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 by(auto simp add:power2_eq_square)
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 topologies as values *}
25 definition "istopology L \<longleftrightarrow> L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))"
26 typedef (open) 'a topology = "{L::('a set) \<Rightarrow> bool. 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_Collect_eq] .
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 (simp add: fun_eq_iff)
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 subsubsection {* 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 mem_Collect_eq
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 subsubsection {* 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 subsubsection {* Subspace topology *}
133 definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
135 lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
138 have "?L {}" by blast
139 {fix A B assume A: "?L A" and B: "?L B"
140 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
141 have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)" using Sa Sb by blast+
142 then have "?L (A \<inter> B)" by blast}
144 {fix K assume K: "K \<subseteq> Collect ?L"
145 have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
147 apply (simp add: Ball_def image_iff)
149 from K[unfolded th0 subset_image_iff]
150 obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
151 have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
152 moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq)
153 ultimately have "?L (\<Union>K)" by blast}
154 ultimately show ?thesis
155 unfolding subset_eq mem_Collect_eq istopology_def by blast
158 lemma openin_subtopology:
159 "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
160 unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
163 lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
164 by (auto simp add: topspace_def openin_subtopology)
166 lemma closedin_subtopology:
167 "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
168 unfolding closedin_def topspace_subtopology
169 apply (simp add: openin_subtopology)
172 apply (rule_tac x="topspace U - T" in exI)
175 lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
176 unfolding openin_subtopology
177 apply (rule iffI, clarify)
178 apply (frule openin_subset[of U]) apply blast
179 apply (rule exI[where x="topspace U"])
182 lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"
183 shows "subtopology U V = U"
186 {fix T assume T: "openin U T" "S = T \<inter> V"
187 from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
188 have "openin U S" unfolding eq using T by blast}
190 {assume S: "openin U S"
191 hence "\<exists>T. openin U T \<and> S = T \<inter> V"
192 using openin_subset[OF S] UV by auto}
193 ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
194 then show ?thesis unfolding topology_eq openin_subtopology by blast
197 lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
198 by (simp add: subtopology_superset)
200 lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
201 by (simp add: subtopology_superset)
203 subsubsection {* The standard Euclidean topology *}
206 euclidean :: "'a::topological_space topology" where
207 "euclidean = topology open"
209 lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
210 unfolding euclidean_def
211 apply (rule cong[where x=S and y=S])
212 apply (rule topology_inverse[symmetric])
213 apply (auto simp add: istopology_def)
216 lemma topspace_euclidean: "topspace euclidean = UNIV"
217 apply (simp add: topspace_def)
219 by (auto simp add: open_openin[symmetric])
221 lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
222 by (simp add: topspace_euclidean topspace_subtopology)
224 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
225 by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
227 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
228 by (simp add: open_openin openin_subopen[symmetric])
230 text {* Basic "localization" results are handy for connectedness. *}
232 lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
233 by (auto simp add: openin_subtopology open_openin[symmetric])
235 lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
236 by (auto simp add: openin_open)
238 lemma open_openin_trans[trans]:
239 "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
240 by (metis Int_absorb1 openin_open_Int)
242 lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
243 by (auto simp add: openin_open)
245 lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
246 by (simp add: closedin_subtopology closed_closedin Int_ac)
248 lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
249 by (metis closedin_closed)
251 lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
252 apply (subgoal_tac "S \<inter> T = T" )
254 apply (frule closedin_closed_Int[of T S])
257 lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
258 by (auto simp add: closedin_closed)
260 lemma openin_euclidean_subtopology_iff:
261 fixes S U :: "'a::metric_space set"
262 shows "openin (subtopology euclidean U) S
263 \<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")
265 assume ?lhs thus ?rhs unfolding openin_open open_dist by blast
267 def T \<equiv> "{x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
268 have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
271 apply (rule_tac x="d - dist x a" in exI)
272 apply (clarsimp simp add: less_diff_eq)
273 apply (erule rev_bexI)
274 apply (rule_tac x=d in exI, clarify)
275 apply (erule le_less_trans [OF dist_triangle])
277 assume ?rhs hence 2: "S = U \<inter> T"
280 apply (drule (1) bspec, erule rev_bexI)
284 unfolding openin_open open_dist by fast
287 text {* These "transitivity" results are handy too *}
289 lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
290 \<Longrightarrow> openin (subtopology euclidean U) S"
291 unfolding open_openin openin_open by blast
293 lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
294 by (auto simp add: openin_open intro: openin_trans)
296 lemma closedin_trans[trans]:
297 "closedin (subtopology euclidean T) S \<Longrightarrow>
298 closedin (subtopology euclidean U) T
299 ==> closedin (subtopology euclidean U) S"
300 by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
302 lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
303 by (auto simp add: closedin_closed intro: closedin_trans)
306 subsection {* Open and closed balls *}
309 ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
310 "ball x e = {y. dist x y < e}"
313 cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
314 "cball x e = {y. dist x y \<le> e}"
316 lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
317 lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
319 lemma mem_ball_0 [simp]:
320 fixes x :: "'a::real_normed_vector"
321 shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
322 by (simp add: dist_norm)
324 lemma mem_cball_0 [simp]:
325 fixes x :: "'a::real_normed_vector"
326 shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
327 by (simp add: dist_norm)
329 lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e" by simp
330 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
331 lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
332 lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
333 lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
334 by (simp add: set_eq_iff) arith
336 lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
337 by (simp add: set_eq_iff)
339 lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
340 "(a::real) - b < 0 \<longleftrightarrow> a < b"
341 "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
342 lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
343 "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b" by arith+
345 lemma open_ball[intro, simp]: "open (ball x e)"
346 unfolding open_dist ball_def mem_Collect_eq Ball_def
347 unfolding dist_commute
349 apply (rule_tac x="e - dist xa x" in exI)
350 using dist_triangle_alt[where z=x]
351 apply (clarsimp simp add: diff_less_iff)
353 apply (erule_tac x="y" in allE)
354 apply (erule_tac x="xa" in allE)
357 lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
358 lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
359 unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
362 assumes "open S" "x\<in>S"
363 obtains e where "e>0" "ball x e \<subseteq> S"
364 using assms unfolding open_contains_ball by auto
366 lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
367 by (metis open_contains_ball subset_eq centre_in_ball)
369 lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
370 unfolding mem_ball set_eq_iff
371 apply (simp add: not_less)
372 by (metis zero_le_dist order_trans dist_self)
374 lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
377 subsection{* Connectedness *}
379 definition "connected S \<longleftrightarrow>
380 ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
381 \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
383 lemma connected_local:
384 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
385 openin (subtopology euclidean S) e1 \<and>
386 openin (subtopology euclidean S) e2 \<and>
387 S \<subseteq> e1 \<union> e2 \<and>
388 e1 \<inter> e2 = {} \<and>
391 unfolding connected_def openin_open by (safe, blast+)
394 fixes P :: "'a set \<Rightarrow> bool"
395 shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
397 {assume "?lhs" hence ?rhs by blast }
399 {fix S assume H: "P S"
400 have "S = - (- S)" by auto
401 with H have "P (- (- S))" by metis }
402 ultimately show ?thesis by metis
405 lemma connected_clopen: "connected S \<longleftrightarrow>
406 (\<forall>T. openin (subtopology euclidean S) T \<and>
407 closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
409 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> {})"
410 unfolding connected_def openin_open closedin_closed
411 apply (subst exists_diff) by blast
412 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> {})"
413 (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis
415 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'))"
416 (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
417 unfolding connected_def openin_open closedin_closed by auto
419 {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)"
421 then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
422 then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
423 then show ?thesis unfolding th0 th1 by simp
426 lemma connected_empty[simp, intro]: "connected {}"
427 by (simp add: connected_def)
430 subsection{* Limit points *}
432 definition (in topological_space)
433 islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where
434 "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
437 assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
439 using assms unfolding islimpt_def by auto
442 assumes "x islimpt S" and "x \<in> T" and "open T"
443 obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
444 using assms unfolding islimpt_def by auto
446 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
448 lemma islimpt_approachable:
449 fixes x :: "'a::metric_space"
450 shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
451 unfolding islimpt_def
453 apply(erule_tac x="ball x e" in allE)
455 apply(rule_tac x=y in bexI)
456 apply (auto simp add: dist_commute)
457 apply (simp add: open_dist, drule (1) bspec)
458 apply (clarify, drule spec, drule (1) mp, auto)
461 lemma islimpt_approachable_le:
462 fixes x :: "'a::metric_space"
463 shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
464 unfolding islimpt_approachable
465 using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
468 text {* A perfect space has no isolated points. *}
470 class perfect_space = topological_space +
471 assumes islimpt_UNIV [simp, intro]: "x islimpt UNIV"
473 lemma perfect_choose_dist:
474 fixes x :: "'a::{perfect_space, metric_space}"
475 shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
476 using islimpt_UNIV [of x]
477 by (simp add: islimpt_approachable)
479 instance euclidean_space \<subseteq> perfect_space
482 { fix e :: real assume "0 < e"
483 def y \<equiv> "x + scaleR (e/2) (sgn (basis 0))"
484 from `0 < e` have "y \<noteq> x"
485 unfolding y_def by (simp add: sgn_zero_iff DIM_positive)
486 from `0 < e` have "dist y x < e"
487 unfolding y_def by (simp add: dist_norm norm_sgn)
488 from `y \<noteq> x` and `dist y x < e`
489 have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
491 then show "x islimpt UNIV" unfolding islimpt_approachable by blast
494 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
496 apply (subst open_subopen)
497 apply (simp add: islimpt_def subset_eq)
498 by (metis ComplE ComplI)
500 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
501 unfolding islimpt_def by auto
503 lemma finite_set_avoid:
504 fixes a :: "'a::metric_space"
505 assumes fS: "finite S" shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
506 proof(induct rule: finite_induct[OF fS])
507 case 1 thus ?case by (auto intro: zero_less_one)
510 from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
511 {assume "x = a" hence ?case using d by auto }
513 {assume xa: "x\<noteq>a"
514 let ?d = "min d (dist a x)"
515 have dp: "?d > 0" using xa d(1) using dist_nz by auto
516 from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
517 with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
518 ultimately show ?case by blast
521 lemma islimpt_finite:
522 fixes S :: "'a::metric_space set"
523 assumes fS: "finite S" shows "\<not> a islimpt S"
524 unfolding islimpt_approachable
525 using finite_set_avoid[OF fS, of a] by (metis dist_commute not_le)
527 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
530 apply (metis Un_upper1 Un_upper2 islimpt_subset)
531 unfolding islimpt_def
532 apply (rule ccontr, clarsimp, rename_tac A B)
533 apply (drule_tac x="A \<inter> B" in spec)
534 apply (auto simp add: open_Int)
537 lemma discrete_imp_closed:
538 fixes S :: "'a::metric_space set"
539 assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
542 {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
543 from e have e2: "e/2 > 0" by arith
544 from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
545 let ?m = "min (e/2) (dist x y) "
546 from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
547 from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
548 have th: "dist z y < e" using z y
549 by (intro dist_triangle_lt [where z=x], simp)
550 from d[rule_format, OF y(1) z(1) th] y z
551 have False by (auto simp add: dist_commute)}
552 then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
556 subsection {* Interior of a Set *}
558 definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
560 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
561 apply (simp add: set_eq_iff interior_def)
562 apply (subst (2) open_subopen) by (safe, blast+)
564 lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
566 lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
568 lemma open_interior[simp, intro]: "open(interior S)"
569 apply (simp add: interior_def)
570 apply (subst open_subopen) by blast
572 lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
573 lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
574 lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
575 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
576 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"
577 by (metis equalityI interior_maximal interior_subset open_interior)
578 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
579 apply (simp add: interior_def)
580 by (metis open_contains_ball centre_in_ball open_ball subset_trans)
582 lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
583 by (metis interior_maximal interior_subset subset_trans)
585 lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
586 apply (rule equalityI, simp)
587 apply (metis Int_lower1 Int_lower2 subset_interior)
588 by (metis Int_mono interior_subset open_Int open_interior open_subset_interior)
590 lemma interior_limit_point [intro]:
591 fixes x :: "'a::perfect_space"
592 assumes x: "x \<in> interior S" shows "x islimpt S"
593 using x islimpt_UNIV [of x]
594 unfolding interior_def islimpt_def
595 apply (clarsimp, rename_tac T T')
596 apply (drule_tac x="T \<inter> T'" in spec)
597 apply (auto simp add: open_Int)
600 lemma interior_closed_Un_empty_interior:
601 assumes cS: "closed S" and iT: "interior T = {}"
602 shows "interior(S \<union> T) = interior S"
604 show "interior S \<subseteq> interior (S\<union>T)"
605 by (rule subset_interior, blast)
607 show "interior (S \<union> T) \<subseteq> interior S"
609 fix x assume "x \<in> interior (S \<union> T)"
610 then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T"
611 unfolding interior_def by fast
612 show "x \<in> interior S"
614 assume "x \<notin> interior S"
615 with `x \<in> R` `open R` obtain y where "y \<in> R - S"
616 unfolding interior_def set_eq_iff by fast
617 from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
618 from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
619 from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
620 show "False" unfolding interior_def by fast
625 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
626 proof (rule interior_unique)
627 show "interior A \<times> interior B \<subseteq> A \<times> B"
628 by (intro Sigma_mono interior_subset)
629 show "open (interior A \<times> interior B)"
630 by (intro open_Times open_interior)
631 show "\<forall>T. T \<subseteq> A \<times> B \<and> open T \<longrightarrow> T \<subseteq> interior A \<times> interior B"
632 apply (simp add: open_prod_def, clarify)
633 apply (drule (1) bspec, clarify, rename_tac C D)
634 apply (simp add: interior_def, rule conjI)
635 apply (rule_tac x=C in exI, clarsimp)
636 apply (rule SigmaD1, erule subsetD, erule subsetD, erule (1) SigmaI)
637 apply (rule_tac x=D in exI, clarsimp)
638 apply (rule SigmaD2, erule subsetD, erule subsetD, erule (1) SigmaI)
643 subsection {* Closure of a Set *}
645 definition "closure S = S \<union> {x | x. x islimpt S}"
647 lemma closure_interior: "closure S = - interior (- S)"
650 have "x\<in>- interior (- S) \<longleftrightarrow> x \<in> closure S" (is "?lhs = ?rhs")
652 let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> - S)"
654 hence *:"\<not> ?exT x"
655 unfolding interior_def
657 { assume "\<not> ?rhs"
659 unfolding closure_def islimpt_def
665 assume "?rhs" thus "?lhs"
666 unfolding closure_def interior_def islimpt_def
674 lemma interior_closure: "interior S = - (closure (- S))"
677 have "x \<in> interior S \<longleftrightarrow> x \<in> - (closure (- S))"
678 unfolding interior_def closure_def islimpt_def
685 lemma closed_closure[simp, intro]: "closed (closure S)"
687 have "closed (- interior (-S))" by blast
688 thus ?thesis using closure_interior[of S] by simp
691 lemma closure_hull: "closure S = closed hull S"
693 have "S \<subseteq> closure S"
694 unfolding closure_def
697 have "closed (closure S)"
698 using closed_closure[of S]
702 assume *:"S \<subseteq> t" "closed t"
705 hence "x islimpt t" using *(1)
706 using islimpt_subset[of x, of S, of t]
709 with * have "closure S \<subseteq> t"
710 unfolding closure_def
711 using closed_limpt[of t]
714 ultimately show ?thesis
715 using hull_unique[of S, of "closure S", of closed]
719 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
720 unfolding closure_hull
721 using hull_eq[of closed, OF closed_Inter, of S]
724 lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
725 using closure_eq[of S]
728 lemma closure_closure[simp]: "closure (closure S) = closure S"
729 unfolding closure_hull
730 using hull_hull[of closed S]
733 lemma closure_subset: "S \<subseteq> closure S"
734 unfolding closure_hull
735 using hull_subset[of S closed]
738 lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
739 unfolding closure_hull
740 using hull_mono[of S T closed]
743 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
744 using hull_minimal[of S T closed]
745 unfolding closure_hull
748 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"
749 using hull_unique[of S T closed]
750 unfolding closure_hull
753 lemma closure_empty[simp]: "closure {} = {}"
754 using closed_empty closure_closed[of "{}"]
757 lemma closure_univ[simp]: "closure UNIV = UNIV"
758 using closure_closed[of UNIV]
761 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
762 using closure_empty closure_subset[of S]
765 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
766 using closure_eq[of S] closure_subset[of S]
769 lemma open_inter_closure_eq_empty:
770 "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
771 using open_subset_interior[of S "- T"]
772 using interior_subset[of "- T"]
773 unfolding closure_interior
776 lemma open_inter_closure_subset:
777 "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
780 assume as: "open S" "x \<in> S \<inter> closure T"
781 { assume *:"x islimpt T"
782 have "x islimpt (S \<inter> T)"
783 proof (rule islimptI)
785 assume "x \<in> A" "open A"
786 with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
787 by (simp_all add: open_Int)
788 with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
790 hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
792 thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
795 then show "x \<in> closure (S \<inter> T)" using as
796 unfolding closure_def
800 lemma closure_complement: "closure(- S) = - interior(S)"
805 unfolding closure_interior
809 lemma interior_complement: "interior(- S) = - closure(S)"
810 unfolding closure_interior
813 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
814 proof (intro closure_unique conjI)
815 show "A \<times> B \<subseteq> closure A \<times> closure B"
816 by (intro Sigma_mono closure_subset)
817 show "closed (closure A \<times> closure B)"
818 by (intro closed_Times closed_closure)
819 show "\<forall>T. A \<times> B \<subseteq> T \<and> closed T \<longrightarrow> closure A \<times> closure B \<subseteq> T"
820 apply (simp add: closed_def open_prod_def, clarify)
822 apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
823 apply (simp add: closure_interior interior_def)
824 apply (drule_tac x=C in spec)
825 apply (drule_tac x=D in spec)
831 subsection {* Frontier (aka boundary) *}
833 definition "frontier S = closure S - interior S"
835 lemma frontier_closed: "closed(frontier S)"
836 by (simp add: frontier_def closed_Diff)
838 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
839 by (auto simp add: frontier_def interior_closure)
841 lemma frontier_straddle:
842 fixes a :: "'a::metric_space"
843 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")
848 let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
850 have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
851 moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
852 unfolding frontier_closures closure_def islimpt_def using `e>0`
853 by (auto, erule_tac x="ball a e" in allE, auto)
854 ultimately have ?rhse by auto
857 { assume "a\<notin>S"
858 hence ?rhse using `?lhs`
859 unfolding frontier_closures closure_def islimpt_def
860 using open_ball[of a e] `e > 0`
861 by simp (metis centre_in_ball mem_ball open_ball)
863 ultimately have ?rhse by auto
869 { fix T assume "a\<notin>S" and
870 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"
871 from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
872 then obtain e where "e>0" "ball a e \<subseteq> T" by auto
873 then obtain y where y:"y\<in>S" "dist a y < e" using as(1) by auto
874 have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
875 using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
877 hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
879 { fix T assume "a \<in> T" "open T" "a\<in>S"
880 then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
881 obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
882 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
884 hence "a islimpt (- S) \<or> a\<notin>S" unfolding islimpt_def by auto
885 ultimately show ?lhs unfolding frontier_closures using closure_def[of "- S"] by auto
888 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
889 by (metis frontier_def closure_closed Diff_subset)
891 lemma frontier_empty[simp]: "frontier {} = {}"
892 by (simp add: frontier_def)
894 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
896 { assume "frontier S \<subseteq> S"
897 hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
898 hence "closed S" using closure_subset_eq by auto
900 thus ?thesis using frontier_subset_closed[of S] ..
903 lemma frontier_complement: "frontier(- S) = frontier S"
904 by (auto simp add: frontier_def closure_complement interior_complement)
906 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
907 using frontier_complement frontier_subset_eq[of "- S"]
908 unfolding open_closed by auto
911 subsection {* Filters and the ``eventually true'' quantifier *}
914 at_infinity :: "'a::real_normed_vector filter" where
915 "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
918 indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
919 (infixr "indirection" 70) where
920 "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
922 text{* Prove That They are all filters. *}
924 lemma eventually_at_infinity:
925 "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
926 unfolding at_infinity_def
927 proof (rule eventually_Abs_filter, rule is_filter.intro)
928 fix P Q :: "'a \<Rightarrow> bool"
929 assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
930 then obtain r s where
931 "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
932 then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
933 then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
936 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
938 lemma trivial_limit_within:
939 shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
941 assume "trivial_limit (at a within S)"
942 thus "\<not> a islimpt S"
943 unfolding trivial_limit_def
944 unfolding eventually_within eventually_at_topological
945 unfolding islimpt_def
946 apply (clarsimp simp add: set_eq_iff)
947 apply (rename_tac T, rule_tac x=T in exI)
948 apply (clarsimp, drule_tac x=y in bspec, simp_all)
951 assume "\<not> a islimpt S"
952 thus "trivial_limit (at a within S)"
953 unfolding trivial_limit_def
954 unfolding eventually_within eventually_at_topological
955 unfolding islimpt_def
957 apply (rule_tac x=T in exI)
962 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
963 using trivial_limit_within [of a UNIV]
964 by (simp add: within_UNIV)
966 lemma trivial_limit_at:
967 fixes a :: "'a::perfect_space"
968 shows "\<not> trivial_limit (at a)"
969 by (simp add: trivial_limit_at_iff)
971 lemma trivial_limit_at_infinity:
972 "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
973 unfolding trivial_limit_def eventually_at_infinity
975 apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
976 apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
977 apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
978 apply (drule_tac x=UNIV in spec, simp)
981 text {* Some property holds "sufficiently close" to the limit point. *}
983 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
984 "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
985 unfolding eventually_at dist_nz by auto
987 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
988 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
989 unfolding eventually_within eventually_at dist_nz by auto
991 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
992 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
993 unfolding eventually_within
994 by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl)
996 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
997 unfolding trivial_limit_def
998 by (auto elim: eventually_rev_mp)
1000 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
1001 unfolding trivial_limit_def by (auto elim: eventually_rev_mp)
1003 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
1004 by (simp add: filter_eq_iff)
1006 text{* Combining theorems for "eventually" *}
1008 lemma eventually_rev_mono:
1009 "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
1010 using eventually_mono [of P Q] by fast
1012 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
1013 by (simp add: eventually_False)
1016 subsection {* Limits *}
1018 text{* Notation Lim to avoid collition with lim defined in analysis *}
1020 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"
1021 where "Lim A f = (THE l. (f ---> l) A)"
1024 "(f ---> l) net \<longleftrightarrow>
1025 trivial_limit net \<or>
1026 (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
1027 unfolding tendsto_iff trivial_limit_eq by auto
1029 text{* Show that they yield usual definitions in the various cases. *}
1031 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
1032 (\<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)"
1033 by (auto simp add: tendsto_iff eventually_within_le)
1035 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
1036 (\<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)"
1037 by (auto simp add: tendsto_iff eventually_within)
1039 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
1040 (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
1041 by (auto simp add: tendsto_iff eventually_at)
1043 lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
1044 unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
1046 lemma Lim_at_infinity:
1047 "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
1048 by (auto simp add: tendsto_iff eventually_at_infinity)
1050 lemma Lim_sequentially:
1051 "(S ---> l) sequentially \<longleftrightarrow>
1052 (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
1053 by (rule LIMSEQ_def) (* FIXME: redundant *)
1055 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
1056 by (rule topological_tendstoI, auto elim: eventually_rev_mono)
1058 text{* The expected monotonicity property. *}
1060 lemma Lim_within_empty: "(f ---> l) (net within {})"
1061 unfolding tendsto_def Limits.eventually_within by simp
1063 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
1064 unfolding tendsto_def Limits.eventually_within
1065 by (auto elim!: eventually_elim1)
1067 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
1068 shows "(f ---> l) (net within (S \<union> T))"
1069 using assms unfolding tendsto_def Limits.eventually_within
1071 apply (drule spec, drule (1) mp, drule (1) mp)
1072 apply (drule spec, drule (1) mp, drule (1) mp)
1073 apply (auto elim: eventually_elim2)
1077 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow> S \<union> T = UNIV
1079 by (metis Lim_Un within_UNIV)
1081 text{* Interrelations between restricted and unrestricted limits. *}
1083 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
1085 unfolding tendsto_def Limits.eventually_within
1086 apply (clarify, drule spec, drule (1) mp, drule (1) mp)
1087 by (auto elim!: eventually_elim1)
1089 lemma eventually_within_interior:
1090 assumes "x \<in> interior S"
1091 shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
1093 from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S"
1094 unfolding interior_def by fast
1096 then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
1097 unfolding Limits.eventually_within Limits.eventually_at_topological
1099 with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
1102 unfolding Limits.eventually_at_topological by auto
1104 { assume "?rhs" hence "?lhs"
1105 unfolding Limits.eventually_within
1106 by (auto elim: eventually_elim1)
1111 lemma at_within_interior:
1112 "x \<in> interior S \<Longrightarrow> at x within S = at x"
1113 by (simp add: filter_eq_iff eventually_within_interior)
1115 lemma at_within_open:
1116 "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"
1117 by (simp only: at_within_interior interior_open)
1119 lemma Lim_within_open:
1120 fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
1121 assumes"a \<in> S" "open S"
1122 shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
1123 using assms by (simp only: at_within_open)
1125 lemma Lim_within_LIMSEQ:
1126 fixes a :: real and L :: "'a::metric_space"
1127 assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
1128 shows "(X ---> L) (at a within T)"
1130 assume "\<not> (X ---> L) (at a within T)"
1131 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"
1132 unfolding tendsto_iff eventually_within dist_norm by (simp add: not_less[symmetric])
1133 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
1135 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"
1136 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"
1137 using r by (simp add: Bex_def)
1138 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"
1140 hence F1: "\<And>n. ?F n \<in> T \<and> ?F n \<noteq> a"
1141 and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
1142 and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
1146 proof (rule LIMSEQ_I, unfold real_norm_def)
1149 (* choose no such that inverse (real (Suc n)) < e *)
1150 then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
1151 then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
1152 show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
1153 proof (intro exI allI impI)
1155 assume mlen: "m \<le> n"
1156 have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
1158 also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
1160 also from nodef have
1161 "inverse (real (Suc m)) < e" .
1162 finally show "\<bar>?F n - a\<bar> < e" .
1165 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`
1166 ultimately have "(\<lambda>n. X (?F n)) ----> L" using F1 by simp
1168 moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
1172 obtain n where "n = no + 1" by simp
1173 then have nolen: "no \<le> n" by simp
1174 (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
1175 have "dist (X (?F n)) L \<ge> r"
1177 with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
1179 then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
1180 with r have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
1181 thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
1183 ultimately show False by simp
1186 lemma Lim_right_bound:
1187 fixes f :: "real \<Rightarrow> real"
1188 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"
1189 assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
1190 shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
1192 assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
1194 assume [simp]: "{x<..} \<inter> I \<noteq> {}"
1196 proof (rule Lim_within_LIMSEQ, safe)
1197 fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"
1199 show "(\<lambda>n. f (S n)) ----> Inf (f ` ({x<..} \<inter> I))"
1200 proof (rule LIMSEQ_I, rule ccontr)
1201 fix r :: real assume "0 < r"
1202 with Inf_close[of "f ` ({x<..} \<inter> I)" r]
1203 obtain y where y: "x < y" "y \<in> I" "f y < Inf (f ` ({x <..} \<inter> I)) + r" by auto
1204 from `x < y` have "0 < y - x" by auto
1205 from S(2)[THEN LIMSEQ_D, OF this]
1206 obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto
1208 assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f ` ({x<..} \<inter> I))) < r)"
1209 moreover have "\<And>n. Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
1210 using S bnd by (intro Inf_lower[where z=K]) auto
1211 ultimately obtain n where n: "N \<le> n" "r + Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
1212 by (auto simp: not_less field_simps)
1213 with N[OF n(1)] mono[OF _ `y \<in> I`, of "S n"] S(1)[THEN spec, of n] y
1219 text{* Another limit point characterization. *}
1221 lemma islimpt_sequential:
1222 fixes x :: "'a::metric_space"
1223 shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
1227 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"
1228 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
1230 have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
1233 { fix e::real assume "e>0"
1234 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
1235 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
1236 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)
1237 moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
1238 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
1240 hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
1241 unfolding Lim_sequentially using f by auto
1242 ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
1245 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
1246 { fix e::real assume "e>0"
1247 then obtain N where "dist (f N) x < e" using f(2) by auto
1248 moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
1249 ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
1251 thus ?lhs unfolding islimpt_approachable by auto
1254 lemma Lim_inv: (* TODO: delete *)
1255 fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
1256 assumes "(f ---> l) A" and "l \<noteq> 0"
1257 shows "((inverse o f) ---> inverse l) A"
1258 unfolding o_def using assms by (rule tendsto_inverse)
1261 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1262 shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
1263 by (simp add: Lim dist_norm)
1265 lemma Lim_null_comparison:
1266 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1267 assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
1268 shows "(f ---> 0) net"
1269 proof (rule metric_tendsto_imp_tendsto)
1270 show "(g ---> 0) net" by fact
1271 show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
1272 using assms(1) by (rule eventually_elim1, simp add: dist_norm)
1275 lemma Lim_transform_bound:
1276 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1277 fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
1278 assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net" "(g ---> 0) net"
1279 shows "(f ---> 0) net"
1280 using assms(1) tendsto_norm_zero [OF assms(2)]
1281 by (rule Lim_null_comparison)
1283 text{* Deducing things about the limit from the elements. *}
1285 lemma Lim_in_closed_set:
1286 assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
1289 assume "l \<notin> S"
1290 with `closed S` have "open (- S)" "l \<in> - S"
1291 by (simp_all add: open_Compl)
1292 with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
1293 by (rule topological_tendstoD)
1294 with assms(2) have "eventually (\<lambda>x. False) net"
1295 by (rule eventually_elim2) simp
1296 with assms(3) show "False"
1297 by (simp add: eventually_False)
1300 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
1302 lemma Lim_dist_ubound:
1303 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
1304 shows "dist a l <= e"
1306 have "dist a l \<in> {..e}"
1307 proof (rule Lim_in_closed_set)
1308 show "closed {..e}" by simp
1309 show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)
1310 show "\<not> trivial_limit net" by fact
1311 show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)
1313 thus ?thesis by simp
1316 lemma Lim_norm_ubound:
1317 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1318 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
1319 shows "norm(l) <= e"
1321 have "norm l \<in> {..e}"
1322 proof (rule Lim_in_closed_set)
1323 show "closed {..e}" by simp
1324 show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)
1325 show "\<not> trivial_limit net" by fact
1326 show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
1328 thus ?thesis by simp
1331 lemma Lim_norm_lbound:
1332 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1333 assumes "\<not> (trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. e <= norm(f x)) net"
1334 shows "e \<le> norm l"
1336 have "norm l \<in> {e..}"
1337 proof (rule Lim_in_closed_set)
1338 show "closed {e..}" by simp
1339 show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)
1340 show "\<not> trivial_limit net" by fact
1341 show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
1343 thus ?thesis by simp
1346 text{* Uniqueness of the limit, when nontrivial. *}
1349 fixes f :: "'a \<Rightarrow> 'b::t2_space"
1350 shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
1351 unfolding Lim_def using tendsto_unique[of net f] by auto
1353 text{* Limit under bilinear function *}
1356 assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
1357 shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
1358 using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
1359 by (rule bounded_bilinear.tendsto)
1361 text{* These are special for limits out of the same vector space. *}
1363 lemma Lim_within_id: "(id ---> a) (at a within s)"
1364 unfolding tendsto_def Limits.eventually_within eventually_at_topological
1367 lemma Lim_at_id: "(id ---> a) (at a)"
1368 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
1371 fixes a :: "'a::real_normed_vector"
1372 fixes l :: "'b::topological_space"
1373 shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
1374 using LIM_offset_zero LIM_offset_zero_cancel ..
1376 text{* It's also sometimes useful to extract the limit point from the filter. *}
1379 netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
1380 "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
1382 lemma netlimit_within:
1383 assumes "\<not> trivial_limit (at a within S)"
1384 shows "netlimit (at a within S) = a"
1385 unfolding netlimit_def
1386 apply (rule some_equality)
1387 apply (rule Lim_at_within)
1388 apply (rule LIM_ident)
1389 apply (erule tendsto_unique [OF assms])
1390 apply (rule Lim_at_within)
1391 apply (rule LIM_ident)
1395 fixes a :: "'a::{perfect_space,t2_space}"
1396 shows "netlimit (at a) = a"
1397 apply (subst within_UNIV[symmetric])
1398 using netlimit_within[of a UNIV]
1399 by (simp add: trivial_limit_at within_UNIV)
1401 lemma lim_within_interior:
1402 "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
1403 by (simp add: at_within_interior)
1405 lemma netlimit_within_interior:
1406 fixes x :: "'a::{t2_space,perfect_space}"
1407 assumes "x \<in> interior S"
1408 shows "netlimit (at x within S) = x"
1409 using assms by (simp add: at_within_interior netlimit_at)
1411 text{* Transformation of limit. *}
1413 lemma Lim_transform:
1414 fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
1415 assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
1416 shows "(g ---> l) net"
1417 using tendsto_diff [OF assms(2) assms(1)] by simp
1419 lemma Lim_transform_eventually:
1420 "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
1421 apply (rule topological_tendstoI)
1422 apply (drule (2) topological_tendstoD)
1423 apply (erule (1) eventually_elim2, simp)
1426 lemma Lim_transform_within:
1427 assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
1428 and "(f ---> l) (at x within S)"
1429 shows "(g ---> l) (at x within S)"
1430 proof (rule Lim_transform_eventually)
1431 show "eventually (\<lambda>x. f x = g x) (at x within S)"
1432 unfolding eventually_within
1433 using assms(1,2) by auto
1434 show "(f ---> l) (at x within S)" by fact
1437 lemma Lim_transform_at:
1438 assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
1439 and "(f ---> l) (at x)"
1440 shows "(g ---> l) (at x)"
1441 proof (rule Lim_transform_eventually)
1442 show "eventually (\<lambda>x. f x = g x) (at x)"
1443 unfolding eventually_at
1444 using assms(1,2) by auto
1445 show "(f ---> l) (at x)" by fact
1448 text{* Common case assuming being away from some crucial point like 0. *}
1450 lemma Lim_transform_away_within:
1451 fixes a b :: "'a::t1_space"
1452 assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1453 and "(f ---> l) (at a within S)"
1454 shows "(g ---> l) (at a within S)"
1455 proof (rule Lim_transform_eventually)
1456 show "(f ---> l) (at a within S)" by fact
1457 show "eventually (\<lambda>x. f x = g x) (at a within S)"
1458 unfolding Limits.eventually_within eventually_at_topological
1459 by (rule exI [where x="- {b}"], simp add: open_Compl assms)
1462 lemma Lim_transform_away_at:
1463 fixes a b :: "'a::t1_space"
1464 assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1465 and fl: "(f ---> l) (at a)"
1466 shows "(g ---> l) (at a)"
1467 using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
1468 by (auto simp add: within_UNIV)
1470 text{* Alternatively, within an open set. *}
1472 lemma Lim_transform_within_open:
1473 assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
1474 and "(f ---> l) (at a)"
1475 shows "(g ---> l) (at a)"
1476 proof (rule Lim_transform_eventually)
1477 show "eventually (\<lambda>x. f x = g x) (at a)"
1478 unfolding eventually_at_topological
1479 using assms(1,2,3) by auto
1480 show "(f ---> l) (at a)" by fact
1483 text{* A congruence rule allowing us to transform limits assuming not at point. *}
1485 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
1487 lemma Lim_cong_within(*[cong add]*):
1488 assumes "a = b" "x = y" "S = T"
1489 assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
1490 shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
1491 unfolding tendsto_def Limits.eventually_within eventually_at_topological
1494 lemma Lim_cong_at(*[cong add]*):
1495 assumes "a = b" "x = y"
1496 assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
1497 shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
1498 unfolding tendsto_def eventually_at_topological
1501 text{* Useful lemmas on closure and set of possible sequential limits.*}
1503 lemma closure_sequential:
1504 fixes l :: "'a::metric_space"
1505 shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
1507 assume "?lhs" moreover
1508 { assume "l \<in> S"
1509 hence "?rhs" using tendsto_const[of l sequentially] by auto
1511 { assume "l islimpt S"
1512 hence "?rhs" unfolding islimpt_sequential by auto
1514 show "?rhs" unfolding closure_def by auto
1517 thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
1520 lemma closed_sequential_limits:
1521 fixes S :: "'a::metric_space set"
1522 shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
1523 unfolding closed_limpt
1524 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]
1527 lemma closure_approachable:
1528 fixes S :: "'a::metric_space set"
1529 shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
1530 apply (auto simp add: closure_def islimpt_approachable)
1531 by (metis dist_self)
1533 lemma closed_approachable:
1534 fixes S :: "'a::metric_space set"
1535 shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
1536 by (metis closure_closed closure_approachable)
1538 text{* Some other lemmas about sequences. *}
1540 lemma sequentially_offset:
1541 assumes "eventually (\<lambda>i. P i) sequentially"
1542 shows "eventually (\<lambda>i. P (i + k)) sequentially"
1543 using assms unfolding eventually_sequentially by (metis trans_le_add1)
1546 assumes "(f ---> l) sequentially"
1547 shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
1548 using assms unfolding tendsto_def
1549 by clarify (rule sequentially_offset, simp)
1551 lemma seq_offset_neg:
1552 "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
1553 apply (rule topological_tendstoI)
1554 apply (drule (2) topological_tendstoD)
1555 apply (simp only: eventually_sequentially)
1556 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
1560 lemma seq_offset_rev:
1561 "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
1562 apply (rule topological_tendstoI)
1563 apply (drule (2) topological_tendstoD)
1564 apply (simp only: eventually_sequentially)
1565 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
1568 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
1570 { fix e::real assume "e>0"
1571 hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
1572 using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
1573 by (metis le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
1575 thus ?thesis unfolding Lim_sequentially dist_norm by simp
1578 subsection {* More properties of closed balls *}
1580 lemma closed_cball: "closed (cball x e)"
1581 unfolding cball_def closed_def
1582 unfolding Collect_neg_eq [symmetric] not_le
1583 apply (clarsimp simp add: open_dist, rename_tac y)
1584 apply (rule_tac x="dist x y - e" in exI, clarsimp)
1585 apply (rename_tac x')
1586 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
1590 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
1592 { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
1593 hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
1595 { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
1596 hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
1598 show ?thesis unfolding open_contains_ball by auto
1601 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
1602 by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
1604 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
1605 apply (simp add: interior_def, safe)
1606 apply (force simp add: open_contains_cball)
1607 apply (rule_tac x="ball x e" in exI)
1608 apply (simp add: subset_trans [OF ball_subset_cball])
1612 fixes x y :: "'a::{real_normed_vector,perfect_space}"
1613 shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
1616 { assume "e \<le> 0"
1617 hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
1618 have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
1620 hence "e > 0" by (metis not_less)
1622 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
1623 ultimately show "?rhs" by auto
1625 assume "?rhs" hence "e>0" by auto
1626 { fix d::real assume "d>0"
1627 have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1628 proof(cases "d \<le> dist x y")
1629 case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1631 case True hence False using `d \<le> dist x y` `d>0` by auto
1632 thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
1636 have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
1637 = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1638 unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
1639 also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
1640 using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
1641 unfolding scaleR_minus_left scaleR_one
1642 by (auto simp add: norm_minus_commute)
1643 also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
1644 unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
1645 unfolding left_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
1646 also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
1647 finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
1651 have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
1652 using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
1654 have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
1655 using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
1656 unfolding dist_norm by auto
1657 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
1660 case False hence "d > dist x y" by auto
1661 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1664 obtain z where **: "z \<noteq> y" "dist z y < min e d"
1665 using perfect_choose_dist[of "min e d" y]
1666 using `d > 0` `e>0` by auto
1667 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1669 using `z \<noteq> y` **
1670 by (rule_tac x=z in bexI, auto simp add: dist_commute)
1672 case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1673 using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
1676 thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
1679 lemma closure_ball_lemma:
1680 fixes x y :: "'a::real_normed_vector"
1681 assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
1682 proof (rule islimptI)
1683 fix T assume "y \<in> T" "open T"
1684 then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
1685 unfolding open_dist by fast
1686 (* choose point between x and y, within distance r of y. *)
1687 def k \<equiv> "min 1 (r / (2 * dist x y))"
1688 def z \<equiv> "y + scaleR k (x - y)"
1689 have z_def2: "z = x + scaleR (1 - k) (y - x)"
1690 unfolding z_def by (simp add: algebra_simps)
1692 unfolding z_def k_def using `0 < r`
1693 by (simp add: dist_norm min_def)
1694 hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
1695 have "dist x z < dist x y"
1696 unfolding z_def2 dist_norm
1697 apply (simp add: norm_minus_commute)
1698 apply (simp only: dist_norm [symmetric])
1699 apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
1700 apply (rule mult_strict_right_mono)
1701 apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
1702 apply (simp add: zero_less_dist_iff `x \<noteq> y`)
1704 hence "z \<in> ball x (dist x y)" by simp
1706 unfolding z_def k_def using `x \<noteq> y` `0 < r`
1707 by (simp add: min_def)
1708 show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
1709 using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
1714 fixes x :: "'a::real_normed_vector"
1715 shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
1716 apply (rule equalityI)
1717 apply (rule closure_minimal)
1718 apply (rule ball_subset_cball)
1719 apply (rule closed_cball)
1720 apply (rule subsetI, rename_tac y)
1721 apply (simp add: le_less [where 'a=real])
1723 apply (rule subsetD [OF closure_subset], simp)
1724 apply (simp add: closure_def)
1726 apply (rule closure_ball_lemma)
1727 apply (simp add: zero_less_dist_iff)
1730 (* In a trivial vector space, this fails for e = 0. *)
1731 lemma interior_cball:
1732 fixes x :: "'a::{real_normed_vector, perfect_space}"
1733 shows "interior (cball x e) = ball x e"
1734 proof(cases "e\<ge>0")
1735 case False note cs = this
1736 from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
1737 { fix y assume "y \<in> cball x e"
1738 hence False unfolding mem_cball using dist_nz[of x y] cs by auto }
1739 hence "cball x e = {}" by auto
1740 hence "interior (cball x e) = {}" using interior_empty by auto
1741 ultimately show ?thesis by blast
1743 case True note cs = this
1744 have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
1745 { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
1746 then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
1748 then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
1749 using perfect_choose_dist [of d] by auto
1750 have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
1751 hence xa_cball:"xa \<in> cball x e" using as(1) by auto
1753 hence "y \<in> ball x e" proof(cases "x = y")
1755 hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
1756 thus "y \<in> ball x e" using `x = y ` by simp
1759 have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
1760 using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
1761 hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
1762 have "y - x \<noteq> 0" using `x \<noteq> y` by auto
1763 hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
1764 using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
1766 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)"
1767 by (auto simp add: dist_norm algebra_simps)
1768 also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1769 by (auto simp add: algebra_simps)
1770 also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
1772 also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
1773 finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
1774 thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
1776 hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
1777 ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
1780 lemma frontier_ball:
1781 fixes a :: "'a::real_normed_vector"
1782 shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
1783 apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
1784 apply (simp add: set_eq_iff)
1787 lemma frontier_cball:
1788 fixes a :: "'a::{real_normed_vector, perfect_space}"
1789 shows "frontier(cball a e) = {x. dist a x = e}"
1790 apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
1791 apply (simp add: set_eq_iff)
1794 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
1795 apply (simp add: set_eq_iff not_le)
1796 by (metis zero_le_dist dist_self order_less_le_trans)
1797 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
1799 lemma cball_eq_sing:
1800 fixes x :: "'a::{metric_space,perfect_space}"
1801 shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
1802 proof (rule linorder_cases)
1804 obtain a where "a \<noteq> x" "dist a x < e"
1805 using perfect_choose_dist [OF e] by auto
1806 hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
1807 with e show ?thesis by (auto simp add: set_eq_iff)
1811 fixes x :: "'a::metric_space"
1812 shows "e = 0 ==> cball x e = {x}"
1813 by (auto simp add: set_eq_iff)
1816 subsection {* Boundedness *}
1818 (* FIXME: This has to be unified with BSEQ!! *)
1819 definition (in metric_space)
1820 bounded :: "'a set \<Rightarrow> bool" where
1821 "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
1823 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
1824 unfolding bounded_def
1826 apply (rule_tac x="dist a x + e" in exI, clarify)
1827 apply (drule (1) bspec)
1828 apply (erule order_trans [OF dist_triangle add_left_mono])
1832 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
1833 unfolding bounded_any_center [where a=0]
1834 by (simp add: dist_norm)
1836 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
1837 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
1838 by (metis bounded_def subset_eq)
1840 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
1841 by (metis bounded_subset interior_subset)
1843 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
1845 from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
1846 { fix y assume "y \<in> closure S"
1847 then obtain f where f: "\<forall>n. f n \<in> S" "(f ---> y) sequentially"
1848 unfolding closure_sequential by auto
1849 have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
1850 hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
1851 by (rule eventually_mono, simp add: f(1))
1852 have "dist x y \<le> a"
1853 apply (rule Lim_dist_ubound [of sequentially f])
1854 apply (rule trivial_limit_sequentially)
1859 thus ?thesis unfolding bounded_def by auto
1862 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
1863 apply (simp add: bounded_def)
1864 apply (rule_tac x=x in exI)
1865 apply (rule_tac x=e in exI)
1869 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
1870 by (metis ball_subset_cball bounded_cball bounded_subset)
1872 lemma finite_imp_bounded[intro]:
1873 fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"
1875 { fix a and F :: "'a set" assume as:"bounded F"
1876 then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
1877 hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
1878 hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
1880 thus ?thesis using finite_induct[of S bounded] using bounded_empty assms by auto
1883 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
1884 apply (auto simp add: bounded_def)
1885 apply (rename_tac x y r s)
1886 apply (rule_tac x=x in exI)
1887 apply (rule_tac x="max r (dist x y + s)" in exI)
1888 apply (rule ballI, rename_tac z, safe)
1889 apply (drule (1) bspec, simp)
1890 apply (drule (1) bspec)
1891 apply (rule min_max.le_supI2)
1892 apply (erule order_trans [OF dist_triangle add_left_mono])
1895 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
1896 by (induct rule: finite_induct[of F], auto)
1898 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
1899 apply (simp add: bounded_iff)
1900 apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
1903 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
1904 by (metis Int_lower1 Int_lower2 bounded_subset)
1906 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
1907 apply (metis Diff_subset bounded_subset)
1910 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
1911 by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
1913 lemma not_bounded_UNIV[simp, intro]:
1914 "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
1915 proof(auto simp add: bounded_pos not_le)
1916 obtain x :: 'a where "x \<noteq> 0"
1917 using perfect_choose_dist [OF zero_less_one] by fast
1918 fix b::real assume b: "b >0"
1919 have b1: "b +1 \<ge> 0" using b by simp
1920 with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
1921 by (simp add: norm_sgn)
1922 then show "\<exists>x::'a. b < norm x" ..
1925 lemma bounded_linear_image:
1926 assumes "bounded S" "bounded_linear f"
1927 shows "bounded(f ` S)"
1929 from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
1930 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)
1931 { fix x assume "x\<in>S"
1932 hence "norm x \<le> b" using b by auto
1933 hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
1934 by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
1936 thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
1937 using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
1940 lemma bounded_scaling:
1941 fixes S :: "'a::real_normed_vector set"
1942 shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
1943 apply (rule bounded_linear_image, assumption)
1944 apply (rule bounded_linear_scaleR_right)
1947 lemma bounded_translation:
1948 fixes S :: "'a::real_normed_vector set"
1949 assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
1951 from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
1952 { fix x assume "x\<in>S"
1953 hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
1955 thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
1956 by (auto intro!: add exI[of _ "b + norm a"])
1960 text{* Some theorems on sups and infs using the notion "bounded". *}
1963 fixes S :: "real set"
1964 shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"
1965 by (simp add: bounded_iff)
1967 lemma bounded_has_Sup:
1968 fixes S :: "real set"
1969 assumes "bounded S" "S \<noteq> {}"
1970 shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
1972 fix x assume "x\<in>S"
1973 thus "x \<le> Sup S"
1974 by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
1976 show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
1977 by (metis SupInf.Sup_least)
1981 fixes S :: "real set"
1982 shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
1983 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal)
1985 lemma Sup_insert_finite:
1986 fixes S :: "real set"
1987 shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
1988 apply (rule Sup_insert)
1989 apply (rule finite_imp_bounded)
1992 lemma bounded_has_Inf:
1993 fixes S :: "real set"
1994 assumes "bounded S" "S \<noteq> {}"
1995 shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
1997 fix x assume "x\<in>S"
1998 from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
1999 thus "x \<ge> Inf S" using `x\<in>S`
2000 by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
2002 show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
2003 by (metis SupInf.Inf_greatest)
2007 fixes S :: "real set"
2008 shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
2009 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)
2010 lemma Inf_insert_finite:
2011 fixes S :: "real set"
2012 shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
2013 by (rule Inf_insert, rule finite_imp_bounded, simp)
2015 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
2016 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
2017 apply (frule isGlb_isLb)
2018 apply (frule_tac x = y in isGlb_isLb)
2019 apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
2023 subsection {* Equivalent versions of compactness *}
2025 subsubsection{* Sequential compactness *}
2028 compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
2029 "compact S \<longleftrightarrow>
2030 (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
2031 (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
2034 assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
2036 unfolding compact_def using assms by fast
2039 assumes "compact S" "\<forall>n. f n \<in> S"
2040 obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
2041 using assms unfolding compact_def by fast
2044 A metric space (or topological vector space) is said to have the
2045 Heine-Borel property if every closed and bounded subset is compact.
2048 class heine_borel = metric_space +
2049 assumes bounded_imp_convergent_subsequence:
2050 "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
2051 \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2053 lemma bounded_closed_imp_compact:
2054 fixes s::"'a::heine_borel set"
2055 assumes "bounded s" and "closed s" shows "compact s"
2056 proof (unfold compact_def, clarify)
2057 fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
2058 obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
2059 using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
2060 from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
2061 have "l \<in> s" using `closed s` fr l
2062 unfolding closed_sequential_limits by blast
2063 show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2064 using `l \<in> s` r l by blast
2067 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
2069 show "0 \<le> r 0" by auto
2071 fix n assume "n \<le> r n"
2072 moreover have "r n < r (Suc n)"
2073 using assms [unfolded subseq_def] by auto
2074 ultimately show "Suc n \<le> r (Suc n)" by auto
2077 lemma eventually_subseq:
2078 assumes r: "subseq r"
2079 shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
2080 unfolding eventually_sequentially
2081 by (metis subseq_bigger [OF r] le_trans)
2084 "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
2085 unfolding tendsto_def eventually_sequentially o_def
2086 by (metis subseq_bigger le_trans)
2088 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
2090 apply (rule_tac x="nat_rec e f" in exI)
2092 apply (rule def_nat_rec_0, simp)
2093 apply (rule allI, rule def_nat_rec_Suc, simp)
2094 apply (rule allI, rule impI, rule ext)
2096 apply (induct_tac x)
2098 apply (erule_tac x="n" in allE)
2102 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
2103 assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
2104 shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N. abs(s n - l) < e"
2106 have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
2107 then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
2108 { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
2110 obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
2111 have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
2112 with n have "s N \<le> t - e" using `e>0` by auto
2113 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 }
2114 hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
2115 hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto }
2116 thus ?thesis by blast
2119 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
2120 assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
2121 shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
2122 using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
2123 unfolding monoseq_def incseq_def
2124 apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
2125 unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
2127 (* TODO: merge this lemma with the ones above *)
2128 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"
2129 assumes "bounded {s n| n::nat. True}" "\<forall>n. (s n) \<le>(s(Suc n))"
2130 shows "\<exists>l. (s ---> l) sequentially"
2132 obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le> a" using assms(1)[unfolded bounded_iff] by auto
2134 have "\<And> n. n\<ge>m \<longrightarrow> (s m) \<le> (s n)"
2135 apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)
2136 apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq) }
2137 hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto
2138 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]
2139 unfolding monoseq_def by auto
2140 thus ?thesis unfolding Lim_sequentially apply(rule_tac x="l" in exI)
2141 unfolding dist_norm by auto
2144 lemma compact_real_lemma:
2145 assumes "\<forall>n::nat. abs(s n) \<le> b"
2146 shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
2148 obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
2149 using seq_monosub[of s] by auto
2150 thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
2151 unfolding tendsto_iff dist_norm eventually_sequentially by auto
2154 instance real :: heine_borel
2156 fix s :: "real set" and f :: "nat \<Rightarrow> real"
2157 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2158 then obtain b where b: "\<forall>n. abs (f n) \<le> b"
2159 unfolding bounded_iff by auto
2160 obtain l :: real and r :: "nat \<Rightarrow> nat" where
2161 r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
2162 using compact_real_lemma [OF b] by auto
2163 thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2167 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $$ i) ` s)"
2168 apply (erule bounded_linear_image)
2169 apply (rule bounded_linear_euclidean_component)
2172 lemma compact_lemma:
2173 fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
2174 assumes "bounded s" and "\<forall>n. f n \<in> s"
2175 shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>
2176 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2178 fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"
2179 have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
2180 hence "\<exists>l::'a. \<exists>r. subseq r \<and>
2181 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2182 proof(induct d) case empty thus ?case unfolding subseq_def by auto
2183 next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto
2184 have s': "bounded ((\<lambda>x. x $$ k) ` s)" using `bounded s` by (rule bounded_component)
2185 obtain l1::"'a" and r1 where r1:"subseq r1" and
2186 lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially"
2187 using insert(3) using insert(4) by auto
2188 have f': "\<forall>n. f (r1 n) $$ k \<in> (\<lambda>x. x $$ k) ` s" using `\<forall>n. f n \<in> s` by simp
2189 obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $$ k) ---> l2) sequentially"
2190 using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
2191 def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
2192 using r1 and r2 unfolding r_def o_def subseq_def by auto
2194 def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"
2195 { fix e::real assume "e>0"
2196 from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially" by blast
2197 from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $$ k) l2 < e) sequentially" by (rule tendstoD)
2198 from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$ i) (l1 $$ i) < e) sequentially"
2199 by (rule eventually_subseq)
2200 have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$ i) (l $$ i) < e) sequentially"
2201 using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def
2202 using insert.prems by auto
2204 ultimately show ?case by auto
2206 thus "\<exists>l::'a. \<exists>r. subseq r \<and>
2207 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2208 apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe
2209 apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe
2210 apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)
2211 apply(erule_tac x=i in ballE)
2212 proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a
2213 assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"
2214 hence *:"i\<ge>DIM('a)" by auto
2215 thus "dist (f (r n) $$ i) (l $$ i) < e" using e by auto
2219 instance euclidean_space \<subseteq> heine_borel
2221 fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"
2222 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2223 then obtain l::'a and r where r: "subseq r"
2224 and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $$ i) (l $$ i) < e) sequentially"
2225 using compact_lemma [OF s f] by blast
2226 let ?d = "{..<DIM('a)}"
2227 { fix e::real assume "e>0"
2228 hence "0 < e / (real_of_nat (card ?d))"
2229 using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
2230 with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))) sequentially"
2233 { fix n assume n: "\<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))"
2234 have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $$ i) (l $$ i))"
2235 apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)
2236 also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
2237 apply(rule setsum_strict_mono) using n by auto
2238 finally have "dist (f (r n)) l < e" unfolding setsum_constant
2239 using DIM_positive[where 'a='a] by auto
2241 ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
2242 by (rule eventually_elim1)
2244 hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
2245 with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
2248 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
2249 unfolding bounded_def
2251 apply (rule_tac x="a" in exI)
2252 apply (rule_tac x="e" in exI)
2254 apply (drule (1) bspec)
2255 apply (simp add: dist_Pair_Pair)
2256 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
2259 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
2260 unfolding bounded_def
2262 apply (rule_tac x="b" in exI)
2263 apply (rule_tac x="e" in exI)
2265 apply (drule (1) bspec)
2266 apply (simp add: dist_Pair_Pair)
2267 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
2270 instance prod :: (heine_borel, heine_borel) heine_borel
2272 fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
2273 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2274 from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
2275 from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
2276 obtain l1 r1 where r1: "subseq r1"
2277 and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
2278 using bounded_imp_convergent_subsequence [OF s1 f1]
2279 unfolding o_def by fast
2280 from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
2281 from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
2282 obtain l2 r2 where r2: "subseq r2"
2283 and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
2284 using bounded_imp_convergent_subsequence [OF s2 f2]
2285 unfolding o_def by fast
2286 have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
2287 using lim_subseq [OF r2 l1] unfolding o_def .
2288 have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
2289 using tendsto_Pair [OF l1' l2] unfolding o_def by simp
2290 have r: "subseq (r1 \<circ> r2)"
2291 using r1 r2 unfolding subseq_def by simp
2292 show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2296 subsubsection{* Completeness *}
2299 "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
2300 unfolding Cauchy_def by blast
2303 complete :: "'a::metric_space set \<Rightarrow> bool" where
2304 "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
2305 --> (\<exists>l \<in> s. (f ---> l) sequentially))"
2307 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
2312 with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
2313 by (erule_tac x="e/2" in allE) auto
2315 assume nm:"N \<le> m \<and> N \<le> n"
2316 hence "dist (s m) (s n) < e" using N
2317 using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
2320 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
2324 unfolding cauchy_def
2328 unfolding cauchy_def
2329 using dist_triangle_half_l
2333 lemma convergent_imp_cauchy:
2334 "(s ---> l) sequentially ==> Cauchy s"
2335 proof(simp only: cauchy_def, rule, rule)
2336 fix e::real assume "e>0" "(s ---> l) sequentially"
2337 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
2338 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
2341 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
2343 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
2344 hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
2346 have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
2347 then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
2348 unfolding bounded_any_center [where a="s N"] by auto
2349 ultimately show "?thesis"
2350 unfolding bounded_any_center [where a="s N"]
2351 apply(rule_tac x="max a 1" in exI) apply auto
2352 apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto
2355 lemma compact_imp_complete: assumes "compact s" shows "complete s"
2357 { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
2358 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
2360 note lr' = subseq_bigger [OF lr(2)]
2362 { fix e::real assume "e>0"
2363 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
2364 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
2365 { fix n::nat assume n:"n \<ge> max N M"
2366 have "dist ((f \<circ> r) n) l < e/2" using n M by auto
2367 moreover have "r n \<ge> N" using lr'[of n] n by auto
2368 hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
2369 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) }
2370 hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast }
2371 hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding Lim_sequentially by auto }
2372 thus ?thesis unfolding complete_def by auto
2375 instance heine_borel < complete_space
2377 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2378 hence "bounded (range f)"
2379 by (rule cauchy_imp_bounded)
2380 hence "compact (closure (range f))"
2381 using bounded_closed_imp_compact [of "closure (range f)"] by auto
2382 hence "complete (closure (range f))"
2383 by (rule compact_imp_complete)
2384 moreover have "\<forall>n. f n \<in> closure (range f)"
2385 using closure_subset [of "range f"] by auto
2386 ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
2387 using `Cauchy f` unfolding complete_def by auto
2388 then show "convergent f"
2389 unfolding convergent_def by auto
2392 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
2393 proof(simp add: complete_def, rule, rule)
2394 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2395 hence "convergent f" by (rule Cauchy_convergent)
2396 thus "\<exists>l. f ----> l" unfolding convergent_def .
2399 lemma complete_imp_closed: assumes "complete s" shows "closed s"
2401 { fix x assume "x islimpt s"
2402 then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
2403 unfolding islimpt_sequential by auto
2404 then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
2405 using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
2406 hence "x \<in> s" using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
2408 thus "closed s" unfolding closed_limpt by auto
2411 lemma complete_eq_closed:
2412 fixes s :: "'a::complete_space set"
2413 shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
2415 assume ?lhs thus ?rhs by (rule complete_imp_closed)
2418 { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
2419 then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
2420 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 }
2421 thus ?lhs unfolding complete_def by auto
2424 lemma convergent_eq_cauchy:
2425 fixes s :: "nat \<Rightarrow> 'a::complete_space"
2426 shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s" (is "?lhs = ?rhs")
2428 assume ?lhs then obtain l where "(s ---> l) sequentially" by auto
2429 thus ?rhs using convergent_imp_cauchy by auto
2431 assume ?rhs thus ?lhs using complete_univ[unfolded complete_def, THEN spec[where x=s]] by auto
2434 lemma convergent_imp_bounded:
2435 fixes s :: "nat \<Rightarrow> 'a::metric_space"
2436 shows "(s ---> l) sequentially ==> bounded (s ` (UNIV::(nat set)))"
2437 using convergent_imp_cauchy[of s]
2438 using cauchy_imp_bounded[of s]
2442 subsubsection{* Total boundedness *}
2444 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
2445 "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
2446 declare helper_1.simps[simp del]
2448 lemma compact_imp_totally_bounded:
2450 shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
2451 proof(rule, rule, rule ccontr)
2452 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)"
2453 def x \<equiv> "helper_1 s e"
2455 have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
2456 proof(induct_tac rule:nat_less_induct)
2457 fix n def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
2458 assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
2459 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
2460 then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
2461 have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
2462 apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
2463 thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
2465 hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
2466 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
2467 from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
2468 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
2470 using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
2471 using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
2472 using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
2475 subsubsection{* Heine-Borel theorem *}
2477 text {* Following Burkill \& Burkill vol. 2. *}
2479 lemma heine_borel_lemma: fixes s::"'a::metric_space set"
2480 assumes "compact s" "s \<subseteq> (\<Union> t)" "\<forall>b \<in> t. open b"
2481 shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
2483 assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"
2484 hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto
2486 have "1 / real (n + 1) > 0" by auto
2487 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 }
2488 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
2489 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)"
2490 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
2492 then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
2493 using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
2495 obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
2496 then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"
2497 using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto
2499 then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"
2500 using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
2502 obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
2503 have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
2504 apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
2505 using subseq_bigger[OF r, of "N1 + N2"] by auto
2507 def x \<equiv> "(f (r (N1 + N2)))"
2508 have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
2509 using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto
2510 have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto
2511 then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto
2513 have "dist x l < e/2" using N1 unfolding x_def o_def by auto
2514 hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)
2516 thus False using e and `y\<notin>b` by auto
2519 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2520 \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"
2522 fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"
2523 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
2524 hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto
2525 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
2526 then obtain bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast
2528 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
2529 then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto
2531 have "finite (bb ` k)" using k(1) by auto
2533 { fix x assume "x\<in>s"
2534 hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3) unfolding subset_eq by auto
2535 hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast
2536 hence "x \<in> \<Union>(bb ` k)" using Union_iff[of x "bb ` k"] by auto
2538 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
2541 subsubsection {* Bolzano-Weierstrass property *}
2543 lemma heine_borel_imp_bolzano_weierstrass:
2544 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'))"
2545 "infinite t" "t \<subseteq> s"
2546 shows "\<exists>x \<in> s. x islimpt t"
2548 assume "\<not> (\<exists>x \<in> s. x islimpt t)"
2549 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
2550 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
2551 obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
2552 using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
2553 from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
2554 { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
2555 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
2556 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 }
2557 hence "inj_on f t" unfolding inj_on_def by simp
2558 hence "infinite (f ` t)" using assms(2) using finite_imageD by auto
2560 { fix x assume "x\<in>t" "f x \<notin> g"
2561 from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
2562 then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
2563 hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
2564 hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto }
2565 hence "f ` t \<subseteq> g" by auto
2566 ultimately show False using g(2) using finite_subset by auto
2569 subsubsection {* Complete the chain of compactness variants *}
2571 lemma islimpt_range_imp_convergent_subsequence:
2572 fixes f :: "nat \<Rightarrow> 'a::metric_space"
2573 assumes "l islimpt (range f)"
2574 shows "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2575 proof (intro exI conjI)
2576 have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"
2577 using assms unfolding islimpt_def
2578 by (drule_tac x="ball l e" in spec)
2579 (auto simp add: zero_less_dist_iff dist_commute)
2581 def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"
2582 have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"
2583 unfolding t_def by (rule LeastI2_ex [OF * conjunct1])
2584 have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"
2585 unfolding t_def by (rule LeastI2_ex [OF * conjunct2])
2586 have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"
2587 unfolding t_def by (simp add: Least_le)
2588 have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"
2589 unfolding t_def by (drule not_less_Least) simp
2590 have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"
2592 apply (erule f_t_neq)
2593 apply (erule (1) less_le_trans [OF f_t_closer])
2595 have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"
2596 by (drule f_t_closer) auto
2597 have t_less: "\<And>e. 0 < e \<Longrightarrow> t e < t (dist (f (t e)) l)"
2598 apply (subst less_le)
2600 apply (rule t_antimono)
2601 apply (erule f_t_neq)
2602 apply (erule f_t_closer [THEN less_imp_le])
2603 apply (rule t_dist_f_neq [symmetric])
2604 apply (erule f_t_neq)
2606 have dist_f_t_less':
2607 "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"
2608 apply (simp add: le_less)
2610 apply (rule less_trans)
2611 apply (erule f_t_closer)
2612 apply (rule le_less_trans)
2613 apply (erule less_tD)
2614 apply (erule f_t_neq)
2615 apply (erule f_t_closer)
2617 apply (erule f_t_closer)
2620 def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"
2621 have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"
2622 unfolding r_def by simp_all
2623 have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"
2624 by (induct_tac n) (simp_all add: r_simps f_t_neq)
2627 unfolding subseq_Suc_iff
2630 apply (simp_all add: r_simps)
2631 apply (rule t_less, rule zero_less_one)
2632 apply (rule t_less, rule f_r_neq)
2634 show "((f \<circ> r) ---> l) sequentially"
2635 unfolding Lim_sequentially o_def
2636 apply (clarify, rule_tac x="t e" in exI, clarify)
2637 apply (drule le_trans, rule seq_suble [OF `subseq r`])
2638 apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)
2642 lemma finite_range_imp_infinite_repeats:
2643 fixes f :: "nat \<Rightarrow> 'a"
2644 assumes "finite (range f)"
2645 shows "\<exists>k. infinite {n. f n = k}"
2647 { fix A :: "'a set" assume "finite A"
2648 hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"
2650 case empty thus ?case by simp
2654 proof (cases "finite {n. f n = x}")
2656 with `infinite {n. f n \<in> insert x A}`
2657 have "infinite {n. f n \<in> A}" by simp
2658 thus "\<exists>k. infinite {n. f n = k}" by (rule insert)
2660 case False thus "\<exists>k. infinite {n. f n = k}" ..
2664 from assms show "\<exists>k. infinite {n. f n = k}"
2668 lemma bolzano_weierstrass_imp_compact:
2669 fixes s :: "'a::metric_space set"
2670 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2673 { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
2674 have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2675 proof (cases "finite (range f)")
2677 hence "\<exists>l. infinite {n. f n = l}"
2678 by (rule finite_range_imp_infinite_repeats)
2679 then obtain l where "infinite {n. f n = l}" ..
2680 hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"
2681 by (rule infinite_enumerate)
2682 then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto
2683 hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2684 unfolding o_def by (simp add: fr tendsto_const)
2685 hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2687 from f have "\<forall>n. f (r n) \<in> s" by simp
2688 hence "l \<in> s" by (simp add: fr)
2689 thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2690 by (rule rev_bexI) fact
2693 with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto
2694 then obtain l where "l \<in> s" "l islimpt (range f)" ..
2695 have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2696 using `l islimpt (range f)`
2697 by (rule islimpt_range_imp_convergent_subsequence)
2698 with `l \<in> s` show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
2701 thus ?thesis unfolding compact_def by auto
2704 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where
2705 "helper_2 beyond 0 = beyond 0" |
2706 "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"
2708 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"
2709 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2712 assume "\<not> bounded s"
2713 then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"
2714 unfolding bounded_any_center [where a=undefined]
2715 apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto
2716 hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"
2717 unfolding linorder_not_le by auto
2718 def x \<equiv> "helper_2 beyond"
2720 { fix m n ::nat assume "m<n"
2721 hence "dist undefined (x m) + 1 < dist undefined (x n)"
2723 case 0 thus ?case by auto
2726 have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"
2727 unfolding x_def and helper_2.simps
2728 using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto
2729 thus ?case proof(cases "m < n")
2730 case True thus ?thesis using Suc and * by auto
2732 case False hence "m = n" using Suc(2) by auto
2733 thus ?thesis using * by auto
2736 { fix m n ::nat assume "m\<noteq>n"
2737 have "1 < dist (x m) (x n)"
2740 hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto
2741 thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith
2743 case False hence "n<m" using `m\<noteq>n` by auto
2744 hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto
2745 thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith
2746 qed } note ** = this
2747 { fix a b assume "x a = x b" "a \<noteq> b"
2748 hence False using **[of a b] by auto }
2749 hence "inj x" unfolding inj_on_def by auto
2753 proof(cases "n = 0")
2754 case True thus ?thesis unfolding x_def using beyond by auto
2756 case False then obtain z where "n = Suc z" using not0_implies_Suc by auto
2757 thus ?thesis unfolding x_def using beyond by auto
2759 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
2761 then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto
2762 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
2763 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"]]
2764 unfolding dist_nz by auto
2765 show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto
2768 lemma sequence_infinite_lemma:
2769 fixes f :: "nat \<Rightarrow> 'a::t1_space"
2770 assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"
2771 shows "infinite (range f)"
2773 assume "finite (range f)"
2774 hence "closed (range f)" by (rule finite_imp_closed)
2775 hence "open (- range f)" by (rule open_Compl)
2776 from assms(1) have "l \<in> - range f" by auto
2777 from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
2778 using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)
2779 thus False unfolding eventually_sequentially by auto
2782 lemma closure_insert:
2783 fixes x :: "'a::t1_space"
2784 shows "closure (insert x s) = insert x (closure s)"
2785 apply (rule closure_unique)
2786 apply (rule conjI [OF insert_mono [OF closure_subset]])
2787 apply (rule conjI [OF closed_insert [OF closed_closure]])
2788 apply (simp add: closure_minimal)
2791 lemma islimpt_insert:
2792 fixes x :: "'a::t1_space"
2793 shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
2795 assume *: "x islimpt (insert a s)"
2797 proof (rule islimptI)
2798 fix t assume t: "x \<in> t" "open t"
2799 show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
2800 proof (cases "x = a")
2802 obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
2803 using * t by (rule islimptE)
2804 with `x = a` show ?thesis by auto
2807 with t have t': "x \<in> t - {a}" "open (t - {a})"
2808 by (simp_all add: open_Diff)
2809 obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
2810 using * t' by (rule islimptE)
2811 thus ?thesis by auto
2815 assume "x islimpt s" thus "x islimpt (insert a s)"
2816 by (rule islimpt_subset) auto
2819 lemma islimpt_union_finite:
2820 fixes x :: "'a::t1_space"
2821 shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
2822 by (induct set: finite, simp_all add: islimpt_insert)
2824 lemma sequence_unique_limpt:
2825 fixes f :: "nat \<Rightarrow> 'a::t2_space"
2826 assumes "(f ---> l) sequentially" and "l' islimpt (range f)"
2829 assume "l' \<noteq> l"
2830 obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
2831 using hausdorff [OF `l' \<noteq> l`] by auto
2832 have "eventually (\<lambda>n. f n \<in> t) sequentially"
2833 using assms(1) `open t` `l \<in> t` by (rule topological_tendstoD)
2834 then obtain N where "\<forall>n\<ge>N. f n \<in> t"
2835 unfolding eventually_sequentially by auto
2837 have "UNIV = {..<N} \<union> {N..}" by auto
2838 hence "l' islimpt (f ` ({..<N} \<union> {N..}))" using assms(2) by simp
2839 hence "l' islimpt (f ` {..<N} \<union> f ` {N..})" by (simp add: image_Un)
2840 hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite)
2841 then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
2842 using `l' \<in> s` `open s` by (rule islimptE)
2843 then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto
2844 with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t" by simp
2845 with `s \<inter> t = {}` show False by simp
2848 lemma bolzano_weierstrass_imp_closed:
2849 fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)
2850 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2853 { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
2855 proof(cases "\<forall>n. x n \<noteq> l")
2856 case False thus "l\<in>s" using as(1) by auto
2858 case True note cas = this
2859 with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto
2860 then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto
2861 thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
2863 thus ?thesis unfolding closed_sequential_limits by fast
2866 text {* Hence express everything as an equivalence. *}
2868 lemma compact_eq_heine_borel:
2869 fixes s :: "'a::metric_space set"
2870 shows "compact s \<longleftrightarrow>
2871 (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2872 --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")
2874 assume ?lhs thus ?rhs by (rule compact_imp_heine_borel)
2877 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
2878 by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])
2879 thus ?lhs by (rule bolzano_weierstrass_imp_compact)
2882 lemma compact_eq_bolzano_weierstrass:
2883 fixes s :: "'a::metric_space set"
2884 shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
2886 assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto
2888 assume ?rhs thus ?lhs by (rule bolzano_weierstrass_imp_compact)
2891 lemma compact_eq_bounded_closed:
2892 fixes s :: "'a::heine_borel set"
2893 shows "compact s \<longleftrightarrow> bounded s \<and> closed s" (is "?lhs = ?rhs")
2895 assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto
2897 assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto
2900 lemma compact_imp_bounded:
2901 fixes s :: "'a::metric_space set"
2902 shows "compact s ==> bounded s"
2905 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')"
2906 by (rule compact_imp_heine_borel)
2907 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
2908 using heine_borel_imp_bolzano_weierstrass[of s] by auto
2910 by (rule bolzano_weierstrass_imp_bounded)
2913 lemma compact_imp_closed:
2914 fixes s :: "'a::metric_space set"
2915 shows "compact s ==> closed s"
2918 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')"
2919 by (rule compact_imp_heine_borel)
2920 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
2921 using heine_borel_imp_bolzano_weierstrass[of s] by auto
2923 by (rule bolzano_weierstrass_imp_closed)
2926 text{* In particular, some common special cases. *}
2928 lemma compact_empty[simp]:
2930 unfolding compact_def
2933 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
2934 unfolding subseq_def by simp (* TODO: move somewhere else *)
2936 lemma compact_union [intro]:
2937 assumes "compact s" and "compact t"
2938 shows "compact (s \<union> t)"
2939 proof (rule compactI)
2940 fix f :: "nat \<Rightarrow> 'a"
2941 assume "\<forall>n. f n \<in> s \<union> t"
2942 hence "infinite {n. f n \<in> s \<union> t}" by simp
2943 hence "infinite {n. f n \<in> s} \<or> infinite {n. f n \<in> t}" by simp
2944 thus "\<exists>l\<in>s \<union> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2946 assume "infinite {n. f n \<in> s}"
2947 from infinite_enumerate [OF this]
2948 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> s" by auto
2949 obtain r l where "l \<in> s" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
2950 using `compact s` `\<forall>n. (f \<circ> q) n \<in> s` by (rule compactE)
2951 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
2952 using `subseq q` by (simp_all add: subseq_o o_assoc)
2953 thus ?thesis by auto
2955 assume "infinite {n. f n \<in> t}"
2956 from infinite_enumerate [OF this]
2957 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> t" by auto
2958 obtain r l where "l \<in> t" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
2959 using `compact t` `\<forall>n. (f \<circ> q) n \<in> t` by (rule compactE)
2960 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
2961 using `subseq q` by (simp_all add: subseq_o o_assoc)
2962 thus ?thesis by auto
2966 lemma compact_inter_closed [intro]:
2967 assumes "compact s" and "closed t"
2968 shows "compact (s \<inter> t)"
2969 proof (rule compactI)
2970 fix f :: "nat \<Rightarrow> 'a"
2971 assume "\<forall>n. f n \<in> s \<inter> t"
2972 hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t" by simp_all
2973 obtain l r where "l \<in> s" "subseq r" "((f \<circ> r) ---> l) sequentially"
2974 using `compact s` `\<forall>n. f n \<in> s` by (rule compactE)
2976 from `closed t` `\<forall>n. f n \<in> t` `((f \<circ> r) ---> l) sequentially` have "l \<in> t"
2977 unfolding closed_sequential_limits o_def by fast
2978 ultimately show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2982 lemma closed_inter_compact [intro]:
2983 assumes "closed s" and "compact t"
2984 shows "compact (s \<inter> t)"
2985 using compact_inter_closed [of t s] assms
2986 by (simp add: Int_commute)
2988 lemma compact_inter [intro]:
2989 assumes "compact s" and "compact t"
2990 shows "compact (s \<inter> t)"
2991 using assms by (intro compact_inter_closed compact_imp_closed)
2993 lemma compact_sing [simp]: "compact {a}"
2994 unfolding compact_def o_def subseq_def
2995 by (auto simp add: tendsto_const)
2997 lemma compact_insert [simp]:
2998 assumes "compact s" shows "compact (insert x s)"
3000 have "compact ({x} \<union> s)"
3001 using compact_sing assms by (rule compact_union)
3002 thus ?thesis by simp
3005 lemma finite_imp_compact:
3006 shows "finite s \<Longrightarrow> compact s"
3007 by (induct set: finite) simp_all
3009 lemma compact_cball[simp]:
3010 fixes x :: "'a::heine_borel"
3011 shows "compact(cball x e)"
3012 using compact_eq_bounded_closed bounded_cball closed_cball
3015 lemma compact_frontier_bounded[intro]:
3016 fixes s :: "'a::heine_borel set"
3017 shows "bounded s ==> compact(frontier s)"
3018 unfolding frontier_def
3019 using compact_eq_bounded_closed
3022 lemma compact_frontier[intro]:
3023 fixes s :: "'a::heine_borel set"
3024 shows "compact s ==> compact (frontier s)"
3025 using compact_eq_bounded_closed compact_frontier_bounded
3028 lemma frontier_subset_compact:
3029 fixes s :: "'a::heine_borel set"
3030 shows "compact s ==> frontier s \<subseteq> s"
3031 using frontier_subset_closed compact_eq_bounded_closed
3035 fixes s :: "'a::t1_space set"
3036 shows "open s \<Longrightarrow> open (s - {x})"
3037 by (simp add: open_Diff)
3039 text{* Finite intersection property. I could make it an equivalence in fact. *}
3041 lemma compact_imp_fip:
3042 assumes "compact s" "\<forall>t \<in> f. closed t"
3043 "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"
3044 shows "s \<inter> (\<Inter> f) \<noteq> {}"
3046 assume as:"s \<inter> (\<Inter> f) = {}"
3047 hence "s \<subseteq> \<Union> uminus ` f" by auto
3048 moreover have "Ball (uminus ` f) open" using open_Diff closed_Diff using assms(2) by auto
3049 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
3050 hence "finite (uminus ` f') \<and> uminus ` f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)
3051 hence "s \<inter> \<Inter>uminus ` f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus ` f'"]] by auto
3052 thus False using f'(3) unfolding subset_eq and Union_iff by blast
3056 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}
3058 lemma bounded_closed_nest:
3059 assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
3060 "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)" "bounded(s 0)"
3061 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
3063 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
3064 from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
3066 then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
3067 unfolding compact_def apply(erule_tac x=x in allE) using x using assms(3) by blast
3070 { fix e::real assume "e>0"
3071 with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
3072 hence "dist ((x \<circ> r) (max N n)) l < e" by auto
3074 have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
3075 hence "(x \<circ> r) (max N n) \<in> s n"
3076 using x apply(erule_tac x=n in allE)
3077 using x apply(erule_tac x="r (max N n)" in allE)
3078 using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto
3079 ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
3081 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
3083 thus ?thesis by auto
3086 text {* Decreasing case does not even need compactness, just completeness. *}
3088 lemma decreasing_closed_nest:
3089 assumes "\<forall>n. closed(s n)"
3090 "\<forall>n. (s n \<noteq> {})"
3091 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
3092 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
3093 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s n"
3095 have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
3096 hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
3097 then obtain t where t: "\<forall>n. t n \<in> s n" by auto
3098 { fix e::real assume "e>0"
3099 then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
3100 { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
3101 hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding subset_eq t by blast+
3102 hence "dist (t m) (t n) < e" using N by auto
3104 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
3106 hence "Cauchy t" unfolding cauchy_def by auto
3107 then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
3109 { fix e::real assume "e>0"
3110 then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded Lim_sequentially] by auto
3111 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
3112 hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
3114 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
3116 then show ?thesis by auto
3119 text {* Strengthen it to the intersection actually being a singleton. *}
3121 lemma decreasing_closed_nest_sing:
3122 fixes s :: "nat \<Rightarrow> 'a::heine_borel set"
3123 assumes "\<forall>n. closed(s n)"
3124 "\<forall>n. s n \<noteq> {}"
3125 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
3126 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
3127 shows "\<exists>a. \<Inter>(range s) = {a}"
3129 obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
3130 { fix b assume b:"b \<in> \<Inter>(range s)"
3131 { fix e::real assume "e>0"
3132 hence "dist a b < e" using assms(4 )using b using a by blast
3134 hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)
3136 with a have "\<Inter>(range s) = {a}" unfolding image_def by auto
3140 text{* Cauchy-type criteria for uniform convergence. *}
3142 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows
3143 "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
3144 (\<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")
3147 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
3148 { fix e::real assume "e>0"
3149 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
3150 { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
3151 hence "dist (s m x) (s n x) < e"
3152 using N[THEN spec[where x=m], THEN spec[where x=x]]
3153 using N[THEN spec[where x=n], THEN spec[where x=x]]
3154 using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto }
3155 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 }
3159 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
3160 then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
3161 using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
3162 { fix e::real assume "e>0"
3163 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"
3164 using `?rhs`[THEN spec[where x="e/2"]] by auto
3165 { fix x assume "P x"
3166 then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
3167 using l[THEN spec[where x=x], unfolded Lim_sequentially] using `e>0` by(auto elim!: allE[where x="e/2"])
3168 fix n::nat assume "n\<ge>N"
3169 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]]
3170 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) }
3171 hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
3175 lemma uniformly_cauchy_imp_uniformly_convergent:
3176 fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"
3177 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"
3178 "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
3179 shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
3181 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"
3182 using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
3184 { fix x assume "P x"
3185 hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
3186 using l and assms(2) unfolding Lim_sequentially by blast }
3187 ultimately show ?thesis by auto
3191 subsection {* Continuity *}
3193 text {* Define continuity over a net to take in restrictions of the set. *}
3196 continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
3197 where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
3199 lemma continuous_trivial_limit:
3200 "trivial_limit net ==> continuous net f"
3201 unfolding continuous_def tendsto_def trivial_limit_eq by auto
3203 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
3204 unfolding continuous_def
3205 unfolding tendsto_def
3206 using netlimit_within[of x s]
3207 by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
3209 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
3210 using continuous_within [of x UNIV f] by (simp add: within_UNIV)
3212 lemma continuous_at_within:
3213 assumes "continuous (at x) f" shows "continuous (at x within s) f"
3214 using assms unfolding continuous_at continuous_within
3215 by (rule Lim_at_within)
3217 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
3219 lemma continuous_within_eps_delta:
3220 "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)"
3221 unfolding continuous_within and Lim_within
3222 apply auto unfolding dist_nz[THEN sym] apply(auto elim!:allE) apply(rule_tac x=d in exI) by auto
3224 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3225 \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
3226 using continuous_within_eps_delta[of x UNIV f]
3227 unfolding within_UNIV by blast
3229 text{* Versions in terms of open balls. *}
3231 lemma continuous_within_ball:
3232 "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3233 f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3236 { fix e::real assume "e>0"
3237 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"
3238 using `?lhs`[unfolded continuous_within Lim_within] by auto
3239 { fix y assume "y\<in>f ` (ball x d \<inter> s)"
3240 hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
3241 apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
3243 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) }
3246 assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
3247 apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
3250 lemma continuous_at_ball:
3251 "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3253 assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3254 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)
3255 unfolding dist_nz[THEN sym] by auto
3257 assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3258 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)
3261 text{* Define setwise continuity in terms of limits within the set. *}
3265 "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
3267 "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
3269 lemma continuous_on_topological:
3270 "continuous_on s f \<longleftrightarrow>
3271 (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
3272 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
3273 unfolding continuous_on_def tendsto_def
3274 unfolding Limits.eventually_within eventually_at_topological
3275 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
3277 lemma continuous_on_iff:
3278 "continuous_on s f \<longleftrightarrow>
3279 (\<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)"
3280 unfolding continuous_on_def Lim_within
3281 apply (intro ball_cong [OF refl] all_cong ex_cong)
3282 apply (rename_tac y, case_tac "y = x", simp)
3283 apply (simp add: dist_nz)
3287 uniformly_continuous_on ::
3288 "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
3290 "uniformly_continuous_on s f \<longleftrightarrow>
3291 (\<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)"
3293 text{* Some simple consequential lemmas. *}
3295 lemma uniformly_continuous_imp_continuous:
3296 " uniformly_continuous_on s f ==> continuous_on s f"
3297 unfolding uniformly_continuous_on_def continuous_on_iff by blast
3299 lemma continuous_at_imp_continuous_within:
3300 "continuous (at x) f ==> continuous (at x within s) f"
3301 unfolding continuous_within continuous_at using Lim_at_within by auto
3303 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
3304 unfolding tendsto_def by (simp add: trivial_limit_eq)
3306 lemma continuous_at_imp_continuous_on:
3307 assumes "\<forall>x\<in>s. continuous (at x) f"
3308 shows "continuous_on s f"
3309 unfolding continuous_on_def
3311 fix x assume "x \<in> s"
3312 with assms have *: "(f ---> f (netlimit (at x))) (at x)"
3313 unfolding continuous_def by simp
3314 have "(f ---> f x) (at x)"
3315 proof (cases "trivial_limit (at x)")
3316 case True thus ?thesis
3317 by (rule Lim_trivial_limit)
3320 hence 1: "netlimit (at x) = x"
3321 using netlimit_within [of x UNIV]
3322 by (simp add: within_UNIV)
3323 with * show ?thesis by simp
3325 thus "(f ---> f x) (at x within s)"
3326 by (rule Lim_at_within)
3329 lemma continuous_on_eq_continuous_within:
3330 "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"
3331 unfolding continuous_on_def continuous_def
3332 apply (rule ball_cong [OF refl])
3333 apply (case_tac "trivial_limit (at x within s)")
3334 apply (simp add: Lim_trivial_limit)
3335 apply (simp add: netlimit_within)
3338 lemmas continuous_on = continuous_on_def -- "legacy theorem name"
3340 lemma continuous_on_eq_continuous_at:
3341 shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
3342 by (auto simp add: continuous_on continuous_at Lim_within_open)
3344 lemma continuous_within_subset:
3345 "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
3346 ==> continuous (at x within t) f"
3347 unfolding continuous_within by(metis Lim_within_subset)
3349 lemma continuous_on_subset:
3350 shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
3351 unfolding continuous_on by (metis subset_eq Lim_within_subset)
3353 lemma continuous_on_interior:
3354 shows "continuous_on s f \<Longrightarrow> x \<in> interior s ==> continuous (at x) f"
3355 unfolding interior_def
3357 by (meson continuous_on_eq_continuous_at continuous_on_subset)
3359 lemma continuous_on_eq:
3360 "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"
3361 unfolding continuous_on_def tendsto_def Limits.eventually_within
3364 text {* Characterization of various kinds of continuity in terms of sequences. *}
3366 (* \<longrightarrow> could be generalized, but \<longleftarrow> requires metric space *)
3367 lemma continuous_within_sequentially:
3368 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3369 shows "continuous (at a within s) f \<longleftrightarrow>
3370 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
3371 --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
3374 { 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"
3375 fix e::real assume "e>0"
3376 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
3377 from x(2) `d>0` obtain N where N:"\<forall>n\<ge>N. dist (x n) a < d" by auto
3378 hence "\<exists>N. \<forall>n\<ge>N. dist ((f \<circ> x) n) (f a) < e"
3379 apply(rule_tac x=N in exI) using N d apply auto using x(1)
3380 apply(erule_tac x=n in allE) apply(erule_tac x=n in allE)
3381 apply(erule_tac x="x n" in ballE) apply auto unfolding dist_nz[THEN sym] apply auto using `e>0` by auto
3383 thus ?rhs unfolding continuous_within unfolding Lim_sequentially by simp
3386 { fix e::real assume "e>0"
3387 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)"
3388 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
3389 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)"
3390 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
3391 { fix d::real assume "d>0"
3392 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
3393 then obtain N::nat where N:"inverse (real (N + 1)) < d" by auto
3394 { fix n::nat assume n:"n\<ge>N"
3395 hence "dist (x (inverse (real (n + 1)))) a < inverse (real (n + 1))" using x[THEN spec[where x="inverse (real (n + 1))"]] by auto
3396 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)
3397 ultimately have "dist (x (inverse (real (n + 1)))) a < d" by auto
3399 hence "\<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < d" by auto
3401 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
3402 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
3403 hence "False" apply(erule_tac x=e in allE) using `e>0` using x by auto
3405 thus ?lhs unfolding continuous_within unfolding Lim_within unfolding Lim_sequentially by blast
3408 lemma continuous_at_sequentially:
3409 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3410 shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
3411 --> ((f o x) ---> f a) sequentially)"
3412 using continuous_within_sequentially[of a UNIV f] unfolding within_UNIV by auto
3414 lemma continuous_on_sequentially:
3415 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3416 shows "continuous_on s f \<longleftrightarrow>
3417 (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
3418 --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
3420 assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
3422 assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
3425 lemma uniformly_continuous_on_sequentially':
3426 "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
3427 ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially
3428 \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
3431 { 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"
3432 { fix e::real assume "e>0"
3433 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"
3434 using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
3435 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
3436 { fix n assume "n\<ge>N"
3437 hence "dist (f (x n)) (f (y n)) < e"
3438 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
3439 unfolding dist_commute by simp }
3440 hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" by auto }
3441 hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding Lim_sequentially and dist_real_def by auto }
3445 { assume "\<not> ?lhs"
3446 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
3447 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"
3448 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
3449 by (auto simp add: dist_commute)
3450 def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
3451 def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
3452 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"
3453 unfolding x_def and y_def using fa by auto
3454 { fix e::real assume "e>0"
3455 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
3456 { fix n::nat assume "n\<ge>N"
3457 hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
3458 also have "\<dots> < e" using N by auto
3459 finally have "inverse (real n + 1) < e" by auto
3460 hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto }
3461 hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto }
3462 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
3463 hence False using fxy and `e>0` by auto }
3464 thus ?lhs unfolding uniformly_continuous_on_def by blast
3467 lemma uniformly_continuous_on_sequentially:
3468 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
3469 shows "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
3470 ((\<lambda>n. x n - y n) ---> 0) sequentially
3471 \<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")
3472 (* BH: maybe the previous lemma should replace this one? *)
3473 unfolding uniformly_continuous_on_sequentially'
3474 unfolding dist_norm tendsto_norm_zero_iff ..
3476 text{* The usual transformation theorems. *}
3478 lemma continuous_transform_within:
3479 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3480 assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
3481 "continuous (at x within s) f"
3482 shows "continuous (at x within s) g"
3483 unfolding continuous_within
3484 proof (rule Lim_transform_within)
3485 show "0 < d" by fact
3486 show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
3487 using assms(3) by auto
3489 using assms(1,2,3) by auto
3490 thus "(f ---> g x) (at x within s)"
3491 using assms(4) unfolding continuous_within by simp
3494 lemma continuous_transform_at:
3495 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3496 assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
3497 "continuous (at x) f"
3498 shows "continuous (at x) g"
3499 using continuous_transform_within [of d x UNIV f g] assms
3500 by (simp add: within_UNIV)
3502 text{* Combination results for pointwise continuity. *}
3504 lemma continuous_const: "continuous net (\<lambda>x. c)"
3505 by (auto simp add: continuous_def tendsto_const)
3507 lemma continuous_cmul:
3508 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3509 shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"
3510 by (auto simp add: continuous_def intro: tendsto_intros)
3512 lemma continuous_neg:
3513 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3514 shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"
3515 by (auto simp add: continuous_def tendsto_minus)
3517 lemma continuous_add:
3518 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3519 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"
3520 by (auto simp add: continuous_def tendsto_add)
3522 lemma continuous_sub:
3523 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3524 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"
3525 by (auto simp add: continuous_def tendsto_diff)
3528 text{* Same thing for setwise continuity. *}
3530 lemma continuous_on_const:
3531 "continuous_on s (\<lambda>x. c)"
3532 unfolding continuous_on_def by (auto intro: tendsto_intros)
3534 lemma continuous_on_cmul:
3535 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3536 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c *\<^sub>R (f x))"
3537 unfolding continuous_on_def by (auto intro: tendsto_intros)
3539 lemma continuous_on_neg:
3540 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3541 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
3542 unfolding continuous_on_def by (auto intro: tendsto_intros)
3544 lemma continuous_on_add:
3545 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3546 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3547 \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
3548 unfolding continuous_on_def by (auto intro: tendsto_intros)
3550 lemma continuous_on_sub:
3551 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3552 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3553 \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
3554 unfolding continuous_on_def by (auto intro: tendsto_intros)
3556 text{* Same thing for uniform continuity, using sequential formulations. *}
3558 lemma uniformly_continuous_on_const:
3559 "uniformly_continuous_on s (\<lambda>x. c)"
3560 unfolding uniformly_continuous_on_def by simp
3562 lemma uniformly_continuous_on_cmul:
3563 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3564 assumes "uniformly_continuous_on s f"
3565 shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
3567 { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
3568 hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"
3569 using tendsto_scaleR [OF tendsto_const, of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
3570 unfolding scaleR_zero_right scaleR_right_diff_distrib by auto
3572 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
3573 unfolding dist_norm tendsto_norm_zero_iff by auto
3577 fixes x y :: "'a::real_normed_vector"
3578 shows "dist (- x) (- y) = dist x y"
3579 unfolding dist_norm minus_diff_minus norm_minus_cancel ..
3581 lemma uniformly_continuous_on_neg:
3582 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3583 shows "uniformly_continuous_on s f
3584 ==> uniformly_continuous_on s (\<lambda>x. -(f x))"
3585 unfolding uniformly_continuous_on_def dist_minus .
3587 lemma uniformly_continuous_on_add:
3588 fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3589 assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"
3590 shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
3592 { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
3593 "((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"
3594 hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"
3595 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
3596 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 }
3597 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
3598 unfolding dist_norm tendsto_norm_zero_iff by auto
3601 lemma uniformly_continuous_on_sub:
3602 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3603 shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s g
3604 ==> uniformly_continuous_on s (\<lambda>x. f x - g x)"
3605 unfolding ab_diff_minus
3606 using uniformly_continuous_on_add[of s f "\<lambda>x. - g x"]
3607 using uniformly_continuous_on_neg[of s g] by auto
3609 text{* Identity function is continuous in every sense. *}
3611 lemma continuous_within_id:
3612 "continuous (at a within s) (\<lambda>x. x)"
3613 unfolding continuous_within by (rule Lim_at_within [OF LIM_ident])
3615 lemma continuous_at_id:
3616 "continuous (at a) (\<lambda>x. x)"
3617 unfolding continuous_at by (rule LIM_ident)
3619 lemma continuous_on_id:
3620 "continuous_on s (\<lambda>x. x)"
3621 unfolding continuous_on_def by (auto intro: tendsto_ident_at_within)
3623 lemma uniformly_continuous_on_id:
3624 "uniformly_continuous_on s (\<lambda>x. x)"
3625 unfolding uniformly_continuous_on_def by auto
3627 text{* Continuity of all kinds is preserved under composition. *}
3629 lemma continuous_within_topological:
3630 "continuous (at x within s) f \<longleftrightarrow>
3631 (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
3632 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
3633 unfolding continuous_within
3634 unfolding tendsto_def Limits.eventually_within eventually_at_topological
3635 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
3637 lemma continuous_within_compose:
3638 assumes "continuous (at x within s) f"
3639 assumes "continuous (at (f x) within f ` s) g"
3640 shows "continuous (at x within s) (g o f)"
3641 using assms unfolding continuous_within_topological by simp metis
3643 lemma continuous_at_compose:
3644 assumes "continuous (at x) f" "continuous (at (f x)) g"
3645 shows "continuous (at x) (g o f)"
3647 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
3648 thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g, unfolded within_UNIV] by auto
3651 lemma continuous_on_compose:
3652 "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
3653 unfolding continuous_on_topological by simp metis
3655 lemma uniformly_continuous_on_compose:
3656 assumes "uniformly_continuous_on s f" "uniformly_continuous_on (f ` s) g"
3657 shows "uniformly_continuous_on s (g o f)"
3659 { fix e::real assume "e>0"
3660 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
3661 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
3662 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 }
3663 thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
3666 text{* Continuity in terms of open preimages. *}
3668 lemma continuous_at_open:
3669 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)))"
3670 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]
3671 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto
3673 lemma continuous_on_open:
3674 shows "continuous_on s f \<longleftrightarrow>
3675 (\<forall>t. openin (subtopology euclidean (f ` s)) t
3676 --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
3679 assume 1: "continuous_on s f"
3680 assume 2: "openin (subtopology euclidean (f ` s)) t"
3681 from 2 obtain B where B: "open B" and t: "t = f ` s \<inter> B"
3682 unfolding openin_open by auto
3683 def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"
3684 have "open U" unfolding U_def by (simp add: open_Union)
3685 moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"
3686 proof (intro ballI iffI)
3687 fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"
3688 unfolding U_def t by auto
3690 fix x assume "x \<in> s" and "f x \<in> t"
3691 hence "x \<in> s" and "f x \<in> B"
3693 with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"
3694 unfolding t continuous_on_topological by metis
3695 then show "x \<in> U"
3696 unfolding U_def by auto
3698 ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto
3699 then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3700 unfolding openin_open by fast
3702 assume "?rhs" show "continuous_on s f"
3703 unfolding continuous_on_topological
3705 fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"
3706 have "openin (subtopology euclidean (f ` s)) (f ` s \<inter> B)"
3707 unfolding openin_open using `open B` by auto
3708 then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f ` s \<inter> B}"
3709 using `?rhs` by fast
3710 then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
3711 unfolding openin_open using `x \<in> s` and `f x \<in> B` by auto
3715 text {* Similarly in terms of closed sets. *}
3717 lemma continuous_on_closed:
3718 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")
3722 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3723 have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
3724 assume as:"closedin (subtopology euclidean (f ` s)) t"
3725 hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
3726 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"]]
3727 unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto }
3732 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3733 assume as:"openin (subtopology euclidean (f ` s)) t"
3734 hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
3735 unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
3736 thus ?lhs unfolding continuous_on_open by auto
3739 text {* Half-global and completely global cases. *}
3741 lemma continuous_open_in_preimage:
3742 assumes "continuous_on s f" "open t"
3743 shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3745 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
3746 have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3747 using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
3748 thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3751 lemma continuous_closed_in_preimage:
3752 assumes "continuous_on s f" "closed t"
3753 shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3755 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
3756 have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3757 using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
3759 using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3762 lemma continuous_open_preimage:
3763 assumes "continuous_on s f" "open s" "open t"
3764 shows "open {x \<in> s. f x \<in> t}"
3766 obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3767 using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
3768 thus ?thesis using open_Int[of s T, OF assms(2)] by auto
3771 lemma continuous_closed_preimage:
3772 assumes "continuous_on s f" "closed s" "closed t"
3773 shows "closed {x \<in> s. f x \<in> t}"
3775 obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3776 using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
3777 thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
3780 lemma continuous_open_preimage_univ:
3781 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
3782 using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
3784 lemma continuous_closed_preimage_univ:
3785 shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
3786 using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
3788 lemma continuous_open_vimage:
3789 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
3790 unfolding vimage_def by (rule continuous_open_preimage_univ)
3792 lemma continuous_closed_vimage:
3793 shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
3794 unfolding vimage_def by (rule continuous_closed_preimage_univ)
3796 lemma interior_image_subset:
3797 assumes "\<forall>x. continuous (at x) f" "inj f"
3798 shows "interior (f ` s) \<subseteq> f ` (interior s)"
3799 apply rule unfolding interior_def mem_Collect_eq image_iff apply safe
3800 proof- fix x T assume as:"open T" "x \<in> T" "T \<subseteq> f ` s"
3801 hence "x \<in> f ` s" by auto then guess y unfolding image_iff .. note y=this
3802 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
3803 apply safe apply(rule_tac x="{x. f x \<in> T}" in exI) apply(safe,rule continuous_open_preimage_univ)
3804 proof- fix x assume "f x \<in> T" hence "f x \<in> f ` s" using as by auto
3805 thus "x \<in> s" unfolding inj_image_mem_iff[OF assms(2)] . qed auto qed
3807 text {* Equality of continuous functions on closure and related results. *}
3809 lemma continuous_closed_in_preimage_constant:
3810 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3811 shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
3812 using continuous_closed_in_preimage[of s f "{a}"] by auto
3814 lemma continuous_closed_preimage_constant:
3815 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3816 shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
3817 using continuous_closed_preimage[of s f "{a}"] by auto
3819 lemma continuous_constant_on_closure:
3820 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3821 assumes "continuous_on (closure s) f"
3822 "\<forall>x \<in> s. f x = a"
3823 shows "\<forall>x \<in> (closure s). f x = a"
3824 using continuous_closed_preimage_constant[of "closure s" f a]
3825 assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
3827 lemma image_closure_subset:
3828 assumes "continuous_on (closure s) f" "closed t" "(f ` s) \<subseteq> t"
3829 shows "f ` (closure s) \<subseteq> t"
3831 have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
3832 moreover have "closed {x \<in> closure s. f x \<in> t}"
3833 using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
3834 ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
3835 using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
3836 thus ?thesis by auto
3839 lemma continuous_on_closure_norm_le:
3840 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3841 assumes "continuous_on (closure s) f" "\<forall>y \<in> s. norm(f y) \<le> b" "x \<in> (closure s)"
3842 shows "norm(f x) \<le> b"
3844 have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
3846 using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
3847 unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
3850 text {* Making a continuous function avoid some value in a neighbourhood. *}
3852 lemma continuous_within_avoid:
3853 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3854 assumes "continuous (at x within s) f" "x \<in> s" "f x \<noteq> a"
3855 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
3857 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"
3858 using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto
3859 { fix y assume " y\<in>s" "dist x y < d"
3860 hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]
3861 apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }
3862 thus ?thesis using `d>0` by auto
3865 lemma continuous_at_avoid:
3866 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3867 assumes "continuous (at x) f" "f x \<noteq> a"
3868 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
3869 using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto
3871 lemma continuous_on_avoid:
3872 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
3873 assumes "continuous_on s f" "x \<in> s" "f x \<noteq> a"
3874 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
3875 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
3877 lemma continuous_on_open_avoid:
3878 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
3879 assumes "continuous_on s f" "open s" "x \<in> s" "f x \<noteq> a"
3880 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
3881 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
3883 text {* Proving a function is constant by proving open-ness of level set. *}
3885 lemma continuous_levelset_open_in_cases:
3886 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3887 shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
3888 openin (subtopology euclidean s) {x \<in> s. f x = a}
3889 ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
3890 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
3892 lemma continuous_levelset_open_in:
3893 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3894 shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
3895 openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
3896 (\<exists>x \<in> s. f x = a) ==> (\<forall>x \<in> s. f x = a)"
3897 using continuous_levelset_open_in_cases[of s f ]
3900 lemma continuous_levelset_open:
3901 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3902 assumes "connected s" "continuous_on s f" "open {x \<in> s. f x = a}" "\<exists>x \<in> s. f x = a"
3903 shows "\<forall>x \<in> s. f x = a"
3904 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
3906 text {* Some arithmetical combinations (more to prove). *}
3908 lemma open_scaling[intro]:
3909 fixes s :: "'a::real_normed_vector set"
3910 assumes "c \<noteq> 0" "open s"
3911 shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
3913 { fix x assume "x \<in> s"
3914 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
3915 have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF `e>0`] by auto
3917 { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
3918 hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
3919 using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
3920 assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
3921 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 }
3922 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 }
3923 thus ?thesis unfolding open_dist by auto
3926 lemma minus_image_eq_vimage:
3927 fixes A :: "'a::ab_group_add set"
3928 shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
3929 by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
3931 lemma open_negations:
3932 fixes s :: "'a::real_normed_vector set"
3933 shows "open s ==> open ((\<lambda> x. -x) ` s)"
3934 unfolding scaleR_minus1_left [symmetric]
3935 by (rule open_scaling, auto)
3937 lemma open_translation:
3938 fixes s :: "'a::real_normed_vector set"
3939 assumes "open s" shows "open((\<lambda>x. a + x) ` s)"
3941 { 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 }
3942 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
3943 ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
3946 lemma open_affinity:
3947 fixes s :: "'a::real_normed_vector set"
3948 assumes "open s" "c \<noteq> 0"
3949 shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
3951 have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
3952 have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
3953 thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
3956 lemma interior_translation:
3957 fixes s :: "'a::real_normed_vector set"
3958 shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
3959 proof (rule set_eqI, rule)
3960 fix x assume "x \<in> interior (op + a ` s)"
3961 then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
3962 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
3963 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
3965 fix x assume "x \<in> op + a ` interior s"
3966 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
3967 { fix z have *:"a + y - z = y + a - z" by auto
3968 assume "z\<in>ball x e"
3969 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
3970 hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"]) }
3971 hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
3972 thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
3975 text {* We can now extend limit compositions to consider the scalar multiplier. *}
3977 lemma continuous_vmul:
3978 fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
3979 shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"
3980 unfolding continuous_def by (intro tendsto_intros)
3982 lemma continuous_mul:
3983 fixes c :: "'a::metric_space \<Rightarrow> real"
3984 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3985 shows "continuous net c \<Longrightarrow> continuous net f
3986 ==> continuous net (\<lambda>x. c(x) *\<^sub>R f x) "
3987 unfolding continuous_def by (intro tendsto_intros)
3989 lemmas continuous_intros = continuous_add continuous_vmul continuous_cmul
3990 continuous_const continuous_sub continuous_at_id continuous_within_id continuous_mul
3992 lemma continuous_on_vmul:
3993 fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
3994 shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R v)"
3995 unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
3997 lemma continuous_on_mul:
3998 fixes c :: "'a::metric_space \<Rightarrow> real"
3999 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
4000 shows "continuous_on s c \<Longrightarrow> continuous_on s f
4001 ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R f x)"
4002 unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
4004 lemma continuous_on_mul_real:
4005 fixes f :: "'a::metric_space \<Rightarrow> real"
4006 fixes g :: "'a::metric_space \<Rightarrow> real"
4007 shows "continuous_on s f \<Longrightarrow> continuous_on s g
4008 ==> continuous_on s (\<lambda>x. f x * g x)"
4009 using continuous_on_mul[of s f g] unfolding real_scaleR_def .
4011 lemmas continuous_on_intros = continuous_on_add continuous_on_const
4012 continuous_on_id continuous_on_compose continuous_on_cmul continuous_on_neg
4013 continuous_on_sub continuous_on_mul continuous_on_vmul continuous_on_mul_real
4014 uniformly_continuous_on_add uniformly_continuous_on_const
4015 uniformly_continuous_on_id uniformly_continuous_on_compose
4016 uniformly_continuous_on_cmul uniformly_continuous_on_neg
4017 uniformly_continuous_on_sub
4019 text {* And so we have continuity of inverse. *}
4021 lemma continuous_inv:
4022 fixes f :: "'a::metric_space \<Rightarrow> real"
4023 shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
4024 ==> continuous net (inverse o f)"
4025 unfolding continuous_def using Lim_inv by auto
4027 lemma continuous_at_within_inv:
4028 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
4029 assumes "continuous (at a within s) f" "f a \<noteq> 0"
4030 shows "continuous (at a within s) (inverse o f)"
4031 using assms unfolding continuous_within o_def
4032 by (intro tendsto_intros)
4034 lemma continuous_at_inv:
4035 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
4036 shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
4037 ==> continuous (at a) (inverse o f) "
4038 using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
4040 text {* Topological properties of linear functions. *}
4043 assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
4045 interpret f: bounded_linear f by fact
4046 have "(f ---> f 0) (at 0)"
4047 using tendsto_ident_at by (rule f.tendsto)
4048 thus ?thesis unfolding f.zero .
4051 lemma linear_continuous_at:
4052 assumes "bounded_linear f" shows "continuous (at a) f"
4053 unfolding continuous_at using assms
4054 apply (rule bounded_linear.tendsto)
4055 apply (rule tendsto_ident_at)
4058 lemma linear_continuous_within:
4059 shows "bounded_linear f ==> continuous (at x within s) f"
4060 using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
4062 lemma linear_continuous_on:
4063 shows "bounded_linear f ==> continuous_on s f"
4064 using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
4066 text {* Also bilinear functions, in composition form. *}
4068 lemma bilinear_continuous_at_compose:
4069 shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
4070 ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
4071 unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
4073 lemma bilinear_continuous_within_compose:
4074 shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
4075 ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
4076 unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
4078 lemma bilinear_continuous_on_compose:
4079 shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
4080 ==> continuous_on s (\<lambda>x. h (f x) (g x))"
4081 unfolding continuous_on_def
4082 by (fast elim: bounded_bilinear.tendsto)
4084 text {* Preservation of compactness and connectedness under continuous function. *}
4086 lemma compact_continuous_image:
4087 assumes "continuous_on s f" "compact s"
4088 shows "compact(f ` s)"
4090 { fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
4091 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
4092 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
4093 { fix e::real assume "e>0"
4094 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
4095 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
4096 { 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 }
4097 hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto }
4098 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 }
4099 thus ?thesis unfolding compact_def by auto
4102 lemma connected_continuous_image:
4103 assumes "continuous_on s f" "connected s"
4104 shows "connected(f ` s)"
4106 { fix T assume as: "T \<noteq> {}" "T \<noteq> f ` s" "openin (subtopology euclidean (f ` s)) T" "closedin (subtopology euclidean (f ` s)) T"
4107 have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
4108 using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
4109 using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
4110 using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
4111 hence False using as(1,2)
4112 using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
4113 thus ?thesis unfolding connected_clopen by auto
4116 text {* Continuity implies uniform continuity on a compact domain. *}
4118 lemma compact_uniformly_continuous:
4119 assumes "continuous_on s f" "compact s"
4120 shows "uniformly_continuous_on s f"
4122 { fix x assume x:"x\<in>s"
4123 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
4124 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 }
4125 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
4126 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)"
4127 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
4129 { fix e::real assume "e>0"
4131 { 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 }
4132 hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto
4134 { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto }
4135 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
4137 { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"
4138 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
4139 hence "x\<in>ball z (d z (e / 2))" using `ea>0` unfolding subset_eq by auto
4140 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`
4141 by (auto simp add: dist_commute)
4142 moreover have "y\<in>ball z (d z (e / 2))" using as and `ea>0` and z[unfolded subset_eq]
4143 by (auto simp add: dist_commute)
4144 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`
4145 by (auto simp add: dist_commute)
4146 ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]
4147 by (auto simp add: dist_commute) }
4148 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 }
4149 thus ?thesis unfolding uniformly_continuous_on_def by auto
4152 text{* Continuity of inverse function on compact domain. *}
4154 lemma continuous_on_inverse:
4155 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
4156 (* TODO: can this be generalized more? *)
4157 assumes "continuous_on s f" "compact s" "\<forall>x \<in> s. g (f x) = x"
4158 shows "continuous_on (f ` s) g"
4160 have *:"g ` f ` s = s" using assms(3) by (auto simp add: image_iff)
4161 { fix t assume t:"closedin (subtopology euclidean (g ` f ` s)) t"
4162 then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto
4163 have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]
4164 unfolding T(2) and Int_left_absorb by auto
4165 moreover have "compact (s \<inter> T)"
4166 using assms(2) unfolding compact_eq_bounded_closed
4167 using bounded_subset[of s "s \<inter> T"] and T(1) by auto
4168 ultimately have "closed (f ` t)" using T(1) unfolding T(2)
4169 using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto
4170 moreover have "{x \<in> f ` s. g x \<in> t} = f ` s \<inter> f ` t" using assms(3) unfolding T(2) by auto
4171 ultimately have "closedin (subtopology euclidean (f ` s)) {x \<in> f ` s. g x \<in> t}"
4172 unfolding closedin_closed by auto }
4173 thus ?thesis unfolding continuous_on_closed by auto
4176 text {* A uniformly convergent limit of continuous functions is continuous. *}
4178 lemma continuous_uniform_limit:
4179 fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"
4180 assumes "\<not> trivial_limit F"
4181 assumes "eventually (\<lambda>n. continuous_on s (f n)) F"
4182 assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"
4183 shows "continuous_on s g"
4185 { fix x and e::real assume "x\<in>s" "e>0"
4186 have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"
4187 using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto
4188 from eventually_happens [OF eventually_conj [OF this assms(2)]]
4189 obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3" "continuous_on s (f n)"
4190 using assms(1) by blast
4191 have "e / 3 > 0" using `e>0` by auto
4192 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"
4193 using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
4194 { fix y assume "y \<in> s" and "dist y x < d"
4195 hence "dist (f n y) (f n x) < e / 3"
4196 by (rule d [rule_format])
4197 hence "dist (f n y) (g x) < 2 * e / 3"
4198 using dist_triangle [of "f n y" "g x" "f n x"]
4199 using n(1)[THEN bspec[where x=x], OF `x\<in>s`]
4201 hence "dist (g y) (g x) < e"
4202 using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
4203 using dist_triangle3 [of "g y" "g x" "f n y"]
4205 hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
4206 using `d>0` by auto }
4207 thus ?thesis unfolding continuous_on_iff by auto
4211 subsection {* Topological stuff lifted from and dropped to R *}
4214 fixes s :: "real set" shows
4215 "open s \<longleftrightarrow>
4216 (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
4217 unfolding open_dist dist_norm by simp
4219 lemma islimpt_approachable_real:
4220 fixes s :: "real set"
4221 shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
4222 unfolding islimpt_approachable dist_norm by simp
4225 fixes s :: "real set"
4226 shows "closed s \<longleftrightarrow>
4227 (\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
4229 unfolding closed_limpt islimpt_approachable dist_norm by simp
4231 lemma continuous_at_real_range:
4232 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4233 shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
4234 \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
4235 unfolding continuous_at unfolding Lim_at
4236 unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
4237 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
4238 apply(erule_tac x=e in allE) by auto
4240 lemma continuous_on_real_range:
4241 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4242 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))"
4243 unfolding continuous_on_iff dist_norm by simp
4245 lemma continuous_at_norm: "continuous (at x) norm"
4246 unfolding continuous_at by (intro tendsto_intros)
4248 lemma continuous_on_norm: "continuous_on s norm"
4249 unfolding continuous_on by (intro ballI tendsto_intros)
4251 lemma continuous_at_infnorm: "continuous (at x) infnorm"
4252 unfolding continuous_at Lim_at o_def unfolding dist_norm
4253 apply auto apply (rule_tac x=e in exI) apply auto
4254 using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
4256 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}
4258 lemma compact_attains_sup:
4259 fixes s :: "real set"
4260 assumes "compact s" "s \<noteq> {}"
4261 shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
4263 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4264 { 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"
4265 have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto
4266 moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
4267 ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using `e>0` by auto }
4268 thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]
4269 apply(rule_tac x="Sup s" in bexI) by auto
4273 fixes S :: "real set"
4274 shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
4275 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def)
4277 lemma compact_attains_inf:
4278 fixes s :: "real set"
4279 assumes "compact s" "s \<noteq> {}" shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
4281 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4282 { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s" "Inf s \<notin> s" "0 < e"
4283 "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"
4284 have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto
4286 { fix x assume "x \<in> s"
4287 hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto
4288 have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
4289 hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto
4290 ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using `e>0` by auto }
4291 thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]
4292 apply(rule_tac x="Inf s" in bexI) by auto
4295 lemma continuous_attains_sup:
4296 fixes f :: "'a::metric_space \<Rightarrow> real"
4297 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4298 ==> (\<exists>x \<in> s. \<forall>y \<in> s. f y \<le> f x)"
4299 using compact_attains_sup[of "f ` s"]
4300 using compact_continuous_image[of s f] by auto
4302 lemma continuous_attains_inf:
4303 fixes f :: "'a::metric_space \<Rightarrow> real"
4304 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4305 \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
4306 using compact_attains_inf[of "f ` s"]
4307 using compact_continuous_image[of s f] by auto
4309 lemma distance_attains_sup:
4310 assumes "compact s" "s \<noteq> {}"
4311 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
4312 proof (rule continuous_attains_sup [OF assms])
4313 { fix x assume "x\<in>s"
4314 have "(dist a ---> dist a x) (at x within s)"
4315 by (intro tendsto_dist tendsto_const Lim_at_within LIM_ident)
4317 thus "continuous_on s (dist a)"
4318 unfolding continuous_on ..
4321 text {* For \emph{minimal} distance, we only need closure, not compactness. *}
4323 lemma distance_attains_inf:
4324 fixes a :: "'a::heine_borel"
4325 assumes "closed s" "s \<noteq> {}"
4326 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
4328 from assms(2) obtain b where "b\<in>s" by auto
4329 let ?B = "cball a (dist b a) \<inter> s"
4330 have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
4331 hence "?B \<noteq> {}" by auto
4333 { fix x assume "x\<in>?B"
4334 fix e::real assume "e>0"
4335 { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
4336 from as have "\<bar>dist a x' - dist a x\<bar> < e"
4337 unfolding abs_less_iff minus_diff_eq
4338 using dist_triangle2 [of a x' x]
4339 using dist_triangle [of a x x']
4342 hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
4345 hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
4346 unfolding continuous_on Lim_within dist_norm real_norm_def
4348 moreover have "compact ?B"
4349 using compact_cball[of a "dist b a"]
4350 unfolding compact_eq_bounded_closed
4351 using bounded_Int and closed_Int and assms(1) by auto
4352 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"
4353 using continuous_attains_inf[of ?B "dist a"] by fastsimp
4354 thus ?thesis by fastsimp
4358 subsection {* Pasted sets *}
4360 lemma bounded_Times:
4361 assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
4363 obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
4364 using assms [unfolded bounded_def] by auto
4365 then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
4366 by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
4367 thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
4370 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
4373 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
4374 unfolding compact_def
4376 apply (drule_tac x="fst \<circ> f" in spec)
4377 apply (drule mp, simp add: mem_Times_iff)
4378 apply (clarify, rename_tac l1 r1)
4379 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
4380 apply (drule mp, simp add: mem_Times_iff)
4381 apply (clarify, rename_tac l2 r2)
4382 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
4383 apply (rule_tac x="r1 \<circ> r2" in exI)
4384 apply (rule conjI, simp add: subseq_def)
4385 apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
4386 apply (drule (1) tendsto_Pair) back
4387 apply (simp add: o_def)
4390 text{* Hence some useful properties follow quite easily. *}
4392 lemma compact_scaling:
4393 fixes s :: "'a::real_normed_vector set"
4394 assumes "compact s" shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
4396 let ?f = "\<lambda>x. scaleR c x"
4397 have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)
4398 show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
4399 using linear_continuous_at[OF *] assms by auto
4402 lemma compact_negations:
4403 fixes s :: "'a::real_normed_vector set"
4404 assumes "compact s" shows "compact ((\<lambda>x. -x) ` s)"
4405 using compact_scaling [OF assms, of "- 1"] by auto
4408 fixes s t :: "'a::real_normed_vector set"
4409 assumes "compact s" "compact t" shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
4411 have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
4412 apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
4413 have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
4414 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
4415 thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
4418 lemma compact_differences:
4419 fixes s t :: "'a::real_normed_vector set"
4420 assumes "compact s" "compact t" shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
4422 have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
4423 apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4424 thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
4427 lemma compact_translation:
4428 fixes s :: "'a::real_normed_vector set"
4429 assumes "compact s" shows "compact ((\<lambda>x. a + x) ` s)"
4431 have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
4432 thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
4435 lemma compact_affinity:
4436 fixes s :: "'a::real_normed_vector set"
4437 assumes "compact s" shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
4439 have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
4440 thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
4443 text {* Hence we get the following. *}
4445 lemma compact_sup_maxdistance:
4446 fixes s :: "'a::real_normed_vector set"
4447 assumes "compact s" "s \<noteq> {}"
4448 shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
4450 have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
4451 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"
4452 using compact_differences[OF assms(1) assms(1)]
4453 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
4454 from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
4455 thus ?thesis using x(2)[unfolded `x = a - b`] by blast
4458 text {* We can state this in terms of diameter of a set. *}
4460 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"
4461 (* TODO: generalize to class metric_space *)
4463 lemma diameter_bounded:
4465 shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4466 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"
4468 let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"
4469 obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto
4470 { fix x y assume "x \<in> s" "y \<in> s"
4471 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) }
4473 { fix x y assume "x\<in>s" "y\<in>s" hence "s \<noteq> {}" by auto
4474 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`
4475 by simp (blast intro!: Sup_upper *) }
4477 { fix d::real assume "d>0" "d < diameter s"
4478 hence "s\<noteq>{}" unfolding diameter_def by auto
4479 have "\<exists>d' \<in> ?D. d' > d"
4481 assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"
4482 hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE)
4483 thus False using `d < diameter s` `s\<noteq>{}`
4484 apply (auto simp add: diameter_def)
4485 apply (drule Sup_real_iff [THEN [2] rev_iffD2])
4489 hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto }
4490 ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4491 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto
4494 lemma diameter_bounded_bound:
4495 "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"
4496 using diameter_bounded by blast
4498 lemma diameter_compact_attained:
4499 fixes s :: "'a::real_normed_vector set"
4500 assumes "compact s" "s \<noteq> {}"
4501 shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
4503 have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
4504 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
4505 hence "diameter s \<le> norm (x - y)"
4506 unfolding diameter_def by clarsimp (rule Sup_least, fast+)
4508 by (metis b diameter_bounded_bound order_antisym xys)
4511 text {* Related results with closure as the conclusion. *}
4513 lemma closed_scaling:
4514 fixes s :: "'a::real_normed_vector set"
4515 assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
4517 case True thus ?thesis by auto
4522 have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
4523 case True thus ?thesis apply auto unfolding * by auto
4526 { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s" "(x ---> l) sequentially"
4527 { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
4528 using as(1)[THEN spec[where x=n]]
4529 using `c\<noteq>0` by auto
4532 { fix e::real assume "e>0"
4533 hence "0 < e *\<bar>c\<bar>" using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
4534 then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
4535 using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto
4536 hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
4537 unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
4538 using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto }
4539 hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto
4540 ultimately have "l \<in> scaleR c ` s"
4541 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"]]
4542 unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto }
4543 thus ?thesis unfolding closed_sequential_limits by fast
4547 lemma closed_negations:
4548 fixes s :: "'a::real_normed_vector set"
4549 assumes "closed s" shows "closed ((\<lambda>x. -x) ` s)"
4550 using closed_scaling[OF assms, of "- 1"] by simp
4552 lemma compact_closed_sums:
4553 fixes s :: "'a::real_normed_vector set"
4554 assumes "compact s" "closed t" shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4556 let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
4557 { fix x l assume as:"\<forall>n. x n \<in> ?S" "(x ---> l) sequentially"
4558 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"
4559 using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
4560 obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
4561 using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
4562 have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
4563 using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
4564 hence "l - l' \<in> t"
4565 using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
4567 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
4569 thus ?thesis unfolding closed_sequential_limits by fast
4572 lemma closed_compact_sums:
4573 fixes s t :: "'a::real_normed_vector set"
4574 assumes "closed s" "compact t"
4575 shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4577 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
4578 apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
4579 thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
4582 lemma compact_closed_differences:
4583 fixes s t :: "'a::real_normed_vector set"
4584 assumes "compact s" "closed t"
4585 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4587 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4588 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4589 thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
4592 lemma closed_compact_differences:
4593 fixes s t :: "'a::real_normed_vector set"
4594 assumes "closed s" "compact t"
4595 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4597 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4598 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4599 thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
4602 lemma closed_translation:
4603 fixes a :: "'a::real_normed_vector"
4604 assumes "closed s" shows "closed ((\<lambda>x. a + x) ` s)"
4606 have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
4607 thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
4610 lemma translation_Compl:
4611 fixes a :: "'a::ab_group_add"
4612 shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
4613 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto
4615 lemma translation_UNIV:
4616 fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
4617 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
4619 lemma translation_diff:
4620 fixes a :: "'a::ab_group_add"
4621 shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
4624 lemma closure_translation:
4625 fixes a :: "'a::real_normed_vector"
4626 shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
4628 have *:"op + a ` (- s) = - op + a ` s"
4629 apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
4630 show ?thesis unfolding closure_interior translation_Compl
4631 using interior_translation[of a "- s"] unfolding * by auto
4634 lemma frontier_translation:
4635 fixes a :: "'a::real_normed_vector"
4636 shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
4637 unfolding frontier_def translation_diff interior_translation closure_translation by auto
4640 subsection {* Separation between points and sets *}
4642 lemma separate_point_closed:
4643 fixes s :: "'a::heine_borel set"
4644 shows "closed s \<Longrightarrow> a \<notin> s ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
4645 proof(cases "s = {}")
4647 thus ?thesis by(auto intro!: exI[where x=1])
4650 assume "closed s" "a \<notin> s"
4651 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
4652 with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
4655 lemma separate_compact_closed:
4656 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4657 (* TODO: does this generalize to heine_borel? *)
4658 assumes "compact s" and "closed t" and "s \<inter> t = {}"
4659 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4661 have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto
4662 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"
4663 using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto
4664 { fix x y assume "x\<in>s" "y\<in>t"
4665 hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto
4666 hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute
4667 by (auto simp add: dist_commute)
4668 hence "d \<le> dist x y" unfolding dist_norm by auto }
4669 thus ?thesis using `d>0` by auto
4672 lemma separate_closed_compact:
4673 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4674 assumes "closed s" and "compact t" and "s \<inter> t = {}"
4675 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4677 have *:"t \<inter> s = {}" using assms(3) by auto
4678 show ?thesis using separate_compact_closed[OF assms(2,1) *]
4679 apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
4680 by (auto simp add: dist_commute)
4684 subsection {* Intervals *}
4686 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows
4687 "{a <..< b} = {x::'a. \<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i}" and
4688 "{a .. b} = {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i}"
4689 by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
4691 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows
4692 "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i)"
4693 "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i)"
4694 using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
4696 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows
4697 "({a <..< b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i))" (is ?th1) and
4698 "({a .. b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i < a$$i))" (is ?th2)
4700 { fix i x assume i:"i<DIM('a)" and as:"b$$i \<le> a$$i" and x:"x\<in>{a <..< b}"
4701 hence "a $$ i < x $$ i \<and> x $$ i < b $$ i" unfolding mem_interval by auto
4702 hence "a$$i < b$$i" by auto
4703 hence False using as by auto }
4705 { assume as:"\<forall>i<DIM('a). \<not> (b$$i \<le> a$$i)"
4706 let ?x = "(1/2) *\<^sub>R (a + b)"
4707 { fix i assume i:"i<DIM('a)"
4708 have "a$$i < b$$i" using as[THEN spec[where x=i]] using i by auto
4709 hence "a$$i < ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i < b$$i"
4710 unfolding euclidean_simps by auto }
4711 hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto }
4712 ultimately show ?th1 by blast
4714 { fix i x assume i:"i<DIM('a)" and as:"b$$i < a$$i" and x:"x\<in>{a .. b}"
4715 hence "a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i" unfolding mem_interval by auto
4716 hence "a$$i \<le> b$$i" by auto
4717 hence False using as by auto }
4719 { assume as:"\<forall>i<DIM('a). \<not> (b$$i < a$$i)"
4720 let ?x = "(1/2) *\<^sub>R (a + b)"
4721 { fix i assume i:"i<DIM('a)"
4722 have "a$$i \<le> b$$i" using as[THEN spec[where x=i]] by auto
4723 hence "a$$i \<le> ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i \<le> b$$i"
4724 unfolding euclidean_simps by auto }
4725 hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto }
4726 ultimately show ?th2 by blast
4729 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows
4730 "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> b$$i)" and
4731 "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"
4732 unfolding interval_eq_empty[of a b] by fastsimp+
4734 lemma interval_sing: fixes a :: "'a::ordered_euclidean_space" shows
4735 "{a .. a} = {a}" "{a<..<a} = {}"
4736 apply(auto simp add: set_eq_iff euclidean_eq[where 'a='a] eucl_less[where 'a='a] eucl_le[where 'a='a])
4737 apply (simp add: order_eq_iff) apply(rule_tac x=0 in exI) by (auto simp add: not_less less_imp_le)
4739 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows
4740 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
4741 "(\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
4742 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
4743 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
4744 unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
4745 by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
4747 lemma interval_open_subset_closed: fixes a :: "'a::ordered_euclidean_space" shows
4748 "{a<..<b} \<subseteq> {a .. b}"
4749 proof(simp add: subset_eq, rule)
4751 assume x:"x \<in>{a<..<b}"
4752 { fix i assume "i<DIM('a)"
4753 hence "a $$ i \<le> x $$ i"
4754 using x order_less_imp_le[of "a$$i" "x$$i"]
4755 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4758 { fix i assume "i<DIM('a)"
4759 hence "x $$ i \<le> b $$ i"
4760 using x order_less_imp_le[of "x$$i" "b$$i"]
4761 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4764 show "a \<le> x \<and> x \<le> b"
4765 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4768 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows
4769 "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th1) and
4770 "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i)" (is ?th2) and
4771 "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th3) and
4772 "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th4)
4774 show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
4775 show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
4776 { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i<DIM('a). c$$i < d$$i"
4777 hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
4778 fix i assume i:"i<DIM('a)"
4779 (** TODO combine the following two parts as done in the HOL_light version. **)
4780 { let ?x = "(\<chi>\<chi> j. (if j=i then ((min (a$$j) (d$$j))+c$$j)/2 else (c$$j+d$$j)/2))::'a"
4781 assume as2: "a$$i > c$$i"
4782 { fix j assume j:"j<DIM('a)"
4783 hence "c $$ j < ?x $$ j \<and> ?x $$ j < d $$ j"
4784 apply(cases "j=i") using as(2)[THEN spec[where x=j]] i
4785 by (auto simp add: as2) }
4786 hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto
4788 have "?x\<notin>{a .. b}"
4789 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4790 using as(2)[THEN spec[where x=i]] and as2 i
4792 ultimately have False using as by auto }
4793 hence "a$$i \<le> c$$i" by(rule ccontr)auto
4795 { let ?x = "(\<chi>\<chi> j. (if j=i then ((max (b$$j) (c$$j))+d$$j)/2 else (c$$j+d$$j)/2))::'a"
4796 assume as2: "b$$i < d$$i"
4797 { fix j assume "j<DIM('a)"
4798 hence "d $$ j > ?x $$ j \<and> ?x $$ j > c $$ j"
4799 apply(cases "j=i") using as(2)[THEN spec[where x=j]]
4800 by (auto simp add: as2) }
4801 hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
4803 have "?x\<notin>{a .. b}"
4804 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4805 using as(2)[THEN spec[where x=i]] and as2 using i
4807 ultimately have False using as by auto }
4808 hence "b$$i \<ge> d$$i" by(rule ccontr)auto
4810 have "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" by auto
4812 show ?th3 unfolding subset_eq and Ball_def and mem_interval
4813 apply(rule,rule,rule,rule) apply(rule part1) unfolding subset_eq and Ball_def and mem_interval
4814 prefer 4 apply auto by(erule_tac x=i in allE,erule_tac x=i in allE,fastsimp)+
4815 { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i<DIM('a). c$$i < d$$i"
4816 fix i assume i:"i<DIM('a)"
4817 from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
4818 hence "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" using part1 and as(2) using i by auto } note * = this
4819 show ?th4 unfolding subset_eq and Ball_def and mem_interval
4820 apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4
4821 apply auto by(erule_tac x=i in allE, simp)+
4824 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows
4825 "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i < c$$i \<or> b$$i < c$$i \<or> d$$i < a$$i))" (is ?th1) and
4826 "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th2) and
4827 "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i < c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th3) and
4828 "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th4)
4830 let ?z = "(\<chi>\<chi> i. ((max (a$$i) (c$$i)) + (min (b$$i) (d$$i))) / 2)::'a"
4831 note * = set_eq_iff Int_iff empty_iff mem_interval all_conj_distrib[THEN sym] eq_False
4832 show ?th1 unfolding * apply safe apply(erule_tac x="?z" in allE)
4833 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4834 show ?th2 unfolding * apply safe apply(erule_tac x="?z" in allE)
4835 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4836 show ?th3 unfolding * apply safe apply(erule_tac x="?z" in allE)
4837 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4838 show ?th4 unfolding * apply safe apply(erule_tac x="?z" in allE)
4839 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4842 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows
4843 "{a .. b} \<inter> {c .. d} = {(\<chi>\<chi> i. max (a$$i) (c$$i)) .. (\<chi>\<chi> i. min (b$$i) (d$$i))}"
4844 unfolding set_eq_iff and Int_iff and mem_interval
4847 (* Moved interval_open_subset_closed a bit upwards *)
4849 lemma open_interval[intro]:
4850 fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
4852 have "open (\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i<..<b$$i})"
4853 by (intro open_INT finite_lessThan ballI continuous_open_vimage allI
4854 linear_continuous_at bounded_linear_euclidean_component
4855 open_real_greaterThanLessThan)
4856 also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i<..<b$$i}) = {a<..<b}"
4857 by (auto simp add: eucl_less [where 'a='a])
4858 finally show "open {a<..<b}" .
4861 lemma closed_interval[intro]:
4862 fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
4864 have "closed (\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i .. b$$i})"
4865 by (intro closed_INT ballI continuous_closed_vimage allI
4866 linear_continuous_at bounded_linear_euclidean_component
4867 closed_real_atLeastAtMost)
4868 also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i .. b$$i}) = {a .. b}"
4869 by (auto simp add: eucl_le [where 'a='a])
4870 finally show "closed {a .. b}" .
4873 lemma interior_closed_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows
4874 "interior {a .. b} = {a<..<b}" (is "?L = ?R")
4875 proof(rule subset_antisym)
4876 show "?R \<subseteq> ?L" using interior_maximal[OF interval_open_subset_closed open_interval] by auto
4878 { fix x assume "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> {a..b}"
4879 then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" by auto
4880 then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto
4881 { fix i assume i:"i<DIM('a)"
4882 have "dist (x - (e / 2) *\<^sub>R basis i) x < e"
4883 "dist (x + (e / 2) *\<^sub>R basis i) x < e"
4884 unfolding dist_norm apply auto
4885 unfolding norm_minus_cancel using norm_basis and `e>0` by auto
4886 hence "a $$ i \<le> (x - (e / 2) *\<^sub>R basis i) $$ i"
4887 "(x + (e / 2) *\<^sub>R basis i) $$ i \<le> b $$ i"
4888 using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]
4889 and e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]
4890 unfolding mem_interval by (auto elim!: allE[where x=i])
4891 hence "a $$ i < x $$ i" and "x $$ i < b $$ i" unfolding euclidean_simps
4892 unfolding basis_component using `e>0` i by auto }
4893 hence "x \<in> {a<..<b}" unfolding mem_interval by auto }
4894 thus "?L \<subseteq> ?R" unfolding interior_def and subset_eq by auto
4897 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"
4899 let ?b = "\<Sum>i<DIM('a). \<bar>a$$i\<bar> + \<bar>b$$i\<bar>"
4900 { fix x::"'a" assume x:"\<forall>i<DIM('a). a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i"
4901 { fix i assume "i<DIM('a)"
4902 hence "\<bar>x$$i\<bar> \<le> \<bar>a$$i\<bar> + \<bar>b$$i\<bar>" using x[THEN spec[where x=i]] by auto }
4903 hence "(\<Sum>i<DIM('a). \<bar>x $$ i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto
4904 hence "norm x \<le> ?b" using norm_le_l1[of x] by auto }
4905 thus ?thesis unfolding interval and bounded_iff by auto
4908 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows
4909 "bounded {a .. b} \<and> bounded {a<..<b}"
4910 using bounded_closed_interval[of a b]
4911 using interval_open_subset_closed[of a b]
4912 using bounded_subset[of "{a..b}" "{a<..<b}"]
4915 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows
4916 "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
4917 using bounded_interval[of a b] by auto
4919 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"
4920 using bounded_closed_imp_compact[of "{a..b}"] using bounded_interval[of a b]
4923 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"
4924 assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
4926 { fix i assume "i<DIM('a)"
4927 hence "a $$ i < ((1 / 2) *\<^sub>R (a + b)) $$ i \<and> ((1 / 2) *\<^sub>R (a + b)) $$ i < b $$ i"
4928 using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
4929 unfolding euclidean_simps by auto }
4930 thus ?thesis unfolding mem_interval by auto
4933 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"
4934 assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
4935 shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
4937 { fix i assume i:"i<DIM('a)"
4938 have "a $$ i = e * a$$i + (1 - e) * a$$i" unfolding left_diff_distrib by simp
4939 also have "\<dots> < e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
4940 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
4941 using x unfolding mem_interval using i apply simp
4942 using y unfolding mem_interval using i apply simp
4944 finally have "a $$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i" unfolding euclidean_simps by auto
4946 have "b $$ i = e * b$$i + (1 - e) * b$$i" unfolding left_diff_distrib by simp
4947 also have "\<dots> > e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
4948 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
4949 using x unfolding mem_interval using i apply simp
4950 using y unfolding mem_interval using i apply simp
4952 finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i < b $$ i" unfolding euclidean_simps by auto
4953 } ultimately have "a $$ 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 }
4954 thus ?thesis unfolding mem_interval by auto
4957 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"
4958 assumes "{a<..<b} \<noteq> {}"
4959 shows "closure {a<..<b} = {a .. b}"
4961 have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto
4962 let ?c = "(1 / 2) *\<^sub>R (a + b)"
4963 { fix x assume as:"x \<in> {a .. b}"
4964 def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
4965 { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
4966 have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
4967 have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
4968 x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
4969 by (auto simp add: algebra_simps)
4970 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
4971 hence False using fn unfolding f_def using xc by auto }
4973 { assume "\<not> (f ---> x) sequentially"
4974 { fix e::real assume "e>0"
4975 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
4976 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
4977 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)
4978 hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto }
4979 hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
4980 unfolding Lim_sequentially by(auto simp add: dist_norm)
4981 hence "(f ---> x) sequentially" unfolding f_def
4982 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]
4983 using tendsto_scaleR [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto }
4984 ultimately have "x \<in> closure {a<..<b}"
4985 using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto }
4986 thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
4989 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"
4990 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a<..<a}"
4992 obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
4993 def a \<equiv> "(\<chi>\<chi> i. b+1)::'a"
4994 { fix x assume "x\<in>s"
4995 fix i assume i:"i<DIM('a)"
4996 hence "(-a)$$i < x$$i" and "x$$i < a$$i" using b[THEN bspec[where x=x], OF `x\<in>s`]
4997 and component_le_norm[of x i] unfolding euclidean_simps and a_def by auto }
4998 thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
5001 lemma bounded_subset_open_interval:
5002 fixes s :: "('a::ordered_euclidean_space) set"
5003 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
5004 by (auto dest!: bounded_subset_open_interval_symmetric)
5006 lemma bounded_subset_closed_interval_symmetric:
5007 fixes s :: "('a::ordered_euclidean_space) set"
5008 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
5010 obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
5011 thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
5014 lemma bounded_subset_closed_interval:
5015 fixes s :: "('a::ordered_euclidean_space) set"
5016 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
5017 using bounded_subset_closed_interval_symmetric[of s] by auto
5019 lemma frontier_closed_interval:
5020 fixes a b :: "'a::ordered_euclidean_space"
5021 shows "frontier {a .. b} = {a .. b} - {a<..<b}"
5022 unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
5024 lemma frontier_open_interval:
5025 fixes a b :: "'a::ordered_euclidean_space"
5026 shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
5027 proof(cases "{a<..<b} = {}")
5028 case True thus ?thesis using frontier_empty by auto
5030 case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
5033 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"
5034 assumes "{c<..<d} \<noteq> {}" shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
5035 unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
5038 (* Some stuff for half-infinite intervals too; FIXME: notation? *)
5040 lemma closed_interval_left: fixes b::"'a::euclidean_space"
5041 shows "closed {x::'a. \<forall>i<DIM('a). x$$i \<le> b$$i}"
5043 { fix i assume i:"i<DIM('a)"
5044 fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). x $$ i \<le> b $$ i}. x' \<noteq> x \<and> dist x' x < e"
5045 { assume "x$$i > b$$i"
5046 then obtain y where "y $$ i \<le> b $$ i" "y \<noteq> x" "dist y x < x$$i - b$$i"
5047 using x[THEN spec[where x="x$$i - b$$i"]] using i by auto
5048 hence False using component_le_norm[of "y - x" i] unfolding dist_norm euclidean_simps using i
5050 hence "x$$i \<le> b$$i" by(rule ccontr)auto }
5051 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
5054 lemma closed_interval_right: fixes a::"'a::euclidean_space"
5055 shows "closed {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i}"
5057 { fix i assume i:"i<DIM('a)"
5058 fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). a $$ i \<le> x $$ i}. x' \<noteq> x \<and> dist x' x < e"
5059 { assume "a$$i > x$$i"
5060 then obtain y where "a $$ i \<le> y $$ i" "y \<noteq> x" "dist y x < a$$i - x$$i"
5061 using x[THEN spec[where x="a$$i - x$$i"]] i by auto
5062 hence False using component_le_norm[of "y - x" i] unfolding dist_norm and euclidean_simps by auto }
5063 hence "a$$i \<le> x$$i" by(rule ccontr)auto }
5064 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
5067 text {* Intervals in general, including infinite and mixtures of open and closed. *}
5069 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
5070 (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i<DIM('a). ((a$$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)"
5072 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)
5073 "is_interval {a<..<b}" (is ?th2) proof -
5074 have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
5075 show ?th1 ?th2 unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
5076 by(meson order_trans le_less_trans less_le_trans *)+ qed
5078 lemma is_interval_empty:
5080 unfolding is_interval_def
5083 lemma is_interval_univ:
5085 unfolding is_interval_def
5089 subsection {* Closure of halfspaces and hyperplanes *}
5091 lemma isCont_open_vimage:
5092 assumes "\<And>x. isCont f x" and "open s" shows "open (f -` s)"
5094 from assms(1) have "continuous_on UNIV f"
5095 unfolding isCont_def continuous_on_def within_UNIV by simp
5096 hence "open {x \<in> UNIV. f x \<in> s}"
5097 using open_UNIV `open s` by (rule continuous_open_preimage)
5098 thus "open (f -` s)"
5099 by (simp add: vimage_def)
5102 lemma isCont_closed_vimage:
5103 assumes "\<And>x. isCont f x" and "closed s" shows "closed (f -` s)"
5104 using assms unfolding closed_def vimage_Compl [symmetric]
5105 by (rule isCont_open_vimage)
5107 lemma open_Collect_less:
5108 fixes f g :: "'a::topological_space \<Rightarrow> real"
5109 assumes f: "\<And>x. isCont f x"
5110 assumes g: "\<And>x. isCont g x"
5111 shows "open {x. f x < g x}"
5113 have "open ((\<lambda>x. g x - f x) -` {0<..})"
5114 using isCont_diff [OF g f] open_real_greaterThan
5115 by (rule isCont_open_vimage)
5116 also have "((\<lambda>x. g x - f x) -` {0<..}) = {x. f x < g x}"
5118 finally show ?thesis .
5121 lemma closed_Collect_le:
5122 fixes f g :: "'a::topological_space \<Rightarrow> real"
5123 assumes f: "\<And>x. isCont f x"
5124 assumes g: "\<And>x. isCont g x"
5125 shows "closed {x. f x \<le> g x}"
5127 have "closed ((\<lambda>x. g x - f x) -` {0..})"
5128 using isCont_diff [OF g f] closed_real_atLeast
5129 by (rule isCont_closed_vimage)
5130 also have "((\<lambda>x. g x - f x) -` {0..}) = {x. f x \<le> g x}"
5132 finally show ?thesis .
5135 lemma closed_Collect_eq:
5136 fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
5137 assumes f: "\<And>x. isCont f x"
5138 assumes g: "\<And>x. isCont g x"
5139 shows "closed {x. f x = g x}"
5141 have "open {(x::'b, y::'b). x \<noteq> y}"
5142 unfolding open_prod_def by (auto dest!: hausdorff)
5143 hence "closed {(x::'b, y::'b). x = y}"
5144 unfolding closed_def split_def Collect_neg_eq .
5145 with isCont_Pair [OF f g]
5146 have "closed ((\<lambda>x. (f x, g x)) -` {(x, y). x = y})"
5147 by (rule isCont_closed_vimage)
5148 also have "\<dots> = {x. f x = g x}" by auto
5149 finally show ?thesis .
5153 assumes "(f ---> l) net" shows "((\<lambda>y. inner a (f y)) ---> inner a l) net"
5154 by (intro tendsto_intros assms)
5156 lemma continuous_at_inner: "continuous (at x) (inner a)"
5157 unfolding continuous_at by (intro tendsto_intros)
5159 lemma continuous_at_euclidean_component[intro!, simp]: "continuous (at x) (\<lambda>x. x $$ i)"
5160 unfolding euclidean_component_def by (rule continuous_at_inner)
5162 lemma continuous_on_inner:
5163 fixes s :: "'a::real_inner set"
5164 shows "continuous_on s (inner a)"
5165 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
5167 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
5168 by (simp add: closed_Collect_le)
5170 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
5171 by (simp add: closed_Collect_le)
5173 lemma closed_hyperplane: "closed {x. inner a x = b}"
5174 by (simp add: closed_Collect_eq)
5176 lemma closed_halfspace_component_le:
5177 shows "closed {x::'a::euclidean_space. x$$i \<le> a}"
5178 by (simp add: closed_Collect_le)
5180 lemma closed_halfspace_component_ge:
5181 shows "closed {x::'a::euclidean_space. x$$i \<ge> a}"
5182 by (simp add: closed_Collect_le)
5184 text {* Openness of halfspaces. *}
5186 lemma open_halfspace_lt: "open {x. inner a x < b}"
5187 by (simp add: open_Collect_less)
5189 lemma open_halfspace_gt: "open {x. inner a x > b}"
5190 by (simp add: open_Collect_less)
5192 lemma open_halfspace_component_lt:
5193 shows "open {x::'a::euclidean_space. x$$i < a}"
5194 by (simp add: open_Collect_less)
5196 lemma open_halfspace_component_gt:
5197 shows "open {x::'a::euclidean_space. x$$i > a}"
5198 by (simp add: open_Collect_less)
5200 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}
5202 lemma eucl_lessThan_eq_halfspaces:
5203 fixes a :: "'a\<Colon>ordered_euclidean_space"
5204 shows "{..<a} = (\<Inter>i<DIM('a). {x. x $$ i < a $$ i})"
5205 by (auto simp: eucl_less[where 'a='a])
5207 lemma eucl_greaterThan_eq_halfspaces:
5208 fixes a :: "'a\<Colon>ordered_euclidean_space"
5209 shows "{a<..} = (\<Inter>i<DIM('a). {x. a $$ i < x $$ i})"
5210 by (auto simp: eucl_less[where 'a='a])
5212 lemma eucl_atMost_eq_halfspaces:
5213 fixes a :: "'a\<Colon>ordered_euclidean_space"
5214 shows "{.. a} = (\<Inter>i<DIM('a). {x. x $$ i \<le> a $$ i})"
5215 by (auto simp: eucl_le[where 'a='a])
5217 lemma eucl_atLeast_eq_halfspaces:
5218 fixes a :: "'a\<Colon>ordered_euclidean_space"
5219 shows "{a ..} = (\<Inter>i<DIM('a). {x. a $$ i \<le> x $$ i})"
5220 by (auto simp: eucl_le[where 'a='a])
5222 lemma open_eucl_lessThan[simp, intro]:
5223 fixes a :: "'a\<Colon>ordered_euclidean_space"
5224 shows "open {..< a}"
5225 by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)
5227 lemma open_eucl_greaterThan[simp, intro]:
5228 fixes a :: "'a\<Colon>ordered_euclidean_space"
5229 shows "open {a <..}"
5230 by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)
5232 lemma closed_eucl_atMost[simp, intro]:
5233 fixes a :: "'a\<Colon>ordered_euclidean_space"
5234 shows "closed {.. a}"
5235 unfolding eucl_atMost_eq_halfspaces
5236 by (simp add: closed_INT closed_Collect_le)
5238 lemma closed_eucl_atLeast[simp, intro]:
5239 fixes a :: "'a\<Colon>ordered_euclidean_space"
5240 shows "closed {a ..}"
5241 unfolding eucl_atLeast_eq_halfspaces
5242 by (simp add: closed_INT closed_Collect_le)
5244 lemma open_vimage_euclidean_component: "open S \<Longrightarrow> open ((\<lambda>x. x $$ i) -` S)"
5245 by (auto intro!: continuous_open_vimage)
5247 text {* This gives a simple derivation of limit component bounds. *}
5249 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5250 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f(x)$$i \<le> b) net"
5251 shows "l$$i \<le> b"
5253 { fix x have "x \<in> {x::'b. inner (basis i) x \<le> b} \<longleftrightarrow> x$$i \<le> b"
5254 unfolding euclidean_component_def by auto } note * = this
5255 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *
5256 using closed_halfspace_le[of "(basis i)::'b" b] and assms(1,2,3) by auto
5259 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5260 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$$i) net"
5261 shows "b \<le> l$$i"
5263 { fix x have "x \<in> {x::'b. inner (basis i) x \<ge> b} \<longleftrightarrow> x$$i \<ge> b"
5264 unfolding euclidean_component_def by auto } note * = this
5265 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *
5266 using closed_halfspace_ge[of b "(basis i)"] and assms(1,2,3) by auto
5269 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5270 assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$$i = b) net"
5272 using ev[unfolded order_eq_iff eventually_conj_iff] using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto
5273 text{* Limits relative to a union. *}
5275 lemma eventually_within_Un:
5276 "eventually P (net within (s \<union> t)) \<longleftrightarrow>
5277 eventually P (net within s) \<and> eventually P (net within t)"
5278 unfolding Limits.eventually_within
5279 by (auto elim!: eventually_rev_mp)
5281 lemma Lim_within_union:
5282 "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>
5283 (f ---> l) (net within s) \<and> (f ---> l) (net within t)"
5284 unfolding tendsto_def
5285 by (auto simp add: eventually_within_Un)
5287 lemma Lim_topological:
5288 "(f ---> l) net \<longleftrightarrow>
5289 trivial_limit net \<or>
5290 (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
5291 unfolding tendsto_def trivial_limit_eq by auto
5293 lemma continuous_on_union:
5294 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
5295 shows "continuous_on (s \<union> t) f"
5296 using assms unfolding continuous_on Lim_within_union
5297 unfolding Lim_topological trivial_limit_within closed_limpt by auto
5299 lemma continuous_on_cases:
5300 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
5301 "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
5302 shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
5304 let ?h = "(\<lambda>x. if P x then f x else g x)"
5305 have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto
5306 hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto
5308 have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto
5309 hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto
5310 ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto
5314 text{* Some more convenient intermediate-value theorem formulations. *}
5316 lemma connected_ivt_hyperplane:
5317 assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
5318 shows "\<exists>z \<in> s. inner a z = b"
5320 assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
5321 let ?A = "{x. inner a x < b}"
5322 let ?B = "{x. inner a x > b}"
5323 have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
5324 moreover have "?A \<inter> ?B = {}" by auto
5325 moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
5326 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
5329 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows
5330 "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)"
5331 using connected_ivt_hyperplane[of s x y "(basis k)::'a" a]
5332 unfolding euclidean_component_def by auto
5335 subsection {* Homeomorphisms *}
5337 definition "homeomorphism s t f g \<equiv>
5338 (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
5339 (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
5342 homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"
5343 (infixr "homeomorphic" 60) where
5344 homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
5346 lemma homeomorphic_refl: "s homeomorphic s"
5347 unfolding homeomorphic_def
5348 unfolding homeomorphism_def
5349 using continuous_on_id
5350 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5351 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5354 lemma homeomorphic_sym:
5355 "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
5356 unfolding homeomorphic_def
5357 unfolding homeomorphism_def
5360 lemma homeomorphic_trans:
5361 assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
5363 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"
5364 using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
5365 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"
5366 using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
5368 { 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 }
5369 moreover have "(f2 \<circ> f1) ` s = u" using fg1(2) fg2(2) by auto
5370 moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
5371 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 }
5372 moreover have "(g1 \<circ> g2) ` u = s" using fg1(5) fg2(5) by auto
5373 moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6) unfolding fg2(5) by auto
5374 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
5377 lemma homeomorphic_minimal:
5378 "s homeomorphic t \<longleftrightarrow>
5379 (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
5380 (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
5381 continuous_on s f \<and> continuous_on t g)"
5382 unfolding homeomorphic_def homeomorphism_def
5383 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
5384 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
5386 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
5387 apply auto apply(rule_tac x="g x" in bexI) apply auto
5388 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
5389 apply auto apply(rule_tac x="f x" in bexI) by auto
5391 text {* Relatively weak hypotheses if a set is compact. *}
5393 lemma homeomorphism_compact:
5394 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5395 (* class constraint due to continuous_on_inverse *)
5396 assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s"
5397 shows "\<exists>g. homeomorphism s t f g"
5399 def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
5400 have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
5401 { fix y assume "y\<in>t"
5402 then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
5403 hence "g (f x) = x" using g by auto
5404 hence "f (g y) = y" unfolding x(1)[THEN sym] by auto }
5405 hence g':"\<forall>x\<in>t. f (g x) = x" by auto
5408 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"])
5410 { assume "x\<in>g ` t"
5411 then obtain y where y:"y\<in>t" "g y = x" by auto
5412 then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
5413 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 }
5414 ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" .. }
5415 hence "g ` t = s" by auto
5417 show ?thesis unfolding homeomorphism_def homeomorphic_def
5418 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
5421 lemma homeomorphic_compact:
5422 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5423 (* class constraint due to continuous_on_inverse *)
5424 shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
5425 \<Longrightarrow> s homeomorphic t"
5426 unfolding homeomorphic_def by (metis homeomorphism_compact)
5428 text{* Preservation of topological properties. *}
5430 lemma homeomorphic_compactness:
5431 "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
5432 unfolding homeomorphic_def homeomorphism_def
5433 by (metis compact_continuous_image)
5435 text{* Results on translation, scaling etc. *}
5437 lemma homeomorphic_scaling:
5438 fixes s :: "'a::real_normed_vector set"
5439 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
5440 unfolding homeomorphic_minimal
5441 apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
5442 apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
5443 using assms apply auto
5444 using continuous_on_cmul[OF continuous_on_id] by auto
5446 lemma homeomorphic_translation:
5447 fixes s :: "'a::real_normed_vector set"
5448 shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
5449 unfolding homeomorphic_minimal
5450 apply(rule_tac x="\<lambda>x. a + x" in exI)
5451 apply(rule_tac x="\<lambda>x. -a + x" in exI)
5452 using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
5454 lemma homeomorphic_affinity:
5455 fixes s :: "'a::real_normed_vector set"
5456 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
5458 have *:"op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
5460 using homeomorphic_trans
5461 using homeomorphic_scaling[OF assms, of s]
5462 using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a] unfolding * by auto
5465 lemma homeomorphic_balls:
5466 fixes a b ::"'a::real_normed_vector" (* FIXME: generalize to metric_space *)
5467 assumes "0 < d" "0 < e"
5468 shows "(ball a d) homeomorphic (ball b e)" (is ?th)
5469 "(cball a d) homeomorphic (cball b e)" (is ?cth)
5471 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5472 show ?th unfolding homeomorphic_minimal
5473 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5474 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5475 using assms apply (auto simp add: dist_commute)
5477 apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)
5478 unfolding continuous_on
5479 by (intro ballI tendsto_intros, simp)+
5481 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5482 show ?cth unfolding homeomorphic_minimal
5483 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5484 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5485 using assms apply (auto simp add: dist_commute)
5487 apply (auto simp add: pos_divide_le_eq)
5488 unfolding continuous_on
5489 by (intro ballI tendsto_intros, simp)+
5492 text{* "Isometry" (up to constant bounds) of injective linear map etc. *}
5494 lemma cauchy_isometric:
5495 fixes x :: "nat \<Rightarrow> 'a::euclidean_space"
5496 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)"
5499 interpret f: bounded_linear f by fact
5500 { fix d::real assume "d>0"
5501 then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
5502 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
5503 { fix n assume "n\<ge>N"
5504 hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding f.diff[THEN sym] by auto
5505 moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
5506 using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
5507 using normf[THEN bspec[where x="x n - x N"]] by auto
5508 ultimately have "norm (x n - x N) < d" using `e>0`
5509 using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto }
5510 hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
5511 thus ?thesis unfolding cauchy and dist_norm by auto
5514 lemma complete_isometric_image:
5515 fixes f :: "'a::euclidean_space => 'b::euclidean_space"
5516 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"
5517 shows "complete(f ` s)"
5519 { fix g assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
5520 then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
5521 using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
5522 hence x:"\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
5523 hence "f \<circ> x = g" unfolding fun_eq_iff by auto
5524 then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
5525 using cs[unfolded complete_def, THEN spec[where x="x"]]
5526 using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
5527 hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
5528 using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
5529 unfolding `f \<circ> x = g` by auto }
5530 thus ?thesis unfolding complete_def by auto
5534 fixes x :: "'a::real_normed_vector"
5535 shows "dist 0 x = norm x"
5536 unfolding dist_norm by simp
5538 lemma injective_imp_isometric: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
5539 assumes s:"closed s" "subspace s" and f:"bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
5540 shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
5541 proof(cases "s \<subseteq> {0::'a}")
5543 { fix x assume "x \<in> s"
5544 hence "x = 0" using True by auto
5545 hence "norm x \<le> norm (f x)" by auto }
5546 thus ?thesis by(auto intro!: exI[where x=1])
5548 interpret f: bounded_linear f by fact
5550 then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
5551 from False have "s \<noteq> {}" by auto
5552 let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
5553 let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
5554 let ?S'' = "{x::'a. norm x = norm a}"
5556 have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by auto
5557 hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
5558 moreover have "?S' = s \<inter> ?S''" by auto
5559 ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
5560 moreover have *:"f ` ?S' = ?S" by auto
5561 ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
5562 hence "closed ?S" using compact_imp_closed by auto
5563 moreover have "?S \<noteq> {}" using a by auto
5564 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
5565 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
5567 let ?e = "norm (f b) / norm b"
5568 have "norm b > 0" using ba and a and norm_ge_zero by auto
5569 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
5570 ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
5572 { fix x assume "x\<in>s"
5573 hence "norm (f b) / norm b * norm x \<le> norm (f x)"
5575 case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
5578 hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
5579 have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def] by auto
5580 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
5581 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"]]
5582 unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
5583 by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq)
5586 show ?thesis by auto
5589 lemma closed_injective_image_subspace:
5590 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
5591 assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
5592 shows "closed(f ` s)"
5594 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
5595 show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
5596 unfolding complete_eq_closed[THEN sym] by auto
5600 subsection {* Some properties of a canonical subspace *}
5602 lemma subspace_substandard:
5603 "subspace {x::'a::euclidean_space. (\<forall>i<DIM('a). P i \<longrightarrow> x$$i = 0)}"
5604 unfolding subspace_def by auto
5606 lemma closed_substandard:
5607 "closed {x::'a::euclidean_space. \<forall>i<DIM('a). P i --> x$$i = 0}" (is "closed ?A")
5609 let ?D = "{i. P i} \<inter> {..<DIM('a)}"
5610 have "closed (\<Inter>i\<in>?D. {x::'a. x$$i = 0})"
5611 by (simp add: closed_INT closed_Collect_eq)
5612 also have "(\<Inter>i\<in>?D. {x::'a. x$$i = 0}) = ?A"
5614 finally show "closed ?A" .
5617 lemma dim_substandard: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
5618 shows "dim {x::'a::euclidean_space. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0} = card d" (is "dim ?A = _")
5620 let ?D = "{..<DIM('a)}"
5621 let ?B = "(basis::nat => 'a) ` d"
5622 let ?bas = "basis::nat \<Rightarrow> 'a"
5623 have "?B \<subseteq> ?A" by auto
5625 { fix x::"'a" assume "x\<in>?A"
5626 hence "finite d" "x\<in>?A" using assms by(auto intro:finite_subset)
5627 hence "x\<in> span ?B"
5628 proof(induct d arbitrary: x)
5629 case empty hence "x=0" apply(subst euclidean_eq) by auto
5630 thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
5633 hence *:"\<forall>i<DIM('a). i \<notin> insert k F \<longrightarrow> x $$ i = 0" by auto
5634 have **:"F \<subseteq> insert k F" by auto
5635 def y \<equiv> "x - x$$k *\<^sub>R basis k"
5636 have y:"x = y + (x$$k) *\<^sub>R basis k" unfolding y_def by auto
5637 { fix i assume i':"i \<notin> F"
5638 hence "y $$ i = 0" unfolding y_def
5639 using *[THEN spec[where x=i]] by auto }
5640 hence "y \<in> span (basis ` F)" using insert(3) by auto
5641 hence "y \<in> span (basis ` (insert k F))"
5642 using span_mono[of "?bas ` F" "?bas ` (insert k F)"]
5643 using image_mono[OF **, of basis] using assms by auto
5645 have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
5646 hence "x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5647 using span_mul by auto
5649 have "y + x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5650 using span_add by auto
5651 thus ?case using y by auto
5654 hence "?A \<subseteq> span ?B" by auto
5656 { fix x assume "x \<in> ?B"
5657 hence "x\<in>{(basis i)::'a |i. i \<in> ?D}" using assms by auto }
5658 hence "independent ?B" using independent_mono[OF independent_basis, of ?B] and assms by auto
5660 have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
5661 hence *:"inj_on (basis::nat\<Rightarrow>'a) d" using subset_inj_on[OF basis_inj, of "d"] by auto
5662 have "card ?B = card d" unfolding card_image[OF *] by auto
5663 ultimately show ?thesis using dim_unique[of "basis ` d" ?A] by auto
5666 text{* Hence closure and completeness of all subspaces. *}
5668 lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"
5670 apply (rule_tac x="{}" in exI, simp)
5672 apply (subgoal_tac "\<exists>x. x \<notin> A")
5674 apply (rule_tac x="insert x A" in exI, simp)
5675 apply (subgoal_tac "A \<noteq> UNIV", auto)
5678 lemma closed_subspace: fixes s::"('a::euclidean_space) set"
5679 assumes "subspace s" shows "closed s"
5681 have *:"dim s \<le> DIM('a)" using dim_subset_UNIV by auto
5682 def d \<equiv> "{..<dim s}" have t:"card d = dim s" unfolding d_def by auto
5683 let ?t = "{x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0}"
5684 have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0} = s \<and>
5685 inj_on f {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0}"
5686 apply(rule subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]])
5687 using dim_substandard[of d,where 'a='a] and t unfolding d_def using * assms by auto
5688 then guess f apply-by(erule exE conjE)+ note f = this
5689 interpret f: bounded_linear f using f unfolding linear_conv_bounded_linear by auto
5690 have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using f.zero using f(3)[unfolded inj_on_def]
5691 by(erule_tac x=0 in ballE) auto
5692 moreover have "closed ?t" using closed_substandard .
5693 moreover have "subspace ?t" using subspace_substandard .
5694 ultimately show ?thesis using closed_injective_image_subspace[of ?t f]
5695 unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
5698 lemma complete_subspace:
5699 fixes s :: "('a::euclidean_space) set" shows "subspace s ==> complete s"
5700 using complete_eq_closed closed_subspace
5704 fixes s :: "('a::euclidean_space) set"
5705 shows "dim(closure s) = dim s" (is "?dc = ?d")
5707 have "?dc \<le> ?d" using closure_minimal[OF span_inc, of s]
5708 using closed_subspace[OF subspace_span, of s]
5709 using dim_subset[of "closure s" "span s"] unfolding dim_span by auto
5710 thus ?thesis using dim_subset[OF closure_subset, of s] by auto
5714 subsection {* Affine transformations of intervals *}
5716 lemma real_affinity_le:
5717 "0 < (m::'a::linordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"
5718 by (simp add: field_simps inverse_eq_divide)
5720 lemma real_le_affinity:
5721 "0 < (m::'a::linordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"
5722 by (simp add: field_simps inverse_eq_divide)
5724 lemma real_affinity_lt:
5725 "0 < (m::'a::linordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"
5726 by (simp add: field_simps inverse_eq_divide)
5728 lemma real_lt_affinity:
5729 "0 < (m::'a::linordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"
5730 by (simp add: field_simps inverse_eq_divide)
5732 lemma real_affinity_eq:
5733 "(m::'a::linordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"
5734 by (simp add: field_simps inverse_eq_divide)
5736 lemma real_eq_affinity:
5737 "(m::'a::linordered_field) \<noteq> 0 ==> (y = m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"
5738 by (simp add: field_simps inverse_eq_divide)
5740 lemma image_affinity_interval: fixes m::real
5741 fixes a b c :: "'a::ordered_euclidean_space"
5742 shows "(\<lambda>x. m *\<^sub>R x + c) ` {a .. b} =
5743 (if {a .. b} = {} then {}
5744 else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
5745 else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
5747 { fix x assume "x \<le> c" "c \<le> x"
5748 hence "x=c" unfolding eucl_le[where 'a='a] apply-
5749 apply(subst euclidean_eq) by (auto intro: order_antisym) }
5751 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])
5752 ultimately show ?thesis by auto
5755 { fix y assume "a \<le> y" "y \<le> b" "m > 0"
5756 hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
5757 unfolding eucl_le[where 'a='a] by auto
5759 { fix y assume "a \<le> y" "y \<le> b" "m < 0"
5760 hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
5761 unfolding eucl_le[where 'a='a] by(auto simp add: mult_left_mono_neg)
5763 { fix y assume "m > 0" "m *\<^sub>R a + c \<le> y" "y \<le> m *\<^sub>R b + c"
5764 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5765 unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
5766 apply(auto simp add: pth_3[symmetric]
5767 intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
5768 by(auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff)
5770 { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
5771 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5772 unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
5773 apply(auto simp add: pth_3[symmetric]
5774 intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
5775 by(auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff)
5777 ultimately show ?thesis using False by auto
5780 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a..b} =
5781 (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})"
5782 using image_affinity_interval[of m 0 a b] by auto
5785 subsection {* Banach fixed point theorem (not really topological...) *}
5788 assumes s:"complete s" "s \<noteq> {}" and c:"0 \<le> c" "c < 1" and f:"(f ` s) \<subseteq> s" and
5789 lipschitz:"\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
5790 shows "\<exists>! x\<in>s. (f x = x)"
5792 have "1 - c > 0" using c by auto
5794 from s(2) obtain z0 where "z0 \<in> s" by auto
5795 def z \<equiv> "\<lambda>n. (f ^^ n) z0"
5797 have "z n \<in> s" unfolding z_def
5798 proof(induct n) case 0 thus ?case using `z0 \<in>s` by auto
5799 next case Suc thus ?case using f by auto qed }
5802 def d \<equiv> "dist (z 0) (z 1)"
5804 have fzn:"\<And>n. f (z n) = z (Suc n)" unfolding z_def by auto
5806 have "dist (z n) (z (Suc n)) \<le> (c ^ n) * d"
5808 case 0 thus ?case unfolding d_def by auto
5811 hence "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
5812 using `0 \<le> c` using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by auto
5813 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]
5814 unfolding fzn and mult_le_cancel_left by auto
5819 have "(1 - c) * dist (z m) (z (m+n)) \<le> (c ^ m) * d * (1 - c ^ n)"
5821 case 0 show ?case by auto
5824 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))))"
5825 using dist_triangle and c by(auto simp add: dist_triangle)
5826 also have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
5827 using cf_z[of "m + k"] and c by auto
5828 also have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
5829 using Suc by (auto simp add: field_simps)
5830 also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
5831 unfolding power_add by (auto simp add: field_simps)
5832 also have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
5833 using c by (auto simp add: field_simps)
5834 finally show ?case by auto
5837 { fix e::real assume "e>0"
5838 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e"
5839 proof(cases "d = 0")
5841 have *: "\<And>x. ((1 - c) * x \<le> 0) = (x \<le> 0)" using `1 - c > 0`
5842 by (metis mult_zero_left real_mult_commute real_mult_le_cancel_iff1)
5843 from True have "\<And>n. z n = z0" using cf_z2[of 0] and c unfolding z_def
5845 thus ?thesis using `e>0` by auto
5847 case False hence "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]
5848 by (metis False d_def less_le)
5849 hence "0 < e * (1 - c) / d" using `e>0` and `1-c>0`
5850 using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"] by auto
5851 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
5852 { fix m n::nat assume "m>n" and as:"m\<ge>N" "n\<ge>N"
5853 have *:"c ^ n \<le> c ^ N" using `n\<ge>N` and c using power_decreasing[OF `n\<ge>N`, of c] by auto
5854 have "1 - c ^ (m - n) > 0" using c and power_strict_mono[of c 1 "m - n"] using `m>n` by auto
5855 hence **:"d * (1 - c ^ (m - n)) / (1 - c) > 0"
5856 using mult_pos_pos[OF `d>0`, of "1 - c ^ (m - n)"]
5857 using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]
5858 using `0 < 1 - c` by auto
5860 have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
5861 using cf_z2[of n "m - n"] and `m>n` unfolding pos_le_divide_eq[OF `1-c>0`]
5862 by (auto simp add: mult_commute dist_commute)
5863 also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
5864 using mult_right_mono[OF * order_less_imp_le[OF **]]
5865 unfolding mult_assoc by auto
5866 also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
5867 using mult_strict_right_mono[OF N **] unfolding mult_assoc by auto
5868 also have "\<dots> = e * (1 - c ^ (m - n))" using c and `d>0` and `1 - c > 0` by auto
5869 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
5870 finally have "dist (z m) (z n) < e" by auto
5872 { fix m n::nat assume as:"N\<le>m" "N\<le>n"
5873 hence "dist (z n) (z m) < e"
5874 proof(cases "n = m")
5875 case True thus ?thesis using `e>0` by auto
5877 case False thus ?thesis using as and *[of n m] *[of m n] unfolding nat_neq_iff by (auto simp add: dist_commute)
5879 thus ?thesis by auto
5882 hence "Cauchy z" unfolding cauchy_def by auto
5883 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
5885 def e \<equiv> "dist (f x) x"
5886 have "e = 0" proof(rule ccontr)
5887 assume "e \<noteq> 0" hence "e>0" unfolding e_def using zero_le_dist[of "f x" x]
5888 by (metis dist_eq_0_iff dist_nz e_def)
5889 then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"
5890 using x[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
5891 hence N':"dist (z N) x < e / 2" by auto
5893 have *:"c * dist (z N) x \<le> dist (z N) x" unfolding mult_le_cancel_right2
5894 using zero_le_dist[of "z N" x] and c
5895 by (metis dist_eq_0_iff dist_nz order_less_asym less_le)
5896 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]]
5897 using z_in_s[of N] `x\<in>s` using c by auto
5898 also have "\<dots> < e / 2" using N' and c using * by auto
5899 finally show False unfolding fzn
5900 using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
5901 unfolding e_def by auto
5903 hence "f x = x" unfolding e_def by auto
5905 { fix y assume "f y = y" "y\<in>s"
5906 hence "dist x y \<le> c * dist x y" using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]
5907 using `x\<in>s` and `f x = x` by auto
5908 hence "dist x y = 0" unfolding mult_le_cancel_right1
5909 using c and zero_le_dist[of x y] by auto
5910 hence "y = x" by auto
5912 ultimately show ?thesis using `x\<in>s` by blast+
5915 subsection {* Edelstein fixed point theorem *}
5917 lemma edelstein_fix:
5918 fixes s :: "'a::real_normed_vector set"
5919 assumes s:"compact s" "s \<noteq> {}" and gs:"(g ` s) \<subseteq> s"
5920 and dist:"\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
5921 shows "\<exists>! x\<in>s. g x = x"
5922 proof(cases "\<exists>x\<in>s. g x \<noteq> x")
5923 obtain x where "x\<in>s" using s(2) by auto
5924 case False hence g:"\<forall>x\<in>s. g x = x" by auto
5925 { fix y assume "y\<in>s"
5926 hence "x = y" using `x\<in>s` and dist[THEN bspec[where x=x], THEN bspec[where x=y]]
5927 unfolding g[THEN bspec[where x=x], OF `x\<in>s`]
5928 unfolding g[THEN bspec[where x=y], OF `y\<in>s`] by auto }
5929 thus ?thesis using `x\<in>s` and g by blast+
5932 then obtain x where [simp]:"x\<in>s" and "g x \<noteq> x" by auto
5933 { fix x y assume "x \<in> s" "y \<in> s"
5934 hence "dist (g x) (g y) \<le> dist x y"
5935 using dist[THEN bspec[where x=x], THEN bspec[where x=y]] by auto } note dist' = this
5936 def y \<equiv> "g x"
5937 have [simp]:"y\<in>s" unfolding y_def using gs[unfolded image_subset_iff] and `x\<in>s` by blast
5938 def f \<equiv> "\<lambda>n. g ^^ n"
5939 have [simp]:"\<And>n z. g (f n z) = f (Suc n) z" unfolding f_def by auto
5940 have [simp]:"\<And>z. f 0 z = z" unfolding f_def by auto
5941 { fix n::nat and z assume "z\<in>s"
5942 have "f n z \<in> s" unfolding f_def
5944 case 0 thus ?case using `z\<in>s` by simp
5946 case (Suc n) thus ?case using gs[unfolded image_subset_iff] by auto
5947 qed } note fs = this
5948 { fix m n ::nat assume "m\<le>n"
5949 fix w z assume "w\<in>s" "z\<in>s"
5950 have "dist (f n w) (f n z) \<le> dist (f m w) (f m z)" using `m\<le>n`
5952 case 0 thus ?case by auto
5955 thus ?case proof(cases "m\<le>n")
5956 case True thus ?thesis using Suc(1)
5957 using dist'[OF fs fs, OF `w\<in>s` `z\<in>s`, of n n] by auto
5959 case False hence mn:"m = Suc n" using Suc(2) by simp
5960 show ?thesis unfolding mn by auto
5962 qed } note distf = this
5964 def h \<equiv> "\<lambda>n. (f n x, f n y)"
5965 let ?s2 = "s \<times> s"
5966 obtain l r where "l\<in>?s2" and r:"subseq r" and lr:"((h \<circ> r) ---> l) sequentially"
5967 using compact_Times [OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def
5968 using fs[OF `x\<in>s`] and fs[OF `y\<in>s`] by blast
5969 def a \<equiv> "fst l" def b \<equiv> "snd l"
5970 have lab:"l = (a, b)" unfolding a_def b_def by simp
5971 have [simp]:"a\<in>s" "b\<in>s" unfolding a_def b_def using `l\<in>?s2` by auto
5973 have lima:"((fst \<circ> (h \<circ> r)) ---> a) sequentially"
5974 and limb:"((snd \<circ> (h \<circ> r)) ---> b) sequentially"
5976 unfolding o_def a_def b_def by (rule tendsto_intros)+
5979 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
5981 have "dist (-x) (-y) = dist x y" unfolding dist_norm
5982 using norm_minus_cancel[of "x - y"] by (auto simp add: uminus_add_conv_diff) } note ** = this
5984 { assume as:"dist a b > dist (f n x) (f n y)"
5985 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"
5986 and "\<forall>m\<ge>Nb. dist (f (r m) y) b < (dist a b - dist (f n x) (f n y)) / 2"
5987 using lima limb unfolding h_def Lim_sequentially by (fastsimp simp del: less_divide_eq_number_of1)
5988 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)"
5989 apply(erule_tac x="Na+Nb+n" in allE)
5990 apply(erule_tac x="Na+Nb+n" in allE) apply simp
5991 using dist_triangle_add_half[of a "f (r (Na + Nb + n)) x" "dist a b - dist (f n x) (f n y)"
5992 "-b" "- f (r (Na + Nb + n)) y"]
5993 unfolding ** by (auto simp add: algebra_simps dist_commute)
5995 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)"
5996 using distf[of n "r (Na+Nb+n)", OF _ `x\<in>s` `y\<in>s`]
5997 using subseq_bigger[OF r, of "Na+Nb+n"]
5998 using *[of "f (r (Na + Nb + n)) x" "f (r (Na + Nb + n)) y" "f n x" "f n y"] by auto
5999 ultimately have False by simp
6001 hence "dist a b \<le> dist (f n x) (f n y)" by(rule ccontr)auto }
6004 have [simp]:"a = b" proof(rule ccontr)
6005 def e \<equiv> "dist a b - dist (g a) (g b)"
6006 assume "a\<noteq>b" hence "e > 0" unfolding e_def using dist by fastsimp
6007 hence "\<exists>n. dist (f n x) a < e/2 \<and> dist (f n y) b < e/2"
6008 using lima limb unfolding Lim_sequentially
6009 apply (auto elim!: allE[where x="e/2"]) apply(rule_tac x="r (max N Na)" in exI) unfolding h_def by fastsimp
6010 then obtain n where n:"dist (f n x) a < e/2 \<and> dist (f n y) b < e/2" by auto
6011 have "dist (f (Suc n) x) (g a) \<le> dist (f n x) a"
6012 using dist[THEN bspec[where x="f n x"], THEN bspec[where x="a"]] and fs by auto
6013 moreover have "dist (f (Suc n) y) (g b) \<le> dist (f n y) b"
6014 using dist[THEN bspec[where x="f n y"], THEN bspec[where x="b"]] and fs by auto
6015 ultimately have "dist (f (Suc n) x) (g a) + dist (f (Suc n) y) (g b) < e" using n by auto
6016 thus False unfolding e_def using ab_fn[of "Suc n"] by norm
6019 have [simp]:"\<And>n. f (Suc n) x = f n y" unfolding f_def y_def by(induct_tac n)auto
6020 { fix x y assume "x\<in>s" "y\<in>s" moreover
6021 fix e::real assume "e>0" ultimately
6022 have "dist y x < e \<longrightarrow> dist (g y) (g x) < e" using dist by fastsimp }
6023 hence "continuous_on s g" unfolding continuous_on_iff by auto
6025 hence "((snd \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
6026 apply (rule allE[where x="\<lambda>n. (fst \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
6027 using lima unfolding h_def o_def using fs[OF `x\<in>s`] by (auto simp add: y_def)
6028 hence "g a = a" using tendsto_unique[OF trivial_limit_sequentially limb, of "g a"]
6029 unfolding `a=b` and o_assoc by auto
6031 { fix x assume "x\<in>s" "g x = x" "x\<noteq>a"
6032 hence "False" using dist[THEN bspec[where x=a], THEN bspec[where x=x]]
6033 using `g a = a` and `a\<in>s` by auto }
6034 ultimately show "\<exists>!x\<in>s. g x = x" using `a\<in>s` by blast
6038 (** TODO move this someplace else within this theory **)
6039 instance euclidean_space \<subseteq> banach ..
6041 declare tendsto_const [intro] (* FIXME: move *)
6043 text {* Legacy theorem names *}
6045 lemmas Lim_ident_at = LIM_ident
6046 lemmas Lim_const = tendsto_const
6047 lemmas Lim_cmul = tendsto_scaleR [OF tendsto_const]
6048 lemmas Lim_neg = tendsto_minus
6049 lemmas Lim_add = tendsto_add
6050 lemmas Lim_sub = tendsto_diff
6051 lemmas Lim_mul = tendsto_scaleR
6052 lemmas Lim_vmul = tendsto_scaleR [OF _ tendsto_const]
6053 lemmas Lim_null_norm = tendsto_norm_zero_iff [symmetric]
6054 lemmas Lim_linear = bounded_linear.tendsto [COMP swap_prems_rl]
6055 lemmas Lim_component = tendsto_euclidean_component
6056 lemmas Lim_intros = Lim_add Lim_const Lim_sub Lim_cmul Lim_vmul Lim_within_id