarrow>
5333 trivial_limit net \<or>
5334 (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
5335 unfolding tendsto_def trivial_limit_eq by auto
5337 lemma continuous_on_union:
5338 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
5339 shows "continuous_on (s \<union> t) f"
5340 using assms unfolding continuous_on Lim_within_union
5341 unfolding Lim_topological trivial_limit_within closed_limpt by auto
5343 lemma continuous_on_cases:
5344 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
5345 "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
5346 shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
5348 let ?h = "(\<lambda>x. if P x then f x else g x)"
5349 have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto
5350 hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto
5352 have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto
5353 hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto
5354 ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto
5358 text{* Some more convenient intermediate-value theorem formulations. *}
5360 lemma connected_ivt_hyperplane:
5361 assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
5362 shows "\<exists>z \<in> s. inner a z = b"
5364 assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
5365 let ?A = "{x. inner a x < b}"
5366 let ?B = "{x. inner a x > b}"
5367 have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
5368 moreover have "?A \<inter> ?B = {}" by auto
5369 moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
5370 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
5373 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows
5374 "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)"
5375 using connected_ivt_hyperplane[of s x y "(basis k)::'a" a]
5376 unfolding euclidean_component_def by auto
5378 subsection {* Homeomorphisms *}
5380 definition "homeomorphism s t f g \<equiv>
5381 (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
5382 (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
5385 homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"
5386 (infixr "homeomorphic" 60) where
5387 homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
5389 lemma homeomorphic_refl: "s homeomorphic s"
5390 unfolding homeomorphic_def
5391 unfolding homeomorphism_def
5392 using continuous_on_id
5393 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5394 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5397 lemma homeomorphic_sym:
5398 "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
5399 unfolding homeomorphic_def
5400 unfolding homeomorphism_def
5403 lemma homeomorphic_trans:
5404 assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
5406 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"
5407 using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
5408 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"
5409 using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
5411 { 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 }
5412 moreover have "(f2 \<circ> f1) ` s = u" using fg1(2) fg2(2) by auto
5413 moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
5414 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 }
5415 moreover have "(g1 \<circ> g2) ` u = s" using fg1(5) fg2(5) by auto
5416 moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6) unfolding fg2(5) by auto
5417 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
5420 lemma homeomorphic_minimal:
5421 "s homeomorphic t \<longleftrightarrow>
5422 (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
5423 (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
5424 continuous_on s f \<and> continuous_on t g)"
5425 unfolding homeomorphic_def homeomorphism_def
5426 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
5427 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
5429 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
5430 apply auto apply(rule_tac x="g x" in bexI) apply auto
5431 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
5432 apply auto apply(rule_tac x="f x" in bexI) by auto
5434 text {* Relatively weak hypotheses if a set is compact. *}
5436 lemma homeomorphism_compact:
5437 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5438 (* class constraint due to continuous_on_inverse *)
5439 assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s"
5440 shows "\<exists>g. homeomorphism s t f g"
5442 def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
5443 have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
5444 { fix y assume "y\<in>t"
5445 then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
5446 hence "g (f x) = x" using g by auto
5447 hence "f (g y) = y" unfolding x(1)[THEN sym] by auto }
5448 hence g':"\<forall>x\<in>t. f (g x) = x" by auto
5451 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"])
5453 { assume "x\<in>g ` t"
5454 then obtain y where y:"y\<in>t" "g y = x" by auto
5455 then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
5456 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 }
5457 ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" .. }
5458 hence "g ` t = s" by auto
5460 show ?thesis unfolding homeomorphism_def homeomorphic_def
5461 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
5464 lemma homeomorphic_compact:
5465 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5466 (* class constraint due to continuous_on_inverse *)
5467 shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
5468 \<Longrightarrow> s homeomorphic t"
5469 unfolding homeomorphic_def by (metis homeomorphism_compact)
5471 text{* Preservation of topological properties. *}
5473 lemma homeomorphic_compactness:
5474 "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
5475 unfolding homeomorphic_def homeomorphism_def
5476 by (metis compact_continuous_image)
5478 text{* Results on translation, scaling etc. *}
5480 lemma homeomorphic_scaling:
5481 fixes s :: "'a::real_normed_vector set"
5482 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
5483 unfolding homeomorphic_minimal
5484 apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
5485 apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
5486 using assms apply auto
5487 using continuous_on_cmul[OF continuous_on_id] by auto
5489 lemma homeomorphic_translation:
5490 fixes s :: "'a::real_normed_vector set"
5491 shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
5492 unfolding homeomorphic_minimal
5493 apply(rule_tac x="\<lambda>x. a + x" in exI)
5494 apply(rule_tac x="\<lambda>x. -a + x" in exI)
5495 using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
5497 lemma homeomorphic_affinity:
5498 fixes s :: "'a::real_normed_vector set"
5499 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
5501 have *:"op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
5503 using homeomorphic_trans
5504 using homeomorphic_scaling[OF assms, of s]
5505 using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a] unfolding * by auto
5508 lemma homeomorphic_balls:
5509 fixes a b ::"'a::real_normed_vector" (* FIXME: generalize to metric_space *)
5510 assumes "0 < d" "0 < e"
5511 shows "(ball a d) homeomorphic (ball b e)" (is ?th)
5512 "(cball a d) homeomorphic (cball b e)" (is ?cth)
5514 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5515 show ?th unfolding homeomorphic_minimal
5516 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5517 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5518 using assms apply (auto simp add: dist_commute)
5520 apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)
5521 unfolding continuous_on
5522 by (intro ballI tendsto_intros, simp)+
5524 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5525 show ?cth unfolding homeomorphic_minimal
5526 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5527 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5528 using assms apply (auto simp add: dist_commute)
5530 apply (auto simp add: pos_divide_le_eq)
5531 unfolding continuous_on
5532 by (intro ballI tendsto_intros, simp)+
5535 text{* "Isometry" (up to constant bounds) of injective linear map etc. *}
5537 lemma cauchy_isometric:
5538 fixes x :: "nat \<Rightarrow> 'a::euclidean_space"
5539 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)"
5542 interpret f: bounded_linear f by fact
5543 { fix d::real assume "d>0"
5544 then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
5545 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
5546 { fix n assume "n\<ge>N"
5547 hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding f.diff[THEN sym] by auto
5548 moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
5549 using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
5550 using normf[THEN bspec[where x="x n - x N"]] by auto
5551 ultimately have "norm (x n - x N) < d" using `e>0`
5552 using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto }
5553 hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
5554 thus ?thesis unfolding cauchy and dist_norm by auto
5557 lemma complete_isometric_image:
5558 fixes f :: "'a::euclidean_space => 'b::euclidean_space"
5559 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"
5560 shows "complete(f ` s)"
5562 { fix g assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
5563 then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
5564 using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
5565 hence x:"\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
5566 hence "f \<circ> x = g" unfolding fun_eq_iff by auto
5567 then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
5568 using cs[unfolded complete_def, THEN spec[where x="x"]]
5569 using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
5570 hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
5571 using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
5572 unfolding `f \<circ> x = g` by auto }
5573 thus ?thesis unfolding complete_def by auto
5577 fixes x :: "'a::real_normed_vector"
5578 shows "dist 0 x = norm x"
5579 unfolding dist_norm by simp
5581 lemma injective_imp_isometric: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
5582 assumes s:"closed s" "subspace s" and f:"bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
5583 shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
5584 proof(cases "s \<subseteq> {0::'a}")
5586 { fix x assume "x \<in> s"
5587 hence "x = 0" using True by auto
5588 hence "norm x \<le> norm (f x)" by auto }
5589 thus ?thesis by(auto intro!: exI[where x=1])
5591 interpret f: bounded_linear f by fact
5593 then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
5594 from False have "s \<noteq> {}" by auto
5595 let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
5596 let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
5597 let ?S'' = "{x::'a. norm x = norm a}"
5599 have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by auto
5600 hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
5601 moreover have "?S' = s \<inter> ?S''" by auto
5602 ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
5603 moreover have *:"f ` ?S' = ?S" by auto
5604 ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
5605 hence "closed ?S" using compact_imp_closed by auto
5606 moreover have "?S \<noteq> {}" using a by auto
5607 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
5608 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
5610 let ?e = "norm (f b) / norm b"
5611 have "norm b > 0" using ba and a and norm_ge_zero by auto
5612 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
5613 ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
5615 { fix x assume "x\<in>s"
5616 hence "norm (f b) / norm b * norm x \<le> norm (f x)"
5618 case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
5621 hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
5622 have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def] by auto
5623 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
5624 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"]]
5625 unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
5626 by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq)
5629 show ?thesis by auto
5632 lemma closed_injective_image_subspace:
5633 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
5634 assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
5635 shows "closed(f ` s)"
5637 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
5638 show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
5639 unfolding complete_eq_closed[THEN sym] by auto
5642 subsection{* Some properties of a canonical subspace. *}
5645 declare euclidean_component.zero[simp]
5647 lemma subspace_substandard:
5648 "subspace {x::'a::euclidean_space. (\<forall>i<DIM('a). P i \<longrightarrow> x$$i = 0)}"
5649 unfolding subspace_def by(auto simp add: euclidean_simps) (* FIXME: duplicate rewrite rule *)
5651 lemma closed_substandard:
5652 "closed {x::'a::euclidean_space. \<forall>i<DIM('a). P i --> x$$i = 0}" (is "closed ?A")
5654 let ?D = "{i. P i} \<inter> {..<DIM('a)}"
5655 let ?Bs = "{{x::'a. inner (basis i) x = 0}| i. i \<in> ?D}"
5658 hence x:"\<forall>i\<in>?D. x $$ i = 0" by auto
5659 hence "x\<in> \<Inter> ?Bs" by(auto simp add: x euclidean_component_def) }
5661 { assume x:"x\<in>\<Inter>?Bs"
5662 { fix i assume i:"i \<in> ?D"
5663 then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::'a. inner (basis i) x = 0}" by auto
5664 hence "x $$ i = 0" unfolding B using x unfolding euclidean_component_def by auto }
5665 hence "x\<in>?A" by auto }
5666 ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" .. }
5667 hence "?A = \<Inter> ?Bs" by auto
5668 thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
5671 lemma dim_substandard: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
5672 shows "dim {x::'a::euclidean_space. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0} = card d" (is "dim ?A = _")
5674 let ?D = "{..<DIM('a)}"
5675 let ?B = "(basis::nat => 'a) ` d"
5676 let ?bas = "basis::nat \<Rightarrow> 'a"
5677 have "?B \<subseteq> ?A" by auto
5679 { fix x::"'a" assume "x\<in>?A"
5680 hence "finite d" "x\<in>?A" using assms by(auto intro:finite_subset)
5681 hence "x\<in> span ?B"
5682 proof(induct d arbitrary: x)
5683 case empty hence "x=0" apply(subst euclidean_eq) by auto
5684 thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
5687 hence *:"\<forall>i<DIM('a). i \<notin> insert k F \<longrightarrow> x $$ i = 0" by auto
5688 have **:"F \<subseteq> insert k F" by auto
5689 def y \<equiv> "x - x$$k *\<^sub>R basis k"
5690 have y:"x = y + (x$$k) *\<^sub>R basis k" unfolding y_def by auto
5691 { fix i assume i':"i \<notin> F"
5692 hence "y $$ i = 0" unfolding y_def
5693 using *[THEN spec[where x=i]] by(auto simp add: euclidean_simps) }
5694 hence "y \<in> span (basis ` F)" using insert(3) by auto
5695 hence "y \<in> span (basis ` (insert k F))"
5696 using span_mono[of "?bas ` F" "?bas ` (insert k F)"]
5697 using image_mono[OF **, of basis] using assms by auto
5699 have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
5700 hence "x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5701 using span_mul by auto
5703 have "y + x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5704 using span_add by auto
5705 thus ?case using y by auto
5708 hence "?A \<subseteq> span ?B" by auto
5710 { fix x assume "x \<in> ?B"
5711 hence "x\<in>{(basis i)::'a |i. i \<in> ?D}" using assms by auto }
5712 hence "independent ?B" using independent_mono[OF independent_basis, of ?B] and assms by auto
5714 have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
5715 hence *:"inj_on (basis::nat\<Rightarrow>'a) d" using subset_inj_on[OF basis_inj, of "d"] by auto
5716 have "card ?B = card d" unfolding card_image[OF *] by auto
5717 ultimately show ?thesis using dim_unique[of "basis ` d" ?A] by auto
5720 text{* Hence closure and completeness of all subspaces. *}
5722 lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"
5724 apply (rule_tac x="{}" in exI, simp)
5726 apply (subgoal_tac "\<exists>x. x \<notin> A")
5728 apply (rule_tac x="insert x A" in exI, simp)
5729 apply (subgoal_tac "A \<noteq> UNIV", auto)
5732 lemma closed_subspace: fixes s::"('a::euclidean_space) set"
5733 assumes "subspace s" shows "closed s"
5735 have *:"dim s \<le> DIM('a)" using dim_subset_UNIV by auto
5736 def d \<equiv> "{..<dim s}" have t:"card d = dim s" unfolding d_def by auto
5737 let ?t = "{x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0}"
5738 have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0} = s \<and>
5739 inj_on f {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0}"
5740 apply(rule subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]])
5741 using dim_substandard[of d,where 'a='a] and t unfolding d_def using * assms by auto
5742 then guess f apply-by(erule exE conjE)+ note f = this
5743 interpret f: bounded_linear f using f unfolding linear_conv_bounded_linear by auto
5744 have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using f.zero using f(3)[unfolded inj_on_def]
5745 by(erule_tac x=0 in ballE) auto
5746 moreover have "closed ?t" using closed_substandard .
5747 moreover have "subspace ?t" using subspace_substandard .
5748 ultimately show ?thesis using closed_injective_image_subspace[of ?t f]
5749 unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
5752 lemma complete_subspace:
5753 fixes s :: "('a::euclidean_space) set" shows "subspace s ==> complete s"
5754 using complete_eq_closed closed_subspace
5758 fixes s :: "('a::euclidean_space) set"
5759 shows "dim(closure s) = dim s" (is "?dc = ?d")
5761 have "?dc \<le> ?d" using closure_minimal[OF span_inc, of s]
5762 using closed_subspace[OF subspace_span, of s]
5763 using dim_subset[of "closure s" "span s"] unfolding dim_span by auto
5764 thus ?thesis using dim_subset[OF closure_subset, of s] by auto
5767 subsection {* Affine transformations of intervals *}
5769 lemma real_affinity_le:
5770 "0 < (m::'a::linordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"
5771 by (simp add: field_simps inverse_eq_divide)
5773 lemma real_le_affinity:
5774 "0 < (m::'a::linordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"
5775 by (simp add: field_simps inverse_eq_divide)
5777 lemma real_affinity_lt:
5778 "0 < (m::'a::linordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"
5779 by (simp add: field_simps inverse_eq_divide)
5781 lemma real_lt_affinity:
5782 "0 < (m::'a::linordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"
5783 by (simp add: field_simps inverse_eq_divide)
5785 lemma real_affinity_eq:
5786 "(m::'a::linordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"
5787 by (simp add: field_simps inverse_eq_divide)
5789 lemma real_eq_affinity:
5790 "(m::'a::linordered_field) \<noteq> 0 ==> (y = m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"
5791 by (simp add: field_simps inverse_eq_divide)
5793 lemma image_affinity_interval: fixes m::real
5794 fixes a b c :: "'a::ordered_euclidean_space"
5795 shows "(\<lambda>x. m *\<^sub>R x + c) ` {a .. b} =
5796 (if {a .. b} = {} then {}
5797 else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
5798 else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
5800 { fix x assume "x \<le> c" "c \<le> x"
5801 hence "x=c" unfolding eucl_le[where 'a='a] apply-
5802 apply(subst euclidean_eq) by (auto intro: order_antisym) }
5804 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])
5805 ultimately show ?thesis by auto
5808 { fix y assume "a \<le> y" "y \<le> b" "m > 0"
5809 hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
5810 unfolding eucl_le[where 'a='a] by(auto simp add: euclidean_simps)
5812 { fix y assume "a \<le> y" "y \<le> b" "m < 0"
5813 hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
5814 unfolding eucl_le[where 'a='a] by(auto simp add: mult_left_mono_neg euclidean_simps)
5816 { fix y assume "m > 0" "m *\<^sub>R a + c \<le> y" "y \<le> m *\<^sub>R b + c"
5817 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5818 unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
5819 apply(auto simp add: pth_3[symmetric]
5820 intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
5821 by(auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff euclidean_simps)
5823 { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
5824 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5825 unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
5826 apply(auto simp add: pth_3[symmetric]
5827 intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
5828 by(auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff euclidean_simps)
5830 ultimately show ?thesis using False by auto
5833 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a..b} =
5834 (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})"
5835 using image_affinity_interval[of m 0 a b] by auto
5837 subsection{* Banach fixed point theorem (not really topological...) *}
5840 assumes s:"complete s" "s \<noteq> {}" and c:"0 \<le> c" "c < 1" and f:"(f ` s) \<subseteq> s" and
5841 lipschitz:"\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
5842 shows "\<exists>! x\<in>s. (f x = x)"
5844 have "1 - c > 0" using c by auto
5846 from s(2) obtain z0 where "z0 \<in> s" by auto
5847 def z \<equiv> "\<lambda>n. (f ^^ n) z0"
5849 have "z n \<in> s" unfolding z_def
5850 proof(induct n) case 0 thus ?case using `z0 \<in>s` by auto
5851 next case Suc thus ?case using f by auto qed }
5854 def d \<equiv> "dist (z 0) (z 1)"
5856 have fzn:"\<And>n. f (z n) = z (Suc n)" unfolding z_def by auto
5858 have "dist (z n) (z (Suc n)) \<le> (c ^ n) * d"
5860 case 0 thus ?case unfolding d_def by auto
5863 hence "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
5864 using `0 \<le> c` using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by auto
5865 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]
5866 unfolding fzn and mult_le_cancel_left by auto
5871 have "(1 - c) * dist (z m) (z (m+n)) \<le> (c ^ m) * d * (1 - c ^ n)"
5873 case 0 show ?case by auto
5876 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))))"
5877 using dist_triangle and c by(auto simp add: dist_triangle)
5878 also have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
5879 using cf_z[of "m + k"] and c by auto
5880 also have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
5881 using Suc by (auto simp add: field_simps)
5882 also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
5883 unfolding power_add by (auto simp add: field_simps)
5884 also have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
5885 using c by (auto simp add: field_simps)
5886 finally show ?case by auto
5889 { fix e::real assume "e>0"
5890 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e"
5891 proof(cases "d = 0")
5893 have *: "\<And>x. ((1 - c) * x \<le> 0) = (x \<le> 0)" using `1 - c > 0`
5894 by (metis mult_zero_left real_mult_commute real_mult_le_cancel_iff1)
5895 from True have "\<And>n. z n = z0" using cf_z2[of 0] and c unfolding z_def
5897 thus ?thesis using `e>0` by auto
5899 case False hence "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]
5900 by (metis False d_def less_le)
5901 hence "0 < e * (1 - c) / d" using `e>0` and `1-c>0`
5902 using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"] by auto
5903 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
5904 { fix m n::nat assume "m>n" and as:"m\<ge>N" "n\<ge>N"
5905 have *:"c ^ n \<le> c ^ N" using `n\<ge>N` and c using power_decreasing[OF `n\<ge>N`, of c] by auto
5906 have "1 - c ^ (m - n) > 0" using c and power_strict_mono[of c 1 "m - n"] using `m>n` by auto
5907 hence **:"d * (1 - c ^ (m - n)) / (1 - c) > 0"
5908 using mult_pos_pos[OF `d>0`, of "1 - c ^ (m - n)"]
5909 using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]
5910 using `0 < 1 - c` by auto
5912 have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
5913 using cf_z2[of n "m - n"] and `m>n` unfolding pos_le_divide_eq[OF `1-c>0`]
5914 by (auto simp add: mult_commute dist_commute)
5915 also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
5916 using mult_right_mono[OF * order_less_imp_le[OF **]]
5917 unfolding mult_assoc by auto
5918 also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
5919 using mult_strict_right_mono[OF N **] unfolding mult_assoc by auto
5920 also have "\<dots> = e * (1 - c ^ (m - n))" using c and `d>0` and `1 - c > 0` by auto
5921 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
5922 finally have "dist (z m) (z n) < e" by auto
5924 { fix m n::nat assume as:"N\<le>m" "N\<le>n"
5925 hence "dist (z n) (z m) < e"
5926 proof(cases "n = m")
5927 case True thus ?thesis using `e>0` by auto
5929 case False thus ?thesis using as and *[of n m] *[of m n] unfolding nat_neq_iff by (auto simp add: dist_commute)
5931 thus ?thesis by auto
5934 hence "Cauchy z" unfolding cauchy_def by auto
5935 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
5937 def e \<equiv> "dist (f x) x"
5938 have "e = 0" proof(rule ccontr)
5939 assume "e \<noteq> 0" hence "e>0" unfolding e_def using zero_le_dist[of "f x" x]
5940 by (metis dist_eq_0_iff dist_nz e_def)
5941 then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"
5942 using x[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
5943 hence N':"dist (z N) x < e / 2" by auto
5945 have *:"c * dist (z N) x \<le> dist (z N) x" unfolding mult_le_cancel_right2
5946 using zero_le_dist[of "z N" x] and c
5947 by (metis dist_eq_0_iff dist_nz order_less_asym less_le)
5948 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]]
5949 using z_in_s[of N] `x\<in>s` using c by auto
5950 also have "\<dots> < e / 2" using N' and c using * by auto
5951 finally show False unfolding fzn
5952 using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
5953 unfolding e_def by auto
5955 hence "f x = x" unfolding e_def by auto
5957 { fix y assume "f y = y" "y\<in>s"
5958 hence "dist x y \<le> c * dist x y" using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]
5959 using `x\<in>s` and `f x = x` by auto
5960 hence "dist x y = 0" unfolding mult_le_cancel_right1
5961 using c and zero_le_dist[of x y] by auto
5962 hence "y = x" by auto
5964 ultimately show ?thesis using `x\<in>s` by blast+
5967 subsection{* Edelstein fixed point theorem. *}
5969 lemma edelstein_fix:
5970 fixes s :: "'a::real_normed_vector set"
5971 assumes s:"compact s" "s \<noteq> {}" and gs:"(g ` s) \<subseteq> s"
5972 and dist:"\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
5973 shows "\<exists>! x\<in>s. g x = x"
5974 proof(cases "\<exists>x\<in>s. g x \<noteq> x")
5975 obtain x where "x\<in>s" using s(2) by auto
5976 case False hence g:"\<forall>x\<in>s. g x = x" by auto
5977 { fix y assume "y\<in>s"
5978 hence "x = y" using `x\<in>s` and dist[THEN bspec[where x=x], THEN bspec[where x=y]]
5979 unfolding g[THEN bspec[where x=x], OF `x\<in>s`]
5980 unfolding g[THEN bspec[where x=y], OF `y\<in>s`] by auto }
5981 thus ?thesis using `x\<in>s` and g by blast+
5984 then obtain x where [simp]:"x\<in>s" and "g x \<noteq> x" by auto
5985 { fix x y assume "x \<in> s" "y \<in> s"
5986 hence "dist (g x) (g y) \<le> dist x y"
5987 using dist[THEN bspec[where x=x], THEN bspec[where x=y]] by auto } note dist' = this
5988 def y \<equiv> "g x"
5989 have [simp]:"y\<in>s" unfolding y_def using gs[unfolded image_subset_iff] and `x\<in>s` by blast
5990 def f \<equiv> "\<lambda>n. g ^^ n"
5991 have [simp]:"\<And>n z. g (f n z) = f (Suc n) z" unfolding f_def by auto
5992 have [simp]:"\<And>z. f 0 z = z" unfolding f_def by auto
5993 { fix n::nat and z assume "z\<in>s"
5994 have "f n z \<in> s" unfolding f_def
5996 case 0 thus ?case using `z\<in>s` by simp
5998 case (Suc n) thus ?case using gs[unfolded image_subset_iff] by auto
5999 qed } note fs = this
6000 { fix m n ::nat assume "m\<le>n"
6001 fix w z assume "w\<in>s" "z\<in>s"
6002 have "dist (f n w) (f n z) \<le> dist (f m w) (f m z)" using `m\<le>n`
6004 case 0 thus ?case by auto
6007 thus ?case proof(cases "m\<le>n")
6008 case True thus ?thesis using Suc(1)
6009 using dist'[OF fs fs, OF `w\<in>s` `z\<in>s`, of n n] by auto
6011 case False hence mn:"m = Suc n" using Suc(2) by simp
6012 show ?thesis unfolding mn by auto
6014 qed } note distf = this
6016 def h \<equiv> "\<lambda>n. (f n x, f n y)"
6017 let ?s2 = "s \<times> s"
6018 obtain l r where "l\<in>?s2" and r:"subseq r" and lr:"((h \<circ> r) ---> l) sequentially"
6019 using compact_Times [OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def
6020 using fs[OF `x\<in>s`] and fs[OF `y\<in>s`] by blast
6021 def a \<equiv> "fst l" def b \<equiv> "snd l"
6022 have lab:"l = (a, b)" unfolding a_def b_def by simp
6023 have [simp]:"a\<in>s" "b\<in>s" unfolding a_def b_def using `l\<in>?s2` by auto
6025 have lima:"((fst \<circ> (h \<circ> r)) ---> a) sequentially"
6026 and limb:"((snd \<circ> (h \<circ> r)) ---> b) sequentially"
6028 unfolding o_def a_def b_def by (rule tendsto_intros)+
6031 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
6033 have "dist (-x) (-y) = dist x y" unfolding dist_norm
6034 using norm_minus_cancel[of "x - y"] by (auto simp add: uminus_add_conv_diff) } note ** = this
6036 { assume as:"dist a b > dist (f n x) (f n y)"
6037 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"
6038 and "\<forall>m\<ge>Nb. dist (f (r m) y) b < (dist a b - dist (f n x) (f n y)) / 2"
6039 using lima limb unfolding h_def Lim_sequentially by (fastsimp simp del: less_divide_eq_number_of1)
6040 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)"
6041 apply(erule_tac x="Na+Nb+n" in allE)
6042 apply(erule_tac x="Na+Nb+n" in allE) apply simp
6043 using dist_triangle_add_half[of a "f (r (Na + Nb + n)) x" "dist a b - dist (f n x) (f n y)"
6044 "-b" "- f (r (Na + Nb + n)) y"]
6045 unfolding ** by (auto simp add: algebra_simps dist_commute)
6047 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)"
6048 using distf[of n "r (Na+Nb+n)", OF _ `x\<in>s` `y\<in>s`]
6049 using subseq_bigger[OF r, of "Na+Nb+n"]
6050 using *[of "f (r (Na + Nb + n)) x" "f (r (Na + Nb + n)) y" "f n x" "f n y"] by auto
6051 ultimately have False by simp
6053 hence "dist a b \<le> dist (f n x) (f n y)" by(rule ccontr)auto }
6056 have [simp]:"a = b" proof(rule ccontr)
6057 def e \<equiv> "dist a b - dist (g a) (g b)"
6058 assume "a\<noteq>b" hence "e > 0" unfolding e_def using dist by fastsimp
6059 hence "\<exists>n. dist (f n x) a < e/2 \<and> dist (f n y) b < e/2"
6060 using lima limb unfolding Lim_sequentially
6061 apply (auto elim!: allE[where x="e/2"]) apply(rule_tac x="r (max N Na)" in exI) unfolding h_def by fastsimp
6062 then obtain n where n:"dist (f n x) a < e/2 \<and> dist (f n y) b < e/2" by auto
6063 have "dist (f (Suc n) x) (g a) \<le> dist (f n x) a"
6064 using dist[THEN bspec[where x="f n x"], THEN bspec[where x="a"]] and fs by auto
6065 moreover have "dist (f (Suc n) y) (g b) \<le> dist (f n y) b"
6066 using dist[THEN bspec[where x="f n y"], THEN bspec[where x="b"]] and fs by auto
6067 ultimately have "dist (f (Suc n) x) (g a) + dist (f (Suc n) y) (g b) < e" using n by auto
6068 thus False unfolding e_def using ab_fn[of "Suc n"] by norm
6071 have [simp]:"\<And>n. f (Suc n) x = f n y" unfolding f_def y_def by(induct_tac n)auto
6072 { fix x y assume "x\<in>s" "y\<in>s" moreover
6073 fix e::real assume "e>0" ultimately
6074 have "dist y x < e \<longrightarrow> dist (g y) (g x) < e" using dist by fastsimp }
6075 hence "continuous_on s g" unfolding continuous_on_iff by auto
6077 hence "((snd \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
6078 apply (rule allE[where x="\<lambda>n. (fst \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
6079 using lima unfolding h_def o_def using fs[OF `x\<in>s`] by (auto simp add: y_def)
6080 hence "g a = a" using tendsto_unique[OF trivial_limit_sequentially limb, of "g a"]
6081 unfolding `a=b` and o_assoc by auto
6083 { fix x assume "x\<in>s" "g x = x" "x\<noteq>a"
6084 hence "False" using dist[THEN bspec[where x=a], THEN bspec[where x=x]]
6085 using `g a = a` and `a\<in>s` by auto }
6086 ultimately show "\<exists>!x\<in>s. g x = x" using `a\<in>s` by blast
6090 (** TODO move this someplace else within this theory **)
6091 instance euclidean_space \<subseteq> banach ..
6093 declare tendsto_const [intro] (* FIXME: move *)
6095 text {* Legacy theorem names *}
6097 lemmas Lim_ident_at = LIM_ident
6098 lemmas Lim_const = tendsto_const
6099 lemmas Lim_cmul = scaleR.tendsto [OF tendsto_const]
6100 lemmas Lim_neg = tendsto_minus
6101 lemmas Lim_add = tendsto_add
6102 lemmas Lim_sub = tendsto_diff
6103 lemmas Lim_mul = scaleR.tendsto
6104 lemmas Lim_vmul = scaleR.tendsto [OF _ tendsto_const]
6105 lemmas Lim_null_norm = tendsto_norm_zero_iff [symmetric]
6106 lemmas Lim_linear = bounded_linear.tendsto [COMP swap_prems_rl]
6107 lemmas Lim_component = euclidean_component.tendsto
6108 lemmas Lim_intros = Lim_add Lim_const Lim_sub Lim_cmul Lim_vmul Lim_within_id