1 (* title: HOL/Library/Topology_Euclidian_Space.thy
2 Author: Amine Chaieb, University of Cambridge
3 Author: Robert Himmelmann, TU Muenchen
4 Author: Brian Huffman, Portland State University
7 header {* Elementary topology in Euclidean space. *}
9 theory Topology_Euclidean_Space
10 imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs"
13 (* to be moved elsewhere *)
15 lemma euclidean_dist_l2:"dist x (y::'a::euclidean_space) = setL2 (\<lambda>i. dist(x$$i) (y$$i)) {..<DIM('a)}"
16 unfolding dist_norm norm_eq_sqrt_inner setL2_def apply(subst euclidean_inner)
17 apply(auto simp add:power2_eq_square) unfolding euclidean_component.diff ..
19 lemma dist_nth_le: "dist (x $$ i) (y $$ i) \<le> dist x (y::'a::euclidean_space)"
20 apply(subst(2) euclidean_dist_l2) apply(cases "i<DIM('a)")
21 apply(rule member_le_setL2) by auto
23 subsection {* General notion of a 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
626 subsection {* Closure of a Set *}
628 definition "closure S = S \<union> {x | x. x islimpt S}"
630 lemma closure_interior: "closure S = - interior (- S)"
633 have "x\<in>- interior (- S) \<longleftrightarrow> x \<in> closure S" (is "?lhs = ?rhs")
635 let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> - S)"
637 hence *:"\<not> ?exT x"
638 unfolding interior_def
640 { assume "\<not> ?rhs"
642 unfolding closure_def islimpt_def
648 assume "?rhs" thus "?lhs"
649 unfolding closure_def interior_def islimpt_def
657 lemma interior_closure: "interior S = - (closure (- S))"
660 have "x \<in> interior S \<longleftrightarrow> x \<in> - (closure (- S))"
661 unfolding interior_def closure_def islimpt_def
668 lemma closed_closure[simp, intro]: "closed (closure S)"
670 have "closed (- interior (-S))" by blast
671 thus ?thesis using closure_interior[of S] by simp
674 lemma closure_hull: "closure S = closed hull S"
676 have "S \<subseteq> closure S"
677 unfolding closure_def
680 have "closed (closure S)"
681 using closed_closure[of S]
685 assume *:"S \<subseteq> t" "closed t"
688 hence "x islimpt t" using *(1)
689 using islimpt_subset[of x, of S, of t]
692 with * have "closure S \<subseteq> t"
693 unfolding closure_def
694 using closed_limpt[of t]
697 ultimately show ?thesis
698 using hull_unique[of S, of "closure S", of closed]
702 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
703 unfolding closure_hull
704 using hull_eq[of closed, OF closed_Inter, of S]
707 lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
708 using closure_eq[of S]
711 lemma closure_closure[simp]: "closure (closure S) = closure S"
712 unfolding closure_hull
713 using hull_hull[of closed S]
716 lemma closure_subset: "S \<subseteq> closure S"
717 unfolding closure_hull
718 using hull_subset[of S closed]
721 lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
722 unfolding closure_hull
723 using hull_mono[of S T closed]
726 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
727 using hull_minimal[of S T closed]
728 unfolding closure_hull
731 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"
732 using hull_unique[of S T closed]
733 unfolding closure_hull
736 lemma closure_empty[simp]: "closure {} = {}"
737 using closed_empty closure_closed[of "{}"]
740 lemma closure_univ[simp]: "closure UNIV = UNIV"
741 using closure_closed[of UNIV]
744 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
745 using closure_empty closure_subset[of S]
748 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
749 using closure_eq[of S] closure_subset[of S]
752 lemma open_inter_closure_eq_empty:
753 "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
754 using open_subset_interior[of S "- T"]
755 using interior_subset[of "- T"]
756 unfolding closure_interior
759 lemma open_inter_closure_subset:
760 "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
763 assume as: "open S" "x \<in> S \<inter> closure T"
764 { assume *:"x islimpt T"
765 have "x islimpt (S \<inter> T)"
766 proof (rule islimptI)
768 assume "x \<in> A" "open A"
769 with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
770 by (simp_all add: open_Int)
771 with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
773 hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
775 thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
778 then show "x \<in> closure (S \<inter> T)" using as
779 unfolding closure_def
783 lemma closure_complement: "closure(- S) = - interior(S)"
788 unfolding closure_interior
792 lemma interior_complement: "interior(- S) = - closure(S)"
793 unfolding closure_interior
797 subsection {* Frontier (aka boundary) *}
799 definition "frontier S = closure S - interior S"
801 lemma frontier_closed: "closed(frontier S)"
802 by (simp add: frontier_def closed_Diff)
804 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
805 by (auto simp add: frontier_def interior_closure)
807 lemma frontier_straddle:
808 fixes a :: "'a::metric_space"
809 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")
814 let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
816 have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
817 moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
818 unfolding frontier_closures closure_def islimpt_def using `e>0`
819 by (auto, erule_tac x="ball a e" in allE, auto)
820 ultimately have ?rhse by auto
823 { assume "a\<notin>S"
824 hence ?rhse using `?lhs`
825 unfolding frontier_closures closure_def islimpt_def
826 using open_ball[of a e] `e > 0`
827 by simp (metis centre_in_ball mem_ball open_ball)
829 ultimately have ?rhse by auto
835 { fix T assume "a\<notin>S" and
836 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"
837 from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
838 then obtain e where "e>0" "ball a e \<subseteq> T" by auto
839 then obtain y where y:"y\<in>S" "dist a y < e" using as(1) by auto
840 have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
841 using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
843 hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
845 { fix T assume "a \<in> T" "open T" "a\<in>S"
846 then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
847 obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
848 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
850 hence "a islimpt (- S) \<or> a\<notin>S" unfolding islimpt_def by auto
851 ultimately show ?lhs unfolding frontier_closures using closure_def[of "- S"] by auto
854 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
855 by (metis frontier_def closure_closed Diff_subset)
857 lemma frontier_empty[simp]: "frontier {} = {}"
858 by (simp add: frontier_def)
860 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
862 { assume "frontier S \<subseteq> S"
863 hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
864 hence "closed S" using closure_subset_eq by auto
866 thus ?thesis using frontier_subset_closed[of S] ..
869 lemma frontier_complement: "frontier(- S) = frontier S"
870 by (auto simp add: frontier_def closure_complement interior_complement)
872 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
873 using frontier_complement frontier_subset_eq[of "- S"]
874 unfolding open_closed by auto
877 subsection {* Filters and the ``eventually true'' quantifier *}
880 at_infinity :: "'a::real_normed_vector filter" where
881 "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
884 indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
885 (infixr "indirection" 70) where
886 "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
888 text{* Prove That They are all filters. *}
890 lemma eventually_at_infinity:
891 "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
892 unfolding at_infinity_def
893 proof (rule eventually_Abs_filter, rule is_filter.intro)
894 fix P Q :: "'a \<Rightarrow> bool"
895 assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
896 then obtain r s where
897 "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
898 then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
899 then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
902 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
904 lemma trivial_limit_within:
905 shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
907 assume "trivial_limit (at a within S)"
908 thus "\<not> a islimpt S"
909 unfolding trivial_limit_def
910 unfolding eventually_within eventually_at_topological
911 unfolding islimpt_def
912 apply (clarsimp simp add: set_eq_iff)
913 apply (rename_tac T, rule_tac x=T in exI)
914 apply (clarsimp, drule_tac x=y in bspec, simp_all)
917 assume "\<not> a islimpt S"
918 thus "trivial_limit (at a within S)"
919 unfolding trivial_limit_def
920 unfolding eventually_within eventually_at_topological
921 unfolding islimpt_def
923 apply (rule_tac x=T in exI)
928 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
929 using trivial_limit_within [of a UNIV]
930 by (simp add: within_UNIV)
932 lemma trivial_limit_at:
933 fixes a :: "'a::perfect_space"
934 shows "\<not> trivial_limit (at a)"
935 by (simp add: trivial_limit_at_iff)
937 lemma trivial_limit_at_infinity:
938 "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
939 unfolding trivial_limit_def eventually_at_infinity
941 apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
942 apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
943 apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
944 apply (drule_tac x=UNIV in spec, simp)
947 text {* Some property holds "sufficiently close" to the limit point. *}
949 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
950 "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
951 unfolding eventually_at dist_nz by auto
953 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
954 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
955 unfolding eventually_within eventually_at dist_nz by auto
957 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
958 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
959 unfolding eventually_within
960 by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl)
962 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
963 unfolding trivial_limit_def
964 by (auto elim: eventually_rev_mp)
966 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
967 unfolding trivial_limit_def by (auto elim: eventually_rev_mp)
969 lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
970 unfolding trivial_limit_def ..
972 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
973 apply (safe elim!: trivial_limit_eventually)
974 apply (simp add: eventually_False [symmetric])
977 text{* Combining theorems for "eventually" *}
979 lemma eventually_rev_mono:
980 "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
981 using eventually_mono [of P Q] by fast
983 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
984 by (simp add: eventually_False)
987 subsection {* Limits *}
989 text{* Notation Lim to avoid collition with lim defined in analysis *}
991 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"
992 where "Lim A f = (THE l. (f ---> l) A)"
995 "(f ---> l) net \<longleftrightarrow>
996 trivial_limit net \<or>
997 (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
998 unfolding tendsto_iff trivial_limit_eq by auto
1000 text{* Show that they yield usual definitions in the various cases. *}
1002 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
1003 (\<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)"
1004 by (auto simp add: tendsto_iff eventually_within_le)
1006 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
1007 (\<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)"
1008 by (auto simp add: tendsto_iff eventually_within)
1010 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
1011 (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
1012 by (auto simp add: tendsto_iff eventually_at)
1014 lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
1015 unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
1017 lemma Lim_at_infinity:
1018 "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
1019 by (auto simp add: tendsto_iff eventually_at_infinity)
1021 lemma Lim_sequentially:
1022 "(S ---> l) sequentially \<longleftrightarrow>
1023 (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
1024 by (rule LIMSEQ_def) (* FIXME: redundant *)
1026 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
1027 by (rule topological_tendstoI, auto elim: eventually_rev_mono)
1029 text{* The expected monotonicity property. *}
1031 lemma Lim_within_empty: "(f ---> l) (net within {})"
1032 unfolding tendsto_def Limits.eventually_within by simp
1034 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
1035 unfolding tendsto_def Limits.eventually_within
1036 by (auto elim!: eventually_elim1)
1038 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
1039 shows "(f ---> l) (net within (S \<union> T))"
1040 using assms unfolding tendsto_def Limits.eventually_within
1042 apply (drule spec, drule (1) mp, drule (1) mp)
1043 apply (drule spec, drule (1) mp, drule (1) mp)
1044 apply (auto elim: eventually_elim2)
1048 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow> S \<union> T = UNIV
1050 by (metis Lim_Un within_UNIV)
1052 text{* Interrelations between restricted and unrestricted limits. *}
1054 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
1056 unfolding tendsto_def Limits.eventually_within
1057 apply (clarify, drule spec, drule (1) mp, drule (1) mp)
1058 by (auto elim!: eventually_elim1)
1060 lemma eventually_within_interior:
1061 assumes "x \<in> interior S"
1062 shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
1064 from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S"
1065 unfolding interior_def by fast
1067 then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
1068 unfolding Limits.eventually_within Limits.eventually_at_topological
1070 with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
1073 unfolding Limits.eventually_at_topological by auto
1075 { assume "?rhs" hence "?lhs"
1076 unfolding Limits.eventually_within
1077 by (auto elim: eventually_elim1)
1082 lemma at_within_interior:
1083 "x \<in> interior S \<Longrightarrow> at x within S = at x"
1084 by (simp add: filter_eq_iff eventually_within_interior)
1086 lemma at_within_open:
1087 "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"
1088 by (simp only: at_within_interior interior_open)
1090 lemma Lim_within_open:
1091 fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
1092 assumes"a \<in> S" "open S"
1093 shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
1094 using assms by (simp only: at_within_open)
1096 lemma Lim_within_LIMSEQ:
1097 fixes a :: real and L :: "'a::metric_space"
1098 assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
1099 shows "(X ---> L) (at a within T)"
1101 assume "\<not> (X ---> L) (at a within T)"
1102 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"
1103 unfolding tendsto_iff eventually_within dist_norm by (simp add: not_less[symmetric])
1104 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
1106 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"
1107 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"
1108 using r by (simp add: Bex_def)
1109 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"
1111 hence F1: "\<And>n. ?F n \<in> T \<and> ?F n \<noteq> a"
1112 and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
1113 and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
1117 proof (rule LIMSEQ_I, unfold real_norm_def)
1120 (* choose no such that inverse (real (Suc n)) < e *)
1121 then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
1122 then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
1123 show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
1124 proof (intro exI allI impI)
1126 assume mlen: "m \<le> n"
1127 have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
1129 also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
1131 also from nodef have
1132 "inverse (real (Suc m)) < e" .
1133 finally show "\<bar>?F n - a\<bar> < e" .
1136 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`
1137 ultimately have "(\<lambda>n. X (?F n)) ----> L" using F1 by simp
1139 moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
1143 obtain n where "n = no + 1" by simp
1144 then have nolen: "no \<le> n" by simp
1145 (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
1146 have "dist (X (?F n)) L \<ge> r"
1148 with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
1150 then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
1151 with r have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
1152 thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
1154 ultimately show False by simp
1157 lemma Lim_right_bound:
1158 fixes f :: "real \<Rightarrow> real"
1159 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"
1160 assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
1161 shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
1163 assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
1165 assume [simp]: "{x<..} \<inter> I \<noteq> {}"
1167 proof (rule Lim_within_LIMSEQ, safe)
1168 fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"
1170 show "(\<lambda>n. f (S n)) ----> Inf (f ` ({x<..} \<inter> I))"
1171 proof (rule LIMSEQ_I, rule ccontr)
1172 fix r :: real assume "0 < r"
1173 with Inf_close[of "f ` ({x<..} \<inter> I)" r]
1174 obtain y where y: "x < y" "y \<in> I" "f y < Inf (f ` ({x <..} \<inter> I)) + r" by auto
1175 from `x < y` have "0 < y - x" by auto
1176 from S(2)[THEN LIMSEQ_D, OF this]
1177 obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto
1179 assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f ` ({x<..} \<inter> I))) < r)"
1180 moreover have "\<And>n. Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
1181 using S bnd by (intro Inf_lower[where z=K]) auto
1182 ultimately obtain n where n: "N \<le> n" "r + Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
1183 by (auto simp: not_less field_simps)
1184 with N[OF n(1)] mono[OF _ `y \<in> I`, of "S n"] S(1)[THEN spec, of n] y
1190 text{* Another limit point characterization. *}
1192 lemma islimpt_sequential:
1193 fixes x :: "'a::metric_space"
1194 shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"
1198 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"
1199 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
1201 have "f (inverse (real n + 1)) \<in> S - {x}" using f by auto
1204 { fix e::real assume "e>0"
1205 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
1206 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
1207 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)
1208 moreover have "\<forall>n\<ge>N. dist (f (inverse (real n + 1))) x < (inverse (real n + 1))" using f `e>0` by auto
1209 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
1211 hence " ((\<lambda>n. f (inverse (real n + 1))) ---> x) sequentially"
1212 unfolding Lim_sequentially using f by auto
1213 ultimately show ?rhs apply (rule_tac x="(\<lambda>n::nat. f (inverse (real n + 1)))" in exI) by auto
1216 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
1217 { fix e::real assume "e>0"
1218 then obtain N where "dist (f N) x < e" using f(2) by auto
1219 moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto
1220 ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto
1222 thus ?lhs unfolding islimpt_approachable by auto
1225 lemma Lim_inv: (* TODO: delete *)
1226 fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
1227 assumes "(f ---> l) A" and "l \<noteq> 0"
1228 shows "((inverse o f) ---> inverse l) A"
1229 unfolding o_def using assms by (rule tendsto_inverse)
1232 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1233 shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
1234 by (simp add: Lim dist_norm)
1236 lemma Lim_null_comparison:
1237 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1238 assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
1239 shows "(f ---> 0) net"
1240 proof (rule metric_tendsto_imp_tendsto)
1241 show "(g ---> 0) net" by fact
1242 show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
1243 using assms(1) by (rule eventually_elim1, simp add: dist_norm)
1246 lemma Lim_transform_bound:
1247 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1248 fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
1249 assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net" "(g ---> 0) net"
1250 shows "(f ---> 0) net"
1251 using assms(1) tendsto_norm_zero [OF assms(2)]
1252 by (rule Lim_null_comparison)
1254 text{* Deducing things about the limit from the elements. *}
1256 lemma Lim_in_closed_set:
1257 assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
1260 assume "l \<notin> S"
1261 with `closed S` have "open (- S)" "l \<in> - S"
1262 by (simp_all add: open_Compl)
1263 with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
1264 by (rule topological_tendstoD)
1265 with assms(2) have "eventually (\<lambda>x. False) net"
1266 by (rule eventually_elim2) simp
1267 with assms(3) show "False"
1268 by (simp add: eventually_False)
1271 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
1273 lemma Lim_dist_ubound:
1274 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
1275 shows "dist a l <= e"
1277 have "dist a l \<in> {..e}"
1278 proof (rule Lim_in_closed_set)
1279 show "closed {..e}" by simp
1280 show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)
1281 show "\<not> trivial_limit net" by fact
1282 show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)
1284 thus ?thesis by simp
1287 lemma Lim_norm_ubound:
1288 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1289 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
1290 shows "norm(l) <= e"
1292 have "norm l \<in> {..e}"
1293 proof (rule Lim_in_closed_set)
1294 show "closed {..e}" by simp
1295 show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)
1296 show "\<not> trivial_limit net" by fact
1297 show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
1299 thus ?thesis by simp
1302 lemma Lim_norm_lbound:
1303 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1304 assumes "\<not> (trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. e <= norm(f x)) net"
1305 shows "e \<le> norm l"
1307 have "norm l \<in> {e..}"
1308 proof (rule Lim_in_closed_set)
1309 show "closed {e..}" by simp
1310 show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)
1311 show "\<not> trivial_limit net" by fact
1312 show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
1314 thus ?thesis by simp
1317 text{* Uniqueness of the limit, when nontrivial. *}
1320 fixes f :: "'a \<Rightarrow> 'b::t2_space"
1321 shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
1322 unfolding Lim_def using tendsto_unique[of net f] by auto
1324 text{* Limit under bilinear function *}
1327 assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
1328 shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
1329 using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
1330 by (rule bounded_bilinear.tendsto)
1332 text{* These are special for limits out of the same vector space. *}
1334 lemma Lim_within_id: "(id ---> a) (at a within s)"
1335 unfolding tendsto_def Limits.eventually_within eventually_at_topological
1338 lemma Lim_at_id: "(id ---> a) (at a)"
1339 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
1342 fixes a :: "'a::real_normed_vector"
1343 fixes l :: "'b::topological_space"
1344 shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
1345 using LIM_offset_zero LIM_offset_zero_cancel ..
1347 text{* It's also sometimes useful to extract the limit point from the filter. *}
1350 netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
1351 "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
1353 lemma netlimit_within:
1354 assumes "\<not> trivial_limit (at a within S)"
1355 shows "netlimit (at a within S) = a"
1356 unfolding netlimit_def
1357 apply (rule some_equality)
1358 apply (rule Lim_at_within)
1359 apply (rule LIM_ident)
1360 apply (erule tendsto_unique [OF assms])
1361 apply (rule Lim_at_within)
1362 apply (rule LIM_ident)
1366 fixes a :: "'a::{perfect_space,t2_space}"
1367 shows "netlimit (at a) = a"
1368 apply (subst within_UNIV[symmetric])
1369 using netlimit_within[of a UNIV]
1370 by (simp add: trivial_limit_at within_UNIV)
1372 lemma lim_within_interior:
1373 "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
1374 by (simp add: at_within_interior)
1376 lemma netlimit_within_interior:
1377 fixes x :: "'a::{t2_space,perfect_space}"
1378 assumes "x \<in> interior S"
1379 shows "netlimit (at x within S) = x"
1380 using assms by (simp add: at_within_interior netlimit_at)
1382 text{* Transformation of limit. *}
1384 lemma Lim_transform:
1385 fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
1386 assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
1387 shows "(g ---> l) net"
1388 using tendsto_diff [OF assms(2) assms(1)] by simp
1390 lemma Lim_transform_eventually:
1391 "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
1392 apply (rule topological_tendstoI)
1393 apply (drule (2) topological_tendstoD)
1394 apply (erule (1) eventually_elim2, simp)
1397 lemma Lim_transform_within:
1398 assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
1399 and "(f ---> l) (at x within S)"
1400 shows "(g ---> l) (at x within S)"
1401 proof (rule Lim_transform_eventually)
1402 show "eventually (\<lambda>x. f x = g x) (at x within S)"
1403 unfolding eventually_within
1404 using assms(1,2) by auto
1405 show "(f ---> l) (at x within S)" by fact
1408 lemma Lim_transform_at:
1409 assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
1410 and "(f ---> l) (at x)"
1411 shows "(g ---> l) (at x)"
1412 proof (rule Lim_transform_eventually)
1413 show "eventually (\<lambda>x. f x = g x) (at x)"
1414 unfolding eventually_at
1415 using assms(1,2) by auto
1416 show "(f ---> l) (at x)" by fact
1419 text{* Common case assuming being away from some crucial point like 0. *}
1421 lemma Lim_transform_away_within:
1422 fixes a b :: "'a::t1_space"
1423 assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1424 and "(f ---> l) (at a within S)"
1425 shows "(g ---> l) (at a within S)"
1426 proof (rule Lim_transform_eventually)
1427 show "(f ---> l) (at a within S)" by fact
1428 show "eventually (\<lambda>x. f x = g x) (at a within S)"
1429 unfolding Limits.eventually_within eventually_at_topological
1430 by (rule exI [where x="- {b}"], simp add: open_Compl assms)
1433 lemma Lim_transform_away_at:
1434 fixes a b :: "'a::t1_space"
1435 assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
1436 and fl: "(f ---> l) (at a)"
1437 shows "(g ---> l) (at a)"
1438 using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
1439 by (auto simp add: within_UNIV)
1441 text{* Alternatively, within an open set. *}
1443 lemma Lim_transform_within_open:
1444 assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
1445 and "(f ---> l) (at a)"
1446 shows "(g ---> l) (at a)"
1447 proof (rule Lim_transform_eventually)
1448 show "eventually (\<lambda>x. f x = g x) (at a)"
1449 unfolding eventually_at_topological
1450 using assms(1,2,3) by auto
1451 show "(f ---> l) (at a)" by fact
1454 text{* A congruence rule allowing us to transform limits assuming not at point. *}
1456 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
1458 lemma Lim_cong_within(*[cong add]*):
1459 assumes "a = b" "x = y" "S = T"
1460 assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
1461 shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
1462 unfolding tendsto_def Limits.eventually_within eventually_at_topological
1465 lemma Lim_cong_at(*[cong add]*):
1466 assumes "a = b" "x = y"
1467 assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
1468 shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
1469 unfolding tendsto_def eventually_at_topological
1472 text{* Useful lemmas on closure and set of possible sequential limits.*}
1474 lemma closure_sequential:
1475 fixes l :: "'a::metric_space"
1476 shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
1478 assume "?lhs" moreover
1479 { assume "l \<in> S"
1480 hence "?rhs" using tendsto_const[of l sequentially] by auto
1482 { assume "l islimpt S"
1483 hence "?rhs" unfolding islimpt_sequential by auto
1485 show "?rhs" unfolding closure_def by auto
1488 thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
1491 lemma closed_sequential_limits:
1492 fixes S :: "'a::metric_space set"
1493 shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
1494 unfolding closed_limpt
1495 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]
1498 lemma closure_approachable:
1499 fixes S :: "'a::metric_space set"
1500 shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
1501 apply (auto simp add: closure_def islimpt_approachable)
1502 by (metis dist_self)
1504 lemma closed_approachable:
1505 fixes S :: "'a::metric_space set"
1506 shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
1507 by (metis closure_closed closure_approachable)
1509 text{* Some other lemmas about sequences. *}
1511 lemma sequentially_offset:
1512 assumes "eventually (\<lambda>i. P i) sequentially"
1513 shows "eventually (\<lambda>i. P (i + k)) sequentially"
1514 using assms unfolding eventually_sequentially by (metis trans_le_add1)
1517 assumes "(f ---> l) sequentially"
1518 shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
1519 using assms unfolding tendsto_def
1520 by clarify (rule sequentially_offset, simp)
1522 lemma seq_offset_neg:
1523 "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
1524 apply (rule topological_tendstoI)
1525 apply (drule (2) topological_tendstoD)
1526 apply (simp only: eventually_sequentially)
1527 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
1531 lemma seq_offset_rev:
1532 "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
1533 apply (rule topological_tendstoI)
1534 apply (drule (2) topological_tendstoD)
1535 apply (simp only: eventually_sequentially)
1536 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
1539 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
1541 { fix e::real assume "e>0"
1542 hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
1543 using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
1544 by (metis le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
1546 thus ?thesis unfolding Lim_sequentially dist_norm by simp
1549 subsection {* More properties of closed balls *}
1551 lemma closed_cball: "closed (cball x e)"
1552 unfolding cball_def closed_def
1553 unfolding Collect_neg_eq [symmetric] not_le
1554 apply (clarsimp simp add: open_dist, rename_tac y)
1555 apply (rule_tac x="dist x y - e" in exI, clarsimp)
1556 apply (rename_tac x')
1557 apply (cut_tac x=x and y=x' and z=y in dist_triangle)
1561 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
1563 { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
1564 hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
1566 { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
1567 hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
1569 show ?thesis unfolding open_contains_ball by auto
1572 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
1573 by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
1575 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
1576 apply (simp add: interior_def, safe)
1577 apply (force simp add: open_contains_cball)
1578 apply (rule_tac x="ball x e" in exI)
1579 apply (simp add: subset_trans [OF ball_subset_cball])
1583 fixes x y :: "'a::{real_normed_vector,perfect_space}"
1584 shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
1587 { assume "e \<le> 0"
1588 hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
1589 have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
1591 hence "e > 0" by (metis not_less)
1593 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
1594 ultimately show "?rhs" by auto
1596 assume "?rhs" hence "e>0" by auto
1597 { fix d::real assume "d>0"
1598 have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1599 proof(cases "d \<le> dist x y")
1600 case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1602 case True hence False using `d \<le> dist x y` `d>0` by auto
1603 thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
1607 have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
1608 = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1609 unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
1610 also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
1611 using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
1612 unfolding scaleR_minus_left scaleR_one
1613 by (auto simp add: norm_minus_commute)
1614 also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
1615 unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
1616 unfolding left_distrib using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
1617 also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
1618 finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
1622 have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
1623 using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
1625 have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
1626 using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
1627 unfolding dist_norm by auto
1628 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
1631 case False hence "d > dist x y" by auto
1632 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1635 obtain z where **: "z \<noteq> y" "dist z y < min e d"
1636 using perfect_choose_dist[of "min e d" y]
1637 using `d > 0` `e>0` by auto
1638 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1640 using `z \<noteq> y` **
1641 by (rule_tac x=z in bexI, auto simp add: dist_commute)
1643 case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
1644 using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
1647 thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
1650 lemma closure_ball_lemma:
1651 fixes x y :: "'a::real_normed_vector"
1652 assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
1653 proof (rule islimptI)
1654 fix T assume "y \<in> T" "open T"
1655 then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
1656 unfolding open_dist by fast
1657 (* choose point between x and y, within distance r of y. *)
1658 def k \<equiv> "min 1 (r / (2 * dist x y))"
1659 def z \<equiv> "y + scaleR k (x - y)"
1660 have z_def2: "z = x + scaleR (1 - k) (y - x)"
1661 unfolding z_def by (simp add: algebra_simps)
1663 unfolding z_def k_def using `0 < r`
1664 by (simp add: dist_norm min_def)
1665 hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp
1666 have "dist x z < dist x y"
1667 unfolding z_def2 dist_norm
1668 apply (simp add: norm_minus_commute)
1669 apply (simp only: dist_norm [symmetric])
1670 apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
1671 apply (rule mult_strict_right_mono)
1672 apply (simp add: k_def divide_pos_pos zero_less_dist_iff `0 < r` `x \<noteq> y`)
1673 apply (simp add: zero_less_dist_iff `x \<noteq> y`)
1675 hence "z \<in> ball x (dist x y)" by simp
1677 unfolding z_def k_def using `x \<noteq> y` `0 < r`
1678 by (simp add: min_def)
1679 show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
1680 using `z \<in> ball x (dist x y)` `z \<in> T` `z \<noteq> y`
1685 fixes x :: "'a::real_normed_vector"
1686 shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
1687 apply (rule equalityI)
1688 apply (rule closure_minimal)
1689 apply (rule ball_subset_cball)
1690 apply (rule closed_cball)
1691 apply (rule subsetI, rename_tac y)
1692 apply (simp add: le_less [where 'a=real])
1694 apply (rule subsetD [OF closure_subset], simp)
1695 apply (simp add: closure_def)
1697 apply (rule closure_ball_lemma)
1698 apply (simp add: zero_less_dist_iff)
1701 (* In a trivial vector space, this fails for e = 0. *)
1702 lemma interior_cball:
1703 fixes x :: "'a::{real_normed_vector, perfect_space}"
1704 shows "interior (cball x e) = ball x e"
1705 proof(cases "e\<ge>0")
1706 case False note cs = this
1707 from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
1708 { fix y assume "y \<in> cball x e"
1709 hence False unfolding mem_cball using dist_nz[of x y] cs by auto }
1710 hence "cball x e = {}" by auto
1711 hence "interior (cball x e) = {}" using interior_empty by auto
1712 ultimately show ?thesis by blast
1714 case True note cs = this
1715 have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover
1716 { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
1717 then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
1719 then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
1720 using perfect_choose_dist [of d] by auto
1721 have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
1722 hence xa_cball:"xa \<in> cball x e" using as(1) by auto
1724 hence "y \<in> ball x e" proof(cases "x = y")
1726 hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
1727 thus "y \<in> ball x e" using `x = y ` by simp
1730 have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
1731 using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
1732 hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
1733 have "y - x \<noteq> 0" using `x \<noteq> y` by auto
1734 hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
1735 using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
1737 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)"
1738 by (auto simp add: dist_norm algebra_simps)
1739 also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
1740 by (auto simp add: algebra_simps)
1741 also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
1743 also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
1744 finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
1745 thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
1747 hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto
1748 ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
1751 lemma frontier_ball:
1752 fixes a :: "'a::real_normed_vector"
1753 shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
1754 apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
1755 apply (simp add: set_eq_iff)
1758 lemma frontier_cball:
1759 fixes a :: "'a::{real_normed_vector, perfect_space}"
1760 shows "frontier(cball a e) = {x. dist a x = e}"
1761 apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
1762 apply (simp add: set_eq_iff)
1765 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
1766 apply (simp add: set_eq_iff not_le)
1767 by (metis zero_le_dist dist_self order_less_le_trans)
1768 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
1770 lemma cball_eq_sing:
1771 fixes x :: "'a::{metric_space,perfect_space}"
1772 shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
1773 proof (rule linorder_cases)
1775 obtain a where "a \<noteq> x" "dist a x < e"
1776 using perfect_choose_dist [OF e] by auto
1777 hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
1778 with e show ?thesis by (auto simp add: set_eq_iff)
1782 fixes x :: "'a::metric_space"
1783 shows "e = 0 ==> cball x e = {x}"
1784 by (auto simp add: set_eq_iff)
1787 subsection {* Boundedness *}
1789 (* FIXME: This has to be unified with BSEQ!! *)
1790 definition (in metric_space)
1791 bounded :: "'a set \<Rightarrow> bool" where
1792 "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
1794 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
1795 unfolding bounded_def
1797 apply (rule_tac x="dist a x + e" in exI, clarify)
1798 apply (drule (1) bspec)
1799 apply (erule order_trans [OF dist_triangle add_left_mono])
1803 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
1804 unfolding bounded_any_center [where a=0]
1805 by (simp add: dist_norm)
1807 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
1808 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
1809 by (metis bounded_def subset_eq)
1811 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
1812 by (metis bounded_subset interior_subset)
1814 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
1816 from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
1817 { fix y assume "y \<in> closure S"
1818 then obtain f where f: "\<forall>n. f n \<in> S" "(f ---> y) sequentially"
1819 unfolding closure_sequential by auto
1820 have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
1821 hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
1822 by (rule eventually_mono, simp add: f(1))
1823 have "dist x y \<le> a"
1824 apply (rule Lim_dist_ubound [of sequentially f])
1825 apply (rule trivial_limit_sequentially)
1830 thus ?thesis unfolding bounded_def by auto
1833 lemma bounded_cball[simp,intro]: "bounded (cball x e)"
1834 apply (simp add: bounded_def)
1835 apply (rule_tac x=x in exI)
1836 apply (rule_tac x=e in exI)
1840 lemma bounded_ball[simp,intro]: "bounded(ball x e)"
1841 by (metis ball_subset_cball bounded_cball bounded_subset)
1843 lemma finite_imp_bounded[intro]:
1844 fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"
1846 { fix a and F :: "'a set" assume as:"bounded F"
1847 then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto
1848 hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto
1849 hence "bounded (insert a F)" unfolding bounded_def by (intro exI)
1851 thus ?thesis using finite_induct[of S bounded] using bounded_empty assms by auto
1854 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
1855 apply (auto simp add: bounded_def)
1856 apply (rename_tac x y r s)
1857 apply (rule_tac x=x in exI)
1858 apply (rule_tac x="max r (dist x y + s)" in exI)
1859 apply (rule ballI, rename_tac z, safe)
1860 apply (drule (1) bspec, simp)
1861 apply (drule (1) bspec)
1862 apply (rule min_max.le_supI2)
1863 apply (erule order_trans [OF dist_triangle add_left_mono])
1866 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
1867 by (induct rule: finite_induct[of F], auto)
1869 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
1870 apply (simp add: bounded_iff)
1871 apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
1874 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
1875 by (metis Int_lower1 Int_lower2 bounded_subset)
1877 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
1878 apply (metis Diff_subset bounded_subset)
1881 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
1882 by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
1884 lemma not_bounded_UNIV[simp, intro]:
1885 "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
1886 proof(auto simp add: bounded_pos not_le)
1887 obtain x :: 'a where "x \<noteq> 0"
1888 using perfect_choose_dist [OF zero_less_one] by fast
1889 fix b::real assume b: "b >0"
1890 have b1: "b +1 \<ge> 0" using b by simp
1891 with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
1892 by (simp add: norm_sgn)
1893 then show "\<exists>x::'a. b < norm x" ..
1896 lemma bounded_linear_image:
1897 assumes "bounded S" "bounded_linear f"
1898 shows "bounded(f ` S)"
1900 from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
1901 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)
1902 { fix x assume "x\<in>S"
1903 hence "norm x \<le> b" using b by auto
1904 hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
1905 by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
1907 thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
1908 using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
1911 lemma bounded_scaling:
1912 fixes S :: "'a::real_normed_vector set"
1913 shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
1914 apply (rule bounded_linear_image, assumption)
1915 apply (rule scaleR.bounded_linear_right)
1918 lemma bounded_translation:
1919 fixes S :: "'a::real_normed_vector set"
1920 assumes "bounded S" shows "bounded ((\<lambda>x. a + x) ` S)"
1922 from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
1923 { fix x assume "x\<in>S"
1924 hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
1926 thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
1927 by (auto intro!: add exI[of _ "b + norm a"])
1931 text{* Some theorems on sups and infs using the notion "bounded". *}
1934 fixes S :: "real set"
1935 shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"
1936 by (simp add: bounded_iff)
1938 lemma bounded_has_Sup:
1939 fixes S :: "real set"
1940 assumes "bounded S" "S \<noteq> {}"
1941 shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
1943 fix x assume "x\<in>S"
1944 thus "x \<le> Sup S"
1945 by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)
1947 show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
1948 by (metis SupInf.Sup_least)
1952 fixes S :: "real set"
1953 shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
1954 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal)
1956 lemma Sup_insert_finite:
1957 fixes S :: "real set"
1958 shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
1959 apply (rule Sup_insert)
1960 apply (rule finite_imp_bounded)
1963 lemma bounded_has_Inf:
1964 fixes S :: "real set"
1965 assumes "bounded S" "S \<noteq> {}"
1966 shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
1968 fix x assume "x\<in>S"
1969 from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
1970 thus "x \<ge> Inf S" using `x\<in>S`
1971 by (metis Inf_lower_EX abs_le_D2 minus_le_iff)
1973 show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
1974 by (metis SupInf.Inf_greatest)
1978 fixes S :: "real set"
1979 shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
1980 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)
1981 lemma Inf_insert_finite:
1982 fixes S :: "real set"
1983 shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
1984 by (rule Inf_insert, rule finite_imp_bounded, simp)
1986 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)
1987 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"
1988 apply (frule isGlb_isLb)
1989 apply (frule_tac x = y in isGlb_isLb)
1990 apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
1994 subsection {* Equivalent versions of compactness *}
1996 subsubsection{* Sequential compactness *}
1999 compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
2000 "compact S \<longleftrightarrow>
2001 (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
2002 (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
2005 assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
2007 unfolding compact_def using assms by fast
2010 assumes "compact S" "\<forall>n. f n \<in> S"
2011 obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
2012 using assms unfolding compact_def by fast
2015 A metric space (or topological vector space) is said to have the
2016 Heine-Borel property if every closed and bounded subset is compact.
2019 class heine_borel = metric_space +
2020 assumes bounded_imp_convergent_subsequence:
2021 "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
2022 \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2024 lemma bounded_closed_imp_compact:
2025 fixes s::"'a::heine_borel set"
2026 assumes "bounded s" and "closed s" shows "compact s"
2027 proof (unfold compact_def, clarify)
2028 fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
2029 obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
2030 using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
2031 from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
2032 have "l \<in> s" using `closed s` fr l
2033 unfolding closed_sequential_limits by blast
2034 show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2035 using `l \<in> s` r l by blast
2038 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
2040 show "0 \<le> r 0" by auto
2042 fix n assume "n \<le> r n"
2043 moreover have "r n < r (Suc n)"
2044 using assms [unfolded subseq_def] by auto
2045 ultimately show "Suc n \<le> r (Suc n)" by auto
2048 lemma eventually_subseq:
2049 assumes r: "subseq r"
2050 shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
2051 unfolding eventually_sequentially
2052 by (metis subseq_bigger [OF r] le_trans)
2055 "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
2056 unfolding tendsto_def eventually_sequentially o_def
2057 by (metis subseq_bigger le_trans)
2059 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
2061 apply (rule_tac x="nat_rec e f" in exI)
2063 apply (rule def_nat_rec_0, simp)
2064 apply (rule allI, rule def_nat_rec_Suc, simp)
2065 apply (rule allI, rule impI, rule ext)
2067 apply (induct_tac x)
2069 apply (erule_tac x="n" in allE)
2073 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
2074 assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
2075 shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N. abs(s n - l) < e"
2077 have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
2078 then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto
2079 { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"
2081 obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
2082 have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
2083 with n have "s N \<le> t - e" using `e>0` by auto
2084 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 }
2085 hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
2086 hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto }
2087 thus ?thesis by blast
2090 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
2091 assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
2092 shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
2093 using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
2094 unfolding monoseq_def incseq_def
2095 apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
2096 unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
2098 (* TODO: merge this lemma with the ones above *)
2099 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"
2100 assumes "bounded {s n| n::nat. True}" "\<forall>n. (s n) \<le>(s(Suc n))"
2101 shows "\<exists>l. (s ---> l) sequentially"
2103 obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le> a" using assms(1)[unfolded bounded_iff] by auto
2105 have "\<And> n. n\<ge>m \<longrightarrow> (s m) \<le> (s n)"
2106 apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)
2107 apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq) }
2108 hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto
2109 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]
2110 unfolding monoseq_def by auto
2111 thus ?thesis unfolding Lim_sequentially apply(rule_tac x="l" in exI)
2112 unfolding dist_norm by auto
2115 lemma compact_real_lemma:
2116 assumes "\<forall>n::nat. abs(s n) \<le> b"
2117 shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
2119 obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
2120 using seq_monosub[of s] by auto
2121 thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
2122 unfolding tendsto_iff dist_norm eventually_sequentially by auto
2125 instance real :: heine_borel
2127 fix s :: "real set" and f :: "nat \<Rightarrow> real"
2128 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2129 then obtain b where b: "\<forall>n. abs (f n) \<le> b"
2130 unfolding bounded_iff by auto
2131 obtain l :: real and r :: "nat \<Rightarrow> nat" where
2132 r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
2133 using compact_real_lemma [OF b] by auto
2134 thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2138 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $$ i) ` s)"
2139 apply (erule bounded_linear_image)
2140 apply (rule bounded_linear_euclidean_component)
2143 lemma compact_lemma:
2144 fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
2145 assumes "bounded s" and "\<forall>n. f n \<in> s"
2146 shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>
2147 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2149 fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"
2150 have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
2151 hence "\<exists>l::'a. \<exists>r. subseq r \<and>
2152 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2153 proof(induct d) case empty thus ?case unfolding subseq_def by auto
2154 next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto
2155 have s': "bounded ((\<lambda>x. x $$ k) ` s)" using `bounded s` by (rule bounded_component)
2156 obtain l1::"'a" and r1 where r1:"subseq r1" and
2157 lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially"
2158 using insert(3) using insert(4) by auto
2159 have f': "\<forall>n. f (r1 n) $$ k \<in> (\<lambda>x. x $$ k) ` s" using `\<forall>n. f n \<in> s` by simp
2160 obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $$ k) ---> l2) sequentially"
2161 using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
2162 def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
2163 using r1 and r2 unfolding r_def o_def subseq_def by auto
2165 def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"
2166 { fix e::real assume "e>0"
2167 from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially" by blast
2168 from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $$ k) l2 < e) sequentially" by (rule tendstoD)
2169 from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$ i) (l1 $$ i) < e) sequentially"
2170 by (rule eventually_subseq)
2171 have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$ i) (l $$ i) < e) sequentially"
2172 using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def
2173 using insert.prems by auto
2175 ultimately show ?case by auto
2177 thus "\<exists>l::'a. \<exists>r. subseq r \<and>
2178 (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"
2179 apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe
2180 apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe
2181 apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)
2182 apply(erule_tac x=i in ballE)
2183 proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a
2184 assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"
2185 hence *:"i\<ge>DIM('a)" by auto
2186 thus "dist (f (r n) $$ i) (l $$ i) < e" using e by auto
2190 instance euclidean_space \<subseteq> heine_borel
2192 fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"
2193 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2194 then obtain l::'a and r where r: "subseq r"
2195 and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $$ i) (l $$ i) < e) sequentially"
2196 using compact_lemma [OF s f] by blast
2197 let ?d = "{..<DIM('a)}"
2198 { fix e::real assume "e>0"
2199 hence "0 < e / (real_of_nat (card ?d))"
2200 using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
2201 with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))) sequentially"
2204 { fix n assume n: "\<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))"
2205 have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $$ i) (l $$ i))"
2206 apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)
2207 also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
2208 apply(rule setsum_strict_mono) using n by auto
2209 finally have "dist (f (r n)) l < e" unfolding setsum_constant
2210 using DIM_positive[where 'a='a] by auto
2212 ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
2213 by (rule eventually_elim1)
2215 hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
2216 with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
2219 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
2220 unfolding bounded_def
2222 apply (rule_tac x="a" in exI)
2223 apply (rule_tac x="e" in exI)
2225 apply (drule (1) bspec)
2226 apply (simp add: dist_Pair_Pair)
2227 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
2230 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
2231 unfolding bounded_def
2233 apply (rule_tac x="b" in exI)
2234 apply (rule_tac x="e" in exI)
2236 apply (drule (1) bspec)
2237 apply (simp add: dist_Pair_Pair)
2238 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
2241 instance prod :: (heine_borel, heine_borel) heine_borel
2243 fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
2244 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
2245 from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
2246 from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
2247 obtain l1 r1 where r1: "subseq r1"
2248 and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
2249 using bounded_imp_convergent_subsequence [OF s1 f1]
2250 unfolding o_def by fast
2251 from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
2252 from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
2253 obtain l2 r2 where r2: "subseq r2"
2254 and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
2255 using bounded_imp_convergent_subsequence [OF s2 f2]
2256 unfolding o_def by fast
2257 have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
2258 using lim_subseq [OF r2 l1] unfolding o_def .
2259 have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
2260 using tendsto_Pair [OF l1' l2] unfolding o_def by simp
2261 have r: "subseq (r1 \<circ> r2)"
2262 using r1 r2 unfolding subseq_def by simp
2263 show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2267 subsubsection{* Completeness *}
2270 "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
2271 unfolding Cauchy_def by blast
2274 complete :: "'a::metric_space set \<Rightarrow> bool" where
2275 "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f
2276 --> (\<exists>l \<in> s. (f ---> l) sequentially))"
2278 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
2283 with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
2284 by (erule_tac x="e/2" in allE) auto
2286 assume nm:"N \<le> m \<and> N \<le> n"
2287 hence "dist (s m) (s n) < e" using N
2288 using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
2291 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
2295 unfolding cauchy_def
2299 unfolding cauchy_def
2300 using dist_triangle_half_l
2304 lemma convergent_imp_cauchy:
2305 "(s ---> l) sequentially ==> Cauchy s"
2306 proof(simp only: cauchy_def, rule, rule)
2307 fix e::real assume "e>0" "(s ---> l) sequentially"
2308 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
2309 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
2312 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
2314 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
2315 hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
2317 have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto
2318 then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
2319 unfolding bounded_any_center [where a="s N"] by auto
2320 ultimately show "?thesis"
2321 unfolding bounded_any_center [where a="s N"]
2322 apply(rule_tac x="max a 1" in exI) apply auto
2323 apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto
2326 lemma compact_imp_complete: assumes "compact s" shows "complete s"
2328 { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
2329 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
2331 note lr' = subseq_bigger [OF lr(2)]
2333 { fix e::real assume "e>0"
2334 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
2335 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
2336 { fix n::nat assume n:"n \<ge> max N M"
2337 have "dist ((f \<circ> r) n) l < e/2" using n M by auto
2338 moreover have "r n \<ge> N" using lr'[of n] n by auto
2339 hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
2340 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) }
2341 hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast }
2342 hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding Lim_sequentially by auto }
2343 thus ?thesis unfolding complete_def by auto
2346 instance heine_borel < complete_space
2348 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2349 hence "bounded (range f)"
2350 by (rule cauchy_imp_bounded)
2351 hence "compact (closure (range f))"
2352 using bounded_closed_imp_compact [of "closure (range f)"] by auto
2353 hence "complete (closure (range f))"
2354 by (rule compact_imp_complete)
2355 moreover have "\<forall>n. f n \<in> closure (range f)"
2356 using closure_subset [of "range f"] by auto
2357 ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
2358 using `Cauchy f` unfolding complete_def by auto
2359 then show "convergent f"
2360 unfolding convergent_def by auto
2363 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
2364 proof(simp add: complete_def, rule, rule)
2365 fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
2366 hence "convergent f" by (rule Cauchy_convergent)
2367 thus "\<exists>l. f ----> l" unfolding convergent_def .
2370 lemma complete_imp_closed: assumes "complete s" shows "closed s"
2372 { fix x assume "x islimpt s"
2373 then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
2374 unfolding islimpt_sequential by auto
2375 then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
2376 using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto
2377 hence "x \<in> s" using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
2379 thus "closed s" unfolding closed_limpt by auto
2382 lemma complete_eq_closed:
2383 fixes s :: "'a::complete_space set"
2384 shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
2386 assume ?lhs thus ?rhs by (rule complete_imp_closed)
2389 { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
2390 then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
2391 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 }
2392 thus ?lhs unfolding complete_def by auto
2395 lemma convergent_eq_cauchy:
2396 fixes s :: "nat \<Rightarrow> 'a::complete_space"
2397 shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s" (is "?lhs = ?rhs")
2399 assume ?lhs then obtain l where "(s ---> l) sequentially" by auto
2400 thus ?rhs using convergent_imp_cauchy by auto
2402 assume ?rhs thus ?lhs using complete_univ[unfolded complete_def, THEN spec[where x=s]] by auto
2405 lemma convergent_imp_bounded:
2406 fixes s :: "nat \<Rightarrow> 'a::metric_space"
2407 shows "(s ---> l) sequentially ==> bounded (s ` (UNIV::(nat set)))"
2408 using convergent_imp_cauchy[of s]
2409 using cauchy_imp_bounded[of s]
2413 subsubsection{* Total boundedness *}
2415 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
2416 "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
2417 declare helper_1.simps[simp del]
2419 lemma compact_imp_totally_bounded:
2421 shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"
2422 proof(rule, rule, rule ccontr)
2423 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)"
2424 def x \<equiv> "helper_1 s e"
2426 have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
2427 proof(induct_tac rule:nat_less_induct)
2428 fix n def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
2429 assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
2430 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
2431 then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto
2432 have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
2433 apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
2434 thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
2436 hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
2437 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
2438 from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
2439 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
2441 using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
2442 using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
2443 using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
2446 subsubsection{* Heine-Borel theorem *}
2448 text {* Following Burkill \& Burkill vol. 2. *}
2450 lemma heine_borel_lemma: fixes s::"'a::metric_space set"
2451 assumes "compact s" "s \<subseteq> (\<Union> t)" "\<forall>b \<in> t. open b"
2452 shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
2454 assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"
2455 hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto
2457 have "1 / real (n + 1) > 0" by auto
2458 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 }
2459 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
2460 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)"
2461 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
2463 then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
2464 using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
2466 obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
2467 then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"
2468 using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto
2470 then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"
2471 using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
2473 obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
2474 have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
2475 apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
2476 using subseq_bigger[OF r, of "N1 + N2"] by auto
2478 def x \<equiv> "(f (r (N1 + N2)))"
2479 have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
2480 using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto
2481 have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto
2482 then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto
2484 have "dist x l < e/2" using N1 unfolding x_def o_def by auto
2485 hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)
2487 thus False using e and `y\<notin>b` by auto
2490 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2491 \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"
2493 fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"
2494 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
2495 hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto
2496 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
2497 then obtain bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast
2499 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
2500 then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto
2502 have "finite (bb ` k)" using k(1) by auto
2504 { fix x assume "x\<in>s"
2505 hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3) unfolding subset_eq by auto
2506 hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast
2507 hence "x \<in> \<Union>(bb ` k)" using Union_iff[of x "bb ` k"] by auto
2509 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
2512 subsubsection {* Bolzano-Weierstrass property *}
2514 lemma heine_borel_imp_bolzano_weierstrass:
2515 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'))"
2516 "infinite t" "t \<subseteq> s"
2517 shows "\<exists>x \<in> s. x islimpt t"
2519 assume "\<not> (\<exists>x \<in> s. x islimpt t)"
2520 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
2521 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
2522 obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
2523 using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
2524 from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
2525 { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
2526 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
2527 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 }
2528 hence "inj_on f t" unfolding inj_on_def by simp
2529 hence "infinite (f ` t)" using assms(2) using finite_imageD by auto
2531 { fix x assume "x\<in>t" "f x \<notin> g"
2532 from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
2533 then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
2534 hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
2535 hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto }
2536 hence "f ` t \<subseteq> g" by auto
2537 ultimately show False using g(2) using finite_subset by auto
2540 subsubsection {* Complete the chain of compactness variants *}
2542 lemma islimpt_range_imp_convergent_subsequence:
2543 fixes f :: "nat \<Rightarrow> 'a::metric_space"
2544 assumes "l islimpt (range f)"
2545 shows "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2546 proof (intro exI conjI)
2547 have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"
2548 using assms unfolding islimpt_def
2549 by (drule_tac x="ball l e" in spec)
2550 (auto simp add: zero_less_dist_iff dist_commute)
2552 def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"
2553 have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"
2554 unfolding t_def by (rule LeastI2_ex [OF * conjunct1])
2555 have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"
2556 unfolding t_def by (rule LeastI2_ex [OF * conjunct2])
2557 have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"
2558 unfolding t_def by (simp add: Least_le)
2559 have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"
2560 unfolding t_def by (drule not_less_Least) simp
2561 have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"
2563 apply (erule f_t_neq)
2564 apply (erule (1) less_le_trans [OF f_t_closer])
2566 have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"
2567 by (drule f_t_closer) auto
2568 have t_less: "\<And>e. 0 < e \<Longrightarrow> t e < t (dist (f (t e)) l)"
2569 apply (subst less_le)
2571 apply (rule t_antimono)
2572 apply (erule f_t_neq)
2573 apply (erule f_t_closer [THEN less_imp_le])
2574 apply (rule t_dist_f_neq [symmetric])
2575 apply (erule f_t_neq)
2577 have dist_f_t_less':
2578 "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"
2579 apply (simp add: le_less)
2581 apply (rule less_trans)
2582 apply (erule f_t_closer)
2583 apply (rule le_less_trans)
2584 apply (erule less_tD)
2585 apply (erule f_t_neq)
2586 apply (erule f_t_closer)
2588 apply (erule f_t_closer)
2591 def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"
2592 have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"
2593 unfolding r_def by simp_all
2594 have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"
2595 by (induct_tac n) (simp_all add: r_simps f_t_neq)
2598 unfolding subseq_Suc_iff
2601 apply (simp_all add: r_simps)
2602 apply (rule t_less, rule zero_less_one)
2603 apply (rule t_less, rule f_r_neq)
2605 show "((f \<circ> r) ---> l) sequentially"
2606 unfolding Lim_sequentially o_def
2607 apply (clarify, rule_tac x="t e" in exI, clarify)
2608 apply (drule le_trans, rule seq_suble [OF `subseq r`])
2609 apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)
2613 lemma finite_range_imp_infinite_repeats:
2614 fixes f :: "nat \<Rightarrow> 'a"
2615 assumes "finite (range f)"
2616 shows "\<exists>k. infinite {n. f n = k}"
2618 { fix A :: "'a set" assume "finite A"
2619 hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"
2621 case empty thus ?case by simp
2625 proof (cases "finite {n. f n = x}")
2627 with `infinite {n. f n \<in> insert x A}`
2628 have "infinite {n. f n \<in> A}" by simp
2629 thus "\<exists>k. infinite {n. f n = k}" by (rule insert)
2631 case False thus "\<exists>k. infinite {n. f n = k}" ..
2635 from assms show "\<exists>k. infinite {n. f n = k}"
2639 lemma bolzano_weierstrass_imp_compact:
2640 fixes s :: "'a::metric_space set"
2641 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2644 { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
2645 have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2646 proof (cases "finite (range f)")
2648 hence "\<exists>l. infinite {n. f n = l}"
2649 by (rule finite_range_imp_infinite_repeats)
2650 then obtain l where "infinite {n. f n = l}" ..
2651 hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"
2652 by (rule infinite_enumerate)
2653 then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto
2654 hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2655 unfolding o_def by (simp add: fr tendsto_const)
2656 hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2658 from f have "\<forall>n. f (r n) \<in> s" by simp
2659 hence "l \<in> s" by (simp add: fr)
2660 thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2661 by (rule rev_bexI) fact
2664 with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto
2665 then obtain l where "l \<in> s" "l islimpt (range f)" ..
2666 have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2667 using `l islimpt (range f)`
2668 by (rule islimpt_range_imp_convergent_subsequence)
2669 with `l \<in> s` show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
2672 thus ?thesis unfolding compact_def by auto
2675 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where
2676 "helper_2 beyond 0 = beyond 0" |
2677 "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"
2679 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"
2680 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2683 assume "\<not> bounded s"
2684 then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"
2685 unfolding bounded_any_center [where a=undefined]
2686 apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto
2687 hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"
2688 unfolding linorder_not_le by auto
2689 def x \<equiv> "helper_2 beyond"
2691 { fix m n ::nat assume "m<n"
2692 hence "dist undefined (x m) + 1 < dist undefined (x n)"
2694 case 0 thus ?case by auto
2697 have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"
2698 unfolding x_def and helper_2.simps
2699 using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto
2700 thus ?case proof(cases "m < n")
2701 case True thus ?thesis using Suc and * by auto
2703 case False hence "m = n" using Suc(2) by auto
2704 thus ?thesis using * by auto
2707 { fix m n ::nat assume "m\<noteq>n"
2708 have "1 < dist (x m) (x n)"
2711 hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto
2712 thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith
2714 case False hence "n<m" using `m\<noteq>n` by auto
2715 hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto
2716 thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith
2717 qed } note ** = this
2718 { fix a b assume "x a = x b" "a \<noteq> b"
2719 hence False using **[of a b] by auto }
2720 hence "inj x" unfolding inj_on_def by auto
2724 proof(cases "n = 0")
2725 case True thus ?thesis unfolding x_def using beyond by auto
2727 case False then obtain z where "n = Suc z" using not0_implies_Suc by auto
2728 thus ?thesis unfolding x_def using beyond by auto
2730 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
2732 then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto
2733 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
2734 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"]]
2735 unfolding dist_nz by auto
2736 show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto
2739 lemma sequence_infinite_lemma:
2740 fixes f :: "nat \<Rightarrow> 'a::t1_space"
2741 assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"
2742 shows "infinite (range f)"
2744 assume "finite (range f)"
2745 hence "closed (range f)" by (rule finite_imp_closed)
2746 hence "open (- range f)" by (rule open_Compl)
2747 from assms(1) have "l \<in> - range f" by auto
2748 from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
2749 using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)
2750 thus False unfolding eventually_sequentially by auto
2753 lemma closure_insert:
2754 fixes x :: "'a::t1_space"
2755 shows "closure (insert x s) = insert x (closure s)"
2756 apply (rule closure_unique)
2757 apply (rule conjI [OF insert_mono [OF closure_subset]])
2758 apply (rule conjI [OF closed_insert [OF closed_closure]])
2759 apply (simp add: closure_minimal)
2762 lemma islimpt_insert:
2763 fixes x :: "'a::t1_space"
2764 shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
2766 assume *: "x islimpt (insert a s)"
2768 proof (rule islimptI)
2769 fix t assume t: "x \<in> t" "open t"
2770 show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
2771 proof (cases "x = a")
2773 obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
2774 using * t by (rule islimptE)
2775 with `x = a` show ?thesis by auto
2778 with t have t': "x \<in> t - {a}" "open (t - {a})"
2779 by (simp_all add: open_Diff)
2780 obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
2781 using * t' by (rule islimptE)
2782 thus ?thesis by auto
2786 assume "x islimpt s" thus "x islimpt (insert a s)"
2787 by (rule islimpt_subset) auto
2790 lemma islimpt_union_finite:
2791 fixes x :: "'a::t1_space"
2792 shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
2793 by (induct set: finite, simp_all add: islimpt_insert)
2795 lemma sequence_unique_limpt:
2796 fixes f :: "nat \<Rightarrow> 'a::t2_space"
2797 assumes "(f ---> l) sequentially" and "l' islimpt (range f)"
2800 assume "l' \<noteq> l"
2801 obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
2802 using hausdorff [OF `l' \<noteq> l`] by auto
2803 have "eventually (\<lambda>n. f n \<in> t) sequentially"
2804 using assms(1) `open t` `l \<in> t` by (rule topological_tendstoD)
2805 then obtain N where "\<forall>n\<ge>N. f n \<in> t"
2806 unfolding eventually_sequentially by auto
2808 have "UNIV = {..<N} \<union> {N..}" by auto
2809 hence "l' islimpt (f ` ({..<N} \<union> {N..}))" using assms(2) by simp
2810 hence "l' islimpt (f ` {..<N} \<union> f ` {N..})" by (simp add: image_Un)
2811 hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite)
2812 then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
2813 using `l' \<in> s` `open s` by (rule islimptE)
2814 then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto
2815 with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t" by simp
2816 with `s \<inter> t = {}` show False by simp
2819 lemma bolzano_weierstrass_imp_closed:
2820 fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)
2821 assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
2824 { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
2826 proof(cases "\<forall>n. x n \<noteq> l")
2827 case False thus "l\<in>s" using as(1) by auto
2829 case True note cas = this
2830 with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto
2831 then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto
2832 thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
2834 thus ?thesis unfolding closed_sequential_limits by fast
2837 text {* Hence express everything as an equivalence. *}
2839 lemma compact_eq_heine_borel:
2840 fixes s :: "'a::metric_space set"
2841 shows "compact s \<longleftrightarrow>
2842 (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)
2843 --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")
2845 assume ?lhs thus ?rhs by (rule compact_imp_heine_borel)
2848 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
2849 by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])
2850 thus ?lhs by (rule bolzano_weierstrass_imp_compact)
2853 lemma compact_eq_bolzano_weierstrass:
2854 fixes s :: "'a::metric_space set"
2855 shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
2857 assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto
2859 assume ?rhs thus ?lhs by (rule bolzano_weierstrass_imp_compact)
2862 lemma compact_eq_bounded_closed:
2863 fixes s :: "'a::heine_borel set"
2864 shows "compact s \<longleftrightarrow> bounded s \<and> closed s" (is "?lhs = ?rhs")
2866 assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto
2868 assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto
2871 lemma compact_imp_bounded:
2872 fixes s :: "'a::metric_space set"
2873 shows "compact s ==> bounded s"
2876 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')"
2877 by (rule compact_imp_heine_borel)
2878 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
2879 using heine_borel_imp_bolzano_weierstrass[of s] by auto
2881 by (rule bolzano_weierstrass_imp_bounded)
2884 lemma compact_imp_closed:
2885 fixes s :: "'a::metric_space set"
2886 shows "compact s ==> closed s"
2889 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')"
2890 by (rule compact_imp_heine_borel)
2891 hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"
2892 using heine_borel_imp_bolzano_weierstrass[of s] by auto
2894 by (rule bolzano_weierstrass_imp_closed)
2897 text{* In particular, some common special cases. *}
2899 lemma compact_empty[simp]:
2901 unfolding compact_def
2904 lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
2905 unfolding subseq_def by simp (* TODO: move somewhere else *)
2907 lemma compact_union [intro]:
2908 assumes "compact s" and "compact t"
2909 shows "compact (s \<union> t)"
2910 proof (rule compactI)
2911 fix f :: "nat \<Rightarrow> 'a"
2912 assume "\<forall>n. f n \<in> s \<union> t"
2913 hence "infinite {n. f n \<in> s \<union> t}" by simp
2914 hence "infinite {n. f n \<in> s} \<or> infinite {n. f n \<in> t}" by simp
2915 thus "\<exists>l\<in>s \<union> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2917 assume "infinite {n. f n \<in> s}"
2918 from infinite_enumerate [OF this]
2919 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> s" by auto
2920 obtain r l where "l \<in> s" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
2921 using `compact s` `\<forall>n. (f \<circ> q) n \<in> s` by (rule compactE)
2922 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
2923 using `subseq q` by (simp_all add: subseq_o o_assoc)
2924 thus ?thesis by auto
2926 assume "infinite {n. f n \<in> t}"
2927 from infinite_enumerate [OF this]
2928 obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> t" by auto
2929 obtain r l where "l \<in> t" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"
2930 using `compact t` `\<forall>n. (f \<circ> q) n \<in> t` by (rule compactE)
2931 hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"
2932 using `subseq q` by (simp_all add: subseq_o o_assoc)
2933 thus ?thesis by auto
2937 lemma compact_inter_closed [intro]:
2938 assumes "compact s" and "closed t"
2939 shows "compact (s \<inter> t)"
2940 proof (rule compactI)
2941 fix f :: "nat \<Rightarrow> 'a"
2942 assume "\<forall>n. f n \<in> s \<inter> t"
2943 hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t" by simp_all
2944 obtain l r where "l \<in> s" "subseq r" "((f \<circ> r) ---> l) sequentially"
2945 using `compact s` `\<forall>n. f n \<in> s` by (rule compactE)
2947 from `closed t` `\<forall>n. f n \<in> t` `((f \<circ> r) ---> l) sequentially` have "l \<in> t"
2948 unfolding closed_sequential_limits o_def by fast
2949 ultimately show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
2953 lemma closed_inter_compact [intro]:
2954 assumes "closed s" and "compact t"
2955 shows "compact (s \<inter> t)"
2956 using compact_inter_closed [of t s] assms
2957 by (simp add: Int_commute)
2959 lemma compact_inter [intro]:
2960 assumes "compact s" and "compact t"
2961 shows "compact (s \<inter> t)"
2962 using assms by (intro compact_inter_closed compact_imp_closed)
2964 lemma compact_sing [simp]: "compact {a}"
2965 unfolding compact_def o_def subseq_def
2966 by (auto simp add: tendsto_const)
2968 lemma compact_insert [simp]:
2969 assumes "compact s" shows "compact (insert x s)"
2971 have "compact ({x} \<union> s)"
2972 using compact_sing assms by (rule compact_union)
2973 thus ?thesis by simp
2976 lemma finite_imp_compact:
2977 shows "finite s \<Longrightarrow> compact s"
2978 by (induct set: finite) simp_all
2980 lemma compact_cball[simp]:
2981 fixes x :: "'a::heine_borel"
2982 shows "compact(cball x e)"
2983 using compact_eq_bounded_closed bounded_cball closed_cball
2986 lemma compact_frontier_bounded[intro]:
2987 fixes s :: "'a::heine_borel set"
2988 shows "bounded s ==> compact(frontier s)"
2989 unfolding frontier_def
2990 using compact_eq_bounded_closed
2993 lemma compact_frontier[intro]:
2994 fixes s :: "'a::heine_borel set"
2995 shows "compact s ==> compact (frontier s)"
2996 using compact_eq_bounded_closed compact_frontier_bounded
2999 lemma frontier_subset_compact:
3000 fixes s :: "'a::heine_borel set"
3001 shows "compact s ==> frontier s \<subseteq> s"
3002 using frontier_subset_closed compact_eq_bounded_closed
3006 fixes s :: "'a::t1_space set"
3007 shows "open s \<Longrightarrow> open (s - {x})"
3008 by (simp add: open_Diff)
3010 text{* Finite intersection property. I could make it an equivalence in fact. *}
3012 lemma compact_imp_fip:
3013 assumes "compact s" "\<forall>t \<in> f. closed t"
3014 "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"
3015 shows "s \<inter> (\<Inter> f) \<noteq> {}"
3017 assume as:"s \<inter> (\<Inter> f) = {}"
3018 hence "s \<subseteq> \<Union> uminus ` f" by auto
3019 moreover have "Ball (uminus ` f) open" using open_Diff closed_Diff using assms(2) by auto
3020 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
3021 hence "finite (uminus ` f') \<and> uminus ` f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)
3022 hence "s \<inter> \<Inter>uminus ` f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus ` f'"]] by auto
3023 thus False using f'(3) unfolding subset_eq and Union_iff by blast
3027 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}
3029 lemma bounded_closed_nest:
3030 assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
3031 "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)" "bounded(s 0)"
3032 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
3034 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
3035 from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
3037 then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
3038 unfolding compact_def apply(erule_tac x=x in allE) using x using assms(3) by blast
3041 { fix e::real assume "e>0"
3042 with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
3043 hence "dist ((x \<circ> r) (max N n)) l < e" by auto
3045 have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
3046 hence "(x \<circ> r) (max N n) \<in> s n"
3047 using x apply(erule_tac x=n in allE)
3048 using x apply(erule_tac x="r (max N n)" in allE)
3049 using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto
3050 ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
3052 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
3054 thus ?thesis by auto
3057 text {* Decreasing case does not even need compactness, just completeness. *}
3059 lemma decreasing_closed_nest:
3060 assumes "\<forall>n. closed(s n)"
3061 "\<forall>n. (s n \<noteq> {})"
3062 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
3063 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
3064 shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s n"
3066 have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
3067 hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
3068 then obtain t where t: "\<forall>n. t n \<in> s n" by auto
3069 { fix e::real assume "e>0"
3070 then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto
3071 { fix m n ::nat assume "N \<le> m \<and> N \<le> n"
3072 hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding subset_eq t by blast+
3073 hence "dist (t m) (t n) < e" using N by auto
3075 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
3077 hence "Cauchy t" unfolding cauchy_def by auto
3078 then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto
3080 { fix e::real assume "e>0"
3081 then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded Lim_sequentially] by auto
3082 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
3083 hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
3085 hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
3087 then show ?thesis by auto
3090 text {* Strengthen it to the intersection actually being a singleton. *}
3092 lemma decreasing_closed_nest_sing:
3093 fixes s :: "nat \<Rightarrow> 'a::heine_borel set"
3094 assumes "\<forall>n. closed(s n)"
3095 "\<forall>n. s n \<noteq> {}"
3096 "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
3097 "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
3098 shows "\<exists>a. \<Inter>(range s) = {a}"
3100 obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
3101 { fix b assume b:"b \<in> \<Inter>(range s)"
3102 { fix e::real assume "e>0"
3103 hence "dist a b < e" using assms(4 )using b using a by blast
3105 hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)
3107 with a have "\<Inter>(range s) = {a}" unfolding image_def by auto
3111 text{* Cauchy-type criteria for uniform convergence. *}
3113 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows
3114 "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
3115 (\<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")
3118 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
3119 { fix e::real assume "e>0"
3120 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
3121 { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
3122 hence "dist (s m x) (s n x) < e"
3123 using N[THEN spec[where x=m], THEN spec[where x=x]]
3124 using N[THEN spec[where x=n], THEN spec[where x=x]]
3125 using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto }
3126 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 }
3130 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
3131 then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
3132 using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
3133 { fix e::real assume "e>0"
3134 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"
3135 using `?rhs`[THEN spec[where x="e/2"]] by auto
3136 { fix x assume "P x"
3137 then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
3138 using l[THEN spec[where x=x], unfolded Lim_sequentially] using `e>0` by(auto elim!: allE[where x="e/2"])
3139 fix n::nat assume "n\<ge>N"
3140 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]]
3141 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) }
3142 hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
3146 lemma uniformly_cauchy_imp_uniformly_convergent:
3147 fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"
3148 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"
3149 "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
3150 shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
3152 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"
3153 using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
3155 { fix x assume "P x"
3156 hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
3157 using l and assms(2) unfolding Lim_sequentially by blast }
3158 ultimately show ?thesis by auto
3162 subsection {* Continuity *}
3164 text {* Define continuity over a net to take in restrictions of the set. *}
3167 continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
3168 where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
3170 lemma continuous_trivial_limit:
3171 "trivial_limit net ==> continuous net f"
3172 unfolding continuous_def tendsto_def trivial_limit_eq by auto
3174 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"
3175 unfolding continuous_def
3176 unfolding tendsto_def
3177 using netlimit_within[of x s]
3178 by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)
3180 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"
3181 using continuous_within [of x UNIV f] by (simp add: within_UNIV)
3183 lemma continuous_at_within:
3184 assumes "continuous (at x) f" shows "continuous (at x within s) f"
3185 using assms unfolding continuous_at continuous_within
3186 by (rule Lim_at_within)
3188 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}
3190 lemma continuous_within_eps_delta:
3191 "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)"
3192 unfolding continuous_within and Lim_within
3193 apply auto unfolding dist_nz[THEN sym] apply(auto elim!:allE) apply(rule_tac x=d in exI) by auto
3195 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3196 \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
3197 using continuous_within_eps_delta[of x UNIV f]
3198 unfolding within_UNIV by blast
3200 text{* Versions in terms of open balls. *}
3202 lemma continuous_within_ball:
3203 "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
3204 f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3207 { fix e::real assume "e>0"
3208 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"
3209 using `?lhs`[unfolded continuous_within Lim_within] by auto
3210 { fix y assume "y\<in>f ` (ball x d \<inter> s)"
3211 hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
3212 apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using `e>0` by auto
3214 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) }
3217 assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
3218 apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
3221 lemma continuous_at_ball:
3222 "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
3224 assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3225 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)
3226 unfolding dist_nz[THEN sym] by auto
3228 assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
3229 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)
3232 text{* Define setwise continuity in terms of limits within the set. *}
3236 "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
3238 "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
3240 lemma continuous_on_topological:
3241 "continuous_on s f \<longleftrightarrow>
3242 (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
3243 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
3244 unfolding continuous_on_def tendsto_def
3245 unfolding Limits.eventually_within eventually_at_topological
3246 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
3248 lemma continuous_on_iff:
3249 "continuous_on s f \<longleftrightarrow>
3250 (\<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)"
3251 unfolding continuous_on_def Lim_within
3252 apply (intro ball_cong [OF refl] all_cong ex_cong)
3253 apply (rename_tac y, case_tac "y = x", simp)
3254 apply (simp add: dist_nz)
3258 uniformly_continuous_on ::
3259 "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
3261 "uniformly_continuous_on s f \<longleftrightarrow>
3262 (\<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)"
3264 text{* Some simple consequential lemmas. *}
3266 lemma uniformly_continuous_imp_continuous:
3267 " uniformly_continuous_on s f ==> continuous_on s f"
3268 unfolding uniformly_continuous_on_def continuous_on_iff by blast
3270 lemma continuous_at_imp_continuous_within:
3271 "continuous (at x) f ==> continuous (at x within s) f"
3272 unfolding continuous_within continuous_at using Lim_at_within by auto
3274 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
3275 unfolding tendsto_def by (simp add: trivial_limit_eq)
3277 lemma continuous_at_imp_continuous_on:
3278 assumes "\<forall>x\<in>s. continuous (at x) f"
3279 shows "continuous_on s f"
3280 unfolding continuous_on_def
3282 fix x assume "x \<in> s"
3283 with assms have *: "(f ---> f (netlimit (at x))) (at x)"
3284 unfolding continuous_def by simp
3285 have "(f ---> f x) (at x)"
3286 proof (cases "trivial_limit (at x)")
3287 case True thus ?thesis
3288 by (rule Lim_trivial_limit)
3291 hence 1: "netlimit (at x) = x"
3292 using netlimit_within [of x UNIV]
3293 by (simp add: within_UNIV)
3294 with * show ?thesis by simp
3296 thus "(f ---> f x) (at x within s)"
3297 by (rule Lim_at_within)
3300 lemma continuous_on_eq_continuous_within:
3301 "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"
3302 unfolding continuous_on_def continuous_def
3303 apply (rule ball_cong [OF refl])
3304 apply (case_tac "trivial_limit (at x within s)")
3305 apply (simp add: Lim_trivial_limit)
3306 apply (simp add: netlimit_within)
3309 lemmas continuous_on = continuous_on_def -- "legacy theorem name"
3311 lemma continuous_on_eq_continuous_at:
3312 shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"
3313 by (auto simp add: continuous_on continuous_at Lim_within_open)
3315 lemma continuous_within_subset:
3316 "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
3317 ==> continuous (at x within t) f"
3318 unfolding continuous_within by(metis Lim_within_subset)
3320 lemma continuous_on_subset:
3321 shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"
3322 unfolding continuous_on by (metis subset_eq Lim_within_subset)
3324 lemma continuous_on_interior:
3325 shows "continuous_on s f \<Longrightarrow> x \<in> interior s ==> continuous (at x) f"
3326 unfolding interior_def
3328 by (meson continuous_on_eq_continuous_at continuous_on_subset)
3330 lemma continuous_on_eq:
3331 "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"
3332 unfolding continuous_on_def tendsto_def Limits.eventually_within
3335 text {* Characterization of various kinds of continuity in terms of sequences. *}
3337 (* \<longrightarrow> could be generalized, but \<longleftarrow> requires metric space *)
3338 lemma continuous_within_sequentially:
3339 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3340 shows "continuous (at a within s) f \<longleftrightarrow>
3341 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
3342 --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
3345 { 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"
3346 fix e::real assume "e>0"
3347 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
3348 from x(2) `d>0` obtain N where N:"\<forall>n\<ge>N. dist (x n) a < d" by auto
3349 hence "\<exists>N. \<forall>n\<ge>N. dist ((f \<circ> x) n) (f a) < e"
3350 apply(rule_tac x=N in exI) using N d apply auto using x(1)
3351 apply(erule_tac x=n in allE) apply(erule_tac x=n in allE)
3352 apply(erule_tac x="x n" in ballE) apply auto unfolding dist_nz[THEN sym] apply auto using `e>0` by auto
3354 thus ?rhs unfolding continuous_within unfolding Lim_sequentially by simp
3357 { fix e::real assume "e>0"
3358 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)"
3359 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
3360 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)"
3361 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
3362 { fix d::real assume "d>0"
3363 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
3364 then obtain N::nat where N:"inverse (real (N + 1)) < d" by auto
3365 { fix n::nat assume n:"n\<ge>N"
3366 hence "dist (x (inverse (real (n + 1)))) a < inverse (real (n + 1))" using x[THEN spec[where x="inverse (real (n + 1))"]] by auto
3367 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)
3368 ultimately have "dist (x (inverse (real (n + 1)))) a < d" by auto
3370 hence "\<exists>N::nat. \<forall>n\<ge>N. dist (x (inverse (real (n + 1)))) a < d" by auto
3372 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
3373 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
3374 hence "False" apply(erule_tac x=e in allE) using `e>0` using x by auto
3376 thus ?lhs unfolding continuous_within unfolding Lim_within unfolding Lim_sequentially by blast
3379 lemma continuous_at_sequentially:
3380 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3381 shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
3382 --> ((f o x) ---> f a) sequentially)"
3383 using continuous_within_sequentially[of a UNIV f] unfolding within_UNIV by auto
3385 lemma continuous_on_sequentially:
3386 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
3387 shows "continuous_on s f \<longleftrightarrow>
3388 (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
3389 --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
3391 assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
3393 assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
3396 lemma uniformly_continuous_on_sequentially':
3397 "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
3398 ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially
3399 \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
3402 { 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"
3403 { fix e::real assume "e>0"
3404 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"
3405 using `?lhs`[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
3406 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
3407 { fix n assume "n\<ge>N"
3408 hence "dist (f (x n)) (f (y n)) < e"
3409 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
3410 unfolding dist_commute by simp }
3411 hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" by auto }
3412 hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding Lim_sequentially and dist_real_def by auto }
3416 { assume "\<not> ?lhs"
3417 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
3418 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"
3419 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
3420 by (auto simp add: dist_commute)
3421 def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
3422 def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
3423 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"
3424 unfolding x_def and y_def using fa by auto
3425 { fix e::real assume "e>0"
3426 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
3427 { fix n::nat assume "n\<ge>N"
3428 hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and `N\<noteq>0` by auto
3429 also have "\<dots> < e" using N by auto
3430 finally have "inverse (real n + 1) < e" by auto
3431 hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto }
3432 hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto }
3433 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
3434 hence False using fxy and `e>0` by auto }
3435 thus ?lhs unfolding uniformly_continuous_on_def by blast
3438 lemma uniformly_continuous_on_sequentially:
3439 fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
3440 shows "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
3441 ((\<lambda>n. x n - y n) ---> 0) sequentially
3442 \<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")
3443 (* BH: maybe the previous lemma should replace this one? *)
3444 unfolding uniformly_continuous_on_sequentially'
3445 unfolding dist_norm tendsto_norm_zero_iff ..
3447 text{* The usual transformation theorems. *}
3449 lemma continuous_transform_within:
3450 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3451 assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
3452 "continuous (at x within s) f"
3453 shows "continuous (at x within s) g"
3454 unfolding continuous_within
3455 proof (rule Lim_transform_within)
3456 show "0 < d" by fact
3457 show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
3458 using assms(3) by auto
3460 using assms(1,2,3) by auto
3461 thus "(f ---> g x) (at x within s)"
3462 using assms(4) unfolding continuous_within by simp
3465 lemma continuous_transform_at:
3466 fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
3467 assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
3468 "continuous (at x) f"
3469 shows "continuous (at x) g"
3470 using continuous_transform_within [of d x UNIV f g] assms
3471 by (simp add: within_UNIV)
3473 text{* Combination results for pointwise continuity. *}
3475 lemma continuous_const: "continuous net (\<lambda>x. c)"
3476 by (auto simp add: continuous_def tendsto_const)
3478 lemma continuous_cmul:
3479 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3480 shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"
3481 by (auto simp add: continuous_def intro: tendsto_intros)
3483 lemma continuous_neg:
3484 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3485 shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"
3486 by (auto simp add: continuous_def tendsto_minus)
3488 lemma continuous_add:
3489 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3490 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"
3491 by (auto simp add: continuous_def tendsto_add)
3493 lemma continuous_sub:
3494 fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
3495 shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"
3496 by (auto simp add: continuous_def tendsto_diff)
3499 text{* Same thing for setwise continuity. *}
3501 lemma continuous_on_const:
3502 "continuous_on s (\<lambda>x. c)"
3503 unfolding continuous_on_def by (auto intro: tendsto_intros)
3505 lemma continuous_on_cmul:
3506 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3507 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c *\<^sub>R (f x))"
3508 unfolding continuous_on_def by (auto intro: tendsto_intros)
3510 lemma continuous_on_neg:
3511 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3512 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
3513 unfolding continuous_on_def by (auto intro: tendsto_intros)
3515 lemma continuous_on_add:
3516 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3517 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3518 \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
3519 unfolding continuous_on_def by (auto intro: tendsto_intros)
3521 lemma continuous_on_sub:
3522 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
3523 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3524 \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
3525 unfolding continuous_on_def by (auto intro: tendsto_intros)
3527 text{* Same thing for uniform continuity, using sequential formulations. *}
3529 lemma uniformly_continuous_on_const:
3530 "uniformly_continuous_on s (\<lambda>x. c)"
3531 unfolding uniformly_continuous_on_def by simp
3533 lemma uniformly_continuous_on_cmul:
3534 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3535 assumes "uniformly_continuous_on s f"
3536 shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
3538 { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
3539 hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"
3540 using scaleR.tendsto [OF tendsto_const, of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
3541 unfolding scaleR_zero_right scaleR_right_diff_distrib by auto
3543 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
3544 unfolding dist_norm tendsto_norm_zero_iff by auto
3548 fixes x y :: "'a::real_normed_vector"
3549 shows "dist (- x) (- y) = dist x y"
3550 unfolding dist_norm minus_diff_minus norm_minus_cancel ..
3552 lemma uniformly_continuous_on_neg:
3553 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3554 shows "uniformly_continuous_on s f
3555 ==> uniformly_continuous_on s (\<lambda>x. -(f x))"
3556 unfolding uniformly_continuous_on_def dist_minus .
3558 lemma uniformly_continuous_on_add:
3559 fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3560 assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"
3561 shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
3563 { fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
3564 "((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"
3565 hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"
3566 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
3567 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 }
3568 thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
3569 unfolding dist_norm tendsto_norm_zero_iff by auto
3572 lemma uniformly_continuous_on_sub:
3573 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3574 shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s g
3575 ==> uniformly_continuous_on s (\<lambda>x. f x - g x)"
3576 unfolding ab_diff_minus
3577 using uniformly_continuous_on_add[of s f "\<lambda>x. - g x"]
3578 using uniformly_continuous_on_neg[of s g] by auto
3580 text{* Identity function is continuous in every sense. *}
3582 lemma continuous_within_id:
3583 "continuous (at a within s) (\<lambda>x. x)"
3584 unfolding continuous_within by (rule Lim_at_within [OF LIM_ident])
3586 lemma continuous_at_id:
3587 "continuous (at a) (\<lambda>x. x)"
3588 unfolding continuous_at by (rule LIM_ident)
3590 lemma continuous_on_id:
3591 "continuous_on s (\<lambda>x. x)"
3592 unfolding continuous_on_def by (auto intro: tendsto_ident_at_within)
3594 lemma uniformly_continuous_on_id:
3595 "uniformly_continuous_on s (\<lambda>x. x)"
3596 unfolding uniformly_continuous_on_def by auto
3598 text{* Continuity of all kinds is preserved under composition. *}
3600 lemma continuous_within_topological:
3601 "continuous (at x within s) f \<longleftrightarrow>
3602 (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>
3603 (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
3604 unfolding continuous_within
3605 unfolding tendsto_def Limits.eventually_within eventually_at_topological
3606 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto
3608 lemma continuous_within_compose:
3609 assumes "continuous (at x within s) f"
3610 assumes "continuous (at (f x) within f ` s) g"
3611 shows "continuous (at x within s) (g o f)"
3612 using assms unfolding continuous_within_topological by simp metis
3614 lemma continuous_at_compose:
3615 assumes "continuous (at x) f" "continuous (at (f x)) g"
3616 shows "continuous (at x) (g o f)"
3618 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
3619 thus ?thesis using assms(1) using continuous_within_compose[of x UNIV f g, unfolded within_UNIV] by auto
3622 lemma continuous_on_compose:
3623 "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
3624 unfolding continuous_on_topological by simp metis
3626 lemma uniformly_continuous_on_compose:
3627 assumes "uniformly_continuous_on s f" "uniformly_continuous_on (f ` s) g"
3628 shows "uniformly_continuous_on s (g o f)"
3630 { fix e::real assume "e>0"
3631 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
3632 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
3633 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 }
3634 thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
3637 text{* Continuity in terms of open preimages. *}
3639 lemma continuous_at_open:
3640 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)))"
3641 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]
3642 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto
3644 lemma continuous_on_open:
3645 shows "continuous_on s f \<longleftrightarrow>
3646 (\<forall>t. openin (subtopology euclidean (f ` s)) t
3647 --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
3650 assume 1: "continuous_on s f"
3651 assume 2: "openin (subtopology euclidean (f ` s)) t"
3652 from 2 obtain B where B: "open B" and t: "t = f ` s \<inter> B"
3653 unfolding openin_open by auto
3654 def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"
3655 have "open U" unfolding U_def by (simp add: open_Union)
3656 moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"
3657 proof (intro ballI iffI)
3658 fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"
3659 unfolding U_def t by auto
3661 fix x assume "x \<in> s" and "f x \<in> t"
3662 hence "x \<in> s" and "f x \<in> B"
3664 with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"
3665 unfolding t continuous_on_topological by metis
3666 then show "x \<in> U"
3667 unfolding U_def by auto
3669 ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto
3670 then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3671 unfolding openin_open by fast
3673 assume "?rhs" show "continuous_on s f"
3674 unfolding continuous_on_topological
3676 fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"
3677 have "openin (subtopology euclidean (f ` s)) (f ` s \<inter> B)"
3678 unfolding openin_open using `open B` by auto
3679 then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f ` s \<inter> B}"
3680 using `?rhs` by fast
3681 then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
3682 unfolding openin_open using `x \<in> s` and `f x \<in> B` by auto
3686 text {* Similarly in terms of closed sets. *}
3688 lemma continuous_on_closed:
3689 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")
3693 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3694 have **:"f ` s - (f ` s - (f ` s - t)) = f ` s - t" by auto
3695 assume as:"closedin (subtopology euclidean (f ` s)) t"
3696 hence "closedin (subtopology euclidean (f ` s)) (f ` s - (f ` s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto
3697 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"]]
3698 unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto }
3703 have *:"s - {x \<in> s. f x \<in> f ` s - t} = {x \<in> s. f x \<in> t}" by auto
3704 assume as:"openin (subtopology euclidean (f ` s)) t"
3705 hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using `?rhs`[THEN spec[where x="(f ` s) - t"]]
3706 unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }
3707 thus ?lhs unfolding continuous_on_open by auto
3710 text {* Half-global and completely global cases. *}
3712 lemma continuous_open_in_preimage:
3713 assumes "continuous_on s f" "open t"
3714 shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3716 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
3717 have "openin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3718 using openin_open_Int[of t "f ` s", OF assms(2)] unfolding openin_open by auto
3719 thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3722 lemma continuous_closed_in_preimage:
3723 assumes "continuous_on s f" "closed t"
3724 shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
3726 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
3727 have "closedin (subtopology euclidean (f ` s)) (t \<inter> f ` s)"
3728 using closedin_closed_Int[of t "f ` s", OF assms(2)] unfolding Int_commute by auto
3730 using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f ` s"]] using * by auto
3733 lemma continuous_open_preimage:
3734 assumes "continuous_on s f" "open s" "open t"
3735 shows "open {x \<in> s. f x \<in> t}"
3737 obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3738 using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
3739 thus ?thesis using open_Int[of s T, OF assms(2)] by auto
3742 lemma continuous_closed_preimage:
3743 assumes "continuous_on s f" "closed s" "closed t"
3744 shows "closed {x \<in> s. f x \<in> t}"
3746 obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
3747 using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
3748 thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
3751 lemma continuous_open_preimage_univ:
3752 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
3753 using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
3755 lemma continuous_closed_preimage_univ:
3756 shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
3757 using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
3759 lemma continuous_open_vimage:
3760 shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)"
3761 unfolding vimage_def by (rule continuous_open_preimage_univ)
3763 lemma continuous_closed_vimage:
3764 shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)"
3765 unfolding vimage_def by (rule continuous_closed_preimage_univ)
3767 lemma interior_image_subset:
3768 assumes "\<forall>x. continuous (at x) f" "inj f"
3769 shows "interior (f ` s) \<subseteq> f ` (interior s)"
3770 apply rule unfolding interior_def mem_Collect_eq image_iff apply safe
3771 proof- fix x T assume as:"open T" "x \<in> T" "T \<subseteq> f ` s"
3772 hence "x \<in> f ` s" by auto then guess y unfolding image_iff .. note y=this
3773 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
3774 apply safe apply(rule_tac x="{x. f x \<in> T}" in exI) apply(safe,rule continuous_open_preimage_univ)
3775 proof- fix x assume "f x \<in> T" hence "f x \<in> f ` s" using as by auto
3776 thus "x \<in> s" unfolding inj_image_mem_iff[OF assms(2)] . qed auto qed
3778 text {* Equality of continuous functions on closure and related results. *}
3780 lemma continuous_closed_in_preimage_constant:
3781 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3782 shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
3783 using continuous_closed_in_preimage[of s f "{a}"] by auto
3785 lemma continuous_closed_preimage_constant:
3786 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3787 shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
3788 using continuous_closed_preimage[of s f "{a}"] by auto
3790 lemma continuous_constant_on_closure:
3791 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3792 assumes "continuous_on (closure s) f"
3793 "\<forall>x \<in> s. f x = a"
3794 shows "\<forall>x \<in> (closure s). f x = a"
3795 using continuous_closed_preimage_constant[of "closure s" f a]
3796 assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto
3798 lemma image_closure_subset:
3799 assumes "continuous_on (closure s) f" "closed t" "(f ` s) \<subseteq> t"
3800 shows "f ` (closure s) \<subseteq> t"
3802 have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
3803 moreover have "closed {x \<in> closure s. f x \<in> t}"
3804 using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
3805 ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
3806 using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
3807 thus ?thesis by auto
3810 lemma continuous_on_closure_norm_le:
3811 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3812 assumes "continuous_on (closure s) f" "\<forall>y \<in> s. norm(f y) \<le> b" "x \<in> (closure s)"
3813 shows "norm(f x) \<le> b"
3815 have *:"f ` s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto
3817 using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
3818 unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
3821 text {* Making a continuous function avoid some value in a neighbourhood. *}
3823 lemma continuous_within_avoid:
3824 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3825 assumes "continuous (at x within s) f" "x \<in> s" "f x \<noteq> a"
3826 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
3828 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"
3829 using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto
3830 { fix y assume " y\<in>s" "dist x y < d"
3831 hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]
3832 apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }
3833 thus ?thesis using `d>0` by auto
3836 lemma continuous_at_avoid:
3837 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)
3838 assumes "continuous (at x) f" "f x \<noteq> a"
3839 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
3840 using assms using continuous_within_avoid[of x UNIV f a, unfolded within_UNIV] by auto
3842 lemma continuous_on_avoid:
3843 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
3844 assumes "continuous_on s f" "x \<in> s" "f x \<noteq> a"
3845 shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
3846 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
3848 lemma continuous_on_open_avoid:
3849 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)
3850 assumes "continuous_on s f" "open s" "x \<in> s" "f x \<noteq> a"
3851 shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
3852 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
3854 text {* Proving a function is constant by proving open-ness of level set. *}
3856 lemma continuous_levelset_open_in_cases:
3857 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3858 shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
3859 openin (subtopology euclidean s) {x \<in> s. f x = a}
3860 ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
3861 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto
3863 lemma continuous_levelset_open_in:
3864 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3865 shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
3866 openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
3867 (\<exists>x \<in> s. f x = a) ==> (\<forall>x \<in> s. f x = a)"
3868 using continuous_levelset_open_in_cases[of s f ]
3871 lemma continuous_levelset_open:
3872 fixes f :: "_ \<Rightarrow> 'b::t1_space"
3873 assumes "connected s" "continuous_on s f" "open {x \<in> s. f x = a}" "\<exists>x \<in> s. f x = a"
3874 shows "\<forall>x \<in> s. f x = a"
3875 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast
3877 text {* Some arithmetical combinations (more to prove). *}
3879 lemma open_scaling[intro]:
3880 fixes s :: "'a::real_normed_vector set"
3881 assumes "c \<noteq> 0" "open s"
3882 shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
3884 { fix x assume "x \<in> s"
3885 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
3886 have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF `e>0`] by auto
3888 { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
3889 hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm
3890 using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
3891 assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)
3892 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 }
3893 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 }
3894 thus ?thesis unfolding open_dist by auto
3897 lemma minus_image_eq_vimage:
3898 fixes A :: "'a::ab_group_add set"
3899 shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
3900 by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
3902 lemma open_negations:
3903 fixes s :: "'a::real_normed_vector set"
3904 shows "open s ==> open ((\<lambda> x. -x) ` s)"
3905 unfolding scaleR_minus1_left [symmetric]
3906 by (rule open_scaling, auto)
3908 lemma open_translation:
3909 fixes s :: "'a::real_normed_vector set"
3910 assumes "open s" shows "open((\<lambda>x. a + x) ` s)"
3912 { 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 }
3913 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
3914 ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
3917 lemma open_affinity:
3918 fixes s :: "'a::real_normed_vector set"
3919 assumes "open s" "c \<noteq> 0"
3920 shows "open ((\<lambda>x. a + c *\<^sub>R x) ` s)"
3922 have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
3923 have "op + a ` op *\<^sub>R c ` s = (op + a \<circ> op *\<^sub>R c) ` s" by auto
3924 thus ?thesis using assms open_translation[of "op *\<^sub>R c ` s" a] unfolding * by auto
3927 lemma interior_translation:
3928 fixes s :: "'a::real_normed_vector set"
3929 shows "interior ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (interior s)"
3930 proof (rule set_eqI, rule)
3931 fix x assume "x \<in> interior (op + a ` s)"
3932 then obtain e where "e>0" and e:"ball x e \<subseteq> op + a ` s" unfolding mem_interior by auto
3933 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
3934 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
3936 fix x assume "x \<in> op + a ` interior s"
3937 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
3938 { fix z have *:"a + y - z = y + a - z" by auto
3939 assume "z\<in>ball x e"
3940 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
3941 hence "z \<in> op + a ` s" unfolding image_iff by(auto intro!: bexI[where x="z - a"]) }
3942 hence "ball x e \<subseteq> op + a ` s" unfolding subset_eq by auto
3943 thus "x \<in> interior (op + a ` s)" unfolding mem_interior using `e>0` by auto
3946 text {* We can now extend limit compositions to consider the scalar multiplier. *}
3948 lemma continuous_vmul:
3949 fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
3950 shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"
3951 unfolding continuous_def by (intro tendsto_intros)
3953 lemma continuous_mul:
3954 fixes c :: "'a::metric_space \<Rightarrow> real"
3955 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3956 shows "continuous net c \<Longrightarrow> continuous net f
3957 ==> continuous net (\<lambda>x. c(x) *\<^sub>R f x) "
3958 unfolding continuous_def by (intro tendsto_intros)
3960 lemmas continuous_intros = continuous_add continuous_vmul continuous_cmul
3961 continuous_const continuous_sub continuous_at_id continuous_within_id continuous_mul
3963 lemma continuous_on_vmul:
3964 fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
3965 shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R v)"
3966 unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
3968 lemma continuous_on_mul:
3969 fixes c :: "'a::metric_space \<Rightarrow> real"
3970 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
3971 shows "continuous_on s c \<Longrightarrow> continuous_on s f
3972 ==> continuous_on s (\<lambda>x. c(x) *\<^sub>R f x)"
3973 unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
3975 lemma continuous_on_mul_real:
3976 fixes f :: "'a::metric_space \<Rightarrow> real"
3977 fixes g :: "'a::metric_space \<Rightarrow> real"
3978 shows "continuous_on s f \<Longrightarrow> continuous_on s g
3979 ==> continuous_on s (\<lambda>x. f x * g x)"
3980 using continuous_on_mul[of s f g] unfolding real_scaleR_def .
3982 lemmas continuous_on_intros = continuous_on_add continuous_on_const
3983 continuous_on_id continuous_on_compose continuous_on_cmul continuous_on_neg
3984 continuous_on_sub continuous_on_mul continuous_on_vmul continuous_on_mul_real
3985 uniformly_continuous_on_add uniformly_continuous_on_const
3986 uniformly_continuous_on_id uniformly_continuous_on_compose
3987 uniformly_continuous_on_cmul uniformly_continuous_on_neg
3988 uniformly_continuous_on_sub
3990 text {* And so we have continuity of inverse. *}
3992 lemma continuous_inv:
3993 fixes f :: "'a::metric_space \<Rightarrow> real"
3994 shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
3995 ==> continuous net (inverse o f)"
3996 unfolding continuous_def using Lim_inv by auto
3998 lemma continuous_at_within_inv:
3999 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
4000 assumes "continuous (at a within s) f" "f a \<noteq> 0"
4001 shows "continuous (at a within s) (inverse o f)"
4002 using assms unfolding continuous_within o_def
4003 by (intro tendsto_intros)
4005 lemma continuous_at_inv:
4006 fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
4007 shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
4008 ==> continuous (at a) (inverse o f) "
4009 using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
4011 text {* Topological properties of linear functions. *}
4014 assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
4016 interpret f: bounded_linear f by fact
4017 have "(f ---> f 0) (at 0)"
4018 using tendsto_ident_at by (rule f.tendsto)
4019 thus ?thesis unfolding f.zero .
4022 lemma linear_continuous_at:
4023 assumes "bounded_linear f" shows "continuous (at a) f"
4024 unfolding continuous_at using assms
4025 apply (rule bounded_linear.tendsto)
4026 apply (rule tendsto_ident_at)
4029 lemma linear_continuous_within:
4030 shows "bounded_linear f ==> continuous (at x within s) f"
4031 using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
4033 lemma linear_continuous_on:
4034 shows "bounded_linear f ==> continuous_on s f"
4035 using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
4037 text {* Also bilinear functions, in composition form. *}
4039 lemma bilinear_continuous_at_compose:
4040 shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
4041 ==> continuous (at x) (\<lambda>x. h (f x) (g x))"
4042 unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto
4044 lemma bilinear_continuous_within_compose:
4045 shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
4046 ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
4047 unfolding continuous_within using Lim_bilinear[of f "f x"] by auto
4049 lemma bilinear_continuous_on_compose:
4050 shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
4051 ==> continuous_on s (\<lambda>x. h (f x) (g x))"
4052 unfolding continuous_on_def
4053 by (fast elim: bounded_bilinear.tendsto)
4055 text {* Preservation of compactness and connectedness under continuous function. *}
4057 lemma compact_continuous_image:
4058 assumes "continuous_on s f" "compact s"
4059 shows "compact(f ` s)"
4061 { fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
4062 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
4063 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
4064 { fix e::real assume "e>0"
4065 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
4066 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
4067 { 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 }
4068 hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto }
4069 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 }
4070 thus ?thesis unfolding compact_def by auto
4073 lemma connected_continuous_image:
4074 assumes "continuous_on s f" "connected s"
4075 shows "connected(f ` s)"
4077 { fix T assume as: "T \<noteq> {}" "T \<noteq> f ` s" "openin (subtopology euclidean (f ` s)) T" "closedin (subtopology euclidean (f ` s)) T"
4078 have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
4079 using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
4080 using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
4081 using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
4082 hence False using as(1,2)
4083 using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
4084 thus ?thesis unfolding connected_clopen by auto
4087 text {* Continuity implies uniform continuity on a compact domain. *}
4089 lemma compact_uniformly_continuous:
4090 assumes "continuous_on s f" "compact s"
4091 shows "uniformly_continuous_on s f"
4093 { fix x assume x:"x\<in>s"
4094 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
4095 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 }
4096 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
4097 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)"
4098 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
4100 { fix e::real assume "e>0"
4102 { 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 }
4103 hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto
4105 { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto }
4106 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
4108 { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"
4109 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
4110 hence "x\<in>ball z (d z (e / 2))" using `ea>0` unfolding subset_eq by auto
4111 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`
4112 by (auto simp add: dist_commute)
4113 moreover have "y\<in>ball z (d z (e / 2))" using as and `ea>0` and z[unfolded subset_eq]
4114 by (auto simp add: dist_commute)
4115 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`
4116 by (auto simp add: dist_commute)
4117 ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]
4118 by (auto simp add: dist_commute) }
4119 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 }
4120 thus ?thesis unfolding uniformly_continuous_on_def by auto
4123 text{* Continuity of inverse function on compact domain. *}
4125 lemma continuous_on_inverse:
4126 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
4127 (* TODO: can this be generalized more? *)
4128 assumes "continuous_on s f" "compact s" "\<forall>x \<in> s. g (f x) = x"
4129 shows "continuous_on (f ` s) g"
4131 have *:"g ` f ` s = s" using assms(3) by (auto simp add: image_iff)
4132 { fix t assume t:"closedin (subtopology euclidean (g ` f ` s)) t"
4133 then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto
4134 have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]
4135 unfolding T(2) and Int_left_absorb by auto
4136 moreover have "compact (s \<inter> T)"
4137 using assms(2) unfolding compact_eq_bounded_closed
4138 using bounded_subset[of s "s \<inter> T"] and T(1) by auto
4139 ultimately have "closed (f ` t)" using T(1) unfolding T(2)
4140 using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto
4141 moreover have "{x \<in> f ` s. g x \<in> t} = f ` s \<inter> f ` t" using assms(3) unfolding T(2) by auto
4142 ultimately have "closedin (subtopology euclidean (f ` s)) {x \<in> f ` s. g x \<in> t}"
4143 unfolding closedin_closed by auto }
4144 thus ?thesis unfolding continuous_on_closed by auto
4147 text {* A uniformly convergent limit of continuous functions is continuous. *}
4149 lemma continuous_uniform_limit:
4150 fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"
4151 assumes "\<not> trivial_limit F"
4152 assumes "eventually (\<lambda>n. continuous_on s (f n)) F"
4153 assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"
4154 shows "continuous_on s g"
4156 { fix x and e::real assume "x\<in>s" "e>0"
4157 have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"
4158 using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto
4159 from eventually_happens [OF eventually_conj [OF this assms(2)]]
4160 obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3" "continuous_on s (f n)"
4161 using assms(1) by blast
4162 have "e / 3 > 0" using `e>0` by auto
4163 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"
4164 using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
4165 { fix y assume "y \<in> s" and "dist y x < d"
4166 hence "dist (f n y) (f n x) < e / 3"
4167 by (rule d [rule_format])
4168 hence "dist (f n y) (g x) < 2 * e / 3"
4169 using dist_triangle [of "f n y" "g x" "f n x"]
4170 using n(1)[THEN bspec[where x=x], OF `x\<in>s`]
4172 hence "dist (g y) (g x) < e"
4173 using n(1)[THEN bspec[where x=y], OF `y\<in>s`]
4174 using dist_triangle3 [of "g y" "g x" "f n y"]
4176 hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
4177 using `d>0` by auto }
4178 thus ?thesis unfolding continuous_on_iff by auto
4182 subsection {* Topological stuff lifted from and dropped to R *}
4185 fixes s :: "real set" shows
4186 "open s \<longleftrightarrow>
4187 (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
4188 unfolding open_dist dist_norm by simp
4190 lemma islimpt_approachable_real:
4191 fixes s :: "real set"
4192 shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
4193 unfolding islimpt_approachable dist_norm by simp
4196 fixes s :: "real set"
4197 shows "closed s \<longleftrightarrow>
4198 (\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
4200 unfolding closed_limpt islimpt_approachable dist_norm by simp
4202 lemma continuous_at_real_range:
4203 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4204 shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
4205 \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
4206 unfolding continuous_at unfolding Lim_at
4207 unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
4208 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
4209 apply(erule_tac x=e in allE) by auto
4211 lemma continuous_on_real_range:
4212 fixes f :: "'a::real_normed_vector \<Rightarrow> real"
4213 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))"
4214 unfolding continuous_on_iff dist_norm by simp
4216 lemma continuous_at_norm: "continuous (at x) norm"
4217 unfolding continuous_at by (intro tendsto_intros)
4219 lemma continuous_on_norm: "continuous_on s norm"
4220 unfolding continuous_on by (intro ballI tendsto_intros)
4222 lemma continuous_at_infnorm: "continuous (at x) infnorm"
4223 unfolding continuous_at Lim_at o_def unfolding dist_norm
4224 apply auto apply (rule_tac x=e in exI) apply auto
4225 using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
4227 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}
4229 lemma compact_attains_sup:
4230 fixes s :: "real set"
4231 assumes "compact s" "s \<noteq> {}"
4232 shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
4234 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4235 { 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"
4236 have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto
4237 moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
4238 ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using `e>0` by auto }
4239 thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]
4240 apply(rule_tac x="Sup s" in bexI) by auto
4244 fixes S :: "real set"
4245 shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
4246 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def)
4248 lemma compact_attains_inf:
4249 fixes s :: "real set"
4250 assumes "compact s" "s \<noteq> {}" shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
4252 from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
4253 { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s" "Inf s \<notin> s" "0 < e"
4254 "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"
4255 have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto
4257 { fix x assume "x \<in> s"
4258 hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto
4259 have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
4260 hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto
4261 ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using `e>0` by auto }
4262 thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]
4263 apply(rule_tac x="Inf s" in bexI) by auto
4266 lemma continuous_attains_sup:
4267 fixes f :: "'a::metric_space \<Rightarrow> real"
4268 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4269 ==> (\<exists>x \<in> s. \<forall>y \<in> s. f y \<le> f x)"
4270 using compact_attains_sup[of "f ` s"]
4271 using compact_continuous_image[of s f] by auto
4273 lemma continuous_attains_inf:
4274 fixes f :: "'a::metric_space \<Rightarrow> real"
4275 shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
4276 \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
4277 using compact_attains_inf[of "f ` s"]
4278 using compact_continuous_image[of s f] by auto
4280 lemma distance_attains_sup:
4281 assumes "compact s" "s \<noteq> {}"
4282 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
4283 proof (rule continuous_attains_sup [OF assms])
4284 { fix x assume "x\<in>s"
4285 have "(dist a ---> dist a x) (at x within s)"
4286 by (intro tendsto_dist tendsto_const Lim_at_within LIM_ident)
4288 thus "continuous_on s (dist a)"
4289 unfolding continuous_on ..
4292 text {* For \emph{minimal} distance, we only need closure, not compactness. *}
4294 lemma distance_attains_inf:
4295 fixes a :: "'a::heine_borel"
4296 assumes "closed s" "s \<noteq> {}"
4297 shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
4299 from assms(2) obtain b where "b\<in>s" by auto
4300 let ?B = "cball a (dist b a) \<inter> s"
4301 have "b \<in> ?B" using `b\<in>s` by (simp add: dist_commute)
4302 hence "?B \<noteq> {}" by auto
4304 { fix x assume "x\<in>?B"
4305 fix e::real assume "e>0"
4306 { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
4307 from as have "\<bar>dist a x' - dist a x\<bar> < e"
4308 unfolding abs_less_iff minus_diff_eq
4309 using dist_triangle2 [of a x' x]
4310 using dist_triangle [of a x x']
4313 hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
4316 hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
4317 unfolding continuous_on Lim_within dist_norm real_norm_def
4319 moreover have "compact ?B"
4320 using compact_cball[of a "dist b a"]
4321 unfolding compact_eq_bounded_closed
4322 using bounded_Int and closed_Int and assms(1) by auto
4323 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"
4324 using continuous_attains_inf[of ?B "dist a"] by fastsimp
4325 thus ?thesis by fastsimp
4329 subsection {* Pasted sets *}
4331 lemma bounded_Times:
4332 assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
4334 obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
4335 using assms [unfolded bounded_def] by auto
4336 then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
4337 by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
4338 thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
4341 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
4344 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
4345 unfolding compact_def
4347 apply (drule_tac x="fst \<circ> f" in spec)
4348 apply (drule mp, simp add: mem_Times_iff)
4349 apply (clarify, rename_tac l1 r1)
4350 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
4351 apply (drule mp, simp add: mem_Times_iff)
4352 apply (clarify, rename_tac l2 r2)
4353 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
4354 apply (rule_tac x="r1 \<circ> r2" in exI)
4355 apply (rule conjI, simp add: subseq_def)
4356 apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
4357 apply (drule (1) tendsto_Pair) back
4358 apply (simp add: o_def)
4361 text{* Hence some useful properties follow quite easily. *}
4363 lemma compact_scaling:
4364 fixes s :: "'a::real_normed_vector set"
4365 assumes "compact s" shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
4367 let ?f = "\<lambda>x. scaleR c x"
4368 have *:"bounded_linear ?f" by (rule scaleR.bounded_linear_right)
4369 show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
4370 using linear_continuous_at[OF *] assms by auto
4373 lemma compact_negations:
4374 fixes s :: "'a::real_normed_vector set"
4375 assumes "compact s" shows "compact ((\<lambda>x. -x) ` s)"
4376 using compact_scaling [OF assms, of "- 1"] by auto
4379 fixes s t :: "'a::real_normed_vector set"
4380 assumes "compact s" "compact t" shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
4382 have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
4383 apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
4384 have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
4385 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
4386 thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
4389 lemma compact_differences:
4390 fixes s t :: "'a::real_normed_vector set"
4391 assumes "compact s" "compact t" shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
4393 have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
4394 apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4395 thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
4398 lemma compact_translation:
4399 fixes s :: "'a::real_normed_vector set"
4400 assumes "compact s" shows "compact ((\<lambda>x. a + x) ` s)"
4402 have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
4403 thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
4406 lemma compact_affinity:
4407 fixes s :: "'a::real_normed_vector set"
4408 assumes "compact s" shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
4410 have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
4411 thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
4414 text {* Hence we get the following. *}
4416 lemma compact_sup_maxdistance:
4417 fixes s :: "'a::real_normed_vector set"
4418 assumes "compact s" "s \<noteq> {}"
4419 shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
4421 have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
4422 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"
4423 using compact_differences[OF assms(1) assms(1)]
4424 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
4425 from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
4426 thus ?thesis using x(2)[unfolded `x = a - b`] by blast
4429 text {* We can state this in terms of diameter of a set. *}
4431 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"
4432 (* TODO: generalize to class metric_space *)
4434 lemma diameter_bounded:
4436 shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4437 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"
4439 let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"
4440 obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto
4441 { fix x y assume "x \<in> s" "y \<in> s"
4442 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) }
4444 { fix x y assume "x\<in>s" "y\<in>s" hence "s \<noteq> {}" by auto
4445 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`
4446 by simp (blast intro!: Sup_upper *) }
4448 { fix d::real assume "d>0" "d < diameter s"
4449 hence "s\<noteq>{}" unfolding diameter_def by auto
4450 have "\<exists>d' \<in> ?D. d' > d"
4452 assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"
4453 hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE)
4454 thus False using `d < diameter s` `s\<noteq>{}`
4455 apply (auto simp add: diameter_def)
4456 apply (drule Sup_real_iff [THEN [2] rev_iffD2])
4460 hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto }
4461 ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"
4462 "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto
4465 lemma diameter_bounded_bound:
4466 "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"
4467 using diameter_bounded by blast
4469 lemma diameter_compact_attained:
4470 fixes s :: "'a::real_normed_vector set"
4471 assumes "compact s" "s \<noteq> {}"
4472 shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
4474 have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
4475 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
4476 hence "diameter s \<le> norm (x - y)"
4477 unfolding diameter_def by clarsimp (rule Sup_least, fast+)
4479 by (metis b diameter_bounded_bound order_antisym xys)
4482 text {* Related results with closure as the conclusion. *}
4484 lemma closed_scaling:
4485 fixes s :: "'a::real_normed_vector set"
4486 assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x) ` s)"
4488 case True thus ?thesis by auto
4493 have *:"(\<lambda>x. 0) ` s = {0}" using `s\<noteq>{}` by auto
4494 case True thus ?thesis apply auto unfolding * by auto
4497 { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c ` s" "(x ---> l) sequentially"
4498 { fix n::nat have "scaleR (1 / c) (x n) \<in> s"
4499 using as(1)[THEN spec[where x=n]]
4500 using `c\<noteq>0` by auto
4503 { fix e::real assume "e>0"
4504 hence "0 < e *\<bar>c\<bar>" using `c\<noteq>0` mult_pos_pos[of e "abs c"] by auto
4505 then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
4506 using as(2)[unfolded Lim_sequentially, THEN spec[where x="e * abs c"]] by auto
4507 hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
4508 unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
4509 using mult_imp_div_pos_less[of "abs c" _ e] `c\<noteq>0` by auto }
4510 hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding Lim_sequentially by auto
4511 ultimately have "l \<in> scaleR c ` s"
4512 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"]]
4513 unfolding image_iff using `c\<noteq>0` apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto }
4514 thus ?thesis unfolding closed_sequential_limits by fast
4518 lemma closed_negations:
4519 fixes s :: "'a::real_normed_vector set"
4520 assumes "closed s" shows "closed ((\<lambda>x. -x) ` s)"
4521 using closed_scaling[OF assms, of "- 1"] by simp
4523 lemma compact_closed_sums:
4524 fixes s :: "'a::real_normed_vector set"
4525 assumes "compact s" "closed t" shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4527 let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"
4528 { fix x l assume as:"\<forall>n. x n \<in> ?S" "(x ---> l) sequentially"
4529 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"
4530 using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
4531 obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
4532 using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
4533 have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
4534 using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
4535 hence "l - l' \<in> t"
4536 using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
4538 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
4540 thus ?thesis unfolding closed_sequential_limits by fast
4543 lemma closed_compact_sums:
4544 fixes s t :: "'a::real_normed_vector set"
4545 assumes "closed s" "compact t"
4546 shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
4548 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
4549 apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
4550 thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp
4553 lemma compact_closed_differences:
4554 fixes s t :: "'a::real_normed_vector set"
4555 assumes "compact s" "closed t"
4556 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4558 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4559 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4560 thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
4563 lemma closed_compact_differences:
4564 fixes s t :: "'a::real_normed_vector set"
4565 assumes "closed s" "compact t"
4566 shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"
4568 have "{x + y |x y. x \<in> s \<and> y \<in> uminus ` t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
4569 apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
4570 thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
4573 lemma closed_translation:
4574 fixes a :: "'a::real_normed_vector"
4575 assumes "closed s" shows "closed ((\<lambda>x. a + x) ` s)"
4577 have "{a + y |y. y \<in> s} = (op + a ` s)" by auto
4578 thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
4581 lemma translation_Compl:
4582 fixes a :: "'a::ab_group_add"
4583 shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
4584 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto
4586 lemma translation_UNIV:
4587 fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
4588 apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto
4590 lemma translation_diff:
4591 fixes a :: "'a::ab_group_add"
4592 shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
4595 lemma closure_translation:
4596 fixes a :: "'a::real_normed_vector"
4597 shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
4599 have *:"op + a ` (- s) = - op + a ` s"
4600 apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
4601 show ?thesis unfolding closure_interior translation_Compl
4602 using interior_translation[of a "- s"] unfolding * by auto
4605 lemma frontier_translation:
4606 fixes a :: "'a::real_normed_vector"
4607 shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
4608 unfolding frontier_def translation_diff interior_translation closure_translation by auto
4611 subsection {* Separation between points and sets *}
4613 lemma separate_point_closed:
4614 fixes s :: "'a::heine_borel set"
4615 shows "closed s \<Longrightarrow> a \<notin> s ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
4616 proof(cases "s = {}")
4618 thus ?thesis by(auto intro!: exI[where x=1])
4621 assume "closed s" "a \<notin> s"
4622 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
4623 with `x\<in>s` show ?thesis using dist_pos_lt[of a x] and`a \<notin> s` by blast
4626 lemma separate_compact_closed:
4627 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4628 (* TODO: does this generalize to heine_borel? *)
4629 assumes "compact s" and "closed t" and "s \<inter> t = {}"
4630 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4632 have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto
4633 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"
4634 using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto
4635 { fix x y assume "x\<in>s" "y\<in>t"
4636 hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto
4637 hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute
4638 by (auto simp add: dist_commute)
4639 hence "d \<le> dist x y" unfolding dist_norm by auto }
4640 thus ?thesis using `d>0` by auto
4643 lemma separate_closed_compact:
4644 fixes s t :: "'a::{heine_borel, real_normed_vector} set"
4645 assumes "closed s" and "compact t" and "s \<inter> t = {}"
4646 shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
4648 have *:"t \<inter> s = {}" using assms(3) by auto
4649 show ?thesis using separate_compact_closed[OF assms(2,1) *]
4650 apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
4651 by (auto simp add: dist_commute)
4655 subsection {* Intervals *}
4657 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows
4658 "{a <..< b} = {x::'a. \<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i}" and
4659 "{a .. b} = {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i}"
4660 by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
4662 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows
4663 "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i)"
4664 "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i)"
4665 using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])
4667 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows
4668 "({a <..< b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i))" (is ?th1) and
4669 "({a .. b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i < a$$i))" (is ?th2)
4671 { fix i x assume i:"i<DIM('a)" and as:"b$$i \<le> a$$i" and x:"x\<in>{a <..< b}"
4672 hence "a $$ i < x $$ i \<and> x $$ i < b $$ i" unfolding mem_interval by auto
4673 hence "a$$i < b$$i" by auto
4674 hence False using as by auto }
4676 { assume as:"\<forall>i<DIM('a). \<not> (b$$i \<le> a$$i)"
4677 let ?x = "(1/2) *\<^sub>R (a + b)"
4678 { fix i assume i:"i<DIM('a)"
4679 have "a$$i < b$$i" using as[THEN spec[where x=i]] using i by auto
4680 hence "a$$i < ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i < b$$i"
4681 unfolding euclidean_simps by auto }
4682 hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto }
4683 ultimately show ?th1 by blast
4685 { fix i x assume i:"i<DIM('a)" and as:"b$$i < a$$i" and x:"x\<in>{a .. b}"
4686 hence "a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i" unfolding mem_interval by auto
4687 hence "a$$i \<le> b$$i" by auto
4688 hence False using as by auto }
4690 { assume as:"\<forall>i<DIM('a). \<not> (b$$i < a$$i)"
4691 let ?x = "(1/2) *\<^sub>R (a + b)"
4692 { fix i assume i:"i<DIM('a)"
4693 have "a$$i \<le> b$$i" using as[THEN spec[where x=i]] by auto
4694 hence "a$$i \<le> ((1/2) *\<^sub>R (a+b)) $$ i" "((1/2) *\<^sub>R (a+b)) $$ i \<le> b$$i"
4695 unfolding euclidean_simps by auto }
4696 hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto }
4697 ultimately show ?th2 by blast
4700 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows
4701 "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> b$$i)" and
4702 "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"
4703 unfolding interval_eq_empty[of a b] by fastsimp+
4705 lemma interval_sing: fixes a :: "'a::ordered_euclidean_space" shows
4706 "{a .. a} = {a}" "{a<..<a} = {}"
4707 apply(auto simp add: set_eq_iff euclidean_eq[where 'a='a] eucl_less[where 'a='a] eucl_le[where 'a='a])
4708 apply (simp add: order_eq_iff) apply(rule_tac x=0 in exI) by (auto simp add: not_less less_imp_le)
4710 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows
4711 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
4712 "(\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
4713 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
4714 "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
4715 unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
4716 by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
4718 lemma interval_open_subset_closed: fixes a :: "'a::ordered_euclidean_space" shows
4719 "{a<..<b} \<subseteq> {a .. b}"
4720 proof(simp add: subset_eq, rule)
4722 assume x:"x \<in>{a<..<b}"
4723 { fix i assume "i<DIM('a)"
4724 hence "a $$ i \<le> x $$ i"
4725 using x order_less_imp_le[of "a$$i" "x$$i"]
4726 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4729 { fix i assume "i<DIM('a)"
4730 hence "x $$ i \<le> b $$ i"
4731 using x order_less_imp_le[of "x$$i" "b$$i"]
4732 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4735 show "a \<le> x \<and> x \<le> b"
4736 by(simp add: set_eq_iff eucl_less[where 'a='a] eucl_le[where 'a='a] euclidean_eq)
4739 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows
4740 "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th1) and
4741 "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i)" (is ?th2) and
4742 "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th3) and
4743 "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th4)
4745 show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
4746 show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
4747 { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i<DIM('a). c$$i < d$$i"
4748 hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
4749 fix i assume i:"i<DIM('a)"
4750 (** TODO combine the following two parts as done in the HOL_light version. **)
4751 { let ?x = "(\<chi>\<chi> j. (if j=i then ((min (a$$j) (d$$j))+c$$j)/2 else (c$$j+d$$j)/2))::'a"
4752 assume as2: "a$$i > c$$i"
4753 { fix j assume j:"j<DIM('a)"
4754 hence "c $$ j < ?x $$ j \<and> ?x $$ j < d $$ j"
4755 apply(cases "j=i") using as(2)[THEN spec[where x=j]] i
4756 by (auto simp add: as2) }
4757 hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto
4759 have "?x\<notin>{a .. b}"
4760 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4761 using as(2)[THEN spec[where x=i]] and as2 i
4763 ultimately have False using as by auto }
4764 hence "a$$i \<le> c$$i" by(rule ccontr)auto
4766 { let ?x = "(\<chi>\<chi> j. (if j=i then ((max (b$$j) (c$$j))+d$$j)/2 else (c$$j+d$$j)/2))::'a"
4767 assume as2: "b$$i < d$$i"
4768 { fix j assume "j<DIM('a)"
4769 hence "d $$ j > ?x $$ j \<and> ?x $$ j > c $$ j"
4770 apply(cases "j=i") using as(2)[THEN spec[where x=j]]
4771 by (auto simp add: as2) }
4772 hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
4774 have "?x\<notin>{a .. b}"
4775 unfolding mem_interval apply auto apply(rule_tac x=i in exI)
4776 using as(2)[THEN spec[where x=i]] and as2 using i
4778 ultimately have False using as by auto }
4779 hence "b$$i \<ge> d$$i" by(rule ccontr)auto
4781 have "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" by auto
4783 show ?th3 unfolding subset_eq and Ball_def and mem_interval
4784 apply(rule,rule,rule,rule) apply(rule part1) unfolding subset_eq and Ball_def and mem_interval
4785 prefer 4 apply auto by(erule_tac x=i in allE,erule_tac x=i in allE,fastsimp)+
4786 { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i<DIM('a). c$$i < d$$i"
4787 fix i assume i:"i<DIM('a)"
4788 from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
4789 hence "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" using part1 and as(2) using i by auto } note * = this
4790 show ?th4 unfolding subset_eq and Ball_def and mem_interval
4791 apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4
4792 apply auto by(erule_tac x=i in allE, simp)+
4795 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows
4796 "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i < c$$i \<or> b$$i < c$$i \<or> d$$i < a$$i))" (is ?th1) and
4797 "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th2) and
4798 "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i < c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th3) and
4799 "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th4)
4801 let ?z = "(\<chi>\<chi> i. ((max (a$$i) (c$$i)) + (min (b$$i) (d$$i))) / 2)::'a"
4802 note * = set_eq_iff Int_iff empty_iff mem_interval all_conj_distrib[THEN sym] eq_False
4803 show ?th1 unfolding * apply safe apply(erule_tac x="?z" in allE)
4804 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4805 show ?th2 unfolding * apply safe apply(erule_tac x="?z" in allE)
4806 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4807 show ?th3 unfolding * apply safe apply(erule_tac x="?z" in allE)
4808 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4809 show ?th4 unfolding * apply safe apply(erule_tac x="?z" in allE)
4810 unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto
4813 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows
4814 "{a .. b} \<inter> {c .. d} = {(\<chi>\<chi> i. max (a$$i) (c$$i)) .. (\<chi>\<chi> i. min (b$$i) (d$$i))}"
4815 unfolding set_eq_iff and Int_iff and mem_interval
4818 (* Moved interval_open_subset_closed a bit upwards *)
4820 lemma open_interval[intro]:
4821 fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
4823 have "open (\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i<..<b$$i})"
4824 by (intro open_INT finite_lessThan ballI continuous_open_vimage allI
4825 linear_continuous_at bounded_linear_euclidean_component
4826 open_real_greaterThanLessThan)
4827 also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i<..<b$$i}) = {a<..<b}"
4828 by (auto simp add: eucl_less [where 'a='a])
4829 finally show "open {a<..<b}" .
4832 lemma closed_interval[intro]:
4833 fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
4835 have "closed (\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i .. b$$i})"
4836 by (intro closed_INT ballI continuous_closed_vimage allI
4837 linear_continuous_at bounded_linear_euclidean_component
4838 closed_real_atLeastAtMost)
4839 also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) -` {a$$i .. b$$i}) = {a .. b}"
4840 by (auto simp add: eucl_le [where 'a='a])
4841 finally show "closed {a .. b}" .
4844 lemma interior_closed_interval[intro]: fixes a :: "'a::ordered_euclidean_space" shows
4845 "interior {a .. b} = {a<..<b}" (is "?L = ?R")
4846 proof(rule subset_antisym)
4847 show "?R \<subseteq> ?L" using interior_maximal[OF interval_open_subset_closed open_interval] by auto
4849 { fix x assume "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> {a..b}"
4850 then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" by auto
4851 then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto
4852 { fix i assume i:"i<DIM('a)"
4853 have "dist (x - (e / 2) *\<^sub>R basis i) x < e"
4854 "dist (x + (e / 2) *\<^sub>R basis i) x < e"
4855 unfolding dist_norm apply auto
4856 unfolding norm_minus_cancel using norm_basis and `e>0` by auto
4857 hence "a $$ i \<le> (x - (e / 2) *\<^sub>R basis i) $$ i"
4858 "(x + (e / 2) *\<^sub>R basis i) $$ i \<le> b $$ i"
4859 using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]
4860 and e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]
4861 unfolding mem_interval by (auto elim!: allE[where x=i])
4862 hence "a $$ i < x $$ i" and "x $$ i < b $$ i" unfolding euclidean_simps
4863 unfolding basis_component using `e>0` i by auto }
4864 hence "x \<in> {a<..<b}" unfolding mem_interval by auto }
4865 thus "?L \<subseteq> ?R" unfolding interior_def and subset_eq by auto
4868 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"
4870 let ?b = "\<Sum>i<DIM('a). \<bar>a$$i\<bar> + \<bar>b$$i\<bar>"
4871 { fix x::"'a" assume x:"\<forall>i<DIM('a). a $$ i \<le> x $$ i \<and> x $$ i \<le> b $$ i"
4872 { fix i assume "i<DIM('a)"
4873 hence "\<bar>x$$i\<bar> \<le> \<bar>a$$i\<bar> + \<bar>b$$i\<bar>" using x[THEN spec[where x=i]] by auto }
4874 hence "(\<Sum>i<DIM('a). \<bar>x $$ i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto
4875 hence "norm x \<le> ?b" using norm_le_l1[of x] by auto }
4876 thus ?thesis unfolding interval and bounded_iff by auto
4879 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows
4880 "bounded {a .. b} \<and> bounded {a<..<b}"
4881 using bounded_closed_interval[of a b]
4882 using interval_open_subset_closed[of a b]
4883 using bounded_subset[of "{a..b}" "{a<..<b}"]
4886 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows
4887 "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
4888 using bounded_interval[of a b] by auto
4890 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"
4891 using bounded_closed_imp_compact[of "{a..b}"] using bounded_interval[of a b]
4894 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"
4895 assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
4897 { fix i assume "i<DIM('a)"
4898 hence "a $$ i < ((1 / 2) *\<^sub>R (a + b)) $$ i \<and> ((1 / 2) *\<^sub>R (a + b)) $$ i < b $$ i"
4899 using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
4900 unfolding euclidean_simps by auto }
4901 thus ?thesis unfolding mem_interval by auto
4904 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"
4905 assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
4906 shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
4908 { fix i assume i:"i<DIM('a)"
4909 have "a $$ i = e * a$$i + (1 - e) * a$$i" unfolding left_diff_distrib by simp
4910 also have "\<dots> < e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
4911 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
4912 using x unfolding mem_interval using i apply simp
4913 using y unfolding mem_interval using i apply simp
4915 finally have "a $$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i" unfolding euclidean_simps by auto
4917 have "b $$ i = e * b$$i + (1 - e) * b$$i" unfolding left_diff_distrib by simp
4918 also have "\<dots> > e * x $$ i + (1 - e) * y $$ i" apply(rule add_less_le_mono)
4919 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
4920 using x unfolding mem_interval using i apply simp
4921 using y unfolding mem_interval using i apply simp
4923 finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $$ i < b $$ i" unfolding euclidean_simps by auto
4924 } ultimately have "a $$ 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 }
4925 thus ?thesis unfolding mem_interval by auto
4928 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"
4929 assumes "{a<..<b} \<noteq> {}"
4930 shows "closure {a<..<b} = {a .. b}"
4932 have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto
4933 let ?c = "(1 / 2) *\<^sub>R (a + b)"
4934 { fix x assume as:"x \<in> {a .. b}"
4935 def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
4936 { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
4937 have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
4938 have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
4939 x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
4940 by (auto simp add: algebra_simps)
4941 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
4942 hence False using fn unfolding f_def using xc by auto }
4944 { assume "\<not> (f ---> x) sequentially"
4945 { fix e::real assume "e>0"
4946 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
4947 then obtain N::nat where "inverse (real (N + 1)) < e" by auto
4948 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)
4949 hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto }
4950 hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
4951 unfolding Lim_sequentially by(auto simp add: dist_norm)
4952 hence "(f ---> x) sequentially" unfolding f_def
4953 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]
4954 using scaleR.tendsto [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto }
4955 ultimately have "x \<in> closure {a<..<b}"
4956 using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto }
4957 thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
4960 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"
4961 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a<..<a}"
4963 obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
4964 def a \<equiv> "(\<chi>\<chi> i. b+1)::'a"
4965 { fix x assume "x\<in>s"
4966 fix i assume i:"i<DIM('a)"
4967 hence "(-a)$$i < x$$i" and "x$$i < a$$i" using b[THEN bspec[where x=x], OF `x\<in>s`]
4968 and component_le_norm[of x i] unfolding euclidean_simps and a_def by auto }
4969 thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
4972 lemma bounded_subset_open_interval:
4973 fixes s :: "('a::ordered_euclidean_space) set"
4974 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
4975 by (auto dest!: bounded_subset_open_interval_symmetric)
4977 lemma bounded_subset_closed_interval_symmetric:
4978 fixes s :: "('a::ordered_euclidean_space) set"
4979 assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
4981 obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
4982 thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
4985 lemma bounded_subset_closed_interval:
4986 fixes s :: "('a::ordered_euclidean_space) set"
4987 shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
4988 using bounded_subset_closed_interval_symmetric[of s] by auto
4990 lemma frontier_closed_interval:
4991 fixes a b :: "'a::ordered_euclidean_space"
4992 shows "frontier {a .. b} = {a .. b} - {a<..<b}"
4993 unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..
4995 lemma frontier_open_interval:
4996 fixes a b :: "'a::ordered_euclidean_space"
4997 shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
4998 proof(cases "{a<..<b} = {}")
4999 case True thus ?thesis using frontier_empty by auto
5001 case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
5004 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"
5005 assumes "{c<..<d} \<noteq> {}" shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
5006 unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
5009 (* Some stuff for half-infinite intervals too; FIXME: notation? *)
5011 lemma closed_interval_left: fixes b::"'a::euclidean_space"
5012 shows "closed {x::'a. \<forall>i<DIM('a). x$$i \<le> b$$i}"
5014 { fix i assume i:"i<DIM('a)"
5015 fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). x $$ i \<le> b $$ i}. x' \<noteq> x \<and> dist x' x < e"
5016 { assume "x$$i > b$$i"
5017 then obtain y where "y $$ i \<le> b $$ i" "y \<noteq> x" "dist y x < x$$i - b$$i"
5018 using x[THEN spec[where x="x$$i - b$$i"]] using i by auto
5019 hence False using component_le_norm[of "y - x" i] unfolding dist_norm euclidean_simps using i
5021 hence "x$$i \<le> b$$i" by(rule ccontr)auto }
5022 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
5025 lemma closed_interval_right: fixes a::"'a::euclidean_space"
5026 shows "closed {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i}"
5028 { fix i assume i:"i<DIM('a)"
5029 fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). a $$ i \<le> x $$ i}. x' \<noteq> x \<and> dist x' x < e"
5030 { assume "a$$i > x$$i"
5031 then obtain y where "a $$ i \<le> y $$ i" "y \<noteq> x" "dist y x < a$$i - x$$i"
5032 using x[THEN spec[where x="a$$i - x$$i"]] i by auto
5033 hence False using component_le_norm[of "y - x" i] unfolding dist_norm and euclidean_simps by auto }
5034 hence "a$$i \<le> x$$i" by(rule ccontr)auto }
5035 thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
5038 text {* Intervals in general, including infinite and mixtures of open and closed. *}
5040 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
5041 (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i<DIM('a). ((a$$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)"
5043 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)
5044 "is_interval {a<..<b}" (is ?th2) proof -
5045 have *:"\<And>x y z::real. x < y \<Longrightarrow> y < z \<Longrightarrow> x < z" by auto
5046 show ?th1 ?th2 unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
5047 by(meson order_trans le_less_trans less_le_trans *)+ qed
5049 lemma is_interval_empty:
5051 unfolding is_interval_def
5054 lemma is_interval_univ:
5056 unfolding is_interval_def
5060 subsection {* Closure of halfspaces and hyperplanes *}
5062 lemma isCont_open_vimage:
5063 assumes "\<And>x. isCont f x" and "open s" shows "open (f -` s)"
5065 from assms(1) have "continuous_on UNIV f"
5066 unfolding isCont_def continuous_on_def within_UNIV by simp
5067 hence "open {x \<in> UNIV. f x \<in> s}"
5068 using open_UNIV `open s` by (rule continuous_open_preimage)
5069 thus "open (f -` s)"
5070 by (simp add: vimage_def)
5073 lemma isCont_closed_vimage:
5074 assumes "\<And>x. isCont f x" and "closed s" shows "closed (f -` s)"
5075 using assms unfolding closed_def vimage_Compl [symmetric]
5076 by (rule isCont_open_vimage)
5078 lemma open_Collect_less:
5079 fixes f g :: "'a::topological_space \<Rightarrow> real"
5080 assumes f: "\<And>x. isCont f x"
5081 assumes g: "\<And>x. isCont g x"
5082 shows "open {x. f x < g x}"
5084 have "open ((\<lambda>x. g x - f x) -` {0<..})"
5085 using isCont_diff [OF g f] open_real_greaterThan
5086 by (rule isCont_open_vimage)
5087 also have "((\<lambda>x. g x - f x) -` {0<..}) = {x. f x < g x}"
5089 finally show ?thesis .
5092 lemma closed_Collect_le:
5093 fixes f g :: "'a::topological_space \<Rightarrow> real"
5094 assumes f: "\<And>x. isCont f x"
5095 assumes g: "\<And>x. isCont g x"
5096 shows "closed {x. f x \<le> g x}"
5098 have "closed ((\<lambda>x. g x - f x) -` {0..})"
5099 using isCont_diff [OF g f] closed_real_atLeast
5100 by (rule isCont_closed_vimage)
5101 also have "((\<lambda>x. g x - f x) -` {0..}) = {x. f x \<le> g x}"
5103 finally show ?thesis .
5106 lemma closed_Collect_eq:
5107 fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
5108 assumes f: "\<And>x. isCont f x"
5109 assumes g: "\<And>x. isCont g x"
5110 shows "closed {x. f x = g x}"
5112 have "open {(x::'b, y::'b). x \<noteq> y}"
5113 unfolding open_prod_def by (auto dest!: hausdorff)
5114 hence "closed {(x::'b, y::'b). x = y}"
5115 unfolding closed_def split_def Collect_neg_eq .
5116 with isCont_Pair [OF f g]
5117 have "closed ((\<lambda>x. (f x, g x)) -` {(x, y). x = y})"
5118 by (rule isCont_closed_vimage)
5119 also have "\<dots> = {x. f x = g x}" by auto
5120 finally show ?thesis .
5124 assumes "(f ---> l) net" shows "((\<lambda>y. inner a (f y)) ---> inner a l) net"
5125 by (intro tendsto_intros assms)
5127 lemma continuous_at_inner: "continuous (at x) (inner a)"
5128 unfolding continuous_at by (intro tendsto_intros)
5130 lemma continuous_at_euclidean_component[intro!, simp]: "continuous (at x) (\<lambda>x. x $$ i)"
5131 unfolding euclidean_component_def by (rule continuous_at_inner)
5133 lemma continuous_on_inner:
5134 fixes s :: "'a::real_inner set"
5135 shows "continuous_on s (inner a)"
5136 unfolding continuous_on by (rule ballI) (intro tendsto_intros)
5138 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
5139 by (simp add: closed_Collect_le)
5141 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
5142 by (simp add: closed_Collect_le)
5144 lemma closed_hyperplane: "closed {x. inner a x = b}"
5145 by (simp add: closed_Collect_eq)
5147 lemma closed_halfspace_component_le:
5148 shows "closed {x::'a::euclidean_space. x$$i \<le> a}"
5149 by (simp add: closed_Collect_le)
5151 lemma closed_halfspace_component_ge:
5152 shows "closed {x::'a::euclidean_space. x$$i \<ge> a}"
5153 by (simp add: closed_Collect_le)
5155 text {* Openness of halfspaces. *}
5157 lemma open_halfspace_lt: "open {x. inner a x < b}"
5158 by (simp add: open_Collect_less)
5160 lemma open_halfspace_gt: "open {x. inner a x > b}"
5161 by (simp add: open_Collect_less)
5163 lemma open_halfspace_component_lt:
5164 shows "open {x::'a::euclidean_space. x$$i < a}"
5165 by (simp add: open_Collect_less)
5167 lemma open_halfspace_component_gt:
5168 shows "open {x::'a::euclidean_space. x$$i > a}"
5169 by (simp add: open_Collect_less)
5171 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}
5173 lemma eucl_lessThan_eq_halfspaces:
5174 fixes a :: "'a\<Colon>ordered_euclidean_space"
5175 shows "{..<a} = (\<Inter>i<DIM('a). {x. x $$ i < a $$ i})"
5176 by (auto simp: eucl_less[where 'a='a])
5178 lemma eucl_greaterThan_eq_halfspaces:
5179 fixes a :: "'a\<Colon>ordered_euclidean_space"
5180 shows "{a<..} = (\<Inter>i<DIM('a). {x. a $$ i < x $$ i})"
5181 by (auto simp: eucl_less[where 'a='a])
5183 lemma eucl_atMost_eq_halfspaces:
5184 fixes a :: "'a\<Colon>ordered_euclidean_space"
5185 shows "{.. a} = (\<Inter>i<DIM('a). {x. x $$ i \<le> a $$ i})"
5186 by (auto simp: eucl_le[where 'a='a])
5188 lemma eucl_atLeast_eq_halfspaces:
5189 fixes a :: "'a\<Colon>ordered_euclidean_space"
5190 shows "{a ..} = (\<Inter>i<DIM('a). {x. a $$ i \<le> x $$ i})"
5191 by (auto simp: eucl_le[where 'a='a])
5193 lemma open_eucl_lessThan[simp, intro]:
5194 fixes a :: "'a\<Colon>ordered_euclidean_space"
5195 shows "open {..< a}"
5196 by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)
5198 lemma open_eucl_greaterThan[simp, intro]:
5199 fixes a :: "'a\<Colon>ordered_euclidean_space"
5200 shows "open {a <..}"
5201 by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)
5203 lemma closed_eucl_atMost[simp, intro]:
5204 fixes a :: "'a\<Colon>ordered_euclidean_space"
5205 shows "closed {.. a}"
5206 unfolding eucl_atMost_eq_halfspaces
5207 by (simp add: closed_INT closed_Collect_le)
5209 lemma closed_eucl_atLeast[simp, intro]:
5210 fixes a :: "'a\<Colon>ordered_euclidean_space"
5211 shows "closed {a ..}"
5212 unfolding eucl_atLeast_eq_halfspaces
5213 by (simp add: closed_INT closed_Collect_le)
5215 lemma open_vimage_euclidean_component: "open S \<Longrightarrow> open ((\<lambda>x. x $$ i) -` S)"
5216 by (auto intro!: continuous_open_vimage)
5218 text {* This gives a simple derivation of limit component bounds. *}
5220 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5221 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f(x)$$i \<le> b) net"
5222 shows "l$$i \<le> b"
5224 { fix x have "x \<in> {x::'b. inner (basis i) x \<le> b} \<longleftrightarrow> x$$i \<le> b"
5225 unfolding euclidean_component_def by auto } note * = this
5226 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *
5227 using closed_halfspace_le[of "(basis i)::'b" b] and assms(1,2,3) by auto
5230 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5231 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$$i) net"
5232 shows "b \<le> l$$i"
5234 { fix x have "x \<in> {x::'b. inner (basis i) x \<ge> b} \<longleftrightarrow> x$$i \<ge> b"
5235 unfolding euclidean_component_def by auto } note * = this
5236 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *
5237 using closed_halfspace_ge[of b "(basis i)"] and assms(1,2,3) by auto
5240 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
5241 assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$$i = b) net"
5243 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
5244 text{* Limits relative to a union. *}
5246 lemma eventually_within_Un:
5247 "eventually P (net within (s \<union> t)) \<longleftrightarrow>
5248 eventually P (net within s) \<and> eventually P (net within t)"
5249 unfolding Limits.eventually_within
5250 by (auto elim!: eventually_rev_mp)
5252 lemma Lim_within_union:
5253 "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>
5254 (f ---> l) (net within s) \<and> (f ---> l) (net within t)"
5255 unfolding tendsto_def
5256 by (auto simp add: eventually_within_Un)
5258 lemma Lim_topological:
5259 "(f ---> l) net \<longleftrightarrow>
5260 trivial_limit net \<or>
5261 (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
5262 unfolding tendsto_def trivial_limit_eq by auto
5264 lemma continuous_on_union:
5265 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"
5266 shows "continuous_on (s \<union> t) f"
5267 using assms unfolding continuous_on Lim_within_union
5268 unfolding Lim_topological trivial_limit_within closed_limpt by auto
5270 lemma continuous_on_cases:
5271 assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
5272 "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
5273 shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
5275 let ?h = "(\<lambda>x. if P x then f x else g x)"
5276 have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto
5277 hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto
5279 have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto
5280 hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto
5281 ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto
5285 text{* Some more convenient intermediate-value theorem formulations. *}
5287 lemma connected_ivt_hyperplane:
5288 assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"
5289 shows "\<exists>z \<in> s. inner a z = b"
5291 assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
5292 let ?A = "{x. inner a x < b}"
5293 let ?B = "{x. inner a x > b}"
5294 have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto
5295 moreover have "?A \<inter> ?B = {}" by auto
5296 moreover have "s \<subseteq> ?A \<union> ?B" using as by auto
5297 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
5300 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows
5301 "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)"
5302 using connected_ivt_hyperplane[of s x y "(basis k)::'a" a]
5303 unfolding euclidean_component_def by auto
5306 subsection {* Homeomorphisms *}
5308 definition "homeomorphism s t f g \<equiv>
5309 (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
5310 (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
5313 homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"
5314 (infixr "homeomorphic" 60) where
5315 homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
5317 lemma homeomorphic_refl: "s homeomorphic s"
5318 unfolding homeomorphic_def
5319 unfolding homeomorphism_def
5320 using continuous_on_id
5321 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5322 apply(rule_tac x = "(\<lambda>x. x)" in exI)
5325 lemma homeomorphic_sym:
5326 "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
5327 unfolding homeomorphic_def
5328 unfolding homeomorphism_def
5331 lemma homeomorphic_trans:
5332 assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
5334 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"
5335 using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
5336 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"
5337 using assms(2) unfolding homeomorphic_def homeomorphism_def by auto
5339 { 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 }
5340 moreover have "(f2 \<circ> f1) ` s = u" using fg1(2) fg2(2) by auto
5341 moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
5342 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 }
5343 moreover have "(g1 \<circ> g2) ` u = s" using fg1(5) fg2(5) by auto
5344 moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6) unfolding fg2(5) by auto
5345 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
5348 lemma homeomorphic_minimal:
5349 "s homeomorphic t \<longleftrightarrow>
5350 (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
5351 (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
5352 continuous_on s f \<and> continuous_on t g)"
5353 unfolding homeomorphic_def homeomorphism_def
5354 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
5355 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
5357 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
5358 apply auto apply(rule_tac x="g x" in bexI) apply auto
5359 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
5360 apply auto apply(rule_tac x="f x" in bexI) by auto
5362 text {* Relatively weak hypotheses if a set is compact. *}
5364 lemma homeomorphism_compact:
5365 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5366 (* class constraint due to continuous_on_inverse *)
5367 assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s"
5368 shows "\<exists>g. homeomorphism s t f g"
5370 def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"
5371 have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
5372 { fix y assume "y\<in>t"
5373 then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto
5374 hence "g (f x) = x" using g by auto
5375 hence "f (g y) = y" unfolding x(1)[THEN sym] by auto }
5376 hence g':"\<forall>x\<in>t. f (g x) = x" by auto
5379 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"])
5381 { assume "x\<in>g ` t"
5382 then obtain y where y:"y\<in>t" "g y = x" by auto
5383 then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto
5384 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 }
5385 ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" .. }
5386 hence "g ` t = s" by auto
5388 show ?thesis unfolding homeomorphism_def homeomorphic_def
5389 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
5392 lemma homeomorphic_compact:
5393 fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
5394 (* class constraint due to continuous_on_inverse *)
5395 shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s
5396 \<Longrightarrow> s homeomorphic t"
5397 unfolding homeomorphic_def by (metis homeomorphism_compact)
5399 text{* Preservation of topological properties. *}
5401 lemma homeomorphic_compactness:
5402 "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
5403 unfolding homeomorphic_def homeomorphism_def
5404 by (metis compact_continuous_image)
5406 text{* Results on translation, scaling etc. *}
5408 lemma homeomorphic_scaling:
5409 fixes s :: "'a::real_normed_vector set"
5410 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
5411 unfolding homeomorphic_minimal
5412 apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
5413 apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
5414 using assms apply auto
5415 using continuous_on_cmul[OF continuous_on_id] by auto
5417 lemma homeomorphic_translation:
5418 fixes s :: "'a::real_normed_vector set"
5419 shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
5420 unfolding homeomorphic_minimal
5421 apply(rule_tac x="\<lambda>x. a + x" in exI)
5422 apply(rule_tac x="\<lambda>x. -a + x" in exI)
5423 using continuous_on_add[OF continuous_on_const continuous_on_id] by auto
5425 lemma homeomorphic_affinity:
5426 fixes s :: "'a::real_normed_vector set"
5427 assumes "c \<noteq> 0" shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
5429 have *:"op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
5431 using homeomorphic_trans
5432 using homeomorphic_scaling[OF assms, of s]
5433 using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a] unfolding * by auto
5436 lemma homeomorphic_balls:
5437 fixes a b ::"'a::real_normed_vector" (* FIXME: generalize to metric_space *)
5438 assumes "0 < d" "0 < e"
5439 shows "(ball a d) homeomorphic (ball b e)" (is ?th)
5440 "(cball a d) homeomorphic (cball b e)" (is ?cth)
5442 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5443 show ?th unfolding homeomorphic_minimal
5444 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5445 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5446 using assms apply (auto simp add: dist_commute)
5448 apply (auto simp add: pos_divide_less_eq mult_strict_left_mono)
5449 unfolding continuous_on
5450 by (intro ballI tendsto_intros, simp)+
5452 have *:"\<bar>e / d\<bar> > 0" "\<bar>d / e\<bar> >0" using assms using divide_pos_pos by auto
5453 show ?cth unfolding homeomorphic_minimal
5454 apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
5455 apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
5456 using assms apply (auto simp add: dist_commute)
5458 apply (auto simp add: pos_divide_le_eq)
5459 unfolding continuous_on
5460 by (intro ballI tendsto_intros, simp)+
5463 text{* "Isometry" (up to constant bounds) of injective linear map etc. *}
5465 lemma cauchy_isometric:
5466 fixes x :: "nat \<Rightarrow> 'a::euclidean_space"
5467 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)"
5470 interpret f: bounded_linear f by fact
5471 { fix d::real assume "d>0"
5472 then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
5473 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
5474 { fix n assume "n\<ge>N"
5475 hence "norm (f (x n - x N)) < e * d" using N[THEN spec[where x=n]] unfolding f.diff[THEN sym] by auto
5476 moreover have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
5477 using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
5478 using normf[THEN bspec[where x="x n - x N"]] by auto
5479 ultimately have "norm (x n - x N) < d" using `e>0`
5480 using mult_left_less_imp_less[of e "norm (x n - x N)" d] by auto }
5481 hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
5482 thus ?thesis unfolding cauchy and dist_norm by auto
5485 lemma complete_isometric_image:
5486 fixes f :: "'a::euclidean_space => 'b::euclidean_space"
5487 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"
5488 shows "complete(f ` s)"
5490 { fix g assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
5491 then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
5492 using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
5493 hence x:"\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
5494 hence "f \<circ> x = g" unfolding fun_eq_iff by auto
5495 then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
5496 using cs[unfolded complete_def, THEN spec[where x="x"]]
5497 using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
5498 hence "\<exists>l\<in>f ` s. (g ---> l) sequentially"
5499 using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
5500 unfolding `f \<circ> x = g` by auto }
5501 thus ?thesis unfolding complete_def by auto
5505 fixes x :: "'a::real_normed_vector"
5506 shows "dist 0 x = norm x"
5507 unfolding dist_norm by simp
5509 lemma injective_imp_isometric: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
5510 assumes s:"closed s" "subspace s" and f:"bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
5511 shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
5512 proof(cases "s \<subseteq> {0::'a}")
5514 { fix x assume "x \<in> s"
5515 hence "x = 0" using True by auto
5516 hence "norm x \<le> norm (f x)" by auto }
5517 thus ?thesis by(auto intro!: exI[where x=1])
5519 interpret f: bounded_linear f by fact
5521 then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
5522 from False have "s \<noteq> {}" by auto
5523 let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
5524 let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
5525 let ?S'' = "{x::'a. norm x = norm a}"
5527 have "?S'' = frontier(cball 0 (norm a))" unfolding frontier_cball and dist_norm by auto
5528 hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
5529 moreover have "?S' = s \<inter> ?S''" by auto
5530 ultimately have "compact ?S'" using closed_inter_compact[of s ?S''] using s(1) by auto
5531 moreover have *:"f ` ?S' = ?S" by auto
5532 ultimately have "compact ?S" using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
5533 hence "closed ?S" using compact_imp_closed by auto
5534 moreover have "?S \<noteq> {}" using a by auto
5535 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
5536 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
5538 let ?e = "norm (f b) / norm b"
5539 have "norm b > 0" using ba and a and norm_ge_zero by auto
5540 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
5541 ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
5543 { fix x assume "x\<in>s"
5544 hence "norm (f b) / norm b * norm x \<le> norm (f x)"
5546 case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
5549 hence *:"0 < norm a / norm x" using `a\<noteq>0` unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
5550 have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def] by auto
5551 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
5552 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"]]
5553 unfolding f.scaleR and ba using `x\<noteq>0` `a\<noteq>0`
5554 by (auto simp add: mult_commute pos_le_divide_eq pos_divide_le_eq)
5557 show ?thesis by auto
5560 lemma closed_injective_image_subspace:
5561 fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
5562 assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 --> x = 0" "closed s"
5563 shows "closed(f ` s)"
5565 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
5566 show ?thesis using complete_isometric_image[OF `e>0` assms(1,2) e] and assms(4)
5567 unfolding complete_eq_closed[THEN sym] by auto
5571 subsection {* Some properties of a canonical subspace *}
5574 declare euclidean_component.zero[simp]
5576 lemma subspace_substandard:
5577 "subspace {x::'a::euclidean_space. (\<forall>i<DIM('a). P i \<longrightarrow> x$$i = 0)}"
5578 unfolding subspace_def by(auto simp add: euclidean_simps) (* FIXME: duplicate rewrite rule *)
5580 lemma closed_substandard:
5581 "closed {x::'a::euclidean_space. \<forall>i<DIM('a). P i --> x$$i = 0}" (is "closed ?A")
5583 let ?D = "{i. P i} \<inter> {..<DIM('a)}"
5584 let ?Bs = "{{x::'a. inner (basis i) x = 0}| i. i \<in> ?D}"
5587 hence x:"\<forall>i\<in>?D. x $$ i = 0" by auto
5588 hence "x\<in> \<Inter> ?Bs" by(auto simp add: x euclidean_component_def) }
5590 { assume x:"x\<in>\<Inter>?Bs"
5591 { fix i assume i:"i \<in> ?D"
5592 then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::'a. inner (basis i) x = 0}" by auto
5593 hence "x $$ i = 0" unfolding B using x unfolding euclidean_component_def by auto }
5594 hence "x\<in>?A" by auto }
5595 ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" .. }
5596 hence "?A = \<Inter> ?Bs" by auto
5597 thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
5600 lemma dim_substandard: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
5601 shows "dim {x::'a::euclidean_space. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0} = card d" (is "dim ?A = _")
5603 let ?D = "{..<DIM('a)}"
5604 let ?B = "(basis::nat => 'a) ` d"
5605 let ?bas = "basis::nat \<Rightarrow> 'a"
5606 have "?B \<subseteq> ?A" by auto
5608 { fix x::"'a" assume "x\<in>?A"
5609 hence "finite d" "x\<in>?A" using assms by(auto intro:finite_subset)
5610 hence "x\<in> span ?B"
5611 proof(induct d arbitrary: x)
5612 case empty hence "x=0" apply(subst euclidean_eq) by auto
5613 thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
5616 hence *:"\<forall>i<DIM('a). i \<notin> insert k F \<longrightarrow> x $$ i = 0" by auto
5617 have **:"F \<subseteq> insert k F" by auto
5618 def y \<equiv> "x - x$$k *\<^sub>R basis k"
5619 have y:"x = y + (x$$k) *\<^sub>R basis k" unfolding y_def by auto
5620 { fix i assume i':"i \<notin> F"
5621 hence "y $$ i = 0" unfolding y_def
5622 using *[THEN spec[where x=i]] by(auto simp add: euclidean_simps) }
5623 hence "y \<in> span (basis ` F)" using insert(3) by auto
5624 hence "y \<in> span (basis ` (insert k F))"
5625 using span_mono[of "?bas ` F" "?bas ` (insert k F)"]
5626 using image_mono[OF **, of basis] using assms by auto
5628 have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
5629 hence "x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5630 using span_mul by auto
5632 have "y + x$$k *\<^sub>R basis k \<in> span (?bas ` (insert k F))"
5633 using span_add by auto
5634 thus ?case using y by auto
5637 hence "?A \<subseteq> span ?B" by auto
5639 { fix x assume "x \<in> ?B"
5640 hence "x\<in>{(basis i)::'a |i. i \<in> ?D}" using assms by auto }
5641 hence "independent ?B" using independent_mono[OF independent_basis, of ?B] and assms by auto
5643 have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
5644 hence *:"inj_on (basis::nat\<Rightarrow>'a) d" using subset_inj_on[OF basis_inj, of "d"] by auto
5645 have "card ?B = card d" unfolding card_image[OF *] by auto
5646 ultimately show ?thesis using dim_unique[of "basis ` d" ?A] by auto
5649 text{* Hence closure and completeness of all subspaces. *}
5651 lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"
5653 apply (rule_tac x="{}" in exI, simp)
5655 apply (subgoal_tac "\<exists>x. x \<notin> A")
5657 apply (rule_tac x="insert x A" in exI, simp)
5658 apply (subgoal_tac "A \<noteq> UNIV", auto)
5661 lemma closed_subspace: fixes s::"('a::euclidean_space) set"
5662 assumes "subspace s" shows "closed s"
5664 have *:"dim s \<le> DIM('a)" using dim_subset_UNIV by auto
5665 def d \<equiv> "{..<dim s}" have t:"card d = dim s" unfolding d_def by auto
5666 let ?t = "{x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x$$i = 0}"
5667 have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0} = s \<and>
5668 inj_on f {x::'a. \<forall>i<DIM('a). i \<notin> d \<longrightarrow> x $$ i = 0}"
5669 apply(rule subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]])
5670 using dim_substandard[of d,where 'a='a] and t unfolding d_def using * assms by auto
5671 then guess f apply-by(erule exE conjE)+ note f = this
5672 interpret f: bounded_linear f using f unfolding linear_conv_bounded_linear by auto
5673 have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using f.zero using f(3)[unfolded inj_on_def]
5674 by(erule_tac x=0 in ballE) auto
5675 moreover have "closed ?t" using closed_substandard .
5676 moreover have "subspace ?t" using subspace_substandard .
5677 ultimately show ?thesis using closed_injective_image_subspace[of ?t f]
5678 unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
5681 lemma complete_subspace:
5682 fixes s :: "('a::euclidean_space) set" shows "subspace s ==> complete s"
5683 using complete_eq_closed closed_subspace
5687 fixes s :: "('a::euclidean_space) set"
5688 shows "dim(closure s) = dim s" (is "?dc = ?d")
5690 have "?dc \<le> ?d" using closure_minimal[OF span_inc, of s]
5691 using closed_subspace[OF subspace_span, of s]
5692 using dim_subset[of "closure s" "span s"] unfolding dim_span by auto
5693 thus ?thesis using dim_subset[OF closure_subset, of s] by auto
5697 subsection {* Affine transformations of intervals *}
5699 lemma real_affinity_le:
5700 "0 < (m::'a::linordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"
5701 by (simp add: field_simps inverse_eq_divide)
5703 lemma real_le_affinity:
5704 "0 < (m::'a::linordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"
5705 by (simp add: field_simps inverse_eq_divide)
5707 lemma real_affinity_lt:
5708 "0 < (m::'a::linordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"
5709 by (simp add: field_simps inverse_eq_divide)
5711 lemma real_lt_affinity:
5712 "0 < (m::'a::linordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"
5713 by (simp add: field_simps inverse_eq_divide)
5715 lemma real_affinity_eq:
5716 "(m::'a::linordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"
5717 by (simp add: field_simps inverse_eq_divide)
5719 lemma real_eq_affinity:
5720 "(m::'a::linordered_field) \<noteq> 0 ==> (y = m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"
5721 by (simp add: field_simps inverse_eq_divide)
5723 lemma image_affinity_interval: fixes m::real
5724 fixes a b c :: "'a::ordered_euclidean_space"
5725 shows "(\<lambda>x. m *\<^sub>R x + c) ` {a .. b} =
5726 (if {a .. b} = {} then {}
5727 else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
5728 else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
5730 { fix x assume "x \<le> c" "c \<le> x"
5731 hence "x=c" unfolding eucl_le[where 'a='a] apply-
5732 apply(subst euclidean_eq) by (auto intro: order_antisym) }
5734 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])
5735 ultimately show ?thesis by auto
5738 { fix y assume "a \<le> y" "y \<le> b" "m > 0"
5739 hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
5740 unfolding eucl_le[where 'a='a] by(auto simp add: euclidean_simps)
5742 { fix y assume "a \<le> y" "y \<le> b" "m < 0"
5743 hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c" "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
5744 unfolding eucl_le[where 'a='a] by(auto simp add: mult_left_mono_neg euclidean_simps)
5746 { fix y assume "m > 0" "m *\<^sub>R a + c \<le> y" "y \<le> m *\<^sub>R b + c"
5747 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5748 unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
5749 apply(auto simp add: pth_3[symmetric]
5750 intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
5751 by(auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff euclidean_simps)
5753 { fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
5754 hence "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}"
5755 unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
5756 apply(auto simp add: pth_3[symmetric]
5757 intro!: exI[where x="(1 / m) *\<^sub>R (y - c)"])
5758 by(auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff euclidean_simps)
5760 ultimately show ?thesis using False by auto
5763 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a..b} =
5764 (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})"
5765 using image_affinity_interval[of m 0 a b] by auto
5768 subsection {* Banach fixed point theorem (not really topological...) *}
5771 assumes s:"complete s" "s \<noteq> {}" and c:"0 \<le> c" "c < 1" and f:"(f ` s) \<subseteq> s" and
5772 lipschitz:"\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
5773 shows "\<exists>! x\<in>s. (f x = x)"
5775 have "1 - c > 0" using c by auto
5777 from s(2) obtain z0 where "z0 \<in> s" by auto
5778 def z \<equiv> "\<lambda>n. (f ^^ n) z0"
5780 have "z n \<in> s" unfolding z_def
5781 proof(induct n) case 0 thus ?case using `z0 \<in>s` by auto
5782 next case Suc thus ?case using f by auto qed }
5785 def d \<equiv> "dist (z 0) (z 1)"
5787 have fzn:"\<And>n. f (z n) = z (Suc n)" unfolding z_def by auto
5789 have "dist (z n) (z (Suc n)) \<le> (c ^ n) * d"
5791 case 0 thus ?case unfolding d_def by auto
5794 hence "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
5795 using `0 \<le> c` using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by auto
5796 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]
5797 unfolding fzn and mult_le_cancel_left by auto
5802 have "(1 - c) * dist (z m) (z (m+n)) \<le> (c ^ m) * d * (1 - c ^ n)"
5804 case 0 show ?case by auto
5807 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))))"
5808 using dist_triangle and c by(auto simp add: dist_triangle)
5809 also have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
5810 using cf_z[of "m + k"] and c by auto
5811 also have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
5812 using Suc by (auto simp add: field_simps)
5813 also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
5814 unfolding power_add by (auto simp add: field_simps)
5815 also have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
5816 using c by (auto simp add: field_simps)
5817 finally show ?case by auto
5820 { fix e::real assume "e>0"
5821 hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e"
5822 proof(cases "d = 0")
5824 have *: "\<And>x. ((1 - c) * x \<le> 0) = (x \<le> 0)" using `1 - c > 0`
5825 by (metis mult_zero_left real_mult_commute real_mult_le_cancel_iff1)
5826 from True have "\<And>n. z n = z0" using cf_z2[of 0] and c unfolding z_def
5828 thus ?thesis using `e>0` by auto
5830 case False hence "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]
5831 by (metis False d_def less_le)
5832 hence "0 < e * (1 - c) / d" using `e>0` and `1-c>0`
5833 using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"] by auto
5834 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
5835 { fix m n::nat assume "m>n" and as:"m\<ge>N" "n\<ge>N"
5836 have *:"c ^ n \<le> c ^ N" using `n\<ge>N` and c using power_decreasing[OF `n\<ge>N`, of c] by auto
5837 have "1 - c ^ (m - n) > 0" using c and power_strict_mono[of c 1 "m - n"] using `m>n` by auto
5838 hence **:"d * (1 - c ^ (m - n)) / (1 - c) > 0"
5839 using mult_pos_pos[OF `d>0`, of "1 - c ^ (m - n)"]
5840 using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]
5841 using `0 < 1 - c` by auto
5843 have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
5844 using cf_z2[of n "m - n"] and `m>n` unfolding pos_le_divide_eq[OF `1-c>0`]
5845 by (auto simp add: mult_commute dist_commute)
5846 also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
5847 using mult_right_mono[OF * order_less_imp_le[OF **]]
5848 unfolding mult_assoc by auto
5849 also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
5850 using mult_strict_right_mono[OF N **] unfolding mult_assoc by auto
5851 also have "\<dots> = e * (1 - c ^ (m - n))" using c and `d>0` and `1 - c > 0` by auto
5852 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
5853 finally have "dist (z m) (z n) < e" by auto
5855 { fix m n::nat assume as:"N\<le>m" "N\<le>n"
5856 hence "dist (z n) (z m) < e"
5857 proof(cases "n = m")
5858 case True thus ?thesis using `e>0` by auto
5860 case False thus ?thesis using as and *[of n m] *[of m n] unfolding nat_neq_iff by (auto simp add: dist_commute)
5862 thus ?thesis by auto
5865 hence "Cauchy z" unfolding cauchy_def by auto
5866 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
5868 def e \<equiv> "dist (f x) x"
5869 have "e = 0" proof(rule ccontr)
5870 assume "e \<noteq> 0" hence "e>0" unfolding e_def using zero_le_dist[of "f x" x]
5871 by (metis dist_eq_0_iff dist_nz e_def)
5872 then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"
5873 using x[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto
5874 hence N':"dist (z N) x < e / 2" by auto
5876 have *:"c * dist (z N) x \<le> dist (z N) x" unfolding mult_le_cancel_right2
5877 using zero_le_dist[of "z N" x] and c
5878 by (metis dist_eq_0_iff dist_nz order_less_asym less_le)
5879 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]]
5880 using z_in_s[of N] `x\<in>s` using c by auto
5881 also have "\<dots> < e / 2" using N' and c using * by auto
5882 finally show False unfolding fzn
5883 using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
5884 unfolding e_def by auto
5886 hence "f x = x" unfolding e_def by auto
5888 { fix y assume "f y = y" "y\<in>s"
5889 hence "dist x y \<le> c * dist x y" using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]
5890 using `x\<in>s` and `f x = x` by auto
5891 hence "dist x y = 0" unfolding mult_le_cancel_right1
5892 using c and zero_le_dist[of x y] by auto
5893 hence "y = x" by auto
5895 ultimately show ?thesis using `x\<in>s` by blast+
5898 subsection {* Edelstein fixed point theorem *}
5900 lemma edelstein_fix:
5901 fixes s :: "'a::real_normed_vector set"
5902 assumes s:"compact s" "s \<noteq> {}" and gs:"(g ` s) \<subseteq> s"
5903 and dist:"\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
5904 shows "\<exists>! x\<in>s. g x = x"
5905 proof(cases "\<exists>x\<in>s. g x \<noteq> x")
5906 obtain x where "x\<in>s" using s(2) by auto
5907 case False hence g:"\<forall>x\<in>s. g x = x" by auto
5908 { fix y assume "y\<in>s"
5909 hence "x = y" using `x\<in>s` and dist[THEN bspec[where x=x], THEN bspec[where x=y]]
5910 unfolding g[THEN bspec[where x=x], OF `x\<in>s`]
5911 unfolding g[THEN bspec[where x=y], OF `y\<in>s`] by auto }
5912 thus ?thesis using `x\<in>s` and g by blast+
5915 then obtain x where [simp]:"x\<in>s" and "g x \<noteq> x" by auto
5916 { fix x y assume "x \<in> s" "y \<in> s"
5917 hence "dist (g x) (g y) \<le> dist x y"
5918 using dist[THEN bspec[where x=x], THEN bspec[where x=y]] by auto } note dist' = this
5919 def y \<equiv> "g x"
5920 have [simp]:"y\<in>s" unfolding y_def using gs[unfolded image_subset_iff] and `x\<in>s` by blast
5921 def f \<equiv> "\<lambda>n. g ^^ n"
5922 have [simp]:"\<And>n z. g (f n z) = f (Suc n) z" unfolding f_def by auto
5923 have [simp]:"\<And>z. f 0 z = z" unfolding f_def by auto
5924 { fix n::nat and z assume "z\<in>s"
5925 have "f n z \<in> s" unfolding f_def
5927 case 0 thus ?case using `z\<in>s` by simp
5929 case (Suc n) thus ?case using gs[unfolded image_subset_iff] by auto
5930 qed } note fs = this
5931 { fix m n ::nat assume "m\<le>n"
5932 fix w z assume "w\<in>s" "z\<in>s"
5933 have "dist (f n w) (f n z) \<le> dist (f m w) (f m z)" using `m\<le>n`
5935 case 0 thus ?case by auto
5938 thus ?case proof(cases "m\<le>n")
5939 case True thus ?thesis using Suc(1)
5940 using dist'[OF fs fs, OF `w\<in>s` `z\<in>s`, of n n] by auto
5942 case False hence mn:"m = Suc n" using Suc(2) by simp
5943 show ?thesis unfolding mn by auto
5945 qed } note distf = this
5947 def h \<equiv> "\<lambda>n. (f n x, f n y)"
5948 let ?s2 = "s \<times> s"
5949 obtain l r where "l\<in>?s2" and r:"subseq r" and lr:"((h \<circ> r) ---> l) sequentially"
5950 using compact_Times [OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def
5951 using fs[OF `x\<in>s`] and fs[OF `y\<in>s`] by blast
5952 def a \<equiv> "fst l" def b \<equiv> "snd l"
5953 have lab:"l = (a, b)" unfolding a_def b_def by simp
5954 have [simp]:"a\<in>s" "b\<in>s" unfolding a_def b_def using `l\<in>?s2` by auto
5956 have lima:"((fst \<circ> (h \<circ> r)) ---> a) sequentially"
5957 and limb:"((snd \<circ> (h \<circ> r)) ---> b) sequentially"
5959 unfolding o_def a_def b_def by (rule tendsto_intros)+
5962 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
5964 have "dist (-x) (-y) = dist x y" unfolding dist_norm
5965 using norm_minus_cancel[of "x - y"] by (auto simp add: uminus_add_conv_diff) } note ** = this
5967 { assume as:"dist a b > dist (f n x) (f n y)"
5968 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"
5969 and "\<forall>m\<ge>Nb. dist (f (r m) y) b < (dist a b - dist (f n x) (f n y)) / 2"
5970 using lima limb unfolding h_def Lim_sequentially by (fastsimp simp del: less_divide_eq_number_of1)
5971 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)"
5972 apply(erule_tac x="Na+Nb+n" in allE)
5973 apply(erule_tac x="Na+Nb+n" in allE) apply simp
5974 using dist_triangle_add_half[of a "f (r (Na + Nb + n)) x" "dist a b - dist (f n x) (f n y)"
5975 "-b" "- f (r (Na + Nb + n)) y"]
5976 unfolding ** by (auto simp add: algebra_simps dist_commute)
5978 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)"
5979 using distf[of n "r (Na+Nb+n)", OF _ `x\<in>s` `y\<in>s`]
5980 using subseq_bigger[OF r, of "Na+Nb+n"]
5981 using *[of "f (r (Na + Nb + n)) x" "f (r (Na + Nb + n)) y" "f n x" "f n y"] by auto
5982 ultimately have False by simp
5984 hence "dist a b \<le> dist (f n x) (f n y)" by(rule ccontr)auto }
5987 have [simp]:"a = b" proof(rule ccontr)
5988 def e \<equiv> "dist a b - dist (g a) (g b)"
5989 assume "a\<noteq>b" hence "e > 0" unfolding e_def using dist by fastsimp
5990 hence "\<exists>n. dist (f n x) a < e/2 \<and> dist (f n y) b < e/2"
5991 using lima limb unfolding Lim_sequentially
5992 apply (auto elim!: allE[where x="e/2"]) apply(rule_tac x="r (max N Na)" in exI) unfolding h_def by fastsimp
5993 then obtain n where n:"dist (f n x) a < e/2 \<and> dist (f n y) b < e/2" by auto
5994 have "dist (f (Suc n) x) (g a) \<le> dist (f n x) a"
5995 using dist[THEN bspec[where x="f n x"], THEN bspec[where x="a"]] and fs by auto
5996 moreover have "dist (f (Suc n) y) (g b) \<le> dist (f n y) b"
5997 using dist[THEN bspec[where x="f n y"], THEN bspec[where x="b"]] and fs by auto
5998 ultimately have "dist (f (Suc n) x) (g a) + dist (f (Suc n) y) (g b) < e" using n by auto
5999 thus False unfolding e_def using ab_fn[of "Suc n"] by norm
6002 have [simp]:"\<And>n. f (Suc n) x = f n y" unfolding f_def y_def by(induct_tac n)auto
6003 { fix x y assume "x\<in>s" "y\<in>s" moreover
6004 fix e::real assume "e>0" ultimately
6005 have "dist y x < e \<longrightarrow> dist (g y) (g x) < e" using dist by fastsimp }
6006 hence "continuous_on s g" unfolding continuous_on_iff by auto
6008 hence "((snd \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
6009 apply (rule allE[where x="\<lambda>n. (fst \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
6010 using lima unfolding h_def o_def using fs[OF `x\<in>s`] by (auto simp add: y_def)
6011 hence "g a = a" using tendsto_unique[OF trivial_limit_sequentially limb, of "g a"]
6012 unfolding `a=b` and o_assoc by auto
6014 { fix x assume "x\<in>s" "g x = x" "x\<noteq>a"
6015 hence "False" using dist[THEN bspec[where x=a], THEN bspec[where x=x]]
6016 using `g a = a` and `a\<in>s` by auto }
6017 ultimately show "\<exists>!x\<in>s. g x = x" using `a\<in>s` by blast
6021 (** TODO move this someplace else within this theory **)
6022 instance euclidean_space \<subseteq> banach ..
6024 declare tendsto_const [intro] (* FIXME: move *)
6026 text {* Legacy theorem names *}
6028 lemmas Lim_ident_at = LIM_ident
6029 lemmas Lim_const = tendsto_const
6030 lemmas Lim_cmul = scaleR.tendsto [OF tendsto_const]
6031 lemmas Lim_neg = tendsto_minus
6032 lemmas Lim_add = tendsto_add
6033 lemmas Lim_sub = tendsto_diff
6034 lemmas Lim_mul = scaleR.tendsto
6035 lemmas Lim_vmul = scaleR.tendsto [OF _ tendsto_const]
6036 lemmas Lim_null_norm = tendsto_norm_zero_iff [symmetric]
6037 lemmas Lim_linear = bounded_linear.tendsto [COMP swap_prems_rl]
6038 lemmas Lim_component = euclidean_component.tendsto
6039 lemmas Lim_intros = Lim_add Lim_const Lim_sub Lim_cmul Lim_vmul Lim_within_id